diff --git "a/spin orbit coupling/1.json" "b/spin orbit coupling/1.json" new file mode 100644--- /dev/null +++ "b/spin orbit coupling/1.json" @@ -0,0 +1 @@ +[ { "title": "1005.2948v4.Anomalous_spin_Hall_effects_in_Dresselhaus__110__quantum_wells.pdf", "content": "arXiv:1005.2948v4 [cond-mat.mes-hall] 27 Sep 2010Anomalous spinHalleffects inDresselhaus (110)quantum we lls\nMing-Hao Liu∗and Ching-Ray Chang†\nDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan\n(Dated: September 14, 2018)\nAnomalous spin Hall effects that belong to the intrinsic typ e in Dresselhaus (110) quantum wells are dis-\ncussed. Fortheout-of-plane spincomponent, antisymmetri ccurrent-induced spinpolarizationinduces opposite\nspin Hall accumulation, even though there is no spin-orbit f orce due to Dresselhaus (110) coupling. A surpris-\ning feature of this spin Hall induction is that the spin accum ulation sign does not change upon bias reversal.\nContribution to the spin Hall accumulation from the spin Hal l induction and the spin deviation due to intrinsic\nspin-orbit force as well as extrinsic spin scattering, can b e straightforwardly distinguished simply by reversing\nthe bias. For the inplane component, inclusion of a weak Rash ba coupling leads to a new type of Syintrinsic\nspinHalleffect solelydue tospin-orbit-force-driven spi nseparation.\nPACS numbers: 72.25.Dc, 71.70.Ej,73.23.Ad\nI. INTRODUCTION\nThe intensive effortson spin Hall effect (SHE) both exper-\nimentally and theoretically during the past decade have suc -\ncessfullybuiltanothermilestoneincondensedmatterphys ics.\nSpin separation in semiconductors is not only possible but\nnatural, so that manipulating spin properties of charge car -\nriers in electronics is promising. The earliest theoretica l idea\nthat up and down spins may laterally separate upon transport\ndue to asymmetric scattering was proposed in 1971.1,2More\nthan three decades later, the power of optical measurements\non high quality mesoscopic samples made SHE in semicon-\nductors no longer an idea but an experimental fact.3Right\nbefore the first observation of Ref. 3in 2004, mechanisms\nof SHE was further extended from spin-dependent scattering\nthat was later categorized as extrinsic, to spin-orbit-cou pled\nband structure that was later categorized as intrinsic.4,5Ex-\nperimentally,mostobservationsso farhave beenattribute dto\ntheextrinsicSHE,3,6–8whileevidenceoftheintrinsicSHE9is\nrelatively few. Nonetheless, intrinsic SHE remains an impo r-\ntantissue thatuntilnowstill receivesenduringefforts.10\nIn the intrinsic SHE, spin separation is solely due to the\nunderlying spin-orbit coupling in the band structure, so th at\nSHE can exist even in systems free of scattering (but of finite\nsizes11,12). In the ballistic limit, the spin separation can be\nvividly visualized by the transverse spin-orbit force13,14de-\nrivedbyusingthe Heisenbergequationofmotion,\nFso=m\ni/planckover2pi1/bracketleftbigg1\ni/planckover2pi1[r,H],H/bracketrightbigg\n, (1)\nwhereris the position operator and His the single-\nparticle Hamiltonian. For well discussed two-dimensional\nsystems with Rashba coupling15described by HR=\n(α//planckover2pi1)(pyσx−pxσy), as well as linear Dresselhaus (001)\ncoupling16,17described by H001\nD= (β//planckover2pi1)(pxσx−pyσy),\nthespin-orbitforceisgivenby13\nFRD001\nso=2m/parenleftbig\nα2−β2/parenrightbig\n/planckover2pi13(p×ez)σz,(2)\nwherepis the momentum and ezis the unit vector of the\nplanenormal. Here αandβareRashbaandDresselhauscou-pling constants, respectively, and σx,σy,σzare Pauli matri-\nces. Equation( 2)clearlydepictsalateralspindeviationofthe\nSz= (/planckover2pi1/2)σzspin component with opposite contributions\nfromRashba andDresselhaus(001)couplings,assketchedin\nFig.1(a)and(b),respectively.\nSHE in Dresselhaus (110) quantum wells (QWs), on the\nother hand, is relatively less discussed theoretically,18al-\nthough a previous experimental effort6had revealed in GaAs\n(110)QWstheexistenceofSHEthatwasattributedtotheex-\ntrinsictype. Inthispaper,anomalousSHEsthatbelongto th e\nintrinsic type in Dresselhaus (110) QWs are discussed. We\nshow that the spin Hall pattern of Szcan be induced when\nthe transport direction is properly oriented, even though t he\nDresselhaus(110)couplingdoesnotresult inspin-orbitfo rce\nto separate opposite Szspins upon transport [see Fig. 1(c)].\nMoreover, we propose a Rashba-coupling-assisted intrinsi c\nSHE inSythat is truly due to spin-orbit force under the in-\nRashba(001)\n(a)\n/Bullet/Bullet\n/Bullet/Bulletx/bardbl[100]y/bardbl[010]z/bardbl[001]\nDresselhaus(001)\n(b)\n/Bullet/Bullet\n/Bullet/Bullet\nDresselhaus(110)\n(c)\n/Bullet/Bullet\n/Bulletx/bardbl[1¯10]y/bardbl[00¯1]z/bardbl[110]\nDresselhaus(110)\n+ WeakRashba\n(d)\n/Bullet/Bullet\n/Bullet/Bullet\nFIG. 1. (Color online) Spin deviation in Szdue to spin-orbit force\nin (a) Rashba and (b) linear Dresselhaus [001] systems. (c) S pin\nHall induction due to antisymmetric CISP along [00 ¯1] in Dressel-\nhaus [110] systems may cause opposite Szaccumulations as well,\neven though there is no spin-orbit force. (d) In the case of li near\nDresselhaus [110] plus weakRashba couplings, a coupled spi n-orbit\nforce given byEq.( 11) mayleadtospindeviation in Sy.2\nteraction of Dresselhaus (110) plus a weak Rashba couplings\n[Fig.1(d)].\nThis paper is organized as follows. In Sec. IIwe briefly\nintroducethe formulasrequiredin the Landauer-Keldyshfo r-\nmalism employedin thenumericalanalysisofSec. III, where\nwe visualize the proposed spin Hall induction and Rashba-\ncoupling-assistedSHE in Sy. Comparison of the present bal-\nlistic calculation with the diffusive experiment of Ref. 6will\nbe discussed and the transportparametersused in our numer-\nical datawill beremarked. We concludeinSec. IV.\nII. FORMULAS\nA. LinearDresselhaus(110) coupling\nTheDresselhaus(110)couplinguptothetermlinearinmo-\nmentumcanbewrittenas\nH110\nD=−β\n/planckover2pi1pxσz, (3)\nwherex,y, andzaxes are chosen along [1 ¯10], [00¯1], and\n[110], respectively. Throughout the present discussion, w e\nwill focus on this Dresselhaus (110) linear term, so that\nwithout ambiguity we use the same notation βto denote its\ncoupling strength. The spin-orbit field subject to Eq. ( 3)\nis depicted in Fig. 2(a). Clearly, when propagating with\nk= (±|kx|,ky)electrons encounter antisymmetric spin-\norbit fields on the left andright sides of the [00 ¯1]axis (or the\nyaxis). Hence the current-induced spin polarization (CISP)\neffect19–21is expected to build opposite Szspin densities at\nthetwo sides,asconceptuallydepictedinFig. 1(c).\nTo better illustrate this spin Hall induction, we will in\nSec.IIIfirst consider a T-bar ballistic nanostructure,attached\nto left, right, and bottom leads (from the top view) that are\nmadeofnormalmetals. Thecentralregionisdescribedbythe\nsquare-latticetight-bindingHamiltonian,\nH= (U+4t0)11/summationdisplay\nnc†\nncn+/summationdisplay\n/angbracketleftnm/angbracketrightc†\nmtm←ncn,(4)\nwhere the sum over /an}bracketle{tnm/an}bracketri}htof the second term is run for the\nsites nearest to each other, satisfying |rm−rn|=a, abeing\nthe lattice grid spacing and rnthe position vector of site n,\nandthehoppingmatrixisgivenby\ntm←n=−t011−itDdxσz. (5)\nHereUistheon-siteenergysetto beconstantoverthewhole\nsample,t0=/planckover2pi12/2ma2is the kinetic hopping energy, 11 is\nthe identity, cm(c†\nm) annihilates (creates) an electron at site\nm,tD=β/2ais the Dresselhaus hopping parameter, and\ndx= (rm−rn)·exis the hopping displacement along x\nfromsite mtositen.\nB. Landauer-Keldyshformalism\nTo image the nonequilibrium charge, charge current, spin,\nand spin current densities under the influence of the biased \n+−\ntD= 0.1t0\ntD= 0tD= 0\n−8−6−4−202468x 10−6\n \n−+\n−8−6−4−202468x 10−6kx/bardbl[1¯10]ky/bardbl[00¯1][110]\n \n/angbracketlefteN/angbracketright− −\n+024\nx 10−6\n \n/angbracketleftσz/angbracketright− −\n+−202\nx 10−7(a)\n(b)\n(c)(d)\n(e)/angbracketleftσz/angbracketright\n/angbracketleftσz/angbracketright\n/Bullet/BoldCircle/Bullet [110]/bardblz x/bardbl[1¯10][00¯1]/bardbly\nFIG. 2. (Color online) (a) Linear Dresselhaus [110] spin-or bit field\ninkspace. (b) Local charge density /angbracketlefteN/angbracketright(color shading) and local\nchargecurrentdensity(arrows)ina 16a×8aDresselhaus[110]sam-\nple subject to three terminals with a weak bias eV0= 10−3t0. (c)\nLocal spin density /angbracketleftσz/angbracketright(color shading) and its corresponding local\nspin current density /angbracketleftJSz/angbracketright(arrows) in the same device as (b) with\nsame conditions. Spin Hall induction in a 40a×40aDresselhaus\n[110] sample with (d) upward bias and (e) downward bias; stro ng\nbiaseV0= 0.4t0isapplied. Notethat in(b)–(e),the regions outside\nthe dashed lines are simulating the leads (zero spin-orbit c oupling\nandconstant on-site energy setequal tothe appliedbias).\nleads, a powerful and convenient approach is the Landauer-\nKeldysh formalism,22,23especially for the present ballistic\ncase free of particle-particle interaction. In this formal ism,\nphysical quantities in a nonequilibrium but steady state ar e\nexpressed in terms of the lesser Green function matrix G<,\nprovided that those physical observables of interest are we ll\ndefined.24Each matrix element G<\nmnin our spin-1/2 electron\nsystemisa 2×2submatrix,sothatthesizeoffull G0. (Note that e=−|e|is the neg-\native electron charge, and hence electrons always flow from\n+to−signs). In the rest of our analysis, we will focus on\nthe nonequilibriumcontribution24of those quantitieslisted in\nEqs. (6)–(9), and hence the integrationrange will be taken as\nEF−eV0/2→EF+eV0/2.\nIII. NUMERICALANALYSIS\nA. SpinHall inductionin Sz: PureDresselhaus(110) coupling\nEmploying the Landauer-Keldysh formalism briefly intro-\nduced above, we now drive electrons in the T-bar nanostruc-\nture from bottom to left and right leads with eV0= 10−3t0,\nas shown in Fig. 2(b), where the backgroundcolorshading is\ndeterminedbythelocal chargedensity /an}bracketle{teN/an}bracketri}htgiveninEq.( 6),\nwhile each arrow indicates the local charge current density\ngivenbyEq.( 8). InFig. 2(c),thecolorshadingisdetermined\nby the local spin /an}bracketle{tSz/an}bracketri}htdensity [Eq. ( 7)] and clearly shows an\nantisymmetric Szpolarization for electrons moving into the\nleft andrightleadsbecauseoftheoppositeDresselhaus(11 0)\nfieldstheyfeel. Thelocalspincurrentdensityindicatedby the\narrows therein is given by Eq. ( 9), which is derived from the\nsymmetrized spin current operator JSi={Jm→n/e,Si}/2\nin a way similar to Ref. 24. A pure spin current from right\nto left is observed;at right side the spin currentis flowing t o-\nward left because of the negative Sztimes the right moving\nparticle current while at left side the left flowing spin curr ent\nstemsfromtheproductof thepositive Szandthe left moving\nparticlecurrent.\nFor a two-terminal device made of Dresselhaus (110) QW\noriented along [00 ¯1] (yaxis), spin Hall accumulation of op-\npositeSzis therefore expected, as shown in Fig. 2(d)–(e),\nwhere we consider a strong bias voltage of eV0= 0.4t0for\na40a×40asample. A striking difference between the spin\nHallinductionintroducedhereandthespinHalldeviationd ue\nto spin-orbit force is the independence of the accumulation\nsignonthebiasdirection. Whetherdrivingtheelectronsfr om\nbottom to top [Fig. 2(d)] or from top to bottom [Fig. 2(e)],\nonealwaysobserveanegative Szaccumulationatrightwhile\npositiveat left.\nContrary to the present ballistic nanostructure here, the\nSHE previously observed in GaAs (110) QWs6was in dif-\nfusiveregimeandattributedto the extrinsictype. Theexpe ri-\nment usedac lock-indetectionreferencedto the frequencyo f\nasquarewavealternatingvoltagewithzerodcbiasoffset,a nd \n−+\nFIG. 4(a)\n↓\n/angbracketleftσz/angbracketright\n−4−2024x 10−6\n \n−+\nFIG. 4(b)\n↓\n/angbracketleftσy/angbracketright\n−2−1012x 10−6 \n− + /angbracketleftσz/angbracketright\nFIG. 4(c) →\n−2 0 2\nx 10−7\n \n− + /angbracketleftσy/angbracketright\nFIG. 4(d) →\n−5 0 5\nx 10−7(a)\n(b)(c)\n(d)\nFIG.3. (Coloronline) Imagingoflocal spindensities /angbracketleftσz/angbracketrightand/angbracketleftσy/angbracketright\nin a60a×60a[110] sample with (a)–(b) top-to-bottom and (c)–\n(d) right-to-left orientations. A strong Dresselhaus [110 ] coupling\ntD= 0.08t0and a weak Rashba coupling tR= 0.02t0are used.\nBiasis set eV0= 0.4t0.\nthe resulting signal is sensitive only to the difference in K err\nrotationbetweenpositiveandnegativebias. Our spinHall i n-\nduction that does not depend on bias direction may therefore\neither hardly contribute or be subtracted in the result of Re f.\n6.\nB. SpinHall deviationin Sy: StrongDresselhaus(110) with\nweak Rashbacouplings\nNext we recall the spin-orbit force Eq. ( 1). For pure Dres-\nselhaus (110) systems given by Eq. ( 3), there is no way to\nobtain a nonvanishing Fsosince eventually the Pauli matrix\nσzwillcommutewithitself,evenifthecubictermthatisstill\nintermsof σzisinvolved. Theonlypossibilityinthiscasefor\na nonvanishing Fsotosurviveis tointroducespin-orbitterms\ninvolving σxorσy. Combination of Rashba coupling with\nthe present linear Dresselhaus (110) term is therefore a nat u-\nral candidate, which is possible for, for example, asymmetr ic\nGaAs (110) QWs, as are the cases of Ref. 6. The spin-orbit\nforceforthisRashba-Dresselhaus(110)QWis\nFRD110\nso=2mα\n/planckover2pi13(p×ez)(ασz−βσy).(10)\nWithoutRashbaterm α,thespin-orbitforcevanishesandzero\nspin current is hence expected. From a gauge viewpoint, the\nexistence of equilibrium spin current in (110) QWs will re-\nquire Rashba term to break the pure gauge.25Note also that\ntheαsquareddependenceforthe σzcomponentinEq.( 10)is\nsimilarto theresultinRef. 26.\nHere of particular interest is the case of weak Rashba cou-\npling,suchthatEq.( 10)becomes4\nFRD110\nso/vextendsingle/vextendsingle\nα≪β≈ −2mαβ\n/planckover2pi13(p×ez)σy,(11)\nwhich predicts a lateral spin Hall deviation in Sythat re-\nquires a weak but nonzero Rashba coupling α. To further\nvisualize the predicted Rashba-assisted SySHE, we con-\nsider a60a×60asample with Dresselhaus (110) hopping\ntD= 0.08t0and Rashba hopping tR≡α/2a= 0.02t0,\nattached to two leads under a bias voltage eV0= 0.4t0. For\nthe [001]-oriented (electron flow along −y) sample, the Sz\nspinHallpatternduetospinHallinductionisobservedinFi g.\n3(a). Meanwhile, an Syspin Hall pattern is also shown in\nFig.3(b), which is a combined consequence of not only the\nspindeviationEq.( 11)butalsoanantisymmetricCISPbythe\nRashbacoupling. Alongthe −yaxis,electronswithwavevec-\ntork= (±|kx|,−|ky|)encounter opposite ycomponent of\ntheclockwiseRashbaspin-orbitfield: negativefor +|kx|and\npositive for −|kx|. Hence a spin Hall induction in Sydue to\nRashba couplingcontributesto Fig. 3(b) as well. In addition,\nthecontributionofthespin-orbitforceEq.( 11)predictsa +Sy\n(−Sy) accumulation at left (right) side of the electron flow,\nforlateraldistanceshorterthanthespinprecessionlengt hLso\n(around15ahere); the accumulation sign reverses when the\nlateral distance exceeds Lso, as is the case in our 60a×60a\nhere. Thereforethetwocontributions,spinHallinduction and\nspin-orbitforceEq.( 11),are additiveinFig. 3(b).\nFor the [ ¯110]-oriented (electron flow along −x) sample,\nthere is a vague spin Hall pattern in Szbecause of the ab-\nsence of the antisymmetric CISP and weak spin-orbit force\n[Fig.3(c)]. The average of /an}bracketle{tSz/an}bracketri}htover the whole sample ba-\nsically reveals the usual CISP effect as observed in Ref. 6.\nTheSypattern, on the other hand, exhibits a clear spin Hall\naccumulation pattern which is solely attributed to the spin -\norbitforceEq.( 11),asshowninFig. 3(d). Asexplained, −Sy\n(+Sy) accumulatesat left (right)side ofthe electronflow be-\ncause the lateral distance has exceeded Lso. Upon the bias\nreversal,the ±Syedgeaccumulationsswap(notshownhere),\nwhichisthegeneralfeatureofthespinHallpatternduetosp in\ndeviation by intrinsic spin-orbit force, as well as by extri nsic\nspin scattering. The spin Hall induction such as that of Szin\nthe Dresselhaus(110)case along ±y, however,doesnot have\nthisfeature. AnotherdifferencebetweenthespinHallpatt erns\ninducedbyantisymmetricCISPandspin-orbit-drivenspind e-\nviationisthatintheformerthesignsofthespinaccumulati on\ndo not change with the increasing sample width, while in the\nlatter they do. This differencecan also be told in Fig. 3: con-\nstant sign in each lateral side in panel (a) but varying sign i n\npanels(b)and(d).\nC. From pureDresselhaus(110) topureRashbacases\nFinally, we laterally scan the local spin densities /an}bracketle{tSz/an}bracketri}htand\n/an}bracketle{tSy/an}bracketri}htin Fig.4at the positions marked by the dashed lines in\nFig.3,forasetofvariousspin-orbitcouplingparametersfrom\npure Dresselhaus (110) (black curves) to pure Rashba (light -\nest gray curves). For the [001]-orientedsample, Szspin Hall−4−2024/angbracketleftσz/angbracketright(10−6)\n−20 0 20−2−1012\nTransverse position x(a)/angbracketleftσy/angbracketright(10−6)−2−1012/angbracketleftσz/angbracketright(10−6)\n−20 0 20−101\nTransverse position y(a)/angbracketleftσy/angbracketright(10−6)+\n−y= 9.5a −+\nx= 9.5atR= 0.1t0,tD= 0tR= 0,tD= 0.1t0(a)\n(b)(c)\n(d)\nFIG. 4. (Color online) Local spin densities /angbracketleftσz/angbracketrightand/angbracketleftσy/angbracketrightin a\n60a×60asample as a function of the transverse position with var-\nious coupling parameters tDandtR. The origin is set at the cen-\nter of the sample. As indicated in panel (a), the coupling par am-\neters from black to the lightest curves are (tR,tD) = (0,0.1)t0,\n(0.02,0.08)t0,(0.04,0.06)t0,(0.06,0.04)t0,(0.08,0.02)t0,and\n(0.1,0)t0. Ineachpanel, the red(dark gray) thickcurve correspond\ntoFig.3.\npatternshowninFig. 4(a)graduallyevolvesfromspinHallin-\nductionduetoDresselhaus(110)couplingtospinHall devia -\ntiondrivenbyspin-orbitforceduetoRashbacoupling. InFi g.\n4(b), the black curve for the pure Dresselhaus (110) shows\nzero everywhere [and so are those for Fig. 4(c)–(d)], while\na weak Rashba coupling assists the formation of the Syspin\nHall pattern; the antisymmetric pattern holds all the way to\npureRashbabecausetheRashbacouplingcontributesthroug h\ntheantisymmetricCISPinthisorientationasexplainedpre vi-\nously. For the [ ¯110]-orientedsample, turning on of the weak\nRashba coupling builds Syspin Hall pattern [Fig. 4(d)] but\nnottoomuchfor Sz[Fig.4(c)]. DowntopureRashba,the Sz\npattern recovers the spin Hall accumulation due to spin-orb it\nforce [Fig. 4(c)], while that for Syshows symmetric CISP\n[Fig.4(d)].\nD. Remark ontransport parameters\nInournumericalanalysisforthepureDresselhauscase,we\nhave settD/t0= 0.1mostly based on an illustrative reason.\nThis coupling ratio allows a direction comparison with Ref.\n22,wheretR/t0= 0.1ischosen,inthelaterpartofcoexisting\nRashba and linear Dresselhaus (110) couplings (such as Fig.\n4).\nComparing with the GaAs (110) QWs used in the exper-\niment of Ref. 6, the coupling ratio tD/t0may be one order\nweakerthanours. Thesampletheyusedbehaveslike a single\n75˚A Al0.1Ga0.9As QW. Using the relation β=γ/an}bracketle{tk2\nz/an}bracketri}htwith\nhard wall approximation /an}bracketle{tk2\nz/an}bracketri}ht ≈(π/w)2andγ≈27eV˚A3\nfor bothGaAs and InAsQWs,27this well width of w= 75˚A5\nTABLE I. Effective mass and Dresselhaus coefficients taken f rom\nRef.27.\nQWtype GaAs AlAs InAs InSb CdTe ZnSe\nm/m00.067 0.15 0.023 0.014 0.09 0.16\nγ(eV˚A3) 27.58 18.53 27.18 760 .1 43.88 14.29\nGaAs AlAsInAsInSbCdTeZnSe00.050.10.15tD/t0\n \nw = 5 nm\nw = 7.5 nm\nw = 10 nm\nFIG. 5. (Online color) Coupling ratios tD/t0estimated by Eq. ( 12)\nfor various QWswithwidths w= 5nm,7.5nm,10nm.\nleads toβ≈4.74×10−2eV˚A. Effectivemass was reported\nto bem= 0.074m0,m0the electron rest mass. The sheet\ndensityis ns= 1.9×1012cm−2,whichallowsustoestimate\nthelocationoftheFermienergy23EF−Eb=π/planckover2pi12ns/m≈6.\n15×10−2eV. In order for the long wavelength limit to\nbe valid, the chosen lattice constant ahas to yield a kinetic\nhopping constant t0that keeps EFclose toEb. Choosing\na= 2nmleadstot0≈0.13eVso thatEF−Eb≈0.12t0is\nsatisfying(recall EF−Eb= 0.2t0inournumericalresultsas\nwell as in Refs. 22and24). The coupling ratio with this ais\ntD/t0≈0.01.The Rashba strengthin Ref. 6was reportedto\nbeα= 0.018eV˚A, leading to α/β≈0.38, which is not too\nfar fromour tR/tD= 0.25in Sec.IIIB. Replacing these pa-\nrametersin ourresults doesnot changesignificantlythe mai n\nfeatureswehaveshown.\nCoupling ratio of tD/t0= 0.1is actually possible for\nQWs with stronger Dresselhaus bulk coefficient γ. For InSb\nQWs,27wehaveγ= 760eV ˚A3. ConsidertheIn 0.89Ga0.11Sb\nQWs with effective mass m= 0.018m0and sheet electron\nconcentration ns= 2.9×1011cm−2reported in Ref. 28,\nwhere the QW width is relatively thick: 30nm. If reducing\nthe QW width to 7.5nm, which is common in GaAs QWs,\nand assuming a= 3nm, the coupling ratio is estimated as\ntD/t0≈9.45×10−2, close to our tD/t0= 0.1. The Fermi\nenergy in units of t0is(EF−Eb)/t0= 2πa2ns≈0.16,\nwhich is also close to our (EF−Eb)/t0= 0.2. Hence the\ntransport parametersused in our calculation are within a re a-\nsonablerange.Ingeneralfora stronger tD/t0,whichcanberewrittenas\ntD\nt0=β/2a\n/planckover2pi12/2ma2≈amγ\n/planckover2pi12/parenleftBigπ\nw/parenrightBig2\n, (12)\na larger product mγ, and a thinner QW width wwill be re-\nquired. The effective mass mand Dresselhaus coefficient\nγfor various QWs taken from Ref. 27are collected in Ta-\nbleI. For these QWs we use Eq. ( 12) witha= 3nm to\nsummarize the coupling ratio tD/t0in Fig.5for QW widths\nw= 5nm,7.5nm,10nm.\nIV. CONCLUSION\nIn conclusion, we have shown that spin Hall induction for\n±[001]transportin (110)QWs due to antisymmetricCISP of\nlinearDresselhauscouplingthatyieldszerospin-orbitfo rceis\npossible to generate a spin Hall accumulation pattern in Sz,\nwhose signsdo not dependon the bias direction. Experimen-\ntal investigations with a dc bias offset in ballistic (or at l east\nquasi-ballistic) III-V (110) symmetric QWs may potentiall y\nidentifyourproposedeffect. Fromthecouplingratiossumm a-\nrized in Fig. 5, InSb (110)QW is promisingfor the presently\nproposed spin Hall induction in Sz, while InAs is less sug-\ngested. A new type of spin Hall deviation in Syis also pre-\ndictedintheDresselhaus(110)QWsinthepresenceofaweak\nRashba coupling. Experimental observation for this Syspin\nHall effect may require a good control over the Rashba and\nDresselhaus couplings, which has been proved possible for\n(001)QWs,29–31andshouldbeachievablealsofor(110)QWs.\nWecategorizethesetwointrinsicspinHallmechanisms—spi n\nHall inductionin Szandspin Halldeviationin Sy, asanoma-\nlousSHEs.\nWe note that the Dresselhaus cubic term, neglected in the\npresent study, will become important when Fermi wave vec-\ntorkFis long or QW width wis thick. In this case the spin\nHall induction in Szalong±[001] axis as discussed above\nshouldremain,whileadditionalspinHallinductionaxes close\nto[1¯11] and [ ¯112] will further emerge; see Fig. 6.20 in Ref.\n27andFig. 2(a) in Ref. 6. Inclusionof the Dresselhauscubic\ntermisleftasa futureextendingwork.\nACKNOWLEDGMENTS\nWe gratefully acknowledgeV. Sih, R. Myers, Y. Kato, and\nD. Awschalomforsharingtheirexperimentalinsight. M.H.L .\nappreciates E. Ya. Sherman for sharing his theoretical view -\npoint. This work is supported by Republic of China National\nScienceCouncilGrantNo. NSC98-2112-M-002-012-MY3.\n∗Current address: Institut f¨ ur Theoretische Physik, Uni-\nversit¨ at Regensburg, D-93040 Regensburg, Germany;\nminghao.liu.taiwan@gmail.com†crchang@phys.ntu.edu.tw\n1M. I.D’yakonov and V.I.Perel’,JETPLett., 13, 467 (1971).\n2M. I. D’yakonov and V. I. Perel’, Phys. Lett. A, 35, 459 (1971),6\nISSN0375-9601.\n3Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom,\nScience,306, 1910 (2004).\n4S. Murakami, N. Nagaosa, and S. C. Zhang, Science, 301, 1348\n(2003).\n5J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, an d\nA.H. MacDonald, Phys.Rev. Lett., 92, 126603 (2004).\n6V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gossard, and\nD.D. Awschalom, Nat.Phys., 1, 31 (2005).\n7S.Valenzuela andM. Tinkham, Nature, 442, 176(2006).\n8T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, and H. Imamura ,\nNat.Mater., 7, 125(2008).\n9J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phy s.\nRev. Lett., 94, 047204 (2005).\n10C. Br¨ une, A. Roth, E. G. Novik, M. K¨ onig, H. Buhmann, E. M.\nHankiewicz, W. Hanke, J. Sinova, and L. W. Molenkamp, Nat.\nPhys.,6, 448 (2010).\n11J.-i.Inoue, G. E.W.Bauer, andL.W.Molenkamp, Phys.Rev. B,\n70, 041303 (2004).\n12O.Chalaev and D.Loss, Phys.Rev. B, 71, 245318 (2005).\n13J. Li,L.Hu, andS.-Q.Shen, Phys.Rev. B, 71, 241305 (2005).\n14B. K. Nikoli´ c, L. P. Zˆ arbo, and S. Welack, Phys. Rev. B, 72,\n075335 (2005).\n15Y. A.Bychkov andE.I. Rashba, JETPLett., 39, 78(1984).\n16G.Dresselhaus, Phys.Rev., 100, 580 (1955).\n17M. I. D’yakonov and V. Y. Kachorovskii, Sov. Phys. Semicond. ,\n20, 110 (1986).18E.M.Hankiewicz,G.Vignale, andM.E.Flatt´ e,Phys.Rev.Le tt.,\n97, 266601 (2006).\n19V. M.Edelstein, SolidStateCommun., 73, 233 (1990).\n20Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom,\nPhys.Rev. Lett., 93, 176601 (2004).\n21M.-H. Liu, S.-H. Chen, and C.-R. Chang, Phys. Rev. B, 78,\n165316 (2008).\n22B. K. Nikoli´ c, S. Souma, L. P. Zarbo, and J. Sinova, Phys. Rev .\nLett.,95, 046601 (2005).\n23S.Datta,ElectronicTransportinMesoscopicSystems (Cambridge\nUniversityPress,Cambridge, 1995).\n24B. K. Nikoli´ c, L. P. Zarbo, and S. Souma, Phys. Rev. B, 73,\n075303 (2006).\n25I.Tokatlyand E.Sherman, Annals of Physics, 325, 1104 (2010).\n26E.Y. Sherman, A. Najmaie, and J.E.Sipe, Appl. Phys.Lett., 86,\n122103 (2005).\n27R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional\nElectronandHole Systems (Springer,Berlin,2003).\n28M. Akabori, V. A. Guzenko, T. Sato, T. Sch¨ apers, T. Suzuki, a nd\nS.Yamada, Phys.Rev. B, 77, 205320 (2008).\n29S. D. Ganichev, V. V. Bel’kov, L. E. Golub, E. L. Ivchenko,\nP. Schneider, S. Giglberger, J. Eroms, J. D. Boeck, G. Borghs ,\nW. Wegscheider, D. Weiss, and W. Prettl, Phys. Rev. Lett., 92,\n256601 (2004).\n30L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch¨ on, and K. E n-\nsslin,Nat. Phys., 3,650 (2007).\n31J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C .\nZhang,S.Mack, andD.D.Awschalom,Nature, 458,610(2009)." }, { "title": "1206.3103v2.Magnetic_ordering_phenomena_of_interacting_quantum_spin_Hall_models.pdf", "content": "Magnetic ordering phenomena of interacting quantum spin Hall models\nJohannes Reuther,1Ronny Thomale,2and Stephan Rachel3, 4\n1Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA\n2Department of Physics, Stanford University, Stanford, CA 94305, USA\n3Department of Physics, Yale University, New Haven, CT 06520, USA\n4Institute for Theoretical Physics, Dresden University of Technology, 01062 Dresden, Germany\nThe two-dimensional Hubbard model de\fned for topological band structures exhibiting a quan-\ntum spin Hall e\u000bect poses fundamental challenges in terms of phenomenological characterization\nand microscopic classi\fcation. In the limit of in\fnite coupling Uat half \flling, the spin model\nHamiltonians resulting from a strong coupling expansion show various forms of magnetic ordering\nphenomena depending on the underlying spin-orbit coupling terms. We investigate the in\fnite U\nlimit of the Kane{Mele Hubbard model with z-axis intrinsic spin-orbit coupling as well as its gener-\nalization to a generically multi-directional spin orbit term which has been claimed to account for the\nphysical scenario in monolayer Na 2IrO3. We \fnd that the axial spin symmetry which is kept in the\nformer but broken in the latter has a fundamental impact on the magnetic phase diagram as we vary\nthe spin orbit coupling strength. While the Kane{Mele spin model shows a continuous evolution\nfrom conventional honeycomb N\u0013 eel to XYantiferromagnetism which avoids the frustration imposed\nby the increased spin-orbit coupling, the multi-directional spin-orbit term induces a commensurate\nto incommensurate transition at intermediate coupling strength, and yields a complex spiral state\nwith a 72 site unit cell in the limit of in\fnite spin-orbit coupling. From our \fndings, we conjecture\nthat in the case of broken axial spin symmetry there is a large propensity for an additional phase\nat su\u000eciently large spin-orbit coupling and intermediate U.\nPACS numbers: 31.15.V-, 75.10.Jm, 03.65.Vf\nI. INTRODUCTION\nThe discovery of the quantum Hall e\u000bect has initial-\nized the era of topological phases in condensed matter\nphysics. For non-interacting band structures with topo-\nlogically unconventional properties, topological indices\ntake over the role of conventional order parameters and\ncan be linked to quantization phenomena of edge modes\nmeasured in experiment. The \frst example of such an in-\ndex was introduced by Thouless, Kohmoto, Nightingale,\nand den Nijs (TKNN) for the integer quantum Hall e\u000bect\n(IQHE).1They could show that the \frst Chern number|\nthe TKNN invariant|is proportional to the transversal\nHall conductivity \u001bxywhich is the integral of the Berry\ncurvature over the Brillouin zone. Nearly a decade ago\nafter Haldane realized that one can de\fne lattice versions\nof IQHE called Chern insulators where complex hopping\nbreaks time-reversal symmetry2, the most recent exam-\nple of a non-interacting topological state of matter is the\ntopological insulator3{5. It is characterized by a Z2topo-\nlogical index6,7.Z2topological insulators (TIs) have not\nonly been proposed theoretically6{8but have also been\nfound in subsequent experiments.9The minimal model\nof aZ2topological insulator is a four{band model pos-\nsessing a \fnite Z2invariant, which in its simplest form\nis a minimal time{reversal invariant generalization of a\nChern insulator. All two{dimensional band structures\nexhibiting a non-trivial Z2invariant can be adiabatically\ntransformed into each other, i.e. without closing the bulk\ngap. In contrast, transforming a Z2TI phase into any\nother topologically trivial phase causes a quantum phase\ntransition where the bulk gap must close. To date, thesetopological band insulators are well understood and sys-\ntematically classi\fed by symmetry.10,11\nAs soon as interactions are taken into account, the\nfull scope of possible scenarios extends to (i) topologi-\ncal band structure phases where the interactions would\nonly renormalize the band parameters but do not change\nthe topology along with (ii) conventional ordering phe-\nnomena where all features of the topologically non-trivial\nphase are gone12{24, and (iii) topological Mott insula-\ntors,12{14,25and (iv) topological bulk order driven by\nstrong interactions along with \fnite quantum dimen-\nsion26, fractionalization of quantum numbers27,28, and\nfractional statistics29as well as interaction-driven topo-\nlogical band structure phases which are not adiabatically\nconnected to the non-interacting limit14,19,25. In analogy\nto the non-interacting counterpart, topological bulk order\nwas \frst discovered in the fractional quantum Hall e\u000bect\n(FQHE)28,30before the concept of topological order was\nestablished by Wen26. Due to the diversity of possible\nphases even in the same symmetry sector, a general clas-\nsi\fcation for interacting topological phases is lacking so\nfar. Aside from many other challenges, it is of particular\ninterest whether the concept of a topological band struc-\nture and topological bulk order can both manifest itself\nin a single microscopic model31. For example, competing\nmagnetic \ructuations originating from a topological band\nstructure model could manage to stabilize a topological\nspin liquid phase. This is the general motif of a class of\nscenarios which we further investigate in this article.\nAs we will show in detail in the following, the pres-\nence or absence of axial spin symmetry stemming from\nthe topological band structure in the interacting casearXiv:1206.3103v2 [cond-mat.str-el] 27 Oct 2012iλσzi˜λσzi˜λσyi˜λσx(a)(b)FIG. 1. Intrinsic spin orbit terms with amplitude \u0015according\n(a) to (1) for the Kane{Mele model and (b) to (2) for the SI\nmodel with multi-directional SOC amplitude ~\u0015.\nwill crucially determine the magnetic order and disorder\nphenomena which appear in the strong coupling limit.\nGenerically, the full SU(2) is broken for interacting topo-\nlogical band structure models because of spin orbit cou-\npling terms. Still, it is both possible that the spin orbit\nterms break SU(2) down to U(1), leaving a continuous\naxial spin symmetry intact, or completely break spin ro-\ntation symmetry. Since its custodial time-reversal sym-\nmetry is una\u000bected, it is irrelevant for the Z2index of\nthe weakly coupled model whether the axial spin sym-\nmetry of the TI is conserved or not: although it has\nbeen shown recently that breaking of axial spin symme-\ntry causes a momentum{dependent rotation of the spin\nquantization axis of the helical edge states,32the topo-\nlogical band structure with conserved spin symmetry can\nstill be transformed into one with broken spin symmetry\nwithout closing of the bulk gap. In contrast, for strong\ninteractions, the resulting phase diagram crucially de-\npends on presence or absence of axial spin symmetry;\nmore speci\fcally, it was claimed that the combination of\nstrong interactions and strong spin orbit coupling might\ngive rise to a topologically ordered phase on the hon-\neycomb lattice when spin is not conserved. This would\nthen be a paradigmatic candidate model which includes\nboth a topological band structure phase and topological\nbulk order in its phase diagram.33Unfortunately, only\nthe conserved U(1) symmetry appears to open up the\npossibility to successfully perform quantum Monte Carlo\n(QMC) simulations for the regime of intermediately cou-\npled topological band structure models; when this sym-\nmetry is absent, we instead have to rely on limited mean{\n\feld, slave{particle, or other approximate methods.\nIn our work, we propose the strategy to \frst gain\ninsight about this kind of models in the limit of in-\n\fnitely large interactions on the footing of an accurate\nmethod adapted to this limit, and to \fnd out which\nof the approximate results at intermediate interaction\nstrength is compatible with it. For this purpose, we\nemploy pseudofermion functional renormalization group\n(PFFRG) which has been recently developed and em-\nployed by two of us in the context of various models offrustrated magnetism34{38. In particular, the anisotropic\nspin terms do not pose additional challenges to the per-\nformance of the PFFRG, which at the same time allows\nus to study large system sizes beyond any other mi-\ncroscopic numerical procedure for two-dimensional spin\nmodels.\nIn this paper, we investigate the strong coupling limit\nof two di\u000berent topological band structures accompanied\nwith Hubbard onsite interactions on the honeycomb lat-\ntice: the Kane{Mele (KM) model6,7preserving axial spin\nsymmetry and a related model which was proposed in the\ncontext of Na 2IrO3by Shitade et al.39which explicitly\nbreaks axial spin symmetry. Because of its connection to\nsodium iridate, it will be referred to as SI model in the\nfollowing. We \fnd that while magnetism in the presence\nof axial spin symmetry can generically avoid the frustra-\ntion e\u000bects caused by the anisotropic spin terms induced\nby spin-orbit coupling and generically yields commensu-\nrate magnetism, the broken axial spin symmetry scenario\nnaturally leads to commensurate-incommensurate transi-\ntions and, as a consequence, a much more complex mag-\nnetic phase diagram. As such, we conjecture that the\nlatter scenario will be most promising to stabilize un-\nconventional, possibly topologically bulk ordered phases\nresulting from anisotropic spin terms. We also discuss\nour \fndings in the context of recent results33for the cor-\nresponding Hubbard models at \fnite coupling.\nThe paper is organized as follows. In Section II, we\nintroduce the KM and SI models and discuss their main\nproperties. The mean \feld phase diagrams of the cor-\nresponding Hubbard models { the Kane{Mele{Hubbard\n(KMH) model as well as the sodium iridate Hubbard\n(SIH) model { are brie\ry reviewed in Section III. We\nsubsequently introduce the corresponding spin models in\nSection IV. In Section V, we elaborate on the PFFRG\nmethod which we employ to investigate the magnetic\nphase diagrams of the KM and SI spin models the results\nof which are presented in Section VI. In Section VII, we\ndraw a line from our \fndings at in\fnite coupling to the\ncorresponding Hubbard models at \fnite coupling in the\ncontext of the recently proposed QSH?phase, a topolog-\nically ordered phase in the SIH model.33In particular,\nwe also point out important generalizations of our study\nwith respect to Rashba coupling, which will generically\nbreak axial spin symmetry. In Section VIII, we conclude\nthat the role of the axial spin symmetry is crucial to\ncharacterize magnetic order and disorder phenomena of\ninteracting topological honeycomb band structures and\nleads to a better understanding of the general theme of\ninteraction e\u000bects in topological insulators.\nThroughout this paper we use the following notations:\nthe non{interacting topological insulators, i.e. the band\nstructures are denoted by hKMandhSI, respectively. The\ncorresponding Hubbard models are called HKMandHSI\nwhile the spin models are denoted by HKMandHSI, re-\nspectively. The real nearest neighbor hopping amplitude\nist; the intrinsic spin orbit couplings are called \u0015for the\nKM model and ~\u0015for the SI model.\n2II. TOPOLOGICAL BAND STRUCTURES\nThe QSH honeycomb models are particularly accessi-\nble from a theoretical perspective: as there are already\ntwo sites per unit cell, it is su\u000ecient to study a single\norbital scenario where complex hoppings generate the\nband inversion giving rise to a non-trivial Z2invariant.\nThere is hope that the QSH e\u000bect on the honeycomb lat-\ntice might be realized, e.g.by doping heavy adatoms in\ngraphene40or by using silicene41which has recently been\naccomplished experimentally42. Depending on the con-\ncise form of the spin-orbit coupling terms, the axial spin\nsymmetry may or may not be broken in the interacting\ncase. In this section we brie\ry introduce the two repre-\nsentative models for both scenarios which are subject to\nfurther investigation in the following.\nA. Kane{Mele model\nKane and Mele6,7proposed the quantum spin Hall\n(QSH) e\u000bect in graphene based on symmetry considera-\ntion. They realized that a mass term /\u001bz\u001cz\u0011zdoes not\nviolate any symmetries of graphene and thus must be al-\nlowed. Here, \u001bis associated with the electron spin, \u001cwith\nthe valleys, and \u0011with the sublattices. The Kane{Mele\nmodel is governed by the tight{binding Hamiltonian\nhKM=\u0000tX\nhiji\u001bcy\ni\u001bcj\u001b+i\u0015X\n\u001cij\u001dX\n\u000b\f\u0017ijcy\ni\u000b\u001bz\n\u000b\fcj\f (1)\nIn principle, there is also the Semeno\u000b mass term which\nwe will ignore for the moment. Similarly, the Rashba spin\norbit term with amplitude \u0015Ris neglected unless noted\notherwise. The \frst term in (1) is the usual nearest{\nneighbor hopping on the honeycomb lattice giving rise\nto the Dirac band structure. The second term in (1) is\nthe lattice version of the \u001bz\u001cz\u0011z{term (a second neigh-\nbor hopping) which corresponds to an intrinsic spin orbit\ncoupling (SOC). The convention of this hopping is illus-\ntrated in Fig. 1a. The nearest neighbor hopping term\npreserves the C6vlattice symmetry of the honeycomb\nlattice as well as SU(2) symmetry of the electron spin.\nThe intrinsic SOC reduces the lattice symmetry to C3v\nand the spin symmetry to U(1). Any \fnite \u0015opens the\ngap of the Dirac band structure and gives rise to QSH\ne\u000bect, i.e. to a topological insulator phase characterized\nby a \fnite Z2invariant, or, in this case, Chern number for\neach spin species. This situation is very special since the\nHamiltonian fully decouples into two independent Chern\ninsulators with opposite Hall conductivity. Generically,\nwe expect the presence of additional terms breaking the\nU(1) spin symmetry and mixing the spin channels. The\nRashba term is such an additional term which will be fur-\nther commented on in Section VII. Even for \fnite Rashba\ncoupling\u0015R, however, the QSH phase is stable as long as\n\u0015R<2p\n3\u0015.6B. Sodium iridate tight binding model\nSoon after Kane and Mele's milestone works, it turned\nout that the spin orbit gap in graphene is vanishingly\nsmall. Therefore other materials with e\u000bective honey-\ncomb structure were considered as candidates for the\nQSH e\u000bect as proposed by Kane and Mele. In 2008, Shi-\ntade et al.39came up with the sodium iridate Na 2IrO3\nas a layered honeycomb system. The authors claimed\nthat the QSH e\u000bect might be realized if Coulomb inter-\nactions are not too strong. A monolayer was shown to\nbe described by a Kane{Mele-type Hamiltonian. The in-\ntrinsic spin orbit coupling was assumed to be relatively\nlarge due to the heavier iridium atoms in contrast to\ngraphene's carbon atoms. Assuming trivial hybridiza-\ntion between nearest neighbor Ir atoms, Shitade et al.\nfound an intrinsic SOC being similar but di\u000berent to the\nKM SOC. It depends on the direction of the spin orbit\nhopping whether the spin degree of freedom is associated\nwith\u001bx,\u001by, or\u001bz. The sodium iridate model is governed\nby the Hamiltonian\nhSI=\u0000tX\nhiji\u001bcy\ni\u001bcj\u001b+i~\u0015X\n\u001cij\u001d\rX\n\u000b\fcy\ni\u000b\u001b\r\n\u000b\fcj\f;(2)\nwhere\r=x;y;z is associated with the di\u000berent next{\nnearest neighbor links on the honeycomb lattice (Fig. 1b).\nThe main di\u000berence of this generalized SOC compared\nto the KM SOC is that axial spin symmetry is not con-\nserved. As for the KM model, in\fnitesimally small ~\u0015\nopens the gap at the Dirac cones and causes QSH e\u000bect.\nThe band structures of hKMandhSIboth belong to\ntheZ2universality class and are thus adiabatically con-\nnected. Both systems exhibit helical edge states on open\ngeometries such as cylinder or disk.\nIII. CORRELATED TOPOLOGICAL\nINSULATORS\nLet us now add Hubbard onsite interactions,\nHI=UX\nini\"ni# (3)\nwhich yields rich phase diagrams for both band struc-\ntures. While the U{\u0015phase diagram of the KMH model\nis well understood13,15,16,19,43,44, theU{~\u0015phase diagram\nof the SIH model is rarely studied33,39, and the available\nresults are controversial. In the following, we will brie\ry\nreview the phase diagrams of both Hubbard-type models.\nA. Kane{Mele{Hubbard model\nThe KMH model is described by a combination of the\nKM and Hubbard model,\nHKM=hKM+HI: (4)\n3XY - AFMTISM\nU/tTIVBS(AFM)QSH*SMU/tλ/t˜λ/t\nNeel051015\n0.60.80.40.20\n(a)(b)0.60.80.40.20024FIG. 2. (color online). (a) phase diagram of the Kane{Mele{Hubbard model as obtained in Ref. 13. The transition from a\ntopological insulator (TI) to XY-plane antiferromagnet (AFM) was derived within slave-rotor theory which underestimates\nUc. (b) mean \feld phase diagram of the sodium{iridate Hubbard model as obtained in Ref. 33. The transition from TI to a\nvalence bond solid (VBS) phase that links to AFM was derived within slave-spin theory which overestimates Uc. The phase\ndiagram of (b) is qualitatively similar to (a) apart from the additional \\QSH?phase\". At \u0015=~\u0015= 0 and not too large Uthe\nsemi-metal (SM) phase of graphene is present. See main text for details.\nIn Ref. 13 the phase diagram shown in Fig. 2a was de-\nrived through slave rotor theory. The semi{metal (SM)\nphase of graphene ( \u0015= 0) as well as the topological in-\nsulator phase ( \u00156= 0) are stable up to moderate inter-\nactions. Above a critical interaction strength Uc, one\n\fnds an antiferromagnetically ordered phase which is of\nN\u0013 eel type ( \u0015= 0) or ofXY{type (\u00156= 0), respectively.\nAt\u0015= 0 and intermediate U, a quantum spin liquid\nphase has been proposed45recently; this conjecture has\nbeen challenged lately.46For very small \u0015it survives but\neventually vanishes for \u0015\u00140:05t15,16,43,44. Since the spin\nliquid is destroyed by \fnite \u0015and just a remnant of the\nnon{topological \u0015= 0 case, we omit the phase here for\nclarity. Also for the strong coupling analysis in this paper\nwe will assume that we are deep in the strong coupling\nregime where this intermediate coupling phenomenon is\nirrelevant for our analysis.\nB. Sodium iridate Hubbard model\nRecently, R uegg and Fiete have studied the SIH\nmodel33governed by the Hamiltonian\nHSI=hSI+HI: (5)\nThey used a Z2slave{spin mean{\feld approach and pro-\nposed an interesting phase diagram (Fig. 2b). It is simi-\nlar to the KMH model, while there is an additional phase\nfor large SOC ~\u0015and largeU, dubbed QSH?phase, which\npresumably extends to the strong coupling regime. Note\nthat this is not a quantum spin Hall phase, but a topo-\nlogical liquid which is characterized by a four{fold de-\ngeneracy on a torus, where the elementary excitations\nare fractional particles obeying Abelian statistics. Re-\ncently it was questioned, however, whether the employed\nZ2slave spin approach is justi\fed.47Also, within the Z2\nslavespin approach one cannot \fnd local moments suchas an antiferromagnetically ordered phase (AFM), but\ninstead obtains a valence bond solid (VBS) phase. In the\nlimit ~\u0015!0 it is obvious that one should \fnd Neel or-\nder instead and that the VBS order is an artifact of the\nspeci\fc slave particle approach.\nRegarding the values of Uc(e.g.for\u0015=~\u0015= 0), one\nshould keep in mind that the microscopic Uc\u00184:3 as\nfound within QMC45is understimated by slave rotor the-\nory (Uc= 1:68) while it is overestimated by the slave spin\napproach (Uc\u00188) (Fig. 2).\nIV. STRONG COUPLING LIMIT\nWe consider the limit of in\fnitely strong electron{\nelectron interactions. As a result, charge \ructuations\nare frozen out and we obtain a pure spin Hamiltonian at\nhalf \flling. Most importantly, the complex next{nearest\nneighbor spin orbit hoppings result in anisotropic and\nmore complicated second neighbor spin exchange terms\nwhich we analyze in the following.\nA. Kane{Mele spin model\nTaking the limit U!1 of the Kane{Mele{Hubbard\nmodel (4) results in the e\u000bective spin model13\nHKM=J1X\nhijiSiSj+J\u0015X\n\u001cij\u001d\u0002\n\u0000Sx\niSx\nj\u0000Sy\niSy\nj+Sz\niSz\nj\u0003\n(6)\nwhereJ1= 4t2=UandJ\u0015= 4\u00152=U. The second neigh-\nbor exchange term (indicated by \u001c\u0001\u001d ) acting merely\non individual, i.e. triangular sublattices partially frus-\ntrates the system. The XY-spin terms prefer ferromag-\nnetic order on the individual sublattices which is consis-\ntent with antiferromagnetic order on the original honey-\n4comb lattice; in contrast, the Ising term Sz\niSz\njfavors an-\ntiferromagnetic order on the sublattice competing with\nboth theXY-terms and the J1term. The magnetiza-\ntion, which might point in any direction for J\u0015= 0 due\nto spin rotational invariance, turns into the XY-plane\nin order to avoid the frustrating part of the J\u0015term13.\nThese \fndings were con\frmed within QMC15,16, varia-\ntional cluster approximation (VCA)43, and cluster dy-\nnamical mean{\feld theory (CDMFT)44calculations at\nintermediate U=t\u00195:::9 and small \u0015. For small J\u0015,\none can thus employ HKMto compare other numerical\napproaches against PFFRG method, which we will use\nin the following.\nB. Sodium iridate spin model\nThe strong coupling limit of the SIH model is given by\nthe spin Hamiltonian\nHSI=J1X\nhijiSiSj\u0000J~\u0015X\n\u001cij\u001dSiSj+2J~\u0015X\n\r\u0000linksS\r\niS\r\nj:(7)\nNote that the \r{links are the second neighbor links (the\ngreen, red, and blue lines in Fig. 1b). It is structurally\nsimilar to the Heisenberg{Kitaev (HK) Hamiltonian37,48\nwhich has been found to adequately describe the A 2IrO3\niridates from a spin-orbit Mott picture (A=Na or Li)38.\nWhereas the SI model assumes the nearest-neighbor hy-\nbridization to be trivial and to be essentially given by real\nIr-Ir hybridization, the kinetic theory underlying the HK\nmodel more carefully resolves the emergent terms from a\nmulti-orbital Ir-O cluster superexchange model48,49. De-\npending on the Ir-O-Ir angle, these terms are either more\nor less relevant than the next nearest neighbor exchange\nterms which are considered in the SI model37. For the\nlinks in vertical direction (links with \u001bz), one obtains the\nsame term as forHKM, while for the links associated with\n\u001bxone \fnds + Sx\niSx\nj\u0000Sy\niSy\nj\u0000Sz\niSz\njand so on. ForHKM\nwe have seen that the magnetization turns into the XY-\nplane. Here, however, the term + Sx\niSx\nj\u0000Sy\niSy\nj\u0000Sz\niSz\nj\nwill force the magnetization into the YZ-plane while the\nterm\u0000Sx\niSx\nj+Sy\niSy\nj\u0000Sz\niSz\njfavors theXZ-plane and\nso on. Since all the terms (links) are equally distributed\nover the lattice, a priori no plane or direction is preferred.\nAs the system sets out to be N\u0013 eel{ordered for J~\u0015= 0,\nit is conceivable that the competing ordering tendencies\nmight at \frst compensate each other and allow for a per-\nsistent N\u0013 eel order at small J~\u0015.\nV. METHOD\nThe PFFRG approach34{37,50starts by reformu-\nlating the spin Hamiltonian in terms of a pseudo\nfermion representation of the spin-1/2 operators S\u0016=\n1=2P\n\u000b\ffy\n\u000b\u001b\u0016\n\u000b\ff\f, (\u000b;\f=\";#,\u0016=x;y;z ) with fermionic\n0.00.20.40.60.81.00.00.51.01.52.02.53.0\nLcmaxLFIG. 3. Characteristic behavior of the \rowing (\u0003-dependent)\nsusceptibility in a magnetically ordered phase. While the RG\n\row is smooth above some critical value \u0003 c\u00190:45, a numeri-\ncally unstable regime is found below that value. This feature\nsignals a magnetic instability which becomes a divergence in\nthe thermodynamic limit. The speci\fc case shown here rep-\nresents the largest component of the susceptibility of HSIat\nJ~\u0015= 0:4J1where the system favors antiferromagnetic order.\noperatorsf\"andf#and Pauli-matrices \u001b\u0016. Such a rep-\nresentation enables us to apply Wick's theorem, lead-\ning to standard Feynman many-body techniques. In this\npseudofermion language, quantum spin models become\nstrongly coupled models with zero fermionic bandwidth\nand \fnite interaction strength.\nA major advancement of the PFFRG35is that it allows\nto tackle this situation by providing a systematic scheme\nfor the in\fnite order self-consistent resummations. The\n\frst conceptual step is the introduction of an infrared fre-\nquency cuto\u000b \u0003 in the fermionic propagator. The FRG\nthen formulates di\u000berential equations for the evolution\nof allm-particle vertex functions under the \row of \u000351.\nHence, one might think of the diagrammatic summations\nas being performed during the RG \row: each discretized\nRG step e\u000bectively increases the amount of diagrams in-\ncluded in the approximation.\nTo reduce the in\fnite hierarchy of coupled equations to\na closed set, a common approach is to restrict oneself to\none-loop diagrams. The PFFRG extends this approach\nby also including certain two-loop contributions35,52to\nretain a su\u000ecient backfeeding of self-energy corrections\nto the two-particle vertex evolution. A crucial property\nof the PFFRG is that the the two-particle vertex includes\nboth graphs that favor magnetic order and those that fa-\nvor disorder in such a way that the method treats both\ntendencies on equal footing35. It is the two-particle ver-\ntex which allows to extract magnetic susceptibility as the\nmain outcome of the PFFRG. The FRG equations are\nsimultaneously solved on the imaginary frequency axis\nand in real space. A numerical solution requires (i) to\ndiscretize the frequency dependencies and (ii) to limit\nthe spatial dependence to a \fnite cluster, thus keeping\ncorrelations only up to some maximal length. In our cal-\nculations, the latter typically extends over distances of\nup to 9 lattice spacings corresponding to a correlation\n5area (cluster size) of 181 lattice sites of the hexagonal\nlattice. The onset of spontaneous long-range order is sig-\nnaled by a sudden breakdown of the smooth RG \row,\nwhile the existence of a stable solution indicates the ab-\nsence of long-range order. (See Refs. 34 and 35 for further\ntechnical details.) Fig. 3 shows an example for the char-\nacteristic \row behavior in a magnetically ordered phase.\nVI. RESULTS\nA. Kane{Mele spin model\nFrom Eq. (6), the Kane{Mele spin model reduces to\nan isotropic nearest neighbor spin system in the limit of\nvanishing spin orbit coupling J\u0015= 0. In this case, the\nsystem exhibits the standard N\u0013 eel state on the honey-\ncomb lattice. Within our PFFRG approach, this type of\norder is signaled by an instability breakdown in the RG\n\row occurring at the K- andK0-points, i.e. the corners\nof the extended (second) Brillouin zone of the honeycomb\nlattice. (Unless stated otherwise, we plot the susceptibil-\nity in the second Brillouin zone of the underlying two-\natomic Bravais lattice because the experimentally con-\nnected unfolded susceptibility has the periodicity of this\nextended zone.) Hence, at an RG scale right before the\nmagnetic order sets in, the momentum resolved suscep-\ntibility shows pronounced peaks at the K- andK0-point\npositions. As a consequence of rotational invariance the\nsusceptibility pro\fle is identical for all directions of exter-\nnal magnetic \felds. Once the spin orbit interaction J\u0015is\nswitched on, the situation changes considerably as shown\nin Fig. 4. While the susceptibility peaks for an external\n\feld inx-direction (or y-direction) become even sharper\nas compared to J\u0015= 0, the peaks in the z-component\ndrop drastically. Already at small J\u0015= 0:1 this e\u000bect is\nrather pronounced, which evidences that for \fnite J\u0015, the\nspins favor the x-yplane. (We set J1= 1 in this section.)\nWith increasing J\u0015, more weight of the z-susceptibility is\ntransferred to the x- andy-components of \u001f. For strong\nenoughJ\u0015, the remnant magnetic \ructuations in \u001fzare\nnot of antiferromagnetic type anymore, which can be seen\nin Fig. 4 for J\u0015= 0:5 showing small maxima at M-point\npositions. We do not observe any particular phase transi-\ntion at\u0015>0. In particular, the magnetic order persists\nin the whole parameter space. The frustration gener-\nated by the J\u0015Sz\niSz\nj-terms has little e\u000bect because the\nspins can circumvent this frustration by avoiding the z-\naxes. With increasing J\u0015, the two sublattices become\ne\u000bectively decoupled such that in the limit J\u0015!1 both\nsublattices exhibit xyferromagnetic order independently.\nB. Sodium iridate spin model\nAs for the KM spin model in the previous section, the\nSI spin model becomes a simple isotropic nearest neigh-\nbor spin system in the limit J~\u0015= 0 and hence shows\n0-4-2\n2402\n-2\nkxyk0.2.4()kz=0.5J\n0-4-2\n2402\n-2\nkxyk01020()kx=0.5J\n0-4-2\n2402\n-2\nkxyk024()kx=0.1J\n0-4-2\n2402\n-2kxyk0.51()kz=0.1JFIG. 4. Magnetic susceptibilities at the critical scale \u0003 = \u0003 c\nfor various values of J\u0015(J1= 1) in the Kane{Mele spin\nmodel, resolved for in plane ( x;y) and out of plane ( z). Top\nrow:\u001fx(k) (left panel) and \u001fz(k) (right panel) for J\u0015= 0:1.\nBottom row: \u001fx(k) =\u001fy(k) (left panel) and \u001fz(k) (right\npanel) forJ\u0015= 0:5. The susceptibility weight along zsignif-\nicantly decreases for large J\u0015. For higher J\u0015, the remainder\nz-susceptibility deviates from the N\u0013 eel AFM structure.\nN\u0013 eel order (upper left plot in Fig. 5). For \fnite but not\ntoo largeJ~\u0015, the antiferromagnetic order persists, i.e.\nthe position of the ordering peaks in the susceptibility\nremains unchanged ( J~\u0015= 0:5 in Fig. 5). As the suscep-\ntibility looses its sixfold rotation symmetry for \fnite J~\u0015,\nthis manifests in the deformation of the ordering peaks as\ncompared to J~\u0015= 0. Note that due to the special connec-\ntion between lattice directions and spin directions in the\nSI spin model, the x-,y- andz-components of the suscep-\ntibility transform into each other under k-space rotations\nof 120\u000ein clockwise direction. Fig. 5 illustrates \u001fzwhich\npreserves the symmetries kx!\u0000kxandky!\u0000ky. Note\nthat regardless of the particular phase, the value of the\nsusceptibility at the six K- andK0-points must always be\nequal. This results from the fact that the three K-points\n(orK0-points) are related by reciprocal lattice vectors\namong each other. Furthermore, since the two sublat-\ntices are equivalent, the K- andK0-points are likewise\ndegenerate.\nAn interesting observation can be made regarding the\norientation of the antiferromagnetic order. Due to the\nequivalence of the x-,y- andz-direction in spin space,\nthe magnetic order can point in each of these directions\nwithout any preference. Even though SU(2) symmetry is\nexplicitly broken, the rotational symmetry of the suscep-\ntibility prevails: consider a magnetic \feld B=vBpoint-\ning in some direction v=P\n\u0016=x;y;zv\u0016e\u0016withjvj= 1.\nThe corresponding susceptibility \u001fv, i.e. the linear re-\n6-4-202402\n-202\nkxyk=0J~\n()k\n-4-202402\n-22\n0\nkx=0.5J~\n()k\nyk\n-4-202402\n-22\n0\nkxyk=0.6J~\n()k\n-4-202402\n-22\n0\nkxyk=0.7J~\n()k\n-4-202402\n-21\n0\nkxyk=1J~\n()k\n4\n2\n0\n2\n4\n2\n0\n2\n0\n.1\n.2\n-4-202402\n-20\nkxyk.2\n.1()k=20J~\n4\n2\n0\n2\n4\n2\n0\n2\n0\n.2\n.4\n-4-202402\n-20\nkxyk.2.4()k=10J~\n.5\n-4-202402\n-20\nkxyk=5J~\n()kFIG. 5. Magnetic susceptibilities of the SI spin model for various values of J~\u0015(J1= 1). All susceptibilities shown refer to\na magnetic \feld in z-direction. The x- andy-components of the susceptibility are obtained by k-space rotations of 120\u000ein\nclockwise or counterclockwise direction, respectively. (See Section VI B for more details.) While the N\u0013 eel peaks initially persist\nfor \fniteJ~\u0015, the peaks start to move due to the onset of incommensurability (Fig. 6). For large J~\u0015, new suceptibility peaks\nemerge which link to the change of unit cell structure of magnetic order.\nsponse to such a perturbation is de\fned as\n\u001fv=@Mv\n@B\f\f\f\nB!0=@(P\n\u0016=x;y;zM\u0016v\u0016)\n@B\f\f\f\nB!0(8)\n=X\n\u00160=x;y;z@(P\n\u0016=x;y;zM\u0016v\u0016)\n@B\u00160@B\u00160\n@B\f\f\f\nB!0\n=X\n\u0016;\u00160=x;y;zv\u0016\u001f\u0016\u00160v\u00160;\nwhere\u001f\u0016\u00160=@M\u0016\n@B\u00160\f\f\nB!0andMis the magnetization.\nSince\u001f\u0016\u00160cannot develop any o\u000b-diagonal elements be-\nfore reaching the magnetic instability in the RG \row,53\nwe have\u001f\u0016\u00160=\u000e\u0016\u00160\u001f\u0016. It follows that\n\u001fv=X\n\u0016=x;y;zv2\n\u0016\u001f\u0016: (9)\nSince\u001fx=\u001fy=\u001fzat allK(0)-points, we obtain\n\u001fv\nK(0)=\u001fz\nK(0)X\n\u0016=x;y;zv2\n\u0016=\u001fz\nK(0): (10)\nHence, in linear response the low energy physics of the\nsystem is rotationally symmetric and the antiferromag-\nnetic order can point in any direction. This is a con-\nsequence of the N\u0013 eel order residing at high-symmetry\npoints of the Brillouin zone as well as the special con-\nnection between lattice directions and spin directions in\nthe SI spin model. However, this argument does not hold\nfor spin \ructuations away from the K- orK0-points. For\n\ructuations at arbitrary momentum, a certain direction\nwill generally be preferred.\n2π\n3\n2π\n32\n32π\n35\n6\n00.20.4 0.6 0.811.21.4Qy\nAF spiral2π\n32\n32π\n3ky\nJ~FIG. 6. Dependence of the ordering vector QyonJ~\u0015inHSI.\nThe inset illustrates the evolution of the ordering peaks in the\nBrillouin zone (thick hexagon: second Brillouin zone, thin\nhexagon: \frst Brillouin zone; see also Fig. 5). In the limit\nJ~\u0015!1 , the system converges again towards a commensurate\nordering vector.\nAsJ~\u0015increases, the deformation of the ordering peaks\nat theK- andK0-points becomes more pronounced. At\nsome coupling J~\u0015\u00190:53, the peaks split and the new\nmaxima move along the ky-direction (Fig. 6). These peak\npositions indicate a phase transition to a spiral phase\nwith incommensurate order. It is important to note, how-\never, that magnetic order persists in the whole parameter\nregime around the transition and we \fnd no magnetically\ndisordered phase. This can be seen from the behavior of\nthe RG \row which always exhibits a characteristic insta-\nbility breakdown. To demonstrate the evolution of the\n7ordering vector in the spiral phase, Fig. 6 shows the peak\nposition as function of J~\u0015. Note that the kx-component\nof the peak position is constant in J~\u0015. With increas-\ningJ~\u0015, the peaks move continuously towards the points\nQ1= (\u00062\u0019\n3;\u00062\n32\u0019p\n3) which lie at two third of the distance\nbetween the K(0)-points and the kx-axis (Fig. 6). Again,\nwith increasing J~\u0015there is no sign of any non-magnetic\nphase.\nThe system at in\fnite spin-orbit coupling is of particu-\nlar interest, as this case represents a model with Kitaev-\nlike interactions on the triangular lattice. As J~\u0015goes to\nin\fnity, the system is e\u000bectively described by decoupled\ntriangular sublattices. Hence, already the \frst Brillouin\nzone, i.e. the Brillouin zone of a triangular sublattice,\ncontains the full information about \u001fink-space. The\nsusceptibility then becomes periodic with respect to this\nsmaller zone. Such a change of periodicity can be seen in\nFig. 5 at large J~\u0015where new peaks at kx= 0 emerge. In\nthe limitJ~\u0015!1 these new peaks reach the same height\nas the ones at Q1and \fnally become identical to them,\nindicating the new periodicity in k-space. Fig. 7a shows\nthe susceptibility in the \frst Brillouin zone of the trian-\ngular sublattice in this limit. From the peak positions,\none can easily derive the corresponding spin pattern in\nreal space. On each triangular sublattice the wave vec-\ntor is half the one of the 120\u000e-N\u0013 eel order residing at the\ncorners of the \frst Brillouin zone. The unit cell contains\n6\u00026 lattice sites as compared to the 3 \u00023 unit cell of the\n120\u000e-N\u0013 eel order. Hence, the order is commensurate and\nthe local magnetic moments along a lattice direction are\nmodulated with a periodicity of 6 sites. Taking into ac-\ncount both sublattices of the honeycomb lattice, we end\nup with a unit cell containing 72 sites.\nOur numerical conclusions for the SI spin model in the\nlimitJ~\u0015!1 can also be reconciled with an analytical\nargument. Performing a transformation in spin space,\nSi!~Si, the system at this point can be mapped to\nan SU(2) invariant antiferromagnetic Heisenberg model\non the triangular lattice, HSI=P\nij~Si~Sj. For this map-\nping, we divide the triangular lattice into four sublattices\ndenoted by \"\u000f\", \"xy\", \"xz\" and \"yz\", each with a dou-\nbled lattice constant (Fig. 7b). The relation between Si\nand~Sidepends on the sublattice,\ni2\"\u000f\" : ~Si= (Sx\ni;Sy\ni;Sz\ni);\ni2\"xy\" : ~Si= (\u0000Sx\ni;\u0000Sy\ni;Sz\ni);\ni2\"xz\" : ~Si= (\u0000Sx\ni;Sy\ni;\u0000Sz\ni);\ni2\"yz\" : ~Si= (Sx\ni;\u0000Sy\ni;\u0000Sz\ni); (11)\ne.i., while on sublattice \" \u000f\" the spins remain unchanged,\non the sublattice \" xy\" thex- andy-components of the\nspin operator acquire a minus sign, and so on (a similar\nmapping for the Heisenberg-Kitaev model at \u000b= 0:5\nis described in Ref. 48). Since the antiferromagnetic\nHeisenberg model on the triangular lattice exhibits mag-\nnetic order via the 120\u000e-N\u0013 eel state54, it follows that the\nSI spin model at J~\u0015!1 is likewise magnetically or-\n2\n1\n0\n1\n2\n2\n1\n0\n1\n2\n4\n2-2-101-2-10124\n02()k\nkxykxy\nyz xzxyxy\nxy\nxzxzxz\nyz\nyzyz(a) (b)FIG. 7. The SI spin model at J~\u0015!1 : (a) Magnetic suscep-\ntibility displayed in the \frst Brillouin zone of the triangular\nsublattice. The two ordering peaks correspond to the peaks\nin Fig. 5 which emerge at J~\u0015&5 andkx= 0. In the limit\nJ~\u0015!1 , these maxima reach the same hight as the ones at\nQ1= (\u00062\u0019\n3;\u00062\n32\u0019p\n3). (b) Mapping of the SI spin model at\nJ~\u0015!1 to the antiferromagnetic Heisenberg model on the\ntriangular lattice: The lattice is divided into four sublattices\ndenoted by \"\u000f\", \"xy\", \"xz\" and \"yz\". As shown in Eq. (11)\nthe transformation from Sito~Sidepends on the sublattice\nwhereiresides. The exchange couplings follow the convention\nshown in Fig. 1.\ndered. The corresponding spin pattern in real space can\nbe found by applying the inverse of the above spin trans-\nformation to the 120\u000e-N\u0013 eel state: Since the structure of\nthe spin rotations (Fig. 7b) has a periodicity of two lat-\ntice sites in each lattice direction, the 3 \u00023 unit cell of\nthe 120\u000e-N\u0013 eel order transforms back into a 6 \u00026 unit cell,\nas found within our PFFRG calculations.\nVII. DISCUSSION\nIn view of our results for the SI model, we specu-\nlate about the implications for the phase diagram at in-\ntermediate U. We \fnd the incommensurate phase for\nJ~\u0015=J1\u00150:53 whereJ~\u0015= 4~\u00152=UandJ1= 4t2=U, imply-\ning a transition at~\u0015\nt\u00190:73 for large U. In Fig. 8, we\nhave replotted the phase diagram of R uegg and Fiete33in\na slightly modi\fed way. Our reasoning is the following:\nsince the spin model corresponds to U!1 , we extrap-\nolated the phase boundary between \\VBS (AFM)\" and\n\\QSH*\" of the phase diagram in Ref. 33 to larger U. As\nthe phase transition occurs for su\u000eciently large Uap-\nproximately at~\u0015\nt\u00190:73 (Fig. 8), we speculate that the\nobserved phase transition from N\u0013 eel to spiral order in the\nspin model is a remnant of the phase transition into the\nQSH* phase at intermediate U. Within this scenario,\nthe QSH* phase would transform into spiral magnetic\norder in the limit U!1 . We note, however, that this\nnecessarily implies that with increasing Uthe gap of the\nQSH* phase closes at some point to form the Goldstone\nmode of the spiral order. In principle, the gap closure\ncan occur at \fnite U(which would imply an additional\n80.2\nTIAFM(Neel)SM\nU/t0.050.50\n0.60.80.40.20051015˜λ/tNeelspiralJ˜λ/J1FIG. 8. (color online) Schematic phase diagram for the SI\nHubbard model as conjectured from our strong coupling re-\nsults of a N\u0013 eel to spiral transition at J~\u0015= 0:53 (upperx\nscale), corresponding to ~\u0015=t\u00190:73 (lowerxscale) . While\nwe cannot ultimately assess the nature of a possible interme-\ndiate phase aside from TI and AFM for strong interaction\nand strong spin-orbit coupling (yellow phase), its existence is\nlikely.\nphase boundary in Fig. 8) or at U!1 . In the latter\nscenario, the QSH* phase could extend up to U!1 .\nFurthermore, one should keep in mind that the QSH*\nphase as found in Ref. 33 might not be the only topo-\nlogical liquid candidate with similar properties, such as adoubled semion spin liquid55. The alternative scenario|\nassuming that a QSH?-type phase does notexist|would\nstill require an additional phase comparable to the yel-\nlow phase of the schematic phase diagram in Fig. 8); in\nthis case, the additional phase would most likely be a\nmagnetically ordered phase ( e.g.spiral order). Whether\nor not this phase is a topological liquid or just another\nmagnetically ordered phase, we conjecture that in either\ncase an additional phase of some kind should be present.\nIn summary, we \fnd that the physics of the SI spin\nmodel is much richer as compared to the KM spin model.\nThis can be traced back to the di\u000berent spin symme-\ntries in both systems. The broken axial symmetry in\nthe SI spin model prevents the spins from forming pla-\nnar antiferromagnetic order and eventually leads to the\nemergence of a spiral phase. Note that this phase does\nnot have any analogue in similar models such as the\nHeisenberg-Kitaev model.37As such, we have identi\fed\nmulti-directional spin orbit terms to be an interesting\nway to create new spin phases in the in\fnite Ulimit and\npossibly even more exotic phases at intermediate Uwhen\ncharge \ructuations enter the picture.\nTo give another direction of further investigation, it\nwill be interesting to study the KM model in the presence\nof Rashba spin orbit coupling\nHR=i\u0015RX\nhijiX\n\u000b\fcy\ni\u000b[^ez(\u001b\u0002dij)]\u000b\fcj\f\nThe Rashba term breaks the remaining U(1) symmetry of\nthe electron spin to Z2, and also a\u000bects the z!\u0000zmir-\nror symmetry as well as particle{hole symmetry. Taking\ninto account the Rashba spin orbit coupling results in a\nmore complicated spin Hamiltonian with some terms be-\ning of Dzyaloshinskii-Moriya type. The full Hamiltonian\nis given by\nHKMR =J\u0015X\n\u001cij\u001d\u0002\n\u0000Sx\niSx\nj\u0000Sy\niSy\nj+Sz\niSz\nj\u0003\n+X\n\u000e1\u0000linksh\n(J1+JR)Sx\niSx\nj+ (J1\u0000JR)(Sy\niSy\nj+Sz\niSz\nj)\u0000p\nJ1JR(Sy\niSz\nj\u0000Sz\niSy\nj)i\n+X\n\u000e2\u0000links\"\nSx\ni=j!\u00001\n2Sx\ni=j\u0000p\n3\n2Sy\ni=jandSy\ni=j!p\n3\n2Sx\ni=j\u00001\n2Sy\ni=j#\n+X\n\u000e3\u0000links\"\nSx\ni=j!\u00001\n2Sx\ni=j+p\n3\n2Sy\ni=jandSy\ni=j!\u0000p\n3\n2Sx\ni=j\u00001\n2Sy\ni=j#\n; (12)\nwhere the third line in (12) is obtained from the second\none by replacing Sx\ni=jby\u00001=2Sx\ni=j\u0000p\n3=2Sy\ni=jand so\non. The di\u000berent links denoted by \u000ei(i= 1;2;3) are the\nthree nearest neighor vectors of the honeycomb lattice.\nWe expect the J\u0015{JRphase diagram to be interesting\nand to host some additional phases, which we defer to afuture publication.\n9VIII. CONCLUSION\nWe have investigated the strong coupling limit of Hub-\nbard models of topological honeycomb band structures.\nWe have considered two band structures both classi\fed\nas two{dimensional Z2topological insulators, where only\nthe Kane{Mele spin model as opposed to the sodium iri-\ndate model preserves axial spin symmetry. For the for-\nmer model at in\fnite coupling, the magnetism tends to\nformXY antiferromagnetic order already at very small\nspin orbit couplings. This way the spins manage to avoid\nthe frustration induced by the spin-orbit anisotropic spin\nterms. As a consequence, frustration e\u000bectively plays no\nrole in the KM spin model. The physical scenario is very\ndi\u000berent for the sodium iridate model with generalized\nspin orbit couplings and hence broken axial spin symme-\ntry. There, the spins cannot form XY,XZorYZorder.\nAs a result, the magnetic phase formation in the strong\ncoupling limit exhibits a commensurate to incommensu-\nrate N\u0013 eel to spiral transition at J~\u0015\u00190:53. In the limit\nof in\fnite spin orbit coupling, the model converges to a\ncommensurate magnetic state with a 6 \u00026-site unit cell\non each of the two decoupled triangular sublattices of the\nunderlying honeycomb model. The emergence of the spi-\nral phase in the in\fnite Ulimit leads us to conjecture thataside from the topological band insulator regime and the\nantiferromagnetic phase, a third phase should exist at \f-\nniteUand \fnite spin orbit coupling. In this respect, our\nresults are not inconsistent with the existence of a frac-\ntionalized QSH* phase as proposed in Ref. 33. Whatever\nthis phase will eventually turn out to be, we \fnd that\nthe breaking of axial spin symmetry is generally vital to\nthe emergence of new phases and an enriched diversity\nof magnetic phases in interacting topological honeycomb\nband structures.\nNote added. Recently, the classical Kitaev-Heisenberg\nmodel has been studied on the triangular lattice within\nMonte Carlo approaches. Interestingly, spiral-like order\nhas been found to be associated with a Z2vortex lat-\ntice.56\nACKNOWLEDGMENTS\nWe acknowledge useful discussions with Karyn Le Hur,\nGregory Fiete, Giniyat Khaliullin, George Jackeli, An-\ndreas R uegg, and Matthias Vojta. JR is supported by\nthe Deutsche Akademie der Naturforscher Leopoldina\nthrough grant LPDS 2011-14. RT is supported by an\nSITP fellowship by Stanford University. SR acknowl-\nedges support from DFG under Grant No. RA 1949/1-1.\n1D. J. Thouless, M. Kohmoto, M. P. Nightingale, and\nM. den Nijs, Phys. Rev. Lett. 49, 405 (1982).\n2F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).\n3M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n4X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n5J. Moore, Nature 464, 194 (2010).\n6C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802\n(2005).\n7C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n8B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science\n314, 1757 (2006).\n9M. K onig, S. Wiedmann, C. Br une, A. Roth, H. Buhmann,\nL. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science\n318, 766 (2007).\n10A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-\nwig, Phys. Rev. B 78, 195125 (2008).\n11A. Y. Kitaev, AIP Conf. Proc. 1134 , 22 (2009).\n12D. A. Pesin and L. Balents, Nature Phys. 6, 376 (2010).\n13S. Rachel and K. Le Hur, Phys. Rev. B 82, 075106 (2010).\n14M. Kargarian, J. Wen, and G. A. Fiete, Phys. Rev. B 83,\n165112 (2011).\n15M. Hohenadler, T. C. Lang, and F. F. Assaad, Phys. Rev.\nLett. 106, 100403 (2011).\n16D. Zheng, G.-M. Zhang, and C. Wu, 84, 205121 (2011).\n17M. Hohenadler, Z. Y. Meng, T. C. Lang, S. Wessel, A. Mu-\nramatsu, and F. F. Assaad, Phys. Rev. B 85, 115132\n(2012).18A. Vaezi, M. Mashkoori, and M. Hosseini, Phys. Rev. B\n85, 195126 (2012).\n19J. C. Budich, R. Thomale, G. Li, M. Laubach, and S.-C.\nZhang, ArXiv:1203.2928.\n20D.-H. Lee, Phys. Rev. Lett. 107, 166806 (2011).\n21C. Griset and C. Xu, Phys. Rev. B 85, 045123 (2012).\n22Z. Wang, X.-L. Qi, and S.-C. Zhang, Phys. Rev. Lett. 105,\n256803 (2010).\n23V. Gurarie, Phys. Rev. B 83, 085426 (2011).\n24W. Witczak-Krempa, T. P. Choy, and Y. B. Kim, Phys.\nRev. B 82, 165122 (2010).\n25S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Phys.\nRev. Lett. 100, 156401 (2008).\n26X. G. Wen, Phys. Rev. B 40, 7387 (1989).\n27W. P. Su, J. R. Schrie\u000ber, and A. J. Heeger, Phys. Rev.\nLett. 42, 1698 (1979).\n28R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).\n29J. M. Leinaas and J. Myrheim, Nuovo Cimento B 37, 1\n(1977).\n30D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev.\nLett. 48, 1559 (1982).\n31G. Fiete, Physica E 44, 845 (2012).\n32T. L. Schmidt, S. Rachel, F. von Oppen, and L. I. Glaz-\nman, Phys. Rev. Lett. 108, 156402 (2012).\n33A. R uegg and G. A. Fiete, Phys. Rev. Lett. 108, 046401\n(2012).\n34J. Reuther and P. W ol\re, Phys. Rev. B 81, 144410 (2010).\n35J. Reuther and R. Thomale, Phys. Rev. B 83, 024402\n(2011).\n36J. Reuther, D. A. Abanin, and R. Thomale, Phys. Rev. B\n84, 014417 (2011).\n1037J. Reuther, R. Thomale, and S. Trebst, Phys. Rev. B 84,\n100406 (2011).\n38Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale,\nW. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett.\n108, 127203 (2012).\n39A. Shitade, H. Katsura, J. Kunes, X.-L. Qi, S.-C. Zhang,\nand N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009).\n40C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu, Phys.\nRev. X 1, 021001 (2011).\n41C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107,\n076802 (2011).\n42P. Vogt, P. De Padova, C. Quaresima, J. Avila,\nE. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and\nG. Le Lay, Phys. Rev. Lett. 108, 155501 (2012).\n43S.-L. Yu, X. C. Xie, and J.-X. Li, Phys. Rev. Lett. 107,\n010401 (2011).\n44W. Wu, S. Rachel, W.-M. Liu, and K. Le Hur, Phys. Rev.\nB85, 205102 (2012).\n45Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and\nA. Muramatsu, Nature 464, 847 (2010).\n46S. Sorella, Y. Otsuka, and S. Yunoki, arXiv:1207.178.\n47R. Nandkishore, M. A. Metlitski, and T. Senthil, Phys.\nRev. B 86, 045128 (2012).48J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev.\nLett. (2010).\n49G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205\n(2009).\n50S. G ottel, S. Andergassen, C. Honerkamp, D. Schuricht,\nand S. Wessel, Phys. Rev. B 85, 214406 (2012).\n51W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and\nK. Sch onhammer, Rev. Mod. Phys. 84, 299 (2012).\n52A. A. Katanin, Phys. Rev. B 70, 115109 (2004).\n53Note that our RG \row is only de\fned in non-magnetic \row\nregimes, i.e., for \u0003 above the critical scale, \u0003 >\u0003c. In such\nnon-magnetic regimes, the susceptibility \u001f\u0016\u00160is a diagonal\nmatrix,\u001f\u0016\u00160=\u000e\u0016\u00160\u001f\u0016. Even though the RG \row cannot\nenter ordered phases, the type of magnetic order can be de-\ntermined from the largest susceptibility component \u001f\u0016(k)\nat the instability \u0003 c.\n54L. Capriotti, A. E. Trumper, and S. Sorella, Phys. Rev.\nLett. 82, 3899 (1999).\n55B. Scharfenberger, R. Thomale, and M. Greiter, Phys.\nRev. B 84, 140404 (2011).\n56I. Rousochatzakis, U. K. R ossler, J. van den Brink, and\nM. Daghofer, arXiv:1209.5895.\n11" }, { "title": "2102.02740v1.Spin_orbit_entangled_electronic_phases_in_4_d__and_5_d__transition_metal_compounds.pdf", "content": "Spin-orbit-entangled electronic phases in 4 dand 5 dtransition-metal compounds\nTomohiro Takayama,1, 2Ji\u0014 r\u0013 \u0010 Chaloupka,3, 4Andrew Smerald,1Giniyat Khaliullin,1and Hidenori Takagi1, 2, 5\n1Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany\n2Institute for Functional Matter and Quantum Technologies,\nUniversity of Stuttgart, Pfa\u000benwaldring 57, 70550 Stuttgart, Germany\n3Department of Condensed Matter Physics, Masaryk University, Brno, Czech Republic\n4Central European Institute of Technology, Masaryk University, Brno, Czech Republic\n5Department of Physics, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan\n(Dated: February 5, 2021)\nComplex oxides with 4 dand 5dtransition-metal ions recently emerged as a new paradigm in cor-\nrelated electron physics, due to the interplay between spin-orbit coupling and electron interactions.\nFor 4dand 5dions, the spin-orbit coupling, \u0010, can be as large as 0.2-0.4 eV, which is comparable\nwith and often exceeds other relevant parameters such as Hund's coupling JH, noncubic crystal\n\feld splitting \u0001, and the electron hopping amplitude t. This gives rise to a variety of spin-orbit-\nentangled degrees of freedom and, crucially, non-trivial interactions between them that depend on\nthed-electron con\fguration, the chemical bonding, and the lattice geometry. Exotic electronic\nphases often emerge, including spin-orbit assisted Mott insulators, quantum spin liquids, excitonic\nmagnetism, multipolar orderings and correlated topological semimetals. This paper provides a se-\nlective overview of some of the most interesting spin-orbit-entangled phases that arise in 4 dand 5d\ntransition-metal compounds.\nI. INTRODUCTION\nIn the 1960's and 70's, correlated-electron physics in\ntransition-metal oxides was already an active \feld of\nresearch, and a major topic in condensed matter sci-\nence. The basic picture of spin and orbital ordering and\nthe interplay between them was unveiled during this pe-\nriod, and collected into the Kanamori-Goodenough rules.\nHowever, the exploration of exotic electronic phases be-\nyond conventional magnetic ordering was stymied by a\nlack of materials and theoretical tools.\nIn 1986, high- Tcsuperconductivity was discovered in\nthe layered 3 dCu oxides, which accelerated both ex-\nperimentally and theoretically the search for novel spin-\ncharge-orbital coupled phenomena produced by electron\ncorrelations. The major arena of such exploration was\ncomplex oxides with 3 dtransition metal ions from Ti\nto Cu, which led to the discoveries of unconventional\nsuperconductivity, colossal magneto-resistance, multifer-\nroics and exotic spin-charge-orbital orderings. 4 dand\n5dtransition metal oxides were also studied, but not as\nextensively as 3 dtransition metal oxides, except for 4 d\nSr2RuO 4, where possible p-wave superconductivity was\ndiscussed. This was at least partially due to the less\nprominent e\u000bect of correlations in 4 dand 5d, which arises\nfrom the wavefunctions being more spatially extended\nthan in 3d.\nIn the late 2000's, the layered perovskite Sr 2IrO4was\nidenti\fed as a weak Mott insulator, and the crucial role\nof spin-orbit coupling (SOC) in stabilizing the Mott state\nawoke a growing interest in 5 dIr oxides and other 4 dand\n5dcompounds. For Ir4+ions, there are \fve 5 d-electrons,\nwhich reside in the t2gmanifold, and therefore have an\ne\u000bective orbital moment L= 1 and form a spin-orbit-\nentangledJ= 1/2 pseudospin. These J= 1/2 pseu-\ndospins behave in some ways like S= 1/2 spins, but havean internal spin-orbital texture where the up/down spin\nstates reside on di\u000berent orbitals. The resulting spin-\norbital entanglement gives rise to non-trivial interactions\nbetween the J= 1/2 pseudospins, and this led to the\nproposal that the Kitaev model, with bond-dependent\nIsing interactions, can be realized on edge-shared honey-\ncomb networks of J= 1/2 pseudospins. In consequence,\nJ= 1/2 honeycomb magnets made out of 5 d5Ir4+and\n4d5Ru3+ions have been extensively studied over the last\nten years, in an e\u000bort to discover the expected quantum\nspin-liquid state and associated Majorana fermions.\nDespite the considerable excitement, the d5J= 1/2\nMott state is not the only spin-orbit-entangled state of\ninterest among the many 4 dand 5dtransition metal com-\npounds. With di\u000berent \flling of d-orbitals and di\u000ber-\nent local structures, a rich variety of spin-orbit-entangled\nstates can be formed, which are characterized not only\nby dipolar moments but also by multipoles such as\nquadrupolar and octupolar moments. Exotic states of\nsuch spin-orbit-entangled matter can be anticipated, re-\n\recting the internal spin-orbital texture and the lattice\nsymmetry, and including excitonic magnetism and mul-\ntipolar liquids.\n4dand 5dtransition metal compounds also form in-\nteresting itinerant states of matter, with one prominent\nexample being the topological semimetal. This arises\nfrom the interplay of lattice symmetry and SOC, and,\nin contrast to typical topological semimetals, 4 dand 5d\nsemimetals tend to also have strong electron correlations.\nThus they provide an arena in which to study the overlap\nbetween topological physics and strong correlation.\nSpin-orbit-entangled phases are also formed in 4 fand\n5felectron systems, which have been extensively studied,\nand it is worth spelling out what makes 4 dand 5delec-\ntron systems distinct. One obvious di\u000berence is that they\ninteract through exchange processes with a much largerarXiv:2102.02740v1 [cond-mat.str-el] 4 Feb 20212\nenergy scale, making them more accessible to experi-\nment, and thus increasing the variety of phenomena that\ncan be e\u000bectively probed. Also, the spin-orbit-entangled\nJstates in 4dand 5dsystems are much less localized\nthan those of 4 f, and can often be itinerant, opening\nup, for example, the exploration of correlated topologi-\ncal semimetals, and SOC driven exotic states formed near\nmetal-insulator transitions. The d- and thef- electron\nsystems thus clearly play a complementary role in the\nexploration of spin-orbit-entangled phases.\nThis review is intended to provide readers with a broad\nperspective on the emerging plethora of 4 dand 5dtransi-\ntion metal oxides and related compounds. We would like\nto address two basic questions: 1) What kind of exotic\nphases of spin-orbit-entangled matter are expected in 4 d\nand 5dcompounds? 2) To what extent are the proposed\nconcepts realized? We limit our discussion to the com-\npounds with octahedrally coordinated 4 dand 5dtransi-\ntion metal ions accommodating less than 6 electrons in\ntheirt2gorbitals (low-spin con\fguration), where the ef-\nfect of the large SOC is prominent due to the smaller\ncrystal \feld splitting of t2gorbitals as compared to eg.\nAs there are many reviews of pseudospin-1/2 d5Mott in-\nsulators, here we will discuss spin-orbit-entangled states\nin 4dand 5dtransition metal compounds from a broader\nperspective, covering, in addition to d5compounds, d1,\nd2andd4insulators, as well as itinerant systems with\nstrong SOC.\nII. CONCEPT OF SPIN-ORBIT-ENTANGLED\nSTATES AND MATERIALS OVERVIEW\nIn Mott insulators, charge \ructuations are frozen, and\nthe low-energy physics is driven by the spin and orbital\ndegrees of freedom of the constituent ions. In 3 dcom-\npounds, orbital degeneracy and hence orbital magnetism\nis largely quenched by noncubic crystal \felds and the\nJahn-Teller (JT) mechanism, and magnetic moments are\npredominantly of spin origin (with some exceptions men-\ntioned below). The large SOC in 4 dand 5dtransition\nmetal ions competes with and may dominate over crys-\ntal \feld splitting and JT e\u000bects, and the revived orbital\nmagnetism becomes a source of unusual interactions and\nexotic phases. We \frst discuss the spin-orbital structure\nof the low-energy states of transition metal ions in a most\ncommon, cubic crystal \feld environment, and then pro-\nceed to their interactions and collective behavior.\nA. Spin, orbital, and pseudospin moments in Mott\ninsulators\nThe valence wavefunctions of 4 dand 5dions are spa-\ntially extended and the Hund's coupling is smaller than\nthe cubic crystal \feld splitting, 10 Dq, betweent2gand\negorbitals. Low-spin ground states are therefore sta-\nbilized for 4 dand 5delectron con\fgurations more fre-quently than in the 3 dcase, where the Hund's coupling\ntypically overcomes the t2g-egcrystal \feld splitting. In\nthe low-spin state, electrons occupy t2gorbitals forming\ntotal spin-1 =2 (d1,d5), spin-1 (d2,d4), and spin-3 =2 (d3).\nIn all of them (except d3orbital singlet not discussed\nhere), the orbital sector is threefold degenerate, formally\nisomorphic to the p-orbital degeneracy, and can thus be\ndescribed in terms of an e\u000bective orbital angular momen-\ntumL= 1. (The e\u000bective orbital angular momentum is\noften distinguished by a special mark, see, e.g., the \\\fcti-\ncious angular momentum\" ~lin the textbook by Abragam\nand Bleaney,1but here we conveniently choose a simpler\nnotationL.) By a direct calculation of matrix elements\nof the physical orbital momentum Ld= 2 ofd-electrons\nwithin the t2gmanifold, one \fnds a relation Ld=\u0000L\nbetween the angular momentum operators.\nBy employing the e\u000bective Loperator, the SOC reads\nasH=\u0007\u0015SL, where the negative (positive) sign refers\nto a less (more) than half-\flled t2gshell ofd1,d2(d4,\nd5) con\fgurations, and \u0015is related to the single-electron\nSOC strength, \u0010, via\u0015=\u0010=2S. The resulting spin-\norbital levels are shown in Fig. 1(a). Apparently, SOC\nbreaks particle-hole symmetry: due to the above sign\nchange, the levels are mutually inverted within the pairs\nof complementary electron/hole con\fgurations such as\nd1/d5andd2/d4. The ground states thus have completely\ndi\u000berent total angular momentum J=S+L. Its con-\nstituents SandLalign in parallel (antiparallel) fashion\nfor the less (more) than half-\flled case. The correspond-\ning \\shapes\" of the ground-state electron densities are\ndepicted in Fig. 1(b). Their nonuniform spin polariza-\ntion with a coherent mixture of spin-up and down densi-\nties clearly shows the coupling between the spin and the\norbital motion of electrons.\nThe observed variety of ionic ground states among dn\ncon\fgurations brings about a distinct physics for each\nof thednions.d1andd2ions withJ > 1=2 host\nhigher order magnetic multipoles, and may lead to uncon-\nventional high-rank order parameters \\hidden\" magneti-\ncally. Nominally nonmagnetic J= 0d4ions may develop\nunusual magnetism due to condensation of the excited\nJ= 1 level.d5ions with Kramers doublet J= 1=2 are\nformally akin to spin one-half quantum magnets which\nare of special interest in the context of various quantum\nground states.\nWhen evaluating the magnetic properties, the orbital\ncomponent of the magnetic moment needs to be incorpo-\nrated. In terms of the e\u000bective L, the magnetic moment\noperator reads as M= 2S\u0000L. In principle, Lcomes\nwith the so-called covalency factor \u0014but we omit it for\nsimplicity. For d1with parallel L= 1 andS= 1=2, one\nhasL= 2Sand thusM= 0, i.e. the J= 3=2 quartet has\nzerog-factor and is thus nonmagnetic. In reality, \u0014<1\nmakes the compensation only partial, resulting in a small\nmagnetic moment. For d2withS=L=J=2, one \fnds\nM= (1=2)J, i.e.g= 1=2.d4withJ= 0 is a nonmag-\nnetic singlet. d5with antiparallel S= 1=2 andL= 1\nhasg-factorg=\u00002, i.e. it is of opposite sign relative to3\nJ=1\nJ=2J=0\nL Sζ21\nζJ=21\n2323\nL Sζ\nJ=\n21J=J=23\n23\nL SζJ=2\nJ=1\nJ=0\nL Sζ\nζ21\n+2 +1 0 −1 −23\n2+3\n2− J =z1\n2+1\n2− spin down spin upd1 3\n2J =, d2J =2 , J =0 d4, d5 1\n2J =,(b)(a)d2d1d5d4\nFIG. 1. (a) Low-energy levels of d1,d2,d4, andd5ions in cubic crystal \feld. The degeneracy of the levels is shown by\nthe number of close lines. For less than half-\flled t2gshell, the SOC aligns the e\u000bective orbital angular momentum Land\nspinSto form larger total angular momentum: J= 3=2 quartet in d1case andJ= 2 quintuplet in d2case, respectively.\nIn the case of more than half-\flled t2gshell,LandSare antialigned, leading to J= 0 singlet ground state for the d4\ncon\fguration while the d5one hosts pseudospin J= 1=2. (b) Orbital shapes corresponding to the ground-state J-levels. Only\nthe angular distribution of the electron density is considered. It is represented by a surface plot where the distance to the\norigin is proportional to the integral density in the corresponding direction. The color of the surface indicates normalized spin\npolarization ( \u001a\"\u0000\u001a#)=(\u001a\"+\u001a#) taking values in the range [ \u00001;+1]. It is shown for electrons in the case of d1andd2states\nand for the holes in the t6\n2gcon\fguration in the case of d4andd5states.\nthe electron g-factor. The above g-factors strongly devi-\nating from pure spin g= 2 are the \fngerprints of large\norbital contribution to magnetism. In the d5case, e.g.,\none \fnds that S= (\u00001=3)Jonly, while L= (4=3)J;\nthat is, magnetism of d5compounds is predominantly of\norbital origin.\nFigure 2 focuses in detail on the particular case of the\nd2con\fguration. The J= 2 ground state is special be-\ncause it is isomorphic to a d-electron with orbital moment\nLd= 2, and thus it has to split under cubic crystal \feld\ninto a triplet of T2gand a doublet of Egsymmetries (of-\nten denoted as \u0000 5and \u0000 3states, respectively). The cor-\nresponding wavefunctions in the basis of Jzeigenstates\njJziare:1j\u00061i, and (j+2i\u0000j\u0000 2i)=p\n2 forT2g, andj0iand\n(j+ 2i+j\u00002i)=p\n2 forEgstates, just like for single elec-\ntrond-orbitals of t2gandegsymmetries, as required by\nthe above isomorphism. Physically, this splitting arises,\ne.g., due to the admixture of the t1\n2ge1\ngcon\fguration into\nt2\n2gby SOC. More speci\fcally, second-order energy cor-\nrections to T2gandEglevels are di\u000berent, which gives a\nsplitting of the J= 2 level by\u0018\u00102=10Dq. With SOC\nparameter\u0010= 0:2 eV and 10 Dq= 3 eV, typical for 4 d\nand 5dions, one obtains a sizable splitting of 20 meV,with theEgdoublet being the lower one. It is evident\nfrom the above wavefunctions that the Egstate has no\ndipolar moment and is therefore magnetically silent. In-\nstead, theEgdoublet hosts quadrupolar and octupolar\nmoments. This is again analogous to egelectrons, which\nare quadrupole active, and it has been discussed in the\ncontext of manganites that they may host an octupolar\nmoment as well.2{4While this e\u000bect was not observed\nin realegorbital systems, the spin-orbit-entangled Eg\ndoublet may show octupolar order driven by intersite ex-\nchange interactions, unless JT coupling to lattice stabi-\nlizes quadrupolar order instead.\nConcerning the JT activity of the ionic ground states,\nthed4singlet and d5Kramers doublet possess no orbital\ndegeneracy and are thus \\JT-silent\" in the \frst approxi-\nmation. However, both d1andd2are JT active ions, and\nstructural phase transitions as in usual 3 dsystems can\nbe expected. As shown in Fig. 1(b), the shapes of the\n\u00063=2 and\u00061=2 states of the d1ion are di\u000berent; there-\nfore, they will split under tetragonal lattice distortions.\nIn fact, these two Kramers doublets can be regarded as\nan e\u000bective egorbital, so JT coupling would read ex-\nactly as for the egorbital, albeit with an e\u000bective JT4\nt2g-eg\nt2\n2g\nt2\n2gt1\n2ge1\ng\nζ2/10∼ Dq\nEgT2gJ=0\nJ=2J=121ζ\nζ\nα+iββα−i Eg(c) (b)\nEgmixing10Dq(a)\nOctupolar order Quadrupolar orderβ\nα\nFIG. 2. (a) Shifts and splitting of the J= 0;1;2 levels ofd2\nion when considering mixing of the ground state t2\n2gcon\fg-\nuration with t1\n2ge1\ngstates by virtue of SOC. Focusing on the\nlowest levels, we that \fnd the originally \fve-fold degenerate\nJ= 2 states split into an Egdoublet and T2gtriplet. Eval-\nuated perturbatively for \u0010\u001c10Dq, the splitting comes out\nproportional to \u00102=10Dq. (b) Tetragonal compression leads\nto an increased repulsion of d-electrons from apical oxygen\nions and further singles out \\planar\" states from EgandT2g\nsets. The quadrupolar moment hosted by the Egdoublet gets\npinned this way. (c) Complex combinations of the Egdoublet\nstates that expose the octupolar moment of \\cubic\" shape.\ncoupling constant reduced by 1 =p\n3, as a result of SOC\nuni\fcation of the Hilbert space. Similarly, the Egdou-\nblet of theJ= 2 manifold in the case of a d2ion should\nexperience JT coupling. Overall, the JT e\u000bect (and re-\nlated structural transition) is still operative, but it a\u000bects\nboth spin and orbital degrees of freedom simultaneously\nas a result of the spin-orbit transformation of the wave-\nfunctions, and conventional JT orbital ordering is con-\nverted into a magnetic quadrupolar ordering of J= 3=2\norJ= 2 moments. An important consequence of spin-orbital entanglement is that, distinct from usual orbital\norder in 3dsystems, magnetic quadrupolar order breaks\nnot only the point-group symmetry of a crystal but also\nthe rotational symmetry in magnetic space, resulting in\nanisotropic, non-Heisenberg-type magnetic interactions,\nsuch as XY or Ising models. In other words, JT coupling\nin spin-orbit-entangled systems has a direct and more\nprofound in\ruence on magnetism.\nThe above discussion is based on the LS-coupling\nscheme, which is adequate for obtaining the ground\nstate quantum numbers. However, the corresponding\nwavefunctions, and hence e\u000bective g-factors, as well as\nexcited-state energy levels, may get some corrections to\nLS-coupling results. This is important for the interpre-\ntation of the experimental data. Similarly, the admix-\nture of the tn\u00001\n2ge1\ngcon\fguration into the ground state\ntn\n2gwavefunctions by SOC and multielectron Coulomb\ninteractions is present for all dn, and may renormalize\ntheg-factors and wavefunctions.5,6However, these e\u000bects\ncannot split the ground state J-levels, except the J= 2\nlevel of the d2con\fguration, as discussed above.\nIn general, the ground state manifold of transition\nmetal ions in Mott insulators is conveniently described\nin terms of e\u000bective spin (\\pseudospin\") ~S, where 2 ~S+ 1\nis the degeneracy of this manifold. For low-spin dnions\nin a cubic symmetry, pseudospin ~Sformally corresponds\nto e\u000bective total angular momentum J(often called Je\u000b),\nwith the exception of the d2case with an Egdoublet host-\ning a pseudospin ~S= 1=2. One has to keep in mind how-\never, that even in the case of cubic symmetry, the pseu-\ndospin wavefunctions are di\u000berent from pure Jstates be-\ncause of various corrections (deviations from LSscheme,\nadmixture of egstates, etc.) discussed above. This is\neven more so when noncubic crystal \felds are present\nand become comparable to SOC. We will occasionally use\nboth ~SandJpseudospin notations, depending on con-\nvenience (e.g., reserving Jfor the Heisenberg exchange\nconstant in some cases).\nIn strong spin-orbit-entangled systems, the notion of\npseudospins remains useful even in doped systems, at\nleast at low doping where Mott correlations, and hence\nthe ionic spin-orbit multiplets, are still intact locally. In\nhighly doped systems, a weakly-correlated regime, a con-\nventional band picture emerges, where SOC operates on\na single-electron level.\nB. Pseudospin interactions in Mott insulators\nThe key element when considering the interactions\namong pseudospins is the entanglement of spin and or-\nbital degrees of freedom. In the pseudospin state, various\njLz;Szicombinations are superposed, forming a compos-\nite object. Figure 3 shows two important examples for\nthed5andd4cases that will be extensively discussed\nlater. Mixing the spins and orbitals in a coherent way,\npseudospins do experience all the interactions that op-\nerate both in the spin and orbital sectors, which have5\n1√\n3/radicalbigg\n2\n3\n/parenrightBigg\n1√\n3/parenleftBigg\nJz=0J=0J=1\n2\nLz=0\nSz=0Lz=−1\nSz= +1Lz= +1\nSz=−1Jz= +1\n2Lz=0\nSz= +1\n2Lz= +1\nSz=−1\n2=+ −(a)\n− + =(b)\nFIG. 3. (a) Decomposition of the J= 1=2 Kramers doublet\nstate ofd5intojLz;Szicomponents of the single hole in t6\n2g\ncon\fguration. The e\u000bective angular momentum is indicated\nby the rotating arrow, spin by the color following the conven-\ntion of Fig. 1. For both contributions, LzandSzsum up to\nJz= +1=2. (b) Similar decomposition of the J= 0 two-hole\nground state of d4. The total orbital angular momentum Lz\nand spinSzmay be combined in three ways here. The inter-\nnal compensation in LandScreates a cubic-shaped object\nshowing no spin polarization.\nvery di\u000berent symmetry properties. Electron exchange\nprocesses conserve total spin, and hence the spin inter-\nactions are of isotropic Heisenberg ( SiSj) form. The\norbital interactions are however far more complex { they\nare anisotropic both in real and magnetic spaces.7{9In\nhigh-symmetry crystals, orbitals are strongly frustrated,\nbecause they are spatially anisotropic and hence can-\nnot simultaneously satisfy all the interacting bond di-\nrections. Via the spin-orbital entanglement, the bond-\ndirectional and frustrating nature of the orbital interac-\ntions are transferred to the pseudospin interactions.10\nMoreover, apart from the exchange interactions driven\nby virtual electron hoppings, there are other contribu-\ntions to the orbital interactions, especially in the d1and\nd2cases. These are mediated by the orbital-lattice JT\ncoupling to the virtual JT-phonons, and by electrostatic\nmultipolar interactions between d-orbitals on di\u000berent\nsites.11,12In low-energy e\u000bective Hamiltonians, these\ninteractions transform into pseudospin multipolar cou-\nplings, driving both structural and \\spin-nematic\" tran-\nsitions breaking cubic symmetry in real and pseudospin\nspaces. The JT-driven interactions are also important\nind4systems, as they split the excited J= 1 levels,\nand hence promote magnetic condensation.13In the case\nofd5systems with Kramers-degenerate J= 1=2 pseu-\ndospins, the e\u000bective Hamiltonians are predominantly of\nexchange origin, as in usual spin-1 =2 systems, although\nJT orbital-lattice coupling still shows up in the \fne de-\nd+1\nd+1pypxtpdπ\ntpdπt(a)z z\nyz\nxy y\nx xyzx(b)\nzx yzzFIG. 4. (a) Hopping via oxygen in the case of 180\u000eM-O-M\nbonds. The orbital label of the two active t2gorbitals (here\nzxandyz) is conserved. The third orbital ( xy, not shown)\ncannot connect to the oxygen pstates. (b) Combining orbitals\nintoL\u000beigenstates with \u000bdetermined by the bond direction,\nthe above rules lead to a conservation of L\u000b=\u00061 while the\nL\u000b= 0xy-orbital is inactive.\ntails of pseudospin dynamics, in the form of pseudospin-\nlattice coupling.13\nIn general, the low-energy pseudospin Hamiltonians\nmay take various forms depending on the electron con\fg-\nurationdnand symmetry of the crystal structure. Sen-\nsitivity of orbital interactions to bonding geometry is a\ndecisive factor shaping the form of the pseudospin Hamil-\ntonians. We illustrate this by considering spin-orbital ex-\nchange processes in two di\u000berent cases { when metal(M)\n-oxygen(O) octahedra MO 6share the corners, and when\nthey share the edges. These two cases are common in\ntransition-metal compounds and are referred to as 180\u000e\nand 90\u000ebonding geometry, re\recting the approximate\nangle of the M-O-M bonds. For simplicity, we limit\nourselves to the case of d5ions with wavefunctions [c.f.\nFig. 3(a)]:\njf~\"i= + sin#j0;\"i\u0000 cos#j+ 1;#i; (2.1)\njf~#i=\u0000sin#j0;#i+ cos#j\u00001;\"i; (2.2)\nwhere we represent the pseudospin-1 =2 Kramers doublet\nbyf-fermion that is associated with a hole in the full t6\n2g\ncon\fguration. The spin-orbit mixing angle is determined\nby tan 2#= 2p\n2=(1+2\u0001=\u0015), where \u0001 is tetragonal split-\nting of the t2gorbital level. In the cubic limit of \u0001 = 0\nshown in Fig. 3(a), one has sin #= 1=p\n3, cos#=p\n2=3.\nThe individual orbital components of the pseudospins\nare subject to distinct hopping processes, as dictated by\nthe orbital symmetry combined with the particular bond-\ning geometry (see Figs. 4 and 5). For the corner-sharing\n180\u000ecase presented in Fig. 4, the nearest-neighbor (NN)\nhopping Hamiltonian takes the form\nH(180\u000e) =\u0000t(ay\ni\u001baj\u001b+by\ni\u001bbj\u001b+ H:c:)\n=\u0000t(dy\n+1i\u001bd+1j\u001b+dy\n\u00001i\u001bd\u00001j\u001b+ H:c:):(2.3)6\nd+1d−1tpdπpz\nxy d0xy d0tpdπ\npz(a) (b)\n−it+it(c) (d)yz\nyx\nxy−t’\nxyzyz\nx yzx\nz\nzx\nFIG. 5. Hopping in the case of 90\u000eM-O-M bonding geom-\netry. (a), (b) Two t2gorbitalszxandyzare interconnected\nby the hopping via oxygen porbitals. They are selected by\nthe orientation of the M 2O2plaquette. (c) The complemen-\ntarity of the orbital labels connected in the M-O-M bridge\nresults in orbital moment non-conserving hopping when con-\nsideringL\u000beigenstates. Here \u000b=zso thatLz= +1 is\n\ripped toLz=\u00001 and vice versa. The corresponding hop-\nping amplitudes are imaginary: \u0006it. (d) Direct overlap of d\norbitals opens an additional hopping channel where the re-\nmainingLz= 0xy-orbital is active.\nA summation over the spin index \u001b=\";#is assumed.\nTwo of the three t2gorbitals participate in oxygen-\nmediated hopping with the amplitude t=t2\npd\u0019=\u0001pd;\nthe active pairfa,bgis selected by the bond direction\n\u000band may be combined into e\u000bective orbital moment\nL\u000b=\u00061 eigenstatesjd\u00061i. For example, the zbond con-\nsidered in Fig. 4 picks up jai\u0011jyziandjbi\u0011jzxithat\nformLz=\u00061 eigenstatesjd\u00061i=\u0007(jyzi\u0006ijzxi)=p\n2.\nThe third orbital jci \u0011 jxyi \u0011 jd0icannot couple to\nthe mediating p-orbitals of oxygen for symmetry rea-\nsons. Since the NN hopping preserves both spin and or-\nbital in this case, pseudospin is also a conserved quantity.\nProjectingH(180\u000e) onto thef-doublet subspace de\fned\nabove, one indeed observes pseudospin-conserving hop-\npingH=\u0000tf(fy\ni\"fj\"+fy\ni#fj#+H:c:), which should there-\nfore lead to isotropic Heisenberg exchange H=J(~Si~Sj)\nwithJ= 4t2\nf=U.\nThe situation in the case of 90\u000ebonding geometry\nis completely di\u000berent. As shown in Fig. 5, two bond-\nselectedt2gorbitalsfa,bgspanning thejd\u00061isubspace\nare again active in the oxygen-mediated hopping t=\nt2\npd\u0019=\u0001pd, but their labels get interchanged during the\nhopping:a$b, i.e.yz$zxforzbond. The third\norbitalccorresponding to jd0iparticipates in direct hop-\npingt0. The two hopping channels are captured by theNN Hamiltonian:10\nH(90\u000e) =t(ay\ni\u001bbj\u001b+by\ni\u001baj\u001b)\u0000t0cy\ni\u001bcj\u001b+ H:c:\n=it(dy\n+1i\u001bd\u00001j\u001b\u0000dy\n\u00001i\u001bd+1j\u001b)\u0000t0dy\n0i\u001bd0j\u001b+ H:c:\n(2.4)\nIn contrast to the 180\u000ecase discussed above, the t-\nhopping term does not conserve Lz, but changes it by\n\u0001Lz=\u00062;jd+1i$jd\u00001i. Due to spin conservation,\nthe total angular momentum projection has to change\nby the same amount, i.e. \u0001 Jz=\u00062. However, such\nhopping cannot connect pseudospin-1 =2 states (maximal\n\u0001Jz=\u00061 can be reached by hopping fy\ni\"fj#+ H:c:);\nindeed, projection of the t-term inH(90\u000e) onto pseu-\ndospinf-space gives simply zero. This implies that\nthe pseudospin wavefunctions cannot form d-p-dbonding\nstates, and thus the conventional pseudospin exchange\nterm 4t2=Udue to hopping tvia oxygen ions is com-\npletely suppressed.10,14The situation is similar to the\negorbital exchange in the 90\u000ebonding geometry, where\negorbitals cannot form d-p-dbonding states and thus\nno spin-exchange process is possible. As in the egcase,\nthe pseudospin interactions in the edge-shared geome-\ntry are generated by various corrections to the above\npicture (a direct overlap of pseudospins due to the t0-\nterm, electron hopping to higher spin-orbital levels, cor-\nrections to pseudospin wavefunctions due to non-cubic\ncrystal \felds, etc.). The resulting pseudospin Hamiltoni-\nans are typically strongly anisotropic, and the most im-\nportant and actually leading term in real compounds is\nbond-dependent Ising coupling. Figure 6 illustrates how\nsuch an interaction emerges due to the t-hopping from\nground state J= 1=2 to higher spin-orbit J= 3=2 levels\nand subsequent Hund's coupling of the excited electrons\nin the virtual state. The resulting exchange interaction\nreads asH=K~Sz\ni~Sz\nj, and the corresponding coupling\nconstantK/\u0000(JH=U) 4t2=Uis of ferromagnetic sign.14\nConsidered on honeycomb lattices, this interaction gener-\nates the famous Kitaev model where the Ising axis is not\nglobal but bond dependent, taking the mutually orthog-\nonal directions x,y, andzon three di\u000berent NN bonds.15\nThis results in strong frustration and a spin-liquid ground\nstate. On the other hand, a direct hopping t0, which con-\nserves both orbital and spin angular momentum, leads to\nconventional AF Heisenberg coupling /t02=U.\nWe will later discuss the pseudospin interactions in\nmore detail in the context of some representative com-\npounds, after a brief materials overview.\nC. Materials overview\nAs discussed above, the interactions between spin-\norbit-entangled pseudospins critically depend on the\nbonding geometry, and the ground states are determined\nby the network of each bonding unit, namely crystal\nstructures. Before discussing the properties of repre-\nsentative materials, it would be instructive to overview7\n↑ ↑ ↓\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2,+1\n2/angbracketrightbiggd−1↓\nd0↓ d−1↑\n d0↑ d+1↓d0↑\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n2,−3\n2/angbracketrightbigg\nd0↑ d+1↓\nJHL∆=±2 zL∆z=0\n~ ~ ~t’t\nFIG. 6. Virtual processes generating the e\u000bective interactions among pseudospins J= 1=2 ofd5con\fgurations in 90\u000ebonding\ngeometry. An M 2O2plaquette perpendicular to the zaxis as in Fig. 5 is assumed. Compared to Fig. 3(a), here we scale the d0,\nd\u00061orbitals to visually hint on their relative contributions to the pseudospin wavefunctions. (left part) Lz-conserving direct\nhoppingt0uses thed0part of the hole wavefunction and leads to a conventional Heisenberg exchange ~Si~Sjfollowing the Pauli\nexclusion principle for the d0orbital. (right part) Hopping via oxygen ttakes thed+1part of the hole wavefunction and by\ntheLz\rip creates a virtual d4con\fguration combining the original d5hole andJz=\u00003=2 quartet hole. The only option to\nreach a \fnal state with two J= 1=2 pseudospins by the second thopping is to remove this Jz=\u00003=2 quartet state again,\nleaving the initial pseudospin directions unchanged during the exchange process. Therefore, the e\u000bective interaction is of Ising\n~Sz\ni~Sz\njtype. Hund's exchange JHbetween the major d\u00061parts of the two holes in the virtual d4con\fguration prefers aligned\npseudospins on the two sites which results in ferromagnetic Kitaev interaction K~Sz\ni~Sz\njwithK < 0.\nthe crystal structures that are frequently seen in the 4 d\nand 5dtransition-metal compounds. We will introduce\ncrystal structures comprising the corner-sharing or edge-\nsharing network of MO 6octahedra.\n1. Corner-sharing network of MO 6octahedra\nThe most representative structure with corner-sharing\nMO6octahedra is the perovskite structure with a chem-\nical formula of ABO 3(A and B are cations). Small\ntransition-metal ions are generally accommodated into\nthe B-site, and the BO 6octahedra form a three-\ndimensional corner-sharing network. The stability of the\nperovskite structure is empirically evaluated by the Gold-\nschmidt tolerance factor t= (rA+rO)/p\n2(rB+rO) where\nrA,rBandrOare the ionic radius of A, B and oxy-\ngen ions, respectively. Note that the perovskite structure\nconsists of alternate stacking of AO layer and BO 2layer.\nt= 1 means that the ionic radii are ideal to form a cubic\nperovskite structure [Fig. 7(a)] with the perfect match-\ning of the spacings of constituent ions for AO and BO 2\nlayers. As the ionic radius rBfor 4dand 5dtransition-\nmetal ions is relatively large, the tolerance factor tof\n4dand 5dperovskites are normally less than 1, giving\nrise to lattice distortions to compensate the mismatch\nbetween AO and BO 2layers. A distorted perovskite\nstructure frequently found in 4 dand 5dtransition-metal\noxides is the orthorhombic GdFeO 3-type (Space group\nPbnm ) [Fig. 7(b)]. The BO 6octahedra rotate about thec-axis and tilt around the [110] direction (Glazer nota-\ntion:a\u0000a\u0000c+16). Because of this distortion, the B-O-B\nangle is smaller than 180\u000e. Many 4dand 5dtransition-\nmetal perovskites such as CaRuO 3, NaOsO 3and SrIrO 3\ncrystallize in the GdFeO 3-type structure.17{19\nIn addition to the three-dimensional network, the\nquasi-two-dimensional analogue with 180\u000ebonding ge-\nometry is realized in the layered derivatives of perovskite\nstructure. The square lattice of octahedrally-coordinated\ntransition-metal ions is seen in the K 2NiF4-type (A 2BO4)\nlayered structure, where the alternate stacking of (AO) 2-\nBO2layers along the c-axis is formed. Generally, the\nBO6octahedra are tetragonally distorted in the layered\nperovskites such as Sr 2VO4. As in the ABO 3-type per-\novskite, the mismatch of ionic radii of A and B cations\nresults in the rotation and the tilting distortion of BO 6\noctahedra, making the B-O-B angle less than 180\u000e. For\nexample, the layered iridate Sr 2IrO4possesses the ro-\ntation of IrO 6octahedra about the c-axis,20whereas\nCa2RuO 4hosts both a rotation and tilting distortion of\nRuO 6octahedra.21\nThe K 2NiF4(A2BO4)-type perovskite is an end mem-\nber of a Ruddlesden-Popper series with a general chemi-\ncal formula of A n+1BnO3n+1. This formula can be rewrit-\nten as AO(ABO 3)n[= AO(AO-BO 2)n], which makes it\neasier to view the crystal structures; there are n-layers of\nBO6octahedra, sandwiched by the double rock-salt-type\nAO layers as illustrated in Fig. 7(c). Generally, with in-\ncreasing number of layers, n, the electronic structure be-\ncomes more three-dimensional and hence the bandwidth\nincreases, which may induce a metal-insulator transition8\nFIG. 7. Crystal structures of perovskite and its derivatives.\n(a) Cubic perovskite ABO 3. (b) Orthorhombic perovskite\nwith the GdFeO 3-type distortion. (c) Ruddlesden-Popper se-\nries layered perovskite A n+1BnO3n+1. (d) Double-perovskite\nA2B0B00O6(left) and A 2MX 6-type halides (right). The crys-\ntal structures are visualized by using VESTA software.22\nas a function of n.\nThe double-perovskite structure is a derivative of per-\novskite, and also called rock-salt-ordered perovskite. In\nthe double-perovskites, two di\u000berent cations, B0and B00,\noccupy the octahedral site alternately and form a rock-\nsalt sublattice. The ordered arrangement of two di\u000berent\nB cations is usually seen when the di\u000berence of valence\nstate of the two cations is more than 2. Both B0and\nB00ions comprise a face-center-cubic (FCC) sublattice.\nNote that there is neither a direct B0-O-B0bond nor a\nB00-O-B00bond. The FCC lattice with d1ord2ions has\nbeen proposed to be a possible realization of multipolar\nordering of delectrons, and the double-perovskite ox-\nides with magnetic B0and nonmagnetic B00ions have\nbeen studied intensively as will be discussed in Section\nV. The FCC lattice of d1ord2ions is also realized in\nthe series of transition-metal halides with a chemical for-\nmula A 2MX6(A+: alkali ion, M4+: transition-metal ion,\nand X\u0000: halogen ion),23where MX2\u0000\n6octahedra and A+\nions form anti-\ruorite-like arrangement. This structurecan be viewed as a B00-site de\fcient double-perovskite\nA2M\u0003X6, where \u0003denotes a vacancy. A 2MX6crys-\ntallizes in a cubic structure with a large A ion such as\nCs+. M4+ions with an ideal cubic environment form a\nFCC sublattice, but also form a distorted structure when\nthe size of A ion is small. A wide variety of 4 dand 5d\ntransition-metal elements can be accommodated into this\nstructure.\nAnother important class of materials with corner-\nsharing MO 6octahedra is the pyrochlore oxide with a\ngeneral formula A 2B2O7(more speci\fcally A 2B2O6O0\nwhere O and O0represent two di\u000berent oxygen sites)\n[Fig. 8(a)]. Transition-metal ions are accommodated\ninto the B-cation site, which forms a BO 6octahedron,\nwhereas the A-cation is surrounded by six O and two O0\natoms in a distorted cubic-like environment. The sub-\nlattice of the B-cations, as well as that of A-cations, is a\nnetwork of corner-shared tetrahedra called the pyrochlore\nlattice [Fig. 8(c)]. The pyrochlore lattice is known to\nprovide geometrical frustration when magnetic moments\nof constituent ions interact antiferromagnetically. There\nare many ways to view the pyrochlore structure as de-\nscribed in Ref. [24]. Most conventionally, the B atom is\nlocated at the origin of unit cell (Wycko\u000b position 16 c)\nfor the space group Fd3m(No. 227, origin choice 2). In\nthis setting, the only tuneable parameters are the lattice\nconstant and the xcoordinate of the O site.25Withx\n= 0.3125, the BO 6forms an ideal octahedron and the\nB-O-B angle is\u0018141\u000e. In 4dand 5dtransition-metal py-\nrochlore oxides, xis usually larger than 0.3125, and the\nBO6octahedra are compressed along the [111] direction\npointing to the center of B-tetrahedra. The compres-\nsive distortion gives rise to a trigonal crystal \feld on B\nions and decreases the B-O-B angle between the neigh-\nboring octahedra from 141\u000e, which reduces the hopping\namplitude and thus bandwidth. When the ionic radius\nof the A atom becomes smaller, the trigonal distortion\nis enhanced. A metal-insulator transition is seen as a\nfunction of the size of A ions in pyrochlore oxides such as\nmolybdates A 2Mo2O7and iridates A 2Ir2O7(A: trivalent\nions such as rare-earth or Y3+).26,27The trigonal crystal\n\feld which splits the t2gmanifold potentially competes\nwith SOC.\n2. Edge-sharing network of MO 6\nAs discussed above, the edge-sharing, namely 90\u000eM-\nO-M, bonding geometry provides magnetic interactions\ndistinct from those in 180\u000ebonds. With the edge-sharing\nnetwork of MO 6octahedra, one can realize a variety of\nlattice structures of interest, such as the triangular lat-\ntice in ABO 2, the honeycomb lattice in A 2BO3and the\npyrochlore lattice in AB 2O4spinels. They can be con-\nstructed from the rock-salt structure.\nTo derive the layered triangular and honeycomb struc-\ntures from the rock-salt-type B002+O2\u0000(B00: transition-\nmetal atom), \frst consider the rock-salt structure viewed9\nFIG. 8. Crystal structures of (a) pyrochlore oxide A 2B2O7\nand (b) spinel oxide AB 2O4. (c) Pyrochlore sublattice com-\nprised by B (or A) atoms of pyrochlore oxide or by B atoms\nof spinel oxide. (d) Hyperkagome sublattice of Ir atoms found\nin Na 4Ir3O8. The pyrochlore sublattice is shared by 3:1 ratio\nof Ir and Na atoms in an ordered manner.\nalong the cubic [111] direction [Fig. 9(a)]. It consists of\nan alternating stack of the triangular B002+planes and the\ntriangular O2\u0000planes. By replacing every pair of adja-\ncent B002+planes with an A+plane and B03+plane, we\nhave the layered AB0O2-type structure with triangular\nlayers of A+and B03+[Fig. 9(b)]. The B0O6octahedra\nform the edge-shared triangular lattice. The trivalent\nB03+can be replaced by a 2:1 ratio of B4+and A+ions.\nThe large di\u000berence of valence states between A+and\nB4+cations facilitates the ordered arrangement of two\ncations in the triangular plane. As a result, the A 1=3B2=3\nlayers contain a honeycomb network of BO 6octahedra\nconnected by three of their six edges [Fig. 9(c)]. The al-\nternate stacking of an A+-cation layer and an A+\n1=3B4+\n2=3\nlayer corresponds to the chemical formula A 2BO3[=\nA(A 1=3B2=3)O2] as found in Na 2IrO3and Li 2RuO 3.28\nThe three-dimensional honeycomb structure of \f-Li2IrO3\nand\r-Li2IrO3can be derived from the rock-salt structure\nas well, but the ordering pattern of Li+and Ir4+ions\nare di\u000berent from the [111] ordering described above.29,30\nThose 4dand 5dtransition-metal oxides with a honey-\ncomb network are attracting attention as a realization of\nexotic quantum magnetism.\nIn the ordered rock-salt structures described above,\nall octahedral voids created by oxygen atoms are \flled\nby cations. The rock-salt structure also has tetrahedral\nvoids that can be occupied by cations. By partially \fll-\ning the tetrahedral and octahedral voids, a spinel struc-\nture AB 2O4can be constructed [Fig. 8(b)]. In the spinel\nstructure, B cations form a network of corner-shared\ntetrahedra as in the pyrochlore oxides. The crucial dif-\nference from A 2B2O7pyrochlore oxides is that the BO 6\nFIG. 9. (a) Rock-salt structure of B00O. (b) AB0O2-type struc-\nture formed by replacing B00with A+and B03+ions which\nstack alternately along the c-axis. (c) Layered honeycomb\nstructure of A 2BO3. The triangular layer of B03+ions in (b)\nis substituted by the 2:1 ratio of B4+and A+ions forming a\nhoneycomb lattice.\noctahedra in the spinel structure are connected by edge-\nsharing bonds. The number of spinel oxides containing\n4dor 5dtransition-metal atoms is rather limited. When\nmultiple cations occupy the pyrochlore B-sublattice, they\nmay form an ordered arrangement. The prominent ex-\nample is hyperkagome iridate Na 4Ir3O8.31In Na 4Ir3O8,\nthe B-site pyrochlore lattice of the spinel is shared in a\n3:1 ratio of Ir and Na atoms. The Ir sublattice is viewed\nas corner-shared triangles in three dimensions, which has\nbeen dubbed the hyperkagome lattice [Fig. 8(d)]. The\nproperties of hyperkagome iridate will be discussed in\nSections III.C and VI.C.\nIII. PSEUDOSPIN-1/2 MAGNETISM IN d5\nCOMPOUNDS\nThe collective behavior of d5ions with Kramers dou-\nblet ground states can be described in terms of a\npseudospin-1 =2 Hamiltonian. Thanks to the spin one-\nhalf algebra, pairwise interactions between pseudospins\nare reduced to various bi-linear terms. While the forms\nof these terms are dictated by lattice symmetry, the\ncorresponding coupling constants may vary broadly, de-\npending on the details of the local chemistry of a\ngiven material.32In Sec. II, we emphasized the di\u000ber-\nence between the 180\u000eand 90\u000ebonding geometry that\nlead to either conventional Heisenberg interaction or\nstrongly frustrated bond-selective interactions of the Ki-\ntaev type. Focusing on these two cases, we now consider\na few representative examples of d5compounds realizing\npseudospin-1 =2 physics.10\nA. 180\u000eM-O-M bonding, perovskites\nThe perovskite iridate Sr 2IrO4has emerged as a\nmodel system for understanding the spin-orbit-entangled\nmagnetism of 5 delectrons. The \frst experimental\nevidence for the J= 1=2 state was provided by\na combination of angle-resolved photoemission spec-\ntroscopy (ARPES), optical spectroscopy, and x-ray ab-\nsorption measurement33and later by resonant elastic x-\nray scattering,34which con\frmed the complex structure\nof thed5hole.\nThe relevant pseudospin-1 =2 model may be anticipated\nbased on the corner-shared IrO 6octahedra in the per-\novskite structure with approximately 180\u000eIr-O-Ir bonds\n(the bonds are not completely straight due to 11\u000ein-\nplane octahedra rotations35). Since the pseudospin wave-\nfunctions overlap well in the d-p-dhopping channel, and\nhopping is pseudospin conserving as discussed in Sec.II.B,\nthe dominant interaction is represented by Heisenberg\ncoupling. However, there are additional terms which\narise due to hoppings to higher level orbitals, tetrago-\nnal distortions and octahedral rotations, and these lead\nto the following NN-interaction Hamiltonian:\nH=J(~Si~Sj)+D(~Si\u0002~Sj)+A(~Sirij)(~Sjrij)+Jz~Sz\ni~Sz\nj:\n(3.1)\nHere, theDterm is an antisymmetric Dzyaloshinskii-\nMoriya (DM) interaction caused by octahedral rotations\nandAandJzrepresent symmetric anisotropy terms.\nTheJzterm is derived from the combined e\u000bect of the\noctahedral rotations and tetragonal \felds, while the A\nterm with dipole-dipole-coupling type bond-directional\nstructure is symmetry allowed even in an ideal cubic\nstructure.36The coupling constants have been calcu-\nlated in Ref. [14], and vary as a function of tetrago-\nnal crystal \feld, rotation angle etc. This Hamiltonian\nnicely accounts for a number of properties of Sr 2IrO4,\nincluding nearly Heisenberg spin dynamics akin to the\ncuprates.37,38\nA closer look in magnon data38reveals that the model\nabove has to be extended, including longer-range interac-\ntionsJ2andJ3, which turn out to be much larger than in\ncuprates. This might be related to the fact that 5 delec-\ntrons are more extended spatially, and to the relatively\nsmall Mott gap.33More recently, it has been shown that\nthe pseudospins in iridates couple to lattice degrees of\nfreedom via a dynamical admixture of higher-lying spin-\norbital levels to the ground state wavefunctions,13which\nexplains the observed in-plane magnon gaps,39and pre-\ndicts sizable magnetostriction e\u000bects breaking tetragonal\nsymmetry below TN.\nIn addition to elastic x-ray scattering,34the spin-\norbit-entangled nature of the d5ions in Sr 2IrO4was de-\ntected by resonant inelastic x-ray scattering (RIXS) ex-\nperiments that observed38transitions from J= 1=2 to\nJ= 3=2 levels, directly con\frming the level structure\nshown in Fig. 1. These excitations, dubbed \\spin-orbit\nexciton\", formally behave as a doped hole in a quantumantiferromagnet moving in a crystal by emitting and ab-\nsorbing magnons. The resulting exciton-magnon contin-\nuum, as well as the expected quasiparticle peak below it,\nhave been indeed observed,40similar to doped holes in\nantiferromagnetic cuprates. Also like in the hole-doped\ncuprates, spin-excitation spectra obtained by RIXS on\nLa-doped Sr 2IrO441,42revealed paramagnons persistent\nwell into the metallic phase.\nEncouraged by the above analogies with cuprates,\ndoped Sr 2IrO4samples have been studied in the search\nfor unconventional superconductivity. However, experi-\nments on doped Sr 2IrO4are severely impeded by di\u000ecul-\nties in obtaining clean samples. Techniques beyond con-\nventional chemical doping such as La-substitution pro-\nducing electron doped Sr 2\u0000xLaxIrO4have to be em-\nployed. For example, surface electron doping achieved by\npotassium deposition on the surface of parent Sr 2IrO4en-\nabled ARPES and scanning tunneling microscopy (STM)\nstudies;43{45another promising route is ionic liquid\ngating.46,47Although no clear evidence for superconduc-\ntivity was so far detected, Fermi surface and pseudo-\ngap phenomena as in cuprates have been observed in\nARPES43,44and STM experiments.45For more a de-\ntailed account on doped Sr 2IrO4, we recommend the re-\ncent review.48\nNext, we brie\ry discuss the bilayer iridate Sr 3Ir2O7.\nBeing \\in-between\" quasi-two-dimensional insulator\nSr2IrO4and three-dimensional metal SrIrO 3, this com-\npound is close to the Mott transition, with a small insu-\nlating gap.49Nonetheless, pseudospin-1 =2 magnons, as\nwell asJ= 3=2 spin-orbit excitons, have been observed\nin RIXS experiments50,51showing that the spin-orbit-\nentangled nature of low-energy states remains largely in-\ntact. A remarkable observation is that the magnetic mo-\nment direction and magnon spectra in this compound\nare radically di\u000berent from those in the sister compound\nSr2IrO4. Possible explanations for this have been o\u000bered\n{ based on enhanced anisotropic pseudospin couplings51\nand on dimer formation on the links connecting the two\nlayers.52\nB. 90\u000eM-O-M bonding, honeycomb lattice\nThe case of J= 1=2 pseudospins on a honeycomb\nlattice has sparked a broad interest after the proposal\nof Ref. [14] that the corresponding materials, such as\nNa2IrO3(see Fig. 9), may realize a Kitaev honeycomb\nmodel.15Since there is already a vast literature on this\ntopic, including several review articles,53{58we will make\njust a few remarks concerning the \\unwanted\" (i.e. non-\nKitaev) exchange terms that are present in the Kitaev-\nmodel candidate materials studied so far.\nAs explained in Sec.II.B above, the pseudospin-1 =2\nwavefunctions cannot communicate with each other via\nthe oxygen ions { the corresponding hopping integral is\nzero in the cubic limit. Finite interactions (albeit not as\nstrong as in 180\u000ecase) do originate from higher order pro-11\ncesses involving spin-orbit J= 3/2 virtual states, or from\ncommunication of the pseudospins via a direct t0hopping,\nas illustrated in Fig. 6. Hoppings to higher lying egstates\nwith subsequent Hund's coupling, as well as pdcharge-\ntransfer excitations do also contribute. Phenomenologi-\ncally, symmetry considerations dictate the following gen-\neral form of NN interactions59{61\nH=K~Sz\ni~Sz\nj+J(~Si~Sj) + \u0000( ~Sx\ni~Sy\nj+~Sy\ni~Sx\nj)\n+ \u00000(~Sx\ni~Sz\nj+~Sz\ni~Sx\nj+~Sy\ni~Sz\nj+~Sz\ni~Sy\nj);(3.2)\nwhich is expressed here for an M 2O2plaquette perpen-\ndicular to the cubic zaxis, as shown in Fig. 5. The\n\frst term represents the Kitaev interaction. Accord-\ning to perturbative calculations,60{62the o\u000b-diagonal ex-\nchange \u0000 arises from combined tandt0hoppings, while\nthe \u00000term is generated by trigonal crystal \felds that\nmodify the pseudospin wavefunctions. Trigonal \feld also\nsuppresses the bond-dependent nature of the pseudospin\ninteractions,10,63so the cubic limit is desired to support\nthe Kitaev coupling Kand to suppress the \u00000term. How-\never, theJand \u0000 terms, generated by a direct hopping t0\nand other possible hopping channels, remain \fnite even\nin the cubic limit. The crucial parameter here is the M-M\ndistance that controls the magnitude of the direct overlap\nof thed-wavefunctions.\nApart from NN J, \u0000, and \u00000terms, the longer-range\n(second/third NN J2=3) interactions are likely present in\nmany Kitaev materials. These couplings, which are detri-\nmental to the Kitaev spin liquid, are also related to the\nspatial extension of the 4 dand 5dorbitals, and to the\nlattice structure \\details\" such as the presence/absence\nof cations (e.g. Li or Na) within or near the honeycomb\nplanes [see Fig. 9(c)], opening additional hopping chan-\nnels.\nNevertheless, the Kitaev-type couplings appear to be\ndominant in 5 d-iridates and also 4 d-ruthenium chloride,\nas evidenced by a number of experiments (see, e.g., the\nreview [57]), although they are not yet strong enough\nto overcome the destructive e\u000bects of non-Kitaev terms\ndiscussed above. So the e\u000borts to design materials with\nsuppressed \\unwanted\" interactions have to be contin-\nued. An interesting perpective in this context may be\nthe employment of J= 1=2 Co2+ions, as has been\nproposed recently.64{66Although the energy scales for\nthe pseudospin interactions are smaller in this case, the\nless-extended nature of 3 dwavefunctions may reduce the\nlonger-range couplings, improving thus the conditions for\nthe realization of the Kitaev model.\nThe non-Heisenberg, bond-dependent nature of the\npseudospin interactions is expected to survive in weakly\ndoped compounds, and a\u000bect their metallic properties.\nIn particular, it has been suggested that they should lead\nto an unconventional p-wave pairing.10,67{70However, ex-\nperimental data on doped d5compounds with 90\u000ebond-\ning geometry is scarce, because a \\clean\" doping of Mott\ninsulators is a challenge in general.C. Pseudospins-1/2 on frustrated lattices\nA combination of geometrical frustration with spin-\norbital frustration may open an interesting pathway\nto exotic magnetism. In fact, the bond-dependent\npseudospin interactions have been \frst discussed in\nthe context of geometrically frustrated triangular10and\nhyperkagome71lattices. Here we discuss some spin-\norbit-entangled pseudospin-1/2 systems with geometri-\ncally frustrated lattices.\n1. Hyperkagome iridate Na 4Ir3O8\nAmong complex iridium oxides, Na 4Ir3O8appears to\nbe the \frst example of an exotic quantum magnet. In\nNa4Ir3O8, Ir atoms share the B-site pyrochlore lattice of\nthe spinel with Na atoms, and form a network of corner-\nshared triangles dubbed the hyperkagome lattice, as de-\nscribed in Sec.II.C.2. The hyperkagome lattice is geo-\nmetrically frustrated if the Ir moments couple antiferro-\nmagnetically.\nIndeed, Na 4Ir3O8displays a strong antiferromagnetic\ninteraction inferred from the large negative Weiss tem-\nperature (j\u0012CWj\u0018650 K) in the magnetic susceptibility\n\u001f(T).31Nevertheless, no sign of magnetic ordering has\nbeen seen down to 2 K both in \u001f(T) and the speci\fc\nheatC(T) as shown in Fig. 10. Na 4Ir3O8thus appeared\nas the \frst candidate for a three-dimensional quantum\nspin liquid.\nIn the\u001f(T) of Na 4Ir3O8, a small bifurcation was\nseen at around 6 K [the inset to Fig. 10(a)]. It was\noriginally interpreted as a glassy behavior due to a\nsmall amount of impurity/defects.31However, it has been\npointed out from the23Na-NMR and \u0016SR measurements\nthat Na 4Ir3O8exhibits a spin-glassy frozen state or qua-\nsistatic spin correlation.72,73We note that the presence\nof such a glassy state may be associated with the disor-\nder of Na ions in the octahedral A-site. Since the suc-\ncessful growth of Na 4Ir3O8single crystals was reported\nrecently,74the understanding of its magnetic ground\nstate is advancing.\nAfter the discovery of the spin-liquid behavior in\nNa4Ir3O8, a plethora of theoretical studies have been\nput forward. In the early days, most of the models\nhave treated the Ir moments as S= 1/2 and considered\nthe antiferromagnetic Heisenberg model on the frustrated\nhyperkagome lattice. The classical model predicted the\npresence of nematic order.75For the quantum limit, the\nground state has been discussed to be either a spin-\nliquid with spinon Fermi surface or a topological Z2spin\nliquid76{78.\nSince the IrO 6octahedra form an edge-sharing net-\nwork, as in honeycomb-based iridates, the presence of\nanisotropic magnetic exchanges such as Kitaev-type cou-\npling is anticipated. The electronic structure calculation\nshowed that SOC of Ir gives rise to a split of the t2g\norbitals into spin-orbit-entangled states with J= 1/212\nFIG. 10. Temperature dependence of (a) inverse magnetic\nsusceptibility \u001f\u00001(T), and (b) magnetic speci\fc heat Cmdi-\nvided by temperature. Insets: (a) Temperature dependence\nof\u001f(T) at various magnetic \felds. (b) Cm=Tat various mag-\nnetic \felds, showing a power law behavior Cm(T)\u0018Tn\n(2< n < 3) at low temperatures. The \fgure is reproduced\nwith permission from Ref. [31] ( ©2007 the American Physical\nSociety).\nand 3/2 characters.79In the pure Kitaev limit on a hy-\nperkagome lattice, a nonmagnetic ground state, possibly\nspin-liquid or valence-bond-solid, has been postulated.80\nIn reality, as in honeycomb iridates, other magnetic ex-\nchanges are present, and the antiferromagnetic Heisen-\nberg coupling seems to be the leading term.81A micro-\nscopic model that takes into account both the Heisenberg\nterm and anisotropic exchanges, such as the DM interac-\ntion, predicts the emergence of q= 0 noncoplanar order\nor incommensurate order depending on the magnitude\nof the anisotropic terms.71,82,83The spin-glass state of\nNa4Ir3O8may be associated with the presence of such\ncompeting magnetic phases.\n2. Pseudospin-1/2 on pyrochlore lattice\nIn addition to the hyperkagome lattice, the pyrochlore\nlattice of pseudospin-1/2 states with an edge-shared\nbonding geometry is found in an A-site de\fcient spinel\nIr2O4.84Theoretically, Ir 2O4is discussed to host spin-\nice-type \\2-in-2-out\" magnetic correlation and a U(1)\nquantum spin-liquid state is predicted under tetragonal\nstrain.85Ir2O4has been obtained only in a thin-\flm form,\nand its magnetic properties remain yet to be investigated.\nThe frustrated magnetism of Ir4+moments is also re-\nalized in pyrochlore oxides A 2Ir2O7(A: trivalent cation).The electronic ground state of A 2Ir2O7depends on the\nsize of the A-ion (ionic radius rA), which likely controls\nthe bandwidth of Ir 5 delectrons.86With the largest rAin\nthe family of A 2Ir2O7, Pr 2Ir2O7exhibits a metallic be-\nhavior down to the lowest temperature measured. With a\nslightly smaller rAsuch as Nd3+, Sm3+or Eu3+, A2Ir2O7\nshows a metal-to-insulator transition as a function of\ntemperature,87accompanied by a magnetic order. For\nan even smaller rAthan that of Eu3+, A2Ir2O7remains\ninsulating up to well above room temperature while mag-\nnetic ordering takes place only at a low temperature,\npointing to the Mott insulating state.27We focus here\non the magnetism of Ir pseudospin-1/2 moments in the\nMott insulating ground state. The metallic states of the\npyrochlore iridates will be discussed in Sec.VI.B.\nFor the pyrochlore iridates in the insulating limit, the\nlocal electronic state of Ir 5 delectrons is primarily of J\n= 1/2 character, but a sizable mixing of the J= 3/2\ncomponents is present.88The mixing was attributed to\nthe presence of a trigonal crystal \feld, which is generated\nnot only by the oxygen cage but also by the surround-\ning cations.89Recently, it turned out that the inter-site\nhopping plays a dominant role in the J= 3/2 mixing. In\nfact, by suppressing the hopping, a nearly pure J= 1/2\nstate can be realized.90\nThe magnetic interaction between the J= 1/2 pseu-\ndospins is predominantly attributed to the antiferromag-\nnetic superexchange interaction via oxygen ions, where\nthe Ir-O-Ir angle is approximately 130\u000e. Although the\nantiferromagnetic Heisenberg model on the pyrochlore\nlattice is predicted to show no magnetic ordering down to\n0 K,91a DM interaction is present in the pyrochlore ox-\nides as there is no inversion symmetry between the NN Ir\natoms. It was shown in the spin Hamiltonian including\nHeisenberg and DM interaction on a pyrochlore lattice\nthat the positive DM term gives rise to the all-in-all-out\n(AIAO) magnetic order, where all the magnetic moments\non a tetrahedron of pyrochlore lattice are pointing inward\nor outward along the local [111] direction [Fig. 11(a)].92\nOn the other hand, when the DM term is negative, non-\ncoplanar XY-type magnetic order appears where the mo-\nments lie in the plane perpendicular to the local [111]\ndirection.\nThe magnetic structure of pyrochlore iridates has been\nstudied by resonant x-ray scattering, and the q= 0\nmagnetic order of Ir moments was revealed in Eu 2Ir2O7\n[Fig. 11(b)]93. Theq= 0 propagation vector sug-\ngests the formation of the AIAO magnetic ordering or\nthe non-coplanar XY antiferromagnetic order, as ex-\npected for the presence of DM interactions. In Sm 2Ir2O7\nand Eu 2Ir2O7, the gapped magnon dispersion revealed\nby RIXS [Fig. 11(c)] supports the AIAO order of Ir\nmoments.94{96The AIAO magnetic order is discussed to\ngive rise to a Weyl semimetallic state in the vicinity of\nthe metal-insulator transition.9713\nFIG. 11. (a) All-in-all-out (AIAO) magnetic ordering on a\npyrochlore lattice. (b) The q= 0 magnetic order revealed by\nresonant x-ray scattering in Eu 2Ir2O7at the IrL3absorp-\ntion edge. The peaks at A(B) correspond to Ir 2 p3=2!t2g\n(2p3=2!eg) excitation, respectively. The strong resonant\nenhancement at A in the \u001b-\u00190channel points to a magnetic\nscattering. The resonant enhancements in the \u001b-\u001b0channel at\nboth A and B originate from anisotropic tensor susceptibility\n(ATS) scattering. The \fgure is reproduced with permission\nfrom Ref. [93] ( ©2013 the American Physical Society). (c)\nThe energy dispersion of magnetic excitation of Sm 2Ir2O7ob-\ntained from RIXS. The black dots are the experimental data\npoints and the blue dotted lines show the calculated magnon\ndispersion assuming AIAO magnetic order. The \fgure is re-\nproduced with permission from Ref. [94] ( ©2016 the Ameri-\ncan Physical Society).\nIV. J= 0 SYSTEMS: EXCITONIC MAGNETISM\nPerhaps the most radical impact of SOC on magnetism\nis realized in compounds of 4 dand 5dions withd4con-\n\fguration, such as Re3+, Ru4+, Os4+, and Ir5+. For\nthese ions with spatially extended d-orbitals, Hund's cou-\npling is smaller than the octahedral crystal \feld splitting\n10Dq, so all four electrons occupy t2glevels. The result-\ningt4\n2gcon\fguration has total spin S= 1 and threefold\norbital degeneracy described by an e\u000bective orbital mo-\nmentL= 1. Despite having well de\fned spin and or-\nbital moments on every lattice site, some d4compounds\nlack any magnetic order. This is because SOC \u0015SLwith\n\u0015>0 binds SandLmoments into a local singlet state\nwith zero total angular momentum J= 0, as shown in\nFig. 1.\nNonetheless, these nominally \\nonmagnetic\" ions\nmay develop a collective magnetism due to interaction\ne\u000bects.98,99Although there are no preexisting local mo-\nments in the ionic ground state, the J= 1 excitations\nbecome dispersive modes in a crystal, and these mobilespin-orbit excitons may condense into a magnetically or-\ndered state. For this to happen, the exchange interac-\ntions should exceed a critical value su\u000ecient to over-\ncome the energy gap \u0015betweenJ= 0 andJ= 1 ionic\nstates. The condensate wavefunction comprises a coher-\nent superposition of singlet and triplet states and carries\na magnetic moment, whose length is determined by the\ndegree of admixture of triplets in the wavefunction. Near\nthe quantum critical point (QCP), the ordered moment\ncan be very small, and the magnetic condensate strongly\n\ructuates both in phase and amplitude (i.e. rotation of\nmoments and their length oscillations). Formally, this is\nanalogous to magnon condensation phenomenon in quan-\ntum dimer systems,100but the underlying physics and\nenergy scales involved here are di\u000berent. While the spin\ngap in dimer models originates from antiferromagnetic\nexchange of two spins forming a dimer, the magnetic gap\nind4systems is of intraionic nature and given by SOC.\nSpin-orbit-entangled J= 0 compounds are interest-\ning for possible novel phases near the magnetic QCP,\nwhich can be driven by doping, pressure, and lattice con-\ntrol. Here, the new element is that J= 1 excitons are\nspin-orbit-entangled objects, and, as we will see shortly,\ntheir interactions can be anisotropic and highly frustrat-\ning. Thus, J= 0 systems with \\soft\" moments can\nnaturally realize interplay between the two phenomena\n{ frustration and quantum criticality { a topic of current\ninterest.101As a general property of orbitally degenerate\nsystems, the symmetry and low-energy behavior of spin-\norbit excitonic models is dictated by chemical bonding\ngeometry, and we discuss two representative cases below.\nA. 180\u000eM-O-M bonding, perovskites\nSimilarly to the pseudospin-1 =2d5case, the straight\n180\u000ebond geometry leads to a nearly isotropic model; in\nthe following we thus \frst focus on the isotropic model of\nO(3) symmetry. When approaching the excitonic mag-\nnet formally, we need to properly re\rect the ionic level\nstructure with nonmagnetic J= 0 ground state and\nlow-lyingJ= 1 excitations. The most natural way\nis to introduce hardcore bosons associated with the lo-\ncal excitation J= 0!1. These bosons, called here\ntriplonsT, come with the energy cost \u0015re\rected by\na local term \u0015nT=\u0015TyTand experience various pro-\ncesses corresponding to the exchange interactions be-\ntween di\u000berent sites. In the second order in Topera-\ntors, they include triplon hopping and creation or anni-\nhilation of triplon pairs in the symmetry-allowed com-\nbination/(T+1T\u00001\u0000T0T0+T\u00001T+1)ij, where the in-\ndices (\u00061;0 ) give the Jzof the triplons. Instead of Jz\neigenstates, it is convenient to use the basis consisting of\nthree triplon operators T\u000bof Cartesian \ravors (\\colors\")\n\u000b=x;y;z de\fned as\nTx=1\nip\n2(T1\u0000T\u00001); Ty=1p\n2(T1+T\u00001); Tz=iT0;\n(4.1)14\nJ=0J=1\nd4\nexchange / λtriplon\ngasJ=0J=1\nd4\nωq ωq(a)\nE(b)\n0 QCPλSOC\ntriplon condensateordered moment\n(c)exchange\n(0,0) (π,π) qtriplon\n(0,0) (π,π) qmagnonHiggsxTyT\nFIG. 12. (a) In the singlet-triplet model for d4systems with\nlarge SOC, each site is supposed to host nonmagnetic J= 0\nground state and low-lying J= 1 triplet excitations at the\nenergy\u0015. Competing with the intrasite SOC gap \u0015are vari-\nous intersite exchange processes such as a transfer of a J= 1\nexcitation to a neighboring site or their pairwise creation and\nannihilation. (b) Ordered moment in an excitonic magnet\ndepending on the ratio of exchange strength versus the local\nSOC gap\u0015. The quantum critical point (QCP) separates the\nlarge-\u0015phase where \\costly\" triplet excitations move in an in-\ncoherent way and the phase where the condensate of triplets\nis established. (c) Schematic dispersions of the elementary\nexcitations. Before the condensation, the elementary excita-\ntions are carried by triplons whose dispersion softens near the\nAF momentum as the QCP is approached. Once the triplon\ncondensate is formed, the oscillations of its amplitude and\nthe moment direction become the new fundamental modes {\n\\Higgs\" mode and magnons, respectively. The red spot at ( \u0019,\n\u0019) represents a magnetic Bragg point.\nwhich form a vector T. The main contribution to the\nexchangeJ-Hamiltonian derived in Ref. [98] then takes\na manifestly O(3) symmetric form:\nH=\u0015X\niTy\niTi+X\nhijiJij\u0010\nTy\niTj\u0000Ty\niTy\nj+ H:c:\u0011\n:(4.2)\n0.00.51.01.52.02.53.0\n-4 -3 -2 -1 0 1 2 3 4E / ζ\n∆ / ζJz = 0\nJz = ±1\nJz = ±2\nJ =0z J=0∆ = 0\nJ=1∆ = +2ζ\nJ =0zJ =z±1(b)(a)\nJ =0z∆ = −2ζFIG. 13. (a) Splitting of d4levels in tetragonal crystal\n\feld (measured in units of \u0010= 2\u0015). Out of the three triplet\nexcitations, MeO 6octahedra elongation/compression selects\nsingleJz= 0 state or the pair of Jz=\u00061 states. Together\nwith theJz= 0 ionic ground state (evolving from cubic J= 0\nstate), they form a local basis for an e\u000bective spin-1/2 (at\n\u0001<0) or spin-1 (at \u0001 >0) low-energy models. (b) Shapes\nof the relevant low-energy states represented in the same way\nas in Fig. 1 (electron density is used, not the hole one).\nThe phase diagram of this model is determined by the\ncompetition of the triplon cost \u0015and superexchange cou-\nplingJas schematically shown in Fig. 12(b). At suf-\n\fcient strength of the superexchange, triplons undergo\nBose-Einstein condensation like in spin-dimer systems.100\nHowever, the physical meaning of triplons is very dif-\nferent here. Since the magnetic moment of a d4ion\nresides primarily on the transition between the J= 0\nandJ= 1 states, as described by T, the presence of a\ntriplon condensate with hTi /ieiQR, where Qis the\nordering vector, directly translates to long-range mag-\nnetic order. The resulting magnetic order is character-\nized also by an unusual excitation spectra, see Fig. 12(c).\nThe condensation is preceded by softening of the three-\nfold degenerate triplon modes near Q. After condensa-\ntion, the modes split, giving rise to a two-fold degener-\nate magnon branch with XY-type dispersion (i.e. with\nmaximum at q= 0) and the amplitude (Higgs) mode of\nthe condensate.98These two hallmarks of soft-spin mag-\nnetism can be probed experimentally, as was done in the\nJ= 0 model system Ca 2RuO 4withd4Ru4+ions.102,10315\nThe perovskite ruthenate Ca 2RuO 4was identi\fed as\na Mott insulator104,105showing a metal-insulator transi-\ntion around 360 K106and antiferromagnetic order below\nTN\u0019110K.104,107,108Early experiments109revealed that\nSOC induces a substantial orbital angular momentum in\nRu 4dlevels, supporting the above J= 0 picture. In\nRu4+ions, the SOC strength is roughly \u0010\u0019150 meV,\ngiving the magnetic gap between J= 0 and 1 states of\nthe order of \u0015=\u0010=2\u001975 meV. Compared to \u0015, ex-\nchange interactions are somewhat smaller in ruthenates\nand would not be able to overcome such a gap. How-\never, as shown in Fig. 13(a), tetragonal distortion and\nthe associated crystal \feld \u0001 splits the J= 1 excitation\nand may reduce the gap signi\fcantly. The lower doublet\nTx=y(for the \u0001>0 case relevant for Ca 2RuO 4) can then\ncondense, giving rise to magnetic order, with the ordered\nmoment in the RuO 2plane. This scenario points to an\ninteresting possibility of a lattice-controlled QCP (e.g.\nby strain) instead of by magnetic \feld or high-pressure\nas in the dimer system TlCuCl 3.110,111This also suggests\nthe importance of the Jahn-Teller e\u000bect even in J= 0\nsystems, which acts through the splitting of triplon levels\nand renormalization of the ground state wavefunction.13\nIn Ca 2RuO 4with \u0001>0, one of the J= 1 states is\nlifted up by the crystal \feld, and we are left with three\nlow-energy states: ground state singlet and the excited\ndoublet, evolving from cubic J= 0 andJz=\u00061 (orTx=y)\nstates, respectively. These three states, whose wavefunc-\ntions are shown in the right hand side of Fig. 13(b), can\nbe used as a local basis for an e\u000bective spin-1. The result-\ning~S= 1 Hamiltonian obtained by mapping Eq. (4.2)\nonto this basis has the form of the XYmodel with a large\nsingle-ion anisotropy:\nH=EX\ni(~Sz\ni)2+JX\nhiji\u0010\n~Sx\ni~Sx\nj+~Sy\ni~Sy\nj\u0011\n; (4.3)\nwhereEdenotes a singlet-doublet excitation gap that is\nsmaller than the singlet-triplet splitting \u0015in Eq. (4.2)\ndue to a crystal \feld e\u000bect. As a result, the exchange\ninteraction Jmay overcome the reduced spin gap Eand\ninduce magnetic order.\nThe expected XY-type of magnon dispersion was\nindeed observed by inelastic neutron scattering on\nCa2RuO 4.102The experimental dispersion presented in\nFig. 14(a) additionally features a large magnon gap due\nto orthorhombicity, which is not included in the above\nsimpli\fed model. The observation of the amplitude\nHiggs mode is to some extent hindered by its strong\ndecay into a 2D two-magnon continuum (as predicted\ntheoretically112,113), which makes it a very broad feature\nin the INS spectra near the AF wavevector Q= (\u0019;\u0019),\nsee Fig. 14(c). On the other hand, the mode is relatively\nwell de\fned away from Qas visible in Figs. 14(a),(b).\nA more direct probe of the Higgs mode that enters INS\nspectra at momentum Qis the Raman scattering in the\nusually magnetically silent Agchannel. In Ref. [103], a\nHiggs mode in the scalar channel, \\unspoiled\" by the\ntwo-magnon continuum, was identi\fed in Raman spec-\nω (meV)\nwavevector q\n 0 10 20 30 40 50 60 70\n(π/2,π/2) (π,0) (π,π) (0,0) (π,0)L\nT T ′intensity (a.u.)\nω (meV)q=(π,π)\n 0 2 4 6 8\n 0 10 20 30 40 50 60T\nLT′intensity (a.u.)q=(0,0)polarized INS, spin-flip channel\nab\nc\n 0 2 4 6 8\nT L\nT′(a)\n(c)(b)FIG. 14. (a) Magnetic excitations of Ca 2RuO 4mapped by\ninelastic neutron scattering. The lines show model dispersions\nobtained within the model of Ref. [102] (its simpler version\nis described in the text). The red line indicates longitudi-\nnal mode L corresponding to the Higgs mode, and the blue\nlines represent the in-plane magnon T (solid line) and out-of-\nplane magnon T' (dashed line). Upper and lower insets are\npictorial representations of the Higgs mode (condensate am-\nplitude oscillations) and magnons (rotations of magnetic mo-\nments), respectively. (b) Magnetic response at zero wavevec-\ntorq= (0;0) obtained by polarized INS. In-plane polariza-\ntion (ab, squares) and out-of-plane polarization ( c, circles)\nwere resolved in the experiment. The experimental data are\noverplotted on top of the theoretical magnetic response that\nis decomposed according to the polarization of the modes. (c)\nThe same as in (b) for the ordering wavevector q= (\u0019;\u0019).\nAll the data are taken from Ref. [102] ( ©2017 The Authors).\ntra of Ca 2RuO 4, and found to couple to phonons, giving\nthem pronounced Fano-like lineshapes. Such an interac-\ntion with lattice modes is natural for triplons, since they\nhave a \\shape\" inherited from orbitals, and hence cou-\nple to lattice vibrations via the Jahn-Teller mechanism\nas mentioned above.\nSpin-orbit exciton condensation and related magnetic16\nQCP are more likely realized in 4 d4compounds such as\nruthenates, where SOC and exchange interactions are of\ncomparable scale and their competition can be tuned\nexperimentally. On the other hand, the 5 d4ion (Ir5+\nor Os4+) compounds are typically nonmagnetic, since\nspin-orbitJ= 1 excitations are too high in energy, as\nevidenced by RIXS experiments in 5 d-electron double\nperovskites.114{116Weak magnetism detected in the 5 d4\niridate has been explained as originating from the Ir4+\nand Ir6+magnetic defects, while the regular Ir5+sites\nremain indeed nonmagnetic.117\nB. 90\u000eM-O-M bonding, honeycomb lattice\nWhen contrasting the 180\u000eand 90\u000ebonding geome-\ntries, we encounter a situation analogous to the J= 1=2\npseudospin case. While the 180\u000ebonding geometry gen-\nerates (in leading order) the isotropic O(3) model of\nEq. (4.2), discussed above, the bonds with 90\u000eoxy-\ngen bridges are highly selective in terms of the active\n\ravors for the triplon interactions. Roughly speaking,\nwhen the oxygen-mediated hopping tdominates, each\nbond allows exchange processes of the type contained\nin Eq. (4.2) for two triplon \ravors only, depending on\nthe bond direction.98For instance, the TxandTybosons\nare equally active in z-type bonds, while the Tzboson\ncannot move in that direction. For the honeycomb lat-\ntice, the resulting pattern of active triplon pairs is pre-\nsented in Fig. 15(a). On the other hand, the dominant\ndirect hopping t0leads to the complementary Kitaev-like\npattern of Fig. 15(b), with a direct correspondence be-\ntween the bond direction \u000band triplon \ravor T\u000bactive on\nthat bond.118,119Each of these two cases is strongly frus-\ntrated; however, the nature of the corresponding ground\nstates is very di\u000berent.\nIn the interaction pattern of Fig. 15(a), each bond\nshows anO(2) symmetry of the triplon exchange Hamil-\ntonian, that is,\nH(z)\nij=JX\n\u000b=x;y(Ty\n\u000biT\u000bj\u0000Ty\n\u000biTy\n\u000bj+ H:c:) (4.4)\nfor az-bondhiji. However, the global symmetry of\nthe model is only the discrete C3one. Namely, there\nare three zigzag chains (colored di\u000berently), along which\nthe individual triplon \ravors can move. This arrange-\nment does not support 2D long-range order but instead\nleads to e\u000bective dimensionality reduction like in compass\nmodels:120C3symmetry is broken by selecting one par-\nticular triplon component, with antiferromagnetic corre-\nlations along the corresponding 1D zigzag. Zigzag chains\ninteract via the hard-core constraint only (triplon density\nchannel), so there are no phase relations and magnetic\norder between di\u000berent chains. The resulting magnetic\ncorrelations are highly anisotropic, both in real and spin\nspaces. This is a combination of an orbital ordered and\nspin-nematic state, made possible due to spin-orbital en-\ntanglement.\n(a) (b)\nxz\nyxyyz\nzxFIG. 15. Patterns indicating active \ravors (colors) for bond-\nselective triplon interactions on honeycomb lattice: (a) Oxy-\ngen mediated t-hopping case { XY-type interactions { two\n\ravors are active for a given bond directions (e.g., TxandTy\nonz-type vertical bonds). Each color forms a system of sep-\narate zigzag chains (one of the zigzag chains for Tzboson is\nmarked by shading). The symmetry resembles famous com-\npass models where each spin component interacts within its\nown 1D chain. (b) Direct t0-hopping case { bond dependent\nIsing-type of interactions. There is one-to-one correspondence\nbetween the active triplon color and the bond direction ( Tz\nonzbond, etc), establishing a bosonic analog of the Kitaev\nmodel.\nThe other limit illustrated by Fig. 15(b) may be called\na bosonic Kitaev model, following the formal similar-\nity of the triplon exchange Hamiltonian, i.e. H(\u000b)\nij=\nJ(Ty\n\u000biT\u000bj\u0000Ty\n\u000biTy\n\u000bj+H:c:) for\u000b-type bonds, to the Kitaev\ninteraction KS\u000b\niS\u000b\nj. As found in Ref. [119], the strong\nfrustration of Kitaev-type prevents a magnetic conden-\nsation at any strength of the exchange coupling Jrel-\native to spin gap \u0015. Interestingly, the model shares a\nnumber of other features with the spin-1 =2 Kitaev model\n{ there is an extensive number of Z2conserved quanti-\nties, magnetic correlations are strictly short-ranged and\ncon\fned to nearest-neighbor sites, and the excitation\nspectrum has a spin gap. However, the strongly cor-\nrelated triplon \\liquid\" ground state found in the large\nexchange limit J\u001d\u0015is smoothly connected to dilute\ntriplon gas119and hence misses the de\fning character-\nistics (long-range entanglement and emergent nonlocal\nexcitations) of genuine spin liquids. Consequently, no\nquasiparticle modes (like Majorana bands in spin-1/2 Ki-\ntaev model) are present within the spin gap. Nonethe-\nless, this strongly correlated paramagnet is far from being\ntrivial { in contrast to what is conventional in pure spin\nsystems, magnetic correlations are highly anisotropic and\nstrictly short-ranged even in the limit where the spin gap\nis very small and the QCP is close by. Magnetic order can\nbe induced by subdominant (non-Kitaev type) triplon in-\nteractions, as well as by doping, which suppresses the spin\ngap. Also, it has been found that triplon excitations ac-\nquire nontrivial band topology and protected edge states\nin a magnetic \feld.118\nBy mixing the above two complementary anisotropic\nlimits with the corresponding couplings J/t2=Uand17\nJ/t02=Uin one-to-one ratio, we recover an isotropic\ntriplon model of Eq. (4.2). Since the honeycomb lat-\ntice is not geometrically frustrated, the model shows the\nsame quantum critical behavior as in the square-lattice\ncase, i.e. dispersing triplons condense at a QCP and give\nrise to long-range antiferromagnetic order. In this con-\ntext, the ratio of the oxygen-mediated and direct hopping\namplitudes t=t0turns out to be an important \\handle\"\ndetermining the degree of frustration (as well as its type)\nof a singlet-triplet system with 90\u000ebonding geometry.\nOn the materials side, the Ru-based honeycomb\nlattice compounds are potential candidates to realize\nfrustrated spin-orbit exciton models. In particular,\nAg3LiRu 2O6,121{123which is derived from Li 2RuO 3by\nsubstituting Ag ions for Li ions between the honeycomb\nplanes, is of interest. While hexagonal symmetry is heav-\nily broken by the structural and spin-orbital dimeriza-\ntion in Li 2RuO 3,124{126Ag3LiRu 2O6avoids this tran-\nsition and thus may serve as a model system to study\nJ= 0 physics in a nearly ideal honeycomb lattice. This\ncompound shows no magnetic order,121{123which im-\nplies that the triplon interactions are either too weak to\novercome the spin-orbit gap, or they are dominated by\nKitaev-type couplings and thus highly frustrated.\nTo summarize this section, we note that physics of\nspin-orbit-entangled J= 0 compounds is still in its in-\nfancy, and indicate below a few directions for future stud-\nies.\n(i) Frustrated spin-orbit exciton models, possible ex-\notic phases and magnetic QCP in these models; topo-\nlogical properties of spin-orbit excitations. Experiments\nin various edge-shared structures and geometrically frus-\ntrated lattices, pressure and strain control of magnetic\nand structural transitions.\n(ii) The nature of metallic states induced by elec-\ntron doping, which injects J= 1=2 fermions moving\nin the background of J= 0 states. Fermion hopping\nis accompanied by creation and annihilation of spin-\norbit excitons, which should give rise to a strongly cor-\nrelated metal. In the case of perovskite lattices with\n180\u000ebonding geometry, Ref. [127] suggested that elec-\ntron doping induces ferromagnetic correlations, and pos-\nsible triplet pairing mediated by spin-orbit excitations.\nOn the experimental side, several studies128{131found\ndoping driven ferromagnetic state in Ca 2RuO 4; interest-\ningly, the recent work has reported also on signatures\nof superconductivity.132In compounds with 90\u000ebonding\ngeometry, the hopping rules are di\u000berent and interactions\nare frustrated; studies of doping e\u000bects in such systems\nmay bring some surprises.\nV. MULTIPOLAR PHYSICS IN d-ELECTRON\nSYSTEMS WITH STRONG SPIN-ORBIT\nCOUPLING\nMultipolar ordering in Mott insulators covers a whole\nhost of phenomena, ranging from the relatively standardquadrupole ordering of egelectrons due to a cooperative\nJahn-Teller e\u000bect7,10to the formation of bond multipoles\nin highly quantum-entangled frustrated magnets.133,134\nIn the absence of signi\fcant SOC, the orbital and spin\ndegrees of freedom typically order at di\u000berent tempera-\ntures. At high temperature the cooperative Jahn-Teller\ne\u000bect drives both a structural distortion of the lattice and\nan associated orbital quadrupole order, while at lower\ntemperature the exchange interaction causes the spins to\norder. Strong SOC ties together the spin and orbital de-\ngrees of freedom, negating the simple picture of separate\ntransitions. As a consequence, the intermediate phase\npicks up a spin contribution to the quadrupole order, and\nat the same time the two transitions tend to get pushed\ncloser together in temperature.\nThe introduction of strong SOC also opens up the\npossibility of unusual types of interactions, in particular\nhigher-order biquadratic and triatic terms in the Hamil-\ntonian. These interactions can drive more unusual types\nof multipolar order, such as an octupolar ground state\nsimilar to those found in f-electron systems,135,136or,\nwhen combined with frustration, cause the multipolar or-\nder to melt away, leaving behind a multipolar spin-liquid.\nA. Quadrupole ordering\nThe idea of tensor order parameters, familiar from the\ntheory of classical multipole ordering, can be readily gen-\neralised to the quantum case. Just as dipolar order is as-\nsociated with a non-zero expectation value of the vector\nhJi, quadrupole order is associated with a \fnite expecta-\ntion value of the rank-2 tensor,\nQ\u0016\u0017\ni=1\n2hJ\u0016\niJ\u0017\nj+J\u0017\niJ\u0016\nji\u0000hJi\u0001Jji\n3\u000e\u0016\u0017; \u0016;\u00172fx;y;zg\n(5.1)\nwhere the site indices i;jcan refer to the same or di\u000berent\nsites.\n1. Quadrupoles in d1systems\nQuadrupole ordering is very common in strongly spin-\norbit-entangled d1systems, since d1ions with a J= 3/2\nground state are Jahn-Teller active, as discussed in Sec-\ntion II.A. The onset of quadrupole order occurs when\nthe degeneracy of the J= 3=2 quadruplet is split, se-\nlecting a low energy Jz=\u00061=2 orJz=\u00063=2 doublet.\nThese doublets have a quadrupolar charge distribution\n(see Fig. 1), as opposed to the cubic charge distribution\nof theJ= 3=2 quadruplet, and thus quadrupolar order-\ning occurs simultaneously with a structural transition in\nwhich the local symmetry of the d1ions is reduced.\nThe driving force for the quadrupole-ordering transi-\ntion comes predominantly from electrostatic and Jahn-\nTeller interactions, with a helping hand from the ex-\nchange interaction. A good way to see this theoretically18\nFIG. 16. Schematic phase diagram proposed for strongly\nspin-orbit coupled d1ions. For a large enough interaction\nVthere are two phase transitions as a function of temper-\nature, with a high-temperature transition into a quadrupole\nordered (QO) phase followed by a low-temperature transition\ninto one of various dipolar phases, including antiferromagnetic\n(AF) and canted antiferrogmanetic (CAF[100], CAF[110]) or-\nders. The \fgure is reproduced with permission from Ref. [137]\n(©2019 The Physical Society of Japan).\nis to project the well-known interactions for the 6-fold de-\ngeneratet2gmanifold of electron con\fgurations into the\nJ= 3=2 quadruplet.138In addition to the usual bilinear\ninteractions, the resulting e\u000bective Hamiltonian also con-\ntains large biquadratic interactions, such as ( Jz\ni)2(Jz\nj)2,\nbetween neighbouring sites. It has been known for a\nlong time that these can be rewritten as quadrupole-\nquadrupole interactions,139{141and so it is not surprising\nthat they favour quadrupole ordering.\nDouble-perovskite oxides [Fig. 7(d)] provide some of\nthe cleanest material realisations of spin-orbit-entangled\nd1Mott insulators. The wide spacing of the magnetic\nions makes them good Mott insulators with small inter-\nsite interactions, and allows a cubic ionic environment\nto be retained to low temperature. Figure 16 illustrates\na generic phase diagram proposed for d1systems with\ndouble-perovskite structure.137\nWhile none of the known double-perovskite materi-\nals have a completely vanishing dipolar magnetic mo-\nment, as would be expected for isolated J= 3=2 ions\n(see Sec.II.A), the magnetic moments are small, indicat-\ning only weak hybridisation with the surrounding oxy-\ngen ions. For example Ba 2NaOsO 6has an e\u000bective mo-\nment of approximately 0 :6\u0016B.143It also shows two tran-\nsitions, with a higher temperature structural transition\natTq= 9:5 K suggestive of the onset of quadrupolar or-\nder, and a lower temperature transition at Tm\u00197:5 K\ninto a magnetically ordered phase.143{145However, since\nthe symmetry above Tqis likely tetragonal rather than\ncubic, as suggested by the approximately Rln2 entropy\nrecovery above Tq,143it is not clear how e\u000bectively the\nsystem explores the full J= 3=2 manifold at higher tem-\nperatures.\nA similar story is found in Ba 2MgReO 6, where there is\n(a)\n(b)FIG. 17. Evidence for the formation of a strongly spin-\norbit-entangled J= 3=2 state. (a) Recovery of Rln4 entropy\nin Ba 2MgReO 6at high temperatures, taken with permission\nfrom Ref. [137] ( ©2019 The Physical Society of Japan). (b)\nTaL3-edge RIXS spectrum showing the splitting of the t2g\nlevel by SOC. Reproduced with permission from Ref. [142]\n(©2019 the American Physical Society).\nan e\u000bective moment of about 0 :7\u0016B, a high-temperature\ntransition at Tq\u001933 K and a low-temperature tran-\nsition atTm\u001918 K to a similar magnetically ordered\nstate to Ba 2NaOsO 6.137,146However, unlike Ba 2NaOsO 6\nthe high temperature structure is cubic, and heat capac-\nity measurements reveal that the full Rln 4 entropy of\ntheJ= 3=2 multiplet is obtained above about 80 K, as\nshown in Fig. 17(a).137A small distortion of ReO 6oc-\ntahedra was observed below Tq, which is consistent with\nquadrupole ordering.147\nThe closely related A 2TaCl 6(A = Cs, Rb) family ap-\npears to provide a particularly good realisation of the\nJ= 3=2 state, as can be seen from the small e\u000bective\nmagnetic moment of 0 :2\u00000:3\u0016B.142The suitability of\ntheJ= 3=2 description has been con\frmed by RIXS ex-\nperiments [Fig. 17(b)] and the recovery of the expected\nRln 4 entropy at high temperature. As with the double\nperovskite oxides, two transitions are observed, with the\nupper transition at Tq\u001930 K for Cs and Tq\u001945 K\nfor Rb and the lower transition at Tm\u00195 K for Cs and\nTm\u001910 K for Rb. The upper transition is associated19\nwith a structural transition from cubic to compressed\ntetragonal, and is suggestive of a ferro-quadrupolar phase\nforming via selection of the Jz=\u00061=2 doublet.\n2. Quadrupoles in d2systems\nQuadrupolar order for d2ions can be expected either\nfrom ordering of the low-lying nonmagnetic Egdoublet\n(see Fig. 2), or driven by a combination of electrostatic,\nJahn-Teller and exchange interactions acting within the\nfullJ= 2 quintuplet.148However, there is currently a\nlack of materials showing the type of double quadrupo-\nlar and magnetic transitions observed in many d1com-\npounds.\nB. Octupole ordering\nOctupole phases involve the ordering of the rank-3 ten-\nsor,\nO\u0016\u0017\u0018\ni=hJ\u0016\niJ\u0017\njJ\u0018\nki; \u0016;\u0017;\u00182fx;y;zg; (5.2)\nin the absence of dipolar or quadrupolar order, where the\nbar indicates symmetrisation over the superscripts.\n1. Octupoles in d1systems\nA candidate to realise octupolar order in the absence\nof any concomitant dipolar order is the material Sr 2VO4\nwith perovskite structure. Although V4+(3d1) is not\nusually thought of as a strongly spin-orbit coupled ion,\nthe combination of SOC and a tetragonal elongation of\nthe oxygen octahedra conspire to select a Jz=\u00063=2\nlowest-energy doublet from the t2gmanifold, as can be\nseen in Fig. 18(a).149Projection of the usual exchange\nHamiltonian for t2gelectrons into this ground state dou-\nblet reveals a checkerboard ground state of alternating\nj i= (j3=2i\u0006j\u0000 3=2i)=p\n2 states, which corresponds to a\nstaggered octupolar order. One possible signature of this\noctupolar order would be a Goldstone mode with vanish-\ning magnetic response at low energies, potentially visible\nin inelastic neutron scattering, as shown in Fig. 18(b).\nExperimental studies are consistent with the local level\nscheme proposed for the V ions, but the question of\nwhether the ground state is octupolar ordered remains\nopen.150{153\n2. Octupoles in d2systems\nOctupolar order has been suggested to be realised in\nthed2double perovskite family Ba 2MOsO 6(M= Zn,\nMg, Ca).154,155While phase transitions are observed at\napproximately 30 K (Zn) and 50 K (Ca, Mg), there is no\nassociated development of dipolar magnetic order.155{157\nFIG. 18. Local states and collective excitations in Sr 2VO4.\n(a) Splitting of the V4+t2glevels by a tetragonal crystal \feld\n\u0001cfand spin-orbit coupling \u0015results in a Jz=\u00063=2 lowest-\nenergy doublet hosting octupolar moment. (b) Prediction for\nthe magnetic response associated with octupolar order. There\nis a sharp dispersive band of octupolar-wave excitations below\nthe continuum, whose spectral weight in magnetic channel\n(shown by line width) disappears approaching the octupolar\nBragg point at M= (\u0019;\u0019), re\recting the absence of dipolar\norder in the ground state. The energy !is in units of J=\nt2=U. The \fgures are taken with permission from Ref. [149]\n(©2009 the American Physical Society).\nFurthermore, the development of quadrupolar order is\nincompatible with the absence of detectable lattice dis-\ntortion. At the same time the recovery of only Rln 2 of\nentropy at temperatures considerably above the transi-\ntion is indicative of a low-lying doublet, and matches the\nexpectedEg-T2gsplitting shown in Fig. 2.\nFrom a theoretical point of view, projection of the in-\nteractions between t2gelectrons into the J= 2 quintu-\nplet shows the importance of bitriatic interactions, such\nas (Jz\ni)3(Jz\nj)3.148,154These can be rewritten as interac-\ntions between octupoles, and, if large enough compared\nto competing bilinear and biquadratic interactions, can\ndrive the formation of octupolar order. This may provide\na mechanism for selecting octupolar order with a ferro-\noctupolar ground-state wavefunction that is a complex\nmix of theEgstates as shown in Fig. 2(c), and in terms\nofJzstates is given by j i=1\n2j2i+ip\n2j0i+1\n2j\u00002i.154,155\nThe breaking of time-reversal symmetry at the transition\nsupports this scenario.156,157\nC. Multipoles and frustration\nOften more interesting than those systems that show\nrobust multipolar order, are those that combine multipo-\nlar order with spin-liquid behaviour, or those that avoid\nmultipole ordering and instead form spin liquids with\nmultipolar correlations. This type of behaviour is as-\nsociated with frustration, which arises in myriad ways in\nstrongly spin-orbit-entangled systems due to the inter-\nplay of lattice geometry with directional-dependent ex-\nchange and higher-order biquadratic and bitriatic inter-20\nactions.\n1.d1on the FCC lattice\nWhile not a spin liquid, the double perovskite\nBa2YMoO 6does form a valence-bond glass, in which\na disordered pattern of spin-singlet dimers freezes at\ntemperatures below about 50 K, as can be seen in\nFig. 19.158{161\nOne suggestion is that this could be associated with a\nhiddenSU(2) symmetry that can emerge from the com-\nplicated and apparently unsymmetric Hamiltonian be-\ntweenJ= 3=2 states.138,162Solving this Hamiltonian\nfor 2 sites, iandj, gives a lowest-energy singlet state\nj i= (j1=2iij\u00001=2ij\u0000j\u00001=2iij1=2ij)=p\n2, and, extend-\ning this to the FCC lattice, results in a degenerate set\nof singlet dimer coverings with lower energy than any\nmagnetically ordered state.162The idea is that in the\nmaterial a small disorder is responsible for selecting one\nof the many degenerate dimer con\fgurations, resulting\nin a dimer glass. This idea is appealing, and, due to the\nnature of the excitations above the dimer states, gives an\nexplanation for the experimentally determined soft gap,\nbut there remains the question of whether Jahn-Teller in-\nteractions, active in d1systems, play an important role.\nFIG. 19. Valence-bond glass formation in Ba 2YMoO 6. Heat\ncapacity measurements show no evidence of a phase transi-\ntion, but do show evidence for a gradual freezing, centred on\na broad maximum at about 50 K. This suggests the forma-\ntion of an amorphous valence-bond state, with a distribution\nof triplet excitation energies. The \fgure is reproduced with\npermission from Ref. [158] ( ©2010 the American Physical\nSociety).\n2.d2on the pyrochlore lattice\nThe material Y 2Mo2O7provides an example of how\nspin-glass and potentially spin-liquid physics can emerge\nout of a quadrupolar phase.163The Mo4+ions form a pyrochlore sublattice and sit in\noxygen octahedra that have a large trigonal distortion at\nall temperatures, with band structure calculations sug-\ngesting that the characteristic energy scale of the trig-\nonal splitting is more than 100 meV164[see Fig. 20(a)].\nWhen combined with SOC this results in a low energy\nJz=\u00062 doublet with quadrupolar symmetry, well sepa-\nrated from higher energy states, and with the zaxis ori-\nentated along the local in/out axes of the Mo tetrahedra,\nas shown in Fig. 20(b).165Since there are no interactions\nthat can transform Jz=\u00062 states into one another, the\nHamiltonian must be dominated by Ising interactions be-\ntween the e\u000bective spins.164{166This would suggest that\neither an all-in-all-out ordered state or a spin-ice-like dis-\nordered 2-in-2-out con\fguration should be realised. How-\never, neutron scattering experiments suggest that spin\ndegrees of freedom alone are insu\u000ecient to describe the\nlow-temperature behaviour of the system.167\nEvidence for what else needs to be taken into account\ncomes from x-ray and neutron pair distribution analyses,\nwhich show that the Mo ions are not forming a perfect\npyrochlore lattice, but instead their positions are shifted\ntowards or away from the tetrahedral centres in a dis-\nordered 2-in-2-out pattern [see Fig. 20(c)].168The ex-\nperiments further show that the oxygen octahedra are\ndragged along by the Mo ions, resulting in very little\nchange in the local crystal-\feld environment, but large\nvariations in the Mo-O-Mo bond angles, with individual\nbond angles dependent on the details of the 2-in-2-out\nlattice displacements. Deviations from the average Mo-\nO-Mo bond angle are expected to result in large changes\nto both the strength and sign of the exchange interactions\n[see Fig. 20(d)], resulting in a large coupling between the\nlattice and spin degrees of freedom and a resulting distri-\nbution in the exchange interactions.164,165As such, these\nmaterials are nice examples of the interplay of SOC with\nstrong magneto-elastic coupling.\nAt low temperatures Y 2Mo2O7shows spin-glass\nbehaviour,163,169and a number of explanations have been\nput forward to explain this.164{167,170One possibility is\nthat the low-temperature spin-glass state freezes out of\nan intermediate-temperature spin-lattice-liquid state, in\nwhich the strong magneto-elastic coupling ties together\nthe spin and lattice degrees of freedom, but the system\nremains dynamic and explores an extensive set of low-\nenergy con\fgurations.165\nVI. SPIN-ORBIT-COUPLED EXOTIC METALS\nAND NON-TRIVIAL TOPOLOGICAL PHASES\nIn the previous sections, we discussed the spin-orbit-\nentangled electronic phases in Mott insulators. However,\nthe Mott insulating state of 4 dand 5dtransition-metal\noxides is not so robust and often close to a metal-insulator\ntransition. In fact, metallic ground states are also fre-\nquently observed. In the itinerant limit, the spin-orbit-\nentangled states form bands which may be understood21\nlongshort\nmedium(a)\nMo\nMo4+\nO2-\nO\n(b) (c) (d)\nFIG. 20. The interplay of spin and lattice degrees of freedom in Y 2Mo2O7. (a) Average positions of Mo and O ions, showing\nthe pyrochlore lattice of Mo ions. (b) Jz=\u00062 states represented as Ising spins pointing along the in/out (local z) axes of\nthe Mo tetrahedra. (c) Mo ions displace into or away from tetrahedral centres, creating long, short and medium length Mo-\nMo separations. (d) Superexchange paths in the neighbouring MoO 6octahedra: (upper part) \\ \u0019-type\" superexchange path\nthat dominates when the Mo's form an undistorted pyrochlore lattice; (lower part) additional \\ \u001b-type\" superexchange path\nthat becomes increasingly important the more the Mo-O-Mo bond angle is changed from its average value. The \fgures are\nreproduced with permission from Ref. [165] ( ©2019 the American Physical Society).\nin the framework of jj-coupling. The strong SOC of 4 d\nand 5delectrons can drastically modify the band struc-\nture and may give rise to exotic metallic states, poten-\ntially with nontrivial topological character. The emer-\ngence of exotic phases such as nodal-line semimetals,\nWeyl semimetals, and topological Mott insulators has\nbeen theoretically discussed. Compared to typical topo-\nlogical semimetals composed of sandpelectrons, the\npresence of electron correlations in these oxides with d-\nelectrons is expected to provide a distinct physics of cor-\nrelated topological materials. We review in this section\nthe exotic metallic states in perovskite and pyrochlore\niridates, as well as in the doped hyperkagome. In addi-\ntion to iridates, the recently veri\fed hidden multipolar\nphase and possible unconventional superconductivity in\nthe pyrochlore rhenate will be discussed.\nA. Orthorhombic perovskite AIrO 3(A = Ca, Sr)\nwith Dirac line node\nAs discussed in Sec.III.A, the layered perovskites\nSr2IrO4and Sr 3Ir2O7are Mott insulators with local-\nizedJ= 1/2 pseudospins. In this series of Ruddlesden-\nPopper perovskite Sr n+1IrnO3n+1 (n = 1, 2,:::), the Ir\n5dbandwidth is expected to increase as a function of the\nnumber of IrO 2planes, n. SrIrO 3, which corresponds to\nthe limit of n =1, crystallizes in an orthorhombic per-\novskite with rotation and tilting distortion of IrO 6oc-\ntahedra (space group Pbnm ), illustrated in Fig. 7(b).19\nThis orthorhombic perovskite is a metastable phase sta-\nbilized under high-pressure or in a thin-\flm form; at am-\nbient pressure, SrIrO 3crystallizes in a distorted 6H-type\nperovskite structure.19\nThe orthorhombic perovskite SrIrO 3was shown to be\nmetallic from the transport and optical properties.49,171\nIt is in fact a semimetal with a small carrier density,\nwhich is produced by an interplay of crystalline symme-\ntry and strong SOC. If there were no rotations and tiltsof IrO 6octahedra, cubic SrIrO 3would have a half-\flled J\n= 1/2 band with a moderate bandwidth. When the rota-\ntions and the tiltings of IrO 6are incorporated, the bands\nare back-folded, and many crossing points in the J= 1/2\nbands show up. The incorporation of SOC opens a gap\nat many of the crossing points, which makes the system\nclose to a band insulator with 20 d-electrons per unit cell\nwith four Ir atoms. In reality, the presence of symmetry-\nprotected band crossing and the overlap of split bands\ngive rise to a semimetallic state.172\nThe semimetallic band structure of SrIrO 3hosts the\nDirac bands near the Fermi energy EF, which prevents\na gap opening. A density functional theory calculation\nand a tight-binding analysis showed the two interpene-\ntrating Dirac dispersions around the U-point of the Bril-\nlouin zone, which yield a nodal-line [Fig. 21(b)].173,174\nThe Dirac nodes are protected by the nonsymmorphic\nsymmetry of the space group Pbnm , which contains two\nglide symmetries, in addition to space- and time-reversal\nsymmetries.175,176The Dirac points are located slightly\nbelowEF, and there are other heavy hole bands cross-\ningEFto retain the charge neutrality. The presence of\nlinearly dispersive electron bands was con\frmed by an\nARPES measurement of thin-\flms.172We note that the\nambient pressure phase of SrIrO 3, crystallizing in a mon-\noclinicC2=cstructure, is also a Dirac semimetal pro-\ntected by the nonsymmorphic symmetry ( c-glide).177\nOne of the characteristic features of Dirac semimet-\nals is the presence of highly mobile carriers, which have\nbeen indeed identi\fed in the perovskite CaIrO 3, isostruc-\ntural to SrIrO 3. A carrier mobility as large as 60,000\ncm2/V\u0001s is observed at low temperatures, as shown in\nFig. 22.178The remarkably high mobility is discussed to\nbe attributed to the proximity of Dirac nodes to EF.178\nBecause of the smaller ionic radius of Ca2+as compared\nto that of Sr2+, CaIrO 3inherits larger rotation and tilt-\ning of IrO 6octahedra, which reduces the bandwidth and\nenhances electron correlations. The strong correlation\nrenormalizes the band structure and places the Dirac22\nFIG. 21. Band structures of SrIrO 3obtained from LDA + U\ncalculation with Hubbard U= 2 eV: (a) without SOC, and (b)\nwith SOC\u0010= 2\u0010at(\u0010atis atomic spin-orbit coupling). The\n\fgure is reproduced with permission from Ref. [173] ( ©2012\nthe American Physical Society).\nnodes near EF.\nIt is important to unravel the key factor determining\nthe evolution from a 3D Dirac semimetal to a magnetic\ninsulator in the series of Sr n+1IrnO3n+1. In bulk form,\nSrn+1IrnO3n+1 with n\u00153 is not stable at ambient pres-\nsure and di\u000ecult to grow. As an alternative approach\nto track the metal-insulator transition, a (001) superlat-\ntice comprising SrIrO 3and nonmagnetic SrTiO 3layers,\ni.e. [(SrIrO 3)m/SrTiO 3], has been designed.179By in-\ncreasing the number of SrIrO 3layers m, the dimension-\nality, and thus the bandwidth, of SrIrO 3layers can be\ncontrolled. The metal-insulator transition takes place at\naround m = 3 as shown in Fig. 23(a). The insulating sam-\nples with m = 1 and 2 show a magnetic transition with\nweak-ferromagnetic moments, which are induced by the\nrotations of IrO 6octahedra about the [001] axis and the\nresultant DM interaction [Fig. 23(d)]. The intimate cor-\nrelation between the metal-insulator transition and the\nappearance of magnetic order suggests that magnetism\nis essential for the occurrence of a metal-insulator tran-\nsition with reducing m.\nThe nodal-line Dirac semimetallic state of SrIrO 3\ncan be potentially exploited as a platform for other\ncorrelated topological phases by the application of\nsymmetry-breaking perturbations such as magnetic \feld\nand strain.176In particular, a variety of superlattice\nstructures has been proposed to realize novel topologi-\nFIG. 22. Transport properties of orthorhombic perovskite\nCaIrO 3. (a) Temperature dependence of longitudinal resis-\ntivity\u001axx(b) Hall conductivity \u001bxyas a function of magnetic\n\feld at several temperatures. (c), (d) Temperature depen-\ndence of carrier mobility \u0016trand carrier density n=3D, respec-\ntively. The huge mobility as large as 60,000 cm2/V\u0001s is seen\nbelow 1 K. The \fgure is reproduced from Ref. [178], CC-BY-\n4.0 (http://creativecommons.org/licenses/by/4.0/).\nFIG. 23. Temperature dependent (a) resistivity \u001a(T), (b)\n\u0000d(ln\u001a)=dT, (c) Hall constant RH, and (d) in-plane magneti-\nzationM(T) of (001) superlattice [(SrIrO 3)m/SrTiO 3] withm\n= 1, 2, 3, 4 and1. The \fgure is reproduced with permission\nfrom Ref. [179] ( ©2015 the American Physical Society).\ncal phases. By introducing a staggered potential that\nbreaks the mirror-symmetry, for example the (001) su-\nperlattice of [(SrIrO 3)/(SrRhO 3)], the appearance of a\ntopological insulator phase is anticipated.173The super-\nlattice of [(SrIrO 3)2/(CaIrO 3)2] has been predicted to be\na topological semimetal hosting a double-helicoid surface\nstate.180\nIn addition to the (001) superlattices, a topological\ninsulator phase was also predicted from fabricating a bi-\nlayer of SrIrO 3along the [111] direction.181{183In a bi-\nlayer of SrIrO 3, the IrO 6octahedra form a buckled hon-23\neycomb lattice. As in the celebrated graphene, electron\nhopping on a honeycomb lattice gives rise to Dirac bands.\nWhen a trigonal crystal \feld is incorporated, it opens a\ngap at the Dirac points, giving rise to a Z2topological\ninsulator.181\nIn fact, the fabrication of a [111] oriented thin-\flm is\ntechnically challenging in perovskite oxides A2+B4+O3,\nsince the (111) surfaces, AO 3or B planes, are polar, in\ncontrast to the (001) surfaces of AO or BO 2.184Addi-\ntional di\u000eculties arise from a size mismatch of SrIrO 3\nwith a standard substrate like SrTiO 3and the stability\nof monoclinic SrIrO 3with a hexagonal motif on a [111]\nsubstrate.185The stabilization of the orthorhombic phase\nby optimizing the A-site ion through Ca substition for Sr\nis quite useful to overcome this di\u000eculty. (111) superlat-\ntices of [(Ca 0:5Sr0:5IrO3)2m/(SrTiO 3)] with m = 1, 2 and\n3 have been successfully fabricated.186In contrast to the\nprediction of a topological insulator, the (111) superlat-\ntices with m\u00143 were found to be magnetic insulators\nand likely trivial. This again points to the importance of\nmagnetism in the superlattices of iridates.182\nB. Potential topological semimetallic state in\npyrochlore iridates\nSince soon after the discovery of spin-orbit-entangled\nphases in iridates, the pyrochlore iridates A 2Ir2O7(A:\ntrivalent cation) have been attracting tremendous inter-\nest, as they provide a unique interplay between SOC,\nelectron correlation and frustration. There have been a\nplethora of theoretical proposals for non-trivial topolog-\nical phases, including Z2topological insulators,187topo-\nlogical Mott insulators,188Weyl semimetals,97and axion\ninsulators.97,189\nThe general trend for the electronic structure of py-\nrochlore iridates has been understood as follows.190\nWhen the on-site electron correlation Uis weak, they\nshow a semimetallic electronic structure. The semimetal-\nlic state may contain small pocket Fermi surfaces\n[Fig. 24(a)] or a quadratic band touching point at the\n\u0000 point near the Fermi energy [Fig. 24(b)], depending\non the hopping parameters. By increasing the elec-\ntron correlation, AIAO order of Ir magnetic moments\ntakes place. When the original nonmagnetic state is a\nsemimetal with quadratic band touching, the AIAO or-\nder splits the degenerate bands and gives rise to crossings\nof linearly-dispersing non-degenerate bands. The resul-\ntant semimetallic phase is a Weyl semimetal with nodes\nof opposite chiralities. There are 4 pairs of Weyl nodes\nalong the [111] or equivalent directions in the Brillouin\nzone. When Uis increased further, the Weyl nodes move\nto the high-symmetry point of the Brillouin zone and the\ndistance between the pair of nodes increases. Eventually,\nthe pair of Weyl nodes with di\u000berent chiralities meets at\nthe zone boundary and annihilates, which renders a gap\nover the whole Brillouin zone and makes the system a\ntrivial AIAO antiferromagnetic insulator.\nFIG. 24. Calculated band structures of pyrochlore iridate\nnear Fermi energy with di\u000berent on-site Hubbard repulsion U.\nThe left and right columns show the results for the di\u000berent\nmagnitude of hopping parameters. The \fgure is taken with\npermission from Ref. [190] ( ©2013 the American Physical\nSociety).\nIn real materials, the relative strength of Ucan be\ntuned e\u000bectively by changing the bandwidth of Ir 5 d\nstates. As described in Sec.III.C.2, the bandwidth is re-\nduced by decreasing the ionic radius of the A-cation, rA,\ni.e. changing the degree of trigonal distortion. Among\nthe family of pyrochlore iridates A 2Ir2O7, Pr 2Ir2O7,\nwhich has the largest rA, remains metallic down to\nthe lowest temperature measured. Pr 2Ir2O7shows a\npoor metallic behavior with a small carrier density of\n\u00181021cm\u00003.191The ARPES measurement revealed that\nPr2Ir2O7has a quadratic band-touching at the \u0000 point as\nshown in Fig. 25.192In this nodal semimetallic state, the\ndensity of states near EFincreases steeply since DOS( E)\n/p\nE, which results in pronounced electron correlations\nand potentially leads to a non-Fermi liquid behavior.193\nAnother interesting behavior of Pr 2Ir2O7is that Pr3+4f\nmoments do not show a long-range magnetic order, but\ninstead a spin-liquid-like behavior.194A \fnite Hall con-\nductivity was observed at zero magnetic \feld despite the\nabsence of hysteresis in the magnetization curve, which\nhas been discussed to originate from the chirality of the\nspin-liquid state.195\nWith decreasing rA, a temperature-driven metal-\ninsulator transition is observed for A = Nd, Sm, and\nEu. The metal-insulator transition accompanies the\nAIAO magnetic order of Ir 5 dmoments as discussed in\nSec.III.C, and thus the low-temperature phase was ex-\npected as a possible realization of a Weyl semimetal.\nHowever, the presence of a charge gap has been seen\nat temperatures well below the magnetic ordering tem-24\nFIG. 25. Quadratic Fermi node of Pr 2Ir2O7revealed by the\nARPES measurement. (a) Energy dispersion along kxdirec-\ntion measured with di\u000berent incidence photon energies. (b)\nThe ARPES data in the kx\u0000k(111) sheet superposed on the\ncalculated band dispersion. The \fgure is reproduced from\nRef. [192], CC-BY-4.0 (http://creativecommons.org/licenses/\nby/4.0/).\nperatureTNfor Ir 5dmoments even in Nd 2Ir2O7,196,197\nwhich is right next to Pr 2Ir2O7. This is incompatible\nwith the Weyl semimetallic state. A Weyl semimetal\nphase might be realized only in the critical vicinity of\na metal-insulator transition and therefore hidden. Fine\ntuning of the metal-insulator transition using pressure\nor doping may help approaching a Weyl semimetal. In-\ndeed, suppression of the metal-insulator transition was\nobserved by the application of pressure198,199or by dop-\ning a small amount of Rh atoms onto the Ir site,196which\nmay stabilize the Weyl semimetallic state.\nAlthough the ground state of Nd 2Ir2O7is unlikely\nto be a Weyl semimetal at ambient conditions, a dras-\ntic magnetic-\feld-induced change of transport properties\nwas discovered, re\recting the modi\fcation of Nd3+mag-\nnetic order.200,201When a magnetic \feld is applied along\nthe [001] direction, the AIAO order of Nd3+4fmoments\nis switched into the 2-in-2-out con\fguration above \u001810\nT. Concomitantly, a drastic drop of resistivity was ob-\nserved as shown in Fig. 26, indicating an insulator to\nsemimetal transition by suppressing the AIAO order of\nIr 5delectrons via the f-dmagnetic exchange. The high-\n\feld semimetallic state has been proposed to be a nodal-\nline semimetal.200,202On the other hand, an application\nof magnetic \feld along the [111] direction induces the 3-\nin-1-out order of Nd3+moments, which is discussed to\nrealize another Weyl semimetallic phase.202\nThe putative Weyl semimetallic state in pyrochlore iri-\ndates is expected to show characteristic features such as\nsurface Fermi arcs and anomalous Hall e\u000bect (AHE). The\nAHE is associated with the fact that the Weyl nodes can\nbe regarded as a source/sink of Berry curvature. In a\nbulk pyrochlore iridate, the anomalous Hall conductivity\nis canceled because of the cubic symmetry.203However,\nin a strained thin-\flm, the cubic symmetry is broken and\nthe emergence of an AHE has been predicted.204Experi-\nmentally, such an AHE was indeed observed in thin-\flm\nFIG. 26. Angle-dependent magnetoresistance in Nd 2Ir2O7.\nThe tables on the top indicate the magnetic con\fguration of\nNd and Ir sublattices such as AIAO (AOAI) order (0-4 and 4-\n0), 2-in-2-out con\fguration (2-2) and 3-in-1-out state (3-1 or\n1-3), respectively. The \fgure is reproduced with permission\nfrom Ref. [200] ( ©2015 the American Physical Society).\npyrochlore iridates. For the Pr 2Ir2O7thin-\flm, this was\nargued to arise from the strain-induced Weyl semimetal-\nlic state with a magnetic order at the surface/interface\nas well as the breaking of cubic-symmetry.205The AHE\nwas observed also in the insulating pyrochlores such as\nEu2Ir2O7and Nd 2Ir2O7, but was attributed to spin-\nchirality206or domain walls207of AIAO magnetic or-\nder, rather than the anomalous conductivity from Weyl\nnodes.\nC. Spin-orbit-coupled semimetal out of the\ncompetition with molecular orbital formation\nA metallic state is realized also by carrier doping\ninto spin-orbit-entangled Mott insulators. In partic-\nular, carrier-doping into Sr 2IrO4has been attempted\nintensively in the search for superconductivity, moti-\nvated by the cuprate physics, as discussed in Sec.III.A.\nA spin-orbit-coupled metallic state induced by carrier-\ndoping was found also in the doped hyperkagome iri-\ndate Na 4Ir3O8. A sister compound Na 3Ir3O8, which\nshares the same hyperkagome sublattice of Ir atoms was\nsynthesized.208The chemical formula indicates that Ir\nhas a valence state of Ir4:33+, i.e. 1/3-hole doped state\nof the Na 4Ir3O8Mott insulator.\nNaively, we would expect the 1/3-hole-doped Mott\ninsulator to be a correlated metal with a large Fermi\nsurface. Na 3Ir3O8, as well as the sister compound\nLi3Ir3O8,209shows a metallic behavior, but turned out\nto be a semimetal with a small number of electrons and\nholes, rather than a large Fermi-surface metal.208The\n\frst-principle calculations indicate that the semimetallic25\nFIG. 27. Calculated band structure of Na 3Ir3O8. (a) Scalar\nrelativistic band structure showing a band insulating state.\nThe right panel illustrates the molecular orbital formation on\nthe Ir hyperkagome lattice. (b) Relativistic band structure in-\ncluding SOC. The bands which form hole and electron pockets\nare colored in red and magenta, respectively. The right panel\nschematically represents the suppression of molecular orbital\nformation by SOC. The \fgures are reproduced from Ref. [208],\nCC-BY-4.0 (http://creativecommons.org/licenses/by/4.0/).\nelectronic structure is produced by an interplay of molec-\nular orbital formation and SOC. The calculation with-\nout SOC yields a band insulator as the ground state of\nNa3Ir3O8, despite the non-integer number of d-electrons\nper Ir atom. The band insulating state can be under-\nstood as the formation of Ir 3trimer molecules with 14\nd-electrons on the triangular unit of the hyperkagome\nlattice [Fig. 27(a)]. The incorporation of SOC sup-\npresses the formation of molecular orbitals by orbital\nmixing. The conduction and valence bands made out\nof the molecular orbitals get broader and overlap, giving\nrise to a semimetallic state with small pockets of Fermi\nsurface. Such a competition between molecular orbital\nformation and SOC is likely a common feature of 4 d\nand 5dtransition-metal oxides with spatially extended\nd-orbitals. Indeed, J= 1/2 magnets often switch into\na dimerized state of transition-metal ions, which can be\nviewed as a molecular orbital formation, for example, in\nhoneycomb-based iridates and in the ruthenium chloride\nunder high pressure.210{213\nD. Spin-orbit-coupled metallic state in Cd 2Re2O7\nThe pyrochlore material Cd 2Re2O7has received much\nattention in recent years due to the spontaneous breaking\nof inversion symmetry, and its impact on superconduc-\ntivity below Tc\u00181 K. The multitudinous experimen-\ntal studies of this compound have been well reviewed inRef. [214], and we brie\ry discuss here the basic physical\nideas.\nThe most popular theoretical framework for describ-\ning Cd 2Re2O7is that of strongly spin-orbit-coupled met-\nals with relatively weak electron-electron interactions.215\nStarting from the high-temperature metal with intact\ntime-reversal and inversion symmetries, one can consider\nthe possible Fermi-surface instabilities. These include\nphases in which inversion symmetry is spontaneously bro-\nken, while time-reversal symmetry remains intact, and\nthe result is a deformation and splitting of the Fermi-\nsurface into spin polarised bands, with momentum-\ndependent spin orientation (see Fig. 28). Many of these\nelectronic order parameters couple to the lattice, and\nshould therefore drive a structural phase transition. The\ninstability that may be relevant to Cd 2Re2O7results in\na quadrupolar order parameter, and can be thought of as\nthe electron analog of chiral nematic liquid crystals.216\nThe inversion symmetry breaking instability may open\nup the possibility of unconventional, odd-parity, topolog-\nical superconductivity in the vicinity of the associated\nquantum critical point.217,218The superconductivity me-\ndiated by the \ructuations of the inversion-symmetry-\nbreaking order parameter can be either pure p-wave or\nmixeds- andp-wave, where the s- andp-mixed state\ncomes from distinct superconducting channels developing\nfrom the weakly-coupled, SOC-split bands (see Fig. 28).\nIn the case that the p-wave channel is dominant, a topo-\nlogically non-trivial state is expected, and the topological\ntransition between the s- andp-wave dominated regions\nis particularly interesting due to the presence of unusual\nvortex defects associated with the enlarged symmetry.218\nExperimentally, an inversion-symmetry-breaking\nstructural transition has been observed in Cd 2Re2O7\natTs1\u0019200 K, while superconductivity sets in at\nTc\u00191 K.214Analysis of second-harmonic generation\nexperiments has been used to tease apart the lattice\nand electronic changes at Ts1, and suggests that an\ninversion-symmetry-breaking electronic nematic phases\nis formed at the transition.219For lower temperatures,\nwhile at ambient pressure the superconductivity appears\nto be essentially s-wave, pressure can be used to tune the\nsystem, increasing both Tcand the upper critical \feld,\nBc2, with the signi\fcant increase of the latter taken\nto indicate the enhancement of the p-wave channel.214\nWhile the agreement between theory and experiment is\nvery encouraging, the experimental phase diagram as a\nfunction of both temperature and pressure is consider-\nably more complicated than the theoretical predictions,\nand much work remains on both the theoretical and\nexperimental fronts.\nVII. CONCLUSION\nCorrelated electrons in the presence of strong SOC\nform a rich variety of localized and itinerant spin-orbit-26\nFIG. 28. Schematic theoretical phase diagram for spin-orbit-\ncoupled metals. The breaking of inversion symmetry drives\nan itinerant multipolar-ordered phase with spin split bands,\nwhere the spin orientation is tied to the momentum.215As a\nfunction of a control parameter, such as pressure, a quantum\ncritical point for inversion-symmetry breaking emerges. At\naround the quantum critical point, a dome-like superconduct-\ning phase may be anticipated. The superconducting dome\nconsists of pure p-wave region and mixed s- andp-wave re-\ngion. Exotic topological properties are expected when the p-\nwave pairing dominates.217,218The \fgure is reproduced with\npermission from Ref. [214] ( ©2018 The Physical Society of\nJapan).\nentangled phases in 4 dand 5dtransition metal com-\npounds. Localized 4 d5and 5d5systems with J= 1/2\npseudospins have been explored extensively in the lastdecade, which has established the 4 dand 5dtransi-\ntion metal oxides and related compounds as an emer-\ngent paradigm in the search for unprecedented quantum\nphases. The Kitaev model has been shown to be relevant\nin a family of d5J= 1/2 honeycomb magnets. Partly\nmotivated by the J= 1/2 physics in the insulating d5\nsystems, the research e\u000bort on d1-d4and itinerant sys-\ntems has become quite active recently. Many attractive\nspin-orbit-entangled states are anticipated to emerge, in-\ncluding multipolar orderings, excitonic magnetism, a cor-\nrelated topological insulator, and a topological supercon-\nductor. As seen in this review, their potential as a mine\nof novel electronic phases has not yet been explored fully,\nparticularly for d1-d4and itinerant systems. Concepts\nhave been put forward, but their realization requires the\ndevelopment of novel materials and approaches. Unusual\nbehaviors have been observed in experiments, but under-\nstanding the physics behind them requires more elabo-\nrate and realistic theories. Besides, many yet unknown\nexotic phases likely remain hidden, and are waiting to be\nunveiled both theoretically and experimentally. We are\nconvinced that the whole family of 4 dand 5dcorrelated\noxides and related compounds with strong SOC consti-\ntutes a rich mine of novel quantum phases and is worthy\nof further exploration.\nACKNOWLEDGMENT\nT.T., A.S. and H.T. were supported by Alexander von\nHumboldt Foundation. J.Ch. acknowledges support\nby Czech Science Foundation (GA \u0014CR) under Project\nNo. GA19-16937S. G.Kh. acknowledges support by\nthe European Research Council under Advanced Grant\n669550 (Com4Com).\n1A. Abragam and B. Bleaney, Electron Paramagnetic Reso-\nnance of Transition Ions (Clanderon Press, Oxford, 1970).\n2A. Takahashi and H. Shiba, J. Phys. Soc. Jpn. 69, 3328\n(2000).\n3R. Maezono and N. Nagaosa, Phys. Rev. B 62, 11576\n(2000).\n4J. van den Brink and D. I. Khomskii, Phys. Rev. B 63,\n140416(R) (2001).\n5J. H. M. Thornley, Journal of Physics C: Solid State\nPhysics 1, 1024 (1968).\n6G. L. Stamokostas and G. A. Fiete, Phys. Rev. B 97,\n085150 (2018).\n7K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231\n(1982).\n8G. Khaliullin, Phys. Rev. B 64, 212405 (2001).\n9G. Khaliullin and S. Okamoto, Phys. Rev. B 68, 205109\n(2003).\n10G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005).\n11J. Kanamori, Journal of Applied Physics 31, S14 (1960).12G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82,\n174440 (2010).\n13H. Liu and G. Khaliullin, Phys. Rev. Lett. 122, 057203\n(2019).\n14G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205\n(2009).\n15A. Kitaev, Ann. Phys. 321, 2 (2006).\n16A. M. Glazer, Acta Crystallographica Section B 28, 3384\n(1972).\n17Y. G. Shi, Y. F. Guo, S. Yu, M. Arai, A. A. Belik, A. Sato,\nK. Yamaura, E. Takayama-Muromachi, H. F. Tian, H. X.\nYang, J. Q. Li, T. Varga, J. F. Mitchell, and S. Okamoto,\nPhys. Rev. B 80, 161104 (2009).\n18W. Bensch, H. W. Schmalle, and A. Reller, Solid State\nIonics 43, 171 (1990).\n19J. Longo, J. Kafalas, and R. Arnott, Journal of Solid\nState Chemistry 3, 174 (1971).\n20M. K. Crawford, M. A. Subramanian, R. L. Harlow, J. A.\nFernandez-Baca, Z. R. Wang, and D. C. Johnston, Phys.\nRev. B 49, 9198 (1994).27\n21O. Friedt, M. Braden, G. Andr\u0013 e, P. Adelmann, S. Nakat-\nsuji, and Y. Maeno, Phys. Rev. B 63, 174432 (2001).\n22K. Momma and F. Izumi, Journal of Applied Crystallog-\nraphy 44, 1272 (2011).\n23R. Armstrong, Physics Report 57, 343 (1980).\n24M. Subramanian, G. Aravamudan, and G. Subba Rao,\nProgress in Solid State Chemistry 15, 55 (1983).\n25J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Rev.\nMod. Phys. 82, 53 (2010).\n26Y. Moritomo, S. Xu, A. Machida, T. Katsufuji, E. Nishi-\nbori, M. Takata, M. Sakata, and S.-W. Cheong, Phys.\nRev. B 63, 144425 (2001).\n27K. Matsuhira, M. Wakeshima, Y. Hinatsu, and S. Tak-\nagi, Journal of the Physical Society of Japan 80, 094701\n(2011).\n28G. C. Mather, C. Dussarrat, J. Etourneau, and A. R.\nWest, J. Mater. Chem. 10, 2219 (2000).\n29T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono,\nL. S. I. Veiga, G. Fabbris, D. Haskel, and H. Takagi,\nPhys. Rev. Lett. 114, 077202 (2015).\n30K. A. Modic, T. E. Smidt, I. Kimchi, N. P. Breznay,\nA. Bi\u000en, S. Choi, R. D. Johnson, R. Coldea, P. Watkins-\nCurry, G. T. McCandless, J. Y. Chan, F. Gandara, Z. Is-\nlam, A. Vishwanath, A. Shekhter, R. D. McDonald, and\nJ. G. Analytis, Nat. Commun. 5, 4203 (2014).\n31Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Tak-\nagi, Phys. Rev. Lett. 99, 137207 (2007).\n32J. B. Goodenough, Magnetism and the Chemical Bond\n(Interscience Publ., New York, 1963).\n33B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park,\nC. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H.\nPark, V. Durairaj, G. Cao, and E. Rotenberg, Phys. Rev.\nLett. 101, 076402 (2008).\n34B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita,\nH. Takagi, and T. Arima, Science 323, 1329 (2009).\n35Q. Huang, J. L. Soubeyroux, O. Chmaissem, I. Natali\nSora, A. Santoro, R. J. Cava, J. J. Krajewski, and W. F.\nPeck Jr., J. Solid State Chem. 112, 355 (1994).\n36J. H. Van Vleck, Phys. Rev. 52, 1178 (1937).\n37S. Fujiyama, H. Ohsumi, T. Komesu, J. Matsuno, B. J.\nKim, M. Takata, T. Arima, and H. Takagi, Phys. Rev.\nLett. 108, 247212 (2012).\n38J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F.\nMitchell, M. van Veenendaal, M. Daghofer, J. van den\nBrink, G. Khaliullin, and B. J. Kim, Phys. Rev. Lett.\n108, 177003 (2012).\n39J. Porras, J. Bertinshaw, H. Liu, G. Khaliullin, N. H.\nSung, J.-W. Kim, S. Francoual, P. Ste\u000bens, G. Deng,\nM. M. Sala, A. E\fmenko, A. Said, D. Casa, X. Huang,\nT. Gog, J. Kim, B. Keimer, and B. J. Kim, Phys. Rev.\nB99, 085125 (2019).\n40J. Kim, M. Daghofer, A. H. Said, T. Gog, J. van den\nBrink, G. Khaliullin, and B. J. Kim, Nature Comm. 5,\n4453 (2014).\n41H. Gretarsson, N. Sung, J. Porras, J. Bertinshaw, C. Di-\netl, J. A. Bruin, A. Bangura, Y. Kim, R. Dinnebier,\nJ. Kim, A. Al-Zein, M. M. Sala, M. Krisch, M. L. Tacon,\nB. Keimer, and B. Kim, Phys. Rev. Lett. 117, 107001\n(2016).\n42X. Liu, M. P. M. Dean, Z. Y. Meng, M. H. Upton, T. Qi,\nT. Gog, Y. Cao, J. Q. Lin, D. Meyers, H. Ding, G. Cao,\nand J. P. Hill, Phys. Rev. B 93, 241102(R) (2016).\n43Y. K. Kim, O. Krupin, J. D. Denlinger, A. Bostwick,\nE. Rotenberg, Q. Zhao, J. F. Mitchell, J. W. Allen, andB. J. Kim, Science 345, 187 (2014).\n44Y. K. Kim, N. H. Sung, J. D. Denlinger, and B. J. Kim,\nNature Phys. 12, 37 (2016).\n45Y. J. Yan, M. Q. Ren, H. C. Xu, B. P. Xie, R. Tao, H. Y.\nChoi, N. Lee, Y. J. Choi, T. Zhang, and D. L. Feng, Phys.\nRev. X 5, 041018 (2015).\n46C. Lu, S. Dong, A. Quindeau, D. Preziosi, N. Hu, and\nM. Alexe, Phys. Rev. B 91, 104401 (2015).\n47J. Ravichandran, C. R. Serrao, D. K. Efetov, D. Yi, Y. S.\nOh, S.-W. Cheong, R. Ramesh, and P. Kim, J. Phys.\nCondens. Matter 28, 505304 (2016).\n48J. Bertinshaw, Y. K. Kim, G. Khaliullin, and B. J. Kim,\nAnnu. Rev. Condens. Matter Phys. 10, 315 (2019).\n49S. J. Moon, H. Jin, K. W. Kim, W. S. Choi, Y. S. Lee,\nJ. Yu, G. Cao, A. Sumi, H. Funakubo, C. Bernhard, and\nT. W. Noh, Phys. Rev. Lett. 101, 226402 (2008).\n50J. W. Kim, Y. Choi, J. Kim, J. F. Mitchell, G. Jackeli,\nM. Daghofer, J. van den Brink, G. Khaliullin, and B. J.\nKim, Phys. Rev. Lett. 109, 037204 (2012).\n51J. Kim, A. H. Said, D. Casa, M. H. Upton, T. Gog,\nM. Daghofer, G. Jackeli, J. van den Brink, G. Khaliullin,\nand B. J. Kim, Phys. Rev. Lett. 109, 157402 (2012).\n52M. M. Sala, V. Schnells, S. Boseggia, L. Simonelli, A. Al-\nZein, J. G. Vale, L. Paolasini, E. C. Hunter, R. S. Perry,\nD. Prabhakaran, A. T. Boothroyd, M. Krisch, G. Monaco,\nH. M. R\u001cnnow, D. F. McMorrow, and F. Mila, Phys. Rev.\nB92, 024405 (2015).\n53J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annu. Rev.\nCondens. Matter Phys. 7, 195 (2016).\n54S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den\nBrink, Y. Singh, P. Gegenwart, and R. Valent\u0013 \u0010, J. Phys.:\nCondens. Matter 29, 493002 (2017).\n55S. Trebst, arXiv e-prints , arXiv:1701.07056.\n56M. Hermanns, I. Kimchi, and J. Knolle, Annu. Rev. Con-\ndens. Matter Phys. 9, 17 (2018).\n57H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and\nS. E. Nagler, Nature Reviews Physics 1, 264 (2019).\n58Y. Motome and J. Nasu, J. Phys. Soc. Jpn. 89, 012002\n(2020).\n59V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoy-\nanova, H. Kandpal, S. Choi, R. Coldea, I. Rousochatzakis,\nL. Hozoi, and J. van den Brink, New J. Phys. 16, 013056\n(2014).\n60J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Phys. Rev. Lett.\n112, 077204 (2014).\n61J. G. Rau and H.-Y. Kee, arXiv e-prints , arXiv:1408.4811.\n62S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent\u0013 \u0010, Phys.\nRev. B 93, 214431 (2016).\n63J. Chaloupka and G. Khaliullin, Phys. Rev. B 92, 024413\n(2015).\n64H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 (2018).\n65R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B 97,\n014408 (2018).\n66H. Liu, J. Chaloupka, and G. Khaliullin, Phys. Rev. Lett.\n125, 047201 (2020).\n67G. Khaliullin, W. Koshibae, and S. Maekawa, Phys. Rev.\nLett. 93, 176401 (2004).\n68T. Hyart, A. R. Wright, G. Khaliullin, and B. Rosenow,\nPhys. Rev. B 85, 140510(R) (2012).\n69Y.-Z. You, I. Kimchi, and A. Vishwanath, Phys. Rev. B\n86, 085145 (2012).\n70S. Okamoto, Phys. Rev. Lett. 110, 066403 (2013).\n71G. Chen and L. Balents, Phys. Rev. B 78, 094403 (2008).28\n72A. C. Shockley, F. Bert, J.-C. Orain, Y. Okamoto, and\nP. Mendels, Phys. Rev. Lett. 115, 047201 (2015).\n73R. Dally, T. Hogan, A. Amato, H. Luetkens, C. Baines,\nJ. Rodriguez-Rivera, M. J. Graf, and S. D. Wilson, Phys.\nRev. Lett. 113, 247601 (2014).\n74H. Zheng, J. Zhang, C. C. Stoumpos, Y. Ren, Y.-S. Chen,\nR. Dally, S. D. Wilson, Z. Islam, and J. F. Mitchell, Phys.\nRev. Materials 2, 043403 (2018).\n75J. M. Hopkinson, S. V. Isakov, H.-Y. Kee, and Y. B. Kim,\nPhys. Rev. Lett. 99, 037201 (2007).\n76Y. Zhou, P. A. Lee, T.-K. Ng, and F.-C. Zhang, Phys.\nRev. Lett. 101, 197201 (2008).\n77M. J. Lawler, H.-Y. Kee, Y. B. Kim, and A. Vishwanath,\nPhys. Rev. Lett. 100, 227201 (2008).\n78M. J. Lawler, A. Paramekanti, Y. B. Kim, and L. Balents,\nPhys. Rev. Lett. 101, 197202 (2008).\n79M. R. Norman and T. Micklitz, Phys. Rev. B 81, 024428\n(2010).\n80I. Kimchi and A. Vishwanath, Phys. Rev. B 89, 014414\n(2014).\n81T. Micklitz and M. R. Norman, Phys. Rev. B 81, 174417\n(2010).\n82R. Shindou, Phys. Rev. B 93, 094419 (2016).\n83T. Mizoguchi, K. Hwang, E. K.-H. Lee, and Y. B. Kim,\nPhys. Rev. B 94, 064416 (2016).\n84H. Kuriyama, J. Matsuno, S. Niitaka, M. Uchida,\nD. Hashizume, A. Nakao, K. Sugimoto, H. Ohsumi,\nM. Takata, and H. Takagi, Applied Physics Letters 96,\n182103 (2010).\n85S. Onoda and F. Ishii, Phys. Rev. Lett. 122, 067201\n(2019).\n86D. Yanagishima and Y. Maeno, Journal of the Physical\nSociety of Japan 70, 2880 (2001).\n87K. Matsuhira, M. Wakeshima, R. Nakanishi, T. Yamada,\nA. Nakamura, W. Kawano, S. Takagi, and Y. Hinatsu,\nJournal of the Physical Society of Japan 76, 043706\n(2007).\n88H. Shinaoka, S. Hoshino, M. Troyer, and P. Werner, Phys.\nRev. Lett. 115, 156401 (2015).\n89L. Hozoi, H. Gretarsson, J. P. Clancy, B.-G. Jeon, B. Lee,\nK. H. Kim, V. Yushankhai, P. Fulde, D. Casa, T. Gog,\nJ. Kim, A. H. Said, M. H. Upton, Y.-J. Kim, and\nJ. van den Brink, Phys. Rev. B 89, 115111 (2014).\n90A. Krajewska, T. Takayama, R. Dinnebier, A. Yaresko,\nK. Ishii, M. Isobe, and H. Takagi, Phys. Rev. B 101,\n121101 (2020).\n91R. Moessner and J. T. Chalker, Phys. Rev. Lett. 80, 2929\n(1998).\n92M. Elhajal, B. Canals, R. Sunyer, and C. Lacroix, Phys.\nRev. B 71, 094420 (2005).\n93H. Sagayama, D. Uematsu, T. Arima, K. Sugimoto, J. J.\nIshikawa, E. O'Farrell, and S. Nakatsuji, Phys. Rev. B\n87, 100403 (2013).\n94C. Donnerer, M. C. Rahn, M. M. Sala, J. G. Vale,\nD. Pincini, J. Strempfer, M. Krisch, D. Prabhakaran,\nA. T. Boothroyd, and D. F. McMorrow, Phys. Rev. Lett.\n117, 037201 (2016).\n95S. H. Chun, B. Yuan, D. Casa, J. Kim, C.-Y. Kim,\nZ. Tian, Y. Qiu, S. Nakatsuji, and Y.-J. Kim, Phys. Rev.\nLett. 120, 177203 (2018).\n96E. K.-H. Lee, S. Bhattacharjee, and Y. B. Kim, Phys.\nRev. B 87, 214416 (2013).\n97X. Wan, A. M. Turner, A. Vishwanath, and S. Y.\nSavrasov, Phys. Rev. B 83, 205101 (2011).98G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).\n99O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi,\nPhys. Rev. B 91, 054412 (2015).\n100T. Giamarchi, C. R uegg, and O. Tchernyshyov, Nature\nPhys. 4, 198 (2008).\n101M. Vojta, Rep. Prog. Phys. 81, 064501 (2018).\n102A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen,\nD. L. Abernathy, J. T. Park, A. Ivanov, J. Chaloupka,\nG. Khaliullin, B. Keimer, and B. J. Kim, Nature Phys.\n13, 633 (2017).\n103S.-M. Souliou, J. Chaloupka, G. Khaliullin, G. Ryu,\nA. Jain, B. J. Kim, M. L. Tacon, and B. Keimer, Phys.\nRev. Lett. 119, 067201 (2017).\n104S. Nakatsuji, S. Ikeda, and Y. Maeno, J. Phys. Soc. Jpn.\n66, 1868 (1997).\n105S. Nakatsuji and Y. Maeno, Phys. Rev. Lett. 84, 2666\n(2000).\n106C. S. Alexander, G. Cao, V. Dobrosavljevic, S. McCall,\nJ. E. Crow, E. Lochner, and R. P. Guertin, Phys. Rev.\nB60, R8422 (1999).\n107G. Cao, S. McCall, M. Shepard, J. E. Crow, and R. P.\nGuertin, Phys. Rev. B 56, R2916 (1997).\n108M. Braden, G. Andr\u0013 e, S. Nakatsuji, and Y. Maeno, Phys.\nRev. B 58, 847 (1998).\n109T. Mizokawa, L. H. Tjeng, G. A. Sawatzky, G. Ghir-\ninghelli, O. Tjernberg, N. B. Brookes, H. Fukazawa,\nS. Nakatsuji, and Y. Maeno, Phys. Rev. Lett. 87, 077202\n(2001).\n110C. R uegg, N. Cavadini, A. Furrer, H.-U. G udel,\nK. Kr amer, H. Mutka, A. Wildes, K. Habicht, and\nP. Vorderwisch, Nature 423, 62 (2003).\n111C. R uegg, B. Normand, M. Matsumoto, A. Furrer, D. F.\nMcMorrow, K. W. Kr amer, H.-U. G udel, S. N. Gvasaliya,\nH. Mutka, and M. Boehm, Phys. Rev. Lett. 100, 205701\n(2008).\n112D. Podolsky, A. Auerbach, and D. P. Arovas, Phys. Rev.\nB84, 174522 (2011).\n113F. Rose, F. L\u0013 eonard, and N. Dupuis, Phys. Rev. B 91,\n224501 (2015).\n114T. Dey, A. Maljuk, D. V. Efremov, O. Kataeva, S. Gass,\nC. G. F. Blum, F. Steckel, D. Gruner, T. Ritschel, A. U. B.\nWolter, J. Geck, C. Hess, K. Koepernik, J. van den Brink,\nS. Wurmehl, and B. B uchner, Phys. Rev. B 93, 014434\n(2016).\n115N. R. Davies, C. V. Topping, H. Jacobsen, A. J. Princep,\nF. K. K. Kirschner, M. C. Rahn, M. Bristow, J. G. Vale,\nI. da Silva, P. J. Baker, C. J. Sahle, Y.-F. Guo, D.-Y. Yan,\nY.-G. Shi, S. J. Blundell, D. F. McMorrow, and A. T.\nBoothroyd, Phys. Rev. B 99, 174442 (2019).\n116B. Yuan, J. P. Clancy, A. M. Cook, C. M. Thompson,\nJ. Greedan, G. Cao, B. C. Jeon, T. W. Noh, M. H. Upton,\nD. Casa, T. Gog, A. Paramekanti, and Y.-J. Kim, Phys.\nRev. B 95, 235114 (2017).\n117S. Fuchs, T. Dey, G. Aslan-Cansever, A. Maljuk,\nS. Wurmehl, B. B uchner, and V. Kataev, Phys. Rev.\nLett. 120, 237204 (2018).\n118P. S. Anisimov, F. Aust, G. Khaliullin, and M. Daghofer,\nPhys. Rev. Lett. 122, 177201 (2019).\n119J. Chaloupka and G. Khaliullin, Phys. Rev. B 100, 224413\n(2019).\n120Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87,\n1 (2015).\n121S. A. J. Kimber, C. D. Ling, D. J. P. Morris, A. Chemsed-\ndine, P. F. Henry, and D. N. Argyriou, J. Mater. Chem.29\n20, 8021 (2010).\n122R. Kumar, T. Dey, P. M. Ette, K. Ramesha,\nA. Chakraborty, I. Dasgupta, J. C. Orain, C. Baines,\nS. T\u0013 oth, A. Shahee, S. Kundu, M. Prinz-Zwick, A. A.\nGippius, N. B uttgen, P. Gegenwart, and A. V. Mahajan,\nPhys. Rev. B 99, 054417 (2019).\n123T. Takayama et al. (unpublished).\n124Y. Miura, Y. Yasui, M. Sato, N. Igawa, and K. Kakurai,\nJ. Phys. Soc. Jpn. 76, 033705 (2007).\n125G. Jackeli and D. I. Khomskii, Phys. Rev. Lett. 100,\n147203 (2008).\n126S. A. J. Kimber, I. I. Mazin, J. Shen, H. O. Jeschke, S. V.\nStreltsov, D. N. Argyriou, R. Valent\u0013 \u0010, and D. I. Khomskii,\nPhys. Rev. B 89, 081408(R) (2014).\n127J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 116,\n017203 (2016).\n128G. Cao, S. McCall, V. Dobrosavljevic, C. S. Alexan-\nder, J. E. Crow, and R. P. Guertin, Phys. Rev. B 61,\nR5053(R) (2000).\n129G. Cao, C. S. Alexander, S. McCall, J. E. Crow, and R. P.\nGuertin, J. Magn. Magn. Mater. 226-230 , 235 (2001).\n130F. Nakamura, M. Sakaki, Y. Yamanaka, S. Tamaru,\nT. Suzuki, and Y. Maeno, Sci. Rep. 3, 2536 (2013).\n131H. Boschker, T. Harada, T. Asaba, R. Ashoori, A. Boris,\nH. Hilgenkamp, C. Hughes, M. Holtz, L. Li, D. Muller,\nH. Nair, P. Reith, X. R. Wang, D. Schlom, A. Soukiassian,\nand J. Mannhart, Phys. Rev. X 9, 011027 (2019).\n132H. Nobukane, K. Yanagihara, Y. Kunisada, Y. Oga-\nsawara, K. Isono, K. Nomura, K. Tanahashi, T. Nomura,\nT. Akiyama, and S. Tanda, Sci. Rep. 10, 3462 (2020).\n133A. Andreev and I. Grishchuk, JETP 87, 467 (1984).\n134N. Shannon, T. Momoi, and P. Sindzingre, Phys. Rev.\nLett. 96, 027213 (2006).\n135Y. Kuramoto, H. Kusunose, and A. Kiss, Journal of the\nPhysical Society of Japan 78, 072001 (2009).\n136P. Santini, S. Carretta, G. Amoretti, R. Caciu\u000bo, N. Mag-\nnani, and G. H. Lander, Rev. Mod. Phys. 81, 807 (2009).\n137D. Hirai and Z. Hiroi, Journal of the Physical Society of\nJapan 88, 064712 (2019).\n138G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82,\n174440 (2010).\n139M. Blume and Y. Hsieh, Journal of Applied Physics 40,\n1249 (1969).\n140H. H. Chen and P. M. Levy, Phys. Rev. Lett. 27, 1383\n(1971).\n141N. Papanicolaou, Nuclear Physics B 305, 367 (1988).\n142H. Ishikawa, T. Takayama, R. K. Kremer, J. Nuss, R. Din-\nnebier, K. Kitagawa, K. Ishii, and H. Takagi, Phys. Rev.\nB100, 045142 (2019).\n143A. S. Erickson, S. Misra, G. J. Miller, R. R. Gupta,\nZ. Schlesinger, W. A. Harrison, J. M. Kim, and I. R.\nFisher, Phys. Rev. Lett. 99, 016404 (2007).\n144L. Lu, M. Song, W. Liu, A. P. Reyes, P. Kuhns, H. O. Lee,\nI. R. Fisher, and V. F. Mitrovi\u0013 c, Nature Communications\n8, 14407 EP (2017).\n145K. Willa, R. Willa, U. Welp, I. R. Fisher, A. Rydh, W.-K.\nKwok, and Z. Islam, Phys. Rev. B 100, 041108 (2019).\n146C. A. Marjerrison, C. M. Thompson, G. Sala, D. D. Ma-\nharaj, E. Kermarrec, Y. Cai, A. M. Hallas, M. N. Wilson,\nT. J. S. Munsie, G. E. Granroth, R. Flacau, J. E. Greedan,\nB. D. Gaulin, and G. M. Luke, Inorganic Chemistry 55,\n10701 (2016).\n147D. Hirai, H. Sagayama, S. Gao, H. Ohsumi, G. Chen, T.-\nh. Arima, and Z. Hiroi, Phys. Rev. Research 2, 022063(2020).\n148G. Chen and L. Balents, Phys. Rev. B 84, 094420 (2011).\n149G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 103, 067205\n(2009).\n150H. D. Zhou, B. S. Conner, L. Balicas, and C. R. Wiebe,\nPhys. Rev. Lett. 99, 136403 (2007).\n151H. D. Zhou, Y. J. Jo, J. Fiore Carpino, G. J. Munoz, C. R.\nWiebe, J. G. Cheng, F. Rivadulla, and D. T. Adroja,\nPhys. Rev. B 81, 212401 (2010).\n152J. Teyssier, R. Viennois, E. Giannini, R. M. Eremina,\nA. G unther, J. Deisenhofer, M. V. Eremin, and D. van der\nMarel, Phys. Rev. B 84, 205130 (2011).\n153J. Teyssier, E. Giannini, A. Stucky, R. \u0014Cern\u0013 y, M. V.\nEremin, and D. van der Marel, Phys. Rev. B 93, 125138\n(2016).\n154A. Paramekanti, D. D. Maharaj, and B. D. Gaulin, Phys.\nRev. B 101, 054439 (2020).\n155D. D. Maharaj, G. Sala, M. B. Stone, E. Kermarrec,\nC. Ritter, F. Fauth, C. A. Marjerrison, J. E. Greedan,\nA. Paramekanti, and B. D. Gaulin, Phys. Rev. Lett. 124,\n087206 (2020).\n156C. M. Thompson, J. P. Carlo, R. Flacau, T. Aharen, I. A.\nLeahy, J. R. Pollichemi, T. J. S. Munsie, T. Medina, G. M.\nLuke, J. Munevar, S. Cheung, T. Goko, Y. J. Uemura,\nand J. E. Greedan, Journal of Physics: Condensed Matter\n26, 306003 (2014).\n157C. A. Marjerrison, C. M. Thompson, A. Z. Sharma, A. M.\nHallas, M. N. Wilson, T. J. S. Munsie, R. Flacau, C. R.\nWiebe, B. D. Gaulin, G. M. Luke, and J. E. Greedan,\nPhys. Rev. B 94, 134429 (2016).\n158M. A. de Vries, A. C. Mclaughlin, and J.-W. G. Bos,\nPhys. Rev. Lett. 104, 177202 (2010).\n159T. Aharen, J. E. Greedan, C. A. Bridges, A. A. Aczel,\nJ. Rodriguez, G. MacDougall, G. M. Luke, T. Imai,\nV. K. Michaelis, S. Kroeker, H. Zhou, C. R. Wiebe, and\nL. M. D. Cranswick, Phys. Rev. B 81, 224409 (2010).\n160J. P. Carlo, J. P. Clancy, T. Aharen, Z. Yamani, J. P. C.\nRu\u000b, J. J. Wagman, G. J. Van Gastel, H. M. L. Noad,\nG. E. Granroth, J. E. Greedan, H. A. Dabkowska, and\nB. D. Gaulin, Phys. Rev. B 84, 100404 (2011).\n161M. A. de Vries, J. O. Piatek, M. Misek, J. S. Lord, H. M.\nR\u001cnnow, and J.-W. G. Bos, New Journal of Physics 15,\n043024 (2013).\n162J. Romh\u0013 anyi, L. Balents, and G. Jackeli, Phys. Rev. Lett.\n118, 217202 (2017).\n163J. Greedan, M. Sato, X. Yan, and F. Razavi, Solid State\nCommunications 59, 895 (1986).\n164H. Shinaoka, Y. Motome, T. Miyake, and S. Ishibashi,\nPhys. Rev. B 88, 174422 (2013).\n165A. Smerald and G. Jackeli, Phys. Rev. Lett. 122, 227202\n(2019).\n166H. Shinaoka, Y. Motome, T. Miyake, S. Ishibashi, and\nP. Werner, Journal of Physics: Condensed Matter 31,\n323001 (2019).\n167H. J. Silverstein, K. Fritsch, F. Flicker, A. M. Hallas, J. S.\nGardner, Y. Qiu, G. Ehlers, A. T. Savici, Z. Yamani, K. A.\nRoss, B. D. Gaulin, M. J. P. Gingras, J. A. M. Paddison,\nK. Foyevtsova, R. Valenti, F. Hawthorne, C. R. Wiebe,\nand H. D. Zhou, Phys. Rev. B 89, 054433 (2014).\n168P. M. M. Thygesen, J. A. M. Paddison, R. Zhang, K. A.\nBeyer, K. W. Chapman, H. Y. Playford, M. G. Tucker,\nD. A. Keen, M. A. Hayward, and A. L. Goodwin, Phys.\nRev. Lett. 118, 067201 (2017).30\n169M. J. P. Gingras, C. V. Stager, N. P. Raju, B. D. Gaulin,\nand J. E. Greedan, Phys. Rev. Lett. 78, 947 (1997).\n170K. Mitsumoto, C. Hotta, and H. Yoshino, Phys. Rev.\nLett. 124, 087201 (2020).\n171J. G. Zhao, L. X. Yang, Y. Yu, F. Y. Li, R. C. Yu, Z. Fang,\nL. C. Chen, and C. Q. Jin, Journal of Applied Physics\n103, 103706 (2008).\n172Y. F. Nie, P. D. C. King, C. H. Kim, M. Uchida, H. I.\nWei, B. D. Faeth, J. P. Ruf, J. P. C. Ru\u000b, L. Xie, X. Pan,\nC. J. Fennie, D. G. Schlom, and K. M. Shen, Phys. Rev.\nLett. 114, 016401 (2015).\n173J.-M. Carter, V. V. Shankar, M. A. Zeb, and H.-Y. Kee,\nPhys. Rev. B 85, 115105 (2012).\n174M. A. Zeb and H.-Y. Kee, Phys. Rev. B 86, 085149 (2012).\n175Y. Chen, H.-S. Kim, and H.-Y. Kee, Phys. Rev. B 93,\n155140 (2016).\n176Y. Chen, Y.-M. Lu, and H.-Y. Kee, Nature Communica-\ntions 6, 6593 (2015).\n177T. Takayama, A. N. Yaresko, and H. Takagi, Journal of\nPhysics: Condensed Matter 31, 074001 (2018).\n178J. Fujioka, R. Yamada, M. Kawamura, S. Sakai, M. Hi-\nrayama, R. Arita, T. Okawa, D. Hashizume, M. Hoshino,\nand Y. Tokura, Nature Communications 10, 362 (2019).\n179J. Matsuno, K. Ihara, S. Yamamura, H. Wadati, K. Ishii,\nV. V. Shankar, H.-Y. Kee, and H. Takagi, Phys. Rev.\nLett. 114, 247209 (2015).\n180C. Fang, L. Lu, J. Liu, and L. Fu, Nature Physics 12,\n936 (2016).\n181D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, and S. Okamoto,\nNature Communications 2, 596 (2011).\n182S. Okamoto and D. Xiao, Journal of the Physical Society\nof Japan 87, 041006 (2018).\n183J. L. Lado, V. Pardo, and D. Baldomir, Phys. Rev. B 88,\n155119 (2013).\n184T. J. Anderson, S. Ryu, H. Zhou, L. Xie, J. P. Pod-\nkaminer, Y. Ma, J. Irwin, X. Q. Pan, M. S. Rzchowski,\nand C. B. Eom, Applied Physics Letters 108, 151604\n(2016).\n185A. Sumi, Y. Kim, N. Oshima, K. Akiyama, K. Saito, and\nH. Funakubo, Thin Solid Films 486, 182 (2005).\n186D. Hirai, J. Matsuno, and H. Takagi, APL Materials 3,\n041508 (2015).\n187H.-M. Guo and M. Franz, Phys. Rev. Lett. 103, 206805\n(2009).\n188D. Pesin and L. Balents, Nat. Phys. 6, 376 (2010).\n189A. Go, W. Witczak-Krempa, G. S. Jeon, K. Park, and\nY. B. Kim, Phys. Rev. Lett. 109, 066401 (2012).\n190W. Witczak-Krempa, A. Go, and Y. B. Kim, Phys. Rev.\nB87, 155101 (2013).\n191Y. Machida, S. Nakatsuji, Y. Maeno, T. Tayama,\nT. Sakakibara, and S. Onoda, Phys. Rev. Lett. 98, 057203\n(2007).\n192T. Kondo, M. Nakayama, R. Chen, J. J. Ishikawa, E.-\nG. Moon, T. Yamamoto, Y. Ota, W. Malaeb, H. Kanai,\nY. Nakashima, Y. Ishida, R. Yoshida, H. Yamamoto,\nM. Matsunami, S. Kimura, N. Inami, K. Ono, H. Ku-\nmigashira, S. Nakatsuji, L. Balents, and S. Shin, Nature\nCommunications 6, 10042 (2015).\n193E.-G. Moon, C. Xu, Y. B. Kim, and L. Balents, Phys.\nRev. Lett. 111, 206401 (2013).\n194S. Nakatsuji, Y. Machida, Y. Maeno, T. Tayama,\nT. Sakakibara, J. v. Duijn, L. Balicas, J. N. Millican,\nR. T. Macaluso, and J. Y. Chan, Phys. Rev. Lett. 96,\n087204 (2006).195Y. Machida, S. Nakatsuji, S. Onoda, T. Tayama, and\nT. Sakakibara, Nature 463, 210 (2010).\n196K. Ueda, J. Fujioka, Y. Takahashi, T. Suzuki, S. Ishiwata,\nY. Taguchi, and Y. Tokura, Phys. Rev. Lett. 109, 136402\n(2012).\n197M. Nakayama, T. Kondo, Z. Tian, J. J. Ishikawa,\nM. Halim, C. Bareille, W. Malaeb, K. Kuroda, T. Tomita,\nS. Ideta, K. Tanaka, M. Matsunami, S. Kimura, N. Inami,\nK. Ono, H. Kumigashira, L. Balents, S. Nakatsuji, and\nS. Shin, Phys. Rev. Lett. 117, 056403 (2016).\n198M. Sakata, T. Kagayama, K. Shimizu, K. Matsuhira,\nS. Takagi, M. Wakeshima, and Y. Hinatsu, Phys. Rev. B\n83, 041102 (2011).\n199F. F. Tafti, J. J. Ishikawa, A. McCollam, S. Nakatsuji,\nand S. R. Julian, Phys. Rev. B 85, 205104 (2012).\n200K. Ueda, J. Fujioka, B.-J. Yang, J. Shiogai, A. Tsukazaki,\nS. Nakamura, S. Awaji, N. Nagaosa, and Y. Tokura,\nPhys. Rev. Lett. 115, 056402 (2015).\n201Z. Tian, Y. Kohama, T. Tomita, H. Ishizuka, T. H. Hsieh,\nJ. J. Ishikawa, K. Kindo, L. Balents, and S. Nakatsuji,\nNature Physics 12, 134 (2016).\n202K. Ueda, T. Oh, B.-J. Yang, R. Kaneko, J. Fujioka, N. Na-\ngaosa, and Y. Tokura, Nature Communications 8, 15515\n(2017).\n203K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B 84,\n075129 (2011).\n204B.-J. Yang and N. Nagaosa, Phys. Rev. Lett. 112, 246402\n(2014).\n205T. Ohtsuki, Z. Tian, A. Endo, M. Halim, S. Katsumoto,\nY. Kohama, K. Kindo, M. Lippmaa, and S. Nakatsuji,\nProceedings of the National Academy of Sciences 116,\n8803 (2019).\n206T. C. Fujita, Y. Kozuka, M. Uchida, A. Tsukazaki,\nT. Arima, and M. Kawasaki, Scienti\fc Reports 5, 9711\n(2015).\n207W. J. Kim, J. H. Gruenewald, T. Oh, S. Cheon, B. Kim,\nO. B. Korneta, H. Cho, D. Lee, Y. Kim, M. Kim, J.-G.\nPark, B.-J. Yang, A. Seo, and T. W. Noh, Phys. Rev. B\n98, 125103 (2018).\n208T. Takayama, A. Yaresko, A. Matsumoto, J. Nuss,\nK. Ishii, M. Yoshida, J. Mizuki, and H. Takagi, Scien-\nti\fc Reports 4, 6818 (2014).\n209T. Takayama, A. N. Yaresko, A. S. Gibbs, K. Ishii,\nD. Kukusta, and H. Takagi, Phys. Rev. Materials 4,\n075002 (2020).\n210L. S. I. Veiga, M. Etter, K. Glazyrin, F. Sun, C. A. Es-\ncanhoela, G. Fabbris, J. R. L. Mardegan, P. S. Malavi,\nY. Deng, P. P. Stavropoulos, H.-Y. Kee, W. G. Yang,\nM. van Veenendaal, J. S. Schilling, T. Takayama, H. Tak-\nagi, and D. Haskel, Phys. Rev. B 96, 140402 (2017).\n211T. Takayama, A. Krajewska, A. S. Gibbs, A. N. Yaresko,\nH. Ishii, H. Yamaoka, K. Ishii, N. Hiraoka, N. P. Fun-\nnell, C. L. Bull, and H. Takagi, Phys. Rev. B 99, 125127\n(2019).\n212V. Hermann, M. Altmeyer, J. Ebad-Allah, F. Freund,\nA. Jesche, A. A. Tsirlin, M. Han\rand, P. Gegenwart, I. I.\nMazin, D. I. Khomskii, R. Valent\u0013 \u0010, and C. A. Kuntscher,\nPhys. Rev. B 97, 020104 (2018).\n213T. Biesner, S. Biswas, W. Li, Y. Saito, A. Pustogow,\nM. Altmeyer, A. U. B. Wolter, B. B uchner, M. Roslova,\nT. Doert, S. M. Winter, R. Valent\u0013 \u0010, and M. Dressel, Phys.\nRev. B 97, 220401 (2018).\n214Z. Hiroi, J.-i. Yamaura, T. C. Kobayashi, Y. Matsub-\nayashi, and D. Hirai, Journal of the Physical Society of31\nJapan 87, 024702 (2018).\n215L. Fu, Phys. Rev. Lett. 115, 026401 (2015).\n216T. C. Lubensky and L. Radzihovsky, Phys. Rev. E 66,\n031704 (2002).217V. Kozii and L. Fu, Phys. Rev. Lett. 115, 207002 (2015).\n218Y. Wang, G. Y. Cho, T. L. Hughes, and E. Fradkin, Phys.\nRev. B 93, 134512 (2016).\n219J. W. Harter, Z. Y. Zhao, J.-Q. Yan, D. G. Mandrus, and\nD. Hsieh, Science 356, 295 (2017)." }, { "title": "1504.03786v1.Spin_diffusion_in_ultracold_spin_orbit_coupled____40__K_gas.pdf", "content": "arXiv:1504.03786v1 [cond-mat.quant-gas] 15 Apr 2015Spin diffusion in ultracold spin-orbit coupled40K gas\nT. Yu and M. W. Wu∗\nHefei National Laboratory for Physical Sciences at Microsc ale,\nKey Laboratory of Strongly-Coupled Quantum Matter Physics and Department of Physics,\nUniversity of Science and Technology of China, Hefei, Anhui , 230026, China\n(Dated: October 13, 2018)\nWe investigate the steady-state spin diffusion for ultracol d spin-orbit coupled40K gas by the\nkinetic spin Bloch equation approach both analytically and numerically. Four configurations, i.e.,\nthe spin diffusions along two specific directions with the spi n polarization perpendicular (transverse\nconfiguration) and parallel (longitudinal configuration) t o the effective Zeeman field are studied.\nIt is found that the behaviors of the steady-state spin diffus ion for the four configurations are\nvery different, which are determined by three characteristi c lengths: the mean free path lτ, the\nZeeman oscillation length lΩand the spin-orbit coupling oscillation length lα. It is analytically\nrevealedandnumerically confirmedthatbytuningthescatte ringstrength, thesystem canbedivided\nintofiveregimes: I, weak scattering regime ( lτ/greaterorsimilarlΩ,lα); II, Zeeman field-dominated moderate\nscattering regime ( lΩ≪lτ≪lα); III, spin-orbit coupling-dominated moderate scatterin g regime\n(lα≪lτ≪lΩ); IV, relatively strong scattering regime ( lc\nτ≪lτ≪lΩ,lα); V, strong scattering\nregime (lτ≪lΩ,lα,lc\nτ), withlc\nτrepresenting the crossover length between the relatively s trong and\nstrong scattering regimes. In different regimes, the behavi ors of the spacial evolution of the steady-\nstate spin polarization are very rich, showing different dep endencies on the scattering strength,\nZeeman field and spin-orbit coupling strength. The rich beha viors of the spin diffusions in different\nregimes are hard to be understood in the framework of the simp le drift-diffusion model or the direct\ninhomogeneous broadening picture in the literature. Howev er, almost all these rich behaviors can be\nwell understood from our modified drift-diffusion model and/or modified inhomogeneous broadening\npicture. Specifically, several anomalous features of the sp in diffusion are revealed, which are in\ncontrast to those obtained from boththe simple drift-diffusion model and the direct inhomogeneo us\nbroadening picture.\nPACS numbers: 67.85.-d, 51.10.+y, 03.75.Ss, 05.30.Fk\nI. INTRODUCTION\nIn recent years, spin dynamics including spin relax-\nation and spin diffusion/transport is extensively studied\nin both Bose1–8and Fermi9–32cold atoms. For the Bose\nsystem, thespindynamicsoftheBose-Einsteincondensa-\ntion has attracted much attention.1–8For the Fermi cold\natoms, the systems without9–25and with26–32spin-orbit\ncoupling (SOC) are extensively investigated. In the ab-\nsence of the SOC, many interesting phenomena, such as\ntheLeggett-Riceeffectin unitarygas17–21andanomalous\nspin segregation in extremely weak scattering limit,22–25\nhave enriched the understanding of the spin dynamics of\nFermions. With the synthetic SOC experimentally re-\nalized by laser control technique in cold atoms,7,8,26,27\nthe spin relaxation for the Fermi cold atoms with\nSOC has been studied both experimentally26–28and\ntheoretically.29–32This is partly motivated by the well-\ncontrolled laser technique, which provides more freedom\nfor the cold atoms than the conventional solids. On one\nhand, rich regimes can be realized by tuning the SOC\nstrength; on the other hand, not only the interatom in-\nteraction can be tuned by the Feshbach resonance,33but\nalso the atom-disorder interaction can be introduced and\ncontrolled by the speckle laser technique.34–37\nThe experimentally realized effective Zeeman field\nand SOC provide an effective magnetic field, whichreads7,26,27\nΩ(k) = (Ω,0,δ+αkx). (1)\nIn above equation, k= (kx,ky,kz) denotes the center-\nof-mass momentum of the atom; Ω acts as an effec-\ntive Zeeman field along the ˆx-direction; δis the Ra-\nman detuning, which is set to be zero in our work;\nΩz(k) =αkxrepresents the k-dependent effective mag-\nnetic field along the ˆz-direction, which is perpendicular\nto the Zeeman field, with |α|being the strength of the\nspin-orbit coupled field. With this specific effective mag-\nnetic field Ω(k) by setting δ= 0, it has been revealed\nthat both the conventional29,30,32and anomalous31,38,39\nD’yakonov-Perel’(DP)40spin relaxations can be realized\nwith∝an}bracketle{t|Ωz(k)|∝an}bracketri}ht/greaterorsimilarΩ and∝an}bracketle{t|Ωz(k)|∝an}bracketri}ht ≪Ω, respectively. For\nthe conventional situation, in the strong (weak) scatter-\ning limit when ∝an}bracketle{t|Ω(k)|∝an}bracketri}htτ∗\nk≪1 (∝an}bracketle{t|Ω(k)|∝an}bracketri}htτ∗\nk/greaterorsimilar1), the spin\nrelaxation time (SRT) τsis inversely proportional (pro-\nportional) to the momentum scattering time τ∗\nk.∝an}bracketle{t...∝an}bracketri}ht\nhere denotes the ensemble average. For the anomalous\nsituation,31,38,39it has been found that by tuning the in-\nteratom interaction, the transverse spin relaxation can\nbe divided into four regimes: the normal weak scatter-\ning (τs∝τ∗\nk), the anomalous DP-like ( τ−1\ns∝τ∗\nk), the\nanomalous Elliott-Yafet (EY)-like41,42(τs∝τ∗\nk) and the\nnormal strong scattering ( τ−1\ns∝τ∗\nk) regimes. Whereas\nthe longitudinal spin relaxation can be divided into two:2\ni.e., the anomalous EY-like and the normal strong scat-\ntering regimes.38\nIn contrast to the spin relaxation, the study for the\nspin diffusion in cold atoms with SOC has not yet been\nreported in the literature. However, the experimen-\ntal configuration realized by Brantut et al.,43for the\n“charge” diffusion of cold atoms can be applied to the\nspin diffusion straightforwardly. In their experiment,\nby adding the “barrier” laser in the middle of the cold\natoms, the system is separatedto the left and rightparts.\nIn the left part, the spin-polarized cold atoms can be\nprepared;26–28,32whereas in the right part, the system\nremains in the equilibrium state. Specifically, the atom\ndensities in the left and right partsarepreparedto be the\nsame. With the SOC introduced to the right part,26–28,32\ni.e., the spin diffusion region, by removing the “barrier”\nlaser, this configuration can be used to study spin diffu-\nsion along one direction for the three dimensional (3D)\nFermi gas with SOC.\nIn the coordinate defined in Eq. (1), there are two\nspecific configurations with the spin diffusions along the\nˆx- andˆy-directions, respectively. Accordingly, in the\nscattering-free situation, the k-dependent spin preces-\nsionfrequenciesinthe spacialdomain, i.e., the inhomoge-\nneous broadening,44,51in the steady-state spin diffusion\nalong the ˆx- andˆy-directions are determined by\nωx(k) =mΩ(k)/kx=/parenleftig\nmΩ/kx,0,mα/parenrightig\n,(2)\nωy(k) =mΩ(k)/ky=/parenleftig\nmΩ/ky,0,mαkx/ky/parenrightig\n,(3)\nrespectively.52,53Here,mis the atom mass. From Eq.(2)\n[Eq. (3)], it can be seen that in contrast to the spin relax-\nation in the time domain, in inhomogeneous broadening\ninthespacialdomain,theoriginal k-independentZeeman\nfield Ω becomes k-dependent, whereas the k-dependent\nspin-orbit coupled field mαkxbecomes k-independent\n(remains k-dependent). Hence, for the spin diffusion\nalong the ˆx-direction, the inhomogeneous broadening\nωx(k) is similar to the one for spin relaxation in time\ndomainΩ(k) with one k-independent magnetic field per-\npendicular to another k-dependent one. Accordingly,\nrich regimes may exit in the steady-state spin diffusion\nalong the ˆx-direction for both the spin polarizations per-\npendicular and parallel to the Zeeman field. Whereas for\nthe spin diffusion along the ˆy-direction, from the differ-\nent inhomogeneous broadenings in Eqs. (2) and (3), its\nbehavior should be very different from the one along the\nˆx-direction.\nIn the present work, we investigate the steady-state\nspin diffusion for the 3D ultracold spin-orbit coupled\n40K gas by the kinetic spin Bloch equation (KSBE)\napproach44both analytically and numerically. Four con-\nfigurations, i.e., the spin diffusion along the ˆx- and\nˆy-directions for the spin polarization Pperpendicular\n(P∝bardblˆz, transverseconfiguration)andparallel( P∝bardblˆx, lon-\ngitudinal configuration) to the Zeeman field are studied.\nIt is shown analytically that the behaviors of the steady-\nstate spin diffusion for the four configurations are verydifferent, which are determined by three characteristic\nlengths: the mean free path lτ=kτk/m, the Zeeman os-\ncillation length lΩ=k/(√\n3mΩ) and the SOC oscillation\nlengthlα= 1/(m|α|). The spin diffusion lengths for the\nspin diffusions in the four configurations are derived in\nthe strong scattering regime, which are then extended to\nthe weak scatteringone. We further find that by dividing\nthe system into different regimes, the complex analytical\nresults can be reduced to extremely simple forms. It is\nrevealed that by tuning the scattering, the system can\nbe divided into fiveregimes: I, weak scattering regime\n(lτ/greaterorsimilarlΩ,lα); II, Zeeman field-dominated moderate scat-\ntering regime ( lΩ≪lτ≪lα); III, SOC-dominated mod-\nerate scattering regime ( lα≪lτ≪lΩ); IV, relatively\nstrong scattering regime ( lc\nτ≪lτ≪lΩ,lα); V, strong\nscatteringregime( lτ≪lΩ,lα,lc\nτ). Here,lc\nτrepresentsthe\ncrossoverlength between the relatively strong and strong\nscattering regimes. In different regimes, the behaviors of\nthe spacialevolution of the steady-statespin polarization\nare very rich, showing different dependencies on the scat-\ntering strength, Zeeman field and SOC strength. These\ndependencies are summarized in Tables I and II for the\nspindiffusionsalongthe ˆx-andˆy-directions,respectively.\nThe rich behaviors of the spin diffusions in differ-\nent regimes are hard to be understood in the frame-\nwork of the previous simple drift-diffusion model45–50\nor the direct inhomogeneous broadening [Eqs. (2) and\n(3)] picture44,51–54in the literature. In the simple drift-\ndiffusion model, there are only two rather than five\nregimes: the strong scattering regime with ls∝1/(m|α|)\nand the weak scattering regime with ls∝k√τk/m. In\nthe present work, it is found that the behaviors of the\nspin diffusions can be analyzed in the situation either\nwith strong Zeeman and weak spin-orbit coupled fields\n(Regimes II and V) or weak Zeeman and strong spin-\norbit coupled fields (Regimes III and IV). Accordingly,\nour previous inhomogeneous broadenings [Eqs. (2) and\n(3)] should be extended to the effective ones. It is found\nthat when the spin polarization is parallel to the larger\nfield between the Zeeman and spin-orbit coupled fields,\nthe spin polarization cannot precess around the effective\ninhomogeneous broadening fields efficiently. In this situ-\nation, the previous drift-diffusion model is applicable but\nτs(k) modified as follows( modified drift-diffusion model).\nIn the strongscatteringregime, τs(k) remainsthe SRT in\nthe conventionalDP mechanism;29,30,32,44whereas in the\nmoderatescatteringregime, τs(k) is replacedbythe helix\nspin-flip rates determined in the helix space.31,39When\nthe spin polarization is perpendicular to the larger field\nbetween the Zeeman and spin-orbit coupled fields, the\nspin polarization can rotate around the effective inho-\nmogeneous broadening fields efficiently. Hence, the be-\nhavior of the spin diffusion is determined by the effec-\ntiveinhomogeneous broadenings together with the spin-\nconserving scattering ( modified inhomogeneous broaden-\ning picture). Based on the modified drift-diffusion model\nand modified inhomogeneous broadening picture, apart\nfrom Regime IV, all the features in different regimes can3\nbe well obtained.\nSeveral anomalous features of the spin diffusion, which\nare in contrast to those obtained from boththe sim-\nple drift-diffusion model and the direct inhomogeneous\nbroadening picture, are revealed. In the scattering\nstrength dependence, it is found that when lα≪lΩ,\nthe longitudinal spin diffusion along the ˆy-direction is\nrobustagainst the scattering in a wide range including\nboth the strong and weakscattering regimes. In the Zee-\nman field dependence, when the system is in Regime II,\nthelongitudinal spin diffusion is enhanced by the Zeeman\nfield. In the SOC strength dependence, we find that the\nspin diffusion length can be also enhanced by the SOC\nin Regime III. All these anomalous behaviors have been\nwell understood from our modified drift-diffusion model\nand/or modified inhomogeneous broadening picture.\nThis paper is organized as follows. In Sec. II, we set\nup the model and KSBEs. In Sec. III, we show the an-\nalytical results for the transverse and longitudinal spin\ndiffusion along the ˆx- andˆy-directions. Different depen-\ndenciesofthe spindiffusion lengthonthemeanfreepath,\nZeeman oscillation length and SOC oscillation length are\nrevealed. In Sec. IV, both the analytical and numer-\nical calculations for the steady-state spin diffusion in\n3D isotropic speckle disorder are presented. Specifically,\nthe disorder strength (Sec. IVA), Zeeman field strength\n(Sec. IVB) and SOC strength (Sec. IVC) dependencies\nare discussed. We conclude and discuss in Sec. V.\nII. MODEL AND KSBES\nWith the 3D disordered speckle potential introduced\nto the spin diffusion region,34–37the Hamiltonian of\nthe spin-orbit coupled ultracold atom, which is com-\nposed by the effective Hamiltonian ˆH0,26,27the disor-\ndered speckle potential U(r),34–37,43and the interatom\ninteraction ˆHint, is written as\nˆH=ˆH0+U(r)+ˆHint. (4)\nThe effective Hamiltonian consists of the kinetic energy\nof the atom and the SOC ( /planckover2pi1≡1),\nH0=k2/(2m)+Ω(k)·σ/2, (5)\nwithσbeing the vector composed of the Pauli matri-\nces. The interaction Hamiltonian ˆHintis approximated\nby the s-wave interatom scattering.30,55–58In our study,\nthe scattering length is tuned to be zero by the Fesh-\nbach resonance,33–37,43and hence ˆHintis absent in our\nfollowing discussion.\nThe KSBEs, derived via the nonequilibrium Green\nfunction method with the generalized Kadanoff-Baym\nAnsatz,44,51,59–61are utilized to study the spin diffusion\nin the ultracold Fermi gas:\n∂tρk(r,t) =∂tρk(r,t)|dif+∂tρk(r,t)|coh+∂tρk(r,t)|scat.\n(6)In theseequations, ρk(r,t) representthe density matrices\nof atom with momentum kat position r= (x,y,z) and\ntimet, in which the diagonal elements ρk,σσdescribe the\natomdistributionfunctionsandtheoff-diagonalelements\nρk,σ−σrepresent the correlation between the spin-up and\ndown states.\nFor the quasi-one dimensional spin diffusion, the diffu-\nsion term is written as\n∂tρk(r,t)|diff=−(kζ/m)∂ζρk(r,t),(7)\nwithζ=xoryfor the spin diffusion along the ˆx- or\nˆy-direction, respectively. The coherent term is given by\n∂tρk(r,t)|coh=−i/bracketleftbig\nΩ(k)·σ/2,ρk(r,t)/bracketrightbig\n,(8)\nwhere [,] denotes the commutator.\nThe scatteringterms ∂tρk(r,t)|scatrepresentthe atom-\ndisorder scattering.37In our study, the effective Zeeman\nsplitting energy and the SOC energy are much smaller\nthan the Fermi energy.62Hence the atom-disorder scat-\ntering reads\n∂tρk|ad\nscat= 2π/summationdisplay\nk′|Uk−k′|2δ(εk−εk′)(ρk′−ρk),(9)\nwhere\n|Uq|2=/integraldisplay /integraldisplay\ndrdr′∝an}bracketle{t[U(r)−U0][U(r′)−U0]∝an}bracketri}hte−iq·(r−r′)\n=/integraldisplay /integraldisplay\ndrdr′C(r−r′)e−iq·(r−r′)≡Cq.(10)\nIn Eq. (10), U0is the average value of the disorder po-\ntential. For the 3D isotropic disordered speckle,34\nCq=π3/2V2\nRσ3\nRexp(−σ2\nRq2/4), (11)\nwhereVRis the potential amplitude and σRdenotes the\nradius of the auto-correlation function of the laser.\nIn our model, with the same atom densities and hence\nthe same chemical potentials in the left and right parts of\nthe system, there is no “charge” diffusion in the system.\nThe spin polarization at the boundary between the left\nand right parts of the system is approximately treated to\nbe fixed.\nIII. ANALYTICAL RESULTS\nIn this section, we analytically study the steady-state\nspin diffusion along the ˆx- andˆy-directions in cold atoms\nwith the atom-disorder scattering [Eq. (9)]. Both the sit-\nuations with spin polarization perpendicular ( P||ˆz) and\nparallel ( P||ˆx) to the Zeeman field are analyzed.\nIn the steady state, the KSBEs are written as\n(kζ/m)∂ζρk(r)+i/bracketleftig\nΩσx/2,ρk(r)/bracketrightig\n+i/bracketleftig\nαkxσz/2,ρk(r)/bracketrightig\n+/summationdisplay\nk′Wkk′/bracketleftbig\nρk(r)−ρk′(r)/bracketrightbig\n= 0, (12)4\nwhereWkk′= 2πCk−k′δ(εk−εk′). In the strong scat-\ntering regime with lτ≪lΩ,lα[i.e.,∝an}bracketle{t|Ω(k)|∝an}bracketri}htτk≪1], the\nsteady-state diffusion lengths for different configurations\nare obtained and extended to the situation with moder-\nate scattering strength ( lΩ≪lτ≪lαorlα≪lτ≪lΩ).\nA. Spin diffusion along the ˆx-direction\nWhen the spin diffusion is along the ˆx-direction, by\ntaking the steady-state condition, the Legendre com-\nponents of the azimuth-angle-averaged density matrix\n[Eq. (A2)] with respect to the zenith angle θkare given\nby\nk\nm∂\n∂x/bracketleftigl¯ρl−1\nk√\n2l−1+(l+1)¯ρl+1\nk√\n2l+3/bracketrightig\n+i/bracketleftigΩ\n2σx,¯ρl\nk√\n2l+1/bracketrightig\n+i/bracketleftigαk\n2σz,l¯ρl−1\nk√\n2l−1+(l+1)¯ρl+1\nk√\n2l+3/bracketrightig\n+¯ρl\nk\nτk,l√\n2l+1 = 0,\n(13)\nin which the momentum relaxation time is denoted by\nτ−1\nk,l=m√\nk\n2π/integraldisplayπ\n0C(cosθ)/bracketleftbig\n1−Pl(cosθ)/bracketrightbig\nsinθdθ,(14)\nwithPl(cosθ) being the Legendre function. By keeping\nboth the zeroth and first orders ( l= 0,1), the analyt-\nical solution for the spin polarization is obtained from\nEq.(13) (refertoAppendix A1). Itisfoundthatforboth\nthe spin polarization in the transverse ( ˆx-T) or longitu-\ndinal (ˆx-L) configuration, the spin polarization is limited\nby one oscillation decay together with one single expo-\nnential decay, i.e.,\nSx\nξ≈Aξexp(−x/Lx\no)cos(x/lx\no)+Bξexp(−x/Lx\ns),(15)\nwithξ=t(transverse) or l(longitudinal). In Eq. (15),\nAξandBξare the amplitudes for the oscillation and\nsingle exponential decays, respectively, which are de-\ntermined by the boundary condition; Lx\noandLx\nsare\nthe decay lengths for the oscillation and single expo-\nnential decays, respectively; lx\nois the oscillation length\nfor the oscillation decay. The integral forms for Lx\ns,Lx\no\nandlx\noare complicated [Eqs. (A9), (A10) and (A11) in\nAppendix A1]. However, when the system is in the\nstrong (lτ≪lΩ,lα) and moderate ( lΩ≪lτ≪lαor\nlα≪lτ≪lΩ) scattering regimes, it is found that the\nanalytical results can be reduced to simple forms, which\ncan describe the behavior of the spin diffusion quite well\n(Sec. IV).\nSpecifically, for the oscillation decay,\nLx\no≈\n\n2lτ/√\n3,whenlΩ≪lτ≪lα √\n2lτlΩ/(√\n3lα),whenlα≪lτ≪lΩ /radicalig\n2lτlΩ/√\n3,whenlτ≪lα&lτ≪lΩ.(16)The corresponding oscillation length\nlx\no≈\n\nlΩ, whenlΩ≪lτ≪lα\nlα, whenlα≪lτ≪lΩ /radicalig\n2lτlΩ/√\n3,whenlτ≪lα&lτ≪lΩ.(17)\nFrom Eq. (17), one further notes that lΩandlαcorre-\nspond to the spacial oscillation length due the Zeeman\nand spin-orbit coupled fields. Because of this, we refer\ntolΩandlαas Zeeman and SOC oscillation lengths, re-\nspectively. For the single exponential decay, the diffusion\nlength reads\nLx\ns≈\n\nlτlα/(√\n3lΩ),whenlΩ≪lτ≪lα\nlτlΩ/(√\n3lα),whenlα≪lτ≪lΩ\nlα, whenlτ≪lα&lτ≪lΩ.(18)\nFrom these simple dependencies of the spin diffusion\nlength on the mean free path, the Zeeman oscillation\nlength and the SOC oscillation length, the behavior of\nthe steady-state spin polarization shown in Eq. (15) can\nbe further simplified. It can be demonstrated that when\nLx\no≈Lx\ns,Aξ≈Bξ. Specifically, only when Lx\no≈Lx\ns,\nboth the oscillation decay and single exponential decay\nare important, but with similar decay length. Other-\nwise, the spin polarization can be approximately reduced\nto one oscillation or single exponential decay, which de-\npends on its amplitude in the spin polarization. In differ-\nent regimes, the different behaviors for the steady-state\nspin polarization are analyzed as follows. In the Zeeman-\nfield (SOC) dominated moderate scattering regime with\nlΩ≪lτ≪lα(lα≪lτ≪lΩ), the condition Lx\no≈Lx\ns\nis never satisfied. Accordingly, when lΩ≪lτ≪lα\n(lα≪lτ≪lΩ), in the transverse/longitudinal situation,\nthe steady-state spin polarization is approximately oscil-\nlation/single exponential (single exponential/oscillation)\ndecay. Inthestrongscatteringregime( lτ≪lα,lΩ), when\nLx\no≈Lx\ns, one obtains\nlc\nτ,x≈√\n3l2\nα/(2lΩ), (19)\nwhich is referred to as the crossover length between the\nrelatively strong and strong scattering regimes. Accord-\ningly, when lτ≫lc\nτ,x(lτ≪lc\nτ,x), the steady-state\nspin polarization is approximated by a single exponen-\ntial/oscillation (oscillation/single exponential) decay in\nthe transverse/longitudinal situation.\nBased on the above analysis, we summarize the be-\nhaviors of the steady-state spin polarization and the spin\ndiffusion lengths in Table I for the two specific config-\nurations ˆx-T andˆx-L in different regimes. As shown\nin the table, the system is divided into five regimes: I,\nweak scattering regime ( lτ≫lΩ,lα); II, Zeeman field-\ndominated moderate scattering regime ( lΩ≪lτ≪\nlα); III, SOC-dominated moderate scattering regime\n(lα≪lτ≪lΩ); IV, relatively strong scattering regime\n(lc\nτ,x≪lτ≪lα,lΩ); V, strong scattering regime ( lτ≪\nlα,lΩ,lc\nτ,x). In different regimes, it can be seen that the5\ndependencies of the spin diffusion length on the scatter-\ning strength, the Zeeman field andSOC strengtharevery\nrich. Specifically, in the Zeeman field- (SOC-) dominated\nmoderate scattering regime with lΩ≪lτ≪lα(lα≪\nlτ≪lΩ), the longitudinal (longitudinal/transverse) spin\ndiffusionisenhancedbytheZeemanfield(SOC); whereas\nin the relatively strong (strong) scattering regime ( lτ≪\nlα,lΩ), the transverse (longitudinal) spin diffusion is de-\ntermined by only the SOC oscillation length, but irrele-\nvant to the Zeeman field with lτ≫lc\nτ,x(lτ≪lc\nτ,x). The\nrich behaviors of the spin diffusions in different regimes\nare hard to be understood in the framework of the previ-\nous simple drift-diffusion model45–50or the direct inho-\nmogeneous broadening [Eqs. (2) and (3)] picture44,51–54\nin the literature. In the simple drift-diffusion model,\nthere are only two rather than fiveregimes: the strong\nscattering regime with ls∝1/(m|α|) and the weak scat-teringregimewith ls∝k√τk/m. Inthe directinhomoge-\nneous broadening picture, from Eq. (2), the Zeeman field\n(SOC) can (cannot) provide the inhomogeneous broad-\nening. Hence, it seems that the spin diffusion can be sup-\npressed by the Zeeman field, but irrelevant to the SOC.\nBelow, we extend our previous inhomogeneous broaden-\nings [Eqs. (2) and (3)] to the effective ones. We will show\nthat based on the effective inhomogeneous broadening,\napart from Regime IV, the above anomalous behaviors\nof the spin diffusion can be understood from the view\npoint of the helix representation.31,39In the helix space,\napart from the spin-conserving scattering [the fifth term\nin the left-hand side of Eq. (A6) in our previous work31],\nadditional terms arise including the helix spin-flip scat-\ntering [the sixth term in the left-hand side of Eq. (A6)\nin Ref. 31] and helix coherence term [the last term in the\nleft-hand side of Eq. (A6) in Ref. 31].31,39\nTABLE I: Behaviors of the steady-state spin polarization in the spatial domain and corresponding spin diffusion lengths for\nconfigurations ˆx-T andˆx-L in different regimes.\nRegime Condition Behavior and Lx\nTinˆx-T Behavior and Lx\nLinˆx-L\nI: weak scattering regime lτ≫lΩ,lα NA NA\nII: Zeeman field-dominated lΩ≪lτ≪lα oscillation decay; single exponential decay;\nmoderate scattering regime 2lτ/√\n3 lτlα/(√\n3lΩ)\nIII: SOC-dominated moderate lα≪lτ≪lΩ single exponential decay; oscillation decay;\nscattering regime lτlΩ/(√\n3lα)√\n2lτlΩ/(√\n3lα)\nIV: relatively strong scattering lτ≪lα,lΩ&lτ≫lc\nτ,x single exponential decay; oscillation decay;\nregime (lτ≪lα≪lΩ) lα/radicalBig\n2lτlΩ/√\n3\nV: strong scattering regime lτ≪lα,lΩ&lτ≪lc\nτ,x oscillation decay; single exponential decay;/radicalBig\n2lτlΩ/√\n3 lα\nlc\nτ,x≈√\n3l2\nα/(2lΩ).\nWhen the system is in Regime II, the Zeeman field-\ndominated moderate scattering regime ( lΩ≪lτ≪lα) or\nRegime V, the strong scattering regime ( lτ≪lα,lΩ≪\nlc\nτ,x), the condition lΩ≪lαis satisfied. Therefore, both\nthe transverse and longitudinal spin diffusions can be\nunderstood in the limit with strong Zeeman and weak\nspin-orbit coupled fields. In this situation, the effective\ninhomogeneous broadening field is given by\nωx\neff(k) = (m/kx)/radicalbig\nΩ2+α2k2xˆx′\n≈/bracketleftbig\nmΩ/kx+mα2kx/(2Ω)/bracketrightbigˆx′,(20)\nwithˆx′=1/radicalbig\n1+(αkx/Ω)2ˆx+αkx/Ω/radicalbig\n1+(αkx/Ω)2ˆzbeing\nnearly parallel to ˆx. Under this effective field, the be-\nhaviors of the spin precession in the spacial domain for\nthe transverseand longitudinal spin diffusions can be ob-\ntained, as schematically shown in Fig. 1. In this figure,for the transverse spin diffusion, the spin polarization\nis perpendicular to the strong Zeeman field and hence\nωx\neff(k) approximately. During the scattering, the spin\nvectorsrotatearoundthe effective inhomogeneousbroad-\nening field ωx\neff(k) fast. Moreover, the scattering can\nalso influence the spin diffusion. In the helix space, as\nmentioned above, apartfrom the originalspin-conserving\nscattering, there exist the helix spin-flip scattering and\nhelix coherence processes. However, with the helix spin-\nflip rate α2k2/(Ω2τk)≪1/τkand helix coherence rate\nαk/(Ωτk)≪1/τkwhen Ω ≫αk, both the helix spin-\nflip scattering and helix coherence can be neglected. In\nthis situation, the effective inhomogeneous broadening\ntogether with the spin-conserving scattering determines\nthe behavior of the spin diffusion. We refer to this pic-\nture as the modified inhomogenous broadening picture.\nFor the longitudinal situation, the spin polarization is6\nT\nkSxˆ/xkm zˆ m˅˄k x\neff\nT\nk\nL\nkS\nFIG. 1: (Color online) Schematic for the spin precession\naround the effective inhomogeneous broadening field ωx\neff(k)\n[Eq. (20)] in the transverse ( ST\nk) and longitudinal ( SL\nk) spin\ndiffusions. With the strong Zeeman and weak spin-orbit cou-\npledfields, for the transverse (longitudinal) situation, t hespin\nvectorST\nk(SL\nk) is perpendicular (parallel) to ωx\neff(k) approx-\nimately. Therefore, the inhomogeneous broadening field can\n(cannot) cause efficient spin precession in the transverse (l on-\ngitudinal) spin diffusion.\nnearly parallel to ωx\neff(k), and hence the effective inho-\nmogeneous broadening cannot cause the spin precession\neffectively. In this situation, the spin diffusion can be\nunderstood from the drift-diffusion model45–50modified\nas follows. The diffusion length ls=/radicalbig\nDτs(k) in which\nD=v2\nFτk/3 is the diffusion coefficient with vFbeing\nthe Fermi velocity. The SRT is analyzed in the helix\nspace. First of all, for the longitudinal situation here,\nthe helix coherence term has no contribution to the spin\nrelaxation. In this situation, there exit two channels\ninfluencing the spin relaxation: (i), the effective inho-\nmogenous broadening together with the spin-conserving\nscattering; (ii), the helix spin-flip scattering.31,39In the\nstrong scattering regime, both channels (i) and (ii) are\nimportant for the spin relaxation, in which τs(k) remains\nthe SRT in the conventional DP mechanism.29,30,32,44In\nthe moderate scattering regime, channel (ii) is dominant\nforthe spin relaxation, and hence τs(k) is replacedby the\nhelix spin-flip rates determined in the helix space.31,39\nWe refer to the above pictures as modified drift-diffusion\nmodel. Accordingly, in the moderate scattering regime,\nthe dependence of the helix spin-flip rate on the scat-\ntering strength, the Zeeman field and SOC strength dis-\nclosed in our previous works31,38,39can also influence the\nspin diffusion.\nSpecifically, in the Zeeman field-dominated moderate\nscattering regime ( lΩ≪lτ≪lα), i.e., Regime II, the\nZeeman oscillation length is the shortest length scale\nin the spin diffusion, which determines the behavior of\nspin precession in the spacial domain during the scat-\ntering. When the spin polarization is perpendicular to\nthe Zeeman field (transverse configuration), the spin vec-\ntor approximately precesses around mΩ/kxˆx′, leading tothe spacial oscillations with the period proportional to\n∝an}bracketle{t|kx|∝an}bracketri}ht/(mΩ) [lx\no≈lΩin Eq. (17)]. Moreover, due to the\nfast spacial oscillations with the strong Zeeman field, the\nspin memory is lost during one spin-conserving scatter-\ning, with the diffusion length being approximately the\nmean free path ( Lx\nT≈2lτ/√\n3 in Table I). When the spin\npolarization is parallel to the Zeeman field (longitudinal\nconfiguration), the effective inhomogeneous broadening\nωx\neff(k) cannot cause spin precession efficiently and the\nsteady-state spin polarization decays without any oscil-\nlation. The spin diffusion can be understood from the\nmodified drift-diffusion model.45–50Specifically, the he-\nlix spin-flip rate is calculated to be α2k2/(2Ω2τk) in the\nmoderate scattering situation.31,38,39Accordingly, the\nspin diffusion length in the modified drift-diffusion model\nis given by ls≈√\n2lτlα/(√\n3lΩ), which is consistent with\nour model shown as Lx\nL≈lτlα/(√\n3lΩ) in Table I. Specif-\nically, one notes that due to the suppression of the spin\nrelaxation by the Zeeman field, the longitudinal spin dif-\nfusion length is enhanced by the Zeeman field.\nIn Regime V, the strong scattering regime ( lτ≪\nlα,lΩ,lc\nτ,x), we consider a limit situation with the Zee-\nman field much stronger than the spin-orbit coupled one\n(lτ≪lα,lΩ≪lc\nτ,x). The effective inhomogeneous broad-\nening is given by Eq. (20). For the transverse spin dif-\nfusion, because the inhomogeneous broadening given by\nthe Zeeman field is dominant, the spin diffusion length is\nsuppressedbytheZeemanfield, butlessinfluencedbythe\nSOC. Moreover, during the diffusion, the atoms experi-\nence several spin-conserving scatterings, which suppress\nthe spin diffusion. Accordingly, we obtain a reasonable\npicture to understand Lx\nL≈/radicalig\n2lτlΩ/√\n3 in Table I. For\nthelongitudinalspindiffusion, theinhomogeneousbroad-\nening cannot cause the spin precession efficiently, with\nthe steady-state spin polarization showing single expo-\nnential decay. Hence, the modified drift-diffusion model\ncan be used.45–50With the SRT in the strong scattering\nregimeτs(k)≈2/(α2k2τk),31,38,39one obtains the spin\ndiffusion length ls≈√\n2lα/√\n3 (Lx\nT≈lαin Table I).\nTherefore, the longitudinal spin diffusion length depends\nonly on the SOC oscillation length.31,45–50\nWhen the system lies in Regime III, the SOC-\ndominated moderate scattering regime ( lα≪lτ≪lΩ)\nand Regime IV, the relatively strong scattering regime\n(lc\nτ,x≪lτ≪lα,lΩ), the spin-orbit coupled field is much\nstronger than the Zeeman one. In this situation, the ef-\nfective inhomogeneous broadening field reads\nω′x\neff(k) = (m/kx)/radicalbig\nα2k2x+Ω2ˆz′\n≈/bracketleftbig\nmα+mΩ2/(2αk2\nx)/bracketrightbigˆz′,(21)\nwhereˆz′=1/radicalbig\n1+[Ω/(αkx)]2ˆz+Ω/(αkx)/radicalbig\n1+[Ω/(αkx)]2ˆxis\nparallel to ˆzapproximately. The analysis is similar to\nthe situation with strong Zeeman and weak spin-orbit\ncoupled fields. One obtains that for the transverse (lon-\ngitudinal) spin diffusion, the spin polarization is nearly\nparallel (perpendicular) to ω′x\neff(k), which cannot (can)7\ncause efficient spin precession. These pictures are sum-\nmarized in Fig. 2. Therefore, the behavior of the trans-\nT\nkS\nxˆ/xkm zˆ m ˅˄k x\neff \nL\nkS\nFIG. 2: (Color online) Schematic for the spin precession\naround the effective inhomogeneous broadening field ω′x\neff(k)\n[Eq. (21)] in the transverse ( ST\nk) and longitudinal ( SL\nk) spin\ndiffusions. With the weak Zeeman and strong spin-orbit cou-\npledfields, for the transverse (longitudinal) situation, t hespin\nvectorST\nk(SL\nk) is parallel (perpendicular) to ω′x\neff(k) approx-\nimately. Therefore, the inhomogeneous broadening field can -\nnot (can) cause efficient spin precession in the transverse (l on-\ngitudinal) spin diffusion.\nverse spin diffusion can be understood from the modified\ndrift-diffusion model;45–50whereas the longitudinal spin\ndiffusion can be analyzed from the features of the preces-\nsion of the spin vectors around ω′x\neff(k).\nSpecifically, in Regime III, i.e., the SOC-dominated\nmoderate scattering regime ( lα≪lτ≪lΩ), in the trans-\nverse configuration, the modified drift-diffusion model is\nused to understand the spin diffusion.45–50In the he-\nlix representation, the helix spin-flip rate is proportional\nto Ω2/(α2k2τk) approximately.31,38,39The corresponding\nspin diffusion length is proportional to lτlΩ/lα, which\nis consistent with Lx\nT≈lτlΩ/(√\n3lα) in Table I. Con-\nsequently, one observes that the spin relaxation is sup-\npressedbythespin-orbitcoupledfield, andthetransverse\nspin diffusion length is enhanced by the SOC. In the lon-\ngitudinal configuration, with m|α| ≫mΩ2/(2|α|k2\nx), the\nspin polarization evolves with oscillations in the spacial\ndomain, whose oscillation length is lx\no≈lα[Eq. (17)].\nFurthermore, only mΩ2/(2αk2\nx)ˆz′inω′x\neff(k) can cause\nthe inhomogeneous broadening. Therefore, it can be ex-\npected that the longitudinal spin diffusion length is in-\nversely proportional (proportional) to the Zeeman field\n(SOC).Moreover,thespin-conservingscatteringcansup-\npress the spin diffusion in the modified inhomogeneous\nbroadening picture. These analysis are consistent with\nLx\nT≈√\n2lτlΩ/(√\n3lα) in Table I.\nIn Regime IV, the relatively strong scattering regime\n(lc\nτ,x≪lτ≪lα,lΩ), for the transverse spin diffu-\nsion, from the modified drift-diffusion model,45–50ls≈√\n2lα/√\n3 (Lx\nT≈lαin Table I). For the longitudinal spin\ndiffusion, one expects that the modified inhomogeneousbroadening picture can be applied. However, this pic-\nture fails to explain the behavior of the longitudinal spin\ndiffusion. It can be seen from Eq. (21) that both the\nspin-orbit coupled and Zeeman fields provide the inho-\nmogeneous broadening [ mΩ2/(2αk2\nx)ˆz′in Eq. (21)]. By\nfurther considering that the spin diffusion is suppressed\nby the spin-conserving scattering, it is obtained that the\nspin diffusion length is proportionalto the SOC strength,\nbut inversely proportional to the Zeeman field strength\nand scattering. However, in Table I, the longitudinal\nspin diffusion length Lx\nL≈/radicalig\n2lτlΩ/√\n3 is irrelevant to\nthe SOC strength. One further notices that Regime IV\nlies in the crossover region between the moderate and\nstrong scattering regimes. When the scattering is rela-\ntively strong, the shortest length scale in the spin diffu-\nsion is the mean free path. However, there still exists\nstrong competition between the effective inhomogeneous\nbroadening and scattering, which makes the behavior of\nthe spin diffusion complicated.50,53,54\nFinally, we address the behavior of the spin diffusion\nalongthe ˆx-directionin the limitsituation wherethe Zee-\nman field is zero. When Ω = 0, lΩis infinite. The corre-\nspondingsteady-statespinpolarizationshowsverydiffer-\nent behaviors compared to the situation with finite Zee-\nman field. In this situation, the transverse (longitudinal)\nspin diffusion length is infinite because ω′x\neff(k) cannot\ncause inhomogeneous broadening.54Specifically, for the\nlongitudinal situation, the spin polarizationis perpendic-\nular to the spin-orbit coupled field, spin helix establishes\nwith the oscillation length being lα.30,54\nB. Spin diffusion along the ˆy-direction\nWhen the spin diffusion is along the ˆy-direction,\nby taking the steady-state condition, the Fourier com-\nponents of the azimuth-angle-averaged density matrix\n[Eq. (A2)] with respect to the zenith angle θkare given\nby\n/planckover2pi1k\n2im∂\n∂y/parenleftbig\n˜ρl−1\nk−˜ρl+1\nk/parenrightbig\n+i/bracketleftigαk\n4σz,˜ρl+1\nk+ ˜ρl−1\nk/bracketrightig\n+i/bracketleftigΩ\n2σx,˜ρl\nk/bracketrightig\n+˜ρl\nk\n˜τk,l= 0, (22)\nin which the momentum relaxation time is denoted by\n˜τ−1\nk,l=m√\nk\n2π/integraldisplayπ\n0C(cosθ)/bracketleftbig\n1−cos(lθ)/bracketrightbig\nsinθdθ.(23)\nOne notes that by choosing the Legendre and Fourier\nexpansions of the density matrix for the spin diffusions\nalong the ˆx- andˆy-directions, the definitions for the mo-\nmentum scattering time τk,l[Eq. (14)] and ˜ τk,l[Eq. (23)]\nare different. However, when l= 1, the two definitions\narethe same. By keeping both the zerothand first orders\n(l= 0,1), the analytical solutions for both the spin po-\nlarization perpendicular ( ˆy-T) and parallel ( ˆy-L) to the\nZeeman field are obtained (refer to Appendix A2).8\nWhenthespinpolarizationisperpendiculartotheZee-\nman field, in the moderate and strongscatteringregimes,\nsimilar to the spin diffusion along the ˆx-direction (Ta-\nble I), the behaviors of the steady-state spin polarization\nare different with different scattering strengths. When\nthe scattering is relatively weak, which satisfies lτ> lc\nτ,y\nwithlc\nτ,y=4lα√\n3lΩ/parenleftig1\nl2\nα−8\n3l2\nΩ/parenrightig−1/2\n, the steady-state spin\npolarizationfor Sy\nTislimited bythebi-exponentialdecay,\ni.e.,\nSy\nT=P+exp(−y/Ly,+\nT)+P−exp(−y/Ly,−\nT),(24)\nwithLy,±\nTbeing the diffusion length. It is further demon-\nstratedthatwhen Ly,+\nT≪Ly,−\nT,P+≪P−andhencethe\nspin polarization [Eq. (24)] reduces to a single exponen-\ntial decay with the decay length being Ly,−\nT. Specifically,\nin the moderate scattering regime with lα≪lτ≪lΩ\n(lΩ≪lτ≪lα), the condition lτ> lc\nτ,yis natu-\nrally (never) satisfied; in the strong scattering regime\n(lc\nτ,y< lτ≪lα,lΩ), it can be obtained that lα≪lΩand\nhencelc\nτ,y≈4l2\nα/(√\n3lΩ). Accordingly, when lτ≫lc\nτ,y,\nthe diffusion length for Sy\nTis written as\nLy,−\nT≈\n\nNA, whenlΩ≪lτ≪lα√\n3lτlΩ/(2lα),whenlα≪lτ≪lΩ√\n3lτlΩ/(2lα),whenlc\nτ,y≪lτ≪lα,lΩ.\n(25)\nWhen the scatteringis relativelystrong, whichsatisfies\nlτ< lc\nτ,y, the transverse spin polarization in the steady\nstate is determined by the oscillation decay, i.e.,\nSy\nT=P0exp(−y/Ly\nT)cos(y/ly\nT). (26)Here, the decay length Ly\nTand oscillation length ly\nTcan\nbe written as\nLy\nT≈\n\n√\n2lτ,whenlΩ≪lτ≪lα\nNA,whenlα≪lτ≪lΩ /radicalbig√\n3lτlΩ,whenlτ≪lα,lΩ,lc\nτ,y(27)\nand\nly\nT≈\n\n/radicalbig\n3/2lΩ,whenlΩ≪lτ≪lα\nNA,whenlα≪lτ≪lΩ /radicalbig√\n3lτlΩ,whenlτ≪lα,lΩ,lc\nτ,y,(28)\nrespectively.\nWhen the spin polarization is parallel to the Zeeman\nfield, it is found that the steady-state spin polarization\nis limited by the single exponential decay, i.e.,\nSy\nL=P0exp(−y/Ly\nL). (29)\nHere,Ly\nL=lα/radicalbig\nl2τ/(3l2\nΩ)+1 is the decay length for Sy\nL.\nIn the moderate and strong scattering regimes,\nLy\nL≈\n\nlτlα/(√\n3lΩ),whenlΩ≪lτ≪lα\nlα, whenlα≪lτ≪lΩ\nlα, whenlτ≪lΩ&lτ≪lα.(30)\nBased on the above results, in different regimes, the\nbehaviors of the steady-state spin polarization and dif-\nfusion lengths are summarized in Table II for the two\nspecific configurations ˆy-T andˆy-L.\nTABLE II: Behaviors of the steady-state spin polarization i n the spacial domain and corresponding spin diffusion length s for\nconfigurations ˆy-T andˆy-L in different regimes.\nRegime Condition Behavior and Ly\nTinˆy-T Behavior and Ly\nLinˆy-L\nI: weak scattering regime lτ≫lΩ,lα NA NA\nII: Zeeman field-dominated lΩ≪lτ≪lα oscillation decay; single exponential decay;\nmoderate scattering regime√\n2lτ lτlα/(√\n3lΩ)\nIII: SOC-dominated moderate lα≪lτ≪lΩ single exponential decay; single exponential decay;\nscattering regime√\n3lτlΩ/(2lα) lα\nIV: relatively strong scattering lτ≪lα,lΩ&lτ≫lc\nτ,y single exponential decay; single exponential decay;\nregime (lτ≪lα≪lΩ)√\n3lτlΩ/(2lα) lα\nV: strong scattering regime lτ≪lα,lΩ&lτ≪lc\nτ,y oscillation decay; single exponential decay;/radicalbig√\n3lτlΩ lα\nlc\nτ,y≈4l2\nα/(√\n3lΩ).\nFrom TableII, it can be seen that the spin diffusion along the ˆy-direction is divided into similar five regimes as the9\nspin diffusion along the ˆx-direction. The anomalous be-\nhaviors for the spin diffusion along the ˆx-direction also\nexist here. Nevertheless, new features arise in the spin\ndiffusion along the ˆy-direction. It is shown in Table II\nthat in Regimes III, IV and V, the longitudinal spin dif-\nfusion length is only determined by the SOC oscillation\nlength, but irrelevant to the scattering. This robustness\nto the scattering for the spin diffusion in a wide range is\nfurther revealed in the scattering dependence of the spin\ndiffusion (Sec. IVA). Below, it is found that based on\nthe modified drift-diffusion model and modified inhomo-\ngeneous broadening picture, apart from Regime IV, all\nthe features in different regimes can be obtained.\nWe first analyze Regime II, the Zeeman field-\ndominated moderate scattering regime ( lΩ≪lτ≪lα)\nand Regime V, the strong scattering regime ( lτ≪\nlα,lΩ≪lc\nτ,y). In these two regimes, with strong Zeeman\nand weak spin-orbit coupled fields, the effective inhomo-\ngeneous broadening field is written as\nωy\neff(k) = (m/ky)/radicalbig\nΩ2+α2k2xˆx′\n≈/bracketleftbig\nmΩ/ky+mα2k2\nx/(2Ωky)/bracketrightbigˆx′,(31)\nwithˆx′nearly parallel to ˆx. In Regime II, i.e., the Zee-\nman field-dominated moderate scattering regime ( lΩ≪\nlτ≪lα), when the spin polarization is perpendicular to\ntheZeemanfield(transverseconfiguration),inthespacial\ndomain, the spin vectors precess approximately around\n(mΩ/ky)ˆx′, which causes spacial oscillations whose pe-\nriod is proportional to ∝an}bracketle{t|ky|∝an}bracketri}ht/(mΩ) [ly\nT≈/radicalbig\n3/2lΩin\nEq. (28)]. Moreover, due to the fast spacial oscillations,\nthe spin memory is lost during one scattering, and hence\nthe spin diffusion length is the mean free path approx-\nimately ( Ly\nT≈√\n2lτin Table II). When the spin po-\nlarization is parallel to the Zeeman field (longitudinal\nconfiguration), the effective inhomogeneous broadening\nfieldωy\neff(k) cannot cause spin precession efficiently and\nthe steady-state spin polarization decays without any os-\ncillation. From the modified drift-diffusion model,45–50\nas calculated in Sec. IIIA, ls≈√\n2lτlα/(√\n3lΩ) [Ly\nL≈\nlτlα/(√\n3lΩ) in Table II].\nIn Regime V, i.e., the strong scattering regime ( lτ≪\nlα,lΩ,lc\nτ,y), we analyze the limit situation with strong\nZeemanandweakspin-orbitcoupledfields( lτ≪lα,lΩ≪\nlc\nτ,y). For the transverse spin diffusion, the inhomoge-\nneous broadening is dominantly determined by the Zee-\nman field [Eq. (31)]. Hence, the transverse spin diffu-\nsion length is suppressed by the Zeeman field, but less\ninfluenced by the SOC. Furthermore, during the diffu-\nsion, the spin-conserving scattering suppresses the spin\ndiffusion ( Ly\nL≈/radicalbig√\n3lτlΩin Table II). For the longitudi-\nnal spin diffusion, the inhomogeneous broadening cannot\ncause the spin precession efficiently (single exponential\ndecay). According to the modified drift-diffusion model,\nthe spin diffusion length depends only on the SOC oscil-\nlation length ( Ly\nT≈lαin Table I).\nWe then analyze Regime III, SOC-dominated moder-\nate scattering regime ( lα≪lτ≪lΩ), and Regime IV,relatively strong scattering regime ( lc\nτ,y≪lτ≪lα,lΩ).\nWith weak Zeeman and strong spin-orbit coupled fields,\nthe effective inhomogeneous broadening field reads\nω′y\neff(k) = (m/ky)/radicalbig\nα2k2x+Ω2ˆz′\n≈/bracketleftbig\nmαkx/ky+mΩ2/(2αkxky)/bracketrightbigˆz′,(32)\nwhereˆz′is parallel to ˆzapproximately. In the SOC-\ndominated moderate scattering regime (Regime III with\nlα≪lτ≪lΩ), in the transverse configuration, the spin\npolarization is nearly parallel to ω′y\neff(k). Therefore,\nthe effective inhomogeneous broadening cannot cause\nthe spin precession effectively (single exponential decay).\nFrom the modified drift-diffusion model, the diffusion\nlength is proportionalto lΩlτ/lα, which is consistent with\nLT\ny≈√\n3lΩlτ/(2lα) in Table II. In the longitudinal con-\nfiguration, the steady-state spin polarization is perpen-\ndicular to ω′y\neff(k) approximately. One notes that in\nω′y\neff(k), the spin-orbit coupled field ( mαkx/ky)ˆz′pro-\nvides the dominant inhomogeneous broadening. More-\nover, this inhomogeneous broadening not only depends\nonkybut also kx, which can be more efficient than the\none depending only on kxorky. Due to this efficient\ninhomogeneous broadening, the steady-state spin polar-\nization decays due to the interference without oscillation.\nMoreover, the spin memory can be lost in the scale of\nSOC oscillation length, which cannot persist in the mean\nfree path ( Ly\nL≈lαin Table II). This is different from the\ntransverse spin diffusion in the Zeeman filed-dominated\nmoderate scattering regime ( lΩ≪lτ≪lα).\nIn the relatively strong scattering regime (Regime IV\nwithlc\nτ,y≪lτ≪lα,lΩ). From Table II, one observes\nthat the behaviors for both the transverse and longitu-\ndinal spin diffusions are similar to the ones in Regime\nIII, i.e., the SOC-dominated moderate scattering regime\n(lα≪lτ≪lΩ). We address that this behavior is hard\nto be understood from both the modified drift-diffusion\nmodel and modified inhomogeneous broadening picture.\nFor the transverse spin diffusion, the spin polarization is\nnearly parallel to ( mαkx/ky)ˆz′, and hence the spin diffu-\nsion length is√\n2lα/√\n3 from the modified drift-diffusion\nmodel. For the longitudinal spin diffusion, the spin po-\nlarization is perpendicular to the inhomogeneous broad-\nening field ( mαkx/ky)ˆz′approximately, and hence the\nSOC can suppress the spin diffusion. Moreover, the spin-\nconserving scattering can suppress the longitudinal spin\ndiffusion. From this analysis, the longitudinal spin dif-\nfusion length is suppressed by the SOC strength and\nscattering, but irrelevant to the Zeeman field (modified\ninhomogeneous broadening picture). However, the pic-\ntures above fail to explain the transverse and longitu-\ndinal spin diffusion lengths for Regime IV in Table II\nwithLy\nT≈√\n3lτlΩ/(2lα) andLy\nL≈lα. This is because\nfor the transverse situation, although the modified drift\ndiffusion model can explain the spin diffusion along the\nˆx-direction, it is too rough to consider the anisotropy be-\ntween the diffusions along the ˆx- andˆy-directions in the\nrelativelystrongscatteringregime. Thiswasfirstpointed10\nout by Zhangand Wu in the study ofthe spin diffusion in\ngraphene.50For the longitudinal spin diffusion, when the\nscattering is strong, the shortest length scale in the spin\ndiffusion is the mean free path. In this situation, there\nexists strong competition between the effective inhomo-\ngeneous broadening and scattering, which makes the be-\nhavior of the spin diffusion complicated.50,53,54\nFinally, we emphasizethat from Table II, forthe trans-\nverse spin diffusion along the ˆy-direction, in Regimes\nIII, IV and V, the spin diffusion lengths are irrelevant\nto the scattering. Therefore, with weak Zeeman and\nstrong spin-orbit coupled fields ( lα≪lΩ), a specific sit-\nuation can be realized where the spin diffusion length is\nrobust against the scattering except with the extremely\nweak scattering. It is noted that in the strong scattering\nregime, this feature was predicted in the simple drift-\ndiffusion model45–49and was also revealed in graphene\nby the KSBE approach.50In this work, we have further\nextended it into the weak scattering regime.\nIV. NUMERICAL RESULTS\nIn the numerical calculation, the KSBEs are solved by\nemploying the double-side boundary conditions,53\n/braceleftigg\nρk(ζ= 0,t) =fk↑+fk↓\n2+fk↑−fk↓\n2σ·ˆn, kζ>0\nρk(ζ=L,t) =f0\nk, k ζ<0,\n(33)\nwherefkσ={exp[(εk−µσ)/(kBT)]+1}−1is the Fermi\ndistribution function at temperature T, withµ↑,↓stand-\ning for the chemical potentials determined by the atom\ndensityna=/summationtext\nkTr[ρk] and the spin polarization P(0) in\nthe left part of the system; ˆndenotes the spin polariza-\ntiondirection; f0\nkistheFermidistributionatequilibrium.\nWhen the system evolves to the steady state, one obtains\nthe diffusion length from the spatial evolution of the spin\npolarization P(ζ) =/summationtext\nkTr[ρk(ζ)σ·ˆn]/na.\nWithin the experimental feasibility by referring to the\nexperiment by Wang et al.,26the parameters are chosen\nas follows. The lowest two magnetic sublevels |9/2,9/2∝an}bracketri}ht\nand|9/2,7/2∝an}bracketri}htare coupled by a pair of Raman beams\nwith wavelength λ= 773 nm and the frequency dif-\nferenceω/(2π) = 10.27 MHz. The Raman detuning\nδ=ωz−ωis set to be zero by choosing the Zeeman shift\nωz/(2π) = 10.27 MHz. The recoil momentum and en-\nergy are set to be kr=k0/10 withk0= 2π/λ, and hence\nEr=k2\nr/(2m) = 2π×83.4 Hz. In our study, the SOC\nstrength varies from 0 .5α0to 6α0withα0=−2kr/m.\nThe strengths of the Zeeman field Ω vary from 10 Erto\n450Er. Furthermore, the Fermi momentum is set to be\nkF= 30kr.26It is noted that with these parameters, the\ncondition that the Zeeman and SOC energies are much\nsmaller than the Fermi energy is satisfied.\nMoreover, the temperature is set to T= 0.3TF\nwithTFbeing the corresponding Fermi temperature.26\nWith these parameters, the thermal deBroglie wave-\nlength Λ dB=h/√2πmkBT≈0.26µm. For the 3Disotropic speckle disorder, VR/kB= 1250 nK and σR=\n0.27µm.34,35With these disorder parameters, the mean\nfreepath lτ≈5µm. In ourstudy, the strengthofthe dis-\norder strength Vis tuned by the laser.34–37,43One notes\nthat when ( V/VR)2/greaterorsimilar20,lτ/lessorsimilarΛdB. According to the\nIoffe-Regel condition for the Anderson localization,63the\nAnderson localization may become relevant. Neverthe-\nless, in our study, to compare with the analytical results\nin different regimes revealed in Sec. III, the numerical\ncalculations are extended to ( V/VR)2≫20.\nA. Scattering strength dependence\nIn this part, we study the scattering strength depen-\ndence of the steady-state spin diffusion of spin-orbit cou-\npled40K gas in the 3D isotropic speckle disorder. The\nSOC strength is set to be α0and the spin polarization is\nchosen to be P= 20%. For the spin diffusion along the\nˆx-direction ( ˆy-direction), both the transverse and longi-\ntudinal spin diffusion lengths Lx\nTandLx\nL(Ly\nTandLy\nL)\nare shown in Figs. 3(a) and (b) [Figs. 3(c) and (d)].\nWefirstanalyzethespindiffusionalongthe ˆx-direction\n[Figs. 3(a) and (b)]. For the transverse spin diffusion, it\ncan be seen from Fig. 3(a) that no matter the Zeeman\nfield is strong with Ω = 450 Er(the blue dashed curve\nwith squares) or weak with Ω = 10 Er(the red solid\ncurve with circles), the transverse spin diffusion length\nLx\nTdecreases with the increase of the disorder strength.\nThe underlyingphysicscanbe understoodasfollows. We\nfirstcalculatethecharacteristiclengthsforthesystemde-\nfined in Sec. III: the SOC oscillation length lα≈0.6µm;\nthe Zeeman oscillation length lΩ≈0.1µm (4µm)\nfor Ω = 450 Er(10Er); when Ω = 450 Er(10Er), the\ncrossoverlength between the relatively strong and strong\nscattering regimes lc\nτ,x≈√\n3l2\nα/(2lΩ)≈3µm (0.08µm),\ni.e., (V/VR)2\nc≈2 [(V/VR)2\nc≈65]. Furthermore, it is cal-\nculated that when lτ≈lα, (V/VR)2≈8; when lτ≈lΩ,\n(V/VR)2≈50 [(V/VR)2≈1] for Ω = 450 Er(10Er). Ac-\ncordingly, with the increase of the disorder strength, the\nsystem experiences several regimes. For Ω = 450 Er, the\nregimes are approximately divided into\n\n\nI :lτ/greaterorsimilarlΩ,lα,when (V/VR)2/lessorsimilar8\nII :lΩ/lessorsimilarlτ/lessorsimilarlα,when 8/lessorsimilar(V/VR)2/lessorsimilar50\nV :lτ≪lΩ,lα,lc\nτ,xwhen (V/VR)2≫50,(34)\nwhich are labelled by the blue Roman numbers with the\nboundaries indicated by the blue crosses at the lower\nframe of Fig. 3(b). Therefore, when ( V/VR)2/lessorsimilar8, the\nsystem is in Regime I, and our calculation shows that\nthe transverse diffusion length decreases with the in-\ncrease of the disorder strength. When 8 /lessorsimilar(V/VR)2/lessorsimilar\n50 [(V/VR)2≫50], the system lies in Regime II\n(V), and from Table I, one comes to Lx\nT≈2lτ/√\n3\n(Lx\nT≈/radicalig\n2lτlΩ/√\n3) with the steady-state spin polariza-\ntionshowingoscillationdecay. Hence, thetransversespin11\ndiffusion length decreases with the increase of the disor-\nder strength. One notes that the corresponding results\ncalculated from the analytical formula Eq. (A10) (the\ngreen dot-dashed curve) agree with the numerical ones\nin Regimes II and V in Fig. 3(a). For Ω = 10 Er, the\nregimes are approximately divided into\n\n\nI :lτ/greaterorsimilarlΩ,lα,when (V/VR)2/lessorsimilar1\nIII :lα/lessorsimilarlτ/lessorsimilarlΩ,when 1/lessorsimilar(V/VR)2/lessorsimilar8\nIV :lc\nτ,x/lessorsimilarlτ/lessorsimilarlΩ,lα,when 8/lessorsimilar(V/VR)2/lessorsimilar65\nV :lτ≪lΩ,lα,lc\nτ,x,when (V/VR)2≫65,\n(35)\nwhich are shown by the red Roman numbers with theboundaries indicated by the red crosses at the upper\nframe of Fig. 3(a). Specifically, in Regime I when\n(V/VR)2/lessorsimilar1, it is shown that the transverse spin dif-\nfusion length decreases with the increase of the scatter-\ning strength. In Regime III (V) with 1 /lessorsimilar(V/VR)2/lessorsimilar\n8 [(V/VR)2≫65], from Table I, it is obtained that\nLx\nT≈lτlΩ/(√\n3lα) (Lx\nT≈/radicalig\n2lτlΩ/√\n3) with the steady-\nstate spin polarization being single exponential (oscilla-\ntion) decay. Hence, the transverse spin diffusion length\ndecreases with the increase of disorder strength. One\nfurther notices that in Fig. 3(a), the result calculated\nfromtheanalyticalformulaEq.(A10)(the orangedashed\ncurve) agrees with the numerical one in Regime V.\n10-1100101102103\n 0.1 1 10 100\n(V/VR)2Lx\nL (µm)\n(b) P || Ωxα=α0\nP=20%\nI II Vnumerical: Ω=450 Er\n10 Er\nanalytical: Ω=450 Er, Ls\n10 Er, Lo\n10 Er, Ls10-1100101102103Lx\nT (µm)\n(a) P⊥ Ωxα=α0\nP=20%I III IV V\nnumerical: Ω=450 Er\n10 Er\nanalytical: Ω=450 Er, Lo\n10 Er, Lo\n10 Er, Ls\n100101\n 0.1 1 10 100\n(V/VR)2Ly\nL (µm)\n(d) P || ΩxP=20%α=α0\nI II Vnumerical: Ω=450 Er\n10 Er\nanalytical: Ω=450 Er, Ls\n10 Er, Ls10-1100101102Ly\nT (µm)\n(c) P⊥ ΩxP=20%α=α0I III IV V\nnumerical: Ω=450 Er\n10 Er\nanalytical: Ω=450 Er, Lo\n10 Er, Ls,o\nFIG. 3: (Color online) Scattering strength dependence of th e steady-state spin diffusion of spin-orbit coupled40K gas with the\n3D isotropic speckle disorder. The SOC strength α=α0and the spin polarization P= 20%. For the spin diffusion along\ntheˆx-direction ( ˆy-direction), the transverse and longitudinal spin diffusio n lengths Lx\nTandLx\nL(Ly\nTandLy\nL) are shown in\nFigs. 3(a) and (b) [Figs. 3(c) and (d)], respectively. Situa tions with strong (Ω = 450 Er) and weak (Ω = 10 Er) Zeeman fields are\ncalculated both analytically and numerically. The blue (re d) crosses on the frames indicate the boundaries between diff erent\nregimes represented by the blue (red) Roman numbers at the lo wer (upper) frame when Ω = 450 Er(10Er). It is shown that\nthe analytical calculations agree with the numerical ones i n the relatively strong/strong scattering regime and cross over region\nbetween the relatively strong and moderate scattering regi mes.\nFor the longitudinal spin diffusion along the ˆx- direction, it can be seen from Fig. 3(b) that no mat-12\nter the Zeeman field is strong (450 Er, blue dashed curve\nwith squares) or weak (10 Er, red solid curve with cir-\ncles), with the increase of the disorder strength, the spin\ndiffusion length decreases first and then becomes insen-\nsitive to the scattering. This can be understood as fol-\nlows. With the increase of the disorder strength, the\ncorresponding regimes can also be divided according to\nEq. (34) [Eq. (35)] when Ω = 450 Er(10Er). Specifi-\ncally, when ( V/VR)2/lessorsimilar8 [(V/VR)2/lessorsimilar1] for Ω = 450 Er\n(10Er), the system is in Regime I, with the longitudi-\nnal spin diffusion suppressed by the scattering. When\n8/lessorsimilar(V/VR)2/lessorsimilar50 [1/lessorsimilar(V/VR)2/lessorsimilar8] for Ω = 450 Er\n(10Er), the system lies in Regime II (III), and one comes\ntoLx\nL≈lτlα/(√\n3lΩ) [Lx\nL≈√\n2lτlΩ/(√\n3lα)] with the\nsteady-state spin polarization showing single exponential\n(oscillation) decay. When ( V/VR)2≫50 [(V/VR)2≫65]\nfor Ω = 450 Er(10Er), the system lies in Regime V, it is\nobtainedthat Lx\nL≈lα(singleexponentialdecay). There-\nfore, the longitudinal spin diffusion is suppressed first\nand then becomes insensitive to the scattering with the\nincrease of the disorder strength. Also in Fig. 3(b), for\nΩ = 450Er, it is shown that the results calculated from\nthe analytical formula Eq. (A9) (the green dot-dashed\ncurve) agrees with the numerical one in both Regimes\nII and V; for Ω = 10 Er, the analytical results (the blue\ndashed curve) agree with the numerical ones in Regime\nV.\nWethenturntothespindiffusionalongthe ˆy-direction\n[Figs. 3(c) and (d)]. For the transverse spin diffusion, it\nis shown in Fig. 3(c) that for both the strong (450 Er,\nblue dashed curve with squares) and weak (10 Er, red\nsolid curve with circles) Zeeman fields, the spin diffu-\nsion length decreases with the increase of the disorder\nstrength. It is calculated that for Ω = 450 Er(10Er),\nlc\nτ,y≈8.3µm (0.2µm), i.e., ( V/VR)2\nc≈0.6 [(V/VR)2\nc≈\n25]. Accordingly, when the Zeeman field is strong (Ω =\n450Er), one can divide the regimesaccordingto Eq. (34).\nSpecifically, when ( V/VR)2/lessorsimilar8, the transverse spin dif-\nfusion is calculated to be suppressed by the scattering.\nWhen 8 /lessorsimilar(V/VR)2/lessorsimilar50 [(V/VR)2≫50], one obtains\nfrom Table II that Ly\nT≈√\n2lτ(Ly\nT≈/radicalbig√\n3lΩlτ) with the\nsteady-state spin polarization showing oscillation decay.\nTherefore, with the increase of the scattering strength,\nthe transverse spin diffusion is suppressed. When the\nZeeman field is weak (10 Er), the regimes are approxi-\nmately divided into\n\n\nI :lτ/greaterorsimilarlΩ,lα,when (V/VR)2/lessorsimilar1\nIII :lα/lessorsimilarlτ/lessorsimilarlΩ,when 1/lessorsimilar(V/VR)2/lessorsimilar8\nIV :lc\nτ,y/lessorsimilarlτ/lessorsimilarlΩ,lα,when 8/lessorsimilar(V/VR)2/lessorsimilar25\nV :lτ≪lΩ,lα,lc\nτ,y,when (V/VR)2≫25,\n(36)\nwhich are labelled by the red Roman numbers with the\nboundaries indicated by the red crosses at the upper\nframe of Fig. 3(c). Specifically, when ( V/VR)2/lessorsimilar1, the\ntransverse spin diffusion length decreases with the in-\ncrease of the disorder strength. When 1 /lessorsimilar(V/VR)2/lessorsimilar8\n[(V/VR)2≫25], it is obtained from Table II that Ly\nT≈√\n3lΩlτ/(2lα) (Ly\nT≈/radicalbig√\n3lΩlτ) with the steady-state\nspin polarization being single exponential (oscillation)\ndecay. Therefore, the transverse spin diffusion length de-\ncreases with the increase of the disorder strength. Fur-\nthermore, it can be seen in Fig. 3(c) that the results\ncalculated from the analytical formulas Eqs. (A19) and\n(A22) agree with the numerical ones in Regime V.\nFor the longitudinal spin diffusion along the ˆy-\ndirection, itisshowninFig.3(d)thatwith theincreaseof\nthe scattering strength, for both the strong (450 Er) and\nweak (10 Er) Zeeman fields, the longitudinal spin diffu-\nsion length is suppressed first and then become insensi-\ntive to the scattering. Specifically, when Ω = 10 Er(lα≪\nlΩ), the longitudinal spin diffusion is robust against the\nscattering except with extremely weak scattering. Here,\nfor the strong (weak) Zeeman field Ω = 450 Er(10Er),\nthe regimes are divided according to Eq. (34) [Eq. (36)].\nFor the strong Zeeman field (450 Er), when ( V/VR)2/lessorsimilar8,\nthe longitudinal spin diffusion is suppressed by the scat-\ntering. When 8 <(V/VR)2/lessorsimilar50 [(V/VR)2≫50], it is\nobtained from Table II that Ly\nL≈lτlα/(√\n3lΩ) (Ly\nL≈lα)\nwith the steady-sate spin polarization being single ex-\nponential decay. Hence, the longitudinal spin diffusion\nlength decreases first and then become insensitive to\nthe scattering with the increase of the disorder strength.\nFor the weak Zeeman field (10 Er), when ( V/VR)2/lessorsimilar1,\nour calculation shows that the longitudinal spin diffusion\nlength decreases slowly with the increase of the disorder\nstrength. When ( V/VR)2/greaterorsimilar1, one obtains from Table II\nthatLy\nL≈lα(single exponential decay). Hence, the lon-\ngitudinal spin diffusion length is insensitive to the scat-\ntering in a wide range. Moreover, it is shown in Fig. 3(d)\nthat the results calculated from the analytical formula\nEq.(A15)agreewiththenumericalonesinboththemod-\nerate and strong scattering regimes.\nFinally, we address the specific features in the scatter-\ning strength dependence of the transverse and longitu-\ndinal spin diffusions along the ˆx- andˆy-directions. Our\ncalculationsshowthatinthe weak( lτ/greaterorsimilarlΩ,lα)scattering\nregime (Regime I), both the transverse and longitudinal\nspin diffusions are suppressed by the scattering. In the\nstrong scattering limit ( lτ≪lΩ,lα,lc\nτ), the longitudinal\nspin diffusions along the ˆxandˆy-directions are insen-\nsitive to the scattering. Specifically, when lα≪lΩ, the\nlongitudinal spin diffusion along the ˆy-direction is robust\nagainst the scattering except with extremely weak scat-\ntering. We emphasize that this robust spin diffusion can-\nnot be obtained from the over-simplified drift-diffusion\nmodel,where ls=/radicalbig\nDτs(k)withD=v2\nFτk/3.45–50From\nthe drift-diffusion model, in Regime V with τs(k)∝τ−1\nk,\none surely obtains that the spin diffusion length is irrel-\nevant to the scattering. However, in Regimes I and II,\ni.e., the weak scattering regime defined in the conven-\ntional DP spin relaxation,38,40,44τs(k)≈τk. Hence it\nis obtained that ls∝τkwith the spin diffusion length\nsuppressed by the scattering. One further notes that the\nexperimental condition can be easily realized to observe\nthis robust spin diffusion. On one hand, the spin diffu-13\nsion is set to be alongthe ˆy-directionwith the initial spin\npolarization parallel to the Zeeman field; on the other\nhand, the Zeeman field is tuned to be much weaker than\nthe spin-orbit coupled one by the laser field.\nB. Zeeman field dependence\nIn this part, we address the Zeeman field dependence\nof the transverse and longitudinal spin diffusions along\ntheˆx- [Fig. 4(a)] and ˆy-directions [Fig. 4(b)]. The SOC\nstrengthissettobe α0andthespinpolarizationischosen\nto beP= 20%. Here, we mainly address the specific\nfeatures in the Zeemanfield dependence when the system\nlies in the moderate and strong scattering regimes, which\ncan be realized by setting ( V/VR)2= 60 and ( V/VR)2=\n10, respectively.\n10-1100101\n 0.5 1 1.5 2 2.5 3 3.5 4 4.5\nΩ/(100Er)Ly (µm)\n(b)P=20%\nα=α0\nIV II(V/VR)2=60, LT\nLL\n(V/VR)2=10, LT\nLL10-1100101Lx (µm)\n(a)IV V II\nP=20%\nα=α0(V/VR)2=60, LT\nLL\n(V/VR)2=10, LT\nLL\nFIG. 4: (Color online) Zeeman field dependence of the trans-\nverse and longitudinal spin diffusions along the (a) ˆx- and\n(b)ˆy-directions. The SOC strength α=α0and the spin po-\nlarization P= 20%. Both the situations with the scattering\nstrength ( V/VR)2= 60 (squares) and ( V/VR)2= 10 (circles)\ncalculated numerically are presented. The green (blue) Ro-\nman numbers at the upper (lower) frame represent the differ-\nent regimes when ( V/VR)2= 10 for the spin diffusion along\ntheˆx-direction ( ˆy-direction) with the boundaries indicated\nby the green (blue) crosses.We first focus on the case with ( V/VR)2= 60. When\n(V/VR)2= 60, one observes from Figs. 4(a) and (b) that\nfor the spin diffusion along both ˆx- andˆy-directions, the\ntransverse diffusion length decreases with the increase\nof the Zeeman field, as shown by the red solid curve\nwith squares. This is because when ( V/VR)2= 60,\nfor the spin diffusion along the ˆx-direction ( ˆy-direction),\nlτ/lessorsimilarlα,lΩ,lc\nτ,x(lτ/lessorsimilarlα,lΩ,lc\nτ,y) is satisfied. Hence, the\nsystem lies in Regime V. Accordingly, for the transverse\nspin diffusion along the ˆx-direction ( ˆy-direction), one ob-\ntains from Table. I (Table. II) that Lx\nT≈/radicalig\n2lτlΩ/√\n3\n[Ly\nT≈√\n3lΩlτ/(2lα)]. Therefore, with the increase of\nthe Zeeman field, the transverse spin diffusion length de-\ncreases. For the longitudinal spin diffusion, it is shown\nin Figs. 4(a) and (b) that no matter the spin diffusion is\nalong the ˆx-direction or the ˆy-direction, the spin diffu-\nsion length is marginally influenced by the Zeeman field\n(blue dashed curve with squares). This anomalous be-\nhavior can be well understood from the analytical results\nin Regime V. For the longitudinal spin diffusion along\ntheˆx-direction ( ˆy-direction), it is obtained that Lx\nL≈lα\n(Ly\nL≈lα). Accordingly, in Regime V, the longitudi-\nnal spin diffusions along both the ˆx- andˆy-directions\nare marginally influenced by the Zeeman field. Based on\nthe drift-diffusion model,45–50it is emphasized that this\nunique feature arises from the insensitivity of diffusion\ncoefficient and SRT to the Zeeman field in the strong\nscattering limit.\nWe then analyze the case with ( V/VR)2= 10. We first\ndivide the regimes for the spin diffusion along the ˆx- and\nˆy-directions, respectively. For the spin diffusion along\ntheˆx-direction, the regimes for the system are divided\ninto\n\n\nIV :lc\nτ,x/lessorsimilarlτ/lessorsimilarlΩ,lα,when Ω/lessorsimilar30Er\nV :lτ/lessorsimilarlΩ,lα,lc\nτ,x,when 30Er/lessorsimilarΩ/lessorsimilar80Er\nII :lΩ/lessorsimilarlτ/lessorsimilarlα,when Ω/greaterorsimilar80Er,\n(37)\nwhich are labelled by the red Roman numbers with the\nboundaries indicated by the red crosses at the upper\nframe of Fig. 4(a). For the spin diffusion along the ˆy-\ndirection, when Ω /lessorsimilar80Erand Ω/greaterorsimilar80Er, the system lies\nin Regimes IV and II, which are represented by the blue\nRoman numbers with the boundaries indicated by the\nblue crossesat the lowerframeofFig. 4(b). It is shownin\nFigs. 4(a) and (b) that in Regime II, the transverse (lon-\ngitudinal) diffusion length decreases slowly (increases)\nwith the increase of the Zeeman field, represented by the\nredsolid(blue dashed)curvewith circles. This isbecause\nfor the transverse spin diffusion along the ˆx-direction ( ˆy-\ndirection), inRegimeII, Lx\nT≈2lτ/√\n3(Ly\nT≈√\n2lτ), with\nthe diffusion length marginally influenced by the Zeeman\nfield. For the longitudinal spin diffusion, in Regime II,\nLx\nL≈lτlα/(√\n3lΩ) [Ly\nL≈lτlα/(√\n3lΩ)], leading to the\nenhancement of the spin diffusion by the Zeeman field.\nWe emphasize that this enhancement of the spin diffu-\nsion arises from the suppression of the longitudinal spin14\nrelaxation by the Zeeman field.\nFinally, we emphasize the unique features in the Zee-\nman field dependence of the spin diffusions along the ˆx-\nandˆy-directions, which can be observed in the experi-\nment. These unique features arise in the longitudinal sit-\nuation, i.e., the initial spin polarization is parallel to the\nZeeman field, for the spin diffusion along both the ˆx- and\nˆy-directions. On one hand, when the scattering is strong\nwith the system in Regime V, the longitudinal spin diffu-\nsion is marginally influenced by the Zeeman field; on the\nother hand, when the scattering is relatively weak and\nthe Zeeman field is strong with the system in Regime\nII, the spin diffusion length is enhanced by the Zeeman\nfield. These unique features can be understood in the\nframework of modified drift-diffusion model addressed in\nSec. III.45–50It isemphasized thatherethe drift-diffusion\nmodel is applicable, which is very different from the lon-\ngitudinal spin diffusion along the ˆy-direction in Regimes\nIII and IV [the red solid curve with circles in Fig. 3(d)]\nwhere the modified inhomogeneous broadening picture is\nused. We readdress that with the strong (weak) Zeeman\nand weak(strong)spin-orbit coupled fields, the condition\nto apply the modified drift-diffusion model/modified in-\nhomogeneous broadening picure is that the initial spin\npolarization is parallel/perpendicular to the larger field.\nC. SOC strength dependence\nIn this part, we analyze the SOC strength dependence\nof the transverse and longitudinal spin diffusions along\ntheˆx- [Fig. 5(a)] and ˆy-directions [Fig. 5(b)]. It has\nbeen well understood from the drift-diffusion model that\nin the strong scattering regime, the spin diffusion length\nis suppressed by the SOC ( ls∝1/|α|).45–50In our study,\nbesides the suppressionofthe spin diffusion length bythe\nSOC, we find two unique features in the SOC strength\ndependence which can not be derived from the over-\nsimplified drift-diffusion model: the spin diffusion length\ncan be either marginally influenced or even enhanced by\nthe SOC. These features can be realized when the sys-\ntem lies in the moderate and strong scattering regimes.\nAccordingly, in our calculation, the Zeeman field is set\nto be Ω = 45 Er; the scattering strengths are set to be\n(V/VR)2= 100 and ( V/VR)2= 20, respectively.\nWe first address the first unique feature, i.e., the\nmarginal influence of the SOC on the spin diffusion. In\nFigs. 5(a) and (b), one observes that when ( V/VR)2=\n100, no matter the spin diffusion is along the ˆx- orˆy-\ndirectionin the transverseconfiguration(the graydashed\ncurve with squares), when α/lessorsimilar4α0, the spin diffu-\nsion length is marginally influenced by the SOC. This\nis because when α/lessorsimilar4α0, the system lies in Regime V\nwithLx\nT≈/radicalig\n2lτlΩ/√\n3 (Lx\nL≈/radicalbig√\n3lτlΩ) for the trans-\nverse spin diffusion along the ˆx-direction ( ˆy-direction).\nIt is emphasized that this robustness of the spin dif-\nfusion to the SOC cannot be obtained from the drift- 0 0.2 0.4 0.6 0.8 1 1.2 1.4\n 1 2 3 4 5 6\nα/α0Ly (µm)(b)\nP=20%\nΩ=45Er\nV III IV(V/VR)2=100, LTLL\n(V/VR)2=20, LTLL 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Lx (µm)(a)V IV III\nP=20%\nΩ=45Er(V/VR)2=100, LTLL\n(V/VR)2=20, LTLL\nFIG. 5: (Color online) SOC dependence of the transverse\nand longitudinal spin diffusions along the (a) ˆx- and (b) ˆy-\ndirections. The Zeeman field Ω = 45 Erand the spin polar-\nizationP= 20%. Both the situations with the scattering\nstrength ( V/VR)2= 100 (squares) and ( V/VR)2= 20 (cir-\ncles) calculated numerically are presented. The red (blue)\nRoman numbers at the upper (lower) frame represent the dif-\nferent regimes for the spin diffusion along the ˆx-direction ( ˆy-\ndirection) when ( V/VR)2= 20 with the boundaries indicated\nby the red (blue) crosses.\ndiffusion model.45–50As we have addressed in Sec. III, in\nthe transverse spin diffusion in Regime V, the inhomo-\ngeneous broadening has dominant influence on the spin\ndiffusion.\nWe then analyze the second unique feature where the\nspin diffusion length can be enhanced by the SOC. For\nthe spin diffusion along the ˆx-direction, it is shown in\nFig. 5(a) by the green (blue) dashed curve with circles\nthat when ( V/VR)2= 20 with α/greaterorsimilar2α0, the transverse\n(longitudinal) spin diffusion is significantly enhanced by\nthe SOC; for the spin diffusion along the ˆy-direction,\nwhen (V/VR)2= 20 with α/greaterorsimilar2α0the transverse spin\ndiffusion length also increases with the increase of the\nSOC (the green dashed curve with circles). This can be\nunderstood as follows. When α/greaterorsimilar2α0, the system lies in\nRegimeIII. Accordingly, for the transverse(longitudinal)15\nspin diffusion along the ˆx-direction, one obtains Lx\nT≈\nlτlΩ/(√\n3lα) [Lx\nL≈√\n2lτlΩ/(√\n3lα)]; for the transverse\nspin diffusion along the ˆy-direction, Lx\nT≈√\n3lτlΩ/(2lα).\nThis unique feature is in contrast to the prediction of the\ndrift-diffusion model.45–50\nIn above sections, we have compared the analytical re-\nsults[Eqs.(A9), (A10), (A15), (A19) and(A22)]with the\nnumerical ones. Now, we address the general condition\nthatthe analyticalresultscanbe applied. It isnotedthat\nour analytical results are derived in the strong scattering\nregime with lτ/lessorsimilarlα,lΩand then extended to the moder-\nate scattering regime. The numerical calculations show\nthatintherelativelystrongandstrongscatteringregimes\n(lτ/lessorsimilarlα,lΩ), the analytical results agree with the numer-\nical ones fairly well; in the moderate scattering regime,\nthe condition to use the analytical results is lΩ/lessorsimilarlτ< lα\norlα/lessorsimilarlτ< lΩ, which are close to the boundary between\nthe moderate and relatively strong scattering regimes.\nHowever, even when the system is away from the bound-\nary between the moderate and strong scattering regimes,\nthe dependencies of the spin diffusion on the scattering\nstrength, Zeeman field and SOC strength are qualita-\ntively correct in the moderate scattering regimes.\nV. CONCLUSION AND DISCUSSION\nIn conclusion, we have investigated the steady-state\nspin diffusion for the 3D ultracold spin-orbit coupled\n40K gas by the KSBE approach44first analytically and\nthen numerically. The spin diffusions along the ˆx- and\nˆy-directions for the transverse ( P||ˆz) and longitudinal\n(P||ˆx)configurationsarestudied. Itisfirstshownanalyt-\nicallythat the behaviorsofthe steady-statespin diffusion\nin the four configurations ( ˆx-T,ˆx-L,ˆy-T andˆy-L) are\ndetermined bythree characteristiclengths: the mean free\npathlτ, the Zeeman oscillation length lΩ, and the SOC\noscillation length lα. We have derived the spin diffusion\nlengthsforthespindiffusionsinthefourconfigurationsin\nthe strong scattering regime, which are then extended to\nthe weak scattering one. We further find that in different\nlimits, the complex analytical reuslts can be reduced to\ndifferent extremely simple forms, and correspodingly, the\nsystem canbe divided into different regimes. Specifically,\nit is revealed that by tuning the scattering strength, the\nsystem can be divided into fiveregimes: I, weak scat-\ntering regime ( lτ/greaterorsimilarlΩ,lα); II, Zeeman field-dominated\nmoderate scattering regime ( lΩ≪lτ≪lα); III, SOC-\ndominated moderate scattering regime ( lα≪lτ≪lΩ);\nIV, relativelystrongscatteringregime ( lc\nτ≪lτ≪lΩ,lα);\nV, strong scattering regime ( lτ≪lΩ,lα,lc\nτ). In different\nregimes, the corresponding behaviors of the spacial evo-\nlution of the spin polarization in the steady state are\nvery rich, showing different dependencies on the scatter-\ning strength, Zeeman field and SOC strength. These de-\npendencies are summarized in Table I (Table II) for the\nspin diffusion along the ˆx-direction ( ˆy-direction). Then\nthe scattering strength, Zeeman field and SOC strengthdependencies of the spin diffusions are numerically cal-\nculated and compared with the analytical ones. It is\nshown that the analytical results agree with the numeri-\ncalonesfairlywell in the relativelystrong/strongscatter-\ning regime and the region close to the boundary between\nthe moderate and relatively strong scattering regimes.\nHowever, it is found that even when the system is away\nfrom the strong scattering regime, the analytical results\nare still qualitatively correct in the moderate scattering\nregimes.\nThe rich behaviors of the spin diffusions in differ-\nent regimes are hard to be understood in the frame-\nwork of the previous simple drift-diffusion model45–50or\nthe direct inhomogeneous broadening [Eqs. (2) and (3)]\npicture44,51–54in the literature. In this work, we extend\nour previous inhomogeneous broadenings [Eqs. (2) and\n(3)] to the effective ones in Eqs. (20) and (21) [Eqs. (31)\nand (32)] for the spin diffusion along the ˆx-direction ( ˆy-\ndirection). In the limit situation, we suggest reasonable\npictures referred to as modified drift-diffusion model and\nmodified inhomogeneous broadening picture to facilitate\nthe understanding of the simple analytical results in Ta-\nblesIandII.Itisshownthatthebehaviorsofthespindif-\nfusionscanbeanalyzedinthesituationeitherwithstrong\nZeeman and weak spin-orbit coupled fields (Regimes II\nand V) or weak Zeeman and strong spin-orbit coupled\nfields (Regimes III and IV). When the spin polarization\nis parallel (perpendicular) to the larger field between the\nZeeman and spin-orbit coupled fields, the spin polariza-\ntion cannot (can) rotate around the effective inhomoge-\nneousbroadeningfieldsefficiently, andhencethe modified\ndrift-diffusion model ( modified inhomogeneous broaden-\ning picture) can be used. In the modified drift-diffusion\nmodel, in the strong scattering regime, τs(k) remains\nthe SRT in the conventional DP mechanism;29,30,32,44\nwhereas in the moderate scattering regime, τs(k) is re-\nplaced by the helix spin-flip rates determined in the he-\nlix space.31,39In the modified inhomogeneous broaden-\ning picture, the behavior of the spin diffusion is deter-\nmined by the effective inhomogeneous broadenings to-\ngether with the spin-conserving scattering. Based on the\nmodified drift-diffusion model and modified inhomoge-\nneous broadening picture, apart from Regime IV, all the\nfeatures in different regimes can be obtained. Below, we\naddress several anomalous features of the spin diffusion,\nwhich are in contrast to boththe simple drift-diffusion\nmodel and the direct inhomogeneous broadening picture.\nIn the scattering strength dependence, it is found that\nwhenlα≪lΩ, the longitudinal spin diffusion along the\nˆy-direction is robustagainstthe scatteringeven when the\nsystem is away from the strong scattering regime, which\nis in contrast to the simple drift-diffusion model. In that\nmodel, in the weak scatteringregime, with ls∝k√τk/m,\nthe spin diffusion length is suppressed by the scatter-\ning. In the Zeeman field dependence, when the system is\nin Regime II, the longitudinal spin diffusion is enhanced\nby the Zeeman field. This is in contrast to the pre-\ndiction from the previous drift-diffusion model and the16\ndirect inhomogeneous broadening picture. In the sim-\nple drift-diffusion model, in the strong (weak) scatter-\ning regime, with ls∝1/(m|α|) (ls∝k√τk/m), it is\nobtained that the diffusion length is irrelevant to the\nZeeman field. In the direct inhomogeneous broadening\npicture, the spin diffusion is suppressed due to the en-\nhancement of the inhomogenous broadening by the Zee-\nman field. Finally, in the SOC strength dependence, we\nfind that the spin diffusion length can also be enhanced\nby the SOC in Regime III. This also goes beyond the\nprediction from the simple drift-diffusion model and the\ndirect inhomogeneous broadening picture. In the sim-\nple drift-diffusion model, in the strong (weak) scattering\nregime, with ls∝1/(m|α|) (ls∝k√τk/m), the spin dif-\nfusion length is suppressed (uninfluenced) by the SOC.\nIn the direct inhomogeneous broadenings picture, the\nspin diffusion is uninfluenced (suppressed) for the spin\ndiffusion along the ˆx-direction ( ˆy-direction). All these\nanomalousbehaviorshavebeen well understood fromour\nmodified drift-diffusion model and/or modified inhomo-\ngeneous broadening picture.\nWe emphasize that for the longitudinal spin diffusion\nalong the ˆy-direction, the robustness against the scatter-\ning strength exists in a wide range including both the\nstrong and weakscattering regimes. It is noted that\nin the strong scattering regime, this feature has been\npredicted in the simple drift-diffusion model45–49and is\nalso revealed in graphene by the KSBE approach.50In\nthis work, we further extend it into the weak scattering\nregime. Moreover, it is found that in a wide range of the\nscattering, the corresponding diffusion length is only de-\ntermined by the SOC strength, and hence also irrelevant\nto the atom density and temperature. One expects sim-\nilar behavior in symmetric (110) quantum wells under a\nweak in-plane magnetic field with similar SOC.38,39,64–68\nMoreover, the enhancement of the longitudinal spin dif-\nfusion by the Zeeman field has not yet been reported in\nthe literature, which is also expected to be observed in\nsymmetric (110) quantum wells when the Zeeman energy\nlarger than the spin-orbit coupled one.\nFinally, wefurthercomparethe picturesofthe spindif-\nfusion providedin this work to understand the calculated\nresults and those in the literature. In the previous works,\nthe spin diffusion in the strongscattering regime have\nbeen extensively studied in the system with SOC, includ-\ning semiconductors,44,69–75graphene50,76–81and recently\nmonolayerMoS 2.82Drift-diffusion model45–49and/or the\ninhomogeneous broadening picture44,51–54were used to\nunderstand the behaviors of the spin diffusion. In this\nwork, the analytical results in the strong scattering\nregime are extended to the weak one and confirmed by\nthe full numerical calculation. For the strong scatter-\ning regime, we further divide it into the relatively strong\nand strongscattering regimes; whereas for the weak scat-\ntering regime, we divide it into the moderate and weak\nscattering regimes. We find all the anomalous behaviors\nrevealed in this work appear in the moderate and rela-\ntively strong scattering regimes. Furthermore, our modi-fieddrift-diffusion model and/or modified inhomogeneous\nbroadening picture are used to understand the behaviors\nof the spin diffusion in all these regimes. It is found that\nin the moderate and strong scattering regimes, these pic-\ntures work well. However, in the relatively strong scat-\ntering regime, which lies in the crossover of the moder-\nate and strong scattering regimes, these pictures fail to\nexplain the behaviors of the spin diffusion along the ˆy-\ndirection. This is because for the drift-diffusion model,\nin the relatively strong scattering regime, it is too rough\nto consider the anisotropy between the diffusions along\ndifferent directions.50Nevertheless, when the scattering\nis strong enough, this anisotropy in the spin diffusion\nbehavior is vanished. Whereas for the inhomogeneous\nbroadening picture, in this regime there exists strong\ncompetition between the effective inhomogeneous broad-\nening and scattering, which makes the behavior of the\nspin diffusion complicated.50,53,54\nAcknowledgments\nThis work was supported by the National Natural Sci-\nence Foundation of China under Grant No. 11334014\nand 61411136001, the National Basic Research Program\nofChinaunderGrantNo. 2012CB922002andthe Strate-\ngic Priority Research Program of the Chinese Academy\nof Sciences under Grant No. XDB01000000.\nAppendix A: Analytical Analysis\nWe analytically derive the transverse and longitudinal\nspin diffusion lengths for the spin diffusions along the ˆx-\nandˆy-directions based on the KSBEs [Eq. (12)].\nGenerally, the density matrix depends on both the\nzenith (between kand ˆx-axis) and azimuth (between k\nand ˆy-axis in the ˆ y-ˆzplane) angles θkandφkin 3D.\nHowever, with the specific form of the SOC [Eq. (1)] and\nisotropic scattering terms [Eq. (9)], we can define the\nquantity\n¯ρk=1\n2π/integraldisplay2π\n0dφkρk, (A1)\nwhich is averaged over the azimuth angle φk, to describe\nthe kinetics ofthe density matrix.31Accordingly, the KS-\nBEs in the steady state become\nkξ\nm∂¯ρk(r)\n∂ξ+i/bracketleftig\nΩσx/2,¯ρk(r)/bracketrightig\n+i/bracketleftig\nαkxσz/2,¯ρk(r)/bracketrightig\n+/summationdisplay\nk′Wkk′/bracketleftbig\n¯ρk(r)−¯ρk′(r)/bracketrightbig\n= 0, (A2)\nin which ¯ ρk(r) only depends on θk.17\n1. Spin diffusion along the ˆx-direction\nFor the spin diffusion along the ˆx-direction, ¯ ρkis ex-\npanded by the Legendre function, which is written as\n¯ρk=/summationdisplay\nl¯ρl\nkC0\nlPl(cosθk), (A3)\nwithC0\nl=/radicalbig\n(2l+1)/(4π). Accordingly, the dynamical\nequation for ¯ ρl\nkis written as Eq. (13). With the spin\nvector defined by ¯Sl\nk= Tr[¯ρl\nkσ], the equations for the\nspin vectors can be obtained. By further keeping the\nzeroth and first orders ( l= 0,1), the equation for the\nvector¯Sk= (¯S0\nk,x,¯S0\nk,y,¯S0\nk,z,¯S1\nk,x,¯S1\nk,y,¯S1\nk,z)Tis written\nas\n∂x¯Sk+Ux¯Sk= 0, (A4)\nwith\nUx=\n0 1/lα0√\n3/lτ0 0\n−1/lα0 0 0√\n3/lτ1/lΩ\n0 0 0 0 −1/lΩ√\n3/lτ\n0 0 0 0 1 /lα0\n0 0 1 /lΩ−1/lα0 0\n0−1/lΩ0 0 0 0\n.\n(A5)\nFrom Eq. (A5), the spin diffusion and oscillation lengths\ncan be found from the eigenvalues of Uxdenoted by λx,\nwhich satisfy\nλ6\nx+aλ4\nx+bλ2\nx+c= 0, (A6)\nwitha= 2/l2\nΩ+2/l2\nα,b= 3/(lτlΩ)2+1/l4\nΩ+2/(lΩlα)2+\n1/l4\nαandc=−3/(lτlΩlα)2. In Eq. (A6), the real\nand imaginary parts of 1 /λxcorrespond to the diffusion\nlength and oscillation length, respectively.\nWith ∆ = ( q/2)2+(p/3)3whereq= 2a3/27−ab/3+c\nandp=b−a2/3, it is demonstrated that in the mod-\nerate and strong scattering regimes, ∆ is always larger\nthan zero. Therefore, there are one real root (Λ re) and\ntwo complex conjugate roots (Λ im,±) forλ2\nxin Eq. (A6),\nwhich are written as\nΛre=3/radicalig\n−q/2−√\n∆+3/radicalig\n−q/2+√\n∆, (A7)\nΛim,±=−(1/2)/parenleftig\n3/radicalig\n−q/2−√\n∆+3/radicalig\n−q/2+√\n∆/parenrightig\n±(√\n3i/2)/parenleftig\n3/radicalig\n−q/2−√\n∆−3/radicalig\n−q/2+√\n∆/parenrightig\n.(A8)\nAccordingly, the spin diffusion length for the single expo-\nnential decay, the spin diffusion length for the oscillation\ndecay and the spin oscillation length are given by\nLx\ns= 1//radicalbig\nΛre, (A9)\nLx\no=√\n2//radicalbigg\nΛre+/radicalig\nΛ2re+|Λim,+−Λim,−|2/3,(A10)\nlx\no= 2√\n3Lx\no/|Λim,+−Λim,−|. (A11)2. Spin diffusion along the ˆy-direction\nFor the spin diffusion along the ˆy-direction, ¯ ρkis ex-\npanded by the Fourier function, which is denoted by\n¯ρk=/summationdisplay\nl˜ρl\nkexp(ilθk). (A12)\nThe corresponding dynamical equation for ˜ ρl\nkis writ-\nten as Eq. (22). By keeping the zeroth and first orders\n(l= 0,1), the equation for the vector ˜Sk=/parenleftbig˜S0\nk,x,˜S1\nk,x−\n˜S−1\nk,x,˜S0\nk,y,˜S0\nk,z,˜S1\nk,y−˜S−1\nk,y,˜S1\nk,z−˜S−1\nk,z/parenrightbigTis written as\n∂y˜Sk+Uy˜Sk= 0, (A13)\nwhere\nUy=\n0i\nlτ0 0 0 0\n−i\nl2\nτ/(3l2\nΩ)+1lτ\nl2\nα0 0 0 0 0\n0 0 0 0i\nlτi√\n3lΩ\n0 0 0 0 −i√\n3lΩi\nlτ\n0 0 −ilτ\nl2\nα−2i√\n3lΩ0 0\n0 02i√\n3lΩ0 0 0\n.\n(A14)\nIn above equation, it is noted that the up-left 2 ×2 and\ndown-right4 ×4blocksUu-l\nyandUd-r\ny, which describe the\nspin diffusion for SxandSy/Sz, are decoupled to each\nother.\nAccordingly, from the eigenvalues of Uu-l\ny, the spin dif-\nfusion length for Sxis found to be\nLy\nL=lα/radicalig\nl2τ/(3l2\nΩ)+1. (A15)\nFromUd-r\ny, it is found that the four eigenvalues λysatisfy\nλ4\ny+/bracketleftbig\n4/(3l2\nΩ)−1/l2\nα/bracketrightbig\nλ2\ny+4/(3l2\nΩl2\nτ)+4/(9l4\nΩ) = 0.(A16)\nFrom Eq. (A16), when 1 /l4\nα−8/(3l2\nΩ)(1/l2\nα+2/l2\nτ)≥0,\nthere are four real roots, among which the two positive\nones read\n|λ±\ny|=/radicaltp/radicalvertex/radicalvertex/radicalbt1\n2l2\nα−2\n3l2\nΩ±/radicaligg\n1\n4l4\nα−2\n3l2\nΩl2\nα−4\n3l2\nΩl2\nτ.(A17)\nAccordingly, the steady-state spin polarization SyorSz\nis limited by the bi-exponential decay, with the diffusion\nlength being\nLy,+\nT= 1/|λ+\ny|, (A18)\nLy,−\nT= 1/|λ−\ny|, (A19)18\nrespectively. Otherwise, when 1 /l4\nα−8/(3l2\nΩ)(1/l2\nα+\n2/l2\nτ)<0, the four roots for λyare complex. Specifi-\ncally, the real part of the roots for Eq. (A16) is identical,\nwhich is written as\nλre\ny=/radicalbigg\n1/(4l2α)−1/(3l2\nΩ)+/radicalig\n1/(9l4\nΩ)+1/(3l2\nΩl2τ).\n(A20)\nThe complex part of the roots (absolute value) for\nEq. (A16) is\n|λim\ny|=/radicalig\n2/(3l2\nΩ)(1/l2α+2/l2τ)−1/(4l4α).(A21)Therefore, the steady-state spin polarization SyorSzis\ndeterminedbythe oscillationdecaywiththedecaylength\nand oscillation length being\nLy\nT= 1/λre\ny, (A22)\nly\nT= 2λre\ny/|λim\ny|, (A23)\nrespectively.\n∗Author to whom correspondence should be addressed;\nElectronic address: mwwu@ustc.edu.cn.\n1J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100,\n153002 (2008).\n2J. Y. Zhang, S. C. Ji, Z. Chen, L. Zhang, Z. D. Du, B.\nYan, G. S. Pan, B. Zhao, Y. J. Deng, H. Zhai, S. Chen,\nand J. W. Pan, Phys. Rev. Lett. 109, 115301 (2012).\n3C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels,\nPhys. Rev. A 88, 021604(R) (2013).\n4L. J. LeBlanc, M. C. Beeler, K. J. Garc´ ıa, A. R. Perry, S.\nSugawa, R. A. Williams, and I. B. Spielman, New J. Phys.\n15, 073011 (2013).\n5M. C. Beeler, R. A. Williams, K. J. Garc´ ıa, L. J. LeBlanc,\nA. R. Perry, and I. B. Spielman, Nature 498, 201 (2013).\n6P. L. Pedersen, M. Gajdacz, F. Deuretzbacher, L. Santos,\nC. Klempt, J. F. Sherson, A. J. Hilliard, and J. J. Arlt,\nPhys. Rev. A 89, 051603(R) (2014).\n7Y. J. Lin, K. J. Garc´ ıa, and I. B. Spielman, Nature (Lon-\ndon)471, 83 (2011).\n8K. J. Garc´ ıa, L. J. LeBlanc, R. A. Williams, M. C. Beeler,\nC. Qu, M. Gong, C. Zhang, and I. B. Spielman, Phys. Rev.\nLett.114, 125301 (2015).\n9A. Sommer, M. Ku, and M. W. Zwierlein, New J. Phys.\n13, 055009 (2011).\n10H. Heiselberg, Phys. Rev. Lett. 108, 245303 (2012).\n11O. Goulko, F. Chevy, and C. Lobo, Phys. Rev. Lett. 111,\n190402 (2013).\n12C. Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka, T.\nSch¨ afer, and J. E. Thomas, Science 331, 58 (2011).\n13A. Sommer, M. Ku, G. Roati, andM. W. Zwierlein, Nature\n472, 201 (2011).\n14G. M. Bruun, New J. Phys. 13, 035005 (2011).\n15T. Enss and R. Haussmann, Phys. Rev. Lett. 109, 195303\n(2012).\n16T. Enss, Phys. Rev. A 88, 033630 (2013).\n17A. J. Leggett and M. J. Rice, Phys. Rev. Lett. 20, 586\n(1968); A. J. Leggett, J. Phys. C 3, 448 (1970).\n18M. Koschorreck, D. Pertot, E. Vogt, and M. K¨ ohl, Nat.\nPhys.9, 405 (2013).\n19A. B. Bardon, S. Beattie, C. Luciuk, W. Cairncross, D.\nFine, N. S. Cheng, G. J. A. Edge, E. Taylor, S. Zhang, S.\nTrotzky, and J. H. Thywissen, Science 344, 722 (2014).\n20S. Trotzky, S. Beattie, C. Luciuk, S. Smale, A. B. Bardon,\nT. Enss, E. Taylor, S. Zhang, and J. H. Thywissen, Phys.\nRev. Lett. 114, 015301 (2015).\n21T. Enss, Phys. Rev. A 91, 023614 (2015).22X. Du, L. Luo, B. Clancy, and J. E. Thomas, Phys. Rev.\nLett.101, 150401 (2008).\n23F. Pi´ echon, J. N. Fuchs, and F. Lalo¨ e, Phys. Rev. Lett.\n102, 215301 (2009).\n24S. S. Natu and E. J. Mueller, Phys. Rev. A 79, 051601(R)\n(2009).\n25X. Du, Y. Zhang, J. Petricka, and J. E. Thomas, Phys.\nRev. Lett. 103, 010401 (2009).\n26P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n27L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n28U. Ebling, J. S. Krauser, N. Fl¨ aschner, K. Sengstock, C.\nBecker, M. Lewenstein, and A. Eckardt, Phys. Rev. X 4,\n021011 (2014).\n29I. V. Tokatly and E. Ya. Sherman, Phys. Rev. A 87,\n041602(R) (2013).\n30S. S. Natu and S. Das Sarma, Phys. Rev. A 88, 033613\n(2013).\n31T. Yu and M. W. Wu, Phys. Rev. A 88, 043634 (2013).\n32J. Radi´ c, S. S. Natu, and V. Galitski, Phys. Rev. Lett.\n112, 095302 (2014).\n33I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.\n80, 885 (2008).\n34S. S. Kondov, W. R. McGehee, J. J. Zirbel, and B. De-\nMarco, Science 334, 66 (2011).\n35F. Jendrzejewski, A. Bernard, K. M¨ uller, P. Cheinet, V.\nJosse, M. Piraud, L. Pezz´ e, L. S. Palencia, A. Aspect, and\nP. Bouyer, Nat. Phys. 8, 398 (2012).\n36D. Cl´ ement, A. F. Var´ on, J. A. Retter, L. S. Palencia, A.\nAspect, and P. Bouyer, New J. Phys. 8, 165 (2006).\n37M. Piraud, L. Pezz´ e, and L. S. Palencia, New J. Phys. 15,\n075007 (2013).\n38Y. Zhou, T. Yu, and M. W. Wu, Phys. Rev. B 87, 245304\n(2013).\n39T. Yu and M. W. Wu, Phys. Rev. B 89, 045303 (2014).\n40M. I. D’yakonov and V. I. Perel’, Zh. Eksp. Teor. Fiz. 60,\n1954 (1971) [Sov. Phys. JETP 33, 1053 (1971)].\n41Y. Yafet, Phys. Rev. 85, 478 (1952).\n42R. J. Elliott, Phys. Rev. 96, 266 (1954).\n43J. P. Brantut, J. Meineke, D. Stadler, S. Krinner, and T.\nEsslinger, Science 31, 337 (2012).\n44M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Rep. 493,\n61 (2010).\n45M. Ziese and M. J. Thornton, eds., Spin Electronics19\n(Springer, Berlin, 2001).\n46Z. G. Yu and M. E. Flatt´ e , Phys. Rev. B 66, 201202(R)\n(2002).\n47I. Zuti´ c , J. Fabian, and S. Das Sarma, Phys. Rev. Lett.\n88, 066603 (2002).\n48J. Fabian, I. Zuti´ c , and S. Das Sarma, Phys. Rev. B 66,\n165301 (2002).\n49J. Fabian, A. M. Abiague, C. Ertler, P. Stano, and I. Zuti´ c,\nActa Phys. Slov. 57, 565 (2007).\n50P. Zhang and M. W. Wu, Phys. Rev. B 84, 045304 (2011).\n51M. W. Wu and C. Z. Ning, Eur. Phys. J. B. 18, 373 (2000);\nM. W. Wu, J. Phys. Soc. Jpn. 70, 2195 (2001).\n52M. Q. Weng and M. W. Wu, Phys. Rev. B 66, 235109\n(2002); J. Appl. Phys. 93, 410 (2003).\n53J. L. Cheng and M. W. Wu, J. Appl. Phys. 101, 073702\n(2007).\n54J. L. Cheng, M. W. Wu, and I. C. da Cunha Lima, Phys.\nRev. B75, 205328 (2007).\n55C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys.\nRev. Lett. 90, 053201 (2003).\n56M. Egorov, B. Opanchuk, P. Drummond, B. V. Hall, P.\nHannaford, and A. I. Sidorov, Phys. Rev. A 87, 053614\n(2013).\n57T. Ozawa, L. P. Pitaevskii, and S. Stringari, Phys. Rev. A\n87, 063610 (2013).\n58R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. J. Garc´ ıa,\nand I. B. Spielman, Phys. Rev. Lett. 111, 095301 (2013).\n59M. W. Wu and H. Metiu, Phys. Rev. B 61, 2945 (2000).\n60H. Haug and A. P. Jauho, Quantum Kinetics in Transport\nand Optics of Semiconductors (Springer, Berlin, 1996).\n61M. Q. Weng and M. W. Wu, Phys. Rev. B 68, 075312\n(2003).\n62J. H. Jiang and M. W. Wu, Phys. Rev. B 79, 125206\n(2009).\n63A. F. Ioffe and A. R. Regel, Prog. Semicond. 4, 237 (1960).\n64Y. Ohno, R. Terauchi, T. Adachi, F. Matsukura, and H.\nOhno, Phys. Rev. Lett. 83, 4196 (1999); Physica E 6, 817\n(2000); T. Adachi, Y. Ohno, F. Matsukura, and H. Ohno,\nPhysica E 10, 36 (2001).\n65M. W. Wu and M. Kuwata-Gonokami, Solid State Com-mun.121, 509 (2002).\n66S. D¨ ohrmann, D. H¨ agele, J. Rudolph, M. Bichler, D.\nSchuh, and M. Oestreich, Phys. Rev. Lett. 93, 147405\n(2004).\n67G. M. M¨ uller, M. R¨ omer, D. Schuh, W. Wegscheider, J.\nH¨ ubner, and M. Oestreich, Phys. Rev. Lett. 101, 206601\n(2008).\n68I. V. Tokatly and E. Y. Sherman, Phys. Rev. B 82,\n161305(R) (2010).\n69T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54,\n437 (1982).\n70F. Meier and B. P. Zakharchenya, Optical Orientation\n(North-Holland, Amsterdam, 1984).\n71Semiconductor Spintronics and Quantum Computation ,\nedited by D. D. Awschalom, D. Loss, and N. Samarth\n(Springer, Berlin, 2002).\n72I.ˇZuti´ c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76,\n323 (2004).\n73J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I.\nˇZuti´ c, Acta Phys. Slov. 57, 565 (2007).\n74Spin Physics in Semiconductors , editedbyM. I.D’yakonov\n(Springer, Berlin, 2008).\n75T. Korn, Phys. Rep. 494, 415 (2010).\n76N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and\nB. J. van Wees, Nature 448, 571 (2007).\n77C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys.\nRev. B80, 041405(R) (2009).\n78K. Pi, W. Han, K. M. McCreary, A. G. Swartz, Y. Li, and\nR. K. Kawakami, Phys. Rev. Lett. 104, 187201 (2010).\n79M. H. D.Guimar˜ aes, A.Veligura, P. J. Zomer, T. Maassen,\nI. J. Vera-Marun, N. Tombros, and B. J. van Wees, Nano\nLett.2012, 3512 (2012).\n80W. Han, K. M. McCreary, K. Pi, W. H. Wang, Y. Li, H.\nWen, J. R. Chen, and R. K. Kawakami, J. Magn. Magn.\nMater.324, 369 (2012).\n81A. Dankert, M. V. Kamalakar, J. Bergsten, and S. P. Dash,\nAppl. Phys. Lett. 104, 192403 (2014).\n82L. Wang and M. W. Wu, Phys. Rev. B 89, 205401 (2014)." }, { "title": "2212.11686v1.The_spin_Hall_effect.pdf", "content": "The spin Hall e\u000bect\nCosimo Gorini\nSPEC, CEA, CNRS, Universit\u0013 e Paris-Saclay, 91191 Gif-sur-Yvette, France\n(Dated: December 23, 2022)\nIn metallic systems with spin-orbit coupling a longitudinal charge current may generate a trans-\nverse pure spin current; vice-versa an injected pure spin current may result in a transverse charge\ncurrent. Such direct and inverse spin Hall e\u000bects share the same microscopic origin: intrinsic\nband/device structure properties, external factors such as impurities, or a combination of both.\nThey allow all-electrical manipulation of the electronic spin degrees of freedom, i.e.without mag-\nnetic elements, and their transverse nature makes them potentially dissipationless. It is customary\nto talk of spin Hall e\u000bects in plural form, referring to a group of related phenomena typical of\nspin-orbit coupled systems of lowered symmetry.\nKeywords: Spin-charge conversion, spin currents, spin Hall e\u000bects, spin-orbit coupling, spinorbitronics, spin-\ntronics, (pseudo)spin-orbit coupled transport\nKey points/objectives\n•History and de\fnition of the e\u000bect(s)\n•Phenomenology and concepts: spin-orbit coupling in solids, charge vs. spin currents, bulk vs. edge e\u000bects\n•Experiments: from low- to room temperature, diversity and complexity of setups\n•Theory: the challenge of complexity, competition between di\u000berent microscopic mechanisms, Onsager reci-\nprocity\n•The broader context: generalisations of the spin Hall e\u000bect(s) and suggested further readings\nI. INTRODUCTION\nA charge-carrying state in a metallic system is a (non-equilibrium) momentum-ordered state of an electronic ensemble:\na collection of quasielectrons1moving preferentially in a given direction, see Fig. 1 (a). However quasielectrons carry\naround their internal spin degrees of freedom, too. For their ensemble to be spin carrying some form of spin order is\nneeded. There are two basic scenarios:\n•Spin order is independent of momentum order. This is the situation in a ferromagnet, where the spins of charge\ncarriers align with the magnetisation via exchange coupling independently of their orbital motion.\n•Spin and momentum orders are correlated. This is possible in the presence of spin-orbit coupling, which quite\ngenerally creates correlations between orbital motion (momentum) and spin.\nConsider the second scenario, and to be de\fnite the somewhat special situation sketched in Fig. 1 (b), where quasi-\nparticles with opposite spin projection along zmove in opposite directions. An ensemble of such particles does not\ncarry any overall charge { the associated charge current is zero, jc= 0 { but it does carry angular momentum: it is a\npure spin-carrying state sustaining a \\pure spin current\" js6= 0\njc\u0011q\u0000\nj\"+j#\u0001\n= 0 (1)\njs\u0011~\n2\u0000\nj\"\u0000j#\u0001\n6= 0; (2)\n1I will restrict my discussion to systems where well-de\fned fermionic quasiparticles exist and make up a Fermi liquid. Words such as\n\\quasielectron\" and \\electron\" or \\quasiparticle\" and \\particle\" will be used interchangeably.arXiv:2212.11686v1 [cond-mat.mes-hall] 22 Dec 2022(a)\njcy xz(b)\njsFIG. 1. Left panel: A charge current jcalongx,i.e.an overall spin-unpolarised ensemble of quasielectrons moving in the y\ndirection. Right panel: A z-polarised pure spin current js\rowing along x.\nwithqthe quasiparticle charge and ~the reduced Planck constant.\nIn the presence of spin-orbit coupling a transverse jscan be generated from a longitudinal jc, or vice-versa, see Fig. 2.\nThese are respectively the spin Hall e\u000bect (sHe)[1, 2] and its reciprocal version, the inverse spin Hall e\u000bect (isHe)[3, 4].\nThe sHe was \frst predicted by Dyakonov and Perel in 1971 [1] but got his current name only in 1999, when Hirsch [2]\nsort of re-discovered it and its fortune really started. The isHe followed a similar path. In the geometry from Fig. 2\none has\njs\ny= (~=q)\u0012sHejc\nx; (3)\njc\nx= (q=~)\u0012isHejs\ny (4)\nwith\u0012sHe;\u0012isHethe spin Hall and inverse spin Hall angles. Note that from now on upper (lower) indices will denote\ncharge/spin (real space) components. The spin Hall/inverse spin Hall angles are crucial quantities used as standard\nmeasure for the spin-charge conversion e\u000eciency of a given setup [5{8]. In Secs. II and IV we will show that their\nexistence can be expected based on crude but general arguments, while their precise origin and values are extremely\nsensitive to the microscopic details of the system.\nThe sHe and isHe exist also in quantised form. Indeed, the \\quantum spin Hall insulator\" is the paradigmatic example\nof a time-reversal symmetric two-dimensional topologically insulating phase [9]. The latter is a phase of matter which\nis insulating in the bulk but hosts two perfectly conducting edge states which are time-reversed partners { up electrons\nmoving in one direction, down in the opposite. In simple terms, a quantum spin Hall phase can be thought of as\ntwo time-reversed copies of a (time-reversal broken) quantum Hall state. Its existence was con\frmed experimentally\nin HgCdTe quantum wells in 2007 [10]. I will not discuss it further here, but refer the interested reader to the\nEncyclopedia Chapter \\Topological E\u000bects\".\n(a)\ny xz(b)\nFIG. 2. Left panel: spin Hall e\u000bect. An x-\rowing charge current is injected, and a z-polarised pure spin current along yis\ngenerated via spin-orbit coupling. Right panel: inverse spin Hall e\u000bect, where the role of the spin and charge currents are\nexchanged. Note that in the sHe/isHe the charge current, spin current and the spin quantisation axis are all orthogonal to each\nother.\nThe sHe and isHe are nowadays routinely employed in metallic systems to inject and/or read out spin signals via\nelectrical means, and their technological relevance is rapidly on the rise since the early 2000s [3, 4, 8, 11]. The rest of\nthe Chapter will focus uniquely on them. In particular I will not discuss the anomalous Hall e\u000bects, closely related\nphenomena which require however to break time-reversal symmetry [3, 12]. The reader should indeed keep in mind\nthat the spin Hall e\u000bect I will deal with in this Chapter actually belongs to a larger class of phenomena which mayappear whenever quasiparticles with internal structure move in a non-trivial medium, i.e.not just in the vacuum2.\nII. PHENOMENOLOGY AND BASIC CONCEPTS\nA. The role of spin-orbit coupling\nTake an electron moving with velocity vin presence of an electric \feld E. In its reference frame the electron sees a\nvelocity-dependent magnetic \feld\nB0(v) =\u0000\rv\nc\u0002E; \r =1p\n1\u0000v2=c2(5)\nwhich couples to its spin s= (~=2)\u001b, with \u001bthe vector of Pauli matrices, as usual\nHso\u0018s\u0001B0\u0018s\u0001(v\u0002E): (6)\nHerecis the speed of light. This is a primitive derivation of the spin-orbit interaction.\nLet us assume that E=\u0000rVextcomes from an external source, e.g.the (random) electrostatic potential from an\nimpurity, see Fig. 3 (a). The quasiparticle spin thus couples to a non-homogeneous \feld B0(v;r), similarly to what\nhappens in a Stern-Gerlach apparatus. This has various consequences, e.g.it causes spin-\rip (Elliot-Yafet) relaxation,\nbut in particular it results in a spin-dependent force F\u0018\u0000r [s\u0001B0] orthogonal to v\nF\"\n?=\u0000F#\n?\u0018\u0000r?B0: (7)\nThis sideway separation of spin-up and spin-down quasiparticles is referred to as Mott skew scattering, and can split\nan incoming \rux of unpolarised electrons into a transverse pure spin current, i.e.it yields a \fnite \u0012sHe. It is an\n\\extrinsic\" mechanism requiring the presence of scattering centres [2{4, 13], but a \fnite \u0012sHemay also have origins\nwhich are \\intrinsic\" to the device and/or band structure. Consider for example E=\u0000rVint(z), withVint(z) an\ninternal potential which de\fnes your structure, see Fig. 3 (b). Such a potential breaks zinversion symmetry and\nconstrains electrons to move in the x-yplane, as in semiconductor quantum wells. One has\nHso\u0018s\u0001(v\u0002ez): (8)\nNowscouples to a homogeneous B0(v) which always points in plane, called a Rashba \feld [14, 15]. The eigenstates of\nthe problem are spinors whose quantisation axis depends on their direction of motion. As sketched in Fig. 3 (b), under\nan electrical bias one expects spin-momentum correlations to arise leading to a transverse, out-of-plane-polarised spin\ncurrent { that is, to a non-vanishing \u0012sHe. Recall that in this intrinsic example inversion symmetry was explicitly\nbroken. Some form of spatial symmetry-breaking is indeed a requirement of intrinsic scenarios beyond the present\nRashba case [4, 15]. On the other hand time-reversal symmetry is preserved by spin-orbit interaction.\nBesides the canonical intrinsic and extrinsic mechanisms just described, other sources for the spin Hall e\u000bects include\ndynamical couplings with phonons [16, 17] or spin \ructuations [18], the interplay of impurity and magnon scattering\n[19], or \ructuations of the Rashba \feld [20]. Irrespectively of the origin, and just as a normal Hall current, the spin\nHall current is transverse with respect to the drift velocity and thus potentially dissipationless.\nB. Size and form of (e\u000bective) spin-orbit coupling\nSince charge carriers in a metallic system move at the Fermi velocity vF\u001cc, should one expect Hsoto be only a\nnegligible relativistic correction? The answer is \\not necessarily\". Electrons in solids do not move in the vacuum, but\nconstantly get close to ionic cores from the lattice where unscreened very strong electric \felds exist, compensating for\ntheir (relativistically) slow velocity. One can indeed show that s-wave Bloch electrons close to the Fermi energy feel\nan e\u000bective spin-orbit interaction which reads\nHso=\u0015\u001b\u0002r\u000eV\u0001p; (9)\n2In condensed matter one customarily talks of \\pseudospin\" internal degrees freedom, such as sublattice or valley pseudospin in graphene,\nwhich may give rise to di\u000berent forms of \\pseudospin Hall e\u000bect\".(a)\nVext(r)xyz\n(b) Vint(z)\n+sz\n −szδvx∼ExFIG. 3. Left panel: Electrons with opposite spins are de\rected di\u000berently by a scattering potential Vext(r). The dashed black\nline is the trajectory without spin-orbit corrections. Right panel: Electrons con\fned to two dimensions by Vint(z). The internal\nvelocity-dependent \feld, see Eq. (8), de\fnes the spin quantisation axis, shown with dotted lines. The two electrons are time-\nreversed partners, and when driven by an electric \feld along xtheir spins will precess around the new (tilted) quantisation\naxis, yielding out-of-plane components \u0006szshown in blue. The result is a z-polarised spin current in the transverse ydirection.\nwith\u000eVany potential other than that of the host lattice, e.g.\u000eV=Vextor\u000eV=Vint(z) from our previous examples.\nWhile the form of Hsois the same as in the vacuum, the coupling constant \u0015is an e\u000bective Compton wavelength\nstrongly renormalised by the lattice: \u0015=\u00150\u0018106, with\u00150the vacuum Compton wavelength. The standard machinery\nbehind such renormalization is k\u0001ptheory, which, together with some auxiliary techniques (L owding partitioning,\nSchrie\u000ber-Wol\u000b transformation,. . . ), is just a systematic way of building low-energy models starting from the band\nstructure (Bloch states) of a given lattice { at the simplest level it leads e.g.to the e\u000bective mass description of\nelectrons in solids [4, 15]. Eq. (9) can actually be generalised to\nHso=b(p)\u0001\u001b; (10)\nwhich looks like Zeeman coupling with a momentum-dependent internal \feld b(p). The latter de\fnes a momentum\nspace spin texture { its vectorial image in reciprocal space { of central importance well beyond spin Hall physics3.\nThis form of e\u000bective spin-orbit interaction is valid beyond the two example scenarios previously introduced, and is\nthe starting point for most discussions of spin-orbit coupled transport phenomena [4, 15]4.\nC. A closer look at spin currents\nThe spin Hall/inverse spin Hall angles, see Eqs. (3), (4), are crucial quantities in spin Hall physics. However to de\fne\nthem one needs to know exactly what a spin current is, which is not a completely trivial task. Let us see why.\nCharge is a conserved quantity respecting the continuity equation\n@tn+r\u0001jc= 0; (11)\nwithnthe charge (particle) density and jc=vnthe current. In formal terms, this is a consequence of gauge invariance:\ncharge (particle number) is the conserved quantity associated with the U(1) gauge symmetry of the system { a case\nof Noether's theorem.\nA purely orbital Hamiltonian H0without spin-orbit interaction cannot a\u000bect the dynamics of the spin degrees of\nfreedom: not only charge, but also spin is conserved for H0. Formally H0is spin-rotation invariant, which for spin\n1/2 particles means that it has SU(2) gauge symmetry. Spin conservation yields the continuity equation\n@tsa+r\u0001ja= 0; a =x;y;z; (12)\n3In the presence of b(p) the Hilbert space of the problem is usually equipped with a non-vanishing Berry curvature, with the potential\nfor hosting non-trivial topological phases\n4There are situations in which an internal \feld b(p) appears for microscopic reasons which have nothing to do with relativistic spin-orbit\ninteraction. The \feld thus couples to some internal pseudospin degree of freedom \u001cof the low-energy quasiparticles, b(p)\u0001\u001c. See\ncomments in the closing of Sec. I.withsathe density of a-polarised electrons, and ja=vsathe corresponding a-spin current density.\nIn the presence of spin-orbit coupling, H0!H=H0+Hso, spin rotation symmetry is broken and spin is not\nconserved anymore. Consider \frst extrinsic spin-orbit due to diluted impurities. In this case spin is not conserved\nduring scattering events, but remains so between them. The de\fnition ja=vsais thus good during \right, but at\neach scattering the a-polarised \rux may rotate, partially split and lose weight due to spin-\rip scattering. The result\nis a spin relaxation rate 1 =\u001ca\ns, with\u001ca\nsthea-spin lifetime, and some further extrinsic spin torque \u0000a\next\n@tsa+r\u0001ja=\u0000sa\n\u001cas+ \u0000a\next: (13)\nThe extra torque describes spin non-conservation e\u000bects beyond simple relaxation, e.g.skew-scattering physics, which\nmay directly couple spin and charge degrees of freedom.\nIf intrinsic spin-orbit coupling is present things are more complicated. First, in this case the velocity v=@pHdoes\nnot in general commute with the spin operator, which is obvious looking at Eq. (10). One can still generalise the spin\ncurrent de\fnition by symmetrization\nja=1\n2fv;sag; (14)\nwithfA;Bg=AB+BAthe anticommutator. However spin is now intrinsically { i.e.everywhere and always { not\nconserved, ergo the continuity equation obeyed by such a current must be modi\fed by an intrinsic spin torque \u0000a\nint\n@tsa+r\u0001ja= \u0000a\nint: (15)\nThe precise form of the torque is \fxed by the internal spin-orbit \feld (10). The intrinsic lack of a conservation law\nmeans that ja, and thus \u0000a, are not uniquely de\fned now. This is no fundamental problem, in the sense that not\nall physically meaningful currents need be conserved. It can however be a delicate operative problem, as changing\nthe de\fnition of jawill change the de\fnitions of the spin Hall/inverse spin Hall angles, which is what experiments\ntypically aim at measuring. Similarly, it will modify the estimates for the spin accumulations generated by the\ncurrent, the accumulation being another popular observable. Indeed, it is not alway obvious how (if) local spin\ncurrents somewhere, say in the bulk, are connected with local spin accumulations elsewhere, e.g.at the sample's edges\n[3, 21{24]. These issues are recurringly discussed in the specialised literature [21{31] and can be dealt with in di\u000berent\nways, for example:\n•One can avoid referring to spin currents within the spin-orbit coupled region, and de\fne them only in the\nmetallic electrodes attached to sample, where Hsois negligible. This is the picture naturally arising in the\nLandauer-B uttiker approach to transport [32]. It is often more or less implicitly assumed in phenomenological\ndiscussions of experiments.\n•One can focus on spin accumulations, i.e.consider the equation of motion for the spin density without assuming\na given form for the spin currents. The appropriate form of the currents may be derived from the density\nequations, notably in the di\u000busive regime [22, 33{35].\n•One can use the non-Abelian gauge properties of the Hamiltonian H,i.e.its spin rotation [ SU(2)] properties, to\nde\fne spin currents much as colour currents are de\fned in high-energy physics [29, 36]. This approach removes\nany ambiguity from the de\fnition of ja;\u0000a, but cannot directly be exploited for any form of b(p). I will comment\nfurther on it in Sec. IV.\n•One can try to de\fne a conserved spin current by combining jaand \u0000a[27, 28].\nThere is arguably no single \\best\" approach. One should decide which way to go based on the speci\fc physical\nsituation, and { if the physics allow { on personal tastes. It should also be kept in mind that both extrinsic and\nintrinsic processes are present in typical setups, and that they may yield further intrinsic-extrinsic crossed processes,\ni.e.the corresponding torques are not simply additive [37, 38]. Moreover { and independently of the spin current\nde\fnition { continuity equations like (12), (13) or (15) must be supplemented with appropriate boundary conditions\ne.g.at interfaces between di\u000berent materials or at the egdes of the system, which may yield additional (local) torques\n[22, 31, 35, 39{43].\nIII. EXPERIMENTS\nThe \frst spin Hall experiments were performed in the 1970s and 1980s in semiconductors [3], but relatively few got\ninterested at the time. The business became fashionable in the early 2000s, and the spin Hall e\u000bects are nowadaysnot only the object of fundamental research [44{48], but also established tools in more application-oriented settings\n[11, 49{51]. In fact, while the \frst experiments were performed at fairly low temperatures (a few to a few tens\nof Kelvins) room temperature measurements are routine today, and large spin Hall angles have been reported in\ndi\u000berent materials. To give a rough idea of the progression, the spin Hall angle reported in a pioneering experiment\nby Valenzuela and Tinkham in 2006 was \u0012sHe\u001810\u00004in Al at T=4.2K, while less than 10 years later room temperature\nmeasurements in Pt, Ta or W reached \u0012sHe\u001810\u00001[6, 7, 49, 52, 53].\n(a)\nVisHejy\nzjxxyz(b)\njxjz\ny\nV\nFIG. 4. Left panel: A broad spot of circularly polarised light creates a non-equilibrium spin polarisation \u000esyat a sample's\nsurface (shaded region), and a a di\u000busion spin current jy\nz\u0018@z\u000esy\rows into the 3D bulk. Here the isHe generates a transverse\ncharge current jx, yielding a \fnite output voltage VisHe. Setups of this kind were proposed already in the early days of spin\nHall physics [3]. Right panel: In a 2D system the charge current jxfrom an applied bias Vis converted into a spin Hall current\njz\ny. The resulting spin accumulation at the edges (shaded region) is measured by Kerr rotation of scattered light (red). The\ntechnique was employed in the \frst experimental observation of the sHe in a 2D electron gas [54], and a similar one in a 2D\nhole gas [55]. The connection between bulk spin currents and edge spin accumulations can however be less direct than this\ncartoon suggests [3, 22{24, 26].\nThe numerous experimental schemes available can roughly be divided into three classes.\n•Optical setups, historically the \frst to be used to detect the sHe/isHe, exploit the interaction between polarised\nlight and the spin of charge carriers. For example, circularly polarised light can be used to generate local\nnon-equilibrium spin accumulations which later di\u000buse through the system. The di\u000busion spin current can\nthen be converted by the isHe into charge signals measured with standard electrodes, see Fig. 4 (a). On the\nother hand, spin accumulations can be measured by circularly polarised electroluminescence or magneto-optical\nKerr and Faraday e\u000bects. An example is shown in Fig. 4 (b), where the spin accumulation at the edge of the\nsystem generated by the sHe is measured by the degree of (Kerr) rotation of light scattered o\u000b the sample.\nAll-optical schemes are also employed [56, 57], allowing in particular time-resolved experiments on ultra-short\n(THz) timescales [56] { which are not accessible to electronic systems.\n•Magneto-electric (spin pumping, spin torque) setups, relying on the interplay between magnetisation and spin\ndynamics [58]. Fig. 5 shows a paradigmatic example: a spin current js\ninis injected from an out-of-equilibrium\nmagnetic element, e.g.a (conducting or insulating) magnet driven by microwaves, and converted into a charge\ncurrent jc\noutcollected at a normal metallic electrode. The latter can actually be used to run the experiment\nunder \\reverse bias\": jc\ninis injected, and one measures the torque that the resulting js\noutexerts on the adjacent\nmagnet. The presence of magnetic elements considerably increases the degree of complexity of the overall\nsystem, and may lead to additional e\u000bects which are however beyond the scope of this short overview [58, 59].\nTime-dependent experiments are also performed, e.g.to measure AC spin Hall e\u000bects [60{62].\n•All-electrical setups, conceptually probably the simplest. Fig. 6 (a) shows a most basic one, without any\nmagnetic element: A charge current jc\ninis injected by a metallic electrode, is converted into a spin signal by the\nsHe, and \fnally re-converted by the isHe into an outgoing jc\noutcollected at some other electrode. A very popular\nscheme requiring a magnetic electrode is instead sketched in Fig. 6 (b). Both are non-local { input electrodes\nare somewhere, output electrodes elsewhere { a common feature in spin Hall setups [5, 63].(a)\nVisHeFM\nn(t)ω\njs\nin jc\nout(b)\nVωFM\nn(t)\njs\nout jc\ninFIG. 5. Left panel: Sketch of a spin pumping setup. Microwaves in a ferro-/ferrimagnet FM at frequency !drive the\nmagnetisation, M=Mn!M(t) =Mn(t). Its precession injects angular momentum into the underlying spin-orbit coupled\nnormal metal, generating a spin current js. The latter is converted by the isHe into a charge current jcand ergo a measurable\nvoltageVisHe, in general with both DC and AC components [61, 62, 64]. Right panel: A reverse-bias scenario, in which a charge\ncurrent at frequency !is injected and converted by the sHe into an AC spin current. The latter exerts a torque on Mand\ndrives its precession. There are numerous DC and AC variations to these schemes, involving many di\u000berent magneto-resistive\ne\u000bects modulated by spin-orbit interaction, see Refs. [8 and 11] for an overview.\n(a)\nV\nVisHejc\ninjsjc\nout(b)\nFM\nV\nVisHejc\nin jsjc\nout\nFIG. 6. Left panel: Hall bar geometry without magnetic elements. The injected charge current jc\ninis converted into a transverse\nspin current jsby the sHe. The latter is converted back into a charge current jc\noutby the isHe in the right arm of the setup,\nyielding a \fnite VisHe. One of the earliest implementations of this setup was used to measure the ballistic sHe in a HgTe\nquantum well [65]. Right panel: non-local setup with a ferromagnetic (FM) electrode, shown in dark grey, deposited on top\nof a T-shaped metallic \flm. The injected current jc\ninis drained to the left contact, since the right metallic arm is at the same\nelectrochemical potential as the FM electrode. The current jc\ninis spin polarised, therefore it creates a non-equilibrium spin\naccumulation underneath the FM contact, shown by the shaded area. Part of it di\u000buses towards the right, yielding a pure spin\ncurrent jswhich is then converted by the isHe into a measurable transverse voltage VisHe. The scheme was \frst employed by\nValenzuela and Tinkham [5].\nThe boundary between classes is clearly blurred, and mixed techniques are often employed. An important general\nobservation is that experimental setups { apart perhaps from the simplest all-electrical ones { are fairly complex,\nconsisting of multiple elements of di\u000berent nature subject to various kinds of drivings. Paired with the vast number\nof spin-charge (or charge-spin) conversion channels available in any given system, this makes for interesting debates\nconcerning possible microscopic interpretations of experiments5.\n5The spin galvanic and inverse spin galvanic e\u000bects are very often crucial \\partners\" of the spin Hall e\u000bects [66{69]. They are another\ncommon channel of spin-charge/charge-spin conversion, discussed in detail in a dedicated Encyclopedia ChapterIV. THEORY\nA. An instructive example and the general framework\nThe spin Hall e\u000bects are non-equilibrium phenomena handled with the usual arsenal of transport theory techniques:\nKeldysh formalism, density matrix and semiclassical kinetics, Kubo formula, Landauer-B uttiker formalism . . . . Irre-\nspective of the techniques employed, a source of substantial theoretical challenges is complexity. In crude terms, the\nHamiltonian of a spin Hall system requires numerous ingredients, recall the discussion from Sec. III. The resulting\nquasiparticle dynamics are in general quite sensitive to the presence/absence of, and competition between, each.\nTo see this in a concrete way it is instructive to start from the barebone model of an ideal Rashba system\nHR=p2\n2m+\u000b\n~[py\u001bx\u0000px\u001by]; (16)\nwheremis the e\u000bective electron mass and \u000b\u0018rVint(z) the Rashba coupling constant, proportional to the (e\u000bective)\nelectric \feld con\fning the electrons to the x-yplane { see the heuristic discussion in Sec. II, Eq. (8).\nGivenHR, the goal is to compute the frequency-dependent spin Hall conductivity \u001bsHe(!), de\fned as\njz\ny(!) =\u001bsHe(!)Ex(!); (17)\nwithjz\nythez-polarised spin current in the ydirection. The standard choice is to take the symmetrised spin current\nde\fnition (14) { other choices are possible, recall the discussion from Sec. II, and the consequences will be addressed\nbelow. The spin Hall conductivity can be written in terms of the spin current-charge current Kubo response function\nhhjz\ny;jxii!\u0011\u0000i\n~Rt\n0\u0002\njz\ny(t);jx(0)\u0003\nei!tdt, with [A;B] =AB\u0000BAthe commutator [70]. Since the charge current couples\nto the vector potential as j\u0001A, andEx(!) =\u0000i!Ax(!), one has\n\u001bsHe(!) =hhjz\ny;jxii!\ni!: (18)\nOnce the DC \u001bsHe\u0011lim!!0\u001bsHe(!) is known the spin Hall angle follows\n\u0012sHe=q\n~\u001bsHe\n\u001bcx; (19)\nwith\u001bc\nxthe longitudinal DC charge conductivity. An explicit computation yields the \\universal\" DC result\n\u001bsHe\nclean =e\n8\u0019~; (20)\nwith the electron charge q=\u0000e<0. The subscript \\clean\" highlights that the system is without any defects. Such\na beautiful result, due to the intrinsic Berry phase of electrons on the Rashba Fermi surface, is unfortunately very\nfragile. If one adds dirt to the model, i.e.a random impurity potential, HR!HR+Vimp(r), the spin Hall conductivity\nexactly vanishes\n\u001bsHe\ndirty= 0: (21)\nThe vanishing is diagrammatically subtle: since it comes from vertex corrections, it cannot be guessed by simply\nintroducing a disorder broadening of the momentum eigenstates in the Kubo response kernel [71{73]. Indeed, it was\noverlooked at \frst in the scienti\fc literature [74]. On the other hand, it is easily understood with kinetic arguments\n[75, 76], since the homogeneous continuity equation for the y-spin component reads\n@tsy=\u00002m\u000b\n~2jz\ny\n|{z}\n\u0000y\nint: (22)\nAt steady state the spin current jz\ny= 0.\nEq. (21) is as fragile as its clean counterpart (20). If one further adds extrinsic spin-orbit interaction, that is spin-orbit\ninteraction with the impurity potential, HR!HR+Vimp(r)+\u0015\u001b\u0002rVimp(r)\u0001p, the spin Hall conductivity is non-zero\n\u001bsHe\nint+ext6= 0; (23)and furthermore depends non-trivially on di\u000berent system parameters. In particular [37, 38]\n\u001bsHe\nint+ext6=\u001bsHe\nint+\u001bsHe\next: (24)\nEquivalent results would have been reached starting from the (linear) Dresselhaus model\nHD=p2\n2m+\f\n~[px\u001bx\u0000py\u001by]; (25)\nwhere the coupling constant \fis now due to bulk inversion asymmetry, i.e.the lack of inversion symmetry of the\nunderlying crystal, as in zincblend compounds [15].\nThe lesson to be learned from this example is not that low-energy e\u000bective models of Rashba or Dresselhaus type are\nunreliable { quite the contrary, they are pillars of spintronics, even if more complex models are often needed e.g.for\nprecise quantitative comparisons with experiments. It is rather that any e\u000bect crucially depending on the coupling\nbetween orbital motion and internal (spin) dynamics is subtler than standard charge-only transport phenomena, even\nwhen their description is based on the simplest models. As a corollary, attacking the problem from di\u000berent angles {\nKubo vs.kinetics in this case { can be a good idea.\nThe Rashba and Dresselhaus scenarios just considered are examples of a standard approach to transport widely em-\nployed throughout condensed matter. The latter starts from some low-energy ( k\u0001p) e\u000bective model whose parameters\ncan be computed with ab-initio methods, or left as symmetry-allowed parameters to be estimated by comparison with\nexperiments. In our case the minimal Hamiltonian for spin 1/2 quasiparticles reads\nH=H0+b(p)\u0001\u001b+\u000eH (26)\nwhereH0describes band-bottom (top) free electrons (holes), and b(p) is the e\u000bective intrinsic spin-orbit \feld, see\nEq. (10). Higher-dimensional models (4 x 4, 6 x 6 . . . ) are employed whenever more than a single s-band lie close\nto the Fermi energy, which is the case e.g.for graphene [77{80], Pt [81] or typically for holes [4, 15]. On the other\nhand there are situations in which the term b(p) is negligible, e.g.in bulk Al or Cu. The last term \u000eHcontains all\nextra ingredients needed in the speci\fc situation, e.g.exchange coupling with a magnetic texture, extrinsic spin-orbit\ncoupling, disorder, phonons and so on.\nThe e\u000bective Hamiltonian (26) is used to study non-equilibrium dynamics with whatever analytical and/or numerical\ntechniques one prefers. The approach is thus very general and \rexible. Alternatively, it is also possible to stick to\nab-initio methods and use an atomistic Hamiltonian throughout. In this case one typically relies on Kubo linear\nresponse formalism to compute the relevant transport coe\u000ecients, e.g.\u001bsHe[82, 83].\nB. Onsager reciprocity and a non-Abelian gauge \feld point of view\nThe sHe and isHe connect spin currents, even under time-reversal, with charge currents, odd under time-reversal.\nFrom the general properties of Kubo response functions [70] there follows\nhhjz\ny;jxii!=\u0000hhjx;jz\nyii!; (27)\nwhich implies\n\u001bsHe=\u0000\u001bisHe: (28)\nThis is the (linear response) Onsager relation between the sHe and isHe. It is evident that changing the de\fnition\nof the spin current changes the value of \u001bsHeand\u001bisHe. This is critical if a direct comparison with experiments is\nseeked: what spin current is being excited in the experimental setup? What spin Hall angle are we talking about? As\ndiscussed in Sec. (II) there are numerous ways to remove any ambiguity from such a comparison. In particular, if one\nis not interested in local quantities such as conductivities, the problem can be bypassed by considering conductances\nbetween metallic leads without spin-orbit interaction [32]. On the other hand a change of spin current de\fnition does\nnot break Onsager reciprocity if done consistently, i.e.if the same de\fnition is used to describe both the direct, say\nsHe, and reverse bias, say isHe, scenario.\nAnother source of concern in the early 2000s was that the standard de\fnition of spin currents, Eq. (14), may yield\nnon-vanishing equilibrium (circulating) currents [25]. Di\u000berent authors highlighted however that there is nothing\nintrinsically unphysical or surprising in this [29, 84]: spin currents are even under time-reversal, so can exist in\nequilibrium, and physical systems hosting di\u000berent kinds of equilibrium currents anyway exist [29, 30, 84, 85]. Indeed,\nadopting a non-Abelian gauge \feld point of view, equilibrium spin currents can be identi\fed with the non-Abelian\nanalogous of dissipationless Landau paramagnetic currents in solids [29].The non-Abelian gauge \feld approach requires to rewrite the spin-orbit interaction in terms of a non-Abelian vector\npotential A,i.e.a tensorAa\niwith both spin ( a) and real space ( i) indices. To be de\fnite, for the Rashba Hamiltonian\n(16) one has\nHR=(p+A)2\n2m+ const: (29)\nwithAy\nx=\u0000Ax\ny= 2m\u000b=~2, whileAa\nj= 0 for all other components. The spin current immediately follows from\nja\ni=@HR=@Aa\ni. It coincides with the standard de\fnition (14) and generally consists of both transport contributions\nand a non-dissipative equilibrium part. Pursuing this route e.g.in a di\u000busive sample, one obtains in particular a\nclear parallel between the standard Hall current jHallin presence of a magnetic \feld Band thea-polarised spin Hall\ncurrent ja\nsHein presence of a non-Abelian pseudomagnetic \feld Bgenerated by A[36]\njHall=q\u001c\nmj\u0002B!ja\nsHe=q\u001c\n4mj\u0002Ba: (30)\nThe non-Abelian gauge \feld approach is based on relatively old ideas [86{88], but was recently revived to describe\nspin-charge coupled transport in di\u000berent settings [36, 38, 89{97]. While its merits are evident, one should realise\nthat a rewriting like Eq. (29) is not always possible. I refer to the relevant literature for details.\nV. CONCLUSIONS\nThe spin Hall e\u000bects are a family of transverse transport phenomena appearing in (pseudo)spin-orbit coupled sys-\ntems. A good chunk of the theory and experimental background was established in the 1970s-1980s, but the e\u000bects\nbecame widely known in condensed matter only starting from the early 2000s, and are nowadays cornerstones of\nboth fundamental and applied spintronic research. Such a late blooming is probably due in good part to two roughly\ncontemporary events. First, the widespread realisation of the importance of geometry/topology-related concepts for\nBloch electrons. Since the latter usually require quasiparticles to have an internal structure, this strongly increased\ninterest for (pseudo)spin-orbit coupled dynamics. Second, technological advances which notably allowed the fabri-\ncation of high-quality semiconductor heterostructures, where spin manipulation became possible with a high level\nof precision, soon after followed by the discovery and functionalisation of graphene and other materials with strong\n(pseudo)spin-orbit interaction.\nIn this short overview I focused on the core, standard forms of the spin Hall e\u000bects, as they exist in normal Fermi\nliquids. In this context they are active in a wide range of parameters, from large samples at room temperature {\nimportant for potential applications { down to mesoscopic samples at low temperatures. However they may also\nappear in e.g.strongly disordered systems [98], superconductors [99], metallic antiferromagnets [100, 101], as \\valley\nHall e\u000bects\" in di\u000berent materials [102{104] or in the propagation of magnons [105, 106] and light [107]. They may also\ncontribute to other transport e\u000bects, such as the spin Hall magnetoresistance [108{110]. In short, they are potentially\npresent in any scenario where transport is due to quasiparticles with some internal structure which couples to a\nnon-trivial background.\nA. Notes on further readings\nThe literature on the spin Hall e\u000bects is vast and rami\fes quickly to neighbouring sub\felds. The bibliography given\nhere is meant to provide barebone directions to the newcomer, but is by no means exhaustive. Numerous review\narticles, each with its own qualities and shortcomings, are available to the interested reader. Refs. [3] and [4] are\nboth must-read works. Ref. [3] provides in particular a thorough historical overview and excellent phenomenological\ndiscussions, while Ref. [4] o\u000bers a high-quality and very compact introduction to the technical background, introducing\nalso modern topological concepts. I also suggest Ref. [11] for a recent, short and more application-oriented discussion,\nand Ref. [111] for its theory part. Finally, Ref. [8] provides an excellent experimental overview.VI. ACKNOWLEDGEMENTS\nI am indebted to Lin Chen for a careful reading of the manuscript, and to Leonid E. Golub for useful comments. I\nalso thank all STherQO members for discussions, in particular Guillaume Weick and R\u0013 emy Dubertrand.\n[1] M. I. Dyakonov and V. I. Perel, Current-induced spin orientation of electrons in semiconductors, Phys. Lett. A 35, 459\n(1971).\n[2] J. E. Hirsch, Spin hall e\u000bect, Phys. Rev. Lett. 83, 1834 (1999).\n[3] M. I. Dyakonov and A. V. Khaetskii, Spin hall e\u000bect, in Spin Physics in Semiconductors (Springer, 2006) p. 211.\n[4] H.-A. Engel, E. I. Rashba, and B. I. Halperin, Theory of spin hall e\u000bects in semiconductors, in\nHandbook of Magnetism and Advanced Magnetic Materials (John Wiley & Sons, 2007) p. 2858.\n[5] S. O. Valenzuela and M. Tinkham, Direct electronic measurement of the spin hall e\u000bect, Nature 442, 176 (2006).\n[6] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Comparative measurements of inverse\nspin hall e\u000bects and magnetoresistance in yig/pt and yig/ta, Phys. Rev. B 87, 174417 (2013).\n[7] M. Obstbaum, M. H artinger, H. G. Bauer, T. Meier, F. Swientek, C. H. Back, and G. Woltersdorf, Inverse spin hall e\u000bect\nin ni 81fe19/normal-metal bilayers, Phys. Rev. B 89, 060407(R) (2014).\n[8] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin hall e\u000bects, Rev. Mod. Phys. 87, 1213\n(2015).\n[9] M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).\n[10] M. K onig, S. Wiedmann, C. Br une, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum spin\nhall insulator state in hgte quantum wells, Science 318, 766 (2007).\n[11] A. Fert and F. N. V. Dau, Spintronics, from giant magnetoresistance to magnetic skyrmions and topological insulators,\nC. R. Physique 20, 817 (2019).\n[12] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall e\u000bect, Rev. Mod. Phys. 82, 1539\n(2010).\n[13] D. Culcer, E. M. Hankiewicz, G. Vignale, and R. Winkler, Side jumps in the spin hall e\u000bect: Construction of the boltzmann\ncollision integral, Phys. Rev. B 81, 125332 (2010).\n[14] Y. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984).\n[15] R. Winkler, Spin-Orbit Coupling E\u000bects in Two-Dimensional Electron and Hole Systems (Springe, 2003).\n[16] C. Gorini, U. Eckern, and R. Raimondi, Spin hall e\u000bects due to phonon skew scattering, Phys. Rev. Lett. 115, 076602\n(2015).\n[17] G. V. Karnad, C. Gorini, K. Lee, T. Schulz, R. L. Conte, A. W. J. Wells, D.-S. Han, K. Shahbazi, J.-S. Kim, T. A. Moore,\nH. J. M. Swagten, U. Eckern, R. Raimondi, and M. Kl aui, Evidence for phonon skew scattering in the spin hall e\u000bect of\nplatinum, Phys. Rev. B 97, 100405 (2018).\n[18] S. Okamoto, T. Egami, and N. Nagaosa, Critical spin \ructuation mechanism for the spin hall e\u000bect, Phys. Rev. Lett.\n123, 196603 (2019).\n[19] Y. Ohnuma, M. Matsuo, and S. Maekawa, Spin transport in half-metallic ferromagnets, Phys. Rev. B 94, 184405 (2016).\n[20] V. K. Dugaev, M. Inglot, E. Y. Sherman, and J. Barna\u0013 s, Robust impurity-scattering spin hall e\u000bect in a two-dimensional\nelectron gas, Phys. Rev. B 82, 121310(R) (2010).\n[21] E. B. Sonin, Proposal for measuring mechanically equilibrium spin currents in the rashba medium, Phys. Rev. Lett. 99,\n266602 (2007).\n[22] I. Adagideli, M. Scheid, M. Wimmer, G. E. W. Bauer, and K. Richter, Extracting current-induced spins: spin boundary\nconditions at narrow hall contacts, New J. Phys. 9, 382 (2007).\n[23] E. B. Sonin, Edge spin accumulation: Spin hall e\u000bect without bulk spin current, Phys. Rev. B 81, 113304 (2010).\n[24] C. Gorini, R. Raimondi, and P. Schwab, Onsager relations in a two-dimensional electron gas with spin-orbit coupling,\nPhys. Rev. Lett. 109, 246604 (2012).\n[25] E. I. Rashba, Spin currents in thermodynamic equilibrium: The challenge of discerning transport currents, Phys. Rev. B\n68, 241315(R) (2003).\n[26] B. K. Nikoli\u0013 c, L. P. Z^ arbo, and S. Souma, Imaging mesoscopic spin hall \row: Spatial distribution of local spin currents\nand spin densities in and out of multiterminal spin-orbit coupled semiconductor nanostructures, Phys. Rev. B 73, 075303\n(2006).\n[27] J. Shi, P. Zhang, D. Xiao, and Q. Niu, Phys. Rev. Lett. 96, 076604 (2006).\n[28] N. Sugimoto, S. Onoda, S. Murakami, and N. Nagaosa, Spin hall e\u000bect of a conserved current: Conditions for a nonzero\nspin hall current, Phys. Rev. B 73, 113305 (2006).\n[29] I. V. Tokatly, Equilibrium spin currents: Non-abelian gauge invariance and color diamagnetism in condensed matter,\nPhys. Rev. Lett. 101, 106601 (2008).\n[30] E. B. Sonin, Spin currents and spin super\ruidity, Advances in Physics 59, 181 (2010).\n[31] A. Khaetskii, Edge spin accumulation in two-dimensional electron and hole systems in a quasiballistic regime, Phys. Rev.\nB89, 195408 (2014).[32] P. Jacquod, R. Whitney, J. Meair, and M. B uttiker, Onsager relations in coupled electric, thermoelectric and spin\ntransport: The ten-fold way, Phys. Rev. B 86, 155118 (2012).\n[33] A. A. Burkov, S. N\u0013 unez, and A. H. MacDonald, Theory of spin-charge-coupled transport in a two-dimensional electron\ngas with rashba spin-orbit interactions, Phys. Rev. B 70, 155308 (2004).\n[34] A. G. Mal'shukov, L. Y. Wang, C. S. Chu, and K. A. Chao, Spin hall e\u000bect on edge magnetization and electric conductance\nof a 2d semiconductor strip, Phys. Rev. Lett. 95, 146601 (2005).\n[35] R. Raimondi, C. Gorini, P. Schwab, and M. Dzierzawa, Quasiclassical approach to the spin hall e\u000bect in the two-\ndimensional electron gas, Phys. Rev. B 74, 035340 (2006).\n[36] C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov, Non-abelian gauge \felds in the gradient expansion: Generalized\nboltzmann and eilenberger equations, Phys. Rev. B 82, 195316 (2010).\n[37] R. Raimondi and P. Schwab, Tuning the spin hall e\u000bect in a two-dimensional electron gas, EPL (Europhysics Letters)\n87, 37008 (2009).\n[38] R. Raimondi, P. Schwab, C. Gorini, and G. Vignale, Spin-orbit interaction in a two-dimensional electron gas: SU(2)\nformulation, Ann. Phys. (Berlin) 524, 153 (2012).\n[39] I. Adagideli and G. E. W. Bauer, Intrinsic spin hall edges, Phys. Rev. Lett. 95, 256602 (2005).\n[40] Y. Tserkovnyak, B. I. Halperin, A. A. Kovalev, and A. Brataas, Boundary spin hall e\u000bect in a two-dimensional semicon-\nductor system with rashba spin-orbit coupling, Phys. Rev. B 76, 085319 (2007).\n[41] A. G. Mal'shukov, L. Y. Wang, and C. S. Chu, Spin-hall interface resistance in terms of landauer-type spin dipoles, Phys.\nRev. B 78, 085315 (2007).\n[42] V. P. Amin and M. D. Stiles, Spin transport at interfaces with spin-orbit coupling: Formalism, Phys. Rev. B 94, 104419\n(2016).\n[43] S. T olle, M. Dzierzawa, U. Eckern, and C. Gorini, Quasiclassical theory of the spin{orbit magnetoresistance of three-\ndimensional rashba metals, New J. Phys. 20, 103024 (2018).\n[44] L. Zhu, L. Zhu, M. Sui, D. C. Ralph, and R. A. Buhrman, Variation of the giant intrinsic spin hall conductivity of pt\nwith carrier lifetime, Science Advances 5, eaav8025 (2019).\n[45] K. Nakagawara, S. Kasai, J. Ryu, S. Mitani, L. Liu, M. Kohda, and J. Nitta, Temperature-dependent spin hall e\u000bect\ntunneling spectroscopy in platinum, Appl. Phys. Lett. 115, 162403 (2019).\n[46] P. Li, L. J. Riddiford, C. Bi, J. J. Wisser, X.-Q. Sun, A. Vailionis, M. J. Veit, A. Altman, X. Li, M. DC, S. X. Wang,\nY. Suzuki, and S. Emori, Charge-spin interconversion in epitaxial pt probed by spin-orbit torques in a magnetic insulator,\nPhys. Rev. Materials 5, 064404 (2021).\n[47] W. Yanez, Y. Ou, R. Xiao, J. Koo, J. T. Held, S. Ghosh, J. Rable, T. Pillsbury, E. G. Delgado, K. Yang, J. Chamorro,\nA. J. Grutter, P. Quarterman, A. Richardella, A. Sengupta, T. McQueen, J. A. Borchers, K. A. Mkhoyan, B. Yan, and\nN. Samarth, Spin and charge interconversion in dirac-semimetal thin \flms, Phys. Rev. Applied 16, 054031 (2021).\n[48] I. Boventer, H. Simensen, A. Anane, M. Kl aui, A. Brataas, and R. Lebrun, Room-temperature antiferromagnetic resonance\nand inverse spin-hall voltage in canted antiferromagnets, Phys. Rev. Lett. 126, 187201 (2021).\n[49] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Current-induced switching of perpendicularly\nmagnetized magnetic layers using spin torque from the spin hall e\u000bect, Phys. Rev. Lett. 109, 096602 (2012).\n[50] C. K. Safeer, J. Ingla-Ayn\u0013 es, N. Ontoso, F. Herling, W. Yan, L. E. Hueso, and F. Casanova, Spin hall e\u000bect in bilayer\ngraphene combined with an insulator up to room temperature, Nano Lett. 20, 4573 (2020).\n[51] S. Karube, D. Sugawara, C. Tang, T. Tanabe, Y. Oyama, and J. Nitta, Enhancement of spin-charge current interconversion\nby oxidation of rhenium, J. Magn. Magn. Mater. 516, 167298 (2020).\n[52] J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W. Savero, J.-P. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M. George, L. Vila,\nand H. Ja\u000br\u0012 es, Spin pumping and inverse spin hall e\u000bect in platinum: The essential role of spin-memory loss at metallic\ninterfaces, Phys. Rev. Lett. 112, 106602 (2014).\n[53] C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Spin transfer torque devices utilizing the giant\nspin hall e\u000bect of tungsten, Appl. Phys. Lett. 101, 122404 (2012).\n[54] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Observation of the spin hall e\u000bect in semiconductors,\nScience 306, 1910 (2004).\n[55] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Experimental observation of the spin-hall e\u000bect in a two-\ndimensional spin-orbit coupled semiconductor system, Phys. Rev. Lett. 94, 047204 (2005).\n[56] L. K. Werake, B. A. Ruzicka, and H. Zhao, Observation of intrinsic inverse spin hall e\u000bect, Phys. Rev. Lett. 106, 107205\n(2011).\n[57] T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. Freimuth, A. Kronenberg, J. Henrizi,\nI. Radu, E. Beaurepaire, Y. Mokrousov, P. M. Oppeneer, M. Jourdan, G. Jakob, D. Turchinovich, L. M. Hayden,\nM. Wolf, M. M unzenberg, M. Kl aui, and T. Kampfrath, E\u000ecient metallic spintronic emitters of ultrabroadband terahertz\nradiation, Nat. Photon. 10, 483 (2016).\n[58] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Nonlocal magnetization dynamics in ferromagnetic\nheterostructures, Rev. Mod. Phys. 77, 1375 (2005).\n[59] A. Manchon, J. \u0015Zelezn\u0013 y, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Current-\ninduced spin-orbit torques in ferromagnetic and antiferromagnetic systems, Rev. Mod. Phys. 91, 035004 (2019).\n[60] C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and J. B. Youssef, Phys. Rev. Lett. 111, 217204 (2013).\n[61] D. Wei, M. Obstbaum, M. Ribow, C. H. Back, and G. Woltersdorf, Spin hall voltages from a.c. and d.c. spin currents,\nNat. Comms. 5, 3768 (2014).[62] M. Weiler, J. M. Shaw, H. T. Nembach, and T. J. Silva, Phase-sensitive detection of spin pumping via the ac inverse spin\nhall e\u000bect, Phys. Rev. Lett. 113, 157204 (2014).\n[63] S. Takahashi and S. Maekawa, Nonlocal spin hall e\u000bect and spin{orbit interaction in nonmagnetic metals, J. Magn. Magn.\nMater. 310, 2067 (2007).\n[64] H. Jiao and G. E. W. Bauer, Spin back\row and ac voltage generation by spin pumping and the inverse spin hall e\u000bect,\nPhys. Rev. Lett. 110, 217602 (2013).\n[65] C. Br une, A. Roth, E. G. Novik, M. K onig, H. Buhmann, E. M. Hankiewicz, W. Hanke, J. Sinova, and L. W. Molenkamp,\nEvidence for the ballistic intrinsic spin hall e\u000bect in hgte nanostructures, Nature Phys. 6, 448 (2010).\n[66] S. D. Ganichev, E. L. Ivchenko, V. V. Bel'kov, S. A. Tarasenko, M. Sollinger, D. Weiss, W. Wegscheider, and W. Prettl,\nSpin-galvanic e\u000bect, Nature 417, 153 (2002).\n[67] S. D. Ganichev, M. Trushin, and J. Schliemann, Spin polarisation by current, arXiv , 1606.02043.\n[68] K. Shen, R. Raimondi, and G. Vignale, Microscopic theory of the inverse edelstein e\u000bect, Phys. Rev. Lett. 112, 096601\n(2014).\n[69] C. Gorini, A. M. Sheikhabad, K. Shen, I. V. Tokatly, G. Vignale, and R. Raimondi, Theory of current-induced spin\npolarization in an electron gas, Phys. Rev. B 95, 205424 (2017).\n[70] L. D. Landau, E. M. Lifschitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Elsevier, 2008).\n[71] P. Schwab and R. Raimondi, Magnetoconductance of a two-dimensional metal in the presence of spin-orbit coupling, Eur.\nPhys. J. B 25, 483 (2002).\n[72] J.-I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Suppression of the persistent spin hall current by defect scattering,\nPhys. Rev. B 70, 041303(R) (2004).\n[73] R. Raimondi and P. Schwab, Spin-hall e\u000bect in a disordered two-dimensional electron system, Phys. Rev. B 71, 033311\n(2005).\n[74] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Universal intrinsic spin hall e\u000bect,\nPhys. Rev. Lett. 92, 126603 (2004).\n[75] O. V. Dimitrova, Spin-hall conductivity in a two-dimensional rashba electron gas, Phys. Rev. B 71, 245327 (2005).\n[76] O. Chalaev and D. Loss, Spin-hall conductivity due to rashba spin-orbit interaction in disordered systems, Phys. Rev. B\n71, 245318 (2005).\n[77] A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev.\nMod. Phys. 81, 109 (2009).\n[78] D. Kochan, S. Irmer, and J. Fabian, Model spin-orbit coupling hamiltonians for graphene systems, Phys. Rev. B 95,\n165415 (2017).\n[79] A. Dyrda l and J. Barna\u0013 s, Spin hall e\u000bect in graphene due to random rashba \feld, Phys. Rev. B 86, 161401(R) (2012).\n[80] M. Milletar\u0013 \u0010, M. O\u000edani, A. Ferreira, and R. Raimondi, Covariant conservation laws and the spin hall e\u000bect in dirac-\nrashba systems, Phys. Rev. Lett. 119, 246801 (2017).\n[81] G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa, Intrinsic spin hall e\u000bect in platinum: First-principles calculations,\nPhys. Rev. Lett. 100, 096401 (2008).\n[82] S. Lowitzer, M. Gradhand, D. K odderitzsch, D. V. Fedorov, I. Mertig, and H. Ebert, Extrinsic and intrinsic contributions\nto the spin hall e\u000bect of alloys, Phys. Rev. Lett. 106, 056601 (2011).\n[83] D. K odderitzsch, K. Chadova, and H. Ebert, Linear response kubo-bastin formalism with application to the anomalous\nand spin hall e\u000bects: A \frst-principles approach, Phys. Rev. B 92, 184415 (2015).\n[84] E. B. Sonin, Phys. Rev. B 76, 033306 (2007).\n[85] J. Heurich, J. K onig, and A. H. MacDonald, Persistent spin currents in helimagnets, Phys. Rev. B 68, 064406 (2003).\n[86] H. Mathur and A. D. Stone, Quantum transport and the electronic aharonov-casher e\u000bect, Phys. Rev. Lett. 68, 2964\n(1992).\n[87] J. Fr ohlich and U. M. Studer, Gauge invariance and current algebra in nonrelativistic many-body theory, Rev. Mod. Phys.\n65, 733 (1993).\n[88] P. W. Brouwer, J. N. H. J. Cremers, and B. I. Halperin, Weak localization and conductance \ructuations of a chaotic\nquantum dot with tunable spin-orbit coupling, Phys. Rev. B 65, 081302(R) (2002).\n[89] I. Adagideli, V. Lutsker, M. Scheid, P. Jacquod, and K. Richter, Spin transistor action from hidden onsager reciprocity,\nPhys. Rev. Lett. 108, 236601 (2012).\n[90] F. S. Bergeret and I. V. Tokatly, Singlet-triplet conversion and the long-range proximity e\u000bect in superconductor-\nferromagnet structures with generic spin dependent \felds, Phys. Rev. Lett. 110, 117003 (2013).\n[91] I. V. Tokatly and E. Y. Sherman, Spin evolution of cold atomic gases in su(2)\nu(1) \felds, Phys. Rev. A 93, 063635\n(2016).\n[92] S. T olle, U. Eckern, and C. Gorini, Spin-charge coupled dynamics driven by a time-dependent magnetization, Phys. Rev.\nB95, 115404 (2017).\n[93] I. V. Bobkova and A. M. Bobkov, Quasiclassical theory of magnetoelectric e\u000bects in superconducting heterostructures in\nthe presence of spin-orbit coupling, Phys. Rev. B 95, 184518 (2017).\n[94] S. H. Jacobsen and J. Linder, Quantum kinetic equations and anomalous nonequilibrium cooper-pair spin accumulation\nin rashba wires with zeeman splitting, Phys. Rev. B 96, 134513 (2017).\n[95] A. G. Mal'shukov, Supercurrent generation by spin injection in an s-wave superconductor{rashba metal bilayer, Phys.\nRev. B 95, 064517 (2017).\n[96] F. Aikebaier, P. Virtanen, and T. Heikkil a, Superconductivity near a magnetic domain wall, Phys. Rev. B 99, 104504\n(2019).[97] E. J. K onig and A. Levchenko, Quantum kinetics of anomalous and nonlinear hall e\u000bects in topological semimetals, Ann.\nPhys. (New York) 435, 168492 (2021).\n[98] D. S. Smirnov and L. E. Golub, Electrical spin orientation, spin-galvanic, and spin-hall e\u000bects in disordered two-\ndimensional systems, Phys. Rev. Lett. 118, 116801 (2017).\n[99] S. Takahashi and S. Maekawa, Spin hall e\u000bect in superconductors, Jpn. J. Appl. Phys. 51, 010110 (2011).\n[100] W. Zhang, M. B. Jung\reisch, W. Jiang, J. E. Pearson, A. Ho\u000bmann, F. Freimuth, and A. Mokrousov, Spin hall e\u000bects in\nmetallic antiferromagnets, Phys. Rev. Lett. 113, 196602 (2014).\n[101] S. A. Gulbrandsen, C. Espedal, and A. Brataas, Spin hall e\u000bect in antiferromagnets, Phys. Rev. B 101, 184411 (2020).\n[102] R. V. Gorbachev, J. C. W. Song, G. L. Yu, A. V. Kretinin, F. Withers, Y. Cao, A. Mishchenko, I. V. Grigorieva, K. S.\nNovoselov, L. S. Levitov, and A. K. Geim, Detecting topological currents in graphene superlattices, Science 346, 448\n(2014).\n[103] K. F. Mak, K. L. McGill, J. Parkand, and P. L. McEuen, The valley hall e\u000bect in mos 2transistors, Science 344, 1489\n(2014).\n[104] Y. D. Lensky, J. C. Song, P. Samutpraphoot, and L. S. Levitov, Topological valley currents in gapped dirac materials,\nPhys. Rev. Lett. 114, 256601 (2015).\n[105] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Observation of the magnon hall e\u000bect, Science\n329, 297 (2010).\n[106] A. Mook, J. Henk, and I. Mertig, Magnon hall e\u000bect and topology in kagome lattices: A theoretical investigation, Phys.\nRev. B 89, 134409 (2014).\n[107] O. Hosten and P. Kwiat, Observation of the spin hall e\u000bect of light via weak measurements, Science 319, 787 (2008).\n[108] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. B. T. Goennenwein, E. Saitoh, and G. E. W. Bauer, Theory\nof spin hall magnetoresistance (smr) and related phenomena, J. Phys.: Condens. Matter 28, 103004 (2016).\n[109] K. Oyanagi, J. M. Gomez-Perez, X.-P. Zhang, T. Kikkawa, Y. Chen, E. Sagasta, A. Chuvilin, L. E. Hueso, V. N. Golovach,\nF. S. Bergeret, F. Casanova, and E. Saitoh, Paramagnetic spin hall magnetoresistance, Phys. Rev. B 104, 134428 (2021).\n[110] L. Chen, F. Matsukura, and H. Ohno, Direct-current voltages in (ga,mn)as structures induced by ferromagnetic resonance,\nNat. Commun. 4, 2055 (2013).\n[111] G. Vignale, Ten years of spin hall e\u000bect, J. Supercond. Nov. Magn. 23, 3 (2010)." }, { "title": "0708.2565v2.Coupling_of_electron_rotation_with_spin_in_semiconductors.pdf", "content": " 1 \nCoupling of electron spin with its rotation in semi conductors. \n \nYuri A. Serebrennikov \nQubit Technology Center \n2152 Merokee Dr., Merrick, NY 11566 \nThe interplay between spin-orbit interaction in sem iconductor valence bands and \nan adiabatic rotational distortion of the wave func tion of a charge carrier leads to the \nscalar spin-orbit-rotation term in the effective-ma ss Hamiltonian of the conduction-band \nelectron. The physical origin of this result lies i n the fact that, similarly to magnetic field \neffects, the motion of a particle and the phase of its wave function may be affected by the \nvector potential of the inertial Coriolis field. He re we present a straightforward derivation \nof this interaction within the multiband envelope f unction approximation. \n72.80.Ey 72.25.Dc 72.25.Hg \n \nIn general, the coupling of the total angular momen tum of a particle with its \nrotation yields “forces” of rotational inertia that work on a quantum level 1. It is well \nknown that the quantum mechanical description of th e motion of a spinless particle in the \nnon-inertial rotating ( R) frame to first order in the angular velocity ω is formally \nidentical to the account of its motion in the prese nce of a weak magnetic field 2 3. To see \nthis one should replace the vector potential of a m agnetic field 2 /rBArrr\n×= with \nωAqcmr\n)/2 (0, where 2 /r Arrr\n×=ωω is the vector potential of the Coriolis field 4 in the \ncorresponding wave equation ( c is the speed of light, q is the charge, and 0m is the mass \nof a particle). This is also apparent if we compare the expression for the canonical 2momentum of a charged particle in the presence of m agnetic field Acqvmpr rr)/(0+= \nwith the corresponding expression in the R-frame ωAmvmprrr\n0 02+= at zero magnetic \nfields, 0=B. \nFor spin bearing particles, the total angular momen tum is the sum of the orbital \nand spin contributions. The non-relativistic R-frame Hamiltonian of a free spin-1/2 \nparticle at zero magnetic fields can be written as 5 \nj mpHRrrh⋅−= ωω 02)(2 / , (1) \nwhere pris the canonical momentum, 2 /σrrr\n+=lj is the total angular momentum, \nprlrrr\nh×= is the orbital momentum, and σr is the 3-vector of Pauli matrices. Notably this \nHamiltonian incorporates the Mashhoon spin-rotation interaction, Srrh⋅−ω, even in the \nabsence of relativistic spin-orbit (SO) coupling. A strong electric field near heavy nuclei \n(Ge, Ga, In, etc.) in common semiconductors leads, however, to a strong intrinsic SO \ninteraction in the valence bands. The interplay bet ween this interaction and an adiabatic \nrotational distortion of the wave function of a cha rge carrier yields the scalar spin-orbit-\nrotation (SOR) coupling term in the effective-mass Hamiltonian of the conduction-band \nelectron. In our recent article 6 we demonstrated that the SOR coupling can be descr ibed \nin purely geometric terms as a consequence of the d ifference in the Berry phase acquired \nby the components of the spin-orbitally mixed Krame rs-doublet during its cyclic \nevolution in the reciprocal momentum space. The phy sical origin of this result lies in the \nfact that, similarly to magnetic field effects, the motion of a particle and the phase of its \nwave function may be affected by the vector potenti al of the inertial Coriolis field. Here 3we present a straightforward derivation of the SOR interaction within the multiband \nenvelope function (EF) approximation 7. \nThe energy band structure of common semiconductors near the center of the first \nBrillouin zone can be well described within the mul tiband EF approximation, which \ngives the following second-order Hamiltonian 8 of a Kramers-degenerate conduction band \nin the absence of external electric and magnetic fi elds \n∑\n′′−′−′+−=\nqqqqqqqq L\nkkDk H\n,1 1)() 1( 2t\n. (2) \nHere 0 , 1,±=′qq and zkk=10, 2/ ) (11 y xikk k ±=±m represent cyclic components of the \ncrystal momentum kr\n, and the superscript ( L) denotes the laboratory L-frame. The dyadic \noperator qqD′t\n acting on the EF-spinors ( slow variables ) is defined by its matrix elements \nin the basis of band-edge Bloch functions ( fast variables ) \n∑−> ><′<− −>=′<′\n′−′ ′\nn nnq q\neqqnn\neq\nqqnpnnpn\nm mnDn\n~ ~1 1\n22 2||~~||\n2) 1(||εεδδh h t\n, (3) \nwhere nεis the energy at the bottom of the n-th band and em is the bare electron mass. In \nthe presence of a k-independent intrinsic SO coupling the Bloch functions >n| are not \nfactorizable into the orbital and spin parts, hence , the total angular momentum, SLJrrr\n+=, \nis required to characterize the basis kets. Within the “spherical approximation” 9, it is \nconvenient to build the corresponding basis from th e spherical spinor functions of the \ncompound L-S system 10 >> =>= ≡> ∑ 1\n,2 / 1 2 / 1 || 0;,||\n11µµ\nµµµµL C kJmLSnJm\nLr\n, \nwhere Jm\nLC\n12 / 1µµis the Clebsch-Gordan coefficient. The matrix eleme nts of the direct \ntensor product qqpp′⋅11 in this basis are well known 10 , which allows to calculate the 4second term in Eq.(3). Within Kane’s eight-band mod el, which takes into account only \nthe coupling of the conduction band ( L = 0, J = ½) to the valence bands ( L = 1, J = 3/2 \nand L = 1, J = ½), after some straightforward algebra, one find s \n∑ ∑\n=−\n′Κ′′\n+− +−+ >==== ===<\n2 / 3 , 2 / 11\n113\n11112 / 1\n2 / 12 / 1 222\n)(\n1\n111 112\n112 / 12 / 1) 12 () 1( ) 2 / 1() 1(22 / 1, 2 / 1, 0||2 / 1, 2 / 1, 0\nJJJ\nqqQqqKQ\nqqm\nKQmKmm\neL\nk\nJKJ kkCC K iPmkmJ SLHmJ SL\nεδh\n \n (4) \nHere )3/(1||||01 em LpLiP >= =<=h is the reduced Kane matrix element, which \ndescribes the coupling of the conduction and valenc e bands, and we set the energy of a \nconduction band to zero. Due to the triangle condit ion for the arguments {½, ½, K} of the \n6-j symbol and Clebsch-Gordan coefficients, only scala r ( K = 0) and vector ( K = 1) terms \nare present in Eq.(4). The sum of the scalar terms on the RHS of Eq.(4) yields the Kane’s \nexpression for the reciprocal effective mass of a c onduction electron \n])(2 [)3 / 2 ( / 1/ 11 1 22 * − −∆++ +=g g e EEP mm h , (5) \nwhereas the vector term gives 3/ ][) 1(])([2 / 1\n12 / 1 11 12∑′\n−− −×−∆+−−\nQm\nQm QQ\ng g Ckk EEiPrr\n. \nHere gE is the band-gap energy and ∆ is the splitting of the valence band determined by \nthe intrinsic SOC. Thus, Eq.(4) yields the followin g effective-mass Hamiltonian of the \nconduction band electron \nσrrr h⋅×∆+∆−= ][)(3ˆ\n22\n*22\n)(kkEEiPI\nmkH\nggL\neff , (6) \nwhere Iˆ is the 2 x 2 unit matrix, and σr is acting on the spinor components of the \nconduction-band EF. 5 The Hamiltonian Eq.(6) represents the model where t he crystal electron is \ntreated as free spin-bearing particle moving in an effective magnetic field kkBeffrrr\n×~ . \nThis field is zero for pure translational motion of a conduction electron and may not \nvanish only if its wave-vector is changing the direc tion in space. Suppose that the crystal \nmomentum of a particle is rotating. The infinitesim al change in the direction of kr\n can be \ndescribed geometrically as ] ˆ[)()( )( L L Lkn kr r\n×=φδδ , where δφ is the angle between \nkr\nand kkrr\nδ+ and )(ˆLn is the unit vector along the instantaneous axis of kr\n-rotation at \ninstance t. In what follows, we choose the axis )(ˆLn to be at the right angle to the plane of \nthe kr\n-rotation. Then 2)( )()(/ ˆ kkknL L Lrr\nδ φδ ×= and one may define the local \ninstantaneous angular velocity of this rotation as 2)()()( )(/ ˆ)/( kkkndtdL L L L\nk&rr r×= =φω . \nWe would like to emphasize here that this expression is purely kinematical, i.e., is \nindependent of the dynamical cause of the kr\n-rotation. \nThe time dependence of kr\n and, hence, the L-frame Hamiltonian Eq.(6), \ncomplicates the mathematical analysis of the proble m. On the other hand, it is physically \nclear that in the reference frame that follows the r otation of kr\n the Hamiltonian of the \nsystem will be time-independent. The unitary transfo rmation ) ()()()( )( )(ttRtL L RΨ=Ψ , \nwhere Ψ is the instantaneous EF spinor, into the R-frame carried along by the \ninfinitesimal rotation of kr\n at the point rr and time t yields the following effective \nHamiltonian j HHkR\neffR\neffrrh⋅−=ω)( )(~, where 1)( )( −=RRHHL\neffR\neff . We recall now that in the \nR-frame the Coriolis vector potential couples to the kinetic momentum of a particle, \nkAm ikeR R\nωr r\nhr\nh 2)( )(−∇−= , and it is easy to see that commutators of its com ponents do 6not vanish, )( )()/2 (][R\nk eRim kk ωrhrr\n=× . Then it follows from Eq.(6) that the R-frame \neffective-mass Hamiltonian of the conduction-band e lectron can be expressed as (to first \norder in kω11 ) \nSOR kR\neff HS I\nmkH +⋅−=rrhhωˆ\n2~\n*22\n)( (7) \nSg Hk SORrrh⋅∆−=ω, (8) \n)](3/[42 2∆+ ∆−=∆gg e EE mPg h . (9) \nRemarkably, the expression for g∆, Eq.(9), coincides with the Roth formula 12 for the \ndeviation of the g-factor of a conduction electron in bulk semiconduc tors from its free \nvalue. The Hamiltonian Eq.(7) depends on the choice of guide that specifies the reference \norientation, i.e. the orientation in which the R-frame coincides with some space-fixed \nframe. At the moment t = 0, the reference orientation may always be chosen such that Mz \ncoincides with Lz, which determines our gauge convention. Note that i f the rotation is \nuniform, the operator ) exp( t ji Rkrrω= maps the L-frame into the actual orientation of the \nR-frame at any time t. \nComparison of Eq.(7) with Eq.(1) shows that the accou nt of SO coupling in the \nsystem yields in addition to Mashhoon spin-rotation interaction in the reciprocal \nmomentum space, Skrrhω−, the term SORH, Eq.(8). It corresponds to the SOR-coupling in \nthe k-space and is the same from the point of view of a r otating as well as inertial \nobservers. Indeed, to describe the evolution of the EF in the local inertial frame we have \nto perform a reverse rotation of the coordinate sys tem compensating for the rotation of \nthe R-frame. This transformation is not associated with a physical change of a state and 7does not affect the isotropic kinetic energy of the carrier. At any moment at time, it is \nmerely the operator of a reverse rotation in the 2x 2 spinor-space, which yields \nSORL\neff HI\nmkH +=ˆ\n2*22\n)(h\n (10) \nThe SOR Hamiltonian Eq.(8) represents the weak-SO-lim it of the effective Hamiltonian \nobtained in Ref.[6] by a more general method. Notab ly, SORH is analogous to the usual \nspin-rotation interaction in molecular systems. The analogy becomes exact if one replace \nkω with the angular velocity of a molecular frame in t he real space. Formally, the same \nspin-Hamiltonian governs the evolution of the Kramers -doublet in an adiabatically \nrevolving external electric field in the limit of a weak SO in teraction in the valence \nbands 13 . Fundamentally, these very diverse physical phenom ena can be described \nuniformly in purely geometric terms as a consequenc e of the difference in the Berry \nphase acquired by the components of the spin-orbita lly mixed Kramers-doublet during its \ncyclic evolution in the relevant parameter space. T he geometric interpretation of the SOR \ninteraction in molecular systems has been given in Ref.[ 14 ] and was extended to \nsemiconductors in Refs.[6, 13]. The Berry phase eff ects in semiconductors emerging \nfrom the SO coupling have been proposed 15 to occur as early as in 1993. Now it is well \nrecognized that an adiabatic change in the directio n of the wave vector of a charge carrier \nleads to non-trivial gauge potentials that appear i n the reciprocal momentum space. The \nassociated covariant gauge field enters the equatio n of motion for the group velocity of a \nwave-packet and may affect the coherent transport pr operties of charge carriers 16 17 18 19 \n20 21 . However, until recently no connection was made betwee n the results of these studies \nand manifestation of SOR interaction in semiconduct ors. 8The straightforward derivation presented here stren gthens the foundation of the \nanalysis made in Ref.[6], which we would like to brief ly summarize here. During the \n“slow” adiabatic ( | |||g kE<<ωrh ) rotation of the envelope wave function the spin of a \nconduction-band electron will follow the rotation of the EF wave-vector with some \nslippage determined by g∆. In the L-frame, this process can be described as a spin \nprecession in an effective magnetic field )/()/( )/(2kkkg g BB k B eff&rr\nhrhr\n×∆−=∆−= µωµ . \nThis field is orthogonal to the plane of particle’s rotation, is related neither to Rashba nor \nto Dresselhaus couplings between the spin of the char ge carrier and its momentum, and \nmay appear in spherically symmetric bulk crystals. Dynamic anisotropy of a system \nlocally in the k-space ( 0≠P) is the fundamental precondition for manifestation of this \ninteraction in the conduction band. Although the SOR interaction in the conduction band \nappears already in the second-order, the small pref actor, sec 10/7⋅≅=GBµh , makes \ndirect gyromagnetic experiment of Barnett- or mecha nical Faraday-type 22 a rather \nchallenging task. On the other hand, an electron can be forced to rotate rapidly by \nexternal and/or internal electric fields. Whereas t he latter, e.g., through collisions with \ncrystal impurities will lead to Elliott spin dephasi ng 23 24 , the former may be used to \ncontrol the spin splitting and precession at B = 0. It is easy to see that if the uniform \nelectric field is the sole source of the electron’s acceleration perpendicular to its \ninstantaneous velocity, then SkEekg HSORrrr\n)/(2×∆= and the magnitude of the SOR \ncoupling can be comparable or larger than Rashba or Dresselhaus interactions 7, e.g., for \n1810−=mk and mVE /105= , meVg HSOR∆= . \n 9 \n1 B. Mashhoon and H. Kaiser, Physica B 385 , 1381 (2006). \n2 S. A. Werner, J.-L. Staudenmann, and R. Colella, P hys. Rev. Lett. 42 , 1103 (1979); D. \nK. Atwood, M. A. Horne, C. G. Shull, and J. Arthur, Phys. Rev. Lett. 52 , 1673 (1984); F. \nHasselbach and M. Nicklaus, Phys. Rev. A 48 , 143 (1993); R. Nutze and F. Hasselbach, \nPhys. Rev. A 58 , 557 (1998). \n3 L. A. Page, Phys. Rev. Lett. 35 , 543 (1975); J. Anandan, Phys. Rev. D 15 , 1448 (1977); \nJ. J. Sakurai, Phys. Rev. D 21 , 2993 (1980). \n4 Y. Aharonov and G. Carmi, Found. Phys. 3, 493 (1973). \n5 B. Mashhoon, Phys. Rev. Lett. 61 , 2639 (1988). \n6 Yu. A. Serebrennikov, Phys. Rev. B 73 , 195317 (2006); Yu. A. Serebrennikov, Phys. \nRev. B 74 , 049901 (E) (2006). \n7 R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electr on and Hole \nSystem (Springer, New York, 2003). \n8 J. M. Luttinger and W. Kohn, Phys. Rev. 97 , 869 (1955). \n9 A. Baldereschi N. and O. Lipari, Phys. Rev. B. 8, 2697 (1973). \n10 D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonsky , Quantum Theory of \nAngular Momentum , (World Scientific, Singapore, 1989) \n11 One may ignore the centrifugal potential energy of a particle in comparison to its \nkinetic energy. Note that the orbit-rotation coupli ng, lkrrhω−, is “hidden” in the kinetic \nenergy of an electron in the R-frame. \n12 L. M. Roth, B. Lax, and S. Zwerdling, Phys. Rev. 114 , 90 (1959). \n13 Yu. A. Serebrennikov, Phys. Rev. B. 70 , 064422 (2004). 10 \n14 U. E. Steiner and Yu. A. Serebrennikov, J. Chem. Phy s. 100 , 7503 (1994); Yu. A. \nSerebrennikov and U. E. Steiner, Mol. Phys. 84 , 627 (1995). \n15 A. G. Aronov and Y. B. Lyanda-Geller, Phys. Rev. Lett . 70 , 343 (1993); A. V. \nBalatsky and B. L. Altshuler, Phys. Rev. Lett. 70 , 1678 (1993) \n16 D. P. Arovas and Y. B. Lyanda-Geller, Phys. Rev. B 57 , 12302 (1998); G. Sundaram \nand Q. Niu, Phys. Rev. B 59 , 14915 (1999); R. Shindou and K. Imura, Nucl. Phys. B. \n720 , 399 (2005). \n17 S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301 , 1348 (2003). \n18 F. Zhou, Phys. Rev. B 70 , 125321 (2004). \n19 Z. F. Jiang, R. D. Li, S.-C. Zhang, and W. M. Liu, Phys. Rev. B 72 , 045201 (2005). \n20 D. Culcer, Y. Yao, and Q. Niu, Phys. Rev. B 72 , 085110 (2005). \n21 S.-Q. Shen, Phys. Rev. Lett. 95 , 187203 (2005). \n22 L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii , Electrodynamics of Continuous \nMedia , 2nd edition (Elsevier, 2006). \n23 R. J. Elliott, Phys. Rev. 96 , 266 (1954). \n24 Yu. A. Serebrennikov, Phys. Rev. Lett. 93 , 266601 (2004). " }, { "title": "1004.1250v1.Coupling_of_bonding_and_antibonding_electron_orbitals_in_double_quantum_dots_by_spin_orbit_interaction.pdf", "content": "arXiv:1004.1250v1 [cond-mat.mes-hall] 8 Apr 2010Coupling of bonding and antibonding electron orbitals in do uble quantum dots by\nspin-orbit interaction\nM.P. Nowak and B. Szafran\nFaculty of Physics and Applied Computer Science,\nAGH University of Science and Technology,\nal. Mickiewicza 30, 30-059 Krak´ ow, Poland\n(Dated: October 29, 2018)\nWe perform a systematic exact diagonalization study of spin -orbit coupling effects for stationary\nfew-electron states confined in quasi two-dimensional doub le quantum dots. We describe the spin-\norbit-interaction induced coupling between bonding and an tibonding orbitals and its consequences\nfor magneto-optical absorption spectrum. The spin-orbit c oupling for odd electron numbers (one,\nthree) opens avoided crossings between low energy excited l evels of opposite spin orientation and\nopposite spatial parity. For two-electrons the spin-orbit coupling allows for low-energy optical tran-\nsitions that are otherwise forbidden by spin and parity sele ction rules. We demonstrate that the\nenergies of optical transitions can be significantly increa sed by an in-plane electric field but only for\nodd electron numbers. Occupation of single-electron orbit als and effects of spin-orbit coupling on\nelectron distribution between the dots are also discussed.\nPACS numbers: 73.21.La\nI. INTRODUCTION\nIn a pair of quantum dots1–4defined in semiconduct-\ning medium the chargecarriersform extended wavefunc-\ntions when their tunneling through the interdot barrier\nbecomes effective enough. In vertically stacked quan-\ntum dots the extended electron and hole orbitals are\nprobed by photoluminescence experiments in external\nelectric field.4The electron single-dot orbitals hybridize\ntobondinggroundstatessimilartotheonesfound innat-\nural covalent molecules. Recent studies5indicated that\nthe hole in artificial molecules of self-assembled quantum\ndots behaves in a different manner forming an antibond-\ning ground-state orbital. This peculiar behavior results5\nof the spin-orbit (SO) coupling induced mixing of light\nand heavy hole states.\nIn the present paper we study the mixing of bonding\nand antibonding electron orbitals that is induced by SO\ninteraction in planar systems of laterally coupled quan-\ntum dots. The coupling between spatial and spin elec-\ntrondegreesoffreedomresultsfrominversionasymmetry\nof the structure6and/or the crystal lattice.7This asym-\nmetry enters into the two-dimensional SO Hamiltonian\nwhich does not conserve the spatial parity and couples\nthe electron spin-up bonding orbitals with spin-down an-\ntibonding orbitals. In order to indicate experimentally\naccessible consequences of this coupling we consider op-\ntical absorption spectra in the external magnetic field for\nuptothreeconfinedelectrons. Inparabolicquantumdots\nthe spin-orbit coupling introduces a distinct dependence\nof the far infrared magneto-optical absorption spectra on\nthe number of confined electrons.8We find that the SO\ninduced modification to the absorption spectra of double\ndots are qualitatively different for even and odd electron\nnumbers.\nLaterallycoupledquantumdots9,10areconsideredcan-\ndidates forrealizationofaquantum gateworkingon elec-tronspins3sincetheheight/widthoftheinterdotbarrier\ncan be tuned by external voltages which is essential for\nthe control of the spin exchange between the electrons\nconfined in adjacent dots. The idea of the spin exchange\nmotivated a number of theoretical investigations on the\nproperties of electron systems in laterally coupled quan-\ntum dots.11–20\nThe SO interaction is one of the issues that are investi-\ngated in the context of spin-based quantum information\nprocessing.18–24,27–35TheSOcouplingallowsforspinma-\nnipulation by the spatial electron motion.32–35Moreover,\nit leads to the spin relaxation20–24mediated by phonons,\nleading to information decay and decoherence. Singlet-\ntriplet induced avoided crossing of two-electron energy\nlevels were observed in electron-transport spectroscopies\nfor gated InAs nanowire quantum dots25as well as for\ndouble InAs quantum dots.26The exchange interaction\nbetween electrons confined in separate dots was found to\ncontain an anisotropic component originating from the\nSO coupling,27which initially motivated a quest for spin\nprocessing procedures28,29minimizing its effects. Later\non, proposals of using the asymmetry of the exchange\ninteraction for construction of universal quantum gates\nthat could work without single spin operations30,31were\nformulated. Recently, it was demonstrated by the exact\ndiagonalization that18the anisotropy of the exchange in-\nteraction – previously discussed within approximate ap-\nproaches– is in fact absent in zero magnetic field. There-\nfore, the treatment of spin-orbit coupling effects for dou-\nbledotsrequiresanexactdiagonalizationapproachwhich\nwe employ below. The SO coupled double quantum dots\nwere so far studied by the exact diagonalization in Ref.\n[36], which provides a detailed analysis of single-electron\nstates and in Refs. [18,37] which deal with the electron\npair in the context of the exchange interaction.2\nII. THEORY\nWe consider an effective mass single-electron Hamilto-\nnian of the form:\nh=/parenleftbiggp2\n2m∗+W(r)/parenrightbigg\n1\n+1\n2gµBBσz+HSIA+HBIA, (1)\nwherep= ¯hk=−i¯h∇+eA,1is the identity ma-\ntrix,W(r) stands for the potential, HSIAandHBIA\nintroduce Rashba6(structure inversion asymmetry) and\nDresselhaus7(bulk inversion asymmetry) spin-orbit in-\nteractions. The vector potential is taken in the symmet-\nric gaugeA=B\n2(−y,x,0). The Rashba and Dresselhaus\nSO interactions have the form\nHSIA=α∇W·(σ×k), (2)\nand\nHBIA=γ/bracketleftbig\nσxkx(k2\nz−k2\ny)+σyky(k2\nx−k2\nz)\n+σzkz(k2\ny−k2\nx)/bracketrightbig\n, (3)\nrespectively. In Eqs. (2) and (3) αandγare bulk SO\ncouplingconstants, σ’sarePaulimatricesand x,y,zaxes\nare oriented parallel to [100], [010] and [001] (growth)\ncrystal directions, respectively.\nWe assume that the confinement potential forming the\nquantum dot is separable into an in-plane Vc(x,y) and a\ngrowth direction Vz(z) components so that the potential\nappearing in the Hamiltonian (1) is\nW(r) =Vc(x,y)+Vz(z)+|e|F·r, (4)\nwhereFis the electric field vector (below we always take\nFy= 0). In the followingwe adopta two-dimensionalap-\nproximation assuming that the electrons occupy a frozen\nlowest-energy state of quantization in the growth direc-\ntion. The two-dimensional SO terms are obtained by\naveragingHSIAandHBIAover the wave function de-\nscribing the electron localization in the growth direction.\nThe two-dimensional Rashba terms are usually37sepa-\nrated into a diagonal\nHdiag\nSIA=ασz/parenleftbigg/bracketleftbigg∂V\n∂y/bracketrightbigg\nkx−/bracketleftbigg∂V\n∂x+|e|Fx/bracketrightbigg\nky/parenrightbigg\n,(5)\nand linear\nHlin\nSIA=α/angb∇acketleft∂W\n∂z/angb∇acket∇ight(σxky−σykx), (6)\nparts. In this formula the average gradient of the po-\ntential calculated for the wave function in the growth\ndirection can be attributed to an effective zcomponent\nof the electric field Fz=1\n|e|/angb∇acketleft∂W\n∂z/angb∇acket∇ight. The two-dimensional\nDresselhaus SO interaction contains the linear\nHlin\nBIA=γ/angb∇acketleftk2\nz/angb∇acket∇ight[σxkx−σyky], (7)and the cubic\nHcub\nBIA=γ/bracketleftbig\nσykyk2\nx−σxkxk2\ny/bracketrightbig\n(8)\nterms. We assume that the quantum dot is made of\nIn0.5Ga0.5As alloy for which we adopt the SO coupling\nconstantsα= 0.572 nm2(after Ref. 40) and γ= 32.2\nmeVnm3(after Ref. 38). The other material parameters\nare taken as arithmetic average41of InAs and GaAs, i.e.\nwe use the electron effective mass m∗= 0.0465m0, Land´ e\nfactorg=−8.97 and dielectric constant ǫ= 13.55. The\nconsidered large value of the gfactor is in the order of\nthe one found for in experimental samples25,26in which\nthe SO coupling effects were studied.\nFor the electron wave function in the growth direction\nidentified with the ground-state of an infinite rectangular\npotentialwellofheight doneobtainsthetwo-dimensional\nlinear Dresselhaus constant γ2D=γ/angb∇acketleftk2\nz/angb∇acket∇ight=γπ2\nd2[see Eq.\n(7)]. In the bulk of our calculations we assume a minimal\nbut still realistic value of d= 5.42 nm, for which γ2D=\n10.8 meVnm.\nThe in-plane confinement potential is taken in form\nVc(x,y) =−V0/parenleftBig\n1+/bracketleftBig\nx2\nR2x/bracketrightBigµ/parenrightBig/parenleftBig\n1+/bracketleftBig\ny2\nR2y/bracketrightBigµ/parenrightBig\n+Vb/parenleftBig\n1+/bracketleftBig\nx2\nR2\nb/bracketrightBigµ/parenrightBig/parenleftBig\n1+/bracketleftBig\ny2\nR2y/bracketrightBigµ/parenrightBig,(9)\nwhereV0= 50 meV is the depth of the dots and Vbis\nthe height of the interdot barrier. We assume µ= 10 for\nwhich the potential profile has a form of a nearly rectan-\ngular potential well, where 2 Rx= 90 nm and 2 Ry= 40\nnm determine the size of the double dot in xandydi-\nrections respectively and 2 Rb= 10 nm is the thickness of\nthe interdot barrier. We consider two values of the bar-\nrier height Vb= 10 meV – for the double dot potential\nandVb= 0 – for a single elongated dot. The elongated\ndot potential on the one hand corresponds to the limit\ncase of strong interdot tunnel coupling and on the other\nit is close in geometry to the nanowire quantum dots,\nin which the spin-orbit coupling induced singlet-triplet\navoided crossing was observed in a single-electron charg-\ning experiment.25The potential Vcis displayed in Fig. 1\nfor both the single and double dot.\nThe single-electron eigenfunctions are found by diago-\nnalization of the two-dimensional version of Hamiltonian\n(1) in a basis of multicenter Gaussian functions42with\nembedded gauge invariance\nψn=/summationdisplay\nkscn\nksχsexp/bracketleftbigg\n−(r−Rk)2\n2a2+ieB\n2¯h(xYk−yXk)/bracketrightbigg\n,\n(10)\nwhere summation over kruns over centers of Gaussian\nRk= (Xk,Yk),s=↑,↓andχsare eigenstates of Pauli σz\nmatrix. The centers Rkare distributed on a rectangular\nmeshof25 ×11pointsspacedby∆ x= ∆y= 5.2nm. The\nvariationally optimal basis function parameter a= 4.7\nnm is used in the calculations.3\n-20 020 y [nm]Vb = 0, F x = 0 Vb = 10 meV, F x = 0 \nN = 1 N = 2 N = 3\n-60 -20 20 60 \nx [nm] -20 020 y [nm]\n-60 -20 20 60 \nx [nm] Vb = 10 meV, F x = 0.5 kV/cm \n-60 -20 20 60 \nx [nm] -20 020 y [nm]- 5 meV \n- 45 meV \nFIG. 1: The shades of blue show the in-plane potential of a\nsingle dot ( Vb= 0 – left column of plots) and of a double\ndot (Vb= 10 meV – central and right columns). In the right\ncolumn of plots an in-plane electric field of Fx= 0.5 kV/cm is\nincluded. Inside the light (darker) blue area the potential falls\nbelow -5 meV (-45 meV). The contours indicate the charge\ndensity for a single (top row), two electrons (middle row) an d\nthree electrons (lowest row of plots) for B= 0.\nThe eigenproblem of of N-electron Hamiltonian\nH=N/summationdisplay\nihi+N/summationdisplay\ni=1,j>ie2\n4πǫ0ǫrij(11)\nis solved using the configuration-interaction approach\nwith a basis constructed of Slater determinants built of\nsingle-electron eigenfunctions (10) of SO-coupled Hamil-\ntonian. Convergence of the energies with a precision bet-\nter than 1µeV is usually reached for inclusion of thirty\none-electron eigenstates.\nThe confinement potential (9) is symmetric with re-\nspect to the origin. In the present work the asymmetry\neffects are introduced by the in-plane electric field Fx.\nForFx= 0andwithoutSOcouplingthestationarystates\npossess a definite spatial parity with respect to point in-\nversionPψn(−r) =±ψn(r), wherePis the inversion\noperator. The eigenvalue +1 corresponds to even parity\nstates and the eigenvalue −1 to the odd parity states.\nWhen SO is introduced the spatial parity eigenvalue is\nno longer a good quantum number even for Fx= 0. For\nsymmetric systems the SO coupled Hamiltonians com-\nmute with the operator Pσz, which implies that the spin-\nup and spin-down components still possess definite but\nopposite spatial parities. We refer to Pσzas the s-parity\noperator. Eigenstates of this operator with eigenvalue\n+1 (-1) are referred to as even (odd) s-parity states or\nforbrevity s-even(s-odd) states. The evens-paritystates\nhaveeven-parityspin-upcomponentandodd-parityspin-\ndown component.\nWe evaluate the optical absorption spectrum using the\nenergies of stationary states and transition probabilities\nfrom statektolthat is proportional to the square of thedipole matrix element\nIkl=/angb∇acketleftΨk|N/summationdisplay\nj=1(xj±iyj)|Ψl/angb∇acket∇ight, (12)\nwhere Ψ kis theN-electron wavefunction for k-th Hamil-\ntonian (11) eigenstate and the signs ±correspond to op-\nposite circular polarization of the exciting light. The op-\ntical transitions conserve the electron spin and invert the\nspatial parity when it is a well-defined quantity. When\nthe SO coupling is introduced the optical transitions can\nonly occur between states of opposite s-parity.\n0 1 2 3\nB [T]-45 -44 -43 -42 -41 E [meV]Vb = 10meV0 1 2 3 4 5\nB [T]-46 -44 -42 -40 E [meV]Vb = 0\n(bonding, ↑)(antibonding, ↑)(antibonding, ↓)\n(bonding, ↓)(a)\n(b)\nΔΕ↑↑\n↓↓\nΔΕ\nFIG. 2: Lowest single-electron energy-levels in function o f the\nmagnetic field for a single elongated dot ( Vb= 0) (a) and for\na double dot ( Vb= 10 meV) (b). Solid (dashed) lines corre-\nspond to the even (odd) s-parity. Black lines show the result s\nwithout SO coupling. The blue curves show the results ob-\ntained with SO coupling for Fz= 0 and the red curves in (a)\nforFz= 188 kV/cm. The spin direction for the energy levels\nwithout SO coupling are marked with arrows. The thin verti-\ncal lines indicate allowed optical transitions from the gro und-\nstate.4wave function\n[arb. units]wave function\n[arb. units]\n-80 -40 0 40 80 \nx [nm]wave function\n[arb. units]\n-80 -40 0 40 80 \nx [nm]↑\n↓↑\n↓Fx = 0 Fx = 0.5 kV/cm\nno SO SO included(a)\n(b)\n(c)\n4× 4×\nFIG. 3: Dashed curves show the potential confinement pro-\nfile forVb= 10 meV calculated for y= 0. (a) The spin-up\ncomponents of the even s-parity ground state (red lines) and\ns-odd parity excited state (blue lines) in the absence of SO\ncoupling (spin-down components exactly vanish). (b) Same\nas (a) but with SO coupling present. Spin-down components\nare presented in (c) The scale for the wave function is the\nsame on all the plots, but in (c) the wave functions were mul-\ntiplied by 4. At right (left) panels an electric field is Fx= 0.5\nkV/cm (zero).\nIII. RESULTS\nA. Single electron\nThe single-electronspectrum for a single elongateddot\nand for the double dot is presented in Fig. 2. For B= 0\nthe ground state and the first excited state are Kramers\ndoublets. In each doublet we find one state of the odd\ns-parity and the other of the even s-parity. At B= 0 the\nelectron in the ground-state (first-excite-state) doublet\noccupies predominantly a bonding (antibonding) orbital.\nWith the solid(dashed) linesweplotted the even (odd) s-\nparity energy levels. Black lines show the results without\nSO coupling. The blue lines correspond to the case of\nSO coupling without the linear Rashba term ( Hlin\nSIA), i.e.\nforFz= 0. The red curves in Fig. 2(a) correspond to\nFz= 188.8 kV/cm, for which the linear two-dimensional\nRashba constant is as large as the linear two-dimensional\nDresselhaus one. Beyond increased width of the avoided\ncrossing no qualitative difference in the results obtained\nfor these two values of Fzis found. Therefore, below we\nassumeFz= 0 unless stated otherwise.\nForillustration ofthe double-dot wavefunctions weas-\nsumed a presence of a residual magnetic field B= 10µT\nwhich lifts the doublet degeneracy and we chose the\nstatesofthegroundandexciteddoubletsthatcorrespond\nto/angb∇acketleftsz/angb∇acket∇ight>0. With the blue lines in Fig. 3 we plotted\nthe spinor components of the even s-parity ground state\nwhich is bonding in its spin-up component with or with-00.2 0.4 0.6 0.8 1oc(e)\n0 1 2 3\nB [T]-0.5 00.5 (a)\n(b)ground-state (s-even)\nfirst excited s-odd statelowest-energy s-odd state\nground-state (s-even)\nlowest-energy s-odd statefirst excited s-odd state\nFIG. 4: Contribution of even-parity orbitals (a) and averag e\nvalue of the zcomponent of the spin (b) (in ¯ hunits) in the\nlowest energy s-even and s-odd parity eigenstates.\nout SO coupling. Its antibonding spin-down component\nappears when the SO coupling is introduced [Fig. 3(c)].\nThe red lines in Fig. 3 correspond to the odd s-parity\nstate of the excited doublet which is antibonding in the\nspin-up component. The SO coupling adds to this state\na bonding spin-down component.\nIn Fig. 2 one observes an avoided crossing of two ex-\ncited energy levels of the odd s-parity stemming of both\nthe ground and the exited doublets. Without the SO\ncoupling the energy level that goes up in the energy\nwithgrowingmagneticfieldcorrespondstothespin-down\nbondingorbital,andtheonethatgoesdown–tothespin-\nup antibonding orbital. The avoided crossing opened by\nthe SO interaction is accompanied by spin and spatial\nparity mixing.\nFor the single electron in ideally symmetric pair of\ndots (Fx= 0) there is a direct correspondence between\nthe SO-coupling-induced mixing of both the spin states\nand the occupation of molecular orbitals of opposite spa-\ntial parity. The occupation of the even parity orbitals\n[oc(e)] is calculated as the norm of this component of the\nspinor that corresponds to the even parity state. Then\nthe average value of the z-component of the electron\nspin is/angb∇acketleftsz/angb∇acket∇ight= ¯h/parenleftbig\noc(e)−1\n2/parenrightbig\nfor the even s-parity and\n/angb∇acketleftsz/angb∇acket∇ight= ¯h/parenleftbig1\n2−oc(e)/parenrightbig\nfor the odd s-parity states. Occupa-\ntion of the even parity orbitals and /angb∇acketleftsz/angb∇acket∇ightis for the double\ndot displayed in Fig. 4 as function of the magnetic field.\nThe ground state at higher field becomes a pure bonding\nspin-up orbital. We notice that the values corresponding\ntothe twoodd s-parityenergylevelsinterchangenear2T\nwhich isrelatedto the energylevelanticrossingpresented\nin Fig. 2(b). At the center of the avoided crossing these\ntwo energy levels correspond to /angb∇acketleftsz/angb∇acket∇ight= 0 and bonding\nand antibonding orbitals are equally occupied.\nThe discussed anticrossing of the odd s-parity energy\nlevels leaves a clear signature on the optical absorption\nspectrum. The energy and probability of excitation from\nthe ground-state are displayed in Fig. 5. The ground-\nstate has the even s-parity hence the absorption is only5\n0 1 2 3 4 5\nB [T]0123ΔE [meV]Vb= 0\nVb = 10meV\n0 1 2 3\nB [T]00.4 0.8 1.2 1.6 2ΔE [meV](a)\n(b)Zeeman splitting\nZeeman splitting\nFIG. 5: The dots show the low-energy absorption from the\nground-state at Fx= 0 for the single dot (a) and for the\ncoupled dots (b) (for the energy spectra see Fig. 2). The area\nof the dots is proportional to the absorption probability. T he\nblack dots show the results without SO coupling. The full\nblue dots correspond to SO coupling with Fz= 0. The open\nblue circles in (b) correspond to height of the dot dincreased\nfrom 5.42 to 7.67 nm which amounts in a two-fold reductionof\nthe 2D Dresselhaus constant. The red dots in (a) correspond\nto a strong linear Rashba coupling present Fz= 188 kV/cm.\nThe dashed grey line indicates the Zeeman splitting gµbB.\nallowed to the odd s-parity final state. The ground-state\nis nearly spin-up polarized (Fig. 4) and since electron\nspin is left unchanged during an optical transition the\nabsorption goes to the s-odd state with spin-up orien-\ntation. When the avoided crossing is opened between\nthe s-odd energy levels both of them possess a non-zero\nspin-up component and the optical transitions to both\nof them from the ground-state are allowed. Outside the\navoided crossing the absorption spectra with or without\nSO coupling are similar.\nThe energy range in which the SO-induced avoided\ncrossing is observed in the absorption spectrum corre-\nsponds to far-infrared or microwave radiation in which\ncyclotron resonance experiments are performed.39One\ncan increase the energy of the avoided crossing twice by\napplying an electric field of 0.5 kV/cm - see Fig. 6(b).\nIn the presence of the electric field the electron in the-45 -44 -43 -42 -41 E [meV]\n0 2 4 6\nB [T]0123ΔE [meV](a)\n(b)\nFx = 0Fx = 0.2 kV/cmFx = 0.5 kV/cm\nFIG. 6: Single-electron energy spectrum (a) and optical ab-\nsorption spectrum (b) in function of the magnetic field for\nFx= 0 (black color), Fx= 0.2 kV/cm (blue) and Fx= 0.5\nkV/cm (green) for coupled quantum dots.\nground-state is pushed to the left dot by Fx>0 while\nthe final state in the absorption process is mainly local-\nized in the right dot [see Fig. 3(b)]. The opposite shifts\nofthe electronwavefunction inthe initial andfinal states\nare translated by the electric field into an increased tran-\nsition energy [see Fig. 6(a) for the energy splitting]. The\nobtained energy increase is accompanied by reduction of\nthe SO-induced avoided crossing.\nFig. 6(b) shows also that for non-zero Fthe absorp-\ntion probabilities vanish at higher B. The separation of\nthe initial and final states [Fig. 3(b)] by the electric field\nis enhanced when the magnetic field is applied, since the\nlatter increases the localization of wave functions near\nthe centers of the dots lifting the interdot tunnel cou-\npling. In consequence - the ground-state becomes totally\nlocalizedin onedot andthe final state ofthe transitionin\nthe other. Vanishing overlapbetween the initial and final\nstatewavefunction impliesvanishingtransitionprobabil-\nity as calculated by formula (12).\nB. Electron pair\nIn the absence of the magnetic field and without SO\ncoupling the first excited state of the electron pair is spin\ntriplet. For B= 0 we find that the first excited state\nis threefold degenerate also with SO coupling present.\nThis applies to both the single elongated dot [Fig. 7(a)]\nand the double dot [Fig. 7(b)]. Without SO coupling6\n-86 -85.5 -85 -84.5 -84 -83.5 -83 E [meV]\n0 1 2 3\nB [T]-84.8 -84.4 -84 -83.6 -83.2 -82.8 E [meV]ST-\nT+T0\nST+T0T-(a)\n(b)Vb = 0\nVb = 10meV\nFIG. 7: Two-electron energy spectrum for a single elongated\ndotVb= 0 (a) and for a couple of dots separated by Vb= 10\nmeV barrier (b). Black (blue) lines show the results without\n(with) SO coupling. For the results without SO coupling we\nadded labels Sfor the singlet and Tfor the triplets (subscript\ndenotesthesignofthe z-componentofthetotalspin). Dashed\n(solid) curves correspond to odd (even) s-parity. Results w ere\nobtained for Fx= 0.\nthe magnetic field induces a singlet-triplet ground-state\ntransition near 1 T for the single dot and near 0.4 T\nfor the double dot. The crossing singlet and triplet en-\nergy levels have the same odd s-parity and an avoided\ncrossing is opened between them when SO coupling is\nintroduced. The calculated width of the avoided cross-\ning is 0.18 and 0.07 meV for the single and double dot,\nrespectively which is within the order of the ones found\nin experiments: 0.25 meV and 0.2 meV for the nanowire\nquantum dot25and for the double dots26.\nFor asymmetric system ( Fx= 0) the optical transition\nfrom the ground-state can only go to the even s-parity\neigenstate. In the absence of the spin-orbit coupling in\nthe considered energy range only the triplet with zero z-\ncomponentofthespin( T0)hastherequiredspatialparity\nto absorb photons. However, this absorption is excluded\nanyway on both sides of the singlet-triplet ground-state\ntransition. For Bbelow this transition the matrix el-\nement (12) vanishes due to opposite symmetry of the\nspatial initial and final wave functions with respect to\nthe electron interchange. For Babove the singlet-triplet\ntransition the ground-state (triplet with sz= ¯hdenoted\nasT+) andT0states have the same symmetry with re-00.4 0.8 1.2 1.6 2ΔE [meV]Vb = 0 Vb = 10meV\n(a) (c)\n0 1 2 3\nB [T](d)\nFx = 0 Fx = 1 kV/cm\n0 1 2 3\nB [T]00.4 0.8 1.2 1.6 2ΔE [meV](b)T+ → T0 T+ → T0\nT+ → T0T+ → T0\nT+ → S S → T+S → T-\nT+ → S\nFIG. 8: Two-electron ground-state absorption spectrum as a\nfunction of the magnetic field for SO coupled single dot (a,b)\nand double dot (c,d). Panels (a,c) correspond to Fx= 0 and\n(b,d)toFx= 1kV/cm. Theareaofthedotsisproportional to\nthe absorption probability. Transitions are denoted by lab els\nof two-electron spin eigenstates which are found without SO\ncoupling. Without SO coupling all the transitions presente d\nin this figure are forbidden.\n-4 -2 0 2 4\nFx[kV/cm]0123chargeintheleftdotone\nelectronthree\nelectrons\ntwo\nelectrons\nFIG. 9: Electron charge localized in the left dot as a functio n\nof the electric field for the double dot at B= 0. Results with\nand without SO coupling are not distinguishable.\nspect to the electron interchange, but the z-components\nof the spin are different. Optical transitions between the\nstates corresponding to energy levels presented in Fig. 7\nare only allowed by the SO coupling. The calculated ab-\nsorption spectrum is shown in Fig. 8. For Fx= 0 [Fig.\n8(a,c)] the absorption probability grows with the mag-\nnetic field after the singlet-triplet ground state avoided\ncrossing. Then, the transition corresponds to T+→T0\nexcitation in terms of states without SO coupling. When\nthe electric field Fxis switched on [Fig. 8(b,d)] the parity\nselection rules no longer apply and we notice appearance7\n-0.6-0.4-0.2 0 0.2 0.4 0.6 \nFx [kV/cm]0.9 0.95 11.05 1.1 charge\nin the left dot\n-0.6-0.4-0.2 0 0.2 0.4 0.6 \nFx [kV/cm]T+SS-84.6 -84.4 -84.2 -84 -83.8 E [meV]B = 0.3 T B = 0.4 T\n(a)\n(b)(c)\n(d) SST+ ST+\n-0.8 -0.4 0 0.4 0.8 \nFx [kV/cm]0.8 0.9 11.1 1.2 charge\nin the left dotT+SS-84.6 -84.3 -84 E [meV]B = 0.6 T\n(e)\n(f)ST+\nFIG. 10: Two-electron energy spectrum of the double dot is sh own in (a,c,e) as a function of the electric field. Plots (b,d, f)\nindicate the charge localized in the left dot. Black (blue) l ines show the results without (with) SO coupling. The labels Sand\nT+correspond to eigenstates without SO coupling.\n0 1 2 3\nB [T]00.02 0.04 0.06 0.08 0.1 probabilityno SO\nground statefirst excited state\nSO\nincluded\nFIG. 11: Probability that both the electrons occupy the same\ndot with and without SO coupling in the ground state and\nfirst excited state.\n00.2 0.4 0.6 0.8 1occupation\n0 1 2 3\nB [T]00.2 0.4 0.6 0.8 1occupationno SO\nSO includedeven spin up\neven spin down\nodd spin downodd spin upeven spin up odd spin up\neven spin down odd spin down(a)\n(b)\nFIG. 12: Occupation of the single-electron orbitals of defin ite\nspatial parity and spin for two-electron ground state with ( b)\nandwithout(a)SOcouplingfortheelectronpairinthedoubl e\ndot.\nof alsoS↔T+andS→T−transitions. The probabili-00.2 0.4 0.6 0.8 1contribution\n0 1 2 3\nB [T]00.2 0.4 0.6 0.8 1contributionno SO\nSO includedodd odd\neven eveneven odd + odd eveneven even\nodd oddeven odd + odd even (a)\n(b)\nFIG. 13: Contributions of the two-electron orbitals to the\nground state with (b) and without (a) SO coupling for the\nelectron pair in the double dot.\nties for the discussed transitions– which areall forbidden\nin the absence of SO coupling – remain very small (less\nthan 0.5%) as compared to the ones found for the single\nand three electrons.\nFor two electrons the role of the electric field for the\nlow-energyoptical absorption is different from the single-\nelectroncase. For N= 1the electricfield distinctly shifts\nthe energy of the absorption lines (Fig. 6). For N= 2\nthe energy shift is very weak, only the transition prob-\nabilities are affected. For the single electron the energy\nshifts resulted from spatial electron-chargedisplacements\nof the initial and final states induced by the electric field.\nFor two electrons these shifts are hampered (see Fig. 1)\nsince the charge shift implies appearance of a double oc-\ncupation of one of the dots. Fig. 9 shows the charge\nlocalized in the left dot in function of the electric field.\nForN= 1 (andN= 3) the dependence of the charge on\nFxis the strongest at zero electric field, while for N= 2\nwe find a plateau centered at Fx= 0.\nForB= 0 we did not find any SO coupling influence8\non the charge distribution as a function of the in-plane\nelectric field. Nevertheless, such an effect is observed in\nthe presence of the external magnetic field - see Fig. 10.\nForB= 0.4 T the ground-state without SO coupling\ncorresponds already to the spin triplet, in which – due\nto the Pauli exclusion – localization of both electrons in\nthe same dot requires occupation of an excited single-dot\nenergy level. The charge of the two-electron system for\nthe triplet ground-stateis evenmoreresistanttoshifts by\nthe electric field than for the singlet state [compare Fig.\n10(b) and (d)]. For B= 0.4 T the ground-state becomes\nsingletagainnear0.4kV/cm. Theelectronsinthesinglet\nstate occupy more easily13the dot made deeper by the\nelectricfield whichrestoresthe singlet ground-statewhen\nFxis applied. We notice [see the dashed line in Fig.\n10(d)] a jump in the occupation of the left dot at the\nsinglet-triplettransition. For B= 0.6Ta similareffect is\nobservedonly at higher Fx[the dashed line in Fig. 10(f)].\nThe SO coupling mixes the singlet and triplet states and\nwe notice that the electron charge in the left dot [blue\nlines in Fig. 10(b,d,f)] becomes a smooth function of\nFx. As a general rule, when the ground-state without\nSO coupling is singlet (triplet) - the SO coupling reduces\n(enhances) the occupation of the deeper dot.\nAt the singlet-triplet transition the SO coupling in-\nfluences also the probability of finding both the elec-\ntrons in the same dot (Fig. 11). Without SO coupling\nthe ground-state probability exhibits a rapid drop at the\nsinglet-triplet transition near 0.4 T. The spin-orbit cou-\npling influences the double occupation probability only\nfor non-zero B.\nIn order to quantify the occupation of the single-\nelectron even and odd parity orbitals we first project\nthe two-electron eigenstates of operator (11) into the ba-\nsis composed of single-electron eigenfunctions obtained\nwithout SO coupling (denoted as ψ′in the following).\nFor a state νwe consider the projection in form\ndν\nkl=1\n2/summationdisplay\ni,j>iCν\nij/angb∇acketleftψi(1)ψj(2)−ψi(2)ψj(1)|\n|ψ′\nk(1)ψ′\nl(2)−ψ′\nk(2)ψ′\nl(1)/angb∇acket∇ight. (13)\nAn eigenfunction ψ′\nkhas a definite spatial parity and\nz-component of the spin associated with a spinor χk\nwhich is the szeigenfunction of eigenvalue ¯ h/2 or−¯h/2\n(χk=| ↑/angb∇acket∇ightorχk=| ↓/angb∇acket∇ight). Hence, the occupation of the\nspin-up even-paritysingle-electron wave functions can be\ncalculated as\noc(e↑) =/summationdisplay\nk,l>k|dkl|2[δp(k,+)δs(k,↑)+δp(l,+)δs(l,↑)],\n(14)\nwhere\nδp(k,±) =1±/integraltext\n(ψ′\nk(r))∗ψ′\nk(−r)dr\n2(15)\nand\nδs(k,↑) =/angb∇acketleftχk| ↑/angb∇acket∇ight. (16)The occupation of the spin-up odd-parity single-electron\nstates is determined by the formula\noc(o↑) =/summationdisplay\nk,l>k|dkl|2[δp(k,−)δs(k,↑)+δp(l,−)δs(l,↑)],\n(17)\nwith an obvious generalization for the spin-down compo-\nnents. The results are displayed in Fig. 12. Without\nSO coupling i) below 0.4 T the ground-state is even par-\nity singlet - the electrons occupy mostly the even par-\nity states ii) above 0.4 T the ground-state is odd parity\ntriplet - the spin-down contributions are removed, one\nof the electrons occupy an even parity and the other an\nodd parity orbital. The jump of the occupations near\n0.4 T that is observed in the results without SO coupling\nis replaced by a smooth transition when SO coupling is\napplied. The values obtained for orbital occupations in\nboth large and zero Blimits are similar.\nNon-conservation of the spatial parity in the presence\nof SO coupling for the two-electron states becomes evi-\ndentwhenoneconsiderscontributionsofthetwo-electron\nbasis elements. The contributions of the elements in\nwhich both electrons occupy orbitals of the same spatial\nparity are calculated as\ncee=/summationdisplay\nk,l>k|dkl|2δp(k,+)δp(l,+), (18)\nfor the even parity orbitals and\ncoo=/summationdisplay\nk,l>k|dkl|2δp(k,−)δp(l,−), (19)\nfor the odd parity orbitals. Contribution of the two-\nelectron basis elements in which the electrons occupy op-\nposite parities is\ncoe+eo=/summationdisplay\nk,l>k|dkl|2[δp(k,−)δp(l,+)+δp(k,+)δp(l,−)].\n(20)\nThe results are displayed in Fig. 13. Without SO cou-\npling forB <0.4T the contributionofthe basiselements\nin which the electrons occupy opposite parity eigenstates\nis zero. In the triplet ground state for B >0.4 T the\nelectrons are bound to occupy orbitals of opposite pari-\nties. When the SO is present for B= 0 there is a nearly\n10% contribution of basis elements in which the electrons\noccupy orbitals of opposite parities. The coe+eogrows\nwith the magnetic field, but it stays below 100% in the\nstudied range of B. This result and the ones presented\naboveindicate that for two electrons the SO coupling has\na noticeable influence on the ground-state properties in\ncontrast to the single electron case.\nC. Three electrons\nForN= 3 in the absence of SO coupling the magnetic\nfield leads to the ground-state spin-polarization transi-\ntion near 3 T in both the single (Fig. 14) and double9\n[Fig. 15(a)] dots. For symmetric dots this transition is\nassociated with energy level crossing even when SO cou-\npling is introduced since the ground-states on both sides\nof the transition correspond to opposite s-parities. The\nin-plane electric field opens an avoided crossing at the\nground-state spin polarization transition [see Fig. 17(a)].\nFor three electrons in a single dot without SO coupling\none observes (Fig. 14) crossings of three s-odd energy\nlevels near 2 T. For the double dot [Fig. 15(a)] the cross-\nings appear in more separated magnetic fields. The three\ncrossing levels have different zprojections of the spin.\nSimilarly as for N= 1 the SO coupling opens avoided\ncrossing in the absorption spectrum, but for N= 3 three\nenergy levels participate in this avoided crossing instead\nof two. These avoided crossings are accompanied by a\nsmooth variation of the spin [Fig. 15(b)].\nFor the spin unpolarized ground-state ( B <3 T) the\nlowest-energy optical transition goes from the ground-\nstate tothe odd s-paritystateswith /angb∇acketleftsz/angb∇acket∇ight ≃¯h/2. Without\nSO coupling and in terms of occupation of single-electron\norbitals we observe (Fig. 16) a transition of one of the\nelectrons occupying a bonding orbital to an occupied an-\ntibonding orbital. One finds a single bright line similar\nto the one found for N= 1. ForB >3 T the principle\nline in the ground-state absorption spectrum disappears\ndue to the ground-state spin polarization.\nThein-planeelectricfield increasesthe energysplitting\nbetween the ground-state and the first excited state lead-\ning to an increase of the energy absorbed at the optical\ntransition [Fig. 17(b)]. The form of the avoided crossing\nis not affected by the field – like in the single electron\ncase.\n0 1 2 3 4\nB [T]-120 -119 -118 -117 -116 -115 E [meV]\n↑↓↑\n↑↓↓↑↓↑↑↓↓↑↑↑↑↓↑↑↓↓ ↓↓↓\nFIG. 14: Three-electron energy spectrum for the single elon -\ngated dot. Black (blue) lines show the results without (with )\nSO coupling. Solid (dashed) lines correspond to even (odd)\ns-parity states. The arrows in the plot indicate the zcompo-\nnent of the spin without SO coupling.\nFor the lowest-energy even s-parity state both occu-\npation of single-electron spin-orbitals [Fig. 18(a,c)] and\ncontribution of three-electron basis elements of definite\nspatial parity [Fig. 19(a,c)] are only weakly affected by\nboth the magnetic field and the spin-orbit coupling. The-117 -116 -115 -114 -113 E [meV]\n↑↓↑ ↑↓↓↑↓↑↑↓↓↑↑↑\n0 1 2 3 4\nB [T]-0.5 00.5 11.5 (a)\n(b)\nFIG. 15: Three-electron energy spectrum (a) for the double\ndot. Black (color) lines show the results without (with) SO\ncoupling. Solid (dashed) lines correspond to even (odd) s-\nparity. The short arrows in the plot indicate the zcomponent\nof the spin without SO coupling and the longer ones show\nthe allowed optical transitions from the ground-state. (b) z\ncomponent of the spin for the lowest even s-parity and three\nlowest odd s-parity energy levels. Type and color of curves\nfor these states is adopted of panel (a).\n0 1 2 3 4\nB [T]00.5 11.5 22.5 E [meV]\n0 1 2 3 4\nB [T]Vb = 0 Vb = 10 meV (a) (b)\nFIG. 16: Optical absorption spectrum for the three-electro n\nsystem in the single dot (a) and in the double dot (b). Black\n(blue) dots correspond to SO coupling absent (present). Are a\nof the dot is proportional to the absorption probability.\ndependence of the studied quantities on the magnetic\nfield is more spectacular for the lowest energy s-odd state\n[Figs. 18(b,d) and 19(b,d)]. Without SO coupling the\nlowest-energys-odd level corresponds to even parity only\nbetween 1.9 and 2.8 T, hence the vanishing contribution\nof the even-parity three-electron basis elements outside\nthisBinterval. In the presence of SO coupling the con-\ntribution of the even-parity basis elements extends over\nthe entire studied range of the magnetic field.10\n0 1 2 3 4\nB [T]00.5 11.5 22.5 ΔE [meV]-118 -116 -114 -112 E [meV](a)\n(b)\nFIG. 17: (a) Three electron energy spectrum for SO-coupled\ndouble dot at Fx= 0 (black dotted lines) and for Fx= 0.5\nkV/cm (blue solid curves). The arrow indicates the ground-\nstate avoided crossing which is opened in presence of non-\nzeroFx. (b) Optical absorption spectrum for the SO-coupled\ndouble dot. Black (blue) dots correspond to Fx= 0 (Fx= 0.5\nkV/cm).\n00.2 0.4 0.6 0.8 1occupationno SO SO included\nlowest\ns-even state\n0 1 2 3 4\nB [T]00.5 11.5 2occupation\n0 1 2 3 4\nB [T]\nlowest\ns-odd stateeven spin up\neven spin downodd spin up\nodd\nspin down(d)(a)\n(b)(c)\neven spin up\nodd\nspin up\nodd\nspin downeven\nspin downeven spin up odd\nspin up\nodd\nspin downeven\nspin down\nodd\nspin downeven spin downeven spin upodd spin up\nFIG. 18: Occupation of the single-electron orbitals of defin ite\nspin orientation and spatial parity without (a,b) and with\n(c,d) SO coupling for the lowest energy s-even (a,c) and s-\nodd (b,d) state for three electrons in the double dot.\nIV. SUMMARY AND CONCLUSIONS\nWe have presented a systematic exact diagonalization\nstudy of one, two and three-electron spin-orbit coupled\nsystems in double quantum dots. We discussed the mix-00.2 0.4 0.6 0.8 1contribution\n0 1 2 3 4\nB [T]00.2 0.4 0.6 0.8 1contribution\nlowest\ns-odd statelowest\ns-even state\n0 1 2 3 4\nB [T]no SO SO included\nodd even even\nodd odd\neven\neven\neven\neven\nodd odd oddodd odd\neven odd even even\nodd odd oddeven\neven\neven(a)\n(b)(c)\n(d)odd odd\nevenodd odd oddeven\neven\nevenodd even even\nodd odd\neven odd odd oddeven\neven\nevenodd even even\nFIG. 19: Contributions of single-electron orbitals of a giv en\nsymmetry to the lowest energy three electron s-even (a,c) an d\ns-odd (b,d) states, with (c,d) and without (a,b) SO coupling .\ning of the bonding and antibonding electron orbitals by\nthe SO coupling. We investigated occupation of even\nand odd parity orbitals, the energy and optical absorp-\ntion spectrain crossedelectric andmagnetic fields aswell\nas the electron distribution.\nFor one and three electrons confined in a pair of iden-\ntical dots we found that the spin-orbit coupling only\nweakly affects the ground-state properties. A strong\nmixing of bonding and antibonding orbitals due to the\nspin-orbit coupling was found in the lowest-energy ex-\ncited states.\nIn contrast to the odd electron numbers, for two elec-\ntrons the spin-orbit interaction affects the properties of\nthe ground-state since the spin-polarization becomes a\nsmooth transition instead of an abrupt singlet-triplet\ntransformation. On the contrary, the spin polarization\nof the three electron system in symmetric dots is not\naffected by the spin-orbit coupling since the low- and\nhigh-spinground-statescorrespondtoopposites-parities.\nFor three electrons the SO coupling makes the spin-\npolarization continuous only when the confinement po-\ntential contains an in-plane asymmetry, e.g. introduced\nby an electric field.\nFor odd electron numbers the spin-orbit-coupling-\ninducedmixingofspatialparitiesofthefirstexcitedstate\nopens characteristic avoided crossings in the optical ab-\nsorption spectrum. An in-plane electric field shifts the\ninitial and final states of the optical transition to oppo-\nsite dots. In consequence it distinctly increases the en-\nergy of the optical transition at an expense of a reduced\nwidth of the avoided crossing.\nThe low-energy optical absorption for two electrons is\nonly allowed by the SO coupling. For two electrons the\nin-planeelectricfieldliftsthespatialparityselectionrules\nbut does not essentially perturb the energy of the optical\ntransitions.11\nAcknowledgments\nThis work was supported by the ”Krakow Interdis-\nciplinary PhD-Project in Nanoscience and AdvancedNanostructures operated within the Foundation for Pol-\nish Science MPD Programme co-financed by the EU Eu-\nropean Regional Development Fund.\n1X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000).\n2A. Harju, S. Siljam¨ aki, and R.M. Nieminen, Phys. Rev.\nLett.88, 226804 (2002).\n3G. Burkard, D. Loss and D.P. DiVincenzo Phys. Rev. B\n59, 2070 (1999).\n4H.J. Krenner, M. Sabahtil, E.C. Clark, A. Kress, D. Schuh,\nM. Bichler, G. Absteiter, and J.J. Finley, Phys. Rev.\nLett.94, 057402 (2005); E.A. Stinaff, M. Scheibner, A.S.\nBracker, I.V. Pomonarev, V.L. Korenev, M.E. Ware, M.F.\nDoty, T.L. Reinecke, and D. Gammon, Science 311, 636\n(2005); H.J. Krenner, E.C. Clark, T. Nakaoka, M. Bich-\nler, C. Scheurer, G. Absteiter, and J.J. Finley, Phys. Rev.\nLett.97, 076403 (2006).\n5J. I. Climente, M. Korkusi´ nski, G. Goldoni, and P. Hawry-\nlak, Phys. Rev. B 78, 115323 (2008); M. F. Doty, J. I.\nClimente, M. Korkusi´ nski, M. Scheibner, A. S. Bracker, P.\nHawrylak, and D. Gammon, Phys. Rev. Lett. 102, 047401\n(2009).\n6Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039\n(1984).\n7G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n8P. Pietil¨ ainen and T. Chakraborty, Phys. Rev. B 73,\n155315 (2006); T. Chakraborty and P. Pietil¨ ainen, Phys.\nRev. Lett. 95, 136603 (2005).\n9J.M. Elzerman, R. Hanson, J.S. Greidanus, L.H. Willems\nvan Beveren, S. De Francheschi, L.M.K Vandersypen,\nS. Tarucha and L.P. Kouwenhoven, Phys. Rev. B 67,\n161308(R) (2003); R. Hanson, L. H. Willems van Beveren,\nI. T. Vink, J. M. Elzerman, W. J. M. Naber, F. H. L.\nKoppens, L. P. Kouwenhoven, and L. M. K. Vandersypen,\nPhys. Rev. Lett. 94, 196802 (2005);\n10J.R. Petta, A.C. Johnson, J.M. Taylor, E.A. Laird, A. Ya-\ncoby, M.D. Lukin, C.M. Markus, M.P. Hanson, and A.C.\nGossards, Science 309, 2180 (2005).\n11X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000).\n12A Harju, S. Siljam¨ aki, and R. M. Nieminen, Phys. Rev.\nLett.88, 226804 (2002).\n13B. Szafran, F.M. Peeters, and S. Bednarek, Phys. Rev. B\n70, 205318 (2004).\n14J. Pedersen, C. Flindt, N.A. Mortensen, and A.-P. Jauho,\nPhys. Rev. B 76, 125323 (2007).\n15A. L. Saraiva, M. J. Calderon, and B. Koiller, Phys. Rev.\nB76, 233302 (2007).\n16M. Stopa, A. Vidan, T. Hatano, S. Tarucha, and R.M.\nWestervelt, Physica E 34, 616 (2006).\n17D.V. Melnikov, J.-P. Leburton, A. Taha, and N. Sobh,\nPhys. Rev. B 74, 041309(R) (2006).\n18F. Baruffa, P. Stano, and J. Fabian, arXiv:0908.2961v2.19S. Gangadharaiah, J. Sun, and O. A. Starykh, Phys. Rev.\nLett.100, 156402 (2008).\n20P.StanoandJ.Fabian, Phys.Rev.Lett. 96, 186602(2006).\n21V.N. Golovach, A. Khaetskii, and D. Loss. Phys. Rev. B\n77, 045328 (2008).\n22K. Shen and M.W. Wu, Phys. Rev. B 76, 235313 (2007).\n23T. Meunier,T. Vink, L. H. Willems van Beveren, K-J. Tiel-\nrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz, W.\nWegscheider, L. P. Kouwenhoven, and L. M. K. Vander-\nsypen, Phys. Rev. Lett. 98, 126601 (2007).\n24J.I. Climente, A. Bertoni, G. Goldoni, M. Rontani, and E.\nMolinari, Phys. Rev. B 75, 081303(R) (2007).\n25C. Fast, A. Fuhrer, L. Samuelson, V.N. Golovach, and D.\nLoss, Phys. Rev. Lett. 98, 266801 (2007).\n26A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys.\nRev. B76, 161308 (R) 2007.\n27K. V. Kavokin Phys. Rev. B 64, 075305 (2001).\n28N. E. Bonesteel, D. Stepanenko, and D. P. DiVincenzo,\nPhys. Rev. Lett. 87, 207901 (2001).\n29G. Burkard and D. Loss, Phys. Rev. Lett. 88, 047903\n(2002).\n30Lian-Ao Wu and Daniel A. Lidar, Phys. Rev. A 66, 062314\n(2002).\n31D. Stepanenko and N. E. Bonesteel, Rhys. Rev. Lett. 93,\n140501 (2004).\n32C. Debald and C. Emary, Phys. Rev. Lett. 94, 226803\n(2005);\n33C. Flindt, A.S. Sorensen, and K. Flensberg, Phys. Rev.\nLett.97, 240501 (2006).\n34S. Bednarek and B. Szafran, Phys. Rev. Lett. 101, 216805\n(2008).\n35P. F¨ oldi, O. Kalman, M. G. Benedict, and F.M. Peeters,\nPhys. Rev. B 73, 155325 (2006); P. F¨ oldi, O. Kalman, M.\nG. Benedict, and F.M. Peeters, Nano Lett. 8, 2556 (2008);\n36P. Stano and J. Fabian, Phys. Rev. B 72, 155410.\n37L. Meza-Montes, C. F. Destefani, and S.E. Ulloa, Phys.\nRev. B78, 205307 (2008).\n38S. Saikin, M. Shen, M. Cheng and V. Privman, J. Appl.\nPhys.94, 1769 (2003).\n39W. Pan, K. Lai, S.P. Bayrakci, N.P. Ong, D.C. Tsui, L.N.\nPfeiffer, and K. West, Appl. Phys. Lett. 83, 3519 (2003).\n40E. A. de Andrada e Silva, G. C. La Rocca and F. Bassani,\nPhys. Rev. B 55, 16293 (1997).\n41M. Willatzen and L. C Lew Yan Voon, J. Phys.: Condens.\nMatter20, 345216 (2008).\n42T. Chwiej and B. Szafran, Phys. Rev. B 78, 245306 (2008)." }, { "title": "2310.05160v1.Vortex_Lattice_Formation_in_Spin_Orbit_Coupled_Spin_2_Bose_Einstein_Condensate_Under_Rotation.pdf", "content": "Springer Nature 2021 L ATEX template\nVortex Lattice Formation in\nSpin–Orbit-Coupled Spin-2 Bose–Einstein\nCondensate Under Rotation\nParamjeet Banger\nDepartment of Physics, Indian Institute of Technology Ropar,\nRupnagar 140001, Punjab, India.\nCorresponding author(s). E-mail(s): 2018phz0003@iitrpr.ac.in;\nAbstract\nWe investigate the vortex-lattice configuration in a rotating spin\norbit-coupled spin-2 Bose-Einstein condensate confined in a quasi-two-\ndimensional harmonic trap. By considering the interplay between rota-\ntion frequency, spin-orbit couplings, and interatomic interactions, we\nexplore a variety of vortex-lattice structures emerging as a ground-\nstate solution. Our study focuses on the combined effects of spin-orbit\ncoupling and rotation, analysed by using the variational method for\nsingle-particle Hamiltonian. We observe that the interplay between rota-\ntion and Rashba spin-orbit coupling gives rise to different effective\npotentials for the bosons. Specifically, at higher rotation frequencies,\nisotropic spin-orbit coupling leads to an effective toroidal potential,\nwhile fully anisotropic spin-orbit coupling results in a symmetric dou-\nble well potential. To obtain these findings, we solve the five coupled\nGross-Pitaevskii equations for the spin-2 BEC with spin-orbit coupling\nunder rotation. Notably, we find that the antiferromagnetic, cyclic,\nand ferromagnetic phases exhibit similar behavior at higher rotation.\nKeywords: Spin-2 BECs, Spin-orbit coupling, Rotation, Vortex-lattice\n1 Introduction\nIn recent decades, advancements in optical traps within cold atom experiments\nhave facilitated investigations into spinor Bose-Einstein condensates (BECs).\n1arXiv:2310.05160v1 [cond-mat.quant-gas] 8 Oct 2023Springer Nature 2021 L ATEX template\n2 Vortex-lattice formation in SO-coupled spin-2 BEC under rotation\nThe experimental realization of spin-orbit (SO) coupling in spinor BECs has\nbeen one of the most important advancements in the last decade [1], thus open-\ning up avenues for numerous noval studies on spin-2 BECs [2, 3]. Spin-2 BECs\nwith SO coupling have also been proposed theoretically [4]. In SO-coupled\nspinor BEC, exotic vortex-lattice configuration has been explored in the pres-\nence of rotation [5]. The combined effect of both SO coupling and rotation\nfrequency facilitates the emergence of a variety of topological configurations in\npseudospin BEC [5–10]. The interplay of isotropic SO coupling and rotation\nresulting in the emergence of half-skyrmion excitations in rotating and rapidly\nquenched spin-1 BECs using stochastic projected Gross-Pitaevskii equations\nhave been studied in [11–15]. Rotating spin-1 BEC also supports many inter-\nesting solutions in the presence of an isotropic [13, 16] as well as anisotropic,\n[11, 12, 16] SO coupling. Moreover, numerical studies on rotating SO-coupled\nBEC has been done in toroidal traps[14], with Weyl SO coupling [15], and\nSU(3) coupling [17]. Recently, by considering the isotropic SO-coupled spin-2\nBEC under rotation topological vortical phase transitions have been studied\ntheoretically [18]. This numerical study has been done for small to moder-\nate rotation frequency. However, SO-coupled spin-2 BECs systems have not\nbeen completely investigated. In this work we fully examine anisotropic as well\nisotropic SO-coupling with moderate to higher rotation frequencies. We inves-\ntigate a quasi-two-dimensional (q2D) harmonically trapped spin-2 condensate,\nfeaturing an anisotropic SO-coupling term proportional to Sxˆpx, as well as an\nisotropic SO coupling term proportional to ( Sxˆpy−Syˆpx). We choose23Na\n[19],87Rb [20], and83Rb [19] BECs as the prototypical examples of systems\nwith antiferromagnetic, cyclic, and ferromagnetic spin-exchange interactions.\nWe compute the rotational energy per particle as a function of rotation fre-\nquency, and at higher rotation, antiferromagnetic, cyclic, and ferromagnetic\nphases exhibit qualitatively similar behavior. To analyze the characteristics\nof the SO-coupled non-interacting Hamiltonian under rotation, we employed\na variational method. During the investigation of the single-particle Hamilto-\nnian, we interpret the effective potentials experienced by bosons. The detailing\nof calculating effective potentials for SO-coupled spin-2 BEC under rotation\nis discussed in Appendix. The paper is as follows. In Section 2, we use a vari-\national method to study the properties of the non-interacting Hamiltonian\nand interpret the effective potential experienced by bosons in the presence of\nrotation and SO coupling. In Section 3, we introduce formalism of the mean-\nfield model for an SO-coupled spin-2 BEC in the rotating frame and discuss\nthe Coupled Gross pitiaviskii Equations (CGPEs). The Numerical results of\nthe interacting systems are discussed in Section 4. Finally, in Section 5, we\nconclude the results of our study.\n2 Single particle Analysis\nIn our study, we examine a spin-2 BEC with SO coupling that is confined in\nthexy-plane. The condensate is subjected to a q2D harmonic trap of the formSpringer Nature 2021 L ATEX template\nVortex-lattice formation in SO-coupled spin-2 BEC under rotation 3\nm(ω2\nxx2+ω2\nyy2+ω2\nzz2)/2, where ωzis significantly larger than ω(=ωx=ωy)\nalong the z-direction. The single particle Hamiltonian for an SO-coupled spin-2\nBEC in rotating frame can be described as [6, 21, 22]\nH=Hlab−ΩLz (1)\nwhere Ω is the angular frequency of rotation, Lz=−i(x∂/∂y −y∂/∂x ) is the\nzcomponent of the angular momentum operator, and Hlabis single particle\nlaboratory frame Hamiltonian, can be written in the dimensionless form\nHlab= \nˆp2\nx+ ˆp2\ny\n2+x2+y2\n2!\n×1+HSOC, (2)\nwhere I represents a 5 ×5 identity matrix, and ˆ pd=−i∂/∂d with d=x, y.\nThe HSOC can be expressed as case I: γxSxˆpxexperimentally achievable\nequal-strength mixture of Rashba and Dresselhaus couplings [1] terming as\nan anisotropic spin-orbit coupling for q2D-system. Alternatively, case II:\nγ(Sxˆpy−Syˆpx), which represents the Rashba spin-orbit coupling [23] between\nthe spin and the linear momentum along the xy-plane. The γxandγare the\nSO coupling strength for respective cases. Sdis the irreducible representations\nof the d-component of angular momentum operators for spin-2 matrix with\nd=x, y.\n(Sx)j′,j=1\n2\u0010p\n(6−j′j)ℏδj′,j+1\n+p\n6−j′j)ℏδj′+1,j\u0011\n, (3)\n(Sy)j′,j=1\n2i\u0010p\n(6−j′j)ℏδj′,j+1\n−p\n(6−j′j)ℏδj′+1,j\u0011\n. (4)\nwith j′andjvaries from −2 to 2. To describe the combined effect of rota-\ntion and spin-orbit coupling, we analyse the single particle Hamiltonian. To\nexamine the Eq. (1) for caseI we consider the following normalized variational\nansatz\nΨ±\nvar=exph\n−(x−x0)2\n2−(y−y0)2\n2+i(kxx+kyy)i\n2√π× \n1\n4,∓1\n2,r\n3\n8,∓1\n2,1\n4!T\n,\n(5)\nwhere x0, y0, kxandkyare the variational parameters. The variational energies\nare\nE±\nvar(x0, y0, kx, ky) =Z\ndxdyΨ±\nvar∗HΨ±\nvar,Springer Nature 2021 L ATEX template\n4 Vortex-lattice formation in SO-coupled spin-2 BEC under rotation\n=1\n2\u0002\nkx(∓4γx+kx+ 2Ω y0) +k2\ny\n−2kyx0Ω +x2\n0+y2\n0+ 2\u0003\n, (6a)\nwhere Ψ±\nvar∗is the conjugate transpose of Ψ±\nvar. The variational parameters can\nbe fixed by minimizing E±\nvarwith respect to ( x0, y0, kx, ky). The location(s) of\nminima thus obtained are\nx0= 0, y0=∓2γxΩ\n1−Ω2, kx=±2γx\n1−Ω2, ky= 0, (7)\nE±\nmin=2γ2\nx+ Ω2−1\nΩ2−1, (8)\nfor Ψ±\nvar. The most generic variational solution for these parameters’ sets is\nc+Ψ+\nvar+c−Ψ−\nvar, where c±are coefficients of superposition with |c+|2+|c−|2=\n1. The resultant density profile reflects the effective two-well potentials with\ntwo minima at ( x0= 0, y0=∓2γxΩ/1−Ω2). The validity of the varia-\ntional method has been checked by considering the set of parameter ( γx,Ω)\nis (1,0.5), (1 ,0.7), and (1 ,0.9). For these parameters set, the variational solu-\ntions Ψ±\nvarare degenerate, and the peak of total variational densities lies at\n±1.33,±2.74,±9.47, respectively. The densities profiles obtained by the\nvariational method for non-interacting systems are in excellent agreement with\nthe numerical results (not shown here) and so as the corresponding energies.\nThe rotating spin-2 BEC in the presence of ansitropic SO-coupling coupling is\nexactly solvable [16]. The ansatz presented here is an exact analytic solution\nto a one-dimensional SO coupling Hamiltonian under rotation.\n−5 0 5\ny05Veff(0,y)\n(a) Ω = 0.5V+2\neff\nV−2\neff\n−5 0 5\ny05\n(b) Ω = 0.7\n−20 0 20\ny020\n(c)Ω = 0.9\n1\nFig. 1 (Color online) (a)-(c) are the curves of effective potentials corresponding to Eq. (22)\nfor Ω rot= 0.5,0.7 and 0.9, respectively. For these curves, SO-coupling strength is γx= 1.\nThe effective potential incurred by the boson can be evaluated by calcu-\nlating the vector and scalar potentials as discussed in ref. [6, 16]. With the\ndefinition given in [6], the effective potential for rotating SO-coupled spin-2\nsystem calculated in Appendix. The effective potential curves for the system\nare\nVj\neff(x, y) =1\n2\u0002\n(1−Ω2)(x2+y2)−j2γ2\nx−2jγxΩy\u0003\n. (9)Springer Nature 2021 L ATEX template\nVortex-lattice formation in SO-coupled spin-2 BEC under rotation 5\nwhere j=±2,±1,0. The minima of effective potential occurred in the curves\ncorresponding to j=±2 depending on the region of y >0 ory <0, respec-\ntively illustrated in Fig. 1. Eq. (9) exhibiting that V+2\neff(x, y) and V−2\neff(x, y)\nare overlap at ( x= 0, y= 0) as shown in Fig. 1. In the region y <0,V−2\neff\ncorresponding to j=−2 curve is lower than other for j= 0,±1,+2 (curves\nforj= 0,±1 are not shown here) and in y > 0,V+2\neffwhich is correspond-\ning to j= +2 is lower than the other curves for j= 0,±1,−2. Fig. 1(a)-(c)\nshow the effective potential curves for rotation frequencies Ω = 0 .5,0.7 and\n0.9, respectively. The potentials incurred by boson are effectively equivalent to\nsymmetric double-well potentials with minima exhibit at ( x= 0, y=∓1.33),\n(x= 0, y=∓2.74), and ( x= 0, y=∓9.47) for Ω = 0 .5,0.7 and 0 .9,\nrespectively.\nIn order to investigate the system with isotropic SO coupling, we consider\nthe following variational ansatz\nΦvar=exp\u0010\nr2\n2σ2\u0011\np\nπσ2n+4Γ(n+ 2)×(A1r|n|einϕ, A2r|n+1|ei(n+1)ϕ, A3r|n+2|ei(n+2)ϕ,\nA4r|n+3|ei(n+3)ϕ, A5r|n+4|ei(n+4)ϕ)T,\n(10)\nwhere, the variational amplitudes are denoted as A1, A2, A3, A4, A5, while σ\nrepresents the variational width of the ansatz, and nis a variational integer.\nThe normalization condition imposes the constraint\n(n+2)σ2\u0000\n(n+ 3)σ2\u0000\nA2\n5(n+ 4)σ2+A2\n4\u0001\n+A2\n3\u0001\n+A2\n2+A2\n1\n(n+ 1)σ2= 1,(11)\non the variational parameters A1, A2, A3, A4, A5, nandσ. The variational\nenergy is\nEvar(A1, A2, A3, A4, A5, n, σ) = [4 A1A2γ(n+1)σ2+(n+1)σ2(2√\n6A2A3γ(n+2)σ2\n+(n+2)σ2(2√\n6A3A4γ(n+3)σ2+(n+3)σ2(4A4A5γ(n+4)σ2+A2\n5(n+4)σ2((n+5)\n(σ4+1)−2(n+4)σ2Ω)+A2\n4((n+4)(σ4+1)−2(n+3)σ2Ω))+ A2\n3((n+3)(σ4+1)−\n2(n+ 2)σ2Ω)) + A2\n2((n+ 2)( σ4+ 1)−2(n+ 1)σ2Ω)) + A2\n1((n+ 1)( σ4+ 1)\n−2nσ2Ω)]/(2(n+ 1)σ4) (12)\nThe variational energy can be minimized with all variational param-\neters subjected to the constraint specified in Eq (11), to decide\nthe variational parameters. To check the validity of the variational\nmethod in this case, we choose ( γ= 1 ,Ω = 0 .9), the variational\nparameters while minimization of (12) are ( A1, A2, A3, A4, A5, n, σ) =\n(−2.3728,0.5012,−0.0645,0.0055,−0.0002,98,0.9491). The comparison of\nvariational density |Φvar(r)|2, and exact numerically evaluated single particleSpringer Nature 2021 L ATEX template\n6 Vortex-lattice formation in SO-coupled spin-2 BEC under rotation\ndensity profiles |Ψ(r)|2, agree with each other. The peak of total variational\ndensity lies along a circle of radius 9 .47 and is related to the effective toroidal\npotential incurred by the boson. For ( γ= 1,Ω = 0 .7) the variational param-\neters while minimization are variational energy ( A1, A2, A3, A4, A5, n, σ) =\n(0.7029,−0.5115,0.2173,−0.0589,0.0094,9,0.8489) for parameters’ sets,\nrespectively. The comparison of |Φvar(r)|2, and|Ψ(r)|2, agree with each other\nand the peak of total variational density lies along a circle of radius 2 .70.\n3 Mean-field interacting Model\nIn q2D trapping potential, a rotating SO-coupled spin-2 BEC under mean-field\napproximation can be described by a set of five CGPEs [21]\ni∂ϕ±2\n∂t=Hϕ±2+c0ρϕ±2+c1{F∓ϕ±1±2Fzϕ±2}+c2Θϕ∗\n∓2√\n5+ Γ±2,(13a)\ni∂ϕ±1\n∂t=Hϕ±1+c0ρϕ±1+c1 r\n3\n2F∓ϕ0+F±ϕ±2±Fzϕ±1!\n−c2Θϕ∗\n∓1√\n5+ Γ±1, (13b)\ni∂ϕ0\n∂t=Hϕ0+c0ρϕ0+c1r\n3\n2{F−ϕ−1+F+ϕ1}+c2Θϕ∗\n0√\n5+ Γ0,(13c)\nwhere, Φ = ( ϕ2, ϕ1, ϕ0, ϕ−1, ϕ−2)Tis a five component order parameter,\nH=−∇2\n2+V−ΩLz,Θ =2ϕ2ϕ−2−2ϕ1ϕ−1+ϕ2\n0√\n5, F z=2X\nj=−2j|ϕj|2\nF−=F∗\n+= 2ϕ∗\n−2ϕ−1+√\n6ϕ∗\n−1ϕ0+√\n6ϕ∗\n0ϕ1+ 2ϕ2ϕ∗\n1,\nandρ=P2\nj=−2|ϕj|2is the total density. The Laplacian, trapping potential\nV, interaction parameters ( c0, c1, c2), are defined as\n∇2=\u0012∂\n∂x2+∂\n∂y2\u0013\n, V=x2+y2\n2, c0=√\n2παz2πN(4a2+ 3a4)\n7aosc,(14a)\nc1=√\n2παz2πN(a4−a2)\n7aosc, c 2=√\n2παz2πN(7a0−10a2+ 3a4)\n7aosc,(14b)\nand the Γ’s for case I are\nΓ±2=−iγx∂ϕ±1\n∂x,Γ±1=−i \nγx∂ϕ±2\n∂x+r\n3\n2γx∂ϕ0\n∂x!\n,Springer Nature 2021 L ATEX template\nVortex-lattice formation in SO-coupled spin-2 BEC under rotation 7\nΓ0=−i r\n3\n2γx∂ϕ1\n∂x+r\n3\n2γx∂ϕ−1\n∂x!\n,\nforcase II are\nΓ±2=−iγ\u0012∂ϕ±1\n∂y±i∂ϕ±1\n∂x\u0013\n,\nΓ±1=−iγ \n∂ϕ±2\n∂y+r\n3\n2∂ϕ0\n∂y∓i∂ϕ±2\n∂x±ir\n3\n2∂ϕ0\n∂x!\n,\nΓ0=−iγ r\n3\n2∂ϕ1\n∂y+r\n3\n2∂ϕ−1\n∂y−ir\n3\n2∂ϕ1\n∂x+ir\n3\n2∂ϕ−1\n∂x!\n.\nwhere a0, a2, a4represent the s-wave scattering lengths in the allowed scat-\ntering channels, and αx, αyandαzare the anisotropy parameters of trapping\nfrequency. All these quantities are dimensionless, which have been calculated\nby expressing lengths ( a0, a2, a4, x, y) in units of aosc≡p\nℏm/ω, energy, den-\nsity and time in the unit of ℏω,a−2\nosc, and ω−1, respectively. The dimensionless\nform of the mean-field model for the SO-coupled spin-2 has two conserved\nquantities: one is the normalization conditionR\nρ(x, y)dxdy = 1. And the other\nis energy per particle defined as\nE=Z\ndxdy\"+2X\nj=−2ϕ∗\njHϕj+c0\n2ρ2+c1\n2|F|2+c2\n2|Θ|2++2X\nj=−2ϕ∗\njΓj#\n,(17)\nfor an SO-coupled BEC. Under the influence of SO coupling longitudinal\nmagnetization\nM=Z\u0000\n2|ϕ+2|2+|ϕ+1|2− |ϕ−1|2−2|ϕ−2|2\u0001\ndxdy, (18)\nis not conserved, although it remain conserved for γx= 0 and γ= 0.\n4 Numerical Results\nWe take a typical example of antiferromagnetic system trapped in a q2D\npotential, consisting of 50 ,000 atoms of an SO-coupled23Na spin-2 BECs with\nω= 2π×10 Hz, ωz= 2π×100 Hz, and so αx=αy= 1 and αz= 10\nwhere αν=ων/ωx. The three scattering lengths considered for this system\narea0= 34.9aB, a2= 45.8aB, a4= 64.5aB[19] and corresponding triplet of\ndimensionless interaction strengths are ( c0, c1, c2) = (340 .45,16.90,−18.25).\nThe oscillator length for this system is 4 .69µm. For the cyclic phase, we exam-\nine a system composed of 50 ,000 atoms of87Rb spin-2 BECs confined in a\nq2D trapping potential with the same trapping frequencies considered for anti-\nferromagnetic phase. The chosen triplet of scattering lengths for this systemSpringer Nature 2021 L ATEX template\n8 Vortex-lattice formation in SO-coupled spin-2 BEC under rotation\n1\nFig. 2 (Color online) (A) and (B) shown the ground state component densities for an\nanisotropic SO coupling and isotropic SO-coupling23Na spin-2 BEC, respectively. The inter-\naction parameters for both cases are c0= 340 .45, c1= 16.90, c2=−18.25, and SO coupling\nstrength for (A) γx= 1 and (B) = γ= 1. Similarly, (C) and (D) represent the same for\n87Rb spin-2 BEC for interaction parameters c0= 1164 .80, c1= 13.88, c2= 0.43.\nconsists of a0= 87 .93aB,a2= 91 .28aB, and a4= 99 .18aB[20], result-\ning in the corresponding dimensionless interaction strengths of ( c0, c1, c2) =\n(1164 .80,13.88,0.43). The oscillator length in this system is 2 .41µm.\nWe investigate the stationary state solutions of these systems in the pres-\nence SO coupling. To solve the set of CGPEs in (13a)-(13c)), we employ the\ntime-splitting Fourier spectral method [24–27]. To obtain the numerical solu-\ntions, a spatial step size of 0 .1 and a temporal step size of 0 .001 to be used.\nIn the absence of a rotation frequency (Ω = 0), the equations are evolved in\nimaginary time propagation to obtain the ground state of the system. Multi-\nple numerical simulations are performed with different initial guesses, and the\nstate with the lowest energy is considered as the ground state solution. When\nΩ = 0, the spin-2 BEC can exhibit various ground state solutions depending\non the interaction parameters and the strength of the spin-orbit coupling. The\nmost general solution is axisymmetric, characterized by ( −2,−1,0,+1,+2)\nphase singularities at densities corresponding to j= +2 ,+1,0,−1,−2 com-\nponents [28]. Additionally, other patterns such as stripes [28], square latticesSpringer Nature 2021 L ATEX template\nVortex-lattice formation in SO-coupled spin-2 BEC under rotation 9\n[28], or triangular lattices [28] can arise through the superposition of counter-\npropagating plane waves, four plane waves with propagation vectors at a right\nangle to each other, or three plane waves with propagation vectors at an angle\nof 2π/3 to each other, respectively. When performing simulation in a rotating\nframe, previously obtained ground state solutions are considered as an ini-\ntial guess. The dynamical stability of each solution has been studied through\nreal-time propagation, with the converged solution obtained from the imagi-\nnary time propagation serving as the initial guess for these dynamical stability\nstudies.\nForcase I In the absence of rotation frequency (Ω = 0), numerically\nobtained ground state solutions for q2D configuration with the aforementioned\ncoupling for γx= 1 having a vertical stripe pattern in the component densities\nshown in Fig. 2A and 2C for antiferromagnetic and cyclic phase, respectively.\nThe vertical stripe are emerge due to the effect of SO-coupling ∝Sxˆpx. Fig.\n2B and 2D, display the ground state solutions for case II , representing hori-\nzontal stripe patterns in the antiferromagnetic and triangular-lattice pattern\nin cyclic phase, respectively, for the aforementioned coupling strength γ= 1.\nThe nature of the solution would depend on the combined effect of type of\ninteraction, type of SO coupling and strength of SO coupling.\nAntiferromagnetic phase\nCase I: For the simulation in the rotating frame, we consider different rota-\ntion frequencies, Ω = 0 .5,0.7,and 0 .9 times the trapping frequency. We select\nthese rotation frequencies to explore all possible vortex lattice configurations\nthat can arise. The anisotropy in the SO coupling leads to a distinction in the\nangular momentum of the components along the x−andy-directions, thereby\nimpacting the rotational symmetry. Fig. 3(A) illustrates the component den-\nsities with a rotation frequency of Ω = 0 .5, where a significant portion of the\nvortices align themselves in a central chain along the x-direction. When consid-\nering higher rotation frequencies, such as Ω = 0 .7, a larger number of vortices\nemerge in the system. Some vortices appear on both sides of the central chain,\nas depicted in Fig. 3(B). The number of vortices in the off-vortices-chain region\nincreases with the rotation frequency. At very large rotation frequency Ω = 0 .9\nall the vortices arrange themselves on both sides of the central chain, as illus-\ntrated in Fig. 3(C). Here, we observed that with strong anisotropic SO coupling\nand large rotation, some local minima about the trap become discernible, and\ntriangular vortex lattices would preferentially form in the regions outside the\nvortex chain.\nCase II: In the rotating frame, we have observed the evolution of the\nsolution already obtained with (Ω = 0) under various rotation frequencies,\n0.5, 0.7, and 0.9. Fig. 4(A) illustrates the solution rotated with a frequency\nof Ω = 0 .5. With this relatively low rotation frequency, a small number of\nvortices have emerged, and all of them tend to arrange themselves in a reg-\nular pattern. Fig. 4(B) depicts the vortex lattice structure obtained with a\nhigher rotation frequency of Ω = 0 .7. In this case, the central region of theSpringer Nature 2021 L ATEX template\n10 Vortex-lattice formation in SO-coupled spin-2 BEC under rotation\n1\nFig. 3 (Color online) (A) shows the ground state component densities for an anisotropic\nSO-coupled23Na spin-2 BEC with c0= 340 .45, c1= 16.90, c2=−18.25,γx= 1 with\nΩ = 0 .5. (B) and (C) represent the same for the rotation frequencies Ω = 0 .7 and 0.9,\nrespectively.\nstructure accommodates singularities with charges (0 ,+1,+2,+3,+4) in the\n(j=−2,−1,0,+1,+2) components, respectively. The charges of these sin-\ngularities have been determined from the phase of the density profiles (not\nshown here). As the rotation frequency increases, the number of vortices also\nincreases. Subsequently, we further increased the rotation frequency to Ω = 0 .9.\nAt this frequency, we observed the formation of a ring-type structure with a\ngiant vortex located near the center of the trap. And triangular vortex lat-\ntices appear in the annular region surrounding the central vortex depicted in\nFig. 4(C). The number of vortices within a circle of radius 9 .47 is consistent\nwith the number obtained from variational analysis. The variational analysis\nfor this case is showing a excellent agreement particularly at higher rotation\nfrequencies.\nCyclic phase\nCase I: The solution reported in Fig. 2(C) evolved in a rotating frame with\nrotation frequencies of Ω = 0 .5, 0.7, and 0 .9. The corresponding results are\ndepicted in Fig. 5(A), (B), and (C), respectively. At a small rotation frequency\nof Ω = 0 .5, the majority of vortices align themselves along a horizontal chain,\nwhile the remaining vortices position themselves either above or below this\nvortex chain. This configuration is illustrated in Fig. 5(A). As the rotationSpringer Nature 2021 L ATEX template\nVortex-lattice formation in SO-coupled spin-2 BEC under rotation 11\n1\nFig. 4 (Color online) (A) present the component densities of the ground state for an\nisotropic SO-coupled23Na spin-2 BEC with c0= 340 .45, c1= 16.90, c2=−18.25,γ= 1\nwith Ω = 0 .5. (B) and (C) showing the same for rotation frequencies Ω = 0 .5,0.7, and 0.9,\nrespectively.\nfrequency increases to Ω = 0 .7, more vortices emerge in the condensate, as\ndepicted in Fig. 5(B). Finally, at a very high rotation frequency of Ω = 0 .9,\na large number of vortices are observed, and they arrange themselves in a\ntriangular vortex lattice pattern, as shown in Fig. 5(C).\nCase II: Fig. 6(A) illustrate the vortex lattice configuration at Ω = 0 .5. In\nFig. 6(B), vortex lattice configuration shows corresponding to Ω = 0 .7, where\nthe central part of the structure hosting (0 ,+1,+2,+3,+4) singularities in\n(j=−2,−1,0,+1,+2) components. At Ω = 0 .9, a vortex lattice structure\nhaving a giant vortex at the center and the triangular vortex lattice pattern\nappear in the annulus as shown in Fig. 6(C).\nThe rotational energy of a scalar BEC in the rotating frame ( EΩ−E0) is\nthe energy of rigid-body rotation −IΩ2/2 where Iis the moment of inertia of\nthe condensate [29]. This energy is proportional to the square of the angular\nfrequency [13, 29], where EΩis given by Eq. (17). In the case of spin-2 spinor\nantiferromagnetic and cyclic BECs, also hold a similar relation. We illustrate\nin Fig.7 the rotational energy for both antiferromagnetic and cyclic spin-2 BEC\nfor isotropic SO coupling. Here, we plot the energy of the ground state versus Ω.\nThe energies of antiferromagnetic and cyclic systems lie in two separate distinct\nlines exhibit similar qualitative behavior. As the rotation frequency increases,\nthe energy decreases due to the negative contribution of the rotational energy\nterm−ΩLzin the energy expression Eq. (17). This behavior is particularlySpringer Nature 2021 L ATEX template\n12 Vortex-lattice formation in SO-coupled spin-2 BEC under rotation\n1\nFig. 5 (Color online) (A) shows the ground state density profiles of the individual com-\nponents of the SO-coupled87Rb spin-2 BEC with c0= 1164 .80, c1= 13.88, c2= 0.43,\nanisotropic SO coupling γx= 1, Ω = 0 .5. Similarly, (B) and (C) present the component\ndensities for Ω = 0 .7 and 0.9, respectively.\nevident in Figure 7, where the rotational energy is behaving ∝ −Ω2as the\nrotation frequency increases specially at higher Ω.\nFerromagnetic phase: Here we have considered 50 ,000 atoms of83Rb in\nan SO-coupled spin-2 BEC with or without rotation frequency. The value\nof three scattering length considered for this system are a0= 83 .0aB, a2=\n82.0aB, a4= 81 .0aB[19] and corresponding interaction parameters are\n(c0, c1, c2) = (980 .33,−1.71,6.86). The numerical results obtained for83Rb\nunder rotation demonstrate qualitative similarities to the cyclic87Rb BEC.\nSo as we have opted not to include the results obtained for the ferromag-\nnetic case in this work. In the actual parameters magnitude of c1andc2is\nvery small comparing to c0, so at moderate to high rotation frequencies the\neffect of interaction is neglible compare to Ω. By adjusting the parameters c1\n(more attractive) or c2(more repulsive) or both, such that the system’s inter-\naction becomes comparable to the rotation frequency, we can effectively study\nthe impact of interaction terms. This will allow us to distinguish between the\nferromagnetic and cyclic phases. [16].\n5 summary\nWe have demonstrated the vortex lattice structures in q2D in SO-coupled spin-\n2 BEC under rotation by employing the variational method for single particle\nand numerical solution of the mean-field model. In this work, we considered\nisotropic as well as fully anisotropic SO-coupling strengths with moderate and\nhigher rotation frequencies. In this work, antiferromagnetic23Na , cyclic phaseSpringer Nature 2021 L ATEX template\nVortex-lattice formation in SO-coupled spin-2 BEC under rotation 13\n1\nFig. 6 (Color online) (A) shows the ground state density profiles of the individual com-\nponents of the SO-coupled87Rb spin-2 BEC with c0= 1164 .80, c1= 13.88, c2= 0.43,\nisotropic SO coupling γ= 1, Ω = 0 .5. Similarly, (B) and (C) present the component densi-\nties for Ω = 0 .7 and 0.9, respectively.\n0 0.2 0.4 0.6 0.8 1\n\n-20-15-10-50\n23Na\n87Rb(E\n\u0000E0)\nFig. 7 (Color online) The plot shows the variation of rotational energy ( EΩ−E0) with the\nΩ for both antiferromagnetic23Na and cyclic87Rb systems.\n87Rb and ferromagnetic83Rb have been discussed in detail. We observed that\nat fast rotational frequencies, all the phases have qualitatively similar behavior\nwhich highlights the negligible role of the spin-exchange interactions. In the\npresence of anisotropic SO coupling γxSxˆpxbosons experienced symmetric\ndouble-well potential, and the central vortices chain appeared along the x-\ndirection in the numerical solutions. Depending on the rotation frequency, the\nmajority of vortices are arranged on both sides of the central chain of vortices.\nWe have reported effective potential curves explored by using single particle\nHamiltonian. With isotropic SO coupling γ(Sxˆpy−Syˆpx) in the presence of\nrotation frequency bosons experienced toroidal shape trapping potential, and\nthe radius of the toroidal trap increased with the rotation frequency. So at\nhigher rotation frequency in an isotropic SO-coupled system giant vortex-typeSpringer Nature 2021 L ATEX template\n14 Vortex-lattice formation in SO-coupled spin-2 BEC under rotation\nsolution observed as a ground state solution. At large rotation, vortices favor\ntriangular vortex lattice patterns in the annual region. All the numerical results\nreported in this work have been explained by using variational analysis for\nsingle particle Hamiltonian.\n6 Appendix\nThe single particle Hamiltonian for case I is\nHrot= \nˆp2\nx+ ˆp2\ny\n2+x2+y2\n2−ΩLz!\n×1+γxSxpx, (19)\nUnder a unitary transformation the Hrotcan be diagonalised, which has the\neigenvalues\nE±2(kx, ky) =1\n2\u0002\nk2\nx+k2\ny±4γxkx\u0003\n−Ω(xky−ykx), (20a)\nE±1(kx, ky) =1\n2\u0002\nk2\nx+k2\ny±2γxkx\u0003\n−Ω(xky−ykx), (20b)\nE0(kx, ky) =1\n2(k2\nx+k2\ny)−Ω(xky−ykx), (20c)\nThe spectrum in Eqs. (20a)-(20b) around a minima can be described by a\nparabola of form ( kx−kxmin)2+ (ky−kymin)2+Emin, which describes the\nparticle moving in an effective gauge field ( A,¯Φ) = ( {kxmin, kymin,0}, Emin),\nwhere Aand¯Φ are vector and scalar potentials, respectively [6]. In the pres-\nence of rotation term kxmin, kyminandEminbecome spatially dependent [6].\nRewriting Eqs. (20a)-(20b) as\nE±2(kx, ky) =1\n2\u0002\n(kx±(2γx±yΩ))2+ (ky−xΩ)2\u0003\n+Emin\n±2(x, y), (21a)\nE±1(kx, ky) =1\n2\u0002\n(kx±(γx±yΩ))2+ (ky−xΩ)2\u0003\n+Emin\n±1(x, y), (21b)\nE0(kx, ky) =1\n2((kx+yΩ)2+ (ky−xΩ)2)\n+Emin\n0(x, y), (21c)\nwhere A±2=∓(2γx±yΩ, xΩ),A±1=∓(γx±yΩ, xΩ),A0= (−yΩ, xΩ),\nand¯Φj=Emin\njwith ( j= 0,±1,±2). The effective potentials, which are the\nsums of trapping and scalar potentials [6], can now be written as\nV±2\neff(x, y) =1\n2\u0002\n(1−Ω2)(x2+y2)−4γ2\nx∓4yγxΩ\u0003\n, (22a)Springer Nature 2021 L ATEX template\nVortex-lattice formation in SO-coupled spin-2 BEC under rotation 15\nV±1\neff(x, y) =1\n2\u0002\n(1−Ω2)(x2+y2)−γ2\nx∓2yγxΩ\u0003\n, (22b)\nV0\neff(x, y) =1\n2\u0002\n(1−Ω2)(x2+y2)\u0003\n. (22c)\nFrom Eqs. (22)a - (22)c, the effective potential curves correspond to V±2\neffare\nthe lowest lying depending on the region.\n7 Acknowledgement\nI thank Sandeep Gautam for the fruitful discussions and comments on the\nmanuscript.\nReferences\n[1] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I.B. Spielman, Nature 471, 83 (2011);\nD.L. Campbell, R.M. Price, A. Putra, A. Vald´ es-Curiel, D. Trypogeorgos,\nand I.B. Spielman, Nature Communications 7, 10897 (2016); X. Luo, L.\nWu, J. Chen, Q. Guan, K. Gao, Zhi-Fang Xu, L. You, and R. Wang,\nScientific Reports 6, 18983 (2016).\n[2] Z.F. Xu, R. L¨ u, and L. You, Phys. Rev. A 83, 053602 (2011).\n[3] T. Kawakami, T. Mizushima, and K. Machida, Phys. Rev. A 84,\n011607(R)(2011) ; Z. F. Xu, Y. Kawaguchi, L. You, and M. Ueda, Phys.\nRev. A 86, 033628 (2012).\n[4] B. M. Anderson,I. B. Spielman, and G. Juzeli¯ unas, Phys. Rev. Lett. 111,\n125301 (2013); Z.-F. Xu, L. You, and M. Ueda, Phys. Rev. Rev. A 87,\n063634 (2013).\n[5] X.-Q. Xu and J. H. Han, Phys. Rev. Lett. 107, 200401 (2011).\n[6] J. Radi´ c, T. A. Sedrakyan, I. B. Spielman, and V. Galitski, Phys. Rev.\nA.84, 063604 (2011); J. Radi´ c, Spin-orbit-coupled quantum gases (2015)\n[Doctoral dissertation, Maryland University].\n[7] X.-F. Zhou, J. Zhou, and C. Wu, Phys. Rev. A 84, 063624 (2011).\n[8] C.-F. Liu, H. Fan, Y.-C. Zhang, D.-S. Wang, and W.-M. Liu, Phys. Rev.\nA86, 053616 (2012).\n[9] A. Aftalion and P. Mason, Phys. Rev. A 88, 023610 (2013).\n[10] H. Wang, L. Wen, H. Yang, C. Shi, and J. Li, J. Phys. B: At. Mol. Opt.\nPhys. 50, 155301 (2017). C. Shi, L. Wen, Q. Wang, H. Yang, and H.\nWang, J. Phys. Soc. Jpn. 87, 094003 (2018); H. Yang, Q. Wang, N. Su,Springer Nature 2021 L ATEX template\n16 Vortex-lattice formation in SO-coupled spin-2 BEC under rotation\nand L. Wen, Eur. Phys. J. Plus, 134, 589 (2019); Q.-B. Wang, H. Yang,\nN. Su, and L.-H. Wen, Chin. Phys. B 29, 116701 (2020); J.-G. Wang and\nY.-Q. Li, Results in Physics 17, 103099 (2020).\n[11] C.-F. Liu, Y.-M. Yu, S.-C. Gou, and W.-M. Liu, Phys. Rev. A. 87, 063630\n(2013).\n[12] Q. Zhao and H. Bi, Int. J .Theor. Phys. 60, 2778 (2021).\n[13] S. K. Adhikari, J. Phys.: Condens. Matter 33, 065404 (2021).\n[14] P. Peng, G.-Q. Li, W.-L. Yang, and Z.-Y. Yanga, Phys. Lett. A 382,\n2493 (2018); X.-F. Zhang, B. Li, and S.-G. Zhang, Laser Phys. 23, 105501\n(2013).\n[15] J. Li, X.-F. Zhang, and W.-M. Liu, ,Ann. Phys. 87396 (2018).\n[16] P. Banger, R K. Kumar, A. Roy, and S. Gautam, J. Phys.: Condens.\nMatter 35, 045401 (2023).\n[17] G.-P. Chen, P. Tu, C.-B. Qiao, J.-X. Zhu, Q. Jia, and X.-F. Zhang,\nFrontiers in Physics, 9, 613 (2021).\n[18] H. Zhu, C.-F. Liu, D.-S. Wang, S.-G. Yin, L. Zhuang, and W.-M. Liu,\nPhys. Rev. A 104, 053325 (2021).\n[19] C V. Ciobanu, S. Yip, T. Ho, Phys. Rev. A 61(2000) 033607.\n[20] A. Widera, F. Gerbier, S. F¨ olling, T. Gericke, O. Tatjana and I. Bloch,\nNew Journal of Physics 8(2006) 152.\n[21] Y. Kawaguchi, M. Ueda, Physics Reports 520(2012) 253-381.\n[22] N. Goldman, G Juzeli¯ unas, P. ¨Ohberg, and I. B. Spielman, Rep. Prog.\nPhys. 77, 126401 (2014).\n[23] I. E. Rashba and Y. A. Bychkov Journal of Physics C 17, 6039 (1984).\n[24] P. Kaur, A. Roy, and S. Gautam, Comp. Phys. Comm. 259, 107671 (2021)\n[25] P. Banger, P. Kaur, S. Gautam, Int. J. Mod. Phys. C 33(4) , 2250046\n(2022).\n[26] R. Ravisankar, D. Vudragovi´ c, P. Muruganandam, A. Balaˇ z, and S. K.\nAdhikari, Comp. Phys. Comm. 259, 107657 (2021); P. Muruganandam,\nA. Balaˇ z , and S. K. Adhikari, Comp. Phys. Comm. 264, 107926 (2021).Springer Nature 2021 L ATEX template\nVortex-lattice formation in SO-coupled spin-2 BEC under rotation 17\n[27] P. Banger, P. Kaur, A. Roy, and S. Gautam, Comp. Phys. Comm. 279,\n108442 (2022).\n[28] P. Kaur, S. Gautam, S.K. Adhikari, Phys. Rev. A, 105, 023303 (2022).\n[29] AL. Fetter, Rev. Mod. Phys. 81(2009) 647." }, { "title": "1412.6441v2.Magnetic_interactions_in_strongly_correlated_systems__spin_and_orbital_contributions.pdf", "content": "arXiv:1412.6441v2 [cond-mat.str-el] 13 Apr 2015Magnetic interactions in strongly correlated systems:\nspin and orbital contributions\nA. Secchia, A. I. Lichtensteinb, M. I. Katsnelsona\naRadboud University, Institute for Molecules and Materials , 6525 AJ Nijmegen, The\nNetherlands\nbUniversitat Hamburg, Institut f¨ ur Theoretische Physik, J ungiusstraße 9, D-20355\nHamburg, Germany\nAbstract\nWe present a technique to map an electronic model with local interac tions (a\ngeneralized multi-orbital Hubbard model) onto an effective model of interacting\nclassical spins, by requiring that the thermodynamic potentials associated t o\nspin rotations in the two systems are equivalent up to second order in the ro-\ntation angles, when the electronic system is in a symmetry-broken p hase. This\nallows to determine the parameters of relativistic and non-relativist ic magnetic\ninteractions in the effective spin model in terms of equilibrium Green’s f unc-\ntions of the electronic model. The Hamiltonian of the electronic syste m in-\ncludes, in addition to the non-relativistic part, relativistic single-par ticle terms\nsuch as the Zeeman coupling to an external magnetic fields, spin-or bit coupling,\nand arbitrary magnetic anisotropies; the orbital degrees of free dom of the elec-\ntrons are explicitly taken into account. We determine the complete r elativistic\nexchange tensors, accounting for anisotropic exchange, Dzyalo shinskii-Moriya\ninteractions, as well as additional non-diagonal symmetric terms ( which may\ninclude dipole-dipole interaction). The expressions of all these magn etic inter-\nactions are determined in a unified framework, including previously dis regarded\nfeatures such as the vertices of two-particle Green’s functions a nd non-local\nself-energies. We do not assume any smallness in spin-orbit coupling, so our\ntreatment is in this sense exact. Finally, we show how to distinguish an d ad-\ndress separately the spin, orbital and spin-orbital contributions to magnetism,\nproviding expressions that can be computed within a tight-binding Dy namical\nMean Field Theory.\nKeywords: Magnetism in strongly correlated systems; Anisotropic exchange\ninteraction; Dzyaloshinskii-Moriya interaction; Green’s functions; Orbital\nproperties.\nEmail addresses: a.secchi@science.ru.nl (A. Secchi), andrea.secchi@gmail.com (A.\nSecchi)\nPreprint submitted to Elsevier June 23, 20211. Introduction\nEstablishing a rigorous connection between magnetic and electronic descrip-\ntions of condensed matter systems is a challenging problem [1], whose formal\nstatement can be formulated as follows: Given a physical system de scribed by\nmeans of a completely known electronic Hamiltonian, what is the spinHamil-\ntonian (supposing that it exists) that most closely reproduces the spectral and\ndynamical features of the system?\nThe answer to this important question is, of course, far from being straight-\nforward. It is well known, e.g., that the spectrum of the lowest ene rgy band of\nthe single-orbital Hubbard model at half filling with nearest-neighbo ur hopping\nTandstrongon-siteCoulombrepulsion Ucanbeeffectivelyrepresentedinterms\nof the antiferromagnetic quantum Heisenberg Hamiltonian, as follow s from per-\nturbation theory in small |T|/U. Dynamics of the electronic system, however,\nmay involve hopping transitions via intermediate higher bands, which a re not\ncaptured in the Heisenberg Hamiltonian alone, as well as real hopping processes\nbecome relevant at other electronic fillings (as a first correction, o ne should con-\nsider theT-Jmodel [2]). A non-Heisenberg character of magnetic interactions\nin itinerant systems wasexplicitly demonstrated, e.g., for the narro w-bandHub-\nbard model on the Bethe lattice beyond half filling [3]. The problem gets much\nmore complicated if one attempts to map more realistic electronic sys tems to\nmagnetic models: for example, the natural extension of the single- orbital Hub-\nbard model is the multi-orbital Hubbard model [4–8], which includes more than\njust one orbital per site, being a more appropriate description of r elevant sys-\ntems such as dandfmaterials. Moreover, when both spin and orbital degrees\nof freedom of the electrons are taken into account, their interpla y gives rise to\nrelativistic interactions such as spin-orbit coupling and anisotropies [9].\nWhen no smallness in some characteristic energy parameters of the system\ncan be assumed (such as |T|<< Uin the Hubbard model), the parameters\ndescribing the magnetic interactions in an electronic system can be definedby\nimposing the equivalence between the response to spin rotations of a quantity\ncharacterizing the system and the analogous response computed for a reference\nclassical spin model [1]. In the case of symmetry-broken phases, the quant ity\nwhich is generally considered is the thermodynamic potential [10–12] computed\nfor anout-of-equilibrium state or statistical superposition, that is, either a pure\nstate which is notan eigenstate of the electronic Hamiltonian, or a statistical\nsuperposition of eigenstates whose weights do notdepend only on their ener-\ngies (which would be the case for the Boltzmann distribution, with weig hts\nWn(β) = e−βEn/Z, whereEnis the eigenenergy of state n,βis the inverse tem-\nperature and Zis the partition function). The idea of using a symmetry-broken\nstate is similar in spirit to the Higgs mechanism: we need first to solve th e non-\nperturbative many-body problem and find the local moments (mass ive Higgs\nfields) and then use the information contained in the single-particle G reen’s\nfunctions and the vertex functions to find perturbatively the sof t modes related\nwith exchange interactions. It has been shown that the expressio ns for the ex-\nchange parameters obtained by applying this approach in the non-r elativistic\n2case, within the framework of time-dependent density functional theory in the\nadiabatic approximation, provide an accurate expression for the s pin-wave stiff-\nness [13], while the computation of static properties requires the int roduction\nof constraining magnetic fields to equilibrate the non-equilibrium spin c onfig-\nuration [14, 15]. However, the corresponding corrections to the e xchange pa-\nrameters [15] are small in the adiabatic approximation, that is, when typical\nmagnon energies are small in comparison with the Stoner splitting [13 ]. This\njustifies our approach. In the non-relativistic case, we have rece ntly extended\nthe treatment of Ref.[10] to systems driven explicitly out of equilibriu m by time-\ndependent external electric fields, by considering the potential a rising from the\nnon-equilibrium Kadanoff-Baym partition function [16] (in Refs.[10–12 , 16] the\nelectronic system was modelled by means of the multi-orbital Hubbar d model).\nMaking a mapping to a classical spin model means that the target of the\nmapping is a Hamiltonian involving a set of interacting unit vectors ei, which\ncanrotateinspaceasclassicalvectors, their componentsvaryin gin acontinuous\ndomain. These vectorscan be called classical spins , and ourgoal is to determine\nthe coefficients of their interactions. It should be noted that thes e effective\nparameters include information related to the magnitudes of the loc al spins,\nsince this is not included into the unit vectors ei.\nUnfortunately, the most correct spin model to be used for the ma pping with\na given electronic system may include interactions between up to an a rbitrary\nnumber of spins, which is not known a priori. In practice, it is generally as-\nsumed that the most relevant magnetic parameters are the exter nal magnetic\nfield, which couples linearly with the spins, and the spin-spin pair intera ctions,\nquantified by the 3 ×3 exchange tensors Hij(one for each pair of spins labelled\nbyiandj). In the non-relativistic case, the exchange tensors are propor tional\nto the identity matrix, Hij→ Jij1, whereJijis the non-relativistic (isotropic)\nexchange parameter. In the relativistic case, the exchange tens ors are general\nmatrices which include anisotropic exchange, Dzyaloshinskii-Moriya, and other\n(symmetric) pair interactions [9]. It should be remarked that higher -order in-\nteractions such as bi-quadratic exchange have been suggested t o be relevant for\nmaterials such as MnO [17], CuO 2plaquettes [18], pnictide superconductors\n[19], and models such as spin ladders [20].\nHowever, in this work we will show that the thermodynamic potential un-\nder small spin rotations of a rather general relativistic electronic s ystem in a\nbroken-symmetry phase is equivalent to that of a quadratic spin mo del with a\ngeneral 3 ×3 exchange tensor, if the equivalence is required up to the second\norder in the rotation angles. We will prove this by determining the complete\nquadratic exchange tensor, which up to now was determined only in t he non-\nrelativistic case [10, 11] or limited to the Dzyaloshinskii-Moriya interac tion in\nthe relativistic case [12]. In addition to that, we will remove two uncon trolled\napproximations that were previously adopted [10, 11], namely neglec ting the\nvertices of two-particle Green’s functions and neglecting non-loca l components\nof the self-energies. The electronic model that we will consider has a completely\ngeneral single-particle Hamiltonian, including terms such as the Zeem an cou-\npling between the magnetic field and the electronic spins, local and no n-local\n3spin anisotropies and spin-orbit couplings, in addition to the non-rela tivistic\nhopping. We emphasize that no smallness will be assumed in anyof the single-\nparticle terms, so that relativistic terms are accounted for in a non -perturbative\nway. We will adopt the only simplification of assuming that the interact ion\nHamiltonian is rotationally invariant (in the same spirit as what was done in\nthe previous literature [10–12, 16]). The electronic model is then a relativis-\ntic multi-orbital Hubbard model . Moreover, in this work we will consider also\nthe contributions to magnetic interactions due to the orbitaldegrees of free-\ndom of the electrons, and we will show how to distinguish them from th e usual\ncontributions due to the intrinsic spin-1 /2.\nWhile the original approach of Ref.[1], based on the non-relativistic mu lti-\nscattering formalism within density-functional theory, was recen tly extended to\naccount for relativistic interactions [21, 22], our approach here is d ifferent, since\nwe apply a consistent many-body treatment of the multi-band Hubb ard model,\nincluding both relativistic and non-relativistic effects on equal footin g, as well\nas fully including contributions to magnetism due to orbital and intrins ic spin\nof the electrons (i.e., we do not assume that the orbital moment is fr ozen).\nThis Article is structured as follows. In Section 2 we introduce our re ference\nelectronic Hamiltonian, specifying the model and the notation; in Sec tion 3 we\nexplain our procedure to probe the response of the electronic sys tem to rota-\ntions of the spin quantization axes; in Sections 4 and 5 we derive the e ffective\nrotational action of the electronic system for small deviations fro m the initial\ndirections of the quantization axes; in Section 6 we derive the effect ive potential\nfor static spin rotations applied to the electronic Hamiltonian; in Sect ion 7 we\nderive the analogous effective potential for a model of classical sp ins, and we\nderive a set of equations connecting the parametersof the spin mo del to Green’s\nfunctions of the electronic model, which allows to identify the magnet ic param-\neters; in Section 8 we solve the system in the general relativistic cas e (for ease\nof reference, the resulting formulas are summarized in Section 8.3) ; in Section\n9 we consider the spin-1 /2 single-orbital Hubbard model in the non-relativistic\nregime, establishing correspondence with the previous literature; in Section 10\nweshowhowtostudyseparatelythe spin, orbital, andspin-orbital contributions\nto magnetism, providing the explicit expressions of the respective c ontributions\nto all the magnetic parameters determined in Section 8; in Section 11 we sum-\nmarize our results and mention possible future developments. In Ap pendix A\nwe discuss the condition for the rotational invariance of the intera ction, which is\nthe initial hypothesisofthis work, while in Appendix B weprovidesomed etails\nrelated to one of the key quantities contributing to the magnetic pa rameters.\n2. The electronic Hamiltonian\nThe Hamiltonian of a general electronic system is written as\nˆH≡ˆHφ\nT+ˆHφ\nV, (1)\nwhereˆHφ\nTandˆHφ\nVare, respectively, the single-particle and interaction Hamil-\ntonians.\n42.1. Single-particle Hamiltonian\nThe single-particle Hamiltonian reads:\nˆHφ\nT=/summationdisplay\n1,2ˆφ†\n1T1\n2ˆφ2, (2)\nwhere 1≡(a1,n1,l1,S1,M1) is a collective index including all the indexes and\nquantumnumbers that specify anatomically-localized(Wannier) sing le-electron\nstate. Namely, ais the atomic index, nis the orbital shell quantum number,\nlis the quantum number of the square orbital angular momentum ˆl2,Sis the\nquantum number of the square total angular momentum ˆS2, withˆS=ˆl+ˆs\n(obviouslys= 1/2), andM∈ {−S,−S+1,...,S−1,S}is the total third com-\nponent of the total angular momentum. It is convenient to group t he indexes\n(a,n,l) by introducing the orbitalindexo1≡(a1,n1,l1) and the magnetic mo-\nmentindexi1≡(o1,S1)≡(a1,n1,l1,S1). Angular momenta associated with a\nsingle-electronstate are measured in a reference frame centere d on its respective\nsite, with a i-specific quantizationaxis ri≡uz(i) (we will employ one of the two\nnotations from case to case). The corresponding reference fra me is then speci-\nfied by the right-handed triple of unit vectors [ ux(i),uy(i),uz(i)]. For a given\norbital index o1≡(a1,n1,l1) there are of course two possible magnetic moment\nindexesi±\n1≡(a1,n1,l1,l1±1/2) ifl1>0, or only one, i1≡(a1,n1,0,1/2) if\nl= 0. Forl1>0, the two fields specified by i±\n1have the same quantization axis,\nwhich defines the common orbital angular momentum and hence their common\nquantum number l1. In the following, for brevity we will refer to the individual\nmagnetic moments as spins, so we will say, e.g., that iis a spin index.\nWe also emphasize that, in the present formulation, the quantizatio n axesri\ncan be chosen arbitrarily, and in particular they may be different for each spin.\nAs we will discuss later, for the purposes of the mapping to the class ical spin\nmodeloneshouldlet ricoincidewiththedirectionoftheexpectationvalueofthe\ni-th local magnetic moment in the reference symmetry-broken non -equilibrium\nstate. The fact that the rivectors are not restricted to be parallel allows us to\ntreat the non-collinear regime.\nEquation (2) can be written more explicitly as\nˆHφ\nT=/summationdisplay\no1,S1,M1/summationdisplay\no2,S2,M2ˆφ†\no1,S1,M1To1,S1,M1\no2,S2,M2ˆφo2,S2,M2. (3)\nIt is convenient to define the spinors\nˆφ†\ni1≡ˆφ†\no1,S1≡/parenleftBig\nˆφ†\no1,S1,S1,ˆφ†\no1,S1,S1−1,...,ˆφ†\no1,S1,−S1+1,ˆφ†\no1,S1,−S1/parenrightBig\n,(4)\nand accordingly we write\nˆHφ\nT=/summationdisplay\no1,S1/summationdisplay\no2,S2ˆφ†\no1,S1·To1,S1\no2,S2·ˆφo2,S2=/summationdisplay\ni1/summationdisplay\ni2ˆφ†\ni1·Ti1\ni2·ˆφi2,(5)\nwhereTo1,S1\no2,S2isamatrixinangularmomentumspace. Notethat ˆφ†\no1,S1has(rows\n×columns) dimensions 1 ×(2S1+1),To1,S1\no2,S2has dimensions (2 S1+1)×(2S2+\n51), andˆφo2,S2has dimensions (2 S2+ 1)×1 in angular momentum space. In\nparticular, it must be noted that the matrix To1,S1\no2,S2is not square but rectangular\nifS1∝ne}ationslash=S2.\nExample: Zeeman Hamiltonian\nThe single-particle Hamiltonian that we are considering is the most gen eral\none: by specifying the form of the matrix To1,S1\no2,S2one can obtain any single-\nelectron term, including of course relativistic contributions in additio n to the\nstandardkinetic hopping. We givetwo examples. First, we considert he Zeeman\nHamiltonian arising from a position-dependent magnetic field Bacoupling with\nelectronic magnetic moments as\nˆHZeeman=µB/summationdisplay\no,SglSBa·ˆSφ\noS, (6)\nwhereo= (a,n,l),µB=|e|/(2mc) for/planckover2pi1= 1 (as we assume throughout this\nArticle),glSis the electron g-factor, and\nˆSφ\noS≡ˆSφ\ni=S/summationdisplay\nM,M′=−Sˆφ†\no,S,M(SoS)M\nM′ˆφo,S,M′≡ˆφ†\ni·Si·ˆφi(7)\nis the angular momentum operator associated with the spin i;SoS=Siis the\nangular momentum matrix of dimension 2 S+ 1 and quantization axis uz(i).\nThis Zeeman term can be obtained in our formalism by specifying a part of the\nsingle-particle matrix Tas:\n(TZeeman)o1,S1,M1\no2,S2,M2=δo1\no2δS1\nS2µBgl1S1Ba1·(So1S1)M1\nM2. (8)\nExample: atomic spin-orbit Hamiltonian\nThe second relativistic example that we consider is the Hamiltonian for\natomic spin-orbit coupling,\nˆHat.SOC=/summationdisplay\na1,n1/summationdisplay\nl1,S1,M1/summationdisplay\nl2,S2,M2ˆφ†\na1n1l1S1M1ξa1n1(l·s)l1,S1,M1\nl2,S2,M2ˆφa1n1l2S2M2\n=/summationdisplay\n1ˆφ†\n11\n2ξa1n1/bracketleftbigg\nS1(S1+1)−3\n4−l1(l1+1)/bracketrightbigg\nˆφ1, (9)\nwhere we have used ˆl·ˆs=/parenleftBig\nˆS2−ˆs2−ˆl2/parenrightBig\n/2 ands= 1/2. The corresponding\nsingle-particle Hamiltonian matrix is then\n(Tat.SOC)1\n2≡δ1\n21\n2ξa1n1/bracketleftbigg\nS1(S1+1)−3\n4−l1(l1+1)/bracketrightbigg\n. (10)\n6Local single-electron Hamiltonian: magnetic field and loca l magnetic anisotropy\nThe general form of the localsingle-electron Hamiltonian can be specified as\nTiM\niM′≡EiδM\nM′+µBgiBi·(Si)M\nM′+AiM\niM′, (11)\nwhere we have included the M-independent energy eigenvalue Eiand a Zeeman\nterm as given in Eq.(8). The last term in Eq.(11), denoted as AiM\niM′, originates\nfrom crystal field effects and can be considered as a single-site con tribution to\nthe magnetic anisotropy. For example, for the case of quadratic s pin anisotropy\n(AQ)iM\niM′=/parenleftbig\nSi·Ai·Si/parenrightbigM\nM′. We note that here (and in the following) we use\nthe term localreferring to a fictitious lattice where each site corresponds to one\nspin, labelled by i. This concept of locality may not correspond to locality in\nposition space (i.e., there can be more than one spin associated with a single\natom constituting the lattice in position space).\n2.2. The interaction Hamiltonian\nThe interaction Hamiltonian is generally written as\nˆHφ\nV≡1\n2/summationdisplay\n1,2,3,4ˆφ†\n1ˆφ†\n2V1,2\n3,4ˆφ3ˆφ4, (12)\nwhere\nV1,2\n3,4≡/integraldisplay\ndv(x)/integraldisplay\ndv(x′)ψ∗\n1(x)ψ∗\n2(x′)V(x−x′)ψ3(x′)ψ4(x).(13)\nWhile we have chosen the single-particle Hamiltonian to be the most gen eral\none, we assume that the interaction Hamiltonian is invariant under ro tation of\nthe electronic quantization axes ridefined for each orbital. In particular, we\nconsider an intra-site (Hubbard) interaction. A more precise form alization of\nthis requirement will be given in the next Section.\n3. Rotation of the spin quantization axes\n3.1. Rotation operator\nWe define the rotation operator for the quantization axis of the i-th individ-\nual spin as:\nRi≡eiδϕi·Si, (14)\nwhere the individual rotation parameter is\nδϕi≡θiui, (15)\nwhereuiis a unit vector and θiis the azimuthal angle of rotation.\n7In the initial Hamiltonian we change the basis according to the spinor t rans-\nformation\nˆφ†\ni1≡ˆψ†\ni1·R†\ni1,\nˆφi2≡Ri2·ˆψi2. (16)\nTo understand the meaning of the transformation (16), we note t hat the expec-\ntation value of the angular momentum on a state of one φfermion is:\n/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsingleˆφi,MˆSφ\niˆ��†\ni,M/vextendsingle/vextendsingle/vextendsingle0/angbracketrightBig\n= (Si)M\nM=Mri=Muz(i), (17)\nwhereˆSφ\niis given by Eq.(7); the expectation value of the same operator on a\nstate of one ψfermion, instead, is given by:\n/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsingleˆψi,MˆSφ\niˆψ†\ni,M/vextendsingle/vextendsingle/vextendsingle0/angbracketrightBig\n=/bracketleftBig\nR†\ni·Si·Ri/bracketrightBigM\nM, (18)\nwhosedirectionisthereforespecified bythe rotationoperators, andin particular\nmay be not parallel to uz(i). If we expand Eq.(18) in powers of small θiup to\nthe second order, we obtain:\n/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsingleˆψi,MˆSφ\niˆψ†\ni,M/vextendsingle/vextendsingle/vextendsingle0/angbracketrightBig\n≈M/bracketleftbigg\nri+ri×δϕi−1\n2ri(δϕi)2+1\n2δϕi(δϕi·ri)/bracketrightbigg\n≡Mei. (19)\nThe rotated quantization axis eisatisfiesei·ei= 1, i.e., it is a unit vector for\nanyδϕi(to the specified order). Therefore, we need only two variables ( θi,ϕi)\nto specify it completely, which are the polar angles with respect to th e initial\nquantization axis ri≡uz(i), which means that one of the three components of\nδϕiis redundant. To determine the necessary and sufficient set of var iables, we\nimpose\nei≡ux(i)sin(θi)cos(ϕi)+uy(i)sin(θi)sin(ϕi)+uz(i)cos(θi)\n≈ux(i)θicos(ϕi)+uy(i)θisin(ϕi)+ri/parenleftbigg\n1−(θi)2\n2/parenrightbigg\n. (20)\nwhere in the last line we have kept up to second-order terms in θi. Recalling\nEq.(15) and comparing Eqs.(19) and (20), we see that\nri×δϕi+1\n2δϕi(δϕi·ri) =ux(i)θicos(ϕi)+uy(i)θisin(ϕi).(21)\nThe RHS of Eq.(21) is of first order in θi, while the term1\n2δϕi(δϕi·ri) on the\nLHS is of second order. Therefore, we need to impose δϕi·ri= 0. From Eq.(21)\nwe then determine δϕiuniquely as:\nδϕi≡θi[sin(ϕi)ux(i)−cos(ϕi)uy(i)]. (22)\n8Example: Spin- 1/2rotations\nIn the particular case of S= 1/2, we have (exactly)\nRo,1/2= cos(θo/2)+i sin(θo/2)uo·σ(o)\n= cos(θo/2)+i sin(θo/2)[sin(ϕo)σx(o)−cos(ϕo)σy(o)],(23)\nwhereσ(o) is the vector of Pauli matrices expressed in the reference frame of\norbitalo. Thischoiceisappropriateforstudyingsystemswithoutorbitalde grees\nof freedom, such as the single-band Hubbard model, or if one is inter ested only\nin the exchange couplings due to the intrinsic spins of the electrons [ 16]. Here\nwe consider the more general case of arbitrary magnetic moments ˆSφ\ni=ˆlφ\ni+ˆsφ\ni,\naccounting for the orbital degrees of freedom.\n3.2. The Hamiltonian after the transformation\nThe single-particle Hamiltonian, Eq.(5), is written in terms of the new ψ\nfields as:\nˆHφ\nT=/summationdisplay\ni1/summationdisplay\ni2ˆψ†\ni1·e−iδϕi1·Si1·Ti1\ni2·eiδϕi2·Si2·ˆψi2. (24)\nWe consider small deviations from the reference quantization axes of the local\ntotalangularmomenta. Therefore,wenowapplyasmall- θexpansionofEq.(24),\nkeeping only the terms of orders θ0,θ1andθ2. The single-particle Hamiltonian\nis then given, to this order, by\nˆHφ\nT≈ˆHψ\nT+ˆHψ,θ\nT+ˆHψ,θ2\nT, (25)\nwith\nˆHψ,θ\nT≡/summationdisplay\niδϕi·ˆVi (26)\nand\nˆHψ,θ2\nT≡1\n2/summationdisplay\nii′δϕi·ˆMii′·δϕi′, (27)\nwhere\nˆViα≡i/summationdisplay\nj/parenleftBig\nˆψ†\nj·Tj\ni·Siα·ˆψi−ˆψ†\ni·Siα·Ti\nj·ˆψj/parenrightBig\n= iTrM/parenleftbigg\nSiα·/bracketleftBig\nˆρ;T/bracketrightBigi\ni/parenrightbigg\n,\n(28)\n9ˆMiα,i′α′≡−δii′1\n2/summationdisplay\nj/bracketleftBig\nˆψ†\ni·(Siα·Siα′+Siα′·Siα)·Ti\nj·ˆψj\n+ˆψ†\nj·Tj\ni·(Siα·Siα′+Siα′·Siα)·ˆψi/bracketrightBig\n+ˆψ†\ni·Siα·Ti\ni′·Si′α′·ˆψi′+ˆψ†\ni′·Si′α′·Ti′\ni·Siα·ˆψi\n=TrM/parenleftBig\nSiα·Ti\ni′·Si′α′·ˆρi′\ni+Si′α′·Ti′\ni·Siα·ˆρi\ni′/parenrightBig\n−δii′1\n2TrM/parenleftbigg\n(Siα·Siα′+Siα′·Siα)·/braceleftBig\nˆρ;T/bracerightBigi\ni/parenrightbigg\n; (29)\nin Eqs.(28) and (29) we have introduced the density matrix operato r\nˆρ1\n2≡ˆψ†\n2ˆψ1, (30)\nwhere 1 and 2 are general indexes for the fermionic fields. We have a lso used the\nnotations/bracketleftBig\nˆA;ˆB/bracketrightBig\n≡ˆAˆB−ˆBˆAand/braceleftBig\nˆA;ˆB/bracerightBig\n≡ˆAˆB+ˆBˆAfor commutators and\nanti-commutators, respectively (if ˆAandˆBare matrices, then matrix products\nare implied). The matrix operator (29) has been defined, for later c onvenience,\nsuch that ˆMiα,i′α′=ˆMi′α′,iα.\nDifferentlyfromthesingle-particleHamiltonian, theinteractionHamilt onian\nisassumed to be rotationally invariant, i.e.,\nˆHφ\nV=ˆHψ\nV. (31)\nIn Appendix A we discuss the conditions for the fulfilment of this requ irement.\n4. Action and partition function\n4.1. Action\nThe derivation of the effective rotational action for the electronic system\nproceeds analogously to our previous treatment of the spin-1 /2 rotations [16].\nWe write the Matsubara action as\nS/bracketleftbig¯φ,φ/bracketrightbig\n=/integraldisplayβ\nεdτ/braceleftBigg\n¯φ(τ)·˙φ(τ−ε)+K/bracketleftBig\n¯φ(τ),φ(τ−ε)/bracketrightBig/bracerightBigg\n,(32)\nwhere¯φ(τ) andφ(τ) are contour Grassmann variables [23], K=H−µNis the\ngrand-canonical potential, with µbeing the chemical potential and Nthe num-\nber of electrons. We then apply the rotation transformation to th eτ-dependent\nGrassmann fields, by introducing i-dependent rotation fields δϕi(τ) along the\ncontour. Keeping into account that the term µNis of course rotationally in-\nvariant, we obtain\nS/bracketleftbig¯φ,φ/bracketrightbig\n=S/bracketleftbig¯ψ,ψ/bracketrightbig\n+S′/bracketleftbig¯ψ,ψ,δϕ/bracketrightbig\n, (33)\n10with\nS′/bracketleftbig¯ψ,ψ,δϕ/bracketrightbig\n≡/integraldisplayβ\nεdτ¯ψ(τ)·∆(τ)·ψ(τ−ε)\n≈/integraldisplayβ\nεdτ¯ψ(τ)·R†(τ)·˙R(τ)·ψ(τ−ε)+/integraldisplayβ\nεdτ/bracketleftBig\nHψ,θ\nT(τ)+Hψ,θ2\nT(τ)/bracketrightBig\n,\n(34)\nwhere ∆(τ) introduced in the first line is a kernel which depends on the fields\nδϕ(τ) and their derivatives δ˙ϕ(τ), in principle to all orders. If we consider the\nregime of small rotations, up to quadratic order, we obtain the exp ression in\nthe second line, where Hψ,θ\nT(τ) andHψ,θ2\nT(τ) correspond, respectively, to the\nexpressions (26) and (27) with the operators ˆψ†andˆψreplaced, respectively,\nby the Grassmann fields ¯ψ(τ) andψ(τ−ε), and we have\nR†(τ)·˙R(τ)≈i[δ˙ϕ(τ)·S]+i\n2[δϕ(τ)×δ˙ϕ(τ)]·S, (35)\nwhere we have used the commutation relations of the spin matrices [ Sα,Sβ] =\ni/summationtext\nγεαβγSγ.\nWe then distinguish the terms which are of the same order in product com-\nbinations of the δϕand theδ˙ϕfields, and accordingly we put\nS′≡∞/summationdisplay\nn=1S(n),S(n)≡/integraldisplayβ\nεdτ¯ψ(τ)·∆(n)(τ)·ψ(τ−ε).(36)\n4.2. Partition function\nThe grand-canonical partition function is written as a path integra l over\nGrassmann variables as:\nZ≡Tr/braceleftBig\ne−β(ˆH−µˆN)/bracerightBig\n≡/integraldisplay\nD/bracketleftbig¯φ,φ/bracketrightbig\ne−S[¯φ,φ], (37)\nwhere the trace is taken over the complete set of many-body eigen states of the\nHamiltonian. It should be emphasized that the states are weighted b y Boltz-\nmann factors, which depend only on the energy and therefore cannot distinguish\nbetween degenerate broken-symmetry states . However,the mappingtoaclassical\nspin model makes sense only if the reference electronic state is not symmetric\nwith respect to rotations of the spins, since this is an essential pro perty of clas-\nsical spin configurations. We therefore define a “broken-symmet ry” partition\nfunction, which we label as Z∗, as\nZ∗≡ ∝an}bracketle{t{ri}|e−β(ˆH−µˆN)|{ri}∝an}bracketri}ht ≡/integraldisplay\nD∗/bracketleftbig¯φ,φ/bracketrightbig\ne−S[¯φ,φ], (38)\nwhere|{ri}∝an}bracketri}htis a reference electronic state which realizes the spin configuration\nspecified by the set of unit vectors {ri}, and the measure D∗of the path integral\nin the last passage is defined formally.\n11We implement the rotations of the spin quantization axes by applying t he\ntransformation discussed above from the/bracketleftbig¯φ,φ/bracketrightbig\nto the/bracketleftbig¯ψ,ψ/bracketrightbig\nfermions, and we\ndefine the functional\nZ[δϕi(τ)]≡/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]e−S′[¯ψ,ψ,δϕ], (39)\nwhereD/bracketleftbig¯φ,φ/bracketrightbig\n≡D∗/bracketleftbig¯φ,φ/bracketrightbig\n/Z∗. We expand Eq.(39) as\nZ[δϕi(τ)]≡/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]/braceleftBigg\n1+∞/summationdisplay\nm=1/parenleftbig\n−S′/bracketleftbig¯ψ,ψ,δϕ/bracketrightbig/parenrightbigm\nm!/bracerightBigg\n≡∞/summationdisplay\nn=0Z(n)[δϕi(τ)], (40)\nwhereZ(n)includes all the terms that require the product of nfieldsδϕorδ˙ϕ.\nHere we focus on the contributions to the broken-symmetry part ition function\ncoming from trajectories δϕi(τ) close toδϕi(τ) =0, i.e., we study the regime\nof small rotations of the spin quantization axes from their initial con figuration.\nSpecifically, we consider Z(0),Z(1)andZ(2); with reference to Eq.(36), these\nare given by\nZ(0)≡/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]= 1,\nZ(1)[δϕi(τ)]≡/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]/braceleftBig\n−S(1)/bracketleftbig¯ψ,ψ,δϕ/bracketrightbig/bracerightBig\n,\nZ(2)[δϕi(τ)]≡/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]/braceleftbigg\n−S(2)/bracketleftbig¯ψ,ψ,δϕ/bracketrightbig\n+1\n2/parenleftBig\nS(1)/bracketleftbig¯ψ,ψ,δϕ/bracketrightbig/parenrightBig2/bracerightbigg\n(41)\n(the definitions used here are slightly different from Eqs.(43) of Ref .[16]).\n5. Effective rotational action for small spin deviations\nWe now derive an effective action for the fields δϕi(τ) in the regime of small\nδϕi(τ) by integrating out the fermionic fields ¯ψi(τ) andψi(τ).\n5.1. Fermionic integration\nTo integrate out the fermionic fields, we use the identities\ni/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]¯ψ2(τ′)ψ1(τ)≡G1\n2(τ,τ′) =−i/angbracketleftBig\nTγˆψ1(τ)ˆψ†\n2(τ′)/angbracketrightBig\n,\ni2/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]¯ψ2(τ2)ψ1(τ1)¯ψ4(τ4)ψ3(τ3)≡χ1,3\n2,4(τ1,τ2,τ3,τ4)\n= i2/angbracketleftBig\nTγˆψ1(τ1)ˆψ†\n2(τ2)ˆψ3(τ3)ˆψ†\n4(τ4)/angbracketrightBig\n, (42)\n12whereGandχrespectively denote single-particle and two-particle Matsubara\nGreen’s functions, ˆψ(τ)≡eτˆKˆψe−τˆK,ˆK=ˆH−µˆN, andTγis the time-ordering\noperator along the Matsubara axis γ. We recall that the Green’s functions\nshould be computed for the electronic state |{ri}∝an}bracketri}ht, as follows from the formal\ndefinition of the measure D/bracketleftbig¯ψ,ψ/bracketrightbig\ngiven in Section 4.2. The two-particle Green’s\nfunction can be written as\nχ1,3\n2,4(τ1,τ2,τ3,τ4)≡G1\n2(τ1,τ2)G3\n4(τ3,τ4)−G1\n4(τ1,τ4)G3\n2(τ3,τ2)\n+/summationdisplay\n1′2′3′4′/integraldisplay\ndτ′\n1/integraldisplay\ndτ′\n2/integraldisplay\ndτ′\n3/integraldisplay\ndτ′\n4G1\n1′(τ1,τ′\n1)G3\n3′(τ3,τ′\n3)\n×Γ1′,3′\n2′,4′(τ′\n1,τ′\n2,τ′\n3,τ′\n4)G4′\n4(τ′\n4,τ4)G2′\n2(τ′\n2,τ2)\n≡/parenleftbig\nχ0/parenrightbig1,3\n2,4(τ1,τ2,τ3,τ4)+/parenleftbig\nχΓ/parenrightbig1,3\n2,4(τ1,τ2,τ3,τ4), (43)\nwhereχΓisthesumofconnectedFeynmandiagrams,dependingontheverte xΓ.\nWhile a number of previous works on magnetic interactions [1, 10, 11, 16] have\nemployed the simplifying approximation of neglecting vertices in two-p article\nGreen’s functions (Γ = 0), we will here remove this assumption and ca rry on\nthe derivation with the full two-particle Green’s functions. In fact , while it has\nbeen shown that neglecting vertices leads nevertheless to the cor rect expression\nfor the spin-wave stiffness if the self-energy is local [24], there is no guaranty\nthat other magnetic properties will be unaffected by that approxim ation, which\nis, as a matter of fact, uncontrolled. Moreover, we consider here the general\ncase of a non-local self-energy.\nAfter the fermionic integration, we obtain:\nZ(1)[δϕi(τ)] =/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]/braceleftBigg\n−/integraldisplayβ\nεdτ¯ψ(τ)·∆(1)(τ)·ψ(τ−ε)/bracerightBigg\n=/summationdisplay\n1/summationdisplay\n2/integraldisplayβ\nεdτ/parenleftBig\n∆(1)(τ)/parenrightBig2\n1iG1\n2(τ−ε,τ)≡ −Tr{i,M}/bracketleftBigg\nρ·/integraldisplayβ\n0dτ∆(1)(τ)/bracketrightBigg\n≡ −S1[δϕi(τ)], (44)\nZ(2)[δϕi(τ)]≡/integraldisplay\nD/bracketleftbig¯ψ,ψ/bracketrightbig\ne−S[¯ψ,ψ]/braceleftBigg\n−/integraldisplayβ\nεdτ¯ψ(τ)·∆(2)(τ)·ψ(τ−)\n+1\n2/integraldisplayβ\nεdτ¯ψ(τ)·∆(1)(τ)·ψ(τ−)/integraldisplayβ\nε′dτ′¯ψ(τ′)·∆(1)(τ′)·ψ(τ′−)/bracerightBigg\n≡ −Tr{i,M}/bracketleftBigg\nρ·/integraldisplayβ\n0dτ∆(2)(τ)/bracketrightBigg\n−1\n2/summationdisplay\n1234/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/parenleftBig\n∆(1)(τ)/parenrightBig2\n1χ1,3\n2,4(τ,τ+,τ′,τ′+)/parenleftBig\n∆(1)(τ′)/parenrightBig4\n3\n≡ −S2[δϕi(τ)]; (45)\n13for the sake of brevity, we do not report here the expressions fo r ∆(1)(τ) and\n∆(2)(τ). In Eqs.(44) and (45) we have introduced the symbol for the sing le-\nparticle density matrix,\nG(τ−ε,τ)≡iρ. (46)\nIt should be noted that the only two-particle Green’s function appe aring in\nEq.(45) is of the form\nχ1,3\n2,4(τ,τ+,τ′,τ′+) =−/angbracketleftbig\nTγˆρ1\n2(τ) ˆρ3\n4(τ′)/angbracketrightbig\n, (47)\nwhich is a correlator between two density-matrix operators [see Eq .(30)] taken\nat different imaginary times. The result depends on imaginary time only via\nthe combination ( τ−τ′).\n5.2. Effective action\nThe contributions to the rotation functional up to the quadratic o rder in the\nδϕfields can be written as\nZ(0)+Z(1)[δϕi(τ)]+Z(2)[δϕi(τ)] = 1−S1[δϕi(τ)]−S2[δϕi(τ)]≈e−S[δϕi(τ)],\n(48)\nwhere the effective action is defined as S[δϕi(τ)] =S1[δϕi(τ)] +S2[δϕi(τ)] +\n1\n2(S1[δϕi(τ)])2. Namely,\nS[δϕi(τ)] = Tr {i,J,M}/braceleftBigg\nρ·/integraldisplayβ\n0dτ/bracketleftBig\n∆(1)(τ)+∆(2)(τ)/bracketrightBig/bracerightBigg\n+1\n2/summationdisplay\n1234/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/parenleftBig\n∆(1)(τ)/parenrightBig2\n1/bracketleftBig\nχ1,3\n2,4(τ,τ+,τ′,τ′+)+ρ1\n2ρ3\n4/bracketrightBig/parenleftBig\n∆(1)(τ′)/parenrightBig4\n3.\n(49)\nWe now write the expression in Eq.(49) explicitly. Using the property\nχ1,3\n2,4(τ,τ+,τ′,τ′+) =χ3,1\n4,2(τ′,τ′+,τ,τ+), (50)\nand defining\nˆViα(τ)≡iTrM/parenleftbigg\nSiα·/bracketleftBig\nˆρ(τ);T/bracketrightBigi\ni/parenrightbigg\n, (51)\nViα≡/angbracketleftBig\nˆViα/angbracketrightBig\n,\nMiα\ni′α′≡/angbracketleftBig\nˆMiα\ni′α′/angbracketrightBig\n=Mi′α′\niα, (52)\n14we can rewrite the action of the electronic model (for small spin rot ations) as:\nS[δϕi(τ)]≡/integraldisplayβ\n0dτ/braceleftBigg\ni/summationdisplay\ni/bracketleftbigg\nδ˙ϕi(τ)+1\n2/parenleftBig\nδϕi(τ)×δ˙ϕi(τ)/parenrightBig/bracketrightbigg\n·/angbracketleftBig\nˆSi/angbracketrightBig\n+/summationdisplay\ni/summationdisplay\nαδϕiα(τ)Viα+1\n2/summationdisplay\ni,i′/summationdisplay\nα,α′δϕiα(τ)δϕi′α′(τ)Miα\ni′α′/bracerightBigg\n+1\n2/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/summationdisplay\ni,i′/summationdisplay\nα,α′/braceleftBigg\nδ˙ϕiα(τ)δ˙ϕi′α′(τ′)/bracketleftBig/angbracketleftBig\nTγˆSiα(τ)ˆSi′α′(τ′)/angbracketrightBig\n−/angbracketleftBig\nˆSiα/angbracketrightBig/angbracketleftBig\nˆSi′α′/angbracketrightBig/bracketrightBig\n−2iδϕiα(τ)δ˙ϕi′α′(τ′)/bracketleftBig/angbracketleftBig\nTγˆViα(τ)ˆSi′α′(τ′)/angbracketrightBig\n−Viα/angbracketleftBig\nˆSi′α′/angbracketrightBig/bracketrightBig\n−δϕiα(τ)δϕi′α′(τ′)/bracketleftBig/angbracketleftBig\nTγˆViα(τ)ˆVi′α′(τ′)/angbracketrightBig\n−ViαVi′α′/bracketrightBig/bracerightBigg\n.\n(53)\nIt must be stressed that, after the integration of the fermionic v ariables, the\nrotational fields δϕi(τ) are the onlydynamical variables left. All the features of\nthe electronic dynamical processes are accounted for by the elec tronic Green’s\nfunctions.\n6. Static potential for the spin rotations\nWe now consider the action (53) in the particular case of static rota tions,\nδϕiα(τ)→δϕiα:\nS[δϕi]≡β\n/summationdisplay\ni/summationdisplay\nαδϕiαViα+1\n2/summationdisplay\ni,i′/summationdisplay\nα,α′δϕiαδϕi′α′/tildewiderMiα\ni′α′\n(54)\nwhere\n/tildewiderMiα\ni′α′≡ Miα\ni′α′−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆViα(τ)ˆVi′α′(τ′)/angbracketrightBig\n+βViαVi′α′.(55)\nIn this case, we can define the effective broken-symmetry partitio n function in\nthe presence of a spin rotation as\nZ∗[δϕi]≡Z∗Z[δϕi]≡e−β(Ω∗\n0+Ω[δϕi]), (56)\nwhere Ω∗\n0=−β−1ln(Z∗), and Ω[δϕi]≡β−1S[δϕi] is the effective thermo-\ndynamic potential associated with spin rotations from the broken- symmetry\nnon-equilibrium configuration |{ri}∝an}bracketri}ht.\n157. Mapping to a classical spin model\n7.1. Classical spin model\nThe Hamiltonian of a classical quadratic spin model is given by\nH[ei]≡/summationdisplay\niei·Bi+1\n2/summationdisplay\ni,i′/summationdisplay\nα,α′ei,αei′,α′Hαα′\nii′, (57)\nwhere the ei’s are unit vectors representing the directions of the classical mag -\nnetic moments, and α,α′∈ {x,y,z}. The vector Biis a local magnetic field,\nwhileHαα′\nii′is the exchange tensor, which can be chosen, without loss of gener -\nality, to satisfy the symmetry property\nHαβ\nij=Hβα\nji. (58)\nFurthermore, the tensor can be decomposed into three vectors Jij,Dij, and\nCijas follows:\nHαβ\nij≡δαβJα\nij+/summationdisplay\nγ/parenleftbig\nεαβγDγ\nij+/vextendsingle/vextendsingleεαβγ/vextendsingle/vextendsingleCγ\nij/parenrightbig\n, (59)\nwhere the parameters\nJα\nij≡ Hαα\nij,Dγ\nij≡1\n2/summationdisplay\nα,βεαβγHαβ\nij,andCγ\nij≡1\n2/summationdisplay\nα,β/vextendsingle/vextendsingleεαβγ/vextendsingle/vextendsingleHαβ\nij(60)\naccount, respectively, for anisotropic exchange, Dzyaloshinskii- Moriya interac-\ntion, and an additional traceless symmetric interaction (which includ es, e.g.,\ndipole-dipole1). One can easily verify that Jij=Jji,Dij=−Dji, and\nCij=Cji, as a consequence of the symmetry given by Eq.(58). We can write\nthe total magnetic Hamiltonian as\nH[ei] =/summationdisplay\niei·Bi+1\n2/summationdisplay\ni,i′/bracketleftbig\nei·Jii′·ei′+Dii′·(ei×ei′)+Cii′·(ei⊗ei′)/bracketrightbig\n,\n(61)\nwhereJii′is the diagonal matrix with elements Jα\nii′on the diagonal, and we\nhave put\nei⊗ei′≡/summationdisplay\nαβγ/vextendsingle/vextendsingleεαβγ/vextendsingle/vextendsingleeiαei′βuγ. (62)\n1For example, the second part of a dipole-dipole interaction term [9, 25] of the form\n1\nR3\nij/bracketleftbigg\nei·ej−3/parenleftbiggei·Rij\nRij/parenrightbigg/parenleftbiggej·Rij\nRij/parenrightbigg/bracketrightbigg\n,\nwhereRijisthe vector connecting the positions of magnetic moments iandj, may be included\ninCij.\n16It should be noted that the exchange tensor has also single-spin te rms corre-\nsponding to i=i′. If the index ilabelling the magnetic moments can be iden-\ntified with a space coordinate, such as an atomic index, then these localterms\nshould be identified with the local anisotropy tensor , having 6 independent com-\nponents: Jα\niiandCα\nii, forα∈ {x,y,z}. We note that the energy contribution\nfrom the diagonal part of this tensor can be written as\n1\n2/summationdisplay\ni/summationdisplay\nα=x,y,z(eiα)2Jα\nii=1\n2/summationdisplay\ni/bracketleftbig\ne2\nix(Jx\nii−Jz\nii)+e2\niy(Jy\nii−Jz\nii)+Jz\nii/bracketrightbig\n,(63)\nwhere we have used the constraint e2\ni= 1. The last term of Eq.(63) is obviously\nrotationally invariant, so that the response of the system to spin r otations will\nnot depend individually on the three parameters Jα\nii, but only on their rela-\ntive differences as expressed, for example, in the combinations ( Jx\nii−Jz\nii) and\n(Jy\nii−Jz\nii).\n7.2. Static potential for the spin rotations\nThe effective potential for the classical spin model, expressing the energy\nchange when the unit vectors are rotated from a given configurat ion{ri}, is\nobtained from Eq.(61) by replacing\nei≈ri+ri×δϕi−1\n2ri(δϕi)2, (64)\nwhich follows from Eq.(19) and the requirement that δϕi·ri= 0. In order to\nmap the effective potential for spin rotations relative to the classic al spin model\nonto the potential derived for the electronic model, we need in fact to require\nthat theδϕifields have the same meaning in the two cases. Therefore, the\nvectors{ri}which specify the classical spin configuration of reference must be\nthe same as the vectors specifying the spin configuration of the ele ctronic state\nof reference |{ri}∝an}bracketri}ht, introduced in Section 4.2.\nThe effective potential for the classical spin model (to second ord er in the\nrotation angles) is:\nΩcl[δϕi] = Ω(1)\ncl[δϕi]+Ω(2)\ncl[δϕi], (65)\nwhere\nΩ(1)\ncl[δϕi] =/summationdisplay\ni/braceleftBigg\nδϕix/bracketleftBigg\nBy\ni+/summationdisplay\ni′(Cx\nii′+Dx\nii′)/bracketrightBigg\n−δϕiy/bracketleftBigg\nBx\ni+/summationdisplay\ni′(Cy\nii′−Dy\nii′)/bracketrightBigg/bracerightBigg\n,\n(66)\nΩ(2)\ncl[δϕi] =1\n2/summationdisplay\ni,i′/ne}ationslash=i/parenleftbig\nδϕixδϕiy/parenrightbig/parenleftbiggJy\nii′−Cz\nii′+Dz\nii′\n−Cz\nii′−Dz\nii′Jx\nii′/parenrightbigg/parenleftbiggδϕi′x\nδϕi′y/parenrightbigg\n+1\n2/summationdisplay\ni/parenleftbigδϕixδϕiy/parenrightbig/parenleftbigg−Bz\ni−/summationtext\njJz\nij+Jy\nii −Cz\nii\n−Cz\nii −Bz\ni−/summationtext\njJz\nij+Jx\nii/parenrightbigg/parenleftbiggδϕix\nδϕiy/parenrightbigg\n.\n(67)\nThe matrices appearing in Eq.(67) are maximally symmetrized.\n177.3. Equations for the effective magnetic parameters\nWe determine the effective magnetic parameters of the classical sp in model\nby putting Ω[ δϕi]≡Ωcl[δϕi] and identifying the terms which depend on the\nsame orders and same combinations of the δϕiαfields. We obtain the following\nthree sets of equations:\nBy\ni+/summationdisplay\ni′(Cx\nii′+Dx\nii′) =Vix,\n−Bx\ni−/summationdisplay\ni′(Cy\nii′−Dy\nii′) =Viy, (68)\n−Bz\ni−/summationdisplay\njJz\nij+Jy\nii=/tildewiderMix\nix,\n−Bz\ni−/summationdisplay\njJz\nij+Jx\nii=/tildewiderMiy\niy,\nCz\nii=−/tildewiderMix\niy, (69)\nJy\nii′=/tildewiderMix\ni′x, i∝ne}ationslash=i′,\nDz\nii′=1\n2/parenleftBig\n/tildewiderMix\ni′y−/tildewiderMiy\ni′x/parenrightBig\n, i∝ne}ationslash=i′,\nCz\nii′=−1\n2/parenleftBig\n/tildewiderMix\ni′y+/tildewiderMiy\ni′x/parenrightBig\n, i∝ne}ationslash=i′,\nJx\nii′=/tildewiderMiy\ni′y, i∝ne}ationslash=i′. (70)\nThe first set, Eqs.(68), is obtained from the identity of the terms o f the effective\npotentials which are of the first order in the rotation angles. The se cond set,\n(69), is obtained from second-order rotations of a single spin, while the third\nset, (70), is obtained from second-order rotations involving differ ent spinsi,i′.\nWhile the equations of the third set are already explicitly solved, we ne ed to\nsolve the first set, Eqs.(68), and two equations of the second set , Eqs.(69). As\nit can be seen, the equations arising from single-spin rotations do no t have a\nunique solution. To obtain the identity of the thermodynamic potent ials up\nto the second order in the rotations, however, we just need onesolution, and\nin the following we will determine it following general principles of physica l\nreasonability and respecting the symmetries of the parameters. F or example,\nthe three components of the effective magnetic field Bimust be proportional to\nthe respective components of the field Bientering the electronic Hamiltonian,\nalthough one could in principle set Bi=0and absorb all the terms depending\nonBiinto the exchange tensor. Obviously this last solution would make no\nsense.\nWe also wish to note that an extension of this analysis to the identity o f\nthe third-order terms in the rotation angles (leaving the spin model unchanged)\nmay provide additional constraints on the quantities which are not c ompletely\ndetermined from the sets (68) and (69). This analysis, however, is beyond the\nscope of the present work.\n188. Solution in the general relativistic regime\nIn this Section we solve the equations needed to determine the rema ining\nparameters of the effective spin model. We will report all the relevan t details,\nso that the derivation can be followed in its entirety. For the sake of clarity, in\nSection 8.3 we will summarize and list all the results.\n8.1. First set - Equations (68)\nFrom the definitions, Eqs.(28) and (52), we write the right-hand-s ides of\nEqs.(68), for α∈ {x,y}, as\nViα= iTrM/bracketleftBig\nSiα·(ρ·T−T·ρ)i\ni/bracketrightBig\n= iTrM/summationdisplay\nj/bracketleftBig\nSiα·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig/bracketrightBig\n= iTrM/bracketleftBig\nρi\ni·/parenleftbig\nTi\ni·Siα−Siα·Ti\ni/parenrightbig/bracketrightBig\n+iTrM/summationdisplay\nj/ne}ationslash=i/bracketleftBig\nSiα·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig/bracketrightBig\n.\n(71)\nIdentifying Eqs.(71) with the corresponding left-hand-sides given in Eqs.(68),\nwe are naturally led to separate local ( j=i) and non-local ( j∝ne}ationslash=i) terms. We\nobtain the following equations:\niTrM/bracketleftBig\nρi\ni·/parenleftbig\nTi\ni·Six−Six·Ti\ni/parenrightbig/bracketrightBig\n=By\ni+Cx\nii,\niTrM/bracketleftBig\nρi\ni·/parenleftbig\nTi\ni·Siy−Siy·Ti\ni/parenrightbig/bracketrightBig\n=−Bx\ni−Cy\nii, (72)\niTrM/summationdisplay\nj/ne}ationslash=i/bracketleftBig\nSix·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig/bracketrightBig\n=/summationdisplay\nj/ne}ationslash=i/parenleftbig\nCx\nij+Dx\nij/parenrightbig\n,\niTrM/summationdisplay\nj/ne}ationslash=i/bracketleftBig\nSiy·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig/bracketrightBig\n=−/summationdisplay\nj/ne}ationslash=i/parenleftbig\nCy\nij−Dy\nij/parenrightbig\n.(73)\nExcept for the presence of the magnetic field Bi, Eqs.(68) were also obtained\nin Ref.[12], which focused on the determination of the Dzyaloshinskii-M oriya\ninteractions. Here we will also discuss the other terms of the excha nge ten-\nsor that can be determined from this set, namely the parameters Cx\nijandCy\nij.\nMoreover, we note that the parameters Cz\nijandDz\nijcannot be determined from\nthe first-order term of the rotational potential, due to the fact thatδϕiz= 0,\nwhich was not discussed in Ref.[12]. These terms are indeed obtained f rom the\nsecond-order terms of the potential, see Eqs.(70) and the last am ong Eqs.(69).\nWe now considerEqs.(72). Using Eq.(11), we compute the LHSs ofEq s.(72),\nfrom which we can separate the contributions due to the magnetic fi eld and to\n19the anisotropy as\nBx\ni=−µBgi/parenleftBig\nBi×/angbracketleftBig\nˆSi/angbracketrightBig/parenrightBig\n·uy\ni,\nBy\ni=µBgi/parenleftBig\nBi×/angbracketleftBig\nˆSi/angbracketrightBig/parenrightBig\n·ux\ni,\nCx\nii= iTrM/bracketleftBig\nρi\ni·/parenleftbig\nAi\ni·Six−Six·Ai\ni/parenrightbig/bracketrightBig\n,\nCy\nii=−iTrM/bracketleftBig\nρi\ni·/parenleftbig\nAi\ni·Siy−Siy·Ai\ni/parenrightbig/bracketrightBig\n. (74)\nSince the local quantization axes are chosen to be parallel to the loc al magnetic\nmoments, i.e.,/angbracketleftBig\nˆSi/angbracketrightBig\n≡Siuz\ni, then it follows that Bx\ni=µBgiSiBx\niandBy\ni=\nµBgiSiBy\ni.\nTo solve Eqs.(73), analogously to Ref.[12] we use the symmetries of t he\nmagnetic parameters under exchange of the indexes i↔j. In general, one can\nalways write\n/summationdisplay\nj/ne}ationslash=ifij=1\n2/summationdisplay\nj/ne}ationslash=i[(fij+fji)+(fij−fji)], (75)\nwherewehavedistinguishedthe functions( fij+fji)/2and(fij−fji)/2, which\nare respectively symmetric and antisymmetric under the permutat ion ofiand\nj. After applying (75) to the LHSs of Eqs.(73), one could think of iden tifying\nthe symmetric and antisymmetric functions of ( i,j), respectively, with the cor-\nresponding components of the vectors CijandDijappearing on the RHSs. We\nwant to note that this procedure is not unique. Indeed, if we have a n identity\nof the form\n/summationdisplay\nj/ne}ationslash=iaij=/summationdisplay\nj/ne}ationslash=ibij,\nwhereaijandbijhavethesamesymmetryunder i↔j, sayaij=±ajiandbij=\n±bji, we can in general only assume that aij=bij+xij, wherexij=±xjiand/summationtext\njxij= 0. The undetermined quantity xijmay be relevant if some unknowns\nappearing in aijorbijmust satisfy constraints imposed also by other equations.\nIn other words, in the most general case we cannot split the equat ions of the\nfirst set into pairs of equations for symmetric and antisymmetric co mponents,\nsincedoingsowouldrequireintroducingadditionalunknowns(the xijquantities\nof the previous example) that cannot be determined uniquely. Howe ver, the\nquantities Cx\nij,Cy\nij,Dx\nijandDy\nijdo not appear in any of the other equations\nthat must be satisfied for the mapping to hold. Therefore, we can j ust take any\n20solution of Eqs.(73) with the correct symmetry properties. We the refore obtain\nDx\nij=i\n2TrM/bracketleftBig\nSix·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig\n−Sjx·/parenleftBig\nρj\ni·Ti\nj−Tj\ni·ρi\nj/parenrightBig/bracketrightBig\n,\nDy\nij=i\n2TrM/bracketleftBig\nSiy·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig\n−Sjy·/parenleftBig\nρj\ni·Ti\nj−Tj\ni·ρi\nj/parenrightBig/bracketrightBig\n,\nCx\nij=i\n2TrM/bracketleftBig\nSix·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig\n+Sjx·/parenleftBig\nρj\ni·Ti\nj−Tj\ni·ρi\nj/parenrightBig/bracketrightBig\n,\nCy\nij=−i\n2TrM/bracketleftBig\nSiy·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig\n+Sjy·/parenleftBig\nρj\ni·Ti\nj−Tj\ni·ρi\nj/parenrightBig/bracketrightBig\n.(76)\nThis completes the determination of the CijandDijvectors. The components\nof the Dzyaloshinskii-Moriyavector Dx\nijandDy\nijare in agreement with Ref.[12],\nexcept that here the parameters are defined in such a way that th ey exhibit\ndefinite symmetry under the permutation of the indexes i,j. Moreover, here we\nhave allowed Sito be in general different from Sj, so Eqs.(76) are valid also in\nthenon-collinear regime. By comparing Eqs.(70) and (76), we observe that the\nexpressions for the zcomponents of the vectors CandDlook formally different\nfrom those that give the xandycomponents of the same vectors. Recalling\nthat the reference frames {ux\ni,uy\ni,uz\ni}are defined locally (they depend on i),\nthis different status of the direction uz\niwith respect to the plane defined by the\ndirections {ux\ni,uy\ni}reflects the fact that uz\niis the direction of the i-th magnetic\nmoment, and our procedure involves rotations in spin space, whose definition is\nnot insensitive the choice of uz\ni. In other terms, the differences in the formulas\nare due to the fact that the definition of the Green’s functions is infl uenced by\nthechoiceofthequantizationaxis,sotheexpressionsbasedonGr een’sfunctions\nexhibit “special” directions, which coincide with the vectors uz\ni≡ri.\n8.2. Second set - Equations (69)\nWe now solve Eqs.(69). The first step is to identify Bz\niwith a corresponding\nterm proportional to Bz\nithat is included in the RHSs of both the first and the\nsecond among Eqs.(69). Thus, we have to find such a term in the exp ression for\n/tildewiderMiα\niα. For brevity, from Eq.(55) we put\n/tildewiderMiα\ni′α′≡ Miα\ni′α′−Wiα\ni′α′+βViαVi′α′, (77)\nwhere we have defined the quantity\nWiα\ni′α′≡1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆViα(τ)ˆVi′α′(τ′)/angbracketrightBig\n, (78)\n21which can be written explicitly as\nWiα\ni′α′≡/summationdisplay\nj,j′/summationdisplay\nM1M2M3M4/bracketleftbigg/parenleftbig\nSiα·Ti\nj/parenrightbigM2\nM1/parenleftBig\nSi′α′·Ti′\nj′/parenrightBigM4\nM3/tildewideχ(jM1)(j′M3)\n(iM2)(i′M4)\n−/parenleftbig\nSiα·Ti\nj/parenrightbigM2\nM1/parenleftBig\nTj′\ni′·Si′α′/parenrightBigM4\nM3/tildewideχ(jM1)(i′M3)\n(iM2)(j′M4)\n−/parenleftBig\nTj\ni·Siα/parenrightBigM2\nM1/parenleftBig\nSi′α′·Ti′\nj′/parenrightBigM4\nM3/tildewideχ(iM1)(j′M3)\n(jM2)(i′M4)\n+/parenleftBig\nTj\ni·Siα/parenrightBigM2\nM1/parenleftBig\nTj′\ni′·Si′α′/parenrightBigM4\nM3/tildewideχ(iM1)(i′M3)\n(jM2)(j′M4)/bracketrightbigg\n,(79)\nwhere we have put\n/tildewideχ1,3\n2,4≡1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′χ1,3\n2,4(τ,τ+,τ′,τ′+), (80)\nwhich has the property /tildewideχ1,3\n2,4=/tildewideχ3,1\n4,2, as follows from Eq.(50). Consequently,\nWiα\ni′α′=Wi′α′\niα. (81)\nAll the three terms appearing in the RHS of Eq.(77) depend explicitly o n the\nmagnetic field, both via the single- and two- electron Green’s functio ns entering\ntheir definitions, and via the explicit dependence of the hopping para meters.\nWhilewewillcommentmoreindetailaboutthisinAppendix B,forthepur pose\nof solving Eqs.(69) we just need to find the term that should be ident ified with\nBz\ni. To this end, we note that\n/bracketleftbig\nSiα,Ti\ni/bracketrightbig\n=/bracketleftbig\nSiα,Ai\ni/bracketrightbig\n+iµBgi(Bi×Si)·uα\ni, (82)\nfrom which it follows that\nViα=µBgi/parenleftBig\nBi×/angbracketleftBig\nˆSi/angbracketrightBig/parenrightBig\n·uα\ni+Viα|Bi=0,\nMiα\niα=−µBgi/parenleftBig\nBi·/angbracketleftBig\nˆSi/angbracketrightBig\n−Bα\ni/angbracketleftBig\nˆSiα/angbracketrightBig/parenrightBig\n+Miα\niα/vextendsingle/vextendsingle\nBi=0, (83)\nwhere with the notation x|Bi=0, here and in the following, we do notmean that\nxdoes not depend explicitly on Bi, we mean instead that xdoes not vanish\nwhenBi=0. In the presence of Bi∝ne}ationslash=0, alsox|Bi=0will depend on Bivia the\nGreen’s functions, as well as via the Peierls phases due to the electr omagnetic\nfield, which have to be included into the hopping parameters.\nUsing the fact that/angbracketleftBig\nˆSi/angbracketrightBig\n=Siuz\ni, a term in the expression of Miα\niαbecomes\nBi·/angbracketleftBig\nˆSi/angbracketrightBig\n−Bα\ni/angbracketleftBig\nˆSiα/angbracketrightBig\n→Bz\niSi, (84)\nand the identification is obvious,\nBz\ni=µBgiSiBz\ni, (85)\n22analogously to the other components of the magnetic field. It shou ld be noted\nthat the whole procedure relies on the existence of non-zero magn etic moments\nSi, which is therefore a requirement for the mapping of the electronic model to\nthe classical spin model via the equivalence of the thermodynamic po tentials for\nspin rotations.\nThe final task is to determine JiiandJz\nij. Let us define\n/tildewiderMiα\niα≡/tildewideNiα\niα−µBgi/parenleftBig\nBi·/angbracketleftBig\nˆSi/angbracketrightBig\n−Bα\ni/angbracketleftBig\nˆSiα/angbracketrightBig/parenrightBig\n, (86)\nso that the first and the second among Eqs.(69) become:\n−/summationdisplay\nj/ne}ationslash=iJz\nij+Jy\nii−Jz\nii=/tildewideNix\nix,\n−/summationdisplay\nj/ne}ationslash=iJz\nij+Jx\nii−Jz\nii=/tildewideNiy\niy, (87)\nor, more compactly,\n−/summationdisplay\nj/ne}ationslash=iJz\nij+J¯α\nii−Jz\nii=/tildewideNiα\niα, (88)\nwhereα,¯α∈ {x,y}andα∝ne}ationslash= ¯α. In the general relativistic case, /tildewideNix\nix∝ne}ationslash=/tildewideNiy\niy, and\nthere seems to be no uniquely defined way of separating the various unknown\nterms. However, we note that one should definitely have /tildewideNix\nix=/tildewideNiy\niyin the non-\nrelativistic regime (more on this in the following Section 9), and consist ently\nJx\nii=Jy\nii=Jz\nii≡ Jii. In this way, in fact, the corresponding Hamiltonian term\nbecomes/summationtext\niJii|ei|2=/summationtext\niJii, which is just a constant term, expressing the\nfact that there is no on-site anisotropy. Therefore, in the relativ istic regime the\nquantity −/summationtext\nj/ne}ationslash=iJz\nijmust be obtained from Eqs.(88), it must be independent of\nα, and it must reduce to/parenleftBig\n/tildewideNiα\niα/parenrightBig\nnrelin the non-relativistic regime. Additionally,\nthe quantities Jz\nijmust be symmetric under i↔j. We can keep into account\nall these requirements by putting\nJz\nij!=1\n2/parenleftBig\n/tildewiderMix\njx+/tildewiderMiy\njy/parenrightBig\n, i∝ne}ationslash=j (89)\nfrom which it follows that\nJy\nii−Jz\nii=/tildewideNix\nix+1\n2/summationdisplay\nj/ne}ationslash=i/parenleftBig\n/tildewiderMix\njx+/tildewiderMiy\njy/parenrightBig\n,\nJx\nii−Jz\nii=/tildewideNiy\niy+1\n2/summationdisplay\nj/ne}ationslash=i/parenleftBig\n/tildewiderMix\njx+/tildewiderMiy\njy/parenrightBig\n. (90)\nThe equations only allow to determine two of the parameters {Jx\nii,Jy\nii,Jz\nii}as\nfunctions of the third one, which is of course understandable since , as discussed\nin Section 7.1, the diagonal part of the local anisotropy tensor pro vides energy\n23contributions under spin rotationsonly depending on the relativediff erences be-\ntweenthethreeelements, becomingrotationallyinvariantwhenthe sedifferences\ndisappear. Hence, the rotationally invariant part is undetermined.\nWe notethatin thenon-relativisticregime, aswewilldiscussinSection 9, we\nhave/tildewiderMix\njxnrel=/tildewiderMiy\njy, andthe sumrule/summationtext\nj/tildewiderMiα\njαnrel= 0. Notingthat /tildewideNiα\njαnrel=/tildewiderMiα\njα,\nwe conclude that, in the non-relativistic regime, our solutions (89) a nd (90)\ncorrectly give Jz\nijnrel=/tildewiderMiα\njαnrel=Jx\nijnrel=Jy\nij, as well as Jx\niinrel=Jy\niinrel=Jz\nii. In the\nrelativistic case, instead, Eqs.(90) account for the non-equivalen ce of the spatial\ndirections.\n8.3. Summary of the formulas for the effective interactions\nFor the convenience of the reader, we here summarize the resultin g for-\nmulas, which completely establish the mapping from the multi-orbital H ub-\nbard model with rotationally invariant interaction (see the related d iscussion\nin Appendix A) to the general quadratic Hamiltonian of classical spins given\nby Eq.(61), under the requirement that the thermodynamic poten tials for spin\nrotations of the two models are the same up to second order in the r otation\nangles. The results are:\nBi=µBgiSiBi; (91)\nDx\nij=i\n2TrM/bracketleftBig\nSix·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig\n−Sjx·/parenleftBig\nρj\ni·Ti\nj−Tj\ni·ρi\nj/parenrightBig/bracketrightBig\n,\nDy\nij=i\n2TrM/bracketleftBig\nSiy·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig\n−Sjy·/parenleftBig\nρj\ni·Ti\nj−Tj\ni·ρi\nj/parenrightBig/bracketrightBig\n,\nDz\nij=1\n2/parenleftBig\n/tildewiderMix\njy−/tildewiderMiy\njx/parenrightBig\n; (92)\nfori∝ne}ationslash=j,\nCx\nij=i\n2TrM/bracketleftBig\nSix·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig\n+Sjx·/parenleftBig\nρj\ni·Ti\nj−Tj\ni·ρi\nj/parenrightBig/bracketrightBig\n,\nCy\nij=−i\n2TrM/bracketleftBig\nSiy·/parenleftBig\nρi\nj·Tj\ni−Ti\nj·ρj\ni/parenrightBig\n+Sjy·/parenleftBig\nρj\ni·Ti\nj−Tj\ni·ρi\nj/parenrightBig/bracketrightBig\n,\nCz\nij=−1\n2/parenleftBig\n/tildewiderMix\njy+/tildewiderMiy\njx/parenrightBig\n; (93)\nfori=j,\nCx\nii= iTrM/bracketleftBig\nρi\ni·/parenleftbig\nAi\ni·Six−Six·Ai\ni/parenrightbig/bracketrightBig\n,\nCy\nii=−iTrM/bracketleftBig\nρi\ni·/parenleftbig\nAi\ni·Siy−Siy·Ai\ni/parenrightbig/bracketrightBig\n,\nCz\nii=−/tildewiderMix\niy; (94)\n24fori∝ne}ationslash=j,\nJx\nij=/tildewiderMiy\njy,\nJy\nij=/tildewiderMix\njx,\nJz\nij=1\n2/parenleftBig\n/tildewiderMix\njx+/tildewiderMiy\njy/parenrightBig\n; (95)\nfori=j,\nJy\nii−Jz\nii=/tildewideNix\nix+1\n2/summationdisplay\nj/ne}ationslash=i/parenleftBig\n/tildewiderMix\njx+/tildewiderMiy\njy/parenrightBig\n,\nJx\nii−Jz\nii=/tildewideNiy\niy+1\n2/summationdisplay\nj/ne}ationslash=i/parenleftBig\n/tildewiderMix\njx+/tildewiderMiy\njy/parenrightBig\n. (96)\nWe recall that the quantities /tildewiderMiα\ni′α′are defined in Eq.(55), in terms of Eqs.(28),\n(29), (51) and (52). The quantities /tildewideNiα\niαare defined in Eq.(86).\nWe alsorecallthat, for i∝ne}ationslash=j, the quantities Jx\nij,Jy\nij,Cz\nijandDz\nijareuniquely\ndetermined in our procedure, as well as Cz\nii. The other magnetic parameters\nwere determined, instead, as particular (physically reasonable) so lutions of a\nset of equations whose number is far smaller than the number of par ameters. It\nmay be that requiring the equivalence of the thermodynamic potent ials for spin\nrotations to higher orders in the rotation angles will lead to modificat ions of\nthe formulas related to these latter parameters, however any mo dification must\nsatisfy Eqs.(68) and (70).\n9. Spin 1 /2 in the non-relativistic regime (single-orbital Hubbard\nmodel)\nA particular case is obtained for the single-orbital Hubbard model, w ith\nS= 1/2 (no orbital exchange, or l= 0), in the non-relativistic regime and in the\nabsence of external magnetic fields. We will label this particular cas e as “soH”\nin the following. In that case, the magnetic moment indexes ( i,i′) coincide with\nthe atomic indexes, the single-particle Hamiltonian is TiM\ni′M′=δM\nM′Ti\ni′, as well as\nTi\ni′=Ti′\ni≡Tii′. Analogously, ρiM\ni′M′=δM\nM′ρM\nii′=δM\nM′ρM\ni′i. The index Massumes\nthe valuesM∈ {+1/2,−1/2} ≡ {↑,↓}. Moreover, Si=s=σ/2, where σis\nthe vector of Pauli matrices, and we have removed the subscript ibecause we\nconsider a collinear spin configuration. As a consequence,\n/bracketleftBig\nSiα,Ti\ni/bracketrightBigsoH= 0. (97)\nIn the soH case, it is easy to compute all the traces and sums over M. Using\nEq.(55), and recalling that α,α′∈ {x,y}, we obtain:\nViαsoH= 0, (98)\n25/tildewiderMiα\ni′α′soH=/parenleftbig\nMiα\ni′α′/parenrightbignrel−/parenleftBig\n/tildewideAiα\ni′α′/parenrightBignrel\nnloc\n(1)=1\n2δαα′Tii′/parenleftBig\nρ↑\nii′+ρ↓\nii′/parenrightBig\n−δii′δαα′1\n2/summationdisplay\njTij/parenleftBig\nρ↑\nij+ρ↓\nij/parenrightBig\n−1\n4/summationdisplay\nj/ne}ationslash=i/summationdisplay\nj′/ne}ationslash=i′TijTi′j′/summationdisplay\nMM′(σα)¯M\nM(σα′)¯M′\nM′/bracketleftBigg\n/tildewideχ(jM)(j′M′)\n(i¯M)(i′¯M′)\n−/tildewideχ(jM)(i′M′)\n(i¯M)(j′¯M′)−/tildewideχ(iM)(j′M′)\n(j¯M)(i′¯M′)+/tildewideχ(iM)(i′M′)\n(j¯M)(j′¯M′)/bracketrightBigg\n(2)=1\n2δαα′Tii′/parenleftBig\nρ↑\nii′+ρ↓\nii′/parenrightBig\n−δii′δαα′1\n2/summationdisplay\njTij/parenleftBig\nρ↑\nij+ρ↓\nij/parenrightBig\n−1\n4/summationdisplay\nj/ne}ationslash=i/summationdisplay\nj′/ne}ationslash=i′TijTi′j′/summationdisplay\nM(σα)¯M\nM(σα′)M\n¯M/bracketleftBigg\n/tildewideχ(jM)(j′¯M)\n(i¯M)(i′M)\n−/tildewideχ(jM)(i′¯M)\n(i¯M)(j′M)−/tildewideχ(iM)(j′¯M)\n(j¯M)(i′M)+/tildewideχ(iM)(i′¯M)\n(j¯M)(j′M)/bracketrightBigg\n(3)=1\n2δαα′/braceleftBigg\nTii′/parenleftBig\nρ↑\nii′+ρ↓\nii′/parenrightBig\n−δii′/summationdisplay\njTij/parenleftBig\nρ↑\nij+ρ↓\nij/parenrightBig\n−1\n2/summationdisplay\nj/ne}ationslash=i/summationdisplay\nj′/ne}ationslash=i′TijTi′j′/summationdisplay\nM=↑,↓/bracketleftBigg\n/tildewideχ(jM)(j′¯M)\n(i¯M)(i′M)\n−/tildewideχ(jM)(i′¯M)\n(i¯M)(j′M)−/tildewideχ(iM)(j′¯M)\n(j¯M)(i′M)+/tildewideχ(iM)(i′¯M)\n(j¯M)(j′M)/bracketrightBigg/bracerightBigg\n,(99)\nwhere we have used the fact that in the non-relativistic regime the H amiltonian\ncannot alter the total number of electrons with a given spin projec tion↑or↓,\ntherefore the only non-vanishing terms of /tildewideχ(i1M1)(i3M3)\n(i2M2)(i4M4)are those with M3=M4\nandM1=M2, or those with M1=M4andM2=M3. This selects the\nterms with M=−M′in going from passage (1) to passage (2) in the previous\nequation. Then, one observes that /tildewideχ1,3\n2,4=/parenleftBig\n/tildewideχ2,4\n1,3/parenrightBig∗\n, and in the non-relativistic\ncase this quantity is real. Thus, the terms with α∝ne}ationslash=α′vanish, and we obtain\nthe last passage (3).\nWe see immediately that in this case Eqs.(95) give (for i∝ne}ationslash=i′)\nJx\nii′soH=Jy\nii′soH=Jz\nii′soH≡ Jii′soH=/tildewiderMix\ni′xsoH=/tildewiderMiy\ni′y\n=1\n2Tii′/parenleftBig\nρ↑\nii′+ρ↓\nii′/parenrightBig\n−1\n4/summationdisplay\nj/ne}ationslash=i/summationdisplay\nj′/ne}ationslash=i′TijTi′j′/summationdisplay\nM=↑↓/bracketleftBigg\n/tildewideχ(jM)(j′¯M)\n(i¯M)(i′M)−/tildewideχ(jM)(i′¯M)\n(i¯M)(j′M)−/tildewideχ(iM)(j′¯M)\n(j¯M)(i′M)+/tildewideχ(iM)(i′¯M)\n(j¯M)(j′M)/bracketrightBigg\n,\n(100)\n26that is, exchangeis isotropic. All the othermagneticparameters, determined for\nthe general relativistic regime, vanish in the soH case. The mapping e quations,\n(68), (69) and (70), reduce to:\nJii′soH=/tildewiderMiα\ni′α, i∝ne}ationslash=i′,\n−/summationdisplay\nj/ne}ationslash=iJijsoH=/tildewiderMiα\niα. (101)\nThe sum rule given by the second among Eqs.(101), combined with the first\nequation, becomes\n/summationdisplay\ni′/tildewiderMiα\ni′αsoH= 0. (102)\nTo check whether this is valid, we use Eq.(99), obtaining:\n/summationdisplay\ni′/summationdisplay\nj/ne}ationslash=i/summationdisplay\nj′/ne}ationslash=i′TijTi′j′/summationdisplay\nM=↑,↓/bracketleftBigg\n/tildewideχ(jM)(j′¯M)\n(i¯M)(i′M)−/tildewideχ(jM)(i′¯M)\n(i¯M)(j′M)−/tildewideχ(iM)(j′¯M)\n(j¯M)(i′M)+/tildewideχ(iM)(i′¯M)\n(j¯M)(j′M)/bracketrightBigg\n= 0, (103)\nwhich is identically true, as follows from the interchange of the dummy indexes\ni′andj′in the second and fourth term on the LHS. Therefore, the sum rule\n(102) is satisfied by the expression for the exchange parameters given in the first\namong Eqs.(101).\nWe compare our results with the previous literature on non-relativis tic ex-\nchange. Most of the previous works on this subject [1, 10, 11, 16] neglected\nthe vertices in the two-electron Green’s functions. This amounts t o putting\nΓ = 0 in Eq.(43), replacing χwithχ0. We will now show what we obtain in\nthe present case when such approximation is performed. We will den ote all\nthe equations derived under this approximation with the equality sym bolΓ=0= .\nThe only two-particle Green’s function that we need is given by Eq.(47 ), which\nbecomes\nχ1,3\n2,4(τ,τ+,τ′,τ′+)Γ=0=/parenleftbig\nχ0/parenrightbig1,3\n2,4(τ,τ+,τ′,τ′+)\n=−ρ1\n2ρ3\n4−G1\n4(τ−τ′−ε)G3\n2(τ′−τ−ε).(104)\nUsing the Matsubara-frequency representation, G(τ)≡1\nβ/summationtext\nωG(iω)e−iωτ, we\nre-write Eq.(80) for Γ = 0, after integrating over d τand dτ′, as:\n/parenleftbig\n/tildewideχ0/parenrightbig1,3\n2,4=−βρ1\n2ρ3\n4−1\nβ/summationdisplay\nωeiω0+G1\n4(iω)G3\n2(iω). (105)\nIn the soH case, we have Giσ\ni′σ′≡δσ\nσ′Gσ\nii′=δσ\nσ′Gσ\ni′i, whereG↑\nii′∝ne}ationslash=G↓\nii′for a\n27symmetry-broken configuration of the system. We obtain\n/tildewiderMiα\ni′αsoH,Γ=0=1\n2/bracketleftBigg\nTii′/parenleftBig\nρ↑\nii′+ρ↓\nii′/parenrightBig\n−δii′/summationdisplay\njTij/parenleftBig\nρ↑\nij+ρ↓\nij/parenrightBig/bracketrightBigg\n+1\n4β/summationdisplay\nσ=↑,↓/summationdisplay\nωeiω0+/braceleftBigg\n[T·Gσ(iω)]i\ni′/bracketleftbig\nT·G¯σ(iω)/bracketrightbigi′\ni−[T·Gσ(iω)·T]i\ni′/bracketleftbig\nG¯σ(iω)/bracketrightbigi′\ni\n−[Gσ(iω)]i\ni′/bracketleftbig\nT·G¯σ(iω)·T/bracketrightbigi′\ni+[Gσ(iω)·T]i\ni′/bracketleftbig\nG¯σ(iω)·T/bracketrightbigi′\ni/bracerightBigg\n.\n(106)\nThe formulas for exchange parameters are often expressed in te rms of self-\nenergies [10, 11, 16], since these are the key quantities for numeric al evaluation\nwithin the framework of Dynamical Mean Field Theory (DMFT) [26–28]. To\ndo so, we use the equations of motion for Matsubara Green’s funct ions, which\nwe write in general matrix notation as\n(ω−iµ)Gσ(iω)+iT·Gσ(iω) = 1−Σσ(iω)·Gσ(iω),\n(ω−iµ)Gσ(iω)+iGσ(iω)·T= 1−Gσ(iω)·Σσ(iω) (107)\n(units have been chosen so that Σ has the dimensions of an energy) . These\nequations allow to express Eq.(106) in terms of single-particle Green ’s functions\nand self-energies Σ, removing the hopping parameters T. After some algebra,\nfori∝ne}ationslash=i′we obtain\nJii′soH,Γ=0=−1\n4β/summationdisplay\nσ=±1/summationdisplay\nωeiω0+/braceleftBigg\n[Σσ(iω)·Gσ(iω)]i\ni′/bracketleftbig\nΣ¯σ(iω)·G¯σ(iω)/bracketrightbigi′\ni\n−[Σσ(iω)·Gσ(iω)·Σσ(iω)]i\ni′/bracketleftbig\nG¯σ(iω)/bracketrightbigi′\ni−[Gσ(iω)]i\ni′/bracketleftbig\nΣ¯σ(iω)·G¯σ(iω)·Σ¯σ(iω)/bracketrightbigi′\ni\n+[Gσ(iω)·Σσ(iω)]i\ni′/bracketleftbig\nG¯σ(iω)·Σ¯σ(iω)/bracketrightbigi′\ni+[Σσ(iω)]i\ni′/bracketleftbig\nG¯σ(iω)/bracketrightbigi′\ni\n+[Gσ(iω)]i\ni′/bracketleftbig\nΣ¯σ(iω)/bracketrightbigi′\ni/bracerightBigg\n. (108)\nEquation (108) is in agreement with Eqs.(185) and (155) from Ref.[16 ] (as it\ncan be seen by using the symmetries of the Green’s functions). The different\npre-factor −2 is due to the different definition of the exchange parameters in\nthe Hamiltonian (57).\nFinally, if we assume the self-energy to be local (“LsoH” assumption ), which\nis a requirement for the direct application of DMFT, by putting Σσ\nii′LsoH=δii′Σσ\ni\nwe obtain the simple formula\nJii′LsoH,Γ=0=2\nβ/summationdisplay\nωeiω0+/bracketleftBig\nG↑\nii′(iω)ΣS\ni′(iω)G↓\ni′i(iω)ΣS\ni(iω)/bracketrightBig\n,(109)\n28where ΣS\ni(iω)≡/bracketleftBig\nΣ↑\ni(iω)−Σ↓\ni(iω)/bracketrightBig\n/2. Equation (109) is in agreement with\nEq.(21) from Ref.[10], Eq.(19) from Ref.[11] and Eq.(191) from Ref.[16 ], again\nup to a factor −2 due to the different definition mentioned above. We have thus\nrecovered the results of the previous literature as particular cas es of our present\nformulation.\n10. Spin, orbital, and spin-orbital contributions to magne tism\nWe now go back to the relativistic regime and to the results summarize d in\nSection 8.3. In this work, as stated in the introduction, we are cons idering as\ndynamicalvariablessomeeffective classical“spins”which arerepre sentedby the\nunit vectors ei. The coefficients of the interactions, that we have determined,\nare related to the response of the system under rotations of the total local\nmagnetic moments expressed by the operators ˆSi=ˆli+ˆsi. It is interesting\nto compare the response of the system under this rotation to the response that\nis obtained when only the spin-1 /2 (ˆsi) or the orbital ( ˆli) components of the\nmagnetic moments are rotated. In order to address this question , we need to\nseparate in the effective magnetic parameters the contributions c oming from\nthe rotation of ˆsifrom the contributions coming from the rotation of ˆli. If\nthese contributions could be decoupled, we could identify them individ ually as\ndistinct contributions to magnetism.\nTo perform this decoupling, we need to switch from the initial basis fo r the\nelectronic fields, where the single-electron wave functions were ch aracterized by\nthe quantum numbers ( a,n,l,S,M ), to the basis characterized by ( a,n,l,m,σ ),\nwheremandσ=±1/2 are the quantum numbers, respectively, of the operators\nˆlzand ˆsz. The change of basis goes via the Clebsch-Gordan transformation ,\nˆψ†\na,n,l,S,M≡l/summationdisplay\nm=−l/summationdisplay\nσ=±1/2Cmσ\nSM(l)ˆψ†\na,n,l,m,σ, (110)\nwhereCmσ\nSM(l)isaClebsch-Gordancoefficient. We note that, up tonow, wehave\nconsidered unit vectors eidepending on the index i≡(a,n,l,S), meaning that\nwe have defined in principle a different spin for each value of Scorresponding\nto a given orbital set o≡(a,n,l). It is not possible to separate spin-1 /2 from\norbital contributions in this situation. To achieve this separation, w e need the\nspins to be independent of S, i.e.,ei→eo. In this way our effective spin\nHamiltonian, from Eq.(57), becomes:\nH[ei] =/summationdisplay\niei·Bi+1\n2/summationdisplay\ni,i′/summationdisplay\nα,α′ei,αei′,α′Hαα′\nii′\n≡/summationdisplay\noeo·Bo+1\n2/summationdisplay\no,o′/summationdisplay\nα,α′eo,αeo′,α′Hαα′\noo′, (111)\n29where\nBo≡/summationdisplay\nSBoS,Hαα′\noo′≡/summationdisplay\nS,S′Hαα′\n(oS)(o′S′). (112)\nNote that this is a particular case ofthe generalprocedurethat we havefollowed\nup tonow, correspondingtothe lessgeneralcaseofthe rotation sdepending only\nonoratherthanon( o,S). Therefore,wesimplyhavetore-definetheparameters\naccording to Eqs.(112), without altering the spin Hamiltonian.\nSumming over the Squantum numbers now allows to separate orbital and\nspin contributions. The most compact way to show how it works is to c onsider\nthe quantities Viαand/tildewiderMiα\ni′α′, since all the magnetic parameters are obtained as\nlinear combinations of these quantities (or parts of them). First, le t us consider\nViα, which in the new Hamiltonian given by the second passage of Eq.(111) will\nbe replaced by [cfr. Eq.(71)]\nVoα=/summationdisplay\nSV(oS)α=/summationdisplay\nSiTrM/summationdisplay\no′S′/bracketleftBig\nS(oS)α·/parenleftBig\nρoS\no′S′·To′S′\noS−ToS\no′S′·ρo′S′\noS/parenrightBig/bracketrightBig\n= iTrS,M/bracketleftBig\nSoα·(ρ·T−T·ρ)o\no/bracketrightBig!= iTrm,σ/bracketleftBig\n(soα+loα)·(ρ·T−T·ρ)o\no/bracketrightBig\n≡ Vspin\noα+Vorb\noα, (113)\nwhere we have defined the separate spin-1 /2 and orbital contributions, respec-\ntively, as\nVspin\noα≡iTrσ/bracketleftBig\nsoα·Trm(ρ·T−T·ρ)o\no/bracketrightBig\n,\nVorb\noα≡iTrm/bracketleftBig\nloα·Trσ(ρ·T−T·ρ)o\no/bracketrightBig\n. (114)\nThe passage marked as!= in Eq.(113) is the step that allows to go from the rep-\nresentation in the ( a,n,l,S,M ) basis to the representation in the ( a,n,l,m,σ )\nbasis, where it is possible to split the total spin matrix into the spin-1 /2 and\nthe orbital contribution. The matrices defined in the new basis, suc h as the\ndensity matrix ρand the hopping parameters T, are obtained from the previous\nrepresentation via the Clebsch-Gordan transformation:\nρanlSM\na′n′l′S′M′=/summationdisplay\nmσ/summationdisplay\nm′σ′ρanlmσ\na′n′l′m′σ′CSM\nmσ(l)Cm′σ′\nS′M′(l′). (115)\nThe equivalence of the traces is a consequence of the completenes s relation\n/summationdisplay\nSMCSM\nmσ(l)Cm′σ′\nSM(l) =δm′\nmδσ′\nσ. (116)\nIt is then clear why we need the sum over Sto split the spin and orbital con-\ntributions, and for this reason our rotation parameters must dep end only on\n(a,n,l).\n30We then consider the term /tildewiderMiα\ni′α′, which in the new Hamiltonian given by\nthe second passage of Eq.(111) will be replaced by [cfr. Eq.(77)]\n/tildewiderMoα\no′α′=/summationdisplay\nS,S′/tildewiderM(oS)α\n(o′S′)α′= TrS,M/parenleftBig\nSoα·To\no′·So′α′·ρo′\no+So′α′·To′\no·Soα·ρo\no′/parenrightBig\n−δo\no′1\n2TrS,M/parenleftBig\n(Soα·Soα′+Soα′·Soα)·{ρ;T}o\no/parenrightBig\n+βVoαVo′α′\n−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVoα(τ)ˆVo′α′(τ′)/angbracketrightBig\n, (117)\nwhere\nˆVoα(τ) = iTrS,M/parenleftBig\nSoα·[ˆρ(τ);T]o\no/parenrightBig\n= iTrσ/parenleftBig\nsoα·Trm[ˆρ(τ);T]o\no/parenrightBig\n+iTrm/parenleftBig\nloα·Trσ[ˆρ(τ);T]o\no/parenrightBig\n≡ˆVspin\noα(τ)+ˆVorb\noα(τ). (118)\nWe can then apply the change of basis, replace Tr S,Mwith Trm,σ, and sepa-\nrate the spin-1 /2 and the orbital contributions by splitting the Soαmatrix into\nsoα+loα. In the case of the quantity /tildewiderMoα\no′α′given by Eq.(117), which depends\non quadratic combinations of the spin matrices, we can distinguish sp in-spin,\norbital-orbital and spin-orbital contributions:\n/tildewiderMoα\no′α′≡/parenleftBig\n/tildewiderMoα\no′α′/parenrightBigspin−spin\n+/parenleftBig\n/tildewiderMoα\no′α′/parenrightBigspin−orb\n+/parenleftBig\n/tildewiderMoα\no′α′/parenrightBigorb−orb\n,(119)\nwhere\n/parenleftBig\n/tildewiderMoα\no′α′/parenrightBigspin−spin\n≡Trm,σ/parenleftBig\nsoα·To\no′·so′α′·ρo′\no+so′α′·To′\no·soα·ρo\no′/parenrightBig\n−δo\no′δα\nα′1\n4Trm,σ{ρ;T}o\no\n+βVspin\noαVspin\no′α′−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\noα(τ)ˆVspin\no′α′(τ′)/angbracketrightBig\n,\n(120)\nwherein the secondline wehaveused the fact that soα·soα′+soα′·soα=1\n2δαα′1,\nwhich is a property of the Pauli matrices,\n/parenleftBig\n/tildewiderMoα\no′α′/parenrightBigorb−orb\n≡Trm,σ/parenleftBig\nloα·To\no′·lo′α′·ρo′\no+lo′α′·To′\no·loα·ρo\no′/parenrightBig\n−δo\no′1\n2Trm/parenleftBig\n(loα·loα′+loα′·loα)·Trσ{ρ;T}o\no/parenrightBig\n+βVorb\noαVorb\no′α′−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVorb\noα(τ)ˆVorb\no′α′(τ′)/angbracketrightBig\n,\n(121)\n31/parenleftBig\n/tildewiderMoα\no′α′/parenrightBigspin−orb\n≡Trm,σ/parenleftBig\nsoα·To\no′·lo′α′·ρo′\no+so′α′·To′\no·loα·ρo\no′\n+loα·To\no′·so′α′·ρo′\no+lo′α′·To′\no·soα·ρo\no′/parenrightBig\n−δo\no′1\n2Trm,σ/parenleftBig\n(soα·loα′+soα′·loα+loα·soα′+loα′·soα)·{ρ;T}o\no/parenrightBig\n+β/parenleftBig\nVspin\noαVorb\no′α′+Vorb\noαVspin\no′α′/parenrightBig\n−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγ/bracketleftBig\nˆVspin\noα(τ)ˆVorb\no′α′(τ′)+ˆVorb\noα(τ)ˆVspin\no′α′(τ′)/bracketrightBig/angbracketrightBig\n.(122)\nThis separation, which we have applied to the quantities Voαand/tildewiderMoα\no′α′,\nhas then to be transferred to the effective magnetic parameters BoandHαα′\noo′\nof Eq.(111), via the solutions of the mapping equations summarized in Section\n8.3. We notice that the magnetic parameters can be separated into two groups:\nthefirst group given by {Bo,Dx\noo′,Dy\noo′,Cx\noo′,Cy\noo′}, and the second group given\nby{Dz\noo′,Cz\noo′,Joo′}. The terms of the first group have the following features:\n•they are expressed in terms of the quantities Voαor parts of them;\n•their evaluation requires computation of single-particle Green’s fun ctions\n(of the density-matrix form);\n•they can be split into spin and orbital contributions.\nThe terms of the second group have the following features:\n•they are expressed in terms of the quantities Voαand/tildewiderMoα\no′α′;\n•their evaluation requires computation of single-particle and two-pa rticle\nGreen’s functions;\n•they can be split into spin-spin, spin-orbital and orbital-orbital con tribu-\ntions.\nIn the latter case, while the spin-spin and orbital-orbital parts obv iously arise\nfrom rotations involving only one of the two contributions to the tot al local\nmagnetic moments, respectively, the spin-orbital term does not a rise in such\nindividual rotations, appearing only when the whole magnetic moment s are\nrotated.\nIn the next Sections we list the explicit formulas for all the paramete rs of\nthe magnetic interactions, separated into spin, orbital and (when applicable)\nspin-orbital parts. For the magnetic parameters of the second g roup, we show\nnot only the complete formulas with the full two-particle Green’s fun ctions,\nbut also the formulas obtained when the vertices are neglected. Th is approx-\nimation, which produces formulas depending only on single-particle Gr een’s\nfunctions, has been routinely applied for computations of isotropic exchange\n32parameters within the framework of the Hubbard model with quenc hed orbital\nmoments [10]. The formulas that we list below are the natural extens ion to the\nunquenched case. The approximated expressions are listed after the exact ones,\nseparated from them by the symbolΓ=0= , analogously to the convention used in\nSection 9. The approximation is achieved by applying Eq.(105). In par ticular,\nwe have\nβVX\noαVY\no′α′−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVX\noα(τ)ˆVY\no′α′(τ′)/angbracketrightBig\nΓ=0=1\nβ/summationdisplay\nωeiω0+Trm,σ/braceleftBig\nSX\noα·[G(iω)·T]o\no′·SY\no′α′·[G(iω)·T]o′\no\n−SX\noα·G(iω)o\no′·SY\no′α′·[T·G(iω)·T]o′\no\n−SX\noα·[T·G(iω)·T]o\no′·SY\no′α′·G(iω)o′\no\n+SX\noα·[T·G(iω)]o\no′·SY\no′α′·[T·G(iω)]o′\no/bracerightBig\n,(123)\nwhere X and Y refer to either spin- or orbital- related terms. Since t he resulting\nexpressions for the magnetic parameters become very long and inv olved, for\nthe sake of readability we make use of the permutation symbol− →Po↔o′, which\nswitches the indexes oando′of any tensor placed on its right side, that is,\nfoo′− →Po↔o′goo′=foo′go′o. (124)\nObviously, the combination 1+− →Po↔o′is then the symmetrization symbol, while\n1−− →Po↔o′is the anti-symmetrization symbol.\nWe stress that the magnetic parameters of the first group do not require\nthe evaluation of two-particle Green’s functions, so their express ions in terms of\nsingle-particle density matrices are exact.\nHere follows the list of all the explicit expressions.\n10.1. Dzyaloshinskii-Moriya interaction\nFrom Eqs.(92), we see that Dα\noo′≡(Dα\noo′)spin+ (Dα\noo′)orbforα=xory,\nwhileDz\noo′≡(Dz\noo′)spin−spin+(Dz\noo′)orb−orb+(Dz\noo′)spin−orb, where\n(Dx\noo′)spin=i\n2/parenleftBig\n1−− →Po↔o′/parenrightBig\nTrσ/bracketleftBig\nsox·Trm/parenleftBig\nρo\no′·To′\no−To\no′·ρo′\no/parenrightBig/bracketrightBig\n,(125)\n(Dx\noo′)orb=i\n2/parenleftBig\n1−− →Po↔o′/parenrightBig\nTrm/bracketleftBig\nlox·Trσ/parenleftBig\nρo\no′·To′\no−To\no′·ρo′\no/parenrightBig/bracketrightBig\n,(126)\n(Dy\noo′)spin=i\n2/parenleftBig\n1−− →Po↔o′/parenrightBig\nTrσ/bracketleftBig\nsoy·Trm/parenleftBig\nρo\no′·To′\no−To\no′·ρo′\no/parenrightBig/bracketrightBig\n,(127)\n(Dy\noo′)orb=i\n2/parenleftBig\n1−− →Po↔o′/parenrightBig\nTrm/bracketleftBig\nloy·Trσ/parenleftBig\nρo\no′·To′\no−To\no′·ρo′\no/parenrightBig/bracketrightBig\n,(128)\n33(Dz\noo′)spin−spin\n=1\n2/parenleftBig\n1−− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·so′y·ρo′\no+so′y·To′\no·sox·ρo\no′/parenrightBig\n+βVspin\noxVspin\no′y−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\nox(τ)ˆVspin\no′y(τ′)/angbracketrightBig/bracerightBigg\nΓ=0=1\n2/parenleftBig\n1−− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·so′y·ρo′\no+so′y·To′\no·sox·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsox·[G(iω)·T]o\no′·so′y·[G(iω)·T]o′\no\n−sox·G(iω)o\no′·so′y·[T·G(iω)·T]o′\no−sox·[T·G(iω)·T]o\no′·so′y·G(iω)o′\no\n+sox·[T·G(iω)]o\no′·so′y·[T·G(iω)]o′\no/bracketrightBigg/bracerightBigg\n, (129)\n(Dz\noo′)orb−orb\n=1\n2/parenleftBig\n1−− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nlox·To\no′·lo′y·ρo′\no+lo′y·To′\no·lox·ρo\no′/parenrightBig\n+βVorb\noxVorb\no′y−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVorb\nox(τ)ˆVorb\no′y(τ′)/angbracketrightBig/bracerightBigg\nΓ=0=1\n2/parenleftBig\n1−− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nlox·To\no′·lo′y·ρo′\no+lo′y·To′\no·lox·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nlox·[G(iω)·T]o\no′·lo′y·[G(iω)·T]o′\no\n−lox·G(iω)o\no′·lo′y·[T·G(iω)·T]o′\no−lox·[T·G(iω)·T]o\no′·lo′y·G(iω)o′\no\n+lox·[T·G(iω)]o\no′·lo′y·[T·G(iω)]o′\no/bracketrightBigg/bracerightBigg\n, (130)\n34(Dz\noo′)spin−orb\n=1\n2/parenleftBig\n1−− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·lo′y·ρo′\no+so′y·To′\no·lox·ρo\no′\n+lox·To\no′·so′y·ρo′\no+lo′y·To′\no·sox·ρo\no′/parenrightBig\n+βVspin\noxVorb\no′y+βVorb\noxVspin\no′y\n−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγ/bracketleftBig\nˆVspin\nox(τ)ˆVorb\no′y(τ′)+ˆVorb\nox(τ)ˆVspin\no′y(τ′)/bracketrightBig/angbracketrightBig/bracerightBigg\nΓ=0=1\n2/parenleftBig\n1−− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·lo′y·ρo′\no+so′y·To′\no·lox·ρo\no′\n+lox·To\no′·so′y·ρo′\no+lo′y·To′\no·sox·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsox·[G(iω)·T]o\no′·lo′y·[G(iω)·T]o′\no\n−sox·G(iω)o\no′·lo′y·[T·G(iω)·T]o′\no−sox·[T·G(iω)·T]o\no′·lo′y·G(iω)o′\no\n+sox·[T·G(iω)]o\no′·lo′y·[T·G(iω)]o′\no+lox·[G(iω)·T]o\no′·so′y·[G(iω)·T]o′\no\n−lox·G(iω)o\no′·so′y·[T·G(iω)·T]o′\no−lox·[T·G(iω)·T]o\no′·so′y·G(iω)o′\no\n+lox·[T·G(iω)]o\no′·so′y·[T·G(iω)]o′\no/bracketrightBigg/bracerightBigg\n. (131)\n10.2. Symmetric out-of-diagonal interactions\nFrom Eqs.(93) and (94), we see that Cα\noo′≡(Cα\noo′)spin+(Cα\noo′)orbforα=xor\ny, whileCz\noo′≡(Cz\noo′)spin−spin+(Cz\noo′)orb−orb+(Cz\noo′)spin−orb, where, in the case\nofo∝ne}ationslash=o′,\n(Cx\noo′)spin=i\n2/parenleftBig\n1+− →Po↔o′/parenrightBig\nTrσ/bracketleftBig\nsox·Trm/parenleftBig\nρo\no′·To′\no−To\no′·ρo′\no/parenrightBig/bracketrightBig\n,(132)\n(Cx\noo′)orb=i\n2/parenleftBig\n1+− →Po↔o′/parenrightBig\nTrm/bracketleftBig\nlox·Trσ/parenleftBig\nρo\no′·To′\no−To\no′·ρo′\no/parenrightBig/bracketrightBig\n,(133)\n(Cy\noo′)spin=−i\n2/parenleftBig\n1+− →Po↔o′/parenrightBig\nTrσ/bracketleftBig\nsoy·Trm/parenleftBig\nρo\no′·To′\no−To\no′·ρo′\no/parenrightBig/bracketrightBig\n,(134)\n(Cy\noo′)orb=−i\n2/parenleftBig\n1+− →Po↔o′/parenrightBig\nTrm/bracketleftBig\nloy·Trσ/parenleftBig\nρo\no′·To′\no−To\no′·ρo′\no/parenrightBig/bracketrightBig\n,(135)\n35(Cz\noo′)spin−spin\n=−1\n2/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·so′y·ρo′\no+so′y·To′\no·sox·ρo\no′/parenrightBig\n+βVspin\noxVspin\no′y−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\nox(τ)ˆVspin\no′y(τ′)/angbracketrightBig/bracerightBigg\nΓ=0=−1\n2/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·so′y·ρo′\no+so′y·To′\no·sox·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsox·[G(iω)·T]o\no′·so′y·[G(iω)·T]o′\no\n−sox·G(iω)o\no′·so′y·[T·G(iω)·T]o′\no−sox·[T·G(iω)·T]o\no′·so′y·G(iω)o′\no\n+sox·[T·G(iω)]o\no′·so′y·[T·G(iω)]o′\no/bracketrightBigg/bracerightBigg\n, (136)\n(Cz\noo′)orb−orb\n=−1\n2/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nlox·To\no′·lo′y·ρo′\no+lo′y·To′\no·lox·ρo\no′/parenrightBig\n+βVorb\noxVorb\no′y−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVorb\nox(τ)ˆVorb\no′y(τ′)/angbracketrightBig/bracerightBigg\nΓ=0=−1\n2/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nlox·To\no′·lo′y·ρo′\no+lo′y·To′\no·lox·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nlox·[G(iω)·T]o\no′·lo′y·[G(iω)·T]o′\no\n−lox·G(iω)o\no′·lo′y·[T·G(iω)·T]o′\no−lox·[T·G(iω)·T]o\no′·lo′y·G(iω)o′\no\n+lox·[T·G(iω)]o\no′·lo′y·[T·G(iω)]o′\no/bracketrightBigg/bracerightBigg\n, (137)\n36(Cz\noo′)spin−orb\n=−1\n2/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·lo′y·ρo′\no+so′y·To′\no·lox·ρo\no′\n+lox·To\no′·so′y·ρo′\no+lo′y·To′\no·sox·ρo\no′/parenrightBig\n+βVspin\noxVorb\no′y+βVorb\noxVspin\no′y\n−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγ/bracketleftBig\nˆVspin\nox(τ)ˆVorb\no′y(τ′)+ˆVorb\nox(τ)ˆVspin\no′y(τ′)/bracketrightBig/angbracketrightBig/bracerightBigg\nΓ=0=−1\n2/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·lo′y·ρo′\no+so′y·To′\no·lox·ρo\no′\n+lox·To\no′·so′y·ρo′\no+lo′y·To′\no·sox·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsox·[G(iω)·T]o\no′·lo′y·[G(iω)·T]o′\no\n−sox·G(iω)o\no′·lo′y·[T·G(iω)·T]o′\no−sox·[T·G(iω)·T]o\no′·lo′y·G(iω)o′\no\n+sox·[T·G(iω)]o\no′·lo′y·[T·G(iω)]o′\no+lox·[G(iω)·T]o\no′·so′y·[G(iω)·T]o′\no\n−lox·G(iω)o\no′·so′y·[T·G(iω)·T]o′\no−lox·[T·G(iω)·T]o\no′·so′y·G(iω)o′\no\n+lox·[T·G(iω)]o\no′·so′y·[T·G(iω)]o′\no/bracketrightBigg/bracerightBigg\n. (138)\nIn the case of o=o′, we have\n(Cx\noo)spin= iTrσ/bracketleftBig\nsox·Trm(ρo\no·Ao\no−Ao\no·ρo\no)/bracketrightBig\n, (139)\n(Cx\noo)orb= iTrm/bracketleftBig\nlox·Trσ(ρo\no·Ao\no−Ao\no·ρo\no)/bracketrightBig\n, (140)\n(Cy\noo)spin=−iTrσ/bracketleftBig\nsoy·Trm(ρo\no·Ao\no−Ao\no·ρo\no)/bracketrightBig\n, (141)\n(Cy\noo)orb=−iTrm/bracketleftBig\nloy·Trσ(ρo\no·Ao\no−Ao\no·ρo\no)/bracketrightBig\n, (142)\n37(Cz\noo)spin−spin=−Trm,σ/parenleftBig\nsox·To\no·soy·ρo\no+soy·To\no·sox·ρo\no/parenrightBig\n−βVspin\noxVspin\noy+1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\nox(τ)ˆVspin\noy(τ′)/angbracketrightBig\nΓ=0=−Trm,σ/parenleftBig\nsox·To\no·soy·ρo\no+soy·To\no·sox·ρo\no/parenrightBig\n−1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsox·[G(iω)·T]o\no·soy·[G(iω)·T]o\no\n−sox·G(iω)o\no·soy·[T·G(iω)·T]o\no−sox·[T·G(iω)·T]o\no·soy·G(iω)o\no\n+sox·[T·G(iω)]o\no·soy·[T·G(iω)]o\no/bracketrightBigg\n, (143)\n(Cz\noo)orb−orb=−Trm,σ/parenleftbigg\nlox·To\no·loy·ρo\no+loy·To\no·lox·ρo\no−1\n2{lox;loy}·{ρ;T}o\no/parenrightbigg\n−βVorb\noxVorb\noy+1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVorb\nox(τ)ˆVorb\noy(τ′)/angbracketrightBig\nΓ=0=−Trm,σ/parenleftbigg\nlox·To\no·loy·ρo\no+loy·To\no·lox·ρo\no−1\n2{lox;loy}·{ρ;T}o\no/parenrightbigg\n−1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nlox·[G(iω)·T]o\no·loy·[G(iω)·T]o\no\n−lox·G(iω)o\no·loy·[T·G(iω)·T]o\no−lox·[T·G(iω)·T]o\no·loy·G(iω)o\no\n+lox·[T·G(iω)]o\no·loy·[T·G(iω)]o\no/bracketrightBigg\n, (144)\n38(Cz\noo)spin−orb\n=−Trm,σ/parenleftBig\nsox·To\no·loy·ρo\no+soy·To\no·lox·ρo\no\n+lox·To\no·soy·ρo\no+loy·To\no·sox·ρo\no/parenrightBig\n+1\n2Trm,σ/bracketleftBig/parenleftBig\n{sox;loy}+{soy;lox}/parenrightBig\n·{ρ;T}o\no/bracketrightBig\n−β/parenleftbig\nVspin\noxVorb\noy+Vorb\noxVspin\noy/parenrightbig\n+1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγ/bracketleftBig\nˆVspin\nox(τ)ˆVorb\noy(τ′)+ˆVorb\nox(τ)ˆVspin\noy(τ′)/bracketrightBig/angbracketrightBig\nΓ=0=−Trm,σ/parenleftBig\nsox·To\no·loy·ρo\no+soy·To\no·lox·ρo\no\n+lox·To\no·soy·ρo\no+loy·To\no·sox·ρo\no/parenrightBig\n+1\n2Trm,σ/bracketleftBig/parenleftBig\n{sox;loy}+{soy;lox}/parenrightBig\n·{ρ;T}o\no/bracketrightBig\n−1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsox·[G(iω)·T]o\no′·lo′y·[G(iω)·T]o′\no\n−sox·G(iω)o\no′·lo′y·[T·G(iω)·T]o′\no−sox·[T·G(iω)·T]o\no′·lo′y·G(iω)o′\no\n+sox·[T·G(iω)]o\no′·lo′y·[T·G(iω)]o′\no+lox·[G(iω)·T]o\no′·so′y·[G(iω)·T]o′\no\n−lox·G(iω)o\no′·so′y·[T·G(iω)·T]o′\no−lox·[T·G(iω)·T]o\no′·so′y·G(iω)o′\no\n+lox·[T·G(iω)]o\no′·so′y·[T·G(iω)]o′\no/bracketrightBigg\n. (145)\n10.3. Exchange interactions\nFrom Eqs.(95) and (96), we see that Jα\noo′≡(Jα\noo′)spin−spin+(Jα\noo′)orb−orb+\n(Jα\noo′)spin−orbfor allα=x,y,z. In the case of o∝ne}ationslash=o′, we obtain\n(Jx\noo′)spin−spin= Trm,σ/parenleftBig\nsoy·To\no′·so′y·ρo′\no+so′y·To′\no·soy·ρo\no′/parenrightBig\n+βVspin\noyVspin\no′y−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\noy(τ)ˆVspin\no′y(τ′)/angbracketrightBig\nΓ=0= Trm,σ/parenleftBig\nsoy·To\no′·so′y·ρo′\no+so′y·To′\no·soy·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsoy·[G(iω)·T]o\no′·so′y·[G(iω)·T]o′\no\n−soy·G(iω)o\no′·so′y·[T·G(iω)·T]o′\no−soy·[T·G(iω)·T]o\no′·so′y·G(iω)o′\no\n+soy·[T·G(iω)]o\no′·so′y·[T·G(iω)]o′\no/bracketrightBigg\n, (146)\n39(Jx\noo′)orb−orb= Trm,σ/parenleftBig\nloy·To\no′·lo′y·ρo′\no+lo′y·To′\no·loy·ρo\no′/parenrightBig\n+βVorb\noyVorb\no′y−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVorb\noy(τ)ˆVorb\no′y(τ′)/angbracketrightBig\nΓ=0= Trm,σ/parenleftBig\nloy·To\no′·lo′y·ρo′\no+lo′y·To′\no·loy·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nloy·[G(iω)·T]o\no′·lo′y·[G(iω)·T]o′\no\n−loy·G(iω)o\no′·lo′y·[T·G(iω)·T]o′\no−loy·[T·G(iω)·T]o\no′·lo′y·G(iω)o′\no\n+loy·[T·G(iω)]o\no′·lo′y·[T·G(iω)]o′\no/bracketrightBigg\n, (147)\n(Jx\noo′)spin−orb=/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsoy·To\no′·lo′y·ρo′\no+loy·To\no′·so′y·ρo′\no/parenrightBig\n+βVspin\noyVorb\no′y−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\noy(τ)ˆVorb\no′y(τ′)/angbracketrightBig/bracerightBigg\nΓ=0=/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsoy·To\no′·lo′y·ρo′\no+loy·To\no′·so′y·ρo′\no/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsoy·[G(iω)·T]o\no′·lo′y·[G(iω)·T]o′\no\n−soy·G(iω)o\no′·lo′y·[T·G(iω)·T]o′\no−soy·[T·G(iω)·T]o\no′·lo′y·G(iω)o′\no\n+soy·[T·G(iω)]o\no′·lo′y·[T·G(iω)]o′\no/bracketrightBigg\n, (148)\n(Jy\noo′)spin−spin= Trm,σ/parenleftBig\nsox·To\no′·so′x·ρo′\no+so′x·To′\no·sox·ρo\no′/parenrightBig\n+βVspin\noxVspin\no′x−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\nox(τ)ˆVspin\no′x(τ′)/angbracketrightBig\nΓ=0= Trm,σ/parenleftBig\nsox·To\no′·so′x·ρo′\no+so′x·To′\no·sox·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsox·[G(iω)·T]o\no′·so′x·[G(iω)·T]o′\no\n−sox·G(iω)o\no′·so′x·[T·G(iω)·T]o′\no−sox·[T·G(iω)·T]o\no′·so′x·G(iω)o′\no\n+sox·[T·G(iω)]o\no′·so′x·[T·G(iω)]o′\no/bracketrightBigg\n, (149)\n40(Jy\noo′)orb−orb= Trm,σ/parenleftBig\nlox·To\no′·lo′x·ρo′\no+lo′x·To′\no·lox·ρo\no′/parenrightBig\n+βVorb\noxVorb\no′x−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVorb\nox(τ)ˆVorb\no′x(τ′)/angbracketrightBig\nΓ=0= Trm,σ/parenleftBig\nlox·To\no′·lo′x·ρo′\no+lo′x·To′\no·lox·ρo\no′/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nlox·[G(iω)·T]o\no′·lo′x·[G(iω)·T]o′\no\n−lox·G(iω)o\no′·lo′x·[T·G(iω)·T]o′\no−lox·[T·G(iω)·T]o\no′·lo′x·G(iω)o′\no\n+lox·[T·G(iω)]o\no′·lo′x·[T·G(iω)]o′\no/bracketrightBigg\n, (150)\n(Jy\noo′)spin−orb=/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·lo′x·ρo′\no+lox·To\no′·so′x·ρo′\no/parenrightBig\n+βVspin\noxVorb\no′x−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\nox(τ)ˆVorb\no′x(τ′)/angbracketrightBig/bracerightBigg\nΓ=0=/parenleftBig\n1+− →Po↔o′/parenrightBig/braceleftBigg\nTrm,σ/parenleftBig\nsox·To\no′·lo′x·ρo′\no+lox·To\no′·so′x·ρo′\no/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/bracketleftBigg\nsox·[G(iω)·T]o\no′·lo′x·[G(iω)·T]o′\no\n−sox·G(iω)o\no′·lo′x·[T·G(iω)·T]o′\no−sox·[T·G(iω)·T]o\no′·lo′x·G(iω)o′\no\n+sox·[T·G(iω)]o\no′·lo′x·[T·G(iω)]o′\no/bracketrightBigg\n, (151)\nand the terms related to Jz\noo′are just obtained as the averages of the respective\nterms related to Jx\noo′andJy\noo′, according to the relation Jz\noo′= (Jx\noo′+Jy\noo′)/2.\n10.4. Magnetic field\nTo separate the magnetic field as Bo≡Bspin\no+Borb\no, from Eq.(91) we notice\nthat\nBi=µBgiBi/angbracketleftBig\nˆSi/angbracketrightBig\n·uz\ni=µBgiBiTrM/parenleftbig\nρi\niSi/parenrightbig\n·uz\ni. (152)\nSubstituting i≡(oS), where we recall that the orbital degree of freedom o=\n(a,n,l), we notice that Bi→Bo→Ba, since the external magnetic field\ndepends only on position, and uz\ni→uz\no, since by hypothesis the separation into\nspin and orbital parts is done under the assumption that the dynam ical vectors\nei(and therefore also their initial values uz\ni) depend only on the orbital degrees\n41of freedom . We then obtain\nBo=µBBa/summationdisplay\nSgoSTrM/parenleftbig\nρoS\noSSoS/parenrightbig\n·uz\no≡µBBaTrM,S(goρo\noSo)·uz\no\n=µBBaTrm,σ(goρo\noSo)·uz\no≡Bspin\no+Borb\no, (153)\nwhere\nBspin\no≡g1/2µBBa[Trσ(soTrmρo\no)·uz\no],\nBorb\no≡glµBBa[Trm(loTrσρo\no)·uz\no], (154)\nwhereg1/2andglare the intrinsic-spin and orbital g-factors, respectively.\n10.5. Local exchange interactions (diagonal anisotropy)\nFrom Eqs.(96) we obtain\nJx\noo−Jz\noo=Bz\no+/tildewiderMoy\noy+1\n2/summationdisplay\no′/ne}ationslash=o(Jx\noo′+Jy\noo′),\nJy\noo−Jz\noo=Bz\no+/tildewiderMox\nox+1\n2/summationdisplay\no′/ne}ationslash=o(Jx\noo′+Jy\noo′), (155)\nwhere the parameters Bz\no,Jx\noo′andJy\noo′foro∝ne}ationslash=o′were determined in the\nprevious paragraphs. The additional terms, for α=xory, are written as\n/tildewiderMoα\noα=/parenleftBig\n/tildewiderMoα\noα/parenrightBigspin−spin\n+/parenleftBig\n/tildewiderMoα\noα/parenrightBigorb−orb\n+/parenleftBig\n/tildewiderMoα\noα/parenrightBigspin−orb\n,\nwhere\n/parenleftBig\n/tildewiderMoα\noα/parenrightBigspin−spin\n≡2Trm,σ(soα·To\no·soα·ρo\no)−1\n4Trm,σ{ρ;T}o\no\n+β/parenleftbig\nVspin\noα/parenrightbig2−1\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\noα(τ)ˆVspin\noα(τ′)/angbracketrightBig\nΓ=0= 2Trm,σ(soα·To\no·soα·ρo\no)−1\n4Trm,σ{ρ;T}o\no\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/braceleftBig\nsoα·[G(iω)·T]o\no·soα·[G(iω)·T]o\no\n−2soα·G(iω)o\no·soα·[T·G(iω)·T]o\no+soα·[T·G(iω)]o\no·soα·[T·G(iω)]o\no/bracerightBig\n,\n(156)\n42/parenleftBig\n/tildewiderMoα\noα/parenrightBigorb−orb\n≡2Trm,σ(loα·To\no·loα·ρo\no)−Trm/parenleftBig\nloα·loα·Trσ{ρ;T}o\no/parenrightBig\n+β/parenleftbig\nVorb\noα/parenrightbig2−1\nβ/integraldisplay��\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVorb\noα(τ)ˆVorb\noα(τ′)/angbracketrightBig\nΓ=0= 2Trm,σ(loα·To\no·loα·ρo\no)−Trm/parenleftBig\nloα·loα·Trσ{ρ;T}o\no/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/braceleftBig\nloα·[G(iω)·T]o\no·loα·[G(iω)·T]o\no\n−2loα·G(iω)o\no·loα·[T·G(iω)·T]o\no+loα·[T·G(iω)]o\no·loα·[T·G(iω)]o\no/bracerightBig\n,\n(157)\n/parenleftBig\n/tildewiderMoα\noα/parenrightBigspin−orb\n≡2Trm,σ(soα·To\no·loα·ρo\no+loα·To\no·soα·ρo\no)\n−Trm,σ/parenleftBig\n{soα;loα}·{ρ;T}o\no/parenrightBig\n+2βVspin\noαVorb\noα−2\nβ/integraldisplayβ\n0dτ/integraldisplayβ\n0dτ′/angbracketleftBig\nTγˆVspin\noα(τ)ˆVorb\noα(τ′)/angbracketrightBig\nΓ=0= 2Trm,σ(soα·To\no·loα·ρo\no+loα·To\no·soα·ρo\no)−Trm,σ/parenleftBig\n{soα;loα}·{ρ;T}o\no/parenrightBig\n+1\nβ/summationdisplay\nωeiω0+Trm,σ/braceleftBig\nsoα·[G(iω)·T]o\no·loα·[G(iω)·T]o\no\n−soα·G(iω)o\no·loα·[T·G(iω)·T]o\no−soα·[T·G(iω)·T]o\no·loα·G(iω)o\no\n+soα·[T·G(iω)]o\no·loα·[T·G(iω)]o\no/bracerightBig\n. (158)\nThe local exchange interaction parameters can then be written as\nJα\noo−Jz\noo≡(Jα\noo−Jz\noo)spin−spin+(Jα\noo−Jz\noo)orb−orb+(Jα\noo−Jz\noo)spin−orb,\nwhere\n(Jα\noo−Jz\noo)spin−spin= (Bz\no)spin+/parenleftBig\n/tildewiderMo¯α\no¯α/parenrightBigspin−spin\n+1\n2/summationdisplay\no′/ne}ationslash=o(Jx\noo′+Jy\noo′)spin−spin,\n(159)\n(Jα\noo−Jz\noo)orb−orb= (Bz\no)orb+/parenleftBig\n/tildewiderMo¯α\no¯α/parenrightBigorb−orb\n+1\n2/summationdisplay\no′/ne}ationslash=o(Jx\noo′+Jy\noo′)orb−orb,\n(160)\n(Jα\noo−Jz\noo)spin−orb=/parenleftBig\n/tildewiderMo¯α\no¯α/parenrightBigspin−orb\n+1\n2/summationdisplay\no′/ne}ationslash=o(Jx\noo′+Jy\noo′)spin−orb,(161)\nwhere (α,¯α) = (x,y) or (y,x), and the various terms are given by Eqs.(154),\n(156), (157), (158), and in Section 10.3.\n4311. Conclusion\nTo conclude, in this work we have established the mapping between a r ela-\ntivistic electronic system with rotationally invariant interactions ont o an effec-\ntiveclassical spin model, via the equivalence of their thermodynamic potentials\nunder rotations of the local total magnetic moments up to the sec ond order in\nthe rotation angles, when the spin configuration of the electrons is symmetry-\nbroken and out of equilibrium. The parameters of the effective spin m odel were\nobtained as functionals of the single- and two-electron Green’s fun ctions of the\nelectronic system. We have removed two approximations which were adopted in\nprevious works on non-relativistic systems [10, 11, 16], namely: (1) here we take\ninto account the vertices of the two-electron Green’s functions, (2) here we in-\nclude the non-local components of the self-energies. Besides, we have extended\nthe theory in order to completely account for relativistic effects, d etermining\nthe complete relativistic exchange tensors in a unified framework. F or two com-\nponents (xandy) of the Dzyaloshinskii-Moriyavectors, which had already been\ndetermined previously [12], we have recovered the known results an d extended\nthem tothe non-collinearcase; moreover,herewehavedetermine d alsothe third\ncomponent ( z), together with the completely new terms describing anisotropic\nexchangeand otherout-of-diagonalsymmetricterms ofthe exc hangetensors. In\nthe particular case of spin-1 /2 (single-band Hubbard model) we have recovered\nthe known expressions for the isotropicexchange parametersbo th in the general\ncase of non-local self-energy [16] and in the particular case of loca l self-energy\n[10, 11]. Having included also an external magnetic field, we have show n how it\ndetermines a renormalization of the exchange tensor via linear and n on-linear\ncontributions (the details can be found in Appendix B). Finally, we hav e shown\nhow to study separately the orbital and spin-1 /2 contributions to magnetism,\nas well as a combined “spin-orbital” contribution which cannot be dec oupled.\nWe remark that our theory should be used to predict spin dynamics in\na given phase of the electronic system, but it cannot be used to pre dict the\nphases of the system themselves. In fact, the application of the t heory for\ncomputations requires fixing the initial spin configuration in a definite out-of-\nequilibrium phase. The subsequent classical spin dynamics is then det ermined\nby the magnetic interactions given by our theory.\nBuilding on this work, we foresee three main possible paths for furth er the-\noretical investigation: (1) study of the response of the thermod ynamic poten-\ntial to higher orders in the rotation angles, (2) extension of the eff ective spin\nmodel to include higher-order spin-spin interactions (the quadrat ic model con-\nsidered here is enough for the second-order response in the angle s of rotation,\nbut may not be enough for higher orders in the angles), (3) inclusion of time-\ndependent external electromagnetic fields, which up to now was do ne only for\na non-relativistic system [16]. This latter extension would be desirable in order\nto realistically describe the manipulation of magnetism and the ultrafa st spin\ndynamics induced by sub-picosecond laser pulses.\n44Acknowledgements\nWe acknowledge useful scientific discussions with Vladimir V. Mazuren ko,\nJohan Mentink, Martin Eckstein and Alexander Chudnovskiy. This wo rk is\nsupported by the European Union Seventh Framework Programme under grant\nagreement No. 281043 (FEMTOSPIN), and by Deutsche Forschun gsgemein-\nschaft under grant SFB-668.\nAppendix A. Considerations on the rotational invariance of the in-\nteraction Hamiltonian\nThe whole treatment has been based on the assumption that the int erac-\ntion term is rotationally invariant. We now discuss this issue more in det ail.\nWe start by assuming that the interaction is local (on-site), in the s pirit of the\nmulti-orbitalHubbardmodel. However,on asinglesite it can mixstate s belong-\ning to different shells and having different angular momenta. The inter action\nHamiltonian is\nˆHφ\nV≡1\n2/summationdisplay\n1,2,3,4ˆφ†\ni1,M1ˆφ†\ni2,M2V{i1,M1},{i2,M2}\n{i3,M3},{i4,M4}ˆφi3,M3ˆφi4,M4,(A.1)\nwhere\nV{i1,M1},{i2,M2}\n{i3,M3},{i4,M4}≡/integraldisplay\ndv(x)/integraldisplay\ndv(x′)φ∗\ni1,M1(x)φ∗\ni2,M2(x′)\n×V(x−x′)φi3,M3(x′)φi4,M4(x). (A.2)\nThe on-site interaction is supposed to be rotationally invariant, i.e., ˆHφ\nV=ˆHψ\nV.\nTo check the conditions under which this condition is fulfilled, we perfo rm the\nrotation of the fermionic fields according to Eq.(16), obtaining\nˆHφ\nV=1\n2/summationdisplay\n1,2,3,4ˆψ†\ni1,M1ˆψ†\ni2,M2ˆψi3,M3ˆψi4,M4\n·/summationdisplay\nM5,M6,M7,M8R†(i1)M1\nM5R†(i2)M2\nM6V{i1,M5},{i2,M6}\n{i3,M7},{i4,M8}R(i3)M7\nM3R(i4)M8\nM4.\n(A.3)\nSuppose that the interaction is intra-atomic ( a1=a2=a3=a4≡a), as in the\nHubbard model, and that\nV{n1,l1,S1,M5},{n2,l2,S2,M6}\na,{n3,l3,S3,M7},{n4,l4,S4,M8}=δM6\nM7δM5\nM8δS2\nS3δS1\nS4V{n1,l1,S1},{n2,l2,S2}\na,{n3,l3,S3},{n4,l4,S4},(A.4)\n45then we obtain:\nˆHφ\nV=1\n2/summationdisplay\na/summationdisplay\n{n}/summationdisplay\n{l}/summationdisplay\n{S}/summationdisplay\n{M}ˆψ†\na,n1,l1,S1,M1ˆψ†\na,n2,l2,S2,M2δS2\nS3δS1\nS4\n·V{n1,l1,S1},{n2,l2,S2}\na,{n3,l3,S3},{n4,l4,S4}/bracketleftBigg/summationdisplay\nM5R†(a,n1,l1,S1)M1\nM5R(a,n4,l4,S1)M5\nM4\n·/summationdisplay\nM6R†(a,n2,l2,S2)M2\nM6R(a,n3,l3,S2)M6\nM3/bracketrightBigg\nˆψa,n3,l3,S3,M3ˆψa,n4,l4,S4,M4,\n(A.5)\nwhere/summationtext\n{n}≡/summationtext\nn1,n2,n3,n4, etcetera. To get rotational invariance of the inter-\nactionHamiltonianwehavenowtwopossibilities: either1)wetakether otations\nto be only site-dependent (i.e., not resolvedwith respect to the she lls and orbital\nangular momenta), so that the rotation matrices are independent ofnandl, or\n2) we further assume that the interaction parameter is\n∝δn1\nn4δn2\nn3δl1\nl4δl2\nl3, (A.6)\nwhich implies that the local Coulomb interaction is spherically symmetric . In\nboth cases, we can perform the summations over M5andM6, obtaining:\nˆHφ\nV=1\n2/summationdisplay\na/summationdisplay\n{n}/summationdisplay\n{l}/summationdisplay\n{S}/summationdisplay\n{M}ˆψ†\na,n1,l1,S1,M1ˆψ†\na,n2,l2,S2,M2\n·δM1\nM4δM2\nM3δS2\nS3δS1\nS4V{n1,l1,S1},{n2,l2,S2}\na,{n3,l3,S3},{n4,l4,S4}ˆψa,n3,l3,S3,M3ˆψa,n4,l4,S4,M4\n=1\n2/summationdisplay\na/summationdisplay\nn1,l1,S1,M1/summationdisplay\nn2,l2,S2,M2/summationdisplay\nn3,l3/summationdisplay\nn4,l4ˆψ†\na,n1,l1,S1,M1ˆψ†\na,n2,l2,S2,M2\n·V{n1,l1,S1},{n2,l2,S2}\na,{n3,l3,S2},{n4,l4,S1}ˆψa,n3,l3,S2,M2ˆψa,n4,l4,S1,M1, (A.7)\nwhich is invariant. Therefore, the assumption (A.4) guarantees th e invariance\nof the interaction Hamiltonian under rotations of the magnetic mome ntssite-\nresolved butnot shell-resolved , while the additional assumption (A.6) allows for\nrotational invariance under shell-resolved rotations.\nAppendix B. Analysis of the matrix /tildewiderMiα\ni′α′\nIn our theory, the matrix /tildewiderMiα\ni′α′[cfr. Eq.(77)] is one of the key quantities in\nterms of which the magnetic parameters are expressed. We now ta ke a closer\nlook at the structure of this matrix, starting with the term Wiα\ni′α′[cfr. Eq.(79)].\nWe see that we can decompose Eq.(79) in order to separate the par ts of\nWiα\ni′α′involving the local single-particle Hamiltonian Ti\nifrom those involving\nthe non-local part. We put\nWiα\ni′α′≡/parenleftbig\nWiα\ni′α′/parenrightbigloc+/parenleftbig\nWiα\ni′α′/parenrightbigloc/nloc+/parenleftbig\nWiα\ni′α′/parenrightbignloc, (B.1)\n46where the three parts originate from terms in the summation over ( j,j′) in\nEq.(79) having different specific values of jandj′in relation with iandi′.\nNamely, the local parts are the terms with j=iandj′=i′,\n/parenleftbig\nWiα\ni′α′/parenrightbigloc≡/summationdisplay\nM1M2M3M4/bracketleftbig\nSiα,Ti\ni/bracketrightbigM2\nM1/tildewideχ(iM1)(i′M3)\n(iM2)(i′M4)/bracketleftBig\nSi′��′,Ti′\ni′/bracketrightBigM4\nM3; (B.2)\nthen, there are terms involving both local and non-local componen ts of the\nsingle-particle Hamiltonian, corresponding to ( j=i,j′∝ne}ationslash=i′) or (j∝ne}ationslash=i,j′=i′),\n/parenleftbig\nWiα\ni′α′/parenrightbigloc/nloc≡/summationdisplay\nM1M2M3M4/braceleftBigg\n/summationdisplay\nj/ne}ationslash=i/bracketleftbigg/parenleftbig\nSiα·Ti\nj/parenrightbigM2\nM1/tildewideχ(jM1)(i′M3)\n(iM2)(i′M4)−/parenleftBig\nTj\ni·Siα/parenrightBigM2\nM1/tildewideχ(iM1)(i′M3)\n(jM2)(i′M4)/bracketrightbigg/bracketleftBig\nSi′α′,Ti′\ni′/bracketrightBigM4\nM3\n+/summationdisplay\nj′/ne}ationslash=i′/bracketleftbig\nSiα,Ti\ni/bracketrightbigM2\nM1/bracketleftbigg\n/tildewideχ(iM1)(j′M3)\n(iM2)(i′M4)/parenleftBig\nSi′α′·Ti′\nj′/parenrightBigM4\nM3−/tildewideχ(iM1)(i′M3)\n(iM2)(j′M4)/parenleftBig\nTj′\ni′·Si′α′/parenrightBigM4\nM3/bracketrightbigg/bracerightBigg\n;\n(B.3)\nfinally, the completely non-local term corresponds to ( j∝ne}ationslash=i,j′∝ne}ationslash=i′),\n/parenleftbig\nWiα\ni′α′/parenrightbignloc≡/summationdisplay\nj/ne}ationslash=i/summationdisplay\nj′/ne}ationslash=i′/summationdisplay\nM1M2M3M4/bracketleftBigg\n/parenleftbig\nSiα·Ti\nj/parenrightbigM2\nM1/parenleftBig\nSi′α′·Ti′\nj′/parenrightBigM4\nM3/tildewideχ(jM1)(j′M3)\n(iM2)(i′M4)\n−/parenleftbig\nSiα·Ti\nj/parenrightbigM2\nM1/parenleftBig\nTj′\ni′·Si′α′/parenrightBigM4\nM3/tildewideχ(jM1)(i′M3)\n(iM2)(j′M4)\n−/parenleftBig\nTj\ni·Siα/parenrightBigM2\nM1/parenleftBig\nSi′α′·Ti′\nj′/parenrightBigM4\nM3/tildewideχ(iM1)(j′M3)\n(jM2)(i′M4)\n+/parenleftBig\nTj\ni·Siα/parenrightBigM2\nM1/parenleftBig\nTj′\ni′·Si′α′/parenrightBigM4\nM3/tildewideχ(iM1)(i′M3)\n(jM2)(j′M4)/bracketrightBigg\n.\n(B.4)\nThe terms/parenleftbig\nWiα\ni′α′/parenrightbiglocand/parenleftbig\nWiα\ni′α′/parenrightbigloc/nlocare completely relativistic terms, since\ntheyvanishwhen/bracketleftbig\nSiα,Ti\ni/bracketrightbig\n= 0, i.e., whenthereisnoexternalmagneticfieldand\nno local anisotropy. The term/parenleftbig\nWiα\ni′α′/parenrightbignloc, on the other hand, survives also in\nthe non-relativistic regime. It should be noted that in our theory, in the general\nrelativistic case, all the terms of the exchange tensor depend on t he magnetic\nfieldandthelocalanisotropynotonlyviatheintrinsicdependenceof theGreen’s\nfunctions, but also via terms which are explicitly linear and even quadr atic in\nsuch parameters, as it is evident by looking at the terms of Eqs.(B.2) , (B.3),\n(B.4), as well as by considering the terms arising from Miα\ni′α′+βViαVi′α′in\nEq.(77).\nTo get some insights into the structure of the parameters, it is inst ructive to\nconsider the case ( iα) = (i′α′), which is relevant for Eqs.(69). Using Eq.(82),\n47we can write\n/parenleftbig\nWiα\niα/parenrightbigloc≡/summationdisplay\nM1M2M3M4/bracketleftbig\nSiα,Ai\ni/bracketrightbigM2\nM1/tildewideχ(iM1)(iM3)\n(iM2)(iM4)/bracketleftbig\nSiα,Ai\ni/bracketrightbigM4\nM3\n+2iµBgi/summationdisplay\nM1M2M3M4[(Bi×Si)·uα\ni]M2\nM1/tildewideχ(iM1)(iM3)\n(iM2)(iM4)/bracketleftbig\nSiα,Ai\ni/bracketrightbigM4\nM3\n−µ2\nBg2\ni/summationdisplay\nM1M2M3M4[(Bi×Si)·uα\ni]M2\nM1/tildewideχ(iM1)(iM3)\n(iM2)(iM4)[(Bi×Si)·uα\ni]M4\nM3,\n(B.5)\n/parenleftbig\nWiα\niα/parenrightbigloc/nloc≡2/summationdisplay\nM1M2M3M4/parenleftBig/bracketleftbig\nSiα,Ai\ni/bracketrightbigM2\nM1+iµBgi[(Bi×Si)·uα\ni]M2\nM1/parenrightBig\n·/summationdisplay\nj/ne}ationslash=i/bracketleftBigg\n/tildewideχ(iM1)(jM3)\n(iM2)(iM4)/parenleftbig\nSiα·Ti\nj/parenrightbigM4\nM3−/tildewideχ(iM1)(iM3)\n(iM2)(jM4)/parenleftBig\nTj\ni·Siα/parenrightBigM4\nM3/bracketrightBigg\n.\n(B.6)\nWe then obtain, for the quantities relevant to Eqs.(69),\n/tildewiderMiα\niα=/tildewiderMiα\niα/vextendsingle/vextendsingle/vextendsingle\nBi=0−µBgi/parenleftBig\nBi·/angbracketleftBig\nˆSi/angbracketrightBig\n−Bα\ni/angbracketleftBig\nˆSiα/angbracketrightBig/parenrightBig\n−2iµBgi/summationdisplay\nM1M2M3M4[(Bi×Si)·uα\ni]M2\nM1/tildewideχ(iM1)(iM3)\n(iM2)(iM4)/bracketleftbig\nSiα,Ai\ni/bracketrightbigM4\nM3\n+µ2\nBg2\ni/summationdisplay\nM1M2M3M4[(Bi×Si)·uα\ni]M2\nM1/tildewideχ(iM1)(iM3)\n(iM2)(iM4)[(Bi×Si)·uα\ni]M4\nM3\n−2iµBgi/summationdisplay\nM1M2M3M4[(Bi×Si)·uα\ni]M2\nM1\n·/summationdisplay\nj/ne}ationslash=i/bracketleftBigg\n/tildewideχ(iM1)(jM3)\n(iM2)(iM4)/parenleftbig\nSiα·Ti\nj/parenrightbigM4\nM3−/tildewideχ(iM1)(iM3)\n(iM2)(jM4)/parenleftBig\nTj\ni·Siα/parenrightBigM4\nM3/bracketrightBigg\n+βµ2\nBg2\ni/bracketleftBig/parenleftBig\nBi×/angbracketleftBig\nˆSi/angbracketrightBig/parenrightBig\n·uα\ni/bracketrightBig2\n+2βµBgiViα|Bi=0/parenleftBig\nBi×/angbracketleftBig\nˆSi/angbracketrightBig/parenrightBig\n·uα\ni,\n(B.7)\nwhere/tildewiderMiα\niα/vextendsingle/vextendsingle/vextendsingle\nBi=0, we recall, is not independent on Bi, but it is a term which\ndoes not vanish when Bi=0. A simplification of the parts which depend\nexplicitly on the local relativistic terms can be achieved after decomp osing the\ntwo-particle Green’s functions as in Eq.(43),\nχ1,3\n2,4(τ,τ+,τ′,τ′+) =/parenleftbig\nχ0/parenrightbig1,3\n2,4(τ,τ+,τ′,τ′+)+/parenleftbig\nχΓ/parenrightbig1,3\n2,4(τ,τ+,τ′,τ′+),(B.8)\nwhere/parenleftbig\nχ0/parenrightbig1,3\n2,4(τ,τ+,τ′,τ′+) =−ρ1\n2ρ3\n4−G1\n4(τ−τ′−ε)G3\n2(τ′−τ−ε) andχΓ\ncontains the vertex corrections. Using the Matsubara-frequen cy representation\n48and Eq.(105), we can then reduce Eq.(B.7) to\n/tildewiderMiα\niα=/tildewiderMiα\niα/vextendsingle/vextendsingle/vextendsingle\nBi=0−µBgi/parenleftBig\nBi·/angbracketleftBig\nˆSi/angbracketrightBig\n−Bα\ni/angbracketleftBig\nˆSiα/angbracketrightBig/parenrightBig\n−µ2\nBg2\ni\nβTrM/braceleftBigg\n[(Bi×Si)·uα\ni]·/summationdisplay\nωeiω0+Gi\ni(iω)·[(Bi×Si)·uα\ni]·Gi\ni(iω)/bracerightBigg\n+µ2\nBg2\ni/summationdisplay\nM1M2M3M4[(Bi×Si)·uα\ni]M2\nM1/parenleftbig\n/tildewideχΓ/parenrightbig(iM1)(iM3)\n(iM2)(iM4)[(Bi×Si)·uα\ni]M4\nM3\n−2iµBgi/summationdisplay\nM1M2M3M4/summationdisplay\nj/bracketleftBigg\n/parenleftbig\n/tildewideχΓ/parenrightbig(iM1)(jM3)\n(iM2)(iM4)/parenleftBig\nSiα·/parenleftbig\nTi\nj/parenrightbig\nBi=0/parenrightBigM4\nM3\n−/parenleftbig\n/tildewideχΓ/parenrightbig(iM1)(iM3)\n(iM2)(jM4)/parenleftbigg/parenleftBig\nTj\ni/parenrightBig\nBi=0·Siα/parenrightbiggM4\nM3/bracketrightBigg\n[(Bi×Si)·uα\ni]M2\nM1\n+2iµBgi\nβTrM/braceleftBigg\n[(Bi×Si)·uα\ni]·/summationdisplay\nωeiω0+/bracketleftBig\nGi\ni(iω)·Siα·[TBi=0·G(iω)]i\ni\n−[G(iω)·TBi=0]i\ni·Siα·Gi\ni(iω)/bracketrightBig/bracerightBigg\n. (B.9)\nIn addition to the first term, which survives in the non-relativistic re gime (and\nalso includes anisotropy contributions), and to the second term in t he first line,\nwhich as discussed should be identified with a component of the effect ive mag-\nnetic field, Eq.(B.9) explicitly shows how the relativistic exchange para meters\nhave a non-trivial dependence on the magnetic field Bi.\nReferences\nReferences\n[1] A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and V. A . Gubanov,\nJ. Magn. Magn. Mater. 67 (1987) 65.\n[2] A. Auerbach, Interacting Electrons and Quantum Magnetism , Springer-\nVerlag New York, Inc. (1994).\n[3] M. I. Auslender and M. I. Katsnelson, Theor. Math. Phys. 51 (1 982) 601;\nSolid State Commun. 44 (1982) 387.\n[4] J. Kanamori, Prog. Theor. Phys. 30 (1963) 275.\n[5] J. Hubbard, Proc. Roy. Soc. A 285 (1965) 542.\n[6] K. I. Kugel and D. I. Khomskii, Sov. Phys. Uspekhi 25 (1982) 23 1.\n[7] A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 57 (1998) 6884.\n49[8] A. Georges, L. de’ Medici, and J. Mravlje, Annu. Rev. Cond. Mat . Phys. 4\n(2013) 137.\n[9] R. M. White, Quantum Theory of Magnetism , 3rd edition, Springer-Verlag\nBerlin Heidelberg (2007).\n[10] M. I. Katsnelson and A. I. Lichtenstein, Phys. Rev. B 61 (2000 ) 8906.\n[11] M. I. Katsnelson and A. I. Lichtenstein, Eur. Phys. J. B: Cond ens. Matter\n30 (2002) 9.\n[12] M. I. Katsnelson, Y. O. Kvashnin, V. V. Mazurenko, and A. I. L ichtenstein,\nPhys. Rev. B 82 (2010) 100403(R) .\n[13] M. I. Katsnelson and A. I. Lichtenstein, J. Phys.: Condens. Ma tter 16\n(2004) 7439-7446.\n[14] G. M. Stocks, B. Ujfalussy, X. Wang, D. M. C. Nicholson, W. A. S helton,\nY. Wang, A. Canning, and B. L. Gy¨ orffy, Philos. Mag. Part B 78 (199 8)\n665-673.\n[15] P. Bruno, Phys. Rev. Lett. 90 (2003) 087205.\n[16] A. Secchi, S. Brener, A. I. Lichtenstein, and M. I. Katsnelson , Ann. Phys.\n333 (2013) 221-271.\n[17] D. S. Rodbell, I. S. Jacobs, J. Owen, and E. A. Harris, Phys. Re v. Lett. 11\n(1963) 10-12.\n[18] Y. Honda, Y. Kuramoto, and T. Watanabe, Phys. Rev. B 47 (19 93) 11329.\n[19] A. L. Wysocki, K. D. Belashchenko, and V. P. Antropov, Natur e Phys. 7\n(2011) 485.\n[20] S. Brehmer, H.-J. Mikeska, M. M¨ uller, N. Nagaosa, and S. Uchid a, Phys.\nRev. B 60 (1999) 329-334.\n[21] L. Udvardi, L. Szunyogh, K. Palot´ as, and P. Weinberger, Phy s. Rev. B 68\n(2003) 104436.\n[22] A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nordstr¨ om , and O.\nEriksson, Phys. Rev. Lett. 111 (2013) 127204.\n[23] A.AltlandandB.Simons, Condensed Matter Field Theory , SecondEdition,\nCambridge University Press, New York (2010).\n[24] A. I. Lichtenstein and M. I. Katsnelson, in Band-Ferromagnetism. Ground\nState and Finite- Temperature Phenomena , edited by K. Baberschke, M.\nDonath, W. Nolting (Springer, Berlin, 2001), p. 75.\n[25] A. I. Akhiezer, V. G. Bar’yakhtar, and S. V. Peletminskii, Spin Waves ,\nNorth-Holland Publishing Company - Amsterdam (1968).\n50[26] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62 (1989) 324- 327.\n[27] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys.\n68 (1996) 13\n[28] G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parc ollet, and\nC. A. Marianetti, Rev. Mod. Phys. 78 (2006) 865.\n51" }, { "title": "1208.5606v1.Spin_orbit_coupled_particle_in_a_spin_bath.pdf", "content": "arXiv:1208.5606v1 [cond-mat.mes-hall] 28 Aug 2012Spin-orbit coupled particle in a spin bath\nPeter Stano1,2, Jaroslav Fabian3, IgorˇZuti´ c4\n1Institute of Physics, Slovak Academy of Sciences, 845 11 Bra tislava, Slovakia\n2Department of Physics, Klingerbergstrasse 82, University of Basel, Switzerland\n3Institute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany\n4Department of Physics, University at Buffalo, NY 14260-1500\nWe consider a spin-orbit coupled particle confined in a quant um dot in a bath of impurity spins.\nWe investigate the consequences of spin-orbit coupling on t he interactions that the particle mediates\nin the spin bath. We show that in the presence of spin-orbit co upling, the impurity-impurity\ninteractions are no longer spin-conserving. We quantify th e degree of this symmetry breaking and\nshow how it relates to the spin-orbit coupling strength. We i dentify several ways how the impurity\nensemble can in this way relax its spin by coupling to phonons . A typical resulting relaxation rate\nfor a self-assembled Mn-doped ZnTe quantum dot populated by a hole is 1 µs. We also show that\ndecoherence arising from nuclear spins in lateral quantum d ots is still removable by a spin echo\nprotocol, even if the confined electron is spin-orbit couple d.\nPACS numbers: 75.30.Et, 76.60.-k, 71.38.-k, 73.21.La, 33. 35.+r\nI. INTRODUCTION\nAsinglyoccupiedquantum dotisthe stateofthe artof\nacontrollablequantumsysteminasemiconductor.1,2Co-\nherent manipulation of the particle spin has been demon-\nstrated in lateral dots, where top gates allow for as-\ntonishing degree of control by electric fields3–6and in\nself-assembled dots, where a weaker control over the dot\nshape and position is compensated by the speed of the\noptical manipulation.7In both of these major groups,\nthere are two main spin-dependent interactions of the\nconfined particle and the semiconductor environment:\nspin-orbit coupling embedded in the band structure, and\nspin impurities, which are either nuclear spins, or mag-\nnetic atoms.8,9\nA particle couples to an impurity spin dominantly\nthrough an exchange interaction, which conserves the\ntotal spin of the pair.10This way the electron spin\nin a lateral GaAs dot will decohere within 10 ns due\nto the presence of nuclei.11–17Typically, such decoher-\nence is considered a nuisance that can be partially re-\nmoved by spin echo techniques prolonging the coherence\nto hundreds microseconds.18–20Whether that decoher-\nence time can be extended further, e.g. by polarizing\nthe impurities,21,22is not clear, as the experimentally\nachieved degree of polarization has been so far insuffi-\ncient, despite great efforts.23–25On the other hand, a\nstrong particle-impurity interaction is desired in magnet-\nically doped quantum dots.26–40Here the confined par-\nticle is central for both supporting energetically, and as-\nsisting in creation, the desired magnetic order of the im-\npurity ensemble. In fact, similar magnetic ordering can\nbe traced to the studies of magnetic polarons in bulk\nsemiconductors, for over fifty years.41The formation of\na magnetic polaron can be viewed as a cloud of local-\nizedimpurityspins, alignedthroughexchangeinteraction\nwith a confined carrier spin.42–45\nThe conservation of the spin by the impurity-particle\ninteraction is a crucial property. For example, the spinrelaxation of the impurity ensemble is impossible with\nonly spin-conserving interactions at hand. This moti-\nvates us to consider possibilities to break this symmetry.\nThe first and obvious candidate is the spin-orbit cou-\npling (SOC).8,9Despite being weak on the scale of the\nparticle orbital energies, it dominates the relaxation of\nthe particle spin in electronic dots, as is well known,46\nbecause it is the dominant spin-non-conserving interac-\ntion. The questions we pose and answer in this work are:\nassuming the particle is weakly spin-orbit coupled, how\nstrong are the effective spin-non-conserving interactions\nwhich appear in the impurity ensemble and what is their\nform? Is the induced particle decoherence still removable\nby spin echo? Is the particle efficient in inducing impu-\nrityensemblespinrelaxation,therebylimitingtheachiev-\nable degree of the dynamical nuclear spin polarization47?\nCan the magnetic orderbe created through the spin-non-\nconserving particle mediated interactions — that is, is\nthis a relevant magnetic polaron creation channel?\nTo address these questions we develop here a frame-\nwork allowing us to treat different particles and impu-\nrity spins in a unified manner. We apply our method\nto two specific systems: a lateral quantum dot in GaAs\noccupied by a conduction electron with nuclear spins\nof constituent atoms as the spin impurities, and a self-\nassembled ZnTe quantum dot occupied by a heavy hole\ndoped with Mn atoms as the spin impurities (readily in-\ncorporated as Mn is isovalent with group-II atom Zn).\nBothofthesesystemsarequasi-twodimensional, thepar-\nticle spin-orbit coupling is weak compared to the particle\norbital level spacing and the particle-impurity interac-\ntion is weak compared to the particle orbital and spin\nlevel spacings.2,32,48,49As it is known,50in this regime\none can derive an effective Hamiltonian for the impu-\nrity ensemble only, in which the particle does not ap-\npear explicitly. This can be done including the particle-\nbath interaction perturbatively in the lowest order, see\nthe scheme in Fig. 1. Our contribution is in show-\ning how the procedure generalizes to a spin-orbit cou-2\npled particle. In addition, we use the resulting Hamil-\ntonian for the calculation of the spin relaxation of the\nimpurities which is phonon-assisted (required to dissi-\npate energy) and particle-mediated (required to dissipate\nspin). We come up with (and evaluate the correspond-\ning rates for) five possible mechanisms how the spin flips\ncan proceed: shifts of the particle by the phonon electric\nfield (Sec. IV.B), position shifts of the impurity atoms\n(Sec. IV.B), relative shifts of bulk bands (Sec. IV.C),\nrenormalizationofthespin-orbitinteractionsduetoband\nshifts (Sec. IV.D) andspin-orbit interactionsarisingfrom\nthe phonon electric field (Sec. IV.E).\nOur main findings are the following: 1. The spin-non-\nconserving interactions couplings are proportional to the\nspin-conserving ones multiplied by some power of small\nparameter(s) which quantify the spin-orbit interaction.\nFor the electron, the small parameter is the dot dimen-\nsion divided by the spin orbit length and the proportion-\nality is linear. For the hole, the small parameters are the\namplitudes of the light hole admixtures into the heavy\nhole states. The proportionality differs (from linear to\nquadratic) depending on which hole excited state medi-\nates the interaction. The interaction form is given in\nEq. (43), our main result. 2. For the electron, the addi-\ntional decoherence is removed by the spin echo, while for\nholes only a partial removal is possible. The latter is be-\ncause, unlike for electrons, the spin-non-conserving cou-\nplingismediatedratherefficientlythroughhigherexcited\nstates. 3. The piezoelectric acoustic phonons are most ef-\nficient in relaxing the impurity spin. The resulting relax-\nation time is unobservably long for nuclear spins, while\nthehole-inducedMnspinrelaxationtimeof1 µsistypical\nfor a 10 nm self-assembled quantum dot, where experi-\nmentally measured times for the polaron creation range\nfrom nano to picoseconds.26,31,32From this we conclude\nthat the interplay of spin-orbit coupling and phonons\ndoesnotgovernthedynamicsofmagneticpolaronforma-\ntion at moderate Mn densities (few percents), but rather\nrepresents the spin-lattice relaxation timescale, similarly\nas is the case in quantum wells.41,51Apart from a very\nlow Mn doping, the analytical formulas presented in this\nwork allow us to identify additional regimes where the\nparticle mediated spin relaxation could be relevant for\nthe polaron creation: an example is a hole located at a\ncharged impurity.\nThe article is organized as follows: In Sec. II we in-\ntroduce the description of the particle focusing on the\nspin-orbit coupling. In Sec. III we specify the particle-\nimpurity interaction, define its important characteristics\nand derive the effective Hamiltonian for the impurity\nensemble. In Sec. IV we calculate the spin relaxation\nrates for the impurity ensemble, after which we conclude.\nWe put numerous technical details into Appendices, with\nwhich the text is self-contained.+1/2−1/2+1/2\n+3/2−3/2\nhh−1/2\nlhb) a)\n∆lh∆\n∆z\nzE\nFIG. 1: (Color online) Effective interaction between impuri -\nties (encircled in red/gray) mediated by a confined particle\n(red/gray lines). a) Electron is excited from the ground sta te\nof spin +1/2 into the closest spin -1/2 state (up by the Zeeman\nenergy ∆ z) upon flipping one of the impurities and deexcited\nback upon flipping another one. b) Hole spectrum is more\ncomplicated, comprising heavy hole (hh) and light hole (lh) -\nlike states, the latter displaced by light-heavy hole split ting\n∆lh.\nII. QUANTUM DOT STATES AND SPIN-ORBIT\nINTERACTION\nA. Electron states\nIn the single band effective mass approximation, that\nwe adopt, the Hamiltonian of a quantum dot electron is\nHdot=p2\n2m+V(r)+p2\nz\n2m+Vz(z)+gµBB·J+Hso.(1)\nThe underlying bandstructure is taken into account as a\nrenormalization of the mass mand the gfactor in the\nelectron kinetic and Zeeman energies, respectively. The\nlatter couples the external magnetic field Bto the elec-\ntron spin through the vector of Pauli matrices σ= 2J.\nWe will below assume a sizeable (above 100 mT) ex-\nternal magnetic field in the electronic case, which is\ntypical in experiments to allow for the electron spin\nmeasurement46,52and to slowdown the impurity dynam-\nics and the resulting decoherence.16On the other hand,\nwe neglect the orbital effects of this field which is justi-\nfied if the confinement length is much smaller than the\nmagnetic length lB=√2/planckover2pi1/eB, where−eis the electron\ncharge. If the field is strong(above 1-2Tesla), the orbital\neffects important here are fully incorporated by a renor-\nmalization of the confinement length l−4→l−4+l−4\nB.\nThe field is applied along s0, a unit vector.\nThe quantum dot is defined by the confinement poten-\ntialV+Vz, which we separated into the in-plane and\nperpendicular components. The corresponding in-plane\nand perpendicular position and momentum components\nreadR={r,z}andP={p,pz}, respectively. When-\never we below need an explicit form of the wavefunction,\nwe assume, for convenience, a parabolic in-plane and a3\nhard-wall perpendicular confinements\nV(r) =/planckover2pi12\n2ml4r2≡1\n2mω2r2, (2a)\nVz(z) =/braceleftbigg\n0, 0< z < w,\n∞, otherwise.(2b)\nThe confinement lengths landwcharacterize a typical\nextent of the wavefunction in the lateral, and perpendic-\nular directions, respectively. Alternatively to the length,\nthe confinement energy /planckover2pi1ωcan be used. However, we\nstress that our results below do not rely on the specific\nconfinement form in any way, as long as the dot is quasi\ntwo dimensional, a condition which for the adopted ex-\nample reads l≫w. Typical values for lateral quantum\ndots in GaAs are l=30 nm, w=8 nm.\nThe last term in Eq. (1) is the spin-orbit interaction8\nHso=/planckover2pi1\n2mld(−σxpx+σypy)+/planckover2pi1\n2mlbr(σxpy−σypx),(3)\ncomprising the Dresselhaus term, which arises in zinc-\nblende structures grown along [001] axis and the\nBychkov-Rashba term, which is a consequence of the\nstrong perpendicular confinement. The interactions are\nparameterized by the spin-orbit lengths lso∈ {ld,lbr},\ntypically a few microns in GaAs heterostructures.\nAssume first that the spin-orbit coupling is absent. To\nbe able to treat the electron and the hole (each referred\nto as the particle) on the same footing, we introduce the\nfollowing notation\n|Ψp∝an}b∇acket∇i}ht=|J∝an}b∇acket∇i}ht⊗|ΦJ\na∝an}b∇acket∇i}ht,(zero SOC) .(4)\nThe complete particle wavefunction, for which we use\nthe Greek letter Ψ, is a two (electron) or four component\n(hole) spatially dependent spinor. Its label p={J,a}\nindicates that the wavefunction is separable into a (posi-\ntion independent) spinor |J∝an}b∇acket∇i}htand a scalarposition depen-\ndent complex amplitude |Φ∝an}b∇acket∇i}ht. The former is labelled by\nthe particle angular momentum in units of /planckover2pi1,J=±1/2\n(electron), J=±3/2,±1/2(hole; alternatively,weuse hh\nfor 3/2 and lhfor 1/2 labels). The set of orbital quan-\ntum numbers adepends on the confinement potential.\nFor the choice in Eqs. (2), it is a set of three numbers\na={nm,k}, withnthe main and mthe orbital quan-\ntum number ( m≡ −m) of a Fock-Darwin state, and k\nthe label of the subband in the perpendicular hard-wall\nconfinement. Finally, for the particle ground state we\nomit the index a, or useG≡ {J,00,0}in place of p. The\nelectron ground state is thus denoted by\n|Ψ1/2∝an}b∇acket∇i}ht=|1/2∝an}b∇acket∇i}ht⊗|ΦG∝an}b∇acket∇i}ht, (5)\nwhere the direction of the spin up spinor |1/2∝an}b∇acket∇i}htis set by\nthe external field along s0.\nLet us now consider the spin-orbit coupling. It turns\nout that for electrons the spin-orbit effects on the wave-\nfunction can be in the leading order written as53,54\n|Ψp∝an}b∇acket∇i}ht=U|J∝an}b∇acket∇i}ht⊗|ΦJ\na∝an}b∇acket∇i}ht. (6)HereUis a unitary 2 by 2 matrix of a spinor rotation,\nU(r) = exp[−inso(r)·J], (7)\nparameterized by an in-plane position dependent vector\nnso(r) =−/parenleftbiggx\nld−y\nlbr,x\nlbr−y\nld,0/parenrightbigg\n.(8)\nA weak spin-orbit coupling allows us to label states with\nthe same quantum numbers as for no spin-orbit case, as\nthereis a clearone toone correspondence. The enormous\nsimplification, thattheunitarymatrixinEq.(6) doesnot\ndepend on the quantum numbers p, is due to the special\nform of the spin-orbit coupling in Eq. (3), what has sev-\neral interesting consequences.55–58To calculate the spin\nrelaxation or spin-orbit induced energy shifts, on which\nUhas no effects, one has to go beyond the leading order\ngiven in Eq. (6).59However, we will see that here Uwill\nresult in spin-non-conserving interactions, and it is thus\nenough to consider the leading order. For the same rea-\nson we neglected the cubic Dresselhaus term in Eq. (3),\nwhat is here, unlike usually,60an excellent approxima-\ntion.\nB. Hole states\nFor holes we restrict to the four dimensional subspace\nof the light and heavy hole subbands. Neglecting the\nspin-orbit coupling, they correspond to the angular mo-\nmentum states J=±3/2, andJ=±1/2, respectively.\nWe use the confinement potential in Eqs. (2), setting the\nconfinementenergyintheheavyholesubbandto20meV,\nwhich gives l≈4 nm and setting the light-heavy hole\nsplitting to ∆ lh= 100 meV, which gives w≈2 nm. The\natomic spin-orbit coupling manifests itself as the orbital\nsplitting of the light and heavy holes from the spin-orbit\nsplit-off band (which is energetically far from the states\nconsidered in this paper), and as a coupling of the light\nand heavy holes at finite momenta. The latter effect we\nrefer to as the hole spin-orbit coupling — we do not con-\nsider higher order effects, which give rise to spin-orbit\ninteractions similar in form to the electronic Dresselhaus\nand Rashba terms. Within this model, we derive the\nspin-orbit coupled wavefunctions from the corresponding\n4 by 4 sector of the Kohn-Luttinger Hamiltonian in Ap-\npendix A, and get the hole ground state as\n|Ψ3/2∝an}b∇acket∇i}ht=|3/2∝an}b∇acket∇i}ht⊗|Φhh\n00,0∝an}b∇acket∇i}ht+λ1|1/2∝an}b∇acket∇i}ht⊗|Φlh\n01,1∝an}b∇acket∇i}ht\n+λ0|−1/2∝an}b∇acket∇i}ht⊗|Φlh\n02,0∝an}b∇acket∇i}ht.(9)\nWe use the notation explained below Eq. (4). The Fock-\nDarwin states in the heavy hole ( hh) and light hole ( lh)\nsubband differ, due to different effective masses. The\nkey quantities are the scalars λ, which quantify the light-\nheavyholemixing. Thespin-non-conservinginteractions,\nas well as the resulting spin relaxation rates, will scale\nwith these scalars. For our parameters, which we list for4\nconvenience in Appendix E, we get λ0≈λ1≈0.05. In\naddition to the ground state, we will need also the lowest\nexcited state in the heavy hole subband, which is the\ntime reversed copy of Eq. (9),\n|Ψ−3/2∝an}b∇acket∇i}ht=|−3/2∝an}b∇acket∇i}ht⊗|Φhh\n00,0∝an}b∇acket∇i}ht−λ1|−1/2∝an}b∇acket∇i}ht⊗|Φlh\n01,1∝an}b∇acket∇i}ht\n+λ0|1/2∝an}b∇acket∇i}ht⊗|Φlh\n02,0∝an}b∇acket∇i}ht,(10)\nand also the lowest state in the light hole subband\n|Ψ1/2∝an}b∇acket∇i}ht=|1/2∝an}b∇acket∇i}ht⊗|Φlh\n00,0∝an}b∇acket∇i}ht+λ′\n1|3/2∝an}b∇acket∇i}ht⊗|Φhh\n01,1∝an}b∇acket∇i}ht\n−λ′\n0|−3/2∝an}b∇acket∇i}ht⊗|Φlh\n02,0∝an}b∇acket∇i}ht,(11)\nwhich will be surprisingly effective in inducing the spin-\nnon-conservingcouplingamongimpurities, aswewillsee.\nWe make a few notes here: First, in the sphericalapprox-\nimation that we adopt, the Kohn-Luttinger Hamiltonian\nconservestheangularmomentum, sothatallcomponents\nin each of Eqs. (9)-(11) have the same value of J+m.\nSecond, the mixing is stronger in the light hole subband,\nλ′\n1≈0.15 andλ′\n0≈0.11. This is because the admixing\nstates are closer in energy—the Fock-Darwin excitation\nenergies add to and subtract from the light-heavy hole\nsplitting in the heavy and light hole case, respectively,\nas evidenced by Eqs. (A9) and (A11). Third, we will be\ninterested in the case of zero external magnetic field for\nholes (we give the Zeeman interaction in Appendix A for\ncompleteness.) Unlike for electrons, such a field is here\nnot required to split the two states in Eqs. (9) and (10),\nas the splitting arises due to the spin impurities. As we\nwill see, this splitting will be of the order of few meV.\nCompared to this, the hole Zeeman energy is negligible\nuptofieldsofseveralTesla. Inaddition, theexternalfield\nsuppresses an interesting feedback between the particle\nand impurities.61Finally, we note that one could relate\nthe first-order and unperturbed hole states analogously\nto the electron case introducing a unitary transformation\nU, whose matrix elements are the coefficients appearing\nin Eqs. (9)-(11). However, since here the transformation\ndoes not have any appealing form similar to the one in\nEq. (7), we do not explicitly construct the matrix Ufor\nholes.\nIII. EFFECTIVE HAMILTONIAN\nIn this section we introduce the particle-impurity ex-\nchange interaction, while providing a unified description\nfor both type of particles and impurities. This interac-\ntion manifests itself as the Knight field acting on the\nimpurities and the Overhauser field acting on the parti-\ncle. (The fields are defined as the expectation value of\nthe exchange interaction in the state of the correspond-\ning subsystem). Historically, the terminology was ini-\ntially applied to nuclei and here we also use it for Mn\nspins. With the help of these fields, we define the unper-\nturbed basis of the particle-impurity system, for which\nwe derive the effective interaction Hamiltonian treatingthe non-diagonalexchange terms perturbatively. Finally,\nwe define the spin-conserving versus spin-non-conserving\ninteraction terms and analyze their relative strength for\nthe electron and hole case.\nOur strategy can be viewed also in the following al-\nternative way. To derive the spin-orbit coupling effects\non the effective impurity interactions, we proceed in two\nsteps: First we unitarily transform the particle basis\nto remove the spin-orbit coupling in the lowest order.\nThe spin-orbitcoupledbasistransformationrenormalizes\nthe particle-impurity exchange interaction and breaks its\nspin-rotationalsymmetry. Thenweintegrateoutthepar-\nticle degrees of freedom by a second unitary transforma-\ntion, using the L¨ owdin (equivalently here, the Schrieffer-\nWolff) transformation, which leaves us with effective in-\nteractions concerning impurities only.\nA. Particle-impurity interaction\nThe particle interacts with impurities by the Fermi\ncontact interaction47,62\nHF=/summationdisplay\nnHn\nF=−/summationdisplay\nnβδ(R−Rn)J·In.(12)\nHeren= 1,2...labels the impurities located at posi-\ntionsRn= (rn,zn) with corresponding spin operators\nInin units of /planckover2pi1. The impurities have spin Iand den-\nsity 1/v0. For the electron-nuclear spins case the impu-\nrity density is in GaAs the atomic density. For holes-Mn\nspins we assume xMnis the fraction of cations replaced\nby Mn atoms, typically xMn= 1%. For impurities with\ndifferent magnetic moments (as for nuclei of different el-\nements) the coupling βshould have the index n, but we\nwill not consider this minor complication. Even though\nthe wavefunction extends to infinity, it makes sense to\nconsider the number of impurities with which the par-\nticle interacts appreciably as N=V/v0, where the dot\nvolumeVis defined by13\n1/V=/integraldisplay\nd3R|ΦG(R)|4. (13)\nThe maximal value of the Hamiltonian HF, if all impu-\nrities are aligned with the particle, is\nE=−/summationdisplay\nnβ|ΦG(Rn)|2JI=−βJI/v 0.(14)\nThe impurity Zeeman energy is\nHnZ=/summationdisplay\nngnµimpIn·B, (15)\nwithgnthe impurity g-factor. For GaAs lateral dots,\nwe consider nuclear spins as impurities, with I= 3/2\nandµimpthe nuclear magneton µN. For ZnTe dots, the\nimpurities are intentionally doped Mn atoms, with I=\n5/2 andµimpthe Bohr magneton µB.5\nB. Knight field\nAssume the particle sits in the ground state G. In the\nlowest order of the the particle-impurity interaction, a\nparticularimpurityspincouplestoalocalfield, calledthe\nKnight field. We define it in units of energy by writing\nKn·In≡ ∝an}b∇acketle{tΨG|Hn\nF|ΨG∝an}b∇acket∇i}ht, (16)\nfrom which, using Eq. (12), we get\nKn=−β∝an}b∇acketle{tΨG|δ(R−Rn)J|ΨG∝an}b∇acket∇i}ht.(17)\nUsing Eq. (6) the Knight field of an electron is\nKn=−β|ΦG(Rn)|2∝an}b∇acketle{t1/2|U†(Rn)JU(Rn)|1/2∝an}b∇acket∇i}ht\n=−β(1/2)|ΦG(Rn)|2∝an}b∇acketle{ts(Rn)∝an}b∇acket∇i}ht.(18)\nIt points along the direction of the electron spin at the\nposition of the n-th impurity, introduced as a unit vector\n∝an}b∇acketle{ts(Rn)∝an}b∇acket∇i}ht ≡RU(Rn)[s0]. The operator RUis defined such\nthat it performs the same rotation on vectors, as Udoes\non spinors. The explicit form of Ris the one in Eq. (7),\nif generators of rotations in three dimensions are used,\n(Jk)lm=−iǫklm. As is apparent from Eq. (18), evalu-\nating the Knight field with perturbed electron wavefunc-\ntions is equivalent to evaluating a unitarily transformed\ninteraction HF→U†HFUwith unperturbed electron\nwavefunctions.\nWe get the Knight field of the hole as (see Appendix\nB),\n[Kn\nx,Kn\ny,Kn\nz] =−β[λ1Refn, λ1Imfn,3|Φhh\n00,0(Rn)|2/2],\n(19)\nwhere we abbreviated fn=√\n3Φlh\n01,1(Rn)Φhh∗\n00,0(Rn) and\nneglected contributions of higher orders in λ. Rather\nthan the exact form, we note that without the spin-\norbit coupling, the direction of the Knight field is fixed\nglobally (along the external magnetic field for electrons,\ns0=B/Band along the z axis — the spin direction of\nheavy holes — for holes, s0=ˆ z). The spin-orbit inter-\naction deflects the Knight field in a position dependent\nway, the deflection beingin the leadingorderlinearin the\nsmallparametercharacterizingthespin-orbitinteraction.\nIn this respect, Eq. (18) and (19) are the same.\nC. Basis\nThe total field aligning the impurity spin is the sum of\nthe Knight field and the external field\nBn=Kn+gnµimpB. (20)\nThe typical energy scale of the Knight field of an electron\nin a lateral dot is tens of peV, which corresponds to the\nimpurity in an external field of 1 mT. For a hole in a self-\nassembled dot the Knight field is of the order of 100 µeV,\ncorresponding to the external field of 0.3 T. Based onthis, in the following analysis we mostly consider typical\nsituations, in which the total field is dominated by the\nexternal field for nuclear spins (electronic case), and the\nKnight field for Mn spins (hole case).\nWenowintroduceforeachimpurityarotated(primed)\ncoordinate system, in which the unit vector ˆ z′is along\nthe total field Bn. Formally, the rotation is performed by\noperator RBndefined by the relation between the unit\nvectors,\nˆ z′=RBn[ˆ s0]. (21)\nThe orientation of the in-plane axes x′,y′in the plane\nperpendicular to ˆ z′is arbitrary, and we denote r′±=\nˆ x′±iˆ y′. We define the impurity ensemble basis states as\ntensor products of states with a definite spin projection\nalong the locally rotated axis z′,\n|I∝an}b∇acket∇i}ht=|I1\nz′∝an}b∇acket∇i}ht⊗|I2\nz′∝an}b∇acket∇i}ht⊗...⊗|IN\nz′∝an}b∇acket∇i}ht. (22)\nThe spin projections take discrete values, In\nz′∈ {I,I−\n1,...,−I}. We use Ias the collective index of the im-\npurities. With this, we are now ready to introduce the\ncomplete system basis, as spanning the states\n|Ψp∝an}b∇acket∇i}ht⊗|I∝an}b∇acket∇i}ht ≡ | Ψp∝an}b∇acket∇i}ht⊗|I1\nz′∝an}b∇acket∇i}ht⊗|I2\nz′∝an}b∇acket∇i}ht⊗...⊗|IN\nz′∝an}b∇acket∇i}ht,(23)\nwith the corresponding energies\nEp,I=Ep+/summationdisplay\nnEIn\nz′=Ep+/summationdisplay\nnBnIn\nz′,(24)\ncomprising the particle energy and the Zeeman energies\nof impurities in the corresponding total fields.\nD. The substantial gap assumption – Overhauser\nfield\nIn addition to the Knight field, another consequence\nof the particle-impurity interaction from Eq. (12) is the\neffective field experienced by the particle spin, known as\nthe Overhauser field2O. To express this field again in\nthe units of energy, it is helpful to consider the matrix\nelements of the particle-impurity interaction within the\nsubspace of the lowest two electron states J,J′∈S≡\n{1/2,−1/2},\n∝an}b∇acketle{tΨJ|−β/summationdisplay\nnδ(R−Rn)J·In|ΨJ′∝an}b∇acket∇i}ht\n=−β/summationdisplay\nn|ΦG(Rn)|2RU†\nn[In]·∝an}b∇acketle{tJ|J|J′∝an}b∇acket∇i}ht.(25)\nWe introduce the field Oas\nHF|S≡O·J, (26)\nwherethe subscript Srefersto the subspace comprisinga\npair of time-reversed particle states and the Overhauser\nfield depends on the choice of S. To quantify the Over-\nhauserfield, wegiveupontryingtotrackthemicroscopic6\nstate of the impurities and instead introduce the averag-\ning (denoted by an overline) over impurity ensembles\nIna= 0,InaIm\nb=δnmδabI(I+1)/3,(27)\nwhich characterizes unpolarized and isotropic ensembles.\nNuclear spins, unless intentionally polarized in dynami-\ncalnuclearpolarizationschemes,22,63,64usuallywellfulfill\nthis. If not polarized by an external field, Mn spins can\nbe considered random initially, before the particle enters\nthe dot and the polarization starts to build up.\nEquation(27) gives azero Overhauserfield on average,\nbut with a finite dispersion, quantifying a typical value.\nFor electrons, we get the well known result,13,65\nO2=β2/summationdisplay\nnm|ΦG(Rn)|2|ΦG(Rm)|2RU†\nn[In]·RU†\nm[Im]\n=β2/summationdisplay\nn|ΦG(Rn)|4I(I+1) =I(I+1)(β/v0)2/N,\n(28)\nstating that the typical value of the Overhauser field is\ninverselyproportionaltothesquarerootofthe numberof\nimpurity spins within the dot. The spin-orbit coupling,\nequivalent to a position dependent spin coordinate frame\nrotation, does not influence the result at all, as Eq. (27)\nassumes isotropicnon-interacting impurities. For our pa-\nrameters, the typical Overhauser field value is 0 .3µeV,\nwhich corresponds to external field of 20 mT. The energy\nsplitting of the electron spin opposite states is therefore\nfor our case dominated by the Zeeman, rather than the\nOverhauser, field.\nFor holes, we will not write the Overhauser field ex-\nplicitly as a vector. Instead we calculate directly the\ntypical matrix elements of the particle-impurity interac-\ntion within the heavy hole subspace with the spin-orbit\nrenormalizedwavefunctions. We leavethe details for Ap-\npendix B and state the results here: the diagonal terms\nare\n|∝an}b∇acketle{tΨ±3/2|HF|Ψ±3/2∝an}b∇acket∇i}ht|2≈(3/4)I(I+1)(β/v0)2/N,(29)\nwhere we neglected small contributions of the spin-orbit\ncoupling. An important difference to an analogous result\nfor the electrons, Eq. (28) is the energy scale. Here the\ntypical value for the diagonal Overhauser field is several\nmeV, which corresponds to huge external fields of many\nTesla. The energy splitting of the hole is thus dominated\nby the Overhauser, rather than Zeeman, field. On the\nother hand, the off-diagonal element is non-zero only in\nthe presence of the spin-orbit coupling,\n|∝an}b∇acketle{tΨ−3/2|HF|Ψ3/2∝an}b∇acket∇i}ht|2∼2I(I+1)(λ0β/v0)2/N.(30)\nThe impurity spins may induce transition (precession) of\nthe heavy hole spin due to the transversal component of\nthe Overhauser field, which is smaller by a factor of λ0compared to the diagonal component. For our param-\neters the transversal component is of the order of tens\nofµeV, so for the heavy hole spin precession to occur,\nthe two spin opposite states have to be degenerate with\nrespect to this energy [which normally does not occur,\nbecause of the diagonal term, Eq. (28)].\nHaving compared the typical energy splittings of the\nparticle induced by the effective Overhauser field, versus\nthe external magnetic field, we are now ready to discuss\nthe crucial assumption for the derivation which will fol-\nlow. It is the assumption that the particle is fixed to\nits ground state by an energy gap, irrespective of the\nevolution of the impurity ensemble. This requires that\nspin flips of impurities cost much less in energy than the\nparticle transitions\n∆EIn≪∆Ep. (31)\nFor electrons, this assumption is guaranteed as both the\nparticle and impurities spin flip costs are dominated by\nthe Zeeman energy, proportional to the magnetic mo-\nment, which is much larger for the electron than for a\nnuclear spin, µimp=µN∼10−3µB. On the other hand,\nfor holes for which the particle and impurity magnetic\nmoments are comparable, the above condition is also ful-\nfilled since the particle spin flip energy cost is dominated\nby the Overhauser field.\nE. The effective Hamiltonian\nOnce the particle is fixed to its ground state (the sub-\nstantial gap assumption), the particle excited states can\nbe integratedout perturbatively50,66,67resulting in an ef-\nfective Hamiltonian for the impurity ensemble Heff. For\nthis purpose we split the interaction Hamiltonian to\nHF≡H0\nF+H′\nF, (32)\nwhere the diagonal part,\nH0\nF=∝an}b∇acketle{tΨG|HF|ΨG∝an}b∇acket∇i}ht=/summationdisplay\nnKn·In,(33)\ntogether with the external field, defines the unperturbed\nHamiltonian H0=Hp+HnZ+H0\nFand the basis, so that\n∝an}b∇acketle{tΨp⊗IA|HnZ+H0\nF|Ψq⊗IB∝an}b∇acket∇i}ht ∝δpqδAB,(34)\nwhereIA,IBdenote arbitrary basis states of the impu-\nrity ensemble. We also note that\n∝an}b∇acketle{tΨG⊗IA|H′\nF|ΨG⊗IB∝an}b∇acket∇i}ht= 0. (35)\nUsing L¨ owdin theory,68,69the matrix elements of the\neffective Hamiltonian, in the lowest order in the non-\ndiagonal part, H′\nF, are7\n∝an}b∇acketle{tIA|Heff|IB∝an}b∇acket∇i}ht=∝an}b∇acketle{tΨG⊗IA|H0+/summationdisplay\np/negationslash=G,I∗/parenleftBig1/2\nEGIA−EpI∗+1/2\nEGIB−EpI∗/parenrightBig\nH′\nF|Ψp⊗I∗∝an}b∇acket∇i}ht∝an}b∇acketle{tΨp⊗I∗|H′\nF|ΨG⊗IB∝an}b∇acket∇i}ht,(36)\nwhere the summation proceeds through the excited par-\nticle states and a complete basis of impurities. We now\nuse the substantial gap assumption, which states that all\nsystem states reachable by the interaction H′\nFhave the\nenergy dominated by the particle energy, so that we can\napproximate EGI−EpI∗≈EG−Ep. By this relation\nthe summation over the impurities gives an identity,\n∝an}b∇acketle{tIA|Heff|IB∝an}b∇acket∇i}ht=\n∝an}b∇acketle{tΨG⊗IA|H0+H′\nF/summationdisplay\np/negationslash=G|Ψp∝an}b∇acket∇i}ht∝an}b∇acketle{tΨp|\nEG−EpH′\nF|ΨG⊗IB∝an}b∇acket∇i}ht.(37)\nSince the impurity states now only sandwich both sides\nof the equation, we can equate the operators\nHeff=∝an}b∇acketle{tΨG|H0|ΨG∝an}b∇acket∇i}ht+/summationdisplay\np/negationslash=G∝an}b∇acketle{tΨG|H′\nF|Ψp∝an}b∇acket∇i}ht∝an}b∇acketle{tΨp|H′\nF|ΨG∝an}b∇acket∇i}ht\nEG−Ep.\n(38)\nEven though this looks like the standard second order\nperturbation theory result, note that even after taking\nmatrix elements with respect to the particle states, the\nexpressions still contain the quantum mechanical opera-\ntors of the impurity spins. On the other hand, by taking\ntheexpectationvalue, the particledegreesoffreedomdis-\nappear from the effective Hamiltonian. The first term is\na sum of the particle ground state energy and the impu-\nrities energy in the Knight field,\n∝an}b∇acketle{tΨG|H0|ΨG∝an}b∇acket∇i}ht=EG+/summationdisplay\nnBn·In.(39)\nTo simplify the notation of the second term in Eq. (38),\nwe introduce\n∝an}b∇acketle{tΨG|H′\nF|Ψp∝an}b∇acket∇i}ht=∝an}b∇acketle{tΨG|HF|Ψp∝an}b∇acket∇i}ht ≡/summationdisplay\nnAn·In,(40)\nso that the p-state dependent complex vector Ais\nAn=−β∝an}b∇acketle{tΨG|δ(R−Rn)J|Ψp∝an}b∇acket∇i}ht. (41)\nWenowtransformvectors Aandspinoperators Iintothe\ncoordinate system along the total field of each impurity\n˜An=R−1\nBn[A],˜In=R−1\nBn[I]. (42)\nThe z-component of a rotated vector is its projection\nalong the direction of the local total field, e.g. ˜Iz=I·ˆ z′\nand similarly for A. Omitting the constant EG, we\nrewrite Eq. (38) with the new notation and arrive at our\nmain result\nHeff=/summationdisplay\nnBn˜In\nz+/summationdisplay\np/negationslash=G/summationdisplay\nn,m1\nEG−Ep(˜An·˜In)(˜Am·˜Im)†.\n(43)The first term defines the spin flip energy cost and the\nspin quantization axis, according to Eq. (21). The in-\nteractions described by the second term can be classified\nasspin-conserving(spin-non-conserving)accordingto ro-\ntated operators ˜Icomponents parallel (perpendicular) to\naglobalaxis ˆ s0,aswewillshowbelow. Tofurtherdemon-\nstrate the usefulness and generality of Eq. (43), we show\nthat known results follow as special limits, and how the\nconsequences of the spin-orbit coupling on the impuri-\nties interactions can be drawn from the formula. We also\nnote that the derivation would proceed in the same way\neven ifGwere not the particle ground state. The only\nrequirement for the validity of Eq. (43) is that the state\nGis far enough in energy from other particle states so\nthat Eq. (31) is valid. For example, thermal excitations\nof the particle would result in a thermal average of the\neffective Hamiltonian (the vectors Aand energies Bdo\ndepend on G). We do not pursue a finite temperature\nregime further here, and assume the thermal energy kBT\nis small such that the particle stays in the ground state.\nBefore we evaluate vectors ˜Ain specific cases, we\nnote an important property of the effective Hamiltonian.\nNamely, for both holes and electrons, the lowest excited\nstate is much closer to the ground state (split by the Zee-\nman energy) compared to higher excited states (split by\norbital excitation energies). If the mediated interactions\nare dominated by this low lying excited state, we can\nwrite\nHeff=/summationdisplay\nnBn˜In\nz+/summationdisplay\nn,m1\nE↑−E↓(˜An\n↑↓·˜In)(˜Am\n↑↓·˜Im)†,(44)\nwhere we denoted G=↑, the sum was restricted to a sin-\ngle term p=↓, and˜An\n↑↓is the vector corresponding to\nthe ground/excited state being the spin up/down state.\nFrom the relation ˜An\n↑↓= (˜An\n↓↑)†it is straightforward to\nsee that had we started with the particle being in the\nspin down state G=↓and restricted to the closest state\nin the spectrum p=↑, the mediated interaction Hamilto-\nnian would be the same as in Eq. (44), but for a minus\nsign in the second term, coming from the denominator.\nThis crucial property, which results in the particle spin\ndecoherencebeingtoalargeextentremovablebythespin\necho protocols,15,16is thus preserved in the presence of\nthe spin-orbit coupling.8\nF. Effective Hamiltonian symmetry and magnitude\nof the spin-non-conserving interactions\nFor electrons, we get from Eq. (18)\n˜An=R−1\nBn[An] =ǫn\npR−1\nBn◦RUn[∝an}b∇acketle{tJG|J|Jp∝an}b∇acket∇i}ht],(45)\nwhere we denoted the position dependent energy\nǫn\np=−βΦ∗\nG(Rn)Φp(Rn)∼ −β/V. (46)\nConsider first that the magnetic field is small such that\nthe total effective field in Eq. (20) is dominated by the\nKnight field, rather than the external field. In other\nwords, the local impurity quantization axis is collinear\nwith the local particle spin direction. Then RBn=RUn\nand we get from Eqs. (43) and (45) the effective Hamil-\ntonian in the following form\nHeff=/summationdisplay\nnBn˜In\nz+/summationdisplay\np∈↑/summationdisplay\nn,mǫn\nGǫm\np\nEG−Ep˜In\nz˜Im\nz\n+/summationdisplay\np∈↓/summationdisplay\nn,mǫn\nGǫm\np\nEG−Ep/parenleftBig\n˜In\n−˜Im\n++˜Im\n−˜In\n+/parenrightBig\n/2.(47)\nWe have split the summation over the particle excited\nstates into those with the same, and the opposite spin as\nis the spin of the ground state, corresponding to the sec-\nond, and the third term in Eq. (47), respectively. Equa-\ntion (47) makes it clear that there is a conservedquantity\neven in the presence of spin-orbit coupling, though it is\nneither the energy nor the total spin along any axis; it\nis the number of impurity spins locally aligned with the\nparticle spin, equal to/summationtext\nn˜In\nz. This result is very general,\nas it is based only on the form of the spin-orbit coupling,\nwhichgivesasingleunitaryoperator Uforthe wholepar-\nticle spectrum. Restricting to the lowest excited state, as\nin Eq. (44), we get the standard result66,67\nHeff=/summationdisplay\nnBn˜In\nz−/summationdisplay\nn,mǫn\nGǫm\nG\nEz/parenleftBig\n˜In\n−˜Im\n++˜Im\n−˜In\n+/parenrightBig\n/2,(48)\ngeneralized to include the effects of the spin-orbit cou-\npling.\nFor the electronic case, we are, however, more inter-\nested in a different regime, where a finite magnetic field\nbreaks the above discussed symmetry and sets a global\nquantization axis for impurities, so that the Zeeman en-\nergy dominates in the total effective field in Eq. (20). We\nthen have RBn≈1and˜I≈I. Equation(45) can be then\nevaluated explicitly, using Eq. (7). Instead, we estimate\nthe effects of weak spin-orbit coupling, which guarantees\nthatRm≪lso, by expanding the rotation operator up\nto the first order as\nU(Rm)≈I+O(rm/lso). (49)\nThe pairwise interaction in the effective Hamiltonian\nthen appear in combinations such as [see Eq. (D1) in\nAppendix D]\n˜In\n+˜Im\n−+γ˜In\n+˜Im\n++γ′˜In\n+˜Im\nz+..., (50)whereγ,γ′=O[l/lso]. This is the most important\nmessage for the electron case, that the spin-orbit cou-\npling results in the spin-non-conserving interactions in\nthe impurity ensemble, which are, compared to the spin-\nconserving ones, suppressed by a position dependent fac-\ntor of the order of the ratio of the confinement and spin-\norbit lengths.\nWe now turn attention to a hole dot, taking the low-\nest state in the heavy hole subband as the ground state\nG= 3/2,00,0. The closest excited state, which gave by\nfar the dominant contribution in the electronic case, is\nthe spin opposite heavy hole state p=−3/2,00,0. The\ncorrespondingvectors Ascale as (see Appendix D for full\nexpressions)\n˜An\n+∼ǫn\npO(λ2\n0),˜An\n−∼ǫn\npλ0,˜An\nz∼ǫn\npλ0.(51)\nTo quantify the prefactor in the second order term, ˜An\n+,\nwe would have to go to the next order in the perturba-\ntion expansion of the wavefunctions. However, this is\nnot necessary as this term does not enter anywhere in\nthe subsequent discussion. We conclude from Eq. (51)\nthat the spin-conserving interactions mediated by the\nlowest heavy hole excited state are proportional to the\nsecond power of parameters λ[through terms such as\n˜A−˜A∗\n−˜I+˜I−], the same as the spin-non-conserving ones\n[e.g.˜A−˜A∗\nz˜I+˜Iz]. This is a drastic difference to the elec-\ntron case, where the spin-conserving interactions domi-\nnate.\nLet us now consider the light hole subband. Taking\np= 1/2,00,0, we get (see Appendix D)\n˜An\n+∼ǫn\np,˜An\n−∼ǫn\npλ0,˜An\nz∼ǫn\npλ′\n1.(52)\nThe light hole excited state does mediate spin-conserving\nimpurity interactions [through ˜A+˜A∗\n+˜I−˜I+]. Com-\npared to these, the leading spin-non-conserving term\n[˜A+˜A∗\nz˜I−˜Iz] is suppressed linearly in λ. The energy de-\nnominator in the effective Hamiltonian is of the order\nof 100 meV for the light hole states (typical light-heavy\nhole band offset) versus a few meV offset of the lowest\nheavy hole excited state. For our parameters, this energy\npenalty is almost exactly compensated by much larger\nmatrix elements for the spin-non-conserving interactions\nandmorethancompensatedforthe spin-conservingones.\nWe conclude that the spin-alike light hole state is the\nmost efficient mediator of the spin-conserving interac-\ntions in the impurity ensemble, and rather efficient medi-\nator of the spin-non-conserving ones. As a direct conse-\nquence, and unlike forelectrons, the decoherenceinduced\nby the hole mediated evolution of the impurity bath will\nnot be removed by the hole spin echo.\nIV. PHONON INDUCED SPIN RELAXATION\nOF THE IMPURITY BATH\nWe now use the results of the previous section to cal-\nculate how fast the impurity ensemble spin relaxes. The9\nfirst and the second term of the effective Hamiltonian,\nEq. (43), induces flip of a single impurity and a pair of\nimpurities, respectively. For the former, terms with in-\nplane components of ˜I, while for the latter terms such\nas the last two terms in Eq. (50) are required for spin-\nnon-conserving transitions. As the initial and final state\nenergies differ, in general, we consider that the transi-\ntion is assisted by phonons, which provide for the energy\nconservation.\nWe consider several possible mechanisms how phonons\ncan couple to the impurity bath and make order of mag-\nnitude estimates for the resulting relaxation rates. We\nfind that the most efficient relaxation is due to the piezo-\nelectricfield spatiallyshifting the particle, leadingtoa µs\nrelaxation time for Mn spins. It is known that phonons\nare ineffective in relaxing nuclear spins,70still we eval-\nuate the resulting rates also for electrons, as we treated\nelectrons and holes on the same footing, the derived for-\nmulas apply for both. We find a 1011s relaxation time\nfor nuclear spins.\nA. Particle-phonon interactions\nThe phonon-impurity interaction Hamiltonian Hiis in\ngeneral a function of the local lattice deformation due to\nthe presence of acoustic phonons,\nδR= i/summationdisplay\nQλ/radicalBigg\n/planckover2pi1\n2V0ρωQλeQλeiQ·R/parenleftBig\naQλ+a†\n−Qλ/parenrightBig\n.(53)\nHere the phonon wavevector is Q, polarization is λ(one\nlongitudinal λ=land two transversal ones λ=t1,2)\nwitheQλa real unit vector ( eQλ=−e−Qλ),V0is the\ncrystal volume, ρis the material density, /planckover2pi1ωQλ=/planckover2pi1cλQ\nis the phonon energy, cλis the phonon velocity, and a†\nQλ\nis the phonon creation operator.\nIn a polar material, such as GaAs, the lattice deforma-\ntion is accompanied by a piezoelectric field, which is the\ngradient of the following potential\nVPZ=−iΞ/summationdisplay\nQλ/radicalBigg\n2/planckover2pi1\nV0ρωQλ1\nQ2eiQ·R/parenleftBig\naQλ+a†\n−Qλ/parenrightBig\n×\n×/parenleftBig\nQxQy(eQλ)z+QzQx(eQλ)y+QyQz(eQλ)x/parenrightBig\n,\n(54)\nwith Ξ the piezoelectric constant.\nIn addition to the previous, the deformation of the\nlattice shifts the electronic bands, quantified as the de-\nformation potential VDP=−σdivδR, which thus reads\nVDP=σ/summationdisplay\nQ/radicalBigg\n/planckover2pi1\n2V0ρωQlQeiQ·R/parenleftBig\naQl+a†\n−Ql/parenrightBig\n,(55)\nwithσthe deformation potential constant.In what follows we will see that a relative shift of the\nimpurity and the particle, which we denote by d, will\ninduce impurity-phonon coupling, leading to the impu-\nrity spin relaxation. Since impurities are tied to atoms,\nthe phonon displacement is obviously such a relative\nshiftd=δR, which we call “geometric”. However,\nthe phonon induced electric fields Ealso lead to shifts.\nNamely, adding the potential of an in-plane field to that\nin Eq. (2) amounts to a shift of the quantum dot position\nbyd=eEl2//planckover2pi1ω(electricallyinducedshiftsalongtheper-\npendicular direction are much smaller, as the wavefunc-\ntion is much stiffer along z due to stronger confinement).\nIf the particle follows these potential changes adiabati-\ncally, which we assume, such a shift is equivalent to the\nshift of the impurities, which are fixed to the lattice, by\n−d. Since the phonon electric fields are proportional to\nthe displacement δR, we can write a general expression\n|d| ∼α|δR|, (56)\nwith a dimensionless factor α. For the geometric shift\nmechanism α= 1 by definition. For the piezoelectric\nfield, comparing Eqs. (53) and (54), we get\nα= 2QlΞl\n/planckover2pi1ω. (57)\nFinally, the deformation potential gives\nα=σ(Ql)2//planckover2pi1ω. (58)\nWe specify the dimensionless factor αin Table I. As it\nenters the relaxation rates in the second power (as we\nwillsee), wecanimmediatelyquantifytherelativeimpor-\ntance of the three consideredchannels. Piezoelectricfield\nis the most effective, for both electron and hole cases, in-\nducing shifts almost two orders of magnitude larger than\nthe geometric shift. The electric field from the defor-\nmation potential is comparable to the piezoelectric for\nholes, and much smaller for electrons, which are deeply\nin the long phonon wavelength limit, Ql≪1. We note\nthat the geometric shifts will be in fact somewhat more\neffective than it seems from the table, as they may (un-\nlike the electric fields) shift the wavefunction along the\nperpendicular direction. This results in an effective en-\nhancement of αby a factor of πl/w, which, however, is\nnot large enough to change the order of importance fol-\nlowing from the Table.\nWe will calculate the relaxation rate Γ due to a general\nphonon-impurity interaction Hiby the Fermi’s golden\nrule. For a given phonon polarization λit reads\nΓ =2π\n/planckover2pi1/summationdisplay\nQ|∝an}b∇acketle{tI′|Hi|I∝an}b∇acket∇i}ht|2δ(EI−EI′−/planckover2pi1ωQλ)NQ,(59)\nwhereIandI′denotetheinitialandfinalstateoftheim-\npurities, and we are interested in transitions where these\ntwo states differ in spin. The phonon occupation factor\nNQ=nQ+1, and NQ=nQ, if the energy of the initial10\nα piezoelectric deformation geometric\nelectron 46 0.0038 1\nhole 38 17 1\nTABLE I: Values for the dimensionless coefficient α, the ratio\nof the induced impurity shift versus the phonon displacemen t\nfor various shift mechanisms (columns) and particles (rows ).\nParameters from Appendix E were used (GaAs and ZnTe for\nthe electron and hole case, respectively), the phonon wavev ec-\ntor for electronic case was specified choosing B= 1 T.\nstate is larger, and smaller than the final state, respec-\ntively, with nQ= 1/[exp(/planckover2pi1ωQλ/kBT)−1]. The energy\nconservation fixes the phonon wavevector magnitude to\n|EI−EI′|=/planckover2pi1cλQ, by which we get\nΓ =V0Q2\nπ/planckover2pi12cλNQ|∝an}b∇acketle{tI′|Hi|I∝an}b∇acket∇i}ht|2. (60)\nThe overline denotes the angular average,\nf(Q) = (1/4π)/integraldisplay\ndΩf(Q), (61)\nover directions of the phonon wavevector Q.\nB. Spin-phonon coupling due to impurity shift\nAssuming the shifts are small, we get the impurity-\nphonon coupling as\nHi=−/summationdisplay\nnd·∂Heff\n∂R|R=Rn. (62)\nTo calculate the spatial derivative of the effective Hamil-\ntonian, Eq. (43), it is easier to first evaluate the deriva-\ntive of vectors BandAin the original coordinate system\nwhich does not depend on the position and then to trans-\nform them into the locally rotated coordinates.\nLet us start with the first term of the effective Hamil-\ntonian, Eq. (43). The finite derivative of the total field,\nEq. (20), is due to the spatial dependence of the Knight\nfield,\n(d·∂R)Bn= (d·∂R)Kn, (63)\ntransversal components of which give the spin increasing\ntransition rate for impurity nas\nΓ(1)=V0Q2\nπ/planckover2pi12cλNQ[r′−·(d·∂R)K]2|∝an}b∇acketle{tIn+1|˜In\n+|In∝an}b∇acket∇i}ht|2,(64)\nand an analogous expression follows swapping the sub-\nscripts±for a spin decreasing transition. Our goal is\nan order of magnitude estimate for the rates as the one\nin Eq. (64), which allows us to simplify: We replace the\nspin dependent matrix elements of the raising/lowering\noperators by\n|∝an}b∇acketle{tIn±1|˜In\n±|In∝an}b∇acket∇i}ht|2=I(I+1)−In(In±1)∼I2,(65)choose the direction of the phonon wavevector that gives\nthe highest contribution, instead of performing the an-\ngular average in Eq. (61), and denote\n∇˜K−≡δ−1r′−·(d·∂R)K. (66)\nFinally, we use Eq. (56) and δR∼√/planckover2pi1/2V0ρcλQto write\nΓ(1)∼α2I2NQQ\n2π/planckover2pi1c2\nλρ(∇˜K−)2, (67)\na general form for the relaxation rate estimate, which we\na few lines below evaluate for specific cases.\nLet us start with the electronic case. The transition\nenergyis dominated by the externalfield /planckover2pi1cλQ≈ |gµNB|\nand is much smaller than the thermal energy, so that\nNQ≈kBT//planckover2pi1cλQ. We evaluate the derivative of the\nKnight field in Appendix C, see Eq. (C6), getting\n∇˜K±∼(β/V)l−1\nso. (68)\nFor the dominant piezoelectric mechanism we get\nΓ(1)∼9I2\n2π3kBT(gµNB)2\n/planckover2pi1E2pzl4\nw2l2so. (69)\nwhere the energy Epz=/radicalbig\n/planckover2pi17c5\nλρ/Ξmβis a material con-\nstant. Evaluating parameters of GaAs, we get a mi-\nnuscule rate Γ(1)∼2×10−11s−1, choosing transversal\nphonons, externalfield 1Teslaandtemperature1Kelvin.\nWe now turn to holes. The transferred energy is now\ngiven by the Knight field, rather than the external field,\n|EI−EI′| ∼Jβ/Vand we again consider a high temper-\nature limit, kBT≥ |EI−EI′|. As we show in Appendix\nC, the formula in Eq. (68) is changed into\n∇˜K±∼(β/V)(√\n3λ1/l), (70)\nshowing that quantities of the form l/λcan be seen as\nan effective spin-orbit length for holes. Equation (69)\ncan be then used putting for the “spin-orbit length” the\none just described and replacing the external Zeeman\nenergy by the Knight field. For convenience, we give the\nrelaxation rate explicitly\nΓ(1)∼35I2J2\n25π4kBT\n/planckover2pi1β2\nE2pzλ2\n1\nw4l2, (71)\nwhich for ZnTe parameters and temperature 10 K yields\na modest rate 0 .35µs−1. The rate is second order in\nthe “spin-orbit strength” and, unlike for electrons, grows\nvery fast upon making the dot smaller, since now the\neffect of Knight field, inversely proportional to the dot\nvolume, dominates overthe effect ofthe shift being larger\nfor softer dot potential.\nComparing the two terms of the effective Hamiltonian,\nEq. (43), and using the results of Appendix D, the pair-\nwise spin transition mediated by the lowest heavy hole\nexcited state relates to the single spin-flip rate by\nNΓ(2)\np=3/2/Γ(1)∼NI2/parenleftbigg\n2λ2\n0\nλ1β/V\nE↓−E↑/parenrightbigg2\n,(72)11\nwhich can be evaluated as 10−2. Similarly, we get for the\nmediation through the light hole state\nNΓ(2)\np=1/2/Γ(1)∼N/parenleftbigg3λ′\n1\n4λ1β/V\n∆lh/parenrightbigg2\n,(73)\nwhere a much larger energy offset, ∆ lh, is partially com-\npensated by a largermatrix element. Note that the num-\nber of pairs available for a flip is of the order of Ntimes\nlarger than the spins themselves, so that when compar-\ning the first and the second order rate, the latter should\nbe multiplied by N, what we did in the previous two\nequations.\nJust for completeness we note that for electrons a sim-\nilar relation between the first and second order rates\nholds,\nΓ(2)∼Γ(1)/parenleftbiggβ/V\nE↓−E↑/parenrightbigg2\nI2≪Γ(1),(74)\nbut the ratio is much smaller, at the external field of 1\nTesla by 10 orders of magnitude.\nWe now consider additional mechanisms of the\nphonon-impurity couplings, through which impurity spin\nrelaxation may arise.\nC. Valence band shifts\nThe phonon induced lattice compression changes the\nbandstructure– the bandsareshifted. Shifts different for\nthe bands of the particle ground and mediating excited\nstatepresult in the impurity-phonon coupling through\nthesecondtermofEq.(43), bychangingthedenominator\nby ∆VDP. As the two states have to belong to different\nbands, such a coupling may arise only for the case of\nholes and takes the form\nHi∼∆VDP\n∆lhHeff, (75)\nexpanding the effective Hamiltonian up to the lowest\norder in the band shift difference ∆ VDP=−(σhh−\nσlh)divδR, which is the difference of the deformation po-\ntentials for the heavy and light hole valence bands. We\ncan thus relate the band shift mechanism to the position\nshift one, comparingEq.(62) with Eq.(75). We find that\nthe latter is described by an effective constant α\nαeff= (σhh−σlh)σQl/∆lh. (76)\nIfweestimatethepotentialdifferencebythetypicalvalue\nof the potential itself, ( σhh−σlh)∼5 eV, which is likely\nan overestimatedvalue, we get αeff= 17, sothat the cou-\npling through band shifts leads to a rate at most com-\nparable to (and most probably much smaller than) that\ndescribed by Eq. (73).D. Renormalization of the spin-orbit length\nPhonon induced renormalization of band offsets influ-\nences the spin-orbit couplings. This is evident from the\nexpressions for the coefficients λwhich are inversely pro-\nportionaltothelight-heavyholeoffset∆ lh,seeEqs.(A9).\nEven though the coupling in the form of the constant\nαis described by the same formula as in Eq. (76), the\nsubstantial difference is that now the phonon-impurity\ncoupling arises also through the first term of the effective\nHamiltonian because fluctuations in spin-orbit fields in-\nduce fluctuations in the Knight field. The corresponding\nαeffis the one given in Eq. (76) and the relaxation rate\nis that in Eq. (67), so that it does not exceed the rate\ndue to the piezoelectric shift mechanism. For the case of\nelectrons, the effective constant follows in an analogous\nform,\nα= (σe−σh)σQlso/∆, (77)\nwhere ∆ is of the order of the conduction-valence band\noffset, which enters the definition of the spin-orbit cou-\nplings 1/lso. The numerical value for αis much less than\none even taking ( σe−σh)∼σe, so that this channel is\nnegligible with respect to the geometrical shifts.\nE. Phonon induced spin-orbit interactions\nFinally, we estimate the influence of spin-phonon cou-\npling through an additional spin-orbit interaction aris-\ning in the presence of a phonon-originated electric field.\nWe assume the new spin-orbit interaction strength re-\nlates to the one we considered in Sec. II, referred to as\n“old”, through the ratio of the internal (interface) elec-\ntric field Eintand the phonon induced electric field, the\nlatter given as the gradient of the appropriate poten-\ntial, Eqs. (54) or (55). Since the phonon-induced spin-\norbit interaction (“new”) arises from an electric field, it\nis of the Rashba functional form, and one can take its\neffects to be additive to the “old” one. For electrons,\nthis means Utot=UoldUnew≈(I+Oold)(I+Onew)≈\nI+(Oold+Onew), which amounts to additive inverse of\nthe spin-orbit lengths 1 /lso→1/lso+ 1/lph−so. By in-\nspecting Eq. (C5), we estimate the effective interaction\nto be described again by Eq. (67) with the constant\nαeff=ΞQl\neEint, (78)\nfor the piezoelectric phonon field (a much smaller defor-\nmation field corresponds to the numerator replaced by\nσQ2l). The numerator evaluates to 106V/m, which is\nnot supposed to be much higher than the internal field,\nso that again, we find that this mechanism is less im-\nportant than the one due to the piezoelectric shift. We\ncome to a similar conclusion for holes (though we do not\nshow calculations in detail here). Namely, even though\nthe phonon induced piezoelectric field is much stronger,12\nof the order of 108V/m, the spin-orbit terms in Kohn-\nLuttinger Hamiltonian areless effective in inducing light-\nheavy hole mixing than the terms we considered explic-\nitly in previous sections (see Appendix F).\nV. CONCLUSIONS\nWe have analyzed the interactions within an ensemble\nof impurity spins, which are mediated by a confined spin-\norbit coupled particle. We have considered two physical\nsystems where such mediated interactions are of great\nimportance: III-V (GaAs) electroniclateralquantum dot\nwith the impurities being nuclear spins, and II-VI (ZnTe)\nself-assembled Mn-doped quantum dot populated by a\nhole. Our focus has been on the consequences of the\nspin-orbit coupling on the character of the mediated in-\nteractions.\nWehavederivedaneffectiveHamiltonianfortheimpu-\nrity ensemble, treating the particle-impurity interaction\nperturbatively. The form of this Hamiltonian allowed us\nto quantify the degree to which the conservation of im-\npurities spin is broken in the presence of the spin-orbit\ncoupling of the particle. We have found that for the elec-\ntron case, the spin-non-conserving terms are suppressed\nrelative to the spin-conservingones by a small factor, the\nratio of the confinement length and the spin-orbit length.\nThe lowestelectronexcited stateis the mosteffective me-\ndiator, what results in a decoherence being removable by\nthe electron spin echo even in the presence of the spin-\norbit coupling. In the case of holes, the spin-conserving\ninteractions are most efficiently mediated by the lowest\nlight hole state, while the spin-non-conserving ones by\nthe lowest heavy hole state. The induced decoherence is\nthen not removable by a hole spin echo anymore.\nAs a direct application of the derived effective Hamil-\ntonian, we have calculated the rates of a phonon-assisted\nimpurity spin relaxation, which arises only in the pres-\nence of the spin-orbit coupling in the particle Hamilto-\nnian. We have considered several coupling mechanisms,\nby which impurity spins couple to phonons. We have\nfound that the most effective is the piezoelectric field in-\nduced shift of the particle wavefunction, with a typical\nrelaxation time of 1 µs in a 10 nm self-assembled strain-\nfree quantum dot. The rate grows upon making the dot\nsmaller, or diminishing the heavy-light hole splitting by\nstrain, possibly to nano seconds for reasonable dot pa-\nrameters.\nWhile we have focused on a single particle, our anal-\nysis of the spin-non-conserving mechanisms already pro-\nvides insights in the magnetic ordering in quantum dots\nwith multiple occupancy. For example, the spin-non-\nconserving mechanisms are the key in understanding the\nprediction of piezomagnetic quantum dots or, equiva-\nlently, nonlinear magneto-electric effects.71Changes in\nthe shape of lateral confinement (from circular to ellip-\ntical) controlled by the pair of the gate electrodes have\nbeen demonstrated experimentally to alter the particleconfiguration from vanishing to finite spin in nonmag-\nnetic quantum dots.72This principle provides intrigu-\ning possibilities for the control of magnetic ordering in\ndots with added Mn impurities where such changes in\nthe particle spin, through exchange interaction, would\nreversibly control the magnetic ordering of the nearby\nMn spins.71The characteristic time scale for the related\nmagneticpolaronformation, similartothe better studied\nquantum well structures, should be largely determined\nby the anisotropic spin-spin interactions of impurities\nwhich explicitly do not conserve the total spin of Mn\natoms.41,51,73,74\nBeyond epitaxial dots that we have presently exam-\nined, recent experimental advances in colloidal quan-\ntum dots warrant also future considerations. Typi-\ncally they are easily synthesized II-VI materials, such\nas ZnTe, ZnSe, CdS, and CdSe,75,76which offer a large\nsize-inducedtunabilityofthetransitionenergiesandlong\nspindecoherencetimes.77Magneticdoping78ofthesecol-\nloidal dots provide an opportunity for a versatile control\nof magnetic order as well as lead to robust magnetic po-\nlaron formation with effective internal magnetic field ap-\nproaching 100 T.41,78–82\nVI. ACKNOWLEDGEMENTS\nWe would like to thank Rafal Oswaldowski for many\ndiscussions, which substantially contributed to this work.\nP.S. would also like to acknowledge useful discussions\nwith Uli Zuelicke. This work was supported by EU\nprojectQ-essence,meta-QUTEITMSNFP26240120022,\nCE SAS QUTE, SCIEX, DOE-BES, ONR, and DFG\nSFB 689.\nAppendix A: Kohn-Luttinger Hamiltonian\nperturbative eigenstates\nHere we derive hole eigenstates in the lowest order per-\nturbation theory. We neglect the influence ofthe conduc-\ntion and spin-orbit split-off subbands and consider only\nthe light ( J=±1/2) and heavy ( J=±3/2) holes in\nthe Kohn-Luttinger Hamiltonian,83HJJ′. The diagonal\nelements are\nH±3/2,±3/2=−/planckover2pi12\n2m0/bracketleftbig\nk2\nz(γ1−2γ2)+(k2\nx+k2\ny)(γ1+γ2)/bracketrightbig\n,\nH±1/2,±1/2=−/planckover2pi12\n2m0/bracketleftbig\nk2\nz(γ1+2γ2)+(k2\nx+k2\ny)(γ1−γ2)/bracketrightbig\n,\n(A1)\nwith/planckover2pi1k=Pthe hole momentum operator, m0the free\nelectron mass, and γ1,γ2, andγ3(below) the Luttinger\nparameters. Together with the in-plane V(r) and het-\nerostructure Vz(z) confinement potentials, assumedto be\nthose in Eq. (2), the kinetic terms in Eq. (A1) define the13\ndot unperturbed eigenstates (normalization omitted)\nΦJ\nnm,k=r|m|e−r2/2l2\nJL|m|\nn(r2/l2\nJ)eimφsin(kπz/w).(A2)\nWe usedcylindricalcoordinates( r,φ,z),Lm\nnarethe asso-\nciated Laguerre polynomials, lJthe in-plane confinement\nlength(whichdiffersforheavyandlightholesduetotheir\ndifferent masses) and wthe heterostructure width. The\nFock-Darwin states are labelled by the principal and or-\nbital quantum numbers nandm, andklabels excitations\nin the perpendicular potential. The corresponding ener-\ngies are\nEJ,nm,k=/planckover2pi1ΩJ(2n+|m|+1)+/planckover2pi12\n2mJw2π2k2,(A3)\nwhere the in-plane excitation energy is parameterized by\nthe mass and confinement length,\n/planckover2pi1ΩJ=/planckover2pi12\nmJl2\nJ. (A4)\nThe in-plane masses are given by Eq. (A1): m±3/2≡\nmhh=m0/(γ1+γ2),m±1/2≡mlh=m0/(γ1−γ2). We\nparameterize the in-plane electrostatic potential choos-\ning a certain value for the heavy hole in-plane excitation\nenergyE3/2,01,0−E3/2,00,0=/planckover2pi1Ωhh, which then specifies\nthe confinement lengths. The light holeexcitation energy\nand confinement length\nΩlh= Ωhh(mhh/mlh)1/2, l lh=lhh(mhh/mlh)1/4,\n(A5)\ndiffer from the corresponding heavy hole quantities due\nto a different in-plane mass. The energies of the hard-\nwall eigenstates also differ for heavy and light holes due\nto different out-of-planemasses, which are m0/(γ1+2γ2),\nandm0/(γ1−2γ2), respectively. We set wby choosing a\ncertain value for the light-heavy hole splitting E3/2,00,0−\nE1/2,00,0= ∆lh.\nTaking the above eigenstates as the basis, we now per-\nturbatively take into account the off-diagonalelements of\nthe Hamiltonian,\nH±3/2,±1/2=±/planckover2pi12\n2m02√\n3γ3k∓kz,\nH±3/2,∓1/2=/planckover2pi12\n2m0√\n3/bracketleftbig\nγ2(k2\nx−k2\ny)∓2iγ3kxky/bracketrightbig\n,\nH±3/2,∓3/2= 0 =H±1/2,∓1/2.(A6)\nWe employ the non-degenerate perturbation theory\n|ΨJa∝an}b∇acket∇i}ht=|J∝an}b∇acket∇i}ht⊗|ΦJ\na∝an}b∇acket∇i}ht+/summationdisplay\nJ′a′/negationslash=Ja∝an}b∇acketle{tΦJ′\na′|HJ′J|ΦJ\na∝an}b∇acket∇i}ht\nEJa−EJ′a′|J′∝an}b∇acket∇i}ht⊗|ΦJ′\na′∝an}b∇acket∇i}ht,\n(A7)\nwhere we use the notation introduced below Eq. (4), so\nthataincludes two quantum numbers of the in-plane\nFock-Darwin state and one of the perpendicular hard-\nwallstate. Toproceed,weneglecthighenergyexcitations\nn >0 andk >1 and adopt the axial approximation\nH±3/2,∓1/2≈/planckover2pi12\n2m0√\n3γk2\n∓, (A8)withγ= (γ2+γ3)/2. With these simplifications, the\notherwiseinfinite sum for |ΨJa∝an}b∇acket∇i}htsimplifies to only a single\nterm for each J′∝ne}ationslash=Jand can be given explicitly.84,85We\nfinally get Eq. (9) with the admixtures\nλ1=/planckover2pi12\nm0llhwγ3√\n3κξ\n∆∗+/planckover2pi1Ωlh, (A9a)\nλ0=/planckover2pi12\nm0l2\nlhγ/radicalbig\n3/2κ\n∆lh+2/planckover2pi1Ωlh, (A9b)\nwhere ∆∗=E1/2,00,1−E3/2,00,0is the z-excited light\nhole offset and\nκ=∝an}b∇acketle{tΦhh\n00,0|Φlh\n00,0∝an}b∇acket∇i}ht=2\n(mhh/mlh)1/4+(mlh/mhh)1/4,\n(A10)\nis the ground state overlap, which differs from one due to\nthe in-plane mass difference. Finally, the dimensionless\nmatrix element ξis defined by ξ=−iw∝an}b∇acketle{t1|kz|0∝an}b∇acket∇i}ht= 8/3.\nTaking a heavy hole 3/2, the admixture of 1/2 light\nhole scales as 1 /wl, costs the in-plane plus perpendicular\norbital energy (the latter is even larger than the light-\nheavy hole offset) and leads to an admixture with a very\ndifferent z-profile (compared to the main wavefunction\ncomponent). The admixture of the -1/2 light hole has a\nsmaller numerator, proportional to 1 /l2, but costs only\nthe in-plane orbital energy (several times smaller than\nthe light-heavy hole offset) and has the same z-profile as\nthe main component.\nAlong the same lines we get Eq. (11) with\nλ′\n1=/planckover2pi12\nm0lhhwγ3√\n3κξ\n∆∗′+/planckover2pi1Ωhh,(A11a)\nλ′\n0=/planckover2pi12\nm0l2\nhhγ/radicalbig\n3/2κ\n∆lh−2/planckover2pi1Ωhh,(A11b)\nwhere ∆∗′=E3/2,00,1−E1/2,00,0is the z-excited heavy\nhole offset.\nFor completeness, we list the hole Zeeman term86\nHhZ= 2κµBJ·B+2qµB/summationdisplay\ni=x,y,zJ3\niBi,(A12)\nwhich can be written for the heavy hole subspace as\nHhh,Z=ghhµB[J/3]z·Bz, (A13)\nwithJ/3≡σ/2 the pseudo spin operator and ghh≡\n6κ+27q/2≈2 for GaAs.85\nAppendix B: Spin matrix elements for holes\nHere we calculate the matrix elements of the spin op-\nerator between perturbative eigenstates of the hole. For\nthe purposes of this appendix, we shorten the expres-\nsion in Eq. (9) introducing Φ = Φhh\n00,0,a=λ1Φlh\n01,1and\nb=λ0Φlh\n02,0to\n|Ψ3/2∝an}b∇acket∇i}ht= Φ|3/2∝an}b∇acket∇i}ht+a|1/2∝an}b∇acket∇i}ht+b|−1/2∝an}b∇acket∇i}ht,(B1)\n|Ψ−3/2∝an}b∇acket∇i}ht= Φ∗|−3/2∝an}b∇acket∇i}ht−a∗|−1/2∝an}b∇acket∇i}ht+b∗|1/2∝an}b∇acket∇i}ht.(B2)14\nWe denote J±=Jx±iJy, so that Jx= (J++J−)/2\nandJy= (J+−J−)/2i and since the orbital operator in\nall the matrix elements is the delta function δ(R−Rn),\nall the complex amplitudes below should be evaluated at\nthe position of the particular impurity, e.g., Φ →Φ(Rn).\nListing only the leading order in small quantities λ0,1, to\nwhichaandbare proportional, we have\n∝an}b∇acketle{tΨ3/2|Jz|Ψ3/2∝an}b∇acket∇i}ht= (3/2)|Φ|2+O(λ2),(B3a)\n∝an}b∇acketle{tΨ3/2|J+|Ψ3/2∝an}b∇acket∇i}ht=√\n3Φ∗a+O(λ2),(B3b)\n∝an}b∇acketle{tΨ3/2|J−|Ψ3/2∝an}b∇acket∇i}ht=√\n3Φa∗+O(λ2),(B3c)\nfrom where Eq. (19) follows directly. The time reversal\nsymmetry gives (using ∝an}b∇acketle{tTa|b∝an}b∇acket∇i}ht=−∝an}b∇acketle{ta|Tb∝an}b∇acket∇i}ht∗, andT2=−1)\n∝an}b∇acketle{tΨ−3/2|J|Ψ−3/2∝an}b∇acket∇i}ht=−∝an}b∇acketle{tΨ3/2|J|Ψ3/2∝an}b∇acket∇i}ht,(B4)\nso that the spin expectation value changes sign upon in-\nverting the hole spin. To evaluate the off-diagonal ele-\nment of the Overhauser field and the vectors Awe need\n∝an}b∇acketle{tΨ−3/2|Jz|Ψ3/2∝an}b∇acket∇i}ht=ab, (B5a)\n∝an}b∇acketle{tΨ−3/2|J+|Ψ3/2∝an}b∇acket∇i}ht= 2b2, (B5b)\n∝an}b∇acketle{tΨ−3/2|J−|Ψ3/2∝an}b∇acket∇i}ht= 2√\n3Φb+O(λ2),(B5c)\nfrom where Eq. (30) follows.\nAnalogously we define short hand notations for the\nlight hole like wavefunctions as Φ′= Φlh\n00,0,a′=λ′\n1Φhh\n01,1,\nandb′=λ′\n0Φhh\n02,0, to write\n|Ψ1/2∝an}b∇acket∇i}ht= Φ′|1/2∝an}b∇acket∇i}ht+a′|3/2∝an}b∇acket∇i}ht−b′|−3/2∝an}b∇acket∇i}ht,(B6)\n|Ψ−1/2∝an}b∇acket∇i}ht= Φ′∗|−1/2∝an}b∇acket∇i}ht−a′∗|−3/2∝an}b∇acket∇i}ht−b′∗|3/2∝an}b∇acket∇i}ht.(B7)\nThe light hole mediated effective interaction is given by\n∝an}b∇acketle{tΨ3/2|Jz|Ψ1/2∝an}b∇acket∇i}ht= (3/2)Φ∗a′+(1/2)a∗Φ′,(B8a)\n∝an}b∇acketle{tΨ3/2|J+|Ψ1/2∝an}b∇acket∇i}ht=√\n3Φ∗Φ′+O(λ2),(B8b)\n∝an}b∇acketle{tΨ3/2|J−|Ψ1/2∝an}b∇acket∇i}ht= 2Φ′b∗+O(λ2). (B8c)\nFinally, the couplingthroughthe spin oppositelight hole\nstate are given by\n∝an}b∇acketle{tΨ3/2|Jz|Ψ−1/2∝an}b∇acket∇i}ht=−(3/2)Φ∗b′∗−(1/2)b∗Φ∗′,(B9a)\n∝an}b∇acketle{tΨ3/2|J+|Ψ−1/2∝an}b∇acket∇i}ht= 2a∗Φ′∗+O(λ2), (B9b)\n∝an}b∇acketle{tΨ3/2|J−|Ψ−1/2∝an}b∇acket∇i}ht=−√\n3b′∗a∗. (B9c)\nAppendix C: Knight field and rotated coordinates\nIn this appendix, we evaluate the Knight field, its spa-\ntial derivative and the corresponding locally rotated co-\nordinate frame, for the case of electrons and holes.1. Electron case\nThe Knight field of an electron in the ground state is\nK=−(β/2)|ΦG(R)|2∝an}b∇acketle{ts(R)∝an}b∇acket∇i}ht, (C1)\nsee Eq. (18). In deriving that, we used the identity\nUJU†=R−1\nU[J], (C2)\nwhere the unitary operator of spinor rotation\nU≡exp(−in·Jφ), (C3)\ncorresponds to a three dimensional rotation around the\nunit vector nby angle φ,\nRU= exp(−in·lφ), (C4)\nwith (lk)mn=−iǫkmn.\nThe spatial derivative of the Knight field, which gives\nthe matrix elements for the impurity spin flips in the\n”first order“ rates, follows from Eq. (C1) as\n∇K=/bracketleftbig\n∇ln(|ΦG(R)|2)/bracketrightbig\nK+(∇nso)×K.(C5)\nThe two terms correspond, respectively, to the position\nchange in the magnitude and direction of vector K.\nFor electrons we consider a regime in which the total\nfield is dominated by the external field. The impurity lo-\ncal coordinate system then coincides with the coordinate\nframe defined by the external magnetic field, RBn≈1\nandˆ z′=s0. This allows us to estimate the components\nof∇Ktransversal to the local coordinate system as\nr′±·∇K∼δ−1(l/lso)K, (C6)\nwhich originate in the first term of Eq. (C5) and where\nthe length δ=lfor a phonon with in-plane polarization\nvector (e⊥ˆ z) andδ=w/πfor a phonon with a po-\nlarization vector along the growth direction ( e||ˆ z). The\nsecond term of Eq. (C5) gives a contribution at most as\nlargeasthe first term, orlower, depending onthe phonon\npolarization.\n2. Holes\nTheKnightfieldofaspin3/2hole, defined byEq.(17),\nKn=−β∝an}b∇acketle{tΨ3/2|δ(R−Rn)J|Ψ3/2∝an}b∇acket∇i}ht,(C7)\nfollows from Eqs. (B3) as\n[Kn\nx,Kn\ny,Kn\nz] =−β[√\n3Re(aΦ∗),√\n3Im(aΦ∗),3/2|Φ|2],\n(C8)\ngiveninthemaintextinEq.(19), hereusingthenotation\nfrom Appendix B, a=λ1Φlh\n01,1(Rn) and Φ = Φhh\n00,0(Rn).\nSince for holes we are interested in the zero magnetic\nfield case (by which Φ is real), the local coordinate frame15\nis defined by a unit vector along the Knight field, ˆ z′=\nK/K,\nˆ z′= [sinφRe(a)/|a|,sinφIm(a)/|a|,cosφ].(C9)\nHereφis the angle between the original and the rotated\nz axes, cos φ=ˆ z·ˆ z′= Φ//radicalbig\n3Φ2/4+|a|2. We choose\nthe remaining two axes of the rotated coordinate system\narbitrarily as\nˆ y′= [−Im(a)/|a|,Re(a)/|a|,0],(C10)\nand\nˆ x′=ˆ y′׈ z′= [cosφRe(a)/|a|,cosφIm(a)/|a|,−sinφ].\n(C11)\nUsing the projectors into the transversal plane of the lo-\ncal coordinate system, r′±, one can evaluate the ampli-\ntudes of the spin-non-conserving terms. As an auxiliary\nresult, we note that for any real vector vwe have\nr′±·v=v+a∗\n|a|cosφ±1\n2+v−a\n|a|cosφ∓1\n2−vzsinφ.\n(C12)\nFor example, close to the dot center, the inequality Φ ≫\n|a|gives sin φ≈ |a|/√\n3/2Φ, cosφ≈1,|a| ∼λ1Φ, and\nthe Knight field derivative follows as\nr′\n±·∇K∼δ−1√\n3λ1β|Φhh\n00,0|2,(C13)\nwhere, again, the length δdepends on the direction of\nthe phonon polarization vector, δ=lforeQ⊥ˆ z, and\nδ=w/πforeQ||ˆ z.\nAppendix D: Interactions in the impurity ensemble:\nvectors A and the effective Hamiltonian terms\nThesecondorderHamiltonianforagivenimpuritypair\nn,mequals 1/(EG−Ep) times the following expression\n(An·In)(Am·Im)†+(Am·Im)(An·In)†=\nIn\nzIm\nz(An\nzAm∗\nz+Am\nzAn∗\nz)+\nIn\n+Im\n−(An\n−Am∗\n−+Am\n+An∗\n+)/4+\nIn\n−Im\n+(An\n+Am∗\n++Am\n−An∗\n−)/4+\nIn\nzIm\n+(An\nzAm∗\n++Am\n−An∗\nz)/2+\nIn\n+Im\nz(An\n−Am∗\nz+Am\nzAn∗\n+)/2+\nIn\nzIm\n−(An\nzAm∗\n−+Am\n+An∗\nz)/2+\nIn\n−Im\nz(An\n+Am∗\nz+Am\nzAn∗\n−)/2+\nIn\n+Im\n+(An\n−Am∗\n++Am\n−An∗\n+)/4+\nIn\n−Im\n−(An\n+Am∗\n−+Am\n+An∗\n−)/4.(D1)\nIf the vectors are expressed in the local coordinate frame\n[that is, all quantities in Eq. (D1) with tildes], the first\nthree terms are spin preserving (connect states with the\nsum of spin projections along the local spin quantizationaxes), the next four terms change the sum by one (repre-\nsenting a single spin flip), and the last two terms induce\ndouble flips. The complex conjugates are defined as\nAm∗\n±≡(r′\n±·Am)∗=Am∗\nx∓iAm∗\ny=r′\n∓·(Am)∗.(D2)\nTo find the effective Hamiltonian, it remains to eval-\nuate the vectors A. For electrons, we assume that the\nmagnetic field dominates the total field for impurities.\nEquation (25) gives ( J= 1/2,J′=−1/2)\nAn=−(β/2)|ΦG(Rn)|2RUn[r′−],(D3)\nwhere we remind that the vectors ˆ x′,ˆ y′, andˆ s0=ˆ z′\nform an orthonormalset, with ˆ s0alongthe external field.\nExpanding the rotation operator in the lowest order in\nthe spin-orbit length we finally find\nAn\n−∼ −(β/2)|ΦG(Rn)|2, An\n+,An\nz∼O(rn/l)An\n−,(D4)\nfrom where Eq. (50) of the main text follows.\nFor holes, we find the vectors Acorresponding to the\ninteraction mediated by the -3/2 state from Eqs. (B5)\nand Eq. (C12) as\nr′+·A=−βb2(a/|a|)(cosφ+1),\nr′\n−·A=−√\n3βΦb(a/|a|)(cosφ+1),\nˆ z′·A=−√\n3βΦb(a/|a|)cosφ.(D5)\nThe largest spin-non-conserving term of the correspond-\ning effective Hamiltonian is\nHnm\n++∼˜In\n+˜Im\n+/parenleftbigg1\nEG−Ep2√\n3(βΦ2)2λ2\n0/parenrightbigg\n.(D6)\nThe vectors Afor the spin alike light hole, using\nEqs. (B8) follow as\nr′+·A=−√\n3βΦ∗Φ′(a∗/|a|)(cosφ+1)/2,\nr′−·A=−βΦ′b∗(cosφ+1),\nˆ z′·A=−(β/2)(3a′Φ∗+a∗Φ′)cosφ.(D7)\nFrom the above results, we see that the largest spin-\nnon-conserving terms in the effective interaction are\nHnm\n++∼˜In\n+˜Im\n+/parenleftbigg1\n∆lh√\n3(βΦ2)2λ0/parenrightbigg\n,(D8)\nand\nHnm\n+z∼˜In\n+˜Im\nz/parenleftbigg1\n∆lh(3/4)√\n3(βΦ2)2λ′\n1/parenrightbigg\n.(D9)\nSimilarly as before, a differentiation with respect to the\nposition, which enters the impurity-phonon rates, brings\nin an additional factor 1 /δin Eqs. (D6)-(D9).16\nquantity electron hole\nV 3×104nm3238 nm3\nN 1.3×106xMn×5278\nβ/V 0.13 neV 64 µeV\nTABLE II: Effective volume, number of impurities and the\ncoupling energy scale.\nAppendix E: Materials parameters\nForthe electroniccaseweassumeaGaAs/AlGaAshet-\nerostructure with the following parameters: electron ef-\nfective mass m= 0.067m0, in-plane confinement length\nl= 30 nm, quantum well width w= 8 nm, spin-orbit\nlengthlso∼1µm, electron-nuclear coupling β= 4µeV\nnm3, material density ρ= 5300 kg/m3, phonon veloci-\ntiescl= 5290 m/s and ct= 2480 m/s, conduction band\npiezoelectricΞ = 1 .4×109eV/m and deformation σ= 10\neV potentials. The g-factors and corresponding energy\nscales are given in Table III.\nFor the hole case, we list the Luttinger parameters\nγ1/γ2/γ3of GaAs:87,887.1/2/2.9, CdTe:894.1/1.1/1.6,\nand ZnTe:903.8/0.7/1.3. We take ZnTe as the mate-\nrial of our choice, with ρ= 5650 kg/m3,cl= 3550 m/s,\nct= 2358 m/s, σ= 5 eV, Ξ = 3 .4×108eV/m. We\nset the heavy hole orbital energy /planckover2pi1Ωhhto 20 meV, which\ngiveslhh= 4.19 nm,llh= 3.82 nm and /planckover2pi1Ωlh= 17 meV.\nWe set the light-heavy hole splitting ∆ lhto 100 meV,\nwhich gives w= 3.24 nm. The hole-impurity interaction\nstrength is β= 1/3 eVa3\n0/4, witha0= 0.61 nm the\nlattice constant. We assume the Mn impurities concen-\ntration is given as xMn, the ratio of cation replaced by\nMn atoms, typically of the order of 1%.\nProperly normalized, the ground state follows from\nEq. (A7) as\nΦJ\n00,0(r,φ,z) =/radicalbigg\n2\nwsin/parenleftBigπz\nw/parenrightBig1√πlJexp/parenleftbigg\n−r2\n2l2\nJ/parenrightbigg\n,\n(E1)\nfrom where the quantum dot volume estimate\nV= 1//integraldisplay\nd3R|Φ(R)|4= (4π/3)wl2\nJ,(E2)\ngives values in Table II.\nFrom the above parameters, the hole admixture coef-\nficients follow as λ0≈0.053,λ1≈0.050,λ′\n0≈0.11, and\nλ′\n1≈0.15.\nAppendix F: Alternative hole mixing mechanisms\nIn this section we estimate two additional light-heavy\nholeadmixturesources,namelythespin-orbitcouplingto\nan external electric field (the Rashba spin-orbit interac-\ntion) and Overhauser impurity field itself. We quantify\nthe resulting light-heavy hole admixture by calculatingquantity nucleus Mn electron hole\ng 1.2 2 -0.44 -2/3\nµ µN µB µB µB\nIorJ 3/2 5/2 1/2 3/2\ngµI 57 neV/T 290 µeV/T 12.7 µeV/T 58 µeV/T\ngµI/k B 660µK/T 3.4 K/T 146 mK/T 670 mK/T\nBeff 66 peV 96 µeV 290 neV 13.8 x1/2\nMnmeV\nBeff/(gµI)1.2 mT 331 mT 23 mT 237 x1/2\nMnT\nTABLE III: g-factor, magnetic moment, spin, the correspond -\ning energy scale and the effective field Beff. For GaAs nuclear\nspins, the g-factor is the average over naturally abundant i so-\ntopes. The effective field for impurities is the Knight field,\nBeff=J(β/V), for the particle it is the Overhauser field,\nBeff=/radicalbig\nNI(I+ 1)(β/V).\nthe corresponding coefficients λ. We find that these al-\nternatives lead to negligible amount of admixture, com-\npared to the terms of the Kohn-Luttinger Hamiltonian\nwe considered in the main text.\nIn the presence of electric field Ethere appear the fol-\nlowing term in the hole Hamiltonian (in the notation of\nRef. 83)\nHr\n8v8v=r8v8v\n41E·J×k. (F1)\nAssuming, for convenience, that the field is perpendic-\nular to the heterostructure interface E=ˆ zEthis term\ntranslates into our notation of Sec. A as\nHso\n±3/2,±1/2=±r8v8v\n41E√\n3k∓/2, (F2)\nwhere the material parameter r8v8v\n41we estimate by its\nZnSe value of −4.1 e˚A2. Taking Eq. (A7) for the heavy\nhole ground state |Ψ3/2∝an}b∇acket∇i}ht, the Rashba spin-orbit inter-\naction leads to an admixture of the light hole state\n|1/2∝an}b∇acket∇i}ht⊗|Φlh\n01,0∝an}b∇acket∇i}htwith a coefficient\nλso=−i\n2√\n3r8v8v\n41E\n∆lh+/planckover2pi1Ωlh. (F3)\nTo quantify its value, it remains put in the value for the\nelectric field E. We estimate it to be of order 107V/m,\nusing a very crude electrostatic model; namely we take\nthe hole to be a point chargein the centerof ahalf sphere\nof a radius l∼5 nm and the electron to be a uniform\nclassicalchargedensity σonthe half-sphere, which corre-\nsponds to the type II-quantum dot populated by a single\nexciton. The resulting internal field E=σ/2ǫis smaller\nthan the piezoelectric field accompanying the phonon,\nEq. (78), which we estimated to be of order 108V/m.\nInsertingthis lattervalue for E, we find λso≈√3×10−3,\nso that both the internal electric field as well as phonon\ninduced fields induce light-heavy hole admixtures much\nsmaller than those we considered in the main text.\nOne may also consider the collective field of the impu-\nrities (the Overhauserfield) as a sourceofthe light-heavy17\nhole mixing. Using Eqs. (B8) we estimate the arising ad-\nmixture of the light hole state |1/2∝an}b∇acket∇i}ht ⊗ |Φlh\n00,0∝an}b∇acket∇i}htinto the\nheavy hole ground state\n|λMn|2≈(β/V)2NI(I+1)\n2∆lh. (F4)\nThe previous is a typical value, the phase being fixed bythe microscopic state of the Mn ensemble. Evaluating\nfor our parameters we get λMn≈7×10−3, typically an\norder of magnitude smaller admixture as that consider\nin the main text, so we can again neglect this admixture\nsource.\n1D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n2R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,\nand L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217\n(2007).\n3J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M.\nMarcus, and M. D. Lukin, Phys. Rev. B 76, 035315 (2007).\n4F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink,\nK. C. Nowack, T. Meunier, L. P. Kouwenhoven, and\nL. M. K. Vandersypen, Nature 442, 766 (2006).\n5K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and\nL. M. K. Vandersypen, Science 318, 1430 (2007).\n6T. Obata, M. Pioro-Ladriere, Y. Tokura, Y.-S. Shin,\nT. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Phys\nRev B81, 085317 (2010).\n7D. Press, T. D. Ladd, B. Zhang, and Y. Yamamoto, Nature\n456, 218 (2008).\n8J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and\nI.ˇZuti´ c, Acta Phys. Slov. 57, 565 (2007), arXiv:0711.1461.\n9I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n10W. A. Coish and J. Baugh, Phys. Stat. Sol. B 246, 2203\n(2009).\n11S. I. Erlingsson, Y. V. Nazarov, and V. I. Fa ˇlko, Phys. Rev.\nB64, 195306 (2001).\n12A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett.\n88, 186802 (2002).\n13I. A. Merkulov, A. L. Efros, and M. Rosen, Phys. Rev. B\n65, 205309 (2002).\n14W. A. Coish, V. N. Golovach, and J. C. Egues, Phys. Stat.\nSol. B243, 3658 (2006).\n15L. Cywinski, W. M. Witzel, and S. Das Sarma, Phys. Rev.\nB79, 245314 (2009).\n16L. Cywinski, W. M. Witzel, and S. Das Sarma, Phys. Rev.\nLett.102, 057601 (2009).\n17I. A. Merkulov, G. Alvarez, D. R. Yakovlev, and T. C.\nSchulthess, Phys. Rev. B 81, 115107 (2010).\n18J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science 309, 2180 (2005).\n19F. H. L. Koppens, K. C. Nowack, and L. M. K. Vander-\nsypen, Phys. Rev. Lett. 100, 236802 (2008).\n20H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu,\nV. Umansky, and A. Yacoby, Nature Phys. 7, 109 (2010).\n21J. Schliemann, A. Khaetskii, and D. Loss, J. Phys: Con-\ndens. Matter 15, R1809 (2003).\n22D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus, M. P.\nHanson, and A. C. Gossard, Science 321, 5890 (2008).\n23A. S. Bracker, E. A. Stinaff, D. Gammon, M. E. Ware, J. G.\nTischler, A. Shabaev, A. L. Efros, D. Park, D. Gershoni,\nV. L. Korenev, et al., Phys. Rev. Lett. 94, 047402 (2005).24C. W. Lai, P. Maletinsky, A. Badolato, and A. Imamoglu,\nPhys. Rev. Lett. 96, 167403 (2006).\n25A. Hgele, M. Kroner, C. Latta, M. Claassen, I. Carusotto,\nC. Bulutay, and A. Imamoglu, arxiv:1110.5524 (unpub-\nlished).\n26J. Seufert, G. Bacher, M. Scheibner, A. Forchel, S. Lee,\nM. Dobrowolska, and J. K. Furdyna, Phys. Rev. Lett. 88,\n027402 (2001).\n27L. Besombes, Y. L´ eger, L. Maingault, D. Ferrand, H. Ma-\nriette, and J. Cibert, Phys. Rev. Lett. 93, 207403 (2004).\n28F. Xiu, Y. Wang, J. Kim, Y. Zhou, X. Kou, W. Han, R. K.\nKawakami, J. Zou, and K. L. Wang, ACS Nano. 4, 4948\n(2010).\n29L. Klopotowski, L. Cywi´ nski, P. Wojnar, V. Volio-\ntis, K. Fronc, T. Kazimierczuk, A. Golnik, M. Ravaro,\nR. Grousson, G. Karczewski, et al., Phys. Rev. B 83,\n081306(R) (2011).\n30A. A. Maksimov, G. Bacher, A. McDonald, V. D. Ku-\nlakovskii, A. Forchel, C. R. Becker, G. Landwehr, and\nL. W. Molenkamp, Phys. Rev. B 62, R7767 (2000).\n31R. Beaulac, L. Schneider, P. I. Archer, G. Bacher, and\nD. R. Gamelin, Science 325, 973 (2009).\n32I. R. Sellers, R. Oszwaldowski, V. R. Whiteside, M. Egin-\nligil, A. Petrou, I. ˇZuti´ c, W.-C. Chou, W. C. Fan,\nA. G.Petukhov, S. J. Kim, et al., Phys. Rev. B 82, 195320\n(2010).\n33A. O. Govorov, Phys. Rev. B 72, 075359 (2005).\n34A. O. Govorov, C.R. Physique 9, 857 (2008).\n35J. Fern´ andez-Rossier and L. Brey, Phys. rev. Lett. 93,\n117201 (2004).\n36F. Qu and P. Hawrylak, Phys. Rev. Lett. 96, 157201\n(2006).\n37N. T. T. Nguyen and F. M. Peeters, Phys. Rev. B 78,\n045321 (2008).\n38R. M. Abolfath, P. Hawrylak, and I. ˇZuti´ c, Phys. Rev.\nLett.98, 207203 (2007).\n39R. Oszwaldowski, I. ˇZuti´ c, and A. G. Petukhov, Phys. Rev.\nLett.106, 177201 (2011).\n40N. Lebedeva, A. Varpula, S. Novikov, and P. Kuivalainen,\nPhys. Rev. B 81, 235307 (2010).\n41D. R. Yakovlev and W. Ossau (Introduction to the Physics\nof Diluted Magnetic Semiconductors edited by J. Kossut\nand J. A. Gaj, Springer, Berlin, 2010).\n42T. Dietl and J. Spalek, Phys. Rev. B 28, 1548 (1983).\n43P. A. Wolff (Semiconductors and Semimetals edited by J.\nK. Furdyna and J. Kossut (Academic Press, San Diego),\nVol. 25, 1988).\n44E. L. Nagaev (Physics of Magnetic Semiconductors (MIR\nPublishers, Moscow), 1983).\n45J. K. Furdyna, J. Appl. Phys. 64, R29 (1988).\n46S. Amasha, K. MacLean, I. P. Radu, D. M. Zumb¨ uhl, M. A.18\nKastner, M. P. Hanson, and A. C. Gossard, Phys. Rev.\nLett.100, 046803 (2008).\n47D. Paget, G. Lampel, B. Sapoval, and V. I. Safarov, Phys.\nRev. B15, 5780 (1977).\n48M. C. Kuo, J. S. Hsu, J. L. Shen, K. C. Chiu, W. C. Fan,\nY. C. Lin, C. H. Chia, W. C. Chou, M. Yasar, R. Mallory,\net al., Appl. Phys. Lett. 89, 263111 (2006).\n49This is further corroborated by applying ZnTe paramaters\nto the theoretical approach from Ref. 91.\n50R. deSousa, in: M. Fanciulli (Ed.), Topics Appl. Physics\n115, 183 (2009).\n51T. Dietl, P. Peyla, W. Grieshaber, and M. d ´Aubign, J.\nMagn. Magn. Mater. 140-144 , 2051 (1995).\n52J. M. Elzerman, R. Hanson, L. H. W. van Beveren,\nB. Witkamp, L. M. K. Vandersypen, and L. P. Kouwen-\nhoven, Nature 430, 431 (2004).\n53I. L. Aleiner and V. I. Fa ˇlko, Phys. Rev. Lett. 87, 256801\n(2001).\n54L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324\n(2003).\n55D. Stepanenko, N. E. Bonesteel, D. P. DiVincenzo,\nG. Burkard, and D. Loss, Phys. Rev. B 68, 115306 (2003).\n56J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett.\n90, 146801 (2003).\n57I. V. Tokatly and E. Y. Sherman, Annals of Physics 325,\n1104 (2010).\n58I. V. Tokatly and E. Y. Sherman, Phys. Rev. B 82, 161305\n(2010).\n59P. Stano and J. Fabian, Phys. Rev. Lett. 96, 186602 (2006).\n60F. Baruffa, P. Stano, and J. Fabian, Phys. Rev. Lett. 104,\n126401 (2010).\n61R. Oswaldowski, P. Stano, A. Pletyukov, and I. ˇZuti´ c (un-\npublished).\n62J. Fischer, W. A. Coish, D. V. Bulaev, and D. Loss, Phys.\nRev. B78, 155329 (2008).\n63A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys.\nRev. Lett. 99, 036801 (2007).\n64M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 99,\n036602 (2007).\n65S. I. Erlingsson and Y. V. Nazarov, Phys. Rev. B 66,\n155327 (2002).\n66N. Shenvi, R. de Sousa, and K. B. Whaley, Phys. Rev. B\n71, 224411 (2005).\n67W. Yao, R.-B. Liu, and L. J. Sham, Phys. Rev. B 74,\n195301 (2006).\n68P. L¨ owdin, J. Chem. Phys. 19, 1396 (1951).\n69G. L. Bir and G. E. Pikus, Symmetry and Strain-induced\nEffects in Semiconductors (Wiley/Halsted Press, 1974).\n70A. Abragam, The Principles of Nuclear Magnetism (Ox-\nford University Press, 1961).71R. M. Abolfath, A. G. Petukhov, and I. ˇZuti´ c, Phys. Rev.\nLett.101, 207202 (2008).\n72D. G. Austing, S. Sasaki, S. Tarucha, S. M. Reimann,\nM. Koskinen, and M. Manninen, Phys. Rev. B 60, 11514\n(1999).\n73F. Henneberger and J. Puls, Diluted Magnetic Quan-\ntum Dots (Introduction to the Physics of Diluted Mag-\nnetic Semiconductors edited by J. Kossut and J. A. Gaj,\nSpringer, Berlin, 2010).\n74D. R. Yakovlev and I. A. Merkulov, Spin and Energy\nTransfer Between Carriers, Magnetic Ions, and Lattice\n(Introduction to the Physics of Diluted Magnetic Semi-\nconductors edited by J. Kossut and J. A. Gaj, Springer,\nBerlin, 2010).\n75V. I. Klimov, Annu. Rev. Phys. Chem. 58, 635 (2007).\n76G. D. Scholes, Adv. Funct. Mater. 18, 1157 (2007).\n77N. P. Stern, M. Poggio, M. H. Bartl, E. L. Hu, G. D.\nStucky, and D. D. Awschalom, Phys. Rev. B 72, 161303\n(2005).\n78R. Beaulac, P. I. Archer, S. T. Ochsenbein, and D. R.\nGamelin, Adv. Funct. Mater. 18, 3873 (2008).\n79D. A. Bussian, S. A. Crooker, M. Yin, M. Brynda, A. L.\nEfros, and V. I. Klimov, Nature Mater. 8, 35 (2009).\n80S. T. Ochsenbein, Y. Feng, K. M. Whitaker, E. Badaeva,\nW. K. Liu, X. Li, and D. R. Gamelin, Nature Nanotech.\n4, 681 (2009).\n81I.ˇZuti´ c and A. G. Petukhov, Nature Nanotech. 4, 623\n(2009).\n82R. Viswanatha, J. M. Pietryga, V. I. Klimov, and S. A.\nCrooker, Phys. Rev. Lett. 107, 067402 (2011).\n83R. Winkler, SpinOrbit Coupling Effects in Two-\nDimensional Electron and Hole Systems (Springer-Verlag,\n2003).\n84A. K. Bhattacharjee, Phys. Rev. B 76, 075305 (2007).\n85J. Fischer and D. Loss, Phys. Rev. Lett. 105, 266603\n(2010).\n86G. Katsaros, V. N. Golovach, P. Spathis, N. Ares, M. Stof-\nfel, F. Fournel, O. G. Schmidt, L. I. Glazman, and S. De\nFranceschi, arxiv:1107.3919 (unpublished).\n87N. Binggeli and A. Baldereschi, Phys. Rev. B 43, 14734\n(1991).\n88B. V. Shanabrook, O. J. Glembocki, D. A. Broido, and\nW. I. Wang, Phys. Rev. B 39, 3411 (1989).\n89T. Friedrich, J. Kraus, G. Schaack, and W. O. G. Schmitt,\nJ. Phys.: Condens. Matter 6, 4307 (1994).\n90H. Wagner, S. Lankes, K. Wolf, W. Kuhn, P. Link, and\nW. Gebhardt, J. Cryst. Growth 117, 303 (1992).\n91K. Vyborny, J. E. Han, R. Oszwa/suppress ldowski, I. ˇZuti´ c, and\nA. G. Petukhov, Phys. Rev. B 85, 155312 (2012)." }, { "title": "2204.07323v2.Quantum_phases_of_spin_orbital_angular_momentum_coupled_bosonic_gases_in_optical_lattices.pdf", "content": "Quantum phases of spin-orbital-angular-momentum coupled bosonic gases in optical\nlattices\nRui Cao,1Jinsen Han,1Jianhua Wu,1,\u0003Jianmin Yuan,2, 1Lianyi He,3,yand Yongqiang Li1,z\n1Department of Physics, National University of Defense Technology, Changsha 410073, P. R. China\n2Department of Physics, Graduate School of China Academy of Engineering Physics, Beijing 100193, P. R. China\n3Department of Physics and State Key Laboratory of Low-Dimensional\nQuantum Physics, Tsinghua University, Beijing 100084, China\n(Dated: June 16, 2022)\nSpin-orbit coupling plays an important role in understanding exotic quantum phases. In this work,\nwe present a scheme to combine spin-orbital-angular-momentum (SOAM) coupling and strong corre-\nlations in ultracold atomic gases. Essential ingredients of this setting is the interplay of SOAM cou-\npling and Raman-induced spin-\rip hopping, engineered by lasers that couples di\u000berent hyper\fne spin\nstates. In the presence of SOAM coupling only, we \fnd rich quantum phases in the Mott-insulating\nregime, which support di\u000berent types of spin defects such as spin vortex and composite vortex with\nantiferromagnetic core surrounded by the outer spin vortex. Based on an e\u000bective exchange model,\nwe \fnd that these competing spin textures are a result of the interplay of Dzyaloshinskii-Moriya\nand Heisenberg exchange interactions. In the presence of both SOAM coupling and Raman-induced\nspin-\rip hopping, more many-body phases appear, including canted-antiferromagnetic and stripe\nphases. Our prediction suggests that SOAM coupling could induce rich exotic many-body phases\nin the strongly interacting regime.\nI. INTRODUCTION\nSpin-orbit coupling, the interplay of particle's spin and\norbital degrees of freedom, plays a crucial role in vari-\nous exotic phenomena in solid-state systems, such as the\nquantum spin Hall e\u000bect [1{4], topological insulators [5],\nand topological superconductors [6]. Ultracold atomic\nsystem, with high controllability degrees of freedom, is\nalso a versatile candidate to investigate these quantum\nphenomena, by overcoming the problem of their neu-\ntrality [7]. One of these schemes relies on two-photon\nRaman transitions between two hyper\fne states (pseu-\ndospin) of atoms [8], which are coupled with the atomic\ncenter-of-mass momentum [9]. Here, propagation direc-\ntions of laser beams are crucial to determine the type of\nspin-orbit coupling in ultracold atoms. When two beams\ncounter-propagate, atom's spin can be coupled with lin-\near momentum of atoms, i.e. spin-linear-momentum\ncoupling [7, 9{11]. Rich exotic quantum states have\nbeen observed in ultracold atomic gases with spin-linear-\nmomentum coupling [12{17].\nAnother fundamental type of spin-orbit coupling is\ncalled SOAM coupling. This coupling can be achieved by\na pair of copropagating Laguerre-Gaussian (LG) lasers,\nwhere LG beam modes carry di\u000berent orbital angular mo-\nmenta along the direction of beam propagation [18, 19].\nThe atomic system obtains orbital angular momentum\nfrom the copropagating LG beams via Raman transitions\namong the internal hyper\fne states of atoms, whereas\nthe transfer of photon momentum into atoms is sup-\n\u0003wujh@nudt.edu.cn\nylianyi@mail.tsinghua.edu.cn\nzliyq@nudt.edu.cnpressed [20, 21]. Within SOAM coupling, several in-\ntriguing quantum phases have been predicted theoreti-\ncally [22{33] and observed experimentally [20, 21, 34, 35].\nIn these studies, however, interactions between atoms\nplay tiny role in the various quantum phases, and one\nmainly focus on the weakly interacting regime.\nIn the paper, we combine SOAM coupling and strong\ncorrelations in ultracold gases, and focus on the response\nof spin degree of freedom to SOAM coupling. To achieve\nthis goal, we propose a setup by introducing a beam with\norbital angular momentum in the third direction ( zdirec-\ntion) for a two-component ultracold bosonic gas loaded\ninto a blue-detuned square lattice, as shown in Fig. 1. By\ncontrolling the frequency di\u000berence between the standing\nwave in the xdirection and the Raman beam in the zdi-\nrection, the two hyper\fne states that match the Raman\nselection rules can be coupled, as shown in Fig 1(b). In\nthis setup, we actually achieve both a SOAM coupling\nin thezdirection [20, 21], and a Raman lattice in the x\ndirection [36]. The competition between SOAM coupling\nand Raman-assisted spin-\rip hopping may give rise to\nvarious quantum many-body phases.\nThis system can be e\u000bectively modeled by an extended\nBose-Hubbard model for a su\u000ecient deep optical lattice.\nWe speci\fcally consider the case of half \flling in the Mott\nregime. To obtain many-body phases of the system, a\nbosonic version of real-space dynamical mean-\feld theory\n(RBDMFT) is implemented. Various competing phases\nare obtained in the Mott-insulating regime, including\ncanted-antiferromagnetism, spin-vortex, and composite\nspin-vortex with nonrotating core. To explain the many-\nbody phases, an e\u000bective spin-exchange model is de-\nrived, and we attribute these competing spin textures to\nthe interplay of Heisenberg exchange and Dzyaloshinskii-\nMoriya interactions. Upon increasing the hopping am-\nplitudes, atoms delocalize, and super\ruid phases appear,arXiv:2204.07323v2 [cond-mat.quant-gas] 15 Jun 20222\nFIG. 1. (Color online) (a) Sketch of Raman couplings, in-\nduced by a plan-wave laser with orbital angular momentum\n2lin the zdirection, and a standing wave in the xdirection.\nTo achieve a two-dimensional square lattice, both a standing\nwave in the y-direction and a strong con\fnement freezing the\nmotional degree of freedom of the atoms in the zdirection are\nadded. (b) Atomic level diagram coupled by the pairs of the\nlaser beams \n 1;2.\nincluding normal super\ruid, rotating super\ruid with vor-\ntex texture, and stripe super\ruid.\nThe paper is organized as follows: in section II, we\nintroduce our setup with SOAM coupling, and the ex-\ntended Bose-Hubbard model. In Section III, we give a\ndetailed description of our RBDMFT approach. Section\nIV covers our results for our model. We summarize with\na discussion in Section V.\nII. MODEL AND HAMILTONIAN\nWe consider two-component bosonic gases trapped in\na conventional two-dimensional (2D) square lattice. A\nplane-wave laser with orbital angular momentum 2 lis\nadded in the z-direction, as shown in Fig. 1(a). The\ntwo spin states are denoted as \u001b=\"and#, which are\ncoupled by Raman transitions induced by the standing\nwave with Rabi frequency \n 1(r) in thexdirection, and\nthe plane-wave laser with Rabi frequency \n 2(r) in the\nzdirection [36{39]. In the large detuning limit \n 1;2\u001c\nj\u0001j, this system can be described by an e\u000bective single-\nparticle Hamiltonian (see Appendix A) [27, 36]\nHs=p2\n2m\u0000l~\nmr2Lz\u001bz+l2~2\n2mr2+Vext(r)\n+\u000e\n2\u001bz+ \n0(r) coskx\u0001\u001bx; (1)\nwhereLzdenotes orbital-angular-momentum operator\nof atoms along the zdirection,\u000ethe e\u000bective Zeeman\n\feld, and \n0(r) coskxthe periodic Raman \feld with\n\n0(r) =\n1(r)\n2(r)\n\u0001being the e\u000bective Raman Rabi cou-\npling.Vextdenotes the external trap potential in the\nx\u0000yplane, and in the following we choose an isotropic\nhard-wall box potential, which has already been realized\nexperimentally [40, 41].\nFor a su\u000eciently deep blue-detuned (\u0001 >0) opti-\ncal lattice, the single-particle states at each site can\nbe approximated by the lowest-band Wannier function!(x\u0000Rj). In this approximation, the single-particle\nHamlitonian (1) can be cast into a tight-binding model\nH0=\u0000X\nhi;ji;\u001b\u0010\nt\u001bcy\ni\u001bcj\u001b+itij(cy\ni\"cj\"\u0000cy\ni#cj#) + H:c:\u0011\n+X\nix(\u00001)ix\n\u0010\ncy\nix\"cix+1#\u0000cy\nix\"cix\u00001#+ H:c:\u0011\n+X\niVi\ntrapni\u001b+mz(ni\"\u0000ni#); (2)\nwherehi;jidenotes nearest neighbors between sites iand\nj, andixis the site in the xdirection.cy\ni\u001bandci\u001bare\ncreation and annihilation operators for site iand spin\u001b,\nrespectively. t\u001bdenotes conventional hopping amplitudes\nbetween nearest neighbors, mzthe Zeeman \feld, ni\u001b=\ncy\ni\u001bci\u001bthe local density, and Vi\ntrapthe external trap with\nthe contribution from centrifugal potentiall2~2\n2mr2being\nabsorbed.tijis the nearest-neighbor hopping induced by\nSOAM coupling, favoring hopping along the azimuthal\ndirection (see Appendix B) [42{45]\ntij=Z\ndx!\u0003(x\u0000Ri)\u0012l~\nmr2Lz\u0013\n!(x\u0000Rj)\n\u0019\u0012xiyj\u0000xjyi\nr02\u0013\ntsoc; (3)\nwheretsoc=\u0000l~2\ndmR\ndx!\u0003(x\u0000d)@x!(x) withdbeing lat-\ntice constant. Here, ( xi;yi) are the coordinates of the ith\nsite with the origin at the trap center, and r0denotes the\nlattice spacing between the midpoint of sites i,j, and\nthe trap center. The Raman-assisted nearest-neighbor\nspin-\rip hopping along the xdirection\n\n =Z\ndx\n0(r)!\u0003(x\u0000Ri)jcoskxj!(x\u0000Rj);(4)\nwhere the Raman-assisted onsite spin-\rip hopping is\nzero, since atoms are symmetrically localized at the nodes\nfor the blue-detuned lattice potential [36, 37].\nFor a deep lattice, interaction e\u000bects should be in-\ncluded. The s-wave contact interaction is given by\nHint=X\ni;\u001b\u001b01\n2U\u001b\u001b0ni\u001b(ni\u001b0\u0000\u000e\u001b\u001b0); (5)\nwhereU\"\";##andU\"#denote the intra- and interspecies\ninteractions, respectively. Additionally, we limit present\nstudy to the situations in which the interactions are re-\npulsive and two hyper\fne components are miscible with\nU\"\"=U##\u00111:01U\"#andt\u0011t\"=t#, which is a good\napproximation for two-hyper\fne-state mixtures of a87Rb\ngas [46]. Thus, the total Hamiltonian of our system reads\nH=H0+Hint\u0000X\ni\u001b\u0016\u001bni\u001b; (6)\nwhere\u0016\u001bis the chemical potential for component \u001b. Due\nto the competition between SOAM coupling and Raman-\ninduced hopping, it is expected that various many-body3\nphases develop in the strongly interacting many-body\nsystem described by Eq. (6). To resolve these quantum\nphases, we apply real-space bosonic dynamical mean-\feld\ntheory (RBDMFT), to obtain the complete phase dia-\ngrams. In the following, we set U\"#\u00111 and optical\nlattice spacing d\u00111 as the units of energy and length,\nrespectively. We focus on the lower \flling case with \fll-\ningni=ni\"+ni#= 1 in the Mott regime (the total\nparticle number N=P\nini= 330), and the lattice size\nNlat= 24\u000224.\nIII. METHOD\nTo resolve the long-range order, we utilize bosonic dy-\nnamical mean-\feld theory (BDMFT) to calculate many-\nbody ground states of the system described by Eq. (6).By neglecting non-local contributions to the self-energy\nwithin BDMFT [47], the N-site lattice problem can be\nmapped to Nsingle-impurity models interacting with\ntwo baths, which correspond to condensing and normal\nbosons, respectively [48{51]. By a self-consistency con-\ndition, we can \fnally obtain the physical information of\ntheN-site model. Note here that, in a real-space system\nwithout lattice-translational symmetry, the self-energy is\nlattice-site dependent, i.e. \u0006 i;j= \u0006i\u000eijwith\u000eijbeing\na Kronecker delta, which motivates us to utilize a real-\nspace version of BDMFT [52{54].\nIn RBDMFT, our challenge is to solve the single-\nimpurity model, and the physics of site iis given by\nthe local e\u000bective action S(i)\nimp. Following the standard\nderivation [47], we can write down the e\u000bective action\nfor impurity site i, which is described by\nS(i)\nimp=\u0000Z\f\n0d\u001cd\u001c0X\n\u001b\u001b0c(i)\n\u001b(\u001c)yG(i)\n\u001b\u001b0(\u001c\u0000\u001c0)\u00001c(i)\n\u001b0(\u001c0) +Z\f\n0d\u001c1\n2X\n\u001b\u001b0U\u001b\u001b0n(i)\n\u001b(\u001c)\u0010\nn(i)\n\u001b0(\u001c)\u0000\u000e\u001b\u001b0\u0011\n+1\nzZ\f\n0d\u001c0\n@\u0000X\nhi;ji;\u001bt\u001b \nc(i)\n\u001b(\u001c) \n\u001e(i)\nj;\u001b(\u001c)\u0003\n\u001e(i)\nj;\u001b(\u001c)!!\n+itij \nc(i)\n\"(\u001c) \n\u001e(i)\nj;\"(\u001c)\u0003\n\u0000\u001e(i)\nj;\"(\u001c)!\n\u0000c(i)\n#(\u001c) \n\u001e(i)\nj;#(\u001c)\u0003\n\u0000\u001e(i)\nj;#(\u001c)!!\n+X\nix;\u001b6=\u001b0(\u00001)ix\n\u0010\nc(ix)\n\u001b(\u001c)\u0003\u0010\n\u001e(ix)\nix+1;\u001b0(\u001c)\u0000\u001e(ix)\nix\u00001;\u001b0(\u001c)\u0011\n+c(ix)\n\u001b(\u001c)\u0010\n\u001e(ix)\nix+1;\u001b0(\u001c)\u0003\u0000\u001e(ix)\ni\u00001;\u001b0(\u001c)\u0003\u0011\u0011\n+X\ni;\u001b6=\u001b0V(i)\ntrapn(i)\n\u001b(\u001c) +mz\u0010\nn(i)\ni\u001b(\u001c)\u0000n(i)\ni\u001b0(\u001c)\u00111\nA: (7)\nHere, G(i)\n\u001b\u001b0(\u001c\u0000\u001c0) is a local non-interacting propagator\ninterpreted as a dynamical Weiss mean \feld which simu-\nlates the e\u000bects of all other sites. To shorten the formula,\nthe Nambu notation is used c(i)\n\u001b(\u001c)\u0011(c(i)\n\u001b(\u001c);c(i)\n\u001b(\u001c)\u0003).\nThe parameter zis the lattice coordination, which is\ntreated as a control parameter within RBDMFT. The\nterms up to subleading order are included in the e\u000bec-\ntive action. The static bosonic mean-\felds are de\fned in\nterms of the bosonic operator cj\u001bas\n\u001e(i)\nj\u001b(\u001c) =D\nc(i)\nj\u001b(\u001c)E\n0; (8)\nwhereh:::i0means the expectation value in the cavity\nsystem without the impurity site.\nInstead of solving the e\u000bective action directly, we nor-\nmally turn to the Hamiltonian representation, i.e. An-\nderson impurity Hamiltonian [49, 55]. By exactly diago-\nnalizing the Anderson impurity Hamiltonian with a \fnite\nnumber of bath orbitals [47, 56], we can \fnally obtain the\nlocal propagator\nG(i)\n\u001b\u001b0;imp(\u001c;\u001c0) =\u0000D\nTc(i)\n\u001b(\u001c)c(i)\ni\u001b0(\u001c0)yE\nS(i)\nimp:(9)Next, we utilize the Dyson equation to obtain site-\ndependent self-energies in the Matsubara frequency rep-\nresentation\n\u0006(i)\n\u001b\u001b0;imp(i!n) =G(i)\n\u001b\u001b0(i!n)\u00001\u0000G(i)\n\u001b\u001b0(i!n):(10)\nIn the framework of RBDMFT, we assume that the\nimpurity self-energy \u0006 imp(i!n) is local (momentum-\nindependent) and coincides with lattice self-energy\n\u0006lattice (i!n), whose assumption is exact in in\fnite di-\nmensions and good approximations in higher dimen-\nsions [47]. Finally, we employ the Dyson equation in the\nreal-space representation to obtain the interacting lattice\nGreen's function\nG\u001b\u001b0;lattice =1\ni!n+\u0016\u0000\"\u0000\u0006imp(i!n);(11)\nwhere boldface quantities denote matrices with site-\ndependent elements. \"denotes a matrix with the el-\nements being nearest-neighbor hopping amplitudes for\na given lattice structure, \u0016represents the onsite hop-\nping amplitudes with the external trap, and \u0006imp(i!n)\ndenotes the self-energy. The self-consistency RBDMFT\nloop is closed by the Dyson equation to obtain a new4\nFIG. 2. (Color online) Nearest-neighbor hopping amplitudes\ntsoc=tas a function of lattice depth V. In the regime with V >\n5ER,tsoc=tscales roughly linearly with the lattice depth V,\nwhere ERis the recoil energy. We choose the orbital angular\nmomentum l= 1.\nlocal non-interacting propagator Gi\n\u001b\u001b0. These processes\nare repeated until the desired accuracy for super\ruid or-\nder parameters and noninteracting Green's functions is\nobtained.\nIV. RESULTS\nA. Spin-orbital-angular-momentum coupling\nBefore exploring the whole model, described by\nEq. (6), we \frst discuss the competition between conven-\ntional nearest-neighbor hopping tand orbital-angular-\nmomentum-induced hopping tsoc. As shown in Fig. 2,\nthe orbital-angular-momentum-induced hopping can be\nthe order of the conventional one even for l= 1, where the\nhopping amplitudes are obtained from band-structure\nsimulations [57]. By neglecting the Raman-induced spin-\n\rip hopping \n, Eq. (6) is reduced to\nH=\u0000X\nhi;ji;\u001b\u0010\ntcy\ni\u001bcj\u001b+itij(cy\ni;\"cj;\"\u0000cy\ni;#cj;#) + H:c:\u0011\n+X\ni;\u001b\u001b01\n2U\u001b\u001b0ni\u001b(ni\u001b0\u0000\u000e\u001b\u001b0) +Vi\ntrapni;\u001b\u0000\u0016i\u001bni\u001b;\n(12)\nwithmz= 0.\nAs shown in Fig. 3, a many-body phase diagram is\nshown as a function of hopping amplitudes tandtsoc\nfor interactions U\"\"=U##= 1:01U\"#, based on RB-\nDMFT. To distinguish quantum phases, we introduce\nsuper\ruid order parameters \u001e\u001b, pseudospin operators\nSz\ni=1\n2(by\ni;\"bi;\"\u0000by\ni;#bi;#),Sx\ni=1\n2(by\ni;\"bi;#+by\ni;#bi;\") and\nSy\ni=1\n2i(by\ni;\"bi;#\u0000by\ni;#bi;\"), and winding number w=\n1\n2\u0019P\nCarg(M\u0003\niMj) withMi=Sx\ni+iSy\niandCbeing a\nclosed loop around the center of the trap [58, 59]. We ob-\nserve \fve di\u000berent quantum phases. When tsoc\u001ct, the\nFIG. 3. (Color online) (a) Many-body phase diagram of two-\ncomponent ultracold bosonic gases in a square lattice in the\npresence of SOAM coupling, described by Eq. (12). The\nsystem favors three Mott phases with ferromagnetism (MI I),\nspin-vortex (MI II), and composite spin-vortex with antiferro-\nmagnetic core (MI III). In the super\ruid regime, two quantum\nphases appear, denoted as conventional super\ruid (SF I) and\nrotating super\ruid (SF II). Inset: winding number was a\nfunction of tsocwitht= 0:015. (b) Contour plots of spin tex-\ntures for phases MI IIand MI IIIin the Mott-insulating regime.\nThe interactions U\"\"=U##= 1:01U\"#.\nsystem demonstrates a ferromagnetic phase (MI I), iden-\ntical with the system without SOAM coupling. With the\ngrowth oftsoc, a spin-vortex phase appears in the Mott-\ninsulating regime with w6= 0 (MI II), since the growth of\ntsocis equivalent to the growth of orbital angular momen-\ntuml. As shown in Fig. 3(b), SOAM-induced spin rota-\ntion appears around the trap center with winding number\nw= 1, indicating that the spin rotates slowly with the\ncorresponding response being mainly around the center of\nthe trap. The physical reason is that the SOAM-induced\nhopping is site-dependent, and pronounced around the\ntrap center, as indicated by Eq. (3). Further increasing\ntsoc, the winding number wgrows as well, as shown in\nthe inset of Fig. 3(a), and \fnally we observe the whole\nsystem rotating in the regime tsoc\u001dt(MIIII). Inter-\nestingly, this spin-vortex phase is actually a composite\nvortex defect, which supports a nonrotating core of an-\ntiferromagnetic spin texture, with the nearest-neighbor\nspins being antiparallel in the trap center, as shown in\nFig. 3(b).5\nFIG. 4. (Color online) Many-body phase diagrams of two-hyper\fne-state mixtures of a87Rb gas in a square lattice, described\nby Eq. (6), in the presence of SOAM coupling and Raman-induced spin-\rip hopping for orbital angular momenta l= 1 (a)\nandl= 2 (b). The system favors Mott phases with ferromagnetic (MI I), vortex (MI II), and canted-antiferromagnetic (MI IV)\norders, and super\ruid phases with vortex (SF II) and stripe (SF III) textures. The interactions U\"\"=U\"#=U##=U\"#= 1:01, and\nthe e\u000bective Zeeman \feld mz= 0.\nTo understand the underlying physics in the Mott\nregime, we treat the hopping as perturbations and de-\nrive an e\u000bective exchange model at half \flling. With\nde\fning the projection operators PandQ= 1\u0000P, we\ncan project the system into the Hilbert space consisting\nof both singly occupied sites and the states being at least\none site with double occupation, and obtain an e\u000bective\nexchange model Heff=\u0000PHtQ(1\nQHUQ\u0000E)QHtP[60{\n62]. The e\u000bective exchange model is \fnally given by:\nHeff=X\nhi;jiJzSz\niSz\nj+J\u0000\nSx\niSx\nj+Sy\niSy\nj\u0001\n+D(Si\u0002Sj)z: (13)\nHere,Jz=\u00004(2\nU\u00001\nU\"#)(t2+t2\nij),J=\u00004(t2\u0000t2\nij)\nU\"#, and\nD=\u00008ttij\nU\"#. The details of derivation are given in the\nAppendix C.\nIn the absence of SOAM coupling, this e\u000bective model\nis reduced to the conventional XXZ model, where the\nsystem prefers ferromagnetic and anti-ferromagnetic or-\nders [63{66]. In the presence of SOAM coupling, a\nDzyaloshinskii-Moriya term [67, 68] appears in the zdi-\nrection. This term competes with the normal Heisenberg\nexchange interactions, resulting spin-vortex defects in the\nMott-insulating regime. Interestingly, the Heisenberg ex-\nchange term Jalso depends on the SOAM-induced hop-\npingtsoc, and dominates in the regime tsoc\u001dt, resulting\nan antiferromagnetic texture. This texture is consistent\nwith our numerical results, as shown in Fig. 3(b). We\nremark that the SOAM-induced Dzyaloshinskii-Moriya\ntermDpreserves rotational symmetry with spin texture\nrotating along the azimuthal direction, in contrast to\nthe spin-linear-momentum coupling by breaking lattice-\ntranslational symmetry [69{73].With the increase of hopping amplitudes, atoms delo-\ncalize and the super\ruid phase appears. We characterize\nthe super\ruid phase with super\ruid order parameters \u001e\u001b.\nIn the super\ruid region, we observe two quantum many-\nbody phases, with one being a phase with phase rotating\n(SFII), and the other with conventional phase (SF I).\nB. Interplay of spin-orbital-angular-momentum\ncoupling and Raman-induced spin-\rip hopping\nNow we turn to study the whole system, described by\nEq. (6), and focus on the stability of spin-vortex tex-\nture in the strongly interacting regime. Generally, SOAM\ncoupling preserves rotational symmetry and favors spin-\nvortex defects [25, 30], whereas the one-dimensional\nRaman-induced spin-\rip hopping prefers the stripe phase\nand breaks translational symmetry [74, 75]. It is ex-\npected that more exotic many-body phases appear, due\nto the competition between SOAM coupling and Raman-\ninduced spin-\rip hopping. Here, we choose two hyper\fne\nstates of a87Rb gas as examples, where all the Hub-\nbard parameters are obtained from band-structure sim-\nulations [57]. To emphasize the in\ruence of SOAM cou-\npling, we consider the orbital angular momenta l= 1\nandl= 2. Note here that t\u0019tsocfor orbital angular\nmomentum l= 1 in the deep lattice, as shown in Fig. 2.\nRich phases are found in Fig. 4, including Mott-\ninsulating phases with ferromagnetic (MI I), vortex\n(MIII), and canted-antiferromagnetic (MI IV) orders [76],\nand super\ruid phases with vortex (SF II) and stripe\n(SFIII) patterns. In the limit \n \u001ctsoc, the many-body\nphases develop spin-vortex textures (MI IIand SF II),\nwhereas the system prefers density-wave orders (MI IV\nand SF III) in the limit \n \u001dtsoc. This conclusion is6\nFIG. 5. (Color online) Phase transitions as a function of\nRaman-induced spin-\rip hopping for di\u000berent lattice depths\nV= 14 ER(t\u00190:011) (a) and V= 11 :5ER(t\u00190:022)\n(b). Inset: spin structure factor (upper) and real-space spin\ntexture (lower) for di\u000berent phases (a), and local phases of su-\nper\ruid order parameter for the spin- \"component (b). The\ninteractions U\"\"=U\"#=U##=U\"#= 1:01, and the orbital an-\ngular momentum l= 2.\nconsistent with our general discussion above, as a result\nof the interplay of SOAM coupling and Raman-induced\nspin-\rip hopping. Note here that the region of the spin-\nvortex phase is enlarged for larger orbital angular mo-\nmentum, as shown in Fig. 4(b), indicating large oppor-\ntunity for observing this spin texture for larger orbital\nangular momentum.\nTo characterize these di\u000berent phases, we choose wind-\ning number, real-space spin texture, spin-structure fac-\ntorS~ q=\f\f\f~Siei~ q\u0001~ ri\f\f\f[69], and local phase of super\ruid\norder parameter, as shown in Fig. 5. Here, we choose the\norbital angular momentum l= 2, and di\u000berent lattice\ndepthsV= 14ERwith hopping t\u00190:011 [Fig. 5(a)],\nandV= 11:5ERwitht\u00190:022 [Fig. 5(b)]. For small \n,\na spin-vortex phase (MI II) develops with winding num-\nFIG. 6. (Color online) Many-body phase diagram of two-\nhyper\fne-state mixtures of a87Rb gas in a square lattice with\nthe depth V= 9ER(t\u00190:045), as a function of Raman-\ninduced spin-\rip hopping \n and e\u000bective magnetic \feld mz.\nInset: local phases of super\ruid order parameters for di\u000berent\nphases. The interactions U\"\"=U\"#=U##=U\"#= 1:01, and the\norbital angular momentum l= 2.\nberw= 4, as shown in Fig. 5(a). Increasing \n, the\nspin texture changes to ferromagnetic (MI I) and canted-\nantiferromagnetic (MI IV) textures with vanishing wind-\ning number, which are characterized both by magnetic\nspin-structure factor S~ qand real-space spin texture, as\nshown in the inset of Fig. 5(a). We remark here that\nthe MI IVphase possesses spin-density-wave in Sz, ferro-\nmagnetic order in Sx, and antiferromagnetic order in Sy.\nThe local phase of super\ruid order parameter is shown\nin Fig. 5(b). For small \n, a nonzero winding number of\nthe local phase develops in the vortex super\ruid (SF II).\nWith \n larger, we \fnd the local phase demonstrates a\nstripe order instead (SF III).\nTo understand the physical phenomena in the Mott\nregime, we derive an e\u000bective exchange model of the sys-\ntem at half \flling, described by Eq. (6),\nHeff=X\nhix;jxiJ0\nzSz\nixSz\njx+J0\nxSx\nixSx\njx+J0\nySy\nixSy\njx\n+X\nhiy;jyiJzSz\niySz\njy+J\u0010\nSx\niySx\njy+Sy\niySy\njy\u0011\n+X\nhi;jiD(Si\u0002Sj)z; (14)\nwhereJ0\nz=\u00004\u0012\n2(t2+t2\nij)\nU+\n2\nU\"#\u0000(t2+t2\nij)\nU\"#\u00002\n2\nU\u0013\n,J0\nx=\n\u00004\u0012\n(t2\u0000t2\nij)\nU\"#+\n2\nU\"#\u0013\n, andJ0\ny=\u00004\u0012\n(t2\u0000t2\nij)\nU\"#\u0000\n2\nU\"#\u0013\n.\nWe observe that the Raman-induced spin-\rip hopping\ndoes not in\ruence Dzyaloshinskii-Moriya interactions,\nbut induce an anisotropy for Heisenberg exchange in-\nteractions in the xandydirections. When \n is large\nenough, the Raman-induced hopping can induce J0\ny;zto\nbe positive and J0\nxnegative. It indicates that a spin-7\ndensity-wave and canted-antiferromagnetic order devel-\nops for large \n, consistent with our numerical results, as\nshown in Fig. 5(a). In the intermediate regime of \n, the\nJ0\nxterm dominates and a ferromagnetic order appears.\nFor small \n, the e\u000bective model reduces to Eq. (13), and\nthe spin vortex pattern dominates.\nIn a realistic system, one can tune the balance of the\ntwo-spin components, which actually acts as an e\u000bective\nmagnetic \feld mz. Here, we can control the chemical\npotential di\u000berence of the two components to study the\ne\u000bect of the magnetic \feld. In Fig. 6, we \fx the depth\nof optical lattice V= 9ERwith hopping t\u00190:045,\nand study the many-body phase diagram as a function\nof the e\u000bective magnetic \feld mzand Raman-induced\nspin-\rip hopping \n. When the e\u000bective magnetic \feld\nis large and negative, the spin- #component supports\na vortex structure, indicated by the local phase of the\nsuper\ruid order parameter, as shown in inset of Fig. 6,\nor vice versa. For large enough Raman coupling, the\ntwo components are mixed, and the system supports a\nstripe pattern in the xdirection. We remake here that\nthe phase diagram is similar to the one achieved by\nthe SOAM experiments in continuous space [21], where\nthe di\u000berence is vortex-antivortex pair phase for larger\nRaman coupling, instead of stripe order [25, 30], since\nwe essentially include a Raman lattice in the xdirection.\nV. CONCLUSION AND DISCUSSION\nIn summary, we propose a scheme to investigate spin-\norbital-angular-momentum coupling in strongly interact-\ning bosonic gases in a two-dimensional square lattice. Us-\ning real-space dynamical mean-\feld theory, we obtain\nvarious quantum phases, including spin-vortex defect,\ncomposite vortex, canted-antiferromagnetic, and ferro-\nmagnetic insulating phases. Based on e\u000bective exchange\nmodels, we \fnd that the spin-vortex texture is a result of\nthe Dzyaloshinskii-Moriya interaction, induced by spin-\norbital-angular-momentum coupling. Due to the compe-\ntition of Dzyaloshinskii-Moriya and Heisenberg exchange\ninteractions, various spin textures develop. In the super-\n\ruid, we \fnd three quantum phases with conventional,\nstripe and vortex orders, characterized by the local phase\nof super\ruid order parameters. Our study would be help-\nful to identify interesting many-body phases in future ex-\nperiments.\nVI. ACKNOWLEDGMENTS\nWe acknowledge helpful discussions with Kaijun Jiang\nand Keji Chen. This work is supported by the Na-\ntional Natural Science Foundation of China under Grants\nNo.12074431, 11774428, and 11974423, and National Key\nRD program, Grant No. 2018YFA0306503. We acknowl-\nedge the Beijing Super Cloud Computing Center (BSCC)for providing HPC resources that have contributed to the\nresearch results reported within this paper.8\nVII. APPENDIX\nA. Single-particle Hamiltonian\nIn our scheme, the Raman transition is \u0003-type con\fguration with \n 1(r) = \n 1cos(kx) along the xdirection and\n\n2(r) = \n 2e\u00002r2=\u001a2e\u0000i2l\u001ealong thezdirection. In the regime j\u0001j\u001d\n1;2, the single-photon transition between the\nground and excited states is suppressed. We can adiabatically remove the excited state, and the system is e\u000bectively\nregarded as two-ground-state mixtures coupled by two-photon Raman processes. Including the two-photon Raman\nprocesses, we can obtain an e\u000bective spin-1 =2 Hamiltonian\nH= \np2\n2m+\u000e\n2\n0(r) coskx\u0001e\u0000i2l\u001e\n\n0(r) coskx\u0001ei2l\u001e p2\n2m\u0000\u000e\n2!\n; (S1)\nwhere\u000eis a two-photon detuning, and \n0(r) = \n 1\n2e\u00002r2=\u001a2=\u0001 denotes the Raman Rabi frequency. After introducing\nthe unity transformation to the single-particle wave function\nU=\u0012\ne\u0000il\u001e0\n0eil\u001e\u0013\n; (S2)\nand Pauli matrix \u001b, we \fnally obtain\nH=\u0000~2\n2mr@\n@r\u0012\nr@\n@r\u0013\n+\u000e\n2\u001bz+(Lz\u0000l~\u001bz)\n2mr22\n+ \n0(r) coskx\u0001\u001bx;\n(S3)\nwhereLz=\u0000i~@zis the orbital angular momentum along the zaxis. Normally, the plan-wave laser is a LG beam\nwith the intensity being suppressed near the trap center, which can in\ruence experimental observations. Here, we\ninstead consider a Gaussian-type Raman beam with orbital angular momentum, where such a Gaussian beam can be\nobtained by a quarter-wave plate [31, 77]. The waist of the plane-wave laser is set to \u001a= 20.\nB. Orbital angular momentum in the Wannier basis\nThe angular momentum tijis given by\nHLz=Z\ndx\ty(x)l~\nmr2Lz\t(x)\n=l~\nmZ\ndx\ty(x)1\nr2Lz\t(x): (S4)\nFor a su\u000ecient deep lattice, the \feld operator \t( x) can be expanded in the lowest-band Wannier basis !(x\u0000Ri).\nEq. (S4) can be rewritten as\nHLz=l~\nmX\nhi;jicy\nicjZ\ndx3!\u0003(x\u0000Ri)1\nr2Lz!(x\u0000Rj): (S5)\nFor nearest neighbors iandj, we have the relation that\n1\nr2\n1Ki;j 1, additional spin-free terms\narise from the transformation of spin-dependent\nHamiltonian due to the picture change effect, which\nare added to the X2C-1e spin-free Hamiltonian\n(ˆHSF=ˆHX2C−1e\nSF+ˆHDKHn\nSF ).\nWhen represented in the form of Eq. (11), the\nmatrix elements of DKH1 spin–orbit Hamiltonian\ncan be expressed as:53,72,97\nFDKH 1,ξ=hDKH 1,ξ+gDKH 1,ξ, (21)\nhDKH 1,ξ=R†\n+X†hξXR +, (22)\ngDKH 1,ξ=R†\n+(GLL,ξ+GLS,ξX+X†GSL,ξ\n+X†GSS,ξX)R+, (23)\nwhere the matrix Xdecouples ΨLandΨSin\nEq. (9) using the X2C-1e approach. The R+ma-\ntrix accounts for the metric renormalization and is\nexpressed as\nR+=S−1\n2\n+(S−1\n2\n+˜S+S−1\n2\n+)−1\n2S1\n2\n+, (24)\n˜S+=S++X†S−X, (25)\nS−=α2\n2T, (26)\nin terms of the non-relativistic overlap ( S+=S)\nand kinetic energy ( T) integrals.\nThe mean-field two-electron term gDKH 1,ξis de-\nfined in terms of the GXY,ξ(X, Y∈ {L, S}) ma-\ntrices53,72,97\nGLL,ξ\nρλ=−X\nµν2Kξ\nµρνλPSS\nµν, (27)\nGLS,ξ\nρλ=−X\nµν(Kξ\nρµνλ+Kξ\nµρνλ)PLS\nµν=−GSL,ξ\nλρ,\n(28)\nGSS,ξ\nρλ=−X\nµν2(Kξ\nρλνµ+Kξ\nρλµν−Kξ\nρµλν)PLL\nµν,\n(29)\nexpressed in the atomic spin-orbital basis labeled\nwith ρ, λ, ν, µ . The two-electron spin–orbit inte-\ngrals\nKξ\nρλνµ=X\noπϵξ\noπ⟨ϕρoϕνπ|ϕλϕµ⟩ (30)\nare related to gξ\nρλνµin Eq. (20) via\ngξ\nρλνµ=−(Kξ\nρλνµ+Kξ\nρλµν). (31)\n5The density matrices PSS,PLS, and PLLappear-\ning in Eqs. (27) to (29) are obtained from the spin-\nfree SA-CASSCF density matrix P(Eq. (15)):\nPSS=XPLLX†, (32)\nPLS=PLLX†, (33)\nPLL=1\n2R+PR†\n+. (34)\nIn Eqs. (27) to (29), the GLS,ξ\nρλandGSL,ξ\nλρmatrices\ndescribe the Coulomb-exchange interactions while\nGLL,ξ\nρλoriginates from the Gaunt-exchange terms.97\nThe GSS,ξ\nρλmatrix represents a mixture of direct\nCoulomb and Gaunt-exchange contributions. Due\nto spin averaging, the direct Gaunt terms vanish.\nThe DKH1 Hamiltonian reduces to the BP Hamil-\ntonian when R+=1andX=1.\nIncorporating the second-order terms gives rise\nto the DKH2 spin–orbit Hamiltonian with matrix\nelements72\nFDKH 2,ξ=hDKH 1,ξ+hDKH 2,ξ+gDKH 1,ξ,\n(35)\nwhere the second-order one-electron spin-\ndependent contribution hDKH 2,ξhas the form:\nhDKH 2,ξ=4\nα4(⃗W×T−1⃗O†+⃗O×T−1⃗W†)ξ\n(36)\nThe components of vectors ⃗Wand⃗Oare defined\nas:\nWξ=α2\n2S+C+wξC†\n−T, (37)\nwξ\npq=−oξ\npq\nE−,q−E+,p, (38)\noξ=C†\n+OξC−, (39)\nOξ=α2\n4R†\n+X†hξR−, (40)\nwhere E+,p/E−,pandC+/C−are the eigenvalues\nand eigenvectors obtained by solving the X2C-1e\nequations for the positive/negative energy states,\nrespectively. The renormalization matrix R−is\ngiven by:\nR−=S−1\n2\n−(S−1\n2\n−˜S−S−1\n2\n−)−1\n2S1\n2\n−, (41)\n˜S−=S−+˜X†S+˜X, (42)\n˜X=−S−1\n+X†S−. (43)As for DKH1, the DKH2 contributions to the two-\ncomponent spin–orbit Hamiltonian are computed\nusing the decoupling matrix Xobtained from the\nX2C-1e procedure. The resulting sf-X2C-1e+so-\nDKH n(n= 1, 2) approach will be termed here as\nDKH nfor brevity.\n2.3 Incorporating Spin–Orbit Coupling\nin QDNEVPT2\nTo incorporate spin–orbit coupling in QD-\nNEVPT2, we augment the perturbation operator\nˆVwith a two-component spin–orbit Hamiltonian\n(ˆV=ˆVee+ˆHSO). The resulting effective Hamilto-\nnian expanded up to the second order in perturba-\ntion theory has the form:\n⟨Ψ(0)\nI|ˆHBP2/DKH 2\neff,SO|Ψ(0)\nJ⟩=E(0)\nIδIJ\n+⟨Ψ(0)\nI|ˆVee+ˆHBP/DKH 2\nSO|Ψ(0)\nJ⟩\n+1\n2⟨Ψ(0)\nI|ˆVee+ˆHBP/DKH 2\nSO|˜Ψ(1)\nJ⟩\n+1\n2⟨˜Ψ(1)\nI|ˆVee+ˆHBP/DKH 2\nSO|Ψ(0)\nJ⟩.(44)\nIn this formulation that consistently treats dy-\nnamic correlation and spin–orbit coupling to sec-\nond order, we choose ˆHBP/DKH 2\nSOto be either the\nBP (Eq. (15)) or DKH2 (Eq. (35)) Hamiltonian in\nthe form of Eq. (11), denoted as BP2-QDNEVPT2\nor DKH2-QDNEVPT2, respectively. Compared\nto conventional QDNEVPT2, the BP2/DKH2-\nQDNEVPT2 effective Hamiltonian contains new\nterms that depend on ˆHBP/DKH 2\nSOand modified\nfirst-order wavefunctions\n|˜Ψ(1)\nI⟩=X\nµ˜t(1)\nµI|ΦµI⟩, (45)\nwhich amplitudes are computed by solving the lin-\near system of equations\nX\nνKµνI˜t(1)\nνI=−⟨ΦµI|ˆVee+ˆHBP/DKH 2\nSO|Ψ(0)\nI⟩\n(46)\nwith KµνIdefined in Eq. (8). Due to mean-field\nspin–orbit approximation, the r.h.s. of Eq. (46) has\nnon-zero contributions from ˆHBP/DKH 2\nSOonly for\nthe semi-internal [0′] and [ ±1′] excitations, making\nthe corresponding ˜t(1)\nνIamplitudes complex-valued.\nFor the remaining excitation classes ([0], [ ±1],\n[±2]), Eq. (46) reduces to Eq. (7), with ˜t(1)\nνI=t(1)\nνI\nwhere t(1)\nνIare the conventional real-valued QD-\n6NEVPT2 amplitudes.\nIn addition to BP2- and DKH2-QDNEVPT2, we\nalso consider two approximations where the spin–\norbit coupling is treated to first order in perturba-\ntion theory using either the BP or DKH1 Hamilto-\nnians, abbreviated as BP1-QDNEVPT2 or DKH1-\nQDNEVPT2, respectively. The corresponding ef-\nfective Hamiltonian has the form:\n⟨Ψ(0)\nI|ˆHBP1/DKH 1\neff,SO|Ψ(0)\nJ⟩=E(0)\nIδIJ\n+⟨Ψ(0)\nI|ˆVee+ˆHBP/DKH 1\nSO|Ψ(0)\nJ⟩\n+1\n2⟨Ψ(0)\nI|ˆVee|Ψ(1)\nJ⟩\n+1\n2⟨Ψ(1)\nI|ˆVee|Ψ(0)\nJ⟩, (47)\nwhere |Ψ(1)\nI⟩is the conventional QDNEVPT2 first-\norder wavefunction with real-valued amplitudes de-\ntermined by solving Eq. (7). We note that the BP1-\nQDNEVPT2 method has been studied in detail in\nRef. 71, while the DKH1-QDNEVPT2 implemen-\ntation is reported for the first time. A summary of\nmethods implemented in this work is provided in\nTable 1.\n3 Implementation and Computa-\ntional Details\nThe two-component relativistic methods outlined\nin Table 1 were implemented in the development\nversion of Prism .98Our implementation utilizes\nfull internal contraction, preserves the degener-\nacy of states with the same total angular momen-\ntum, and avoids the calculation of four-particle re-\nduced density matrices using the techniques de-\nveloped in Ref. 71. All integrals and the SA-\nCASSCF reference wavefunctions were computed\nusing the Pyscf package.99In addition to Pyscf ,\nPrism was interfaced with Socutils ,100which\nprovided the matrix elements of DKH1 Hamil-\ntonian for the DKH1-QDNEVPT2 calculations.\nThe DKH2 Hamiltonian matrix elements used in\nDKH2-QDNEVPT2 were implemented in a local\nversion of Socutils .\nWe benchmarked the performance of spin–orbit\nQDNEVPT2 methods for a variety of atomic and\nmolecular systems. First, in Section 4.1, we as-\nsess their accuracy for calculating zero-field split-\nting in main group elements and diatomics against\nthe reference data from experiments and theoret-\nical calculations. For this study, all calculationswere performed using the uncontracted ANO-RCC\nand ANO-RCC-VTZP basis sets.101Other com-\nputational parameters (geometries, active spaces,\nnumber of states averaged in SA-CASSCF) are pro-\nvided in the Supplementary Material.\nNext, in Section 4.2, we use the spin–orbit\nQDNEVPT2 methods to calculate the ground-\nor excited-state zero-field splittings in transition\nmetal atoms, namely: Sc, Y, La, Ag, and Au. For\nall of these atoms, the all-electron X2C-TZVPall-\n2c basis set was used.102The calculations of Sc, Y\nand La in their2Dground states were performed\nwith 3 electrons in 9 active orbitals (3e, 9o), which\nincluded the ns,np, and ( n−1)dshells with n=\n4, 5, and 6, respectively. For Ag and Au, we com-\nputed the excited2Dzero-field splitting utilizing\nthe (11e, 6o) active space corresponding to the ns\nand ( n−1)dorbitals with n= 5 and 6, respec-\ntively. Additional details of these calculations can\nbe found in the Supplementary Material.\nFinally, in Section 4.3, we test the performance\nof our two-component QDNEVPT2 methods for\nthree chemical systems with strong relativistic ef-\nfects: U5+, NpO2+\n2, and UO+\n2. The calculations\nof U5+in its2Fground electronic term utilized the\nSARC-DKH2 basis set103and (1e, 7o) active space,\nwhich incorporated the 5 forbitals. For NpO2+\n2,\nthe uncontracted ANO-RCC-VTZP and cc-pVTZ\nbasis sets104were used for the Np and O atoms,\nrespectively. In the case of UO+\n2, the contracted\nANO-RCC-VTZP basis set was employed for all\natoms. Calculations of both molecules utilized the\n(7e, 10o) active space, as shown in the Supplemen-\ntary Material. The NpO2+\n2and UO+\n2structures\nhave linear geometries with the Np–O bond dis-\ntance of 1.70 ˚A and the U–O bond distance of 1.802\n˚A.\n4 Results and Discussion\n4.1 Main Group Elements and Di-\natomics\nWe begin by investigating the accuracy of spin–\norbit QDNEVPT2 methods for simulating the\nzero-field splitting (ZFS) in open-shell atoms and\ndiatomic molecules consisting of main group ele-\nments ( p-block of periodic table), for which accu-\nrate theoretical and experimental reference data is\navailable. Our first benchmark set consists of 9\natoms and 8 diatomics shown in Table 2. These\natoms and molecules possess either the2Por2Π\n7Table 1: Two-component methods implemented in this work. For each method, dynamic correlation\n(DC) and spin–orbit coupling (SO) are expanded to the order specified in the second and third column,\nrespectively. Also indicated are the spin-free (SF) and SO Hamiltonians employed in each method.\nMethod DC order SO order SF Hamiltonian SO Hamiltonian\nBP1-QDNEVPT2 2 1 X2C-1e BP\nDKH1-QDNEVPT2 2 1 X2C-1e DKH1\nBP2-QDNEVPT2 2 2 X2C-1e BP\nDKH2-QDNEVPT2 2 2 X2C-1e + DKH2 DKH2\nTable 2: Spin–orbit zero-field splitting (cm−1) in the2Pground term of atoms and2Π ground term of\ndiatomics computed using the spin–orbit QDNEVPT2 methods. Results are compared to the reference\ndata from the SO-EOM-CCSD method with relaxed amplitudes105and experiments.106–118All methods\nemployed the uncontracted ANO-RCC basis set.\nSystem BP1- BP2- DKH1- DKH2- SO- Experiment\nQDNEVPT2 QDNEVPT2 QDNEVPT2 QDNEPVT2 EOM-CCSD105\nB 15.0 14.5 15.0 14.5 13.7 15.3116\nAl 107.6 109.9 106.8 109.4 107.5 112109\nGa 887.4 867.9 840.4 818.8 797.6 826117\nIn 2560.8 2859.2 2205.2 2219.0 2103.6 2213118\nTl 12475.8 8655.5 7745.1 8113.3 6794.1 7793106\nF 401.5 405.7 400.5 405.0 396.8 404106\nCl 789.7 867.8 779.5 858.6 872.8 882107\nBr 3574.4 3926.0 3329.4 3625.0 3555.4 3685106\nI 8149.9 10343.7 6824.7 7581.0 7288.8 7603108\nOH 152.5 123.4 152.3 123.2 136.3 139110\nSH 375.6 381.7 371.4 378.2 373.8 377110\nSeH 1836.7 1930.1 1719.5 1793.2 1716.8 1763113\nTeH 4293.5 5238.1 3637.4 3956.5 3751.7 3816111\nFO 180.0 189.5 179.5 189.2 193.6 197114\nClO 299.7 326.6 297.0 324.4 318.7 322119\nBrO 961.9 1085.4 903.5 1012.0 984.2 975115\nIO 2303.8 2924.2 1959.7 2237.5 2143.6 2091112\nground electronic term, which split into2P1/2and\n2P3/2or2Π1/2and2Π3/2energy levels upon incor-\nporating spin–orbit coupling, respectively. In this\nbenchmark, we employ the uncontracted ANO-\nRCC basis set and compare the performance of\nspin–orbit QDNEVPT2 methods to that of spin–\norbit equation-of-motion coupled cluster theory\nwith single and double excitations developed by\nCheng and co-workers (SO-EOM-CCSD).105The\nSO-EOM-CCSD method is a two-component per-\nturbative approach that utilizes the X2C-1e treat-\nment of scalar relativistic effects and mean-field\nX2C description of spin–orbit coupling, which has a\nclose relationship with the DKH1/DKH2 approach\ndescribed herein.\nThe performance of spin–orbit QDNEVPT2 and\nSO-EOM-CCSD methods in predicting ZFS iscompared in Figure 1, where mean absolute errors\n(MAE, %) relative to experimental data are com-\nputed for atoms and molecules in Table 2 across\neach group (a) or period (b) of periodic table. All\nfour QDNEVPT2 methods show very similar per-\nformance for the second period with errors of ∼\n5 %. Significant differences in computed MAE\nare observed already for the third period where\nBP2- and DKH2-QDNEVPT2 show smaller errors\n(∼2 %) compared to that of BP1- and DKH1-\nQDNEVPT2 ( ∼5 to 6 %). For the fourth pe-\nriod, a large increase in MAE is observed from\nBP1- to BP2-QDNEVPT2, highlighting the well-\nknown problems of Breit–Pauli Hamiltonian in de-\nscribing the spin–orbit coupling of elements with\nheavier nuclei. This trend continues for period 5\nwhere BP1- and BP2-QDNEVPT2 exhibit MAE\n8/uni00000025/uni00000033/uni00000014/uni00000010\n/uni00000003/uni00000034/uni00000027/uni00000031/uni00000028/uni00000039/uni00000033/uni00000037/uni00000015/uni00000025/uni00000033/uni00000015/uni00000010\n/uni00000003/uni00000034/uni00000027/uni00000031/uni00000028/uni00000039/uni00000033/uni00000037/uni00000015/uni00000027/uni0000002e/uni0000002b/uni00000014/uni00000010\n/uni00000003/uni00000034/uni00000027/uni00000031/uni00000028/uni00000039/uni00000033/uni00000037/uni00000015/uni00000027/uni0000002e/uni0000002b/uni00000015/uni00000010\n/uni00000003/uni00000034/uni00000027/uni00000031/uni00000028/uni00000039/uni00000033/uni00000037/uni00000015/uni00000036/uni00000032/uni00000010/uni00000028/uni00000032/uni00000030/uni00000010\n/uni00000026/uni00000026/uni00000036/uni00000027/uni00000013/uni00000011/uni00000013/uni00000015/uni00000011/uni00000018/uni00000018/uni00000011/uni00000013/uni0000001a/uni00000011/uni00000018/uni00000014/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000011/uni00000018/uni00000014/uni00000018/uni00000011/uni00000013/uni00000014/uni0000001a/uni00000011/uni00000018/uni00000015/uni00000013/uni00000011/uni00000013/uni00000008/uni00000003/uni00000030/uni00000024/uni00000028/uni00000003/uni00000059/uni00000056/uni00000011/uni00000003/uni00000048/uni0000005b/uni00000053/uni00000048/uni00000055/uni0000004c/uni00000050/uni00000048/uni00000051/uni00000057/uni00000044/uni0000004f/uni00000003/uni00000047/uni00000044/uni00000057/uni00000044/uni00000025/uni0000000f/uni00000003/uni00000024/uni0000004f/uni0000000f/uni00000003/uni0000002a/uni00000044/uni0000000f/uni00000003/uni0000002c/uni00000051/uni0000000f/uni00000003/uni00000037/uni0000004f\n/uni00000029/uni0000000f/uni00000003/uni00000026/uni0000004f/uni0000000f/uni00000003/uni00000025/uni00000055/uni0000000f/uni00000003/uni0000002c\n/uni00000032/uni0000002b/uni0000000f/uni00000003/uni00000036/uni0000002b/uni0000000f/uni00000003/uni00000036/uni00000048/uni0000002b/uni0000000f/uni00000003/uni00000037/uni00000048/uni0000002b\n/uni00000029/uni00000032/uni0000000f/uni00000003/uni00000026/uni0000004f/uni00000032/uni0000000f/uni00000003/uni00000025/uni00000055/uni00000032/uni0000000f/uni00000003/uni0000002c/uni00000032(a)\n* * (b)\nFigure 1: Mean absolute errors (MAE, %) in zero-field splitting for the main group elements and\ndiatomics calculated using the spin–orbit QDNEVPT2 methods and SO-EOM-CCSD105relative to the\nexperimental data. MAE are calculated for the chemical systems across each (a) group and (b) period of\nthe periodic table. Bars that exceed the scale of the plot are indicated with asterisks. See Table 2 for\ndata on individual systems.\nlarger than 10 %. The DKH-based methods per-\nform reliably for periods 2 to 5, with MAE of\n∼5 % for DKH1-QDNEVPT2 and ≲2.5 % for\nDKH2-QDNEVPT2, the latter being very close to\nthe MAE of SO-EOM-CCSD. For the only element\nfrom period 6 in this benchmark set (Tl), the best\nresults are shown by DKH1-QDNEVPT2 (0.6 % er-\nror) and DKH2-QDNEVPT2 (4.1 % error), while\nSO-EOM-CCSD shows a large error of 12.8 %.\nIn Table 3 and Figure 2, we compare the accuracy\nof spin–orbit QDNEVPT2 methods in calculating\nZFS to that of the spin–orbit EOM-CCSD method\n(EOM-CCSD(SOC)) developed by Cao et al.120In\nEOM-CCSD(SOC), the dynamic correlation and\nspin–orbit coupling effects are incorporated by self-\nconsistently solving the coupled cluster equations\nutilizing the same two-component Hamiltonian as\nthe one employed in DKH1-QDNEVPT2 (sf-X2C-\n1e+so-DKH1). The calculations for this bench-\nmark set were performed using the uncontracted\nANO-RCC-VTZP basis to enable direct compar-\nison with the EOM-CCSD(SOC) results. Com-\npared to Table 2, the data in Table 3 includes ZFS\nfor At (period 6) and group 14 hydrides (OH, SH,\nSeH, TeH), but does not contain data for the group\n17 oxides.\nAs illustrated in Figure 2, the performance\nof DKH2-QDNEVPT2 is similar to EOM-\nCCSD(SOC), which shows somewhat smaller MAE\nfor periods 2 to 5 (by ∼1 to 1.5 %), but a larger\nerror for period 6 (by ∼1 %). Meanwhile, DKH1-\nQDNEVPT2 exhibits significantly larger errors(by a factor of ∼3) when compared to EOM-\nCCSD(SOC) for periods 3 to 5, despite using the\nsame two-component Hamiltonian. This suggests\nthat the second-order effects in the description of\ndynamic correlation and spin-orbit coupling incor-\nporated in DKH2-QDNEVPT2 are important to\nachieve accuracy similar to that of self-consistent\ntwo-component relativistic methods such as EOM-\nCCSD(SOC).\nOverall, our results demonstrate that for the\nmain group elements and their diatomic molecules\nwith predominantly single-reference electronic\nstructure DKH2-QDNEVPT2 shows the highest\naccuracy for calculating ZFS out of all spin–orbit\nQDNEVPT2 methods considered in this work.\nThe DKH1-QDNEVPT2 method exhibits some-\nwhat larger errors in ZFS, but performs reliably\nfor elements across the entire p-block of periodic\ntable. The BP1- and BP2-QDNEVPT2 implemen-\ntations start to deteriorate in quality for period 4\nand are unreliable for periods 5 and 6. The accu-\nracy of DKH2-QDNEVPT2 is comparable to that\nof spin–orbit equation-of-motion coupled cluster\nmethods based on the X2C-type Hamiltonians.\nAlthough all chemical systems in Tables 2 and 3\nhave single-reference electronic structure, the QD-\nNEVPT2 methods considered in this work are mul-\ntireference in nature and are expected to be more\nreliable than coupled cluster theory for electronic\nstates with strong multiconfigurational character.\n9Table 3: Spin–orbit zero-field splitting (cm−1) in the2Pground term of atoms and2Π ground term of\ndiatomics computed using the spin–orbit QDNEVPT2 methods. Results are compared to the reference\ndata calculated using the EOM-CCSD(SOC) method120and experiments.106–111,113,116–118All methods\nemployed the uncontracted ANO-RCC-VTZP basis set.\nSystem BP1- BP2- DKH1- DKH2- EOM- Experiment\nQDNEVPT2 QDNEVPT2 QDNEVPT2 QDNEVPT2 CCSD(SOC)120\nB 15.0 14.5 15.0 14.5 13.7 15.3116\nAl 107.6 109.9 106.8 109.4 107.5 112109\nGa 887.4 867.9 840.4 818.8 797.6 826117\nIn 2560.8 2859.2 2205.2 2219.0 2103.6 2213118\nTl 12475.8 8655.5 7745.1 8113.3 6794.1 7793106\nF 401.5 405.7 400.5 405.0 397.7 404106\nCl 789.7 867.8 779.5 858.6 876.0 882107\nBr 3574.4 3926.0 3329.4 3625.0 3648.8 3685106\nI 8150.0 10343.7 6824.7 7581.0 7754.6 7603108\nAt 34153.5 345491.0 19970.1 23002.4 24880.5 –\nCH 29.0 27.4 29.0 27.3 27.4 27121\nSiH 128.0 136.6 127.0 135.6 139.3 142121\nGeH 864.1 910.2 815.4 854.9 882.9 892110\nSnH 2286.3 2713.0 1961.6 2103.7 2187.0 2178110\nOH 152.6 123.4 152.3 123.2 140.1 139110\nSH 375.6 381.7 371.4 378.2 375.3 377110\nSeH 1835.2 1931.3 1718.1 1793.3 1742.9 1763113\nTeH 4281.9 5212.2 3626.1 3942.6 3913.4 3816111\n4.2 Transition Metal Elements\nIn contrast to the main group elements, most tran-\nsition metals are known to exhibit significant mul-\ntireference effects in the ground or excited elec-\ntronic states. In Tables 4 and 5, we apply the\nspin–orbit QDNEVPT2 methods to the Sc, Y, and\nLa atoms with the ground2Dterm ( nd1configura-\ntion, n= 3, 4, 5) and to the Ag and Au atoms with\nthe excited2Dterm ( nd9(n+1)s2configuration, n\n= 4, 5). We compare our results to the available\nZFS data from experiments123,125and variational\nrelativistic electronic structure calculations.122,124\nAll theoretical ZFS were computed using the X2C-\nTZVPall-2c basis set (see Section 3 for details).\nIncorporating spin–orbit coupling in Sc, Y, and\nLa spits their ground2Dterm into the2D3/2and\n2D5/2levels. Simulating this ZFS accurately is\nchallenging even for variational electronic structure\nmethods, as demonstrated by the two-component\nX2C-MRCISD results122in Table 4 that exhibit\nlarge errors relative to the experimental data123\n(up to 11.2 %). Similarly, the ZFS computed using\nBP2- and DKH2-QDNEVPT2 deviate significantly\nfrom the experimental data with errors ranging\nfrom 14.9 to 20.4 %. Although we cannot quantify\nthe source of these errors, the poor performance ofvariational X2C-MRCISD method for Sc and La\nsuggests that they are at least in part due to high-\norder dynamic correlation effects, such as triple\n(and higher) excitations in non-active orbitals, and\ntheir interplay with spin–orbit coupling. Lowering\nthe level of theory to BP1- and DKH1-QDNEVPT2\nfortuitously improves agreement with the experi-\nment, producing errors smaller than those of X2C-\nMRCISD for Sc and La.\nTable 5 presents the spin–orbit QDNEVPT2 re-\nsults for the ZFS in excited2Dterm of Ag and\nAu. Here, we use the experimental results125\nas the reference and present the data from two-\nand four-component CASSCF calculations per-\nformed by Sharma et al.124(X2C-CASSCF and\n4C-CASSCF, respectively) that did not incorpo-\nrate dynamic correlation effects outside the ac-\ntive space. The highest accuracy is demonstrated\nby DKH2-QDNEVPT2, which predicts the2D3/2\n–2D5/2splitting in Ag and Au with 2.5 % and\n0.1 % errors, respectively, relative to experiment.\nThe accuracy of QDNEVPT2 methods decreases\nin the order DKH2 >DKH1 >BP2 >BP1,\nwith the BP1-QDNEVPT2 errors reaching 7.5% for\nAu. Except for BP1-QDNEVPT2, all QDNEVPT2\nmethods agree better with experiment than X2C-\nCASSCF and 4C-CASSCF, suggesting that includ-\n10/uni00000025/uni00000033/uni00000014/uni00000010\n/uni00000034/uni00000027/uni00000031/uni00000028/uni00000039/uni00000033/uni00000037/uni00000015/uni00000025/uni00000033/uni00000015/uni00000010\n/uni00000034/uni00000027/uni00000031/uni00000028/uni00000039/uni00000033/uni00000037/uni00000015/uni00000027/uni0000002e/uni0000002b/uni00000014/uni00000010\n/uni00000034/uni00000027/uni00000031/uni00000028/uni00000039/uni00000033/uni00000037/uni00000015/uni00000027/uni0000002e/uni0000002b/uni00000015/uni00000010\n/uni00000034/uni00000027/uni00000031/uni00000028/uni00000039/uni00000033/uni00000037/uni00000015/uni00000028/uni00000032/uni00000030/uni00000010\n/uni00000026/uni00000026/uni00000036/uni00000027/uni0000000b/uni00000036/uni00000032/uni00000026/uni0000000c/uni00000013/uni00000011/uni00000013/uni00000015/uni00000011/uni00000018/uni00000018/uni00000011/uni00000013/uni0000001a/uni00000011/uni00000018/uni00000014/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000011/uni00000018/uni00000014/uni00000018/uni00000011/uni00000013/uni00000014/uni0000001a/uni00000011/uni00000018/uni00000015/uni00000013/uni00000011/uni00000013/uni00000008/uni00000003/uni00000030/uni00000024/uni00000028/uni00000003/uni00000059/uni00000056/uni00000011/uni00000003/uni00000048/uni0000005b/uni00000053/uni00000048/uni00000055/uni0000004c/uni00000050/uni00000048/uni00000051/uni00000057/uni00000044/uni0000004f/uni00000003/uni00000047/uni00000044/uni00000057/uni00000044/uni00000025/uni0000000f/uni00000003/uni00000024/uni0000004f/uni0000000f/uni00000003/uni0000002a/uni00000044/uni0000000f/uni00000003/uni0000002c/uni00000051/uni0000000f/uni00000003/uni00000037/uni0000004f\n/uni00000026/uni0000002b/uni0000000f/uni00000003/uni00000036/uni0000004c/uni0000002b/uni0000000f/uni00000003/uni0000002a/uni00000048/uni0000002b/uni0000000f/uni00000003/uni00000036/uni00000051/uni0000002b\n/uni00000032/uni0000002b/uni0000000f/uni00000003/uni00000036/uni0000002b/uni0000000f/uni00000003/uni00000036/uni00000048/uni0000002b/uni0000000f/uni00000003/uni00000037/uni00000048/uni0000002b\n/uni00000029/uni0000000f/uni00000003/uni00000026/uni0000004f/uni0000000f/uni00000003/uni00000025/uni00000055/uni0000000f/uni00000003/uni0000002c(a)\n* * (b)\nFigure 2: Mean absolute errors (MAE, %) in zero-field splitting for the main group elements and\ndiatomics calculated using the spin–orbit QDNEVPT2 methods and EOM-CCSD(SOC)120relative to\nthe experimental data. MAE are calculated for the chemical systems across each (a) group and (b)\nperiod of the periodic table. Bars that exceed the scale of the plot are indicated with asterisks. See\nTable 2 for data on individual systems.\nTable 4: Spin–orbit zero-field splitting (cm−1) in the2Dground term of transition metal atoms computed\nusing the spin–orbit QDNEVPT2 methods. Results are compared to the reference data calculated using\nthe X2C-MRCISD method122and experiment.123Shown in parentheses are the % errors with respect to\nexperimental results. All methods employed the X2C-TZVPall-2c basis set.\nSystem BP1- BP2- DKH1- DKH2- X2C-MRCISD122Experiment123\nQDNEVPT2 QDNEVPT2 QDNEPVT2 QDNEVPT2\nSc 174.3 (3.6) 139.8 (16.9) 174.3 (3.6) 140.9 (16.3) 185.5 (10.2) 168.3\nY 494.2 (6.8) 422.0 (20.4) 488.2 (7.9) 428.4 (19.2) 524.3 (1.1) 530.3\nLa 999.9 (5.1) 882.9 (16.2) 965.6 (8.3) 896.6 (14.9) 935.6 (11.2) 1053.2\ning dynamic correlation is quite important for com-\nputing accurate ZFS of Ag and Au.\n4.3 Heavy Elements and Molecules\nFinally, we consider U5+, NpO2+\n2, and UO+\n2,\nwhich contain actinide elements that are chal-\nlenging for perturbative two-component relativis-\ntic theories due to strong spin–orbit coupling and\nnearly degenerate partially filled f-orbitals in their\nelectronic states.53,122,128–130\nTable 6 presents the spin–orbit QDNEVPT2 re-\nsults for the2Fground term of U5+originating\nfrom the 5 f1electronic configuration. As a refer-\nence, we employ the experimental ZFS reported by\nKaufman et al.123and the theoretical data from\nvariational X2C-MRCISD calculations by Hu et\nal.122For this system, all computations were per-\nformed using the SARC-DKH2 basis set. DKH2-\nQDNEVPT2 shows the best agreement with ex-\nperiment out of all perturbative methods, under-\nestimating the experimental ZFS by 3.8 %, whichis similar to the error of X2C-MRCISD (3.4 %).\nAs for Ag and Au, the accuracy of spin–orbit QD-\nNEVPT2 methods decreases in the order DKH2\n(3.8 % error) >DKH1 (5.7 %) >BP2 (6.1 %) >\nBP1 (7.4 %), demonstrating that the second-order\ndescription of dynamical correlation and spin–orbit\ncoupling using the DKH2 Hamiltonian is essential\nfor achieving accuracy similar to X2C-MRCISD.\nNext, we use spin–orbit QDNEVPT2 to com-\npute the energies of excited states originating from\nthe zero-field splitting in the2Φ and2∆ terms\nof NpO2+\n2, which exhibit strong electron corre-\nlation and spin–orbit coupling effects (Table 7).\nIn this study, we benchmark against the recently\npublished results of SO-SHCI calculations57that\nutilized the variational two-component treatment\nof relativistic effects with the DKH1 Hamiltonian.\nWe note that the SO-SHCI calculations were per-\nformed in the (13e, 60o) active space, while our\nspin–orbit QDNEVPT2 methods correlated all 107\nelectrons in 433 molecular orbitals thus providing\na more accurate description of dynamic correla-\n11Table 5: Spin–orbit zero-field splitting (meV) in the2Dexcited term of Ag and Au computed using\nthe spin–orbit QDNEVPT2 methods. Results are compared to the data from the X2C-CASSCF and\n4C-CASSCF calculations124and experiment.125Shown in parentheses are the % errors with respect to\nexperimental results. All methods employed the X2C-TZVPall-2c basis set.\nSystem BP1- BP2- DKH1- DKH2- X2C- 4C- Experiment125\nQDNEVPT2 QDNEVPT2 QDNEVPT2 QDNEVPT2 CASSCF124CASSCF124\nAg 542 (2.1) 545 (1.6) 532 (3.9) 540 (2.5) 584 (5.4) 586 (5.7) 554\nAu 1636 (7.5) 1569 (3.1) 1522 (0.1) 1519 (0.1) 1571 (3.2) 1601 (5.2) 1521\nTable 6: Spin–orbit zero-field splitting (cm−1) in the2Fground term of U5+computed using the spin–orbit\nQDNEVPT2 methods. Results are compared to the reference data from the X2C-MRCISD calculations122\nand experiment.123All methods employed the SARC-DKH2 basis set.\nSystem BP1- BP2- DKH1- DKH2- X2C- Experiment123\nQDNEVPT2 QDNEVPT2 QDNEPVT2 QDNEVPT2 MRCISD122\nU(V) 8170.8 7144.1 8038.2 7316.4 7863.9 7605.8\ntion. Using the same basis set and molecular ge-\nometry as in the SO-SHCI study, the best agree-\nment with the reference data is achieved by the\nDKH2-QDNEVPT2 method with the largest er-\nror of 6.2 % (444 cm−1) for2Φ7/2u. The error\nin2Φ7/2uexcitation energy increases when using\nDKH1-QDNEVPT2 (10.4 %) or BP1-QDNEVPT2\n(12.5 %). The BP2-QDNEVPT2 method yields\nseverely underestimated excitation energies despite\nusing the same reference SA-CASSCF wavefunc-\ntion as the other spin–orbit QDNEVPT2 calcula-\ntions.\nIn Table 8, we also report the excited-state ener-\ngies for UO+\n2, which has the same electronic states\nand configuration as NpO2+\n2. We compare the\nspin–orbit QDNEVPT2 results to the data from\nexperimental measurements for the2∆3/2ustate127\nand perturbative CASPT2-SO calculations126uti-\nlizing the same basis set and structural parameters.\nInterestingly, we find that for this system BP2-\nand DKH2-QDNEVPT2 yield similar results, in a\ncloser agreement to the experimental2∆3/2uenergy\nthan BP1- and DKH1-QDNEVPT2, despite BP2-\nQDNEVPT2 showing large errors for NpO2+\n2.\nThis uneven performance of BP2-QDNEVPT2 is\nlikely associated with the low- Znature of approx-\nimations in the Breit–Pauli Hamiltonian and war-\nrants further investigation.\n5 Conclusions\nIn this work, we developed a formulation of\nquasidegenerate N-electron valence perturba-\ntion theory (QDNEVPT) that enables consistentsecond-order treatment of dynamic correlation and\nspin-orbit coupling for chemical systems with mul-\nticonfigurational electronic structure. Utilizing\nthe Breit–Pauli (BP) and exact two-component\nDouglas–Kroll–Hess (DKH) relativistic Hamil-\ntonians, the resulting approaches termed BP2-\nand DKH2-QDNEVPT2 have computational cost\nsimilar to that of conventional non-relativistic\nQDNEVPT2. Although derived from perturba-\ntion theory, the BP2- and DKH2-QDNEVPT2\nmethods compute the energies and wavefunctions\nof electronic states by diagonalizing an effective\nHamiltonian, which delivers the exact eigenvalues\nand eigenstates of BP and DKH2 Hamiltonians in\nthe limit of full configuration interaction. By ex-\npanding the treatment of dynamic correlation and\nspin-dependent relativistic effects to second order,\nBP2- and DKH2-QDNEVPT2 allow to obtain the\naccurate energies and wavefunctions of spin–orbit-\ncoupled states with compact non-relativistic rep-\nresentations of effective Hamiltonian. To quantify\nthe importance of second-order effects, we also con-\nsidered QDNEVPT2 with the first-order BP and\nDKH treatment of spin–orbit coupling, denoted as\nBP1- and DKH1-QDNEVPT2, respectively.\nOur results demonstrate that, out of four spin–\norbit QDNEVPT2 approaches studied in this work,\nDKH2-QDNEVPT2 provides the most accurate\nand reliable description of zero-field splitting for\na variety of chemical systems, including main\ngroup elements, transition metal atoms, actinides,\nand their compounds. For the main group ele-\nments with single-reference electronic structures,\nthe accuracy of DKH2-QDNEVPT2 is similar to\nthat of two-component equation-of-motion coupled\n12Table 7: Excited-state energies (cm−1) of NpO2+\n2computed using the spin–orbit QDNEVPT2 methods.\nResults are compared to the reference data from the SO-SHCI calculations.57For all methods, the uncon-\ntracted ANO-RCC-VTZP and cc-pVTZ basis sets were used for the Np and O atoms, respectively.\nElectronic BP1- BP2- DKH1- DKH2- SO-SHCI57\nstate QDNEVPT2 QDNEVPT2 QDNEPVT2 QDNEVPT2\n2Φ5/2u 0.0 0.0 0.0 0.0 0.0\n2∆3/2u 3603.7 3025.8 3570.5 3595.1 3429\n2Φ7/2u 8057.2 3162.6 7916.3 7608.6 7165\n2∆5/2u 9238.4 3288.4 9100.7 8956.7 8868\nTable 8: Excited-state energies (cm−1) of UO+\n2computed using the spin–orbit QDNEVPT2 methods.\nResults are compared to the data from CASPT2-SO calculations126and experiment.127The contracted\nANO-RCC-VTZP basis set was employed in all calculations.\nElectronic BP1- BP2- DKH1- DKH2- CASPT2- Experiment127\nstate QDNEVPT2 QDNEVPT2 QDNEPVT2 QDNEVPT2 SO126\n2Φ5/2u 0.0 0.0 0.0 0.0 0.0 0\n2∆3/2u 2912.1 2838.2 2922.9 2862.0 2616 2658\n2Φ7/2u 6471.7 6187.3 6429.7 6136.4 6679 –\n2∆5/2u 7905.2 7668.8 7918.8 7653.1 7889 –\ncluster theory with single and double excitations.\nWhen applied to the Ag and Au transition metal\natoms, DKH2-QDNEVPT2 shows higher accuracy\nthan exact two-component (X2C-) complete ac-\ntive space self-consistent field method, but exhibits\nlarger errors than the X2C implementation of mul-\ntireference configuration interaction with singles\nand doubles (X2C-MRCISD) for Sc, Y, and La.\nFor heavier elements and their compounds (U5+,\nNpO2+\n2, and UO+\n2), DKH2-QDNEVPT2 deliv-\ners results of the quality similar to that of X2C-\nMRCISD and spin–orbit implementation of semis-\ntochastic heat-bath CI (SO-SHCI). The DKH1-\nQDNEVPT2 method tends to show larger er-\nrors than DKH2-QDNEVPT2 by ∼2 to 3 %\nrelative to experimental results. The BP1- and\nBP2-QDNEVPT2 implementations exhibit accu-\nrate performance for the second- and third-period\nelements, but become increasingly inaccurate and\nunreliable for heavier atoms and molecules.\nOverall, the DKH2-QDNEVPT2 method devel-\noped in this work shows promise as an accu-\nrate electronic structure approach that incorpo-\nrates multireference effects, dynamic correlation,\nand spin–orbit coupling with affordable computa-\ntional cost. Applications of DKH2-QDNEVPT2 to\nchemical systems larger than the ones presented in\nthis study necessitate its efficient computer imple-\nmentation. Other developments of this approach\ncan be envisioned, such as extensions to simu-\nlate spin-dependent and magnetic properties, high-energy states, and nonradiative decay rates.\nSupporting Information Available\nStructural parameters, active spaces, and the\nstates included in SA-CASSCF calculations.\nAcknowledgement\nThis work was supported by the National Science\nFoundation under Grant No. CHE-2044648. Com-\nputations were performed at the Ohio Supercom-\nputer Center under Project No. PAS1583.131The\nauthors would like to thank Professor Lan Cheng\nfor insightful discussions.\nReferences\n(1) Kaszuba, J. P.; Runde, W. H. The aqueous\ngeochemistry of neptunium: Dynamic con-\ntrol of soluble concentrations with applica-\ntions to nuclear waste disposal. Environ. Sci.\nTechnol. 1999 ,33, 4427–4433.\n(2) Woodruff, D. N.; Winpenny, R. E.; Lay-\nfield, R. A. Lanthanide single-molecule mag-\nnets. Chem. Rev. 2013 ,113, 5110–5148.\n(3) McAdams, S. G.; Ariciu, A. M.; Kostopou-\nlos, A. K.; Walsh, J. P.; Tuna, F. Molecular\nsingle-ion magnets based on lanthanides and\n13actinides: Design considerations and new\nadvances in the context of quantum tech-\nnologies. Coord. Chem. Rev. 2017 ,346, 216–\n239.\n(4) Lv, K.; Fichter, S.; Gu, M.; M¨ arz, J.;\nSchmidt, M. An updated status and trends\nin actinide metal-organic frameworks (An-\nMOFs): From synthesis to application. Co-\nord. Chem. Rev. 2021 ,446, 214011.\n(5) Lee, N.; Petrenko, T.; Bergmann, U.;\nNeese, F.; Debeer, S. Probing valence or-\nbital composition with iron K βx-ray emis-\nsion spectroscopy. J. Am. Chem. Soc. 2010 ,\n132, 9715–9727.\n(6) Kasper, J. M.; Lestrange, P. J.;\nStetina, T. F.; Li, X. Modeling L2,3-\nEdge X-ray Absorption Spectroscopy with\nReal-Time Exact Two-Component Rela-\ntivistic Time-Dependent Density Functional\nTheory. J. Chem. Theory Comput. 2018 ,\n14, 1998–2006.\n(7) Maganas, D.; Kowalska, J. K.; Nooijen, M.;\nDebeer, S.; Neese, F. Comparison of mul-\ntireference ab initio wavefunction method-\nologies for X-ray absorption edges: A case\nstudy on [Fe(II/III)Cl 4]2–/1–molecules. J.\nChem. Phys. 2019 ,150, 104106.\n(8) Carbone, J. P.; Cheng, L.; Myhre, R. H.;\nMatthews, D.; Koch, H.; Coriani, S. An\nanalysis of the performance of coupled clus-\nter methods for K-edge core excitations and\nionizations using standard basis sets. Adv.\nQuantum Chem. 2019 ,79, 241–261.\n(9) Stetina, T. F.; Kasper, J. M.; Li, X.\nModeling L 2,3-edge X-ray absorption spec-\ntroscopy with linear response exact two-\ncomponent relativistic time-dependent den-\nsity functional theory. J. Chem. Phys. 2019 ,\n150, 234103.\n(10) Vidal, M. L.; Coriani, S.; Pokhilko, P.;\nKrylov, A. I. Equation-of-motion coupled-\ncluster theory to model l-edge x-ray ab-\nsorption and photoelectron spectra. J. Phys.\nChem. Lett. 2020 ,11, 8314–8321.\n(11) Jensen, H. J. A.; Dyall, K. G.; Saue, T.;\nFægri, K. Relativistic four-component mul-\nticonfigurational self-consistent-field theoryfor molecules: Formalism. J. Chem. Phys.\n1996 ,104, 4083–4097.\n(12) Liu, W. Ideas of relativistic quantum chem-\nistry. Mol. Phys. 2010 ,108, 1679–1706.\n(13) Saue, T. Relativistic Hamiltonians for\nChemistry : A Primer. ChemPhysChem\n2011 ,12, 3077–3094.\n(14) Fleig, T. Invited review: Relativistic wave-\nfunction based electron correlation methods.\nChem. Phys. 2012 ,395, 2–15.\n(15) Pyykk¨ o, P. Relativistic Effects in Chemistry:\nMore Common Than You Thought. Ann.\nRev. Phys. Chem. 2012 ,63, 45–64.\n(16) Reiher, M.; Wolf, A. Exact decoupling of the\nDirac Hamiltonian. II. The generalized Dou-\nglas–Kroll–Hess transformation up to ar-\nbitrary order. J. Chem. Phys. 2004 ,121,\n10945.\n(17) Kutzelnigg, W. Solved and unsolved prob-\nlems in relativistic quantum chemistry.\nChem. Phys. 2012 ,395, 16–34.\n(18) Gao, J.; Liu, W.; Song, B.; Liu, C. Time-\ndependent four-component relativistic den-\nsity functional theory for excitation energies.\nJ. Chem. Phys. 2004 ,121, 6658–6666.\n(19) Paquier, J.; Toulouse, J. Four-component\nrelativistic range-separated density-\nfunctional theory: Short-range exchange\nlocal-density approximation. J. Chem.\nPhys. 2018 ,149, 174110.\n(20) Yanai, T.; Nakajima, T.; Ishikawa, Y.; Hi-\nrao, K. A new computational scheme for the\nDirac–Hartree–Fock method employing an\nefficient integral algorithm. J. Chem. Phys.\n2001 ,114, 6526–6538.\n(21) Sun, S.; Stetina, T. F.; Zhang, T.; Hu, H.;\nValeev, E. F.; Sun, Q.; Li, X. Efficient Four-\nComponent Dirac-Coulomb-Gaunt Hartree-\nFock in the Pauli Spinor Representation.\nJ. Chem. Theory Comput. 2021 ,17, 3388–\n3402.\n(22) Fleig, T.; Sørensen, L. K.; Knecht, S.; Mar-\nian, C. M. Four-component relativistic cou-\npled cluster and configuration interaction\n14calculations on the ground and excited states\nof the RbYb moleeule. J. Phys. Chem. A\n2009 ,113, 12607–12614.\n(23) Liu, J.; Cheng, L. Relativistic coupled-\ncluster and equation-of-motion coupled-\ncluster methods. Wiley Interdiscip. Rev.\nComput. Mol. Sci. 2021 ,11, e1536.\n(24) Fleig, T.; Jensen, H. J.; Olsen, J.; Viss-\ncher, L. The generalized active space con-\ncept for the relativistic treatment of elec-\ntron correlation. III. Large-scale configu-\nration interaction and multiconfiguration\nself-consistent-field four-component meth-\nods with application to UO2. J. Chem. Phys.\n2006 ,124, 104106.\n(25) Abe, M.; Gopakmar, G.; Nakajima, T.;\nHirao, K. Relativistic Multireference\nPerturbation Theory: Complete Active-\nSpace Second-Order Perturbation Theory\n(CASPT2) With The Four-Component\nDirac Hamiltonian. Challenges Adv. Com-\nput. Chem. Phys. 2008 ,5, 157–177.\n(26) Abe, M.; Nakajima, T.; Hirao, K. The\nrelativistic complete active-space second-\norder perturbation theory with the four-\ncomponent Dirac Hamiltonian. J. Chem.\nPhys. 2006 ,125, 234110.\n(27) Shiozaki, T.; Mizukami, W. Relativistic\nInternally Contracted Multireference Elec-\ntron Correlation Methods. J. Chem. Theory\nComput. 2015 ,11, 4733–4739.\n(28) Reynolds, R. D.; Shiozaki, T. Zero-Field\nSplitting Parameters from Four-Component\nRelativistic Methods. J. Chem. Theory\nComput. 2019 ,15, 1560–1571.\n(29) Hoyer, C. E.; Lu, L.; Hu, H.; Shu-\nmilov, K. D.; Sun, S.; Knecht, S.; Li, X.\nCorrelated Dirac-Coulomb-Breit multicon-\nfigurational self-consistent-field methods. J.\nChem. Phys. 2023 ,158, 44101.\n(30) Van Lenthe, E.; Baerends, E. J.; Sni-\njders, J. G. Relativistic regular two-\ncomponent Hamiltonians. J. Chem. Phys.\n1993 ,99, 4597–4610.(31) Dyall, K. G. Interfacing relativistic and non-\nrelativistic methods. I. Normalized elimina-\ntion of the small component in the modified\nDirac equation. J. Chem. Phys. 1997 ,106,\n9618–9626.\n(32) Kutzelnigg, W. Relativistic one-electron\nHamiltonians ‘for electrons only’ and the\nvariational treatment of the Dirac equation.\nChem. Phys. 1997 ,225, 203–222.\n(33) Kutzelnigg, W.; Liu, W. Quasirelativistic\ntheory equivalent to fully relativistic theory.\nJ. Chem. Phys. 2005 ,123, 241102.\n(34) Barysz, M.; Sadlej, A. J.; Snijders, J. G.\nNonsingular two/one-component relativis-\ntic Hamiltonians accurate through arbitrary\nhigh order in α2.Int. J. Quant. Chem. 1997 ,\n65, 225–239.\n(35) Reiher, M.; Wolf, A. Exact decoupling of\nthe Dirac Hamiltonian. I. General theory. J.\nChem. Phys. 2004 ,121, 2037–2047.\n(36) Peng, D.; Liu, W.; Xiao, Y.; Cheng, L.\nMaking four- and two-component relativis-\ntic density functional methods fully equiv-\nalent based on the idea of ”from atoms\nto molecule”. J. Chem. Phys. 2007 ,127,\n104106.\n(37) Nakajima, T.; Hirao, K. The Douglas-Kroll-\nHess approach. Chem. Rev. 2012 ,112, 385–\n402.\n(38) Li, Z.; Xiao, Y.; Liu, W. On the spin sepa-\nration of algebraic two-component relativis-\ntic Hamiltonians. J. Chem. Phys. 2012 ,137,\n154114.\n(39) Cheng, L.; Gauss, J. Perturbative treatment\nof spin-orbit coupling within spin-free ex-\nact two-component theory. J. Chem. Phys.\n2014 ,141, 164107.\n(40) Breit, G. Dirac’s Equation and the Spin-Spin\nInteractions of Two Electrons. Phys. Rev.\n1932 ,39, 616–624.\n(41) Bearpark, M. J.; Handy, N. C.; Palmieri, P.;\nTarroni, R. Molecular Physics An Inter-\nnational Journal at the Interface Between\nChemistry and Physics Spin-orbit interac-\ntions from self consistent field wavefunctions\n15Spin-orbit interactions from self consistent\nfield wavefunctions. Mol. Phys. 1993 ,80,\n479–502.\n(42) Berning, A.; Schweizer, M.; Werner, H.-J.;\nKnowles, P. J.; Palmieri, P. Spin-orbit ma-\ntrix elements for internally contracted mul-\ntireference configuration interaction wave-\nfunctions Spin-orbit matrix elements for in-\nternally contracted mult. Mol. Phys. 2000 ,\n98, 1823–1833.\n(43) Van Lenthe, E.; Baerends, E. J.; Sni-\njders, J. G. Relativistic regular two-\ncomponent Hamiltonians. J. Chem. Phys.\n1993 ,99, 4597–4610.\n(44) Van Lenthe, E.; Baerends, E. J.; Sni-\njders, J. G. Relativistic total energy us-\ning regular approximations. J. Chem. Phys.\n1994 ,101, 9783–9792.\n(45) Van Lenthe, E.; Snijders, J. G.;\nBaerends, E. J. The zero-order regular\napproximation for relativistic effects: The\neffect of spin–orbit coupling in closed shell\nmolecules. J. Chem. Phys. 1996 ,105,\n6505–6516.\n(46) Douglas, M.; Kroll, N. M. Quantum electro-\ndynamical corrections to the fine structure\nof helium. Ann. Phys. 1974 ,82, 89–155.\n(47) Hess, B. A. Relativistic electronic-structure\ncalculations employing a two-component no-\npair formalism with external-field projection\noperators. Phys. Rev. A 1986 ,33, 3742.\n(48) Jansen, G.; Hess, B. A. Revision of the\nDouglas-Kroll transformation. Phys. Rev. A\n1989 ,39, 6016.\n(49) Sadlej, A. J.; Snijders, J. G.; Van Lenthe, E.;\nBaerends, E. J. Four component regular rela-\ntivistic Hamiltonians and the perturbational\ntreatment of Dirac’s equation. J. Chem.\nPhys. 1995 ,102, 1758–1766.\n(50) Kutzelnigg, W.; Liu, W. Quasirelativis-\ntic theory I. Theory in terms of a quasi-\nrelativistic operator. Mol. Phys. 2006 ,104,\n2225–2240.\n(51) Ilias, M.; Saue, T. An infinite-order two-\ncomponent relativistic Hamiltonian by asimple one-step transformation. J. Chem.\nPhys. 2007 ,126, 064102.\n(52) Ganyushin, D.; Neese, F. A fully varia-\ntional spin-orbit coupled complete active\nspace self-consistent field approach: Appli-\ncation to electron paramagnetic resonance g-\ntensors. J. Chem. Phys. 2013 ,138, 104113.\n(53) Mussard, B.; Sharma, S. One-Step Treat-\nment of Spin-Orbit Coupling and Elec-\ntron Correlation in Large Active Spaces. J.\nChem. Theory Comput. 2018 ,14, 154–165.\n(54) Jenkins, A. J.; Liu, H.; Kasper, J. M.;\nFrisch, M. J.; Li, X. Variational Relativis-\ntic Two-Component Complete-Active-Space\nSelf-Consistent Field Method. J. Chem.\nTheory Comput. 2019 ,15, 2974–2982.\n(55) Lu, L.; Hu, H.; Jenkins, A. J.; Li, X.\nExact-Two-Component Relativistic Mul-\ntireference Second-Order Perturbation The-\nory. J. Chem. Theory Comput. 2022 ,18,\n2983 – 2992.\n(56) Guo, Y.; Zhang, N.; Liu, W. SOiCISCF:\nCombining SOiCI and iCISCF for Varia-\ntional Treatment of Spin-Orbit Coupling.\nJ. Chem. Theory Comput. 2023 ,19, 6668–\n6685.\n(57) Wang, X.; Sharma, S. Relativistic Semis-\ntochastic Heat-Bath Configuration Interac-\ntion. J. Chem. Theory Comput. 2023 ,19,\n848–855.\n(58) Fedorov, D. G.; Finley, J. P. Spin-orbit mul-\ntireference multistate perturbation theory.\nPhys. Rev. A 2001 ,64, 042502.\n(59) Roos, B. O.; Malmqvist, P. Relativistic\nquantum chemistry: the multiconfigura-\ntional approach. Phys. Chem. Chem. Phys.\n2004 ,6, 2919–2927.\n(60) Kleinschmidt, M.; Tatchen, J.; Mar-\nian, C. M. SPOCK.CI: A multireference\nspin-orbit configuration interaction method\nfor large molecules. J. Chem. Phys. 2006 ,\n124, 124101.\n(61) Meitei, O. R.; Houck, S. E.; Mayhall, N. J.\nSpin-Orbit Matrix Elements for a Combined\nSpin-Flip and IP/EA approach. J. Chem.\nTheory Comput. 2020 ,16, 3597–3606.\n16(62) Neese, F.; Solomon, E. I. Calculation of\nZero-Field Splittings, g-Values, and the Rel-\nativistic Nephelauxetic Effect in Transition\nMetal Complexes. Application to High-Spin\nFerric Complexes. Inorg. Chem. 1998 ,37,\n6568–6582.\n(63) Malmqvist, P.; Roos, B. O.; Schimmelpfen-\nnig, B. The restricted active space (RAS)\nstate interaction approach with spin–orbit\ncoupling. Chem. Phys. Lett. 2002 ,357, 230–\n240.\n(64) Neese, F. Efficient and accurate approxi-\nmations to the molecular spin-orbit cou-\npling operator and their use in molecular g-\ntensor calculations. J. Chem. Phys. 2005 ,\n122, 034107.\n(65) Sayfutyarova, E. R.; Chan, G. K. L. A\nstate interaction spin-orbit coupling density\nmatrix renormalization group method. J.\nChem. Phys. 2016 ,144, 234301.\n(66) Angeli, C.; B., S.; Cestari, M.; Cimiraglia, R.\nA quasidegenerate formulation of the second\norder n-electron valence state perturbation\ntheory approach. J. Chem. Phys. 2004 ,121,\n4043–4049.\n(67) Sokolov, A. Y. Multireference Perturbation\nTheories Based on the Dyall Hamiltonian.\n2024; arXiv:2401.11262.\n(68) Lang, L.; Neese, F. Spin-dependent proper-\nties in the framework of the dynamic correla-\ntion dressed complete active space method.\nJ. Chem. Phys. 2019 ,150, 104104.\n(69) Lang, L.; Atanasov, M.; Neese, F. Improve-\nment of Ab Initio Ligand Field Theory by\nMeans of Multistate Perturbation Theory. J.\nPhys. Chem 2020 ,2020, 1025–1037.\n(70) Lang, L.; Sivalingam, K.; Neese, F. The com-\nbination of multipartitioning of the Hamil-\ntonian with canonical Van Vleck perturba-\ntion theory leads to a Hermitian variant of\nquasidegenerate N-electron valence pertur-\nbation theory. J. Chem. Phys. 2020 ,152,\n14109.\n(71) Majumder, R.; Sokolov, A. Y. Simulating\nSpin-Orbit Coupling with QuasidegenerateN-Electron Valence Perturbation Theory. J.\nPhys. Chem. A 2023 ,127, 546–559.\n(72) Li, Z.; Xiao, Y.; Liu, W. On the spin sepa-\nration of algebraic two-component relativis-\ntic Hamiltonians: Molecular properties. J.\nChem. Phys. 2014 ,141, 054111.\n(73) Hinze, J. MC-SCF. I. The multi-\nconfiguration self-consistent-field method.\nJ. Chem. Phys. 1973 ,59, 6424–6432.\n(74) Werner, H. J.; Meyer, W. A quadratically\nconvergent MCSCF method for the simul-\ntaneous optimization of several states. J.\nChem. Phys. 1980 ,74, 5794.\n(75) Roos, B. O.; Taylor, P. R.; Sigbahn, P. E.\nA complete active space SCF method\n(CASSCF) using a density matrix formu-\nlated super-CI approach. Chem. Phys. 1980 ,\n48, 157–173.\n(76) Werner, H. J.; Knowles, P. J. A second order\nmulticonfiguration SCF procedure with op-\ntimum convergence. J. Chem. Phys. 1985 ,\n82, 5053.\n(77) Siegbahn, P. E.; Alml¨ of, J.; Heiberg, A.;\nRoos, B. O. The complete active space SCF\n(CASSCF) method in a Newton–Raphson\nformulation with application to the HNO\nmolecule. J. Chem. Phys. 1981 ,74, 2384.\n(78) Kirtman, B. Simultaneous calculation of sev-\neral interacting electronic states by gener-\nalized Van Vleck perturbation theory. J.\nChem. Phys. 1981 ,75, 798.\n(79) Kirtman, B. Variational Form of Van Vleck\nDegenerate Perturbation Theory with Par-\nticular Application to Electronic Structure\nProblems. J. Chem. Phys. 2003 ,49, 3890.\n(80) Certain, P. R.; Hirschfelder, J. O. New Parti-\ntioning Perturbation Theory. I. General For-\nmalism. J. Chem. Phys. 2003 ,52, 5977.\n(81) Shavitt, I.; Redmon, L. T. Quasidegenerate\nperturbation theories. A canonical van Vleck\nformalism and its relationship to other ap-\nproaches. J. Chem. Phys. 2008 ,73, 5711.\n(82) Sharma, S.; Jeanmairet, G.; Alavi, A. Quasi-\ndegenerate perturbation theory using matrix\n17product states. J. Chem. Phys. 2016 ,144,\n034103.\n(83) Dyall, K. G. The choice of a zeroth-order\nHamiltonian for second-order perturbation\ntheory with a complete active space self-\nconsistent-field reference function. J. Chem.\nPhys. 1995 ,102, 4909.\n(84) Angeli, C.; Cimiraglia, R.; Evangelisti, S.;\nLeininger, T.; Malrieu, J. P. Introduction of\nn-electron valence states for multireference\nperturbation theory. J. Chem. Phys. 2001 ,\n114, 10252.\n(85) Angeli, C.; Cimiraglia, R.; Malrieu, J. P.\nn-electron valence state perturbation the-\nory: A spinless formulation and an efficient\nimplementation of the strongly contracted\nand of the partially contracted variants. J.\nChem. Phys. 2002 ,117, 9138.\n(86) Angeli, C.; Bories, B.; Cavallini, A.; Cimi-\nraglia, R. Third-order multireference pertur-\nbation theory: The n-electron valence state\nperturbation-theory approach. J. Chem.\nPhys. 2006 ,124, 054108.\n(87) Nishimoto, Y. Locating conical intersec-\ntions using the quasidegenerate partially\nand strongly contracted NEVPT2 methods.\nChem. Phys. Lett. 2020 ,744, 137219.\n(88) Kenneth G. Dyall, K. F. J. Introduction\nto Relativistic Quantum Chemistry ; Oxford\nUniversity Press Inc.: New York, 1995.\n(89) Markus Reiher, A. W. Relativistic Quan-\ntum Chemistry: The Fundamental Theory of\nMolecular Science ; Wiley-VCH: New York,\n2014.\n(90) Pyykk¨ o, P. Relativistic effects in chemistry:\nMore common than you thought. Annu. Rev.\nPhys. Chem. 2012 ,63, 45–64.\n(91) Liu, W. Advances in relativistic molecular\nquantum mechanics. Phys. Rep. 2014 ,537,\n59–89.\n(92) Liu, W.; Peng, D. Exact two-component\nHamiltonians revisited. J. Chem. Phys.\n2009 ,131, 031104.(93) Wolf, A.; Reiher, M.; Heß, B. A. The\ngeneralized Douglas-Kroll transformation. J.\nChem. Phys. 2002 ,117, 9215–9226.\n(94) Heß, B. A.; Marian, C. M.; Wahlgren, U.;\nGropen, O. A mean-field spin-orbit method\napplicable to correlated wavefunctions.\nChem. Phys. Lett. 1996 ,251, 365–371.\n(95) Foldy, L. L.; Wouthuysen, S. A. On the Dirac\nTheory of Spin 1/2 Particles and Its Non-\nRelativistic Limit. Phys. Rev. 1950 ,78, 29.\n(96) Stanton, R. E.; Havriliak, S. Kinetic balance:\nA partial solution to the problem of varia-\ntional safety in Dirac calculations. J. Chem.\nPhys. 1984 ,81, 1910–1918.\n(97) Cao, Z.; Li, Z.; Wang, F.; Liu, W. Combin-\ning the spin-separated exact two-component\nrelativistic Hamiltonian with the equation-\nof-motion coupled-cluster method for the\ntreatment of spin–orbit splittings of light\nand heavy elements. Phys. Chem. Chem.\nPhys. 2017 ,19, 3713–3721.\n(98) Moura, C. E. V.; Sokolov, A. Y. Prism,\nan implementation of electronic struc-\nture theories for simulating spectro-\nscopic properties, for current version see\nhttps://github.com/sokolov-group/prism.\n(99) Sun, Q.; Zhang, X.; Banerjee, S.; Bao, P.;\nBarbry, M.; Blunt, N. S.; Bogdanov, N. A.;\nBooth, G. H.; Chen, J.; Cui, Z. H. et al.\nRecent developments in the PySCF program\npackage. J. Chem. Phys. 2020 ,153, 024109.\n(100) Wang, X. Xubwa/Socutils; github, 2022.\nhttps://github.com/xubwa/socutils.\n(101) Roos, B. O.; Lindh, R.; Malmqvist, P.;\nVeryazov, V.; Widmark, P. O. Main Group\nAtoms and Dimers Studied with a New Rel-\nativistic ANO Basis Set. J. Phys. Chem. A\n2004 ,108, 2851–2858.\n(102) Pollak, P.; Weigend, F. Segmented Con-\ntracted Error-Consistent Basis Sets of\nDouble- and Triple- ζValence Quality for\nOne- and Two-Component Relativistic All-\nElectron Calculations. J. Chem. Theory\nComput. 2017 ,13, 3696–3705.\n18(103) Rolfes, J. D.; Neese, F.; Pantazis, D. A. All-\nelectron scalar relativistic basis sets for the\nelements Rb–Xe. J. Comput. Chem. 2020 ,\n41, 1842–1849.\n(104) Dunning, T. H. Gaussian basis sets for use\nin correlated molecular calculations. I. The\natoms boron through neon and hydrogen. J.\nChem. Phys. 1989 ,90, 1007.\n(105) Cheng, L.; Wang, F.; Stanton, J. F.;\nGauss, J. Perturbative treatment of spin-\norbit-coupling within spin-free exact two-\ncomponent theory using equation-of-motion\ncoupled-cluster methods. J. Chem. Phys.\n2018 ,148, 044108.\n(106) Moore, C. E., Atomic Energy Levels ; Cir-\ncular of the National Bureau of Standard\n(NBS, Washington, DC), 1949; Vol. I.\n(107) Radziemski, L. J.; Kaufman, V. Wave-\nlengths, Energy Levels, and Analysis of Neu-\ntral Atomic Chlorine (Cl i). JOSA 1969 ,59,\n424–443.\n(108) Luc-Koenig, E.; Morillon, C.; Verg` es, J.;\nLuc-Koenig, E.; Morillon, C.; Verg` es, J.\nEtude Exp´ erimentale et Th´ eorique de l’Iode\nAtomique. Observation du Spectre d’Arc In-\nfrarouge, Classification et Structure Hyper-\nfine. PhyS 1975 ,12, 199–219.\n(109) Martin, W. C.; Zalubas, R. Energy levels of\naluminum, Al I through Al XIII. J. Phys.\nChem. Ref. Data 1979 ,8, 817–864.\n(110) Kerr, J. A. K.P. Huber and G. Herzberg,\nmolecular spectra and molecular structure:\nIV constants of diatomic molecules. Anal.\nChim. Acta. 1982 ,144, 298.\n(111) Fink, E. H.; Setzer, K. D.; Ramsay, D. A.;\nVervloet, M. Near-infrared emission bands\nof TeH and TeD. J. Mol. Spectrosc. 1989 ,\n138, 19–28.\n(112) Gilles, M. K.; Polak, M. L.;\nLineberger, W. C. Photoelectron spec-\ntroscopy of IO-.J. Chem. Phys. 1991 ,95,\n4723–4724.\n(113) Ram, R. S.; Bernath, P. F. Fourier Trans-\nform Infrared Emission Spectroscopy of SeH.\nJ. Mol. Spectrosc. 2000 ,203, 9–15.(114) Miller, C. E.; Drouin, B. J. The X12Π3/2\nand X22Π1/2 Potential Energy Surfaces of\nFO.J. Mol. Spectrosc. 2001 ,205, 312–318.\n(115) Drouin, B. J.; Miller, C. E.; M¨ uller, H. S.;\nCohen, E. A. The Rotational Spectra, Iso-\ntopically Independent Parameters, and In-\nteratomic Potentials for the X12Π3/2 and\nX22Π1/2 States of BrO. J. Mol. Spectrosc.\n2001 ,205, 128–138.\n(116) Kramida, A. E.; Ryabtsev, A. N. A criti-\ncal compilation of energy levels and spectral\nlines of neutral boron. Phys. Scr. 2007 ,76,\n544–557.\n(117) Shirai, T.; Reader, J.; Kramida, A. E.;\nSugar, J. Spectral data for gallium: Ga\ni through Ga XXXI. J. Phys. Chem. Ref.\nData 2007 ,36, 509–615.\n(118) Deverall, G. V.; Meissner, K. W.; Zis-\nsis, G. J. Hyperfine structures of the res-\nonance lines of indium (In115). Phys. Rev.\n1953 ,91, 297–299.\n(119) Drouin, B. J.; Miller, C. E.; Cohen, E. A.;\nWagner, G.; Birk, M. Further Investigations\nof the ClO Rotational Spectrum. J. Mol.\nSpectrosc. 2001 ,207, 4–9.\n(120) Cao, Z.; Li, Z.; Wang, F.; Liu, W. Combin-\ning the spin-separated exact two-component\nrelativistic Hamiltonian with the equation-\nof-motion coupled-cluster method for the\ntreatment of spin–orbit splittings of light\nand heavy elements. Phys. Chem. Chem.\nPhys. 2017 ,19, 3713–3721.\n(121) Kerr, J. A. K.P. Huber and G. Herzberg,\nmolecular spectra and molecular structure:\nIV constants of diatomic molecules. Anal.\nChim. Acta. 1982 ,144, 298.\n(122) Hu, H.; Jenkins, A. J.; Liu, H.;\nKasper, J. M.; Frisch, M. J.; Li, X.\nRelativistic Two-Component Multirefer-\nence Configuration Interaction Method\nwith Tunable Correlation Space. J. Chem.\nTheory Comput. 2020 ,16, 2975–2984.\n(123) Kaufman, V.; Radziemski Jr., L. F. The\nsixth spectrum of uranium (U VI). J. Opt.\nSoc. Am. 1976 ,66, 599–600.\n19(124) Sharma, P.; Jenkins, A. J.; Scalmani, G.;\nFrisch, M. J.; Truhlar, D. G.; Gagliardi, L.;\nLi, X. Exact-Two-Component Multiconfig-\nuration Pair-Density Functional Theory. J.\nChem. Theory Comput. 2022 ,18, 2947–\n2954.\n(125) Kramida, A.; Ralchenko, Y.; Reader, J.;\nand NIST ASD Team, NIST Atomic\nSpectra Database (ver. 5.10), Available\nhttps://physics.nist.gov/asd , National\nInstitute of Standards and Technology,\nGaithersburg, MD., 2022.\n(126) Gendron, F.; Pritchard, B.; Bolvin, H.;\nAutschbach, J. Magnetic resonance prop-\nerties of actinyl carbonate complexes and\nplutonyl(VI)-tris-nitrate. Inorg. Chem.\n2014 ,53, 8577–8592.\n(127) Merritt, J. M.; Han, J.; Heaven, M. C. Spec-\ntroscopy of the UO2+ cation and the delayed\nionization of UO2. J. Chem. Phys. 2008 ,\n128, 84304.\n(128) Ruip´ erez, F.; Danilo, C.; R´ eal, F.; Fla-\nment, J. P.; Vallet, V.; Wahlgren, U. An\nab initio theoretical study of the electronic\nstructure of UO 2+ and [UO2(CO3)3] 5-. J.\nPhys. Chem. A 2009 ,113, 1420–1428.\n(129) Gendron, F.; P´ aez-Hern´ andez, D.; Not-\nter, F. P.; Pritchard, B.; Bolvin, H.;\nAutschbach, J. Magnetic Properties and\nElectronic Structure of Neptunyl(VI) Com-\nplexes: Wavefunctions, Orbitals, and\nCrystal-Field Models. Chem. – A Eur. J.\n2014 ,20, 7994–8011.\n(130) Knecht, S.; Keller, S.; Autschbach, J.; Rei-\nher, M. A Nonorthogonal State-Interaction\nApproach for Matrix Product State Wave\nFunctions. J. Chem. Theory Comput. 2016 ,\n12, 5881–5894.\n(131) Center, O. S. Ohio Supercomputer Center.\nhttp://osc.edu/ark:/19495/f5s1ph73 .\n20TOC Graphic\n* *Second-Order\nDKH Spin–Orbit\nCouplingQDNEVPT2\nElectron\nCorrelation\nDKH2-QDNEVPT2Consistent Treatment of Spin–Orbit Coupling \nand Dynamic Correlation \n21" }, { "title": "1302.5465v1.Anderson_Localization_of_cold_atomic_gases_with_effective_spin_orbit_interaction_in_a_quasiperiodic_optical_lattice.pdf", "content": "arXiv:1302.5465v1 [cond-mat.quant-gas] 22 Feb 2013Anderson Localization of cold atomic gases with effective sp in-orbit interaction in a\nquasiperiodic optical lattice\nLu Zhou1,2, Han Pu2and Weiping Zhang1\n1Quantum Institute for Light and Atoms, Department of Physic s,\nEast China Normal University, Shanghai 200062, China and\n2Department of Physics and Astronomy, and Rice Quantum Insti tute,\nRice University, Houston, TX 77251-1892, USA\nWe theoretically investigate the localization properties of a spin-orbit coupled spin-1/2 particle\nmoving in a one-dimensional quasiperiodic potential, whic h can be experimentally implemented us-\ning cold atoms trapped in a quasiperiodic optical lattice po tential and external laser fields. We\npresent the phase diagram in the parameter space of the disor der strength and those related to the\nspin-orbit coupling. The phase diagram is verified via multi fractal analysis of the atomic wavefunc-\ntions and the numerical simulation of diffusion dynamics. We found that spin-orbit coupling can\nlead to the spectra mixing (coexistence of extended and loca lized states) and the appearance of\nmobility edges.\nPACS numbers: 03.75.Lm, 71.70.Ej, 72.15.Rn, 03.75.Kk\nI. INTRODUCTION\nAnderson localization (AL) is considered as a funda-\nmental physical phenomenon, which was first studied in\nthe system of noninteracting electrons in a crystal with\nimpurities [1]. AL predicts the absence of diffusion of\nelectronic spin, which stems from the disorder of crystal\nand is the result of quantum interference. Since disorder\nis ubiquitous, AL is rather universal and can occur in a\nvariety of other physical systems including light waves[2]\nand atomic matter-waves [3–5]. Due to its intimate re-\nlation with metal-insulator transition, many interesting\ntopics in AL such as the interplay between nonlinearity\nand disorder [6–9] are still under intense study.\nIn condensed-matter physics, spin-orbit (SO) coupling\noriginates from the interaction between the intrinsic spin\nofanelectronandthemagneticfieldinduced byitsmove-\nment. It connectsthe electronicspin toits orbitalmotion\nand thus the electron transport becomes spin-dependent.\nSO interaction can significantly affect AL and this prob-\nlem had been addressed by a few works in the electronic\ngas system [10, 11].\nThe experimental realization of ultracold quantum\ngases, together with the technique of optical lattice po-\ntential, have provided a powerful playground for the sim-\nulation of condensed-matter systems. In this compos-\nite system, one can achieve unprecedented control over\nalmost all parameters by optical or magnetical means.\nSpecifically, pseudo-disorder can be generated by super-\nimposing two standing optical waves of incommensurate\nwavelengths together. As a consequence, AL of atomic\nmatter wave can take place, which had been experimen-\ntally observed [5, 7, 8] and extensively studied in theory\n[6, 12].\nThis work is motivated by the recent experimental re-\nalization of SO coupling in ultracold atomic gas [13–16].\nWe investigate the impact of SO coupling on Anderson\nlocalizationofaspin-1/2particleusingthe systemofcoldatomic gases trapped in a quasiperiodic one-dimensional\n(1D) optical lattice potential and simultaneously subject\nto the laser-induced SO interaction. The similar topic\nhad also been addressed in [17, 18], with the focus on the\nlocalization properties of relativistic Dirac particles with\ncold atoms in a light-induced gauge field. Our model and\nmethod are different from theirs and we do not consider\nthe relativistic region.\nThe paper is organized as follows. Sec. II introduces\nthe theoretical model and tight-binding approximation\nis applied in Sec. III. In Sec. IV we present the phase\ndiagram and discuss its implications. Sec. V is devoted\nto the multifractal analysis of the atomic wavefunction,\nfrom which the proposed phase diagram is verified. The\ndiffusion dynamics is studied in Sec. VI for a initially\nlocalized Gaussian wavepacket. Finally we conclude in\nSec. VII.\nII. MODEL AND HAMILTONIAN\nWe consider the following model depicted in Fig. 1,\ncold atomic gas with internal spin states |↑/an}bracketri}htand|↓/an}bracketri}ht\nconfined in a spin-independent 1D quasiperiodic optical\nlattice potential s1ER1sin2(k1x) +s2ER1sin2(k2x+φ)\nalong thex-direction, which is formed by combining\ntwo incommensurate optical lattice together [5]. Here\nki= 2π/λiis the lattice wavenumber, siis the height of\nthe lattice in unit of the recoil energy ER1=h2/2mλ2\n1\nandφistherelativephasebetweenthetwostanding-wave\nmodes (without loss ofgenerality, weassume φ= 0in the\nfollowing discussion). It is assumed that the depth of the\nlattice with wavevector k1is deep enough to serve as the\ntight-binding primary lattice. In the meanwhile, a pair\nof counter propagating Raman beams couple the atomic\nstates|↑,kx=q/an}bracketri}htand|↓,kx=q+2kR/an}bracketri}ht, which creats the\neffective SO coupling [13].\nIn the basis composed of atomic pseudo-spin states2\n1 2 \n!\"#\n1 2 \nx\nFIG. 1: (Color online) Schematic diagram showing the system\nunder consideration.\n{|↑/an}bracketri}ht,|↓/an}bracketri}ht}, the single-particle Hamiltonian reads\nˆh=/bracketleftbiggp2\nx\n2m+s1ER1sin2(k1x)+s2ER1sin2(k2x)/bracketrightbigg\nˆI\n+Ω\n2/parenleftbigg\n0e2ikRx\ne−2ikRx0/parenrightbigg\n,\nwith Ω the effective Raman coupling strength. Here\nwe have assumed that the Raman two-photon detun-\ning is 0. Then introduce the dressed pseudospin states/braceleftbig\n|↑/an}bracketri}ht′=|↑/an}bracketri}hte−ikRx,|↓/an}bracketri}ht′=|↓/an}bracketri}hteikRx/bracerightbig\n, which is equivalent\nto performing a pseudo-spin rotation with the operator\nˆR(kR) = exp( −ikRxˆσz), and perform a global pseudo-\nspin rotation ˆ σz→ˆσy, ˆσy→ˆσx, ˆσx→ˆσz[13, 16]. In\nthe new basis, Hamiltonian ˆhcan then be rewritten as\nˆh=ˆhSO+V=/parenleftig\npx−ˆA/parenrightig2\n2m+Ω\n2ˆσz+s1ER1sin2(k1x)\n+s2ER1sin2(k2x), (1)\nin which ˆhSOdescribes single-particle motion in\nthe presence of the effective SO coupling, whichis embodied in the vector potential ˆA=−mλˆσy\n(λ=/planckover2pi1kR/mcharacterize SOC strength) and the effec-\ntive Zeeman field Ω /2.\nˆheffectively describes a spin-1 /2 particle moving in a\n1D quasiperiodic potential and subject to both the Zee-\nman field and equal Rashba-Dresselhaus SO coupling,\nwhich can be used to simulate the corresponding con-\ndensed matter system such as the motion of electron in\none dimentional semiconductor nanowire with disorder\nand SO interactions.\nIII. TIGHT-BINDING APPROXIMATION\nIn the language of quantum field theory, the total\nHamiltonian describing our system reads\nˆH(x) =/integraldisplay\ndxˆΨ†(x)ˆh(x)ˆΨ(x), (2)\nwithˆΨ =/parenleftig\nˆψ↑,ˆψ↓/parenrightigT\nthe atomic field operators.\nIn the tight-binding limit, the atomic field opera-\ntor can be expanded as ˆΨ(x) =/summationtext\njwj(x)ˆcj, where\nwj(x) =w(x−xj) is the Wannier state of the primary\nlattice at the j-th site and ˆ cj= (ˆcj↑,ˆcj↓)Tare annihila-\ntion operators. By considering that the tunneling takes\nplace between nearest neighbour sites and retaining only\nthe onsite contribution of the secondary lattice, one can\nachieve the following description of (2) (with the energy\nmeasured in units of ER1and length scaled in units of\nk−1\n1)\nˆH=/summationdisplay\nj/bracketleftbigg\n−/parenleftig\nˆc†\njˆTˆcj+1+H.c./parenrightig\n−∆cos(2πβj)ˆc†\njˆcj+Ω\n2ˆc†\njˆσzˆcj/bracketrightbigg\n=/summationdisplay\nj/braceleftbigg\n−J/bracketleftig\nˆc†\nj(cosπγ−iˆσysinπγ)ˆcj+1+H.c./bracketrightig\n−∆cos(2πβj)ˆc†\njˆcj+Ω\n2ˆc†\njˆσzˆcj/bracerightbigg\n,\nwhere the tunneling amplitude ˆT=Jexp/parenleftig\n−i//planckover2pi1/integraltextˆAdl/parenrightig\nis obtained through Peierls subsititution [19, 20], which was\nalso used in recent works to study the impact of SO coupling on two-d imensional Bose-Hubbard model [21–23]./integraltextˆAdl=ˆA(xj−xj+1) =/planckover2pi1πγˆσyis the integral of the vector potential along the hopping path with γ=kR/k1.Jis\nthe tunneling amplitude without SO coupling, β=k2/k1.Jand ∆ can be calculated as\nJ=−/integraldisplay\ndxwj+1(x)/bracketleftbigg\n−d2\ndx2+s1sin2x/bracketrightbigg\nwj(x),\n∆ =s2β2\n2/integraldisplay\ndxcos(2βx)|w(x)|2.3\nBy writing |ψ/an}bracketri}ht=/summationtext\nj,σψj,σˆc†\nj,σ|0/an}bracketri}ht, the stationary Schr¨ odinger equation ˆH|ψ/an}bracketri}ht=E|ψ/an}bracketri}htlead to\n−Jcos(πγ)(ψj+1,↑+ψj−1,↑)−Jsin(πγ)(ψj+1,↓−ψj−1,↓)−∆cos(2πβj)ψj,↑+Ω\n2ψj,↑=Eψj,↑,(3a)\n−Jcos(πγ)(ψj+1,↓+ψj−1,↓)+Jsin(πγ)(ψj+1,↑−ψj−1,↑)−∆cos(2πβj)ψj,↓−Ω\n2ψj,↓=Eψj,↓.(3b)\nThe second terms on the left hand side of Eqs. (3),\nwhich are proportional to Jsin(πγ), represent the spin-\nflipping tunneling which arises from the effective SO in-\nteraction. In the absence of SO coupling ( γ= 0, Ω = 0),\nspin-↑and↓components are decoupled and we have\n−J(ψj+1+ψj−1)−∆cos(2πβj)ψj=Eψj,(4)\nwhich represents the typical Harper equation [24] or the\nAubry-Andr´ e model [25]. Eq. (4) satisfies Aubry duality,\nas can be seen by performing the transformation ψj=/summationtext\nm˜ψmeim(2πβj), insert it into Eq. (4) will lead to\n−∆\n2/parenleftig\n˜ψm+1+˜ψm−1/parenrightig\n−2Jcos(2πβm)˜ψm=E˜ψm,(5)\nEqs. (4) and (5) are identical at ∆ /J= 2. Since thetransformationmadeaboverepresentsthetypicalFourier\ntransform which transforms localized states to extended\nstates and vice versa, then the critical point ∆ /J= 2\nis identified as the transition point between the local-\nizedstatesandextendedstates, i.e., allthe single-particle\nstates are extended when ∆ /J <2 and localized when\n∆/J >2.\nThe properties of the Aubry-Andr´ e model, as repre-\nsented by Eq.(4), havebeen theoreticallystudied [26, 27]\nand it can be implemented in systems of Bloch electrons\n[20] and cold atoms [5]. We have also studied this model\nwith ∆ dressed by a cavity field through nonlinear feed-\nback [9].\nA similar transformation ψj,σ=ǫσ/summationtext\nm˜ψm,σeim(2πβj)\n(ǫ↑= 1,ǫ↓=i) made to Eqs. (3) will lead to\n−∆\n2/parenleftig\n˜ψm+1,↑+˜ψm−1,↑/parenrightig\n+2Jsin(πγ)sin(2πβm)˜ψm,↓−2Jcos(πγ)cos(2πβm)˜ψm,↑+Ω\n2˜ψm,↑=E˜ψm,↑,(6a)\n−∆\n2/parenleftig\n˜ψm+1,↓+˜ψm−1,↓/parenrightig\n+2Jsin(πγ)sin(2πβm)˜ψm,↑−2Jcos(πγ)cos(2πβm)˜ψm,↓−Ω\n2˜ψm,↓=E˜ψm,↓.(6b)\nA comparison between Eqs. (3) and Eqs. (6) shows that\nthe presence of the spin-flipping tunneling terms breaks\nthe duality. This distinguishes our current work from\nthat reported in Ref. [11], in which the authors studied a\nsystemoftwo-dimensional(2D) electronsonasquarelat-\ntice subject to Rashba spin-orbit coupling and immersed\ninaperpendicularuniformmagneticfield. Inthissystem,\nit has been shown [11] that a gneralized Aubry duality\nis preserved when tunneling along the two perpendicular\nlattice directions are exchanged. Such an operation is\nnot available in our system as ours is an intrinsically 1D\nmodel.\nDue to the lack of duality in the current model, it is\nnot clear whether there exists a phase transition between\nthe localized and extended states in the present system.\nIn addition, what effect will SO coupling take? Will it\nenhance the tendency to localization or delocalization?\nWe focus on these problems in the following discussion.IV. PHASE DIAGRAM ANALYSIS\nHere we follow the method in [11] to map the phase\ndiagram using a quantity called the total width of all the\nenergy bands B, which had been proved to be useful in\ninvestigating phase transition in a quasiperiodic system\n[11, 26, 27]. In order to observe its property, as people\nusually do, we first choose the optical lattice wavelength\nratioβto beβn=Fn/Fn+1=p/q, in whichFnis the\nn-th Fibonacci number defined by the recursion relation\nFn+1=Fn+Fn−1withF0=F1= 1. When n→ ∞,\nβn→/parenleftbig√\n5−1/parenrightbig\n/2, which is the inverse of the golden\nratio.\nSincepandqare integers prime to each other,\nthe system is periodic with the period q. Under\nthe periodic boundary condition, according to Bloch’s\ntheorem,ψi+q,σ=eikxqψi,σ. Eqs. (3) then re-\nduce to an eigenvalue problem Hψ=Eψwith\nψ= (ψ1,↑,ψ1,↓,ψ2,↑,ψ2,↓,···,ψq−1,↑,ψq−1,↓,ψq,↑,ψq,↓)\nand the 2q×2qmatrixHtakes the following form4\nH=\nH1L0··· 0e−ikxqL†\nL†H2L0 0\n0L†H3L0\n0···\n···L0\n0 0 L†Hq−1L\neikxqL0 0 L†Hq\n\nwith\nHj=/parenleftbigg\n−∆cos(2πβj+φ)+Ω\n20\n0 −∆cos(2πβj+φ)−Ω\n2/parenrightbigg\nandL=/parenleftbigg\n−Jcos(πγ)−Jsin(πγ)\nJsin(πγ)−Jcos(πγ)/parenrightbigg\n. The Hermite\nmatrixHcan be diagonalized with 2 qreal eigenvalues\nEi(kx), which form 2 qenergy bands as the function of\nkxin the first Brillouin zone q|kx| ≤π.\nIn the absence of SO coupling, the energy bands are\ndegenerate for spin- ↑and↓, with the band edges locate\natkx= 0 andkx=±π/q, soBcan be calculated by B=/summationtext2q\ni=1|Ei(0)−Ei(±π/q)|. It was first demonstrated in\n[26]thatforextendedstateswith∆ /J <2,Bapproaches\na finite value for q→ ∞; while forlocalized states ∆ /J >\n2,Brapidly tends to zero as q→ ∞; at the critical point\n∆/J= 2,B≈q−1. In this manner, the critical value\n∆ = ∆ csignaling AL transition can be determined by\nobserving the property of Bas a function of the period\nq.\n/s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32/s32\nΩ\n∆/s99/s114/s105/s116/s105/s99/s97/s108\n/s108/s111/s99/s97/s108/s105/s122/s101/s101/s120/s116/s101/s110/s100\nFIG. 2: (Color online) Phase diagram for Anderson localization\nof SO coupled BEC in 1D quasiperiodic lattice in the paramete r\nspace∆-Ω.γ= 0.7,∆andΩare estimated in units of J.\nNow taking SO coupling into account, we anticipate\nthat the phase diagram is composed of three different\nphases: (i) Extended phasein which the energyspectrum\nis purely continuous and all the eigenstates are extended;\n(ii)Localizedphaseischaracterizedbypurelydensepoint\nand all the wavefunctions are localized; (iii) The energyspectrum is mixed in the critical phase with extended\neigenstatescoexistwithlocalizedones. Usingthemethod\ndescribed above, ∆ cis determined as a function of Ω,\nwhich is indicated in the phase diagram of Fig. 2 by\nthe line separating the regions signaling localized phase\nand critical phase. Calculation is performed with the\nparameterof γ= 0.7, whichisused throughoutthe paper\nand can be experimentally realized for87Rb atoms by\nadjusting the angle between Raman beams [13]. At Ω =\n0, AL transition occurs at ∆ /J= 2, reminiscent of the\nsituation without SO coupling. This can be understood\nfromEq. (1) by that SO interactioncanbe removedfrom\nthe Hamiltonian through a unitary transformation with\nthe operator ˆS= exp(−ixˆσy/2ξ) when Ω = 0. Examples\nof data are shown in Fig. 3(a). At ∆ /J= 2.02,Btends\nto zero for Ω <Ω(∆c) andBtends to a finite value for\nΩ>Ω(∆c).\nDue to that duality is broken by the SO coupling here,\nthe boundary between extended phase and critical phase\nis not related to that separates localized phase and crit-\nical phase, which is different from [11]. The extended\nphase and critical phase cannot be differentiated by ex-\naming the properties of B, since the energy spectrum\ncontain absolutely continuous parts in both these two\nphases, which leads Bto a finite value in the quasiperi-\nodic limit, as shown in Fig. 3(b).\n/s49 /s50 /s51/s45/s52/s45/s51/s45/s50/s45/s49/s48\n/s49 /s50 /s51/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53\n/s32Ω /s47/s74/s61/s48/s46/s48/s48/s49\n/s32Ω /s47/s74/s61/s48/s46/s48/s49\n/s32Ω /s47/s74/s61/s48/s46/s50\n/s32/s32/s108/s111/s103\n/s49/s48/s66\n/s108/s111/s103\n/s49/s48/s113/s40/s97/s41∆ /s47/s74/s61/s50/s46/s48/s50\n/s32/s32\n/s108/s111/s103\n/s49/s48/s113/s32Ω /s47/s74/s61/s48/s46/s48/s48/s49\n/s32Ω /s47/s74/s61/s48/s46/s48/s49\n/s32Ω /s47/s74/s61/s48/s46/s50∆ /s47/s74/s61/s49/s46/s57/s56\n/s40/s98/s41\nFIG. 3: Total bandwidth versus period q. Atq→ ∞the system\nbecomes quasiperiodic. The parameters are specified in the fi gure.\nThe localization property of the atomic wavefunction\ncan be characterized with the inverse participation ratio\n(IPR) which is defined as\nP−1=N/summationdisplay\nj=1/parenleftig\n|ψj,↑|4+|ψj,↓|4/parenrightig\n,\nin whichNis the number of lattice sites, ψj,↑(↓)are so-\nlutions of Eqs. (3) and fulfil the normalization condition/summationtext\nj/parenleftig\n|ψj,↑|2+|ψj,↓|2/parenrightig\n= 1. IPRreflectsthe inverseofthe\nnumber of the lattice sites being occupied by the atoms.5\nFor extended states, P−1→1/Nand approach 0 for\nlargeN. While for localized states, IPR tends to a finite\nvalue and a larger value of P−1means that the atoms\nare more localized in space.\n1 2 3−4−2024\n∆/JEnergy(a)\n1 2 3−4−2024\n∆/J(b)\n \nFIG. 4: Eigenenergies obtained from numerical diagonalization\nof Hamiltonian matrix, as a function of ∆/J. Calculations are\nperformed under periodic condition with β= 610/987,N= 987,\nγ= 0.7and (a)Ω = 0.1; (b)Ω = 1.\nWe use IPR to determine the boundary separating the\nextended phase and critical phase, which is identified by\nthe turning point of IPR as a function of ∆ /J, as those\nhadbeendoneinmanypreviousworks[9,12,28,30]. The\ncalculation is performed with β=F14/F15= 610/987\nandN=F15= 987 under periodic boundary condition.\nThis give the extend-critical phase boundary shown in\nFig. 2, which indicate that with the increase of the Rabi\nfrequency Ω the system is more likely to start become lo-\ncalized. In order to understand this, we plot the energy\nspectrum as a function of ∆ /J. When Ω is relatively\nsmall, the spectrum shown in Fig. 4(a) possesses similar\nproperties as that in the absence of SO coupling: Along\nwiththeincreaseof∆ /J,twomajorgapsdevidethespec-\ntrum into three parts, each of which in turn devides into\nthree smaller parts, and so on. This is because the value\nof 1/βlies between 2 and 3. The spectrum of localized\nstates is then characterized by the presence of an infinite\nnumber of gaps and bands. The effective Zeeman term\nwhich is propotional to Ω, in combination with SO cou-\npling and the lattice structure, take the effect of opening\ngaps between different energy bands, as shown in Fig.\n4(b). So the critical value ∆ ctakes a smaller value with\nthe increase of Ω.V. MULTIFRACTAL ANALYSIS OF\nWAVEFUNCTIONS\nTo check the proposed phase diagram, we investi-\ngate the scaling property of the wavefunctions using the\nmethod of multrifractal analysis described in [11]. Take\nthe period of the lattice to be Fibonacci number Fn,\nfrom the wavefunctions {ψj,σ}we have the probability\npj=|ψj,↑|2+|ψj,↓|2, which is normalized as/summationtextFn\nj=1pj= 1.\nThe scaling index αforpjis defined as pj=F−α\nn. We\nthen assume that the number of sites satisfying the scal-\ning is propotional to Ffn(α)\nn,f(α) can be calculated as\nf(α) = lim n→∞fn(α).\nThe localization properties of wavefunctions are char-\nacterized by f(α) in the following manner: For extended\nwavefunctions, all the lattice satisfy pj∼F−1\nn, sof(α)\nis fixed at f(α= 1) = 1. On the other hand, a local-\nized wavefunction has a nonvanishing probability only\non a finite number of sites. These sites have α= 0\n[f(0) = 0] and other sites with probability zero have\nα=∞[f(∞) = 1]. For critical wavefunctions, αhas\na distribution, which means that f(α) is a smooth func-\ntion defined on a finite interval [ αmin,αmax].\nNumericallywecalculate fn(α) forFibonacci indices n\nand extrapolate them to n→ ∞. One can then discrimi-\nnate extended, localized and critical wavefunction by ex-\naming the minimum value of αin the following manner\nextended wavefunction αmin= 1,\ncritical wavefunction αmin/ne}ationslash= 0,1,\nlocalized wavefunction αmin= 0.\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s32/s32\nα\n/s109/s105/s110\n/s49/s47/s110/s40/s97/s41\n/s32/s32\n/s49/s47/s110/s40/s98/s41\n/s32/s32\n/s49/s47/s110/s40/s99/s41\nFIG. 5:αminversus1/nfor the wavefunctions of the lowest band\n/circlecopyrt(i= 1)and the centre band /square(i=Fn). (a)Ω/J= 0.2,\n∆/J= 1.5; (b)Ω/J= 0.4,∆/J= 1.5; (c)Ω/J= 2.5,∆/J=\n0.1corresponds to extend phase, critical phase and localize ph ase,\nrespectively.\nαminis calculated for example wavefunctions and the\nresults are shown in Fig. 5. The wavefunction of the\nlowest band is denoted by i= 1, while that at the centre6\nof the energy spectrum is denoted by i=Fn. First, for\nΩ/J= 0.2, ∆/J= 1.5 at which the system is in the ex-\ntendedphaseaccordingtothephasediagramFig. 2, αmin\nextrapolates to 1 for both i= 1 andi=Fn, as shown in\nFig. 5(a), indicating that the energy spectrum is purely\ncontinuous and all the wavefunctions are extended. The\npoint Ω/J= 0.4, ∆/J= 1.5 corresponds to the critical\nphase in Fig. 2, and in Fig. 5(b) one can find out that\nαminextrapolates to 0 for i= 1 andαminextrapolates\nto 1 fori=Fn. This suggests that the wavefunction at\nthe edge of the energy spectrum is localized while that\nat the centre is extended, which indicates the existance\nof mobility edges.\nThe appearance of mobility edges here can be under-\nstood as the result of the breaking of original self-duality\nvia SO interaction, which can also be aroused by other\neffects such as hopping beyond neighbouring lattice sites\n[29, 30]. We would like to note that, even if the duality is\npreserved, SO coupling can also lead to the appearance\nof critical phase and mobility edges, as those had been\ndemonstrated in [11].\nVI. DIFFUSION DYNAMICS\nIn realistic experiment, localization properties can be\ninvestigated by loading the SO-coupled BEC into the\nquasiperiodic potential and observing its transportation\nalong the lattice [5]. The equations-of-motion associated\nwith Eqns. (3) are\ni∂Ψj\n∂t=−Je−iπγˆσy(Ψj+1+Ψj−1)\n+/bracketleftbigg\n−∆cos(2πβj)+Ω\n2ˆσz/bracketrightbigg\nΨj,(7)\nwhere Ψ j= (ψj,↑,ψj,↓)T. We study the diffusion of ul-\ntracold atomic gas in quasiperiodic optical lattice with\nEq. (7) by taking the initial atomic wavefunction to be\na localized Gaussian wavepacket with width a\nΨj(t= 0) =/parenleftbig\n2a√π/parenrightbig−1/2e−j2/2a2/parenleftbigg1\ni/parenrightbigg\n,\nin which we take a= 5 in the following calculation. Here\nweassumethattheatomicwavepacketsinitiallylieinthe\ncentre atj= 0 with the system size of 2000 lattice sites.\nThe numerical simulation is performed with vanishing\nboundary condition and during the time evolution the\natomic wavepacket never reaches the boundaries so that\nthe effect of boundary condition does not appear.\nTo measure the localization, we use the width of the\nwave-packet defined as\nw=/radicalbigg/angbracketleftig\n(∆x)2/angbracketrightig\n=\n\n/summationdisplay\njj2/parenleftig\n|ψj,↑|2+|ψj,↓|2/parenrightig\n\n1/2\n.\nIn the absence of SO coupling, the time evolution of w(t)\ncanbe parametrizedas w(t)∼tη[6,31], anditspropertyis intimately related to the localization properties of the\nsystem:\n(i) in the extended phase of ∆ /J <2, the wavepacket\nwill experience ballistic expansion with η= 1;\n(ii) at the critical point of ∆ /J= 2 the wavepacket\nsubject to subdiffusion with η∼0.5;\n(iii) the wavepacket is localized when ∆ /J >2 and\nη= 0.\n/s48 /s49 /s50 /s51/s48/s49/s50\n/s48 /s49 /s50 /s51/s119/s105/s116/s104/s32/s83/s79/s32/s105/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110\n/s50/s46/s53\n/s32/s32/s108/s111/s103\n/s49/s48/s40/s119/s41\n/s108/s111/s103\n/s49/s48/s40/s74/s116/s41/s40/s97/s41/s119/s105/s116/s104/s111/s117/s116/s32/s83/s79/s32/s105/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110\n∆ /s47/s74/s61/s49/s46/s53\n/s50\n/s50/s46/s49\n/s50/s46/s53/s50/s46/s49∆ /s47/s74/s61/s49/s46/s53\n/s32/s32\n/s108/s111/s103\n/s49/s48/s40/s74/s116/s41/s50/s40/s98/s41\nFIG. 6: Timeevolution of the wavepacket width w(t). (a) without\nSO interaction; (b) in the presence of SO interaction with γ= 0.7\nandΩ/J= 0.2.\nThe above time-evolution properties are demonstrated\nin Fig. 6(a). In Fig. 6(b) we present the results in\nthe presence of SO interaction. One can find out that,\nfor ∆/J= 2.1, which corresponds to the critical phase\nshowninFig. 2, thewavepacketstill subdiffuse with time\nevolution. In addition, the time evolution of wavepacket\nat sample time are shown in Fig. 7. At ∆ /J= 1.5\ncorrespond to the system in the extended phase, the\nwavepacket rapidly diffuses and almost all the lattice\nsites become populated. While at ∆ /J= 2.5 for the\nlocalized phase of the system, there are no diffusion be-\ncause in this regime the initial Gaussian wavepacket can\nbe decomposed into the superposition of several single-\nparticle localized eigenstates. For the system in the crit-\nical phase at ∆ /J= 2.1, the wavepacket diffusion is ac-\ncompanied with solitonlike structures in the centre and\nspreading sideband, which reflects that extended and lo-\ncalized eigenstates coexist in the system.\nBesides that, the nature of localized states can also\nbe extracted from the momentum distribution of the\natom stationary states. This is because a more localized\natomic wavefunction corresponds to a wider momentum\ndistribution via Fourier transformation. The momentum\ndistribution can be measured by turning off the Raman\nlasers, releasing the atoms from the lattice and perform-\ning time-of-flight imaging. Since our above discussions\nare for the dressed spin states {|↑,k/an}bracketri}htd,|↓,k/an}bracketri}htd}, their mo-\nmentum distribution can be mapped from that of the\nbare spins |↑,k+kR/an}bracketri}htand|↓,k−kR/an}bracketri}htin the following7\n/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48 /s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s74/s116/s61/s49/s48/s48/s48/s32\n/s32\n∆ /s47/s74/s61/s50/s46/s53∆ /s47/s74/s61/s50/s46/s49\n/s108/s97/s116/s116/s105/s99/s101/s32/s115/s105/s116/s101/s115∆ /s47/s74/s61/s49/s46/s53\n/s108/s97/s116/s116/s105/s99/s101/s32/s115/s105/s116/s101/s115 /s108/s97/s116/s116/s105/s99/s101/s32/s115/s105/s116/s101/s115/s74/s116/s61/s53/s48 /s74/s116/s61/s53/s48/s48\nFIG. 7: Time evolution of wavepackets |ψj,↑|2correspond to Fig. 6(b) at specific times.\nmanner\n|↑,k/an}bracketri}htd=1+i\n2|↑,k+kR/an}bracketri}ht+−1+i\n2|↓,k−kR/an}bracketri}ht,\n|↓,k/an}bracketri}htd=1+i\n2|↑,k+kR/an}bracketri}ht+1−i\n2|↓,k−kR/an}bracketri}ht.\nVII. CONCLUSION\nIn conclusion, we have studied the system of a SO-\ncoupled spin-1/2 particle moving in a one-dimensional\nquasiperiodic potential. We mapped out the system\nphase diagram in the tight-binding regime and accord-\ningly discussed the localization properties. In the ab-\nsence of SO interaction the system can be mapped into\nthe AA model and self-dual if ∆ /J= 2, SO interac-\ntion breaks the duality and leads to the appearance of\ncritical phase, in which the extened and localized states\ncoexist in the energy spectra. We also verified the phase\ndiagram via multifractal analysis of the wavefunctions\nand diffusion dynamics of a initially localized Gaussian\nwavepacket. Experimentaldetectionoflocalizationprop-\nerties are discussed. We proposed an experimental re-alization of the system using cold atomic gas trapped\nin a quasiperiodic optical lattice potential and external\nlaser fields. Since the two ingredients of our proposed\nscheme, the quasiperiodic optical lattice potential [5] and\nSO coupling [13, 15, 16], had already been achieved for\ncold atoms, it is expected that the localization properties\ndiscussed in this work can be readily observed in experi-\nment.\nAcknowledgments\nThis work is supported by the National Basic Re-\nsearch Program of China (973 Program) under Grant\nNo. 2011CB921604, the National Natural Science Foun-\ndation of China under Grant Nos. 11234003, 11129402,\n11004057 and 10828408, the ”Chen Guang” project sup-\nported by Shanghai Municipal Education Commission\nand Shanghai Education Development Foundation under\ngrant No. 10CG24, and sponsored by Shanghai Rising-\nStar Program under Grant No. 12QA1401000. H.P. is\nsupported by the US NSF, the Welch Foundation (Grant\nNo. C-1669), and the DARPA OLE program.8\n[1] P. W. Anderson, Phys. Rev. 109, 1492 (1958).\n[2] T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Na-\nture446, 52 (2006); Y. Lahini, A. Avidan, F. Pozzi,\nM. Sorel, R. Morandotti, D. N. Christodoulides, and Y.\nSilberberg, Phys. Rev. Lett. 100, 013906 (2008); I. V.\nShadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and\nY. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010); J.\nCheng and G. Huang, Phys. Rev. A 83, 053847 (2011).\n[3] L. Sanchez-Palencia, D. Cl´ ement, P. Lugan, P. Bouyer,\nG. V. Shlyapnikov, and A. Aspect, Phys. Rev. Lett. 98,\n210401 (2007).\n[4] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P.\nLugan, D. Cl´ ement, L. Sanchez-Palencia, P. Bouyer, and\nA. Aspect, Nature (London) 453, 891 (2008).\n[5] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M.\nZaccanti, G. Modugno, M. Modugno, and M. Inguscio,\nNature (London) 453, 895 (2008).\n[6] M. Larcher, F. Dalfovo, and M. Modugno, Phys. Rev. A\n80, 053606 (2009).\n[7] B. Deissler, M. Zaccanti, G. Roati, C. D’Errico, M. Fat-\ntori, M. Modugno, G. Modugno, and M. Inguscio, Nat.\nPhys.6, 354 (2010).\n[8] E. Lucioni, B. Deissler, L. Tanzi, G. Roati, M. Zaccanti,\nM. Modugno, M. Larcher, F. Dalfovo, M. Inguscio, and\nG. Modugno, Phys. Rev. Lett. 106, 230403 (2011).\n[9] L. Zhou, H. Pu, K. Zhang, X.-D. Zhao, and W. Zhang,\nPhys. Rev. A 84, 043606 (2011).\n[10] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor.\nPhys.63, 707 (1980).\n[11] M. Kohmoto and D. Tobe, Phys. Rev. B 77, 134204\n(2008).\n[12] M. Modugno, New J. Phys. 11, 033023 (2009).\n[13] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Natur e\n(London) 471, 83 (2011).\n[14] for a review, see J. Dalibard, F. Gerbier, G. Juzeli¯ una s,and P.¨Ohberg, Rev. Mod. Phys. 83, 1523 (2011) and\nreferences therein.\n[15] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n[16] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[17] S.-L. Zhu, D.-W. Zhang, and Z. D. Wang, Phys. Rev.\nLett.102, 210403 (2009).\n[18] M. J. Edmonds, J. Otterbach, R. G. Unanyan, M. Fleis-\nchhauer, M. Titov, and P. ¨Ohberg, New J. Phys. 14,\n073056 (2012).\n[19] R. E. Peierls, Z. Phys. 80, 763 (1933).\n[20] D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).\n[21] K. Osterloh, M. Baig, L.Santos, P.Zoller, andM. Lewen-\nstein, Phys. Rev. Lett. 95, 010403 (2005).\n[22] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi,\nPhys. Rev. Lett. 109, 085302 (2012).\n[23] J. Radi´ c, A. Di Ciolo, K. Sun, and V. Galitski, Phys.\nRev. Lett. 109, 085303 (2012).\n[24] P. G. Harper, Proc. Phys. Soc. Lond. A 68, 874 (1955).\n[25] S. Aubry S and G. Andr´ e, Ann. Isr. Phys. Soc. 3, 33\n(1980).\n[26] M. Kohmoto, Phys. Rev. Lett. 51, 1198 (1983).\n[27] D. J. Thouless, Phys. Rev. B 28, 4272 (1983).\n[28] G. Dufour and G. Orso, Phys. Rev. Lett. 109, 155306\n(2012).\n[29] D. J. Boers, B. Goedeke, D. Hinrichs, and M. Holthaus,\nPhys. Rev. A 75, 063404 (2007).\n[30] J. BiddleandS.DasSarma, Phys.Rev.Lett. 104, 070601\n(2010).\n[31] H. Hiramoto and S. Abe, J. Phys. Soc. Jpn. 57, 1365\n(1988)." }, { "title": "1810.06878v1.Induced_spin_orbit_coupling_in_silicon_thin_films_by_bismuth_doping.pdf", "content": "1 \n Induced spin -orbit coupling in silicon thin films by bismuth doping \n \nF.Rortais1*, S.Lee1, R.Ohshima1, S.Dushenko1, Y.Ando1 and M.Shiraishi1 \n1Department of Electronic Science and Engineering, Kyoto University, Kyoto, Kyoto 615 -8510 \nE-mail: rortais.fabien.85z@st.kyoto -u.ac.jp \n \n \nWe demonstrate an enhancement of the spin -orbit coupling in silicon ( Si) thin films by doping with \nbismuth ( Bi), a heavy metal , using ion implantation . Quantum corrections to conductance at low temperature in \nphosphorous -doped Si before and after Bi implantation is measured to probe the increase of the spin -orbit \ncoupling , and a clear modification of magnetoconductance signals is observed : Bi doping changes \nmagnetoconductance from weak localization to the crossover between weak localization and weak \nantilocalization. The elastic diffusion length, phase coherence length and spin -orbit coupling length in Si with \nand without Bi implantation are estimate d, and the spin-orbit coupling length after the Bi doping becomes the \nsame order of magnitude ( Lso = 54 nm) with the phase coherence length ( Lϕ = 35 nm) at 2 K . This is an \nexperimental proof that the spin -orbit coupling strength in Si thin film is tun able by doping with heavy metals. \n 2 \n Silicon (Si) is an attracting material for spintronics1–11 because its spin-orbit coupling is intrinsically \nweak , and spin lifetime in it can be longer than one nanosecond2,10 even at room temperatur e. Almost all the \nrecent spintronics studies of Si used lateral spin valves for spin injection , detection and transport2–10. The \ndifficulty and low efficiency of spin injection into semiconductor has been one of the central issues in \nsemiconductor spintronics : the so -called conductivity mismatch12 hinders the spin injection from ferromagnet ic \nmetals into the semiconductor channel. Although insertion of tunneling barrier between ferromagnetic metals \nand semiconductors allows to inject spins into semiconductors , spin injection/detection using the ordina ry and \nthe inverse spin Hall effects (SHE and ISHE) —instead of electrical spin injection from ferromagnetic \nmetals —can be the other pathway to avoid the conductivity mismatch13–18. \nSi possesses a low spin -orbit coupling because of the small electric charge of its nucleus and a lack of \nan internal electric field due to the lattice inversion symmetry , which allows devices with spin transport but \nlimits new device -designing possibilities with the spin-charge conversion effects —the SHE and the ISHE \nmentioned above . However, if it can be engineered to possess a significant spin -orbit coupling, the SHE and the \nISHE in Si can be used for spin injection/detection. Such additional functionality will pave the way to create \nall-Si spin devices consisting of injector, detector and transport medium made of Si —a breakthrough for \nsemiconductor spintronics and significant advance in applied physics. The purpose of this study is to create a \nsizable spin -orbit interaction in Si by implantation of a heavy element, bismuth ( Bi). To probe the strength of the \nspin-orbit coupling, quantum correction s to the conductivity at low tem perature s were measured in Si with and \nwithout Bi implantation. The e nhancement of the spin -orbit interaction is discussed using the spin-orbit coupling \nlength extracted from the measurements . \nSamples were fabricated using an intrinsic silicon -on-insulator (SOI) substrate with a 100 nm -thick Si \nchannel isolated from the backside bulk Si by 200 nm-thick layer of SiO 2. The Si channel was doped with \nphosphorus (P) by ion implantation to assure ohmic contact s for electrical measurements . The doses and energies \nof ion beam necessary to reach a homogeneous profile of 1×1020 cm-3—which exceeds the degenerate limit of \n3.5×1018 cm-3 for P-doped Si19—were calculated from the ion implantation profile simulated with the Transp ort \nof Ions in Matter (TRIM ) program . The three implantation steps are summarized in Table I, and the simulated \nimplantation profile is shown in Fig. 1(a) . To activate P and decrease the number of defects in the Si channel \ninduced by the implantation , the substrate was annealed using rapid thermal annealing at 500° C for 10 s for the \nfirst step , and at 900° C for 1 s for the last step , with a 4 s temperature ramp up in -between . A part of the 3 \n substrate was used as reference sample s (SOI:P ), while t he rest was additionally doped by Bi to form a double \ndonor system (SOI:P:Bi ). Energies and doses for Bi implantation were chosen to realize homogeneous doping \nconcentration of 5.0 ×1019 cm-3 and are summarized in Table I. The doping profile simulated by TRIM is shown \nin Fig. 1(a) (blue -filled squares). It was previously demonstrate d that such doping concentration is above the \ncritical level for metal -semiconductor transition for Si doped with Bi20. The implantation was followed by a \nthermal annea ling at 600 ° C for 30 min under argon atmosphere to electrically activate the Bi dopants21. \nTable I. Energies and doses used for P and Bi doping . \n P doping Bi doping \nStep n ° Energ y \n(keV) Dose \n(×1013 cm-2) Energ y \n(keV) Dose \n(×1013 cm-2) \n1 5 8 35 5.4 \n2 15 13 70 7.4 \n3 35 50 120 11.0 \n4 250 19.0 \nThe electron beam lithography and argon milling processes were used to pattern the Si channel into Hall bar \ndevice s as shown in FIG. 1(b). It should be noted that the Bi concentration is very low in the top 20 nm of Si \nchannel (see Fig. 1 (a)) and , therefore , we etched the top 50 nm to use the uniformly doped region as a channel . \nThe etching was performed on both types of samples (SOI:P and SOI:P:Bi ) to ensure the same channel structure. \nThe Hall bar s were 20 µm in width and 110 µm in length in -between the voltage measurement branches \n(terminal s 3 and 5 in Fig. 1 (b)). The electrical contacts on top of the Si channel were formed in the two -step \nprocess: deposition of 30 nm -thick AuSb followed by annealing for 30 s at 300 ° C, and then deposition of T i(3 \nnm)/Au( 70 nm ) bilayer . 4 \n Figure 1(c) demonstrates the linear I-V characteristic s measured from the SOI:P:Bi sample, confirming \nsuccessful fabrication of the ohmic contacts with the Si channel. The linear I-V characteristic s were measured in \nthe whole temperature range from 300 K down to 2 K, which rules out contribution of non -linearities in I-V \ncharacteristic on the magnetoconductance measurements at low temperature s. Figure 2(a) shows t he temperature \ndependence of resistivity for the SOI:P and the SOI:P:Bi samples . For the SOI:P , the resistivity increase d from \n1.3×10-3 Ω⋅cm at 2 K to 2.4 ×10-3 Ω⋅cm at 300 K . The metallic behavior confirm s the degenerate state of the Si \nchannel after the P doping . On the contrary , in the Bi-doped sample (SOI:P:Bi ), a decrease of resistivity from \n8.2×10-3 Ω⋅cm at 2 K to 4.2 ×10-3 Ω⋅cm at 300 K was observed . To clarify its origin , we measured the Hall effect \nto extract carrier concentration in the channel . Consequently, the carrier concentration s at 2K were estimated to \nbe 4.0×1019 cm-3 and 5.0 ×1019 cm-3 for the SOI:P and the SOI:P:Bi , respectively . As expected, t he carrier \nconcentration increased after the Bi doping in SOI:P:Bi samples compared with the SOI:P samples. H owever, \nthe measured carrier densit ies for both P and Bi are smaller than those predicted by the TRIM simulations, which \nindicates that only a part of the implanted ions is electrically activated . Figure 2(b) shows the mobilities of the Si \nchannel extracted from the Hall effect measurements . The c arrier mobilit ies were strongly reduced after Bi \nFIG. 1. (a) Doping profile s of Bi and P in Si simulated using the Transport of Ions in Matter (TRIM) program \nwith the irradiation beam parameters specified in Table I . Solid lines are guide to the eye. (b) Schematic \nlayout of the Hall bar devices used to carry out measurements . (c) Temperature dependence of I-V \ncharacteristic s measured between injection contacts (terminals 1 and 6) of SOI:P:Bi sample . Sample exhibited \nlinear I-V characteristic s in the whole measured temperature range. \n5 \n doping from 118 cm2⋅V-1⋅s-1 (SOI:P ) to 14 cm2⋅V-1⋅s-1 (SOI:P:Bi ) at 2 K and from 64 cm2⋅V-1⋅s-1 (SOI:P ) to 18 \ncm2⋅V-1⋅s-1 (SOI:P:Bi ) at 300 K. This decrease of the mobilit ies is attributed to the increase d number of scattering \ncenters in the Si channel with the implantation of Bi ions and the additional crystal defects produced during the \nimplantation , as was also reported in the previous studies22–24. However, we stress that despite the decrease in the \nmobilit ies, linear I-V characteristics were still obtained for SOI:P and SOI :P:Bi samples at all temperatures. \n \n \nFIG. 2. Temperature dependence of (a) resistivity and (b) mobility of the P-doped Si channel without Bi doping \n(SOI:P , open symbols) and with Bi doping (SOI:P:Bi , filled symbols ). Solid lines are guide to the eye. \nTo compare the spin-orbit coupling strength in the Si channel with and without Bi doping (samples \nSOI:P :B and SOI:P , respectively) , we measure d quantum correction s to the conductance . At low temperature , a \nconduction electron can be scattered by impurities several times without losing its phase coherence , which \ncreate s interference on self -crossing paths and leads to localization of the wave function (i.e. decreased \nconductance) —the phenomenon known as weak localization (WL)25–28. The phase coherence length , Lϕ , defines \nthe characteristic length during which the phase is conserved . The spin-orbit coupling —which can be \ncharacterized by spin-orbit coupling length Lso—also influen ces the relative phase between interfering waves by \ncoupling spin of the electron to its momentum . This results in destructive interference on the self-crossing \npaths —the weak antilocalization effect (WAL) that manifests itself in the increased conductance 27,26. \nApplication of a magnetic field introduces an additional phase factor that destroy s interference and leads to \nsuppression of the conductance change due to the WL and WAL effects. Thus , both effects result in \nmagnetoconductance , however, its polarity is reversed between systems with a strong spin-orbit coupling (WAL \n6 \n case) and a week spin-orbit coupling (WL case)26,29 –31. The crossover betw een the WAL and the WL is governed \nby the ratio Lϕ/Lso27—making magnetoconductance a sensitive tool to determine the spin orbit coupling \nstrength27, as was already demonstrated in various system s 30–34. \nThe magneto conductance measurements were carried out using Physical Property Measurement System \n(PPMS , Quantum Desi gn): the longitudinal voltage between terminal s 3 and 5 (see FIG. 1(b)) was measured \nunder an application of the out -of-plane magnetic field and the injection current of 10 µA. The linear \nbackground, which probably originates from the thermal drift, was subtracted from the data before further \nprocessing. We calculated t he normalized conductance ∆G/G0`=(G(B)-G(B=0))/G0`, where G(B) is the \nconductance under the application of the magnetic field B; G(B=0) is the zero -magnetic field conductance ; and \nG0`=G0/(2π)=e2/(2π2\u0000), where G0 is conductance quantum, e is the elementary charge, \u0000 is the reduced Planck \nconstant. Figures 3(a) and (b) show the normalized conductance as a function of the magnetic field for the SOI:P \nand the SOI:P:Bi samples, respectively . The sample without Bi doping (SOI:P ) shows increasing of the \nconductance with increasing the magnetic field —a signature of the WL effect . The amplitude of the quantum \ncorrections to the conductance by the WL can be extracted by removing the parabolic contribution due to the \nclassical magnetoconductance (not shown) . The WL signal monotonically increased from 2 0 K to 2 K , with an \nampli tude of 1.18±0.04% of the zero-field conductance at 2 K . Our results confirm previous observation of the \nWL in P-doped Si10. The sample with Bi doping (SOI:P:Bi) exhibited strikingly different behavior of \nmagnetoconductance . The WL in SOI:P:Bi sample at 2 K was more than five times smaller than in SOI:P sample \nwithout Bi doping . Furthermore , the amplitude of the WL decreased from 5 K (0.28 ±0.01%) to 2 K \n(0.22 ±0.01%) in the SOI:P:Bi, in contrast to the increase from 5 K (0.88 ±0.04%) to 2 K (1.18 ±0.04%) in the \nSOI:P. The change of the shape of the magnetoconductance curve around the zero magnetic field from narrow \ndip (Fig. 3(a) ) to broad dip (Fig. 3(b)) —along with the strong decrease in the amplitude of the WL—indicate s \ntransition from the WL to the WAL in the SOI:P:Bi sample . Such behavior is expected from \nmagnetoconductance of a material that possesses a significant spin-orbit coupling , leading to the spin-orbit \ncoupling length and the phase coherence length s to have the same order of magnitude27. The characteristics \nlengths Lso and Lϕ can be extracted from the magnetoconductance data by using the Hikami -Larkin -Nagaoka \nmodel35: 7 \n \nwhere Ψ is the digamma function defined as: Ψ𝑥= Γ!𝑥/Γ𝑥 with Γ𝑥= 𝑦!!!e!! d𝑦!\n!; the \ncharacteristic field s Bϕ, Bso and Be are connected to the characteristic lengths Lϕ, Lso and Le, respectively 27 by \n𝐵!=ℏ\n!!!!!, where the suffix i stands for ϕ, SO, e; p is the coefficient describing strength of the parabolic \ncontribution due to the classical magnetoconductance . We stress that the classical magnetoconductance was \nconsidered even at low magnetic field : it was evaluated using magnetoconductance data at the magnetic field |B| \n> 2 T at each temperature and then included as a fixed parameter in the fitting functio n (Eq. (1)) . The resulting \nfitting curves are plotted as black solid lines in Figs. 3(a) and (b) . The elastic diffusion length, phase coherence \nlength and spin -orbit coupling length extracted from the fitting are summarized in Figs. 3( c) and (d) for the \nSOI:P and the SOI:P:Bi . The elastic diffusion length has the same order of magnitude (10 nm -25 nm) in the \nsamples with and without Bi doping . The p hase coherence length , however, decreased after the Bi doping (with \nthe strongest decrease from 177 nm to 35 nm at 2 K ), indicating that Bi atoms in the Si are mainly acting as \ninelastic scattering centers . The use of the Hikami -Larkin -Nagaoka model —a 2D model —is justified because the \nextracted phase coherence length has the same order of magnitude or is superior to the thickness of the Si \nchannel . The fitting results for the SOI:P:Bi demonstrated that Bi doping is capable of inducing the spin-orbit \ncoupling length of the same order of magnitude with the phase coherence length (Lso= 54 nm and Lϕ= 35 nm at \n2K). \nAs expected , the implantation of heavy elements like Bi in the Si channel as new scattering center s \ndecrease d the phase coherence length . This implantation -induced disorder in the Si channel can be seen from the \nmodification of the mobility and its temperature dependence (Fig. 2(b)) . The temperature dependence of the \nphase coherence length itself was also modified by the Bi doping (FIG. 3(c) and (d)). In the temperature range \nfrom 2 K to 5 K, the reference SOI:P sample presents a decrease of 46%, while the Bi doped sample , the \nSOI:P:B i, shows only 29% decrease and a less steeper decreasing law, which is characteristic to a different \nscattering mechanism36. The ratio Bso/Bϕ ∝ Lϕ2/Lso2 that describes the crossover between the WL/WAL27 is \nmodified by the addition of Bi: the increase d spin-orbit coupling strength led to the observed crossover in the ∆𝐺\n𝐺!`=𝐺𝐵−𝐺0 \n𝑒!\n2𝜋!ℏ \n=ln𝐵!\n𝐵−ψ1\n2+𝐵!\n𝐵+2ln𝐵!\"+𝐵!\n𝐵−ψ1\n2+𝐵!\"+𝐵!\n𝐵\n−3ln4\n3𝐵!\"+𝐵!\n𝐵−ψ1\n2+4\n3𝐵!\"+𝐵!\n𝐵+𝑝𝐵!, \n \n(1) 8 \n magnetoconductance close to the zero magnetic field . The crossover between the WL and the WAL was \nobserved —rather than full polarity reversal of magnetoconductance correction —since the spin-orbit coupling \nlength ( Lso= 54 nm at 2K ) is still higher than phase coherence length (35 nm), as was also shown in the previous \nstudies27. Increasing number of electrically activated Bi atoms can provide a way to further en hance t he \nspin-orbit couplin g without increase in Bi doping concentration . \n \n \nFIG. 3. (a) The normalized magnetoconductance ∆G/G0`=(G(B)-G(B=0))/G0` of (a) P-doped Si channel \nwithout Bi doping (SOI:P) and (b) P-doped Si channel with Bi doping (SOI:P:Bi) measured at different \ntemperatures . Inset in panel (b) shows full magnetic field B scan range data measured at T=2 K. The curves \nare offset along the y-axis for clarity . Solid black lines show fitting by the Eq. (1), as described in the text. \nTemperature dependence of the mean free path length Le (squares) , the phase coherence length Lϕ (circles) and \nthe spin -orbit coupling length Lso (triangles) of (c) P-doped Si channel without Bi doping (SOI:P) and (d) \nP-doped Si channel with Bi doping (SOI:P:Bi) . Solid lines are guide to the eye. \n \n9 \n For spintronics application s, one of the key parameters that characterizes material is the spin diffusion \nlength —the mean distance between spin -flipping event s. Long spin diffusion length characterizes how far spin \ncan carry information and is crucial for device application. The exact relation between the spin diffusion length \nand the spin -orbit coupling length in Si is complicated because of the anisotropic band structure34,37,38. In \naddition, in Si with Bi doping , the effective mass theory is expected to be less applicable because of the large \nionization energy and small Bohr radius22. Furthermore, the temperature dependence of the phase coherence \nlength is modified by the Bi implantatio n—a proof of the modification of the scattering mechanism —which \ncomplicat es the comparison between the spin life time in two systems . However, the appearance of the WAL and \nthe observable spin -orbit coupling length in Bi -doped samples signifies the enhance ment of the spin -orbit \ncoupling in Si, which inevitably leads to the decrease in the spin diffusion length . In this sense , the increased \nspin-orbit coupling provides an advantage of utilizing a spin-charge conversion mechanism , but, at the same \ntime, introduces a new limitation on the size of spintronics devices.39 The interp lay between these two effects \ncan be controlled by tuning Bi doping concentration —a pathway to create a device that integrates spin -charge \nconversion and spin transport. \nIn conclusion, we demonstrated a control over the strength of the spin -orbit coupling in Si channel \nusing Bi doping. Magnetoconductance measurements were carried out on the Hall -bar-shaped P -doped Si \ndevices with and without Bi doping. Strong modification of electrical transport properties after Bi doping —an \nincrease in resistivity despite increased carrier concentration and decrease of mobility and modification of its \ntemperature behavior —indicated enhanced impurity scattering. Quantum corrections to conductance were \nstrongly modified by Bi doping: a WL signal of 1.18±0.0 4% at 2 K was suppressed to 0.22±0.01% after the \ndoping . Also the temperature dependence of WL amplitude is reversed between 2 K and 5 K for SOI:P and \nSOI:P:Bi respectively. More over, close to the zero magnetic field, we observe d a beginning of polarity reversal \nof magnetoconductance in Bi -doped samples —a signature of a significant spin -orbit coupling in our Si devices . \nBy using the magnetoconductance fitting by the Hikami -Larkin -Nagaoka model , we demonstrate d a spin -orbit \ncoupling length of 54 nm at 2 K in SOI:P:Bi . Our result paves the way for control over the spin-orbit coupling in \nSi by using doping with heavy elements , which opens new possibilities in designing Si spintronics devices with \nenhanced spin -orbit coupling enabling a use of spin-charge conversion via the spin Hall effects . \n 10 \n F.R. acknowledges support by JSPS (Japan Society for the Promotion of Science) Postdoctoral Fellowship and \nJSPS KAKENHI Grant No. 820170900047 . A part of this study is supported by Grant -in-Aid for Scientific \nResearch from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan \n(Innovative Area “Nano Spin Conversion Science” No. 26103002 and Grant -in-Aid for Scientific Research (S) \n“Semiconductor Spincurren tronics” No. 16H06330 ). 11 \n 1 I. Appelbaum, B. Huang, and D. Monsma, Nature 447, 295 (2007). \n2 A. Spiesser, H. Saito, Y. Fujita, S. Yamada, K. Hamaya, S. Yuasa, and R. Jansen, Phys. Rev. Appl. 8, 64023 \n(2017). \n3 T. Sasaki, Y. Ando, M. Kameno, T. Tahara, H. Koike, T. Oikawa, T. Suzuki, and M. Shiraishi, Phys. Rev. \nAppl. 2, 34005 (2014). \n4 T. Tahara, H. Koike, M. Kameno, T. Sasaki, Y. Ando, K. Tanaka, S. Miwa, Y. Suzuki, and M. Shiraishi, Appl. \nPhys. Express 8, 113 004 (2015). \n5 O.M.J. van ’t Erve, A.T. Hanbicki, M. Holub, C.H. Li, C. Awo -Affouda, P.E. Thompson, and B.T. Jonker, \nAppl. Phys. Lett. 91, 212109 (2007). \n6 Y. Saito, T. Tanamoto, M. Ishikawa, H. Sugiyama, T. Inokuchi, K. Hamaya, and N. Tezuka, J. Appl. Phys . \n115, 17C514 (2014). \n7 S. Sato, R. Nakane, T. Hada, and M. Tanaka, Phys. Rev. B 96, 235204 (2017). \n8 T. Sasaki, T. Oikawa, M. Shiraishi, Y. Suzuki, and K. Noguchi, Appl. Phys. Lett. 98, 12508 (2011). \n9 Y. Aoki, M. Kameno, Y. Ando, E. Shikoh, Y. Suzuki, T. Shinjo, M. Shiraishi, T. Sasaki, T. Oikawa, and T. \nSuzuki, Phys. Rev. B 86, 81201 (2012). \n10 S. Lee, N. Yamashita, Y. Ando, S. Miwa, Y. Suzuki, H. Koike, and M. Shiraishi, Appl. Phys. Lett. 110, \n192401 (2017). \n11 F. Rortais, C. Vergnaud, C. Ducruet, C. Beigné, A. Marty, J. -P. Attané, J. Widiez, H. Jaffrès, J. -M. George, \nand M. Jamet, Phys. Rev. B 94, 174426 (2016). \n12 E.I. Rashba, Phys. Rev. B 62, R16267 (2000). \n13 J. Sinova, S.O. Valenzuela, J. Wunderlich, C.H. B ack, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). \n14 V. Sih, W.H. Lau, R.C. Myers, V.R. Horowitz, A.C. Gossard, and D.D. Awschalom, Phys. Rev. Lett. 97, \n96605 (2006). 12 \n 15 T. Jungwirth, J. Wunderlich, and K. Olejník, Nat. Mater. 11, 382 (2012). \n16 J. Wunderlich, B. -G. Park, A.C. Irvine, L.P. Zarbo, E. Rozkotova, P. Nemec, V. Novak, J. Sinova, and T. \nJungwirth, Science (80 -. ). 330, 1801 (2010). \n17 T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). \n18 S.O. Val enzuela and M. Tinkham, Nature 442, 176 (2006). \n19 T.F. Rosenbaum, K. Andres, G.A. Thomas, and R.N. Bhatt, Phys. Rev. Lett. 45, 1723 (1980). \n20 A. Ferreira da Silva, B.E. Sernelius, J.P. de Souza, H. Boudinov, H. Zheng, and M.P. Sarachik, Phys. Rev. B \n60, 15824 (1999). \n21 E. Abramof, A. Ferreira da Silva, B.E. Sernelius, J.P. de Souza, and H. Boudinov, Phys. Rev. B 55, 9584 \n(1997). \n22 A. Ferreira da Silva, B.E. Sernelius, J.P. de Souza, and H. Boudinov, J. Appl. Phys. 79, 3453 (1996). \n23 P.W. Chapman, O.N. Tufte, J.D. Zook, and D. Long, J. Appl. Phys. 34, 3291 (1963). \n24 R.A. Logan and A.J. Peters, J. Appl. Phys. 31, 122 (1960). \n25 G. Bergmann, Zeitschrift Für Phys. B Condens. Matter 48, 5 (1982 ). \n26 G. Bergmann, Solid State Commun. 42, 815 (1982). \n27 G. Bergmann, Phys. Scr. T14, 99 (2007). \n28 S. Chakravarty and A. Schmid, Phys. Rep. 140, 193 (1986). \n29 S.I. Dorozhkin, a. a. Kapustin, and S.S. Murzin, JETP Lett. 97, 149 (2013). \n30 R.L. Kallaher and J.J. Heremans, Phys. Rev. B - Condens. Matter Mater. Phys. 79, 75322 (2009). \n31 T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi, Phys. Rev. Lett. 89, 46801 (2002). \n32 F. Rortais, S. Oyarzún, F. Bottegoni, J. -C. Rojas -Sánchez, P. Laczkow ski, A. Ferrari, C. Vergnaud, C. \nDucruet, C. Beigné, N. Reyren, A. Marty, J. -P. Attané, L. Vila, S. Gambarelli, J. Widiez, F. Ciccacci, H. Jaffrès, 13 \n J.-M. George, and M. Jamet, J. Phys. Condens. Matter 28, 165801 (2016). \n33 P.J. Newton, R. Mansell, S.N. Hol mes, M. Myronov, and C.H.W. Barnes, Appl. Phys. Lett. 110, 62101 \n(2017). \n34 Y. Niimi, D. Wei, H. Idzuchi, T. Wakamura, T. Kato, and Y. Otani, Phys. Rev. Lett. 110, 62502 (2013). \n35 S. Hikami, A.I. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 63, 707 (1980). \n36 J.J. Lin and J.P. Bird, J. Phys. Condens. Matter 14, 201 (2002). \n37 I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). \n38 Y. Niimi and Y. Otani, Reports Prog. Phys. 78, 124501 (2015). \n39 Y. Niimi, M. Morota, D.H. Wei, C. Deranlot, M. Basletic, A. Hamzic, A. Fert, and Y. Otani, Phys. Rev. Lett. \n106, 1 (2011). \n \n " }, { "title": "2104.06912v1.Spin_orbit_gaps_in_the_s_and_p_orbital_bands_of_an_artificial_honeycomb_lattice.pdf", "content": "Spin-orbit gaps in the sand porbital bands of an artificial honeycomb lattice\nJ.J. van den Broekey, I. Swartx, C. Morais Smithy, and D. Vanmaekelberghx\nyInstitute for Theoretical Physics, Utrecht University, Utrecht 3584 CC, Netherlands and\nxDebye Institute for Nanomaterials Science, Utrecht University, Utrecht 3584 CC, Netherlands\n(Dated: April 15, 2021)\nMuffin-tin methods have been instrumental in the design of honeycomb lattices that show, in\ncontrast to graphene, separated sand in-plane pbands, aporbital Dirac cone, and a porbital\nflat band. Recently, such lattices have been experimentally realized using the 2D electron gas on\nCu(111). A possiblenext avenueisthe introduction of spin-orbit coupling to these systems. Intrinsic\nspin-orbit coupling is believed to open topological gaps, and create a topological flat band. Although\nRashba coupling is straightforwardly incorporated in the muffin-tin approximation, intrinsic spin-\norbit coupling has only been included either for a very specific periodic system, or only close to the\nDirac point. Here, we introduce general intrinsic and Rashba spin-orbit terms in the Hamiltonian\nfor both periodic and finite-size systems. We observe a strong band opening over the entire Brillouin\nzone between the porbital flat band and Dirac cone hosting a pronounced edge state, robust against\nthe effects of Rashba spin-orbit coupling.\nI. INTRODUCTION\nEver since the prediction of the quantum spin Hall\neffect in graphene as a result of intrinsic spin-orbit cou-\npling by Kane and Mele [1], efforts have been made to\nobserve the predicted state. Not only would the intrin-\nsic spin-orbit coupling turn graphene into a bulk insu-\nlator, it would also create conducting edge modes that\nwould be protected from scattering by the topology of\nthe system [1]. Unfortunately, the gap turned out to\nbe too small for practical applications [2]. Although\nefforts have been made to enhance the intrinsic spin-\norbit coupling [3, 4], no convincing method has been\nfound so far. An alternative route to study intrinsic\nspin-orbit effects in a honeycomb lattice is by creating\nan artificial lattice by placing energy barriers on top of\na (heavy) metal with a 2D electronic surface gas using a\nscanning tunnelling microscope. This method was pio-\nnereed by Gomes et al. [5] using CO molecules as energy\nbarriers on top of Cu(111). Later, Gardenier et al. [6]\nextended the method and created artificial honeycomb\nlattices on Cu(111) in which the sorbital and porbital\nDirac bands were separated, and the porbital bands\nincluded a Dirac cone and a flat band, as predicted by\ntight-binding methods [7–12].\nThe system of in-plane porbital bands is of high in-\nterest to study the effects of intrinsic spin-orbit cou-\npling. Not only is the effect of the coupling shown to\nbe larger in these systems [9, 12, 13] because it is onsite\ninstead of next-nearest neighbour, as would be the case\nforsandpzorbitals, but it is also predicted to generate\ntopological flat bands [10]. Flat bands are particularly\ninteresting to study interactions, as the kinetic energy\nis quenched. Thus, interaction driven phenomena like\nsuperconductivity and charge density waves become ac-\ncessible. Moreover, flat bands can be even more inter-\nestingwhentheyaretopological[14]. Thus, introducingintrinsic spin-orbit coupling in artificial electron lattices\ncould open up exciting new possibilities for the field.\nPatterned lattices are usually described as a two-\ndimensional (2D) free-electron gas confined to the lat-\ntice using a modulated (muffin-tin) potential. This\nmuffin-tin method has yielded remarkably accurate pre-\ndictions for effectively spinless systems, and has proven\nto be a vital tool in the design of artificial electronic\nlattices [6, 15–20]. For these systems, the muffin-tin ap-\nproach is often more convenient than the tight-binding\nmethod because there are only a few parameters in-\nvolved, namely the potential landscape V(x;y)and the\nelectron effective mass. These parameters only have to\nbe determined once for a material and patterning tech-\nnique, whereas the tight-binding approach requires new\nfitting for each design. The muffin-tin method has the\nadditional advantage that it does not require any as-\nsumption about the orbital character of the system. It\ncan be complemented with a tight-binding parametriza-\ntion, enabling one to understand which lattice orbitals\nare involved in the band formation. Besides this, the\nmuffin-tin method is also well suited for finite-size cal-\nculations, and thus very useful for the study of edge\nstates in practical systems.\nUnfortunately, the inclusion of intrinsic spin-orbit\ncoupling in muffin-tin models is not straightforward.\nThe theoretical approaches proposed so far vary sub-\nstantially [13, 21], and are only valid for a specific setup\nor only describe the physics near the Dirac point. In\naddition, these techniques are not easily extended to\nfinite-size calculations.\nHere, we propose a heuristic method using text book\nspin-orbit Hamiltonian terms that have very few input\nparameters and reproduce the defining features of other\napproaches. Additionally, this method allows for calcu-\nlations on finite-size systems, as is vital for topological\napplications.\nThe outline of this paper is the following: in Sec. II,arXiv:2104.06912v1 [cond-mat.mes-hall] 14 Apr 20212\nwe review the muffin-tin model and adapt it to incorpo-\nrate Rashba and intrinsic spin-orbit coupling. We then\ninvestigatetheinfluenceofspin-orbitcouplingonahon-\neycombtoymodelinSec.III,anpresentourconclusions\nin Sec. IV.\nII. THE MODEL\nLet us consider an artificial lattice, created by\nadatomsarrangedtoformananti-latticetotheunderly-\ning electrons from the the surface state of the substrate.\nThese can be approximated as a 2D electron gas with\nan effective mass m\u0003, and the system can be described\nby the one-electron time-independent 2D Schrödinger\nequation,\n\u0012\u0000~2\n2m\u0003r2+V\u0013\n\t =E\t; (1)\nwhereVis the potential created by the adatoms pat-\nterning the surface. Thus, the only freedom in the in-\nput parameters is in the shape of the potential. When\nmodeling a patterned potential V(x;y)as a collection\nof disk shaped protrusions, also called a muffin-tin po-\ntential, only two parameters remain, namely the disk\nheight and the disk width.\nIn order to study the effect of spin-orbit coupling in\nartificial lattices, we start with the spin-orbit coupling\nthat originates from the Dirac equation as a relativistic\ncorrection to the Schrodinger equation, given by\nHSO=~\n4m2c2(rV\u0003\u0002p)\u0001\u001b; (2)\nwheremis the real electron mass, V\u0003is the full po-\ntential, pis the vector momentum, \u001bis the vector of\nPauli matrices, and cis the speed of light. Here, we see\nthat spin-orbit coupling is proportional to the gradient\nof the potential V\u0003. The Rashba spin-orbit coupling\noriginates from Eq. (2) at the surface of materials, due\nto the large change in the potential at the interface. As\ninversion symmetry is not present, spin degeneracy is\nnot required and is indeed broken. We can obtain the\nRashba term for the muffin-tin model by considering a\n2D material with a potential change in the out-of-plane\ndirection:\nHR=\u000b1(px\u001by\u0000py\u001bx): (3)\nHere,pi(i=x;y)is the momentum component, \u001biare\nthe Pauli matrices, and \u000b1is the effective strength of\nthe Rashba coupling. As a check, we add this term\nto Eq. (1), and use \u000b1andm\u0003as experimentally mea-\nsured in Ref. [22] for the Au(111) surface state, which\nis known to have a large Rashba splitting. It can be\nseen that our calculation reproduces the previously ob-\nserved Rashba splitting of 0.26 nm\u00001between the two\nFigure1. Thegoldsurfacestate, calculatedasafreeelectron\ngas (a) without and (b) with Rashba spin-orbit coupling.\nHere, an effective mass of 0.25 em and \u000b1= 6:02\u0002104m/\ns were used, as measured in Ref. [22]\nparabolaminima[22], asshowninFig.1. Therefore, the\nmuffin-tin method can accurately describe the Rashba\nspin-orbit coupling.\nNext, we consider intrinsic spin-orbit coupling, which\nis a consequence of the coupling between the magnetic\nmoment of the orbital angular momentum and the spin\nof the electron. In atomic systems, the intrinsic spin-\norbit termHIscales supralinear with the atomic num-\nber [23]. Thus, HItends to be much larger for heavier\nelements. In the case of the muffin-tin technique how-\never, the substrate is approximated as a 2D electron\ngas with an effective mass m\u0003and a scattering poten-\ntial originating from the patterned adatoms. Thus, de-\ntails on the precise potential landscape, like the size of\nthe nuclei in the substrate, that give rise to the intrinsic\nspin-orbit coupling, are lost. Here, we propose a heuris-\ntic solution to this issue that maps the intrinsic spin-\norbit coupling coming from Eq. (2) to the muffin-tin\ncalculations by assuming an effective coupling \u000b2be-\ntween the patterned muffin-tin potential and the spin-\norbit term,\nHI=\u000b2(rV\u0002p)\u0001\u001b: (4)\nBy allowing this effective parameter \u000b2to be system de-\npendent, the method can be fitted to any substrate ma-\nterial. Additionally, because of the relative simplicity\nof this approach, it easily translates to both finite and\nperiodic calculations, which is highly convenient when\nworking with topological materials. Please note that \u000b1\nand\u000b2do not have the same units and HR\u0019HIdoes\nnot mean that \u000b1\u0019\u000b2, as the potential derivative is\nonly absorbed in \u000b1, cf. Eq. (3) and (4).\nAs the electrons are confined to the x;yplane, only\nthezcomponent of the cross product survives, and the\nintrinsic spin-orbit contribution becomes,\nHI=\u000b2\u0012@V\n@xpy\u0000@V\n@ypx\u0013\n\u001bz: (5)\nIt is hoped that with appropriate fitting for the effec-\ntive parameter \u000b2, Eq. (5) will yield adequate predic-\ntions for spin-orbit coupling in an artificial lattice. In-\ndeed, as shown in the next section, Eq. (5) reproduces3\nthe main features found using other methods. Adding\nHRandHIto Eq. (1), the full time-independent one-\nelectron Schrödinger equation becomes:\n\u0014\u0000~2\n2mr2\u0000i~\u000b2\u0012@V\n@x@\n@y\u0000@V\n@y@\n@x\u0013\n\u001bz\n\u0000i~\u000b1\u0012@\n@x\u001by\u0000@\n@y\u001bx\u0013\n+V\u0015\n\t\u001b=E\t\u001b:(6)\nForfinite-sizesystems, solvingthisequationisnotmuch\ndifferent from solving the spinless system. Neverthe-\nless, there is one important point to consider. Due to\nthe presence of a derivative of the potential, the pre-\ncise shape of the potential becomes important. For the\nmuffin-tin potential, we would encounter infinities in\nthe spin-orbit term. We solve this by using Gaussian\npotentials instead. As shown in Appendix A, a change\nin potential shape from muffin-tin to Gaussian does not\nyield significantly different results in the case without\nspin-orbit coupling, and is therefore an appropriate ap-\nproximation.\nInthecaseofaperiodicsystem, carefulFouriertrans-\nformation is required to incorporate the spin-orbit cou-\nplings. We first Fourier transform the wave function:\n\t\u001b(x) =1p\nAX\nkeik\u0001x\t\u001b(k); (7)\nwhereA=L2is the system size in which the wave func-\ntionisperiodic, and k=2\u0019\nL(lx;ly), withlirangingfrom\n\u00001to1. Meanwhile, we also Fourier transform the\npotentialV. However, as Vhas the unit cell periodicity\nwe have\nV(x) =X\nKeiK\u0001xVK; (8)\nwhere Kare the reciprocal lattice vectors. Applying\nthese transformations to Eq. (6), the resulting equation\nalso has to hold for a single Fourier component q,\n~2\n2mq2\t\u001b(q)\u0000~\u000b1(qy\u001bx\u0000qx\u001by) \t\u001b(q)\n\u0000i~\u000b2X\nKV\u0000K(Kxqy\u0000Kyqx)\u001bz\t\u001b(q+K)\n+X\nKV\u0000K\t\u001b(q+K) =Eq\t\u001b(q):(9)\nOnly wave functions of the shape \t(q)and\t(q+K)\nappear in this equation. We can therefore apply a shift\nq!q+K0andK!K\u0000K0to obtain a coupledsystem of equations for each qin the Brillouin zone,\n~2\n2m(q+K0)2\t\u001b(q+K0)\n\u0000~\u000b1\u0002\n(qy+K0\ny)\u001bx\u0000(qx+K0\nx)\u001by\u0003\n\t\u001b(q+K0)\n\u0000i~\u000b2X\nKVK0\u0000K\u0002\n(Kx\u0000K0\nx)qy\u0000(Ky\u0000K0\ny)qx+\nKxK0\ny\u0000KyK0\nx\u0003\n\u001bz\t\u001b(q+K)\n+X\nKVK0\u0000K\t\u001b(q+K) =Eq+K0\t\u001b(q+K0):(10)\nIn principle, Eq. 10 is an infinite set of equations, one\nfor each K0, and can therefore not be solved. How-\never, we are only interested in the lowest bands. As VK\nbecomes exponentially small for large values of K2for\nboth Gaussian and muffin-tin potentials (see also Ap-\npendix A), we can introduce a cutoff in the values of\nK0that we consider. We can then solve the system of\nequations for arbitrary qin the Brillouin zone. In this\nwork, a square grid iK1+jK2withK1;2the reciprocal\nprimitive vectors and iandjintegers ranging from -4\nto 4, was used.\nIII. HONEYCOMB STRUCTURES\nIn order to see the effect of the spin-orbit terms intro-\nduced above on the band structure of artificial lattices,\nit is instructive to first investigate a test system. For\nthis, we will consider a honeycomb lattice, as spin-orbit\ncoupling has been extensively studied in graphene-like\nlattices through other methods [1, 3, 10]. As a start-\ning point, the first artificial graphene lattice realized by\nGomes et al. [5] using CO on the copper (111) surface\nmight appear as a good choice. However, more elabo-\nrated designs of honeycomb lattices have recently been\nshown to lead to interesting features, like the appear-\nance of a flat pband [6]. Additionally, for patterned\nquantum wells, intrinsic spin-orbit coupling is predicted\nto open a larger band gap between these higher bands\nthan at the Dirac cone between the lower (i.e. predom-\ninantlysorbital) energy bands [13]. We therefore use\nthe system described in Ref. [6] as a reference system.\nThis system also uses CO molecules on copper (111),\nbut instead of positioning single CO molecules in a tri-\nangular lattice as in Ref. [5], clusters of CO molecules\nare used. The clusters consist of two highly symmetri-\ncal rings. This added structure gives more confinement\nto the surface electrons without breaking the symmetry,\nwhich leads to a clear separation of sandporbitals and\nthe appearance of not only sbands, as in Ref. [5], but\nalso (nearly flat) pbands and a pcharacter Dirac cone.\nThe cluster arrangement of the CO molecules on the\nCu (111) surface is shown in Fig. 2 (a). In Ref. [6], a4\nFigure 2. Periodic system calculations, using Cu (111)\nparameters as a test case (effective mass m\u0003=0.42 em,\nCO molecules as Gaussians with a height of 0.45eV and\na FWHM of 0.6 nm). (a) shows the arrangement of CO\nmolecules on Cu (111) in the reference system realised ex-\nperimentally in Ref. [6] without spin-orbit coupling. The\nband structure with sandporbital bands plotted in red\nand blue respectively are shown (b) without any spin-orbit\ncoupling; (c) with only Rashba spin-orbit coupling ( \u000b1=\n1:6\u0002104m/s); (d,e) with only intrinsic spin-orbit coupling\n(\u000b2= 0:8\u00021015s/kg,\u000b2= 2\u00021015s/kg) and (f) with both\nRashba and intrinsic spin-orbit coupling ( \u000b1= 1:6\u0002104m/s,\n\u000b2= 2\u00021015s/kg).\nmuffin-tin potential with a height of 0.9 eV and a diam-\neter of 0.6 nm is used. When switching to Gaussians,\nthe choice was made for Gaussians with a full width\nat half maximum of 0.6 nm. With an adjustment of\nthe potential height to 0.45 eV, this setup fully repro-\nduces the muffin-tin results from Ref. [6], as shown in\nAppendix A.\nThe bare band structure of the reference system is\nshown in Fig 2 (b). Here, we see the two lowest energy\nbandsformingthewellknownDiracconeattheKpoint,\nlike in graphene. The pbands start with a (nearly) flat\nband, connected to two bands forming a porbital Dirac\ncone at the K point, which is connected to a fourth por-\nbital band. Upon inclusion of the Rashba coupling, spin\ndegeneracy is lifted everywhere except at the \u0000point,\nas shown in Fig. 2 (c). This result is analogous to that\nof previous tight-binding studies on single-orbital hon-\neycomb lattices [24, 25]. Additionally, the splitting of\nDirac cones under the influence of Rashba spin-orbitcoupling [24] is recovered, as shown in Appendix B.\nIf only the intrinsic spin-orbit coupling is included, as\nshown in Fig. 2 (d,e), the spin degeneracy remains, and\ninstead we see gaps opening up between the original\nband touching points. Indeed, this is also the result of\nintrinsic spin-orbit coupling in numerous other theoreti-\ncal studies on honeycomb systems [9, 10, 13, 21, 24–26].\nThere is both theoretical and experimental evidence for\nthese gaps to be topological and harbor protected edge\nstates [9, 10, 12, 13, 21, 26]. Notably, the gap open-\ning up between the first two porbital bands at the \u0000\npoint is much larger than the gaps opening up at the\nKpoints. In previous works, a similar trend of larger\ngaps between the porbital bands than between the s\norbital ones is observed as a consequence of the same\nintrinsic spin-orbit coupling [9, 13, 26]. This effect can\nbe explained by the angular momentum of porbitals,\nmaking intrinsic spin-orbit coupling an onsite effect. In\nsorbitals that have no angular momentum, the spin-\norbit coupling can only emerge through next-nearest-\nneighbour coupling, which connect the same sublattice\nin a honeycomb geometry. On the other hand, for p\norbitals the intrinsic spin-orbit coupling can couple px\nandpyorbitals on the same site, thus rendering the ef-\nfectmorerobust[9]. Additionally, weseeanunexpected\neffect, namely, if \u000b2is large enough, the porbital flat\nband is no longer isolated as in Fig. 2 (d), but the gap\nbetween the sandptype bands seemingly closes to\nform a Dirac cone at the \u0000point, as shown in Fig. 2\n(e). However, a small gap can be observed between the\ntwo bands. A zoom in on the gap can be found in Ap-\npendixC.Thisphenomenonisinterestingandithasnot\nbeen observed before, as far as the authors are aware.\nFinally, we can also include both Rashba and intrinsic\nspin-orbit coupling, as shown in Fig. 2 (d). Here, we see\nthat the Rashba coupling can close the gaps opened by\nthe intrinsic spin-orbit coupling, diminishing the pro-\ntection of possible topological states. However, the gap\nbetween the first two porbitals is remarkably robust to\nthe Rashba coupling. This robustness against Rashba\nspin-orbit coupling is of high importance in applications\nof topological materials, as Rashba spin orbit coupling\nis to a certain degree always present in devices based\non 2D materials.\nIn order to further study the topological nature of\nband gaps opened by the intrinsic spin-orbit coupling,\nwe calculated the local density of states (LDOS) of the\nfinite lattice. This is very helpful, as it allows to detect\nthe existence of in-gap edge localized states, as shown\nin Fig. 3. The design used for the finite-size system is\nshown in Fig. 3 (a). We study two types of locations\nhere, onsite locations indicated by the pink dot, and\nbridge locations, indicated by the green dot for the bulk\nand blue dot for the edge. Along the top edge, onsite\nedge locations have been marked with purple crosses,\nand edge and sub-edge bridge sites have been marked5\nFigure 3. Finite-size system calculations using Cu (111) parameters as a test case as in Fig. 2. Here, a spectral broadening\nof 40 meV realistic for the CO on Cu (111) system has also been included. (a) The locations of CO molecules (red) on\nthe Cu (111) grid (gray). Along the top edge, the locations of lattice sites, edge bridge sites and sub edge bridge sites\nhave been indicated with purple crosses, yellow disks and open disks respectively. (b) The spectra calculated for the spots\nindicated in (a). The top and bottom curves correspond to the system without and with intrinsic spin-orbit coupling\n(\u000b2= 2\u00021015s/kg), respectively. (c), (d) A map of the calculated LDOS on the energy indicated by a vertical line in (b)\natE= 0:027eV without and with intrinsic spin-orbit coupling , respectively.\nwith yellow and open circles, respectively. This design\nis different from the periodic case only at the boundary,\nwhere blocker potentials have been placed to separate\nthe lattice from the surrounding 2D electron gas. The\nintroduction of these blockers is crucial as without them\ntherewouldbenoclearboundaryandthereforeitwould\nnot be possible to study edge states. The location of\nblocking potentials is non trivial, as the introduction of\nout of lattice potentials can change the onsite energy\nof nearby sites. They have, therefore, been chosen in\nsuch a way as to not shift the LDOS spectra at the\nedge sites with respect to the bulk, as can be seen by\ncomparing the blue and green lines in the top graph of\nFig. 3 (b). A triangular design was chosen to optimise\nthe distance between the boundaries and at the same\ntime have edges as uniform as possible, given the small\nsystem size. The system was studied without and with\nintrinsic spin-orbit coupling. The LDOS spectra of the\ntwo systems are mostly similar in the bulk, as shown inFig. 3 (b). The pink and green spectra representing the\nbulk both with and without spin-orbit coupling display\ntwo peaks from the sorbitals at\u0019-0.3 and\u0019-0.2 eV,\nand both show a peak corresponding to the flat pband\naround 0 eV. These peak locations are inline with the\nband structure in Fig. 2 (b) and (e). However, some\ndifferences are visible. In the case without spin-orbit\ncoupling (Fig. 3 (b) top), there is very little mixing be-\ntween thesandporbitals resulting in an on-site (pink)\ndip at the flat band energy due to negative interference\nbetween the porbitals in the flat band. In the intrin-\nsic spin-orbit case, there is more mixing, as evidenced\nby the band touching between the highest sand lowest\nporbital in Fig. 2 (e). Therefore, there is some onsite\ndensity of state in this case, as evidenced by a pink peak\naround 0 eV (Fig. 3 (b) bottom). At a slightly higher\nenergy(0-0.06eV)weseeadipinthebulkspectraofthe\nintrinsic spin-orbit case. This corresponds to the band\ngap between the first two porbitals in Fig. 2 (e) from6\n0 till 0.05 eV. Notably, the edge bridge site where the\nspectrum was calculated (blue) has an increased LDOS\nin this range (the shoulder is a signature of an addi-\ntional peak, which cannot be separated from the flat\nband one), signaling a possible edge state. Indeed, if we\nlook at LDOS maps at 0.027 eV, we see a state clearly\nlocalized on the (sub) edge bridge sites of the system\nindicated in Fig. 3 (a) for the intrinsic spin-orbit sys-\ntem, as shown in Fig. 3 (d), whereas none is present for\nthe system shown in Fig. 3 (c), without spin-orbit cou-\npling. Thus, as expected, intrinsic spin-orbit coupling\nresults in a strong decrease in the DOS over the entire\nbulk of the finite system in the gap between the third\nand fourth band, accompanied with an increase in the\nintensity at the bridge sites at the system edge, point-\ning to a protected edge state. The intensity is maximal\non the bridge sites, in accordance with the porbital\nbands. We cannot test the precise topological nature\nof the edge state within the present muffin tin model;\nbut an atomistic tight-binding study on a semiconduc-\ntor system showed that this state is a helical quantum\nspin Hall state [10].\nIV. CONCLUSION\nWe have presented an effective muffin-tin model by\nintroducingtheintrinsicandRashbaspin-orbitcoupling\ninto the Schrödinger equation and tested the adequacy\nof the model by comparison with established results.\nThen, we have studied the effect of spin-orbit coupling\non a honeycomb system with separated sandporbital\nbands, which allows us to study porbital physics in the\nhoneycomb system. Besides the expected band open-ings at the Dirac points, intrinsic spin-orbit coupling\nshifts theporbital flat band downwards, causing hy-\nbridization with the sbands. As a result, a broad gap\narises between the third and fourth band of the system;\nour results on a finite lattice show the emergence of an\nedge state in this gap. We see that Rashba spin-orbit\ncoupling reduces the spin-orbit gaps at the K and K’\npoints of the Brillouin zone, but the broad gap between\nthe third and fourth band remains robust. We should\nalso remark that the model that we developed can be\nused to study the effects of spin-orbit coupling on any\ntype of artificial lattice.\nExperimentally, strong spin-orbit coupling might be\nrealized in artificial lattices by using a metallic surface\nstate on a heavy element metal such as rhenium, lead or\nbismuth, and/or using heavy adatoms as potential bar-\nriers or attractive sites for the surface electrons. A sim-\nilar concept has been reported for graphene, by placing\nIn and Tl atoms in the hollow sites of a graphene mono-\nlayer [3]. Real devices, applicable in electronics, can\nbe achieved by nanoscale patterning of heavy-element\nsemiconductor quantum wells, such as Ge, GaAs and\nInSb, with a honeycomb or another geometry of inter-\nest [20, 27, 28].\nACKNOWLEDGEMENTS\nFinancialsupportfromtheEuropeanResearchCoun-\ncil (Horizon 2020 “FIRSTSTEP”, 692691 and \"FRAC-\nTAL\", 865570 ) is gratefully acknowledged. We would\nlike to thank S. E. Freeney for help with visualising the\ndesign, and S. J. M. Zevenhuizen and R. Ligthart for\nhelp in developing an earlier non spin resolved version\nof the muffin-tin code.\n[1] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[2] J. Sichau, M. Prada, T. Anlauf, T. Lyon, B. Bosnjak,\nL.Tiemann, andR.Blick,Phys.Rev.Lett. 122,046403\n(2019).\n[3] C.Weeks, J.Hu, J.Alicea, M.Franz, andR.Wu,Phys.\nRev. X1, 021001 (2011).\n[4] L. Kou, Y.Ma, Z.Sun, T.Heine, andC. Chen,J.Phys.\nChem. Lett. 8, 1905 (2017).\n[5] K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C.\nManoharan, Nature 483, 306 (2012).\n[6] T. S. Gardenier, J. J. van den Broeke, J. R. Moes,\nI. Swart, C. Delerue, M. R. Slot, C. M. Smith, and\nD. Vanmaekelbergh, ACS Nano 14, 13638 (2020),\npMID: 32991147.\n[7] C. Wu, D. Bergman, L. Balents, and S. D. Sarma,\nPhys. Rev. Lett. 99, 070401 (2007).[8] C. Wu and S. D. Sarma, Phys. Rev. B 77, 235107\n(2008).\n[9] W. Beugeling, E. Kalesaki, C. Delerue, Y.-M. Niquet,\nD. Vanmaekelbergh, and C. M. Smith, Nat. Commun.\n6, 1 (2015).\n[10] J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. Vergniory,\nC. Felser, M. Aroyo, and B. A. Bernevig, Phys. Rev.\nLett.120, 266401 (2018).\n[11] G. van Miert, V. Juričić, and C. M. Smith, Phys. Rev.\nB90, 195414 (2014).\n[12] E. Kalesaki, C. Delerue, C. M. Smith, W. Beugeling,\nG. Allan, and D. Vanmaekelbergh, Phys. Rev. X 4,\n011010 (2014).\n[13] H. Scammell and O. Sushkov, Phys. Rev. B 99, 085419\n(2019).\n[14] D.-S. Ma, Y. Xu, C. S. Chiu, N. Regnault, A. A. Houck,\nZ. Song, and B. A. Bernevig, Phys. Rev. Lett. 125,\n266403 (2020).7\n[15] S. Li, W.-X. Qiu, and J.-H. Gao, Nanoscale 8, 12747\n(2016).\n[16] C.-H. Park and S. G. Louie, Nano lett. 9, 1793 (2009).\n[17] M. R. Slot, T. S. Gardenier, P. H. Jacobse, G. C. van\nMiert, S. N. Kempkes, S. J. Zevenhuizen, C. M. Smith,\nD. Vanmaekelbergh, and I. Swart, Nat. Phys. 13, 672\n(2017).\n[18] S. Kempkes, M. Slot, J. van den Broeke, P. Ca-\npiod, W. Benalcazar, D. Vanmaekelbergh, D. Bercioux,\nI.Swart, andC.M.Smith,Nat.Mater. 18,1292(2019).\n[19] M. Slot, S. Kempkes, E. Knol, W. Van Weerdenburg,\nJ. van den Broeke, D. Wegner, D. Vanmaekelbergh,\nA. Khajetoorians, C. M. Smith, and I. Swart, Phys.\nRev. X9, 011009 (2019).\n[20] N. A. Franchina Vergel, L. C. Post, D. Sciacca,\nM. Berthe, F. Vaurette, Y. Lambert, D. Yarekha,\nD. Troadec, C. Coinon, G. Fleury, et al., Nano Lett.\n(2020).\n[21] P. Ghaemi, S. Gopalakrishnan, and T. L. Hughes,\nPhys. Rev. B 86, 201406 (2012).\n[22] M. Hoesch, M. Muntwiler, V. Petrov, M. Hengsberger,\nL. Patthey, M. Shi, M. Falub, T. Greber, and J. Os-\nterwalder, Phys. Rev. B 69, 241401 (2004).\n[23] P. Atkins, J. De Paula, and J. Keeler, Physical Chem-\nistry(Oxford University Press, UK, 2018).\n[24] R. van Gelderen and C. M. Smith, Phys. Rev. B 81,\n125435 (2010).\n[25] M. Zarea and N. Sandler, Phys. Rev. B 79, 165442\n(2009).\n[26] F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass,\nW. Hanke, R. Thomale, J. Schäfer, and R. Claessen,\nScience357, 287 (2017).\n[27] S. Wang, D. Scarabelli, L. Du, Y. Y. Kuznetsova, L. N.\nPfeiffer, K. W. West, G. C. Gardner, M. J. Manfra,\nV. Pellegrini, S. J. Wind, et al., Nat. Nanotechnol. 13,\n29 (2018).\n[28] L. Post, T. Xu, N. F. Vergel, A. Tadjine, Y. Lambert,\nF. Vaurette, D. Yarekha, L. Desplanque, D. Stiévenard,\nX. Wallart, et al., Nanotechnology 30, 155301 (2019).\nAppendix A: Gaussian potentials\nRecent works have been modeling the experimen-\ntal results of artificial lattices built using CO on cop-\nper (111) by using a Muffin-tin calculation. Here, the\nSchrödinger equation is solved in two dimensions for a\npotential landscape where the CO molecules are mod-\neled as positive disk shaped protrusions. This approach\nis convenient in Fourier space, as the Fourier trans-\nformed form of this potential is analytically known.\nHowever, the intrinsic spin-orbit coupling term con-\ntains a derivative with respect to the potential, making\nthe muffin-tin approach less ideal. We therefore turn\nto modeling the CO molecules as Gaussian potential\nbarriers. We find that this reproduces the muffin-tin\nresults. Furthermore, there appears to be a lot of free-\ndom in choosing the width of the Gaussians, as long as\nthe height is adjusted as well (the broader the Gaus-\nFigure 4. Finite-size system comparison. Top images are\nmade using Gaussians with a full width at half maximum\n(FWHM) of 0.6 nm and a heigth of 0.45eV and the bottom\nimages were created using a classical muffin-tin calculation\nwith a diameter of 0.6 nm and a heigth of 0.9eV.\n●\n●\n●\n0.2 0.3 0.4 0.5 0.602468\nFWHM(nm)Height(eV)\nFigure 5. Relation between the height of the Gaussians and\ntheir full width at half maximum (FWHM) that reproduce\nthe muffin-tin results.\nsian, the lower it should be). The classical muffin-tin\nresults, along with the results for a Gaussian potential\nlandscape, are shown in Fig. 4. The relation between\nthe width of the Gaussian and its height in order to re-\nproduce the classical muffin-tin results is shown in Fig.\n5.\nIn case of a periodic system, we run into another im-\nportant point. Gaussian potentials are only periodic\nby approximation. It is therefore not possible to ana-\nlytically calculate the Fourier transform. However, as\nthe standard deviation of the Gaussians used is several\ntimes smaller than the lattice vector, we can approxi-\nmate the Fourier transform of the Gaussian in the unit\ncell as an infinite Fourier transform. The error is mini-\nmal, as the Gaussian potential exponentially decreases\naway from the center and is thus very small outside of8\nFigure 6. Band structure of double ring design calculated\nusing (a) Muffin-tin and (b) Gaussian shaped potentials.\nThe position of CO molecules on a copper lattice is shown\nin Fig. 2 (a).\nthe unit cell. We have\nVK=1\nAZ\nunitcelle\u0000ajxj2eiK\u0001xdx\n\u00191\nAZ1\n\u00001e\u0000ajxj2eiK\u0001xdx;(A1)\nwhereAis the unit cell surface. The integral on the\nright is a known integral, and thus we get\nVK\u0019\u0019\nae\u0000jKj2=4a: (A2)\nIndeed, the potential becomes exponentially small for\nlarge K, thus making it possible to introduce a cutoff\non the Kvalues included in Eq. 10 to calculate the\nband structure.\nIn classical mufin-tin calculations, the potential is\nfully periodic and the Fourier transformed potential is\nanalytically known,\nVK=\u0019d\nAjKjJ1\u0012\njKjd\n2\u0013\nV0: (A3)\nHere,dis the diameter of the muffin-tin potential disks,\nV0is the height of the potentials, and J1is the Bessel\nfunction. Just as for the Gaussian potential, VKbe-\ncomes small for large K.\nWhen the spin-orbit coupling is tuned to zero, the\nmuffin-tin and Gaussian potential can be compared.Using the same parameters as before, we indeed see\nthe same band structures for both potentials. This is\nshown in Fig. 6.\nAppendix B: Rashba modified Dirac point\nAs mentioned in the main text, the Rashba cou-\nplingcreatesadditionalDiracconesaroundthe sorbital\nDirac cone in the muffin-tin method, as shown in Fig. 7.\nThis is in agreement with tight-binding calculations for\nRashba coupling in graphene [24].\nFigure 7. A zoom in on the sorbital Dirac cone in Fig. 2\n(c). The location of the zoom is indicated in (a). (b) shows\nthe zoomed in image.\nAppendix C: \"Dirac\" point between sandpbands\nIn Fig. 2 (e) the sandpbands seem to hybridize to\nform a Dirac cone. However upon close inspection we\nsee that the gap between the bands does not close, as\nshown in Fig. 8.\nFigure 8. A zoom in on the apparent Dirac cone between the\nsandpbands in Fig. 2 (e) of the main text. The location\nof the zoom is indicated in (a). (b) shows the zoomed in\nimage." }, { "title": "1903.07049v1.Sensing_Kondo_correlations_in_a_suspended_carbon_nanotube_mechanical_resonator_with_spin_orbit_coupling.pdf", "content": "Sensing Kondo correlations in a suspended carbon nanotube mechanical resonator\nwith spin-orbit coupling\nDong E. Liu\nState Key Laboratory of Low-Dimensional Quantum Physics and\nDepartment of Physics, Tsinghua University, Beijing 100084, China\n(Dated: March 19, 2019)\nWe study electron mechanical coupling in a suspended carbon nanotube (CNT) quantum dot\ndevice. Electron spin couples to the \rexural vibration mode due to spin-orbit coupling in the electron\ntunneling processes. In the weak coupling limit, i.e. electron-vibration coupling is much smaller\nthan electron energy scale, the damping and resonant frequency shift of the CNT resonator can be\nobtained by calculating the dynamical spin susceptibility. We \fnd that strong spin-\rip scattering\nprocesses in Kondo regime signi\fcantly a\u000bect the mechanical motion of the carbon nanotube: Kondo\ne\u000bect induces strong damping and frequency shift of the CNT resonator.\nPACS numbers:\nI. INTRODUCTION AND SHORT SUMMARY\nCarbon nanotubes (CNTs) have been considered as an\nideal platform for quantum dot (QD) devices,1,2which\nshow Coulomb blockade oscillations, Luttinger liquid be-\nhavior,3Kondo e\u000bects,4,5and phase transitions.6,7CNTs\nis also emerging as a promising material for high-quality\nquantum nanomechanical applications8{15due to their\nlow mass and high sti\u000bnesses, and thus is useful in quan-\ntum sensing,16{18and in quantum information process-\ning.19{22The strong coupling between mechanical vibra-\ntions and the electronic degree of freedom was achieved\nin high quality suspended CNT QD resonators,12,13when\nsingle electron tunnelings through the CNT QD are\nturned on. This electron-vibration coupling provides an\nopportunity to study the electron correlations and quan-\ntum noise through the measurement of the vibration\nof CNT resonator, and provides a way to achieve the\nelectron-induced cooling of the resonator. Indeed, strong\ndamping, frequency shift, and their nonlinearity e\u000bects\nof the CNT resonator are observed.12,13\nIn those observations, the electron-vibration coupling\nis induced by gate capacitance dependence of the res-\nonator displacement, which only results in the cou-\npling between the electron density and the vibrations.\nTherefore, Kondo e\u000bects, caused by spin-\rip scatter-\nings due to strong e\u000bective spin exchange coupling in\nlow temperature,23seem to be irrelevant when consid-\nering mechanical e\u000bects. However, a recent theoretical\nproposal24shows that the coupling between the \rexural\nvibration mode and the electron spin can be achieved in\nCNT, because the spin-orbit coupling in CNT25{28tends\nto align the spin with the tangent direction of CNT. Most\ninterestingly, Kondo e\u000bects become relevant due to such\nspin-vibration coupling (\u0001 SO): Strong spin-\rip scatter-\ning processes in Kondo regime might signi\fcantly a\u000bect\nthe CNT vibrations. Therefore, CNT resonator may also\nprovide a way to \"sensing\" those quantum many body\ncorrelations.\nIn this work, we study a suspended CNT QD cou-\npled to source-drain leads in the Kondo regime. Both\nsupport CNT QD \nGate ܸௌ \nA \nܸீ \nsupport S D Side gates for barrier control \nFIG. 1: (color online) Schematics of the experimetal setup: a\nsuspended doubly-clamped semiconducting CNT QD is con-\nnected by source and drain leads. Back gate is used to control\nthe energy level of the QD, and the two side gates are for tun-\nneling barrier control.\nthe electron density-vibration coupling due to gate ca-\npacitance change and the spin-vibration coupling due to\nspin-orbit coupling are considered. When those couplings\nare much smaller than an electron energy scale (Kondo\ntemperature), a perturbation treatment shows that the\ndamping\rand resonant frequency ( !0) shift of the CNT\nresonator due to electron (both density and spin) vibra-\ntion coupling can be directly connected to the dynami-\ncal charge and spin susceptibilities. In the experimental\nrealizable regime !0\u0018TSU(2)\nK\u001c\u0001KK0\u001c\u0001SO(\u0001KK0\nis intervalley scattering), those dynamical susceptibilities\ncan be obtained from non-crossing approximation.23,29{31\nWe show that the strong spin-\rip scatterings in Kondo\nregime induce strong damping and frequency shift of the\nCNT resonator. Those e\u000bects can be detected by using\na \fnite frequency noise measurement.14\nII. MODEL AND HAMILTONIAN\nWe consider a suspended doubly-clamped semicon-\nducting CNT QD shown in Fig. 1. Due to the\ntwofold real spin and twofold isospin symmetries, all the\neigenenergies of the CNT quantum dot becomes fourfold\ndegenerate.32,33For each eigen-energy, the eigenstatesarXiv:1903.07049v1 [cond-mat.mes-hall] 17 Mar 20192\ncan be written as j\u001csi=j\u001ci\njsi, wherej\u001ci=j1\n2i;j\u00001\n2i\nrepresents the isospin and jsi=j1\n2i;j\u00001\n2irepresents the\nreal spin. The CNT also includes a spin-orbit coupling\nterm and an intervalley scattering term, and the Hamil-\ntonian is as follows\nHCNT=H0+\u0001SO\n2\u001c3(s\u0001^t) + \u0001KK0\u001c1 (1)\nwhereH0j\u001csi=E0j\u001csi, \u0001SOand \u0001KK0are the spin-\norbit coupling and the intervalley scattering, respectively.\n^t= (tx;ty;tz) is the local tangent vector along the CNT\naxis, and we assume the CNT is along the z-direction\nwithout deformation. For the vibration, we focus on the\nlowest \rexural mode, which can be described by a har-\nmonic oscillator25,26\nHvib=p2\n2M+1\n2M!2\n0q2; (2)\nwith CNT mass M, resonant frequency !0, and the dis-\nplacementq.\nThe \rexural vibration of CNT QD couples to the\nelectronic degree of freedom through two mechanisms:\n1) the in\ruence to the tangent vector which results\nin spin-vibration coupling24; 2) the in\ruence to the\ngate capacitance which induces the density-vibration\ncoupling.8,12,13For the \frst case, the \rexural vibra-\ntion (along x-direction) changes the tangent vector ^t;\nand the vector becomes coordinate dependent ^t(z) =\n(@zq=(p\n1 + (@zq)2);0;1=(p\n1 + (@zq)2)). Up to second\norder of the small quantity @zq, one has\n^t(z)w(1\u0000(@zq)2=2)^z+@zq^x: (3)\nThe derivative of the displacement can be obtained:\nh@zqi\u0018RL=2\n\u0000L=2dz\u001a(z)(df(z)=dz), where\u001a(z) is the elec-\ntron density, f(z) is the dimensionless wave function\nform, andLis the length of the CNT. For symmetric\nquantum dot, the integral vanishes for even vibration\nmode. This cancellation can be avoided by introduc-\ning an asymmetric potential, considering odd vibration\nmode, or con\fning the electron only in part of the sus-\npended CNT.19We choose last choice19for simplicity's\nsake and obtain h@zqiwq=L andh(@zq)2iwq2=L2(a\nconstant pre-factor \u00181 is dropped), the similar result can\nbe obtained for other two choices. The nonlinear term in\n^t(z) can be neglected since q\u001cL. For the second case,\nthe gate capacitance becomes a function of the displace-\nmentq:Cg(q) =C0\ng+@qC0\ngq+@2\nqC0\ngq2=2 +\u0001\u0001\u0001.8,12,13\nSince (@qC0\ngq)=(@2\nqC0\ngq2=2)\u0018h=q\u001d1 (his the distance\nbetween CNT and the gate), the second and higher or-\nder terms can be neglected. We use a capacitor model\nto describe the electron-electron interaction, and in the\npresence of the vibration, we have\nHINT=Ec(N\u0000N0\ng)2\u0000Ec2Vg\ne@qC0\ngNq (4)\nwhereEc=e2=(2CP) is the Coulomb charging energy\nwith total capacitance CP=CL+CR+Cg,Nis theelectron number operator in QD, and Ng=VgC0\ng=2 de-\nnotes the background charge with gate voltage Vg. For\nT;VSD\u001c\u0001< Ec(T,VSD, and \u0001 are temperature,\nsource-drain voltage, and level spacing of QD), only a\nsingle energy level ( d) near the Fermi level is relevant.\nThe Hamiltonian of the CNT QD can be written as\nHD=Ec(N\u0000N0\ng)2+p2\n2M+1\n2M!2\n0q2+\u0001SO\n2\u001c3sz\n+\u0001KK0\u001c1+\u0015DNq+\u0015SO\u001c3sxq (5)\nwhereN=P\n\u001b=fs;\u001cgdy\n\u001bd\u001b,\u0015SO= \u0001SO=L, and\u0015D=\n\u0000Ec2Vg@qC0\ng=e. The operator d\u001bannihilates a spin \u001b\nelectron in the CNT QD.\nThe CNT QD is coupled to two CNT leads, and the\nHamiltonian for the whole system shown in Fig. 1 is\nH=X\n\u000b=L=RX\nkX\n\u001b=fs;\u001cg\u000fk;\u001bcy\n\u000bk;\u001bc\u000bk;\u001b+HD\n+X\n\u000bX\nkX\n\u001b=fs;\u001cgV\u000bk\u0010\ncy\n\u000bk\u001bd\u001b+h:c:\u0011\n(6)\nwherec\u000bk;\u001b annihilates an electron with momentum k\nin the\u000blead with spin \u001b.V\u000bkdescribes the tunneling\nstrength between CNT QD and the \u000blead, and can be\ncontrolled by two side gate shown in Fig. 1.\nIII. PERTURBATION TREATMENT FOR\nWEAK ELECTRON-VIBRATION COUPLING\nWe want to study how the electron dynamics a\u000bect\nthe physics of the resonator. For weak electron-vibration\ncoupling (\u0015SO;\u0015D\u001c\u0000e), we treat \u0015SOand\u0015Das small\nparameter. Here, \u0000 eindicates the electron energy scale in\nthe system, i.e. Kondo temperature in the Kondo regime,\nhybridization \u0000 = \u0019\u001a0jVLj2+\u0019\u001a0jVRj2(\u001a0is the electron\ndensity of state in the leads) in the mixed valence regime\n(i.e. the energy level of the dot \u000fdis closed to the fermi\nlevelj\u000fdj\u001c\u0000).\nThe electron vibration coupling can be written in a\ngeneral form He\u0000v=h(1)q, where the linear coupling\nh(1)=\u0015DN+\u0015SO\u001c3sxin our problem. For small cou-\npling, the linear response theory results in\nh(1)(t) =h(1)\n0(t)\u0000Z1\n\u00001\u000b(1)\nh(t0)q(t\u0000t0)dt0(7)\nwhere\u000b(1)\nh(t\u0000t0) =i\u0012(t\u0000t0)h[h(1)(t); h(1)(t0)]i0and\nh\u0001\u0001\u0001i 0indicates the average for system without electron-\nvibration coupling. By solving the Heisenberg equation\nof motion, we can obtain the equation describing the dy-\nnamics of the resonator in the linear response limit34\nq+ 2\r_q+!2\n0q=F\nMcos(!Ft)\u0000h(1)(t)\nM(8)\nwhere we include a periodic driving force Fwith fre-\nquency!F, and a bare damping \r=!0=Q0with quality3\nfactorQ0. In the limit \r;j!F\u0000!0j\u001c!0, one can ana-\nlyze the problem in a rotating frame using the following\ntransformation34\nq(t) =u(t)ei!Ft+u\u0003(t)e\u0000i!Ft\n_q(t) =i!F\u0000\nu(t)ei!Ft\u0000u\u0003(t)e\u0000i!Ft\u0001\n(9)\nCombining Eq. (7), (8) and (9), and then neglect the fast\noscillating terms (like e\u0006i!Ft,e\u00062i!Ft,\u0001\u0001\u0001), the equation\nof motion for the resonator becomes34\n_u=\u0000ih\n!F\u0000!0+Re\u0010\n\u000b(1)\nh(!F)\u0011\n2M!Fi\nu (10)\n\u0000h\n\r+Im\u0010\n\u000b(1)\nh(!F)\u0011\n2M!Fi\nu\u0000iF\n4M!F+ih(1)\n0(t)\n2M!Fe\u0000i!Ft:\nOne then obtain the damping ( !F\u0019!0) due to electron\nvibration coupling\n\re\u0000v=\rs\u0000v+\rd\u0000v=Imh\n\u000b(1)\nh(!0)i\n2M! 0(11)\n=\u00152\nSOIm[\u001f\u001c3sx(!0)]\n2M! 0+\u00152\nDIm [\u001fN(!0)]\n2M! 0;\nwhere\u001f\u001c3sx(!) and\u001fN(!) represent the dynamical\nspin susceptibility and density susceptibility respec-\ntively, which are the Fourier transforms of the func-\ntions\u001f\u001c3sx(t) =i\u0012(t)h[\u001c3sx(t);\u001c3sx(0)]i0and\u001fN(t) =\nih[N(t); N(0)]i0. We choose ~=kB= 1 throughout\nthe paper. The frequency shift due to electron resonator\ninteraction is\n\u0001!s\u0000v=Reh\n\u000b(1)\nh(!0)i\n2M! 0; (12)\ncorresponding to the real part of the sum of the dynami-\ncal spin susceptibility and density susceptibility. The last\nterm in Eq. (11) describes the noise.\nIn equilibrium, those susceptibilities are directly re-\nlated to the spin and change \ructuations via \ructuation\ndissipation theorem. The \ructuations show di\u000berent be-\nhaviors in di\u000berent regimes. In the mixed valence regime\n(j\u000fdj\u001c\u0000), large charge and spin \ructuations12,13,31(i.e.\nelectrons hop onto and o\u000b the CNT QD) induce large\ndamping and frequency shift of the CNT resonator. The\ndamping and frequency shift e\u000bects become weaker as\ntemperature decreases. If energy level \u000fdlays in the mid-\ndle of conductance valley, electron tunnelings are block-\naded. In low Tlimit, this middle valley regime can be\nclassi\fed into two cases: 1) The total spin of QD is a\nsinglet, 2) the total spin is a non-singlet. For the \frst\ncase, no spin exchange coupling can be generated in low\nenergy; and thus the \ructuations will be suppressed in\nlowT(with very small leftover due to quantum mechan-\nical co-tunneling processes). For the second case, spin\nexchange coupling is generated in low Tand results in\nȁͳ ՝\u0003ۄ ȁʹ ՝\u0003ۄ ȁͳ ՛\u0003ۄ \nȁʹ ՛\u0003ۄ ȟௌை \nȟ\nௌை\nȟᇲ \nDrain Source \nȁͳ ՝\u0003ۄ ȁʹ ՝\u0003ۄ ȁͳ ՛\u0003ۄ \nȁʹ ՛\u0003ۄ ȟᇲ \n߬ଷݏ௫ \nȁͳ ՝\u0003ۄ ȁʹ ՝\u0003ۄ ȁͳ ՛\u0003ۄ \nȁʹ ՛\u0003ۄ ܵ௫ǣ \u0003֛\u0003֝ ܶௌ ଶا ȟᇲ (a) \n(b) FIG. 2: (color online) (a) Energy splitting due to spin orbit\ncoupling. The blue cross arrows indicate the intervalley scat-\nterings. (b) In the limit TSU(2)\nK\u001c\u0001KK0, the operator \u001c3sz\nis equivalent to the operator Sxin the lower subspace in the\nlow energy. The coupling becomes\u0015SOq\u0001KK 0\n\u0001SO=\u0015KK0q.\nKondo e\u000bects23whenT 100, and Ps= 0.34 atN= 40 and\nPs= 0.5atN= 80. Theseresultsarequantitativelycon-\nsistent with the experiment [11]. In fact, the dephasing/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s61/s48/s46/s48/s48/s53\n/s61/s48/s46/s53\n/s61 /s47/s53/s40/s97/s41\n/s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s69/s110/s101/s114/s103/s121/s61 /s47/s50\n/s61/s48/s40/s99/s41/s71 /s32/s40/s101/s50\n/s47/s104/s41/s44/s32 /s71 /s32/s40/s101/s50\n/s47/s104/s41/s44 /s32/s80\n/s83\n/s32\n/s32/s40/s98/s41\n/s61/s48\nFIG. 2: (color online). (a) Energy-dependence of conduc-\ntanceG↑(solid line), G↓(dotted line), and spin polarization\nPs(dashed line) for realistic situation. (b) and (c) show G↑\nandPsin the absence of the dephasing and of the helical\nsymmetry, respectively. Here N=30.\nhas two effects: (i) it promotes the openness of the two-\nterminal device and produces the spin polarization [28];\n(ii) it makes the charge lose its phase and spin memories\nandthen Psisdecreasedbyfurther increasingΓ. Accord-\ningly, for large Γ with the phase coherence length Lφ[22]\nshorter than the dsDNA length, Pswill be quite small.\nPs<0.05 for Γ = 0 .5 andPs→0 if Γ→ ∞. Due to the\ninterplay between the above two effects, a small Γ, rang-\ning from 0.0002 to 0.01, where Lφis much larger than\n100, is optimal for large Ps. In addition, Fig. 3(c) shows\nthe averaged conductance ∝angb∇acketleftG↑∝angb∇acket∇ightvsN.∝angb∇acketleftG↑∝angb∇acket∇ightis declined\nby increasing Nor Γ, because large Nor Γ will enhance\nthe scattering. However, ∝angb∇acketleftG↑∝angb∇acket∇ightremains quite large for\nN= 100 and Γ = 0 .012. Therefore, the dsDNA is a well\nspin filter due to the large Psand∝angb∇acketleftG↑∝angb∇acket∇ight.\nLet us further study the spin polarization by varying\nother model parameters. Figures 4(a) and 4(b) show Ps\natE= 0.488 with N= 80 and ∝angb∇acketleftPs∝angb∇acket∇ightwithN= 30, re-\nspectively, as functions of the SOC tsoand the dephasing\nstrength Γ. Psand∝angb∇acketleftPs∝angb∇acket∇ightare zero exactly when Γ = 0 or\ntso= 0. Of course, tsois a key factor for the spin polar-\nization or equivalently tsois “the driving force” of Ps. If\nthere is no SOC, no spin polarization would appear for\nwhatever other parameters are. In general, strong SOC\nusually lead to large ∝angb∇acketleftPs∝angb∇acket∇ight[Fig.4(b)]. However, for a fixed\nenergy,Pswill not increase monotonically with tso, as\nseen in Fig. 4(a). A large Pscan be obtained for long ds-\nDNA even for quite small tso, because the spin polarized\nelectrons will accumulate gradually when electrons are4\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s46/s48/s48/s48/s46/s48/s54/s48/s46/s49/s50\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48/s46/s48/s48/s54 /s48/s46/s48/s49/s50 /s48/s46/s48/s49/s56/s50/s53/s53/s48/s55/s53/s80\n/s83\n/s78\n/s80\n/s83\n/s61/s48/s46/s48/s48/s48/s49\n/s61/s48/s46/s48/s48/s48/s52\n/s61/s48/s46/s48/s48/s52\n/s61/s48/s46/s48/s49/s50/s78/s71 /s32/s40/s101/s50\n/s47/s104/s41\n/s78\n/s78\n/s67/s40/s97/s41\n/s40/s98/s41\n/s40/s99/s41 /s40/s100/s41\nFIG. 3: (color online). Length-dependence (a) of PsatE=\n0.488, (b) of ∝angbracketleftPs∝angbracketright, and (c) of ∝angbracketleftG↑∝angbracketrightfor different values of the\ndephasing parameter. (d) The critical length Ncvs Γ. Here\nNcis extracted from the curve of ∝angbracketleftPs∝angbracketright-Nand the solid line is\nthe fitting curve with Nc∝Γ−1. The legends in (d) are for\npanels (a), (b), and (c).\n/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48 /s47\n/s32/s32\n/s48\n/s48/s46/s49/s51/s40/s99/s41/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s40/s97/s41\n/s32/s32\n/s116\n/s115/s111/s48\n/s48/s46/s54\n/s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s116\n/s50\n/s116\n/s49/s45/s48/s46/s49/s53\n/s48/s46/s49/s53/s40/s100/s41/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s116\n/s115/s111/s48\n/s48/s46/s49/s51/s40/s98/s41\nFIG. 4: (color online). (a) Psvs the SOC tsoand the dephas-\ning Γ at E= 0.488 with N= 80. (b), (c), and (d) show ∝angbracketleftPs∝angbracketright\nwithN= 30 as functions of tsoand Γ, of Γ andθ\nπ, and of t1\nandt2, respectively.\ntransmitting along the dsDNA. In addition, we observe a\nlarge area with red color in Fig. 4(b), where ∝angb∇acketleftPs∝angb∇acket∇ightexceeds\n0.1 for short dsDNA. This implies that the dsDNA would\nbe an efficient spin filter in a wide parameters range.\nFigure 4(c) plots the averaged spin polarization ∝angb∇acketleftPs∝angb∇acket∇ight\nvs Γ andθ\nπby fixing the radius Rand the arc length\nlato account for the rigid sugar-phosphate backbones.\nThe helix angle θcan be changed by stretching the DNAmolecule[29]. Itisobviousthat ∝angb∇acketleftPs∝angb∇acket∇ightiszerointheabsence\nof the helical symmetry ( θ=π\n2) and∝angb∇acketleftPs∝angb∇acket∇ightis increased\nmonotonically by decreasing θ. This indicates that the\nhelix of the dsDNA plays a vital role to the existence of\nthe spin polarization. Finally, we present the influence of\nthe hopping integrals t1andt2on the spin polarization,\nas illustrated in Fig. 4(d). We can see that ∝angb∇acketleftPs∝angb∇acket∇ightis small\nwhent1andt2haveidenticalsignandbecomelargewhen\nt1andt2haveoppositesign. Sincethesignofthehopping\nintegralissensitivetothetypeofneighboringnucleobases\n[27] and to the twist angle ∆ ϕ[30], the spin polarization\ncouldbe improvedbysynthesizingspecificDNAmolecule\nand putting force along the helix axis of the dsDNA.\nIn summary, we propose a model Hamiltonian to sim-\nulate the quantum spin transport through the dsDNA.\nThis two-terminal dsDNA-based device would exhibit\nhigh spin polarization by considering the SOC, the de-\nphasing, and the helical symmetry, although no spin po-\nlarization exists in the ssDNA. The spin polarization in-\ncreases with increasing the DNA length. Additionally,\nthespinpolarizationcouldbeimprovedbyproperlymod-\nifyingthehoppingintegralanddecreasingthehelixangle.\nThis work was supported by China-973 program and\nNSF-China under Grants No. 10974236 and 10821403.\n∗Electronic address: sunqf@iphy.ac.cn\n[1] A. R. Rocha, V. M. Garc´ ıa-su´ arez, S. W. Bailey, C. J.\nLambert, J. Ferrer, and S. Sanvito, Nature Mater. 4, 335\n(2005).\n[2] A. Fert, Rev. Mod. Phys. 80, 1517 (2008).\n[3] V. A. Dediu, L. E. Hueso, I. Bergenti, and C. Taliani,\nNature Mater. 8, 707 (2009).\n[4] M. Urdampilleta, S. Klyatskaya, J-P. Cleuziou, M.\nRuben, and W. Wernsdorfer, Nature Mater. 10, 502\n(2011).\n[5] Z. H. Xiong, D. Wu, Z. V. Vardeny, and J. Shi, Nature\n(London) 427, 821 (2004).\n[6] C. Barraud, P. Seneor, R. Mattana, S. Fusil, K. Bouze-\nhouane, C. Deranlot, P. Graziosi, L. Hueso, I. Bergenti,\nV. Dediu, F. Petroff, and A. Fert, Nature Phys. 6, 615\n(2010).\n[7] J. Brede, N. Atodiresei, S. Kuck, P. Lazi´ c, V. Caciuc, Y.\nMorikawa, G. Hoffmann, S. Bl¨ ugel, and R. Wiesendan-\nger, Phys. Rev. Lett. 105, 047204 (2010).\n[8] N. Atodiresei, J. Brede, P. Lazi´ c, V. Caciuc, G. Hoff-\nmann, R. Wiesendanger, and S. Bl¨ ugel, Phys. Rev. Lett.\n105, 066601 (2010).\n[9] S. Schmaus, A. Bagrets, Y. Nahas, T. K. Yamada, A.\nBork, M. Bowen, E. Beaurepaire, F. Evers, and W.\nWulfhekel, Nature Nanotech. 6, 185 (2011).\n[10] F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen,\nNature (London) 452, 448 (2008).\n[11] B. G¨ ohler, V. Hamelbeck, T. Z. Markus, M. Kettner,\nG. F. Hanne, Z. Vager, R. Naaman, and H. Zacharias,\nScience331, 894 (2011).\n[12] Z. Xie, T. Z. Markus, S. R. Cohen, Z. Vager, R. Gutier-\nrez, and R. Naaman, Nano Lett. 11, 4652 (2011).5\n[13] S. Yeganeh, M. A. Ratner, E. Medina, and V. Mujica, J.\nChem. Phys. 131, 014707 (2009).\n[14] R. Gutierrez, E. D´ ıaz, R. Naaman, and G. Cuniberti,\narXiv: 1110.0354 (2011).\n[15] G. L. J. A. Rikken, Science 331, 864 (2011).\n[16] M. Di Ventra and Y. V. Pershin, Nature Nanotech. 6,\n198 (2011).\n[17] G. Cuniberti, E. Maci´ a, A. Rodriguez, and R. A. R¨ omer,\ninCharge Migration in DNA: Perspectives from Physics,\nChemistry and Biology , edited by T. Chakraborty\n(Springer-Verlag, Berlin, 2007).\n[18] D. Hochberg, G. Edwards, and T. W. Kephart, Phys.\nRev. E55, 3765 (1997).\n[19] In other words, here we consider the SOC induced by the\nboundary-confining potential. All the results are qualita-\ntively same if other kinds of SOC are considered.\n[20] Q.-F. Sun, X. C. Xie, and J. Wang, Phys. Rev. B 77,\n035327 (2008).\n[21] X.-Q. Li and Y.-J. Yan, Appl. Phys. Lett. 79, 2190\n(2001).\n[22] Y. Xing, Q.-F. Sun, and J. Wang, Phys. Rev. B 77,115346 (2008).\n[23] H. Jiang, S. Cheng, Q.-F. Sun, and X. C. Xie, Phys. Rev.\nLett.103, 036803 (2009).\n[24] Q.-F. Sun, J. Wang, and H. Guo, Phys. Rev. B 71,\n165310 (2005).\n[25]Electronic Transport in Mesoscopic Systems , edited by\nS. Datta (Cambridge University Press, Cambridge, U.K.,\n1995).\n[26] Y. J. Yan and H. Y. Zhang, J. Theor. Comp. Chem. 1,\n225 (2002).\n[27] K. Senthilkumar, F. C. Grozema, C. F. Guerra, F. M.\nBickelhaupt, F. D. Lewis, Y. A. Berlin, M. A. Ratner,\nand L. D. A. Siebbeles, J. Am. Chem. Soc. 127, 14894\n(2005).\n[28] Q.-F. SunandX.C. Xie, Phys.Rev.B 71, 155321 (2005).\n[29] J. Gore, Z. Bryant, M. N¨ ollmann, M. U. Le, N. R. Coz-\nzarelli, and C. Bustamante, Nature (London) 442, 836\n(2006).\n[30] R. G. Endres, D. L. Cox, and R. R. P. Singh, Rev. Mod.\nPhys.76, 195 (2004)." }, { "title": "1110.1225v1.Spin_and_pseudospin_symmetry_along_with_orbital_dependency_of_the_Dirac_Hulthen_problem.pdf", "content": "arXiv:1110.1225v1 [quant-ph] 6 Oct 2011Spin and pseudospin symmetry along with orbital dependency of\nthe Dirac-Hulth´ en problem\nSameer M. Ikhdair,1,∗C¨ uneyt Berkdemir,2,†and Ramazan Sever3,‡\n1Physics Department, Near East University, Nicosia, North C yprus, Turkey\n2Physics Department, Erciyes University, 38039 Kayseri, Tu rkey\n3Physics Department, Middle East Technical University, 065 31 Ankara, Turkey\n(Dated: October 8, 2018)\nAbstract\nThe role of the Hulth ´ en potential on the spin and pseudospin symmetry solutions is investigated\nsystematically by solving the Dirac equation with attracti ve scalar S(/vector r) and repulsive vector V(/vector r)\npotentials. The spin and pseudospin symmetry along with orb ital dependency (pseudospin-orbit\nand spin-orbit dependent couplings) of the Dirac equation a re included to the solution by introduc-\ning the Hulth ´ en-square approximation. This effective approach is based on f orming the spin and\npseudo-centrifugal kinetic energy term from the square of t he Hulth ´ en potential. The analytical\nsolutions of the Dirac equation for the Hulth ´ en potential with the spin-orbit and pseudospin-\norbit-dependent couplings are obtained by using the Nikifo rov-Uvarov (NU) method. The energy\neigenvalue equations and wave functions for various degene rate states are presented for several\nspin-orbital, pseudospin-orbital and radial quantum numb ers under the condition of the spin and\npseudospin symmetry.\nKeywords: Spin and pseudospin symmetry; orbital dependenc y; Dirac equation; Hulth ´ en poten-\ntial; Nikiforov-Uvarov Method.\nPACS numbers: 03.65.Ge; 03.65.Pm; 11.30.Pb; 21.60.Cs; 31.30.Jv\n∗E-mail: sikhdair@neu.edu.tr\n†E-mail: berkdemir@erciyes.edu.tr\n‡E-mail: sever@metu.edu.tr\n1I. INTRODUCTION\nThe spin and pseudospin symmetry [1,2] observed originally almost 40 y ears ago as a\nmechanism to explain different aspects of the nuclear structure is o ne of the most interest-\ning phenomena in the relativistic quantum mechanics. It plays a crucia l role for a Dirac\nhamiltonian with realistic scalar S(/vector r) and vector V(/vector r) potentials, for nucleon spectrum in\nnuclei, for the existence of identical bands in superdeformed nucle i, etc [3]. The key feature\nof the pseudospin symmetry is based on the small energy difference between single-nucleon\ndoublets with quantum numbers nr,ℓ,j=ℓ+1/2 andnr−1,ℓ+2,j=ℓ+3/2, wherenr,ℓ\nandjare the single nucleon radial, orbital and total angular quantum num bers, respectively.\nThese quantum numbers are relabelled as pseudospin doublets; ℓ+ 1 =˜ℓis the ”pseudo”\norbital angular momentum, ˜ s= 1/2 is the ”pseudo” spin and j=˜ℓ±˜sis the total ”pseudo”\nangular momentum for the two states in the doublet [4]. For example, ”nrs1/2,(nr−1)d3/2”\nis valid for ˜ℓ= 1, ”nrp3/2,(nr−1)f5/2is valid for ˜ℓ= 2, etc. Another key feature is the\nsingle-particle Hamiltonian of the oscillator shell model. This means tha t the pseudospin\nconcept in the nuclear theory is a division of the single-particle total angular momentum\nintopseudorather than normalorbital and spin parts. The shell model implies that nucle-\nons move in a relativistic mean field produced by the interactions betw een nucleons. The\nrelativistic dynamics of nucleons moving in the relativistic mean field are described by using\nthe Dirac equation and not the Schr¨ odinger equation.\nThe pseudospin symmetry concept is investigated by the framewor k of the Dirac equation\nand occurs as a symmetry of the Dirac hamiltonian when an attractiv e scalarS(/vector r) and a\nrepulsive vector V(/vector r) potentials near equal to each other in magnitude, but opposite in s ign,\ni.e.,S(/vector r)∼ −V(/vector r). On the other hand, the sum of the vector and scalar potentials in the\nDirac equation is a constant, i.e.,V(/vector r)+S(/vector r) =constant, for the solution of the pseudospin\nsymmetry in nuclei. This condition has been found by Ginocchio [5] and a pplied to the case\nof the spherical harmonic oscillator [6]. Meng et al[7] showed that the pseudospin symmetry\nis exact under the condition of d(V(/vector r)+S(/vector r))/dr= 0. Lisboa et al.studied the generalized\nharmonic oscillator for spin-1 /2 particles by setting either Σ( /vector r) =V(/vector r) +S(/vector r) = 0 or\n∆(/vector r) =V(/vector r)−S(/vector r) = 0 [8]. A necessary condition for occurrence of the pseudospin\nsymmetry in nuclei is to consider the case Σ( /vector r) = 0 [3,5,9,10]. For more realistic nuclear\nsystems, the quality of the pseudospin symmetry is increased in the framework of the single-\n2particlerelativistic modelsandhence thecompetitionbetween theps eudo-centrifugal barrier\nand the pseudospin-orbital potential is completed in the onset of p seudospin symmetry\n[11]. The Dirac equation with the pseudospin symmetry is solved numer ically for nucleons\nwhich move independently in the relativistic mean field with external sc alar and vector\npotentials [12,13]. In addition to the numerical solutions, some analyt ical solutions are\nalso discussed for solving the Dirac equation for some realistic poten tials [14-17] with the\npseudospin symmetry. The analytical solutions show that under th e condition of pseudospin\nsymmetry, the exact solution of the Dirac equation gives the bound -state energy spectra and\nspinor wave functions [18-20].\nThe aim of this paper is to present an analytical bound state solution s of the Dirac\nequation for the Hulth ´ en potential under the conditions of the exact pseudospin symme-\ntry and exact spin symmetry. To obtain a general solution for all va lues of the pseu-\ndospin (spin) quantum numbers, the pseudospin (spin) symmetry a nd orbital dependency,\npseudospin-orbit (spin-orbit) dependent coupling are included to t he lower component of\nthe Dirac equation as an integer quantum number. This component h as the structure of the\nSchr¨ odinger-like equations with the pseudo-centrifugal (spin-c entrifugal) kinetic energy term\nand its solution is analyzed by using some algebraic methods and effect ive approaches. One\nof these effective approaches is applied to the pseudo-centrifuga l (spin-symmetry) kinetic en-\nergy term in the case of ˜ℓ>0 (ℓ>0) and also an effective potential suggested in the form of\nthe square of the Hulth ´ en potential is taken into account instead of the pseudo-centrifug al\nkinetic energy term. For small values of the radial coordinate r, this effective potential\ngives a centrifugal energy term in the first approximation. Theref ore, the pseudo-centrifugal\n(spin-centrifugal) kinetic energy term is accepted as an effective t erm in this region. It is\nworthy to state that Jia et al[21,22] have proposed an improved new approximation scheme\nto deal with the centrifugal kinetic energy term in the solution of th e Schr¨ odinger-Hulth ´ en\nproblem. Using this approximation scheme, Jia et al[23,24] have obtained approximate\nanalytical solutions for the Dirac-generalized P¨ oschl-Teller and Kle in-Gordon-P¨ oschl-Teller\nproblems including the centrifugal kinetic energy term. Recently, I khdair [25] has applied\nthe approximation scheme to deal with the orbital centrifugal ter m in the Schr¨ odinger-\nManning-Rosen problem using the Nikiforov-Uvarov method. Furth er, the approximation\nhas also been applied to the Schr¨ odinger–Hulth ´ en problem using the improved quantization\nrule [26].\n3In the present work, the Dirac equation for the Hulth ´ en potential is arranged under\nthe condition of the exact pseudospin (spin) symmetry and it’s solut ion is obtained sys-\ntematically by using the Nikiforov-Uvarov (NU) method [27-31]. As an application of the\nDirac-Hulth ´ en problem with the pseudospin (spin) symmetry, the relativistic eigen value\nspectrum for various degenerate states is presented for sever al pseudo-orbital (spin-orbital)\nand pseudospin (spin) quantum numbers.\nThe structure of the paper is as follows. In Sec. 2, the basic ideas o f the Nikiforov-Uvarov\n(NU) methodareoutlined inshort. InSec. 3, the Diracequation isbr iefly introduced forthe\nspin andpseudospin symmetry solutions. InSec. 4, theHulth ´ enpotentialis substituted into\nthe lower component of the Dirac equation and the pseudo-centrif ugal (or spin-centrifugal)\nkinetic energy term is replaced by the square of the Hulth ´ en potential to apply the Hulth ´ en\nsquare approximation. The main results obtained in previous section s are connected by\nmeans of the main equation of the NU method. Lastly, the general p rocedures of the\nsolution method are followed to obtain the energy eigenvalue equatio n and two-spinor wave\nfunctions. Results and conclusions are performed in Sec. 5.\nII. BASIC IDEAS OF THE NIKIFOROV-UVAROV (NU) METHOD\nIt is especially well known that the solutions of the Schr¨ odinger and Schr¨ odinger-like\nequations including the centrifugal barrier and/or the spin-orbit c oupling terms have not\nbeen obtained straightforwardly for the exponential-type poten tials such as Morse, Hulth ´ en,\nWoods-Saxon, etc [32]. Although the exact solution of the Schr¨ od inger equation for the\nexponential-type potentials has been obtained for ℓ= 0, anyℓ-state solutions have been\ngivenapproximatelybyusingsomeanalyticalmethodsunder acerta innumber ofrestrictions\n[33,34]. One of the calculational tools utilized in these studies is the NU m ethod. This\ntechnique is based on solving the hypergeometric type second-ord er differential equations\nby means of the special orthogonal functions [35]. For a given pote ntial, the Schr¨ odinger or\nSchr¨ odinger-like equations in spherical coordinates are reduced to a generalized equation of\nhypergeometric type with an appropriate coordinate transforma tionr→sand then they\nare solved systematically to find the exact or particular solutions. T he main equation which\n4is closely associated with the method is given in the following form [27]\nψ′′(s)+/tildewideτ(s)\nσ(s)ψ′(s)+/tildewideσ(s)\nσ2(s)ψ(s) = 0, (1)\nwhereσ(s) and/tildewideσ(s) are polynomials at most second-degree, /tildewideτ(s) is a first-degree polynomial\nandψ(s) is a function of the hypergeometric type.\nLet us now try to reduce Eq.(1) to a comprehensible form by taking ψ(s) =φ(s)y(s) and\nchoosing an appropriate function φ(s):\ny′′(s)+/parenleftbigg\n2φ′(s)\nφ(s)+/tildewideτ(s)\nσ(s)/parenrightbigg\ny′(s)+/parenleftbiggφ′′(s)\nφ(s)+φ′(s)\nφ(s)/tildewideτ(s)\nσ(s)+/tildewideσ(s)\nσ2(s)/parenrightbigg\ny(s) = 0.(2)\nAt the first stage, Eq.(2) can be seen to be more complicated than t he main equation,\nEq.(1). To ensure the reasonable understanding, the coefficient o fy′(s) is taken in the form\nτ(s)/σ(s), whereτ(s) is a polynomial of degree at most one, i.e.,\n2φ′(s)\nφ(s)+/tildewideτ(s)\nσ(s)=τ(s)\nσ(s), (3)\nand hence the most regular form is obtained as follows,\nφ′(s)\nφ(s)=π(s)\nσ(s), (4)\nwhere\nπ(s) =1\n2[τ(s)−/tildewideτ(s)]. (5)\nThe most useful demonstration of Eq. (5) is\nτ(s) =/tildewideτ(s)+2π(s). (6)\nThe new parameter π(s) is a polynomial of degree at most one. In addition, the term\nφ′′(s)/φ(s) which appears in the coefficient of y(s) in Eq.(2) is arranged as follows\nφ′′(s)\nφ(s)=/parenleftbiggφ′(s)\nφ(s)/parenrightbigg′\n+/parenleftbiggφ′(s)\nφ(s)/parenrightbigg2\n=/parenleftbiggπ(s)\nσ(s)/parenrightbigg′\n+/parenleftbiggπ(s)\nσ(s)/parenrightbigg2\n. (7)\nIn this case, the coefficient of y(s) is transformed into a more suitable arrangement by taking\nthe form in Eq.(4);\nφ′′(s)\nφ(s)+φ′(s)\nφ(s)/tildewideτ(s)\nσ(s)+/tildewideσ(s)\nσ2(s)=¯σ(s)\nσ2(s)(8)\nwhere\n¯σ(s) =/tildewideσ(s)+π2(s)+π(s)[/tildewideτ(s)−σ′(s)]+π′(s)σ(s). (9)\n5Substituting the right-hand sides of Eq.(3) and Eq.(8) into Eq.(2), a n equation of the same\ntype as Eq.(1) is obtained as\ny′′(s)+τ(s)\nσ(s)y′(s)+¯σ(s)\nσ2(s)y(s) = 0. (10)\nAs a consequence of the above algebraic transformations, the fu nctional form of Eq.(1)\nis protected by following a systematic way. Therefore, the transf ormations allow us to\nreplace the function of the hypergeometric type ψ(s) by the substitution φ(s)y(s), where\nφ(s) satisfies Eq.(4) whit an arbitrary linear polynomial π(s). If the polynomial ¯ σ(s) in\nEq.(10) is divisible by σ(s),i.e.,\n¯σ(s) =λσ(s), (11)\nwhereλis a constant, Eq.(10) is reduced to an equation of hypergeometric type\nσ(s)y′′+τ(s)y′+λy= 0, (12)\nand also its solution is given as a function of hypergeometric type [35]. To determine the\npolynomial π(s), Eq.(9) is compared with Eq.(11) and then a quadratic equation for π(s) is\nobtained as follows,\nπ2(s)+π(s)[/tildewideτ(s)−σ′(s)]+/tildewideσ(s)−kσ(s), (13)\nwhere\nk=λ−π′(s). (14)\nThe solution of this quadratic equation for π(s) yields the following equality\nπ(s) =σ′(s)−/tildewideτ(s)\n2±/radicalBigg/parenleftbiggσ′(s)−/tildewideτ(s)\n2/parenrightbigg2\n−/tildewideσ(s)+kσ(s). (15)\nIn order to obtain the possible solutions according to the plus and min us signs of Eq.(15),\nthe parameter kwithin the square root sign must be known explicitly. To provide this\nrequirement, the expression under the square root sign has to be the square of a polynomial,\nsinceπ(s) is a polynomial of degree at most one. In this case, an equation of t he quadratic\nform is available for the constant k. Setting the discriminant of this quadratic equal to\nzero, the constant kis determined clearly. After determining k, the polynomial π(s) is\nobtained from Eq.(15), and then τ(s) andλare also obtained by using Eq.(5) and Eq.(14),\nrespectively.\n6A common trend which is followed to generalize the solutions of Eq.(12) is to show that\nall the derivatives of functions of hypergeometric type are also of hypergeometric type. For\nthis purpose, Eq.(12) is differentiated by using the representation v1(s) =y′(s)\nσ(s)v′′\n1(s)+τ1(s)v′\n1(s)+µ1v1(s) = 0, (16)\nwhereτ1(s) =τ(s) +σ′(s) andµ1=λ+τ′(s).τ1(s) is a polynomial of degree at most\none andµ1is independent of the variable s. It is clear that Eq.(16) is an equation of\nhypergeometric type again. By taking v2(s) =y′′(s) as a new representation, the second\nderivation of Eq.(12) becomes\nσ(s)v′′\n2(s)+τ2(s)v′\n2(s)+µ2v2(s) = 0, (17)\nwhere\nτ2(s) =τ1(s)+σ′(s) =τ(s)+2σ′(s), (18)\nµ2=µ1+τ′\n1(s) =λ+2τ′(s)+σ′′(s). (19)\nIn a similar way, an equation of hypergeometric type for vn(s) =y(n)(s) is constructed as a\nfamily of particular solutions of Eq.(12) corresponding to a given λ;\nσ(s)v′′\nn(s)+τn(s)v′\nn(s)+µnvn(s) = 0, (20)\nand here the general recurrence relations for τn(s) andµnare found as follows, respectively,\nτn(s) =τ(s)+nσ′(s), (21)\nµn=λ+nτ′(s)+n(n−1)\n2σ′′(s). (22)\nWhenµn= 0, Eq.(22) becomes as follows\nλ=λn=−nτ′(s)−n(n−1)\n2σ′′(s),(n= 0,1,2,...) (23)\nand then Eq.(20) has a particular solution of the form\ny(s) =yn(s) =Bn\nρ(s)dn\ndrn[σn(s)ρ(s)],\nwhich is the Rodrigues relation of degree nandρ(s) is the weight function satisfying the\ndifferential equation\n[σ(r)ρ(r)]′=τ(r)ρ(r).\nTo obtain an eigenvalue solution through the NU method, the relation ship between λand\nλnmust be set up by means of Eq.(14) and Eq.(23).\n7III. DIRAC EQUATION\nIn the relativistic description, the Dirac equation of a single-nucleon with the mass µ\nmoving in an attractive scalar potential S(/vector r) and a repulsive vector potential V(/vector r) can be\nwritten as/bracketleftBig\n/vector α.c/vectorP+β(µc2+S(/vector r))+V(/vector r)/bracketrightBig\nψnrκ(/vector r) =Enrκψnrκ(/vector r), (24)\nwhere\n/vectorP=−i/planckover2pi1/vector∇, /vector α =\n0/vector σ\n/vector σ0\n, β =\n0I\n−I0\n, (25)\nwith/vector σis the vector Pauli spin matrix and Iis the identity matrix. /vectorPis the three momentum\noperators,/vector αandβare the usual 4 ×4 Dirac matrices [36], cis the velocity of light in vacuum\nand/planckover2pi1isthePlanck’s constant divided by2 π.Enrκdenotes therelativistic energyeigenvalues\noftheDiracparticle. Fornucleiwithsphericalsymmetry, S(/vector r)andV(/vector r)potentialsinEq.(24)\nrepresent only the radial coordinates, i.e.,S(/vector r) =S(r) andV(/vector r) =V(r), whereris the\nmagnitude of /vector r. The spinor wave functions ψnrκ(/vector r) can be written in the following form\nψnrκ(/vector r) =1\nr\nFnrκ(r)/bracketleftbig\nYℓ(θ,φ)χ±/bracketrightbig(j)\nm\niGnrκ(r)/bracketleftbig\nY˜ℓ(θ,φ)χ±/bracketrightbig(j)\nm\n, (26)\nwhereYℓ(θ,φ) (Y˜ℓ(θ,φ)) andχ±are the spin (pseudospin) spherical harmonic and spin wave\nfunction which are coupled to angular momentum jwith projection m, respectively. Fnrκ(r)\nandGnrκ(r) are the radial wave functions for the upper and lower component s, respectively.\nThe labelκhas two explanations; the aligned spin j=ℓ+1/2 (s1/2,p3/2,etc.) is valid for the\ncaseofκ=−(j+1/2)andthen ˜ℓ=ℓ+1, whiletheunalignedspin j=ℓ−1/2(p1/2,d3/2,etc.)\nisvalidfor thecase of κ= (j+1/2) andthen ˜ℓ=ℓ−1. Thus, the quantum number κandthe\nradial quantum number nrare sufficient to label the Dirac eigenstates. The Dirac equation\ngiven in Eq.(24) may be reduced to a set of two coupled ordinary differ ential equations (in\nunits ofc=/planckover2pi1= 1):\n/parenleftbiggd\ndr+κ\nr/parenrightbigg\nFnrκ(r) = (µ+Enrκ−∆(r))Gnrκ(r), (27)\n/parenleftbiggd\ndr−κ\nr/parenrightbigg\nGnrκ(r) = (µ−Enrκ+Σ(r))Fnrκ(r), (28)\n8where ∆(r) =V(r)−S(r) and Σ(r) =V(r)+S(r) are the difference and the sum potentials,\nrespectively. By substituting\nFnrκ(r) =1\n(µ−Enrκ+Σ(r))/parenleftbiggd\ndr−κ\nr/parenrightbigg\nGnrκ(r),\ninto Eq.(27), the following second order Schr¨ odinger-like different ial equation for Gnrκ(r)\ncan be obtained as\n/parenleftBigg\nd2\ndr2−κ(κ−1)\nr2−(µ+Enrκ−∆(r))(µ−Enrκ+Σ(r))−dΣ\ndr/parenleftbigd\ndr−κ\nr/parenrightbig\nµ−Enrκ+Σ(r)/parenrightBigg\nGnrκ(r) = 0,\n(29)\nwhereEnrκ/negationslash=µwhen Σ(r) = 0 (exact pseudospin symmetry). Further, a similar equation\nforFnrκ(r) can be obtained as follows\n/parenleftBigg\nd2\ndr2−κ(κ+1)\nr2−(µ+Enrκ−∆(r))(µ−Enrκ+Σ(r))+d∆\ndr/parenleftbigd\ndr+κ\nr/parenrightbig\nµ+Enrκ−∆(r)/parenrightBigg\nFnrκ(r) = 0,\n(30)\nwhereEnrκ/negationslash=−µwhen ∆(r) = 0 (exact spin symmetry). Under the condition of exact spin\nsymmetry, ( d∆(r)/dr= 0,i.e., ∆(r) =C=constant), Eq. (30) turns out to be\n/parenleftbiggd2\ndr2−ℓ(ℓ+1)\nr2−(µ+Enrκ−C)Σ(r)+E2\nnrκ−µ2+C(µ−Enrκ)/parenrightbigg\nFnrκ(r) = 0,(31)\nwhereℓ(ℓ+1) comes from κ(κ+ 1) andℓ(ℓ+1)/r2is the spin-centrifugal kinetic en-\nergy term. On the other hand, under the condition of the exact ps eudospin symmetry\n(dΣ(r)/dr= 0,i.e., Σ(r) =C=constant), Eq. (29) is reduced to the form\n/parenleftBigg\nd2\ndr2−˜ℓ(˜ℓ+1)\nr2+(µ−Enrκ+C)∆(r)+E2\nnrκ−µ2−C(µ+Enrκ)/parenrightBigg\nGnrκ(r) = 0,(32)\nwhere˜ℓ(˜ℓ+1) comes from κ(κ−1) and˜ℓ(˜ℓ+1)/r2is the pseudo-centrifugal kinetic energy\nterm. According to the original definition of the pseudo-orbital an gular momentum, the\ncases˜ℓ=κ−1 and˜ℓ=−κare valid for κ >0 andκ <0, respectively. Therefore, the\ndegenerate states come into existence with the same ˜ℓbut different κ, generating pseudospin\nsymmetry. Another important point which is necessary to be said on Eq.(32) is that the\nradial part of the spinor wave function ψnrκ(/vector r) must satisfy the boundary conditions that\nGnrκ(r)/rbecomes zero when r→ ∞, andGnrκ(r)/ris finite atr= 0.\n9IV. BOUND STATE SOLUTION BY MEANS OF THE NU METHOD\nA. Hulth´ en Square Approximation\nIn this section, we shall involve the Hulth ´ en potential to solve the Dirac equation given\nin Eq.(32), meaning that the potential ∆( r) is exponential in rand the pseudo-centrifugal\nkinetic energytermisquadraticin1 /r. Theexponential potentialin risthefamousHulth ´ en\npotential [37,38];\n∆(r) =−∆0e−δr\n1−e−δr, (33)\nwhereδis the screening parameter which is used for determining the range o f the Hulth ´ en\npotential. The parameter ∆ 0representsδZe2, whereZeis the charge of the nucleon [39].\nThe intensity of the Hulth ´ en potential is denoted by ∆ 0under the condition of δ >0. This\npotential has been used in several branches of physics and its disc rete and continuum states\nhave been studied by a variety of techniques such as the algebraic p erturbation calculations\nwhich are based upon the dynamical group structure SO(2,1) [40], t he formalism of super-\nsymmetric quantum mechanics within the framework of the variation al method [41], the\nsupersymmetry and shape invariance property [42], the asymptot ic iteration method [43,44]\nandtheapproachproposedbyBiedenharn fortheDirac-Coulombp roblem[45,46]. Withthis\npotential in place, Eq.(32) has to be solved numerically because the e xponential behavior of\n∆(r) is not compatible with the quadratic behavior of the pseudo-centr ifugal kinetic energy\nterm. However, Eq.(32) is analytically solvable only for the zero value of the pseudo-orbital\nangular momentum, i.e.,˜ℓ= 0 (κ= 1). In order to obtain more realistic results relating to\nthe degenerate states, the Dirac equation should be solved for an y˜ℓ-states. In one of the\nmethods used for solving Eq.(32), Hulth ´ en square approximation can be introduced as an\neffective approximationtothepseudo-centrifugal kineticenergy terminthecaseof ˜ℓ>0and\nsmallr. Following the original work of Filho et al[41] for this approximation, an effective\npotential term can be considered as follows\n˜ℓ(˜ℓ+1)δ2e−2δr\n(1−e−δr)2=˜ℓ(˜ℓ+1)δ2\n([1+δr+...]−1)2≃˜ℓ(˜ℓ+1)\nr2. (34)\nThe exponential numerator in Eq.(34) is expanded for small values o frand higher-order\nterms are ignored up to first-order term. Recently, the authors of [42,44,46] have been used\na more efficient approximation than that of Eq.(34) instead of the ps eudo-centrifugal kinetic\n10energy term ˜ℓ(˜ℓ+1)/r2. This approximation has the advantage that it is only valid for small\nvalues ofδand˜ℓ. Whereas the present approximation in Eq.(34) can also be used for small\nvalues ofδand˜ℓ..\nWhen ∆(r) is taken as the Hulth ´ en potential and the approximation of the centrifugal\nterm as in Eq.(34), Eq.(32) yields\n/bracketleftBigg\nd2\ndr2−˜ℓ(˜ℓ+1)δ2e−2δr\n(1−e−δr)2−(µ−Enrκ+C)∆0e−δr\n1−e−δr+E2\nnrκ−µ2−C(µ+Enrκ)/bracketrightBigg\nGnrκ(r) = 0,\n(35)\nwhereκ=˜ℓ+ 1 forκ >0 andκ=−˜ℓforκ <0 and the wave function has to satisfy\nthe boundary conditions, i.e., Gnrκ(r= 0) = 0 and Gnrκ(r→ ∞) = 0.It is convenient to\nintroduce the following variable and parameters:\ns=e−δr, r∈(0,∞)→s∈[0,1] (36)\nν2\n1=(µ−Enrκ+C)∆0\nδ2, (37)\nω2\n1=E2\nnrκ−µ2−Cµ−CEnrκ\nδ2, (38)\nA1=ω2\n1+ν2\n1−˜ℓ(˜ℓ+1), (39)\nB1= 2ω2\n1+ν2\n1, (40)\nwhich allows us to rewrite Eq.(35) in the simple form\n/parenleftbiggd2\nds2+1−s\ns(1−s)d\nds+A1s2−B1s+ω2\n1\ns2(1−s)2/parenrightbigg\nGnrκ(s) = 0, (41)\nwhere the finiteness of our solution requires that Gnrκ(s= 1) = 0 for r→0 andGnrκ(s=\n0) = 0 for r→ ∞.The above equation can be solved by using a special solution method\nmentioned in Ref.[27] and following a short-cut procedure given in Sec tion 2. First of all,\nbefore starting the procedure of the solution, Eq.(41) is compare d with the hypergeometric\ntype differential equation given in Eq.(1) and consequently Eq.(41) is solved analytically due\nto the fact that the solution is still subjected to a methodology usin g algebra and calculus.\nThis part will be treated in the next subsection partially.\nB. Pseudospin Symmetry Solution\nBy applying the basic ideas of Ref.[27] and imposing the theory of orth ogonal functions\nwhich are known as a generalization of the Rodrigues formula [35], the comparison of the\n11differential equations in Eq.(41) and Eq.(1) gives us the following polyn omials;\n/tildewideτ(s) = 1−s, σ(s) =s(1−s),/tildewideσ(s) =A1s2−B1s+ω2\n1. (42)\nIn the present case, if we want to substitute the polynomials given b y Eq.(42) into Eq.(15),\nthe following equality for the polynomial π(s) is obtained\nπ(s) =−s\n2±1\n2/radicalBig\n(1−4A1−4k)s2+4(B1+k)s−4ω2\n1. (43)\nThe expression under the square root of the above equation must be the square of a poly-\nnomial of first degree. This is possible only if its discriminant is zero and the constant\nparameterkcan be determined from the condition that the expression under th e square\nroot has a double zero. Hence, kis obtained as k+,−= 2ω2\n1−B1±iω1/parenleftBig\n2˜ℓ+1/parenrightBig\n. In that\ncase, it can be written in the four possible forms of π(s);\n\n\nπ(s) =−s\n2±1\n2/parenleftBig\n−/bracketleftBig\n2˜ℓ+1−2iω1/bracketrightBig\ns−2iω1/parenrightBig\n,fork+= 2ω2\n1−B1+iω1/parenleftBig\n2˜ℓ+1/parenrightBig\n,\nπ(s) =−s\n2±1\n2/parenleftBig\n−/bracketleftBig\n2˜ℓ+1+2iω1/bracketrightBig\ns+2iω1/parenrightBig\n,fork−= 2ω2\n1−B1−iω1/parenleftBig\n2˜ℓ+1/parenrightBig\n.\n(44)\nOne of the four possible forms of π(s) must be chosen to obtain an eigenvalue equation.\nTherefore, its most suitable form can be established by\nπ(s) =iω1−/parenleftBig\niω1+˜ℓ+1/parenrightBig\ns,\nfork−. The trick in this selection is to find the negative derivative of τ(s) given in Eq.(6).\nHence,τ(s) andτ′(s) are obtained as\nτ(s) = 1+2iω1−(2iω1+2˜ℓ+3)s, τ′(s) =−(2iω1+2˜ℓ+3)<0. (45)\nIn this case, a new eigenvalue equation for the Dirac equation becom es\nλnr=n2\nr+2nr/parenleftBig\n˜ℓ+1/parenrightBig\n+2nriω1, (46)\nwhere it is beneficial to invite the quantity λnr=−nrτ′(s)−nr(nr−1)\n2σ′′(s) in Eq.(23). An\nother eigenvalue equation is obtained from the equality λ=k−+π′in Eq.(14),\nλ=−ν2−/parenleftBig\n˜ℓ+1/parenrightBig\n(1+2iω). (47)\n12In order to find an eigenvalue equation, the right-hand sides of Eq.( 46) and Eq.(47) must\nbe compared with each other. In this case the result obtained will de pend onEnrκin the\nclosed form:\n−ω2\n1=\n(1+2nr)/parenleftBig\n˜ℓ+1/parenrightBig\n+n2\nr+ν2\n1\n2/parenleftBig\nnr+˜ℓ+1/parenrightBig\n2\n. (48)\nSubstituting the terms of right-hand sides of Eqs.(37) and (38) int o Eq.(48), the energy\neigenvalue equation for Enrκcan be immediately obtained;\n/parenleftBigg\n1+/parenleftbigg∆0\nYδ/parenrightbigg2/parenrightBigg\nE2\nnrκ−/parenleftbigg\nC+2T∆0\nY2/parenrightbigg\nEnrκ+/parenleftbiggTδ\nY/parenrightbigg2\n−µ2−Cµ= 0,(49)\nwhere\nU= (1+2nr)/parenleftBig\n˜ℓ+1/parenrightBig\n+n2\nr, (50)\nY= 2/parenleftBig\nnr+˜ℓ+1/parenrightBig\n, (51)\nT=U+(C+µ)∆0\nδ2. (52)\nThe energy spectrum of the Dirac equation for ∆( r) =V(r)−S(r) =−∆0e−δr\n1−e−δris obtained\nby means of Eq.(49). In this case, the states with the same nrand˜ℓwill be degenerate. The\ntwo energy solutions of the quadratic equation can be obtained as\nE±\nnrκ=δ2(2∆0T+CY2)±/radicalBig\n4δ2(∆0(C+µ)−δ2T)(∆0µ+δ2T)Y2+δ4(C+2µ)2Y4\n2(∆2\n0+Y2δ2).\n(53)\nFor a given value of nrandκ(or/tildewideℓ), the above equation provides two distinct positive and\nnegative energy spectra related with E+\nnrκorE−\nnrκ, respectively. One of the distinct solutions\nis only valid to obtain the negative-energy bound states in the limit of t he pseudospin\nsymmetry. Before seeking the acceptable solution, it is useful to p resent some analogy\nabout the energy spectra.\nNow, we are going to find the corresponding wave functions for the present potential\nmodel. Firstly, we calculate the weight function defined as [47-51]\nρ(s) =1\nσ(s)exp/parenleftbigg/integraldisplayτ(s)\nσ(s)ds/parenrightbigg\n=s2iω1(1−s)2˜ℓ+1, (54)\nand the first part of the wave function in Eq. (4):\nφ(s) = exp/parenleftbigg/integraldisplayπ(s)\nσ(s)ds/parenrightbigg\n=siω1(1−s)˜ℓ+1. (55)\n13Hence, the second part of the wave function which is the solution of Eq.(20) can be obtained\nby means of the so called Rodrigues representation\nynr(s) =cnrκs−2iω1(1−s)−(2˜ℓ+1)dnr\ndsnr/bracketleftBig\nsnr+2iω1(1−s)nr+2˜ℓ+1/bracketrightBig\n∼P(2iω1,2˜ℓ+1)\nnr(1−2s), s∈[0,1], (56)\nwhere the Jacobi polynomial P(µ,ν)\nnr(x) is defined for Re( ν)>−1 and Re(µ)>−1 for\nthe argument x∈[−1,+1] andcnrκis the normalization constant .By usingGnrκ(s) =\nφ(s)ynr(s),in this way we may write the lower-spinor wave function in the following f ashion\nGnrκ(r) =cnrκ(exp(−iω1δr))(1−exp(−δr))˜ℓ+1P(2iω1,2˜ℓ+1)\nn (1−2exp(−δr))\n=cnrκ(2iω1+1)nr\nnr!(exp(−iω1δr))(1−exp(−δr))˜ℓ+1\n×2F1/parenleftBig\n−nr,nr+2/parenleftBig\niω1+˜ℓ+1/parenrightBig\n;1+2iω1;exp(−δr)/parenrightBig\n, (57)\nwhere\niω1δ=/radicalBig\nC(µ+Enrκ)+µ2−E2\nnrκ>0. (58)\nThe hypergeometric series2F1/parenleftBig\n−nr,nr+2/parenleftBig\niω1+˜ℓ+1/parenrightBig\n;1+2iω1;exp(−δr)/parenrightBig\nterminates\nfornr= 0 and thus converges for all values of real parameters ω1>0 and˜ℓ>0.In case if\nC= 0,theniω1δ=/radicalbig\n(µ+Enrκ)(µ−Enrκ) with the following restriction Enrκ<µrequired\nto obtain bound state (real) solutions for both positive and negativ e solutions of Enrκin\nEq. (53).Now, before presenting the corresponding upper-component Fnrκ(r),let us recall\na recurrence relation of hypergeometric function\nd\nds/bracketleftBig\n2F1(a;b;c;s)/bracketrightBig\n=/parenleftbiggab\nc/parenrightbigg\n2F1(a+1;b+1;c+1;s), (59)\nwith which the corresponding upper component Fnrκ(r) can be given by solving Eq. (28) as\nfollows\nFnrκ(r) =dnrκ(exp(−iω1δr))(1−exp(−δr))˜ℓ+1\n(µ−Enrκ+C)\n/parenleftBig\n˜ℓ+1/parenrightBig\nδexp(−δr)\n(1−exp(−δr))−iω1δ−κ\nr\n\n×2F1/parenleftBig\n−nr,nr+2/parenleftBig\niω1+˜ℓ+1/parenrightBig\n;1+2iω1;exp(−δr)/parenrightBig\n+dnrκ\nnr/bracketleftBig\nnr+2/parenleftBig\niω1+˜ℓ+1/parenrightBig/bracketrightBig\nδ(exp(−δr))iω1+1(1−exp(−δr))˜ℓ+1\n(1+2iω1)(µ−Enrκ+C)\n\n14×2F1/parenleftbigg\n1−nr,nr+2/parenleftbigg\niω1+˜ℓ+3\n2/parenrightbigg\n;2(1+iω1);exp(−δr)/parenrightbigg\n, (60)\nwhereEnrκ/negationslash=µwhenC= 0, exact pseudospin symmetry and dnrκis the normalization\nfactor.\nC. Spin Symmetry Solution\nThis symmetry arises from the near equality in magnitude of an attra ctive scalar, S(/vector r),\nand repulsive vector, V(/vector r),relativistic mean field, S(/vector r)∼V(/vector r) in which the nucleon move\n[48-51]. Therefore, we simply take the sum potential equal to the is otonic potential model,\ni.e.,\nΣ(r) =−Σ0e−δr\n1−e−δr. (61)\nalong with the approximation given by Eq.(34) to deal with the spin-or bit centrifugal term\nℓ(ℓ+1)/r2. In the last equation, the choice of Σ( r) = 2V(r)→V(r) allows us to reduce the\nresulting solutions of the Dirac equation into their non-relativistic limit s under appropriate\nchoice of parameter transformations [51]. Therefore, the spin-s ymmetry Dirac equation (31)\nbecomes\n/braceleftbiggd2\ndr2−ℓ(ℓ+1)δ2e−2δr\n(1−e−δr)2+(µ+Enrκ−C)Σ0e−δr\n1−e−δr−/bracketleftbig\nµ2−E2\nnrκ−C(µ−Enrκ)/bracketrightbig/bracerightbigg\nFnrκ(r) = 0,\n(62)\nwhereκ=ℓforκ>0 andκ=−(ℓ+1) forκ<0.It is convenient to introduce the following\nnew variable and parameters:\ns=e−δr, r∈(0,∞)→s∈[0,1] (63)\nν2\n2=(µ+Enrκ−C)Σ0\nδ2, (64)\nω2\n2=E2\nnrκ−µ2+Cµ−CEnrκ\nδ2, (65)\nA2=ω2\n2−ν2\n2−ℓ(ℓ+1), (66)\nB2= 2ω2\n2−ν2\n2, (67)\nwhich allow us to rewrite Eq.(62) in a more simple form as\n/parenleftbiggd2\nds2+1−s\ns(1−s)d\nds+A2s2−B2s+ω2\n2\ns2(1−s)2/parenrightbigg\nFnrκ(s) = 0, (68)\n15where the finiteness of our solutions require that Fnrκ(1) = 0 and Fnrκ(0)→0.We apply the\nNUmethodfollowingthesamestepsofsolutioninprevioussectiontoo btaintheexpressions:\n/tildewideτ(s) = 1−s, σ(s) =s(1−s),/tildewideσ(s) =A2s2−B2s+ω2\n2. (69)\nTo avoid repition, the functions required by the method for π(s), kandτ(s) can be estab-\nlished as\nπ(s) =iω2−(iω2+ℓ+1)s, (70)\nk= 2ω2\n2−B2−iω2(2ℓ+1), (71)\nand\nτ(s) = 1+2iω2−(2iω2+2ℓ+3)s, τ′(s) =−(2iω2+2ℓ+3)<0. (72)\nrespectively, with prime denotes the derivative with respect to s.Also, the parameters λ\nandλntake the forms:\nλnr=n2\nr+2nr(ℓ+1)+2nriω2andλ=ν2\n2−(ℓ+1)(1+2iω2). (73)\nUsing the basic condition λ=λnfollowed by simple algebra ,we obtain\n−ω2\n2=/parenleftbigg(1+2nr)(ℓ+1)+n2\nr−ν2\n2\n2(nr+ℓ+1)/parenrightbigg2\n, nr,ℓ= 1,2,3,··· (74)\nand then the energy eigenvalue equation is immediately obtained\n/parenleftBigg\n1+/parenleftbiggΣ0\nZδ/parenrightbigg2/parenrightBigg\nE2\nnrκ−/parenleftbigg\nC+2SΣ0\nZ2/parenrightbigg\nEnrκ+/parenleftbiggSδ\nZ/parenrightbigg2\n−µ2+Cµ= 0,(75)\nwhere\nW= (1+2nr)(ℓ+1)+n2\nr, (76)\nZ= 2(nr+ℓ+1), (77)\nS=W+(C−µ)Σ0\nδ2. (78)\nThe two energy solutions of the quadratic equation (75) can be obt ained as\nE±\nnrκ=δ2(2Σ0S+CZ2)±/radicalBig\n4δ2(Σ0(C−µ)−δ2S)(−Σ0µ+δ2S)Z2+δ4(C−2µ)2Z4\n2(Σ2\n0+Z2δ2).\n(79)\n16For a given value of nrandκ(orℓ), the above equation provides two distinct positive and\nnegative energy spectra related with E+\nnrκorE−\nnrκ, respectively. One of the distinct solutions\nis only valid to obtain the positive-energy bound states in the limit of th e spin symmetry.\nIn our calculations for the spin symmetry wave functions, we firstly find the weight\nfunction:\nρ(s) =s2iω2(1−s)2ℓ+1, (80)\nand from which the second part of the wave function by means of Ro drigues formula as\nynr(s) =anrκs−2iω2(1−s)−(2ℓ+1)dnr\ndsnr/bracketleftBig\nsnr+2iω2(1−s)nr+2ℓ+1/bracketrightBig\n∼P(2iω2,2ℓ+1)\nnr(1−2s), s∈[0,1], (81)\nwhere the Jacobi polynomial P(µ,ν)\nnr(x) is defined for Re( ν)>−1 and Re(µ)>−1 for the\nargumentx∈[−1,+1] andanrκis the normalization constant .Further, the first part of the\nwave function is being calculated as\nφ(s) =siω2(1−s)ℓ+1. (82)\nBy usingFnrκ(s) =φ(s)ynr(s),in the spin symmetry case, we may write down the upper-\nspinor wave function in the following fashion\nFnrκ(r) =anrκ(exp(−iω2δr))(1−exp(−δr))ℓ+1P(2iω2,2ℓ+1)\nnr(1−2exp(−δr))\n=anrκ(2iω2+1)nr\nnr!(exp(−iω2δr))(1−exp(−δr))ℓ+1\n×2F1(−nr,nr+2(iω2+ℓ+1);1+2iω2;exp(−δr)), (83)\nwhere\niω2δ=/radicalBig\nC(Enrκ−µ)+µ2−E2\nnrκ>0. (84)\nItisnotedthatthehypergeometricseries2F1(−nr,nr+2(iω2+ℓ+1);1+2iω2;exp(−δr))\nterminates for nr= 0 and thus it converges for all values of real parameters ω2>0 and\nℓ>0.In case when C= 0,theniω1δ=/radicalbig\n(µ+Enrκ)(µ−Enrκ) with a restriction for real\nbound states that Enrκ< µfor both positive and negative solutions of Enrκin Eq. (79) .\nThus, the corresponding spin-symmetric lower-component Gnrκ(r) can be found as follows\nGnrκ(r) =bnrκ(exp(−iω2δr))(1−exp(−δr))ℓ+1\n(µ+Enrκ−C)/bracketleftbigg(ℓ+1)δexp(−δr)\n(1−exp(−δr))−iω2δ+κ\nr/bracketrightbigg\n17×2F1(−nr,nr+2(iω2+ℓ+1);1+2iω2;exp(−δr))\n+bnrκ/bracketleftBigg\nnrδ[nr+2(ℓ+1+iω2)](exp(−δr))iω2+1(1−exp(−δr))ℓ+1\n(1+2iω1)(µ+Enrκ−C)/bracketrightBigg\n×2F1/parenleftbigg\n1−nr,nr+2/parenleftbigg\niω2+ℓ+3\n2/parenrightbigg\n;2(1+iω2);exp(−δr)/parenrightbigg\n, (85)\nwhereEnrκ/negationslash=−µwhenC= 0, exact spin symmetry and bnrκis the normalization constant.\nLet us finally remark that a careful inspection to our present spin- symmetric solution\nshows that it can can be easily recovered by knowing the relationship between the present\nset of parameters ( ω2\n2,ν2\n2,A2,B2) and the previous set of parameters ( ω2\n1,ν2\n1,A1,B1).This\ntells us that the positive energy solution for spin symmetry (negativ e energy solution for\npseudospin symmetry) can be obtained directly from those of the n egative energy solution\nfor pseudospin symmetry (positive energy solution for spin symmet ry) by performing the\nfollowing replacements [48-51]:\nFnrκ(r)↔Gnrκ(r), V(r)→ −V(r) (or Σ 0↔ −∆0), ℓ(ℓ+1)↔˜ℓ(˜ℓ+1)\n, E+\nnrκ↔ −E−\nnrκ, ω2\n2↔ω2\n1andν2\n2↔ −ν2\n1. (86)\nThat is, with the above replacements, Eqs. (49) and (57) yield Eqs. (75) and (83) and vice\nversa is true.\nLet us now present the non-relativistic limit. This can be achieved whe n we setC= 0,\nΣ0=δand using the mapping Enrκ−µ→EnrℓandEnrκ+µ→2µin Eqs.(64), (65) and\n(74),then the resulting energy eigenvalues (in /planckover2pi1=c=e= 1 units) are\nEnrℓ=−1\n2µ/bracketleftbigg(1+2nr)(ℓ+1)δ+n2\nrδ−2µ\n2(nr+ℓ+1)/bracketrightbigg2\n, nr,ℓ= 0,1,2,3,···.(87)\nAlso, the wave functions in Eqs.(83) and (84) turns out to become\nRnrℓ(r) =anrℓr−1/parenleftBig\nexp(−/radicalbig\n−2µEnrℓr)/parenrightBig\n(1−exp(−δr))ℓ+1P(2√\n−2µEnrℓ/δ,2ℓ+1)\nnr (1−2exp(−δr))\n=anrℓ/parenleftbig\n2/radicalbig\n−2µEnrℓ/δ+1/parenrightbig\nnr\nnr!r−1/parenleftBig\nexp(−/radicalbig\n−2µEnrℓr)/parenrightBig\n(1−exp(−δr))ℓ+1\n×2F1/parenleftBig\n−nr,nr+2/parenleftBig/radicalbig\n−2µEnrℓ/δ+ℓ+1/parenrightBig\n;1+2/radicalbig\n−2µEnrℓ/δ;exp(−δr)/parenrightBig\n, Enrℓ<0.\n(88)\n18V. RESULTS AND CONCLUSIONS\nInthepresent study, theDiracequationfor theHulth ´ enpotential isapproximately solved\nunder the condition of the exact spin and pseudospin symmetry with in the framework of\nthe relativistic mean field theory. By using the basic ideas of the NU me thod, the energy\neigenvalue expression for the arbitrary pseudo-orbital angular m omentum ˜ℓis obtained ap-\nproximately. The second-order differential equation given in Eq.(32 ) is solved by applying\nthe Hulth ´ en square approximation to deal with the pseudospin–orbit and spin- orbit cen-\ntrifugal and kinetic energy terms ˜ℓ(˜ℓ+1)/r2andℓ(ℓ+1)/r2. The energy spectrum for any ˜ℓ\nstates is obtained analytically. Under the condition of the exact pse udospin and spin sym-\nmetry limitations, the energy relations in the Dirac equation with equa l scalar and vector\nHulth´ en potentials are recovered to see degenerate states.\nThe results obtained for this motivation show the orbital dependen cy of the Dirac equation\nfor the Hulth ´ en potential. Certainly, an analysis detailed by solving Dirac equation in r el-\nativistic mean field theories needs to use a very large scale ( ∼660 MeV) comparing to the\nnuclear physics scale (few MeV) ) in point of the intensity of potentia ls [6]. For this reason,\nthe intensity of the potential, ∆ 0, used in Eq.(33) is considered as 3.4 fm−1. The units\nof/planckover2pi1=c=e= 1 are used throughout the present work for the sake of simplicity . Hence,\nthe energy eigenvalue expression given in Eq.(53) can be simply discus sed by using a set of\nphysical parameter values. In the below explanations, although th e energy spectrums can\nbe calculated in dimensionless or arbitrary units, the calculations are preferably made in\nfm−1for the energy, mass, Cand intensity of the potential.\nIt is noted that the energy spectrum given by Eq.(53) indicates a fa mily of the pseu-\ndospin symmetry Hulth ´ en potential. Moreover, the analytical expression for Eq.(53) can b e\nconfronted with the results of [44] which is slightly in agreement with t he result presented\nin Eq.(47) for the pseudospin symmetry solution. The results are on ly valid for small values\nofδand˜ℓ(orκ). This spectrum changes with the relevant quantum numbers as we ll as the\nscreening parameter δ. The variation of the energy spectrums ( E+\nnrκandE−\nnrκ) according\nto the screening parameter δis shown in Fig.1a and Fig.1b, with the choices of parameters\nC=−4.9fm−1andµ= 5.0fm−1, which is in the range of nucleon mass value ( ∼1\nGeV). Figure 1a indicates the positive-energy bound states, i.e.,E+\nnrκ, while Fig.1b shows\nthe negative-energy bound states, i.e.,E−\nnrκ. For a given value of nrand/tildewideℓ, it is seen that an\n19increment on δleads to an increment on E−\nnrκalong the negative-energy direction whereas\nthesame increment on δresults witha reduction on E+\nnrκalong thepositive-energy direction.\nThe results presented in Fig.1a show that the energy difference bet ween the states is still\nsmall although the values of screening parameter δincreases. Figure 1b has two interesting\nresults: The first one indicates that the negative-energy bond st ates appear with the large\nvalues ofδand˜ℓ. The reason of this aspect comes from the approximation mentione d in the\nprevious sections. The second one belongs to the small values of δ. For instance, E−\nnrκfor\n1s1/2, 1d7/2and 1g9/2is valid under the condition of δ/greaterorsimilar0.09,δ/greaterorsimilar0.05 andδ/greaterorsimilar0.03, respec-\ntively. Therefore, E−\nnrκstill represents the negative-energy bound states for small valu es of\nδwhen/tildewideℓincreases. These results can be also expanded on the other state s of the Hulth ´ en\npotential with the pseudospin symmetry.\nIt is well-known that for the finite nuclei the constant Cis adjusted to zero because\neach potential goes to zero at large distances. If the difference b etween the scalar S(r) and\nvectorV(r) potentials equals to a given constant C, this is equivalent to adding the relevant\nconstant to the relativistic energy and mass. The constant for th e energy is unimportant\nbecause it does not affect the energy difference. Whereas the var iation ofCis equivalent\nto the variation of mass. This is more physically transparent for the pseudospin-orbit de-\npendency of the Dirac equation. Moreover, in the case of the infinit e nuclear matter, the\nconstantCcould be non-zero. The energy spectrum versus the mass µis plotted by setting\nC=−4.9fm−1, as shown in Fig.2. The variation of the energy spectrum for ˜ℓ= 1,˜ℓ= 3,\n˜ℓ= 5 and ˜ℓ= 7 is presented by using δ= 0.25. The radial quantum number is fixed to\n1 (nr= 1). In Fig.2, the dashed (red lines, see colour online) and solid (blue lin es, see\ncolour online) lines represent E+\nnrκandE−\nnrκ, respectively. According to Fig.2, there are two\ndifferent regions of energy spectrum versus the mass. For 0 < µ≤2.3fm−1the energy\nspectrum is in the negative region completely. With the ˜ℓincreasing, E+\nnrκfan out along the\npositive part of the energy spectrum whereas E−\nnrκoverlaps in going from ˜ℓ= 1 to˜ℓ= 7.\nThe case of E−\nnrκin the interval of 0 < µ≤2.3fm−1represents the degenerate states in\nthe different values of ˜ℓfor a given value of nr. A similar trend is seen when δ≥3.2fm−1\nin the several values of ˜ℓforE+\nnrκby crossing the zero axis toward the positive direction of\nthe energy spectrum. However, the numerical results of E+\nnrκover the axis are not relevant\nfor the negative-energy bound states. Meanwhile, the results of E−\nnrκare also valid under\nthe conditions of δ≥3.9,3.5,3.3,3.2fm−1for˜ℓ= 1,3,5,7, respectively, but only\n20relevant in the large values of ˜ℓ. Furthermore, the energy spectrum versus the constant C\nis plotted by taking µ= 5.0fm−1andδ= 0.25 as shown in Fig.3. According to Fig.3, it is\nseen that the negative values of Cshow more strongly binding energies under the condition\nofC≤ −11.0fm−1forE+\nnrκ(dashed lines) and E−\nnrκ(blue line) in the whole values of ˜ℓ.\nMoreover,E+\nnrκ(dashed line) still shows the negative-energy bound states on con dition that\n−8< C <−5fm−1up to the zero axis. In the case of E−\nnrκ(blue lines) the situation\nbecomes different than that of E+\nnrκ. The axis is crossed with changing ˜ℓforE−\nnrκ. The\ncrossing points of axis from C=−8fm−1become large with the ˜ℓincreasing. In Table 1,\nthe degenerate states are presented for a Dirac particle within th e Hulth´ en potential, with\nC=−4.9fm−1,µ= 5.0fm−1and ∆ 0= 3.4fm−1. The several pseudo-orbital and radial\nquantum numbers are used in the numerical calculations to predict t he orbital dependency\noftheDiracequation under theconditionoftheexact pseudospin s ymmetry. Asanexample,\nthe Dirac eigenstate 1 s1/2withnr= 1 andκ=−1 will have a partner which is denoted by\nthe 0d3/2withnr−1 = 0 and κ= 2. These states are called the pseudospin partner and\ndegenerated with each other.\nAcknowledgments\nThe partial support provided by the Scientific and Technical Resea rch Council of Turkey\n(T¨UB˙ITAK) is highly appreciated. One of the authors (C.B.) acknowledges the financial\nsupport provided by the Science Foundation of Erciyes University.\n21[1] A. Arima, M. Harvey, K. Shimizu, Pseudo LS coupling and ps eudo SU3 coupling schemes,\nPhys. Lett. B 30 (1969) 517-522.\n[2] K.T. Hecht, A. Adler, Generalized seniority for favored J/negationslash= 0 pair in mixed configurations,\nNucl. Phys. A 137 (1969) 129-143.\n[3] J.N. Ginocchio, A relativistic symmetry in nuclei, Phys . Reports 315 (1999) 231-240.\n[4] A. Leviatan, Supersymmetric Patterns in the Pseudospin , Spin, and Coulomb Limits of the\nDirac Equation with Scalar and Vector Potentials, Phys. Rev . Lett. 92 (2004) 202501.\n[5] J.N. Ginocchio, Pseudospin as a relativistic symmetry, Phys. Rev. Lett. 78 (3) (1997) 436-439.\n[6] J.N. Ginocchio, Relativistic symmetries in nuclei and h adrons, Phys. Rep. 414 (4-5) (2005)\n165-261.\n[7] J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring, A. Arima , Pseudospin symmetry in rela-\ntivistic mean field theory, Phys. Rev. C 58 (2) (1998) R628-R6 31.\n[8] R. Lisboa, M. Malheiro, A.S. de Castro, P. Alberto and M. F iolhais, Pseudospin symmetry\nand the relativistic harmonic oscillator, Phys. Rev. C 69 (2 004) 024319. 15pp.\n[9] P. Alberto, M. Fiolhais, M. Malheiro, A. Delfino, M. Chiap parini, Isospin Asymmetry in the\nPseudospin Dynamical Symmetry, Phys. Rev. Lett. 86 (2001) 5 015-5018.\n[10] P. Alberto, M. Fiolhais, M. Malheiro, A. Delfino, M. Chia pparini, Pseudospin symmetry as a\nrelativistic dynamical symmetry in the nucleus, Phys. Rev. C 65 (2002) 034307. 9pp.\n[11] C. Ti-Sheng, L. Hong-Feng, M. Jie, Z. Shuang-Quan and Z. Shan-Gui, Pseudospin Symmetry\nin Relativistic Framework with Harmonic Oscillator Potent ial and Woods-Saxon Potential,\nChin. Phys. Lett. 20 (2003) 358-361.\n[12] J.Y. Guo, X.Z. Fang, X.F Xu, Pseudospin symmetry in the r elativistic harmonic oscillator,\nNucl. Phys. A 757 (2005) 411-421.\n[13] Q.XuandS.J.Zhu, Pseudospinsymmetryandspinsymmetr yintherelativistic Woods-Saxon,\nNucl. Phys. A 768 (2006) 161-169.\n[14] C. Berkdemir, A. Berkdemir and R. Sever, Polynomial sol utions of the Schr¨ odinger equation\nfor the generalized Woods-Saxon potential, Phys. Rev. C 72 ( 2005) 027001. 4pp. [Editorial\nNote, 74 (2006) 039902(E)].\n[15] J.Y. Guo, J.C. Han and R.D. Wang, Pseudospin symmetry an d the relativistic ring-shaped\n22non-spherical harmonic oscillator, Phys. Lett. A 353 (2006 ) 378-382.\n[16] C. Berkdemir, A. Berkdemir and R. Sever, Systematical a pproach to the exact solution of the\nDirac equation for a deformed form of the Woods-Saxon potent ial, J. Phys. A: Math. Gen. 39\n(2006) 13455-13464.\n[17] C.Berkdemir, Pseudospinsymmetryintherelativistic Morsepotential includingthespin-orbit\ncoupling term, Nucl. Phys. A 770 (2006) 32-39.\n[18] Y. Xu, S. He and C.-S. Jia, Approximate analytical solut ions of the Dirac equation with the\nP¨ oschl-Teller potential including the spin-orbit coupli ng term, J. Phys. A: Math. Theor. 41\n(2008) 255302-255309.\n[19] C.S. Jia, P. Guo and X.L. Peng, Exact solution of the Dira c-Eckart problem with spin and\npseudospin symmetry, J. Phys. A: Math. Theor. 39 (2006) 7737 -7744.\n[20] C.S. Jia, J.Y. Wang, S. He and L.T. Sun, Shape invariance and the supersymmetry WKB\napproximation for a diatomic molecule potential, J. Phys. A : Math. Gen. 33 (2000) 6993-6998.\n[21] C.S. Jia, J.Y. Liu and P.Q. Wang, A new approximation sch eme for the centrifugal term and\nthe Hulth ´ en potential, Phys. Lett. A 372 (2008) 4779-4782.\n[22] C.S. Jia, Y.F. Diao, L.Z. Yi and T. Chen, Arbitrary l-wav e solutions of the Schr¨ odinger\nequation with the Hulth ´ en potential, Int. J. Mod. Phys. A 24 (24) (2009) 4519-4528.\n[23] C.S. Jia, T. Chen and L.G. Cui, Approximate analytical s olutions of the Dirac equation with\nthe generalized P¨ oschl-Teller potential including the ps eudo-centrifugal term, Phys. Lett. A\n373 (2009) 1621-1626.\n[24] Y. Xu, S. He and C.S. Jia, Approximate analytical soluti ons of the Klein-Gordon equation\nwith the P¨ oschl-Teller potential including the centrifug al term, Phys. Scr. 81 (2010) 045001.\n[25] S.M. Ikhdair, On the bound-state solutions of the Manni ng-Rosen potential including an\nimproved approximation to the orbital centrifugal term, Ph ys. Scr. 83 (2011) 015010.\n[26] S.M. Ikhdair and J. Abu-Hasna, Quantization rule solut ion to the Hulth ´ en potential in arbi-\ntrary dimension with a new approximate scheme for the centri fugal term, Phys. Scr. 83 (2011)\n025002.\n[27] A.F. Nikiforov, V.B. Uvarov, Special Functions of Math ematical Physics, Birkhauser, Basel,\n1988.\n[28] A. Berkdemir, C. Berkdemir and R. Sever, Eigenvalues an d eigenfunctions of Woods-Saxon\npotential in PT-symmetric quantum mechanics, Mod. Phys. Le tt. A 21 (2006) 2087-2097.\n23[29] C.Berkdemir, Relativistictreatment ofaspin-zeropa rticlesubjecttoaKratzer-typepotential,\nAm. J. Phys. 75 (2007) 81-86.\n[30] Y.F. Cheng and T.Q Dai, Exact solution of the Schr¨ oding er equation for the modified Kratzer\npotential plus a ring-shaped potential by the Nikiforov-Uv arov method, Phys. Scr. 75 (2007)\n274-277.\n[31] C. Berkdemir and R. Sever, Pseudospin symmetry solutio n of the Dirac equation with an\nangle-dependent potential, J. Phys. A: Math. Theor. 41 (200 8) 045302. 11pp.\n[32] S. Fl¨ ugge, Practical Quantum Mechanics I and II, Sprin ger-Verlang, Berlin, 1971.\n[33] C. Berkdemir and J. Han, Any l-state solutions of the Mor se potential through the Pekeris\napproximation and Nikiforov-Uvarov method, Chem. Phys. Le tt. 409 (2005) 203-207.\n[34] S. Haouat and L. Chetouani, Bound states of Dirac partic le subjected to the pseudoscalar\nHulth´ en potential, J. Phys. A: Math. Theor. 40 (34) (2007) 10541-10 548.\n[35] G. Szego, Orthogonal Polynomials, American Mathemati cal Society, New York, 1959.\n[36] W. Greiner, B. M¨ uller and J. Rafelski, Quantum Electro dynamics of Strong Fields: With an\nIntroduction into Modern Relativistic Quantum Mechanics, Springer, 2nd edition, New York,\n1985.\n[37] C.S. Lai and W.C. Lin, Energies of the Hulth´ en potentia l forl/negationslash= 0, Phys. Lett. A 78 (4)\n(1980) 335-337.\n[38] Y.P. Varshni, Eigenenergies and oscillator strengths for the Hulth´ en potential, Phys. Rev A\n41 (9) (1990) 4682-4689.\n[39] U. Myhrman, A recurrence formula for obtaining certain matrix elements in the base of\neigenfunctions of the Hamiltonian for a particular screene d potential, J. Phys. A: Math. Gen.\n16 (2) (1983) 263-270.\n[40] B. Roy and R. Roychoudhury, Dirac equation with Hulth ´ en potential: an algebraic approach,\nJ. Phys. A: Math. Gen. 23 (21) (1990) 5095-5102.\n[41] E.D. Filho and R.M. Ricotta, Supersymmetry, variation al method and Hulth ´ en potential,\nMod. Phys. Lett. A 10 (1995) 1613-1618.\n[42] S.W. Qian, B.W. Huang and Z.Y. Gu, Supersymmetry and sha pe invariance of the effective\nscreened potential, New J. Phys. 4 (2002) 13.1.\n[43] H. Ciftci, R. L. Hall, and N. Saad, Asymptotic iteration method for eigenvalue problems, J.\nPhys. A: Math. Gen. 36 (48) (2003) 11807. 10pp\n24[44] A. Soylu, O. Bayrak and I. Boztosun, An approximate solu tion of Dirac-Hulth ´ en problem\nwith pseudospin and spin symmetry for any κstate, J. Math. Phys. 48 (2007) 082302. 9 pp.\n[45] L.C. Biedenharn, Remarks on the Relativistic Kepler Pr oblem, Phys. Rev. 126 (1962) 845-851.\n[46] S. Haouat and L. Chetouani, Approximate solutions of Kl ein-Gordon and Dirac equations in\nthe presence of the Hulth ´ en potential, Phys. Scr. 77 (2) (2008) 025005.\n[47] S.M. Ikhdair, Rotation and vibration of diatomic molec ule in the spatially-dependent mass\nSchr¨ odinger equation with generalized q-deformed Morse p otential, Chem. Phys. 361 (2009)\n9-17.\n[48] S.M. Ikhdair and R. Sever, Solutions of the spatially-d ependent mass Dirac equation with the\nspin and pseudospin symmetry for the Coulomb-like potentia l, Appl. Math. Comp. 216 (2)\n(2010) 545-555.\n[49] S.M. IkhdairandR. Sever, Approximate boundstate solu tions of Dirac equation with Hulth ´ en\npotential including Coulomb-like tensor potential, Appl. Math. Comp. 216 (3) (2010) 911-923.\n[50] S.M. Ikhdair, Approximate solutions of the Dirac equat ion for the Rosen-Morse potential\nincluding the spin-orbit centrifugal term, J. Math. Phys. 5 1 (2) (2010) 023525. 16pp.\n[51] S.M. Ikhdair and R. Sever, Approximate analytical solu tions of the generalized Woods-Saxon\npotentials including the spin-orbit coupling term and spin symmetry, Cent. Eur. J. Phys. 8\n(4) (2010) 652-666.\n25FIG. 1: The variation of the energy spectrum in units of fm−1versus the screening parameter δ.\nFIG. 2: The variation of the energy spectrum versus the mass µ. All parameters are in units of\nfm−1.\nFIG. 3: The variation of the energy spectrum versus the const antC. All parameters are in units\noffm−1.\n26TABLE I: The negative-energy degenerate states in units of fm−1of the pseudospin-symmetry\nHulth´ en potential for various values of nr,˜ℓandδ. For a special case, µ= 5fm−1, ∆0= 3.4fm−1\nandC=−4.9fm−1.\n˜ℓ nrδDegenerate States Enr,κ˜ℓ nrδDegenerate States Enr,κ\n1 1 0 .025 (1s1/2,0d3/2) 0.0963638 1 2 0 .025 (2s1/2,1d3/2) 0.0928939\n0.100 0 .0425738 0 .100 −0.0103694\n0.175 −0.0710009 0 .175 −0.2174930\n0.250 −0.2346580 0 .250 −0.4920870\n2 1 0 .025 (1p3/2,0f5/2) 0.0912282 2 2 0 .025 (2p3/2,1f5/2) 0.0863238\n0.100 −0.0363590 0 .100 −0.1078600\n0.175 −0.2930130 0 .175 −0.4732160\n0.250 −0.6351320 0 .250 −0.9131390\n3 1 0 .025 (1d5/2,0g7/2) 0.0839128 3 2 0 .025 (2d5/2,1g7/2) 0.0775818\n0.100 −0.1447100 0 .100 −0.2316110\n0.175 −0.5760950 0 .175 −0.7705370\n0.250 −1.0984500 0 .250 −1.3540100\n4 1 0 .025 (1f7/2,0h9/2) 0.0744360 4 2 0 .025 (2f7/2,1h9/2) 0.0666955\n0.100 −0.2784550 0 .100 −0.3771030\n0.175 −0.8953110 0 .175 −1.0870200\n0.250 −1.5671200 0 .250 −1.7758200\n27" }, { "title": "2005.05468v1.Effect_of_the_spin_orbit_interaction_on_thermodynamic_properties_of_liquid_uranium.pdf", "content": "arXiv:2005.05468v1 [cond-mat.mtrl-sci] 11 May 2020Effect of the spin-orbit interaction on thermodynamic prope rties of liquid uranium\nMinakov D.V.,∗Paramonov M.A.,†and Levashov P.R.‡\nJoint Institute for High Temperatures, Izhorskaya 13 bldg 2 , Moscow 125412, Russia and\nMoscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia\n(Dated: May 13, 2020)\nWe present the first quantum molecular dynamics calculation of zero-pressure isobar of solid and\nliquid uranium that account for spin-orbit coupling. We dem onstrate that inclusion of spin-orbit\ninteraction leads to higher degree of the thermal expansion of uranium, especially in the liquid\nphase. Full accounting of relativistic effects for valence e lectrons, particularly spin-orbital splitting\nof the 5fband, is substantial for the reproduction of the experiment al density of molten uranium at\nthe melting temperature. Influence of the spin-orbit intera ction on the thermodynamic properties\nat high temperatures and pressures is also analyzed.\nWhile influence of spin-orbit coupling (SOC) on the\nproperties of solid uranium is intensively studied us-\ning first-principle methods [1–9] and vigorously dis-\ncussed [10, 11], effect on the liquid uranium is still un-\nclear. The reason is extreme computational complexity\nof such calculations for an unordered phase that requires\ntaking into account dynamics of the atoms. Quantum\nmolecular dynamics (QMD) simulation of uranium is es-\nsentially complex due to a large number of valence elec-\ntrons, and a calculation that account for the spin-orbit\n(SO) interaction requires about an order of magnitude\nmore time than a scalar-relativistic one [12].\nIt was shown earlier [1] that the SO splitting of the\n5fband explains the experimentally observed anoma-\nlously high room-temperature thermal expansion of light\nactinides, in particular, uranium, neptunium, and pluto-\nnium.\nIn this Letter we demonstrate that SOC is responsi-\nble for the effect of significant increase of pressure for\nuranium in the vicinity of melting mostly in the liquid\nphase. As a consequence, lower densities along the zero\nisobar are predicted by QMD with SOC. Full account-\ning of relativistic effects for valence electrons, especially\nSO splitting, is substantial for the reproduction of the\nmolten uranium density. In Fig. 1 we demonstrate avail-\nable data of static [13–18] and dynamic [19–21] experi-\nments on thermal expansion of uranium in the vicinity of\nmelting as well as results of our QMD calculations of the\nzero-pressureisobarwith andwithout SOCofthe valence\nelectrons. As can be seen from the figure, accounting of\nSOC provides excellent agreement with data on thermal\nexpansion of solid uranium. We can also notice discrep-\nancybetween calculationswith and without SOC astem-\nperature rises. However, the most significant difference is\nobservedforthe calculationsinthe liquidphase. We have\nfound out that account of SOC increases pressure in the\nsystem by 7-8 kbar for solid uranium and by more than\n10.5 kbar for liquid uranium near melting. Meanwhile,\nSOC makesit possible to describe with high accuracythe\nrelative density of molten uranium at melting tempera-\nture measured in static experiments by Rohr et al.[17]\nand Shpil’rain [15]. This value is generally accepted asa reference density of liquid uranium [22]. However, the\nslope of the thermal expansion curve in liquid uranium\nis still a subject of debate [23]. As can be seen from\nFig. 1, experimental data on liquid uranium obtained by\ndifferent authors are very contradictory. Nevertheless,\nthe slope of our curve are in excellent agreement with\nmeasurements by Rohr et al.[17], Grosse et al.[16], and\nDrottning [18]. Satisfactory agreement is also observed\nbetween our curve and the slope of the isobar dynami-\ncally measured using a pulse-heating technique by Shel-\ndon and Mulford [20].\nThermodynamic properties of solid and liquid ura-\nnium are derived from QMD simulations using Vienna\nab initio simulation package (VASP) [24–26]. Calcula-\ntions were carried out using finite–temperature density\nfunctional theory (FT-DFT) within the Perdew-Burke-\n/s49 /s50 /s51 /s52 /s53 /s54/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s84\n/s109/s101/s108/s116/s47\n/s48\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s107/s75/s41/s32/s81/s77/s68/s32/s119/s105/s116/s104/s32/s83/s79/s67\n/s32/s81/s77/s68/s32/s119/s105/s116/s104/s111/s117/s116/s32/s83/s79/s67\n/s32/s84/s111/s117/s108/s111/s117/s107/s105/s97/s110/s44/s32/s49/s57/s55/s53\n/s32/s76/s97/s119/s115/s111/s110/s44/s32/s49/s57/s56/s56\n/s32/s66/s111/s105/s118/s105/s110/s101/s97/s117/s44/s32/s49/s57/s57/s51\n/s32/s83/s104/s101/s108/s100/s111/s110/s44/s32/s49/s57/s57/s49\n/s32/s83/s104/s112/s105/s108/s39/s114/s97/s105/s110/s44/s49/s57/s56/s56\n/s32/s71/s97/s116/s104/s101/s114/s115/s44/s32/s49/s57/s56/s54\n/s32/s68/s114/s111/s116/s116/s110/s105/s110/s103/s44/s32/s49/s57/s56/s50\n/s32/s82/s111/s104/s114/s44/s32/s49/s57/s55/s48\n/s32/s71/s114/s111/s115/s115/s101/s44/s32/s49/s57/s54/s49/s85/s84\n/s109/s101/s108/s116\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s56/s56/s48/s46/s57/s50/s48/s46/s57/s54\nFIG.1. Phase diagram ofuraniuminthevicinityofmeltingin\nthe relative density versus temperature plane. Results of o ur\nQMD calculations of the zero-pressure isobar with account o f\nSOC are red dots and without SOC are open grey dots. Red\nand grey lines are polynomial approximations of liquid stat e\ncalculations. Experimental data are from [13–21]. The inse t\nis a closer look at the melting region.2\n!\"#$%&'(#)*+&,\"-#$.*/!012*+3-+,-3)#&$4 \n!\"#$ %&\n'()!*+\n*,-!%!./0 1!\"#$ )!*\nFIG. 2. Technique of calculation of the SOC correction to\npressure for a QMD calculation. Blue dots are SOC calcula-\ntions for configurations denoted as red dots.\nErnzerhof generalized gradient approximation [27, 28] as\nimplemented in VASP. The plane-wave basis set was ex-\npanded to a cutoff energy Ecutof 500 eV. We employ a\nprojector augmented wave (PAW) [29, 30] pseudopoten-\ntial with 14 valence electrons. Electronic states were cal-\nculated at the Baldereschi mean–value point [31] for the\nliquid phaseandusinga2 ×2×2Monkhorst–Packgridfor\nthe solid phase. The FT-DFT electronic structure calcu-\nlations are performed within a collinear formulation of\nspin states, in other words, with spin polarization. Su-\npercells of 108 atoms for α-U, 128 atoms for γ-U, and\n54 atoms for liquid uranium were used for simulations.\nThe convergence with respect to the number of atoms\nandk-point sampling was checked. Finite–temperature\neffects on the electrons were taken into account by using\nthe Fermi–Dirac smearing. The ionic temperature was\ncontrolled by the Nos´ e-Hoover thermostat [32].\nAll QMD simulations were performed in the NVTen-\nsemble, the zero isobar was restored from calculations\nalong isotherms in the solid phase and along isochors in\nthe liquid phase using linear regression as described in\nour previous works [33, 34]. Since the calculated den-\nsity at normal conditions slightly differs from the exper-\nimental value, it is reasonable to compare the results of\ncalculations and measurements in the units of relative\ndensityρ/ρ0, whereρ0is a density at normal conditions\n(see Fig. 1). In case of QMD ρ0= 19.386 g/cm3and\n19.48 g/cm3for the calculations with and without SOC,\ncorrespondingly.\nIt should be mentioned, that in the conventional mode\nVASP performs a fully relativistic calculation for the\ncore-electrons and treats valence electrons in a scalar rel-\nativisticapproximation[35]. Forconvenience,wewillfur-\nther denote such calculations as “noSOC”. Meanwhile,\nSOC may be also switched on for valence electrons.\nIn VASP, the explicit implementation of SOC is based\non the zeroth-order regular approximation [36] and de-\nscribed in detail in [37]. We will designate such calcula-\ntions as “SOC”./s50/s52/s54\n/s84/s61/s55/s48/s48/s48/s32/s75/s44/s32 /s61/s49/s48/s32/s103/s47/s99/s109/s51/s84/s61/s50/s53/s48/s48/s32/s75/s44/s32 /s61/s49/s55/s32/s103/s47/s99/s109/s51/s32/s83/s79/s67\n/s32/s110/s111/s83/s79/s67/s84/s61/s51/s48/s48/s32/s75/s44/s32 /s61/s49/s57/s46/s53/s52/s32/s103/s47/s99/s109/s51\n/s50/s52/s54\n/s45/s52/s53 /s45/s52/s48 /s45/s51/s53 /s45/s51/s48 /s45/s50/s53 /s45/s50/s48 /s45/s49/s53 /s45/s49/s48 /s45/s53 /s48/s48/s50/s52/s54/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s47/s97/s116/s111/s109/s41\n/s69/s45/s69\n/s70/s32/s40/s101/s86/s41/s40/s100/s41/s40/s99/s41/s40/s98/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121\n/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s115/s40/s97/s41\nFIG. 3. Spectroscopic experimental data and electronic DOS\nfrom QMD simulations. (a)—experiments [38, 39]; (b)—\nQMD DOS for α-U atT= 300 K and ρ= 19.54 g/cm3;\n(c)—QMD DOS for liquid U at T= 2500 K and ρ=\n17 g/cm3; (d)—QMD DOS for liquid U at T= 7000 K and\nρ= 10 g/cm3. All energies with respect to the Fermi level.\nSince QMD simulation with SOC is tremendously\ntime–consuming and memory demanding we have devel-\noped and applied a special correction technique. Our\ncalculation method consists of the following steps: 1)\nwe perform a QMD simulation at a given temperature\nand density with spin polarization; 2) we choose several\nconfigurations on the trajectory and perform full rela-\ntivistic calculations (SOC) for the chosen configurations;\n3) we retrieve the pressure/energy difference between the\nSOC and noSOC calculations for each configuration; 4)\nwe determine the correction to the QMD-calculated pres-\nsure/energy by averaging the differences for dozens of\nconfigurations. The approach is schematically shown in\nFig. 2 for pressure evolution. We usually perform no less\nthan 20 SOC calculations for each QMD run. We have\ncheckedstatistical significance by carryingout more than\n100 SOC calculations for several long simulations. Our\nanalysis shows that SOC corrections to pressure and en-\nergy are described well by the normal distribution and\naveraging over 20 configurations provides a mean value\nthat agree with averaging over 100 configurations within\nthe standard error.\nIn order to understand the reason for significant effect\nof the SO interaction on the thermal expansion of solid\nand liquid uranium we analyze the electronic density of3\n/s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s45/s56/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s80\n/s83/s79/s67/s32/s45/s32/s80\n/s110/s111/s83/s79/s67/s32/s40/s107/s98/s97/s114/s41\n/s69/s108/s101/s99/s116/s114/s111/s110/s32/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s107/s75/s41/s32 /s45/s85/s44/s32/s49/s57/s46/s52/s56/s32/s103/s47/s99/s109/s51\n/s32 /s45/s85/s32/s40/s98/s99/s99/s41/s44/s32/s49/s56/s46/s49/s50/s55/s32/s103/s47/s99/s109/s51\n/s32/s102/s99/s99/s44/s32/s49/s55/s32/s103/s47/s99/s109/s51\nFIG. 4. Pressure difference between SOC and noSOC static\nFT-DFT calculations as a function of electron temperature\nfor different lattices and densities.\nstates (DOS). DOS restored from QMD simulations for\nsolid uranium at T= 300 K and ρ= 19.54 g/cm3is\nshown in Fig. 3(b), and for liquid uranium at T= 2500\nandT= 7000 K in Fig. 3(c) and Fig. 3(d), respectively.\nSmooth curves were obtained by averaging over QMD\nsnapshots and applying a Gaussian smearing of 0.1 eV\nto the band energies. Experimental data [38, 39] are also\nshown in Fig. 3(a). X-ray photoemission spectroscopy\n(XPS) [38, 40, 41] revealed spin-orbit splitting of the U\n6pcore-states(9.5 eV). Bremsstrahlungisochromatspec-\ntroscopy (BIS) allowed to investigate the electronic DOS\nabove the Fermi level and revealed the SO splitting of\n5fstates with a separation of 1.15 eV [39]. As can be\nseen from the figure, both effects of splitting of 6 pand\n5fstates, as well as the valence band spectrum near the\nFermi level obtained by ultraviolet photoemission spec-\ntrosopy (UPS) [38], are described very well by our QMD-\nSOC calculations. The splitting remains at higher tem-\nperatures and lower densities.\nWeinvestigatetheinfluenceofSOCduringthethermal\nexcitation of valence electrons more precisely using static\nDFT calculations. In this case we calculate the pressure\ndifference between SOC and noSOC calculations for 3\ndifferent crystal lattices: α-U,γ-U (bcc), and fcc. Fcc\nis a close-packed structure with high coordination num-\nber (12), which is close to our estimate of the first co-\nordination number of uranium liquid near melting (14).\nDensities are chosen to correspond to α,γ, and liquid\nphases of U from QMD calculations, respectively. Static\ncalculations were performed with a higher energy con-\nvergence criterion for the electronic loop (10−7eV) and\nfinerk-point grid (15 ×15×15). Figure 4 shows that the\npressuredifferencebetweenSOCandnoSOCcalculations\nhas an explicit maximum in the range of 2–3 kK for all\n/s32/s100 /s45/s110/s111/s83/s79/s67/s69\n/s70/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s47/s97/s116/s111/s109/s41\n/s69/s45/s69\n/s70/s32/s40/s101/s86/s41/s69\n/s70\nFIG. 5. Total and 6 dand 5fpartial DOS in α-U atρ=\n19.48 g/cm3from static FT-DFT calculation. All energies\nwith respect to the Fermi level. The inset shows occupied\nDOS in the vicinity of the Fermi level from SOC and noSOC\ncalculations as thin red and blue lines, respectively.\nconsidered lattices. We note here that the peak is sig-\nnificantly higher for the fcc lattice. We also present the\ntotal andpartial electronicDOS for α-Ufrom static DFT\ncalculation in Fig. 5 to study the effect of thermal excita-\ntion of valence electrons in more detail. It can be clearly\nseen that the SO splitting of 5 fstates is responsible for\nthe redistribution of electronic states from higher ener-\ngies closer to the region of the Fermi level. We demon-\nstrate the occupied states at 3 kK in the inset in Fig. 5.\nFor convenience we fill the area where the occupied DOS\nfrom the SOC calculation is above (below) the one from\nthe noSOC calculation with red (blue). Apparently, the\nSO splitting leads to a higher electron occupancy in the\nvicinity of the Fermi level and above, which in turn re-\nsults in higher electronic pressure in a certain tempera-\nture range compared to noSOC calculation. At higher\ntemperatures the occupancies of the excited electrons for\nenergies above 1 eV is lower in SOC calculations than in\nnoSOC ones due to the gap between the shoulders of the\nsplit 5fband, so the opposite effect of negative pressure\ndifference is observed.\nAs can be seen from Fig. 4 the effect of the spin-orbit\ninteraction on pressure depends not only upon temper-\nature but structure as well, and it may be stronger for\ndenser-packed structures. This cumulative effect seems\nto be able to explain noticeably more intensive influence\nofSOConthermalexpansionofliquid uraniumthanthat\nof solid in the vicinity of melting, as well as further di-\nminishing of the discrepancy between SOC and noSOC\ncalculations at higher temperatures.\nIn order to provide a broader picture of the influence\nof the SO interaction on the thermodynamic properties4\n/s50/s48/s32/s103/s47/s99/s109/s51/s32/s81/s77/s68/s44/s32/s83/s79/s67\n/s32/s81/s77/s68/s44/s32/s110/s111/s83/s79/s67/s80/s114/s101/s115/s115/s117/s114/s101/s32/s40/s107/s98/s97/s114/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s107/s75/s41\nFIG. 6. QMD-calculated pressure with respect to tempera-\nture along isochors for liquid uranium. Open black dots are\nresults of QMD simulations with SOC corrections to pres-\nsure, solid black dots are results without corrections. Sol id\nand dashed lines are linear fits for isochors from SOC and\nnoSOC calculations, respectively. Color contour map indi-\ncates smoothed values of SOC corrections to pressure.\nofliquid uranium at higher temperatureand pressure, we\npresentthedependenceofthepressuredifferencebetween\nQMD SOC and noSOC calculations upon temperature\nalong the set of isochors in Fig. 6. The color contour\nmap illustrates a complex effect of pressure and temper-\nature. Nevertheless, it is consistent with our previous\nstatic DFT analysis in general. Interestingly, there is an\narea of the maximum impact of the SOC near the melt-\ning line at low pressures. On the other hand for higher\ntemperatures influence of the SOC becomes stronger as\npressure rises.\nIn conclusion, this study demonstrates, for the first\ntime, that relativistic effects have substantial impact on\nthermodynamic properties of liquid uranium, especially\non density at the melting point. The reason is the SO\nsplitting of the 5 fband and the thermal excitation of\nvalence electrons. We have analyzed the influence of the\nSOC on the thermodynamic properties of liquid uranium\nat high temperatures and pressures and revealed param-\neters where the effect of SOC can be neglected. Finally,\nthis study become possible due to the technique of QMD\nsimulation with account of SOC that was developed and\nsuccessfully applied in this Letter.\nWe appreciate sincerely Prof Igor Iosilevskiy for the\nmotivationforthiswork. Weacknowledgethe JIHTRAS\nSupercomputer Center, the Joint Supercomputer Center\nof the Russian Academy of Sciences, and the Shared Re-\nsource Centre “Far Eastern Computing Resource” IACP\nFEB RAS for for providing computing time.\nThe work is supported by the Russian Science Foun-dation (grant No.18-79-00346).\n∗minakovd@ihed.ras.ru\n†mikhail-paramon@mail.ru\n‡pasha@ihed.ras.ru\n[1] P. S¨ oderlind, L.Nordstr¨ om, L.Yongming, andB. Johans -\nson, Phys. Rev. B 42, 4544 (1990).\n[2] M. D. Jones, J. C. Boettger, R. C. Albers, and D. J.\nSingh, Phys. Rev. B 61, 4644 (2000).\n[3] L. Nordstr¨ om, J. M. Wills, P. H.Andersson, P. S¨ oderlin d,\nand O. Eriksson, Phys. Rev. B 63, 035103 (2000).\n[4] P. S¨ oderlind, Phys. Rev. B 66, 085113 (2002).\n[5] J. G. Tobin, K. T. Moore, B. W. Chung, M. A. Wall,\nA. J. Schwartz, G. van der Laan, and A. L. Kutepov,\nPhys. Rev. B 72, 085109 (2005).\n[6] S.Xiang, H.Huang,andL.M.Hsiung,JournalofNuclear\nMaterials 375, 113 (2008).\n[7] P. S¨ oderlind, G. Kotliar, K. Haule, P. M. Oppeneer, and\nD. Guillaumont, MRS Bulletin 35, 883 (2010).\n[8] P. S¨ oderlind, B. Grabowski, L. Yang, A. Landa,\nT. Bj¨ orkman, P. Souvatzis, and O. Eriksson,\nPhys. Rev. B 85, 060301 (2012).\n[9] W. Xie, W. Xiong, C. A. Marianetti, and D. Morgan,\nPhys. Rev. B 88, 235128 (2013).\n[10] P. S¨ oderlind, A. Landa, and P. E. A. Turchi,\nPhys. Rev. B 90, 157101 (2014).\n[11] W. Xie, C. A. Marianetti, and D. Morgan,\nPhys. Rev. B 93, 157101 (2016).\n[12] N. A. Smirnov, Phys. Rev. B 97, 094114 (2018).\n[13] Y. S. Touloukian, R. K. Kirby, R. E. Taylor, and\nP. D. Desai, Thermal Expansion: Metallic Elements and\nAlloys, Thermophysical Properties of Matter Vol. 12\n(Plenum, New York, 1975).\n[14] A. C. Lawson, C. E. Olsen, J. W. Richardson, M. H.\nMueller, and G. H. Lander, Acta Crystallographica Sec-\ntion B: Structural Science 44, 89 (1988).\n[15] E. Shpil’rain, V. A. Fomin, and V. V. Kachalov, High\nTemperature 26, 690 (1988).\n[16] A. V. Grosse, J. A. Cahill, and A. D. Kirshenbaum, Jour-\nnal of the American Chemical Society 83, 4665 (1961).\n[17] W. G. Rohr and L. J. Wittenberg, The Journal of Phys-\nical Chemistry 74, 1151 (1970).\n[18] W. D. Drotning, High Temperatures–High Pressures 14,\n253 (1982).\n[19] G.R.Gathers,Reports on Progress in Physics 49, 341 (1986).\n[20] R. I. Sheldon and R. N. Mulford,\nJournal of Nuclear Materials 185, 297 (1991).\n[21] M. Boivineau, L. Arl` es, J. M.\nVermeulen, and T. Th´ evenin,\nPhysica B: Physics of Condensed Matter 190, 31 (1993).\n[22] D. R. Lide, ed., CRC Handbook of Chemistry and\nPhysics, Internet Version 2005 (CRC Press, Boca Ra-\nton, FL, 2005).\n[23] I. L. Iosilevskiy and V. K. Gryaznov,\nJournal of Nuclear Materials 344, 30 (2005).\n[24] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).\n[25] G. Kresse and J. Hafner, Phys. Rev. B 49, 14251 (1994).\n[26] G. Kresse and J. Furthm¨ uller,\nPhys. Rev. B 54, 11169 (1996).\n[27] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.5\nLett.77, 3865 (1996).\n[28] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.78, 1396 (1997).\n[29] P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).\n[30] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n[31] A. Baldereschi, Phys. Rev. B 7, 5212 (1973).\n[32] S.Nos´ e,The Journal of Chemical Physics 81, 511 (1984).\n[33] D. V. Minakov, M. A. Paramonov, and P. R. Levashov,\nAIP Advances 8, 125012 (2018).\n[34] D. V. Minakov, M. A. Paramonov, and P. R. Levashov,\nHigh Temperatures–High Pressures 49, 211 (2020).\n[35] J. Hafner, Journal of computational chemistry 29, 2044\n(2008).[36] E. v. Lenthe, E. J. Baerends, and J. G. Snijders,\nThe Journal of Chemical Physics 99, 4597 (1993).\n[37] S. Steiner, S. Khmelevskyi, M. Marsmann, and\nG. Kresse, Phys. Rev. B 93, 224425 (2016).\n[38] C. P. Opeil, R. K. Schulze, M. E. Manley, J. C. Lash-\nley, W. L. Hults, R. J. Hanrahan, J. L. Smith, B. Mi-\nhaila, K. B. Blagoev, R. C. Albers, and P. B. Littlewood,\nPhys. Rev. B 73, 165109 (2006).\n[39] Y. Baer and J. K. Lang, Phys. Rev. B 21, 2060 (1980).\n[40] J. Fuggle, A. Burr, L. Watson, D. Fabian, and W. Lang,\nJournal of Physics F: Metal Physics 4, 335 (1974).\n[41] W. McLean, C. A. Colmenares, R. L. Smith, and G. A.\nSomorjai, Phys. Rev. B 25, 8 (1982)." }, { "title": "1601.07687v2.Interfacial_spin_orbit_torque_without_bulk_spin_orbit_coupling.pdf", "content": "Interfacial Spin-Orbit Torque without Bulk Spin-Orbit Coupling\nSatoru Emori,\u0003Tianxiang Nan,yAmine M. Belkessam, Xinjun Wang, Alexei D.\nMatyushov, Christopher J. Babroski, Yuan Gao, Hwaider Lin, and Nian X. Sun\n1Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115\n(Dated: February 10, 2016)\nAn electric current in the presence of spin-orbit coupling can generate a spin accumulation that\nexerts torques on a nearby magnetization. We demonstrate that, even in the absence of materials\nwith strong bulk spin-orbit coupling, a torque can arise solely due to interfacial spin-orbit coupling,\nnamely Rashba-Eldestein e\u000bects at metal/insulator interfaces. In magnetically soft NiFe sandwiched\nbetween a weak spin-orbit metal (Ti) and insulator (Al 2O3), this torque appears as an e\u000bective\n\feld, which is signi\fcantly larger than the Oersted \feld and sensitive to insertion of an additional\nlayer between NiFe and Al 2O3. Our \fndings point to new routes for tuning spin-orbit torques by\nengineering interfacial electric dipoles.\nAn electric current in a thin \flm with spin-orbit cou-\npling can produce a spin accumulation [1{3], which can\nthen exert sizable torques on magnetic moments [4{\n7]. First demonstrated in a ferromagnetic semicon-\nductor [8], \\spin-orbit torques\" are nowadays studied\nin room-temperature ferromagnetic metals (FMs) inter-\nfaced with heavy metals (HMs) with strong spin-orbit\ncoupling, such as Pt, Ta, and W [9{24]. These torques\ncan arise from (1) spin-dependent scattering of conduc-\ntion electrons in the bulk of the HM, i.e., the spin-Hall\ne\u000bect [2, 3, 9{12], and (2) momentum-dependent spin\npolarization at the HM/FM interface, i.e., the Rashba-\nEdelstein e\u000bect [1, 5, 13{16]. Since a HM/FM system\ncan exhibit either or both of these spin-orbit e\u000bects,\nit can be a challenge to distinguish the spin-Hall and\nRashba-Edelstein contributions [3, 6, 7, 17, 18]. Spin-\norbit torques may be further in\ruenced by spin scatter-\ning [25] or proximity-induced magnetization [26] at the\nHM/FM interface. Moreover, in many cases [9{24], the\nFM interfaced on one side with a HM is interfaced on the\nother with an insulating material, and the electric dipole\nat the FM/insulator interface [27, 28] may also give rise\nto a Rashba-Edelstein e\u000bect. Recent studies [20{24] in-\ndeed suggest nontrivial in\ruences from insulating-oxide\ncapping layers in perpendicularly-magnetized HM/FM\nsystems. However, with the FM only <\u00181 nm thick [20{\n24], changing the composition of the capping layer may\nmodify the ultrathin FM and hence the HM/FM inter-\nface. The points above make it di\u000ecult to disentangle\nthe contributions from the HM bulk, HM/FM interface,\nand FM/insulator interface, thereby posing a challenge\nfor coherent engineering of spin-orbit torques.\nIn this Letter, we experimentally show a spin-orbit\ntorque that emerges exclusively from metal/insulator in-\nterfaces in the absence of materials with strong bulk spin-\norbit coupling. Our samples consist of magnetically soft\nNi80Fe20(NiFe) sandwiched between a weak spin-orbit\nlight metal (Ti) and a weak spin-orbit insulator (Al 2O3).\nWe observe a \\\feld-like\" spin-orbit torque that appears\nas a current-induced e\u000bective \feld, which is signi\fcantlylarger than the Oersted \feld. This torque is conclusively\nattributed to the Rashba-Edelstein e\u000bect, i.e., spin accu-\nmulation at the NiFe/Al 2O3interface exchange coupling\nto the magnetization in NiFe [4, 5]. We also observe a\n\\nonlocal\" torque with Cu inserted between NiFe and\nAl2O3due to spin accumulation at the Cu/Al 2O3inter-\nface. Our \fndings demonstrate simple systems exhibiting\npurely interfacial spin-orbit coupling, which are free from\ncomplications caused by strong spin-orbit HMs, and open\npossibilities for spin-orbit torques enabled by engineered\nelectric dipoles at interfaces.\nThin-\flm heterostructures are sputter-deposited on Si\nsubstrates with a 50-nm thick SiO 2overlayer. All layers\nare deposited at an Ar pressure of 3 \u000210\u00003Torr with\na background pressure of <\u00182\u000210\u00007Torr. Metallic lay-\ners are deposited by dc magnetron sputtering, whereas\nAl2O3is deposited by rf magnetron sputtering from a\ncompositional target. The deposition rates are calibrated\nby X-ray re\rectivity. For each structure, unless otherwise\nnoted, a 1.2-nm thick Ti seed layer is used to promote\nthe growth of NiFe with narrower resonance linewidth\nand near-bulk saturation magnetization. Devices are\npatterned and contacted by Cr(3 nm)/Au(100 nm) elec-\ntrodes by photolithography and lifto\u000b.\nWe \frst examine the current-induced \feld in a tri-\nlayer of Ti(1.2 nm)/NiFe(2.5 nm)/Al 2O3(1.5 nm) by us-\ning the second-order planar Hall e\u000bect (PHE) voltage\ntechnique devised by Fan et al. [10, 11]. As illustrated\nin Fig. 1(a), a dc current Idcalong the x-axis gener-\nates a planar Hall voltage VPHalong the y-axis in a\n100-\u0016m wide Hall bar, which is placed in the center\nof a two-axis Helmholtz coil. The second-order planar\nHall voltage \u0001 VPH=VPH(+Idc) +VPH(\u0000Idc) is mea-\nsured while sweeping the external \feld Hx(Fig. 1(b)).\nThe total current-induced in-plane transverse \feld HI\n(which includes the Oersted \feld) pulls the magnetiza-\ntion away from the x-axis at an angle \u0012. WhenjHxjis\nsu\u000eciently large ( >\u001810 Oe),\u0012is small and \u0001 VPHis pro-\nportional to I2\ndcH\u00001\nxdHI=dIdc[10]. Following the proce-\ndure in Ref. [11], we apply a constant transverse bias \feldarXiv:1601.07687v2 [cond-mat.mtrl-sci] 12 Feb 20162\n0 2 4 6 8-1.0-0.50.00.51.0\nHOe,TiHOe,max\n HI (Oe)\nIdc (mA)\n-60 -40 -20 020 40 60-2-1012\nIdc=8 mA\n VPH (mV)\nHx (Oe)Hy=+1Oe\nHy=0 \nHy=-1Oe(a) \n(b) (c) \nHyHx\nM\nθxy\nHII-\nI+V-V+\nAl2O3\nNiFe\nTi\nFigure 1. (a) Schematic of the second-order PHE measure-\nment. (b) Second-order planar Hall voltage \u0001 VPHcurves at\ndi\u000berent transverse bias \felds Hy. (c) Current-induced \feld\nHIversusIdc. The dotted line shows HOe ;Tibased on the\nestimated fraction of Idcin Ti. The shaded area is bounded\nby the maximum possible Oersted \feld HOe ;max.\njHyj= 1 Oe (Fig. 1(a),(b)) and extrapolate the critical\nHyrequired to cancel HI, i.e., to null the \u0001 VPHspec-\ntrum. For the data in Fig. 1(b), Hy=-0.75 Oe would\nnull \u0001VPH, soHI= 0.75 Oe at Idc= 8 mA.\nAs shown in Fig. 1(c), HIscales linearly with Idcwith\nslope dHI/dIdc= 0.095 Oe per mA. To estimate the Oer-\nsted \feld contribution to HI, the current is assumed to\nbe uniform within each conductive layer, such that the\nOersted \feld comes only from the current in the Ti layer,\nHOe ;Ti=fTiIdc=2w, wherefTiis the fraction of Idcin\nTi andwis the Hall bar width. The sheet resistances\n2000 \n/sq for Ti(1.2 nm) and 350 \n/sq for NiFe(2.5 nm),\nfound from four-point resistance measurements, yield fTi\n= 0.15 andjHOe ;Tij= 0:009 Oe per mA. The net HIis\ntherefore an order of magnitude larger than HOe ;Ti, and\nmoreover, the direction of HIopposesHOe ;Ti.\nThe actual Oersted \feld may deviate from HOe ;Tibe-\ncause of nonuniform current distribution within each con-\nductive layer and interfacial scattering, both of which are\ndi\u000ecult to quantify. However, we can place the upper\nbound on the Oersted \feld, jHOe ;maxj=jIdcj=2w, by as-\nsuming that the entireIdc\rows above or below the mag-\nnetic layer. In Fig. 1(c), we shade the range bounded by\njHOe ;maxj. The magnitude of HIstill exceeds HOe ;max,\ncon\frming the presence of an additional current-induced\n\feld with a component collinear with the Oersted \feld.\nWe also measure HIwith a technique based on spin-\ntorque ferromagnetic resonance (ST-FMR) [29, 30]. As\nillustrated in Fig. 2(a), the rf excitation current is in-\njected into a 5- \u0016m wide, 25- \u0016m long strip through a\nground-signal-ground electrode. While the in-plane ex-\nternal \feld His swept at an in-plane angle \u0012, the rec-\n200 300 400 500 600-40-2002040\nIdc= -1.5 mA\nIdc= 0 mA\nIdc= -1.5 mA\n Vmix (V)\nH (Oe)\n-180 -90 0 90 180-2-1012\n HFMR/Idc (Oe/mA)\n (deg.)\n-180 -90 0 90 180-1012\nHOe,Ti\n HI/HOe,max\n (deg.)ST-FMR\nPHE(e) (a) \n(b) \n390400410\n (c) \n(d) \n-2 -1 0 1 2398400402404\nHOe,max\nHOe,Ti\n HFMR (Oe)\nIdc (mA)\nVmix~ ref\nsignallock-in\nIrfIdc\nHI\nxy\nθM\nHAl2O3\nNiFe\nTi\nFigure 2. (a) Schematic of the ST-FMR setup. (b) ST-FMR\nspectra at di\u000berent dc bias currents Idc, with rf current exci-\ntation at 5 GHz and +8 dBm and external \feld Hat\u0012= 40\u000e.\nInset:Idc-induced shift of ST-FMR spectra. (c) Shift of reso-\nnance \feld HFMR due toIdcat\u0012= 40\u000e. The error bar is the\nstandard deviation of 5 measurements. The dotted line shows\nthe estimated Oersted \feld from Ti, HOe ;Ti. The shaded area\nis bounded by the maximum possible Oersted \feld, HOe ;max.\n(d) Angular dependence of Idc-inducedHFMR shift. The solid\ncurve indicates the \ft to sin \u0012. (e) Transverse current-induced\n\feldHI=\u0000\u0001HFMR=sin\u0012normalized by HOe ;maxat various\n\u0012. The error bar is the error in linear \ft of HFMR versus\nIdc. The solid line indicates the average of the ST-FMR data\npoints. The dotted line indicates estimated HOe ;Ti. The PHE\ndata point at \u0012= 0 is the average of three devices.\nti\fed mixing voltage Vmixacross the strip is acquired\nwith a lock-in ampli\fer. The resulting spectrum is well\n\ft to a Lorentzian curve Vmix=VsFs+VaFacon-\nsisting of the symmetric component Fs=W2=((H\u0000\nHFMR)2+W2) and antisymmetric component Fa=\nW(H\u0000HFMR)=((H\u0000HFMR)2+W2), whereWis the\nresonance linewidth and HFMR is the resonance \feld. We\ninject a small dc bias current jIdcj\u00142 mA to measure the\nshift inHFMR caused by the net Idc-induced \feld HI[31].\nAlthough the scatter in the ST-FMR data is greater than\nthe PHE data (Fig. 1(c)), Fig. 2(c) shows that the ob-\nserved shift in HFMR is signi\fcantly larger than (and op-\nposes) the contribution from HOe ;Ti, and its magnitude\nexceeds the maximum possible shift from HOe ;max.\nFig. 2(d) shows the Idc-induced shift \u0001 HFMR as a func-\ntion of in-plane magnetization angle, equal to the applied\n\feld angle\u0012for the soft NiFe layer. This angular depen-3\ndence is well described by a sin \u0012relation, which implies\nthatHIis transverse to the current axis. Fig. 2(e) shows\nthat the constant HI=\u0000\u0001HFMR=sin\u0012indeed agrees\nwell with the PHE data measured at \u0012\u00190. This \fnding\ncon\frms that HI, including the non-Oersted contribu-\ntion, is entirely transverse to the current and is indepen-\ndent of the magnetization orientation.\nFor a wide range of NiFe thickness tNiFe, as shown in\nFig. 3(a), we observe HIthat cannot be accounted for\nby the Oersted \feld alone. The observed HIopposes\nHOe ;Tiin all samples, and HIis more than a factor of 2\nlarger than HOe ;maxattNiFe\u00192 nm. The drop in HI\nfortNiFe<\u00182 nm is caused by the increasing magnitude\nofHOe ;Ti, as NiFe becomes more resistive and a larger\nfraction of current \rows through Ti with decreasing tNiFe.\nThe anomalous portion of HI, which cannot be ex-\nplained by the classical Oersted \feld, may be due to a\nspin-orbit torque that acts as a \\spin-orbit \feld\" HSO.\nIn Fig. 3(b), we plot the estimated HSO=HI\u0000HOe ;Ti\nnormalized by the current density in NiFe, JNiFe. This\nnormalized HSOscales inversely with tNiFe, implying that\nthe source of HSOis outside or at a surface of the NiFe\nlayer. Therefore, HSOdoes not arise from spin-orbit ef-\nfects within the bulk of NiFe [32], i.e., the reciprocal of\nthe recently reported inverse spin-Hall e\u000bect in FMs [33{\n36]. Moreover, any possible spin-orbit toques arising\nfrom the bulk of NiFe would depend on the magnetiza-\ntion orientation [32] and are thus incompatible with the\nobserved symmetry of HSO(Fig. 2(e)). It is unlikely that\nHSOis generated by the spin-Hall e\u000bect in Ti, because\nits spin-Hall angle is small ( <0.001) [37, 38] and only a\nsmall fraction of Idcis expected to be in the resistive\nultrathin Ti layer. In Ti/NiFe/Al 2O3, we also do not\nobserve a damping-like torque that would be expected to\narise from the spin-Hall e\u000bect [6, 39]; the linewidth W\nis invariant with Idcwithin our experimental resolution\n<0.2 Oe/mA [31].\nWith spin-orbit e\u000bects in the bulk of NiFe and Ti ruled\nout as mechanisms behind HSO, the only known mecha-\nnism that agrees with the observed HSOis the Rashba-\nEdelstein e\u000bect [1, 4, 5]: an interfacial spin accumula-\ntion (polarized transverse to the current) exchange cou-\nples to the magnetization in NiFe. Indeed, tight-binding\nRashba model calculations reveal a \feld-like torque, but\nno damping-like torque, in the \frst order of spin-orbit\ncoupling due to transverse spin accumulation indepen-\ndent of the magnetization orientation [40].\nWe now gain further insight into the origin of HSOby\nexamining its dependence on the layer stack structure, as\nsummarized in Fig. 4(a-f). In the symmetric Al 2O3(1.5\nnm)/NiFe(2.3 nm)/Al 2O3(1.5 nm) trilayer (Fig. 4(a)),\nHIvanishes, which is as expected because the Oersted\n\feld should be nearly zero and the two nominally iden-\ntical interfaces sandwiching NiFe produces no net spin\naccumulation. Breaking structural inversion symmetry\nwith the Ti(1.2 nm) seed layer results in an uncompen-\n(a) \n1 2 3 4 5 60246\n HSO/JNiFe (Oe/1011 A/m2)\ntNiFe (nm)\n-10123\n HOe,Ti\n HI/HOe,maxPHE\nST-FMR\n(b) Figure 3. (a) NiFe-thickness tNiFedependence of HInormal-\nized byHOe ;max. The dotted curve indicates the estimated\nOersted \feld from Ti, HOe ;Ti. Each ST-FMR data point is\nthe mean of results at several frequencies 4-7 GHz at \u0012= 45\u000e\nand\u0000135\u000e.HI=HOe ;max>0 is de\fned as HI==+ywhenIdc==+x\n(illustrated in Figs. 1(a) and 2(a)). (b) Estimated spin-orbit\n\feldHSOper unit current density in NiFe, JNiFe. The solid\ncurve indicates the \ft to tNiFe\u00001.\nsated interfacial spin accumulation that generates a \fnite\nHSO=HI\u0000HOe ;Ti(Fig. 4(b)).\nInserting Pt(0.5 nm) between the NiFe and Al 2O3lay-\ners suppresses HSO, such that the estimated Oersted \feld\nHOe ;NMfrom the nonmagnetic Ti and Pt layers entirely\naccounts for HI(Fig. 4(c)). This may seem counterin-\ntuitive since Pt exhibits strong spin-orbit coupling and\na large Rashba-Edelstein e\u000bect may be expected at the\nPt surface [41]. However, Pt is also a strong spin scat-\nterer, as evidenced by an increase in the Gilbert damping\nparameter from\u00190.013 for Ti/NiFe/Al 2O3to\u00190.03 for\nTi/NiFe/Pt/Al 2O3. The accumulated spins may quickly\nbecome scattered by Pt, such that there is no net \feld-\nlike torque mediated by exchange coupling [4, 5] between\nthese spins and the magnetization in NiFe. Based on the\nsuppression of HSOby Pt insertion, we infer that the\nRashba-Edelstein e\u000bect at the NiFe/Al 2O3interface is\nthe source of HSO.\nInserting Cu(1 nm) at the NiFe/Al 2O3interface re-\nverses the direction of HSO=HI\u0000HOe ;NM(Fig. 4(d)).\nWe deduce a Rashba-Edelstein e\u000bect (opposite in sign to\nthat of NiFe/Al 2O3) at the Cu/Al 2O3interface, rather\nthan the NiFe/Cu interface, because (1) if NiFe/Cu gen-\nerates the reversed HSO, we should see an enhanced HSO\nfor NiFe sandwiched between Cu (bottom) and Al 2O3\n(top), but this is not the case (Fig. 4(e)); and (2) insert-\ning a spin-scattering layer of Pt(0.5 nm) between Cu and\nAl2O3suppressesHSO(Fig. 4(f)).4\n-1012\n \n HI/HOe,m ax\n HOe,NM/HOe,m axHI/HOe,m ax, HOe,NM/HOe,m ax(g) \n(h) \n-2-1012\nHOe,NM\n HI/HOe,max\nPHE\nST-FMR\n0 2 4 6 8 10 12-4-202\n HSO/JCu (Oe/1011 A/m2)\ntCu (nm)Al2O3 \nTi NiFe Cu Pt \nTi NiFe Al2O3 \nCu Al2O3 \nTi NiFe Pt \nTi NiFe Al2O3 \nAl2O3 NiFe Al2O3 (a) \nTi NiFe Al2O3 \nCu (b) (c) (d) (e) (f) \nFigure 4. (a-f) Structural dependence of HI(mean of mea-\nsurements on three PHE devices) normalized by HOe ;max.\nHOe ;NMis the Oersted \feld from current in the nonmagnetic\nmetal layers (Ti, Cu, Pt). The nominal layer thicknesses are\nNiFe: 2.3 nm, Al 2O3: 1.5 nm, Ti: 1.2 nm, Cu: 1.0 nm, and\nPt: 0.5 nm. (g) Cu-thickness tCudependence of HInormal-\nized byHOe ;maxat NiFe thickness 2.5 nm. The blue dotted\ncurve indicates HOe ;NM. (h) Estimated spin-orbit \feld HSO\nper unit current density in Cu, JCu.\nFig. 4(g) plots the dependence of HIon Cu thickness\ntCu. In the limit of large tCu(\u001910 nm),HIapproaches\nHOe ;NMthat is predominantly due to the current in the\nhighly conductive Cu layer. From the estimated current\ndistribution, we obtain HSO=HI\u0000HOe ;NMnormalized\nby the current density in the Cu layer, JCu. As shown in\nFig. 4(h),HSO=JCu\u00191-2 Oe/1011A/m2exhibits little\ndependence on tCu. This is consistent with the Rashba-\nEdelstein e\u000bect at the Cu/Al 2O3interface that is present\nirrespective of tCu. Persistence of HSOeven at large tCu\nimplies a nonlocal Rashba-Edelstein \feld: the spin accu-\nmulation at the Cu/Al 2O3interface couples to the mag-\nnetization in NiFe across the Cu layer. However, further\nstudies are required to elucidate the mechanism involv-\ning Cu, since we do not observe any apparent oscillation\ninHSOwithtCuthat would be expected for exchange\ncoupling across Cu [42].AttCu\u00192 nm,HIvanishes because HSOandHOe ;NM\ncompensate each other (Fig. 4(g)). Fan et al. also show\nnear vanishing of HIin NiFe(2 nm)/Cu( tCu)/SOi 2(3.5\nnm) attCu\u00193 nm [10], and Avci et al. report a current-\ninduced \feld in Co(2.5 nm)/Cu(6 nm)/AlO x(1 nm) that\nis well below the estimated Oersted \feld [19]. In each\nof these studies [10, 19], a spin-orbit \feld due to the\nRashba-Edelstein e\u000bect at the Cu/oxide interface may\nhave counteracted the Oersted \feld.\nIn summary, we have shown a current-induced\nspin-orbit torque due to Rashba-Edelstein e\u000bects at\nNiFe/Al 2O3and Cu/Al 2O3interfaces. This torque is\ndistinct from previously reported spin-orbit torques\nbecause it arises even without spin-orbit coupling in the\nbulk of the constituent materials. The origin of this\ntorque is purely interfacial spin-orbit coupling, which\nlikely emerges from the electric dipoles that develop at\nthe metal/insulator interfaces [27, 28]. This mechanism\nis supported by recent theoretical predictions of current-\ninduced spin polarization at metal/insulator interfaces\nin the absence of bulk spin-orbit coupling [43, 44].\nRashba-Edelstein e\u000bects at metal/insulator interfaces\nmay be universal and should motivate the use of various\npreviously-neglected materials as model systems for\ninterfacial spin-dependent physics and as components\nfor enhancing spin-orbit torques, perhaps combined\nwith gate-voltage tuning [20, 21, 45]. One possibility is\nto apply interfacial band alignment techniques, similar\nto those for semiconductor heterostructures [46], to\nengineer giant dipole-induced Rashba-Edelstein e\u000bects.\nThis work was supported by the AFRL through\ncontract FA8650-14-C-5706, the W.M. Keck Foundation,\nand the NSF TANMS ERC Award 1160504. X-ray\nre\rectivity was performed in CMSE at MIT, and lithog-\nraphy was performed in the George J. Kostas Nanoscale\nTechnology and Manufacturing Research Center. We\nthank Geo\u000brey Beach, Carl Boone, Xin Fan, Adrian\nFeiguin, Chi-Feng Pai, and Kohei Ueda for helpful\ndiscussions. We give special thanks to Mairbek Chshiev,\nSergey Nikolaev, and Noriyuki Sato for their comments\nand sharing of unpublished results.\n\u0003Current Address: Geballe Laboratory for Advanced Ma-\nterials and Department of Applied Physics, Stanford Uni-\nversity, Stanford, CA 94305 USA ; satorue@stanford.edu\nyCurrent Address: Department of Materials Science and\nEngineering, University of Wisconsin Madison, Madison,\nWI 53706 USA\n[1] V. Edelstein, Solid State Commun. 73, 233 (1990).\n[2] A. Ho\u000bmann, IEEE Trans. Magn. 49, 5172 (2013).\n[3] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).5\n[4] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n[5] P. Gambardella and I. M. Miron, Philos. Trans. A. Math.\nPhys. Eng. Sci. 369, 3175 (2011).\n[6] P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D. Stiles, Phys. Rev. B 87, 174411 (2013).\n[7] A. Brataas and K. M. D. Hals, Nat. Nanotechnol. 9, 86\n(2014).\n[8] A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna,\nY. Lyanda-Geller, and L. P. Rokhinson, Nat. Phys. 5,\n656 (2009).\n[9] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and\nR. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012).\n[10] X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, and J. Q.\nXiao, Nat. Commun. 4, 1799 (2013).\n[11] X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V. O. Lorenz,\nand J. Q. Xiao, Nat. Commun. 5, 3042 (2014).\n[12] C.-F. Pai, M.-H. Nguyen, C. Belvin, L. H. Vilela-Le~ ao,\nD. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 104,\n082407 (2014).\n[13] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Au\u000bret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n[14] T. D. Skinner, M. Wang, A. T. Hindmarch, A. W. Rush-\nforth, A. C. Irvine, D. Heiss, H. Kurebayashi, and A. J.\nFerguson, Appl. Phys. Lett. 104, 062401 (2014).\n[15] M. Kawaguchi, T. Moriyama, T. Koyama, D. Chiba, and\nT. Ono, J. Appl. Phys. 117, 17C730 (2015).\n[16] G. Allen, S. Manipatruni, D. E. Nikonov, M. Doczy, and\nI. A. Young, Phys. Rev. B 91, 144412 (2015).\n[17] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Bl ugel, S. Au\u000bret, O. Boulle,\nG. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8,\n587 (2013).\n[18] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi,\nS. Fukami, T. Suzuki, S. Mitani, and H. Ohno, Nat.\nMater. 12, 240 (2013).\n[19] C. O. Avci, K. Garello, M. Gabureac, A. Ghosh,\nA. Fuhrer, S. F. Alvarado, and P. Gambardella, Phys.\nRev. B 90, 224427 (2014).\n[20] R. H. Liu, W. L. Lim, and S. Urazhdin, Phys. Rev. B\n89, 220409 (2014).\n[21] S. Emori, U. Bauer, S. Woo, and G. S. D. Beach, Appl.\nPhys. Lett. 105, 222401 (2014).\n[22] X. Qiu, K. Narayanapillai, Y. Wu, P. Deorani, D.-H.\nYang, W.-S. Noh, J.-H. Park, K.-J. Lee, H.-W. Lee, and\nH. Yang, Nat. Nanotechnol. 10, 333 (2015).\n[23] M. Akyol, G. Yu, J. G. Alzate, P. Upadhyaya, X. Li,\nK. L. Wong, A. Ekicibil, P. Khalili Amiri, and K. L.\nWang, Appl. Phys. Lett. 106, 162409 (2015).\n[24] N. Sato, A. El-Ghazaly, R. M. White, and S. X. Wang,to be pubished IEEE. Trans. Magn. (2016).\n[25] K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602\n(2015).\n[26] W. Zhang, M. B. Jung\reisch, W. Jiang, Y. Liu, J. E.\nPearson, S. G. E. te Velthuis, A. Ho\u000bmann, F. Freimuth,\nand Y. Mokrousov, Phys. Rev. B 91, 115316 (2015).\n[27] L. Xu and S. Zhang, J. Appl. Phys. 111, 07C501 (2012).\n[28] F. Ibrahim, H. X. Yang, A. Hallal, B. Dieny, and\nM. Chshiev, Phys. Rev. B 93, 014429 (2016).\n[29] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n[30] D. Fang, H. Kurebayashi, J. Wunderlich, K. V\u0013 yborn\u0013 y,\nL. P. Z^ arbo, R. P. Campion, A. Casiraghi, B. L. Gal-\nlagher, T. Jungwirth, and A. J. Ferguson, Nat. Nan-\notechnol. 6, 413 (2011).\n[31] T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm,\nJ. G. Jones, B. M. Howe, G. J. Brown, and N. X. Sun,\nPhys. Rev. B 91, 214416 (2015).\n[32] T. Taniguchi, J. Grollier, and M. D. Stiles, Phys. Rev.\nAppl. 3, 044001 (2015).\n[33] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys.\nRev. Lett. 111, 066602 (2013).\n[34] A. Tsukahara, Y. Ando, Y. Kitamura, H. Emoto,\nE. Shikoh, M. P. Delmo, T. Shinjo, and M. Shiraishi,\nPhys. Rev. B 89, 235317 (2014).\n[35] A. Azevedo, O. Alves Santos, R. O. Cunha, R. Rodr\u0013 \u0010guez-\nSu\u0013 arez, and S. M. Rezende, Appl. Phys. Lett. 104,\n152408 (2014).\n[36] H. Wang, C. Du, P. Chris Hammel, and F. Yang, Appl.\nPhys. Lett. 104, 202405 (2014).\n[37] C. Du, H. Wang, F. Yang, and P. C. Hammel, Phys. Rev.\nB90, 140407 (2014).\n[38] K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Mu-\nrakami, and E. Saitoh, J. Phys. Condens. Matter 26,\n343202 (2014).\n[39] F. Freimuth, S. Bl ugel, and Y. Mokrousov, Phys. Rev. B\n90, 174423 (2014).\n[40] A. Kalitsov, S. A. Nikolaev, J. Velev, W. H. Butler,\nM. Chshiev, and O. Mryasov, unpublished (2016).\n[41] H. J. Zhang, S. Yamamoto, Y. Fukaya, M. Maekawa,\nH. Li, A. Kawasuso, T. Seki, E. Saitoh, and K. Takanashi,\nSci. Rep. 4(2014).\n[42] S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 (1991).\n[43] X. Wang, J. Xiao, A. Manchon, and S. Maekawa, Phys.\nRev. B 87, 081407 (2013).\n[44] J. Borge, C. Gorini, G. Vignale, and R. Raimondi, Phys.\nRev. B 89, 245443 (2014).\n[45] U. Bauer, L. Yao, A. J. Tan, P. Agrawal, S. Emori, H. L.\nTuller, S. van Dijken, and G. S. D. Beach, Nat. Mater.\n14, 174 (2015).\n[46] H. Kroemer, Rev. Mod. Phys. 73, 783 (2001)." }, { "title": "1712.02942v2.Spin_orbit_coupling_induced_valley_Hall_effects_in_transition_metal_dichalcogenides.pdf", "content": "Spin-orbit coupling induced valley Hall e\u000bects in transition-metal dichalcogenides\nBenjamin T. Zhou1,\u0003Katsuhisa Taguchi2,\u0003Yuki Kawaguchi2, Yukio Tanaka2, and K. T. Law1y\n1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China\n2Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan\nIn transition-metal dichalcogenides, electrons in the K-valleys can experience both\nIsing and Rashba spin-orbit couplings. In this work, we show that the coexistence of\nIsing and Rashba spin-orbit couplings leads to a special type of valley Hall e\u000bect, which\nwe call spin-orbit coupling induced valley Hall e\u000bect. Importantly, near the conduction\nband edge, the valley-dependent Berry curvatures generated by spin-orbit couplings\nare highly tunable by external gates and dominate over the intrinsic Berry curvatures\noriginating from orbital degrees of freedom under accessible experimental conditions.\nWe show that the spin-orbit coupling induced valley Hall e\u000bect is manifested in the\ngate dependence of the valley Hall conductivity, which can be detected by Kerr e\u000bect\nexperiments.\nIntroduction\nValley degrees of freedom emerge from local extrema\nin electronic band structures of two-dimensional Dirac\nmaterials. When spatial inversion symmetry is broken in\nsuch systems, valley-contrasting e\u000bective magnetic \felds\ncan arise in momentum space, known as Berry curva-\nture \felds[1, 2]. Upon application of in-plane electric\n\felds, the Berry curvature drives carriers from opposite\nvalleys to \row in opposite transverse directions, lead-\ning to valley Hall e\u000bects (VHEs)[3, 4]. It was \frst pre-\ndicted that VHEs can exist in gapped graphene ma-\nterials, where global inversion breaking is introduced\nbyh-BN substrates[5] or external electric \felds[6, 7].\nMore recently, valley Hall phenomena were proposed in\nmonolayer transition-metal dichalcogenides (TMDs)[8],\nin which nontrivial Berry curvatures result from intrin-\nsically broken inversion symmetry in the trigonal pris-\nmatic structure of their unit cells[9]. Because of its ver-\nsatility to couple to optical[10{17], magnetic[18{22] and\nelectrical[23, 24] controls, valley Hall physics in TMD-\nbased materials have been under intensive theoretical and\nexperimental studies in recent years.\nBesides Berry curvature \felds, broken inversion sym-\nmetry in monolayer TMDs also induces e\u000bective Zeeman\n\felds in momentum space[8, 25{32], referred to as Ising\nspin-orbit coupling(SOC) \felds[33{35]. In the conduc-\ntion band, the energy splitting due to Ising SOCs ranges\nfrom a few to tens of meVs[31, 32], while in the valence\nbands it can be as large as 400 meV in tungsten-based\nTMDs[25, 26]. Originating from in-plane mirror sym-\nmetry breaking and atomic spin-orbit interactions, the\nIsing SOC pins electron spins near opposite K-valleys\nto opposite out-of-plane directions. Due to its special\nroles in extending valley lifetimes[8], integrating spin and\nvalley degrees of freedom[36, 37], and enhancing upper\ncritical \felds in Ising superconductors[33, 34, 38, 39],\nIsing SOCs have attracted extensive interests in stud-\nies of both valleytronics[10] and novel superconducting\nstates[35, 40{44] in TMD materials. In gated TMDsor polar TMDs[45, 46], Rashba-type SOCs[47] also arise\nnaturally[33, 48{51]. Despite its wide existence, the ef-\nfect of Rashba SOCs in TMDs has only been studied very\nlately, with focuses on superconducting states[33, 40] and\nspintronic applications[50, 52{54].\nIn this work, we show that the coexistence of Ising\nand Rashba SOCs in gated/polar TMDs results in novel\nvalley-contrasting Berry curvatures and a special type\nof valley Hall e\u000bect, which we call spin-orbit coupling\ninduced valley Hall e\u000bects (SVHEs). In contrast to con-\nventional Berry curvatures due to inversion-asymmetric\nhybridization of di\u000berent d-orbitals[8], the new type of\nBerry curvatures originates from inversion-asymmetric\nspin-orbit interactions. To distinguish their physical ori-\ngins, we refer to the Berry curvature induced by SOCs as\nspin-type Berry curvatures, and the conventional Berry\ncurvatures/valley Hall e\u000bects from orbital degrees of free-\ndom as orbital-type Berry curvatures/orbital VHEs. Im-\nportantly, under experimentally accessible gating[33, 38],\nspin-type Berry curvatures near the conduction band\nedge can reach nearly ten times of its orbital counterpart.\nThus, in gated or polar TMDs the SVHE can dominate\nover the orbital VHE, which signi\fcantly enhances the\nvalley Hall e\u000bects in a wide class of TMDs and enriches\nthe valley Hall phenomena in 2D Dirac materials. In ad-\ndition, the SVHE proposed in this study provides a novel\nscheme to manipulate valley degrees of freedom of TMD\nmaterials.\nResults\nMassive Dirac Hamiltonian and spin-type Berry\ncurvatures from Ising and Rashba SOCs\nTo illustrate the spin-type Berry curvature and spin-\norbit coupling induced valley Hall e\u000bect(SVHE) in\nTMDs, we consider gated monolayer MoS 2as an ex-\nample throughout this section, but the predicted e\u000bects\ngenerally exists in the whole class of gated TMDs orarXiv:1712.02942v2 [cond-mat.mes-hall] 1 Mar 20192\nK -K \n(a) (b) \n(c) (d) \nx y z \nGated TMD \n𝐄𝐆 \nE \n𝛼soc (𝑚𝑒𝑉 ∙Å) \n-0.4 -0.2 00.2 0.404080120\n |𝛀spinc,±| \n|𝛀orbc| \n𝑘𝑥𝑎 (Å2) 𝛀spinc,− 𝛀spinc,− \n0 10 20 30050100150200\n \n|𝛀𝐬𝐩𝐢𝐧𝐜,±| (Å2) \n-K K \nFIG. 1: Spin-orbit coupling induced valley Hall e\u000bects. (a)\nSchematics for the Ising spin-orbit coupling (SOC) (orange\narrows), the Rashba SOC (golden arrows) and the spin-type\nBerry curvatures \nc;\u0000\nspin(red/blue arrows) in the lower spin\nbands represented by red/blue pockets above K/\u0000K-points.\n(b) Valley Hall e\u000bects due to \nc\nspin. White arrows indicate\nout-of-plane gating \felds/electric polarization labelled by EG.\n(c) Magnitudes of spin-type Berry curvature j\nc;\u0006\nspinjnear the\nconduction band edge (red solid curve) and orbital-type Berry\ncurvaturej\nc\norbj(black solid curve) near K-points. Rashba\ncoupling strength is set to be \u000bc\nso= 21:4 meV\u0001\u0017Aaccording to\n\u000bc\nsokF\u00193 meV[33], comparable to 2 j\fc\nsoj= 3 meV[31, 32].\nParameters forj\nc\norbjare set to be: \u0001 = 0 :83 eV,VF=\n3:51 eV\u0001\u0017A[8]. Clearly,j\nc;\u0006\nspinjis nearly ten times of j\nc\norbj.\n(d)j\nc;\u0006\nspinjas a function of \u000bc\nsoat theK-points. Evidently,\nj\nc;\u0006\nspinjscales quadratically with \u000bc\nso.\npolar TMDs. In recent experiments, upon electrostatic\ngating the conduction band minima near the K-valleys\ncan be partially \flled[33, 38], where the electron bands\noriginate predominantly from the 4 dz2-orbitals of Mo-\natoms[30, 32]. Under the basis formed by spins of dz2-\nelectrons, the e\u000bective Hamiltonian near the K-valleys\nfor gated MoS 2can be written as[33, 40, 54]:\nHspin(k+\u000fK) =\u0018c\nk\u001b0+\u000bc\nso(ky\u001bx\u0000kx\u001by) +\u000f\fc\nso\u001bz:(1)\nHere,\u0018c\nk=jkj2\n2m\u0003\nc\u0000\u0016denotes the usual kinetic energy\nterm,m\u0003\ncis the e\u000bective mass of the electron band, \u0016\nis the chemical potential, k= (kx;ky) is the momentum\ndisplaced from K(\u0000K)-valleys,\u000f=\u0006is the valley index.\nThe\fc\nso-term refers to the Ising SOC which pins electron\nspins to out-of-plane directions (depicted by the orange\narrows in Fig.1a). The origin of Ising SOC is the breaking\nof an in-plane mirror symmetry (mirror plane perpendic-\nular to the 2D lattice plane) as well as the atomic SOC\nfrom the transition metal atoms. The \u000bc\nso-term describes\nthe Rashba SOC which pins electron spins in in-plane\ndirections with helical spin textures (indicated by the\ngolden arrows in Fig.1a). Rashba SOC will arise whenthe out-of-plane mirror symmetry (mirror plane paral-\nlel to the lattice plane) is broken by gating or by lattice\nstructure (as in the case of polar TMDs [45, 46]). Clearly,\nHspinhas the form of a massive Dirac Hamiltonian (by\nneglecting the kinetic term which has no contribution to\nBerry curvatures), and the Ising SOC plays the role of a\nvalley-contrasting Dirac mass, which is on the order of a\nfew to tens of meVs[32].\nImportantly, the Pauli matrices \u001b= (\u001bx;\u001by;\u001bz) in\nEq.1 act on spin degrees of freedom. This stands in con-\ntrast to the massive Dirac Hamiltonian in Refs.[8]:\nHorb(k+\u000fK) =VF(\u000fkx\u001cx+ky\u001cy) + \u0001\u001cz: (2)\nwhere the Pauli matrices \u001c= (\u001cx;\u001cy;\u001cz) act on the sub-\nspace formed by di\u000berent d-orbitals. The VF-term results\nfrom electron hopping, and the Dirac mass \u0001 is generated\nby the large band gap ( \u00182\u0001) on the order of 1 \u00002eVs\nin monolayer TMDs[8].\nAs shown in Fig.1a, Ising and Rashba SOCs re-\nsult in non-degenerate spin sub-bands near the con-\nduction band minimum. The energy spectra of up-\nper/lower spin-subbands are given by E\u000f\nc;\u0006(k) =\u0018c\nk\u0006p\n(\u000bcsok)2+ (\u000f\fcso)2. The Berry curvatures generated by\nSOCs in the lower spin-bands with energy E\u000f\nc;\u0000(k) is\ngiven by:\n\nc;\u0000\nspin(k+\u000fK) =(\u000bc\nso)2\u000f\fc\nso\n2[(\u000bcsok)2+ (\fcso)2]3=2: (3)\nNote that \nc;\u0000\nspinhas valley-dependent signs due to the\nvalley-contrasting Dirac mass generated by Ising SOCs.\nAs a result, under an in-plane electric \feld, \nc;\u0000\nspincan\ndrive electrons in the lower spin-bands at opposite valleys\nto \row in opposite transverse directions, which leads to\ntransverse valley currents(Fig.1b). To distinguish this\nnovel phenomenon from the intrinsic VHE in monolayer\nTMDs[8], we call this special type of VHE the spin-orbit\ncoupling induced valley Hall e\u000bect (SVHE) due to its\nphysical origin in spin degrees of freedom. Likewise, the\nBerry curvatures generated by Ising and Rashba SOCs\nare called spin-type Berry curvatures to distinguish it\nfrom the orbital-type Berry curvatures due to inversion-\nasymmetric mixing of di\u000berent d-orbitals[8].\nWe note that for the upper spin-band with energy\nE\u000f\nc;+(k), we have \nc;+\nspin=\u0000\nc;\u0000\nspin. Therefore, valley cur-\nrents from upper and lower spin-bands can partially can-\ncel each other when both of them are occupied. However,\nnon-zero valley currents can still be generated due to the\npopulation di\u000berence in the spin-split bands.\nBased on Eq.3, \nc;\u0006\nspinhas a formal similarity with its\norbital counterpart[8]:\n\nc\norb(k+\u000fK) =\u0000\u000fV2\nF\u0001\n2[(VFk)2+ \u00012]3=2: (4)\nHowever, we point out that \nc;\u0006\nspinoriginates from a very\ndi\u000berent physical mechanism from \nc\norband has impor-\ntant implications in valleytronic applications.3\nOn one hand, the magnitude of \nc\norbin TMDs is gen-\nerally small (\u001810\u0017A2) due to the large Dirac mass from\nthe band gap 2\u0001 \u00181\u00002 eV[8]. In contrast, for \nc;\u0006\nspin,\nthe Dirac mass \fc\nsois on the order of a few meVs near the\nconduction band edges[32]. For gated MoS 2, the Rashba\nenergy can reach \u000bc\nsokF\u00193 meV at the Fermi energy[33]\n(see Supplementary Note 1 for details), which is compa-\nrable to the energy-splitting 2 j\fc\nsoj\u00193 meV caused by\nIsing SOCs[32]. In this case, j\nc;\u0006\nspinjnear the conduction\nband minimum can be nearly ten times of j\nc\norbj(Fig.1c).\nTherefore, the SVHE is expected to generate pronounced\nvalley Hall signals in gated/polar TMDs.\nOn the other hand, the strength of \nc\norbis determined\nby parameters intrinsic to the material, thus can hardly\nbe tuned. However, \nc;\u0006\nspinhas a quadratic dependence on\nthe Rashba coupling strength \u000bc\nso(Eq.3), which can be\ncontrolled by external gating \felds. As shown in Fig.1d,\nj\nc;\u0006\nspinjcan be strongly enhanced by increasing \u000bc\nsowithin\nexperimentally accessible gating strength[33]. This sug-\ngests that the SVHE can serve as a promising scheme for\nelectrical control of valleys in TMD-based valleytronic\ndevices.\nWe note that the form of e\u000bective Hamiltonian in Eq.1\nalso applies to the K-valleys in the valence band (see Sup-\nplementary Note 2), thus SVHEs can also occur in the\nvalence band. Unfortunately, as we demonstrate below,\nthe spin-type Berry curvature is much weaker in the va-\nlence band due to the giant Ising SOC strength \fv\nso\u0018100\nmeV near the valence band edge[8, 25{30, 32].\nInterplay between spin-type and orbital-type Berry\ncurvatures\nIn real gated/polar TMDs, the spin-type Berry cur-\nvature \n spinalways coexist with the orbital-type Berry\ncurvature \n orb. In this section, we demonstrate the inter-\nplay between \n spinand \n orbnear theK-valleys (shown\nschematically in Fig.2a-b). Speci\fcally, using monolayer\nMoS 2as an example, we study the total Berry curvatures\nat theK-points based on a realistic tight-binding(TB)\nmodel[32] which takes both \n spinand \n orbinto account.\nThe TB Hamiltonian is presented in the Method section\nand detailed model parameters are presented in the Sup-\nplementary Note 3.\nFor simplicity, we focus on Berry curvatures at K=\n(4\u0019=3a;0), and the physics at \u0000Kfollows naturally\nfrom the requirement imposed by time-reversal symme-\ntry: \n(\u0000K) =\u0000\n(K).\nFirst, we study the conduction band case where the\nIsing SOC strength \fc\nsois small. In the absence of gating,\nthe total Berry curvatures \nc;\u0006\ntot:in both spin-subbands at\nK= (4\u0019=3a;0) consist of orbital-type contributions \nc;\u0006\norb\nonly, both pointing to the negative z-direction[8]. By\ngradually turning on the Rashba coupling strength, \nc;\u0006\nspin\ncome into play and change \nc;\u0006\ntot:dramatically. In partic-\n(Å2) \n(Å2) (a) (b) \n(c) (d) \n𝛼sov (𝑚𝑒𝑉 ∙Å) 𝛼soc (𝑚𝑒𝑉 ∙Å) \nK -K \nK -K 2𝛽sov 2𝛽soc \n𝛀𝐬𝐩𝐢𝐧𝐜, − \n𝛀𝐬𝐩𝐢𝐧𝐜,− \n 𝛀𝐬𝐩𝐢𝐧𝐜,+ \n 𝛀𝐬𝐩𝐢𝐧𝐜,+ 𝐜,+ \n𝐜,− 𝐯,+ \n𝐯,− \n𝛀𝐨𝐫𝐛𝐜 𝛀𝐨𝐫𝐛𝐯 𝛀𝐨𝐫𝐛𝐯 \n𝛀𝐬𝐩𝐢𝐧𝐯,− 𝛀𝐬𝐩𝐢𝐧𝐯,+ \n𝛀𝐬𝐩𝐢𝐧𝐯,+ 𝛀𝐬𝐩𝐢𝐧𝐯,− \n0 10 20 30050100\n \n𝛀𝐭𝐨𝐭.𝐯, −(𝑲) \n𝛀𝐭𝐨𝐭.𝐯, +(𝑲) \n0 10 20 30-1000100200\n 𝛀𝐭𝐨𝐭.𝐜, −(𝑲) \n𝛀𝐭𝐨𝐭.𝐜, +(𝑲) 𝛀𝐨𝐫𝐛𝐜 FIG. 2: Interplay between \n spinand \n orb. (a) The case\nnear the conduction band edge. (b) The case near the\nvalence band edge. (c) Total Berry curvatures \nc;\u0006\ntot:of\nupper(c;+)/lower(c;\u0000) spin-subbands at + K-point versus\n\u000bc\nsonear the conduction band edge. As \u000bc\nsoincreases, \nc;\u0006\nspin\nbecomes dominant and changes \nc;\u0006\ntot:dramatically. (d) To-\ntal Berry curvatures \nv;\u0006\ntot:of upper(v;+)/lower(v;\u0000) spin-\nsubbands at + K-point as a function of \u000bv\nsoat the valence\nband edge. Obviously, \nv;\u0006\ntot:are insensitive to \u000bv\nsoand remain\nclose to \nv;\u0006\norbat\u000bv\nso= 0.\nular, for the lower spin-subband, \nc;\u0000\nspinalso points to the\nnegativez-direction (Fig.2a). This is due to the fact that\n\fc\nso<0 in molybdenum(Mo)-based TMDs[31, 32], which\nleads to a negative value of \nc;\u0000\nspin(Eq.3). As a result,\n\nc;\u0000\ntot:keeps increasing its magnitude as \u000bc\nsoincreases (red\nsolid curve in Fig.2c). For the upper spin-subband, how-\never, \nc;+\nspin=\u0000\nc;\u0000\nspin, which is anti-parallel to its orbital\ncounterpart \nc;+\norb, thus they compete against each other\nas shown in Fig.2a. As \u000bc\nsogrows up, \nc;+\nspinbecomes com-\nparable to \nc;+\norb, resulting in a zero total Berry curvature\n\nc;+\ntot:= 0 at certain \u000bc\nso(indicated by the intersection be-\ntween the blue curve and zero in Fig.2c). As \u000bc\nsoincreases\nfurther, \nc;\u0006\nspindominates, leading to di\u000berent signs of to-\ntal Berry curvatures in upper and lower spin bands, with\n\nc;+\ntot:>0 and \nc;\u0000\ntot:<0.\nNotably, in tungsten(W)-based TMDs Ising SOCs in\nthe conduction band have a di\u000berent sign \fc\nso>0[31, 32].\nIn this case, \nc;\u0000\nspin competes with \nc;\u0000\norb, while \nc;+\nspin\naligns with \nc;+\norb. This is contrary to the behaviors in\nmolybdenum(Mo)-based materials. As a result, \nc;\u0000\ntot:in\nW-based materials can change its sign when \nc;\u0000\nspindom-\ninates, similar to \nc;+\ntot:in Mo-based case (blue curve in\nFig.2c). As we discuss in the next section, this can re-\nverse the direction of total valley currents. Similar plots\nas shown in Fig.2 for W-based TMDs are presented in4\nSupplementary Note 4.\nIn contrast to conduction bands, valence band edges\nin TMDs exhibit extremely strong Ising SOC with \fv\nso\u0018\n100\u0000200 meV[8, 25{30]. This leads to very weak\nspin-type Berry curvatures \nv;\u0006\nspin(shown schematically\nin Fig.2b). This is because electron spins near the va-\nlence band edges are strongly pinned by the Ising SOCs\nto the out-of-plane directions, and the Rashba SOCs due\nto gating or electric polarization cannot compete with\nIsing SOCs. As a result, the Berry phase acquired dur-\ning an adiabatic spin rotation driven by Rashb SOC \felds\nbecomes negligible. Therefore, in the valence band the\norbital-type contribution \nv;\u0006\norbgenerally dominates. As\nshown clearly in Fig.2d, \nv;\u0006\ntot:are almost insensitive to\n\u000bv\nsoand remain close to \nv;\u0006\norbat\u000bv\nso= 0.\nIt is worth noting that the behavior of total Berry cur-\nvatures in Fig.2(c)-(d) can be understood by consider-\ning spin-type and orbital-type contributions separately.\nThis is due to the fact that the total Berry curvature\n\nn\ntot:at theK-points for a given band ncan be writ-\nten as the algebraic sum of \nn\nspinand \nn\norb: \nn\ntot:(\u000fK) =\n\nn\nspin(\u000fK)+\nn\norb(\u000fK). Detailed derivations can be found\nin Supplementary Note 5.\nDetecting spin-orbit coupling induced valley Hall\ne\u000bects\nIn this section, we discuss how to detect unique exper-\nimental signatures of SVHEs in n-type monolayer TMDs\nusing Kerr e\u000bect measurements. In particular, we study\nthe cases of molybdenum(Mo)-based and tungsten(W)-\nbased TMDs separately.\nAs demonstrated in the previous section, for Mo-based\nmaterials the total Berry curvature in the lower spin-\nband \nc;\u0000\ntot:can be signi\fcantly enhanced by \nc;\u0000\nspinun-\nder gating(Fig.2c). Therefore, when only the lower spin-\nbands are \flled, the extra contribution from SVHEs can\nstrongly enhance the total valley Hall conductivity \u001bV\nxy\nin gated/polar TMDs, which is expected to far exceed\nthe intrinsic \u001bV\nxyfrom orbital VHEs.\nMoreover, when \nc\nspindominates, \nc;\u0000\ntot:has a di\u000berent\nsign (Fig.2c). When both spin-bands are \flled, valley\ncurrents from upper and lower spin bands partially can-\ncel each other, with \fnite valley currents generated from\ntheir population di\u000berence. In this case \u001bV\nxyis expected\nto increase at a lower rate as doping level increases. This\nbehavior is very di\u000berent from the orbital valley Hall ef-\nfect for electron-doped samples: since \nc;\u0006\norbhave the same\nsign[8](See Fig.2(c) at \u000bc\nso= 0), when both spin sub-\nbands are \flled, \u001bV\nxyis expected to increase at a higher\nrate as a function of doping level.\nTo study this unique signature of \u001bV\nxydue to SVHEs,\nwe calculate \u001bV\nxyforn-type monolayer MoS 2using the\ntight-binding model[32] presented in the Method section.\n(a) / \n0 2 400.0050.01\n 𝜎𝑥𝑦V(MoS 2) 𝜎𝑥𝑦V(Gated MoS 2) \n𝜇/|𝛽soc| \n𝜎𝑥𝑦V(WSeTe) \n𝜎𝑥𝑦V(WSe 2) \n𝜇/|𝛽soc| (c) (d) \nx y z \nGated MoS 2 \n Pristine MoS 2 E \n K \nK \nK \nK \nK \nK \nK \n-K \n-K \n-K \n-K \n-K \n-K \n-K E \n𝜃K 𝜃K (b) / \nx y z \nWSeTe Pristine WSe 2 E \n E \n𝜃K 𝜃K \nK \n-K -K -K \nK K \nK -K \nFIG. 3: Detecting spin-orbit coupling induced valley Hall ef-\nfects(SVHEs). (a) Total valley Hall conductivity \u001bV\nxyas a\nfunction of chemical potential \u0016for gated MoS 2(red curve)\nand pristine sample (black curve). The blue dashed line in-\ndicates the location where \u0016= 2j\fc\nsoj. (b) Polar Kerr ef-\nfect measurements to detect SVHEs in Mo-based transition-\nmetal dichalcogenides(TMDs). For Mo-based TMDs, SVHEs\nstrongly enhance \u001bV\nxyin the regime \u0016\u00182j\fc\nsojcomparing to\nthe intrinsic value. This creates a signi\fcant valley imbal-\nancenVand valley magnetization at the sample boundaries,\nwhich can be signi\fed by a large Kerr angle \u0012K. (c) Total\n\u001bV\nxyversus chemical potential \u0016for polar TMD WSeTe (red\ncurve) and pristine WSe 2(black curve). Clearly, in the regime\n\u0016<2j\fc\nsojthe sign of\u001bV\nxyin WSeTe is reversed due to SVHEs.\n(d) Schematics for polar Kerr experiments to detect SVHEs\nin tungsten-based polar TMD WSeTe. The reversed valley\ncurrent is signi\fed by the sign reversal of \u0012K.\nThe\u001bV\nxyfor electron-doped TMDs is given by (see Sup-\nplementary Note 6 for details):\n\u001bV\nxy=\u00002e2\n~Zd2k\n(2\u0019)2[fc;+(k)\nc;+\ntot:(k) +fc;\u0000(k)\nc;\u0000\ntot:(k)]:(5)\nHere, the integral is calculated near the K-point, and\nfc;\u0006(k) =f1+exp[(E\u000f=+\nc;\u0006(k)\u0000\u0016)=kBT]g\u00001are the Fermi\nfunctions associated with the upper/lower spin-bands\nnear the conduction band edge. In the limit T!0,\nthe calculated \u001bV\nxyas a function of chemical potential\n\u0016for gated (red solid curve) and pristine (black solid\ncurve) monolayer MoS 2are shown in Fig.3a. The chem-\nical potential \u0016is measured from the conduction band\nminimum.\nWhen\u0016<2j\fc\nsoj, only the lower spin-band is occupied,\ni.e.,fc;+(k) = 0. It is evident from Fig.3a that as \u0016\nincreases, the net \u001bV\nxyfor gated MoS 2(red solid curve in\nFig.3a) grows much more rapidly than the intrinsic \u001bV\nxy\n(black solid curve in Fig.3a). For \u0016>2j\fc\nsoj, the intrinsic\n\u001bV\nxystarts increasing at a slightly higher rate, while the\n\u001bV\nxyfor gated sample increases at a lower rate.5\nTo detect this distinctive \rattening behavior in the\n\u001bV\nxy\u0000\u0016curve due to SVHEs, we propose polar Kerr e\u000bect\nexperiments (Fig.3b) which can directly map out the spa-\ntial pro\fle of net magnetization in a 2D system[24, 55].\nIn the steady state, valley currents Jv=\u001bV\nxyE\u0002^zgener-\nated by the electric \feld E(green arrows in Fig.3b) are\nbalanced by valley relaxations at the sample boundaries,\nwhich establishes a \fnite valley imbalance nV/\u001bV\nxy\nnear the sample edges. Due to valley-contrasting Berry\ncurvatures, the valley imbalance nVinduces a nonzero\nout-of-plane orbital magnetization Mz/nV[1], which\ncan be measured by the Kerr rotation angle \u0012K, with\n\u0012K/nV/\u001bV\nxy[24]. Therefore, \u0012Kas a function of doping\nlevel for intrinsic/gated monolayer TMDs are expected to\nexhibit similar features as the \u001bV\nxy\u0000\u0016curves in Fig.3a. In\nprevious Kerr measurements on MoS 2, the Kerr angle due\nto valley imbalance generated by orbital VHE is roughly\n\u0012K\u001960\u0016rad when both subbands are \flled[24]. Accord-\ning to Fig.3a, the SVHE can enhance \u001bV\nxyby nearly 4-5\ntimes when \u0016>2j\fc\nsoj, hence we expect \u0012K\u0019200\u0000300\n\u0016rad at the same doping level for gated MoS 2.\nNow we discuss the distinctive signature of SVHE in\nn-type tungsten(W)-based TMDs. As we pointed out in\nthe last section, \nc;\u0000\nspincompetes with \nc;\u0000\norbin W-based\nmaterials due to \fc\nso>0. Therefore, when \nc;\u0000\nspindomi-\nnates, \nc;\u0000\ntot:has an opposite sign to \nc;\u0000\norb. When only the\nlower spin-band is \flled ( i.e.,\u0016<2j\fc\nsojandfc;+(k) = 0),\nthis changes the sign of \u001bV\nxy(Eq.5), and the direction of\nvalley currents is reversed.\nTo demonstrate the reversal of valley current direc-\ntions due to SVHEs in W-based TMDs, we compare the\n\u001bV\nxyof a pristine monolayer tungsten-diselenide(WSe 2)\nand a polar TMD tungsten-selenide-telluride(WSeTe)\nwhere strong band-splitting induced by Rashba SOCs\nis predicted[50]. Using \ftted values of \u000bc\nsoand\fc\nsofor\nWSeTe (details presented in Supplementary Note 4), we\ncalculate the \u001bV\nxy\u0000\u0016curves for WSe 2and WSeTe as\nshown in Fig.3c. Clearly, for \u0016 < 2j\fc\nsoj,\u001bV\nxyof WSeTe\nhas a di\u000berent sign from that of WSe 2. As a result, the\nvalley currents in WSe 2and WSeTe under applied elec-\ntric \feld \row in opposite transverse directions(Fig.3d).\nThis leads to opposite valley magnetization on the same\nboundaries, which can be signi\fed by the sign di\u000berence\nin\u0012Kcorrespondingly[24].\nWe note that the relation \u0012K/nVdiscussed above re-\nlies on the fact that the valley orbital magnetic moments\nremain almost una\u000bected by \n spin. This is explained in\ndetails in Supplementary Note 7.\nDiscussion\nWe discuss a few important aspects on SVHEs studied\nabove. First, the novel SVHE as well as its unique signa-\ntures studied above applies in general to the whole classof monolayer TMDs. In particular, for molybdenum-\nbased materials, strong Ising and Rashba SOC e\u000bects in\nthe conduction band, such as MoSe 2and MoTe 2[32, 50],\nhave sizable band-splitting of 20 \u000030 meV near the band\nedge and exhibit pronounced signals of SVHEs. Detailed\ncalculations of SVHEs in MoTe 2are presented in Suppl-\nmentary Note 8.\nSecond, a strong gating \feld is not necessarily re-\nquired to induce strong Rashba SOCs in TMDs. As\nwe mentioned above, in polar transition-metal dichalco-\ngenides MXY (M=Mo,W; X,Y=S, Se, Te and X 6=\nY)[45, 46, 50, 51], out-of-plane electric polarizations are\nbulit-in from intrinsic mirror symmetry breaking in the\ncrystal structure. Thus, Ising and Rashba SOCs nat-\nurally coexist in these polar TMD materials without\nany further experimental design. This is very di\u000ber-\nent from graphene-based devices where valley currents\nare generated by inversion breaking from substrates[5{7]\nor strains[56]. On the other hand, in heterostructures\nformed by TMD and other materials, interfacial Rashba\nSOCs can also emerge. For example, strong Rashba\nSOC has been reported recently in graphene/TMD hy-\nbrid structures[57]. In the cases mentioned above, one\ncan use moderate gating to tune the Fermi level in the\nrange\u0016\u00182j\fc\nsojand study the unique \u001bV\nxy\u0000\u0016curve due\nto SVHEs (Fig.3).\nThird, the Berry phase in Eq.5 for one K valley can also\nbe generated in two-dimensional Rashba systems with\nlarge perpendicular magnetic \feld if orbital e\u000bects are\nignored[58]. However, to generate a Zeeman splitting of\na few meVs, an external magnetic \feld on the order of\n100 Tesla is needed[33]. In such a strong magnetic \feld,\nthe orbital e\u000bects cannot be ignored. Therefore, the fam-\nily of TMDs with large Ising SOCs is very unique for\ndemonstrating the novel SVHE.\nLastly, due to Ising SOC, valley Hall e\u000bects are generi-\ncally accompanied by spin Hall e\u000bects in TMDs[8], which\ncan also establish \fnite out-of-plane spin imbalance near\nthe edges and contributes to polar Kerr e\u000bects. However,\nspin magnetic moments .\u0016B(\u0016B: Bohr magneton) are\ngenerally small compared to the orbital valley magnetic\nmoments\u00183\u00004\u0016Bin TMDs. Thus, polar Kerr e\u000bects\nare expected to be dominated by orbital magnetization.\nMethods: Tight-binding Hamiltonian\nIn the Bloch basis of the following d-orbitals\nfjdz2;\"i;jdxy;\"i;j;dx2\u0000y2;\"i;jdz2;#i;jdxy;#i;jdx2\u0000y2;#ig,\nthe tight-binding Hamiltonian for gated/polar monolayer\nTMD is given by[32]:\nHTB(k) =HTNN(k)\n\u001b0\u0000\u0016I6\u00026+1\n2\u0015Lz\n\u001bz(6)\n+HR(k) +Hc\nI(k):6\nwhere\nHTNN(k) =0\n@V0V1V2\nV\u0003\n1V11V12\nV\u0003\n2V\u0003\n12V221\nA; Lz=0\n@0 0 0\n0 0\u00002i\n0 2i01\nA(7)\nrefer to the spin-independent term and the atomic\nspin-orbit coupling term, respectively. \u0016is the chemi-\ncal potential, and I6\u00026is the 6\u00026 identity matrix. The\nRashba SOC term is given by\nHR(k) = diag[2\u000b0;2\u000b2;2\u000b2]\n(fy(k)\u001bx\u0000fx(k)\u001by):(8)\nAnd the Ising SOC in the conduction band is given by\nHc\nI(k) = diag[\f(k);0;0]\n\u001bz: (9)\nDetails of the matrix elements above can be found in\nSupplementary Note 3.\nData availability\nThe data supporting the \fndings of this study are\navailable within the paper, its Supplementary Informa-\ntion \fles, and from the corresponding author upon rea-\nsonable request.\nAcknowledgement\nThe authors thank Noah F. Yuan, C. Xiao and K. F.\nMak for illuminating discussions. Y.T. was supported by\nGrant-in-Aid for Scienti\fc Research on Innovative Ar-\neas \\Topological Materials Science\" (KAKENHI Grants\nNo.JP15H05851, No.JP15H05853, and No.JP15K21717),\nand for Challenging Exploratory Research (KAKENHI\nGrant No.JP15K13498). K.T.L acknowledges the sup-\nport of Croucher Foundation, Dr. Tai-chin Lo Founda-\ntion and HKRGC through 16309718, 16307117, 16324216\nand C6026-16W.\nAuthor contributions\nK.T.L conceived the ideas. B.T.Z. and K.T.L. pre-\npared the manuscript. B.T.Z. and K.T. carried out the\ntheoretical calculations. B.T.Z., K.T., Y.K. and Y.T.\ncontributed to the analysis and interpretation of the re-\nsults. All the authors discussed the results and com-\nmented on the manuscript.\nAdditional information\nCompeting interests: The authors declare no compet-\ning interests.\u0003These authors contributed equally to this work.\nyCorresponding author.\nphlaw@ust.hk\n[1] Xiao, D., Chang, M. and Niu, Q. Berry phase e\u000bects\non electronic properties. Rev. Mod. Phys. 82, 1959-2007\n(2010).\n[2] Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H.\nand Ong, N. P. Anomalous Hall e\u000bect Rev. Mod. Phys.\n82, 1539 (2010).\n[3] Xiao, D., Yao, W. and Niu, Q. Valley-Contrasting\nPhysics in Graphene: Magnetic Moment and Topolog-\nical Transport, Phys. Rev. Lett. 99, 236809 (2007).\n[4] Yamamoto, M., Shimazaki, Y., Borzenets, I.V., Tarucha,\nS. Valley Hall E\u000bect in Two-Dimensional Hexagonal Lat-\ntices. J. Phys. Soc. Jpn. 84, 121006 (2015).\n[5] Gorbachev, R. V. et al. Detecting topological currents in\ngraphene superlattices. Science 346, 448-451 (2014).\n[6] Sui, M. et al. Gate-tunable topological valley transport\nin bilayer graphene. Nat. Phys. 11, 1027-1031 (2015).\n[7] Shimazaki, Y. et al. Generation and detection of pure\nvalley current by electrically induced Berry curvature in\nbilayer graphene. Nat. Phys. 11, 1032-1036 (2015).\n[8] Xiao, D., Liu, G., Feng, W., Xu, X., Yao, W. Coupled\nspin and valley physics in monolayers of MoS 2and other\ngroup-VI dichalcogenides. Phys. Rev. Lett. 108, 196802\n(2012).\n[9] Mattheiss, L. F. Band Structures of Transition-Metal-\nDichalcogenide Layer Compounds, Phys. Rev. B 8, 3719\n(1973).\n[10] Xu, X., Yao, W., Xiao, D. and Heinz, T. F. Spin and\npseudospins in layered transition metal dichalcogenides.\nNat. Phys. 10, 343-350 (2014).\n[11] Zhang, Y.J., Oka, T., Suzuki, R., Ye, J.T. and Iwasa,\nY. Electrically switchable chiral lightemitting transistor.\nScience 344, 725-728 (2014).\n[12] Mak, K.F., McGill, K.L., Park, J. and McEuen, P.L. The\nvalley Hall e\u000bect in MoS 2transistors. Science 344, 1489-\n1492 (2014).\n[13] Kim, J. et al. Ultrafast generation of pseudo-magnetic\n\feld for valley excitons in WSe 2monolayers. Science 346,\n1205-1208 (2014).\n[14] Sie, E.J. et al. Valley-selective optical Stark e\u000bect in\nmonolayer WS2. Nat. Mater. 14, 290-294 (2015).\n[15] Ubrig, N. et al. Microscopic Origin of the Valley Hall\nE\u000bect in Transition Metal Dichalcogenides Revealed\nby Wavelength-Dependent Mapping. Nano Lett. ,179,\n57195725 (2017).\n[16] Onga, M. et al. Exciton Hall e\u000bect in monolayer MoS 2.\nNat. Mater. 16, 11931197 (2017).\n[17] Cao, T. et al. Valley-selective circular dichroism of mono-\nlayer molybdenum disulphide. Nature Communications ,\n3, 887 (2012).\n[18] Li, Y. L. et al. Valley splitting and polarization by the\nZeeman e\u000bect in monolayer MoSe 2.Phys. Rev. Lett. 113,\n266804 (2014).\n[19] MacNeill, D. et al. Breaking of valley degeneracy by mag-\nnetic \feld in monolayer MoSe 2.Phys. Rev. Lett. 114,\n037401 (2015).\n[20] Aivazian, G. et al. Magnetic control of valley pseudospin\nin monolayer WSe 2.Nat. Phys. 11, 148-152 (2015).\n[21] Srivastava, A. et al. Valley Zeeman e\u000bect in elementary7\noptical excitations of monolayer WSe 2.Nat. Phys. 11,\n141-147 (2015).\n[22] Tong, W.-Y., Gong, S.-J., Wan, X. and Duan, C.-G.,\nConcepts of ferrovalley material and anomalous valley\nHall e\u000bect. Nature Communications 7, 13612 (2016).\n[23] Ye, Y. et al. Electrical generation and control of the valley\ncarriers in a monolayer transition metal dichalcogenide.\nNat. Nano-tech. 11, 598-602 (2016).\n[24] Lee, J., Mak, K. F. & Shan, J. Electrical control of the\nvalley Hall e\u000bect in bilayer MoS2 transistors. Nat. Nano-\ntech.11, 421-425 (2016).\n[25] Yuan, H. T. et al. Zeeman-type spin splitting controlled\nby an electric \feld, Nat. Phys. 9, 563-569 (2013).\n[26] Zeng, H. L., Liu, G.-B., Dai, J. F., Yan, Y. J., Zhu, B.\nR., He, R. C., Xie, L., Xu, S. J., Chen, X. H., Yao, W. &\nCui, X. D. Optical signature of symmetry variations and\nspin-valley coupling in atomically thin tungsten dichalco-\ngenides, Sci. Rep. 3, 1608 (2013).\n[27] Zhu, Z. Y., Cheng, Y. C. and Schwingenschlogl U. Gi-\nant spin-orbit-induced spin splitting in two-dimensional\ntransition-metal dichalcogenide semiconductors. Phys.\nRev. B 84, 153402 (2011).\n[28] Kormanyos, A., Zolyomi, V., Drummond, N. D., Rakyta,\nP., Burkard G. and Falko, V. I. Monolayer MoS 2: Trigo-\nnal warping, the \u0000-valley, and spin-orbit coupling e\u000bects.\nPhys. Rev. B 88, 045416 (2013).\n[29] Zahid, F., Liu, L., Zhu, Y., Wang, J. and Guo, H. A\nGeneric Tight-Binding Model for Monolayer, Bilayer and\nBulk MoS 2.AIP Adv. 3, 052111 (2013).\n[30] Cappelluti, E., Roldan, R., Silva-Guillen, J. A., Orde-\njon, P. and Guinea, F. Tight-binding model and direct-\ngap/indirect-gap transition in single-layer and multilayer\nMoS 2.Phys. Rev. B 88, 075409 (2013).\n[31] Ko\u0013 smider, K., Gonz\u0013 alez J. W. and Fern\u0013 andez-Rossier,\nJ. Large spin splitting in the conduction band of transi-\ntion metal dichalcogenide monolayers, Phys. Rev. B 88,\n245436 (2013).\n[32] Liu, G.-B., Shan, W.-Y., Yao, Y., Yao, W. & Xiao, D.\nThree-band tight-binding model for monolayers of group-\nVIB transition metal dichalcogenides, Phys. Rev. B 88,\n085433 (2013).\n[33] Lu, J. M., Zheliuk, O., Leermakers, I., Yuan, N. F. Q.,\nZeitler, U., Law, K. T. and Ye, J. T. Evidence for two-\ndimensional Ising superconductivity in gated MoS 2,Sci-\nence350, 1353-1357 (2015).\n[34] Xi, X., Wang, Z., Zhao, W., Park, J.-H., Law, K. T.,\nBerger, H., Forro, L., Shan, J. and Mak, K. F. Ising pair-\ning in superconducting NbSe 2atomic layers, Nat. Phys.\n12, 139-143 (2016).\n[35] Zhou, B. T., Yuan, Noah. F. Q., Jiang, H.-L. and Law,\nK. T. Ising Superconductivity and Majorana Fermions\nin Transition Metal Dichalcogenides, Phys. Rev. B 93,\n180501(R) (2016).\n[36] Wang, G. et al. Spin-orbit engineering in transition metal\ndichalcogenide alloy monolayers. Nat. Commun. 6:10110\ndoi: 10.1038/ncomms10110 (2015).\n[37] Xie, L. and Cui, X. Manipulating spin-polarized\nphotocurrents in 2D transition metal dichalcogenides.\nProc. Natl. Acad. Sci. USA 113(14):3746-50. doi:\n10.1073/pnas.1523012113. (2016).\n[38] Saito, Y., Nakamura, Y., Bahramy, M. S., Kohama, Y.,\nYe, J., Kasahara, Y., Nakagawa, Y., Onga, M., Toku-\nnaga, M., Nojima, T., Yanase, Y. and Iwasa, Y. Super-\nconductivity protected by spin-valley locking in ion-gatedMoS 2,Nat. Phys. 12, 144-149 (2016).\n[39] Ili, S., Meyer, J. S. and Houzet, M. Enhancement of\nthe Upper Critical Field in Disordered Transition Metal\nDichalcogenide Monolayers. Phys. Rev. Lett. 119, 117001\n(2017).\n[40] Yuan, N. F. Q., Mak, K. F. and Law, K. T. Possible\nTopological Superconducting Phases of MoS 2,Phys. Rev.\nLett.113, 097001 (2014).\n[41] He, W.-Y., Zhou, B. T., He, J. J., Zhang, T. and Law,\nK. T. Magnetic Field Driven Nodal Topological Super-\nconductivity in Monolayer Transition Metal Dichalco-\ngenides, Communications Physics ,1, 40 (2018).\n[42] Sosenko, E., Zhang J. and Aji, V. Unconventional super-\nconductivity and anomalous response in hole-doped tran-\nsition metal dichalcogenides, Phys. Rev. B 95, 144508\n(2017).\n[43] Hsu, Y.-T., Vaezi, A., Fischer, M. H. and Kim, E.-\nA. Topological superconductivity in monolayer transi-\ntion metal dichalcogenides. Nat. Commun. 8, 14985 doi:\n10.1038/ncomms14985 (2017).\n[44] Liu, C.-X. Unconventional Superconductivity in Bilayer\nTransition Metal Dichalcogenides. Phys. Rev. Lett. 118,\n087001 (2017).\n[45] Lu A.-Y. et al. Janus monolayers of transition metal\ndichalcogenides. Nat. Nano-tech. 12, 744749 (2017).\n[46] Zhang, J. et al. Janus Monolayer Transition-Metal\nDichalcogenides. ACS Nano ,11(8), 81928198 (2017).\n[47] Rashba, E. I. Symmetry of bands in wurzite-type crys-\ntals. 1. Symmetry of bands disregarding spin-orbit inter-\naction, Sov. Phys. Solid. State 1, 368 (1959).\n[48] Ochoa, H. and Roldan, R. Spin-orbit-mediated spin re-\nlaxation in monolayer MoS 2.Phys. Rev. B 87, 245421\n(2013).\n[49] Kormanyos, A., Zolyomi, V., Drummond, N. D. and\nBurkard, G. Spin-Orbit Coupling, Quantum Dots, and\nQubits in Monolayer Transition Metal Dichalcogenides.\nPhys. Rev. X 4, 011034 (2014).\n[50] Yao, Q.-F. et al. Manipulation of the large Rashba\nspin splitting in polar two-dimensional transition-metal\ndichalcogenides. Phys. Rev. B 95, 165401 (2017).\n[51] Cheng, Y. C., Zhu, Z. Y., Tahir, M. and Schwingen-\nschlogl, U. Europhys. Lett. 102, 57001 (2013).\n[52] Klinovaja J. and Loss, D. Spintronics in MoS 2monolayer\nquantum wires, Phys. Rev. B 88, 075404 (2013).\n[53] Shao, Q. et al. Strong Rashba-Edelstein E\u000bect-Induced\nSpinOrbit Torques in Monolayer Transition Metal\nDichalcogenide/Ferromagnet Bilayers. Nano Lett. 16\n(12), 7514-7520 (2016).\n[54] Taguchi, K., Zhou, B. T., Kawaguchi, Y. , Tanaka, Y.\nand Law, K. T. Valley Edelstein e\u000bect in monolayer tran-\nsition metal dichalcogenides, Phys. Rev. B 98, 035435\n(2017).\n[55] Lee, J., Wang, Z., Xie, H., Mak, K. F. and Shan, J. Val-\nley Magnetoelectricity in Single-Layer MoS 2,Nat. Mat. ,\ndoi:10.1038/nmat4931 (2017).\n[56] Firoz Islam, S. K. and Benjamin, C. A scheme to re-\nalize the quantum spin-valley Hall e\u000bect in monolayer\ngraphene, Carbon 110, 304 (2016).\n[57] Yang, B. et al. Strong electron-hole symmetric Rashba\nspin-orbit coupling in graphene/monolayer transition\nmetal dichalcogenide heterostructures. Phys. Rev. B 96,\n041409(R)(2017).\n[58] Culcer, D., MacDonald, A. and Niu, Q. Anomalous Hall\ne\u000bect in paramagnetic two-dimensional systems. Phys.8\nRev. B 68, 045327 (2003)." }, { "title": "1802.00132v1.Spin_Seebeck_effect_and_thermal_spin_galvanic_effect_in_Ni80Fe20_p_Si_bilayers.pdf", "content": " 1 Spin Seebeck effect and thermal spin galvanic effect in Ni80Fe20/p-Si bilayers Ravindra G. Bhardwaj1, Paul C Lou1 and Sandeep Kumar1,2* 1 Department of Mechanical Engineering, University of California, Riverside, CA 92521, USA 2 Materials Science and Engineering Program, University of California, Riverside, CA 92521, USA 2 Abstract The development of spintronics and spin-caloritronics devices need efficient generation, detection and manipulation of spin current. The thermal spin current from spin-Seebeck effect has been reported to be more energy efficient than the electrical spin injection methods. But, spin detection has been the one of the bottlenecks since metals with large spin-orbit coupling is an essential requirement. In this work, we report an efficient thermal generation and interfacial detection of spin current. We measured a spin-Seebeck effect in Ni80Fe20 (25 nm)/p-Si (50 nm) (polycrystalline) bilayers without heavy metal spin detector. The p-Si, having the centosymmetric crystal structure, has insignificant intrinsic spin-orbit coupling leading to negligible spin-charge conversion. We report a giant inverse spin-Hall effect, essential for detection of spin-Seebeck effect, in the Ni80Fe20/p-Si bilayer structure, which originates from Rashba spin orbit coupling due to structure inversion asymmetry at the interface. In addition, the thermal spin pumping in p-Si leads to spin current from p-Si to Ni80Fe20 layer due to thermal spin galvanic effect and spin-Hall effect causing spin-orbit torques. The thermal spin-orbit torques leads to collapse of magnetic hysteresis of 25 nm thick Ni80Fe20 layer. The thermal spin-orbit torques can be used for efficient magnetic switching for memory applications. These scientific breakthroughs may give impetus to the silicon spintronics and spin-caloritronics devices. 3 The performance of thermoelectric semiconductors, especially commercially available, has been stagnant for years. The materials that show increase in thermoelectric performance require complex and scarce (rare earth) elements. An innovative approach to improving thermoelectric energy storage and conversion is the spin dependent thermoelectric energy conversion using spin Seebeck effect (SSE), anomalous Nernst effect (ANE) and spin Nernst effect (SNE), which will bring efficiencies because pure spin current, as opposed to charge current, is believed to be dissipationless1. The discovery of Spin Seebeck effect (SSE) by Uchida et. al. has led to significant progress in ongoing research on generation of pure spin current, a precession of spins or flow of electrons with opposite spins in opposite directions, over a large distance in spintronic devices due to applied temperature gradient in ferromagnetic (FM) materials2-4. The SSE can be an efficient way to produce low cost and large memory spintronics devices5. The SSE is observed in ferromagnetic metals3,6-11, semiconductors12-15, insulators16-22 and even in half metallic Heusler compounds23. In the spin caloritronics studies, homogenous temperature gradient as well as length scale dependent temperature gradient is established to study the interplay of spin degrees of freedom and temperature gradient in the magnetic structures22. There are two universal SSE device configuration, longitudinal spin Seebeck effect (LSSE) and transverse spin Seebeck effect (TSSE) in which in-plane external magnetic field and temperature gradient is applied in the plane of the sample to measure the SSE22. In LSSE11, a spin current is generated parallel to the temperature gradient as opposed to the spin current is perpendicular to the temperature gradient in TSSE4,5,21. The spin current generated in a FM material is detected by inverse spin-Hall effect (ISHE) in a high spin orbit coupling metals (Pt, W and Ta) in contact with FM 3,5,21. The ISHE voltage 𝐸\"#$% generated perpendicularly to the magnetization 𝑀 is given by equation, 4 𝐸\"#$%=𝜃#$𝜌𝐽#×𝜎 (1) Where, 𝜃#$,𝜌,𝐽# and 𝜎 are spin-Hall angle, electrical resistivity of paramagnetic metal, longitudinal spin current due to SSE, and spin polarization vector parallel to 𝑀 3,7. The thermoelectric energy conversion from spin current depends on efficient spin to charge conversion. Currently, the primary material for spin to charge conversion is Pt due to its large spin Hall angle, which inhibits the further scientific research in spin thermoelectric conversion behavior. The SSE is enhanced due to phonon drag24 and phonons drive the spin redistribution 13. The spin-phonon coupling can provide an able platform to engineer spin dependent thermoelectric conversion. To make the spin mediated thermoelectric energy conversion a reality, we need to discover earth abundant material/interfaces for giant SSE/ANE/SNE and efficient spin to charge conversion. In this work, we report the experimental measurement of giant SSE and thermal spin galvanic effect (SGE25-27) in the Ni80Fe20/p-Si (poly) bilayers. The spin-phonon coupling in p-Si leads to giant enhancement in SSE at the Ni80Fe20/p-Si (poly) bilayer and SHE in p-Si leads to giant spin-orbit torque (SOT), which can be used for SOT based memory applications. We developed an experimental setup to measure the longitudinal SSE. In the experimental setup, we use Pt heater to create the temperature gradient across the Ni80Fe20/p-Si (poly) bilayer specimen as shown in the Figure 1-a. This temperature gradient will lead to spin current in the bilayer and will allow us to measure the spin mediated thermoelectric behavior. An AC bias across the Pt heater creates the temperature gradient. We measure the first harmonic and third harmonic response across the heater to quantify the temperature gradient between the heater and the Si substrate. The SSE, ANE and SNE are measured from the second harmonic response across the Ni80Fe20/p-Si bilayer specimen. We use the micro/nanofabrication techniques to make 5 the experimental setup as shown in Figure 1 b. To fabricate the experimental setup, we take a Si wafer and deposit 300 nm of silicon oxide using plasma enhanced chemical vapor deposition (PECVD). We, then, deposit the Ni80Fe20/p-Si (poly) bilayer (blue color) using the RF sputtering as shown in Figure 1 b. The p-Si target is B-doped having resistivity of 0.005-0.01 W-cm. We sputter 50 nm MgO (green color) to electrically isolate the heater and the specimen. We then deposit Ti (10 nm)/Pt (100 nm) (pink color), which acts as a heater. (Figure 1) The experimental measurement is carried inside a quantum design physical property measurement system (PPMS). For energy conversion applications, the thermoelectric behavior should be robust at higher temperatures. We applied a 20 mA-5 Hz of heating current across the outer two electrodes of Pt heater starting at 400 K. We then measured the second harmonic response as a function of applied magnetic field in z-direction and y-direction as shown in Figure 1 c. For the magnetic field in y-direction, the field is perpendicular to the direction of temperature gradient and we observe a large second harmonic response, which may be related to the ANE/SSE. But, we observe an equally large signal when the magnetic field is applied along z-direction (field parallel to the temperature gradient). The Ni80Fe20 thin films have in-plane magnetic easy axis and out of plane hard axis28,29, which are verified from the magnetoresistance measurement as shown in Supplementary Figure S1 and anisotropic magnetoresistance (AMR) in Supplementary Figure S2. The second harmonic response in z-direction is attributed to the hard axis magnetization. We, then, measured the second harmonic response as a function of heating power at 400 K as shown in Figure 1 d. We observe linear relationship between the heating power and the second harmonic response30 as expected. (Figure 2) 6 We then measured the second harmonic responses as a function of magnetic field (from 1000 Oe to -1000 Oe) and applied current of 15 mA to 50 mA at 300 K as shown in Figure 2 a-d. We observe linear second harmonic response (comprising of ANE and SSE) as a function of heating power. Surprisingly, we observe that the magnetic hysteresis of second harmonic response for field along z-direction collapses as the heating current is raised to 50 mA. At 30 mA of heating current at 300 K, we estimate the increase in temperature at heater to be ~20.84 K from the third harmonic measurement. This lead us to believe that the observed collapse of hysteresis for z-direction at 300 K cannot arise due to heating effect only since the collapse of hysteresis in z-direction is not observed at 400 K. This behavior indicates existence of additional spin current from p-Si (Poly) to Ni80Fe20 layer. This additional spin current leads to spin-orbit torque and resulting change in the hysteresis behavior. For the magnetization perpendicular to the plane of interface, ANE and SSE will not be observed since M||DT and Js||s respectively. In order to decouple the contributions of ANE, SSE and to discover the origin of SOT, we measured the second harmonic response for an applied magnetic field (2T) rotated in the xy, zy and zx-plane as shown in Figure 3 a, where temperature gradient is along the z-direction. We observe a sine behavior attributed to the SSE in the xy-rotation. The angular dependence in zx-plane is observed to be cosine and zy-plane shows combined sine and cosine behavior. These measurements led us to believe there is a second thermoelectric effect in the bilayer thin films that is giving rise to the cosine second harmonic response in addition to the out of plane magnetic field dependent behavior reported in Figure 2. This behavior is similar to the SGE response reported in Fe/GaAs structures due to Rashba effect25. The thermal SGE, in this study, will lead to charge current across the bilayer specimen causing the second harmonic response. We will like state that the ANE coefficient in Ni80Fe20 is extremely small (4.8 nV/K31). In 7 addition, the specimen size in our measurement is 160µm X 12 µm, which is relatively very small as compared to sample area for the ANE measurements reported31,32. The ordinary Nernst effect (ONE) is not considered in this study because the ONE does not give rise to the switching behavior observed in this work. These measurements lead to two challenges in the interpretation of the results. First, the SSE measurement requires inverse spin-Hall effect (ISHE) to convert the spin current into voltage. While the SHE has been reported in p-Si33,34 but the spin-Hall angle of p-Si is negligible and may not lead to observable signal. To address the first challenge, we hypothesize that the ISHE occurs due to Rashba spin orbit coupling at the Ni80Fe20/p-Si interface. The second challenge is to uncover the origin of SOT observed in this study. As stated earlier that cosine behavior is attributed to the thermal SGE, which causes the second harmonic response while the magnetization and temperature gradient are parallel to each other. But, this behavior should not lead to SOT observed in Figure 2. We propose a two-step process that will lead to SOT observed in this study. The first step is thermal SGE where tunneling of spin polarized electrons across the interface in z-direction, having polarization in z-direction as well, lead to charge current parallel to the interface in x-direction due to inverse Rashba-Edelstein effect, which can be written as: 𝐽./=𝜆#1%𝐽23,45 25,35 (2) where 𝜆#1% is “effective thickness” of spin orbit layer25,35. Since the spin current is a function of temperature gradient, this equation can be written as: 𝐽./∝𝜆#1%(𝑇9:−𝑇#<), (3) where 𝑇9:\tand\t𝑇#< are temperature of ferromagnetic layer and temperature of Si layer respectively. The Rashba potential leads to spin precession causing a projection of the 8 polarization in y-direction, which then leads to ISHE and charge current in x-direction25,35. The field like SOT acts along 𝑚×𝜎 and damping like torque acts along 𝑚×(𝑚×𝜎), where 𝑚 and 𝜎 are the unit vectors of magnetization and spin polarization respectively36. For 𝑚 acting in the z-direction, the spin polarization (𝜎) vector has to be in the plane of the thin film for the SOT. In the second step, the interfacial charge current leads to SHE due to Rashba spin-orbit coupling causing an inverse spin current from p-Si to Ni80Fe20 layer having spin polarization in the plane of the thin film. The charge to spin conversion relationship can be written as: 𝐽23,4C=𝜃#$%𝐽./, (4) where 𝜃#$% is spin-Hall angle. The spin current entering the Ni80Fe20 layer can be considered as magnetization entering and exiting the ferromagnetic layer, which will cause the spin orbit torque. The spin current causing the SOT can be related to the temperature gradient through the following approximate equation: 𝑆𝑂𝑇∝\t𝐽23,4C∝𝜃#$%𝜆#1%(𝑇9:−𝑇#<) (5) The SOT characterization requires application of electric current across the specimen and measurement of first and second harmonic Hall responses. In this study, the Ni80Fe20 thin film is two orders of magnitude more conducting than the p-Si (poly) layer. Hence, the SOT observed in this study is not quantifiable with current techniques since it is of thermal origin. But, the SOT leads to collapse of hysteresis in a 25 nm Ni80Fe20 thin film as compared to the few nanometer films used in the SOT studies37-40 and only earth abundant materials are used. While we propose that the second harmonic response for out of plane magnetic field is due to Rashba effect mediated thermal SGE but other mechanisms may also be present, which can lead to better understanding of the observed measurements. (Figure 3) 9 Now, we needed to quantify the LSSE at the Ni80Fe20/p-Si (poly) interface. The efficiency of converting spin current-voltage at interface of bilayer in a LSSE device is given by 41, 𝑆F##%=%GHIJ∇L=MGHIJNOPQRP∆L (6) Where, 𝑉\"#$% is the electric voltage measured due to ISHE by paramagnetic metal or normal metal (NM), 𝑡9: is thickness of FM material, 𝑤W: is the distance between electrical contact in NM and ∆𝑇 is the temperature gradient across the sample. For thin film structures, the temperature gradient is difficult to find out. We estimate the temperature gradient between heater and substrate using 3w method and temperature gradient across the specimen is estimated using finite element modelling (FEM) (COMSOL). The temperature gradient between heater and far field temperature using 3w method42 is given by, Δ𝑇=YMZ[\\]\"^_` (7) Where 𝑉ab is the third harmonic response, 𝑅dis the resistance as a function of temperature and 𝐼fg2 is the heating current. The measured 𝑅d is 0.07 W/K (Supplementary Figure S3). Using the 3w method, we calculated the temperature gradient at heater to be 4.98 K, 10.9 K and 20.84 K for 15 mA, 20 mA and 30 mA of heating current respectively. Using FEM, we estimated the temperature gradient across the specimen to be ~14.08 mK corresponding to 20 mA of heating current. For modeling the temperature gradient, we assumed the 𝜅ij#<=25\tW/mK 43,44 and 𝜅W 0 is the on-site attractive interaction between\nthe quasi-particles, h:::idenotes the thermal average, f(\u000f)\nis the Fermi-Dirac distribution, and we have inserted\nthe Bogoliubov transformation ci\u001c=P\nn[un\u001c(i)\rn+\nv\u0003\nn\u001c(i)\ry\nn], where\ry\nn(\rn) are the Bogoliubov quasi-\nparticle creation (destruction) operators, which repre-\nsent a complete set of energy eigenstates: H=Eg+P\nn\u000fn\ry\nn\rn.Egis the groundstate energy; the summa-\ntion runs over positive energy eigenstates with an energy\nsmaller than the cut-o\u000b energy \u0016 h!Dset by the Debye\nfrequency!D.\nThe Hamiltonian (7) is transformed to BdG Hamilto-\nnian by using the Bogoliubov transformation36, which is\nthen iteratively solved together with the self-consistency\ncondition (8)42until the Euclidean norm of the pair\npotential (k\u0001k=pP\nij\u0001ij2) reaches a relative error\non the order 10\u00005. In the following, the Hamiltonian\n(7) is scaled with the hopping energy ~tand the chem-\nical potential, the pairing strength, the Rashba (Dres-\nselhaus) SOC, the Zeeman splitting, the thermal energy\nkBT, and the Debye frequency are set to: \u0016=~t=\u00004,\nV=~t= 5, ~\u000bR(D)=~t= 0:5,h0=~t= 0:1,kBT=~t= 0:001,\nand \u0016h!D=~t= 2:0, respectively. In Eq. (7), we use open\nboundary conditions. The hopping and Rashba energies\nin the tight-binding Hamiltonian (7) are related to a cen-\ntral di\u000berence discretization of the corresponding con-\ntinuum model via the relationships ~t= \u0016h2=2ma2and\n~\u000bR(D)=~t=ma\u000bR(D)=\u0016h2, whereais the spacing between\nthe grid points and \u000bR(D)is the SOC parameter in the\ncontinuum model. The parameter values given above\nmodel a lightly hole-doped semiconductor in proximity\nto a conventional s-wave superconductor, in which the\n0.2 \n-0.2 0\n-0.1 0.1 0.3 0.5 Rashba SOC Dresselhaus SOC\n0.2 \n0\n-0.2 \n0.5 \n0.3 \n0.1 \n-0.1 \n××\n× ×××\n× ×a\nb\nc fed\nx\nxx\nxy y\ny yFIG. 2: (color online). (a) The equilibrium state for a sys-\ntem with Rashba SOC and a Zeeman splitting \feld along\nx. The color represents the phase \u001eiof the pair potential\n\u0001i=j\u0001ijexp(i\u001ei), while the black arrows illustrate the lo-\ncal spin density S(i) = (\u0016h=2)hcy\ni\u001bcii. (b) System (a) with\nan enforced superconducting phase di\u000berence of \u0019=2 between\ntwo of the sample edges. (c) Symmetry plot of the induced\nspin density for a Rashba system. The \fgure shows the stereo-\ngraphic projection of the C2vpoint group and the blue arrows\nillustrate the orientation of the induced spin density for a su-\npercurrent along di\u000berent crystallographic directions. (d)-(f)\nShow corresponding plots for the case with Dresselhaus SOC\nand a Zeeman splitting \feld along y. In (a)-(b) and (d)-(e),\nthe size of the system is 31 \u000227 grid points.\ne\u000bective mass is m= 0:6me(meis the electron mass),\nthe SOC is \u000bR(D)= 0:21 eV \u0017A, the Fermi energy is\nEF= 2:47 meV when measured from the bottom of the\nlowest subband, and the Fermi wavelength is \u0015F\u001820\nnm, which is much larger than the discretization con-\nstanta= 3 nm.43\nB. Results and discussion\nFirst, we study the equilibrium spin density S(i) =\n(\u0016h=2)hcy\ni\u001bciiof the superconducting condensate. We\nconsider the two cases with Rashba and Dresselhaus\nSOC separately. Fig. 2a shows the self-consistent solu-\ntion for a Rashba system with an exchange \feld along\nx. The black arrows represent the spin density, while5\nI/Imax\nP/Pmax\n0.20.2\n0.40.6 0.8 1.0 00.40.60.81.0\nFIG. 3: (color online). Current-phase relation and the in-\nduced spin-polarization of the Cooper pairs for the Rashba\nsystem in Figs. 2a,b. The lines represent piecewise polyno-\nmial \fts of the data points.\nthe color illustrates the phase \u001eiof the pair potential\n\u0001i=j\u0001ijexp(i\u001ei). The phase variation perpendicular\nto the exchange \feld (i.e., along y) is a signature of the\nhelical phase. We see that the condensate has a net spin\npolarization anti-parallel to the exchange \feld. This is\nalso the case for a system with Dresselhaus SOC of the\nform \u0011so= ~\u000bD\u001bzand an exchange \feld along y(Fig. 2d).\nNote that in this case, the pair potential has a phase vari-\nation parallel to the exchange \feld, which is in agreement\nwith Eq. (2) when \u0014ij/(\u001bz)ij.\nNext, we investigate the e\u000bects of a supercurrent. A\nsupercurrent is induced along the x-axis by enforcing the\npair potential to have a constant phase in a small region\nclose to each of the two boundaries along x. We set\nthe widths of these two regions to three lattice points.\nThus, the pair potential is solved self-consistently for the\nentire sample except for the two regions at the boundaries\nwhere the phase \u001eiis kept \fxed (however, the magnitude\nj\u0001ijis allowed to optimize itself). These two regions will\ntherefore act as sinks/sources for the supercurrent.\nFig. 2b,e shows the solution for the Rashba and Dres-\nselhaus systems with a phase di\u000berence of \u0019=2 between\nthe two boundaries. In both cases, the spin density is\ntilted away from the equilibrium value. In other words:\nthe supercurrent induces a spin-density Sind. A simi-\nlar inverse spin-galvanic e\u000bect has been theoretically pre-\ndicted for superconductors with Rashba SOC in the ab-\nsence of magnetization.44,45\nSindis solely an e\u000bect of the SOC, and its orientation\nis determined by the direction of the supercurrent rel-\native to the crystallographic axes. Fig. 2c,f shows the\nstereographic projection of the C2vpoint group, and the\nblue arrows illustrate the orientation of Sindfor di\u000berent\ndirections of the supercurrent ( r\u001e >0 along the di\u000ber-\nent directions). Generally, the supercurrent results in a\nspin density Sind;i/\u0014ij\u0003j. Via the exchange coupling,\nSindproduces a torque on the magnetization and is the\n0.1\n-0.10.0\n0.0 0.5 1.0 1.5 2.0\n×10-2 \n16 \n8\n0\n00.080.16812 16 ×10-2 \n00.08 0.16a\nb cFIG. 4: (color online). (a) A microscopic calculation of\nthe anisotropic part Fme(\u0012) =Fs(\u0012)\u0000Fs(0) of the super-\nconductor's free energy Fsfor di\u000berent directions of h=\nh0[cos(\u0012);sin(\u0012);0]. (b) The magnetoelectric anisotropy con-\nstantKme= (jFme(\u0019=2)j+jFme(3\u0019=2)j)=2 for di\u000berent values\nofh0. (c) The average energy gap for di\u000berent values of h0.\nIn all \fgures, the size of the system is 25 \u000223 grid points and\nthe phase di\u000berence between the two boundaries is \u001e= 0:8\u0019.\nThe squares represent the calculated values, while the line in\n(a) is a piecewise polynomial \ft. In (a) the free energy was\ncalculated for h0=~t= 0:1.\nphysical origin of the SOT \feld in Eq. (4): Hso/Sind.\nThe polarization of the condensate originates from\nspin-triplet correlations. Let gT+(i) =h~ci\"~ci\"i(gT\u0000(i) =\nh~ci#~ci#i) denote the amplitude for triplet pair correla-\ntions with spin up (down) along an arbitrary quan-\ntization axis, which is determined by the unitary ro-\ntation operator U\u001c\u001c0. Here, ~ci\u001c=U\u001c\u001c0ci\u001c0are the\nfermionic operators in the rotated frame. The quantity\nP=P\ni[jgT+(i)j2\u0000jgT\u0000(i)j2] represents a measure of the\nspin polarization of the Cooper pairs along the quanti-\nzation axis. In Fig. 3, we consider the Rashba system\nin Fig. 2a-b and plot Pand the supercurrent Ialongx\nas a function of the phase di\u000berence \u001ebetween the left\nand right boundaries. The spin quantization axis is along\ny. It is clear from Fig. 3 that Pis proportional to the\nsupercurrent. We obtain a similar relationship between\nthe current and Pfor the Dresselhaus system in Fig. 2d-\ne when the polarization is measured along x. Thus, we\nconclude that the underlying physical mechanism of the\nSOT \feld (4) is current-induced spin-triplet correlations.\nThe strength of the SOT \feld can be investigated by\nself-consistently calculating the free energy of the con-6\ndensate for di\u000berent directions of h=h0[cos(\u0012);sin(\u0012);0].\nHere,\u0012is the angle with the x-axis, which is parallel to\nthe direction of the supercurrent. The anisotropic part\nof the free energy is then a direct measure of the Lifshitz\ninvariant (2).\nWe consider a system with Rashba SOC. The free en-\nergy of an inhomogeneous superconductor is46\nFs=\u00001\n\fX\nnln\u0014\n2 cosh\u0012\f\u000fn\n2\u0013\u0015\n+1\nVZ\ndrj\u0001(r)j2;(9)\nwhere the sum is over the positive energy eigenstates and\n\f= 1=kBT.\nFig. 4a shows the anisotropic part Fme(\u0012) =Fs(\u0012)\u0000\nFs(0) of the free energy. The angular dependence of Fme\nfollows the functional form Fme\u0018(^z\u0002h)\u0001\u0003, which is\nconsistent with the Lifshitz invariant (2) when a current\nis applied along the x-axis (with extrema at \u0012=\u0019=2 and\n\u0012= 3\u0019=2). The di\u000berent extremum values at Fme(\u0019=2)\nandFme(3\u0019=2) is caused by a change in the momentum\ndensity due to the helical modulation (along the x-axis)\nof the order parameter \feld.\nThe e\u000bect of the Zeeman splitting h0on the SOT is\ntwofold. Firstly, it determines the coupling strength be-\ntween the spin system and the condensate and thus en-\nhances the magnetoelectric coupling Fme. Secondly, it\nsuppresses superconductivity and thus reduces the super-\ncurrent/momentum density. The competition between\nthese two counteracting e\u000bects implies that there ex-\nists an intrinsic limitation for the maximum achievable\nSOT. Fig. 4b shows the magnetoelectric anisotropy con-\nstantKme= (jFme(\u0019=2)j+jFme(3\u0019=2)j)=2. A maximum\nSOT is achieved for h0=~t\u00180:125 withKme\u00180:16~t=\n1:13 meV and corresponds the point where the Zeeman\nsplitting is comparable to the pair potential, i.e., h0\u0018\u0001.\nFor larger values of h0, the suppression of the supercon-\nductivity becomes stronger (Fig. 4c), which leads to a\nlowering of Kme.\nThe e\u000bective SOT \feld induced by the supercurrent is\nHso\u0018Kme=VMs, whereVis the volume of the ferro-\nmagnetic system. Assuming Ms= 70:8 e.m.u. cm\u00003,7\nV= 23\u000225a3, andKme= 1:13 meV, yields an SOT\n\feld on the order of Hso\u00180:16 mT. In the ferromag-\nnetic semiconductor (Ga,Mn)As, current-driven magne-\ntization switching has been observed for e\u000bective SOT\n\felds on the order of 0 :14\u00000:35 mT.2Therefore, it is\nreasonable to believe that the supercurrent-induced SOT\nis strong enough to manipulate the magnetization of the\nferromagnetically ordered spins.\nIV. SUMMARY\nIn summary, we have studied the magnetization dy-\nnamics of a two-dimensional lattice of spins in contactwith a conventional superconductor and have formulated\na phenomenological description of the coupled dynamics\nof the superconducting condensate and the magnetiza-\ntion. Interestingly, we found that supercurrents induce a\nreactive SOT \feld that originates from current-induced\nspin-triplet correlations and whose spatial orientation is\ndetermined by the symmetry of the SOC. Furthermore,\nwe showed that there exists an intrinsic limitation for\nthe maximum achievable SOT, which is determined by\nthe coupling strength between the condensate and the\nspin system. Based on material parameters for a prox-\nimitized hole-doped semiconductor, we estimated the in-\nduced SOT \feld to be on the order of 0 :16 mT.\nAppendix A: Expressions for spin-density, pair\ncorrelations and current density\nThe Hamiltonian (7) can be diagonalized by using the\nBogoliubov transformation36\nci\u001c(r) =X\nn\u0000\nun\u001c(i)\rn+v\u0003\nn\u001c(i)\ry\nn\u0001\n: (A1)\nHere,\ry\nnand\rnare the Bogoliubov quasi-particle cre-\nation and destruction operators, which satisfy fermionic\nanti-commutation relations and represent a complete set\nof energy eigenstates:\nH=Eg+X\nn\u000fn\ry\nn\rn: (A2)\nEgis the ground state energy, and the summation runs\nover positive energy eigenstates with an energy lower\nthan the cut-o\u000b energy set by the Debye frequency. The\nthermal averages of the Bogoliubov quasi-particle ex-\ncitations are given by h\ry\nn\rni=f(\u000fn), wheref(\u000f) =\n1=(exp(\f\u000f) + 1) is the Fermi-Dirac distribution. It is\nalso useful to introduce the distribution function of the\ncorresponding hole states: fh(\u000f) = 1\u0000f(\u000f).\nBy using the Bogoliubov transformation (A1), the spin\ndensity S(i) = (\u0016h=2)hcy\ni\u001bciican be expressed as\nS\u000b(i) =\u0016h\n2X\n\u001c\u001c0(\u001b\u000b)\u001c\u001c0\u001a\u001c\u001c0(i);\n\u001a\u001c\u001c0(i)\u0011X\nn[(u\u0003\nn\u001c(i)un\u001c0(i)\u0000vn\u001c(i)v\u0003\nn\u001c0(i))f(\u000fn)\n+vn\u001c(i)v\u0003\nn\u001c0(i)]:\nThe charge density \u001ai=qhniiat site iis given by the\nthermal average of the number operator ni=cy\nici, where\nqis the charge of the quasi-particles. An expression for\nthe current density js(i) is found from the Heisenberg\nequationdni=dt= (i=\u0016h)[H;n i], which yields7\n(js(i)))k=2q~t\n\u0016hX\nn\u001cIm [u\u0003\nn\u001c(i)Dkun\u001c(i)f(\u000fn) +vn\u001c(i)Dkv\u0003\nn\u001c(i)fh(\u000fn)] +\n2q\n\u0016hX\nn\u001c\u001c0h\nu\u0003\nn\u001c(i)Ak;\u001c\u001c0un\u001c0(i)f(\u000fn) +vn\u001c(i)Ak;\u001c\u001c0v\u0003\nn\u001c0(i)fh(\u000fn)i\n+2q\n\u0016hX\n\u001c\u001c0Imh\n\u0001ii\u001by;\u001c\u001c0hcy\ni\u001ccy\ni\u001c0ii\n:(A3)\nHere,Dkun\u001c(i) = [un\u001c(i+ak)\u0000un\u001c(i\u0000ak)]=2 and\nAk;\u001c\u001c0=\u0010\n\u001b\u0001\u0011so\u0001^di(i\u0000ak)\u0011\n\u001c\u001c0, where akis the lattice\nvector along k2fx;y;zg. Note that the last term van-\nishes when the pair potential satis\fes the self-consistency\ncondition. Otherwise, the term acts as a sink/source.\nEq. (A3) is used to calculate the current-phase relation,\nwhich is shown in Fig. 3 of this article.\nThe pair correlations are given by\nhci\u001cci\u001c0i=X\nn[(v\u0003\nn\u001c(i)un\u001c0(i)\u0000un\u001c(i)v\u0003\nn\u001c0(i))f(\u000fn)\n+un\u001c(i)v\u0003\nn\u001c0(i)]: (A4)These correlation functions can be expressed in an ar-\nbitrary reference frame by transforming the fermionic op-\nerators: ~ci\u001c=U\u001c\u001c0ci\u001c0.U\u001c\u001c0is the unitary rotation op-\nerator, which maps the z-axis to the quantization axis in\nthe new reference frame.\n1For reviews see, e.g., A. Brataas, A. D. Kent, and H. Ohno,\nNature Mat. 11, 372 (2012); D. C. Ralph and M. Stiles, J.\nMagn. Mat. 320, 1190 (2008).\n2A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y.\nLyanda-Geller and L. P. Rokhinson, Nat. Phys. 5, 656\n(2009).\n3I. M. Miron, G. Gaudin, S. Au\u000bret, B. Rodmacq, A.\nSchuhl, S. Pizzini, J. Vogel and P. Gambardella Nat. Mat.\n9, 230 (2010).\n4D. Fang, H. Kurebayashi, J. Wunderlich, K. Vyborny, L.\nP. Zarbo, R. P. Campion, A. Casiraghi, B. L. Gallagher,\nT. Jungwirth and A. J. Ferguson, Nat. Nanotech. 6, 413\n(2011).\n5X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, and J. Q.\nXiao, Nat. Commun. 4, 1799 (2013).\n6H. Kurebayashi et al. , Nat. Nanotech. 9, 211 (2014).\n7C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y.\nTserkovnyak, H. Kurebayashi, A. Brataas, and A. Fergu-\nson, Nat. Nanotech. 10, 50 (2015).\n8For reviews see, A. Brataas and K. M. D. Hals, Nat. Nan-\notech. 9, 86 (2014); P. Gambardella and I. M. Miron, Phil.\nTrans. R. Soc. A 369, 3175 (2011).\n9B. A. Bernevig and O. Vafek, Phys. Rev. B 72, 033203\n(2005).\n10A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n11I. Garate and A. H. MacDonald, Phys. Rev. B 80, 134403\n(2009).\n12K. M. D. Hals, A. Brataas, and Y. Tserkovnyak, Europhys.\nLett. 90, 47002 (2010).\n13D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86,\n014416 (2012).\n14E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406\n(2012).\n15X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201(2012).\n16F. Freimuth, S. Bl ugel, Y. Mokrousov, Phys. Rev. B 90,\n174423 (2014).\n17K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423\n(2013).\n18S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yaz-\ndani, Phys. Rev. B 88, 020407(R) (2013).\n19J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys.\nRev. Lett. 111, 186805 (2013).\n20M. M. Vazifeh and M. Franz, Phys. Rev. Lett. 111, 206802\n(2013).\n21B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202\n(2013).\n22S. Nadj-Perge et al. , Science 346, 602 (2014).\n23M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. B 82,\n134521 (2010).\n24J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,\nPhys. Rev. Lett. 104, 040502 (2010).\n25L. Mao and C. Zhang, Phys. Rev. B 82, 174506 (2010).\n26K. Bj ornson and A. M. Black-Scha\u000ber, Phys. Rev. B 88,\n024501 (2013).\n27J. Li, T. Neupert, Z. J. Wang, A. H. MacDonald, A. Yaz-\ndani, B. A. Bernevig, arXiv:1501.00999.\n28X. Waintal and P. W. Brouwer, Phys. Rev. B 65, 054407\n(2002).\n29E. Zhao and J. A. Sauls, Phys. Rev. B 78, 174511 (2008).\n30F. Konschelle and A. Buzdin, Phys. Rev. Lett. 102, 017001\n(2009).\n31J. Linder, A. Brataas, Z. Shomali, and M. Zareyan, Phys.\nRev. Lett. 109, 237206 (2012).\n32I. Kulagina and J. Linder, Phys. Rev. B 90, 054504 (2014).\n33K. Halterman, O. T. Valls, and C. T. Wu, Phys. Rev. B\n92, 174516 (2015).\n34For a review see J. Linder and J. W. A. Robinson, Nat.\nPhys. 11, 307 (2015) and references therein.8\n35L. P. Gor'kov and E. I. Rashba, Phys. Rev. Lett. 87,\n037004 (2001).\n36P. G. de Gennes, Superconductivity of metals and alloys\n(W. A. Benjamin, INC., New York, 1966).\n37For a symmetry transformation \u0013r=Rr, the tensorial\nform of an invariant polar tensor Tij:::k of rankkis de-\ntermined by the equations Tij:::k =Ri\u000bRj\f:::Rk\rT\u000b\f:::\r ,\nwhereas an invariant axial tensor is determined by Tij:::k =\njRjRi\u000bRj\f:::Rk\rT\u000b\f:::\r . Here,jRjrepresents the deter-\nminant of the matrix R, which is an element of the sys-\ntem's point group.\n38V. P. Mineev and K. V. Samokhin, Zh. Eksp. Teor. Fiz.\n105, 747 (1994); V. M. Edelstein, J. Phys.: Condens. Mat-\nter8, 339 (1996).\n39S. S. Pershoguba, K. Bj ornson, A. M. Black-Scha\u000ber, and\nA. V. Balatsky, Phys. Rev. Lett. 115, 116602 (2015); K.Bj ornson, S. S. Pershoguba, A. V. Balatsky, and A. M.\nBlack-Scha\u000ber, Phys. Rev. B 92, 214501 (2015).\n40D. F. Agterberg and R. P. Kaur, Phys. Rev. B 75,\n064511(2007); O. Dimitrova and M. V. Feigel'man, Phys.\nRev. B 76, 014522 (2007).\n41S. Hart et al., arXiv:1509.02940.\n42P. D. Sacramento, V. K. Dugaev, and V. R. Vieira, Phys.\nRev. B. 76, 014512 (2007).\n43H. L. Stormer, Z. Schlesinger, A. Chang, D. C. Tsui, A.\nC. Gossard, and W. Wiegmann, Phys. Rev. Lett. 51, 126\n(1983).\n44V. M. Edelstein, Phys. Rev. Lett. 75, 2004 (1995).\n45V. M. Edelstein, Phys. Rev. B 72, 172501 (2005).\n46I. Kosztin, S. Kos, M. Stone, and A. J. Leggett, Phys. Rev.\nB58, 9365 (1998)." }, { "title": "1504.04786v2.Hydrodynamics_of_Normal_Atomic_Gases_with_Spin_orbit_Coupling.pdf", "content": "arXiv:1504.04786v2 [cond-mat.quant-gas] 19 Jan 2016Hydrodynamics ofNormalAtomicGaseswithSpin-orbit Coupl ing\nYan-Hua Hou and Zhenhua Yu∗\nInstitute for Advanced Study, Tsinghua University, Beijin g 100084, China\nSuccessful realization of spin-orbit coupling in atomic ga ses by the NIST scheme opens the prospect of\nstudyingthe effectsof spin-orbit coupling onmany-body ph ysics inanunprecedentedly controllable way. Here\nwe derive the linearized hydrodynamic equations for the nor mal atomic gases of the spin-orbit coupling by the\nNIST scheme with zero detuning. We show that the hydrodynami cs of the system crucially depends on the\nmomentum susceptibilities which canbe modifiedby the spin- orbit coupling. We reveal the effects of the spin-\norbitcouplingonthesoundvelocities andthe dipolemode fr equency ofthegases byapplying ourformalismto\nthe ideal Fermigas. Wealsodiscuss the generalization of ou r results toother situations.\nThe persisting quest to simulate charged particles in solid state systems by neutral atoms1–3stimulated the pioneering ex-\nperimental achievement of realizing “spin-orbit” couplin g in atomic gases through the Raman process by the NIST group4. In\nthe NIST scheme, a small magneticfield in the zdirectionwas used to openthe degeneracyof atomichyperfine spins, andtwo\nRamanlasersaligninginthe xdirectionshedonagasofBoseatomsgaverisetoacouplingbi linearinmomentumand(pseudo-)\nspin of atoms in one direction. After a unitary transformati on5, the resulting single atom Hamiltonian has the form (we take\n/planckover2pi1= 1throughout)\nH0=(k+krσzˆx)2\n2m−δ\n2σz+Ω\n2σx, (1)\nwheremis the atomic mass, σiare the atomic (pseudo-) spin operators, kris the Raman laser wave vector, δis the detuning\ntunablebychangingthefrequencydifferencebetweenthetw oRamanlasers,and ΩistheRabifrequencyfortheRamanprocess.\nNote that the spin-obit coupling in Eq. ( 1) is different from the Rashba and the Dresselhaus forms; whe n the lasers are turned\noff, i.e.,Ω = 0, the coupling kxσzin the kinetic energy can be eliminated by a gauge transforma tion5. Only nonzero Ωgives\nrise to nontrivial spin-orbit coupling. Experimentally kris fixed by the wavevector of the Raman lasers and Ωcan be tuned by\nthelaser intensities. Directdiagonalizationof H0yieldsthesingleatomdispersions\nǫ±,k=k2+k2\nr\n2m±/radicalBigg/parenleftbiggkrkx\nm−δ\n2/parenrightbigg2\n+/parenleftbiggΩ\n2/parenrightbigg2\n, (2)\nwith+(−)standingfortheupper(lower)branch. Thesame schemewasl ater appliedtoFermigases6,7.\nThe ground state of a Bose gas subject to the spin-orbit coupl ing realized by the NIST scheme can be either the stripe, the\nmagnetic,orthe non-magneticstates dependingon thespin- orbitcouplingsandinteratomicinteractions4,5,8. Recently thefinite\ntemperaturephasediagramoftheBosegaswasdeterminedexp erimentallyforzerodetuning δ= 09. Asubsequentperturbative\ncalculation reproduced the correct trend of the thermal eff ects on the phase boundary between the stripe and the magneti c\nphases10.\nTheeffectsof thespin-orbitcouplingof the NIST schemeont he dynamicsofatomic gaseswere first experimentallyinvest i-\ngatedthroughthecollectivedipoleoscillationofaBose-E insteincondensateinaharmonictrap11. Intheabsenceofthespin-orbit\ncoupling,theatomicgashastheGalileaninvariance,which guaranteesthedipoleoscillationfrequency ωdequaltotheharmonic\ntrappingfrequency ω012. The dispersions( 2) given rise to by the NIST scheme apparentlybreak down the Ga lilean invariance,\nwhichindicatesthat ωdcan bedifferentfrom ω0. Nevertheless,since duringthe small dipoleoscillationt he Bose-Einstein con-\ndensate accesses mostlythe lowerbranchstates arounda min imumofǫ−,kat momentum k0, it is sufficient to approximatethe\ndispersion ǫ−,k≈ǫ−,k0+(k−k0)2/2m∗,whoseformrestorestheGalileaninvariancethoughthe“ef fectivemass” m∗depends\non the spin-orbit coupling. The experimentaldata for the di pole frequencyof the Bose-Einstein condensate with the spi n-orbit\ncouplingcanbemainlyexplainedbythe“effectivemass”app roximation ωd∼ω0/radicalbig\nm/m∗11.\nUptonow,thestudyofthecollectivemodesandthehydrodyna micsofthespin-obitcoupledatomicgasesmainlyfocuseson\ntheBose-Einsteincondensates,inwhichcasetheexistence ofasinglecondensatewavefunctiongreatlysimplifiesthet heoretical\ntreatment13–18. However, to describe the dynamics of Bose gases with a subst antial normal fraction and Fermi gases in the\npresenceofspin-orbitcouplingrequiresamoregeneralfra mework. Inthiswork,weconsideratomicgasesinthehydrody namic\nregimeandderivethelinearizedhydrodynamicequationsfo rthenormalatomicgaseswiththeNISTspin-orbitcouplingf orzero\ndetuning δ= 0. We showthat in the absence ofthe Galileaninvariance,the h ydrodynamicsof the system cruciallydependson\nthe momentumsusceptibilities, which can be modifiedby the s pin-orbitcoupling. We apply our general formalism to the id eal\n∗Correspondence to huazhenyu2000@gmail.com2\nFermi gases and reveal the effects of the spin-orbit couplin gon the sound velocities and the dipole oscillation frequen cyof the\ngases.\nResults\nLinearized Hydrodynamic Equations. The normal atomic gases with the NIST spin-orbit coupling fo r zero detuning δ= 0\nconserve the number of atoms N, the energy E, and the (pseudo-) momentum K[cf. Eq. (1)]. The three conservation laws in\ntheirdifferentialformsare\n∂ǫ\n∂t+∇·jǫ= 0, (3)\n∂g\n∂t+∇·Π= 0, (4)\n∂n\n∂t+∇·jn= 0. (5)\nHereǫ,gandnarethe localdensitiesofenergy,momentumandnumberofato msrespectively,and jǫ,Πandjnarethe energy\ncurrent, momentum current tensor and number current respec tively. We assume that the inter-particle collision is so fr equent\nthat the system is in the hydrodynamic regime. The density of any physical quantity Ocan be calculated by the local grand\ncanonicalensemblewiththedistribution exp[−β(H−µN−K·v)],whereβ,µandvaretheinverseoftemperature T(wetake\nkB= 1throughout),chemical potential and velocity fields which a re functions of position and time. The total Hamiltonian H\nincludesH0andinteratomicinteractions. Notethatduetothespin-orb itcoupling,thespindegreesoffreedomarenotconserved.\nSpin dynamicsin the presenceof spin-orbitcouplinghasbee n studied in the contextof bothelectron gases19and atomic Fermi\ngases20.\nWe focus on the dissipationless limit. The condition that th e entropy change of the total system is zero determines the\nconstitutiverelations21,22\njn=nv (6)\nΠab=Pδab+vagb (7)\njǫ= (ǫ+P)v, (8)\nwherePis the pressure and the subscript stands for the component in dex. If the atoms are subject to an additional external\npotentialUext(r), themomentumconservationequationbecomes\n∂gb\n∂t+/summationdisplay\na∇a(Pδab+vagb)+n∇bUext= 0. (9)\nThe absence of the Galilean invariance manifest through Eq. (2) is a key feature of the atomic gas with spin-orbit coupling\nengineeredbytheRamanprocessesdrivenbyexternallasers . FornormalatomicgaseswiththeGalileaninvariance,them omen-\ntumdensity gisalwaysequalto mnvforarbitraryvelocityfield v. However,thisequalitygenerallybreaksdowninthepresen ce\nofspin-orbitcoupling. TheconsequenceofthelackoftheGa lileaninvariancehasbeenshowntoaffectthesuperfluidity andthe\nbrightsolitonsinBose-Einsteincondensates23,24. Nevertheless,if weare interestedinsmall variationsof β,µandvawayfrom\ntheglobalequilibrium,we cankeeptothefirst orderoftheva riationsandhave\nga(β,µ,v) =χabvb, (10)\nǫ(β,µ,v) =1\n2/summationdisplay\na,bχabvavb+ǫ(β,µ,v= 0), (11)\nwherethemomentumsusceptibilityis\nχab=δga/δvb|v=0. (12)\nFor simplicity, we have assumed that g(β,µ,v= 0) = 0 , i.e., when the gas is not moving, its momentum is zero. This\nassumption is true for the NIST scheme with zero detuning. Ge neralization of our results to cases without this assumptio n is\nstraightforward. Apparently χabmust be symmetric. We choosethe coordinatesystem such that χab= ˜χaδab. Using Eqs. ( 10)\nand(11),we distill thelinearizeddissipationlesshydrodynamic equationsinto\n∂2δn\n∂t2=/summationdisplay\na∂a/braceleftbiggn0\n˜χa/bracketleftbigg\n∂a/parenleftbigg/parenleftbigg∂P\n∂n/parenrightbigg\n¯sδn/parenrightbigg\n+δn∂aUext/bracketrightbigg/bracerightbigg\n, (13)3\nwhereδnisthevariationofnumberdensity nawayfromitsglobalequilibriumvalue n0and(∂P/∂n)¯sistakenatfixedentropy\nperparticle. At T= 0,Eq.(13) furthersimplifiesinto\n∂2δn\n∂t2=/summationdisplay\na∂a/bracketleftbiggn2\n0\n˜χa∂a/parenleftbigg∂µ\n∂nδn/parenrightbigg/bracketrightbigg\n. (14)\nThephenomenologicalhydrodynamicequations( 13)and(14)shallbeapplicabletobothnormalBoseandFermigaseswith the\nspin-obitcoupling.\nMomentumSusceptibility. Thelinearizedhydrodynamicequationsforatomicgaseswit hspin-orbitcouplingdependexplicitly\nonthemomentumsusceptibility χabwhichcanbecalculatedbytheformula22\nχab=1\nV/summationdisplay\nf,iρi/angb∇acketlefti|Kb|f/angb∇acket∇ight/angb∇acketleftf|Ka|i/angb∇acket∇ight\nEf−Ei, (15)\nwhereEiand|i/angb∇acket∇ightare the eigenvaluesand eigenstatesof the many-bodyHamilt onianH,ρiis the distributionfunction,and Vis\nthegasvolume.\nTo manifestthe effectsofthe spin-orbitcouplingon χab, we assume that the inter-atomicinteractionis weak enough that we\ncanevaluate χabusingtheidealgasHamiltonian,andfind\nχab=/summationdisplay\nα=±∂\n∂µ/integraldisplaydDk\n(2π)Dkbkaf(ǫα,k), (16)\nwheref(ǫα,k)is the Fermi (Bose) distribution function for fermionic (bo sonic) atoms, Dis the dimension of the gas. The\ninformation of the spin-orbit coupling is encoded in the sin gle atom dispersion ǫα,k. For degenerate Fermi gases, χabshall be\ndominatedbythecontributionfromclose totheFermisurfac e.\nWe calculate χxx(˜χx) by Eq. ( 16) for1D,2D and3D Fermi gases with spin-orbit coupling generated in the xdirection by\nthe NIST schemewith δ= 0. The (quasi-) 1D or2D gasescan be achievedwhenthere is strong confinementin the ydirection\norinboththe yandzdirections. Forthe 1DFermigasofchemicalpotential µatzerotemperature,we find\n1)µ/ǫr≥1+Ω/2ǫr,\nχxx\nmn= 1+µ/ǫr+1−/radicalbig\n(µ/ǫr−1)2−(Ω/2ǫr)2\n2[µ/ǫr+(Ω/4ǫr)2], (17)\n2)1−Ω/2ǫr≤µ/ǫr<1+Ω/2ǫr,\nχxx\nmn= 1+1/radicalbig\nµ/ǫr+(Ω/4ǫr)2, (18)\n3)−(Ω/4ǫr)2≤µ/ǫr<1−Ω/2ǫr,\nχxx\nmn= 1+µ/ǫr+1+/radicalbig\n(µ/ǫr−1)2−(Ω/2ǫr)2\n2[µ/ǫr+(Ω/4ǫr)2], (19)\nwheretherecoilenergyis ǫr=k2\nr/2m.\nFigure (1) shows at zero temperature how χxxfor the1D Fermi gas changes with Ωfor different atomic densities n. The\nplateausappearingat small Ωarepeculiartothis 1D case andcanbe understoodinthe followingway. When δ= 0andΩ = 0,\nthetwo energybandsgivenbyEq.( 2)touchat energy ǫrwhenkx= 0. TheFermisurfacecrossesthe lowerbandat fourpoints\nforn/kr<2/π(≈0.64),andcrossesbothbandseachat twopointsfor n/kr>2/π. WhenΩis increasedfromzero,anenergy\ngap∼Ωopens atkx= 0. One can show analytically that for small densities nbefore the point of the lower band at kx= 0\nbecomeslowerthanthe Fermisurface,orforlargedensities nbeforetheupperbandbottomat kx= 0becomeshigherthanthe\nFermi surface, χxx/mn= 1+(2kr/πn)2. The furtheraway the density nis from2/π, the larger Ωthe plateau persists to. Of\ncoursefor ourhydrodynamicapproachto be applicable, Ωmust be sufficientlylargerthanlocal equilibrationrates. In the large\nΩlimit,theFermisurfacecrossesthelowerbandattwo points andχxx/mn→(1−4ǫr/Ω)−1[cf.Eq.(2)].\nWealsoplot χxxversusatomicdensity nforthe1DFermigaswithdifferent ΩinFig.(2). Whenthedensityishigh,theFermi\nsurface lies at high momenta where the spin-orbit coupling h as negligible effects, and χxxapproaches mnas expected. When\nΩ/ǫr= 2,ǫ−,kxhas minima at two distinct kx. ForΩ/ǫr= 4,ǫ−,kxhas zero curvature at its single minimum. These features4\nofǫ−,kxgive rise to the correspondingdivergencesof χxx/mnin the low density limit shown in Fig. ( 2). WhenΩ/ǫr= 5, the\nratioχxx/mn, though finite, is enhanced to be substantially larger than u nity by the spin-orbit coupling. Similar behavior of\nχxxwouldbefoundin the 2Dand3DFermigasesaswell.\nIn Fig. (3) is shown the finite temperature behavior of χxxfor the Fermi gases with various Rabi frequency Ωin different\ndimensions. Thespin-orbitcouplingmoves χxx/mnmoreawayfromunityatlowertemperaturesorinlowerdimens ions. When\nTis finite, the low density limit corresponds to the chemical p otentialµ→ −∞. In this limit, to zero order, the distribution\nfunctionfin Eq. (16) can be approximatedby the Boltzmann distributionfunctio n;χxx/mnacquiresthe same value for fixed\nΩandTin differentdimensions.\nSound Velocities. Inthe case that there areno externalpotentials,i.e., Uext= 0, we can readoffthe soundvelocitiesalongthe\nprincipalaxesfromEq.( 13) as\nc2\na=n0\n˜χa/parenleftbigg∂P\n∂n/parenrightbigg\n¯s, (20)\nwhere¯sis the entropy per particle. Besides the momentum susceptib ility˜χa, the sound velocities also depend on the equation\nofstate viatheadiabaticcompressibility κs= (∂n/∂P)¯s/n. Forconvenience,we recastthe soundvelocitiesinto\nc2\na=1\n˜χaκTCP\nCV, (21)\nby using the thermodynamic identity κs/κT=CV/CP, whereκT= (∂n/∂µ)T,N/n2is the isothermal compressibility,\nCV= (∂E/∂T)V,NandCP= (∂E/∂T)P,N=CV+VTκT(∂P/∂T)2\nV,Narethespecific heats.\nWe calculate cxby Eq.(21) combinedwithourpreviousresultsfor χxxfor1D,2D and3D ideal Fermi gaseswith spin-orbit\ncoupling generated in the xdirection by the NIST scheme with δ= 0. For convenience, the sound velocity is normalized by\nc0which is the correspondingsoundvelocity of the ideal Fermi gas withoutthe spin-orbitcouplingat the same density. Fig ure\n(4) shows that at finite temperatures cxfor the 1D case changesnon-monotonicallyas the density of t he Fermi gas varies. This\nnon-monotonicbehaviorcanbeunderstoodfromthezerotemp eraturelimit. Atzerotemperature,accordingtotheGibbs- Duhem\nrelation we have dP/dn=ndµ/dn; from Eq. ( 20) the sound velocities depend on the density of states at the e nergy scale of\nthechemicalpotential µ. In1D,whenthechemicalpotentialapproachesthe minimumo ftheNIST dispersion ǫ+,k, thedensity\nof statesis divergent. Thisdivergencesuppressesthe soun dvelocityto zero asshownin Fig. ( 4) forΩ/ǫr= 4, andgivesrise to\na discontinuity there. Therefore at finite temperatures, th e thermal effects smear out the singular behavior of cxatT= 0and\nresult in the non-monotonicone shown in Fig. ( 4). When the dimensionof the gasis raised up to 2D and 3D, the ef fectsof the\ndensityofstateson cxpersistandcauseafinitejumpin2DshowninFig.( 5)andacuspin3DasinFig.( 6)atzerotemperature.\nThusthefinitetemperaturebehaviorof cxfollowsthe generaltrendofitszerotemperatureone.\nDipoleModeinHarmonicTraps. Collectivemodesofatomicgasesconfinedinharmonictraps Uext=mω2\n0r2/2areimportant\nphysical observables. To investigate how the spin-orbit co upling affects the dipole mode frequency ωdof the normal atomic\ngaseswiththeNISTspin-orbitcoupling,insteadofsolving Eq.(13),weadoptavariationalformalismwhichisequivalenttoth e\nlinearizedhydrodynamicequation( 13)25–27. We start withthe action A=/integraltext\ndtLandtheLagrangianis\nL=/integraldisplay\ndDr/braceleftBigg\n1\n2/summationdisplay\na˜χav2\na−ε−nUext+φ/bracketleftbigg∂n\n∂t+∇·(nv)/bracketrightbigg\n+η/bracketleftbigg∂s\n∂t+∇·(sv)/bracketrightbigg/bracerightbigg\n, (22)\nwheresis the entropydensity,and φandηare the Lagrangianfactors introducedto enforcethe number conversation ∂n/∂t+\n∇·(nv) = 0andthedissipationlesscondition ∂s/∂t+∇·(sv) = 025,26.\nTo estimate the frequency of the dipole mode oscillating in t he principal axis xdirection, we assume the ansatz n(r,t) =\nn0(r−x0(t)ˆx)ands(r,t) =s0(r−x0(t)ˆx)withn0ands0the functionsat equilibrium. This ansatz is motivatedby th e fact\nthat the dipolemode is mainlythe “centerof mass” motionoft he gas. The numberconservationgives v=∂x0(t)ˆx/∂t, which\nalso maintainsthedissipationlesscondition. Aftersubst itutingtheaboveansatzintotheLagrangian,we have\nL=L0+1\n2/bracketleftbigg/integraldisplay\ndDr˜χx/bracketrightbigg\nv2\nx−1\n2/bracketleftbigg/integraldisplay\ndDrn0/bracketrightbigg\nmω2\n0x2\n0, (23)\nwhereL0istheLagrangianat equilibrium. FromEq.( 23),we obtain\n∂2x0\n∂t2=−m/integraltext\ndDrn0/integraltext\ndDr˜χxω2\n0x0. (24)5\nThedipolemodefrequencyis\nωd\nω0=/parenleftbiggm/integraltext\ndDrn0/integraltext\ndDr˜χx/parenrightbigg1/2\n. (25)\nThe independenceof ωdon the equationof state originatesfrom the ansatz we use. It is worth mentioningthat within the local\ndensity approximation, the result ( 25) agrees with the one given by the sum rule approach28. Figure ( 7) plots the dipole mode\nfrequencyof 1D, 2D and 3D normal Fermi gases of Ntotnumberof fermionswith the spin-orbit couplingby the NIST s cheme\nwithδ= 0. Due to the enhancement of ˜χxcompared to mnshown above, the dipole mode frequency ωd/ω0is generally\nsuppressedbelowunity.\nDiscussion\nThe hydrodynamicequations ( 13) and (14) are valid for normal atomic gases with any spin orbit coupli ng under the condition\nthat the momentum K= 0if the velocity v= 0. Apparently even within the NIST scheme, if δ/negationslash= 0, this condition does not\nhold. Equation( 10) shall includefirst ordervariationof µandβ, and Eq.( 11) shall changecorrespondingly;the generalization\nto Eqs. (13) and (14) shall be straightforward. We have revealed the effects of s pin-orbit coupling on the hydrodynamics of\natomic gases by explicitly applying our general formalism t o ideal Fermi gases. Interatomic interactions are expected to make\nquantitative changes to the results presented above. Howev er, to take into account the interaction effects, one needs r eliable\ndetermination of the susceptibilities χaband the equation of state of the gases, which is beyond the sco pe of our work. The\ngeneralization of our hydrodynamic equations to the cases i n which there is condensation requires correct treatment of the\n“superfluid” part. Since the form of the “superfluid” part dep ends on the exact structure of the order parameter, we leave t he\ngeneralizationtoa futurestudy.\nOur calculation manifests the effects of the spin-orbit cou pling on physical observables such as sound velocities and d ipole\nmode frequency. Previously the sound velocity of a unitary F ermi gas has been measured by creating a local density variat ion\nthrougha thinslice ofgreenlaserandmonitoringthe propag ationofthedensityvariationafterwards29. Recentmeasurementof\nsecondsoundvelocityinthesuperfluidphaseofunitaryFerm igaseshasalsobeenachieved30. ThedipoleoscillationofaBose-\nEinsteincondensatewithspin-orbitcouplingwasexcitedb yasuddenchangeoftheRamandetuningandyieldedclearviol ation\nofKohntheorem11. We expectthatsimilarexperimentaltechniquescanbeempl oyedtoconfirmourtheoreticalpredictions.\n1Dalibard,J.,Gerbier,F.,Juzeliunas, G.and ¨Ohberg,P.,Colloquium: Artificialgauge potentialsforneu tral atoms. Rev.Mod. Phys. 83, 1523\n(2011).\n2Zhai,H.,Spin-orbit coupled quantum gases. Int. J. Mod. Phys.B 26, 1230001 (2012).\n3Zhai,H.,Degenerate quantum gases withspin-orbit couplin g: a review. Rep.Prog. Phys. 78, 026001 (2015).\n4Lin,Y.-J. Jimenez-Garcia, K.,andSpielman, I.B.,Spin-or bit-coupled Bose-Einsteincondensates. Nature471, 83(2011).\n5Ho, T.and Zhang, S.,Bose-Einsteincondensates withspin-o rbit Interaction. Phys. Rev. Lett. 107, 150403 (2011).\n6Wang, P.etal, Spin-orbitcoupled degenerate Fermigases. Phys.Rev. Lett. 109, 095301 (2012).\n7Cheuk, L.W.et al,Spin-injectionspectroscopy of a spin-or bit coupled Fermi gas. Phys. Rev.Lett. 109, 095302 (2012).\n8Li, Y., Pitaevskii, L.P., and Stringari, S., Quantum tricri ticality and phase transitions in spin-orbit coupled Bose- Einstein condensates.\nPhys. Rev. Lett. 108, 225301 (2012).\n9Ji, S. et al, Experimental determination of the finite-tempe rature phase diagram of a spin-orbit coupled Bose gas. Nature Physics 10, 314\n(2014).\n10Yu, Z., Equation of state and phase transition in spin-orbit -coupled Bose gases at finite temperature: A perturbation ap proach.\nPhys. Rev. A 90, 053608 (2014).\n11Zhang, J. etal, Collective dipole oscillations of a spin-or bit coupled Bose-Einsteincondensate. Phys.Rev. Lett. 109, 115301 (2012).\n12Stringari,S.,Collective excitations of a trapped Bose-co ndensed gas. Phys. Rev.Lett. 77, 2360 (1996).\n13Zheng, W., and Li, Z., Collective modes of a spin-orbit-coup led Bose-Einstein condensate: A hydrodynamic approach. Phys. Rev. A 85,\n053607 (2012).\n14Chen, Z.,and Zhai,H.,Collective-mode dynamics ina spin-o rbit-coupled Bose-Einsteincondensate. Phys. Rev. A 86, 041604(R) (2012).\n15Martone, G.I. Li, Y., Pitaevskii, L.P., and Stringari, S., A nisotropic dynamics of a spin-orbit-coupled Bose-Einstei n condensate.\nPhys. Rev. A 86, 063621 (2012).\n16Xu, X.Q., and Han, J.H., Emergence of chiral magnetism in spi nor Bose-Einstein condensates with Rashba coupling Phys. Rev. Lett. 108,\n185301 (2012).\n17Price,H.M.,and Cooper, N.R.,Effectsof Berrycurvature on the collective modes of ultracoldgases. Phys. Rev.Lett. 111, 220407 (2013).\n18Zhang, Y., Chen, C., and Zhang, C., Tunable spin-orbit coupl ing and quantum phase transition in a trapped Bose-Einstein condensate,\nSci. Rep. 3, 1937 (2013).\n19Tokatly, I.V., and Sherman, E.Y., Gauge theory approach for diffusive and precessional spin dynamics in a two-dimensio nal electron gas,\nAnn. Phys. 325, 1104 (2010).\n20Tokatly, I.V.,and Sherman, E.Y.,Spindynamics of coldferm ions withsynthetic spin-orbit coupling, Phys.Rev. A 87, 041602(R) (2013).\n21Martin, P.,Parodi, O., and Pershan, P.S.,Unified hydrodyna mic theory for crystals, liquid crystals, and normal fluids, Phys. Rev. A 6, 2401\n(1972).6\n22Chaikin, P.M.,andLubensky, T.C., Principles of condensed matterphysics (Cambridge UniversityPress,Cambridge, 1995).\n23Zhu, Q.,Zhang, C.,andWu, B.,Exoticsuperfluidity inspin-o rbit coupled Bose-Einsteincondensates. EPL100, 50003 (2012).\n24Xu, Y.,Zhang, Y., andWu, B.,Brightsolitons inspin-orbit- coupled Bose-Einsteincondensates. Phys. Rev. A 87, 013614 (2013).\n25Zilsel,P.R.,Liquidhelium II:The hydrodynamics of the two -fluidmodel. Phys.Rev. 79, 309 (1950).\n26Taylor, E.,and Griffin,A.,Two-fluidhydrodynamic modes ina trapped superfluidgas. Phys. Rev.A 72, 053630 (2005).\n27Gao, C.and Yu,Z.,Breathingmode of two-dimensional atomic Fermigases inharmonic traps. Phys.Rev. A 86, 043609 (2012).\n28Li, Y., Martone, G., and Stringari, S., Sum rules, dipole osc illation and spin polarizability of a spin-orbit coupled qu antum gas. EPL99,\n56008 (2012).\n29Joseph, J.et al,Measurement of sound velocityina Fermigas near a Feshbach resonance, Phys. Rev.Lett. 98, 170401 (2007).\n30Leonid, A.S.etal, Secondsound andthe superfluid fractioni na Fermi gaswithresonant interactions, Nature498, 78(2013).\nAcknowledgements We thank Zeng-Qiang Yu and Hui Zhai for helpful discussions. This work is supported by Tsinghua\nUniversityInitiativeScientific ResearchProgram,NSFCun derGrantNo.11474179andNo.11204152.\nAuthorContributions Y.H. andZ.Y.wrotethemainmanuscripttextandY.H.prepare dFigures1-7.\nAdditionalinformation\nCompetingfinancialinterests: Theauthorsdeclarenocompetingfinancialinterests.7\n/s48 /s52 /s56 /s49/s50 /s49/s54 /s50/s48/s49/s50/s51/s52/s53/s54/s120/s120/s109/s110\n/s47\n/s114/s32/s110/s47/s107\n/s114/s61/s48/s46/s51\n/s32/s110/s47/s107\n/s114/s61/s48/s46/s52\n/s32/s110/s47/s107\n/s114/s61/s48/s46/s53\n/s32/s110/s47/s107\n/s114/s61/s48/s46/s56\n/s32/s110/s47/s107\n/s114/s61/s49/s46/s48\nFIG. 1: Momentum susceptibility χxxof the 1D ideal Fermi gas at zero temperature versus Rabi freq uencyΩ. Dependence of the\nmomentum susceptibility χxxofthe 1D ideal Fermi gason Ωfor different gas densities n.8\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s50/s52/s54/s56\n/s32\n/s114/s61/s50\n/s32\n/s114/s61/s52\n/s32\n/s114/s61/s53\n/s32/s32/s120/s120/s47/s109/s110\n/s110/s47/s107\n/s114\nFIG. 2: Momentum susceptibility χxxof the 1D ideal Fermi gas at zero temperature versus atomic de nsityn. Dependence of the\nmomentum susceptibility χxxof the 1D ideal Fermi gas on the Rabi frequency Ωand the gas density n. The grey dash line is χxx=mnfor\ncases inwhichthe spin-orbit coupling has negligible effec ts.9\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s32/s49/s68/s44 /s32\n/s114/s84/s47\n/s114/s61/s49/s41\n/s32/s50/s68/s44 /s32\n/s114/s84/s47\n/s114/s61/s49\n/s32/s51/s68/s44 /s32\n/s114/s84/s47\n/s114/s61/s49\n/s32/s49/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s49\n/s32/s50/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s49\n/s32/s51/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s49\n/s32/s49/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s50\n/s32/s50/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s50\n/s32/s51/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s50\n/s32/s49/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s50\n/s32/s50/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s50\n/s32/s51/s68 /s44/s32\n/s114/s84/s47\n/s114/s61/s50/s32/s32/s120/s120/s47/s110/s109\n/s110/s47/s107/s68\n/s114\nFIG.3:Momentumsusceptibility χxxof the1D,2D and3D idealFermi gas atfinitetemperatureversu s atomic density n. Dependence\nofχxx/mnonthe temperature Tandthe density nfor the 1D,2D and 3D ideal Fermigases.10\n/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s47\n/s114\n/s32/s32/s99\n/s120/s47/s99\n/s48\n/s110/s47/s107\n/s114/s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s32/s32/s68/s111/s115\nFIG. 4:Sound velocity cxof the 1D ideal Fermi gas versus atomic density n. Dependence of sound velocity cxof the 1D ideal Fermi\ngas on the temperature Tand the density n. The sound velocity is normalized by c0the corresponding sound velocity of 1D ideal Fermi gas\nwithoutthe spin-orbitcoupling atthesame density. Thegre ensolidlineisfor T= 0,theblue short–dash linefor T= 0.5ǫr,the reddashline\nforT= 1.0ǫr, the purple short–dash–dot line for T= 2.0ǫrall withΩ = 4ǫr. The blacksolid line is for T= 1.0ǫrwithΩ = 5ǫr. Inset is the\ndensity ofstates inunit of V1Dkr/ǫratΩ = 4ǫr.11\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57 /s49/s46/s50 /s49/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s47\n/s114\n/s32/s32/s99\n/s120/s47/s99\n/s48\n/s110/s47/s107/s50\n/s114/s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s32\n/s32/s32/s68/s111/s115\nFIG. 5:Soundvelocity cxof the 2D ideal Fermi gas versus atomic density n. Dependence of sound velocity cxof the 2D ideal Fermi gas\nwithΩ = 4ǫron the temperature Tand the density n. Herec0is the finite temperature sound velocity of the 2D ideal Fermi gas without the\nspin-orbit coupling. Seethe caption of Fig. 4for the temperatures for each curves. Inset is the densityof states inunit of mV2DatΩ = 4ǫr.12\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s47\n/s114\n/s32/s32/s99\n/s120/s47/s99\n/s48\n/s110/s47/s107/s51\n/s114/s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s32/s32/s68/s111/s115\nFIG. 6:Soundvelocity cxof the 3D ideal Fermi gas versus atomic density n. Dependence of sound velocity cxof the 3D ideal Fermi gas\nwithΩ = 4ǫron the temperature Tand the density n. Herec0is the finite temperature sound velocity of the 3D ideal Fermi gas without the\nspin-orbit coupling. Seethe caption of Fig. 4for the temperatures for each curves. Inset is the densityof states inunit of mV3DkratΩ = 4ǫr.13\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s54/s53/s48/s46/s55/s48/s48/s46/s55/s53/s48/s46/s56/s48/s48/s46/s56/s53/s48/s46/s57/s48\n/s32 /s47\n/s114/s61/s53/s44/s78/s116/s111/s116/s61/s49/s48/s50\n/s40/s49/s68/s41\n/s32 /s47\n/s114/s61/s53/s44/s78/s116/s111/s116/s61/s49/s48/s52\n/s40/s50/s68/s41\n/s32 /s47\n/s114/s61/s53/s44/s78/s116/s111/s116/s61/s49/s48/s54\n/s40/s51/s68/s41\n/s32 /s47\n/s114/s61/s52/s44/s78/s116/s111/s116/s61/s49/s48/s50\n/s40/s49/s68/s41\n/s32 /s47\n/s114/s61/s52/s44/s78/s116/s111/s116/s61/s49/s48/s52\n/s40/s50/s68/s41\n/s32 /s47\n/s114/s61/s52/s44/s78/s116/s111/s116/s61/s49/s48/s54\n/s40/s51/s68/s41/s32\n/s32/s32\n/s84/s47\n/s114/s100/s47\n/s48\nFIG. 7:Dipole mode frequency ωdversus temperature T. Dependence of dipole mode frequency at finite temperature o n the temperature\nandthe Raman coupling forthe 1D, 2D and3D systems. Wehave ta kenω0= 2π×164Hzandǫr= 2π×8340Hz6and the totalnumber of\nparticlesNtot= 102,104,106,respectively." }, { "title": "1209.4770v2.Adiabatic_pumping_through_an_interacting_quantum_dot_with_spin_orbit_coupling.pdf", "content": "Adiabatic pumping through an interacting quantum dot with spin-orbit coupling\nStephan Rojek,1J urgen K onig,1and Alexander Shnirman2\n1Theoretische Physik, Universit at Duisburg-Essen and CENIDE, 47048 Duisburg, Germany\n2Institut f ur Theorie der Kondensierten Materie and DFG Center for Functional Nanostructures (CFN),\nKarlsruhe Institute of Technology, 76128 Karlsruhe, Germany\n(Dated: February 20, 2020)\nWe study adiabatic pumping through a two-level quantum dot with spin-orbit coupling. Using\na diagrammatic real-time approach, we calculate both the pumped charge and spin for a periodic\nvariation of the dot's energy levels in the limit of weak tunnel coupling. Thereby, we compare the\ntwo limits of vanishing and in\fnitely large charging energy on the quantum dot. We discuss the\ndependence of the pumped charge and pumped spin on gate voltages, the symmetry in the tunnel-\nmatrix elements and spin-orbit coupling strength. We identify the possibility to generate pure spin\ncurrents in the absence of charge currents.\nPACS numbers: 72.25.-b,85.75.-d,73.23.Hk,72.10.Bg\nI. INTRODUCTION\nA central issue in the \feld of spintronics is the design\nof spin-based electronic devices.1,2They may involve fer-\nromagnets or external magnetic \felds to control the spin\ndegree of freedom.3But recently, all-electric spintronic\ndevices also have gained interest.4They rely on spin-\norbit (SO) interaction, the strength of which is tunable\nvia external gates in semiconductor heterostructures,5,6a\nbasic requirement for the realization of a spin \feld-e\u000bect\ntransistor.7{10\nA spin-polarized current in a semiconductor can be\ngenerated by spin injection.11{15Here we focus on an al-\nternative route that relies on pumping. By varying the\nparameters of a mesoscopic system periodically in time,\na \fnite charge or spin current can be sustained. Experi-\nmental studies have investigated charge pumping in sev-\neral mesoscopic devices.16{20Spin pumping has been ex-\nperimentally realized in the presence of an external mag-\nnetic \feld.21Theoretical studies of spin pumping involve\nexternal magnetic \felds,22ferromagnetic leads,23{25and\nalso SO coupling.26{28\nIn the present paper, we consider the minimal model\nthat contains SO interaction: a quantum dot with two\nspin-degenerate orbital levels. Such a two-level quantum\ndot with more than two leads has been suggested as a\nspin \flter.29We focus on the adiabatic limit of pumping,\ni.e., the parameters are varied slowly in time compared\nto the dwell time of the mesoscopic system.30Adiabatic\npumping of charge and spin through such a two-level dot\nhas been considered in the limit of vanishing charging\nenergy.28It was found that this system can act as an\nall-electric spin battery , i.e., a \fnite spin current can be\nachieved without ferromagnets by electrically controlling\nthe dot parameters. For speci\fc symmetries in the tun-\nnel coupling of the dot to the leads even pure spin cur-\nrents have been suggested. From the analysis of Ref. 28,\nwhich was based on a scattering-matrix approach,31{33\nit is not clear whether and how the conclusions can be\ntransferred to quantum dots with non vanishing Coulomb\ninteraction. To answer this question is the main goal ofthe present paper.\nIn order to take the Coulomb interaction into account,\nwe use a diagrammatic real-time approach34{36that al-\nlows for arbitrary strengths of the Coulomb interaction.\nWe focus on the limit of weak tunnel coupling, for which\nwe perform a systematic perturbation expansion to low-\nest order. To emphasize the role of Coulomb interac-\ntion, we compare the limit of vanishing Coulomb interac-\ntion with the limit of an in\fnitely large charging energy.\nThe paper is organized as follows. In Sec. II we in-\ntroduce the model that describes the SO interaction in\na two-level quantum dot with Coulomb interaction. Sec-\ntion III deals with the technique to calculate the pumped\ncharge and pumped spin during one pumping cycle. To\nstudy the dependence of the pumped charge (spin) on\nthe four tunnel-matrix elements in a transparent way, we\nintroduce in Sec. IV an isospin representation of the or-\nbital degree of freedom. Finally, in Sec. V we present the\nresults for the pumped charge and pumped spin.\nII. MODEL\nWe consider a quantum dot with two spin-degenerate\norbital levelsj\u000b\u001bi(with labels \u000b= 1;2 for the orbital and\n\u001b=\";#for the spin), tunnel coupled to the left (L) and\nthe right (R) lead (see Fig. 1). The system is described\nby the Hamiltonian\nH=Hdot+Hlead+Htun: (1)\nHere,Hdotis the Hamiltonian of the isolated dot, Hlead\nof the leads, and Htunof the tunneling between dot and\nleads.\nThe Hamiltonian for the isolated quantum dot con-\ntains two parts. The single-particle contribution for the\ntwo orbitals \u000bwith energy \u000f\u000b, which are coupled by SO\ninteraction, can be cast in the 4 \u00024 matrix\n\u0012\n\u000f1\u001b0\u0000i\u000bso\u0001\u001b\ni\u000bso\u0001\u001b\u000f2\u001b0\u0013\n; (2)arXiv:1209.4770v2 [cond-mat.mes-hall] 8 Feb 20132\n/epsilon12/epsilon11\nVL1VR1VR2VL2LR\nFIG. 1. (Color online) Energy scheme of the two-level quan-\ntum dot. The two orbital, spin-degenerate levels can be varied\nin time. They are tunnel coupled to the left (L) and the right\n(R) lead, with tunnel-matrix elements V\u0015\u000b. The leads have\nthe same chemical potential \u0016.\nfor the basisfj1\"i;j1#i;j2\"i;j2#ig, where the spin\nquantization axis is chosen arbitrarily. Here, \u001bdenotes\nthe vector of Pauli matrices, \u001b0is the identity matrix,\nand\u000bsois a real vector describing the SO coupling. The\nmatrix in Eq. (2) has the most general form that al-\nlows time-reversal symmetry. It has been used in the\ncontext of pumping28and was also recently applied to\nelectron-transport in the presence of a magnetic \feld37\nand to study the Josephson current through a double-dot\nstructure.38In the following, we choose the spin quanti-\nzation axis parallel to \u000bsoso the matrix becomes diagonal\nin spin space.\nThe second part of the dot Hamiltonian accounts for\nthe charging energy EC(N\u0000ng)2, whereNis the total\nnumber of dot electrons and ngan external gate charge.\nWithout loss of generality, we can choose ng= 1=2 (any\nother value can be achieved by a constant shift of the\nenergies\u000f\u000b). This leads (up to an additive constant) to\nthe dot Hamiltonian\nHdot=X\n\u001b\u000b\u000f\u000bdy\n\u000b\u001bd\u000b\u001b+X\n\u001bi\u001b\u000bso\u0010\ndy\n2\u001bd1\u001b\u0000h.c.\u0011\n+UX\n\u000bn\u000b\"n\u000b#+UX\n\u001b\u001b0n1\u001bn2\u001b0; (3)\nwhere the operator dy\n\u000b\u001bcreates an electron in state j\u000b\u001bi\nand the corresponding number operator is n\u000b\u001b=dy\n\u000b\u001bd\u000b\u001b.\nWe used the notation \u001b=\u00061 for spin parallel (antipar-\nallel) to\u000bso,\u000bso=j\u000bsoj, andU= 2EC.\nThe leads are modeled as reservoirs of noninteracting\nelectrons,\nHlead=X\n\u001bk\u0015\u000fkcy\nk\u001b\u0015ck\u001b\u0015; (4)\nwherecy\nk\u001b\u0015is the creation operator for an electron with\nspin\u001band momentum kin lead\u0015. Tunneling betweendot and leads is described by the Hamiltonian\nHtun=X\n\u001b\u000bk\u0015V\u0015\u000bcy\nk\u001b\u0015d\u000b\u001b+ h.c.; (5)\nwith (spin-independent) tunnel-matrix elements V\u0015\u000bfor\ntunneling between lead \u0015and orbital \u000b.\nPumping is achieved by varying system parameters pe-\nriodically in time. In this paper, we assume that the en-\nergy levels\u000f\u000b(t) can be changed in time via external gates\ncapacitively coupled to the system. In principle, the ex-\nternal gates also may a\u000bect the SO coupling, the tunnel\ncouplings, and the electro chemical potential of the leads\n(via parasitic capacitances). To simplify the discussion,\nhowever, we assume for the following these parameters\nto be constant in time.\nWe focus on the regime of adiabatic pumping, which\nis achieved for pumping frequencies \n smaller than the\ninverse of the dwell time. This is valid for \n \u001c\u0000, where\n\u0000 is the tunnel-coupling strength, \u0000 =P\n\u0015\u000b\u0000\u0015\u000b\u000b, with\n\u0000\u0015\u000b\u000b0= 2\u0019\u001aV\u0015\u000b0V\u0003\n\u0015\u000b. The density of states \u001ais assumed\nto be \rat and equal for the left and right leads. We choose\na gauge where all four tunnel-matrix elements are real.\nTo study the e\u000bect of Coulomb interaction, we com-\npare results for the limit of noninteracting ( U= 0) and\nin\fnitely strong interacting ( U=1) electrons on the\ndot. In the latter case, the total number of electrons in\nthe quantum dot can only be zero or 1.\nIII. METHOD\nTo calculate the pumped charge and pumped spin,\nwe use a diagrammatic real-time approach to adia-\nbatic pumping through quantum-dot systems.39For the\npresent context, we extend the analysis of Ref. 39 to allow\nfor a time-dependent transformation of the basis states.\nThis is necessary since the SO coupling couples time-\ndependent orbital levels, which, in turn, makes the dot\neigenstates time dependent.\nWe start in Sec. III A with the kinetic equation for\nthe reduced density matrix in its general form, which de-\nscribes the time evolution of the dot's degrees of freedom.\nSubsequently, we perform both an adiabatic expansion,\ni.e., a perturbation expansion in the pumping frequency\n(Sec. III B) and a perturbation expansion in the tunnel-\ncoupling strength (Sec. III C) to describe the limit of\nweak tunnel coupling. The pumped charge and pumped\nspin currents to lowest order in \u0000 and \n are derived in\nSec. III D. Finally, in Sec. III E, we perform the limit\nof weak pumping which assumes small amplitudes of the\npumping parameters.\nA. Kinetic equation\nThe main idea of the diagrammatic real-time technique\nis based on the fact that the leads are described as large3\nreservoirs of noninteracting electrons which can be inte-\ngrated out in order to arrive at a reduced density ma-\ntrixpfor the dot degrees of freedom only. For a matrix\nrepresentation with matrix elements p\u001f1\u001f2=h\u001f1j\u001adotj\u001f2i\n(for the diagonal elements we introduce the notation\np\u001f\u0011p\u001f\n\u001f), it is convenient to use the eigenstates j\u001fiiwith\ncorresponding eigenenergies E\u001fias a basis. For this, we\nemploy a time-dependent unitary transformation Tact-\ning on the dot Hamiltonian Hdot, such that TyHdotTis\ndiagonal.\nThe time evolution of the reduced density matrix is\ngiven by the kinetic equation\nd\ndtp(t) =\u0000i\n~\u0001E(t)p(t)\u0000h\nTy_T(t);p(t)i\n+Zt\n\u00001dt0W(t;t0)p(t0): (6)\nThe bold face indicates tensor notation. The reduced\ndensity matrix pandTy_Tare tensors of rank 2, while\n\u0001EandWare tensors of rank 4, i.e.,\n(W(t;t0)p(t0))\u001f1\n\u001f2=X\n\u001f0\n1;\u001f0\n2W\u001f1\u001f0\n1\n\u001f2\u001f0\n2(t;t0)p\u001f0\n1\n\u001f0\n2(t0):(7)\nThe kernel element W\u001f1\u001f0\n1\n\u001f2\u001f0\n2(t;t0) describes the transition\nfromp\u001f0\n1\n\u001f0\n2(t0) at timet0top\u001f1\u001f2(t) at timet. It is given\nby the sum over all irreducible blocks on the Keldysh\ncontour which correspond to the described transition.\nThe elements of \u0001Eare di\u000berences of the eigenenergies\nde\fned as ( \u0001E(t))\u001f1\u001f0\n1\n\u001f2\u001f0\n2= (E\u001f1(t)\u0000E\u001f2(t))\u000e\u001f1\u001f0\n1\u000e\u001f2\u001f0\n2.\nThe second termh\nTy_T;p(t)i\noriginates from the time\ndependence of the transformation T, and _Tdenotes the\ntime-derivative of T. In the following adiabatic expan-\nsion and the expansion in the tunnel-coupling strength,\nwe follow the lines of Ref. 39.\nB. Adiabatic expansion\nIn the limit of slow variation of the system param-\neters, such that the duration of one pumping cycle,\nT= 2\u0019=\n, is much larger than the dwell time of an\nelectron in the quantum dot, we can perform an adi-\nabatic expansion of Eq. (6), which is equivalent to an\nexpansion of all time dependencies around the \fnal time\ntand to systematically keep all contributions that con-\ntain one time derivative. For this, we \frst do a Taylor\nexpansion of the reduced density matrix around the \f-\nnite timet, i.e.,p(t0)!p(t) + (t0\u0000t)d\ndtp(t). We then\nexpand the kernel and the density matrix in the pump-\ning frequency, i.e., p(t)!p(i)\nt+p(a)\ntandW(t;t0)!\nW(i)\nt(t\u0000t0) +W(a)\nt(t\u0000t0). The instantaneous order,\nindicated by the index ( i), describes the limit where all\nsystem parameters are frozen at time t. The adiabatic\ncorrection, labeled by ( a), contains one time derivative,i.e., it collects all contributions to \frst order in the pump-\ning frequency \n. The di\u000berence in the eigenenergies of\nthe isolated dot, \u0001E(t), is of instantaneous order, while\nTy_T(t) belongs to the adiabatic correction.\nSince both W(i)andW(a)depend only on the dif-\nferencet\u0000t0, it is convenient to perform the Laplace\ntransformF(z) =Rt\n\u00001dt0e\u0000z(t\u0000t0)F(t\u0000t0). Using\nthe short notations W(i=a)\nt =W(i=a)\nt(z= 0+) and\n@W(i)\nt= (@W(i)\nt(z)=@z)jz=0+, the kinetic equation reads\n0 =\u0012\nW(i)\nt\u0000i\n~\u0001E\u0013\np(i)\nt; (8)\nin instantaneous order, and\nd\ndtp(i)\nt=\u0012\nW(i)\nt\u0000i\n~\u0001E\u0013\np(a)\nt\u0000h\nTy_T;p(i)\nti\n+W(a)\ntp(i)\nt+@W(i)\ntd\ndtp(i)\nt (9)\nfor the adiabatic correction. The normalization condition\nfor the density matrix is expressed as Tr p(i)\nt= 1 and\nTrp(a)\nt= 0.\nC. Expansion in the tunnel-coupling strength\nIn addition to the adiabatic expansion, we perform a\nperturbation expansion in the tunnel-coupling strength\n\u0000. For a systematic expansion of the kinetic equations,\nwe need to analyze the term \u0001Ep(i=a). It vanishes for all\ndiagonal matrix elements of p(i=a). The o\u000b-diagonal ma-\ntrix elements, associated with coherent superpositions,\nare only nonzero when the superposition is not forbidden\nby conserved quantum numbers and when the energy dif-\nference of the corresponding states is smaller or of the\norder of \u0000. Therefore, we count all contributing matrix\nelements of \u0001Eto be of the order of \u0000.\nThe expansion of the kernels, W(i=a)\nt =P1\nn=1W(i=a;n )\nt , starts to \frst order in \u0000 and the\ninstantaneous order of the reduced density matrix to\nzeroth order, p(i)\nt=P1\nn=0p(i;n)\nt. To properly match\nthe powers of \u0000 in Eq. (9), the adiabatic correction of\nthe reduced density matrix, p(a)\nt=P1\nn=\u00001p(a;n)\nt, has to\nstart to minus \frst order.39\nIn the following, we consider the limit of weak tunnel\ncoupling, \u0000\u001ckBT, for which we restrict ourselves to\nthe lowest-order contributions in \u0000. The instantaneous\npart of the kinetic equation starts to \frst order in \u0000,\n0 =\u0012\nW(i;1)\nt\u0000i\n~\u0001E\u0013\np(i;0)\nt; (10)\nwith normalization Tr p(i;0)\nt= 1. For the adiabatic cor-\nrection, the expansion of Eq. (9) to lowest (zeroth) order\nin \u0000 yields\nd\ndtp(i;0)\nt=\u0012\nW(i;1)\nt\u0000i\n~\u0001E\u0013\np(a;\u00001)\nt; (11)4\nwith Trp(a;0)\nt = 0. All other terms appearing on the\nright-hand side of Eq. (9) are of higher order in \u0000 and\ndrop out. This is immediately obvious for the last two\nterms in Eq. (9). But alsoh\nTy_T;p(i;0)\nti\ndrops out in the\nabsence of any bias voltage. In this case, p(i;0)\ntis given\nby the equilibrium distribution, which is diagonal with\nmatrix elements being determined by Boltzmann factors.\nSince energy di\u000berences, \u0001E, of states for which coher-\nent superpositions are allowed are of the order of \u0000, the\ndi\u000berence of the corresponding occupation probabilities\nfor these states is also of the order of \u0000 and, therefore,\nvanishes in the perturbation expansion. This means that\nthe matrix elements\n\u0010h\nTy_T;p(i;0)\nti\u0011\u001f1\n\u001f2=\u0010\np(i;0)\nt \u001f2\u0000p(i;0)\nt \u001f1\u0011\u0010\nTy_T\u0011\u001f1\n\u001f2(12)\nvanish for all combinations of \u001f1and\u001f2which are needed\nin the kinetic equation.\nD. Pumped charge and pumped spin\nThe pumped current and the pumped spin current\nfrom the dot into the left lead are given by\nIL(t) =eZt\n\u00001dt0Trh\nWL\nQ(t;t0)p(t0)i\n; (13)\nSL(t) =~\n2Zt\n\u00001dt0Trh\nWL\nS(t;t0)p(t0)i\n; (14)\nrespectively. Here, we have introduced WL\nQ=S(t;t0) =P\nq(q\"\u0006q#)WLq\"q#(t;t0) where WLq\"q#(t;t0) only con-\ntains those diagrams of W(t;t0) in which q\u001belectrons\nwith spin\u001benter the left lead, i.e., in which the number\nof lines for lead Land spin\u001bgoing from the upper to the\nlower contour minus the number of lines from the lower\nto the upper contour is q\u001b.\nAnalogously to the expansion of the kinetic equation,\nwe perform the adiabatic expansion and the perturba-\ntion expansion in the tunnel-coupling strength for the\npumped charge and spin current. To lowest order we get\nI(a;0)\nL(t) =eTr\u0014\u0010\nWL\nQ;t\u0011(i;1)\np(a;\u00001)\nt\u0015\n; (15)\nS(a;0)\nL(t) =~\n2Tr\u0014\u0010\nWL\nS;t\u0011(i;1)\np(a;\u00001)\nt\u0015\n: (16)\nThe pumped charge and the pumped spin per pump-\ning cycle is obtained by integration, Q=RT\n0dtI(a;0)\nL(t)\nand \u0006 =RT\n0dt S(a;0)\nL(t). The diagrammatic rules\nto calculate analytically W(i;1)can be found in Ap-\npendix A.34{36,39,40After having determined W(i;1), we\nobtain the adiabatic correction to the reduced density\nmatrix,p(a;\u00001)\nt , by solving the kinetic equations (10)\nand (11). Those, then, enter Eqs. (15) and (16) for the\npumped charge and spin currents.E. Weak pumping\nWe split the energy of the orbital levels into the time-\naveraged part, \u0016 \u000f\u000b=1\nTRT\n0dt \u000f\u000b(t), and the deviation\n\u000e\u000f\u000b(t):\n\u000f1(t) = \u0016\u000f1+\u000e\u000f1(t); (17)\n\u000f2(t) = \u0016\u000f2+\u000e\u000f2(t): (18)\nIn the limit of weak pumping, the time-dependent part of\nthe pumping parameters is small compared to other en-\nergy scales of the system such as tunnel-coupling strength\nand temperature, \u000e\u000f\u000b(t)\u001c\u0000;kBT. Hence, we can ex-\npand the pumped charge, Q, and pumped spin, \u0006, to\nlowest (bilinear) order in \u000f1(t) and\u000f2(t). For adiabatic\npumping, the pumped charge (spin) is proportional to the\narea enclosed by the path of ( \u000f1(t);\u000f2(t)) in the parameter\nspace during one pumping cycle. Therefore, a phase dif-\nference is necessary to gain \fnite pumped charge (spin).\nThe enclosed area is given by \u0011=RT\n0dt \u000e\u000f1(t)@t\u000e\u000f2(t).\nAll results in Sec. V are calculated in the weak-pumping\nlimit.\nIV. ISOSPIN TRANSFORMATION\nFor each matrix element p\u001f1\u001f2that needs to be consid-\nered (all diagonal ones and those o\u000b-diagonal ones that\ndescribe possible coherent superpositions), there is one\nkinetic equation. It is often convenient to transform the\nreduced density matrix such that only linear combina-\ntions of the p\u001f1\u001f2's appear, which allows for a straight-\nforward physical interpretation. In the context of spin\ntransport through a single-level quantum dot with ferro-\nmagnetic leads, it is advantageous to formulate the ki-\nnetic equations separately for the occupation probability\nof zero, one or two electrons on the quantum dot, and\nthe three components of the spin on the dot.24,41,42The\nvector character of the spin accounts for both a spin im-\nbalance along a given axis and the coherent dynamics of\nthe accumulated spin. One virtue of such a transforma-\ntion lies in the fact that it is possible to write the kinetic\nequation in a (spin-)coordinate-free form, which does not\ndepend on the choice of the spin quantization axis.\nA similar transformation can also be used for the or-\nbital degree of freedom in systems in which coherent su-\nperpositions of the occupation of di\u000berent orbitals ap-\npear. These superpositions are conveniently described by\nde\fning an isospin. This has been done before for sev-\neral double-dot systems.25,43{45We will introduce such\nan isospin description now for the system under consid-\neration.\nIn this paper, we focus on the limits of U= 0 and\nU=1. In the \frst case, U= 0, the Hilbert space is 16-\ndimensional, i.e., the reduced density matrix is a 16 \u000216\nmatrix. However, since we choose the spin-quantization\naxes along the direction of the SO \feld, the Hamiltonian\ndivides into two independent spin channels. As a result,5\nthe reduced density matrix can be written as a direct\nproduct of the 4 \u00024 density matrices for spin up and\nspin down,pU=0= (pU=0)\"\n(pU=0)#. In the basis\nfj0i;j1i;j2i;jdig\u001bthat corresponds, for each spin \u001b, to\nthe occupation of none of the orbitals, of orbital 1, of\norbital 2, and of both orbitals, respectively, the reduced\ndensity matrix reads\n(pU=0)\u001b=0\nB@p00 0 0\n0p1p1\n20\n0p2\n1p20\n0 0 0pd1\nCA\n\u001b: (19)\nNote that, in order to keep the notation simple, we put\nthe index\u001bonly once at the matrix indicating the \u001b-\ndependence of each of the matrix elements.\nForU=1the dot is either singly occupied or empty,\ni.e., the Hilbert space is \fve-dimensional. The reduced\ndensity matrix takes the form\npU=1=0\nBBBBB@p00 0 0 0\n0p1\"p1\"\n2\"0 0\n0p2\"\n1\"p2\"0 0\n0 0 0 p1#p1#\n2#\n0 0 0 p2#\n1#p2#1\nCCCCCA: (20)\nNote that, here, p0is the probability that the dot is not\noccupied with either spin, while for U= 0 we used ( p0)\u001b\nfor the probability that the dot is not occupied with spin\n\u001b, irrespective of the occupation of spin \u0000\u001b.\nTo describe the coherent superposition associated with\nthe o\u000b-diagonal matrix elements, it is convenient to in-\ntroduce, for each physical spin, an isospin operator ^I\u001b\nwith quantum-statistical expectation value I\u001b=h^I\u001bi.\nChoosing the coordinate system for the isospin such\nthatj1\u001biandj2\u001biare the eigenstates of ^I\u001b\nz, we get\nI\u001b\nx=\u0000\np1\u001b\n2\u001b+p2\u001b\n1\u001b\u0001\n=2,I\u001b\ny=i\u0000\np1\u001b\n2\u001b\u0000p2\u001b\n1\u001b\u0001\n=2, andI\u001b\nz=\n(p1\u001b\u0000p2\u001b)=2. Since ultimately we aim at a coordinate-\nfree form of the kinetic equations, we abbreviate the z-\naxis chosen above by the normalized vector n, i.e.,j1\u001bi\nandj2\u001biare the eigenstates of ^I\u001b\u0001n.\nThe isospin direction ncharacterizes the eigenstates of\nthe isolated quantum dot in the absence of SO coupling.\nThe SO coupling, however, couples the two orbitals. As\na consequence, the dot eigenstates\n\u0012\nj+\u001bi\nj\u0000\u001bi\u0013\n=T\u001b\u0012\nj1\u001bi\nj2\u001bi\u0013\n(21)\nfor single occupation with spin \u001bare linear combinations\nof the two orbitals j1\u001biandj2\u001bi, given by the transfor-\nmation\nT\u001b=1p\n2\u0018(\u0018+ \u0001\u000f)\u0012\n\u0018+ \u0001\u000f i\u001b\u000b so\ni\u001b\u000bso\u0018+ \u0001\u000f\u0013\n: (22)\nThe corresponding eigenenergies are E\u0006=\u000f\u0006\u0018, with the\nmean dot level \u000f= (\u000f1+\u000f2)=2 and\u0018=p\n\u0001\u000f2+\u000b2so. The\ntransformation depends on the spin \u001b, the level spacingof the both orbitals, \u0001 \u000f= (\u000f1\u0000\u000f2)=2, and the strength of\nthe SO coupling, \u000bso. We consider the regime where \u000bso\nand \u0001\u000fare of order \u0000. Therefore, the level spacing 2 \u0018of\nthe eigenenergies E\u0006is also of order \u0000. As we pump on\nboth energies, \u000f1(t) and\u000f2(t), the transformation T\u001b(t)\nand the eigenenergies E\u0006(t) are time dependent.\nThe unitary transformation T\u001bcorresponds to a rota-\ntion about the x-axis with the spin-dependent angle\n\u0012\u001b=\u0000\u001barcsin\u0012\u000bso\n\u0018\u0013\n(23)\nin isospin space. This means that the dot eigenstates\nj\u0006\u001biare eigenstates to the isospin projection ^I\u001b\u0001~n\u001balong\nthe direction ~n\u001bthat is obtained from nby the above\nmentioned rotation (see Fig. 2).\nThe tunneling Hamiltonian couples the lead-electron\nstates to both orbitals, i.e., to a linear combination of\nj1\u001biandj2\u001bi. To diagonalize the tunneling from and to\nlead\u0015, we employ the unitary transformation\nF\u0015=1p\nV2\n\u00151+V2\n\u00152\u0012\nV\u00151V\u00152\n\u0000V\u00152V\u00151\u0013\n: (24)\nIn isospin space, this transformation corresponds to a\nrotation about the y-axis with angle\n\u001e\u0015=\u0000arcsin\u00122V\u00151V\u00152\nV2\n\u00151+V2\n\u00152\u0013\n: (25)\nApplying this rotation on ngenerates the direction m\u0015\n(see Fig. 2) which has the following physical interpreta-\ntion: Only dot electrons with j+i^I\u001b\u0001m\u0015isospin projection\nalong ^I\u001b\u0001m\u0015couple to reservoir \u0015, while thej\u0000i^I\u001b\u0001m\u0015\nisospin projection is decoupled from the lead.44There-\nfore, in a ferromagnetic analogy, the leads are full isospin\npolarized with polarization along m\u0015.\nFirst, we write the kinetic equations (10) and (11) in\nthe basisfj0i;j+i;j\u0000i;jdig\u001bfor theU= 0 limit and\nfj0i;j+\"i;j\u0000\"i;j+#i;j\u0000#ig forU=1. Those ki-\nnetic equations are treated perturbatively to \frst order\nin the tunnel-coupling strength \u0000. As described above,\nwe count both \u000bsoand \u0001\u000fas one order in \u0000. The ele-\nments of the kernel W(i;1)are calculated by the rules\nin Appendix A. Including the isospin in the formula-\ntion of the kinetic equation, the system is fully described\nby the occupation probabilities of the dot and the ex-\npectation values of the isospins. In particular, we per-\nform the transformation from fp0;p+;p\u0000;p\u0000\n+;p+\n\u0000;pdg\u001bto\nfp0;ps;pd;Ig\u001bin the limit of vanishing Coulomb inter-\naction. The probabilities describing the occupation of\nthe dot with spin \u001bare (p0;ps;pd)\u001bfor empty,p0, single,\nps=p1+p2, and double occupation, pd. In the limit of\nstrong Coulomb interaction, U=1, the transforma-\ntion readsfp0;p+\";p\u0000\";p\u0000\"\n+\";p+\"\n\u0000\";p+#;p\u0000#;p\u0000#\n+#;p+#\n\u0000#gto\nfp0;p\";p#;I\";I#g. The relevant occupation probabilities\nare (p0;p\";p#) withp\u001b=p1\u001b+p2\u001bbeing the possibility\nthat the dot is occupied by a single electron with spin \u001b.\nWe identify in the resulting kinetic equations the vectors6\nxyz\n˜n↑˜n↓nmLmR\nφLφR\nθ↑θ↓\nFIG. 2. (Color online) Scheme of di\u000berent relevant isospin\nquantization axes. The vector nrepresents the quantization\nwhere the orbital levels j1\u001biandj2\u001biare the eigenstates of\nthe^Izoperator of the isospin. The two axes ~n\u001bare the quan-\ntization axes where the eigenstates of ^Izare the eigenstates\nofHdotfor single occupation. In a ferromagnetic analogy, the\nleads are fully isospin polarized along the axes m\u0015.\nm\u0015and~n\u001band get, thus, a representation that is inde-\npendent of the choice of basis. In the limit of U= 0, we\nget\nd\ndt0\n@p0\nps\npd1\nA\n\u001b=\u0000\n~0\n@\u0000f1\u0000f\n20\nf\u00001\n21\u0000f\n0f\n2\u0000(1\u0000f)1\nA0\n@p0\nps\npd1\nA\n\u001b\n+\u0000\n~0\n@1\u0000f\n2f\u00001\n\u0000f1\nA(I\u001b\u0001\u0016m); (26a)\nd\ndtI\u001b=\u0000\n~\u0012f\n2p0+2f\u00001\n4ps\u00001\u0000f\n2pd\u0013\n\u001b\u0016m\n\u0000\u0000\n2~I\u001b+I\u001b\u0002B\u001b; (26b)\nwheref=f(\u000f) is the Fermi function at energy \u000f. As the\ndi\u000berence of the eigenenergies, 2 \u0018, is of order \u0000, we have\nto drop 2\u0018in terms which are already linear in \u0000. There-\nfore, the Fermi function, f, depends here only on the\nmean level position, \u000f, since every term which includes\nthe Fermi function is linear in \u0000. In the equations for the\nprobabilities, the isospin projections along the directions\nde\fned by the leads enter in the weighted average\n\u0016m=\u0000L\n\u0000mL+\u0000R\n\u0000mR; (27)\nwith \u0000\u0015=P\n\u000b\u0000\u0015\u000b\u000b. The isospin projection direction\ngiven by the SO coupling, on the other hand, gives rise\nto a precession term about the e\u000bective \feld\nB\u001b=\u00002\u0018\n~~n\u001b (28)\nin the equation for the isospin. This e\u000bective \feld is the\nonly place where the SO coupling enters the kinetic equa-tions. Equations (26a) and (26b) represent both the in-\nstantaneous order and the adiabatic correction of the ki-\nnetic equation. For the \frst case, one needs to set the left-\nhand side to zero and add the index ( i;0) to the isospin\nand the occupation probabilities on the right-hand side.\n(Note that the instantaneous part of the isospin vanishes\nin lowest order in \u0000, I(i;0)\n\u001b= 0.) For the the second case,\nwe need to add the index ( i;0) on the left-hand side and\n(a;\u00001) on the right-hand side, respectively.\nIn the limit of strong Coulomb interaction, U=1,\nthe kinetic equations read\nd\ndt0\n@p0\np\"\np#1\nA=\u0000\n~0\n@\u00002f(1\u0000f)=2 (1\u0000f)=2\nf\u0000(1\u0000f)=2 0\nf 0\u0000(1\u0000f)=21\nA0\n@p0\np\"\np#1\nA\n+\u0000\n~(1\u0000f)0\n@(I\"\u0001\u0016m) + (I#\u0001\u0016m)\n\u0000(I\"\u0001\u0016m)\n\u0000(I#\u0001\u0016m)1\nA; (29a)\nd\ndtI\u001b=\u0000\n~\u0012f\n2p0\u00001\u0000f\n4p\u001b\u0013\n\u0016m\n\u0000\u0000\n~1\u0000f\n2I\u001b+I\u001b\u0002(B\u001b+BU): (29b)\nIn addition to the e\u000bective \feld B\u001bgenerated by the\nSO coupling, we identi\fed here another e\u000bective \feld BU\nacting on the isospin. The latter appears as a conse-\nquence of the interplay between tunneling and Coulomb\ninteraction. It is formally identical to the exchange \feld\nacting on the physical spin in quantum dots attached to\nferromagnetic leads.24,41,42In the limit of U=1, it is\ngiven by\nBU=\u0000\n2\u0019~\u0016mReZ\nd!f(!)\n\u000f\u0000!+i0+\n=\u0000\n2\u0019~\u0014\nln\fUcuto\u000b\n2\u0019\u0000Re \t\u00121\n2+i\f\u000f\n2\u0019\u0013\u0015\n;(30)\nwhere \t is the digamma function and we used \f=\n1=kBT. The high-energy cuto\u000b Ucuto\u000b appearing in\nthe second line guarantees convergence of the energy\nintegral.34{36Physically it is provided by the smaller of\nthe band width of the leads and the charging energy.\nFor practical calculations it is not necessary to use the\nbasis-independent form of this isospin representation. It\nallows for a better physical understanding of the sys-\ntems dynamics but for evaluating the pumped charge\nand spin as described in Sec. III, it is convenient to\nuse the basisfj0i;j1i;j2i;jdig\u001bin theU= 0 limit and\nfj0i;j+\"i;j\u0000\"i;j+#i;j\u0000#ig forU=1.\nV. RESULTS\nIn this section, we present the results for the adiabati-\ncally pumped charge (spin) in the weak-pumping regime.\nTo calculate those, we use the formalism that has been\nintroduced in Sec. III. We integrate the pumped charge7\nand spin currents, Eqs. (15) and (16), over one pumping\ncycle and obtain the pumped charge and pumped spin\nper pumping cycle. In order to simplify the time depen-\ndence of the pumped currents, we make use of the weak\npumping limit (see Sec. III E) and expand the integrand\nup to bilinear order in the pumping parameters, \u000f1(t) and\n\u000f2(t). In this case, all results are proportional to the area,\n\u0011, enclosed in the pumping-parameter space. We normal-\nize our results by \u0011and, thus, they are independent of\nthe exact path in parameter space.\nTo analyze the e\u000bect of Coulomb interaction, we com-\npare results for noninteracting electrons, U= 0, with the\nlimit of strong Coulomb interaction. The latter is real-\nized by setting U=1in the Hamiltonian and, thereby,\nsuppressing occupation of the quantum dot with more\nthan one electron. Furthermore, \fnite Coulomb interac-\ntion in\ruences the amplitude of the exchange \feld, BU,\nvia the high-energy cuto\u000b, Ucuto\u000b . In all calculations,\nwe setUcuto\u000b = 100kBT. We assume weak tunnel cou-\npling between quantum dot and leads, \u0000 \u001ckBT, i.e.,\nwe restrict the calculation to lowest order in the tunnel-\ncoupling strength \u0000. If not stated otherwise, the SO-\ncoupling strength is \u000bso= \u0000=10.\nForU= 0, the results of this paper can be compared\nwith calculations that include higher orders in \u0000. For ex-\nample, Brosco et al. have studied the two-level quantum\ndot with SO coupling and vanishing Coulomb interac-\ntion in the limit of zero temperature.28Calculations to\nall orders in \u0000 can be done, e.g., with a scattering matrix\napproach,31{33which is equivalent to an approach that is\nbased on a formula relating the pumped current to the\ninstantaneous dot Greens functions.46The latter is, in\ngeneral, extendable to \fnite interaction.\nIn this section, we study the dependence of the pumped\ncharge (spin) on various parameters: the strength of the\nSO coupling, \u000bso, the tunnel coupling to the leads, V\u0015\u000b,\nand the time-averaged dot levels, \u0016 \u000f\u000b. It is convenient\nto parametrize the latter by the time-averaged mean dot\nlevel,\u000f= (\u000f1+\u000f2)=2, and the averaged spacing of both\norbital levels, \u0001\u000f= (\u000f1\u0000\u000f2)=2.\nIn the regime under consideration, the temperature ap-\npears only in two ways. First, since the mean energy level\nonly appears in the combination \f\u000f, the temperature pro-\nvides the energy scale on which variation of the mean\nlevel energy changes the pumped charge (spin). Second,\nthe absolute value of the pumped charge and pumped\nspin are proportional to ( kBT)\u00001. Therefore, all plots are\nnormalized accordingly. The dependences of the pumped\ncharge (spin) on the other parameters are not a\u000bected by\ntemperature.\nThe tunnel coupling of the two dot orbitals to the left\nand the right lead is de\fned by four real tunnel-matrix\nelements,V\u0015\u000b. If the tunnel-matrix elements are equal\nfor the coupling to the left and right lead, VL\u000b=VR\u000b,\nthen, for symmetry reasons, there will be no pumping\ntransport via variation of the quantum dot's levels. To\nachieve pumping, the left-right symmetry needs to be\nbroken by changing either the magnitude or the signof one the tunnel couplings. We \fnd it convenient to\nparametrize the tunnel-matrix elements by angles \u001e\u0015,\nwhich have been introduced in the previous section (see\nEq. (25)). The tunnel-matrix elements then are given by\nthe relations V\u00151=q\n\u0000\u0015\n2\u0019\u001acos\u001e\u0015\n2andV\u00152=q\n\u0000\u0015\n2\u0019\u001asin\u001e\u0015\n2.\nFor\u001e\u0015=\u0019=2 both orbital levels are coupled symmetri-\ncally to lead \u0015, i.e.,V\u00151=V\u00152, and for\u001e\u0015=\u0000\u0019=2 the\norbitals are coupled antisymmetrically, V\u00151=\u0000V\u00152. The\nnecessary condition to get a \fnite pumped charge (spin)\nis\u001eL6=\u001eR, since\u001eL=\u001eR(even for \u0000 L6= \u0000R) leads auto-\nmatically to an e\u000bective one parameter pumping without\nany \fnite pumped charge (spin) in the adiabatic limit.\nA. Charge and spin pumping\nMotivated by the previous discussion, we \frst focus\non a tunnel-coupling con\fguration with \u0000 L= \u0000 Rbut\n\u001eL6=\u001eR, where the pumped charge and pumped spin are,\nin general, \fnite. Both depend on the mean dot-level po-\nsitions,\u000fand\u0001\u000f, which is shown in Fig. 3. Those orbital\nenergies of the quantum dot can, in principal, be adjusted\nby capacitively coupled gate votages. Figures 3(a)-3(d)\nillustrate the pumped charge (spin) for U= 0 andU=1\nand for a tunnel-coupling con\fguration where the cou-\npling to the left lead is symmetric regarding the orbitals,\n\u001eL=\u0019=2, while the coupling to the right lead is given\nby\u001eR=\u0019=4, i.e.,VR1=VR2= cot\u0019=8. In the case where\norbital 1 is symmetrically ( VL1=VR1) and orbital 2 is\nantisymmetrically ( VL2=\u0000VR2) coupled to the left and\nright leads, the pumped spin is in general \fnite while\nthe pumped charge vanishes for this con\fguration. In\nFigs. 3(e) and 3(f) the pumped spin is exemplarily cal-\nculated for\u0000\u001eL=\u001eR=\u0019=4, which is equivalent to\nVL1=VR1=\u0000VL2cot\u0019=8 =VR2cot\u0019=8.\nEach plot in Fig. 3 shows a maximum and a minimum\nvalue. For no Coulomb interaction, the maximum value\nis located at \u000f= 0 (relative to the chemical potential\n\u0016of the leads). In the limit of strong Coulomb inter-\naction, the extrema positions are shifted to values of \u000f,\nwhose order of magnitude is given by the temperature.\nThe\u0001\u000f-position of the maximum pumped charge (spin)\ndepends on the tunnel coupling to the leads and the SO-\ncoupling strength. Increasing \u000bsoalso increases the max-\nimum's position with respect to \u0001\u000f. Furthermore, the\npumped charge is in general larger for no Coulomb inter-\naction apart from special tunnel-coupling con\fgurations\ndiscussed in detail in the next section. That is not sur-\nprising since the Coulomb interaction reduces the pos-\nsible transport channels through the dot by suppressing\noccupations of the dot with more than one electron.\nB. Exchange-\feld interaction\nBoth limits U= 0 andU=1show di\u000berent sym-\nmetries with respect to \u0001\u000f!\u0000 \u0001\u000f. In the limit U= 0,8\n−4−3−2−101234\n/epsilon1/kBT−0.4−0.200.20.4∆/epsilon1/ΓQ[eη/kBT]\n−0.64−0.48−0.32−0.1600.160.320.480.64\n(a) Pumped charge for U= 0\n−2−10123456\n/epsilon1/kBT−0.4−0.200.20.4∆/epsilon1/ΓQ[eη/kBT]\n−0.80−0.64−0.48−0.32−0.1600.16 (b) Pumped charge for U=1\n−4−3−2−101234\n/epsilon1/kBT−0.4−0.200.20.4∆/epsilon1/ΓΣ/bracketleftBig\n¯h\n2η/kBT/bracketrightBig\n−1.8−1.2−0.600.61.21.8\n(c) Pumped spin for U= 0\n−2−10123456\n/epsilon1/kBT−0.4−0.200.20.4∆/epsilon1/ΓΣ/bracketleftBig\n¯h\n2η/kBT/bracketrightBig\n−0.18−0.12−0.0600.060.120.18 (d) Pumped spin for U=1\n−4−3−2−101234\n/epsilon1/kBT−0.4−0.200.20.4∆/epsilon1/ΓΣ/bracketleftBig\n¯h\n2η/kBT/bracketrightBig\n−0.64−0.48−0.32−0.160.000.160.320.480.64\n(e) Pure pumped spin for U= 0\n−2−10123456\n/epsilon1/kBT−0.4−0.200.20.4∆/epsilon1/ΓΣ/bracketleftBig\n¯h\n2η/kBT/bracketrightBig\n−0.24−0.18−0.12−0.0600.060.120.180.24 (f) Pure pumped spin for U=1\nFIG. 3. (Color online) Pumped charge (spin) with \fnite SO coupling, \u000bso= \u0000=10, in theU= 0 and the U=1limit depending\non the time-averaged orbital level positions. There are two sets of coupling parameters: First, \u001eL=\u0019=2 and\u001eR=\u0019=4 for\npanels (a)-(d), and second, the antisymmetric combination, \u001eL=\u0000\u0019=4 and\u001eR=\u0019=4, for panels (e) and (f). For all panels we\nchose \u0000 L= \u0000 R. The latter, antisymmetric combination leads to vanishing pumped charge.\nthe pumped charge (spin) is exactly antisymmetric in \u0001\u000f.\nThe antisymmetry with respect to ( \u000f;\u0001\u000f)!(\u0000\u000f;\u0000\u0001\u000f)\noriginates from the particle-hole symmetry. The anti-\nsymmetry in \u0001\u000falone, on the other hand, is a non-trivial\nresult and only valid for the lowest order contribution\nin \u0000. In the limit of strong Coulomb interaction, the\nsymmetry in \u0001\u000fdi\u000bers from the U= 0 limit. The ex-change \feld BU, which interacts with the isospin, leads\nto a contribution of the pumped charge (spin) that is\nnot antisymmetric in \u0001\u000f. Therefore, the antisymmetry\nis, in general, broken. To point out the symmetry char-\nacteristics, we study the pumped charge (spin) in two\ndi\u000berent tunnel-coupling con\fgurations (1) and (2), for9\n−0.20.00.20.40.6Q[eη/kBT](1), U= 0\n(1), U=∞\n(2), U= 0\n(2), U=∞\n−0.6−0.4−0.2 0 0.2 0.4 0.6\n∆/epsilon1/Γ−1.5−1.0−0.50.00.51.01.5Σ/bracketleftbig¯h\n2η/kBT/bracketrightbig\nFIG. 4. (Color online) Pumped charge (spin) with \fnite\nSO coupling, \u000bso= \u0000=10, in theU= 0 and the U=1limit\ndepending on the averaged level-spacing of both orbital lev-\nels. In theU= 0 limit we choose \u000f= 0, which is the position\nof the maximum value. In the limit of strong Coulomb inter-\naction, we use \u000f=kBTas an approximation to the position\nof the maximum value. The two sets of coupling parameters\n(1) and (2) are the ones given in the text (see Eq. (31)).\nThe vertical dotted lines indicate pure spin pumping . In the\nweak-coupling limit, this is only possible for pumping with\nCoulomb interaction.\nboth \u0000 L= \u0000R,\n(1) :\u001eL=\u0019\n4; \u001eR=2\u0019\n3;\n(2) :\u001eL=\u0000\u0019\n5; \u001eR=\u0019\n4; (31)\nwhich is equivalent to (1): VL1=VL2= cot\u0019=8,\nVR1=VR2= 1=p\n3 in the \frst case, and (2): VL1=VL2=\n\u0000cot\u0019=10,VR1=VR2= cot\u0019=8 in the second one. Tunnel\ncouplings (1) and (2) show that the exchange \feld can af-\nfect the pumped charge and the pumped spin di\u000berently,\nand the e\u000bect, thus, depends on the tunnel-coupling pa-\nrameters. That is accounted for by Fig. 4, where the\ncut through the contour plot (of Fig. 3 but with coupling\ncon\fgurations (1) and (2)) for \fxed \u000fis shown. The \fxed\nvalue of\u000fis\u000f= 0 in the limit of vanishing Coulomb in-\nteraction and, for comparison, \u000f=kBTin the limit of\nstrong Coulomb interaction.\nFor con\fguration (1), the exchange \feld leads to a peak\nlocated at \u0001\u000f= 0 which has a nearly symmetric behav-\nior in \u0001\u000f. The pumped spin, on the other hand, is still\napproximately antisymmetric in \u0001\u000f. Furthermore, with-\noutBU, the pumped charge (spin) is usually smaller for\nU=1, compared to U= 0, because of the reduced\nnumber of transport channels through the dot, but the\nexchange \feld can enhance the pumped charge. There\nare sets of parameters where the charge transport is even\nlarger for \fnite Coulomb interaction than for U= 0.\nFor tunnel coupling (2), the symmetric part of theexchange-\feld contribution is less important. The\npumped charge, in this case, is not dominated by a sym-\nmetric behavior as we observed for coupling (1). It is,\nrather, a shift of the point of zero pumped charge to a\n\fnite value of \u0001\u000fsimilar to the pumped spin.\nComparing the exchange-\feld contribution for con\fg-\nurations (1) and (2), the contribution to the pumped\nspin reaches its maximum where the contribution to the\npumped charge vanishes, and it is approximately half\nof its absolute maximum value where the contribution\nto the pumped charge has its maximum. For large val-\nues of exchange-\feld contribution, near its maximum, the\npumped charge has a dominant symmetric contribution\nwhile the exchange-\feld contribution to the pumped spin\nis, in general, too small to generate a peak at \u0001\u000f= 0.\nC. Pure spin pumping\nPure spin pumping is achieved whenever the pumped\ncharge vanishes but the pumped spin remains \fnite. To\n\fnd such points it is helpful that the pumped charge\nand pumped spin behave di\u000berently in the presence of\nCoulomb interaction, as discussed in the previous section,\nand that the pumped charge is more sensitive to symme-\ntry in the tunnel-matrix elements than the pumped spin.\nThis de\fnes the two strategies to obtain pure spin pump-\ning: to tune either the orbital energy levels of the dot or\nthe tunnel-matrix elements.\n1. pure spin pumping by tuning orbital energies\nFor \fxed tunnel couplings, we try to tune the orbital\nenergies such that the pumped charge vanishes but the\npumped spin remains \fnite. As discussed above, this is\neasily possible for strong Coulomb interaction, because\nin this case, the value of \u0001\u000fat which the pumped charge\nchanges its sign is shifted away from \u0001\u000f= 0 due to\nthe exchange \feld BU. In absence of Coulomb interac-\ntion (and to lowest order in the tunnel coupling strength),\nthis does, in general, not work apart from special cou-\npling con\fgurations, where the pumped charge vanishes\nindependently of the orbital energies, as discussed in the\nnext section. The reason is that both the pumped charge\nand the pumped spin are, to lowest order in \u0000, exactly\nantisymmetric in \u0001\u000f, i.e., the pumped charge and spin\nvanish simultaneously. The comparison between the two\nlimits is shown in Fig. 4. The points of pure spin pumping\nare indicated by the vertical dotted lines. Another inter-\nesting feature of the \fnite di\u000berence between the zero-\npoints for the pumped charge and the pumped spin is\nthe possibility to change the sign of the pumped spin,\nwhile charge is pumped in the same direction.10\n2. pure spin pumping by tuning tunnel couplings\nThere are cases in which pure spin current is not only\npossible for special, \fne-tuned orbital energies but for all\nvalues of\u000fand\u0001\u000f. This is illustrated in Fig. 5, which\nshows the maximum absolute value of the pumped charge\n(spin) in the ( \u000f;\u0001\u000f) parameter space, as a function of the\ncoupling parameters \u001e\u0015for \u0000 L= \u0000Rand for both limits\nU= 0 andU=1. The plots can be periodically contin-\nued. The dotted lines represent coupling con\fgurations\nwhere the pumped charge and the pumped spin are zero.\nAlong the middle dotted line, \u001eR=\u001eL, pumped charge\nand pumped spin vanish due to left-right symmetry as\nmentioned previously. Here, the tunnel-matrix elements\nare equal for the coupling to the left and the right lead,\nVL\u000b=VR\u000b. The dotted zero-lines \u001eR=\u001eL\u0006\u0019for zero\npumped charge (spin) only exist for lowest order in \u0000;\nhigher-order corrections would lead, in general, to a \f-\nnite pumped charge (spin). The latter conclusion can be\ndrawn by comparing with calculations for U= 0 which\nare exact in \u0000, e.g., by means of a scattering matrix\napproach,28,31{33and it is self-evident that even \fnite\nCoulomb interaction does not change that signi\fcantly.\nAlong these lines the tunnel-matrix elements are given\nbyVL1VR1=\u0000VL2VR2. In any case, these dotted lines\ndo no mark good candidates for pure spin pumping since,\nthere, charge and spin behave similar.\nThe situation di\u000bers along the dashed lines. The mid-\ndle dashed line, \u001eR=\u0000\u001eL, represents a con\fguration\nwhere for each orbital the absolute value of the tunnel-\nmatrix elements is the same, but one element of all four\nhas an opposite sign, i.e. VL1=VR1andVL2=\u0000VR2\n(or equivalently 1 $2). Here, we \fnd (to lowest order\nin \u0000) pure spin pumping for both vanishing and strong\nCoulomb interaction. This generalizes the result found in\nRef. 28 for the U= 0 limit to the limit of strong Coulomb\ninteraction. The dependence of the pure pumped spin for\n\u001eR=\u0000\u001eL=\u0019=4 on\u000fand\u0001\u000fin both Coulomb regimes\nis shown in Figs. 3(e) and 3(f).\nThe dashed lines \u001eR=\u0000\u001eL\u0006\u0019(equivalent to\nVL1VR1=VL2VR2) indicate a further scenario for pure\nspin pumping to lowest order in \u0000 in the U= 0 limit.\nFor higher orders in \u0000, however, the pumped charge be-\ncomes \fnite. It also becomes \fnite for U=1(and\nlowest order in \u0000) as a consequence of the exchange \feld\nacting on the isospin.\nHow important is the symmetry \u0000 L= \u0000R? To answer\nthis question, we calculate the pumped charge and spin\nfor \u0000 L= 2\u0000 R; see Fig. 6. As we see, the dependence of\nthe pumped charge and spin on \u001e\u0015changes substantially\nfor the pumped charge but not so much for the pumped\nspin. In particular, there are no straight lines with pure\nspin pumping anymore. For U= 0 (and to lowest order\nin \u0000), pure spin pumping is still possible on curved lines\nin the\u001e\u0015parameter space but not for U=1. There-\nfore, \u0000 L= \u0000 Ris a necessary requirement for pure spin\npumping.D. Spin-orbit coupling strength\nThe dependence of the pumped charge and pumped\nspin on the SO-coupling strength is visualized in Fig. 7.\nHere, the di\u000berent functions again show the maximum\nvalue of the absolute pumped charge (spin) in the ( \u000f;\u0001\u000f)\nparameter space. As can be seen from the upper plot, the\npumped charge decreases with increasing SO coupling.\nIt also decreases with increasing \u0001\u000f. In both cases, the\npumping is suppressed since the di\u000berence of the eigenen-\nergies of the dot Hamiltonian becomes large.\nIn general, the Coulomb interaction reduces the\namount of pumped charge and pumped spin. For small\nvalues of\u000bsocompared to \u0000, however, the Coulomb in-\nteraction has the opposite e\u000bect on the pumped charge.\nIn this regime, the Coulomb interaction increases the\npumped charge compared to the limit of U= 0. The\nlatter is an e\u000bect of the exchange \feld: Without the ex-\nchange \feld, the pumped charge would be reduced due\nto the Coulomb interaction. Increasing \u000bsodecreases\nthe in\ruence of the exchange \feld, i.e., for large \u000bsothe\nCoulomb interaction again reduces the pumped charge.\nFor the pumped spin, the situation di\u000bers: The exchange\n\feld reduces the pumped spin even further.\nThe pumped spin, in contrast to the pumped charge,\nvanishes for \u000bso= 0. Therefore, there is an optimal value\nof\u000bsothat maximizes the pumped spin (see Fig. 7). This\nvalue is smaller than \u0000 and it depends on the tunnel\ncoupling.\nVI. CONCLUSION\nWe analyze the possibility to build an all-electric spin\nbattery and to generate a pure spin current with a two-\nlevel quantum dot in the presence of Coulomb interac-\ntion. In the limit of vanishing Coulomb interaction, both\nare possible, as has been demonstrated in Ref. 28. Here,\nwe show that this is also possible for the experimentally\nrelevant case of a quantum dot with large Coulomb in-\nteraction. The Coulomb interaction changes the pump-\ning characteristics substantially. In particular, symme-\ntries with respect to the orbital energies change as a\nconsequence of an e\u000bective exchange \feld acting on an\nisospin de\fned by the orbital level index. The nonvanish-\ning Coulomb interaction opens the possibility to achieve\na pure spin current by tuning the orbital levels in the\nweak tunnel-coupling limit. Furthermore, we \fnd that\na pure spin current is obtained independently of the or-\nbital level energies for a certain con\fguration of tunnel\ncouplings, where one level is symmetrically and the other\none antisymmetrically coupled to the left and right lead,\nVL1=VR1andVL2=\u0000VR2in terms of tunnel-matrix\nelements.11\n−π−π\n20π\n2π\nφL−π−π\n20π\n2πφRmax\n/epsilon1,∆/epsilon1(|Q|) [eη/kBT]\n00.160.320.480.640.800.961.121.28\n(a) Pumped charge for U= 0\n−π−π\n20π\n2π\nφL−π−π\n20π\n2πφRmax\n/epsilon1,∆/epsilon1(|Q|) [eη/kBT]\n00.160.320.480.640.800.961.12 (b) Pumped charge for U=1\n−π−π\n20π\n2π\nφL−π−π\n20π\n2πφRmax\n/epsilon1,∆/epsilon1(|Σ|)/bracketleftBig\n¯h\n2η/kBT/bracketrightBig\n00.30.60.91.21.51.82.12.4\n(c) Pumped spin for U= 0\n−π−π\n20π\n2π\nφL−π−π\n20π\n2πφRmax\n/epsilon1,∆/epsilon1(|Σ|)/bracketleftBig\n¯h\n2η/kBT/bracketrightBig\n00.040.080.120.160.200.240.280.32 (d) Pumped spin for U=1\nFIG. 5. (Color online) Pumped charge and pumped spin in the U= 0 andU=1limits depending on the orbital-coupling\ncon\fguration for \fxed \u0000 L= \u0000 R. The illustrated function shows the maximum value of the pumped charge (spin) in the\n(\u000f;\u0001\u000f) parameter space for \u000bso= \u0000=10. The dotted lines represent coupling con\fgurations where the pumped charge (spin) is\nzero. Along the dashed lines, for U= 0, the pumped charge is always zero while the spin is still \fnite. For strong Coulomb\ninteraction, U=1, only the line \u001eL=\u0000\u001eRleads to vanishing pumped charge.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the DFG via\nSPP 1285 and the EU under Grant No. 238345 (GE-\nOMDISS).\nAppendix A: Diagrammatic rules\nWe now specify the diagrammatic rules to calculate\nthe diagrams of the kernels W(i;n)\u001f1\u001f0\n1\nt \u001f 2\u001f0\n2withntunneling\nlines based on Refs. 35, 36, 39, and 40. Throughout\nthe presented calculations, only the diagrams with one\ntunneling line W(i;1)\ntare necessary.\n1. Draw all topologically di\u000berent irreducible dia-\ngrams with ntunneling lines and the dot eigen-states\u001f2 fj0i;j1i;j2i;jdig\u001b, forU= 0, and\n\u001f2 fj0i;j+\"i;j\u0000\"i;j+#i;j\u0000#ig , forU=1,\ncontributing to W(i;n)\u001f1\u001f0\n1\nt \u001f 2\u001f0\n2. Each segment of the\nupper and lower contour separated by vertices is\nassigned with the corresponding eigenenergy E\u001f(t).\nEach tunneling line is labeled with the lead \u0015, spin\n\u001band energy !.\n2. Each time segment of the diagram between two ver-\ntices at the times tjandtj+1leads to a contribution\n1=(\u0001Ej+i0+), where \u0001Ejis the di\u000berence of left\ngoing energies minus right going energies.\n3. Each tunneling line that goes forward or back-\nward with respect to the Keldysh contour con-\ntributes with a factor (1 \u0000f(!)) orf(!), respec-\ntively, where f(!) is the Fermi function. Fur-12\n−π−π\n20π\n2π\nφL−π−π\n20π\n2πφRmax\n/epsilon1,∆/epsilon1(|Q|) [eη/kBT]\n00.20.40.60.81.01.21.4\n(a) Pumped charge for U= 0\n−π−π\n20π\n2π\nφL−π−π\n20π\n2πφRmax\n/epsilon1,∆/epsilon1(|Q|) [eη/kBT]\n00.160.320.480.640.800.96 (b) Pumped charge for U=1\n−π−π\n20π\n2π\nφL−π−π\n20π\n2πφRmax\n/epsilon1,∆/epsilon1(|Σ|)/bracketleftBig\n¯h\n2η/kBT/bracketrightBig\n00.30.60.91.21.51.82.12.4\n(c) Pumped spin for U= 0\n−π−π\n20π\n2π\nφL−π−π\n20π\n2πφRmax\n/epsilon1,∆/epsilon1(|Σ|)/bracketleftBig\n¯h\n2η/kBT/bracketrightBig\n00.040.080.120.160.200.240.280.32 (d) Pumped spin for U=1\nFIG. 6. (Color online) Pumped charge and pumped spin in the U= 0 andU=1limits depending on the orbital-coupling\ncon\fguration for \fxed \u0000 L= 2\u0000 R. The illustrated functions show the maximum value of the pumped charge (spin) in the ( \u000f;\u0001\u000f)\nparameter space for \u000bso= \u0000=10.\nthermore, a tunneling line that begins at a ver-\ntex containing a dot operator d\r\u001b, with\r=\u0006,\nand ends at a vertex containing dy\n\r0\u001bintroduces a\nfactor ~\u0000\u001b\u0015\r0\r=2\u0019. The matrix elements ~\u0000\u001b\u0015\r0\rare\nobtained from the transformation ~\u0000\u001b\u0015=Ty\n\u001b\u0000\u0015T\u001b,\nwith \u0000\u0015\u000b\u000b0= 2\u0019\u001aV\u0015\u000b0V\u0003\n\u0015\u000b, where\u000b;\u000b0= 1;2.\n4. Each vertex in the U= 0 limit that connects state\nj\u0000iwith statejdigives rise to a minus sign.5. The overall prefactor is \u0000i\n~(\u00001)b+c, wherebis the\nnumber of vertices on the lower contour line and c\nthe number of crossings in the tunneling lines.\n6. Integrate over all energies of the tunneling lines and\nsum over\u0015and\u001b. Sum up all contributing dia-\ngrams.\n1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n2D. D. Awschalom and M. E. Flatte, Nat. Phys. 3, 153\n(2007).3I.\u0014Zuti\u0013 c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76,\n323 (2004).\n4D. Awschalom and N. Samarth, Physics 2, 50 (2009).\n5J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997).13\n00.20.40.60.81.01.21.4max\n/epsilon1,∆/epsilon1(|Q|) [eη/kBT]U= 0\nU=∞\nU=∞withBU→0\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\nαso/Γ00.51.01.52.0max\n/epsilon1,∆/epsilon1(|Σ|)/bracketleftbig¯h\n2η/kBT/bracketrightbig\nFIG. 7. (Color online) Pumped charge and pumped spin in\ntheU= 0 andU=1limits depending on the strength of\nthe SO coupling. Additionally, the green dotted line shows\nthe pumped charge (spin) for U=1in the case that the\nexchange \feld is turned o\u000b by hand. The illustrated func-\ntions are the maximum value of the pumped charge (spin) for\ngiven SO strength, \u000bso, in the (\u000f;\u0001\u000f) parameter space. The\ncoupling parameters are: \u0000 L= \u0000 R,\u001eL=\u0019=2, and\u001eR=\u0019=4.\n6S. J. Papadakis, E. P. De Poortere, H. C. Manoharan,\nM. Shayegan, and R. Winkler, Science 283, 2056 (1999).\n7Y. A. Bychkov and E. I. Rashba, Eksp. Teor. Fiz. 39, 66\n(1984) [JETP Lett. 39, 78 (1984)].\n8Y. A. Bychkov and E. I. Rashba, J. Phys. C: Solid State\nPhys. 17, 6039 (1984).\n9S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n10H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and\nM. Johnson, Science 325, 1515 (2009).\n11A. G. Aronov and G. E. Pikus, Fiz. Tekh. Poluprovodn. 10,\n1177 (1976) [Sov. Phys. Semicond., Vol. 10, 698 (1976)].\n12P. R. Hammar, B. R. Bennett, M. J. Yang, and M. John-\nson, Phys. Rev. Lett. 83, 203 (1999).\n13F. Monzon and M. Roukes, J. Magn. Magn. Mater. 198-\n199, 632 (1999).\n14A. T. Filip, B. H. Hoving, F. J. Jedema, B. J. van Wees,\nB. Dutta, and S. Borghs, Phys. Rev. B 62, 9996 (2000).\n15H. J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier,\nH. Sch onherr, and K. H. Ploog, Phys. Rev. Lett. 87,\n016601 (2001).\n16H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H.\nDevoret, Europhys. Lett. 17, 249 (1992).\n17M. Switkes, C. M. Marcus, K. Campman, and A. C. Gos-\nsard, Science 283, 1905 (1999).18A. Fuhrer, C. Fasth, and L. Samuelson, Appl. Phys. Lett.\n91, 052109 (2007).\n19M. R. Buitelaar, V. Kashcheyevs, P. J. Leek, V. I. Talyan-\nskii, C. G. Smith, D. Anderson, G. A. C. Jones, J. Wei,\nand D. H. Cobden, Phys. Rev. Lett. 101, 126803 (2008).\n20B. Kaestner, V. Kashcheyevs, G. Hein, K. Pierz, U. Sieg-\nner, and H. W. Schumacher, Appl. Phys. Lett. 92, 192106\n(2008).\n21S. K. Watson, R. M. Potok, C. M. Marcus, and V. Uman-\nsky, Phys. Rev. Lett. 91, 258301 (2003).\n22E. R. Mucciolo, C. Chamon, and C. M. Marcus, Phys.\nRev. Lett. 89, 146802 (2002).\n23J. Wu, B. Wang, and J. Wang, Phys. Rev. B 66, 205327\n(2002).\n24J. Splettstoesser, M. Governale, and J. K onig, Phys. Rev.\nB77, 195320 (2008).\n25R. Riwar and J. Splettstoesser, Phys. Rev. B. 82, 205308\n(2010).\n26P. Sharma and P. W. Brouwer, Phys. Rev. Lett. 91, 166801\n(2003).\n27M. Governale, F. Taddei, and R. Fazio, Phys. Rev. B 68,\n155324 (2003).\n28V. Brosco, M. Jerger, P. San-Jos\u0013 e, G. Zar\u0013 and, A. Shnir-\nman, and G. Sch on, Phys. Rev. B. 82, 041309 (2010).\n29M. Eto and T. Yokoyama, J. Phys. Soc. Jpn. 79, 123711\n(2010).\n30D. J. Thouless, Phys. Rev. B 27, 6083 (1983).\n31M. B uttiker, H. Thomas, and A. Pr^ etre, Z. Phys. B 94,\n133 (1994).\n32P. W. Brouwer, Phys. Rev. B 58, R10135 (1998).\n33J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, Phys.\nRev. B. 62, R10618 (2000).\n34H. Schoeller and G. Sch on, Phys. Rev. B 50, 18436 (1994).\n35J. K onig, J. Schmid, H. Schoeller, and G. Sch on, Phys.\nRev. B 54, 16820 (1996).\n36J. K onig, H. Schoeller, and G. Sch on, Phys. Rev. Lett. 76,\n1715 (1996).\n37S. Grap, V. Meden, and S. Andergassen, Phys. Rev. B 86,\n035143 (2012).\n38S. Droste, S. Andergassen, and J. Splettstoesser, J. Phys.:\nCondens. Matter 24, 415301 (2012).\n39J. Splettstoesser, M. Governale, J. K onig, and R. Fazio,\nPhys. Rev. B 74, 085305 (2006).\n40B. Sothmann and J. K onig, Phys. Rev. B. 82, 245319\n(2010).\n41J. K onig and J. Martinek, Phys. Rev. Lett. 90, 166602\n(2003).\n42M. Braun, J. K onig, and J. Martinek, Phys. Rev. B 70,\n195345 (2004).\n43B. Wunsch, M. Braun, J. K onig, and D. Pfannkuche, Phys.\nRev. B. 72, 205319 (2005).\n44S. Legel, J. K onig, G. Burkard, and G. Sch on, Phys. Rev.\nB76, 085335 (2007).\n45D. Urban and J. K onig, Phys. Rev. B. 79, 165319 (2009).\n46J. Splettstoesser, M. Governale, J. K onig, and R. Fazio,\nPhys. Rev. Lett. 95, 246803 (2005)." }, { "title": "2401.05703v1.Orbital_Hanle_Magnetoresistance_in_a_3d_Transition_Metal.pdf", "content": "Orbital Hanle magnetoresistance in a 3 dtransition metal\nGiacomo Sala, Hanchen Wang, William Legrand, and Pietro Gambardella\nDepartment of Materials, ETH Zurich, H¨ onggerbergring 64, 8093 Zurich, Switzerland\nThe Hanle magnetoresistance is a telltale signature of spin precession in nonmagnetic conductors,\nin which strong spin-orbit coupling generates edge spin accumulation via the spin Hall effect. Here,\nwe report the existence of a large Hanle magnetoresistance in single layers of Mn with weak spin-\norbit coupling, which we attribute to the orbital Hall effect. The simultaneous observation of a\nsizable Hanle magnetoresistance and vanishing small spin Hall magnetoresistance in BiYIG/Mn\nbilayers corroborates the orbital origin of both effects. We estimate an orbital Hall angle of 0.016,\nan orbital relaxation time of 2 ps and diffusion length of the order of 2 nm in disordered Mn.\nOur findings indicate that current-induced orbital moments are responsible for magnetoresistance\neffects comparable to or even larger than those determined by spin moments, and provide a tool to\ninvestigate nonequilibrium orbital transport phenomena.\nThe original manuscript has been published with DOI\n10.1103/PhysRevLett.131.156703 by the American Phys-\nical Society, which ows the copyright of the published ar-\nticle. Publication on a free-access e-print server of files\nprepared and formatted by the authors is allowed by the\nAmerican Physical Society. The authors own the copy-\nright of these files.\nThe angular momentum induced by electric currents in\nthin films is at the origin of several types of magnetoresis-\ntance. In ferromagnet/nonmagnet bilayers, the spin cur-\nrent generated by the spin Hall effect in the nonmagnetic\nconductor interacts with the magnetization in the ferro-\nmagnet, giving rise to the spin Hall magnetoresistance\n(SMR) [1–9]. In the SMR scenario, the bilayer resistance\nis maximum (minimum) when the magnetization Mis\nperpendicular (parallel) to the polarization ζof the spin\ncurrent. When M∥ζ, the spin current is reflected at\nthe interface and is converted into a transverse charge\ncurrent by the inverse spin Hall effect. This conversion\nyields a lower resistance. In contrast, when M⊥ζ, the\nspin current is absorbed, the spin-to-charge conversion\nis inhibited, and the resistance is maximum. A similar\nmagnetoresistance occurs when the spin polarization is\ngenerated by the Rashba-Edelstein effect [10–12].\nSimilar to the SMR, the Hanle magnetoresistance\n(HMR) also originates from the modulation of the spin-\ncharge interconversion by the inverse spin Hall effect [13–\n17]. However, contrary to the SMR, the HMR appears in\nsingle nonmagnetic layers when the interfacial exchange\nfield due to the magnetization is replaced by an exter-\nnal magnetic field B. As sketched in Fig. 1, the spins\naccumulated at the edges of the nonmagnetic layer by\nthe spin Hall effect precess about B, and the net spin\npolarization decreases because of the combination of the\nprecession and electron diffusion. The consequent atten-\nuation of the inverse spin Hall effect results in a higher\nresistance. Since the Larmor precession and spin dephas-\ning are the core ingredients of the Hanle effect, the HMR\nis large when ωτ > 1, with ωandτthe Larmor frequency\nand the spin relaxation time, respectively. If τ≈1 ps,\nmagnetic fields of the order of a few Tesla are requiredto observe the HMR. Overall, the SMR and HMR pro-\nvide a powerful means to understand the generation and\ntransport of angular momentum in thin films and across\ninterfaces.\nBoth magnetoresistances scale as the square of the spin\nHall angle θ2, which quantifies the strength of the charge-\nto-spin interconversion. Therefore, heavy elements with\nlarge spin-orbit coupling such as Pt and Ta are known to\ngenerate strong SMR and HMR. However, recent theo-\nries [18–23] and experiments [24] have shown that electric\ncurrents can induce a large orbital accumulation in both\nlight and heavy metals as a consequence of the orbital\nHall and orbital Rashba-Edelstein effects. The orbital\nangular momentum can source spin-orbit torques [25–31]\nand contribute to the magnetoresistance [32–34]. How-\never, whether the orbital momentum can induce the or-\nbital analog of the SMR and HMR is an open question\n[33]. More generally, the physics underlying the bulk\nand interfacial transport of orbital momentum, the in-\nteraction of orbital moments with the magnetization and\nmagnetic fields, and the associated time- and lengthscale\nremain poorly understood.\nHere, we demonstrate the existence of a sizable HMR\nin single layers of Mn. The longitudinal and transverse\nHMR of Mn can be as high as 6 .5·10−5and 2·10−5, re-\nspectively, which are comparable to or even larger than\nthe HMR of Pt and Ta [15–17]. The magnitude of the\nHMR and the weak spin-orbit coupling of Mn suggest\nthat this magnetoresistance is driven by orbital moments.\nThe orbital origin of the HMR is further supported by the\nvery small SMR ≈6.4·10−6found in BiYIG/Mn bilayers,\nin which orbital moments do not interact directly with\nthe magnetization of BiYIG [35]. The analysis of the\nfield and thickness dependence of the HMR in Mn pro-\nvides insight into the orbital relaxation time and diffu-\nsion length. Our findings reveal the existence of a sizable\nmagnetoresistance of orbital origin in a light transition-\nmetal element, which provides a new tool to investigate\nnonequilibrium orbital transport phenomena.\nWe studied the HMR in Mn( tMn) layers with variable\nthickness tMn= 4−40 nm by performing both angular\nmeasurements of the magnetoresistance at constant mag-\nnetic field and field sweeps along orthogonal directionsarXiv:2401.05703v1 [cond-mat.mes-hall] 11 Jan 20242\n(a) (b)\nBζ\njc\njζ\nxyz\nB\nζjc\nFIG. 1. (a) Schematic representation of the Hanle effect. The\nangular momentum ζinduced by an electric current jcvia the\nspin or orbital Hall effect precesses about the magnetic field\nB. The net ζdecreases because electrons travelling along dif-\nferent paths accumulate different phases. (b) Hanle effect in\nthree different configurations. Left: when B∥ζ, no preces-\nsion occurs, and the spin (orbital) current jζpolarized along\n−yis converted by the inverse spin (orbital) Hall effect into\na charge current jcalong x. The resistance is thus minimum.\nMiddle and right: Binduces the precession of ζ, reducing\nζy, and suppressing both jζandjc. The resistance is max-\nimum. When B∥z(middle), an additional charge current\nflows along ybecause of the field-induced ζx. This current\ninduces the transverse HMR.\n0 90 180 270 360769.80769.85769.90R (W)\nb (°)2 T\n3 T\n4 T\n5 T\n6 T\n7 T\n0 90 180 270 360769.80769.82769.84769.86\n zx zyR (W)\nb, g (°)β\nαγ\nxyz(a) (b)\n(c) (d)\nR\nRH\n50 µmjc\nxy\nz\nFIG. 2. (a) Optical image of a Hall bar device and sketch of\nthe measurement configuration. RandRHare the longitu-\ndinal and transverse resistance, respectively. (b) Coordinate\nsystem showing the angle of the applied field in the xy,zx,\nandzyplanes.(c) Longitudinal resistance measured in Mn(6)\nbyzxandzyangle scans in a field of 7 T. (d) Longitudinal re-\nsistance in the same device during zyangle scans at increasing\nmagnetic fields. The solid lines in (c, d) are fits to a cos2(β)\nfunction. The angle scans are limited to the [25◦−355◦] range\nbecause of technical constraints.\n[Fig. 2(a,b)]. The Mn layers were grown by magnetron\nsputtering on Si/SiO 2substrates, capped with SiN(8),\nand patterned in Hall bars by Ar ion etching [36]. In the\nfollowing, we generalize the parameters ζandθto the\norbital polarization and orbital Hall angle, respectively,\nnoting that the spin and orbital degrees of freedom are\nnot entirely separable in the presence of spin-orbit cou-\npling [29].Figure 2(c,d) shows the room-temperature longitudinal\nresistance measured in Mn(6) while rotating the mag-\nnetic field in the zxandzyplanes. The resistance re-\nmains constant when the field rotates in the zxplane,\nwhich is perpendicular to the current-induced ζ(∥y). In\ncontrast, the resistance shows a cos2βdependence when\nthe field is rotated in the zyplane with the angles defined\nin Fig. 2(b). In particular, the resistance is maximum\nwhenB⊥ζand minimum when B∥ζ. This angular de-\npendence is typical of the HMR, whose longitudinal and\ntransverse components depend on the magnetic field vec-\ntorb=B/B= [bx, by, bz] = [cos αsinδ,sinαsinδ,cosδ],\nwith δ=β, γ, as\nR=R0+ ∆R(1−b2\ny), (1)\nRH= ∆RHbz+ ∆R bxby, (2)\nwhere R0is the field-independent longitudinal resis-\ntance, and ∆ Rand ∆ RHare determined by the diffusion\nand precession of the spin and orbital moments, as de-\nscribed below. The angular symmetry defined by Eqs.\n1-2 is the same as that of the SMR, however the depen-\ndence of the longitudinal resistance on the field strength\n[Fig. 2(d)] and the absence of magnetic layers rule out\nthe SMR as a cause of the observed effects. The ordi-\nnary Lorentz magnetoresistance is also excluded by the\nflat response measured in the zxangle scans [Fig. 2(c)],\nthe non-parabolic dependence of ∆ Rand nonlinear de-\npendence of ∆ RHon the magnetic field (see below, Fig.\n3), and the nonmonotonic dependence of���R\nR0on the film\nthickness (see below, Fig. 4(a)). Because these measure-\nments are performed at room temperature, weak antilo-\ncalization effects cannot influence the sample resistance\n[15]. Finally, the magnetoresistance in Fig. 2(c,d) can-\nnot be associated with a magnetically-ordered state of\nMn because the N´ eel temperature of bulk Mn is far be-\nlow room temperature [43, 44], and our samples do not\nshow any indication of antiferromagnetism [36].\nTo confirm the Hanle origin of the observed magnetore-\nsistance, we performed field scans along the three coordi-\nnate axes, as shown in Fig. 3(a,b). The longitudinal re-\nsistance increases monotonically with the magnetic field\nonly if the latter is applied along xorz, and remains sta-\nble when the field is along y. The transverse resistance,\ninstead, vanishes when the field is oriented along xor\nyand follows a sigmoidal curve when the field is along\nz. The dependence of both the longitudinal and trans-\nverse resistances on the direction and amplitude of the\nmagnetic field are characteristic fingerprints of the HMR\n(Fig. 1 and Eqs. 1-2) [13–17]. As a reference, we show in\nFig. 3(c,d) similar measurements performed in a Pt(5)\ndevice fabricated with the same procedure as for Mn.\nThe two materials show the same response to external\nmagnetic fields, although the parabolic (non-saturating)\nbehavior of the longitudinal (transverse) magnetoresis-\ntance of Pt indicates different scattering parameters com-\npared to Mn (see below and Table I in Ref. 36). Im-\nportantly, the normalized longitudinal HMR, defined as3\n-8-6-4-202468-0.02-0.010.000.010.02RH (W)\nB (T)\n-8-6-4-202468231.66231.68231.70231.72R (W)\nB (T)\n-8-6-4-202468-0.010-0.0050.0000.0050.010RH (W)\nB (T)\n-8-6-4-202468498.76498.78498.80 x axis\n y axis\n z axisR (W)\nB (T)(a)\n(b)(c)\n(d)\nFIG. 3. (a-b) Longitudinal and transverse resistance of Mn(9)\nmeasured during field scans along three orthogonal directions.\nThe contribution from the ordinary Hall effect was subtracted\nfrom the transverse resistance measured in the z-field-scan.\nThe solid line in (a) is a fit with shared parameters of Eq. 3 to\nboth the x- and z-field-scan magnetoresistance. The solid line\nin (b) is a fit of Eq. 4 to the z-field-scan magnetoresistance.\n(c-d) Same as (a-b) for Pt(5).\n∆R\nR0= [R(7 T)−R(0)]/R(0), is 6 .5·10−5and 9 .0·10−5\nin Mn(9) and Pt(5), respectively. Whereas the HMR of\nPt is in good agreement with earlier reports [15–17], the\nlarge HMR of Mn is unexpected for an element with neg-\nligible spin-orbit coupling [45]. As we discuss below, this\nHMR provides evidence of physics beyond the spin Hall\neffect.\nTo obtain a quantitative insight into the HMR of Mn,\nwe performed systematic measurements of the magne-\ntoresistance as a function of tMn, as shown in Fig. 4(a).\nHere, the normalized transverse HMR is calculated as\n∆RH\nR0= [RH(7 T)−RH(0)]/R(0). Both the longitudinal\nand transverse HMR show a nonmonotonic dependence\non the film thickness that is typical of diffusive phenom-\nena occurring on the lengthscale of the diffusion length\nλ[14, 15, 17]. This dependence is described by\n∆R\nR0= 2θ2\u0014λ\nttanh\u0012t\n2λ\u0013\n−Re\u001aΛ\nttanh\u0012t\n2Λ\u0013\u001b\u0015\n,\n(3)\n∆RH\nR0= 2θ2Im\u001aΛ\nttanh\u0012t\n2Λ\u0013\u001b\n, (4)\nwhere tis the film thickness and Λ−1=p\n1/λ2+i/λ2m.\nHere, λm=p\nD/ω =p\nDℏ/gµBBis an effective length\nthat quantifies the interplay between the electron diffu-\nsion and field-induced precession of the angular momen-\ntum ( D,ℏ,g, and µBare the diffusion coefficient, the\nreduced Planck constant, the Land´ e g-factor, and the\n0 10 20 30 400.02.0×10-54.0×10-56.0×10-58.0×10-5\n Longitudinal\n TransverseNormalized HMR\ntMn (nm)(a) (b)\n050100 150 200 250 3000.02.0x10-54.0x10-56.0x10-58.0x10-5\nT (K)Normalized HMR\n500520540\nR (W)FIG. 4. (a) Thickness dependence of the normalized longitu-\ndinal and transverse HMR. (b) Temperature dependence of\nthe longitudinal HMR (left axis) and longitudinal resistance\n(right axis) in Mn(9).\nBohr magneton, respectively). Equations 3-4 describe\nthe field and thickness dependence of the HMR. In partic-\nular, they predict that in the low-field regime ( ≲2 T) the\nlongitudinal and transverse resistance increase quadrat-\nically and linearly with the magnetic field, respectively,\nwhich is consistent with the experimental curves in Fig.\n3.\nFitting Eqs. 3-4 to the field dependence in Fig. 3 (both\nlongitudinal and transverse HMR) and the thickness de-\npendence in Fig. 4(a) yields consistent results, as sum-\nmarized in Table I in Ref. 36. A most striking finding\nis the similar θof Mn and Pt, which are about 0.016\nand 0.033, respectively. Whereas θof Pt is in line with\nprevious measurements [15, 17, 46, 47], that of Mn is 10\ntimes larger than found in spin-pumping measurements\nof YIG/Mn bilayers [48, 49]. The apparent contradiction\nbetween this result and the weak charge-spin conversion\nefficiency expected for an element with small spin-orbit\ncoupling may be reconciled if we consider the orbital po-\nlarization. Theory shows that the orbital Hall effect is\nmuch stronger than the spin Hall effect in 3 dtransition-\nmetal elements [50, 51]. Because both spin and orbital\nmoments precess in a magnetic field, the orbital momen-\ntum should also give rise to a finite HMR. Therefore, the\nunexpected large HMR in the absence of heavy elements\nsuggests the presence of an important orbital Hall effect.\nThe parameters extracted from the fits give insight\ninto the physics of the orbital transport, which has re-\nmained so far elusive. We find an orbital conductivity\nσL≈55ℏ\ne(Ωcm)−1, diffusion length λ≈2 nm, diffusion\ncoefficient D≈2.5·10−6m2/Vs, and orbital relaxation\ntime τ=λ2/D≈2 ps. We note that the estimate of D\nandτis affected by uncertainties on the Land´ e factor g.\nWhereas g= 2 for an electron with spin-only magnetic\nmoment, the Land´ e factor of an orbital-carrying conduc-\ntion electron depends on the orbital part of the wavefunc-\ntion. We thus expect g̸= 2, but the exact value remains\nunknown. The orbital diffusion length that we estimate\nis five times shorter than the spin diffusion length deter-\nmined from spin pumping measurements in YIG/Mn [48].\nRemarkably, λis significantly smaller than the orbital\ndiffusion length estimated from spin-orbit torque mea-\nsurements in other 3 dmetals [27, 29, 31]. The estimated4\norbital Hall conductivity is also significantly smaller than\nthe value predicted by theory for bcc Mn [50, 51], which\nis about 9000ℏ\ne(Ωcm)−1. In the absence of a clear theo-\nretical understanding of the mechanisms responsible for\nthe orbital quenching and relaxation, we hypothesize a\nrelation between the small orbital Hall conductivity, the\nshort diffusion length, and the disordered crystal struc-\nture of sputtered Mn, as discussed below.\nMn has the largest and most complex unit cell of\nall transition-metal elements [52, 53], and can grow in\ndifferent crystalline phases. X-ray diffraction and re-\nsistivity measurements indicate that our Mn films are\npolycrystalline and disordered [36]. The unusual tem-\nperature dependence of the Mn resistance in Fig. 4(b)\nis another indication of the glass-like metallic behavior\nof Mn thin films, consistently with previous work [54–\n56]. Because the transport of orbital momentum de-\npends strongly on the coherence of the orbital part of\nthe electronic wavefunction, electron scattering at grain\nboundaries and crystal defects can be major sources of\nthe orbital quenching and relaxation. In agreement with\nthis hypothesis, the thickness dependence of the Mn re-\nsistivity reported in Ref. 36 suggests an electron scat-\ntering length shorter than 5 nm. Therefore, the com-\nplex and disordered crystalline structure of Mn may be\nthe limiting factor of the orbital generation and diffu-\nsion. The anomalous resistive behavior associated with\nthe structural complexity of Mn is likely also the reason\nfor the temperature dependence of the HMR reported\nin Fig. 4(b). Differently from the HMR of Pt [15], the\nHMR of Mn decreases at low temperature. Such a de-\ncrease may be ascribed, at least in part, to the reduced\nlow temperature conductivity typical of disordered metal\nsystems [56, 57]. In future studies, relating the HMR,\nits temperature dependence, and the orbital parameters\nto the crystalline structure and disorder of metal films\nmay help to understand the mechanisms favoring orbital\ntransport. In this respect, we note that σLestimated\nfrom magneto-optical measurements in a single nonmag-\nnetic Ti layer is also two orders of magnitude smaller than\npredicted by theory [24]. This discrepancy and the scal-\ning of the orbital Hall angle with the longitudinal con-\nductivity [31] suggest that extrinsic mechanisms could\npossibly be responsible for the orbital generation. So far,\nhowever, there are no predictions of an extrinsic orbital\nHall effect nor estimates of the influence of the electronic\nscattering on the orbital transport.\nIn comparison to Mn, we find that Pt has a slightly\nhigher diffusion coefficient ( D≈4.5·10−6m2/Vs), and\na shorter relaxation time ( τ <1 ps). These values are\nin agreement with previous results [15], and explain the\nparabolic dependence of the magnetoresistance in Fig.\n3(c). The combination of a large diffusion coefficient and\na small relaxation time implies that stronger magnetic\nfields are necessary to cause the dephasing of the angular\nmomentum and, hence, modulate the sample resistance.\nIn addition to these considerations, further evidence of\nthe orbital nature of the magnetoresistance in Mn is pro-\n0 90 180 270 360615.046615.048615.050615.052615.054 xy zx zyR (W)\na, b, g (°)\n0 90 180 270 360336.85336.90336.95R (W)\na, b, g (°)(a) (b)FIG. 5. (a) Longitudinal resistance of BiYIG/Mn(10) mea-\nsured by rotating a constant magnetic field of 1 T in the xy,\nzx, and zyplanes. The red and grey lines are fits of cos2(α, β)\nto the data. (b) Same as (a) in BiYIG/Pt(5).\nvided by SMR measurements in bilayers of BiYIG/Mn\nand BiYIG/Pt [36]. Because θof Mn and Pt differ by\na factor ≈2-3, the SMR in BiYIG/Mn should be 5-10\ntimes smaller than in BiYIG/Pt if only spin accumula-\ntion occurred at the interface. This prediction, however,\nis at odds with our observation [Fig. 5]. In BiYIG/Pt(5),\nthe normalized SMR is of the order of 2.6 ·10−4, which\nis similar to earlier reports [4]. In contrast, the SMR in\nBiYIG/Mn(10) is much weaker. Fitting a cos2βfunction\nto the zyangle scan yields an SMR of about 6.4 ·10−6,\nwhich is 40 times smaller than in BiYIG/Pt(5). In com-\nparison, the HMR in BiYIG/Mn(10) ( ≈4.5·10−5) and\nBiYIG/Pt(5) ( ≈1.5·10−4) are similar to those found\nin Mn and Pt single layers grown on SiO 2[36]. In prin-\nciple, the different SMR amplitude in BiYIG/Mn and\nBiYIG/Pt could be assigned to a significant variation\nof the spin-mixing conductance between the two sam-\nples. However, previous works have found similar mix-\ning conductance at the YIG/Mn and YIG/Pt interfaces\n[48, 49, 58]. Therefore, the similar and large HMR but\nvery different SMR in Mn and Pt indicate that different\nactors are at play in the two elements. In Pt, the spin\naccumulation determined by the strong spin-orbit cou-\npling is responsible for both the HMR and SMR because\nthe spin angular momentum couples to both the external\nmagnetic field (HMR) and the interfacial exchange field\n(SMR). In contrast, orbital moments precess and dephase\nin the magnetic field, but do not interact directly with the\nmagnetization [35]. This means that a pure orbital accu-\nmulation can generate a finite HMR, but cannot produce\nany SMR. Therefore, we ascribe the magnetoresistance of\nMn to an orbital-driven effect. In this scenario, a small\nbut non-zero spin orbit coupling may generate the tiny\nSMR detected in BiYIG/Mn(10).\nIn conclusion, we have observed a strong Hanle effect in\nMn thin films. The combination of an unexpected large\nHMR with an almost zero SMR in a 3 delement indi-\ncates that this magnetoresistance originates from orbital\nmoments rather than spin moments. Similar orbital mag-\nnetoresistance effects may exist in other transition met-\nals. Additionally, we find an orbital Hall angle of about\n0.016, an orbital relaxation time of about 2 ps and an\norbital diffusion length of the order of 2 nm. The small5\norbital Hall conductivity and short diffusion length are\nat odds with the expected strength and lengthscale of or-\nbital transport and associated to the disordered structure\nof Mn thin films. The Hanle effect therefore provides a\nnew means to explore the orbital physics in nonmagnetic\nelements.ACKNOWLEDGMENTS\nWe thank Dongwook Go, Mingu Kang, Paul N¨ oel, and\nRichard Schlitz for providing useful comments. This\nresearch was supported by the Swiss National Science\nFoundation (Grant No. 200020-200465). Hanchen Wang\nacknowledges the support of the China Scholarship Coun-\ncil (CSC, Grant No. 202206020091). William Legrand\nacknowledges the support of the ETH Zurich Postdoc-\ntoral Fellowship Program (21-1 FEL-48).\nI. BIBLIOGRAPHY\n[1] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr¨ ags,\nM. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B.\nGoennenwein, and E. Saitoh, Spin Hall Magnetoresis-\ntance Induced by a Nonequilibrium Proximity Effect,\nPhysical Review Letters 110, 206601 (2013).\n[2] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer,\nTheory of spin Hall magnetoresistance, Physical Review\nB87, 144411 (2013).\n[3] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and\nJ. Ben Youssef, Spin-Hall magnetoresistance in platinum\non yttrium iron garnet: Dependence on platinum thick-\nness and in-plane/out-of-plane magnetization, Physical\nReview B 87, 184421 (2013), arXiv:1301.3266.\n[4] M. Althammer, S. Meyer, H. Nakayama, M. Schreier,\nS. Altmannshofer, M. Weiler, H. Huebl, S. Gepr¨ ags,\nM. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M.\nSchmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen,\nG. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein,\nQuantitative study of the spin Hall magnetoresistance in\nferromagnetic insulator/normal metal hybrids, Physical\nReview B 87, 224401 (2013), arXiv:1304.6151.\n[5] M. Isasa, A. Bedoya-Pinto, S. V´ elez, F. Golmar,\nF. S´ anchez, L. E. Hueso, J. Fontcuberta, and\nF. Casanova, Spin Hall magnetoresistance at Pt/CoFe\n2 O 4 interfaces and texture effects, Applied Physics Let-\nters105, 142402 (2014), arXiv:1307.1267.\n[6] C. O. Avci, K. Garello, J. Mendil, A. Ghosh,\nN. Blasakis, M. Gabureac, M. Trassin, M. Fiebig,\nand P. Gambardella, Magnetoresistance of heavy and\nlight metal/ferromagnet bilayers, Applied Physics Let-\nters107, 192405 (2015), arXiv:1510.06285.\n[7] C. O. Avci, K. Garello, A. Ghosh, M. Gabureac, S. F.\nAlvarado, and P. Gambardella, Unidirectional spin Hall\nmagnetoresistance in ferromagnet-normal metal bilayers,\nNature Physics 11, 570 (2015), arXiv:1502.0689.\n[8] J. Kim, P. Sheng, S. Takahashi, S. Mitani, and\nM. Hayashi, Spin Hall Magnetoresistance in Metallic\nBilayers, Physical Review Letters 116, 097201 (2016),\narXiv:1503.08903.\n[9] X.-P. Zhang, F. S. Bergeret, and V. N. Golovach,\nTheory of Spin Hall Magnetoresistance from a Mi-\ncroscopic Perspective, Nano Letters 19, 6330 (2019),arXiv:1903.10558.\n[10] S. S.-L. Zhang, G. Vignale, and S. Zhang, Anisotropic\nmagnetoresistance driven by surface spin-orbit\nscattering, Physical Review B 92, 024412 (2015),\narXiv:1504.03310.\n[11] V. L. Grigoryan, W. Guo, G. E. W. Bauer, and J. Xiao,\nIntrinsic magnetoresistance in metal films on ferromag-\nnetic insulators, Physical Review B 90, 161412 (2014),\narXiv:1407.3571.\n[12] H. Nakayama, Y. Kanno, H. An, T. Tashiro, S. Haku,\nA. Nomura, and K. Ando, Rashba-Edelstein Magnetore-\nsistance in Metallic Heterostructures, Physical Review\nLetters 117, 116602 (2016), arXiv:1609.04122.\n[13] M. Johnson and R. H. Silsbee, Interfacial charge-spin\ncoupling: Injection and detection of spin magnetization\nin metals, Physical Review Letters 55, 1790 (1985).\n[14] M. I. Dyakonov, Magnetoresistance due to Edge Spin Ac-\ncumulation, Physical Review Letters 99, 126601 (2007),\narXiv:0705.2738.\n[15] S. V´ elez, V. N. Golovach, A. Bedoya-Pinto, M. Isasa,\nE. Sagasta, M. Abadia, C. Rogero, L. E. Hueso,\nF. S. Bergeret, and F. Casanova, Hanle Magnetore-\nsistance in Thin Metal Films with Strong Spin-Orbit\nCoupling, Physical Review Letters 116, 016603 (2016),\narXiv:1502.04624.\n[16] H. Wu, X. Zhang, C. H. Wan, B. S. Tao, L. Huang, W. J.\nKong, and X. F. Han, Hanle magnetoresistance: The role\nof edge spin accumulation and interfacial spin current,\nPhysical Review B 94, 174407 (2016).\n[17] J. Li, A. H. Comstock, D. Sun, and X. Xu, Comprehen-\nsive demonstration of spin Hall Hanle effects in epitaxial\nPt thin films, Physical Review B 106, 184420 (2022).\n[18] D. Go, J.-P. Hanke, P. M. Buhl, F. Freimuth,\nG. Bihlmayer, H.-W. Lee, Y. Mokrousov, and S. Bl¨ ugel,\nToward surface orbitronics: giant orbital magnetism from\nthe orbital Rashba effect at the surface of sp-metals, Sci-\nentific Reports 7, 46742 (2017).\n[19] D. Go, D. Jo, C. Kim, and H. W. Lee, Intrinsic Spin\nand Orbital Hall Effects from Orbital Texture, Physical\nReview Letters 121, 86602 (2018), arXiv:1804.02118.\n[20] L. Salemi, M. Berritta, A. K. Nandy, and P. M. Op-\npeneer, Orbitally dominated Rashba-Edelstein effect in\nnoncentrosymmetric antiferromagnets, Nature Commu-\nnications 10, 5381 (2019), arXiv:1905.08279.6\n[21] S. Bhowal and S. Satpathy, Intrinsic orbital and spin\nHall effects in monolayer transition metal dichalco-\ngenides, Physical Review B 102, 035409 (2020),\narXiv:2006.07754.\n[22] A. Johansson, B. G¨ obel, J. Henk, M. Bibes, and I. Mertig,\nSpin and orbital Edelstein effects in a two-dimensional\nelectron gas: Theory and application to SrTiO3 in-\nterfaces, Physical Review Research 3, 013275 (2021),\narXiv:2006.14958.\n[23] L. Salemi and P. M. Oppeneer, First-principles theory\nof intrinsic spin and orbital Hall and Nernst effects in\nmetallic monoatomic crystals, Physical Review Materials\n6, 095001 (2022), arXiv:2203.17037.\n[24] Y.-G. Choi, D. Jo, K.-h. Ko, D. Go, K.-H. Kim, H. G.\nPark, C. Kim, B.-C. Min, G.-M. Choi, and H.-w. Lee,\nObservation of the orbital Hall effect in a light metal Ti,\nNature 619, 52 (2023).\n[25] S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth,\nS. Becker, F. Kammerbauer, J. Yang, G. Jakob,\nY. Mokrousov, and M. Kl¨ aui, Harnessing Orbital-to-Spin\nConversion of Interfacial Orbital Currents for Efficient\nSpin-Orbit Torques, Physical Review Letters 125, 177201\n(2020).\n[26] J. Kim, D. Go, H. Tsai, D. Jo, K. Kondou, H.-w. Lee, and\nY. Otani, Nontrivial torque generation by orbital angular\nmomentum injection in ferromagnetic-metal Cu/Al2O3\ntrilayers, Physical Review B 103, L020407 (2021).\n[27] S. Lee, M.-G. Kang, D. Go, D. Kim, J.-H. Kang, T. Lee,\nG.-H. Lee, J. Kang, N. J. Lee, Y. Mokrousov, S. Kim,\nK.-J. Kim, K.-J. Lee, and B.-G. Park, Efficient conver-\nsion of orbital Hall current to spin current for spin-orbit\ntorque switching, Communications Physics 4, 234 (2021),\narXiv:2106.02286.\n[28] D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun,\nD. Jo, S. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park,\nB.-C. Min, H. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee,\nOrbital torque in magnetic bilayers, Nature Communica-\ntions 12, 6710 (2021).\n[29] G. Sala and P. Gambardella, Giant orbital Hall effect\nand orbital-to-spin conversion in 3d, 5d, and 4f metal-\nlic heterostructures, Physical Review Research 4, 033037\n(2022).\n[30] S. Dutta and A. A. Tulapurkar, Observation of nonlocal\norbital transport and sign reversal of dampinglike torque\nin Nb/Ni and Ta/Ni bilayers, Physical Review B 106,\n184406 (2022).\n[31] H. Hayashi, D. Jo, D. Go, T. Gao, S. Haku,\nY. Mokrousov, H.-W. Lee, and K. Ando, Observation\nof long-range orbital transport and giant orbital torque,\nCommunications Physics 6, 32 (2023), arXiv:2202.13896.\n[32] H.-W. Ko, H.-J. Park, G. Go, J. H. Oh, K.-W.\nKim, and K.-J. Lee, Role of orbital hybridization in\nanisotropic magnetoresistance, Physical Review B 101,\n184413 (2020), arXiv:2004.04370.\n[33] S. Ding, Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu,\nS. Becker, W. Yang, H. Du, C. Wang, Y. Yang, G. Jakob,\nM. Kl¨ aui, Y. Mokrousov, and J. Yang, Observation of the\nOrbital Rashba-Edelstein Magnetoresistance, Physical\nReview Letters 128, 067201 (2022), arXiv:2105.04495.\n[34] S. Ding, P. No¨ el, G. K. Krishnaswamy, and P. Gam-\nbardella, Unidirectional orbital magnetoresistance in\nlight-metal–ferromagnet bilayers, Physical Review Re-\nsearch 4, L032041 (2022).\n[35] D. Go, D. Jo, H.-W. Lee, M. Kl¨ aui, and Y. Mokrousov,Orbitronics: Orbital currents in solids, EPL (Europhysics\nLetters) 135, 37001 (2021), arXiv:2107.08478.\n[36] See Supplemental Material [URL] for the sample charac-\nterization, fits of the HMR, and HMR measurements in\nBiYIG/Mn and BiYIG/Pt samples, which includes Refs.\n[37-42], .\n[37] A. M. Miller, M. Lemon, M. A. Choffel, S. R. Rich,\nF. Harvel, and D. C. Johnson, Extracting information\nfrom X-ray diffraction patterns containing Laue oscilla-\ntions, Zeitschrift f¨ ur Naturforschung B 77, 313 (2022).\n[38] S. Kahl and A. M. Grishin, Enhanced Faraday rotation\nin all-garnet magneto-optical photonic crystal, Applied\nPhysics Letters 84, 1438 (2004).\n[39] L. Caretta, S.-H. Oh, T. Fakhrul, D.-K. Lee, B. H. Lee,\nS. K. Kim, C. A. Ross, K.-J. Lee, and G. S. D. Beach, Rel-\nativistic kinematics of a magnetic soliton, Science 370,\n1438 (2020).\n[40] F. Boakye and A. D. Grassie, The influence of deposition\nparameters on the low temperature resistivity of α-Mn\nthin films, Thin Solid Films 221, 224 (1992).\n[41] L. Skuja, M. Hirano, H. Hosono, and K. Kajihara, De-\nfects in oxide glasses, physica status solidi (c) 2, 15\n(2005).\n[42] E. H. Poindexter and W. L. Warren, Paramagnetic Point\nDefects in Amorphous Thin Films of SiO2 and Si3 N 4:\nUpdates and Additions, Journal of The Electrochemical\nSociety 142, 2508 (1995).\n[43] S. Murayama and H. Nagasawa, Magnetoresistance in\nAntiferromagnetic α-Mn Metal, Journal of the Physical\nSociety of Japan 43, 1216 (1977).\n[44] F. Boakye and K. G. Adanu, The Neel temperature of\na-Mn thin films, Thin Solid Films 279, 29 (1996).\n[45] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel,\nand F. Y. Yang, Scaling of Spin Hall Angle in 3d, 4d,\nand 5d Metals Y3FeO12/Metal Spin Pumping, Physical\nReview Letters 112, 197201 (2014).\n[46] X. Tao, Q. Liu, B. Miao, R. Yu, Z. Feng, L. Sun, B. You,\nJ. Du, K. Chen, S. Zhang, L. Zhang, Z. Yuan, D. Wu,\nand H. Ding, Self-consistent determination of spin Hall\nangle and spin diffusion length in Pt and Pd: The role\nof the interface spin loss, Science Advances 4, eaat1670\n(2018).\n[47] A. Manchon, J. ˇZelezn´ y, I. M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nCurrent-induced spin-orbit torques in ferromagnetic and\nantiferromagnetic systems, Reviews of Modern Physics\n91, 035004 (2019), arXiv:1801.09636.\n[48] C. Du, H. Wang, F. Yang, and P. C. Hammel, System-\natic variation of spin-orbit coupling with d-orbital fill-\ning: Large inverse spin Hall effect in 3d transition metals,\nPhysical Review B 90, 140407 (2014), arXiv:1410.1590.\n[49] D. Qu, T. Higo, T. Nishikawa, K. Matsumoto, K. Kon-\ndou, D. Nishio-Hamane, R. Ishii, P. K. Muduli, Y. Otani,\nand S. Nakatsuji, Large enhancement of the spin Hall ef-\nfect in Mn metal by Sn doping, Physical Review Materials\n2, 102001 (2018).\n[50] D. Jo, D. Go, and H.-W. Lee, Gigantic intrinsic orbital\nHall effects in weakly spin-orbit coupled metals, Physical\nReview B 98, 214405 (2018), arXiv:1808.05546.\n[51] L. Salemi and P. M. Oppeneer, Theory of magnetic\nspin and orbital Hall and Nernst effects in bulk fer-\nromagnets, Physical Review B 106, 024410 (2022),\narXiv:2203.17037.\n[52] J. A. Oberteuffer and J. A. Ibers, A refinement of the7\natomic and thermal parameters of α-manganese from a\nsingle crystal, Acta Crystallographica Section B Struc-\ntural Crystallography and Crystal Chemistry 26, 1499\n(1970).\n[53] V. Sliwko, P. Mohn, and K. Schwarz, The electronic\nand magnetic structures of alpha - and beta -manganese,\nJournal of Physics: Condensed Matter 6, 6557 (1994).\n[54] A. Grassie and F. Boakye, The low temperature resistiv-\nity of α-manganese films and its relationship to deposi-\ntion conditions, Thin Solid Films 57, 169 (1979).\n[55] F. Boakye, Temperature dependence of the resistivity\nof amorphous Mn thin films, Journal of Non-CrystallineSolids 249, 189 (1999).\n[56] J. H. Mooij, Electrical conduction in concentrated disor-\ndered transition metal alloys, Physica Status Solidi (a)\n17, 521 (1973).\n[57] M Kaveh and N F Mott, Universal dependences of the\nconductivity of metallic disordered systems on tempera-\nture, magnetic field and frequency, Journal of Physics C:\nSolid State Physics 15, L707 (1982).\n[58] C. Du, H. Wang, P. C. Hammel, and F. Yang, Y3Fe5O12\nspin pumping for quantitative understanding of pure spin\ntransport and spin Hall effect in a broad range of mate-\nrials (invited), Journal of Applied Physics 117, 172603\n(2015), arXiv:1410.1597." }, { "title": "1508.00932v1.Spin_structure_of_harmonically_trapped_one_dimensional_atoms_with_spin_orbit_coupling.pdf", "content": "arXiv:1508.00932v1 [cond-mat.quant-gas] 4 Aug 2015Spin structure of harmonically trapped one-dimensional at oms with spin-orbit\ncoupling\nQ. Guan1and D. Blume1\n1Department of Physics and Astronomy, Washington State Univ ersity, Pullman, Washington 99164-2814, USA\n(Dated: June 10, 2021)\nWe introduce a theoretical approach to determine the spin st ructure of harmonically trapped\natoms with two-body zero-range interactions subject to an e qual mixture of Rashba and Dresselhaus\nspin-orbit coupling created through Raman coupling of atom ic hyperfine states. The spin structure\nof bosonic and fermionic two-particle systems with finite an d infinitetwo-body interaction strength g\nis calculated. Taking advantage of the fact that the N-boson and N-fermion systems with infinitely\nlarge coupling strength gare analytically solvable for vanishing spin-orbit coupli ng strength kso\nand vanishing Raman coupling strength Ω, we develop an effect ive spin model that is accurate to\nsecond-order in Ω for any ksoand infinite g. The three- and four-particle systems are considered\nexplicitly. It is shown that the effective spin Hamiltonian, which contains a Heisenberg exchange\nterm and an anisotropic Dzyaloshinskii-Moriya exchange te rm, describes the transitions that these\nsystems undergo with the change of ksoas a competition between independent spin dynamics and\nnearest-neighbor spin interactions.\nPACS numbers:\nI. INTRODUCTION\nSpin-orbit (more precisely, spin-momentum) coupled\nsystems continue to attract a great deal of attention\ndue to the rich physics of the spin-Hall effect, topo-\nlogical insulators, and Majorana fermions [1–5]. While\nthese topics fall traditionally into condensed matter ter-\nritory, recent advances in cold atomic gases have led to a\nfruitful cross fertilization of atomic and condensed mat-\nter physics. On the experimental side, synthetic gauge\nfieldshavebeenrealizedinultracoldatomsystems[6–15].\nOn the theoretical side, systems with different kinds of\nspin-orbit coupling have been studied at the many- and\nfew-body levels [6, 16–24]. Understanding the effects of\nspin-orbit coupling in few-atom systems opens the door\nto a bottom-up understanding of many-body systems.\nNowadays, ultracold few-atom systems can be prepared\nand manipulated in experiments [25, 26]. For example,\ntaking advantage of a Fano-Feshbach resonance, the in-\nteraction between the atoms can be tuned [27]. More-\nover, the confinement geometry can be changed from\nthree-dimensional to effectively two-dimensional to ef-\nfectively one-dimensional [28]. These experimental ad-\nvances were guided by and stimulated a good number\nof theoretical few-body studies [29–42]. Much analyti-\ncal work has been done by approximating the true alkali\natom-alkali atom potential by a zero-range contact po-\ntential [29, 32–34, 36, 40–45]. This approximation cap-\ntures the low energy regime reliably but fails to repro-\nduce high-energy properties (such as the characteristics\nof deeply bound states). Assuming zero-range interac-\ntions, the energy spectra of two one-dimensional bosons\nand two one-dimensional fermions with arbitrary two-\nbody coupling constant and spin-orbit coupling strength\nhave been calculated and the interplay between the spin-\norbit coupling term, Raman coupling term, and the two-\nbody potential has been analyzed [20].Extending earlier work [20, 21], this paper investi-\ngates the spin structure of harmonically trapped atoms\nwith two-body zero-range interactions subject to an\nequal mixture of Rashba and Dresselhaus spin-orbit cou-\npling created through Raman coupling of atomic hyper-\nfine states. The spin structure of two identical one-\ndimensional bosons and two identical one-dimensional\nfermions with finite and infinite two-body interaction\nstrengthgis calculated for weak to strong spin-orbit\ncoupling strength. The bosonic and fermionic systems\ndisplay, in general, different behaviors as the spin-orbit\ncoupling strength ksoand the two-body coupling con-\nstantgare changed. For infinite g, however, the bosonic\nand fermionic systems display the same spin structure.\nAn interesting transition of the spin structure is found in\ngoing from small to large kso. To understand the system\nbehaviors for finite and infinite g, an effective Hamilto-\nnian forany ksothat is accurateup to secondorderin the\nRaman coupling strength Ω is derived. Effective Hamil-\ntonian have been utilized in various areas of physics and\nthe Hubbardmodel [46], the Born-Oppenheimerapproxi-\nmation[47], the Isingmodel [48], andthe Heisenbergspin\nchain [49] are prominent examples. Various approaches\nto generate effective Hamiltonian and the connection be-\ntween perturbation theory and selected effective Hamil-\ntonian have been discussed in Refs. [50–55].\nFor infinitely large gand arbitrary N, we integrate out\nthe spatial degrees of freedom and recast the resulting\neffective Hamiltonian in terms of spin operators. Single\nspin terms are proportional to Ω while spin-spin interac-\ntions are proportional to Ω2. Effective spin Hamiltonian\nhavebeenderivedpreviouslyforone-dimensionalsystems\nwithout spin-orbit and Raman coupling [56–59]. In those\ncases, spin-spin interactions were introduced by allowing\nfor small deviations from |g|=∞; in essence, this in-\ntroduces a tunneling term. In our case, the single spin\nand the spin-spin terms are introduced by the Raman2\ncoupling. The single spin term is proportional to Ω and\nhas been discussed in Ref. [21]. In essence, the Raman\ncoupling creates an effective local /vectorB-field that the spins\nfollow. The spin-spin interaction term has, to the best of\nour knowledge, not been discussed before in this context.\nThis term contains two contributions. The first contri-\nbution is of the type of the “usual” Heisenberg exchange\nterm [45], i.e., it is a /vector σj·/vector σkterm, where /vector σjis the spin\noperatorof spin j. The second contribution is of the type\nof the anisotropic Dzyaloshinskii-Moriya exchange term,\ni.e., it is a one-dimensional analog of the /vectorD·(/vector σj×/vector σk)\nterm [60–62], where /vectorDis a constant vector.\nThe remainder of this paper is organized as follows:\nSection II defines the system Hamiltonian, discusses\nits symmetries and introduces an effective low-energy\nHamiltonian. Section III determines the spin structure\nof the two-particle system for different gandkso. Using\nthe effective low-energy Hamiltonian, Sec. IV derives an\neffective spin Hamiltonian for infinitely large gand ap-\nplies this Hamiltonian to determine the spin correlations\nofthe three-and four-particlesystems. Atransition from\na regime where the dynamics is governed by single spin\nphysics to a regime where the dynamics is governed by\nspin-spin interactions is obtained. Finally, Sec. V sum-\nmarizesandconcludes. Theappendicescontainanumber\nof technical details, including the evaluation of the ma-\ntrix elements for infinite g, which are needed to construct\nboth the effective Hamiltonian and the full Hamiltonian.\nThe analytical techniques and results are expected to be\nuseful for other one-dimensional studies.\nII. SYSTEM HAMILTONIAN AND GENERAL\nCONSIDERATIONS\nWe consider None-dimensional atoms of mass min\na harmonic trap with angular trapping frequency ωand\nzero-rangetwo-body interactions gδ(xj−xk), wherexjis\nthe position of the jth particle with respect to the center\nof the trap. We assume that each particle feels an equal\nmixture of Rashba and Dresselhaus spin-orbit coupling\nof strength ksoand Raman coupling of strength Ω. The\nsystem Hamiltonian ˜Hreads\n˜H=HsrˆI+N/summationdisplay\nj=1/planckover2pi1kso\nmpjσ(j)\ny+Ω\n2σ(j)\nx,(1)\nwhere\nHsr=N/summationdisplay\nj=1−/planckover2pi12\n2m∂2\n∂x2\nj+1\n2mω2x2\nj+/summationdisplay\nj/ h _\n-0.2-0.100.10.2 / h _\n-3 -2 -1 0 1 2 3\nx/aho-0.04-0.03-0.02-0.010/ h _\n-3 -2 -1 0 1 2 3\nx/aho-0.02-0.0100.010.02/ h _(a) (b)\n(c) (d)\nFIG. 4: Benchmarking the effective Hamiltonian approach for\ntheN= 2 ground state for Ω = /planckover2pi1ω/2 andg=∞. Panels (a)\nand (b) show /angbracketleftSx(x)/angbracketrightand/angbracketleftSz(x)/angbracketrightforksoaho= 1/5; panels\n(c) and (d) show /angbracketleftSx(x)/angbracketrightand/angbracketleftSz(x)/angbracketrightforksoaho= 4. The\nsolid lines show the spin structure obtained from the exact\ndiagonalization while the dashed lines show the spin struct ure\nobtained within the effective Hamiltonian approach. On the\nscale shown, the solid and dashed lines nearly coincide.\ncoefficients Cf\n1andCf\n2in Eq. (24) vanish, yielding\n/angbracketleftSb\nz(x)/angbracketright=/angbracketleftSf\nx(x)/angbracketright=/angbracketleftSf\nz(x)/angbracketright= 0 and /angbracketleftSb\nx(x)/angbracketright /negationslash= 0.\nFor a non-vanishing spin-orbit coupling strength, the sin-\nglet and triplet states are coupled, leading to non-zero\n/angbracketleftSb\nz(x)/angbracketright,/angbracketleftSf\nx(x)/angbracketright,and/angbracketleftSf\nz(x)/angbracketright. With increasing g, the\nenergy difference between the singlet and triplet states\nforkso= 0 decreases. As a consequence, for a fixed and\nrelatively small kso, the spin structures for two identical\nbosons and two identical fermions are more similar for\nlargergthan for smaller g[see Figs. 2(a) and 2(c) for\nksoaho/lessorsimilar1.5]. However, the coupling between the sin-\nglet and triplet states is weakened with increasing kso.\nSpecifically, for fixed g, the spin structures for bosons\nand fermions differ more as ksoincreases [see Figs. 2(a)\nand 2(c) for ksoaho>2]. For a larger kso, a largerg\nis needed to get the triplet (singlet) state mixed signif-\nicantly into the ground state for two identical fermions\n(bosons), resulting in similar spin structures for bosons\nand fermions. For an infinitely large g, the relative even\nand odd parity states have the same energy and the spin\nstructures for bosonic and fermionic systems are identi-\ncal for allksoand Ω (see the solid lines in Fig. 2). This\neffect can be attributed to the Bose-Fermi duality (see\nnext section for more details).\nOur discussion so far has been based on the effective\nHamiltonian approach. To benchmark this approach, we\ncompare the effective Hamiltonian results with those ob-\ntained from the exact diagonalization for various g,kso,\nand Ω combinations. The agreement is good for all cases\nconsidered. As an example, Fig. 4 compares the two-\nparticle spin structure for infinite gand Ω = /planckover2pi1ω/2. Fig-\nures 4(a) and 4(b) are for small kso(ksoaho≪1) and\nFigs. 4(c) and 4(d) are for large kso(ksoaho≫1). The\nagreement between the effective Hamiltonian approach7\nresults (solid lines) and the exact diagonalization ap-\nproach results (dashed lines) is very convincing.\nIV. N-PARTICLE SYSTEM WITH g=∞\nA. Formulation\nWhen the two-body coupling constant gis infinitely\nlarge, the particles cannot pass through each other.\nIn the absence of the spin-orbit and Raman coupling\nterms, the atomic gas behaves like a Tonks-Girardeau\ngas [30, 63]. The Tonks-Girardeau gas has a large de-\ngeneracy and bosons fermionize [31, 63, 64]. The fact\nthat the particles cannot pass through each other im-\nplies that the particles can be ordered. Since there are\nN! ways to order the particles, the degeneracy of each\neigenstate of Hsrwithg=∞isN! if the particle ex-\nchange symmetry is not being enforced. We thus pursue\nan approach where we first determine the eigenstates of\nHsrˆIfor a fixed particle ordering. The resulting eigen-\nstates are then used either to calculate the eigenstates\nand eigenenergies of Hthrough exact diagonalization or\nto calculate the eigenstates and eigenenergies of Heffby\nidentifying an appropriate subspace HL. The fact that\nthe particles can be ordered also allows us to derive an\neffective spin Hamiltonian Hspin\neffthat is independent of\nthe spatial coordinates.\nFor infinite g, the eigenstates of Hsrcan be written\nas [31]\nφn1,n2,...,nN(/vector x) =D(n1,n2,···,nN)Θxj10, i.e., the states\nwith|Ms|>0 are two-fold degenerate. The first-order\nterm breaks the degeneracy of the eigenstates with Ms\nand−Ms. As a result, the eigenstates of the full Hamil-\ntonianHspin\neff(including zero-, first-, and second-order\nterms) are, for kso→ ∞, approximately superpositions\nof states that have the same absolute value of the Ms\nquantum number. The σ(j)\n−σ(k)\n+andσ(j)\n+σ(k)\n−terms cor-\nrespond to nearest neighbor spin hopping. These terms\nlead to a lowering of the energy. The more possibility\nfor the nearest neighbor spin hopping a state has, theN kso→0 kso→ ∞\n2P0= 1/2,P2= 1/2 P0= 1,P2= 0\n3P1= 3/4,P3= 1/4 P1= 1,P3= 0\n4P0= 3/8,P2= 1/2,P4= 1/8P0= 1,P2= 0,P4= 0\nTABLE I: Spin correlations for the ground state in the limit-\ningcases kso→0andkso→ ∞forvarious N. Theprobability\nP|Ms|that the ground state has the absolute value of Msis\nreported. For small kso, all spin states are equally weighted,\nwhich means that P|Ms|is proportional to the number of spin\nstates that have the same |Ms|. For large kso, the spin states\nare not equally weighted. In this case, the ground state con-\ntains only spin states with the minimum allowed |Ms|.\nlower the energy associated with that state is. For ex-\nample, the states | ↑↓/angbracketrightyand| ↓↑/angbracketrightyare “connected” via\nnearest neighbor hoppings while the states | ↑↑/angbracketrightyand\n| ↓↓/angbracketrightyare not connected with each other or with | ↑↓/angbracketrighty\nor| ↓↑/angbracketrightyvia nearest neighbor hopping. Correspondingly,\ntheN= 2 ground state is a linear combination of the\n| ↑↓/angbracketrightyand| ↓↑/angbracketrightystates. For N= 3, e.g., the state | ↑↑↓/angbracketrighty\nis connected to | ↑↓↑/angbracketrightyviaσ(−)\n2σ(+)\n3and to| ↓↑↑/angbracketrightyvia\nσ(−)\n1σ(+)\n3while the state | ↓↓↑/angbracketrightyis connected to | ↓↑↓/angbracketrighty\nviaσ(+)\n2σ(−)\n3and to| ↑↓↓/angbracketrightyviaσ(+)\n1σ(−)\n3. Correspond-\ningly, theN= 3 ground state is a linear combination of\nall|Ms|= 1 states. Table I shows the values of P|Ms|=m\nin the limits kso→0 andkso→ ∞forN= 2−4.\nFigure 5 shows the probability to find the system in a\nstate with a given |Ms|as a function of ksofor the two-\n, three-, and four-particle systems with infinitely large\ng(see the next subsection for the calculational details).\nFor largekso,P|Ms|=0= 1 for the two-particle system,\nP|Ms|=1= 1 for the three-particle system, and P|Ms|=0=\n1 for the four-particle system.\nB. Application to systems with N= 2−4\nThis section evaluates the expectation values of the\nspin operators defined in Eqs. (7)-(9) for the three- and\nfour-particle systems with infinite g. ForN= 2, the\nexpectation values of Sx(x) andSz(x) for infinitely large\nghave been calculated in Sec. III. For N= 3, the non-\nsymmetrized ground state wave function of the effective\nHamiltonian Heffis\nψgr=D(0,1,2)√\n2Θx1/planckover2pi1ω, making\na perturbative approach meaningless. However, the full\nperturbativeexpressionalsocontainsthematrixelements\nand energy denominator. It turns out that the product\nof three matrix elements is highly suppressed compared\nto the product of two matrix elements. We conclude that\nthe second-order effective spin Hamiltonian capture the\nphysics, including the spin structure up to, at first sight,\nsurprisingly large Ω /(/planckover2pi1ω) for relatively large kso.\nA key ingredient that went into deriving the effective\nlow-energy spin Hamiltonian is that the particles in one-\ndimensional space can be ordered if the two-body cou-\npling constant gis infinitely large. For finite g, this is\nnot the case, i.e., particles are allowed to pass through\neach other. In this case, a low-energy Hamiltonian that\ndepends on the spatial and spin degrees of freedom was\nderived (in fact, the effective spin Hamiltonian for g=∞\nwas derived by taking this Hamiltonian and integrat-\ning out the spatial degrees of freedom). The low-energy\nHamiltonianwastestedfortwoparticlesandshowntore-\nproduce the full Hamiltonian dynamics well. Moreover,\nit was shown to provide a powerful theoretical frame-\nwork within which to interpret the full Hamiltonian re-\nsults. We believe that the formalism can be applied to\nlarger one-dimensional system and extended to higher-\ndimensional systems.\nVI. ACKNOWLEDGEMENT\nSupport by the National Science Foundation through\ngrant number PHY-1205443 and discussions with X. Y.\nYin, S. E. Gharashi and T.-L. Ho are gratefully acknowl-\nedged.\nAppendix A: Calculation of involved integrals\nThis section contains the evaluation of the integral given in Eq. (26) . We denote the nth harmonic oscillator\neigenstate by ϕn(x),ϕn(x) =NnHn(x/aho)e−x2/(2a2\nho), whereNn= 1//radicalbig√πn!2nahois the normalization constant13\nandHn(x) thenth Hermite polynomial. Throughout this appendix, we set aho= 1, i.e., we work with dimensionless\nspatial coordinates. Expanding the Slater determinant D(n1,n2,···,nN), we have\nD(n1,n2,···,nN) =/summationdisplay\np1,p2,···,pN(−1)Pp1,p2,···,pNΠN\nl=1ϕnpl(xl), (A1)\nwherep1,p2,···,pNdenotes a permutation of 1 ,2,···,NandPp1,p2,···,pNthe number of permutations needed to obtain\nthe orderp1,p2,· · ·,pNfrom the ordinary order 1 ,2,· · ·,N. The sum in Eq. (A1) contains N! terms. Note that\nsince the eigenstates φn1,n2,···,nN(/vector x) in Eq. (25) are only non-zero for a particular particle ordering, we do not need\na prefactor of 1 /√\nN! in front of the Slater determinant to normalize the eigenstates. E quation (26) contains two\ndifferent Slater determinants ( Dfunctions), one with arguments n1,···,nNand the other with arguments n′\n1,···,n′\nn.\nTo simplify the notation, we use m1,···,mNinstead ofn′\n1,···,n′\nnin what follows. The corresponding permutations\nare denoted by q1,···,qN. With these conventions, Eq. (26) becomes\nDj\nn1,n2,···,nNm1,m2,···,mN=/integraldisplay∞\n−∞/parenleftBigg/summationdisplay\np1,p2,···,pN(−1)Pp1,p2,···,pNΠN\nl=1ϕnpl(xl)/parenrightBigg\n/parenleftBigg/summationdisplay\nq1,q2,···,qN(−1)Pq1,q2,···,qNΠN\nl=1ϕmql(xl)/parenrightBigg\nΘx1jI(2)\nnpl,mql(xj)/parenrightBig/bracketrightBigg\n(A4)\nwith\nI(1)\nnpk,mqk(xj) =/integraldisplayxj\n−∞ϕnpk(x)ϕmqk(x)dx (A5)\nand\nI(2)\nnpl,mql(xj) =/integraldisplay∞\nxjϕnpl(x)ϕmql(x)dx. (A6)\nAccording to the generalized Feldheim identity, the product of any n umber of Hermite polynomials can be expanded\ninto a finite sum of Hermite polynomials [72],\nHN1(x)···HNm(x) =/summationdisplay\nν1···νm−1aν1···νm−1HM(x), (A7)\nwhere\nM=m−1/summationdisplay\nl=1(Nl−2νl)+Nm (A8)\nand\naν1···νm−1= Πm−1\nl=1/parenleftbiggNl+1\nνl/parenrightbigg/parenleftbigg/summationtextl−1\nk=1(Nk−2νk+Nl)\nνl/parenrightbigg\n2νlνl!. (A9)14\nThe limits of the summation indices νkin Eq. (A7) are given by\n0≤ν1≤min(N1,N2),0≤ν2≤min(N3,N1+N2−2ν1),···,0≤νm−1≤min/parenleftBigg\nNm,m−2/summationdisplay\nk=1(Nk−2νk)+Nm−1/parenrightBigg\n.\n(A10)\nUsing Eqs. (A7)-(A10) to rewrite the product of the two Hermite p olynomials contained in I(1)(xj) andI(2)(xj), we\nobtain\nI(1)\nnpk,mqk(xj) =NnpkNmqkmin(npk,mqk)/summationdisplay\nν(k)\n1=0aν(k)\n1/integraldisplayxj\n−∞Hnpk+mqk−2ν(k)\n1(x)exp(−x2)dx (A11)\nand\nI(2)\nnpl,mql(xj) =NnplNmqlmin(npl,mql)/summationdisplay\nν(l)\n1=0aν(l)\n1/integraldisplay∞\nxjHnpl+mql−2ν(l)\n1(x)exp(−x2)dx. (A12)\nInstead of working with Eqs. (A11) and (A12) directly, we replace t he Hermite polynomials in the integrals using the\ngenerating function g(x,t) of the Hermite polynomials,\ng(x,t) =e−t2+2tx=∞/summationdisplay\nn=0Hn(x)tn\nn!. (A13)\nFrom Eq. (A13), we have\nHn(x) =∂ng(x,t)\n∂tn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0. (A14)\nInserting Eq. (A14) into Eqs. (A11)-(A12), we find\nI(1)\nnpk,mqk(xj) =NnpkNmqkmin(npk,mqk)/summationdisplay\nν(k)\n1=0aν(k)\n1∂npk+mqk−2ν(k)\n1\n∂tnpk+mqk−2ν(k)\n1/bracketleftbigg/integraldisplayxj\n−∞exp(−t2+2tx−x2)dx/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0(A15)\nand\nI(2)\nnpl,mql(xj) =NnplNmqlmin(npl,mql)/summationdisplay\nν(l)\n1=0aν(l)\n1∂npl+mql−2ν(l)\n1\n∂tnpl+mql−2ν(l)\n1/bracketleftBigg/integraldisplay∞\nxjexp(−t2+2tx−x2)dx/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0.(A16)\nThe key point is that the Gaussian integrals can be calculated analytic ally. This yields\nI(1)\nnpk,mqk(xj) =NnpkNmqkmin(npk,mqk)/summationdisplay\nν(k)\n1=0aν(k)\n1/bracketleftBigg/parenleftbigg√π\n2−√π\n2erf(xj)/parenrightbigg\nδnpk+mqk−2ν(k)\n1,0\n+exp(−x2\nj)Hnpk+mqk−2ν(k)\n1−1(xj)(1−δnpk+mqk−2ν(k)\n1,0)/bracketrightBigg\n(A17)\nand\nI(2)\nnpl,mql(xj) =NnplNmqlmin(npl,mql)/summationdisplay\nν(l)\n1=0aν(l)\n1/bracketleftBigg/parenleftbigg√π\n2+√π\n2erf(xj)/parenrightbigg\nδnpl+mql−2ν(l)\n1,0\n−exp(−x2\nj)Hnpl+mql−2ν(l)\n1−1(xj)(1−δnpl+mql−2ν(l)\n1,0)/bracketrightBigg\n,(A18)15\nwhere erf(x) is the error function,\nerf(x) =2√π/integraldisplayx\n0exp(−t2)dt, (A19)\nandδn,mthe Kronecker delta function. Plugging Eqs. (A17) and (A18) into E q. (A4) and rearranging the order of\nthe sums and products, Eq. (A4) can be written as a finite sum over one-dimensional integrals,\nIj\nnp1,np2,···,npNmq1,mq2,···,mqN=ΠN\nk=1NpkNmqk\n(j−1)!(N−j)!min(np1,mq1)/summationdisplay\nν(1)\n1=0/braceleftBigg\naν(1)\n1×···min(npj−1,mqj−1)/summationdisplay\nν(j−1)\n1=0/braceleftBigg\naν(j−1)\n1×min(npj+1,mqj+1)/summationdisplay\nν(j+1)\n1=0/braceleftBigg\naν(j+1)\n1×\n···min(npN,mqN)/summationdisplay\nν(N)\n1=0/braceleftBigg\naν(N)\n1Fj\nnp1,np2,···,npNmq1,mq2,···,mqN(ν(1)\n1,···,ν(j−1)\n1,ν(j+1)\n1,···,ν(N)\n1)/bracerightBigg\n···/bracerightBigg/bracerightBigg\n···/bracerightBigg\n,(A20)\nwhere\nFj\nnp1,np2,···,npNmq1,mq2,···,mqN(ν(1)\n1,···,ν(j−1)\n1,ν(j+1)\n1,···,ν(N)\n1) =/integraldisplay∞\n−∞Hnpj(xj)Hmqj(xj)exp(2iksoxj−x2\nj)×\nΠkj/bracketleftbigg/parenleftbigg√π\n2+√π\n2erf(xj)/parenrightbigg\nδnpl+mql−2ν(l)\n1,0−exp(−x2\nj)Hnpl+mql−2ν(l)\n1−1(xj)(1−δnpl+mql−2ν(l)\n1,0)/bracketrightbigg\ndxj.(A21)\nTo simplify the notation, we define the index function d(k),\nd(k) =npk+mqk−2ν(k)\n1. (A22)\nForkj, we usek1,k2,···,kKandl1,l2,···,lL, respectively, to indicate all the k′sandl′sthat maked(k)\nandd(l) non-zero (0 ≤ K ≤j−1 and 0≤ L ≤N−j). Then Eq. (A21) becomes\nFj\nnp1,np2,···,npNmq1,mq2,···,mqN(ν(1)\n1,···,ν(j−1)\n1,ν(j+1)\n1,···,ν(N)\n1) =\n/integraldisplay∞\n−∞Hnpj(xj)Hmqj(xj)exp/bracketleftbig\n2iksoxj−(K+L+1)x2\nj/bracketrightbig\n(−1)L/parenleftbigg√π\n2−√π\n2erf(xj)/parenrightbiggj−1−K/parenleftbigg√π\n2+√π\n2erf(xj)/parenrightbiggN−j−L\n×\nHd(k1)−1(xj)···Hd(kK)−1(xj)Hd(l1)−1(xj)···Hd(lL)−1(xj)dxj.(A23)\nExpanding the/parenleftBig√π\n2−√π\n2erf(xj)/parenrightBigj−1−K\nand/parenleftBig√π\n2+√π\n2erf(xj)/parenrightBigN−j−L\nterms, Eq. (A23) becomes\nFj\nnp1,np2,···,npNmq1,mq2,···,mqN(ν(1)\n1,···ν(j−1)\n1,ν(j+1)\n1,···ν(N)\n1) =\n/parenleftbigg√π\n2/parenrightbiggN−L−K− 1\n(−1)Lj−1−K/summationdisplay\nr=0N−j−L/summationdisplay\ns=0/parenleftbiggj−1−K\nr/parenrightbigg/parenleftbiggN−j−L\ns/parenrightbigg\n(−1)r×/integraldisplay∞\n−∞exp/bracketleftbig\n2iksoxj−(K+L+1)x2\nj/bracketrightbig\n[erf(xj)]r+s×\nHnpj(xj)Hmqj(xj)Hd(k1)−1(xj)···Hd(kK)−1(xj)Hd(l1)−1(xj)···Hd(lL)−1(xj)dxj.(A24)\nEquation (A24) contains a product of Hermite polynomials, which can be converted into a finite sum of Hermite\npolynomials according to the Feldheim identity,\nHnpj(xj)Hmqj(xj)Hd(k1)−1(xj)···Hd(kK)−1(xj)Hd(l1)−1(xj)···Hd(lL)−1(xj) =/summationdisplay\nν1···νL+K+1aν1···νL+K+1HM(xj),(A25)\nwhere the coefficients aν1···νL+K+1and the relationship between Mand the indices npj,mqj,d(k1),···,d(lL) are defined\nin Eqs. (A8)-(A10).16\nPlugging Eq. (A25) into Eq. (A24), we find\nFj\nnp1,np2,···,npNmq1,mq2,···,mqN(ν(1)\n1,···,ν(j−1)\n1,ν(j+1)\n1,···,ν(N)\n1) =\n/parenleftbigg√π\n2/parenrightbiggN−L−K− 1\n(−1)Lj−1−K/summationdisplay\nr=0N−j−L/summationdisplay\ns=0/parenleftbiggj−1−K\nr/parenrightbigg/parenleftbiggN−j−L\ns/parenrightbigg\n(−1)r/summationdisplay\nν1···νL+K+1aν1···νL+K+1GM\n(r+s;K+L),(A26)\nwhere\nGM\n(r+s;K+L)=/integraldisplay∞\n−∞exp/bracketleftbig\n2iksoxj−(K+L+1)x2\nj/bracketrightbig\n[erf(xj)]r+sHM(xj)dxj. (A27)\nWe callGM\n(r+s;K+L)theGintegral of the order r+s. TheGintegral of order 0 can be evaluated analytically,\nGM\n(0,K+L)=√π√\nK+L+1exp/parenleftbigg\n−k2\nso\nK+L+1/parenrightbigg\nHM/parenleftBigg\n2ikso/radicalbig\n(K+L+1)(K+L)/parenrightBigg/parenleftbiggK+L\nK+L+1/parenrightbiggM/2\n. (A28)\nTo evaluate GM\n(r+s;K+L)of higher order, we develop an iterative procedure, in which the Gintegral of order r+sis\nwritten in terms of Gintegrals of order r+s−1. Since the r+s= 0 integral is known, this allows for the evaluation\nof theGintegral of arbitrary order. Using the generating function of the Hermite polynomials, we have\nGM\n(r+s;K+L)=/braceleftbigg∂M\n∂tM/integraldisplay∞\n−∞exp/bracketleftbig\n2(ikso+t)xj−(K+L+1)x2\nj−t2/bracketrightbig\n[erf(xj)]r+sdxj/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0. (A29)\nUsing integration by parts, Eq. (A29) becomes\nGM\n(r+s;K+L)=/braceleftBigg\n∂M\n∂tM/bracketleftBigg\ng(1)\n(r+s,K+L)−g(2)\n(r+s,K+L)/bracketrightBigg/bracerightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0, (A30)\nwhere\ng(1)\n(r+s,K+L)=H(kso,K,L;t,xj)[erf(xj)]r+s|xj=∞\nxj=−∞ (A31)\nand\ng(2)\n(r+s,K+L)= (r+s)2√π/integraldisplay∞\n−∞H(kso,K,L;t,xj)[erf(xj)]r+s−1exp(−x2\nj)dxj (A32)\nwith\nH(kso,K,L;t,xj) =/integraldisplayxj\n0exp/bracketleftbig\n2(ikso+t)x−(K+L+1)x2−t2/bracketrightbig\ndx. (A33)\nThe Gaussian integral in Eq. (A33) can be calculated analytically,\nH(kso,K,L;t,xj) =√π\n2√\nK+L+1exp/bracketleftbigg\n−k2\nso−2iksot+(K+L)t2\nK+L+1/bracketrightbigg\nerf/bracketleftbigg−ikso−t+(K+L+1)xj√\nK+L+1/bracketrightbigg\n.(A34)\nUsing that the value of the error function erf( x) at infinity is known, erf( ±∞) =±1,g(1)\n(r+s,K+L)evaluates to\ng(1)\n(r+s,K+L)=√π\n2√\nK+L+1exp/bracketleftbigg\n−k2\nso−2iksot+(K+L)t2\nK+L+1/bracketrightbigg/bracketleftbig\n1−(−1)r+s+1/bracketrightbig\n. (A35)\nInserting Eq. (A34) into Eq. (A32), we have\ng(2)\n(r+s,K+L)= (r+s)exp/bracketleftBig\n−k2\nso−2iksot+(K+L)t2\nK+L+1/bracketrightBig\n√\nK+L+1/braceleftbigg/integraldisplay∞\n−∞erf/bracketleftbigg−ikso−t+(K+L+1)xj√\nK+L+1/bracketrightbigg\n[erf(xj)]r+s−1exp(−x2\nj)dxj/bracerightbigg\n.\n(A36)17\nNext, we expand g(1)\n(r+s,K+L)andg(2)\n(r+s,K+L)in terms of tup to power Mand evaluate the Mth derivative with\nrespect totatt= 0. For K+L /negationslash= 0, the exponential in Eq. (A35) can be interpreted as a generatin g function of the\nHermite polynomials. Expanding this term into a sum of Hermite polynom ials, we find\ng(1)\n(r+s,K+L)=\n√π\n2√\nK+L+1/bracketleftbig\n1−(−1)r+s+1/bracketrightbig\nexp\n−k2\nso\nK+L+1+2ikso/radicalbig\n(K+L+1)(K+L)/parenleftBigg/radicalbigg\nK+L\nK+L+1t/parenrightBigg\n−/parenleftBigg/radicalbigg\nK+L\nK+L+1t/parenrightBigg2\n\n=√π\n2√\nK+L+1/bracketleftbig\n1−(−1)r+s+1/bracketrightbig\nexp/parenleftbigg\n−k2\nso\nK+L+1/parenrightbigg\n∞/summationdisplay\nn=0Hn/parenleftBigg\n2ikso/radicalbig\n(K+L+1)(K+L)/parenrightBigg/parenleftBig/radicalBig\nK+L\nK+L+1t/parenrightBign\nn!\n.(A37)\nForK+L= 0, the power series of g(1)\n(r+s,K+L)in terms of tis\ng(1)\n(r+s,K+L)=√π\n2√\nK+L+1/bracketleftbig\n1−(−1)r+s+1/bracketrightbig\nexp/parenleftbig\n−k2\nso/parenrightbig/bracketleftBigg∞/summationdisplay\nn=0(2iksot)n\nn!/bracketrightBigg\n. (A38)\nThis can be understood as the limiting result of Eq. (A37),\nlim\nK+L→0Hn/parenleftBigg\n2ikso/radicalbig\n(K+L+1)(K+L)/parenrightBigg/parenleftbiggK+L\nK+L+1/parenrightbiggn/2\n= (2ikso)n. (A39)\nIn what follows, we take this limit when K+L= 0. The derivative of g(1)\n(r+s;K+L)with respect to tatt= 0 is then\n/parenleftbigg∂M\n∂tMg(1)\n(r+s;K+L)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=\n√π\n2√K+L+1exp/parenleftbigg\n−k2\nso\nK+L+1/parenrightbigg\nHM/parenleftBigg\n2ikso/radicalbig\n(K+L+1)(K+L)/parenrightBigg/parenleftbiggK+L\nK+L+1/parenrightbiggM/2/bracketleftbig\n1−(−1)r+s+1/bracketrightbig\n.(A40)\nTo evaluate g(2)\n(r+s,K+L), we notice that it can be written in the form g(1)\n(r+s,K+L)/integraltext\nerf(···)dxj. The expansion of\ng(1)\n(r+s,K+L)is given in Eq. (A37). An additional t-dependence enters through the error function in the integrand in\nEq. (A36). Rewriting the error function in a series in t, we have\nerf/parenleftBigg\n−ikso−t+(K+L+1)xj√\nK+L+1/parenrightBigg\n= erf/parenleftBigg\n√\nK+L+1xj−ikso√\nK+L+1/parenrightBigg\n−2√πexp/parenleftbiggk2\nso\nK+L+1/parenrightbigg\n×\n∞/summationdisplay\nn=1exp/bracketleftbig\n2iksoxj−(K+L+1)x2\nj/bracketrightbig\nHn/parenleftbigg√\nK+L+1xj−ikso√\nK+L+1/parenrightbigg/parenleftBigg\nt√\nK+L+1/parenrightBiggn\n1\nn!.(A41)\nUsing the multiplication theorem [73]\nHn(αx) =[n\n2]/summationdisplay\nv=0αn−2v(α2−1)v/parenleftbiggn\n2v/parenrightbigg(2v)!\nv!Hn−2v(x) (A42)\nand the addition theorem [74]\nHn(x+y) =n/summationdisplay\nw=0/parenleftbiggn\nw/parenrightbigg\nHw(x)(2y)n−w, (A43)18\nwhere [n\n2] denotes the integerpart of n/2, the Hermite polynomial on the righthand side ofEq.(A41) can be e xpanded\ninto a finite sum over products of Hermite polynomials in which the depe ndence onxjhas been “isolated”,\nHn/parenleftbigg√\nK+L+1xj−ikso√\nK+L+1/parenrightbigg\n=\n[n\n2]/summationdisplay\nv=0n−2v/summationdisplay\nw=0(K+L+1)n−2v\n2(K+L)v/parenleftbiggn\n2v/parenrightbigg(2v)!\nv!/parenleftbiggn−2v\nw/parenrightbigg\nHw/parenleftbigg2ikso\nK+L+1/parenrightbigg\nHn−2v−w(xj).(A44)\nForK+L= 0, Eq. (A44) contains an indeterminate term 00which is understood to be 1. In this case, Eq. (A44)\nbecomes\nHn(xj−ikso) =n/summationdisplay\nw=0/parenleftbiggn\nw/parenrightbigg\nHw(2ikso)Hn−w(xj). (A45)\nUsing Eqs. (A37), (A41) and (A44), Eq. (A36) becomes\ng(2)\n(r+s;K+L)=\n∞/summationdisplay\nn=0/braceleftBigg\nr+s√K+L+1exp/parenleftbigg\n−k2\nso\nK+L+1/parenrightbigg/bracketleftBigg\nHn/parenleftBigg\n2ikso/radicalbig\n(K+L+1)(K+L)/parenrightBigg/parenleftbiggK+L\nK+L+1/parenrightbiggn/2\nI(r+s−1,K+L)/bracketrightBigg\n+\n2(r+s)\n√π(K+L+1)n+1\n2n/summationdisplay\nu=0[u\n2]/summationdisplay\nv=0u−2v/summationdisplay\nw=0Hn−u/parenleftBigg\n2ikso/radicalbig\n(K+L+1)(K+L)/parenrightBigg\n(K+L)n−u\n2+v(K+L+1)u−2v\n2×\n/parenleftbiggn\nu/parenrightbigg/parenleftbiggu\n2v/parenrightbigg/parenleftbiggu−2v\nw/parenrightbigg(2v)!\nv!Hw/parenleftbigg2ikso\nK+L+1/parenrightbigg\nG(u−2v−w)\n(r+s−1,K+L+1)/bracerightBigg\ntn\nn!,(A46)\nwhere\nI(r+s−1,K+L)=/integraldisplay∞\n−∞erf/parenleftBigg\n√\nK+L+1xj−ikso√\nK+L+1/parenrightBigg\n[erf(xj)]r+s−1exp(−x2\nj)dxj. (A47)\nTheMth derivative of g(2)\n(r+s;K+L)with respect to tatt= 0 is then\n/parenleftbigg∂M\n∂tMg(2)\n(r+s;K+L)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=g(2,1)\n(r+s,K+L)+g(2,2)\n(r+s,K+L), (A48)\nwhere\ng(2,1)\n(r+s,K+L)=\nr+s√\nK+L+1exp/parenleftbigg\n−k2\nso\nK+L+1/parenrightbigg\nHM/parenleftBigg\n2ikso/radicalbig\n(K+L+1)(K+L)/parenrightBigg/parenleftbiggK+L\nK+L+1/parenrightbiggM/2\nI(r+s−1,K+L)(A49)\nand\ng(2,2)\n(r+s,K+L)=2(r+s)\n√π(K+L+1)M+1\n2M/summationdisplay\nu=0[u\n2]/summationdisplay\nv=0u−2v/summationdisplay\nw=0HM−u/parenleftBigg\n2ikso/radicalbig\n(K+L+1)(K+L)/parenrightBigg\n(K+L)M−u\n2+v(K+L+1)u−2v\n2×\n/parenleftbiggM\nu/parenrightbigg/parenleftbiggu\n2v/parenrightbigg/parenleftbiggu−2v\nw/parenrightbigg(2v)!\nv!Hw/parenleftbigg2ikso\nK+L+1/parenrightbigg\nG(u−2v−w)\n(r+s−1,K+L+1).(A50)\nTo evaluate the integral I(r+s−1,K+L), we integrate by parts. Using\n/integraldisplayxj\n0[erf(x)]r+s−1exp(−x2)dx=√π[erf(xj)]r+s\n2(r+s)(A51)19\nand\n/bracketleftBigg\nerf/parenleftBigg\n√\nK+L+1xj−ikso√\nK+L+1/parenrightBigg√π[erf(xj)]r+s\n2(r+s)/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglexj=∞\nxj=−∞=√π\n2(r+s)/bracketleftbig\n1−(−1)r+s+1/bracketrightbig\n,(A52)\nwe find\nI(r+s−1,K+L)=√π\n2(r+s)/bracketleftbig\n1−(−1)r+s+1/bracketrightbig\n−√\nK+L+1exp/parenleftBig\nk2\nso\nK+L+1/parenrightBig\nr+sJ(r+s,K+L), (A53)\nwhere\nJ(r+s,K+L)=/integraldisplay∞\n−∞[erf(xj)]r+sexp[−(K+L+1)x2\nj+2iksoxj]dxj. (A54)\nUsing Eq. (A53) in Eq. (A49) and using Eq. (A40), we find\nGM\n(r+s;K+L)=J(r+s,K+L)−g(2,2)\n(r+s,K+L). (A55)\nOur manipulations have reduced GM\n(r+s;K+L)to an expression that contains two types of one-dimensional integ rals.\nThe first one, J(r+s,K+L), only depends on kso,KandL, implying that only a few of these integrals need to be\ncalculated for a given kso. Forr+s= 1,J(1,K+L)can be calculated analytically,\nJ(1,K+L)=/radicalbiggπ\nK+L+1exp/parenleftbigg\n−k2\nso\nK+L+1/parenrightbigg\nerf/bracketleftBigg\nikso/radicalbig\n(K+L+1)(K+L+2)/bracketrightBigg\n. (A56)\nForr+s >1,J(1,K+L)can be calculated numerically with essentially arbitrary accuracy. Th e second integral,\nGu−2v−w\n(r+s−1;K+L+1)(theGintegral of order r+s−1), is contained in g(2,2)\n(r+s,K+L). To see the iterative structure of our\nresult more clearly, we rewrite Eq. (A55) as\nGM\n(r+s;K+L)=J(r+s,K+L)−/summationdisplay\njcjGu−2v−w\n(r+s−1;K+L+1), (A57)\nwherejruns over all combinations of allowed indices and cjcontains all the prefactors [the cjcan be read off\nEq. (A50)]. Thus, to determine the Gintegral of order r+s, a finite number of Gintegrals of order r+s−1 is needed.\nTo evaluate the Gintegral of order r+s−1, a finite number of Gintegrals of order r+s−2 is needed, and so on.\nSince the Gintegral of order 0 is known analytically [see Eq. (A28)], the Gintegral of arbitrary order can be obtained\niteratively. It should be noted that the Gintegral of order 1 has an analytical result since G(0,K+L)andJ(1,K+L)are\nknown analytically.\nIn summary, first we calculate, using Eq. (A54), all the Jintegrals needed for the evaluation of the Gintegrals.\nSecondly, using the values of the Jintegrals, the required Gintegrals are calculated iteratively based on Eq. (A55).\nThirdly, plugging the values of the Gintegrals into Eq. (A26), the Fintegrals with arguments ν(1)\n1,···,ν(j−1)\n1,ν(j+1)\n1,··\n·,ν(N)\n1are obtained. Forthly, plugging the values of the Fintegrals into Eq. (A20), the Iintegrals can be calculated.\nFinally, plugging the values of the Iintegrals into Eq. (A3), we obtain the values of the Dintegrals. The sums in the\nabove expressions are all finite. As a result, the only error in evalua ting the Dintegrals comes from the numerical\nevaluation of the one-dimensional Jintegrals. Everything is analytical for N= 2 while one and three Jintegrals\nhave to be evaluated numerically for N= 3 and 4, respectively. The reader may wonder why we choose the o utlined\niterative procedure for evaluating the one-dimensional integral g iven in Eq. (A27) over an evaluation of Eq. (A27) by\ndirect numerical integration. The answer is two-fold. First, Eq. (A 27) has to be evaluated for Hermite polynomials\nof different orders; the numerical integrals required in our iterativ e procedure, in contrast, are independent of the M\nquantum number. Second, for large M, the integrand in Eq. (A27) is highly oscillatory, making the one-dimen sional\nintegration somewhat non-trivial.\nTo be concrete, we discuss how to apply the formalism to the three p article system. In this case, the Dintegral\ndepends on j(j= 1,2,3) and the quantum numbers n1,n2,n3,m1,m2,andm3. For a fixed j, there are 3! ×3! = 36\nterms (Iintegrals) in the sum on the right hand side of Eq. (A3). For each Iintegral, we have a double sum on\nthe right hand side of Eq. (A20). For j= 2, e.g., the summation dummies are ν(1)\n1andν(3)\n1. Correspondingly,\ntheFintegrals have two arguments. The indices KandLin Eq. (A23) satisfy 0 ≤ K ≤1 and 0 ≤ L ≤1. As20\na result, the indices randsdefined in Eq. (A24) satisfy 0 ≤r≤1 and 0 ≤s≤1 with the additional constraint\nr+s≤N−1−K −L,which is due to the limits r≤j−K −1 ands≤N−L−jin the sums on the right hand\nside of Eq. (A24). Thus, the allowed ( r+s,K+L) combinations are (2 ,0),(1,0),(0,0),(1,1),(0,1),(0,2). Since the\nnumber of Hermite polynomials in the product in Eq. (A24) is K+L+2,which is less or equal to 4 (since 0 ≤ K ≤1\nand 0≤ L ≤1), the coefficients on the right hand side of Eq. (A25) have at most three indices.\nAppendix B: Expressions for /angbracketleftSx(x)/angbracketrightand/angbracketleftSz(x)/angbracketrightforN= 3−4\nForN= 3, the expressions for C1x,C2x,andCzin Eqs. (47)-(48) in terms of the coefficients C1−C4in Eq. (46)\nare\nC1x= (C1−C3)∗(C2+C4)+(C1−C3)(C2+C4)∗, (B1)\nC2x=C∗\n1C3+C1C∗\n3−|C2|2−|C4|2, (B2)\nand\nCz=i[(C1−C3)∗(C2−C4)−(C1−C3)(C2−C4)∗]. (B3)\nThe expressions for n1x(x),n2x(x),andnz(x) in Eqs. (47)-(48) are\nn1x(x) =/integraldisplay∞\n−∞|D(0,1,2)|2Θx1¯hω2.Cr(ω)is a rotated ellipse with semi-axis of lengths (major)\nka(ω) = ¯hω/2|α−β|and(minor) kb(ω) = ¯hω/2|α+β|orientedalongthe (1,1)and(−1,1)directionsrespectively.\nThe sample parameters usedhere are n= 5×1011cm−2,α= 1.6×10−9eVcm,β= 0.5αandm∗= 0.055m.\nare concentric circles. If α=±βthe spin-splitting along the ( ±1,1) direction vanishes and the spin-\nsplit dispersionbranchesare two circleswith the same radi usanddisplacedfromthe origin(along( ∓1,1)\ndirection).\nWithinthelinearresponseKuboformalismthespinsuscepti bilityisgivenby\nχµµ′(ω) =i\n¯h/integraldisplay∞\n0dtei(ω+iη)t/an}b∇acketle{t[σµ(t),σµ′(0)]/an}b∇acket∇i}ht, µ,µ′=x,y (3)\nwhere the symbol /an}b∇acketle{t···/an}b∇acket∇i}ht= Σλ/integraltext\nd2kf(ǫλ(k))(···)indicates quantum and thermal averaging, f(ǫ)is the\nFermidistributionfunction,and η→0+. Thisisaspin-spinresponsefunctionforaspatiallyhomog eneous\n(in-plane)perturbationoscillatingat frequency ω.\nInthelimitofvanishingtemperature,eq. (3)takestheform\nχµµ′(ω) =1\nπ2/integraldisplay′\nd2k/an}b∇acketle{t−|σµ|+/an}b∇acket∇i}ht/an}b∇acketle{t+|σµ′|−/an}b∇acket∇i}htε+(k)−ε−(k)\n[ε+(k)−ε−(k)]2−[¯h(ω+iη)]2, (4)\nthe prime on the integral indicates that integration is rest ricted to the region between the Fermi contours,\nq+(θ)< k < q −(θ), forwhich ε−(k)< εF< ε+(k), (Fig. 1).\nUsingtheresult\n/an}b∇acketle{t−|σµ|+/an}b∇acket∇i}ht=−/an}b∇acketle{t+|σµ|−/an}b∇acket∇i}ht=i\n∆(θ)[δµx(αcosθ−βsinθ)+δµy(αsinθ−βcosθ)] (5)\nthesusceptibilitytensorbecomes\nχµµ′(ω) =1\nπ2/integraldisplay2π\n0dθgµµ′(θ)\n∆(θ)/integraldisplayq−(θ)\nq+(θ)dkk2\n4k2∆2(θ)−[¯h(ω+iη)]2, (6)\nwhere\ngµµ′(θ) =δµµ′[δµx(αcosθ−βsinθ)2+δµy(αsinθ−βcosθ)2]\n+(1−δµµ′)(αcosθ−βsinθ)(αsinθ−βcosθ).\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheim4 C.L´ opez-Bastidas etal.:Interplayof the Rashba andDres selhaus spin-orbit coupling\nIt can be shown that χxx(ω) =χyy(ω)andχxy(ω) =χyx(ω). Note also that for β= 0,χxy(ω) =\nχyx(ω) = 0.\nWe canwritethe susceptibilityin theform χµµ′=χ′\nµµ′+iχ′′\nµµ′. Therealpartis\nχ′\nµµ′(ω) =χµµ′(0)+¯hω\n16π2/integraldisplay2π\n0dθgµµ′(θ)\n∆4(θ)log/vextendsingle/vextendsingle/vextendsingle/vextendsingle[ω+Ω+(θ)][ω−Ω−(θ)]\n[ω+Ω−(θ)][ω−Ω+(θ)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle(7)\nwhere¯hΩ±=|εF−ε∓(q±(θ),θ)|= 2q±(θ)∆(θ)andthestatic valueis\nχµµ′(0) =ν0\n2/bracketleftbigg\nδµµ′−(1−δµµ′)/parenleftbiggβ\nαΘ(α2−β2)+α\nβΘ(β2−α2)/parenrightbigg/bracketrightbigg\n, (8)\nν0=m∗/π¯h2is the density of states of a spin-degenerate 2DEG, and Θ(x)is the unit step function,\nΘ(x) = 1ifx >0andΘ(x) = 0ifx <0.\nFortheimaginarypartwe have\nχ′′\nµµ′(ω) =π/integraldisplay′d2k\n(2π)2/an}b∇acketle{t−|σµ|+/an}b∇acket∇i}ht/an}b∇acketle{t+|σµ′|−/an}b∇acket∇i}htδ(ε+(k)−ε−(k)−¯hω) (9)\n=¯hω\n16π/integraldisplay\ndθgµµ′(θ)\n∆4(θ)Θ[¯hω−¯hΩ+(θ)]Θ[¯hΩ−(θ)−¯hω]. (10)\nThese equations express the fact that the only transitions a llowed between spin-split subbands ελdue to\nphotonabsorptionat energy ¯hωare thoseforwhich ¯hΩ+(θ)≤¯hω≤¯hΩ−(θ). Thatis, fora given ωonly\nthose anglessatisfying this conditionmust be consideredi n the integral (10), see Fig.2c. This is different\nto the pure Rashba (or Dresselhaus) case, where the whole int erval[0,2π]contributes to the integral for\neach allowed photon energy. The non-isotropic spin-splitt ing originated by the simultaneous presence of\nbothcouplingstrengths,forcestheopticalexcitationtob ek−selective.\nIn Fig.2 we show χxx(ω)as obtained from eqs. (7)-(10), the xycomponent behaves similarly. The\nresult is remarkably different from that of the pure Rashba o r Dresselhaus case, where the spin-splitting\nis isotropic in the momentum space. For example, if β= 0,α/ne}ationslash= 0, thenχ′′\nµµ′(ω) =χRδµµ′only for\n2αk+≤¯hω≤2αk−, otherwise it vanishes, where χR= ¯hω/16α2, withk±=q±(β= 0)being\nindependent of angle θ; (see Fig.3). Thus, in this case the width ∆Eof the spectrum is ∆ER= 4εR\n(or4εDifα= 0,β/ne}ationslash= 0);εR=m∗α2/¯h2andεD=m∗β2/¯h2are the SO characteristic energy\nscales for the Rashba and Dresselhaus coupling. As was discu ssed in Ref.[22], ∆ER,Dcan be about an\norder of magnitude smaller than the width of the spectrum sho wn in Fig.2b. Assuming that α > βand\n(kso(θ)/k0)2≪1,we have ∆E= 4βk0+∆ER+∆ED(ifα < βthefirst termchangesto 4αk0). Thus,\ntheabsorptionbandwidthcouldbemanipulatedbytuningthe couplingstrength αand/orthroughvariations\noftheelectrondensity n=k2\n0/2π. Thisfactcouldalso beusedtodeterminethesignof α−β.[22]\nTo understand the structure of the spectra of Fig.2, we furth er note that, according to eq. (9), the\nminimum(maximum)photonenergy ¯hω+(¯hω−) requiredto induceoptical transitionsbetween the initia l\nλ=−and the final λ= +subband corresponds to the excitation of an electron with wa ve vector lying\non theq+(q−) Fermi line at θ+=π/4or5��/4(θ−= 3π/4or7π/4), giving ¯hω±= ¯hΩ±(θ±) =\n2k0|α∓β|∓2m∗(α∓β)2/¯h2. TheabsorptionedgesinthespectrumofFig.2bcorresponde xactlyto¯hω±.\nThefunction χ′′\nµµ′(ω)canalsobewrittenasalineintegralalongthearcsoftheres onantcurve Cr(ω)lying\nwithin the region enclosed by the Fermi lines qλ(θ); see Fig.1. The peaks observedin Fig.2b correspond\nto electronic excitations involving states with allowed wa ve vectors on Cr(ω)such that |∇k(ε+−ε−)|\ntakes its minimum value. The first (second) peak is at a photon energy¯hωa(¯hωb) for which the major\n(minor) semi-axis of the ellipse Cr(ω)(Fig.1) coincides with the Fermi line q−(θ+) (q+(θ−)), hence\n¯hωa= ¯hΩ−(θ+) = 2k0|α−β|+2m∗(α−β)2/¯h2and¯hωb= ¯hΩ+(θ−) = 2k0|α+β|−2m∗(α+β)2/¯h2.\nThespectrumof χ′′\nxx(ω)looksverysimilartothejointdensityofstatesforthespin -splitbands ε±.[22]The\nunequalsplittingattheFermilevelalongthesymmetry (1,1)and(−1,1)directionsisthusresponsiblefor\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheimpss header willbe provided by the publisher 5\n1.5\n1\n0.57/4\n5/4\n3/4\n1/4\n1.5\n1.0\n0.5\n 0\n-0.5\n-1.0\n-1.5\n 0 2 4 6 8 10\n−hω (meV)ω+ωaωbω-Ω-Ω+\n(a)(b)(c)χ'xx/ν0χ''xx/ν0 θ/π\nFig. 2(c) Angular region (shaded) in k−space available for direct transitions as a function of phot on energy. Only\nthe shaded region contribute to the optical absorption [eq. (10)]. The energy boundaries are given by ¯hΩ+(θ) =\nεF−ε−(q+(θ),θ)and¯hΩ−(θ) =ε+(q−(θ),θ)−εF. (b)Imaginaryand(a)Realpartoftheopticalspinsuscepti bility\nχxx(ω),ν0=m∗/π¯h2. For the frequencies ω+= Ω+(π/4),ωa= Ω−(π/4),ωb= Ω+(3π/4),ω−= Ω−(3π/4),\nsee the text. The sample parameters are the same as inFig.1.\nthe peaks at photon energies ¯hωaand¯hωbrespectively, giving meaning to the structure of the spectr um.\nThe overall magnitude and the asymmetric shape of the spectr um are due to the factor gµµ′(θ)/∆4(θ)in\neq.(10). Theresultsforseveralvaluesof β/αareshowninFig.3.\nThe real part of χµµ′(ω)presents additionalspectral features. For photonenergie sin the range ¯hωa≤\n¯hω≤¯hωbwefindnumericallythatittakestheconstantvalues χ′\nxx(ω) =ν0/2andχ′\nxy(ω) =−(ν0/2)(α2+\nβ2)/2αβ. The spectral characteristics of the response displayed in Fig.2a shows that the magnitude and\nthedirectionofthedynamicspinmagnetizationcouldbemod ifiedviaelectricalgatingand/orbyadjusting\ntheexcitingfrequency. Thissuggestsnewpossibilitiesof electricalmanipulationofthespin orientationin\na 2DEGinthepresenceofcompetingRashba andDresselhausSO couplings.\nFollowing Ref.[36] we have also obtained the static value of χµµ′(ω)for finite momentum relaxation\nrateη >0(see eq. (6)). This parameter accounts phenomenologically for dissipation effects due to\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheim6 C.L´ opez-Bastidas etal.:Interplayof the Rashba andDres selhaus spin-orbit coupling\n 0 2 4 6 8\n 2 3 4 5 6 7 8 9 10χ''xx/ν0\n−hω (meV)β/α=0.50.250.1β=0\nFig. 3Imaginary part of the spin susceptibility χxx(ω)for several values of the ratio β/α. Other parameters are as\nthose used inFig.1.\nimpurityscattering. We foundthat,tolinearorderin εR,D/¯hη,it vanishesas( α/ne}ationslash=β/ne}ationslash= 0)\nχxx(0;η)≈4ν0/parenleftbiggεF\n¯hη/parenrightbigg /parenleftbiggεR+εD\n¯hη/parenrightbigg\nχxy(0;η)≈ −4ν0/parenleftbiggεF\n¯hη/parenrightbigg2√εRεD\n¯hη.(11)\nItisalsopossibletorelatethespincurrentresponsetothe spindensityresponse. Thedefinitionofthespin\nconductivity σs,z\niy(ω)describinga z−polarized-spincurrentflowinginthe i−directionasaresponsetothe\nfieldE(ω)ˆ yinvolvesthecommutator [Jz\ni(t),jy(0)],whereji=eviandJz\ni= ¯h{σz,vi}/4arethecharge\nandspincurrentoperators,respectively. Usingtheveloci tyoperator v(k) =∇kH/¯h= ¯hk/m∗+ˆ x(βσx+\nασy)/¯h−ˆ y(ασx+βσy)/¯h, this commutator can be written in terms of the correlators [σi(t),σj(0)],\n(i=x,y),whichdeterminesthespinsusceptibility(3). Thus,thef ollowingrelationscanbederived\nσs,z\nxy(ω)\ne/8π=/parenleftbiggα2+β2\nα2−β2/parenrightbiggχxx(ω)\nν0/2+/parenleftbigg2αβ\nα2−β2/parenrightbiggχxy(ω)\nν0/2(12)\nσs,z\nyy(ω)\ne/8π=/parenleftbigg2αβ\nα2−β2/parenrightbiggχxx(ω)\nν0/2+/parenleftbiggα2+β2\nα2−β2/parenrightbiggχxy(ω)\nν0/2. (13)\nThese expressions are formally equivalent to eqs. (39) and ( 40) of Ref.[34]. This connection is very\nconvenientbecausea spinpolarizationismoreexperimenta llyaccesiblethana spincurrent.\nIn summary, we have calculated the finite frequency spin susc eptibility tensor of a two-dimensional\nelectron gas with competing Rashba and Dresselhaus spin-or bit interaction. We find that the angular\nanisotropy of the energy spin-splitting introduced by the i nterplay between both SO coupling strengths\nyields a finite-frequencyresponse with spectral featurest hat are significantly differentfrom that of a pure\nRashba (Dresselhaus) coupling case. As a consequence, an op tically modulable spin density response is\nthenachievablein suchsystemswhichmaybeusefulforspint ronicsapplications.\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheimpss header willbe provided by the publisher 7\nThis work was supported by CONACyT-Mexico grants J40521F, J 41113F, and by DGAPA-UNAM\nIN114403-3.\nReferences\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka-\nnova, andD. M.Treger, Science 294, 1488 (2001).\n[2] E.I.Rashba, Physica E 20, 189 (2004).\n[3] I.ˇZuti´ c,J. Fabian,and S.Das Sarma,Rev. Mod. Phys. 76, 323 (2004).\n[4] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron a nd Hole Systems , (Springer, Berlin,\n2003).\n[5] S.Dattaand B.Das, Appl. Phys.Lett. 56, 665 (1990).\n[6] J. Schliemann, J.C.Egues, andD.Loss, Phys.Rev. Lett. 90, 146801 (2003).\n[7] J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T. Jungwirth , and A.H. MacDonald, Phys. Rev. Lett. 92, 126603\n(2004).\n[8] J. Sinova,S.Murakami, S.-Q.Shen, and M.-S.Choi,Solid StateCommun. 138, 214 (2006).\n[9] J. Schliemann, Int.J.Mod. Phys.B 20, 1015 (2006).\n[10] J. Wunderlich, B.Kaestner, J.Sinova, and T.Jungwirth , Phys.Rev. Lett. 94, 047204 (2005).\n[11] Y.K.Kato, R.C.Myers A.C.Gossard, and D.D.Awschalom, Science306, 1910 (2004).\n[12] V. Sih,R.C.Myers, Y.K.Kato, W.H.Lau, A.C.Gossard, an dD.D.Awschalom, Nature 1, 31(2005).\n[13] S.O.Valenzuela and M.Tinkham, Nature 442, 176 (2006).\n[14] V.M. Edelstein,Solis StateCommun. 73, 233 (1990).\n[15] L.I.Magarilland M.V. Entin,JETPLett. 72, 134 (2000)\n[16] A.V.Chaplik, M.V. Entin,and L.I.Magarill,Physica E 13, 744 (2002).\n[17] Y. Kato,R.C.Myers, A.C.Gossard, and D.D.Awschalom, P hys. Rev. Lett. 93, 176601 (2004).\n[18] M. Duckheim andD. Loss,Nature Phys. 2,195 (2006).\n[19] T.O.Cheche and E.Barna,Appl. Phys.Lett. 89, 042116 (2006).\n[20] E.G.Mishchenko and B.I.Halperin, Phys. Rev. B 68, 045317 (2003).\n[21] C.Zhang andZ.Ma, Phys. Rev. B 71, 121307(R) (2005).\n[22] J.A.Maytorena, C.L´ opez-Bastidas, andF.Mireles, to be published inPhysicalReview B;cond-mat/0603722.\n[23] L.I.Magarill,A.V.Chaplik, and M.V. ´Entin,JETP 92, 153(2001).\n[24] E.I.Rashba, Phys.Rev. B 70161201(R) (2004).\n[25] E.Ya.Sherman, A.Najmaie, andJ.E.Sipe,Appl. Phys. Le tt.86, 122103 (2005).\n[26] D.W.Yuan, W.Xu,Z.Zeng, and F.Lu,Phys.Rev. B 72, 033320 (2005).\n[27] C.M.Wang, S.Y. Liu,andX.L.Lei,Phys.Rev. B 73, 035333 (2006).\n[28] A.Shekhter, M. Khodas, A.M.Finkel’stein, Phys.Rev. B 71, 165329 (2005).\n[29] O.Dimitrova, Phys.Rev. B 71, 245327 (2005).\n[30] C.Grimaldi,E.Cappelluti, and F.Marsiglio, Phys.Rev . Lett.97, 066601 (2006).\n[31] W.Xu, Appl. Phys.Lett. 82, 724(2003).\n[32] X.F.Wang,Phys. Rev. B 72, 085317 (2005).\n[33] M. Pletyukhov andV. Gritsev, Phys.Rev. B 74, 045307 (2006).\n[34] S.I.Erlingsson, J.Schliemann, and Daniel Loss,Phys. Rev. B71, 035319 (2005).\n[35] E.I.Rashba, J.Supercond. 18, 137 (2005).\n[36] J. Schliemannand D.Loss,Phys. Rev. B 69, 165315 (2004)\nc/circlecopyrt2003WILEY-VCH VerlagGmbH&Co. KGaA, Weinheim" }, { "title": "0708.3618v1.Diffusive_Ballistic_Crossover_and_the_Persistent_Spin_Helix.pdf", "content": "arXiv:0708.3618v1 [cond-mat.mes-hall] 27 Aug 2007Diffusive-Ballistic Crossover and the Persistent Spin Heli x\nB. Andrei Bernevig\nPrinceton Center for Theoretical Physics, Princeton Unive rsity, Princeton, NJ 08544 and\nDepartment of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544\nJiangping Hu\nDepartment of Physics, Purdue University, West Lafayette, IN 47907\n(Dated: August 9, 2021)\nConventional transport theory focuses on either the diffusi ve or ballistic regimes and neglects the\ncrossover region between the two. In the presence of spin-or bit coupling, the transport equations\nare known only in the diffusive regime, where the spin precess ion angle is small. In this paper,\nwe develop a semiclassical theory of transport valid throug hout the diffusive - ballistic crossover of\na special SU(2) symmetric spin-orbit coupled system. The theory is also valid in the physically\ninteresting regime where the spin precession angle is large . We obtain exact expressions for the\ndensity and spin structure factors in both 2 and 3 dimensiona l samples with spin-orbit coupling.\nPACS numbers: 72.25.-b, 72.10.-d, 72.15. Gd\nThe physics of systems with spin-orbit coupling has\ngenerated great interest from both academic and practi-\ncal perspectives [1]. Spin-orbit coupling allows for purely\nelectric manipulation of the electron spin [2,3,4,5,6], and\ncould be of practical use in areas from spintronics to\nquantum computing. Theoretically, spin-orbit coupling\nis essential to the proposal of interesting effects and new\nphases of matter such as the intrinsic and quantum spin\nHall effect [7,8,9,10,11,12[.\nWhile the diffusive transport theory for a system with\nspin-orbit coupling has recently been derived [13,14], the\nanalysis of diffusive-ballistic transport - where the spin\nprecession angle during a mean free path is compara-\nble to (or larger than) 2 π- has so far remained con-\nfined to numerical methods [15]. This situation is exper-\nimentally relevant since the momentum relaxation time\nτin high-mobility GaAs or other semiconductors can\nbe made large enough to render the precession angle\nφ=αkFτ >2π, whereα,kFare the spin-orbit coupling\nstrength and Fermi momentum respectively. The math-\nematical difficulty in obtaining the crossover transport\nphysics rests in the fact that one has to sum an infinite\nseries of diagrams which, due to the spin-orbit coupling,\nare not diagonal in spin-space. In this paper we obtain\nthe explicit transport equations for a the series of models\nwith spin-orbitcouplingwhereaspecial SU(2)symmetry\nhas recently been discovered [16].\nWe first consider a two-dimensional electron gas with-\nout inversion symmetry for which the most general form\nof linear spin-orbit coupling includes both Rashba and\nDresselhaus contributions:\nH=k2\n2m+α(kyσx−kxσy)+β(kxσx−kyσy),(1)\nwherekx,yis the electron momentum along the [100] and\n[010]directionsrespectively, α, andβarethe strengthsof\nthe Rashba, and Dresselhaussspin-orbit couplingsand m\nis the effective electron mass. At the point α=β, which\nmay be experimentally accessible through tuning of theRashba coupling via externally applied electric fields [2],\na newSU(2) finite wave-vector symmetry was theoreti-\ncally discovered [16]. The Dresselhauss [110] model, de-\nscribing quantum wells grown along the [110] direction,\nexhibits the above symmetry without tuning to a partic-\nular point in the spin-orbit coupling space. At the sym-\nmetry point, the spin relaxation time becomes infinite\ngiving rise to a Persistent Spin Helix. The energy bands\nin Eq.[1] at the α=βpoint have an important shifting\nproperty:ǫ↓(/vectork) =ǫ↑(/vectork+/vectorQ), whereQ+= 4mα,Q −= 0\nfor theH[ReD]model and Qx= 4mα,Q y= 0 for the\nH[110]model. The exact SU(2) symmetry discovered in\n[16] is generated by the spin operators (written here in a\ntransformed basis as):\nS−\nQ=/summationtext\n/vectorkc†\n/vectork↓c/vectork+/vectorQ↑, S+\nQ=/summationtext\n/vectorkc†\n/vectork+/vectorQ,↑c/vectork↓\nSz\n0=/summationtext\n/vectorkc†\n/vectork↑c/vectork↑−c†\n/vectork↓c/vectork↓, (2)\nwithck↑,↓being the annihilationoperatorsofspin-upand\ndown particles. These operators obey the commutation\nrelations for angular momentum, [ Sz\n0,S±\nQ] =±2S±\nQand\n[S+\nQ,S−\nQ] =Sz\n0. Early spin-grating experiments on GaAs\nexhibit phenomena consistent with the existence of such\na symmetry point [17].\nIn [16] the spin-charge transport equations for the\nHamiltonian Eq.[1] have been obtained in the diffusive\nlimit in which αkFτ <<1. However the regions αkFτ∼\n1 andαkFτ >>1 are also experimentally accessible, and\nno theory is yet available to deal with these regimes. We\nnow present the exact spin and charge structure factors\nat the exact symmetry point for any value of the param-\neterαkFτ.\nWe first obtain the spin and charge structure factors\nin the absence of spin-orbit coupling, but valid in both\ntheτ→0 and inτ→ ∞regimes. One should think of\nthe structure factor obtained this way as a generalization\nof the classic Lienhard formulas in the presence of disor-\nder. We then use a non-abelian gauge transformation2\n+++++XY\n−1−a a\n+++++\n− − − −− − − −\nFIG. 1: The sketch of the branch cut and the integral contour\nin the calculation of S(t,q).\nintroduced in [16] to obtain the structure factors for the\nspin-orbit coupling problem described above.\nWe start by formulating the problem in the language\nof the Keyldish formalism [14,18]. Assuming isotropic\nscattering with momentum lifetime τ, the retarded and\nadvanced Green’s functions are:\nGR,A(k,ǫ) = (ǫ−H±i\n2τ)−1. (3)\nWe introduce a momentum, energy, and position depen-\ndent charge-spin density which is a 2 ×2 matrixg(k,r,t).\nSumming over momentum:\nρ(r,t)≡/integraldisplayd2k\n(2π)3νg(k,r,t), (4)\ngives the real-space spin-charge density ρ(r,t) =n(r,t)+\nSi(r,t)σi, wheren(r,t) andSi(r,t) are the charge and\nspin density and ν=m/2πis the density of states in\ntwo-dimensions. ρ(r,t) andg(k,r,t) satisfy a Boltzman-\ntype equation 14,18:\n∂g\n∂t+1\n2/braceleftbigg∂H\n∂ki,∂g\n∂ri/bracerightbigg\n+i[H,g] =−g\nτ+i\nτ(GRρ−ρGA).(5)\nthat we now solve for a free electron gas Hamiltonian.\nTo obtain the spin-charge transport equations, we follow\nthe general sequence of technical manipulations: time-\nFourier transform the above equation; find a general so-\nlution forg(k,r,t) involving ρ(r,t) and thek-dependent\nspin-orbit coupling; perform a gradient expansion of that\nsolution (assuming ∂r<< k FwherekFis the Fermi\nwavevector) to second order; and, finally, integrate over\nthe momentum. The formalism is valid even through the\ndiffusive-ballistic boundary. For the diffusive limit, when\nτis small, we need to keep only the second order term in\nthe gradient expansion which gives rise to the usual spin\nand charge propagator ( iω−Dq2)−1. Asτincreases, we\nneed to keep higher order terms in the gradient expan-\nsion to accurately describe the transport physics. The\nballistic limit requires infinite summation over the gra-\ndient expansion. This can be easiest seen in the regimeof zero spin-orbit coupling, in which the sums can be ex-\nactly performed. It is then fortuitous that our spin-orbit\ncoupled problem can be mapped into a free electron plus\ndisorder problem where we can obtain the structure fac-\ntor exactly. By Fourrier transforming in time we obtain\nthe following recursive equation:\n−iωρ(r,t) =−i/integraldisplaydθkdk\n(2π)2mΩ∞/summationdisplay\nn=1gn(k,r,t) (6)\nwere Ω =ω+i/τand then-th order term reads:\ngn(k,r,t) =∂r1...∂rn/parenleftbigg\n(−ki1\nm)...(−kin\nm)(i\nΩ)ng0(k,r,t)/parenrightbigg\n(7)\nwhereg0(k,r,t) contains a term which fixes the momen-\ntum at the Fermi surface:\ng0(k,r,t) =i\nΩ2π\nτδ(ǫF−k2\n2m) (8)\nSince the initial Hamiltonian and the transport equa-\ntions are rotationally invariant we can assume propaga-\ntion only on [100] and with the use of the identities:\n/integraldisplay2π\n0dθ(cos(θ))n=(1+(−1)n)√πΓ(1+n\n2)\nΓ(1+n\n2)(9)\n∞/summationdisplay\nn=1(1+(−1)n)√πΓ(1+n\n2)\nΓ(1+n\n2)1\n2πan=1−√\n1−a2\n√\n1−a2(10)\nwe can integrate over the Fermi surface angles to obtain\nthe structure factor pole:\nS(ω,q) =1\niω−1\nτ+1\nτ1s\n1−v2\nFq2\n(ω+i\nτ)2(11)\nThecorrectinterpretationofourstructurefactorrequires\nconsistently picking a branch of the square-root function\nin the denominator. We pick the branch cut along the\npositivex-axis. The pole in the structure factor repre-\nsents the characteristic frequencies of the system:\nω1,2=−i\nτ±/radicalbigg\nq2v2\nF−1\nτ2(12)\nwhich in the diffusive and ballistic limits reduces to the\nwell known expressions:\nτ→ ∞ ⇒ω1,2≈ ±vFq\nτ→0⇒ω≈ −iDq2(13)\nwhereD=v2\nFτ/2. The presence of only one (exponen-\ntially decaying) solution in the diffusive limit follows di-\nrectly from correctly treating the branch-cut singularity\nin our structure factor. It can then be seen that the ex-\nponentially divergent solution ω≈iDq2is a false pole of\nEq[11].3\nAlthough not of immediate interest to the present pa-\nper, we also present the structure factor for a bulk Fermi\ngasin the presenceofdisorder. With the densityofstates\ndefined as ν=(2m)3/2E1/2\nF\n4π2the transport equation be-\ncomes:\n−iωρ=−i/integraldisplay /integraldisplay /integraldisplaydφsinθdθk2dk\n(2π)4ντΩ∞/summationdisplay\nn=1gn(14)wheregnandΩareasbeforeandΩ = ω+i/τ. Rotational\ninvariance allows us to take ki=kzand we obtain:\n−iΩρ=mkF\n(2π)2ντ∞/summationdisplay\nn=0/parenleftBigvFq\nΩ/parenrightBig/integraldisplay1\n−1xndx=mkF\n(2π)2ντΩ\nvFqln/parenleftBigg\n1+q\nvFΩ\n1−q\nvFΩ/parenrightBigg\nρ (15)\nIntroducingthethree-dimensionaldensityofstatesatthe\nFermi surface, as well as a δ-function source term, the\nstructure factor reads:\nρ=1\niΩ+Ω\n2τvFqln/parenleftbigg\n1+q\nvFΩ\n1−q\nvFΩ/parenrightbigg (16)\nTo see the diffusive pole we need to carefully expand the\nlogarithm:\nτ→0 :ρ=1\niω−v2\nFτ\n3q2(17)\nWhich is the right diffusive pole in 3 D. For the ballistic\npole we solve the equation (the one below is valid for any\nτ):\nω=vFqe−ivFqτ+eivFqτ\ne−ivFqτ−eivFqτ−i\nτ(18)\nIn the ballistic limit τ→ ∞the exponentials in the frac-\ntion are oscillating wildly and must be regularized. De-\npending on on the regularization q→q+0±the charac-\nteristic frequencies are:\nω=±vFq (19)\nwhich are the ballistic poles.\nHaving solved the free-Fermi gas case, we now add\nspin-orbit coupling at the special SU(2) symmetric point\nof the Persistent Spin Helix. Following [16], we express\nthe spin-orbit coupling Hamiltonian Eq.[1] in the form\nof a background non-abelian gauge potential HReD=\nk2\n−\n2m+1\n2m(k+−2mασz)2+const.where the field strength\nvanishes identically for α=β. Therefore, we can elimi-\nnate the vector potential by a non-abelian gauge trans-\nformation: Ψ ↑(x+,x−)→exp(i2mαx+)Ψ↑(x+,x−),\nΨ↓(x+,x−)→exp(−i2mαx+)Ψ↓(x+,x−). Under this\ntransformation, the spin-orbit coupled Hamiltonian is\nmapped to that of the free Fermi gas, but, while diagonal\noperatorssuch asthe charge nandSzremainunchanged,0 1 2 3 4 0 1 2 3 4t(a) (b) \nt Im(S(t,q)) Re(S(t,q)) \nFIG.2: (a)Theimaginary partand(b)thereal partof S(t,q).\nWesetτ= 1. Forbothfigures, from bottomtotop, thecurves\nare corresponding to a= 2.2,2.6,3,3.4,3.8,4.2.\noff-diagonal operators, such as S−(/vector x) =ψ†\n↓(/vector x)ψ↑(/vector x)\nandS+(/vector x) =ψ†\n↑(/vector x)ψ↓(/vector x) are transformed: S−(/vector x)→\nexp(−i/vectorQ·/vector r)S−(/vector x),S+(/vector x)→exp(i/vectorQ·/vector r)S+(/vector x). Here\n/vectorQis the shifting wavevector of the spin-orbit coupled\nHamiltonian. Since in the gauge transformed basis, all\nthree components of the spin and charge have the struc-\nture factor derived above, in the original (experimentally\nmeasurable)basis, the SxandSyhavethefollowingform:\nS±(ω,/vector q) =1\niω−1\nτ+1\nτ1s\n1−v2\nF(/vector q±/vectorQ)2\n(ω+i\nτ)2(20)\nThe above result represents the exact form factor for\na spin-orbit coupled system valid everywhere from the\ndiffusive to ballistic regimes. The Persistent Spin He-\nlix is clearly maintained for any values of τ,α,vfsince\nS(ω,/vectorQ) = 1/iωwhich renders the spin life-time infinite.\nThe transient grating experiments [17,19] measure\ntheωFourrier transform of S(ω,q), i.e.S(t,q) =\n1\n2π/integraltext\ndte−iωtS(ω,q).S(ω,q) is analytic in the upper half\ncomplex plane. Thus, S(t,q) is zero for t<0. Fort>0,4\n0 0.5 1 1.5 2 2.5 3 3.500.10.20.30.40.50.60.7\na Ω\nFIG. 3: The oscillation frequency Ω in the imaginary part of\nS(t,q) as a function of a=νF|/vector q±/vectorQ|τ.\nby selecting the integral contour as shown in fig.(1), we\nobtain its real part and imaginary part as follows:\nIm(S(t,q))\ne−t\nτ=a\n1+a2+P/integraldisplay∞\na2\nπ√\nx2−a2cos(xt\nτ)\nx(x2−1−a2)\nRe(S(t,q))\ne−t\nτ=−a2+cos(√\n1+a2t\nτ)\n1+a2(21)\nwherea=vF|/vector q±/vectorQ|τandPindicates the principal value\nof the integral.\nIn Fig.(2), we plot the real and imaginary part of\nS(t,q) for different values of a. In the figure, we setτ= 1andfrombottomto top, the curvesarecorrespond-\ning toa= 2.2,2.6,3,3.4,3.8,4.2. Although the real part\nis clearly an oscillating function of twith an oscillation\nfrequency,√\n1+a2\nτ, the oscillation is not easily seen in the\nfigure. However, the imaginary part has a much larger\noscillation amplitude than the real part and the oscilla-\ntion becomes clear as increasing a, reflecting the ballistic\nnature of the sample. The oscillation frequency Ω in the\nimaginary part is linearly dependent on aas shown in\nFig.(3).\nIn this paper we have obtained the exact transport\nequations valid in the diffusive, ballistic, and crossover\nregimes of a special type of spin-orbit coupled system\nwhich enjoys an SU(2) gauge symmetry. We obtained\nthe exact form of the structure factors, and found the\ndependence of the spin-density as would be observed in a\ntransient-grating experiment. It would be interesting to\nwork out the transport equations in the diffusive-ballistic\nregime in perturbation theory away from the Persistent\nSpin Helix.\nB.A.B. wishes to acknowledge the hospitality of the\nKavli Institute for Theoretical Physics at University of\nCalifornia at Santa Barbara, where part of this work was\nperformed. BAB acknowledges fruitful discussions with\nJoe Orenstein, C.P. Weber, Jake Koralek and Shoucheng\nZhang. This work is supported by the Princeton Cen-\nter for Theoretical Physics and by the National Science\nFoundation under grant number: PHY-0603759.\n1S. A. Wolf et. al., Science 294, 1488 (2001).\n2J. Nitta et. al., Phys. Rev. Lett. 78, 1335 (1997).\n3D. Grundler, Phys. Rev. Lett. 84, 6074 (2000).\n4Y. Kato et. al., Phys. Rev. Lett. 93, 176601 (2004).\n5Y. Kato et. al., Nature 427, 50 (2004).\n6C.P. Weber et. al, Nature 437, 1330 (2005).\n7S.Murakami, N.Nagaosa, andS.Zhang, Science 301, 1348\n(2003).\n8J. Sinova et. al., Phys. Rev. Lett. 92, 126603 (2004).\n9C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n10C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95, 146802\n(2005).\n11B.A. Bernevig and S.C. Zhang, Phys. Rev. Lett. 96,\n106802 (2006).12B.A. Bernevig, T.L.Hughes, and S.C. Zhang, Science 314,\n1757 (2006).\n13A. Burkov, A. Nunez, and A. MacDonald, Phys. Rev. B\n70, 155308 (2004).\n14E.G. Mishchenko, A.V.Shytov, and B.I. Halperin, Phys.\nRev. Lett. 93, 226602 (2004).\n15K. Nomura and et. al., Phys. Rev. B 72, 245330 (2005).\n16B.A. Bernevig, J. Orenstein, and S.C. Zhang, Phys. Rev.\nLett.97, 236601 (2006).\n17C.P. Weber and J. Orenstein et. al, Phys. Rev. Lett. 98,\n076604 (2007).\n18J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).\n19N. Gedik et. al., Science 300, 1410 (2003)." }, { "title": "0910.5931v2.Gravitational_waveforms_from_unequal_mass_binaries_with_arbitrary_spins_under_leading_order_spin_orbit_coupling.pdf", "content": "Gravitational waveforms from unequal-mass binaries with arbitrary spins under\nleading order spin-orbit coupling\nManuel Tessmer\u0003\nTheoretisch-Physikalisches Institut, Friedrich-Schiller-Universit at Jena, Max-Wien-Platz 1, 07743 Jena, Germany\n(Dated: May 29, 2022)\nThe paper generalizes the structure of gravitational waves from orbiting spinning binaries under\nleading order spin-orbit coupling, as given in the work by K onigsd or\u000ber and Gopakumar [PRD 71,\n024039 (2005)] for single-spin and equal-mass binaries, to unequal-mass binaries and arbitrary spin\ncon\fgurations. The orbital motion is taken to be quasi-circular and the fractional mass di\u000berence is\nassumed to be small against one. The emitted gravitational waveforms are given in analytic form.\nPACS numbers: 04.30.Db, 04.25.Nx\nI. INTRODUCTION\nAs already stated in many publications before, grav-\nitational waves from inspiralling compact binaries are\nthe most promising sources for ground based planned\nand already operating gravitational wave (GW) detec-\ntors. To guarantee successful search for GWs, one needs\nto obtain promising search templates incorporating all\nimportant physical e\u000bects that have an in\ruence on the\nform of the signal. Ground-based detector networks\nlikeLIGO (USA) ,VIRGO (France/Italy) and GEO 600\n(Germany/UK) have the sensitivity to be able to see the\nlast seconds or minutes of the binary's inspiral, where\nthe corrections, coming from general relativity, to the\nNewtonian orbital motion get important, depending on\ntheir masses. In order to detect GWs from inspiralling\ncompact binaries without spin in quasi-circular orbits, a\nlarge library of ready-to-use inspiral templates has been\nput up [1]. Eccentric inspiral models without spin have\nalso been developed [2, 3] and are well understood. Re-\ncently, Yunes and collaborators obtained a formalism for\nfrequency domain GW \flters for eccentric binaries [4].\nAll of them heavily employ the post Newtonian (PN) ap-\nproximation to general relativity.\nIt has been shown by several authors, that for a suc-\ncessful detection, e\u000bects of spin have to be included and\nfoundations for the detection of spins have been laid down\n[5, 6, 7, 8, 9]. During the inspiral phase, before reaching\nthe last stable orbit, those e\u000bects are long-term modu-\nlations of the GW signal in a comparison with the time\nscale of only one orbit. They can lead to substantially dif-\nferent shapes of the signal compared to those ones show-\ning up if the spins are neglected. The foundations for\nthe motion of spins in curved spacetimes are given in\n[10]. In harmonic coordinates, the spin dependent EOM\nwere derived up to next-to-leading order in the spin-orbit\ncoupling by Faye et al. [11] and Blanchet et al. [12],\nwhere velocities have been used to characterize the or-\nbits. In Arnowitt-Deser-Misner coordinates [13], higher\n\u0003Electronic address: M.Tessmer@uni-jena.deorder Hamiltonians dictating the equations of motion for\norbits and spin (from this point on referred to as EOM)\nhave recently been derived by Damour et al. [14] and\nSteinho\u000b et al. [15, 16]. The spin-independent part of\nthe binary Hamiltonian is known to 3PN order [17].\nThe solution to \\simple precession\" of the leading or-\nder spin-orbit interaction, which was the case for single\nspin or equal mass, was discussed in [9] and later in [18],\nwhere the GW polarizations h\u0002andh+were derived as\na PN accurate analytic solution for eccentric orbits. The\nlatter has heavily inspired this work, which will give an\napproach to the more general case of unequal masses and\narbitrary two-spin con\fgurations.\nThe paper will be organized as follows. Section II will\npresent the involved Hamiltonians and the associated\nEOM for the binary in the center-of-mass frame. In\nsection III, the geometry and the coordinates relating\nthe generic reference frame with the orientation of the\nspins and the angular momentum vector are provided and\ncharacterized by rotation matrices. The time derivatives\nof these rotation matrices will be compared by Poisson\nbrackets in section IV and \frst order time derivatives of\nthe associated rotation angles will be obtained. A \frst-\norder perturbative solution to the EOM for the spins\nis worked out in section V. The orbital motion will be\ncomputed, for quasi-circular orbits (circular orbits in the\nprecessing orbital plane), in section VI. As an applica-\ntion, the resulting GW polarizations, h\u0002andh+in the\nquadrupolar restriction, are given in section VII.\nII. THE CONSERVATIVE EQUATIONS OF\nMOTION FOR THE SPINS\nIn this section the dynamics of spinning compact bina-\nries is investigated where the spin contributions are re-\nstricted to the leading order gravitational coupling. The\nHamiltonian associated therewith reads\nH=HN+H1PN+H2PN+HSO; (2.1)\nwithHN,H1PNandH2PNrespectively are the Newto-\nnian, \frst and second PN order contributions to the con-\nservative point particle dynamics (e.g., [19] and refer-arXiv:0910.5931v2 [gr-qc] 21 Dec 20092\nences therein) and HSOis the leading order spin-orbit\nHamiltonian [20].\nIn the following computations, use will be made of the\nfollowing scalings to convert the quantities in calligraphic\nletters to dimensionless ones on the rhs:\nH=H\u0016c2; (2.2)\nR=rGm\nc2; (2.3)\nP=p\u0016c; (2.4)\nSa=SaGma\nc2(mac); (2.5)wheremais the mass of the athobject (a= 1;2),mis\nthe total mass, m=m1+m2,\u0016is the reduced mass de-\n\fned asm1m2=mand the symmetric mass ratio is given\nby\u0011:=m1m2=m2. The variables pandrare the scaled\nlinear canonical momentum and position vectors, respec-\ntively, and commute with the spins Sa. Explicitly, the\ncontributions to the scaled version of Eq. (2.1) read\nH(r;p;S1;S2) =HN(r;p) +H1PN(r;p) +H2PN(r;p)\n+HSO(r;p;S1;S2); (2.6)\nwith\nHN(r;p) =p2\n2\u00001\nr; (2.7a)\nH1PN(r;p) =1\nc2\u001a1\n8(3\u0011\u00001)\u0000\np2\u00012\u00001\n2\u0002\n(3 +\u0011)p2+\u0011(n\u0001p)2\u00031\nr+1\n2r2\u001b\n; (2.7b)\nH2PN(r;p) =1\nc4\u001a1\n16\u0000\n1\u00005\u0011+ 5\u00112\u0001\u0000\np2\u00013+1\n8h\u0000\n5\u000020\u0011\u00003\u00112\u0001\u0000\np2\u00012\u00002\u00112(n\u0001p)2p2\u00003\u00112(n\u0001p)4i1\nr\n+1\n2\u0002\n(5 + 8\u0011)p2+ 3\u0011(n\u0001p)2\u00031\nr2\u00001\n4(1 + 3\u0011)1\nr3\u001b\n; (2.7c)\nHSO(r;p;S1;S2) =1\nc2r3(r\u0002p)\u0001Se\u000b; (2.7d)\nwherer\u0011jrjandSe\u000bis the so-called e\u000bective spin ,\nSe\u000b\u0011\u000e1S1+\u000e2S2; (2.8a)\n\u000e1\u0011\u0011\n2+3\n4\u0010\n1 +p\n1\u00004\u0011\u0011\n; (2.8b)\n\u000e2\u0011\u0011\n2+3\n4\u0010\n1\u0000p\n1\u00004\u0011\u0011\n: (2.8c)\nConsidering only the spin-independent part of the Hamil-\ntonian, the orbital angular momentum vector is a con-\nserved quantity. The motion of the reduced mass \u0016will,\nwithout SO interactions, take place in a plane that is\nperpendicular to Land that is invariant in time. Adding\nthe spin-orbit term will, in general, lead to a precession\nof the orbital angular momentum. The EOM for L, de-\n\fned by L:=r\u0002pand the individual spins S1&S2\ncan be deduced from the equations\ndL\ndt=fL;HSOg=1\nc2r3Se\u000b\u0002L; (2.9a)\ndS1\ndt=fS1;HSOg=\u000e1\nc2r3L\u0002S1; (2.9b)\ndS2\ndt=fS2;HSOg=\u000e2\nc2r3L\u0002S2: (2.9c)\nEquation (2.9a) describes the precession of Lw.r.t.\nthe total angular momentum vector J, de\fned as\nJ\u0011L+S1+S2. The key idea in the next sections is\nto compute time dependent rotation matrices for L,S1andS2for a number of rotation axes and angles that are\nto be introduced in the next section. Let us state that\nthe magnitudes L,S1andS2of the vectors L,S1and\nS2are conserved,\ndL2\ndt=d\ndt(L\u0001L) =2\nc2r3L\u0001(Se\u000b\u0002L) = 0; (2.10a)\ndS2\n1\ndt=d\ndt(S1\u0001S1) =2\u000e1\nc2r3S1\u0001(L\u0002S1) = 0;(2.10b)\ndS2\n2\ndt=d\ndt(S2\u0001S2) =2\u000e2\nc2r3S2\u0001(L\u0002S2) = 0:(2.10c)\nEquations (2.9) show that _L=\u0000(_S1+_S2) and, thus,\nthe total angular momentum vector Jsatis\fes\ndJ\ndt= 0 , givingdjJj\ndt= 0: (2.11)\nThe magnitudes of SandSe\u000bbehave as follows,\ndS2\ndt=\u00003p1\u00004\u0011\nc2r3L\u0001(S1\u0002S2); (2.12a)\ndS2\ne\u000b\ndt=\u00003p1\u00004\u0011(12 +\u0011)\u0011\n4c2r3L\u0001(S1\u0002S2):(2.12b)\nNotice the conservation of S2\ne\u000bin both the test-mass\n(\u0011= 0) and equal-mass ( \u0011= 1=4) cases. Using above\nequations, we will be able to compute the evolution equa-\ntions for the rotation angles. The associated geometry is\nintroduced next.3\nIII. GEOMETRY OF THE BINARY\nAs done in [18], it is very useful to use a \fxed or-\nthonormal frame ( eX;eY;eZ) and to set eZalong the\n\fxed vector J. The invariable plane perpendicular to\nJwill then be spanned by the vectors ( eX;eY). The\nmotion of the reduced mass will take place in the or-\nbital plane perpendicular to the unit vector k:=L=L.\nFor a clear understanding of the following, please take a\nlook at Fig. 1. First, the vector kis inclined to eZby\nthe ( time-dependent ) angle \u0002, which was also the con-\nstant precession cone of Laround Jfor the single-spin\nand equal-mass case of [18]. As before, the orbital plane,\nitself spanned by the vectors ( i;j), where j=k\u0002i, in-\ntersects the invariable plane at the line of nodes i, with\nthe longitude \u0007 measured in the invariable plane from\neX.\nThe geometry of the binary will be completed by the\nspin related coordinate system ( is;js;ks). This frame\nis constructed from the system ( i;j;k) to be rotated\naround the axis ito point from the top of Lto the top\nofJwith the new direction ks. In other words, this spin\ncoordinate system is chosen in such a way that the total\nspin,S1+S2, has only a kscomponent and is\u0011iholds.\nIf \u0002 is known, the spins are left with an additional free-\ndom to rotate around ksby an angle \u001es(the index \\s\"\nis a hint for positions in the spin system). This angle is\nmeasured from isto the projection of S1to the ( is;js)\nplane, similar to \u0007's function in the reference frame.\nThere exist simple geometrical relations that will re-\nduce the freedom to choose rotation angles arbitrarily, aswill be shown in the next subsection.\nA. Geometrical issues\nAs mentioned already, in this geometry the spins and\nangular momenta { being \fxed in their magnitudes {\nonly have three degrees of freedom: the angles \u0002, \u0007 and\n\u001es. Once \u0002 is determined, also \u000bks(the angle between L\nandS) is \fxed and so is magnitude SofS=S1+S2by\ntriangular relations. Calling \u000b12the angle between the\nspinsS1andS2, the following equations list the rotation\nangles and magnitudes as functions of \u0002, where also use\nis made of the sin relations,\nS(\u0002) =p\nJ2\u00002JLcos \u0002 +L2; (3.1a)\n\u000b12(\u0002) = cos\u00001\u0012S(\u0002)2\u0000S2\n1\u0000S2\n2\n\u00002S1S2\u0013\n; (3.1b)\n\u000bks(\u0002) =\u0019\u0000sin\u00001\u0012Jsin(\u0002)\nS(\u0002)\u0013\n; (3.1c)\n~s(\u0002) = sin\u00001\u0012S2sin\u000b12(\u0002)\nS(\u0002)\u0013\n: (3.1d)\nThese relations will be used extensively to simplify the\nangles evolution equations. How they are incorporated\nand applied will be shown next.\nB. Coordinate bases and associated transformation\nmatrixes\nThis section introduces the coordinate transformations\nfrom the reference system to the orbital triad and the spin\nsystem. To construct the EOM for the 3 physical angles\n\u0002;\u0007 and\u001es, the idea is to compare the evolution of these\nrotation angles - as arguments for rotation matrices -\nwith the Poisson brackets, Eqs. (2.9a) - (2.9c). Let us\nbegin with the explicit computation of the transformed\ncoordinate bases.\n1. The orbital triad ( i;j;k) can be, not surprisingly,\nconstructed by only 2 rotations from the reference\nsystem. In terms of rotation matrices, we have\n0\n@i\nj\nk1\nA=0\n@1 0 0\n0 cos \u0002 sin \u0002\n0\u0000sin \u0002 cos \u00021\nA0\n@cos \u0007 sin \u0007 0\n\u0000sin \u0007 cos \u0007 0\n0 0 11\nA(3.2)\n\u00020\n@eX\neY\neZ1\nA:2. The spin system is constructed, simply by another\nrotation of\u000bksaround the vector i, from the orbital\ntriad,\n0\n@is\njs\nks1\nA=0\n@1 0 0\n0 cos\u000bks\u0000sin\u000bks\n0 sin\u000bks cos\u000bks1\nA0\n@i\nj\nk1\nA; (3.3)\nsuch that is\u0011iholds. Important note: the angle\n\u000bkshas negative sign relative to \u0002. That's because\nkshas to be moved \\backwards\" to point to J!\nHaving transformed the unit vectors with these ma-\ntrices, the coordinates transform by their transposed in-\nverses, which are { in case of rotations { the matrices\nthemselves.\nNow, we have everything under control to construct\nthe set of all the physical vectors. I will list all of them\nbelow. First, let me de\fne some shorthands for rotation\nmatrices:\n[\u0002]\u00110\n@1 0 0\n0 cos \u0002 sin \u0002\n0\u0000sin \u0002 cos \u00021\nA; (3.4)4\nk\nii\ni\njeXeYjsks\nk=L/LS1S2J=JeZ\nΘ\nΥαksα12φs ˜s\nInvariable planeOrbital plane\nFIG. 1: Binary geometry completed by a rotating spin coordinate system. The usual reference frame is ( eX;eY;eZ) having\nchosen eZto be aligned with J. The vectors L;S1;S2describe the orbital angular momentum and the individual spins,\nrespectively. The angle \u0002 denotes the inclination angle of Lw.r.t. J, which is { of course { to be taken as a time dependent\nquantity. The orbital plane, being perpendicular to Lby construction, is spanned by the orthonormal vectors jandi, where\nthe latter one intersects the invariable plane at the angle \u0007 measured from eX. The spin-coordinate system is constructed\nout of the orbital dreibein ( i;j;k) by a rotation of \u000bksaround i, such that the vector pointing from LtoJis the total spin\nS1+S2. The angle \u000b12is measured between S1andS2. The spin S1, projected into the ( js;is\u0011i) plane is rotated by an\nangle\u001esfromi, andS1itself is moving on the circle (with variable radius) embedded in the \fgure.\n[\u0007]\u00110\n@cos \u0007 sin \u0007 0\n\u0000sin \u0007 cos \u0007 0\n0 0 11\nA; (3.5)\n[\u000bks]\u00110\n@1 0 0\n0 cos\u000bks\u0000sin\u000bks\n0 sin\u000bks cos\u000bks1\nA: (3.6)\nThe orbital angular momentum Lin the reference system\n(indices labeled inv) arises from two rotations from the\norbital triad ( ot) where it has only one component:\nL=f[\u0002(t)] [\u0007(t)]g\u00001(0;0;L); (3.7)\nor, in components,\n(L)inv\ni=\u001a\n[\u0002(t)] [\u0007(t)]\u001b\u00001\nij(L)ot\nj: (3.8)The spins, in the spin system ( s), where the ksis aligned\nwithS=S1+S2, have the following form,\nS1=S1(cos\u001essin ~sis+ sin\u001essin ~sjs+ cos ~sks);\n(3.9a)\nS2=Sks\u0000S1; (3.9b)\nS=Sks: (3.9c)\nIV. THE TIME DERIVATIVES OF \u0007,\u0002AND\u001es\nTo obtain an EOM for the angle \u0002, one possibility is\nto use the time derivative of jSj2= (S\u0001S), Eq. (2.12a),\nto apply this, for example, in the spin system and to\ncompare the result with the time derivative of Eq. (3.1a)5\nwith \u0002 = \u0002( t). The result is\n_\u0002 =\u0000CSS1S\n2Jsin\u000bkscos\u001essin ~scsc \u0002 (4.1)\nwith\nCS=\u00003p1\u00004\u0011\nc2r3: (4.2)The same result will be obtained by computing the time\nderivative of the orbital angular momentum Lin the in-\nvariable system. Therefore, take Eq. (3.7), compute its\ntime derivative and \fnally compare the result with (2.9a).\nBecause the angular velocities appear in relatively simple\nrelations, it is easy to extract them from the eXandeY\nentry. The results are\n_\u0007 =\u0000CLcsc \u0002 [S1(\u000e2\u0000\u000e1) cos\u000bkssin\u001essin ~s+ sin\u000bks(S1(\u000e1\u0000\u000e2) cos ~s+S\u000e2)]; (4.3)\n_\u0002 =CLS1(\u000e1\u0000\u000e2) cos\u001essin ~s; (4.4)\nwithCL:= (c2r3)\u00001. The functional dependencies\nof\u000bks, ~sandSon \u0002 are implicated. Inserting the geo-\nmetrical relations, Eqs. (3.1), it turns out that Eqs. (4.1)\nand (4.4) are equivalent. Also, the allegedly worrying\nasymmetric appearance of the quantity S1can be stu-\ndiously avoided by replacing ~ sby its function of \u0002.1\nAlso note that, if the relations \u0011= 1=4 orSi= 0 (i=\n1 or 2) are inserted in Eq. (4.3), one recovers Eq. (4.32)\nof [18].\nNow, let us turn to the last quantity to be determined,\nthe angle\u001es. The geometry o\u000bers various possibilities to\ncalculate the time derivative of this angle. The easy way\nis to compute S1in the invariable system. In compo-\nnents, we have\n(S1)inv\ni=\u001a\n[\u000bks(t)] [\u0002(t)] [\u0007(t)]\u001b\u00001\nij(S1)s\nj: (4.5)\nThe time derivative of (4.5) might be compared with Eq.\n(2.9b). The result will be given in terms of the angles\nalready determined: since we already know _\u0002 and _\u0007 on\nthe one hand and \u000bksas a function of \u0002 on the other, we\nhave the expression under full control.\nThe other way is to take the Leibniz product rule for\nS, namely@tS=@tSiei+Si@teiwithei= (is;js;ks).\nWe we already know that is\u0011i,ksjjSandjs\u0011is\u0002ks,\nwhose time derivatives are already known. Both consid-\nerations result in\n_\u001es=C\u0002(_\u0002\u0000_\u000bks) +C~s_~s+ \n 0; (4.6a)\nC\u0002= tan\u001escot(\u000bks\u0000\u0002) + sec\u001ecot ~s; (4.6b)\nC~s=\u0000sec\u001escot(\u000bks\u0000\u0002)\u0000tan\u001ecot ~s; (4.6c)\n\n0=\u0000C1Lsin \u0002 csc(\u000bks\u0000\u0002): (4.6d)\nFor the case of equal masses ( \u000e1=\u000e2=\u000e= 7=8), one\nobtains for _\u001es,_\u0007 and _\u0002 a very simple system of EOM,\n_\u0002 = 0; (4.7a)\n1The angular velocities, (4.3) and (4.4), are in complete agreement\nwith Eqs. (5.11a) and (5.11b) of [21]._\u0007 =7J\n8c2r3; (4.7b)\n_\u001es=\u0000L\u000esin \u0002\nc2r3csc\u001a\nsin\u00001\u0012Jsin \u0002\nS(\u0002)\u0013\n+ \u0002\u001b\n;(4.7c)\nwhich can be integrated immediately, giving\n\u0002(t) = \u0002 0; (4.8a)\n\u0007(t) = \n \u0007t+ \u0007 0; (4.8b)\n\u001es(t) = \n\u001et+\u001es0; (4.8c)\nwith the angular velocities\n\n\u0007\u00117J\n8c2r3; (4.9a)\n\n\u001e\u0011\u0000L\u000esin \u0002 0\nc2r3csc\u001a\nsin\u00001\u0014Jsin \u0002 0\nS(\u00020)\u0015\n+ \u0002 0\u001b\n:(4.9b)\nSummarizing the EOM for the coordinate transformation\nangles, Eqs. (4.3), (4.4) and (4.6), this system of EOM\ncan be written in a compact manner. Calling the vector\nof constants, C=fE;S 1;S2;L;k\u0001Se\u000bg{ whereEandL\nare related in the case of quasi-circular orbits { and the\nvector of dynamic variables, associated with spins and\nangular momentum, X=f\u0002;\u0007;\u001esg, we may write\nd\ndtX=YC(X): (4.10)\nA perturbative solution will be given in the next section.\nV. FIRST ORDER PERTURBATIVE SOLUTION\nTO THE EOM FOR THE NON-EQUAL MASS\nCASE\nThe EOM for (\u0002 ;\u0007;\u001es) can also be solved by a simple\nreduction scheme. We assume that the deviation from\nthe equal-mass case is small compared to unity,\n\u000e1\u0000\u000e2\n\u000e1+\u000e2\u001c1: (5.1)6\nThen, having the equal-mass case under full analytic con-\ntrol, we can construct a perturbative solution to the non-\nequal mass case. The proceeding is as follows: Imagine a\nsystem of EOM for a number Nof dependent variables\nX:\n_X=Y(X): (5.2)\nThe time domain solution to this system is denoted by\nthe superscript \\0\", viz\nX(t) =X(0)(t): (5.3)\nLet us assume that the EOM, Eq. (5.2), are perturbed by\nsome terms of the order \u000f(\u000fis a dimensionless ordering\nparameter),\n_X=Y(X) +\u000fP(X): (5.4)\nThe solution at the \frst order in \u000fcan be obtained by\nadding a small perturbed quantity to be determined to\nthe solution of the homogeneous equation,\nX(1)\ni(t) =X(0)\ni(t) +\u000fSi(t): (5.5)\nInserting this into Eq. (5.4), one obtains\n_X(1)\ni=_X(0)\ni+\u000f_Si\n=Yi(X(0)\nj+\u000fSj) +\u000fPi(X(0)\nj+\u000fSj)\n=Yi(X(0)\nj) +\u000fNX\nj=1@Yi\n@XjSj+\u000fPi(X(0)\nj) +O(\u000f2) (5.6)\nComparing the coe\u000ecients of the two orders of \u000fgives\n0 :_X(0)\ni=Yi(X(0)\nj); (5.7)\n1 : _Si=NX\nj=1@Yi\n@XjSj+Pi(X(0)\nj): (5.8)\nThe \frst equation is solved via de\fnition, and what re-\nmains is the second, having inserted the unperturbed so-\nlution in the perturbing function P. For our purposes,\nN= 3 with X=f\u0007;\u0002;\u001esgis a small number of EOMs,\nbut complicated functional dependencies are included.\nThe matrix appearing in Eq. (5.8) does not mean a prob-\nlem to us, because fortunately, the only dependency of\nthe sources is on \u0002.\nFor our computation, we need to divide the EOM into\na non-perturbative and a perturbative part. In the fol-\nlowing, we use the de\fnitions\n\u001f1=\u000e1+\u000e2\n2; (5.9)\n\u001f2=\u000e1\u0000\u000e2\n2: (5.10)\nRewriting the EOM for the angles in terms of \u001f1and\u001f2,\nlabeling all \u001f2contributions with the order parameter \u000f\nas well as inserting the non-perturbative solution, Eqs.\n(4.8) to these terms, one obtains\n_\u0002(1)=\u000f_S\u0002=\u000fCLS12\u001f2cos(t\n\u001e+\u001e0) sin ~s(\u00020); (5.11a)\n_\u0007(0)+\u000f_S\u0007=CLS(\u0002)\u001f1sin\u000bks(\u0002) csc \u0002|{z}\n\u0011CLJ\u001f1=const.+\u000f\u0002\nCL\u001f2csc \u0002 0\u0000\n2S1cos\u000bks(\u00020) sin ~s(\u00020) sin (t\n\u001e+\u001e0)\n\u0000sin\u000bks(\u00020) (S(\u00020)\u00002S1cos ~s(\u00020))\u0001\u0003\n; (5.11b)\n_\u001e(0)\ns+\u000f_S\u001e=\u0000C1Lsin \u0002 csc\u000bks(\u0002) +\u000f\u0014\nC~s(\u0002;\u001es)@~s\n@\u0002+C\u0002(\u0002;\u001es)\u0012\n1\u0000@\u000bks\n@\u0002\u0013\u0015\n\u0002=\u0002 0\n\u001es=t\n\u001e+\u001es0_\u0002; (5.11c)\n=\u0000(\u001f1+\u000f\u001f2)L\nc2r3sin \u0002 csc\u000bks(\u0002) +\u000f\u0014\nC~s(\u0002;\u001es)@~s\n@\u0002+C\u0002(\u0002;\u001es)\u0012\n1\u0000@\u000bks\n@\u0002\u0013\u0015\n\u0002=\u00020\n\u001es=t\n\u001e+\u001es0_\u0002:\n(5.11d)\nThe parameter \u000fsimply counts the order of the perturbative contribution and is later set to one. The \frst term for\n\u0007 is constant and thus does not have to be expanded in powers of \u000f, but the associated \frst term for \u001esdoes, such\nthat the perturbative solution for \u0002 has to be included. Taylor expanding this term, removing all contributions to\nthe unperturbed problem, what remains is a system of EOM for S\u0002;S\u0007;S\u001ethat can be simply integrated, because\nas soon asS\u0002(t) is known, all the other contributions are straightforwardly evaluated. Requiring that the perturbing\nsolutions vanish at t= 0, the solutions are simply given by\nS\u0002(t) =Zt\n0_S\u0002dt; (5.12a)\nS\u0007(t) =Zt\n0_S\u0007dt; (5.12b)7\nS\u001e(t) =Zt\n0_S\u001edt; (5.12c)\nand explicitly read\nS\u0002(t) =\u0000CLS1S22\u001f2\nS(0)\n\u001esin\u000b12(0)(sin\u001es0\u0000sin (t\n\u001e+\u001es0)); (5.13a)\nS\u0007(t) =CL\u001f2csc \u0002 0\n\n\u001e\u0014\n2S1cos\u000bks(0)sin ~s(0)(cos\u001es0\u0000cos (t\n\u001e+\u001es0))\n\u0000t\n\u001esin\u000bks(0)\u0000\nS(0)\u00002S1cos ~s(0)\u0001\u0015\n; (5.13b)\nS\u001e(t) =Cstatt+C0(t) +C~s(t) +C\u0002(t) +C\u000b(t); (5.13c)\nwith the shorthands\nCstat=\u001f2\n\u001e\n\u001f1; (5.14a)\nC0(t) =CLJS1S2\u001f2sin \u0002 0sin\u000b12(0)(t\n\u001esin\u001es0+ cos (t\n\u001e+\u001es0)\u0000cos\u001e0)\u0010\n\u00002c2Jr3\n\u0003\u00002\u001f1S2\n(0)\u0011\n\u001f1S3\n(0)\n\u001eq\nS2\n(0)\u0000J2sin2\u00020;(5.14b)\nC~s(t) =\u00002CLJ\u001f2\u0010\nS2\n(0)cot\u000b12(0)\u0000S1S2sin\u000b12(0)\u0011\n\u001f1S2\n(0)\n\u001eq\nS2\n(0)\u0000S2\n2sin2\u000b12(0)\u0002\n\u0000\nc2r3t\n\u001esin ~s(0)\n\u0003+L\u001f1sin \u0002 0cos ~s(0)(cos\u001es0\u0000cos (t\n\u001e+\u001es0))\u0001\n; (5.14c)\nC\u0002(t) =2S1t\u001f2cos ~s(0)\nc2r3+2S1\u001f2\n\u0003csc \u0002 0sin ~s(0)(cos\u001es0\u0000cos (t\n\u001e+\u001es0))\nL\u001f1\n\u001e;\nC\u000b(t) =C\u0002(t)J\u0010\nS2\n(0)cos \u0002 0\u0000JLsin2\u00020\u0011\nS2\n(0)q\nS2\n(0)\u0000J2sin2\u00020; (5.14d)\nthe initial values of the functions (3.1a) - (3.1d)\nS(0)\u0011S(\u00020); (5.15a)\n\u000bks(0)\u0011\u000bks(\u00020); (5.15b)\n\u000b12(0)\u0011\u000b12(\u00020); (5.15c)\n~s(0)\u0011~s(\u00020); (5.15d)\n(5.15e)\nand the de\fnition\n\n\u0003\u0011\n\u001es\n1\u0000L2\u001f2\n1sin2\u00020\nc4r6\n2\n\u001e: (5.15f)\nVI. THE ORBITAL MOTION\nThe motion of the spins is only half of the physical con-\ntent of the spin-orbit dynamics. Once we fully have themotion of all the spin-related angles under control, we\nmight turn to the orbital dynamics, i.e.the motion of\nthe reduced mass in the orbital plane. It will turn out\nthat employing coordinate transformations will be very8\nhelpful here, too.\nThe aim is to solve the orbital EOM to the full Hamil-\ntonian,\nH=HN+H1PN+H2PN+HSO: (6.1)\nAt this point, we can do a useful simpli\fcation. As long\nas we incorporate only leading order spin dynamics, only\nNewtonian point particle and spin dependent contribu-\ntions will mix at the end, higher order PN terms cou-\npling with the spins will be neglected consequently. For\nthe computation of the spin dependent part of the orbital\nphase, therefore, we only have to take HN;SO=HN+HSO\nand add the 1PN and 2PN (spinless) terms for the point\nparticle afterwards.\nH=HN;SO+H1PN+H2PN; (6.2)\n_'= _'N;SO+ _'1PN+ _'2PN: (6.3)\nThe Newtonian and spin orbit part of eq. (6.1) reads\nHN;SO=p2\n2\u00001\nr+1\nc2r3(r\u0002p)\u0001Se\u000b: (6.4)\nand can be handled with the method described in [18].\nThe aim there was to introduce advantageous spher-\nical coordinates, ( r;\u0012;\u001e );with their associated ONS\n(n;e\u0012;e\u001e) with eZ\u0001n= cos\u0012,n\u0001eX= cos\u001esin\u0012, as\ncan be seen in Fig. (2). First, we de\fne the normalized\nrelative separation vector according to\nn= sin\u0012cos\u001eeX+ sin\u0012sin\u001eeY+ cos\u0012eZ:(6.5)\nThe time derivative of r, the linear momentum p, its de-\ncomposition in radial components and the corresponding\northogonal ones can be written as\nr=rn; (6.6a)\n_r= _rn+r_\u0012e\u0012+rsin\u0012_\u001ee\u001e; (6.6b)\np=prn+p\u0012e\u0012+p\u001ee\u001e; (6.6c)\np2=p2\nr+p2\n\u0012+p2\n\u001e= (n\u0001p)2+ (n\u0002p)2\n=p2\nr+L2\nr2: (6.6d)\nInsertingp2into Eq. (6.4), computing p\u001e=p\u0001e\u001eand\nusing the orthogonality relation of the used triad, one\nobtains\np2\nr= 2E+2\nr\u0000L2\nr2\u00002(L\u0001Se\u000b)\nc2r3; (6.7a)\np\u001e=Lz\nrsin\u0012; (6.7b)\np2\n\u0012=L2\nr2\u0000p2\n\u001e=1\nr2\u0012\nL2\u0000L2\nz\nsin2\u0012\u0013\n; (6.7c)\nIn [18], it was possible to reduce these equations by some\nalgebraic relations and the fact that the angle \u0002 was\naks\neX=peYL=LkS1S2\nqNJ=JeZ\ni\njnΘ\nΥϕθ\nφi\ni0i0\nInvariable planeOrbital plane\nplane of skyFIG. 2: The geometry of the binary, having added the ob-\nserver related frame ( p;q;N) (in dashed and dotted lines)\nwithNas the line{of{sight vector, after removing the angles\nin the spin frame. The line{of{sight vector is chosen to lie in\ntheeY{eZ{plane, and measures an angle i0(associated with\nthe rotation around eX) from eZ, such that p=eX, and this\nis the point where the orbital plane meets the plane of the\nsky. Because of this rotation, the angle i0is also found be-\ntween the vector q, itself positioned in ( eY;eZ), too, and eY.\nThe grey area in the graphics completely lies in the orbital\nplane, spanned by ( i;j) and'measures the angle between\nthe separation vector randi. The polarization vectors pand\nqspan the plane of the sky. The inclination of this plane with\nrespect to the orbital plane is the orbital inclination i. The\ninclination of the orbital plane with respect to the invariable\nplane is denoted by \u0002. Please note that Ldoes notlie on the\nunit sphere, only kdoes!\nconstant in time - here, it is more complicated. It is still\nallowed to express Lz, the projection of LontoeZ, in\nEq. (6.7b) and (6.7c) over \u0002 with the help of\np\u001e=L\nrcos \u0002\nsin\u0012; (6.8a)\np2\n\u0012=L2\nr2\u0012\n1\u0000cos2\u0002\nsin2\u0012\u0013\n: (6.8b)\nAbove equations are, for our purposes, the most simpli-\n\fed versions of the pcomponents and will enter in the\ndynamics of the angle 'in their current form.\nOur aim is now to connect the coordinate velocities,\nnamely _r;_\u001eand _\u0012, to conserved quantities associated\nwith the Hamiltonian of Eq. (6.4). Computation of the\nvelocity in spherical coordinates, Eq. (6.5), gives follow-\ning formulae using Hamilton's EOM, _r=@HNSO=@p,\nn\u0002e\u0012=e\u001eandn\u0002e\u001e=\u0000e\u0012.\n_r=n\u0001_r=pr; (6.9a)\nr_\u0012=e\u0012\u0001_r=p\u0012+e\u001e\u0001Se\u000b\nc2r2; (6.9b)\nrsin\u0012_\u001e=e\u001e\u0001_r=p\u001e\u0000e\u0012\u0001Se\u000b\nc2r2: (6.9c)9\nOf course, in the case of quasi-circular motion, _ r= 0 =pr\nholds for all times. Remembering the geometry of Fig.\n2, we recall that ris lying in the plane orthogonal to L,\nwhich itself is spanned by the vectors iandj. Calling'\n(the orbital phase) the measure for the angular distance\nfromi, we can write\nr=rcos'i+rsin'j: (6.10)\nThe comparison of r, given by Eqs. (6.6a) and (6.5), with\nthe one in the new angular variables, Eq. (6.10) with\nEqs. (3.2), implies the transformation\n(\u0012;\u001e)!(\u0007;') :8\n><\n>:cos\u0012= sin'sin \u0002\nsin(\u001e\u0000\u0007) sin\u0012= sin'cos \u0002\ncos(\u001e\u0000\u0007) sin\u0012= cos':\n(6.11)\nTime derivation of the \frst equation will give an expres-\nsion for _\u0012, which can be simpli\fed using the third one.\nThe \fnal expression is\n_\u0012=\u0000sin \u0001 _\u0002\u0000r\n1\u0000cos2\u0002\nsin2\u0012_' (6.12)\nwith \u0001\u0011\u001e\u0000\u0007. Setting \u0002 constant, one naturally recov-\ners Eq. (4.28a) of [18]. Using this equation to eliminate\n_\u0012in (6.9b) and after substition \u0006p\u0012from (6.8b), one ob-\ntains a solution for _ 'and _\u0007\n_'=\u0007L\nr2\u0000~S\u001eq\n1\u0000cos2\u0002\nsin2\u00121\nc2r3\u0000sin \u0001q\n1\u0000cos2\u0002\nsin2\u0012_\u0002;(6.13)where ~S\u001eis a shorthand for Se\u000b\u0001e\u001e. The ambiguity of\nthe sign in the \frst term can be removed if one takes the\nrotation sense of the reduced mass, or equivalently, the\ndirection of Linto account. Having (initially) the vector\nLin the northern hemisphere, one should choose \\+\" in\nabove equation. This condition then holds anytime as\nlong asS1+S2 \n8mT \n0mT (b) \n-10 +10 B (mT) t(1,1)=1µs \n(c) mixed states \n[1] [2] [3] \n[4] \ncycle evolution (time) energy \n[5] B>0 \n(0,1) (1,1) (0,2) (0,1) S(0,2) S(1,1) \n(0,1) T (1,1) \nT (1,1) +-\nT (1,1) 0\n(0,1) \nFIG. 3: (color online). (a)Average number of pumped elec-\ntrons per cycle /angbracketleftN/angbracketrightas a function of the time t(1,1) for dif-\nferent magnetic fields. (b)Dependence of the long time limit\nof/angbracketleftN/angbracketright(t(1,1) = 1µs) on the external magnetic field B.(c)\nScheme of the energy levels along the pumping cycle for a\nmagnetic field B >0. The system evolves along the thick\nlines (labels [1]-[5], gray areas represent waiting times) : [1]\nstart in (0 ,1); [2] tunneling of an electron into one of the\n(1,1) states (arrows); [3] evolution and relaxation in the ( 1,1)\nsubspace; [4] transition along the detuning axis ε, [5] tunnel-\ning out. Only electrons coming from S(0,2) give rise to a\npumped current (lowest arrow at [5]). Electrons coming from\n(1,1)-states are only shuttled back and forth (empty arrows).\nAt the transition [4], hyperfine interaction or SOI hybridiz e\ndifferent spin states (avoided crossings). During step [3], evo-\nlution between the mixed states and relaxation to T+(1,1)\ncan occur.\nthe charge stability diagram - provided the transition is\nnot forbidden by spin selection rules. Beyond that, the\npumping efficiency depends on the size of the pulse loop\nin Fig.1(b). For example, if the (0 ,2)-corner is chosen at\na too high detuning, the transition from (1 ,1) to (0,2)\noccurs by electron escape via (0 ,1) [3]. We adjusted the\npulsing parameters so that these processes are minimal.\nThe pumping scheme allows to study the time evo-\nlution of the quantum states involved in the SB. For a\ntolerable signal-to-noise ratio of the pumped current, the\ntotal cycle times should be shorter than ≈2µs. Within\nthis limit, we observe no dependence of the pumping ef-\nficiency when varying separately the times t(0,1) and\nt(0,2) (not shown). However, t(1,1) has a strong influ-\nence on the pumped current.\nIn Fig.3(a), the average number of pumped electrons\nper cycle /angbracketleftN/angbracketrightis plotted as a function of t(1,1). To main-\ntain a well detectable signal, the total cycle period is\nfixed to 1 .2µs. Since /angbracketleftN/angbracketrightonly depends on t(1,1) for\nthese timescales, we fix t(0,2) = 100ns and compensatethe time spent in (1 ,1) by shortening the time in (0 ,1)\ncorrespondingly. A monotonic long-time increase of /angbracketleftN/angbracketright\nis found for times >200ns [24]. At finite field, this effect\nis much more pronounced than at B= 0T.\nThe long-time limit of /angbracketleftN/angbracketrightis studied as a function of\nB-field in Fig.3(b). For t(1,1) = 1µs,/angbracketleftN/angbracketrightis sensitive to\nmagnetic fields of a few mT. This behavior is in line with\nthe field dependence of the current through SB at finite\nbias (Fig.1(d)).\nIn order to analyze the behavior of the pumped cur-\nrent, we use the singlet-triplet model for SB [25]. The\nvalues of the pumped currents in Fig.2 are related to the\nspin-transition rules between the corners of the pumping\nloop. For the anti-clockwise cycle (lower round inset),\nthe transition from (0 ,2) to (1,1) is always allowed and\none electron is transfered from right to left during each\nroundtrip. In the opposite direction (upper round inset),\nthe transition from (1 ,1) to (0,2) is spin selective. The\ntripletsT0,±(1,1) are blocked and only the singlet can\npass, which reduces the pumped current. At B= 1T,\nthe excited triplet T+(0,2) comes close in energy to the\nground state S(0,2) and both are mixed by SOI [8]. This\nway SB is lifted and the full pumping current is recov-\nered.\nTo understand the decay in Fig.3(a), we analyze the\nspin-selective transition (1 ,1)-(0,2) for different mag-\nnetic fields. A contribution to the pumped current is\ngenerated only by those (1 ,1)-states, which are trans-\nfered into a singlet during the pulse. In other states, the\nelectron is blocked.\nAtB= 0T, all (1 ,1)-states are close in energy [2, 21]\nand becomemixed bydifferent spin coupling mechanisms\nduringthe time t(1,1). The pumped currentthen reflects\nthe overlap with the singlet. In Fig.3(a), the curve for\nB= 0T shows only a weak time dependence. This sup-\nports that there is no preferential evolution towards a\ncertain state, but mixing between all states.\nFor finite field, the level evolution along the triangular\npumping cycle is sketched in Fig.3(c). Between the (1 ,1)\nand (0,2) corners, triplets and singlet levels would cross\nat two points (label [4] in Fig.3(c)). In the presence of\nSOIorhyperfineinteraction, hybridizationofstatesleads\nto avoided crossings at these points [21, 23].\nZeeman splitting lowers the energy of the state with\nT+(1,1)-character. Relaxation to this new ground state\noccursduringthetime t(1,1). Thisincreasesthepumped\ncurrent, because T+(1,1)is admixedto the singlet during\nthe charge transition (label [4] in Fig.3(c)). We estimate\na relaxation time T1(1,1)≈300ns by fitting with an ex-\nponential curve. A comparable relaxation process is not\nreported in GaAs DQDs, where SB is generally restored\nwith finite magnetic fields [2, 3, 21].\nThe B-dependence of /angbracketleftN/angbracketrightfor long t(1,1) (Fig.3(b))\nsuggests a SOI mediated relaxation. The relaxation rate\nfor these processes generally increases with the splitting\nof the involved states [17, 18, 19]. In contrast, spin state4\n100mT (a) \n \n0mT \n0.4 0.6 \n \n(b) \n0 20 80 0.4 0.6 \n \ninit in (0,1) \ninit in (1,2) • \nt(1,1) (ns) \nFIG. 4: (color online). Number of pumped electrons per cycle\nas a function of t(1,1).(a)Dependent on the external field,\n/angbracketleftN/angbracketrightshows oscillations with a period 9 .4ns and characteristic\ndecay time of 25ns (for 0mT) and 45ns (100mT). (b)When\nchanging the initialization state of the cycle, the phase of the\noscillations changes by π. Cycles are (0 ,1)-(1,1)-(0,2)-(0,1)\n(blue dots) and (1 ,2)-(1,1)-(0,2)-(1,2) (black rhombs).\ndecay due to hyperfine interaction with the nuclei is sup-\npressed in a field which splits T±(1,1) [21].\nFor times shorter than the relaxation time, the curves\ninFig.3(a)showup-turnswhicharenotfullyunderstood.\nHowever, high resolution measurements in this region re-\nveal striking oscillations of /angbracketleftN/angbracketrightas a function of the time\nt(1,1), as shown in Fig.4(a). As above, the total cycle\ntime is constant (140ns) in a regime, where the signal\nonly depends on t(1,1) (t(0,2) = 20ns fixed). The os-\ncillation period does not vary with magnetic field, but\nthe decay is changed. A purely exponentially decaying\nfunction cannot be fitted to the amplitude. Neverthe-\nless it allows to estimate a decay time, which increases\nmonotonically from 25ns at 0T to 45ns at 100mT.\nThe oscillations as a function of t(1,1) are robust\nagainst variation of the two other waiting times and the\ntotal cycle period. The period corresponds to an energy\nsplitting of h/9.4ns = 0.44µeV, which is consistent with\nthe energy scales for exchange coupling, hyperfine inter-\naction and spin-orbit interaction (at small fields) in the\nsystem[8]. Theseenergyscales,themagneticfielddepen-\ndence of the decay and the selective time dependence on\nt(1,1) suggest coherent evolution in the (1 ,1) subspace\nas the origin of the oscillations.\nA detection of coherent oscillations in the pumping\nscheme would imply a selective state preparation. In\nFig.4(b) we observe a striking dependence of the phase\nof the oscillations on the way the two-electron state is\nloaded. Moving the initial state from (0 ,1) to (1,2) in\nthe chargestability diagram(Fig.1(b)) results in a phase\nshift ofπ(in both cases, the charge is pumped in the\ndirection of SB). These observations suggest that the na-\nture and the coupling of spin-states in DQDs are signifi-\ncantly changed by the SOI compared to the well under-stood situation in GaAs dots.\nBy pumping single electrons through a spin-blockaded\nInAs DQD, westudied the dynamics oftwocoupled spins\nin the presence of strong SOI. Beyond the spin-selection\nrulesleadingtoSB,SOImediatedrelaxationtothe(1 ,1)-\ntriplet ground state was observed at finite magnetic field.\nFor times shorter than the relaxation time, oscillations\nwere detected in the pumped current. These processes\ncan influence the operation of two-qubit gates in systems\nwith strong SOI.\nWe thank B. Altshuler, A. Imamoglu, D. Klauser,\nD. Loss, C. Marcus, Y. Meir, L. Vandersypen and\nA. Yacoby for stimulating discussions, M. Borgstr¨ omand\nE. Gini for advice in nanowire growth. We acknowledge\nfinancial support from the ETH Zurich.\n[1] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120\n(1998).\n[2] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science 309, 2180 (2005).\n[3] A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby,\nM. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C.\nGossard, Nature 435, 925 (2005).\n[4] E. A. Laird, J. R. Petta, A. C. Johnson, C. M. Marcus,\nA. Yacoby, M. P. Hanson, and A. C. Gossard, Phys. Rev.\nLett.97, 056801 (2006).\n[5] F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink,\nK. C. Nowack, T. Meunier, L. P. Kouwenhoven, and\nL. M. K. Vandersypen, Nature 442, 766 (2006).\n[6] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and\nL. M. K. Vandersypen, Science 318, 1430 (2007).\n[7] C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and\nD. Loss, Phys. Rev. Lett. 98, 266801 (2007).\n[8] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq,\nPhys. Rev. B 76, 161308 (2007).\n[9] F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen,\nNature452, 448 (2008).\n[10] G. Burkard and D. Loss, Phys. Rev. Lett. 88, 047903\n(2002).\n[11] D. Stepanenko, N. E. Bonesteel, D. P. DiVincenzo,\nG. Burkard, and D. Loss, Phys. Rev. B 68, 115306\n(2003).\n[12] M. Trif, V. N. Golovach, and D. Loss, Phys. Rev. B 75,\n085307 (2007).\n[13] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H.\nDevoret, Europhys. Lett. 17, 249 (1992).\n[14] A. Fuhrer, C. Fasth, and L. Samuelson, Appl. Phys. Lett.\n91, 052109 (2007).\n[15] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Sci-\nence297, 1313 (2002).\n[16] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq,\nPhys. Rev. Lett. 99, 036801 (2007).\n[17] S. Amasha, K. MacLean, I. P. Radu, D. M. Zumbuhl,\nM. A. Kastner, M. P. Hanson, and A. C. Gossard, Phys.\nRev. Lett. 100, 046803 (2008).\n[18] T. Meunier, I. T. Vink, L. H. W. van Beveren, K.-J.\nTielrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz,5\nW. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-\ndersypen, Phys. Rev. Lett. 98, 126601 (2007).\n[19] R. Hanson, L. H. W. van Beveren, I. T. Vink, J. M. Elz-\nerman, W. J. M. Naber, F. H. L. Koppens, L. P. Kouwen-\nhoven, and L. M. K. Vandersypen, Phys. Rev. Lett. 94,\n196802 (2005).\n[20] W. G. van der Wiel, S. D. Franceschi, J. M. Elzerman,\nT. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev.\nMod. Phys. 75, 1 (2002).\n[21] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Han-\nson, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz,\nW. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-\ndersypen, Science 309, 1346 (2005).\n[22] H. O. H. Churchill, D. Marcos, F. Kuemmeth, S. K. Wat-\nson, and C. M. Marcus, Contribution to INTNAN8 con-ference (2008).\n[23] D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus,\nM. P. Hanson, and A. C. Gossard (2008).\n[24] The minimum at t(1,1)≈160ns is also observed for\ndifferent total cycle periods and in schemes, where only\nt(1,1) is varied. The origin is not fully understood, but\nit does not affect the analysis of the relaxation process\nabove 200ns.\n[25] Since the (0 ,2) singlet-triplet splitting is much larger\nthen the energy scale of the SOI [8], the (0 ,2)-states\nare reasonably described as triplets T0,±(0,2) and sin-\ngletS(0,2). The nature of the (1 ,1)-levels could however\nbe strongly modified by SOI." }, { "title": "1110.5366v1.Manipulation_of_single_electron_spin_in_a_GaAs_quantum_dot_through_the_application_of_geometric_phases__The_Feynman_disentangling_technique.pdf", "content": "arXiv:1110.5366v1 [cond-mat.mes-hall] 24 Oct 2011Manipulation of single electron spin in a GaAs quantum dot th rough the application\nof geometric phases: The Feynman disentangling technique\nSanjay Prabhakar,1,2James Raynolds,1Akira Inomata,3and Roderick Melnik2,4\n1College of Nanoscale Science and Engineering, University a t Albany, State University of New York\n2M2NeT Laboratory, Wilfrid Laurier University, Waterloo, ON, N2L 3C5 Canada\n3Department of Physics, University at Albany, State Univers ity of New York\n4BCAM, Bizkaia Technology Park, 48160 Derio, Spain\n(Dated: September 12, 2018)\nThe spin of a single electron in an electrically defined quant um dot in a 2DEG can be manipu-\nlated by moving the quantum dot adiabatically in a closed loo p in the 2D plane under the influence\nof applied gate potentials. In this paper we present analyti cal expressions and numerical simula-\ntions for the spin-flip probabilities during the adiabatic e volution in the presence of the Rashba\nand Dresselhaus linear spin-orbit interactions. We use the Feynman disentanglement technique to\ndetermine the non-Abelian Berry phase and we find exact analy tical expressions for three special\ncases: (i) the pure Rashba spin-orbit coupling, (ii) the pur e Dresselhause linear spin-orbit coupling,\nand (iii) the mixture of the Rashba and Dresselhaus spin-orb it couplings with equal strength. For\na mixture of the Rashba and Dresselhaus spin-orbit coupling s with unequal strengths, we obtain\nsimulation results by solving numerically the Riccati equa tion originating from the disentangling\nprocedure. We find that the spin-flip probability in the prese nce of the mixed spin-orbit couplings\nis generally larger than those for the pure Rashba case and fo r the pure Dresselhaus case, and that\nthe complete spin-flip takes place only when the Rashba and Dr esselhaus spin-orbit couplings are\nmixed symmetrically.\nI. INTRODUCTION\nGeometric phases abound in physics and their study\nhas attracted considerable attention since the seminal\nwork of Berry.1,2In recent years a number of researchers\nhave shown their interest in the geometric phases associ-\nated with single- and few-spin systems for potential ap-\nplications in the field of quantum computing and non-\ncharge based logic.3–5One interesting proposal is the no-\ntion that the spin of a single electron trapped in an elec-\ntrostaticallydefined2Dquantumdotcanbemanipulated\nthrough the application of gate potentials by moving the\ncenter of mass of a quantum dot adiabatically in a closed\nloop and inducing a non-Abelian matrix Berry phase.6A\nrecent work shows that the Berry phases can be changed\ndramatically by the applications of gate potentials and\nmay be detected in an interference experiment.7\nIn the present paper, we study the non-Abelian uni-\ntary operator of the spin states during the adiabatic mo-\ntion of a single electron spin. The non-Abelian nature\nhere stems from the spin-orbit coupling of an electron\nin two dimensions. The evolution operator which gives\nrise to the Berry phase is not easy to evaluate as it con-\ntains non-commuting operators. In 1951, Feynman8de-\nveloped an operator calculus for quantum electrodynam-\nics, in which he devised a way to disentangle the evo-\nlution operator involving non-commuting operators. In\n1958, Popov9applied the operator calculus, combined\nwith group-theoretical considerations, to the spin rota-\ntion for a particle with a magnetic moment in an exter-\nnal magnetic field to obtain exact transition probabilities\nbetween the initial and final spin states. In a way similar\nto Popov’s we employ the Feynman technique to disen-\ntangle the evolution operator for a quantum dot withthe Rashba and Dresselhaus spin-orbit couplings and de-\nrive analytical expressions for spin transition probabili-\nties. In particular, we obtain exact closed form expres-\nsions for three specific cases: (i) the pure Rashba spin-\norbit coupling10(ii) the pure linear Dresselhaus spin-\norbit coupling,11and (iii) the symmetric combination of\nthe Rashba and Dresselhaus spin-orbit couplings. This\napproach provides us a convenient numerical scheme for\nan arbitrary mixing of the two types of spin-orbit cou-\nplings via a Riccati equation.12An interesting result we\nfind is that the spin-flip probability for the case of an ar-\nbitrary mixture of the Rashba and the Dresselhaus spin-\norbit couplings is generally greater than that for the case\nwhere either the Rashba or the Dresselhaus interaction\nacts alone. Furthermore, we see that the complete spin\nprecession occurs only when the Rashba spin-orbit cou-\npling and the Dresselhaus spin-orbit coupling are equal\nin strength.\nTheworkofBerryteachesthatifparameterscontained\nin the Hamiltonian of a quantal system are adiabatically\ncarried around a closed loop an extra geometric phase\n(Berry phase) is induced in addition to the familiar dy-\nnamical phase.1,2A slow variation of such parameters\nalong a closed path Cwill return the system to its orig-\ninal energy eigenstate with an additional phase factor\nexp{iγn(C)}. More specifically, the state acquires phases\nafter a period of the cycle Tas\n|Ψn(T)/an}b∇acket∇i}ht= exp/braceleftBigg\n−i\n¯h/integraldisplayT\n0En(t)dt/bracerightBigg\n·exp{iγn(C)} |ψn/an}b∇acket∇i}ht.\n(1)\nHowever this equation applies only to non-degenerate\nstates. The detailed numerical and analytical calcula-\ntionsofBerryphase γn(C)fortheHamiltonianofaquan-2\ntum dot in 2D plane for different non-degenerate eigen\nstates are explained in Ref. 13. The system of interest\nhere (a single spin in a 2D electrically defined quantum\ndot) is degenerate14,15for which (1) is not directly ap-\nplicable. In the formulation developed by Wilczek and\nothers2,16for degenerate cases, the geometric phase fac-\ntor is replaced by a non-Abelian unitary operator Uab\nacting on the initial states within the subspace of degen-\neracy. The evolution equation of the state is modified in\nthe form,\n|Ψn,a(t)/an}b∇acket∇i}ht= exp/braceleftbigg\n−i\n¯h/integraldisplayt\n0E(t)dt/bracerightbigg/summationdisplay\nbUab(t)|ψn,b/an}b∇acket∇i}ht,(2)\nwhereaandbare the labels for degeneracy. The non-\nAbelian unitary operator can be expressed in the form,\nUab(t) =Texp/braceleftbigg\n−i\n¯h/integraldisplayt\n0Aab(t′)·˙Rdt′/bracerightbigg\n,(3)\nwhereTsignifies the time-ordering, and\nAab=−i¯h/an}b∇acketle{tψn,a|∇R|ψn,b/an}b∇acket∇i}ht, (4)\nRand∇Rbeing a vector and the gradient in parame-\nter space, respectively. In general, the geometric phase\ntransformation Uab(t) of (3) in parameter space contains\nnon-commuting operators and time-dependent parame-\nters. It is possible to view the parameter-dependent evo-\nlution in the subspace of degeneracy as a non-Abelian\nlocal gauge transformation. Correspondingly Aabin (4)\nmay be seen as a non-Abelian gauge connection (or the\nYang-Mills fields).\nAlthough it is not straightforward to construct the\nnon-Abelian gauge connection, we consider the following\nobservation instructive for the case where the parameter\nspace coincides with the configuration space. Suppose\nthe Hamiltonian of a system is given by\nH=1\n2m(P−A)2+V(r). (5)\nThe energy eigenequation H|ψn/an}b∇acket∇i}ht=En|ψn/an}b∇acket∇i}htremains in-\nvariantunderthe local(position-dependent) gaugetrans-\nformation,\n|ψn/an}b∇acket∇i}ht → |ψ′\nn/an}b∇acket∇i}ht=¯U|ψn/an}b∇acket∇i}ht,A→A′=¯UA¯U†+i¯h¯U∇¯U†.\n(6)\nIfwechoosesuchagaugethat thetransformedvectorpo-\ntential vanishes, that is, A′= 0, then the transformation\noperator is to be of the form,\n¯U= exp/braceleftbigg\n−i\n¯h/contintegraldisplay\ncA·dr/bracerightbigg\n. (7)\nIn other words, this transformation will “gauge away”\nthe vector potential from the Hamiltonian (5). Con-\nversely, if the state with the vanishing gauge is taken\nto be the initial state, the final state with an arbitrary\ngaugeAis obtained by the inversegaugetransformation,|ψn/an}b∇acket∇i}ht=¯U−1|ψ′\nn/an}b∇acket∇i}ht. Moreover, if the inverse gauge process\nis time-dependent via the variation of position, then the\nevolution operator is given by\nU(t) =¯U−1(t) =Texp/braceleftbiggi\n¯h/integraldisplayt\n0A·˙rdt/bracerightbigg\n.(8)\nThis observation will be useful for our discussion on the\nBerry phase associated with the spin-orbit coupling.\nA matrix element of the evolution operator gives the\ntransition amplitude (propagator) from an initial state\nto the final state, which is usually evaluated by approx-\nimation. For instance, the propagator for the spin-orbit\ninteraction has been calculated semiclassically in a dif-\nferent context by Feynman’s path integral represented in\ncoherent states.17\nIn Sec. 2, we treat the phase transformation (3) as a\ngauge transformation, and employ Feynman’s disentan-\ngling technique, rather than Feynman’s path integral, to\nevaluate the time-ordered exponential for the spin-orbit\ncoupling Hamiltonian. Use of Feynman’s disentangling\nmethod in Popov’s version18enables us to obtain analyt-\nical and numerical results for the spin transition proba-\nbilities without approximation. In Sec. 3, we plot the\nspin-flip probability versus the rotation angle, and com-\npare the data for the pure Rashba, the pure Dresselhaus,\nand mixed cases. Sec. 4 is devoted in deriving analytical\nexpressions of the non-Abelian Berry phase (the adia-\nbatic evolution operator as a 2 ×2 matrix) for the pure\nRashba and the pure Dresselhaus coupling.\nII. SPIN TRANSITION PROBABILITIES VIA\nFEYNMAN DISENTANGLING METHOD\nTo discuss the revolution of spin that induces a geo-\nmetric phase, we consider a GaAs quantum dot formed\nin the plane of a two-dimensional electron gas (2DEG),\nthe center of mass of which moves adiabatically along a\nclosed path under the influence of applied potentials.6\nThe single-electron Hamiltonian in 2DEG (in the xy\nplane) may be written in the form,\nH=1\n2mP2+HSO, (9)\nwheremistheeffectivemass. Thefirsttermisthekinetic\nenergyin two dimensions. Evidently, P2=P2\nx+P2\ny. The\nsecond term is the spin-orbit (SO) coupling Hamiltonian\nin linear approximation,\nHSO= 2α(PySx−PxSy)−2β(PxSx−PySy).(10)\nHereSis the spin operator whose components obey the\nSU(2) algebra (see, e.g., Ref. 19):\n[S+,S−] = 2S0,[S0,S±] =±S±,(11)\nwhereS±=Sx±iSyandS0=Sz. The spin-orbit\nHamiltonian (10) consists of the Rashba coupling whose3\nFIG. 1. (color online) Transition probability, w1/2,−1/2vs.θfor three cases: (a) pure Rashba ( β= 0), (b) pure Dresselhaus\n(α= 0), and (c) mixed (non-zero αandβ) spin-orbit interactions. The orbital radius is 60 nm. The t hree curves represent\nthe following electric field strengths: 1 ×105V/cm (solid black line), 5 ×105V/cm (dashed red line), and 1 ×106V/cm\n(dotted-dashed blue line) respectively.\nFIG. 2. (color online) Transition probability w1/2,−1/2vs.θfor the following cases: (a) pure Rashba ( β= 0), (b) pure\nDresselhaus ( α= 0), and (c) mixed (non-zero αandβ). The orbit radius is chosen to be 250 nm and the following val ues of the\nelectric field are considered: 1 ×105V/cm (solid black line), 5 ×105V/cm (dashed red line), and 1 ×106V/cm (dotted-dashed\nblue line)\nFIG. 3. (color online) Transition probability w1/2,−1/2vs.θfor the following cases: (a) pure Rashba ( β= 0), (b) pure\nDresselhaus ( α= 0), and (c) mixed (non-zero αandβ). The orbit radius was chosen to be 500 nm and the following va lues of\nthe electric field were chosen: 1 ×105V/cm (solid black line), 5 ×105V/cm (dashed red line), and 1 ×106V/cm (dotted-dashed\nblue line).4\nFIG. 4. (color online) Transition probability w1/2,−1/2vs.θ\nfor the following cases: pure Rashba ( β= 0: dotted-dashed\nblue line), pure Dresselhaus ( α= 0: dashed red line) and\nmixed (non-zero αandβ: solid black line). The orbit ra-\ndius was chosen to be 250 nm and the following values of the\nelectric field were chosen: (a) E= 1×105V/cm and, (b)\nE= 5×105V/cm.\nFIG. 5. (color online) Transition probability w1/2,−1/2vs.θ\nforα=β. Physically, this situation occurs for electric field\nstrength given by E= 3.02×106V/cm. Here the solid black\nline represents for both Rashba and Dresselhaus spin-orbit\ncoupling effects whereas the dashed red line represents only\nfor Dresselhaus spin-orbit coupling effect and open red circ les\nrepresents only for Rashba spin-orbit coupling effect. Here we\nchoose 60 nm orbit radius.\nstrength is characterized by parameter αand the linear\nDresselhaus coupling with β. These coupling parameters\nare dependent on the electric field Eof the quantum well\nconfining potential (i.e., E=−∂V/∂z) along z-direction\nat the interface in a heterojunction as\nα=e\n¯haRE, β =0.7794γc\n¯h/parenleftbigg2me\n¯h2/parenrightbigg2/3\nE2/3,(12)\nwhereaR= 4.4˚A2andγc= 26eV˚A3for the GaAs quan-\ntum dot.14The quantum well confining potential (i.e.,\nE=−∂V/∂z)alongz-directionis not symmetric in III-V\ntype semiconductor.15It means, the formation of quan-\ntum dot at the interface of III-V type semiconductor in\nthe plane of 2DEG is asymmetric.\nNow we look for the evolution operator (8) for the case\nof spin-orbit coupling. It has been known that the linear\nspin-orbit term in (9) can be gauged away.20,21In fact,\nFIG. 6. (color online) Transition probability w1/2,−1/2vs.θ\nforα=β. Physically, this situation occurs for electric field\nstrength given by E= 3.02×106V/cm. The following orbit\nradii were chosen: 60 nm (solid black line), 175 nm (dashed\nred line), and, 250 nm (dotted-dashed blue line).\nthe Hamiltonian (9) may be expressed as\nH=1\n2m(P−A)2−V0, (13)\nwhere\nA= 2m/parenleftbigg\nαSy+βSx\n−αSx−βSy/parenrightbigg\n(14)\nand\nV0=m¯h2(α2+β2). (15)\nIf the semiclassical momentum P=m˙ris used for the\nadiabatic evolution, then the spin-orbit gauge connection\nis related to the SO Hamiltonian (10),\nA·˙r=−HSO. (16)\nAssuming that the spin-orbit coupling is adiabatically\nintroduced into the initial state, we obtain via (8) the\nevolution operator of the form,\nU(t) =Texp/braceleftbigg\n−i\n¯h/integraldisplayt\n0HSO(t′)dt′/bracerightbigg\n,(17)\nwhich we shall evaluate by utilizing the Feynman disen-\ntangling method. This form of the evolution operator\nis commonly employed for Berry’s phase associated with\nthe spin-orbit interaction.6,15\nBeforedisentangling, wenote that the SOHamiltonian\n(10) may also be expressed as\nHSO=H+S++H−S− (18)\nwith\nH±= (αPy−βPx)∓i(βPy−αPx).(19)5\nSuppose the quantum dot orbits around a closed circular\npath of radius R0in thex−yplane under the influence\nof gate potentials, so that r=R0(cosωt,sinωt,0). Then\nthe semiclassical momentum P=m˙rhas components,\nPx=−R0mωsinωt, Py=R0mωcosωt, Pz= 0.(20)\nSubstitution of (20) into (19) yields\nH±=R0mω(αe∓iωt∓iβe±ωt). (21)\nSinceS+andS−do not commute, the evaluation of the\ntime-ordered exponential for the evolution operator (17)\nis cumbersome.\nWe now turn to a discussion of the Feynman disentan-\ngling technique and its application to the present prob-\nlem. For the case where the Hamiltonian is given by\nH=α(t)A+β(t)B+γ(t)C+···,(22)\nwhereA,B,C, ... are noncommuting operators, and\nα,β,γ, ... are time-dependent parameters, Feynman8\ndevised an operator calculus by which the time-ordered\nexponential can be disentangled in the form\nU(t) =ea(t)Aeb(t)Bec(t)C···, (23)\nwherea(t),b(t),c(t), ... are time-dependent coefficients\nwhich can be determined by solving relevant differential\nequations. This procedure is referred to as the Feynman\ndisentangling method.18\nHereweapplyFeynman’smethodfordisentanglingthe\ntime-ordered exponential in (17) with the Hamiltonian\n(10). First we rewrite the Hamiltonian (10) as\nHSO=ξS++(H+−ξ)S++H−S−,(24)\nwhereξis a time-dependent function to be determined\nappropriately. According to Feynman’s procedure, the\nevolution operator may be put into the form,\nU(t) =ea(t)S+exp/braceleftbigg1\ni¯h/integraldisplayt\n0dt′[(H+−ξ)S′\n++H−S′\n−]/bracerightbigg\n,\n(25)\nwhere\na(t) =1\ni¯h/integraldisplayt\n0ξ(t′)dt′, (26)\nS′\n+=e−aS+S+eaS+=S+ (27)\nand\nS′\n−=e−aS+S+eaS+=S−−2aS0−a2S+.(28)\nSubstituting (27) and (28) into (25) and choosing ξ(t)\nsuch that the coefficient of S+in the integrand vanishes,\nwe get\nU(t) =ea(t)S+Texp/braceleftbigg1\ni¯h/integraldisplayt\n0dt′[−2aH−S0+H−S−]/bracerightbigg\n,\n(29)in which the term containing S+is disentangled. In a\nsimilar fashion, we disentangle the time-ordered expo-\nnential involving the mutually non-commuting operators\nS0andS−by letting\nU(t) =ea(t)S+eb(t)S0\nTexp/braceleftbigg1\ni¯h/integraldisplayt\n0dt′[(−2aH−−η)S′′\n0+H−S′′\n−]/bracerightbigg\n,(30)\nwhere\nb(t) =1\ni¯h/integraldisplayt\n0η(t′)dt′, (31)\nS′′\n0=e−bS0S0ebS0=S0 (32)\nand\nS′′\n−=e−bS0S−ebS0=S−eb. (33)\nAgain choosing η(t) =−2aH−, we reduce the evolution\noperator (25) into the completely disentangled form,\nU(t) =ea(t)S+eb(t)S0ec(t)S−, (34)\nwhere\na(t) =1\ni¯h/integraldisplayt\n0[H+(t′)−a2(t′)H−(t′)]dt′,(35)\nb(t) =−2\ni¯h/integraldisplayt\n0a(t′)H−(t′)dt′(36)\nand\nc(t) =1\ni¯h/integraldisplayt\n0H−(t′)eb(t′)dt′. (37)\nAlthough the time-ordered exponential is disentangled,\nthe evaluation of the evolution operator remains incom-\nplete until the coefficients a(t),b(t) andc(t) are deter-\nmined. In general, the integral equations (35)-(37) or\nthe equivalent differential equations are difficult to solve.\nIn Sec. 4, we shall determine the coefficients and the\nevolution operator for the pure Rashba, and the pure\nDresselhaus coupling.\nAs it is seen in Appendix A, the spin transition prob-\nability depends only on a(t). Therefore the full form of\nthe evolution operator is not needed. To determine a(t),\nwe convert the integral equation (35) together with (19)\ninto a Riccati equation of the form,\nda\ndt=−Rω[f(t)+f∗(t)a2(t)], (38)\nwhereR=mR0/¯h,\nf(t) =βiωt+iαe−iωt, (39)6\nand\nf∗(t) =β−iωt−iαeiωt. (40)\nSolving (38) for a(t), we can obtain the spin transition\nprobabilities, ws,s′. In particular, the transition proba-\nbilities from spin 1 /2 to±1/2 are calculated by\nw1/2,1/2=1\n1+|a|2, w 1/2,−1/2=|a|2\n1+|a|2.(41)\nIII. NUMERICAL ANALYSIS\nAs it is shown in Appendix B, exact solutions of the\nRiccati equation (38) can be obtained only for special\ncases, which include those for (i) the Rashba limit\n(β= 0), (ii) the Dresselhaus limit ( α= 0) and (iii)\nthe symmetric mixture of the two couplings ( α=β).\nThe spin-flip probabilities obtained in Appendix B for\nexactly solvable cases (with θ=ωt) are:\n(i)The Rashba limit (α/ne}ationslash= 0, β= 0):\nwR\n1/2,−1/2=4R2α2\n1+4R2α2sin2/parenleftbigg1\n2/radicalbig\n1+4R2α2θ/parenrightbigg\n; (42)\n(ii)The Dresselhaus limit (α= 0, β/ne}ationslash= 0):\nwD\n1/2,−1/2=4R2β2\n1+4R2β2sin2/braceleftbigg1\n2/radicalbig\n1+4R2β2θ/bracerightbigg\n; (43)\n(iii)The symmetric Rashba-Dresselhaus limit (α=\nβ/ne}ationslash= 0):\nwsym\n1/2,−1/2= sin2/braceleftBig√\n2αR(sinθ−cosθ+1)/bracerightBig\n.(44)\nFor an arbitrarily mixed Rashba-Dresselhaus coupling\n(mixed R-D), the Riccati equation (38) is not exactly\nsolvable. Therefore numerical analysis is needed. In the\nbelow we treat the mixed R-D coupling ( α/ne}ationslash=β) and the\nsymmetric R=D coupling ( α=β) separately.\nComparison of the Rashba coupling, the Dres-\nselhaus coupling and the mixed R-D coupling :\nFigs. 1, 2 and 3 plot the spin-flip probability w1/2,−1/2\nversus the rotation angle θ=ωtin the unit of 2 πfor\nthe orbit radius R0=60 nm, 250 nm, and 500 nm, re-\nspectively. The plots of (a), (b) and (c) in these figures\ncorrespond to (a) the pure Rashba case ( β= 0), (b) the\npure Dresselhaus case ( α= 0) and (c) the mixed R-D\ncase (α/ne}ationslash= 0,β/ne}ationslash= 0), respectively. The three different\nvalues of the electric field E= 1×105V/cm, 5 ×105\nV/cm, and 1 ×106V/cm, are chosen for the curves in\neach figure, solid black, dashed red, and dotted-dashed\nblue, respectively. The symmetric case (R=D) will be\nexamined separately with Figs. 5 and 6.\nThe curves for (a) the pure Rashba case and (b) the\npure Dresselhauscase areobtained from the exact results(42) and (43). As it is obvious from these equations,\nthe spin-flip probability increases as the electric field in-\ncreases via the coupling parameter but remains to be less\nthan unity. Another observation we can make from these\nplots is that the periods of spin-flip for the pure Rashba\ncoupling and the pure Dresselhaus coupling are different.\nThis is also expected from the analytical results (42) and\n(43).\nThe curves in Figs. 1(c), 2(c) and 3(c) show the spin-\nflip probability for (c) the mixed R-D case where both\nαandβare not zero and not equal. Note that they\narenotthe results from the exact formula (44) for the\nsymmetric R-D coupling. Since the Riccati equation (38)\nfor arbitrary non-zero αandβis not solvable, we carry\nout numerical simulations by using numerical solutions\nof (38) in (41). The spin-flip probability for the mixed\ncaseis generallylargerthan the pure cases. Furthermore,\nit does not reach unity if α/ne}ationslash=β. In other words, the\ncomplete spin-flip is not likely to occur during the entire\nperiod of the adiabatic motion along the closed orbit. In\nthevicinityofthesymmetrypoint( α=β), thetransition\nprobability becomes very close to unity at certain angles.\nFig. 4 gives a further comparison study of the\ntransition probability for the pure Rashba, the pure\nDresselhaus, and the mixed case. In Fig. 4(a), when\nthe electric field is weak, the curve for the mixed case\nappears to be a superposition of those for the two pure\ncases. As the electric field increases, the superposition\neffect becomes obscure as is seen in Fig. 4(b). As the\nRiccati equation is nonlinearin nature, there is no reason\nto expect that the mixed case is a superposition of the\ntwo pure cases. It is interesting to observe that the\nmixed case has a better chance to achieve the spin-flip\nthan the pure cases during the period of evolution.\nAnalysis of the symmetric R-D coupling :-Thesym-\nmetric mixture of the Rashba and Dresselhaus couplings\nhas been discussed in connection with the persistent spin\nhelix.22,23Bernevig et al.22found an exact SU(2) sym-\nmetry in the symmetric mixture and predicted the per-\nsistentspin helixwhich isa helicalspindensity wavewith\nconserved amplitude and phase. Recently spin life time\nenhancement of two orders of magnitude near the sym-\nmetry point ( α=β) has been reported experimentally.24\nThe coupling parameters αandβof the Rashba and\nDresselhaus interactions are given by (12) for the GaAs\nquantum dot. The two parameters become equal at\nE= 3.02×106V/cm. For the situation in which the\ntwo couplings have equal strength (i.e., α=β), the Ric-\ncatiequation(38)isexactlysolvedandthecorresponding\ntransitionprobabilityisgivenby(44). InFig. 5,thespin-\nflip probability versus the angle of rotation along the or-\nbit of radius 60 nm is plotted at E= 3.02×106V/cm for\nthe pure Rashba case (open red circles), the pure Dres-\nselhaus case (dashed red line), and the symmetric case\n(solid black line). We see that the symmetric Rashba-\nDresselhausspin-orbitcouplingdefinitelyachievesaspin-\nflip during the adiabatic process whereas the two pure7\ncases have less chances. Fig. 6 plots the transition prob-\nability of the symmetric R-D case for three different radii\nof the orbit of the quantum dot: 60 nm (solid black line),\n175nm(dashedredline)and250nm(dotted-dashedblue\nline). It shows that the chance of being in the spin-flip\nstate is enhanced by increasing the orbit radius.\nIt is important to notice that the complete spin flip\ntakes place only in the symmetric R-D coupling. This\nmay be an indication of the persistent spin helix. Al-\nthough the assumed orbit of motion is circular, we can\nregard the motion for a small angle of rotation as lin-\near. Letθ=ε≈0 orθ= 3π/2−ε. Ifεis small, then\nsinθ−cosθ+1≈ε, and the exact formula (44) may be\napproximated by\nwsym\n1/2,−1/2= sin2/braceleftBig√\n2αRε/bracerightBig\n. (45)\nAsεvaries from 0 to π/(2√\n2αR), the spin-flip probabil-\nity moves from zero to unity, that is, the spin completes\na full precession. For instance, if R= 60 nm, the range\n0≤√\n2αRε≤π/2 corresponds to the portion of the\nsolid black curve for 0 ≤θ/2π <0.2 in Fig. 5. Let\nεs=π/(2√\n2αR). Then the Rεsis the distance the elec-\ntron progresses while the spin precesses by 2 π. There-\nfore, we may be able to identify this distance with the\nspin diffusion length Lsas\nLs=Rε0/π=1\n2√\n2α. (46)\nIV. ANALYTICAL EXPRESSION FOR THE\nNON-ABELIAN BERRY PHASE\nApplying the Feynman disentangling method, we have\nbeen able to reduce the time-ordered evolution opera-\ntor (17) to the disentangled form (34) with the time-\ndependent scalarfunctions a(t),b(t) andc(t) obeying the\nintegral equations (35)-(37). It is sometimes convenient\nto express the evolution operator as a 2 ×2 matrix in\nthe spin representation of SU(2). Evidently the SU(2)\nalgebra (11) is satisfied by\nS+=/parenleftbigg\n0 1\n0 0/parenrightbigg\n, S0=1\n2/parenleftbigg\n1 0\n0−1/parenrightbigg\n, S−=/parenleftbigg\n0 0\n1 0/parenrightbigg\n.\n(47)\nUsing the properties S2\n±= 0 andS2\n0= 1/4, we can write\n(34) as\nU(t) =/parenleftbigg\n1a\n0 1/parenrightbigg/parenleftbigg\neb/20\n0e−b/2/parenrightbigg/parenleftbigg\n1 0\nc1/parenrightbigg\n,(48)\nfrom which immediately follows that\nU(t) =/parenleftbigg\neb/2+ace−b/2ae−b/2\nce−b/2e−b/2/parenrightbigg\n.(49)\nThis is the desired matrix representation for the Berry\nphase, and is used for calculating the spin-flip probabili-\nties in Appendix A.The expressions (34) and (49) remain formal until the\ntime-dependent functions a(t),b(t) andc(t) are speci-\nfied. Eq. (35) for a(t) is equivalent to a Riccati equation\nwithout whose solution, (36) and (37) cannot be solved\nforb(t) andc(t). In Appendix B, we show that the Ric-\ncati equation can be solved exactly if the function h(t)\ndefined by\nh(t) =β2−α2\n2R(α2+β2)3/2/bracketleftbigg\n1+2αβ\nα2+β2sin(2ωt)/bracketrightbigg−3/2\n(50)\nbecomes time-independent ( h(t) =h0). The last restric-\ntion (50) is fulfilled only when one of the following con-\nditions is met: α= 0,β= 0 orα=β. This implies that\nthe function a(t) can be determined only for the pure\nDresselhaus coupling, the pure Rashba coupling and the\nsymmetric Rashba-Dresselhaus coupling. The result we\nfind fora(t) is\na(t) =if\n|f|eiφ(t)−1\nn1eiφ(t)−n2, (51)\nwhere\nf(t) =βeiωt+iαe−iωt, (52)\nφ(t) =Rω(n1−n2)/integraldisplayt\n0|f(t′)|dt′.(53)\nHere\nn1,n2=h0±/radicalBig\nh2\n0+1. (54)\nFor convenience, we choose n1> n2. A closed form ex-\npression for φ(t) is given in (B12).\nIn calculating the spin transition probability, all we\nneed isa(t). However, for completing the evolution op-\nerator we have also to determine other functions b(t) and\nc(t) by solving (36) and (37) for the already determined\nfunctiona(t). As has been mentioned above, the Riccati\nequation can be solved exactly for the pure Rashba cou-\npling, the pure Dresselhaus coupling and the symmetric\nRashba-Dresselhaus coupling. In the two pure couplings,\nthe phase function φ(t) can be expressed in the form,\nφ(t) =ϕt, (55)\nwhereϕ=√\n1+4α2R2ωfor the Rashba coupling and\nϕ=/radicalbig\n1+4β2R2ωfor the Dresselhaus coupling. For\nthe symmetric R-D coupling, it cannot be simplified in\nthe form of (55). Therefore, it is difficult to carry out\nintegration in (36) and (37). This means that we have\nanalyticalexpressionsof the adiabatic evolution operator\n(49) only for the pure Rashba and the pure Dresselhaus\ncases. For the symmetric R-D coupling, even though we\nhave no analytical expressions for b(t) andc(t), we can\ncalculate the spin-flip probability since a(t) is found in\nclosed form.8\nIn what follows we provide the results of integration\nfor the pure Rashba coupling and the pure Dresselhaus\ncoupling.\n(i)The pure Rashba coupling (α/ne}ationslash= 0, β= 0):\nIn this case, (B2), (B4), (B5) and (B11) yield,\nif/|f|=−eiωt, h 0=−1\n2αR\nand\nφ(t) =ϕt, ϕ =ω/radicalbig\n1+4α2R2.\nUpon substitution of these results into (B13) we arrive\nat\na(t) =−eiωteiϕt−1\nn1eϕt−n2, (56)\nwhere\nn1, n2=−1\n2αR±1\n2αR/radicalbig\n1+4α2R2.\nFrom Eq.(36), using\nH−=αR¯hωeiωt,\ntogether with (56), we obtain\neb(t)=(n1−n2)2ei(ϕ−ω)t\n(n1eiϕt−n2)2, (57)\nand\nc(t) =1−eiϕt\nn1eiϕt−n2. (58)\n(ii)The pure Dresselhaus coupling (α= 0, β/ne}ationslash= 0):\nIn this case, we have\nif/|f|=ieiωt, h 0=1\n2βR\nand\nφ(t) =ϕt, ϕ =ω/radicalbig\n1+4β2R2.\nHence we get\na(t) =ieiωteiϕt−1\nn1eϕt−n2, (59)\nwhere\nn1, n2=−1\n2βR±1\n2βR/radicalbig\n1+4β2R2.\nUse of\nH−=iβR¯hωe−iωt,and (59) leads to\neb(t)=(n1−n2)2ei(ϕ+ω)t\n(n1eiϕt−n2)2, (60)\nand\nc(t) =−i1−eiϕt\nn1eiϕt−n2. (61)\nV. CONCLUSION\nIn the present paper we have considered spin manip-\nulation via the non-Abelian Berry phase induced by\nan adiabatic transport of a single spin along a circular\npath in the 2D plane in the presence of the Rashba\nand Dresselhaus spin-orbit couplings. We have adopted\nthe Feynman disentangling technique to calculate the\nspin-flip probability. We have shown that the problem\ncan be solved exactly in three cases: (i) the pure Rashba\ncoupling, (ii) the pure Dresselhaus coupling, and (iii)\nthe symmetric combination of Rashba and Dresselhaus\ncouplings. For an arbitrary combination of the two\ncouplings, we have carried out numerical simulations.\nWe have plotted the spin-flip probability versus the\nangle of the adiabatic rotation with various values of\nthe electric field and the radius of the circular path\nin the 2D plane. We have observed that a complete\nspin-flip (a complete spin precession) occurs only when\nthe strength of the two couplings becomes equal. The\nrelation between the complete spin precession and the\npersistent spin helix will be discussed in detail elsewhere.\nWe have also obtained analytical expressions of the\nnon-Abelian Berry phase for the pure Rashba case and\nthe pure Dresselhaus case.\nAppendix A: The spin transition probabilities\nFollowing Popov’s procedure,9,18we show that the\nspin-flip probability can be expressed in the form of (41).\nSince the time-evolution of the spin state can be achieved\nby a time-dependent rotation, the transition amplitude\nfor spinσtoσ′is given by\n/an}b∇acketle{tσ|U(t)|σ′/an}b∇acket∇i}ht=Ds\nσ,σ′(ϕ,ϑ,φ) = exp[−i(σϕ+σ′φ)]dσσ′(ϑ).\n(A1)\nHereϕ,ϑ,φare the time-dependent Eulerian angles,\nDs\nσ,σ′(ϕ,ϑ,φ) are the elements of the Wigner D-matrix\nbeing the irreducible unitary representations of SU(2)\ngroup, and dσσ′(ϑ) is Wigner’s d-function.\nThe corresponding transition probability along the z-\naxis is\nwσσ′=|ds\nσσ′(ϑ)(t)|2. (A2)9\nIn particular, the transition probability from spin 1 /2 to\n±1/2 is\nw1/2,1/2= cos2/parenleftbiggϑ(t)\n2/parenrightbigg\n, (A3)\nand\nw1/2,−1/2= sin2/parenleftbiggϑ(t)\n2/parenrightbigg\n, (A4)\nbecause\nd1/2\n1/2,1/2(ϑ) = cosϑ\n2, d1/2\n1/2,−1/2(ϑ) =isinϑ\n2.(A5)\nFor spins= 1/2, the rotation matrix is given in the\nstandard form,18,19\nD(ϕ,ϑ,ψ) =/parenleftbigg\n˜α−˜β∗\n˜β˜α∗/parenrightbigg\n, (A6)\nwhere\n˜α= cosϑ\n2exp/bracketleftbigg\niψ+ϕ\n2/bracketrightbigg\n,˜β=isinϑ\n2exp/bracketleftbigg\niψ−ϕ\n2/bracketrightbigg\n.\n(A7)\nComparison of the evolution operator for the spin 1/2\ntransition expressed in the matrix form,\nU=/parenleftbigg\neb/2+ace−b/2ae−b/2\nce−b/2e−b/2/parenrightbigg\n,(A8)\nand the rotation matrix yields\n|a|2= tan2ϑ\n2. (A9)\nAgain comparing this result with (A3) and (A4), we ar-\nrive at\nw1/2,1/2=1\n1+|a|2, w 1/2,−1/2=|a|2\n1+|a|2.(A10)\nNote thatw1/2,1/2+w1/2,−1/2= 1.\nAppendix B: Special solutions of the α−βRiccati\nequation\nHere we wish to solve under a special condition the\nRiccati equation (38):\nda(t)\ndt=−Rω{f(t)+f∗(t)a2(t)},(B1)\nwhere\nf(t) =βeiωt+iαe−iωt, f∗(t) =βe−iωt−iαeiωt.\n(B2)This equation contains the Rashba limit ( α/ne}ationslash= 0, β= 0),\nand the Dresselhaus limit ( α= 0, β/ne}ationslash= 0), both of which\nhave exact solutions.\nFirst we let a(t) =g(t)X(t) in (B1). If we further let\ng(t) =−if/|f|, then we see that X(t) obeys\ndX\ndt=iRω|f(t)|/braceleftbig\nX2−2h(t)X−1/bracerightbig\n,(B3)\nwhere\n|f(t)|=/bracketleftbig\nα2+β2+2αβsin(2ωt)/bracketrightbig1/2(B4)\nand\nh(t) =β2−α2\n2R(α2+β2)3/2/bracketleftbigg\n1+2αβ\nα2+β2sin(2ωt)/bracketrightbigg−3/2\n.\n(B5)\nNowweconsideraspecialcasewhere h(t)isaconstant,\nsay,h0. In this case, (B3) can be expressed as\ndX\n(X−n1)(X−n2)=iRω|f(t)|dt, (B6)\nwheren1andn2are roots of\nX2−2h0X−1 = 0, (B7)\nthat is,\nn1, n2=h0±/radicalBig\nh2\n0+1. (B8)\nNote that\nn1n2=−1, n1+n2= 2h0, n1−n2= 2/radicalBig\nh2\n0+1.\n(B9)\nUponintegration,weobtainwiththecondition X(0) = 0,\nX(t) =−1−eiφ(t)\nn2−n1eiφ(t). (B10)\nThe phase function φ(t) is\nφ(t) =Rω(n1−n2)/integraldisplayt\n0|f(τ)|dτ (B11)\nwhich can be expressed in closed form,\nφ(t) = 2Rω/radicalBig\nh2\n0+1(α+β)\n/braceleftbigg\nE/parenleftbigg\nωt−π\n4,2√αβ\nα+β/parenrightbigg\n−E/parenleftbigg\n−π\n4,2√αβ\nα+β/parenrightbigg/bracerightbigg\n,(B12)\nwhereE(ϕ,k) is the elliptic function of the second kind\ndefined by\nE(ϕ,k) =/integraldisplayϕ\n0/radicalbig\n1−k2sin2θdθ.10\nConsequently, for the case where h(t) =h0, the start-\ning Riccati equation (B1) is exactly solved, the result\nbeing of the form,\na(t) =if\n|f|eiφ−1\nn1eiφ−n2. (B13)\nSinceφ(0) = 0, it is evident that a(0) = 0. Using (B8)\nin (B13), we obtain\n|a(t)|2=sin2(φ/2)\nh2\n0+1−sin2(φ/2). (B14)\nThe transition probabilities from spin 1 /2 to±1/2 are\ngiven by\nw1/2,1/2= 1−1\nh2\n0+1sin2/parenleftbiggφ(t)\n2/parenrightbigg\n(B15)\nand\nw1/2,−1/2=1\nh2\n0+1sin2/parenleftbiggφ(t)\n2/parenrightbigg\n,(B16)\nwhich are characterized only by the constant h0and the\nphase function φ(t).\nAlthough the above results are exact under the as-\nsumption that h(t) =h0is a constant, they are approxi-\nmate results when h(t)≈h0.\nFinally, specifying the values of h0andφ(t), we shall\nobtain the exact results for the Rashba, the Dresselhaus\nand the symmetric cases.\n(i)The Rashba limit (α/ne}ationslash= 0, β= 0): In this case,\nfrom (B5) follows\nh0=−1\n2αRω. (B17)\nFurthermore the right-hand side of (B11) can be easily\nintegrated, so that\nφ(t) =/radicalbig\n1+4α2R2ωt. (B18)\nThus the spin flip probability is obtained in the form,\nwR\n1/2,−1/2=4α2R2\n1+4α2R2sin2/braceleftbigg1\n2/radicalbig\n1+4α2R2θ/bracerightbigg\n,\n(B19)whereθ=ωt.\n(ii)The Dresselhaus limit (α= 0, β/ne}ationslash= 0): In this\ncase, (B5) leads to\nh0=1\n2βRω. (B20)\nThe integral of (B11) yields\nφ(t) =/radicalbig\n1+4β2R2ωt. (B21)\nHence the spin-flip probability is\nwD\n1/2,−1/2=4β2R2\n1+4β2R2sin2/braceleftbigg1\n2/radicalbig\n1+4β2R2θ/bracerightbigg\n.(B22)\n(iii)The symmetric case (α=β/ne}ationslash= 0): In this partic-\nular case,\nh0= 0. (B23)\nThe phase factor becomes\nφ(t) = 2√\n2αR[sin(ωt)−cos(ωt)+1].(B24)\nThe corresponding spin-flip probability as a function of\nθ=ωtis\nwsym\n1/2,−1/2= sin2/braceleftBig√\n2αR[sinθ−cosθ+1]/bracerightBig\n.(B25)\nACKNOWLEDGMENTS\nThis work was supported by the NRI INDEX center,\nUSA, NSERC and CRC program, Canada.\n1M. V. Berry, Proc. Roy. Soc. London, Series A, Math.\nPhys. Sci., 392, 45 (1984).\n2F. Wilczek and A. Shapere Geometric Phases in Physics,\nWold Scientific (1989).\n3D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120(1998);\nG. Burkard, D. Loss and D. P. DiVincenzo, Phys. Rev. B\n59, 2070 (1999).4J. A. Jones, V. Vedral, A. Ekert and G. Castagnoli, Nature\n403, 869 (2000);G. Falci, R. Fazio, G. M. Palma, J. Siewert\nad V. Vedral, Nature 407, 355(2000).\n5X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000).\n6P. San-Jose, G. Zarand, A. Shnirman and G. Sch¨ on, Phys.\nRev. Lett. 97, 076803 (2006); P. San-Jose, G. Sch¨ on, A.\nShnirmanandG.Zarand, PhysicaE:Low-dimensional Sys-11\ntems and Nanostructures 40, 76 (2007); P. San-Jose, B.\nScharfenberger, G. Sch¨ on, A. Shnirman, G. Zarand, Phys.\nRev. B 77, 045305 (2008).\n7H. Wang, K.-D. Zhu, EPL 82, 60006 (2008).\n8R. P. Feynman, Phys. Rev. 84, 108(1951).\n9V. S. Popov, J. Exptl. Theoret. Phys. (U.S.S.R.) 35, 985\n(1958);[Sov. Phys. JETP 8, 687 (1959)].\n10Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039\n(1984).\n11G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n12W. T. Reid, Riccati Differential Equations (Academic\nPress, New York, 1972).\n13S. Prabhakar, J. E. Raynolds, A. Inomata, SPIE 7720,\n77020V (2010).\n14R.deSousa, S.DasSarma, Phys.Rev.B68, 155330 (2003).\n15S. Prabhakar and J. E. Raynolds, Phys. Rev. B 79, 195307\n(2009).\n16F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984).17M. Pletyukhov and O. Zaitsev, Journal of Physics A:\nMathematical and General 36, 5181 (2003);O. Zaitsev, D.\nFrustaglia and K. Richter, Phys. Rev. B 72, 155325 (2005).\n18V. S. Popov, Phys.-Usp. 50, 1217 (2007).\n19A. Inomata, H. Kuratsuji and C. C. Gerry, Path Integrals\nand Coherent States of SU(2) and SU(1,1), World Scien-\ntific, Singapore (1992).\n20I. L. Aleiner and V. I. Fal’ko, Phys. Rev. Lett. 87, 256801\n(2001).\n21S.-H. Chen and C.-R. Chang, Phys. Rev. B 77, 045324\n(2008).\n22B. A. Bernevig, J. Orenstein and S.-C. Zhang, Phys. Rev.\nLett. 97, 236601 (2006).\n23M.-H. Liu, C.-R. Chang and S.-H. Chen, Phys. Rev. B 71,\n153305 (2005); M.-H. Liu, K.-W. Chen, S.-H. Chen and\nC.-R. Chang, Phys. Rev. B 74, 235322 (2006).\n24J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,\nS.-C. Zhang, S. Mack and D. D. Awschalom, Nature 458,\n610 (2009)." }, { "title": "1606.07654v1.Low_energy_physics_of_three_orbital_impurity_model_with_Kanamori_interaction.pdf", "content": "Low-energy physics of three-orbital impurity model with Kanamori interaction\nAlen Horvat,1Rok \u0014Zitko,1, 2and Jernej Mravlje1\n1Jo\u0014 zef Stefan Institute, Jamova 39, Ljubljana, Slovenia\n2Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia\nWe discuss the low-energy physics of the three-orbital Anderson impurity model with the Coulomb\ninteraction term of the Kanamori form which has orbital SO(3) and spin SU(2) symmetry and\ndescribes systems with partially occupied t2gshells. We focus on the case with two electrons in the\nimpurity that is relevant to Hund's metals. Using the Schrie\u000ber-Wol\u000b transformation we derive an\ne\u000bective Kondo model with couplings between the bulk and impurity electrons expressed in terms\nof spin, orbital, and orbital quadrupole operators. The bare spin-spin Kondo interaction is much\nsmaller than the orbit-orbit and spin-orbital couplings or is even ferromagnetic. Furthermore, the\nperturbative scaling equations indicate faster renormalization of the couplings related to orbital\ndegrees of freedom compared to spin degrees of freedom. Both mechanisms lead to a slow screening\nof the local spin moment. The model thus behaves similarly to the related quantum impurity\nproblem with a larger SU(3) orbital symmetry (Dworin-Narath interaction) where this was \frst\nobserved. We \fnd that the two problems actually describe the same low-energy physics since\nthe SU(3) symmetry is dynamically established through the renormalization of the splittings of\ncoupling constants to zero. The perturbative renormalization group results are corroborated with the\nnumerical-renormalization group (NRG) calculations. The dependence of spin Kondo temperatures\nand orbital Kondo temperatures as a function of interaction parameters, the hybridization, and the\nimpurity occupancy is calculated and discussed.\nI. INTRODUCTION\nThe theoretical work of recent years has led to a con-\nsiderably better understanding of the origin of electronic\ncorrelations in materials with wide bands and relatively\nweak Coulomb interactions, such as iron-based supercon-\nductors and ruthenates. Based on the dynamical mean-\n\feld theory calculations (DMFT)1it has been realized\nthat a small multiplet splitting coming from the Hund's\nrule part of the Coulomb interaction ( J\u001cU 0 are positive\nconstants.\nIt is interesting to look at the spin and orbit Kondo\ntemperatures also as a function of hybridization. In\nFig. 5(d) we present the logarithms of Tspin\nKandTorb\nKas\na function of \u0000\u00001for zero and non-zero value of Hund's\nrule coupling. In the \frst case, the spin and orbit Kondo\nscales are the same for all \u0000. Conversely, in the sec-\nond case, the spin Kondo temperature is below the orbit\nKondo temperature for all \u0000. The leading exponential\ndependence on \u0000 is the same for both Tspin\nKandTorb\nK, as\nseen from equal slopes of the lines. The slopes depend\non the repulsion and are \u0000Ue\u000b=cwith (atNd= 2)c\u00193\nfor the zero- Jcase andc\u00194 for the \fnite- Jcase. The\ndi\u000berence is due to increased degeneracy of multiplets in\ntheJ= 0 case.C. Kanamori results away from integer \flling\nWe now turn to the results away from integer \flling.\nIn Fig. 6(a){(f) we display the Kondo temperatures for\nseveral \u0000 and J, still keeping Ue\u000b= 2 \fxed, as a func-\ntion of the impurity occupancy Ndin an interval around\n2. The spin and orbital Kondo temperatures behave dif-\nferently.Tspin\nKexhibit an overall diminishing trend as Nd\nis increased towards half-\flling ( Nd= 3) with a shallow\nminimum at Nd= 2 that becomes less pronounced for\nlarger \u0000 where log Tspin\nKis roughly linear in Nd. Con-\nversely,Torb\nKincreases when occupancy is changed from\nNd= 2 in both directions for all values of \u0000.\nThe di\u000berent behavior of both Kondo temperatures\non approaching half-\flling is due to the lowest states at\nNd= 3 having large spin but vanishing orbital moment,\nL= 0;S= 3=2, thus the screening of the spin is strongly\nsuppressed because of its large size14,21, while the orbital\nmoment is screened at a higher temperature. At half \fll-\ning, the notion of orbital Kondo temperature becomes\nmeaningless, as the orbital moment is zero also in the\nlimit of vanishing hybridization. This distinction disap-\npears forJ= 0, see Fig. 6(g) where the results for zero\nand non-zero Jare shown in a broader range of Nd. For\nJ= 0 the spin and orbit Kondo temperatures are the\nsame.\nOn approaching small occupancies, Nd.1, the Kondo\ntemperatures rapidly increase and no distinction is seen\nbetween zero and non-zero Jcases in panel (g). When\nthere is on average a single electron in the impurity the\nHund's coupling has no e\u000bect.\nIn Fig. 6(h) the ratio between the spin and orbital\nKondo temperatures is shown. One sees that Torb\nK=Tspin\nK\nrapidly increases as Ndis increased and at the occupancy\nNd= 2 this ratio is about 10 and is further increasing as\nwe approach half-\flling.\nV. CONCLUSION\nWe investigated the low-energy behavior of the\nKanamori model in the RG and NRG approaches. We\nderived the appropriate Kondo model that is described\nin terms of spin, orbital, and quadrupole degrees of free-\ndom. At low energies the splitting between the orbital\nand quadrupole coupling constants becomes insigni\fcant,\ntherefore similar behavior as for a Hamiltonian with a\nlarger SU(3)25symmetry can be expected. The NRG re-\nsults con\frm these poor-man's scaling \fndings. In partic-\nular, both models have the same strong-coupling Fermi-\nliquid stable \fxed point at low energies and approach this\n\fxed point in a similar way (in the physically relevant pa-\nrameter range). We calculated the dependence of the spin\nand orbital Kondo temperatures on interaction parame-\nters, hybridization, and impurity occupancy. The orbital\nKondo temperature is larger, thus orbital moments are\nquenched \frst as the temperature is lowered. This behav-\nior starts to occur as soon as the Hund's rule coupling8\n−7−6−5−4−3−2−1log(Tspin\nK)\n(a)Γ = 0.075\n(b)Γ = 0.125\n(c)Γ = 0.2\n1.4 1.6 1.8 2.0 2.2 2.4 2.6\nNd−4−3−2−1log(Torb\nK)\n(d)\n1.4 1.6 1.8 2.0 2.2 2.4 2.6\nNd(e)\n1.4 1.6 1.8 2.0 2.2 2.4 2.6\nNd(f)J= 0.1\nJ= 0.2\nJ= 0.3\nJ= 0.4J= 0.5\nJ= 0.6\nJ= 0.7\nJ= 0.8\n0.5 1.0 1.5 2.0 2.5 3.0 3.5\nNd−6−5−4−3−2−10log10(TK)\n(g)J= 0\nJ= 0.4\nTorb\nK\nTspin\nK\n1.4 1.6 1.8 2.0 2.2 2.4 2.6\nNd0.00.10.20.30.40.5Tspin\nK/Torb\nK\n(h)1.4 1.6 1.8 2.0 2.2 2.4\nNd050100Torb\nK/Tspin\nKJ= 0.1\nJ= 0.2\nJ= 0.3\nJ= 0.4\nJ= 0.5\nJ= 0.6\nJ= 0.8\nFigure 6. Kanamori model. (a-f) Spin and orbit Kondo temperatures as a function of the impurity occupancy Ndfor di\u000berent\nvalues of the Hund's coupling Jat \fxedUe\u000b= 2, (g) Spin and orbit Kondo temperatures in a larger region of impurity \flling\nand for zero and non-zero Hund's coupling. At J= 0 the spin and orbit Kondo temperature are the same. \u0000 = 0 :1. (h) Ratio\nbetween the spin and orbit Kondo temperatures. The arrows indicate the direction of increasing J.\nis increased above the Kondo temperature of the prob-\nlem without the Hund's rule coupling. The screening of\nthe spin-moments occurs at a temperature that is about\nan order of magnitude smaller38. The ratio of the or-\nbital Kondo temperature to the spin Kondo temperature\nbecomes particularly large as the impurity occupancy is\nincreased towards half-\flling. Our results demonstrate\nthat the NRG is capable of treating problems with real-\nistic three-orbital interactions. This method could hence\nbe used in the DMFT calculations, too. Another inter-\nesting line of investigation is the analysis of the derived\nKondo impurity model for parameters that do not cor-\nrespond to the Anderson-type model. Our preliminary\nresults reveal a rich phase diagram with several distinct\nnon-Fermi-liquid phases.\nACKNOWLEDGMENTS\nWe acknowledge the support of the Slovenian Research\nAgency (ARRS) under P1-0044.Appendix A: Kondo Hamiltonian Derivation\nIn this appendix we derive the Kondo Hamiltonian\nfrom the AIM with either Dworin-Narath or Kanamori\ninteraction using the Schrie\u000ber-Wol\u000b transformation.\nKondo Hamiltonian having SO(3) orbital and SU(2) spin\nsymmetry was earlier written in terms of unit tensor op-\nerators in Ref.39. Kondo Hamiltonian having SU(M) or-\nbital and SU(N) spin symmetry was derived in Ref.25.\nThe Schrie\u000ber-Wol\u000b transformation reads:\nHK=\u0000PnHhyb X\naPa\nn+1\n\u0001Ea\nn+1+X\nbPb\nn\u00001\n\u0001Eb\nn\u00001!\nHhybPn:\n(A1)\nThe projector operator Pnprojects onto the atomic\nground state multiplet with occupancy n=Nd. The pro-\njectorsPa\nn\u00061project onto the high-energy atomic multi-\nplets having energy Ea\nn\u00061(indicesa;bdenote the di\u000berent\ninvariant subspaces with respect to Himpas presented in\nthe main text) and the virtual excitation energies are\n\u0001Ea\nn\u00061=Ea\nn\u00061\u0000En,Enbeing the ground-state energy.9\nWe adopt the Einstein summation notation and for the\nsake of clarity we at \frst disregard all the constants (e.g.\nV2=\u0001E). The projection operators to atomic multiplets\ntransform as an identical representation under all symme-\ntry transformations of the problem, hence the multiplet\nsplitting of the excited states a\u000bects only the coupling\nconstants (we write \u0000 = Hhyb):\nX\nahnj\u0000Pa\nn+1\n\u0001Ea\nn+1\u0000jni=X\na1\n\u0001Ea\nn+1hnj\u0000\u0000jni:(A2)\njni=Pnj\tLSiis the ground state with valence n, orbital\nmomentLand spinS. The virtual charge excitation\nprocess conserves the impurity charge, thus Pndy\njdy\niPn=\n0. The non-zero terms in the Kondo Hamiltonian are of\nthe form:\nH0\nK=Pn\u0000\u0000Pn=Pn(cy\ni\u001bidi\u001bidy\nk\u001bkck\u001bk+ h:c:)Pn(A3)\nNext we insert an identity:\ncy\ni\u001bidi\u001bidy\nk\u001bkck\u001bk= (cy\ni\u001bi\u000ei;l\u000e\u001bi;\u001bldl\u001bl)(dy\nk\u001bk\u000ek;j\u000e\u001bk;\u001bjcj\u001bj);\n(A4)\nand use the following group-theoretical relations40,41:\n\u000ei;l\u000ek;j=1\nm\u000ei;j\u000ek;l+1\na(\u001cb)i;j(\u001cb)k;l;SU(m);(A5)\n\u000ei;l\u000ek;j=\u000ei;k\u000ej;l+2\na(Tb)i;j(Tb)k;l;SO(m):(A6)\nThe generators \u001c;T live in the de\fning (fundamen-\ntal) representation of the SU( m), SO(m) symmetric Lie\ngroup, respectively. The constant adepends on the nor-\nmalization of the generators Tr( TbTc) =a\u000eb;c(typically\na= 2). In the SU(2) case \u001care the Pauli matrices and\nin the SU(3) case \u001care the Gell-Mann matrices.\nTo obtain the Kondo Hamiltonian from the AIM with\nthe Dworin-Narath interaction in terms of spin and or-\nbital operators, we insert the identity (A5) into equa-\ntion (A4) for the spin and orbital degrees of freedom\n(since both have SU symmetry). The relation (A5) leads\nto a result in which the dummy indices associated with\nthe bulk operators ci;jare independent from the in-\ndices associated with the impurity operators, and can\nbe summed over to yield spin/orbital momentum opera-\ntors. The Kondo Hamiltonian with the Dworin-Narath\ninteraction reads:\nHDN\nK=JpNf+JsS\u0001s+JtT\u0001t+\nJts(T\nS)\u0001(t\ns): (A7)\nBath operators are de\fned as:\ns=X\nmcy\nm\u001b\u00121\n2\u001b\u001b\u001b0\u0013\ncm\u001b0;\nt=X\n\u001bcy\nm\u001b\u001cmm0cm0\u001b:(A8)\n\u001c;\u001bare the Pauli and Gell-Mann matrices, respectively.\nSandTare the generators of spin-1 representation of\nSU(2) and the fundamental representation of SU(3).On the other hand the relation (A6) does not decou-\nple the bulk/impurity dummy indices due to the term\n\u000ei;k\u000ej;l. However, this problematic term can be, for the\n3-dimensional SO(3) symmetric group, rewritten as\n\u000ei;l\u000ek;j=1\n3\u000ei;j\u000ek;l+1\n2Tc\ni;jTc\nk;l+1\n2Qde\ni;jQde\nk;l; (A9)\nwhich does lead to the desired decoupling. Above we\nused the orbital quadrupole operators de\fned as\nQbc\ni;j=1\n2\u0000\nTb\ni;mTc\nm;j+Tc\ni;mTb\nm;j\u0001\n\u00002\n3\u000eb;c\u000ei;j;(A10)\nTr(Q\u000bQ\f) = 2\u000e\u000b;\f;(A11)\nwhich are symmetric and traceless. We derive the iden-\ntity (A9) by calculatingP\nb;cQbc\nijQbc\nkland using the iden-\ntity (A6). By inserting the identity (A9) for orbital\nand (A5) for spin degrees of freedom into the Hamilto-\nnian (A4), we express the Kondo Kanamori Hamiltonian\nas:\nHK=JpNf+JsS\u0001s+JlL\u0001l+JqQ\u0001q+\nJls(L\nS)\u0001(l\ns) +Jqs(Q\nS)\u0001(q\ns):(A12)\nS;L;Q(s;l;q) are total impurity (bath) spin, orbit,\norbital-quadrupole operators respectively.42\nAppendix B: RG \row\nIn the second order of the perturbation theory we inte-\ngrate out the scattering events to the states close to the\nband edges,\u0006\u000f2[D\u0000\u000eD;D ]. The \frst correction to\nthe renormalized Kondo interaction is\n\u0001HK\u00191\n\u0001EHKPHK: (B1)\nThe projector Pdescribes all the scattering events of\nelectrons from the impurity to the band edges. The pref-\nactor is 1=\u0001E=\u001aj\u000eDj(E\u0000D+\u000fk)\u00001\u0019\u001aj\u000eDjD\u00001.\nWe assume that the conduction band is wide. Dis the\nhalf-bandwidth, Eis the energy measured relative to the\nground state of the conduction electron gas and can be\nneglected,\u000fkis the energy of electrons near the Fermi\nsurface and can also be neglected relative to D.\nIn the following we present a convenient way for calcu-\nlating the second order corrections to the renormalized\nHamiltonian using the completeness relations from the\nprevious section. We will illustrate the procedure on the\ncase of the spin-spin Kondo interaction term JS\u0001\u001bfor\na single orbital model with S= 1=2. First, we write the\nimpurity operators in terms of the fermionic operators\nS\u000b!dy\ni\u001b\u000b\nijdj; (B2)\nwith additional constraint dy\n\"d\"+dy\n#d#= 1.dy\ni;dicre-\nates/annihilates an electron on the impurity with spin10\ni2f\";#g,\u001b\u000bare the Pauli matrices. The bulk electron\nspin operator is:\n\u001b\u000b!cy\ni\u001b\u000b\nijcj; (B3)\ncy\ni;cicreates/annihilates an electron with spin iin the\nbulk. The spin-spin operators may be expressed in terms\nof Kronecker \u000esymbols using the following completeness\nrelation:\nX\n\u000b(\u001b\u000b)i;j(\u001b\u000b)k;l= 2\u000ei;l\u000ek;j\u0000\u000ei;j\u000ek;l: (B4)\n[For other operators, such as orbital, quadrupole, and\nmixed operators, one can derive similar expressions from\nEqs. (A5),(A6),(A10).] After inserting the completeness\nrelation we obtain:\nJ2X\nijkl(2\u000ei;l\u000ek;j\u0000\u000ei;j\u000ek;l)dy\nidjcy\nkclP\u0002 (B5)\n\u0002X\nmnop(2\u000em;p\u000eo;n\u0000\u000em;n\u000eo;p)cy\nocp=\n=J2X\nijklX\nmnopAijkl\nmnopPdy\nidjdy\nmdncy\nkclcy\nocp:(B6)\nThe projector Pconsists of two contributions:\nP=\u000ejm(\u000elo+\u000ekp): (B7)\nThe \frst term \u000ejmfollows from the single-occupancy\nconstraint of auxiliary fermions, while the second term\n\u000elo+\u000ekpdescribes the processes that involve scattering\nof electrons/holes to the upper/lower band edge. In theexpressions one can use cy\n\u001bkc\u001bk= 0 for the electron states\nkin the upper band edge that are assumed empty and\ncy\n\u001bkc\u001bk= 1 for the electron states kat the lower band\nedge that are assumed \flled.\nNow we sum over the indices m;o to eliminate Kro-\nnecker\u000esymbols that come from the projection operator.\nThe contribution of the electron scattering to the upper\nband edge reads:\nJ2X\nijklX\nnpAijkl\njnlpdy\nidncy\nkcp: (B8)\nNext we sum over the dummy indices j;l. The correction\nto the Kondo exchange reads:\nJ2X\niknp(\u00004\u000eip\u000ekn+ 5\u000ein\u000ekp)dy\nidncy\nkcp= (B9)\n=\u00002J2S\u0001\u001b+ 3J2X\niknp\u000ein\u000ekpdy\nidncy\nkcp:(B10)\nThis result has the same form as the initial exchange in-\nteraction with an additional potential scattering term. A\ncontribution from the scattering to the lower band edge\nis obtained in a similar fashion; the exchange term is the\nsame, while the potential scattering term has an oppo-\nsite sign and therefore cancels out that in Eq. (B9) since\nwe have assumed a particle-hole symmetric conduction\nband. We recover the standard \ffunction of the S= 1=2\nKondo model.\nSimilar approach can be used to tackle the multi-\norbital problem. The scaling functions for a \rat band,\ngeneral number of orbitals MandN= 2 are:\n\fs=M\u0000\nJls2\u0000M\u0000\nJls2+ 2Js2\u0001\u0001\n\u0000Jqs2\u0000\nM2+M\u00002\u0001\n2M2; (B11)\n\fl=1\n16\u0000\n\u00004Jl2(M\u00002)\u00003Jls2(M\u00002)\u0000(M+ 2)\u0000\n4Jq2+ 3Jqs2\u0001\u0001\n; (B12)\n\fq=\u00001\n8M(4JlJq+ 3JlsJqs); (B13)\n\fls=\u0000Jls\u0000\nM(Jl(M\u00002) + 4Js) +Jqs\u0000\nM2\u00004\u0001\u0001\n+JqJqsM(M+ 2)\n2M; (B14)\n\fqs=\u00002JqsM(JlM+ 4Js) +JlsM(Jls(M\u00002) + 2JqM) +Jqs2\u0000\nM2+ 2M\u00008\u0001\n4M: (B15)\nWhen\u000b= 0;Jq=Jl;Jqs=Jlsand results are the same\nas obtained in Ref.17,25for the model with SU(M) orbital\nsymmetry.\nAppendix C: Rescaled Kondo Hamiltonian\nIn the Coqblin-Schrie\u000ber model the coupling constants\nare related to each other: 3 Jp;s= 2Jl;q=Jls;qs. We in-troduce rescaled coupling constants: ~Jp;s= 3Jp;s;~Jl;q=\n2Jl;q;~Jls;qs=Jls;qs. The Kondo Hamiltonian in terms of\nrescaled couplings reads:\nHK=~Jp=3Nf+~Js=3S\u0001s+~Jl=2L\u0001l+~Jq=2Q\u0001q+\n~Jls(L\nS)\u0001(l\ns) +~Jqs(Q\nS)\u0001(q\ns):(C1)11\nHence the rescaled Kondo couplings are written in a more\nsymmetric form:\n~Jp=V2\n6\u00126\n\u0001E1\u00004\n\u0001Ea\n3\u00005\n\u0001Eb\n3\u00003\n\u0001Ec\n3\u0013\n;(C2)\n~Js=V2\n6\u00126\n\u0001E1\u00002\n\u0001Ea\n3+5\n\u0001Eb\n3+3\n\u0001Ec\n3\u0013\n;(C3)\n~Jl=V2\n6\u00126\n\u0001E1+8\n\u0001Ea\n3\u00005\n\u0001Eb\n3+3\n\u0001Ec\n3\u0013\n;(C4)\n~Jq=V2\n6\u00126\n\u0001E1+8\n\u0001Ea\n3+1\n\u0001Eb\n3\u00003\n\u0001Ec\n3\u0013\n;(C5)\n~Jls=V2\n6\u00126\n\u0001E1+4\n\u0001Ea\n3+5\n\u0001Eb\n3\u00003\n\u0001Ec\n3\u0013\n;(C6)\n~Jqs=V2\n6\u00126\n\u0001E1+4\n\u0001Ea\n3\u00001\n\u0001Eb\n3+3\n\u0001Ec\n3\u0013\n:(C7)\nNotice that in the limit of vanishing Hund's coupling J=\n0, \u0001Ei= \u0001E, and all the couplings are the same and soare the scaling functions:\n~\fp= 0; (C8)\n~\fs=\u00001\n3\u0010\n3~J2\nls+ 5~J2\nqs+~J2\ns\u0011\n; (C9)\n~\fl=\u00001\n8\u0010\n~J2\nl+ 3~J2\nls+ 5\u0010\n~J2\nq+ 3~J2\nqs\u0011\u0011\n;(C10)\n~\fq=\u00003\n4(~Jl~Jq+ 3~Jls~Jqs); (C11)\n~\fls=\u00001\n12(3~Jl~Jls+ 10 ~Jls~Jqs+ 8~Jls~Js+ (C12)\n+15~Jq~Jqs);\n~\fqs=\u00001\n12(~Jqs(9~Jl+ 7~Jqs+ 8~Js) + (C13)\n+3~J2\nls+ 9~Jls~Jq):\nAppendix D: Comparison between Kanamori and\nDworin-Narath models\nUsing parameter \u000b(Eq. (5) in the main text) the im-\npurity interaction can be continuously tuned between the\nDworin-Narath ( \u000b= 0) and the Kanamori ( \u000b= 1) form.\nEven though the SO(3) orbital symmetry is dynamically\nrestored to SU(3) at low energies and hence the behav-\nior of the two models is similar there are quantitative\ndi\u000berences that we illustrate here.\nIn Fig. 7 we present the spin and the orbit Kondo tem-\nperatures as a function of Hund's coupling for di\u000berent\nvalues of\u000b. Overall a qualitatively similar behavior is\nfound. At small hybridizations up to an order of magni-\ntude di\u000berence is found for large J. For small hybridiza-\ntion the spin Kondo temperature for Dworin-Narath is\nnon-monotonic at large Jwhich is not the case for the\nKanamori model. The calculated Kondo temperatures\nthere di\u000ber by an order of magnitude between the two\nmodels which can be important for realistic DMFT cal-\nculations where the quantitative agreement with exper-\niments is desired. Despite the overall similarity of the\nDworin-Narath and Kanamori results, the more realistic\nKanamori interaction needs to be used there.\n1A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n2K. Haule and G. Kotliar, New J. Phys. 11, 025021 (2009).\n3P. Werner, E. Gull, M. Troyer, and A. J. Millis, Phys.\nRev. Lett. 101, 166405 (2008).\n4J. Mravlje, M. Aichhorn, T. Miyake, K. Haule, G. Kotliar,\nand A. Georges, Phys. Rev. Lett. 106(2011).\n5P. Hansmann, R. Arita, A. Toschi, S. Sakai, G. Sangio-\nvanni, and K. Held, Phys. Rev. Lett. 104, 197002 (2010).\n6A. Georges, L. d. Medici, and J. Mravlje, Annu. Rev.\nCondens. Matter Phys. 4, 137 (2013).\n7Z. P. Yin, K. Haule, and G. Kotliar, Nat. Mater. 10, 932\n(2011).8L. deMedici, in Iron-Based Superconductivity , Springer Se-\nries in Materials Science, Vol. 211, edited by P. D. Johnson,\nG. Xu, and W.-G. Yin (Springer International Publishing,\n2015) pp. 409{441.\n9L. Fanfarillo and E. Bascones, Phys. Rev. B 92, 075136\n(2015).\n10A. Hewson, The Kondo Problem to Heavy Fermions (Cam-\nbridge University Press, 1993).\n11A. A. Khajetoorians, M. Valentyuk, M. Steinbrecher,\nT. Schlenk, A. Shick, J. Kolorenc, A. I. Lichtenstein, T. O.\nWehling, R. Wiesendanger, and J. Wiebe, Nature Nan-\notech 10, 958 (2015).\n12H. T. Dang, M. dos Santos Dias, A. Liebsch, and S. Lounis,12\n−5−4−3−2−1log10(Tspin\nK)\nΓ = 0.1\nα= 0.0\nα= 0.1\nα= 0.2\nα= 0.4\nα= 0.6\nα= 0.8\nα= 0.9\nα= 1.0\nlog10(Torb\nK)\nΓ = 0.1\n−5−4−3−2−1log10(Tspin\nK)\nΓ = 0.2\nlog10(Torb\nK)\nΓ = 0.2\n0.00.10.20.30.40.50.60.70.80.9\nJ−5−4−3−2−1log10(Tspin\nK)\nΓ = 0.4\n0.00.10.20.30.40.50.60.70.80.9\nJlog10(Torb\nK)\nΓ = 0.4\nFigure 7. Spin and orbital Kondo temperatures as a function of Hund's coupling Jfor di\u000berent values of parameter \u000b(\u000b= 0\nDN interaction, \u000b= 1 Kanamori interaction). Model parameters are Ue\u000b= 2;Nd= 2:\nPhys. Rev. B 93, 115123 (2015).\n13L. deMedici, J. Mravlje, and A. Georges, Phys. Rev. Lett.\n107(2011).\n14J. R. Schrie\u000ber, J. Appl. Phys. 38, 1143 (1967).\n15C. Jayaprakash, H. R. Krishna-murthy, and J. W. Wilkins,\nPhys. Rev. Lett. 47, 737 (1981).\n16B. A. Jones and C. M. Varma, Phys. Rev. Lett. 58, 843\n(1987).\n17Y. Kuramoto, The European Physical Journal B-\nCondensed Matter and Complex Systems 5, 457 (1998).\n18H. Kusunose and K. Miyake, J. Phys. Soc. Jpn. 66, 1180\n(1997).\n19S. Yotsuhashi, H. Kusunose, and K. Miyake, J. Phys. Soc.\nJpn.70, 186 (2001).\n20T. Pruschke and R. Bulla, Eur. Phys. J. B 44, 217 (2005).\n21A. H. Nevidomskyy and P. Coleman, Phys. Rev. Lett. 103,\n147205 (2009).\n22Y. Nishikawa and A. C. Hewson, Physical Review B 86,\n245131 (2012).\n23L. Dworin and A. Narath, Phys. Rev. Lett. 25, 1287 (1970).\n24Z. P. Yin, K. Haule, and G. Kotliar, Phys. Rev. B 86,\n195141 (2012).\n25C. Aron and G. Kotliar, Phys. Rev. B 91, 041110 (2015).\n26K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and\nA. Weichselbaum, Phys. Rev. Lett. 115(2015).\n27I. Okada and K. Yosida, Prog. Theor. Phys. 49, 1483\n(1973).\n28J. Schrie\u000ber and P. Wol\u000b, Phys. Rev. 149, 491 (1966).\n29A. L. Fetter and J. D. Walecka, Quantum theory of many-particle systems (Courier Corporation, 2003).\n30L. de Leo, Ph.D. thesis, SISSA (2004).\n31P. W. Anderson, J. Phys. C: Solid State Phys. 3, 2436\n(1970).\n32T. Kuzmenko, K. Kikoin, and Y. Avishai, Physical Review\nLetters 89, 156602 (2002).\n33K. Kikoin and Y. Avishai, Physical Review B 65, 115329\n(2002).\n34L. Borda, G. Zarand, W. Hofstetter, B. Halperin, and\nJ. von Delft, Physical Review Letters 90, 026602 (2003).\n35K. Kikoin, M. Kiselev, and M. Wegewijs, Physical Review\nLetters 96, 176801 (2006).\n36A. J. Keller, S. Amasha, I. Weymann, C. P. Moca,\nI. G. Rau, J. A. Katine, H. Shtrikman, G. Zar\u0013 and, and\nD. Goldhaber-Gordon, Nature Physics 10, 1 (2013).\n37R.\u0014Zitko, \\NRG Ljubljana,\" nrgljubljana.ijs.si/.\n38The precise value depends on the parameters. The domi-\nnant exponential dependence on the Coulomb interaction\nparameters is, however, the same for the spin and orbital\nKondo temperature.\n39L. Hirst, Adv. Phys. 27, 231 (1978).\n40P. Cvitanovic, Phys. Rev. D 14, 1536 (1976).\n41D. A. Varshalovich, Quantum theory of angular momentum\n: Irreducible tensors, spherical harmonics, vector coupling\ncoe\u000ecients 3 nj symbols (World Scienti\fc Pub, Singapore\nPhiladelphia, 1989).\n42L. C. Biedenharn and J. D. Louck, Angular momentum in\nquantum physics (Cambridge University Press, 1984)." }, { "title": "1211.2097v3.Spin_orbit_Coupled_Bose_Einstein_Condensates_in_Spin_dependent_Optical_Lattices.pdf", "content": "arXiv:1211.2097v3 [cond-mat.quant-gas] 10 Jan 2013Spin-orbit Coupled Bose-Einstein Condensates in Spin-dep endent Optical Lattices\nWei Han,1,2Suying Zhang,2and Wu-Ming Liu1\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2Institute of Theoretical Physics, Shanxi University, Taiy uan 030006, China\nWeinvestigate theground-statepropertiesofspin-orbitc oupledBose-Einstein condensatesinspin-\ndependent optical lattices. The competition between the sp in-orbit coupling strength and the depth\nof the optical lattice leads to a rich phase diagram. Without spin-orbit coupling, the spin-dependent\noptical lattices separate the condensates into alternatin g spin domains with opposite magnetization\ndirections. With relatively weak spin-orbit coupling, the spin domain wall is dramatically changed\nfrom N´ eel wall toBloch wall. For sufficiently strongspin-or bit coupling, vortex chains andantivortex\nchainsare excitedinthespin-upandspin-downdomains resp ectively, correspondingtotheformation\nof a lattice composed of meron-pairs and antimeron-pairs in the pseudospin representation. We also\ndiscuss how to observe these phenomena in real experiments.\nPACS numbers: 03.75.Lm, 03.75.Mn, 05.30.Jp, 67.85.Fg\nIntroduction.— In recent years, the experimental con-\ntrol on ultracold atomic gases has reached truly unprece-\ndented levels. By employing two lasers with different\nfrequencies and polarizations plus a non-uniform vertical\nmagneticfield, experimentalistshaveproducedspin-orbit\n(SO) coupling, which couples the internal states and the\norbit motion of the atoms [ 1–4]. The SO coupled ultra-\ncold atomic gases have attracted great interests of re-\nsearchers [ 5–15]. It has been indicated that the interplay\namong SO coupling, interatomic interaction and exter-\nnal potential leads to rich ground-state phases, such as\nplane wave, density stripe, fractional vortex and various\nvortex lattices [ 16–26]. The SO coupled ultracold atomic\ngases open a new window for quantum simulation, and\nprovide opportunities to study SO coupling phenomena\nin a highly controllable impurity-free environment.\nAll the existing studies on SO coupled Bose-Einstein\ncondensates (BECs) only refer to the case that different\ninternal states of the atoms are trapped in an identical\nexternal potential. However, by using two counterpropa-\ngating laserswith the same frequency but different polar-\nizations, the experimentalists have been able to produce\nspin-dependent optical lattices that allow different inter-\nnal states of the atoms experience drastically different\nexternal potentials [ 27–31]. The spin-dependent optical\nlattices bring more complicated geometry of condensates\nand have potential applications in quantum computation\n[27], cooling and thermometry [ 28], and quantum simu-\nlation [29–32]. It is natural to ask what new structures\ncan be formed due to the competition between the SO\ncoupling and the spin-dependent optical lattices.\nIn this Letter, we investigate the ground-state phase\ndiagram of SO coupled BECs in spin-dependent optical\nlattices. In the absence of SO coupling, the ground state\nof the system is characterized by the formation of al-\nternating spin domains. However, such a structure can\nbe dramatically changed due to the SO coupling effects.\nRelatively weak SO coupling basically changes the ori-entation of the spins in the domain walls, causing the\ntransformation from N´ eel wall to Bloch wall. Sufficiently\nstrong SO coupling excites meron-pairs and antimeron-\npairs in the spin-up and spin-down domains respectively\nand generates a meron-pair lattice. This is essentially\ndifferent from the mechanism of generating a meron-pair\nlattice by bulk rotation [ 33]. Our findings provide a new\nway to create and manipulate topological excitations in\nSO coupled systems.\nEnergy band structure.— We consider SO coupled\nBECs confined in the combined potential of a quasi-2D\nharmonic trap and 1D spin-dependent optical lattices.\nThe Hamiltonian of this system is given by\nH=/integraldisplay\ndr/bracketleftBig\nΨ†(−¯h2∇2\n2m+Vso)Ψ+V1(r)n↑+V2(r)n↓\n+g11n2\n↑+g22n2\n↓+g12n↑n↓/bracketrightBig\n, (1)\nwhereΨ= [Ψ↑(r),Ψ↓(r)]Tdenotes the two-component\nwave functions and is normalized as/integraltext\ndrΨ†Ψ=Nwith\n0510152025300150300450600750900(a)\nI\nIIA\nκ[units of ahω⊥]V0[units of ¯ hω⊥]\nIIB\n0246810020406080100(b)\ndensity stripe\nrectangular latticetriangular lattice\nκ[units of ahω⊥]V0[units of ¯ hω⊥]\nFIG. 1: (Color online) (a) Single-particle phase diagram\nspanned by the SO coupling strength κand the depth V0of\nthe optical lattice. (b) Many-body phase diagram spanned by\nκandV0with the effective interaction parameter ˜ g= 6000.\nThe dashed line in (b) indicates the phase boundary between\nphase I and phase II in (a) for comparison.2\nFIG. 2: (Color online) Band structures induced by the compet ition between the SO coupling strength κand the depth V0of\nthe optical lattice corresponding to phase I (a), phase IIA ( b), and phase IIB (c) of the single-particle phase diagram in Fig.\n1(a). The superposition of the momenta at the minima of the ba nds produces a density stripe in (a), a lattice in (b) and (c).\nNthe total particle number. n↑= Ψ∗\n↑Ψ↑andn↓= Ψ∗\n↓Ψ↓\ndescribe the particle number density of each component.\ngij= 4π¯h2aij/m(i,j= 1,2) represent the interatomic\ninteraction strengths characterized by the s-wave scat-\ntering lengths aijand the atomic mass m.\nWe consider a RashbaSO coupling Vso=−i¯hκ(σx∂x+\nσy∂y), where σx,yare the Pauli matrices and κde-\nnotes the SO coupling strength. The combined exter-\nnal potential Vi(r) =VH(r) +VOLi(x), where VH=\n1\n2mω2\n⊥[(x2+y2) +λ2z2] is the harmonic trapping po-\ntential with λ=ωz/ω⊥≫1, andVOL1=V0sin2(νx)\nandVOL2=V0cos2(νx) describe the 1D spin-dependent\noptical lattice potentials, which are experienced by the\ntwo components, respectively. Approximating the zde-\npendence of the wave functions by the single-particle\nground state in a harmonic potential, one can obtain\nthe 2D dimensionless effective interaction parameters\n˜gij= 2√\n2πλNa ij/ah, withah=/radicalbig\n¯h/(mω⊥) [34].\nThe single-particle energy bands are critically impor-\ntant to understand the ground-state properties of the\ncondensates. Without considering the harmonic trap,\nthe 2D single-particle wave functions in the k-space\nobey the secular equations¯h2\n2m[(kx+2sν)2+k2\ny]Us−\nV0\n4(Us+1−2Us+Us−1)+¯hκ(kx+2sν−iky)Ws=εUs\nand¯h2\n2m[(kx+2sν)2+k2\ny]Ws+V0\n4(Ws+1+2Ws+Ws−1)+\n¯hκ(kx+2sν+iky)Us=εWs, where Us=\nU(kx+2sν,ky) andWs=W(kx+2sν,ky) with\ns= 0,±1,±2,...,±∞represent the wave functions at\nthe point ( kx+2sν,ky). Typically, we choose ν=a−1\nh\nfor our present discussion. By numerical exact diagonal-\nization, we can solve the secular equations and obtain\nthe energy band structure.\nWe find that there exist three different kinds of energy\nband structures depending on the competition between\nthe SO coupling strength κand the depth V0of the op-\ntical lattices. Fig. 1(a) presents the single-particle phase\ndiagram spanned by κandV0. In phase I, the minima of\nthe energy bands locate in a set of k points with ky= 0\nandkx∈K1={±ν,±3ν,±5ν,...}[See Fig. 2(a)]. In\nphase II, the minima of the energy bands locate in a set\nof k points with ky=±δ/parenleftbig\n0< δ≤mκ\n¯h/parenrightbig\n, andkx∈K1for\nphase IIA ( kx∈K2={0,±2ν,±4ν,±6ν,...}for phaseIIB) [See Figs. 2(b) and2(c)].\nThe single-particle ground state in phase I is nonde-\ngenerate and can be expressed as a linear superposition\nof the plane waves with wave vectors ( kx∈K1,ky= 0).\nThis yields alternating spin domains with opposite mag-\nnetization directions (density stripe). In both phase IIA\nand phase IIB, the single-particle ground state is double-\ndegenerate. Each degenerate state can be expressed as a\nlinear superposition of the plane waveswith wave vectors\n(kx∈K,ky=δ) or (kx∈K,ky=−δ), where K=K1\nfor phase IIA and K=K2for phase IIB. For an arbi-\ntrary nonzero superposition of the two degenerate states,\nlattice will be formed as the single-particle ground state.\nPhase diagram.— By using the imaginary time evolu-\ntion method, we can solve Eq. ( 1) to obtain the many-\nbody ground state. Considering that the spin-exchange\nintegrationsareveryweakintypicalexperiments, wejust\ndiscuss the case that ˜ gij= ˜g. The many-body phase di-\nagram spanned by κandV0with ˜g= 6000 is presented\nin Fig.1(b). We find that the competition between the\nSO coupling strength and the depth of the optical lattice\nleads to three distinct phases—density stripe, triangular\nvortex lattice and rectangular vortex lattice.\nIn the density stripe phase, the spin-up and spin-down\ncomponents are arranged alternately and form alternat-\ning spin domains [See Figs. 3(a1-a6)]. Comparing the\nphase diagrams in Figs. 1(a) and1(b), we find that the\ninteraction has no significant influence on the phase re-\ngion of the density stripe.\nIn the vortex lattice phases, both the triangular\nand rectangular lattices are composed of alternately ar-\nranged vortex and antivortex chains, which are excited\nin the spin-up and spin-down domains respectively [See\nFigs.3(b1-b6) and 3(c1-c6)]. The only difference is that\nthe vortices of the neighboring chains are staggered for\nthe triangular lattice, but are parallel for the rectan-\ngular lattice [See Figs. 3(b3) and 3(c3)]. These two\ndifferent arrangements of vortices correspond to odd-\nparity and even-parity distributions of the particles in\nkxdirection of the k-space. [See Figs. 3(b6) and 3(c6)].\nThese correspond to the single particle band structures\ndescribed in Figs. 2(c) and (b), where the minima of3\na1\nx[units of ah]y[units of ah]\n−10 0 010\n0\n−10\na2\nx[units of ah]−10 0 010\n0\n−10\na3\nx[units of ah]−10 0 010\n0\n−10\na4\nx[units of ah]−2π 0 2π2π\n0\n−2π\na5\nx[units of ah]−2π 0 2π2π\n0\n−2π\na6\nkx[units of a−1\nh]ky[units of a−1\nh]\n−6 0 66\n0\n−6\nb1\nx[units of ah]y[units of ah]\n−10 0 010\n0\n−10\nb2\nx[units of ah]−10 0 010\n0\n−10\nb3\nx[units of ah]−10 0 010\n0\n−10\nb4\nx[units of ah]−2π 0 2π2π\n0\n−2π\nb5\nx[units of ah]−2π 0 2π2π\n0\n−2π\nb6\nkx[units of a−1\nh]ky[units of a−1\nh]\n−6 0 66\n0\n−6\nc1\nx[units of ah]y[units of ah]\n−10 0 010\n0\n−10\nc2\nx[units of ah]−10 0 010\n0\n−10\nc3\nx[units of ah]−10 0 010\n0\n−10\nc4\nx[units of ah]−2π 0 2π2π\n0\n−2π\nc5\nx[units of ah]−2π 0 2π2π\n0\n−2π\nc6\nkx[units of a−1\nh]ky[units of a−1\nh]\n−6 0 66\n0\n−6\nFIG. 3: (Color online) Ground state as a function of the SO cou pling strength κand the depth V0of the optical lattice with\nκ= 4ahω⊥,V0= 40¯hω⊥(a1-a6), κ= 4ahω⊥,V0= 20¯hω⊥(b1-b6), κ= 4ahω⊥,V0= 15¯hω⊥(c1-c6). The effective interaction\nparameter is fixed at ˜ g= 6000. The spin-up, spin-down, and total density profiles ar e shown in (a1, b1, c1), (a2, b2, c2), and\n(a3, b3, c3), respectively. The phases of the spin-up and spi n-down wave functions, with values ranging from −2πto 2π(blue\nto red), are shown in (a4, b4, c4) and (a5, b5, c5). The momentu m distributions are depict in (a6, b6, c6).\nthe bands also show odd-parity and even-parity distribu-\ntions respectively, although their phase regions are not\nconsistent due to the influence of the interatomic inter-\nactions [See Figs. 1(a) and 1(b)]. As discussed above,\nthe single-particle ground state in phase II is double-\ndegenerate. From Figs. 3(b6) and 3(c6), we can see that\nthe interaction removes the degeneracy and chooses an\nequal weighted linear superposition of the two degener-\nate states as the many-body ground state.\nThe alternating arrangement of the vortex and an-\ntivortexchainsleadstoalternating-directionplanewaves,\nwhich propagating on two sides of each chain [See\nFigs.3(b4,b5) and 3(c4,c5)]. The vortex line density\nnvand the wave number of the plane waves kysatisfy\nnv=ky\nπ. Numerical simulations indicate that for a given\nSO coupling strength κ, as the lattice depth V0increases\nfrom 0,kygradually decreases frommκ\n¯hand eventually\nbecomes 0 on the boundary of the vortex lattice phase.\nThis implies that by adjusting the depth of the optical\nlattice, one can continuously control the vortex line den-\nsity from 0 tomκ\nπ¯h.\nSpin domain wall.— The separation between the spin-\nup and spin-down domains is not sharp, but requires the\nspin density vector varying gradually across the oppo-\nsite domains and forming a spin domain wall [ 35]. There\nare two basic types of domain walls, N´ eel wall and Blochwall. In the N´ eel wall spin flip occurs in a plane, while\nin the Bloch wall the spin flip occurs by tracing a he-\nlix [See Figs. 4(b1) and 4(b2)]. An intriguing finding\nof the present work is that the SO coupling dramati-\ncally changed the domain wall from N´ eel wall to Bloch\nwall. Figs. 4(a1) and 4(a2) show the spin density vector\nS=Ψ†σΨ\n|Ψ|2without and with SO coupling. We can see\nthat in the absence of SO coupling, the spin density vec-\ntor across the opposite domains forms a N´ eel wall, while\nin the presence of SO coupling it forms a Bloch wall.\nThis phenomenon can be understood as follows. The\ndirection of the spin flip in the domain wall only depends\non the relative phase, and can be represented by an az-\nimuthal angle α= arctan( Sy/Sx) =θ↓−θ↑, whereθ↑and\nθ↓are the phases of the wave functions. When the SO\ncoupling is absent, there is a constant phase difference 0\norπ[See the solid line of Fig. 4(c)], so the spin in the\nwallsjust flipsalongthe x-directionandformsN´ eelwalls.\nWhen the SO coupling is present, the phase difference is\nchanged into ±π\n2[See the dashed line of Fig. 4(c)], so\nthe spin in the walls just flips along the y-direction and\nforms Bloch walls.\nMeron-pair lattice.— The regular triangular or rectan-\ngularvortexlattice obtained in Fig. 3can be equivalently\ndescribed by the spin density vector Sin the pseudospin\nrepresentation. Fig. 5(a) presents the vectorial plots of S4\n(b1)\n(b2)Néel wall\nBloch wall\n−3π −2π −π 0 π 2π 3π−π/20π/2π\nx [units of ah]θ↑−θ↓ [units of rad](c)\nFIG. 4: (Color online) (a) The vectorial plots of the pseu-\ndospinSprojected onto the x-yplane with ˜ g= 6000, V0=\n40¯hω⊥, andκ= 0 (a1), κ= 4ahω⊥(a2). The colors ranging\nfrom blue to red describe the values of the axial spin Szfrom\n−1 to 1. (b) 3D renderings of N´ eel wall and Bloch wall. (c)\nSection views of the relative phase θ↑−θ↓along the xaxis\nwithκ= 0 (solid line) and κ= 4ahω⊥(dashed line).\nunder apseudo-spin rotation, which correspondingto the\nstate represented in Figs. 3(c1-c6), and the correspond-\ning topological charge density q(r) =1\n8πǫijS·∂iS×∂jS\nis plotted in Fig. 5(b). One can see that the spin tex-\nture in Fig. 5(a) represents a lattice composed of meron\npairs and antimeron pairs [ 36,37]. Either a meron pair\nor an antimeron pair has a “circular-hyperbolic” struc-\nture [See Figs. 5(c1) and 5(c2)], and the only difference\nis that they have exactly opposite spin orientations. The\nspatial integral of q(r) indicates that a meron pair just\ncarries topological charge 1, while an antimeron pair car-\nries topological charge −1. Previous studies indicated\nthat stable meron-pair lattice can be obtained in a rotat-\ning system [ 33]. Our results show that meron-pair lattice\ncan also be stabilized in alternating spin domains by SO\ncoupling without rotation.\nExperimental proposal.— In real experiments, we\ncan choose a two-level87Rb BEC system with\n|F= 1,mf= 1/an}bracketri}htand|F= 1,mf=−1/an}bracketri}ht. About N=\n1.7��105atoms are confined in a harmonic trap with the−π−π/2 0π/2π−π−π/20π/2π\nx [units of ah]y [units of ah](a)\n \n(b)\nx [units of ah]y [units of ah]\n−π−π/2 0π/2ππ\nπ/2\n0\n−π/2\n−π\n(c1) meron pair\n (c2) antimeron pair\n \ncircular antimeron hyperbolic antimeron circular meron hyperbolic meron c2\nc1\nFIG. 5: (Color online) (a) The vectorial plots of the pseu-\ndospinSprojected onto the x-yplane under a pseudo-spin\nrotation σx→ −σzandσz→σx. (b) The topological charge\ndensityq(r). (c) The amplification of the two kinds of ele-\nments in (a).\ntrapping frequencies ω⊥≈2π×40 Hz and ωz≈2π×200\nHz. It is convenient to produce spin-dependent optical\nlattices with a large lattice spacing by using a CO 2laser\noperated at a wavelength of 10.6 µm. Under typical\nexperimental conditions, the s-wave scattering lengths\naij≈100aB(aBis the Bohr radius). Based on these\nexperimental parameters, we can calculate that the effec-\ntive interaction parameter ˜ g≈6000 and the wave vector\nν=a−1\nh, which are consistent with our present calcula-\ntion.\nFor a given SO coupling strength κ= 4/radicalbig\n¯hω⊥/m,\nadjusting the lattice depth from V0=kB×80 nK to\nV0=kB×40 nK, then to V0=kB×30 nK (kBis\nthe Boltzmann’s constant), one can directly observe the\nphase transitions from density stripe to triangular vortex\nlattice, then to rectangle vortex lattice by monitoring in\nsitu the density profile. And for a given lattice depth\nV0=kB×80 nK, adjusting the SO coupling strength\nfromκ= 0 toκ= 4/radicalbig\n¯hω⊥/m, one may indirectly ob-\nserve the transition of the spin domain wall from N´ eel\nwalltoBlochwallbydualstate imagingtechnique, which\ncan spatially resolve the relative phase [ 38].\nConclusion.— In summary, we have investigated the\nground-state phase diagram of spin-orbit coupled BECs\nin spin-dependent optical lattices. We observe novel\nspin-orbit coupling effects on alternating spin domains\nproduced by the spin-dependent optical lattices. Actu-\nally, alternating spin domains exist in many physical sys-\ntems, such as magnetic materials and condensed matter\nsystems [ 39,40], thus similar spin-orbit coupling effects5\nwould be discovered in those systems. We hope that our\nfindings will deepen the understanding of spin-orbit cou-\npling phenomena and provide thoughts on engineering\nnew quantum states in spin-orbit coupled systems.\nWe are grateful to Shih-Chuan Gou for helpful dis-\ncussions. This work was supported by the NKBRSFC\nunder Grants No. 2011CB921502, No. 2012CB821305,\nNo. 2009CB930701, and No. 2010CB922904, NSFC un-\nder Grants No. 10972125 and No. 10934010, NSFC-\nRGC under Grants No. 11061160490 and No. 1386-\nN-HKU748/10, NSFSP under Grant No. 2010011001-2,\nand SFRSP.\n[1] Y. J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Natur e\n(London) 471, 83 (2011).\n[2] P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n[3] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[4] J. Y. Zhang, S. C. Ji, Z. Chen, L. Zhang, Z. D. Du, B.\nYan, G. S. Pan, B. Zhao, Y. J. Deng, H. Zhai, S. Chen,\nand J. W. Pan, Phys. Rev. Lett. 109, 115301 (2012).\n[5] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg,\nRev. Mod. Phys. 83, 1523 (2011).\n[6] A. M. Dudarev, R. B. Diener, I. Carusotto, and Q. Niu,\nPhys. Rev. Lett. 92, 153005 (2004).\n[7] T. D. Stanescu, B. Anderson, andV. Galitski, Phys. Rev.\nA78, 023616 (2008).\n[8] X.F.Zhou, J.Zhou, andC.Wu, Phys.Rev.A 84, 063624\n(2011).\n[9] Z. Cai, X. Zhou, and C. Wu, Phys. Rev. A 85, 061605(R)\n(2012).\n[10] X. Q. Xu and J. H. Han, Phys. Rev. Lett. 107, 200401\n(2011).\n[11] R. Liao, Y. Yi-Xiang, and W. M. Liu, Phys. Rev. Lett.\n108, 080406 (2012).\n[12] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Let t.\n108, 225301 (2012).\n[13] T. Ozawa and G. Baym, Phys. Rev. Lett. 109, 025301\n(2012).\n[14] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi,\nPhys. Rev. Lett. 109, 085302 (2012).\n[15] J. Radi´ c, A. Di Ciolo, K. Sun, and V. Galitski, Phys.\nRev. Lett. 109, 085303 (2012).\n[16] C. Wang, C. Gao, C. M. Jian, and H. Zhai, Phys. Rev.\nLett.105, 160403 (2010).\n[17] T. L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403\n(2011).\n[18] Z. F. Xu, R. L¨ u, and L. You, Phys. Rev. A 83, 053602(2011).\n[19] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107,\n270401 (2011).\n[20] Y. Zhang, L. Mao, and C. Zhang, Phys. Rev. Lett. 108,\n035302 (2012).\n[21] H. Hu, B. Ramachandhran, H. Pu, and X. J. Liu, Phys.\nRev. Lett. 108, 010402 (2012).\n[22] B. Ramachandhran, B. Opanchuk,X. J. Liu, H.Pu, P. D.\nDrummond, and H. Hu, Phys. Rev. A 85, 023606 (2012).\n[23] Y. Deng, J. Cheng, H. Jing, C. P. Sun, and S. Yi, Phys.\nRev. Lett. 108, 125301 (2012).\n[24] S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C.\nGou, Phys. Rev. A 86, 023601 (2012).\n[25] Z. F. Xu, Y. Kawaguchi, L. You, and M. Ueda, Phys.\nRev. A86, 033628 (2012).\n[26] E. Ruokokoski, J. A. M. Huhtam¨ aki, and M. M¨ ott¨ onen,\nPhys. Rev. A 86, 051607(R) (2012).\n[27] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W.\nH¨ ansch, andI.Bloch, Phys.Rev.Lett. 91, 010407 (2003);\nNature (London) 425, 937 (2003).\n[28] D. McKay and B. DeMarco, New J. Phys. 12, 055013\n(2010).\n[29] C. Becker, P. Soltan-Panahi, J. Kronj¨ ager, S. D¨ orsch er,\nK. Bongs and K. Sengstock, New J. Phys. 12, 065025\n(2010).\n[30] P. Soltan-Panahi, J. Struck, P. Hauke, A. Bick, W.\nPlenkers, G. Meineke, C. Becker, P. Windpassinger, M.\nLewenstein, and K. Sengstock, Nature Phys. 7, 434\n(2011).\n[31] P. Soltan-Panahi, D. L¨ uhmann, J. Struck, P. Wind-\npassinger, and K. Sengstock, Nature Phys. 8, 71 (2012).\n[32] P. Hauke, O. Tieleman, A. Celi, C. ¨Olschl¨ ager, J. Si-\nmonet, J. Struck, M. Weinberg, P. Windpassinger, K.\nSengstock, M. Lewenstein, and A. Eckardt, Phys. Rev.\nLett.109, 145301 (2012).\n[33] K. Kasamatsu, M. Tsubota, and M. Ueda, Phys. Rev.\nLett.93, 250406 (2004).\n[34] K. Kasamatsu, M. Tsubota, and M. Ueda, Phys. Rev. A\n67, 033610 (2003).\n[35] B. A. Malomed, H. E. Nistazakis, D. J. Frantzeskakis,\nand P. G. Kevrekidis, Phys. Rev. A 70, 043616 (2004).\n[36] G. E. Volovik, The Universe in a Helium Droplet (Oxford\nUniversity, New York, 2003).\n[37] F. Zhou, Phys. Rev. B 70, 125321 (2004).\n[38] R. P. Anderson, C. Ticknor, A. I. Sidorov, and B. V.\nHall, Phys. Rev. A 80, 023603 (2009).\n[39] B. Vodungbo, J. Gautier, G. Lambert, A. B. Sardinha,\nM. Lozano, S. Sebban, M. Ducousso, W. Boutu, K.\nLi, B. Tudu, M. Tortarolo, R. Hawaldar, R. Delaunay,\nV. L´ opez-Flores, J. Arabski, C. Boeglin, H. Merdji, P.\nZeitoun, and J. L¨ uning, Nature Commun. 3, 999 (2012).\n[40] S. Erdin, I. F. Lyuksyutov, V. L. Pokrovsky, and V. M.\nVinokur, Phys. Rev. Lett. 88, 017001 (2001)." }, { "title": "0710.5316v1.Some_symmetry_properties_of_spin_currents_and_spin_polarizations_in_multi_terminal_mesoscopic_spin_orbit_coupled_systems.pdf", "content": "arXiv:0710.5316v1 [cond-mat.mes-hall] 28 Oct 2007Some symmetry properties of spin currents and spin polariza tions in multi-terminal\nmesoscopic spin-orbit coupled systems\nYongjin Jiang\nDepartment of physics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China\nLiangbin Hu\nDepartment of physics and Laboratory of photonic informati on technology,\nSouth China Normal University, Guangdong 510631, P. R. Chin a\nWe study theoretically some symmetry properties of spin cur rents and spin polarizations in multi-\nterminal mesoscopic spin-orbit coupled systems. Based on a scattering wave function approach, we\nshow rigorously that in the equilibrium state no finite spin p olarizations can exist in a multi-terminal\nmesoscopic spin-orbit coupled system ( both in the leads and in the spin-orbit coupled region )\nand also no finite equilibrium terminal spin currents can exi st. By use of a typical two-terminal\nmesoscopic spin-orbit coupled system as the example, we sho w explicitly that the nonequilibrium\nterminal spin currentsin amulti-terminal mesoscopic spin -orbitcoupled system are non-conservative\nin general. This non-conservation of terminal spin current s is not caused by the use of an improper\ndefinition of spin current but is intrinsic to spin-dependen t transports in mesoscopic spin-orbit\ncoupled systems. We also show that the nonequilibrium later al edge spin accumulation induced by\na longitudinal charge current in a thin strip of finitelength of a two-dimensional electronic system\nwith intrinsic spin-orbit coupling may be non-antisymmetr ic in general, which implies that some\ncautions may need to be taken when attributing the occurrenc e of nonequilibrium lateral edge spin\naccumulation induced by a longitudinal charge current in su ch a system to an intrinsic spin Hall\neffect.\nPACS numbers: 72.25.-b, 73.23.-b, 75.47.-m\nI. INTRODUCTION\nThe efficient generation of finite spin polarizations\nand/or spin-polarized currents in paramagnetic semi-\nconductors by all-electrical means is one of the princi-\npal challengesin semiconductor spin-basedelectronics.1,2\nFor this purpose several interesting ideas have been pro-\nposed based on the spin-orbit ( SO ) interaction char-\nacter of electrons in some semiconductor systems.3,4,5,6\nOne such interesting idea is the so-called intrinsic spin\nHall effect ,5,6which is the generation of a finite spin\ncurrent perpendicular to an applied charge current in a\nparamagnetic semiconductor with intrinsic SO coupling.\nSuch ideas have attracted much theoretical interests\nrecently7−24and substantial achievements do have been\nobtained along these lines,25,26while at the same time\nthey also raised a lot of debates and controversies.8,9,10\nA central problem related to these debates and contro-\nversies is that, what is the correct definition of spin cur-\nrent in a system with strong SO coupling and what is\nthe actual relation between the spin current and the in-\nduced spin-accumulation in such a system.27In most re-\ncent studies the conventional ( standard ) definition of\nspin current ( i.e., the expectation value of the product\nof spin and velocity operators ) have been applied. As\nis well know, this conventional definition of spin current\ncan describe properly spin-polarized transport in a sys-\ntem without intrinsic SO coupling. However, since spin\nis not a conserved quantity in a system with intrinsic\nSO coupling, the physical meanings of spin current cal-\nculated based on the conventional definition are some-what ambiguous and the actual relations between the\nspin current and the induced spin-accumulation are not\nmuch clear. In fact, as has been noticed in several recent\npapers,28,29,30,31there may exist some serious problems\nwith this conventional definition of spin current when\nusing it to describe spin-polarized transport in a system\nwith intrinsic SO coupling. In order to avoid such serious\nproblems, several alternative definitions of spin current\nwere proposed in these papers based on different theoret-\nical considerations, which are significantly different from\nthe conventional one and also significantly different from\neach other.28,29,30,31Another possible way to circumvent\nthis problem is to study a mesoscopic SO coupled system\nattached to external leads. If no SO couplings present\nin the leads or the SO couplings in the leads are much\nweak, then the conventionaldefinition ofspin currentcan\nbe well applied without ambiguities in the leads. Several\nrecent works have adopted this strategy14,15,16,17,18and\nsome interesting results were also obtained. Of course,\nthe study of spin transport in mesoscopic SO coupled\nsystems is not only of theoretical interest but also might\nfind some practical applications in the design of spin-\nbased electronic devices.3\nIn this paper we study theoretically some interesting\nproblems related to spin-dependent transports in multi-\nterminal mesoscopic SO coupled systems. We focus our\nstudy on the symmetry properties of equilibrium and\nnonequilibrium spin currents and spin polarizations in\nsuch mesoscopic structures. As is well known, symme-\ntry analysis is usually of great theoretical importance\nin the study of many physical phenomena, including2\nthe spin-dependent transport phenomena in SO coupled\nsystems.32Based on the analyses of symmetry proper-\nties of equilibrium and nonequilibrium spin currents and\nspin polarizations in multi-terminal mesoscopic SO cou-\npled systems, some controversial issues related to spin-\ndependent transports in mesoscopic SO coupled systems\nwill be investigated in some detail in this paper. Some\nsymmetry properties discussed in this paper might also\nbe helpful for clarifyingsome controversialissues encoun-\nteredin the study ofspin-dependent transportsin macro-\nscopic SO coupled systems. The study carried out in\nthis paper is based on a scattering wave function ap-\nproach within the framework of the standard Landauer-\nB¨ uttiker’s formalism. From the theoretical points of\nview, this scattering wave function approach is in prin-\nciple exactly equivalent to the more frequently employed\nGreen’s function approach in literature33. The main\nmerit of this scatteringwavefunction approachis its con-\nceptual simplicity, and due to its conceptual simplicity,\nsome symmetry properties of equilibrium and nonequi-\nlibrium spin currents and spin polarizations in a multi-\nterminal mesoscopic SO coupled system can be more ex-\nplicitly shown.\nThe paper is organized as follows: In section II we\nwill first give a brief introduction of the structure con-\nsidered and the approach applied. In section III we will\nuse the approach introduced in section II to investigate\nwhether there can exist nonvanishing equilibrium spin\npolarizations or nonvanishing equilibrium terminal spin\ncurrents in a multi-terminal mesoscopic SO coupled sys-\ntem. In sectionIVwewill studythe symmetryproperties\nof nonequilibrium spin polarizations and nonequilibrium\nterminal spin currents in a typical two-terminal meso-\nscopic structure with both Rashba and Dresselhaus SO\ncoupling. Finally in Section V a brief summary of the\nmain conclusions obtained in the paper will be given.\nII. DESCRIPTION OF THE STRUCTURE AND\nTHE SCATTERING WAVE FUNCTION\nAPPROACH\nWeconsiderageneralmulti-terminalmesoscopicstruc-\nture as shown in Fig.1, where a SO coupled mesoscopic\nsystem is attached to several ideal leads. In a discrete\nrepresentation, both the SO coupled region and the ideal\nleads are described by a tight-binding ( TB ) Hamilto-\nnian, and the total Hamiltonian for the entire structure\nreads:\nˆH=Hleads+Hsys+Hs−l. (1)\nHereHleads = Σ pHpandHp=\n−tpΣσ(ˆC†\npiσˆCpjσ+H.C.) is the Hamilto-\nnian for an isolated lead p, withˆCpjσdenoting the\nannihilation operator of electrons with spin index σat a\nlattice site pjin leadpandtpthe hopping parameter\nbetween two nearest-neighbored lattice sites piandpj/s76/s101/s97/s100/s32/s51 /s76/s101/s97/s100/s32/s49/s76/s101/s97/s100/s32/s50\n/s76/s101/s97/s100/s32/s52/s83/s79 /s32/s99/s111/s117/s112/s108/s101/s100\n/s50/s68/s69/s71\nFIG. 1: Schematic geometry of a multi-terminal mesoscopic\nSO coupled system.\nin the lead. We assume that the external leads are ideal\nand nonmagnetic, i.e., no any SO couplings ( or other\nkinds of spin-flip processes rather than that induced by\nthe scatterings from the central SO coupled region )\npresent in the leads. In such ideal cases the standard\ndefinition of spin current can be well applied without\nambiguities in the leads. Usually ( if not specified )\nwe will choose the zaxis ( normal to the 2DEG plane\n) as the quantization axis of spin. Hsys=H0+Hso\nis the Hamiltonian for the isolated SO coupled region,\nin whichHsodescribes the SO coupling of electrons\nandH0=−tΣσ(ˆC†\nriσˆCrjσ+H.C.) describes\nthe spin-independent hopping of electrons between\nnearest-neighbored lattice sites ( denoted by /an}bracketle{tri,rj/an}bracketri}ht) in\nthe region. The discrete version of Hsowill depend on\nthe actual form of the SO interaction, e.g., for the usual\nRashba and k-linear Dresselhaus SO coupling, one has\nHR\nso=−tR/summationdisplay\nri[i(ˆΨ†\nriσxˆΨri+∆y−ˆΨ†\nriσyˆΨri+∆x)+H.C.],\n(2a)\nHD\nso=−tD/summationdisplay\nri[i(ˆΨ†\nriσyˆΨri+∆y−ˆΨ†\nriσxˆΨri+∆x)+H.C.],\n(2b)\nwheretRandtDarethe RashbaandDresselhausSOcou-\npling strength, respectively, ˆΨri= (ˆCri,↑,ˆCri,↓) denotes\nthe spinor annihilation operators, and ∆xand∆yde-\nnote the lattice vectors between two nearest-neighbored\nlattice sites along the xandydirections, respectively.\nThe last term in the Hamiltonian (1) describes the cou-\npling between the leads and the SO coupled region,\nHs−l=−Σptps/summationtext\nn(ˆC†\npnσˆCrnσ+H.C.), wherepndenotes\na boundary lattice site in lead pconnected directly to a\nboundary lattice site rnin the SO coupled region and\ntpsthe hopping parameter between lead pand the SO\ncoupled region. It should be noticed that in general the\nTB Hamiltonian will also contain an on-site energy term,\nwhich is not explicitly shown above.3\nNow we consider the scattering of a conduction elec-\ntron incident on a lead pby the SO coupled region. For\nconveniences, we will adopt a separate local coordinate\nframe in each lead, i.e., we will use a double coordinate\nindex (xp,yp) to denote a lattice site in lead p, where\nxp= 1,2,...,∞( away from the border between the lead\nand the SO coupled region ) and yp= 1,...,Np(Np\nis the width of lead p). In the local coordinate frame,\nthe spatial wave function of a conduction electron inci-\ndent on lead pwill be given by e−ikp\nmxpχp\nm(yp), where\nkp\nmdenotes the longitudinal wave vector and χp\nm(yp)\nthe transverse spatial wave function and mthe label\nof the transverse mode. The longitudinal wave vector\nwill be determined by the following dispersion relation,\n−2tcos(kp\nm) +εp\nm=E, whereεp\nmis the eigen-energy\nof themth transverse mode and Ethe energy of the\nincident electron. It should be noted that, due to the\npresence of SO coupling in the central scattering region,\nspin-flip processes ( e.g., the spin-flip reflection ) will be\ninduced in the leads when a conduction electron is scat-\ntered or reflected by the central scattering region, even if\nthe leads are ideal and nonmagnetic ( which is the just\ncase assumed in the present paper ). Due to this fact, for\na conduction electron incident from the mth transverse\nchannel of lead pand with a givenspin indexσ, both the\nscatteringwavefunction |ψpmσ(r)/an}bracketri}htin the centralSOcou-\npled region and the scattering wave function |ψpmσ(xp′)/an}bracketri}ht\n(xp′≡(xp′,yp′) ) in a lead p′will be inherently a su-\nperposition of a spin-up and a spin-down components,\n|ψpmσ(r)/an}bracketri}ht=/summationdisplay\nσ′ψpmσ\nσ′(r)ˆC†\nri,σ′|0/an}bracketri}ht,(3a)\n|ψpmσ(xp′)/an}bracketri}ht=/summationdisplay\nσ′ψpmσ\nσ′(xp′)ˆC†\nxp′,σ′|0/an}bracketri}ht,(3b)\nwhere|0/an}bracketri}htstands for the vacuum state. The spin-resolved\ncomponents ψpmσ\nσ′(xp′) of the scattering wave function in\nleadp′can be expressed in the following general form,\nψpmσ\nσ′(xp′) =δpp′δσσ′e−ikp\nmxpχp\nm(yp)\n+/summationdisplay\nm′∈p′φpmσ\np′m′σ′eikp′\nm′xp′χp′\nm′(yp′),(4)\nwhereφpmσ\np′m′σ′stands for the scattering amplitude from\nthe (mσ) channel of lead p( theincident mode ) to\nthe (m′σ′) channel of lead p′( theout-going mode ).\nIfp′=p, the second term on the right-hand side of\nEq.(4) will denote actually the spin-resolved reflected\nwaves in lead pandφpmσ\np′m′σ′denote the spin-flip ( σ/ne}ationslash=σ′\n) and non-spin-flip ( σ=σ′) reflection amplitudes. The\nscattering amplitudes φpmσ\np′m′σ′can be obtained by solving\nthe Schr¨ odinger equation for the entire structure. Since\nEq.(4) is just a linear combination of all out-going modes\nwith the same energy Ein leadp′, the Schr¨ odinger equa-\ntion is satisfied automatically in lead p′except at those\nboundary lattice sites in lead p′connected directly to\nthe SO coupled region. Due to the coupling between theleads and the SO coupled region, the amplitudes of the\nwave function at these boundary lattice sites ( which are\ndetermined by the scattering amplitudes φpmσ\nqnσ′) must be\nsolved simultaneously with the wave function ψpmσ(r)\ninside the SO coupled region. From the discrete version\nof the Schr¨ odinger equation, one can show that the am-\nplitudes of the wave function in the SO coupled region\nand at those boundary lattice sites of the external leads\nconnected directly to the SO coupled region will satisfy\nthe following coupled equations:\nEψpmσ\nσ′(rs) =/summationdisplay\nr′\ns,σ′′Hsys(rsσ′,r′\nsσ′′)ψpmσ\nσ′′(r′\ns)\n−/summationdisplay\np′,yp′tp′sδrs,np′y′ψpmσ\nσ′(1,y′\np′),(5a)\nEψpmσ\nσ′(x′\np′) =/summationdisplay\nx′′\np′Hp′(x′\np′σ′,x′′\np′σ′)ψpmσ\nσ′(x′′\np′)\n−/summationdisplay\nrstp′sδrs,np′y′ψpmσ\nσ′(rs),(5b)\nwhere (rs,r′\ns) denote two nearest-neighbored lattice sites\nin the SO coupled region and ( x′\np′,x′′\np′) two nearest-\nneighbored boundary lattice sites in lead p′. ( For sim-\nplicity of notation, in the subscript of the Kronecker\nδ−function we have used simply a symbol np′y′to de-\nnote a boundary lattice site in the SO coupled region\nwhich is connected directly to a boundary lattice site\nx′\np′= (1,y′\np′) in leadp′. ) The matrix elements\nHsys(rsσ′,r′\nsσ′′) andHp′(x′\np′σ′,x′′\np′σ′) can be written\ndown directly from the Hamiltonian (1). Eq.(5a) and\n(5b)arethematchconditionsofthescatteringwavefunc-\ntion on the borders between the SO coupled region and\nthe leads, from which both the scattering wave function\nin the entire structure and all scattering amplitudes can\nbe obtained simultaneously. Some details of the deriva-\ntions are given in the appendix.\nIII. SOME RIGOROUS PROPERTIES OF\nEQUILIBRIUM STATES\nA controversial issue encountered in the study of spin-\npolarized transports in intrinsically SO coupled systems\nis that wether there can exist nonvanishing equilibrium\nbackground spin currents in such systems. Recently\nRashba pointed out that, in a bulk two-dimensional elec-\ntrongas(2DEG)withRashbaSOcoupling,afiniteequi-\nlibrium background spin current could be obtained if the\nconventional definition of spin current is applied to such\nsystems.8If this equilibrium background spin current\ndoes exist, it would imply that nonvanishing equilibrium\nspin polarizationsshould also exist near the edges of such\na system due to the flow of the equilibrium background\nspin current. It was argued in Ref.[8] that such equilib-\nrium background spin currents are an artefact caused by4\nthe improper use of the conventional definition of spin\ncurrent to an intrinsically SO coupled system, i.e., the\nconventional definition of spin current cannot be applied\nin the presence of intrinsic SO coupling. Since it seemed\nthat no consensus had been arrived on whether there is\na uniquely correct definition for spin current in a SO\ncoupled system28,29,30,31, it would be meaningful if this\ncontroversialissue can be investigated in a somewhat dif-\nferent way. In this section we will use the scattering\nwave function approach introduced in section II to inves-\ntigate wether there can exist nonvanishing equilibrium\nbackground spin currents and/or nonvanishing equilib-\nrium spin polarizations in a multi-terminal mesoscopic\nSO coupled system. Based on some simple but rigorous\narguments, we will show that no finite equilibrium spin\npolarizations and/or finite equilibrium terminal spin cur-\nrentscanexistinamulti-terminalmesoscopicSOcoupled\nsystem.\nA. Absence of equilibrium spin polarizations\nIn the tight-binding representation, the operator for\nthe local spin density at a lattice site ireads\nˆ/vectorS(i) =/planckover2pi1\n2/summationdisplay\nαβˆC†\niα/vector σαβˆCiβ. (6)\n( For simplicity of notation, from now on we will use\nsimply a symbol ito denote a lattice site in the en-\ntire structure, i.e., both in the SO coupled region and\nin the external leads. ) Under time reversal transfor-\nmation, the local spin density operator will transform as\nˆ/vectorS(i)→ˆTˆ/vectorS(i)ˆT−1=−ˆ/vectorS(i) and the spin operator trans-\nform as/vector σαβ→ˆT/vector σαβˆT−1= (−1)α+β/vector σ∗\n¯α¯β=−/vector σαβ, where\n¯α≡ −α,¯β≡ −β,ˆT≡iσyˆKdenotes the time-reversal\ntransformation operator and ˆKthe conjugate operator.\nWithin the framework of the standard Landauer-\nB¨ uttiker’s formalism, any physical quantities of a meso-\nscopic system are contributed to by all scattering states\nof conduction electrons incident from all contacts. These\nscattering states constitute an ensemble which can be\nspecified by a chemical potential µpfor each con-\ntact through their separate Fermi distribution function\nf(E,µp). For the problems discussed in the present pa-\nper, this ensemble consists of all scattering states de-\nscribed by the scattering wave functions {ψpmσ}given\nby Eqs.(3-4). In order that there is only one particle\nfeeding into each incident channel33, when we use this\nensemble of scattering wave functions to calculate the\nexpectation value of an operator, one should first nor-\nmalize the scattering wave function ψpmσby a factor of\n1/√\nL(L→ ∞is the length of lead p), correspond-\ning to that one changes the incident wave functions from\neikp\nmxptoeikp\nmxp/√\nL33. By use of the ensemble of the\nnormalized scattering wave functions {ψpmσ}and notic-\ning that the density of states ( DOS ) for the mth trans-\nverse mode of lead pis given byL\n2πdk\ndE=L\n2π/planckover2pi1vpm, wherevpm=2tp\n/planckover2pi1sin(kp\nm) is the longitudinal velocity of the mth\ntransversemode of lead p, then one can see that the local\nspin density at a lattice site i( either in the SO coupled\nregion or in the external leads ) will be given by\n/an}bracketle{tˆ/vectorS(i)/an}bracketri}ht=/summationdisplay\npmσ/integraldisplaydE\n2πf(E,µp)1\n/planckover2pi1vpm\n×/summationdisplay\nα,β[ψpmσ∗\nα(i)(/planckover2pi1\n2/vector σ)αβψpmσ\nβ(i)+H.C.],(7)\nwhereψpmσis the scattering wave function correspond-\ning to an incident electron from the ( mσ) channel of lead\npwith a given energy E.This formula is valid both in\nthe equilibrium and in the nonequilibrium states. If the\nsystem is in an equilibrium state, the chemical potential\nµpwill be independent of the lead label, i.e., µp≡µ\nandf(E,µp)≡f(E,µ). Then in Eq.(7) the summation/summationtext\npmσ[...] can be performed first before carrying out the\nintegration over energy E, and the result for this sum-\nmation can be expressed as\n/summationdisplay\nαβ/summationdisplay\npmσ[ψpmσ∗\nα(i)(/planckover2pi1\n2/vector σ)αβψpmσ\nβ(i)//planckover2pi1vpm+H.C.]\n=/summationdisplay\nαβ[Aβα(i,i;E)(/planckover2pi1\n2/vector σ)αβ+H.C.]\n=i/summationdisplay\nαβ{[GR(E)−GA(E)]iβ,iα(/planckover2pi1\n2/vector σ)αβ−H.C.}.(8)\nHereGR,A(E) = [EI−ˆH±i0+]−1is just the re-\ntarded and advanced Green’s functions for the sys-\ntem, whose explicit spin-resolved matrix forms are given\nbyGR,A\niα,jβ(E) =/summationtext\np′m′σ′ψp′m′σ′∗\nα(i)ψp′m′σ′\nβ(j)/[E−E′±\ni0+]−1(E′is the incident energy correspond-\ning to a scattering wave function ψp′m′σ′); and\nAβα(j,i;E)≡/summationtext\np′m′σ′ψp′m′σ′∗\nα(i)ψp′m′σ′\nβ(j)//planckover2pi1vp′m′=\ni[GR(E)−GA(E)]jβ,iαis the spin-resolved spectral func-\ntion. If the total Hamiltonian ˆHfor the entire system\nis time-reversal invariant, the retarded and advanced\nGreen’s functions can be related by the time reversal\ntransformation as\nGA\niα,jβ= (ˆTGRˆT−1)iα,jβ= (−1)α+βGR∗\ni¯α,j¯β(9)\nCombining Eq.(8) and Eq.(9) and taking into account\nthe fact that ˆT/vector σαβˆT−1= (−1)α+β/vector σ∗\n¯α¯β=−/vector σαβ, one gets\nimmediately that the right-hand side of Eq.(8) should\nvanish exactly, thus the spin density given by Eq.(7) van-\nishes exactly in the equilibrium state, suggesting that no\nfinite equilibrium spin polarizations can survive at any\nlattice siteiin the entire structure ( both in the SO cou-\npled region and in the leads ). It should be noted that in\narriving at this conclusion we have only made use of the\nassumption that the total Hamiltonian ˆHfor the entire\nsystem is time-reversal invariant ( which should be the5\ncase in the absence of magnetic fields ) and did not in-\nvolve the actual form of the SO coupling in the system,\nso it is a much general conclusion.\nB. Absence of equilibrium terminal spin currents\nIn this subsection we discuss whether there can ex-\nist nonvanishing equilibrium terminal spin currents in\na multi-terminal mesoscopic SO coupled system. Since\nwe have assumed that the leads are ideal and nonmag-\nnetic ( i.e., described by a simple Hamiltonian ˆHp=\n−tpΣσ(ˆC†\npiσˆCpjσ+H.C.) ), the conventional defi-\nnitions of charge and spin currents can be well applied in\nthe leads without ambiguities. According to the conven-\ntional definitions and in the lattice representation, the\ncharge current and spin current ( with spin parallel to\ntheαaxis34) flowing from a lattice site pito a nearest-\nneighbored lattice site pjin leadpcan be given by the\nthe corresponding particle density current as following,\nˆIp,pi→pj=e[ˆJ+\npi→pj+ˆJ−\npi→pj],(10a)\nˆIα\np,pi→pj=/planckover2pi1\n2[ˆJ+\npi→pj−ˆJ−\npi→pj],(10b)\nwhereˆIp,pi→pjdenotes the charge current operator and\nˆIα\np,pi→pjthe spin current operator ( with spin parallel to\ntheαaxis ) and ˆJσ\npi→pjthe spin-resolved particle den-\nsity current operator and σ=±denotes the spin-up\nand spin-down states with respect to the αaxis. From\nthe Heisenberg equationof motion for the on-site particle\ndensity:d\ndtˆNpi=1\ni/planckover2pi1[ˆNpi,ˆHp], where ˆNpi=ˆC†\npiσˆCpiσis\nthethe on-siteparticledensityoperatorinlead p, onecan\nshoweasilythatthespin-resolvedparticledensitycurrent\nflowing from a lattice site pito a nearest-neighbored lat-\nticepjin leadpwill be given by\nˆJσ\npi→pj=itp\n/planckover2pi1(ˆC†\npjσˆCpiσ−ˆC†\npiσˆCpjσ).(11)\nNow we calculate the terminal charge and spin cur-\nrents flowing along the longitudinal direction of a lead q.\nFirstly we consider the contribution of an incident elec-\ntron from the ( mσ) channel of lead pto the longitudinal\nchargeand spin currents( with spin parallelto the αaxis\n) flowing through a transversecross-section( saying, e.g.,\nthe cross-sectionat x=xq) ofleadq, which bydefinition\nwill be given by\n/an}bracketle{tˆIq/an}bracketri}htpmσ=1\nL/summationdisplay\nyq/an}bracketle{tψpmσ(xq+1,yq)|\n׈Iq,(xq,yq)→(xq+1,yq)|ψpmσ(xq,yq)/an}bracketri}ht\n=e\nL\n\n/summationdisplay\nn,σ′vqn|φpmσ\nqnσ′|2−vpmδpq\n\n,(12a)/an}bracketle{tˆIα\nq/an}bracketri}htpmσ=1\nL/summationdisplay\nyq/an}bracketle{tψpmσ(xq+1,yq)|\n׈Iα\nq,(xq,yq)→(xq+1,yq)|ψpmσ(xq,yq)/an}bracketri}ht\n=h\n4πL/braceleftBigg/summationdisplay\nnvqn/bracketleftbig\n|φpmσ\nqn+|2−|φpmσ\nqn−|2/bracketrightbig\n−σvpmδpq/bracerightBigg\n,\n(12b)\nwhere (xq,yq) and (xq+ 1,yq) denote two nearest-\nneighbored lattice sites along the longitudinal direction\nof leadq,1√\nL|ψpmσ(xq,yq)/an}bracketri}htdenotes the normalized scat-\ntering wave function in lead qcorresponding to the in-\ncident electron from the ( mσ) channel of lead p( which\nis given by Eqs.(3–4) ), and vqn=2tp\n/planckover2pi1sin(kq\nn) denotes\nthe longitudinal velocity of the nth transverse mode in\nleadqandvpm=2tp\n/planckover2pi1sin(kp\nm) the longitudinal velocity\nof themth transverse mode in lead p. The summation\nover the transverse coordinate yqruns over from 1 to Nq\n(Nqis the width of lead q)35, and the following or-\nthogonality relations for transverse modes in lead qhave\nbeen applied in obtaining the last lines of Eq.(12a) and\n(12b):/summationtext\nyqχq\nm(yq)χq\nn(yq) =δmn. It should be noted\nthat, ifp=q, the results given by Eq.(12a) and (12b)\nwill denote actually the contribution of an incident elec-\ntron from lead qto the charge and spin currents flowing\nin the same lead and φpmσ\nqnσ′(σ′=±) denote actually\nthe spin-flip ( σ/ne}ationslash=σ′) and non-spin-flip ( σ=σ′)\nreflection amplitudes. ( See the explanations given to\nEqs.(3-4) in section II ). In such cases, the results given\nby Eq.(12a) and (12b) can be expressed as the subtrac-\ntion of the contributions due to the incident wave ( i.e.,\nthe terms proportional to δpqin Eq.(12a) and (12b) )\nand the contributions due to the spin-flip and non-spin-\nflip reflected waves.\nThe total terminal charge current Iqand the total ter-\nminal spin current Iα\nqflowing in lead qwill be obtained\nby summing the contributions of all incident electrons\nfrom all leads with the corresponding density of states (\nsee the explanations given above Eq.(7) ). Then we get\nthat\nIq=e\nh/summationdisplay\npmσ/integraldisplay\ndEf(E,µp)[/summationdisplay\nn,σ′|φpmσ\nqnσ′|2vqn\nvpm−δpq]\n=e\nh/summationdisplay\npσσ′/integraldisplay\ndEf(E,µp)[Tpσ\nqσ′(E)−δpqδσσ′Nq(E)]\n=e\nh/summationdisplay\npσσ′/integraldisplay\ndE[f(E,µp)Tpσ\nqσ′(E)−f(E,µq)Tqσ\npσ′(E)],\n(13a)\nIα\nq=/summationdisplay\npmσ/integraldisplay\ndEf(E,µp)[/summationdisplay\nnvqn\n4πvpm(|φpmσ\nqn+|2−|φpmσ\nqn−|2)]\n=1\n4π/summationdisplay\npσ/integraldisplay\ndEf(E,µp)[Tpσ\nq+(E)−Tpσ\nq−(E)],(13b)6\nwhereTpσ\nqσ′(E)≡/summationtext\nm,n/vextendsingle/vextendsingle/vextendsingleφpmσ\nqnσ′/vextendsingle/vextendsingle/vextendsingle2vqn\nvpmdenotes ( by definition\n) the transmission probability from lead pwith spinσto\nleadqwithspinσ′(seeRef.[33]andalsotheexplanations\ngiven in the appendix A ), and in obtaining the last line\nof Eq.(13a) the following relation has been applied33:\n/summationdisplay\npσTpσ\nqσ′(E) =/summationdisplay\npσTpσ\nq¯σ′(E) =Nq(E),(14)\nwhereNq(E) is the total number of conducting trans-\nverse modes in lead qcorresponding to a given energy\nE. As was discussed in detail in Ref.[33], this relation\nfollows directly from the unitarity of the S-matrix, which\nis essential for the particle number conservation.\nEq.(13a) is just the usual Landauer-B¨ uttiker formula\nfor terminal charge currents in a multi-terminal meso-\nscopic system33. The second line in Eq.(13a) indicates\nclearly that the terminal charge current flowing in lead q\ncan be expressed as the subtraction of the contributions\ndue to all incident modes ( corresponding to the terms\nproportionalto δpq) and the contributionsdue to all out-\ngoing modes ( corresponding to the terms proportional\ntoTpσ\nqσ′), which include both the transmitted waves from\nother leads ( p/ne}ationslash=q) and the reflected waves in lead q\n(p=q), noticing that Tpσ\nqσ′denotes actually the spin-\nflip or non-spin-flip reflection probabilities from lead q\nto leadqifp=q. Eq.(13b) is somewhat different from\nEq.(13a) in appearance, but the terminal spin current\ngiven by Eq.(13b) can still be divided into two different\nkinds of contributions, namely the contributions due to\nthe transmitted waves from other leads ( corresponding\nto those terms with p/ne}ationslash=qin the summation/summationtext\npσ[...] )\nand the contributions due to the spin-flip and non-spin-\nflip reflected waves in lead q( corresponding to those\nterms with p=qin the summation/summationtext\npσ[...] ).36\nEq.(13a) and (13b) are valid both in the equilibrium\nand in the nonequilibrium states. In the equilibrium\nstate, since µp≡µ( independent of the lead label )\nandf(E,µp)≡f(E,µ), in Eq.(13a) and (13b) the sum-\nmation/summationtext\npσ[...] can be performed first before carrying\nout the energy integration and we get that\nIq=e\nh/integraldisplay\ndEf(E,µ)/summationdisplay\npσσ′[Tpσ\nqσ′(E)−Tqσ\npσ′(E)],(15a)\nIα\nq=1\n4π/integraldisplay\ndEf(E,µ)/summationdisplay\npσ[Tpσ\nq+(E)−Tpσ\nq−(E)].(15b)\nThen by use of Eq.(14) one can see clearly that both ter-\nminalchargecurrentsandterminalspincurrentswillvan-\nish exactly in the equilibrium state. It should be stressed\nthat in arriving at this conclusion we did not involve\nthe controversial issue of what is the correct definition of\nspin current in the central SO coupled region at all, so\nthose ambiguities that might be caused by the use of an\nimproper definition of spin current to the SO coupled re-\ngion have been eliminated in our derivations. Though wecannot prove that the spin current also vanishes exactly\ninside the SO coupled region based on the approach ap-\nplied above, however, for a mesoscopic system only the\nterminal ( charge or spin ) currents are the real useful\nquantities from the practical point of view ( i.e., one\nneed to add external contacts to induct the charge or\nspin currents out of a mesoscopic sample ). We note\nthat a similar conclusion as was obtained above has also\nbeen derived in Ref.[37] based on some somewhat differ-\nent arguments. Compared with the derivations given in\nRef.[37], theargumentsgivenaboveseemto be moresim-\nple and more transparent in principle. It also should be\nnoted that, based on a similar Landauer-B¨ uttikerformal-\nism, it was argued in Ref.[38] that the equilibrium termi-\nnal spin currents should indeed take place in a three ter-\nminal system with spin-orbit coupling38, in contradiction\nto the conclusion obtained in the present paper and in\nRef.[37]. To our understandings, this contradiction was\ncaused by the fact that the contributions due to the spin-\nflip and non-spin-flip reflections in the leads ( induced by\nthe scatterings from the central SO coupled region ) was\nneglected in the calculations of terminal spin currents\nperformed in Ref.[38]. In contrast, in the calculations of\nterminal spin currents performed in the present paper,\nthe contributions due to the spin-flip and non-spin-flip\nreflections in the leads induced by the scatterings from\nthe central SO coupled regionhavebeen treated in an ac-\ncurate and strict way, assuming that the leads are ideal\nand nonmagnetic.\nIV. SOME SYMMETRY PROPERTIES OF\nNONEQUILIBRIUM SPIN CURRENTS AND\nSPIN POLARIZATIONS IN TWO-TERMINAL\nMESOSCOPIC SO COUPLED SYSTEMS\nWhen a multi-terminal mesoscopic SO coupled system\nis driven into a nonequilibrium state ( i.e., there is charge\ncurrent flow between different leads ), nonequilibrium\nspin polarizations and/or terminal spin currents may be\ninduced by the charge current flow. In this section we\ndiscusssomesymmetrypropertiesofsuchnonequilibrium\nspin currents and spin polarizations. For clarity, we take\na typical two-terminal mesoscopic structure as shown in\nFig.??as the example, where a ballistic two-dimensional\nelectron gas ( 2DEG ) with Rashba and/or k-linear Dres-\nselhaus SO coupling is attached to two ideal leads.\n/s108/s101/s97/s100/s32/s50\n/s50/s68/s69/s71/s108/s101/s97/s100/s32/s49\n/s120/s121\n/s111\nFIG. 2: Schematic geometry of a two-terminal mesoscopic SO\ncoupled system.7\nA. Non-antisymmetric lateral edge spin\naccumulations\nThe study of nonequilibrium lateral edge spin accu-\nmulation induced by a longitudinal charge current in a\nthin strip of a two-dimensionalelectron gas with intrinsic\nSO coupling is of great theoretical interest because of its\nclose relations with the intrinsic spin Hall effect in such\nsystems. It was generally believed that the principal ob-\nservable signature of the intrinsic spin Hall effect in a SO\ncoupled system is that, when a longitudinal charge cur-\nrent circulates through such a system with a thin strip\ngeometry, antisymmetric lateral edge spin accumulation\n( polarized perpendicular to the 2DEG plane ) will be\ninduced at the two lateral edges of the strip due to the\nflow of the transverse spin Hall current. Several recent\nnumerical calculations had demonstrated that a longitu-\ndinal charge current circulating through a thin strip of\na ballistic two-dimensional electron gas with Rashba SO\ncoupling does can lead to the generation of antisymmet-\nric edge spin accumulation at the two lateral edges of the\nstrip, and the antisymmetric character of the transverse\nspatial distribution of the lateral edge spin accumulation\n( i.e.,/an}bracketle{tSz(x,y)/an}bracketri}ht=−/an}bracketle{tSz(x,−y)/an}bracketri}ht) had been argued to be\na strong support of the existence of intrinsic spin Hall\neffect in such mesoscopic SO coupled systems15. Here\nwe discuss this issue from a different point of view. We\nwill show that, when a longitudinal charge current circu-\nlates through a thin strip of a ballistic two-dimensional\nelectron gas with both Rashba and Dresselhaus SO cou-\npling, the transverse spatial distribution of the induced\nnonequilibrium lateral edge spin accumulation ( polar-\nized perpendicular to the 2DEG plane ) are in general\nnon-antisymmetric. The non-antisymmetric character\nof the lateral edge spin accumulation contradicts seri-\nously with the usual physical pictures of spin Hall effect,\nthough according to some theoretical predictions, intrin-\nsic spin Hall effect should also survive in the presence of\nboth Rashba and Dresselhaus SO coupling12. The non-\nantisymmetric character of the lateral edge spin accumu-\nlation implies that, in addition to the intrinsic spin Hall\neffect, there may exist some other physical reasons that\nmight also lead to the generation of nonequilibrium lat-\neral edge spin accumulation in a SO coupled system (\nwith a thin strip geometry ) when a longitudinal charge\ncurrent circulates through it.23,24\nFirstly let us look at what symmetry relations can be\nobtained for the nonequilibrium lateral edge spin accu-\nmulation induced by a longitudinal charge current based\non the symmetry analysis of the Hamiltonian of the sys-\ntem under study ( sketched in Fig.2 ). If only Rashba (\nor only Dresselhaus ) SO coupling presents, based on the\nsymmetry properties of the Hamiltonian of the structure\nunder study, one can show rigorous that the nonequi-\nlibrium lateral edge spin accumulation does should be\nantisymmetric at the two lateral edges. Let us consider\nfirst the case in which only Rashba SO coupling presents\n( i.e., the Dresselhaus SO coupling strength is zero ).If only Rashba SO coupling presents, from Eq.(2a) and\nFig.2 one can see that the Hamiltonian of the entire\nstructure is invariant under the combined transforma-\ntion of the real space reflection y⇒ −yand the spin\nspace rotation around the Syaxis ( with an angle π\n). From this invariance one can get immediately that\n/an}bracketle{tSz(x,y)/an}bracketri}htI=−/an}bracketle{tSz(x,−y)/an}bracketri}htI, where /an}bracketle{tSz/an}bracketri}htIdenotes the\nnonequilibrium spin density induced by a longitudinal\ncharge current flowing from lead 1 to lead 2. It is inter-\nesting to note that this antisymmetric relation can be de-\nduced directly from the symmetry of the structure under\nstudy but without need to resort to the concept of spin\nHall effect at all. Next, let us consider the case in which\nonly Dresselhaus SO coupling presents ( i.e., the Rashba\nSO coupling strength is zero ). If only Dresselhaus SO\ncoupling presents, then from Eq.(2b) and Fig.2 one can\nsee that the Hamiltonian of the entire structure is invari-\nant under the combined transformation of the real space\nreflectiony⇒ −yand the spin space rotation around the\nSxaxis ( with an angle π). From this invariance one also\ngets immediately that /an}bracketle{tSz(x,y)/an}bracketri}htI=−/an}bracketle{tSz(x,−y)/an}bracketri}htI, i.e.,\nthe nonequilibrium lateral edge spin accumulation still\nshould be antisymmetric at the two lateral edges if only\nDresselhaus SO coupling presents.\nIf both Rashba and Dresselhaus SO couplings are\npresent, then fromEqs.(2a-2b)andFig.2onecanseethat\nthe total Hamiltonian of the entire structure is invariant\nunder the combined transformationof the real space cen-\nter inversion r⇒ −rand the spin space rotation around\ntheSzaxis ( with an angle π). From this invariance\none can get that /an}bracketle{tSz(x,y)/an}bracketri}htI=/an}bracketle{tSz(−x,−y)/an}bracketri}ht−I, where\n/an}bracketle{tSz/an}bracketri}ht−Idenotes the nonequilibrium spin density induced\nby a longitudinal charge current flowing from lead 2 to\nlead 1. On the other hand, from Eq.(7) one can see that\nin the linear response regime one has\n/an}bracketle{tSz(x,y)/an}bracketri}htI=−/an}bracketle{tSz(x,y)/an}bracketri}ht−I. (16)\nCombining the two relations /an}bracketle{tSz(x,y)/an}bracketri}htI=\n/an}bracketle{tSz(−x,−y)/an}bracketri}ht−Iand/an}bracketle{tSz(x,y)/an}bracketri}htI=−/an}bracketle{tSz(x,y)/an}bracketri}ht−I,\nthen the following symmetry relation can be obtained\nfor the nonequilibrium spin accumulation induced by\na longitudinal charge current flowing from lead 1 to\nlead 2:/an}bracketle{tSz(x,y)/an}bracketri}htI=−/an}bracketle{tSz(−x,−y)/an}bracketri}htI. This symmetry\nrelation implies that the transverse spatial distribution\nof the nonequilibrium lateral edge spin accumulation\nwill be antisymmetric in the center cross-section of the\nstrip, i.e., /an}bracketle{tSz(0,y)/an}bracketri}htI=−/an}bracketle{tSz(0,−y)/an}bracketri}htI. Due to the\nexistence of this symmetry relation, one can deduce that\nin aninfinitestrip ( i.e, the length of the strip tends to\ninfinity and hence the effects of the contacted leads can\nbe neglected ), the transverse spatial distribution of the\nnonequilibrium lateral edge spin accumulation will still\nbe antisymmetric ( i.e., /an}bracketle{tSz(x,y)/an}bracketri}htI=−/an}bracketle{tSz(x,−y)/an}bracketri}htIfor\nallx) in the presence of both Rashba and Dresselhaus\nSO coupling. However, unlike the case in which only\nRashba ( or only Dresselhaus ) SO coupling presents,\nfor a thin strip of finite length, in the presence of both\nRashba and Dresselhaus SO coupling, one cannot de-8\nduce a general antisymmetric relation for the transverse\nspatial distribution of the nonequilibrium lateral edge\nspin accumulation based on the symmetry analysis of\nthe total Hamiltonian of the entire structure under\nstudy. From the theoretical points of view, this is\ndue to the fact that, in the presence of both Rashba\nand Dresselhaus SO coupling, the total Hamiltonian of\nthe entire structure under study ( i.e., a thin strip of\nfinite length contacted to two ideal leads ) is no longer\ninvariant under the combined transformation of the real\nspace reflection y⇒ −yand the spin space rotation\naround the Sy( orSx) axis with an angle π. Indeed, as\nwill be shown below, this non-antisymmetric character\ncan be verified by detailed numerical calculations.\nOne particular interesting case is that the Rashba and\nthe Dresselhaus SO coupling strengths are equal ( i.e.,\ntR=tDortR=−tD). In this particular case, the\ntotal Hamiltonian is invariant under the following uni-\ntary transformation in spin space ( while the real space\ncoordinate rremain unchanged ): ˆU+ˆHˆU+=ˆH( if\ntR=tD) orˆU−ˆHˆU−=ˆH( iftR=−tD), where\nˆU+= (ˆσx+ˆσy)/√\n2 andˆU−= (ˆσx−ˆσy)/√\n2. Under this\nunitary transformation, the spin operatorswill transform\nas following: ˆ σz→ −ˆσz, ˆσx⇋ˆσy( iftR=tD) or\nˆσx⇋−ˆσy( iftR=−tD). Since the real space coordi-\nnaterremain unchanged under this symmetry manipu-\nlation, from the above symmetry properties in spin space\none gets immediately that /an}bracketle{tSz(x,y)/an}bracketri}htI=−/an}bracketle{tSz(x,y)/an}bracketri}htI,\nsuggesting that /an}bracketle{tSz(x,y)/an}bracketri}htIshould vanish everywhere if\nthe Rashba and the Dresselhaus SO coupling strengths\nare equal. This conclusion is in exact agreement with\nthe corresponding numerical results obtained based on\nthe scattering wave approach introduced in section II-\nIII, which shows that /an}bracketle{tSz(x,y)/an}bracketri}htIdoes vanish everywhere\nin the particular case of tR=tDortR=−tD.\nToshowmoreexplicitlythenon-antisymmetriccharac-\nterofthelateraledgespinaccumulationinducedbyalon-\ngitudinal charge current in a 2DEG strip of finite length\nwith both Rashba and Dresselhaus SO coupling, in Fig.3\nwe plotted a typical pattern of the two-dimensional spa-\ntial distribution of the nonequilibrium spin density /an}bracketle{tSz/an}bracketri}ht\nin the strip obtained by numerical calculations with the\nscattering wave function approach introduced in Sec.II-\nIII. In our numerical calculations we take the typical val-\nues of the electron effective mass m= 0.04me, the lattice\nconstanta= 3nm, and the 2DEG strip contains 120 ×40\nlattice sizes. The chemical potentials in the two leads\nare set by fixing the longitudinal charge current to flow\nfrom lead 1 to lead 2 as shown in Fig.2 and fixing the\nlongitudinal charge current density to 100 µA/1.5µm( as\nreportedin Ref.[26]). FromFig.3onecanseeclearlythat\nthe transverse spatial distribution of the nonequilibrium\nspindensity /an}bracketle{tSz/an}bracketri}htinthestripisnon-antisymmetricingen-\neral( i.e., /an}bracketle{tSz(x,y)/an}bracketri}htI/ne}ationslash=−/an}bracketle{tSz(x,−y)/an}bracketri}htIforgeneralx), ex-\ncept in the center cross-section ( i.e., x= 0 ) of the strip.\nThe non-antisymmetric character of the transverse spa-\ntial distribution of the nonequilibrium spin density /an}bracketle{tSz/an}bracketri}ht\ncan be more clearly seen from Fig.4(a), where we plotted−1.5 −1 −0.5 0 0.5 1 1.5\nx 10−3\n−40−30 −20 −10 0 10 20 30 40 50−15 −10−505101520〈 SZ(x,y) 〉I (h/4π)\n x/ay/a \nFIG. 3: (Color online)A typical pattern of the two-\ndimensional spatial distribution of the current induced\nnonequilibrium spin density /angbracketleftSz/angbracketrightin a two-terminal structure (\nsketchedinFig.2)inthepresenceofbothRashbaandDressel -\nhaus SO coupling. The Rashba and Dresselhaus SO coupling\nstrength is set to tR/t= 0.08 andtD/t= 0.02.\nseveral typical patterns of the profiles of the transverse\nspatial distributions of the nonequilibrium spin density\n/an}bracketle{tSz/an}bracketri}htin a cross-section of the strip at x/ne}ationslash= 0. ( For\ncomparison, the corresponding results obtained in the\ncase that only Rashba or only Dresselhaus SO coupling\npresents were also plotted in Fig.4(b) ). The three typi-\ncal patterns shown in Fig.4(a) are obtained by fixing the\nDresselhaus SO coupling strength to tD= 0.02t(tis the\nspin-independent hopping parameter ) and varying the\nRashba SO coupling strength tR. From Fig.4(a) one can\nsee clearly that, the transverse spatial distribution of the\nnonequilibrium spin density /an}bracketle{tSz(x,y)/an}bracketri}htIcan have either\nthe same signs or opposite signs at the two lateral edges\nofthestrip,dependingontheratiosof tR/tD. Eveninthe\ncase that /an}bracketle{tSz/an}bracketri}hthas opposite signs at the two lateral edges,\nthe transverse spatial distributions of /an}bracketle{tSz/an}bracketri}htmay still not\nbe antisymmetric ( i.e., /an}bracketle{tSz(x,y)/an}bracketri}ht /ne}ationslash=−/an}bracketle{tSz(x,−y)/an}bracketri}ht), con-\ntradictingsignificantlywiththe usualphysicalpicturesof\nspin Hall effect. The non-antisymmetric character of the\nlateral edge spin accumulation suggests that some cau-\ntionsmayneedtobetakenwhenattributingthenonequi-\nlibrium lateral edge spin accumulation induced by a lon-\ngitudinal charge current in a thin strip of a SO coupled\nsystem to a spin Hall effect, especially in the mesoscopic\nregime.\nThe results shown in Figs.3-4 are obtained in the ab-\nsence of impurity scatterings. One can show that the\nsymmetry properties shown in Fig.3-4 are robust against\nspinless weak impurity scatterings. To model spinless\nweak disorder scatterings, we assume that the on-site\nenergy at lattice sites in the 2DEG strip are randomly\ndistributed in a narrow energy region [ −W,W], where9\n/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s48/s46/s48/s48/s49/s53/s45/s48/s46/s48/s48/s49/s48/s45/s48/s46/s48/s48/s48/s53/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s53/s48/s46/s48/s48/s49/s48/s48/s46/s48/s48/s49/s53\n/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s48/s46/s48/s48/s48/s56/s45/s48/s46/s48/s48/s48/s54/s45/s48/s46/s48/s48/s48/s52/s45/s48/s46/s48/s48/s48/s50/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s50/s48/s46/s48/s48/s48/s52/s48/s46/s48/s48/s48/s54/s48/s46/s48/s48/s48/s56/s60/s83\n/s90/s40/s120/s61/s49/s48/s44/s121/s41/s62\n/s73/s40/s104/s47/s52 /s41\n/s40/s98/s41\n/s121 /s47/s97/s121/s47/s97/s60/s83\n/s90/s40/s120/s61/s49/s48/s44/s121/s41/s62\n/s73/s40/s104/s47/s52 /s41/s32/s116\n/s82/s61/s48/s46/s48/s50/s116/s44/s32/s116\n/s68/s61/s48/s46/s48/s49/s116\n/s32/s116\n/s82/s61/s48/s46/s48/s50/s116/s44/s32/s116\n/s68/s61/s48/s46/s48/s52/s116\n/s32/s116\n/s82/s61/s48/s46/s48/s50/s116/s44/s32/s116\n/s68/s61/s48/s46/s48/s56/s116/s40/s97/s41\n/s32/s116\n/s82/s61/s48/s44/s116\n/s68/s61/s48/s46/s48/s50/s116\n/s32/s32/s116\n/s82/s61/s48/s46/s48/s50/s116/s44/s116\n/s68/s61/s48\nFIG. 4: (Color online)(a) Some typical profiles of the trans-\nverse spatial distributions of the nonequilibrium spin den sity\n/angbracketleftSz/angbracketrightinduced by a longitudinal charge current in a 2DEG strip\nwith bothRashbaandDresselhaus SOcouplings, whichshows\nclearly that the transverse spatial distributions of /angbracketleftSz/angbracketrightare\nnon-antisymmetric in general at both edges of the strip in\nthe presence of both Rashba and Dresselhaus SO coupling.\n/angbracketleftSz/angbracketrightvanishes everywhere in the particular case of tR=tDor\ntR=−tD(notshownexplicitlyinthefigure). (b)Thecorre-\nspondingresults obtained inthecase thatonly Rashbaoronl y\nDresselhaus SO coupling presents, which shows clearly that\nthe transverse spatial distributions of /angbracketleftSz/angbracketrightare antisymmetric\nat both edges of the strip if only Rashba or only Dresselhaus\nSO coupling presents.\nWis the amplitude of the on-site energy fluctuations\ncharacterizing the disorder strength. ( In the absence\nof disorder scatterings, the on-site energy at each lattice\nsite can be set simply to zero. ) We calculate the spin\ndensity for a number of random impurity configurations\nand then do impurity average. In Fig.5 we show the\nvariations of the transverse spatial distributions of the\nnonequilibrium spin density /an}bracketle{tSz/an}bracketri}htin a cross-section of the\nstrip as the disorder strength increases, from which one\ncan see that the symmetry properties of the transverse\nspatial distributions of the nonequilibrium spin density\nare robust against spinless weak disorder scatterings.\n/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s54/s45/s48/s46/s48/s48/s52/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s60/s83\n/s90/s40/s120/s61/s49/s48/s44/s121/s41/s62\n/s73/s40/s104/s47/s52 /s41\n/s121/s47/s97/s32/s87 /s61/s48/s46/s50/s116\n/s32/s87 /s61/s48/s46/s52/s116\n/s32/s87 /s61/s48/s46/s56/s116/s116\n/s82/s47/s116/s61/s48/s46/s48/s56/s44/s32/s116\n/s68/s47/s116/s61/s48/s46/s48/s50\nFIG. 5: (Color online)The profiles of the transverse spatial\ndistributions of the nonequilibrium spin density /angbracketleftSz/angbracketrightin the\npresence of disorder. We have done impurity average over\n1000 random impurity configurations for each case.B. Non-conservative terminal spin currents\nAs has been mentioned in the introduction section, a\nmuch controversial issue related to the study of spin-\npolarized transports in SO coupled systems is that\nwhether spin currents should be a conserved quantity\nand what is the correct definition of spin currents in such\nsystems. It was now well established that, in the pres-\nence of SO coupling, spin current calculated based on\nthe conventional definition is not a conserved quantity,\nand in order to make spin current a conserved quantity\nin the presence of SO coupling, significant modifications\nto the conventional definition will be needed.28,30Never-\ntheless, it seemed that no consensus had been arrived on\nwhether spin current should be a conserved quantity in\na SO coupled system or whether there is a uniquely cor-\nrect definition for spin current in such a system.28,29,30,31\nBelow we will discuss this controversial issue from a dif-\nferent point of view, i.e., we do not consider the problem\nthat what is the correct definition of spin current in a\nSO coupled system but focus our discussion on the ques-\ntion that whether the terminal spin currents in a multi-\nterminal mesoscopicSO coupled system are conservative.\nAs mentioned earlier, for a mesoscopic SO coupled sys-\ntem, only the terminal spin currents are the real useful\nquantities. By use of the two-probe mesoscopic struc-\nture shown in Fig.2 as the example, we will show explic-\nitly that the terminal spin currents in a multi-terminal\nmesoscopic SO coupled system are non-conservative in\ngeneral, i.e., the total spin currents flowing into the SO\ncoupled region are not equal to the total spin currents\nflowing out of the same region. To illustrate this point\nclearly, we take a two-terminal mesoscopic system with\nboth Rashba and Dresselhaus SO coupling as the exam-\nple. In Fig.6(a) and (b) we plotted the terminal spin\ncurrentsIz\n1andIz\n2in the two leads ( with spin parallel\nto thezaxis) andthe terminalspin currents Iy\n1andIy\n2in\nthe two leads ( with spin parallelto the yaxis ) as a func-\ntion of the Rashba SO coupling strength tR, respectively.\nIn our calculations we fix the Dresselhaus SO coupling\nstrength to tD= 0.02tand fix the longitudinal charge\ncurrent density to 100 µA/1.5µm. The lattice constant\na= 3nmand the lattice size of the 2DEG strip is taken\nto be 100 ×40. The positive direction of the spin current\nflow is defined to be from lead 1 to lead 2. From Fig.6(a)\none can see that the terminal spin currents Iz\n1andIz\n2in\nthe two leads have the same signs, which means that the\nterminal spin currentswith spin parallel to the zaxis will\nflowfrom lead 1 into the SO coupled regionand then flow\nout of the SO coupled region into lead 2, similar to the\nusual charge current transport. From Fig.6(b) one can\nsee that the terminal spin currents Iy\n1andIy\n2in the two\nleads have opposite signs, which means that the terminal\nspin currents with spin parallel to the yaxis will flow\nout of the SO coupled region in both leads and hence\nare non-conservative ( i.e., the spin current flowing into\nthe SO coupled region does not equal to the spin current\nflowing out of the same region ). Similarly one can show10\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s48/s46/s48/s48/s50/s48/s45/s48/s46/s48/s48/s49/s53/s45/s48/s46/s48/s48/s49/s48/s45/s48/s46/s48/s48/s48/s53/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s53/s48/s46/s48/s48/s49/s48/s48/s46/s48/s48/s49/s53/s48/s46/s48/s48/s50/s48/s48/s46/s48/s48/s50/s53\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s48/s46/s48/s49/s48/s45/s48/s46/s48/s48/s56/s45/s48/s46/s48/s48/s54/s45/s48/s46/s48/s48/s52/s45/s48/s46/s48/s48/s50/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s32/s84/s101/s114/s109/s105/s110/s97/s108/s32/s115/s112/s105/s110/s32/s99/s117/s114/s114/s101/s110/s116/s32/s47/s98/s105/s97/s115/s32/s118/s111/s108/s116/s97/s103/s101/s32/s32 /s40/s101/s47/s52 /s32/s32/s122\n/s49/s47/s32/s86\n/s32/s45/s122\n/s50/s47/s32/s86/s40/s97/s41\n/s40/s98/s41\n/s116\n/s82/s47/s116/s116\n/s82/s47/s116\n/s32/s121 \n/s49/s47/s32/s86\n/s32/s121 \n/s50/s47/s32/s86\nFIG. 6: (Color online)(a) The terminal spin currents Iz\n1and\n−Iz\n2( divided by the voltage ) as a function of the Rashba SO\ncoupling strenght tR( in units of t). (b) The terminal spin\ncurrents Iy\n1andIy\n2( divided by the voltage ) as a function of\nthe Rashba SO coupling strenght tR. The figures show that\nthe terminal spin currents Iz\n1andIz\n2in the two leads have the\nsame signs and the terminal spin currents Iy\n1andIy\n2in the\ntwo leads have opposite signs. ( Note that for clarity a minus\nsign is added before Iz\n2in Fig.6(a) ). The parameters used\nare given in the text or shown in the figures.\nthat the terminal spin currents with spin parallel to the\nxaxis have also opposite signs in the two leads, similar\nto the case shown in Fig.6(b). This simple example il-\nlustrates explicitly that the terminal spin currents in a\nmulti-terminal mesoscopic SO coupled system are non-\nconservative in general. It should be stressed that this\nnon-conservation of terminal spin currents is not caused\nby the use ofan improper definition of spin current but is\nintrinsic to spin-dependent transports in mesoscopic SO\ncoupled systems. In fact, in our calculations we did not\ninvolve the controversial issue of what is the correct def-\ninition of spin current in the SO coupled region at all, so\nthe ambiguities that may be caused by the use of an im-\nproperdefinition ofspin currentto the SO coupled region\nhave been eliminated in our calculations.\nV. CONCLUSION\nIn summary, based on a scattering wave function ap-\nproach, in this paper we have studied theoretically some\nsymmetry properties of spin currents and spin polariza-\ntions in a multi-terminal mesoscopic structure in which\na spin-orbit coupled system is contacted to several ideal\nand nonmagnetic external leads. Some interesting new\nresults were obtained based on the symmetry analysis\nof spin currents and spin polarizations in such a multi-\nterminal mesoscopic structure. First, we showed that inthe equilibrium state no finite spin polarizations can ex-\nist both in the leads and in the central SO coupled region\nand also no finite equilibrium terminal spin currents can\nsurvive. Second, we showed that the lateral edge spin ac-\ncumulation induced by a longitudinal charge current in a\nthin strip of a ballistic two-dimensional electron gas with\nboth Rashba and Dresselhaus SO coupling may be non-\nantisymmetric in general, which implies that some cau-\ntionsmayneedtobetakenwhenattributingthenonequi-\nlibrium lateral edge spin accumulation induced by a lon-\ngitudinal charge current in a thin strip of such a system\nto a spin Hall effect, especially in the mesoscopic regime.\nFinally, by use of a typical two-probe structure as the\nexample, we showed explicitly that the nonequilibrium\nterminal spin currents in a multi-terminal mesoscopic\nSO coupled system may be non-conservative in general.\nSome symmetry properties discussed in the present pa-\nper might alsobe helpful for clarifyingsome controversial\nissues related to the study of spin-dependent transports\nin macroscopic SO coupled systems.\nAcknowledgments\nY. J. Jiang was supported by the Natural Science\nFoundation of Zhejiang province ( Grant No.Y605167 ).\nL. B. Hu was supported by the National Science Founda-\ntion of China ( GrantNo.10474022) and the NaturalSci-\nence Foundation of Guangdong province ( No.05200534\n).\nAPPENDIX A: SOME DETAILS FOR THE\nDERIVATIONS OF THE SCATTERING\nAMPLITUDES AND THE TRANSMISSION\nPROBABILITIES\nIn this appendix we give some details on how to derive\nthe scattering amplitudes and the transmission probabil-\nities from the scattering wave function approach intro-\nduced in Sec.II. For convenience of notation, we arrange\nthe scattering wave function ψpmσ(ri) inside the SO cou-\npled region into a column vector Ψ swhose dimension is\n2N(Nis the total number of lattice sites in the SO\ncoupled region ) and arrange the scattering amplitudes\nφpmσ\nqnσ′into a column vector Φ whose dimension is 2 M(\nM=/summationtext\npNpandNpis the width of lead p). Substi-\ntuting Eqs.(3-4) into Eqs.(5a-5b) and making use of the\northogonality relations for the transverse modes in the\nleads, one can show that the two column vectors Ψ sand\nΦ will satisfy the following relations:\nAΨs=b+BΦ,CΦ =d+DΨs,(A1)\nwhereA,B,C,Dare four rectangular matrices with the\ndimensions of 2 N×2N, 2N×2M, 2M×2M, and\n2M×2N, respectively; banddare two column vec-\ntors with the dimensions of 2 Nand 2M, respectively.11\nThe elements of these matrices and column vectors can\nbe written down explicitly as\nA=EI−Hsys,\nB(np′′yσ′′,p′m′σ′) =−δp′′,p′δσ′′,σ′tp′sχp′\nm′(yp′′)eikp′\nm′,\nD(p′m′σ′,np′′yσ′′) =−δp′′,p′δσ′′,σ′tp′sχp′\nm′(yp′′),\nC(p′m′σ′,p′′m′′σ′′) =−δp′′,p′δσ′′,σ′δm′′m′tp′,\nb(np′yσ′) =−δpp′δσσ′tpsχp\nm(yp′)e−ikp\nm,\nd(p′m′σ′) =δpp′δmm′δσσ′tp, (A2)\nwhereIstands for the identity matrix. The indices for\nleads, transverse modes, lattice sites and spins can take\nall possible values. ( For simplicity of notation, we have\nused simply a symbol np′y′to denote a boundary lattice\nsite in the SO coupled region which is connected directly\nto a boundary lattice site x′\np′= (1,y′\np′) in leadp′. )\nEq.(A1) is just a compact form of the match conditions\n(5a-5b)on thebordersbetweenthe leadsandthe SOcou-\npled region, from which both the scattering amplitudes\n{φpmσ\nqnσ′}and the transmission probabilities {Tpσ\nqσ′}can be\nobtained readily.\nToderiveacompactformulaforthetransmissionprob-\nabilities between two leads, we define an auxiliary matrix\nΣR≡BC−1D. By use of Eq.(A2) the matrix elements\nof ΣRcan be written down readily as following,\nΣR(np′y1σ′,np′y2σ′) =−/summationdisplay\nm′t2\np′s\ntp′χp′\nm′(y1)χp′\nm′(y2)eikp′\nm′,\n(A3)\nand all other matrix elements not shown explicitly above\nare zero. With the help of this auxiliary matrix, from\nEq.(A1) one can get that\nΨs= (A−ΣR)−1(b+BC−1d) =GRg,(A4)\nwheregis a column vector defined by g≡b+BC−1d\nandGRis a matrix defined by GR≡(A−ΣR)−1=[EI−Hsys−ΣR]−1, which is just the usual retarded\nGreen’s function. By use of Eq.(A2), the elements of\nthe column vector gcan also be written down readily as\nfollowing,\ng(np′yσ′) = 2iδpp′δσσ′tpssin(kp\nm)χp\nm(y).(A5)\nBy substituting Eq.(A4) into Eq.(A1) one gets that Φ =\nC−1d+C−1DGRg. Inserting Eq.(A5) into this formula\nand making use of Eq.(A2), one can show readily that\nthe scattering amplitudes φpmσ\nqnσ′will be given by\nφpmσ\np′m′σ′=−δpp′δmm′δσσ′+2it−1\np′tp′s/summationdisplay\nyp,yp′tpssin(kp\nm)\n×χp′\nm′(yp′)GR\nσ′σ(nyp′,nyp)χm(yp).(A6)\nThe total transmission probability of a conduction elec-\ntron from lead p( with spin index σ) to leadp′( with\nspin indexσ′) is defined by Tpσ\np′σ′=/summationtext\nm,m′/vextendsingle/vextendsingle/vextendsingleφpmσ\np′m′σ′/vextendsingle/vextendsingle/vextendsingle2vp′m′\nvpm,\nwherevp′m′=1\n/planckover2pi12tp′sin(kp′\nm′) is the longitudinal velocity\nof the mode m′in leadp′. Substituting Eq.(A6) into this\nformula, for p/ne}ationslash=p′one can get that\nTpσ\np′σ′=Tr(ΓpGA\nσσ′Γp′GR\nσ′σ), (A7)\nwhereGR\nσσ′andGA\nσσ′(≡GR†\nσ′σ) are the spin-resolved re-\ntarded and advanced Green’s functions, respectively, and\nΓp(yp,yp) is defined by\nΓp(yp,yp) =/summationdisplay\nm(tps\ntp)2χm(yp)vpmχm(yp).(A8)\nThe transmission probabilities given by Eq.(A7) have ex-\nactly the sameform aswas obtained by the usual Green’s\nfunction approach33.\n1I. Zutic, J. Fabian, and S. Sarma, Rev. Mod. Phys. 76,\n323(2004).\n2D. Awschalom, D. Loss, and N. Samarth, Semiconductor\nSpintronics and Quantum Computation ( Springer, Berlin,\n2002).\n3S. Datta and B. Das, Appl. Phys. Lett. 56, 665(1990).\n4J. E. Hirsch, Phys. Rev. Lett. 83, 1834(1999). S. Zhang,\nPhys. Rev. Lett. 85, 393(2000).\n5S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301,\n1348(2003).\n6J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth,\nand A. H. MacDonald, Phys. Rev. Lett. 92, 126603(2004).\n7J. P. Hu, B. A. Bernevig, and C. J. Wu, Int. J. Mod. Phys.\nB17, 5991(2003).\n8E. I. Rashba, Phys. Rev. B 68, (R)241315(2003).\n9J. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys.\nRev. B70, 041303(R)(2004).10E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys.\nRev. Lett. 93, 226602(2004).\n11S. -Q. Shen, M. Ma, X. C. Xie, and F. C. Zhang, Phys.\nRev. Lett. 92, 256603 (2004).\n12N.A.Sinitsyn,E. M.Hankiewicz, W.Teizer, andJ.Sinova,\nPhys. Rev. B 70, 081312(2004).\n13L. B. Hu, J. Gao, and S. -Q. Shen, Phys. Rev. B 70,\n235323(2004); X. H. Ma, L. B. Hu, R. B. Tao, and S. -\nQ. Shen, Phys. Rev. B 70, 195343(2004).\n14L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett.\n94, 016602(2005).\n15B. K. Nikolic, S. Souma, L. P. Zarbo and J. Sinova, Phys.\nRev.Lett. 95, 046601(2005); B. K.Nikolic, L.P.Zarboand\nS. Souma, Phys. Rev. B 72, 075361(2005); B. K. Nikolic,\nL. P. Zarbo and S. Souma, Phys. Rev. B 73, 075303(2006).\n16M. Onoda and N. Nagaosa, Phys. Rev. B 72, 081301\n(2005).12\n17E. M. Hankiewicz, J. Li, T. Jungwirth, Q. Niu, S. -Qing\nShen, J. Sinova, Phys. Rev. B 72, 155305 (2005).\n18J. Li, L. B. Hu, and S. Q. Shen, Phys. Rev. B 71,\n241305(R)(2005).\n19A. G. Malshukov, L. Y. Wang, C. S. Chu, and K. A. Chao,\nPhys. Rev. Lett. 95, 146601(2005).\n20H. A. Engel, B. I. Halperin, and E. I. Rashba, Phys. Rev.\nLett.95, 166605 (2005).\n21J. Yao and Z. Yang, Phys. Rev. B 73, 033314(2006).\n22W. K. Tse and S. Das Sarma, Phys. Rev. Lett. 96, 056601\n(2006).\n23A. Reynoso, G. Usaj, and C. A. Balseiro, Phys. Rev. B 73,\n115342 (2006).\n24Y. J. Jiang and L. B. Hu, Phys. Rev. B 74, 075302 (2006)\n25Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Science 306, 5703(2004).\n26J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth,\nPhys. Rev. Lett. 94, 047204(2005).\n27J. Sinova, S. Murakami, S. -Q. Shen, M. S. Choi, Solid\nState Comm. 138, 214 (2006).\n28S. Murakami, N. Nagaosa, and S.-C. Zhang, Phys. Rev. B\n69, 235206 (2004).\n29Q. F. Sun and X. C. Xie, Phys. Rev. B 72, 245305(2005).30J. R. Shi, P. Zhang, D. Xiao, Q. Niu, Phys. Rev. Lett. 96,\n076604 (2006).\n31P.-Q. Jin, Y.-Q. Li and F.-C. Zhang, cond-mat/0502231.\n32F. Zhai and H. Q. Xu, Phys. Rev. Lett. 94, 246601(2005).\n33S. Datta, Electronic transport in mesoscopic systems,\nChapter 3 ( Cambridge University Press, Cambridge, 1997\n).\n34When we calculate the terminal spin currents with spin\nparallel to the α( =x,y,z) axis, we will choose the α\naxis as the quantization axis of spins in the leads. This\nconvention will be assumed throughout the paper.\n35The expectation values of ˆIqandˆIα\nqgiven by Eq.(12a) and\n(12b) are independent of the longitudinal coordinate xqin\nthe steady state.\n36Since we have assumed that the leads are ideal and non-\nmagnetic, the contributions due to incident waves from\nleadqwith opposite spin indexes ( given by the terms\nproportional to δpqin Eq.(12b) ) cancel exactly with each\nother in Eq.(13b).\n37A. A. Kiselev and K. W. Kim, Phys. Rev. B 71,\n153315(2005).\n38T.P.Pareek, Phys. Rev. Lett. 92, 076601(2004)." }, { "title": "1408.6353v1.Spin_orbit_Coupling_and_Multiple_Phases_in_Spin_triplet_Superconductor_Sr__2_RuO__4_.pdf", "content": "arXiv:1408.6353v1 [cond-mat.supr-con] 27 Aug 2014Typesetwith jpsj3.cls FULLPAPER\nSpin-orbit Coupling andMultiplephases inSpin-triplet Su perconductor Sr 2RuO4\nYouichiYanase1,2∗, ShuheiTakamatsu2,andMasafumiUdagawa3\n1Department of Physics, NiigataUniversity,8050 Ikarashi, Nishi-ku,Niigata950-2181, Japan\n2Garaduate School of Science andTechnology, NiigataUniver sity,8050 Ikarashi, Nishi-ku,Niigata950-2181, Japan\n3Department of AppliedPhysics, University of Tokyo, 7-3-1 H ngo, Bunkyo-ku, Tokyo 113-8656, Japan\nWestudythe spin-orbit coupling inspin-tripletCooper pai rsand clarifymultiplesuperconducting (SC)\nphases in Sr 2RuO4. Based on the analysis of the three-orbital Hubbard model wi th atomic LS coupling,\nwe show some selection rules of the spin-orbit coupling in Co oper pairs. The spin-orbit coupling is small\nwhenthe two-dimensional γ-band isthe maincause of the superconductivity, although t heLScoupling is\nmuchlargerthantheSCgap.Consideringthiscase,weinvest igatemultipleSCtransitionsinthemagnetic\nfieldsforboth H/bardbl[001]andH/bardbl[100]usingtheGinzburg-Landau theoryandthequasi-classical t heory.\nRich phase diagrams are obtained because the spin degree of f reedom inCooper pairs is not quenched by\nthe spin-orbit coupling. Experimental indications for the multiple phases inSr 2RuO4are discussed.\nKEYWORDS: spin-tripletsuperconductivity,spin-orbitco upling,multiplephases,Sr 2RuO4\n1. Introduction\nIn this short review, we study the spin-orbit coupling and\nmultiple phases in spin-triplet superconductors. The spin -\norbit coupling plays a crucial role in determining the spin-\ntriplet pairing state. Discussions are particularly focus ed on\nSr2RuO41whichisanestablishedcandidateofthespin-triplet\nsuperconductor2,3in addition to the superfluid3He,4heavy\nfermion superconductor UPt 3,5–7and ferromagnetic super-\nconductors,UGe 2,URhGe,andUCoGe.8\nBecause of the simple electronic structure of Sr 2RuO4\ncompared with the U-based heavy fermion superconductors,\nstudieson Sr 2RuO4for these two decadeshave madenotice-\nableprogressinthemicroscopicunderstandingofspin-tri plet\nsuperconductivity.LowenergyquasiparticlesinSr 2RuO4are\ndescribedbythetwo-dimensionaltight-bindingmodelfort he\nthreet2g-orbitalsinRuionsonthetetragonalcrystalwith D4h\npoint group symmetry. Indeed, electronic and SC properties\nof Sr2RuO4have been elucidated on the basis of the three-\norbitalHubbardmodel.3Thosemicroscopictheoriesprovided\nseveral clear understandingsof spin-triplet superconduc tivity\nwhichhavenotbeenobtainedinthestudiesoff-electronsys -\ntems.5,6\nIn the first part of this article ( §3), we elucidate how the\nspin-orbitcouplinginspin-tripletCooperpairsarisesfr omthe\natomicLScouplingofelectrons.Analysisofthethree-orbi tal\nHubbardmodel shows that the spin-orbit couplingin Cooper\npairsissmall,inspiteofthelargeLScouplingcomparedwit h\nthe energy scale of superconductivity. We also demonstrate\nsome selection rules which derive from the symmetry of lo-\ncal electron orbital. The selection rules sometimes determ ine\nthed-vector, namely, the order parameter of spin-triplet su-\nperconductivity.In the second part ( §4 and§5), we study the\nmultiple SC phases in the magnetic field on the basis of the\nGinzburg-Landau(GL) theory and quasi-classical theory de -\nrived from the three-orbital Hubbard model. When the spin-\norbit coupling is small but finite, multiple SC phases appear\nin the magnetic-field-temperature( H-T) phase diagram. We\nclarify the pairing state in Sr 2RuO4and discuss the experi-\nmental results. Several indications for the multiple phase s as\n∗E-mail address: yanase@phys.sc.niigata-u.ac.jpwell assomeunresolvedissuesarediscussed.\n2. Spin-tripletSuperconductivityinTetragonalCrystals\nFirst, we review the general aspects of spin-triplet super-\nconductivity and define the “spin-orbit coupling in Cooper\npairs”.SincetheCooperpairshavetotalspin S= 1,theorder\nparameter of spin-triplet superconductorsis described by the\nthreecomponentvector, d= (dx,dy,dz),4,9\n/parenleftbigg\n∆↑↑∆↑↓\n∆↓↑∆↓↓/parenrightbigg\n=/parenleftbigg\n−dx+idydz\ndzdx+idy/parenrightbigg\n.(1)\nThep-wave superconductivity in the tetragonal crystal also\nhas two orbital components,that is pxandpy, and therefore,\nthe SC state is represented by the 2×3 = 6component or-\nderparameters.Inthepresenceofthespin-orbitcoupling, the\nspinisentangledwiththeorbital,andtheSCstatesareclas si-\nfiedonthebasisofthepointgroup.9FortheD4hpointgroup\nsymmetry, the SC state belongs to the two-dimensional irre-\nducible representation Eu, or to four one-dimensional repre-\nsentations,A1u,A2u,B1u, andB2u, as summarized in Ta-\nble I. Some experiments of Sr 2RuO4, such asµSR10and\nKerr rotation,11observe the spontaneous time-reversal sym-\nmetry breaking (TRSB) in the SC state2,3and indicate the\nSC state belongingto the Eurepresentation.This means that\nthe spin-orbit coupling favors the “chiral SC state”, namel y\nd= (px±ipy)ˆz. On the other hand, the other SC states be-\nlonging toA1u,A2u,B1u, orB2urepresentation are called\n“helicalSCstate”.\nWe would like to stress that the spin-orbit coupling partic-\nularlyplaysa crucialroleinthetetragonalcrystal,incon trast\ntothecubiccrystalsandtherotationally-symmetricsuper fluid\n3He. TheB-phase of superfluid3He is stable at low tempera-\nturesevenintheabsenceofthespin-orbitcouplingsothatt he\ncondensationenergyismaximizedthroughtheisotropicexc i-\ntation gap.4Then, the weak spin-orbit coupling arising from\nthe dipole interaction plays a minor role. On the other hand,\nthe spin-orbit couplingplays an essential role in determin ing\nthe SC state of Sr 2RuO4even when the spin-orbit coupling\nis small, because the condensation energy is equivalent be-\ntweenthe SCstates inTable Iin theabsenceofthespin-orbit\n12 J.Phys.Soc.Jpn. FULLPAPER Author Name\ncoupling. Therefore, it is crucial to investigate the spin- orbit\ncouplingin Cooperpairsforthestudyofspin-tripletSC sta te\nin Sr2RuO4. This is also the case in the other spin-triplet su-\nperconductorsexceptforthoseinthecubiccrystals.\nIrreducible representation Order parameter Dimension\nA1u d=pxˆx+pyˆy 1\nB1u d=pxˆx−pyˆy 1\nA2u d=pyˆx−pxˆy 1\nB2u d=pyˆx+pxˆy 1\nEu d= (px±ipy)ˆz 2\nTable I. Spin-triplet SC states in tetragonal crystals with D4hpoint group\nsymmetry. We show the order parameters belonging to the irre ducible\nrepresentation, A1u,A2u,B1u,B2u, andEu. Their dimension is also\nshown.\nFor the aim of a coherent discussion, we here define the\nspin-orbitcouplinginCooperpairs.Thatisdenotedas ηΓΓ′=\n(TΓ\nc−TΓ′\nc)/Tc, whereΓandΓ′label the irreducible repre-\nsentation. Thus, the spin-orbit coupling in Cooper pairs re p-\nresents the difference of transition temperature between i r-\nreducible representations. Clearly, ηΓΓ′= 0when the spin\nSU(2)symmetry is conserved.On the other hand, the viola-\ntion ofSU(2)symmetry gives rise to a finite spin-orbit cou-\npling,ηΓΓ′.\nWhen the transition temperature is highest for the pairing\nstate belonging to an irreducible representation Γ0, such SC\nstate is stabilized below Tc=TΓ0c. Among four indepen-\ndent spin-orbit couplings in the D4hpoint group symmetry,\nthe most important one is η=ηΓ0Γ1whereΓ1is the irre-\nducible representation having the second highest transiti on\ntemperature. The magnitude of ηrepresents the anisotropy\nof Cooper pairs in the spin space. Generally speaking, mul-\ntiple SC phases appear in the H-Tphase diagram when the\n“anisotropy” ηis small. Other spin-orbit couplings also play\nimportantrolesindeterminingthemultiplephases(see §4).\n3. Spin-orbitCoupling in Spin-tripletCooperPairs\nNext, we discuss the microscopic aspects of the spin-orbit\ncoupling in Cooper pairs. Although we see some similarities\nbetween the superfluid3He and spin-triplet superconductors,\nthe origin and properties of the spin-orbit coupling are qui te\ndifferent between them. It has been clarified that the releva nt\nspin-orbit coupling in3He is the dipole interaction.4On the\nother hand, electrons in the crystals are affected by the so-\ncalled LS couplingwhich originatesfromthe relativistic m o-\ntion of electrons near nuclei.12It has been established that\nthe spin anisotropy of electrons mainly originates from the\nLS coupling in the solid state physics.13Therefore, it is rea-\nsonable that the LS coupling gives rise to the leading spin-\norbitcouplinginspin-tripletCooperpairs.However,ther ela-\ntion between the LS coupling and the spin-orbit coupling in\nCooperpairsisnon-trivial.Inthissection,weclarifyhow the\nspin-orbit coupling in Cooper pairs arises from the LS cou-\nplingofelectrons.14,153.1 Three-orbitalHubbardmodel\nOur discussions are based on the theoretical analysis of\ntwo-dimensional three-orbital Hubbard model which repro-\nduces the band structure of Sr 2RuO4.2,3,16,17We here focus\nSr2RuO4asatypicalexample,however,thefollowingresults\non the spin-orbit coupling in Cooper pairs are valid for othe r\n3dand4delectronsystemstoo.Themodelis\nH=Hkin+Hhyb+HCEF+HLS+HI,(2)\nwhereHkin=/summationtext\nk/summationtext\nm=1,2,3/summationtext\ns=↑,↓εm(k)c†\nkmsckmsis the\nkinetic energy, Hhyb=/summationtext\nk/summationtext\ns[V(k)c†\nk1sck2s+ h.c.]de-\nscribes the intersite hybridization between the d yz- and d zx-\norbitals,HCEF= ∆/summationtext\nk/summationtext\nsc†\nk3sck3sis the crystal elec-\ntric field term, and HLS=λ/summationtext\niLi·Sirepresents the LS\ncoupling. The d yz-, dzx-, and d xy-orbitals are denoted by\nthe indices m=1, 2, and 3, respectively. We will show\nthat the absence of intersite hybridization between d xy- and\ndyz/dzx-orbitals plays an important role in the spin-orbit\ncoupling of Cooper pairs. Taking account of the symmetry\noft2g-orbitals, we adopt the tight-binding form, ε1(k) =\n−2t4coskx−2t3cosky,ε2(k) =−2t3coskx−2t4cosky,\nε3(k) =−2t1(coskx+ cosky)−4t2coskxcosky, and\nV(k) = 4t5sinkxsinky.\n−π0π\n−π 0 πky\nkxγ-band\nβ-bandα-band\nFig. 1. Fermi surfaces of the α-,β-, andγ-bands in the three-orbital Hub-\nbard model. The dashed lines show the Fermi surfaces in the ab sence of\nthe LS coupling HLSand intersite hybridization term Hhyb. The band\ngap opens along k/bardbl[110]owing to HLSandHhyb(solid lines). Tight-\nbinding parameters are shown in Ref. 14. These Fermi surface s reproduce\nthecylindrical Fermisurfaces of Sr 2RuO4.2,3,16,17\nThis model appropriatelyreproducesthe band structure of\nSr2RuO4which has been elucidated by the first principle\nband structure calculation16,17as well as by the de Haas-van\nAlphen oscillation measurements and angle-resolved photo -\nemission spectroscopy.2,3As shown in Fig. 1, we see the\nquasi-two-dimensional Fermi surface of the d xy-orbital (γ-\nband) and two quasi-one-dimensional Fermi surfaces con-\nsisting of the (d yz, dzx)-orbitals (α- andβ-bands). The elec-\ntron correlation effect renormalizesthe band structure,2,3but\nhardlychangestheFermi surfaces.\nAlthough the BCS theory assumed the s-wave supercon-\nductivity induced by the electron-phonon coupling, the un-\nconventional non- s-wave superconductivity occurs through\nthe Coulomb interactions in the strongly correlated electr on\nsystems.18Thus, we take into account the on-site Coulomb\ninteraction term HIwhich consists of the intraorbital repul-\nsionU, interorbital repulsion U′, Hund’s rule coupling J,J.Phys.Soc.Jpn. FULLPAPER Author Name 3\nandpairhopping J′.Twospin-tripletSCstateshavebeenob-\ntainedbythetheoreticalanalysisofthethree-orbitalHub bard\nmodel. One is the p-wave SC state which is mainly caused\nby the quasi-two-dimensional γ-band.19–21The other is the\np- orf-wave state mainly due to the quasi-one-dimensional\n(α,β)-bands.22,23The partial density of states (DOS) of the\nbandsdetermineswhichSC state isstable. Indeed,ourcalcu -\nlation showed the crossover from the quasi-two-dimensiona l\nsuperconductivitytothequasi-one-dimensionalsupercon duc-\ntivitybytuningthetight-bindingparameterssoastodecre ase\nthe partial DOS of γ-band.14Thus, the superconductivity is\nmainly inducedby the “active orbital” as proposedby Agter-\nbergetal.24Whenwechoosetherealisticparameterssoasto\nreproducethe 57%partialDOS inthe γ-band,2,3bothpertur-\nbation theory14,19and functional renormalization group the-\nory21support the superconductivity driven by the γFermi\nsurface. In all cases, the spin-triplet SC states summarize din\nTable I are degeneratein the absence of the LS coupling,be-\ncausethespin SU(2)symmetryisconserved.\n3.2 Orderestimationofspin-orbitcoupling\nNow we move on to the role of LS coupling, which is the\nmain topic of this article. Although it is not difficult to non -\nperturbativelydeal with the LS couplingterm, we here adopt\nthe perturbation expansion for λby which we obtain some\nselection rules in the following way. The discussion is base d\non the hierarchy of energy scales in the 3d and 4d electron\nsystems,\nTc≪λ≪EF. (3)\nIn the case of Sr 2RuO4, the LS coupling λ∼100K is\nmuch larger than the transition temperature of superconduc -\ntivityTc∼1K, but much smaller than the Fermi energy\nEF= 1000∼10000K. Now let us consider the perturba-\ntionexpansionofthespin-orbitcouplinginCooperpairs,\nη=∞/summationdisplay\nn=1∞/summationdisplay\nm=1Anm(λ/Tc)n(λ/EF)m.(4)\nWhen the coefficients Anm(n≥1)are finite, this expan-\nsion is unreliable because the expansion parameter λ/Tcis\nhuge. However,we find Anm= 0forn≥1.14,15This prop-\nerty is guaranteed by the inversion symmetry of the system\nas we discuss in §3.5. Thus, we obtain the quantitatively re-\nliable perturbation expansion of ηfor the small parameter\nλ/EF≪1, asη=/summationtext∞\nm=1A0m(λ/EF)m. At the same time\nwe understand that the spin-orbit coupling in Cooper pairs η\nis small when λ/EF≪1, even though the LS coupling is\nmuchlargerthantheenergyscale ofsuperconductivity.\n3.3 Selectionrules\nNext, we show the selection rules of the spin-orbit cou-\npling in Cooper pairs. We here discuss the following two SC\nphases in a separate way; (1) superconductivityin the quasi -\none-dimensional ( α,β)-bands and (2) that in the quasi-two-\ndimensional γ-band. Indeed, all bands are superconducting\nowing to the inter-band proximity effect, and therefore, th e\nspin-orbitcouplinginCooperpairsareobtainedbyaddingt he\ncontributionsof α,β, andγbands. However, the SC proper-\nties are mainly determined by the active band having a large\nSC gap, since the orbital dependent SC phases24are likelystabilizedinSr 2RuO4(see§3.1).\n6-fold Eu 6-foldor\nA1u, A2uB1u, B2u\nEu\nB1u, B2uA1u, A2u\nFig. 2. (Color online) Illustration of the selection rule fo r the SC state\ndriven by the (d yz, dzx)-orbitals. We show the energy levels of 6 spin-\ntriplet SC states. The 6-fold degeneracy is lifted by the firs t order term\nof LS coupling. One of the doublet, ( A1u,A2u) or (B1u,B2u), has the\nlowest energy. The2-fold degeneracy in the doublet is lifte d by the higher\norder terms.\nWe begin with the discussion of the case (1). In this case,\nthe leading order terms of the spin-orbit coupling ηΓΓ′are\nfirst order in λ/EF. Analyzing the Eliashberg equation18for\nthe three-orbital Hubbard model, it is shown that the first\norder terms of the irreducible vertex in the particle-parti cle\nchannel have the particular symmetry.14,15Those terms have\nthedxysymmetry in the momentum space, and conserves\nthez-component of the total spin. The selection rule which\nis schematically shown in Fig. 2 arises from this symmetry.\nThe first order terms do not lift the degeneracy between the\nA1uandA2ustates and between the B1uandB2ustates. On\nthe other hand, the degeneracy between the two doublet is\nlifted, and one of the doublet has the lowest energy. This se-\nlection rule is explicitly described as ηA1uEu=ηA2uEu=\n−ηB1uEu=−ηB2uEu=O(λ/EF). The degeneracyof A1u\nandA2ustates (B1uandB2ustates) is slightly lifted by the\nsecond order term, as ηA1uA2u=O(λ2/E2\nF)(ηB1uB2u=\nO(λ2/E2\nF)). Although the signs of ηA1uEuandηA1uA2ude-\npend on the electronic structure, we obtain an exact conclu-\nsion; Oneof thehelical SCstates isstabilized bythe LS cou-\npling.Inotherwords,the chiralSCstate belongingto the Eu\nrepresentation can not be stable when the ( α,β)-bands are\nmainlysuperconducting.Thisfeaturewasalsopointedoutb y\nthesemi-microscopiccalculation.25\nImportantly, these selection rules are independent of the\nCoulomb interactions. Indeed, we confirmed that the selec-\ntion rules for ηΓΓ′are satisfied in all order of perturbation\nterms for Coulomb interactions U,U′,J, andJ′.14,15This\nfeatureisnotalteredevenwhenwetakeintoaccountthelong -\nrange Coulomb interaction. Thus, the leading order term of\nspin-orbit coupling in Cooper pairs obeys the selection rul e\nwhich is independent of the electron correlation. This is in\nsharp contrast to the fact that the pairing interaction lead ing\nto the unconventionalsuperconductivitydependsonthe ban d\nstructures,Coulombinteractions,andsoon.18Thismeansthat\nthe spin-orbit coupling in Cooper pairs is not closely relat ed\nwith themechanismofCooperpairing.\nWe turn to the discussion of the case (2). When the super-\nconductivity is mainly caused by the quasi-two-dimensiona l\nγ-band, the first order term with respect to the LS coupling\nvanishes. We find this feature by analyzing the Eliashberg\nequation. The first order terms of the irreducible vertex lea d\ntotheinter-bandCooperpairing,andtheyarenegligiblewh en\nTc≪EF. This is also the selection rule and comes form the\nfactthattheintersitehybridizationbetweend xy-andd yz/dzx-4 J.Phys.Soc.Jpn. FULLPAPER Author Name\n0.007 0.0075 0.008\nT0.960.981.001.02λeA1u, B1u\nA2u, B2u\nEu\n1.2 1.4 1.6 1.8nγ0.350.400.45t2/t1\nA1u, B1u\nEu(a) (b)\nFig. 3. (Color online) (a) Phase diagram of the three-orbita l\nHubbard model14for a tight-binding parameter t2/t1and\nthe electron density in the γ-band,nγ. The other parame-\nters are chosen to be (t1,t3,t4,t5,λ,∆,U,U′,J,J′) =\n(1,1.25,0.1,0,0.2,−0.3,5,1.5,1,1). Circles show the Eustate,\nand triangles show the ( A1u,B1u) state. (b) Temperature dependence of\neigenvalues ofEliashberg equation forthe( A1u,B1u)state(dashed line),\n(A2u,B2u)state(dot-dashed line), and Eustate(solid line),respectively.\nWeassume t2/t1= 0.4andnγ= 1.37.\norbitals vanishes owing to the mirror symmetry along the c-\naxis. Although such hybridization terms appear in the three -\ndimensionalmodel,theydonotaltertheselectionrule.\nAs the leading order term is roughly estimated as ηΓΓ′=\nO(λ2/E2\nF)∼0.01forλ/EF∼0.1, the spin-orbit couplings\nin Cooper pairs ηΓΓ′are small when the γ-band is mainly\nsuperconducting. In order to investigate this small spin-o rbit\ncoupling, we are required to solve the three-orbital Hubbar d\nmodel with use of some approximatetreatments of Coulomb\ninteractions. We do not find any selection rule for the second\nordertermsin λexceptfortheaccidentaldegeneracybetween\ntheA1uandB1ustates and between the A2uandB2ustates.\nUsing the perturbationtheoryfor Coulombinteractionsup t o\nthe third order, we solved the linearized Eliashberg equati on\nand obtained the results in Fig. 3.14Figure 3(a) shows the\nphase diagram against the tight-binding parameter t2/t1and\nthenumberdensityofelectronsinthe γ-band,nγ.Itisshown\nthatthespin-tripletSCstatedependsontheserelevantpar am-\neters.ForrealisticparametersofSr 2RuO4,namelynγ∼1.33\nandt2/t1∼0.4, theEustate is stable, although the A1uor\nB1ustate isstabilizedin apartofthephasediagram.\nFigure 3(b) shows the temperature dependence of the\neigenvalue of the linearized Eliashberg equation λefor the\nEu, (A1u,B1u), and (A2u,B2u) states. The Tcof each su-\nperconducting state is obtained by the criterion, λe= 1. We\nsee that the Eustate has the highest Tc, but the splitting\nofTcis smallηEuA1u= 0.013. Thus, the spin-orbit cou-\npling in Cooper pairs is small as we expected from the or-\nder estimation, although the LS coupling ( λ= 0.2) is much\nlarger than the transition temperature of superconductivi ty\n(Tc= 0.0073).\nAs we have shown above,when the spin-tripletSC state is\ninducedbythe γ-band“asmallspin-orbitcouplinginCooper\npairs favors the chiral SC state d= (px±ipy)ˆz(Eustate)”.\nThe same result was obtainedbythe recent calculationbased\non the functional renormalization group theory.21We show\nthis result in the Table II, although it is not obtained by the\nselection rule. On the other hand, it has been shown that the\nhelicalSCstateisstabilizedwhentheCoulombinteraction on\nOxygen ions is large.26It is reasonable that the pairing state\ndepends on the electron interaction, because we do not findanyselectionrulein thiscase.\nTable II summarizes the d-vector of spin-triplet Cooper\npairs which we obtained.15We also show the case of the\nhexagonalcrystalwith D6hpointgroupsymmetry.Itisshown\nthat the anisotropy ηand the direction of d-vector are deter-\nmined by the symmetries of crystal lattice, local electron o r-\nbital,andsuperconductivity.Interestingly,wefindasimi larity\nbetween the tetragonal crystal and hexagonal crystal. When\nthe SC is induced by the A1g-orbital in the latter, the ηis in\nthe second order of λ/EFas in the case of d xy-orbital in the\nformer. On the other hand, the first order term in λ/EFsta-\nbilizes the d-vector parallel to the ab-plane when the p-wave\nSCoccursinthe Eg-orbitalsofthehexagonalcrystal.Whatis\ndifferentfromthetetragonallatticeappearsinthelastco lumn\nof Table II. In contrast to the tetragonal crystal, the f-wave\nsuperconductivityisdistinguishedfromthe p-wavesupercon-\nductivityinthehexagonalcrystal.Inthe f-waveSCstate,the\nfirst order term in λ/EFvanishes, and we can not determine\nthe d-vector by the selection rule. Thus, the orbital symme-\ntry of superconductivity also plays an important role for th e\nspin-orbitcouplinginCooperpairs.\n3.4 Spin-orbitcouplinginSr 2RuO4\nWe discusstheexperimentalresultsindicatingtheSCstate\nof Sr2RuO4. First, the spontaneous TRSB observed in the\nµSR10and Kerr rotation11measurementsimplies that the Eu\nstate is stabilized at zero magneticfield. This finding is com -\npatiblewithourresultsonthesuperconductivityinthequa si-\ntwo-dimensional γ-band.14,21On the other hand, the TRSB\nis incompatible with the selection rule for the quasi-one-\ndimensional ( α,β)-bands, which does not allow the Eustate\nto bestabilizedatzeromagneticfield.\nAlthough the interpretation of the µSR10and Kerr rota-\ntion11data are still under the discussion,27the magnitude of\nthe spin-orbit coupling is also consistent with the superco n-\nductivityin the quasi-two-dimensional γ-band.A small spin-\norbit coupling below η <0.01is indicated by several ex-\nperiments.Thenuclearmagneticresonance(NMR) measure-\nments have shown the temperature-independent Knight shift\nthrough the SC transition temperature in both magnetic field\ndirections along the ab-plane and along the c-axis.28–30This\nobservationshowsthatthespin-orbitcouplinginCooperpa irs\nis so small thatthe d-vectorrotatesinthe magneticfield.The\nmagnitudeofspin-orbitcouplingisestimatedtobe η∼0.001\naccordingtothetemperatureindependentKnightshiftdata at\nHc= 0.02T.29Such a tiny spin-orbit coupling is not incom-\npatible with our calculation for the superconductivity in t he\nγ-band. We obtained η= 0.01for the LS coupling λ= 50\nmeVin§3.4,buttheLScouplingmaybesmaller,becausethe\nLS coupling of Sr 2RuO4is reduced by the strong hybridiza-\ntionofRuandOionsasdemonstratedbyanotherNMRmea-\nsurement.31Furthermore,the spin-orbitcoupling ηdecreases\nbecause of the competitive contributions between the activ e\nγ-band and the passive ( α,β)-bands. Thus, the superconduc-\ntivity which is mainly caused by the quasi-two-dimensional\nγ-band may be accompanied by a tiny spin-orbit coupling\nη∼0.001,consistentwiththeNMRdata.\nA small spin-orbit coupling is also indicated by the obser-\nvation of the half-quantum vortex.32The half-quantum vor-\ntex is formed by the π-rotation ofd-vector aroundthe vortex\ncore.33This intriguing topological defect is unstable unlessJ.Phys.Soc.Jpn. FULLPAPER Author Name 5\nthe spin-orbit coupling is small.34Indeed, theoretical studies\nofthehalf-quantumvortexinSr 2RuO4haveassumedasmall\nspin-orbit coupling.34–36Such a small spin-orbit coupling is\ncompatible with the superconductivity in the γ-band, but in-\ncompatiblewiththequasi-one-dimensionalsuperconducti vity\ndriven by the ( α,β)-bands. A moderate spin-orbit coupling\nη∼0.1is expectedin the later (see Table II). Thus, not only\nthethermodynamicandtransportproperties3,37,38butalsothe\nfeatures of the spin-orbit coupling indicate the supercond uc-\ntivitymainlycausedbythequasi-two-dimensional γ-band.\n3.5 Colossaleffectof brokeninversionsymmetry\nOur discussions in this section have been based on the in-\nversion symmetry of the crystal structure, as we mentioned\nin§3.2. When the inversionsymmetryis broken,for instance\nnear the surface, the spin-orbit coupling in Cooper pairs dr a-\nmaticallychanges.\nA simple way to describe the spin-orbit coupling in non-\ncentrosymmetric systems is to adopt the antisymmetric spin -\norbit coupling (such as the Rashba spin-orbit coupling),\nHASOC=α/summationtext\nkg(k)S(k).39The antisymmetric spin-orbit\ncouplinggivesrisetoalargeanisotropyinspin-tripletCo oper\npairs,η=O(1), when|α|> Tc, and it stabilizes the\npairing state with d-vector parallel to the g-vector, that is,\nd(k)/ba∇dblg(k).40\nFrom the microscopic point of view, the antisymmetric\nspin-orbit coupling arises from the combination of the LS\ncoupling and the parity mixing in local electron orbitals.41\nThe latter is taken into account in the three-orbital Hubbar d\nmodelbyaddingtheparitymixingterm,42\nHodd=/summationdisplay\nks[Vx(k)c†\nk1sck3s+Vy(k)c†\nk2sck3s+h.c.].(5)\nFor the extended model H′=H+Hoddcoefficients Anm\n(n≥1) in Eq. (4) are finite, and therefore, the perturbation\nexpansion with respect to the LS coupling is unreliable. The\nnon-perturbative calculation shows that a small parity mix -\ning due to the broken inversion symmetry stabilizes the A2u\nstate.42Thus, the spin-triplet SC state is sensitive to the bro-\nkeninversionsymmetry.\nThe randomnessyielding the locally non-centrosymmetric\nstructurealsoremarkablyaffectsthespin-tripletSCstat e.For\ninstance,weinvestigatedtherolesoftherandomRashbaspi n-\norbit coupling induced by stacking faults, and found that a\nsmallmeansquarevalueofRashbaspin-orbitcoupling, ¯α= 2\nK,stabilizesthe A2ustate.43SuchSCstatemayappearinthe\neutectic crystal Sr 2RuO4/Sr3Ru2O7which are indeed influ-\nencedbystackingfaults.44\n4. Superconductingphasesin Sr 2RuO4forH/bardbl[001]\nThe spin-triplet Cooper pairs in Sr 2RuO4seem to be af-\nfected by a small but finite spin-orbit coupling η= 0.001∼\n0.01, as indicated by both theoretical estimations and experi-\nmental data (see §3). Such a small spin-orbitcouplingallows\nthe multiple SC transitions to occur. In the following part,\nwe theoretically demonstratemultiple SC phases in the mag-\nnetic field. We consider the magnetic field along the crystal-\nlographic c-axis in this section, and study the SC state in the\nmagneticfieldalongthe ab-planein §5.\nWe assume that the spin-orbit coupling in Cooper pairs\nis so small that a moderate magnetic field below Hc2sup-presses the d-vector parallel to the magnetic field through\nthe paramagnetic depairing effect. This is likely the case o f\nSr2RuO4as we discussed in §3.4. When the magnetic field\nis parallel to the c-axis, the chiral state d= (px±ipy)ˆz\n(Eustate) is destabilized, and other two spin components\ndxanddymay appear. We describe these order parameters\nin the spin basis ∆↑↑(r,k)and∆↓↓(r,k)instead of the d-\nvectorform.Thequasi-classical formis used forthe studyo f\nspatially inhomogeneous SC state (vortex state). Each spin\ncomponent is divided into the two orbital components, as\n∆σσ(r,k) = ∆σσ,x(r)φx(k)+∆σσ,y(r)φy(k),whereφx(k)\nandφy(k)stand for pairing functions with the px- andpy-\nwave symmetry, respectively. In this way, the SC state is\ndescribed by the 2×2 = 4component order parameters,\n(∆↑↑,x(r),∆↑↑,x(r),∆↓↓,x(r),∆↓↓,y(r)).\nSC state is investigated on the basis of the following GL\nmodel,45\nf=/summationdisplay\nσ=↑,↓f0\nσ+fSO1+fSO2, (6)\nwherethefirsttermistheordinarypartoftheGLfreeenergy\ndensity for the two orbital component ( px,py)-wave super-\nconductorsin thetetragonallattice,46,47\nf0\nσ=α0(T−T0\nc)/parenleftbig\n|∆σσ,x|2+|∆σσ,y|2/parenrightbig\n+β1/parenleftbig\n|∆σσ,x|2+|∆σσ,y|2/parenrightbig2/2\n+β2(∆σσ,x∆∗\nσσ,y−c.c.)2/2+β3|∆σσ,x|2|∆σσ,y|2\n+ξ2\n1/bracketleftbig\n|Dx∆σσ,x|2+|Dy∆σσ,y|2/bracketrightbig\n+ξ2\n2/bracketleftbig\n|Dx∆σσ,y|2+|Dy∆σσ,x|2/bracketrightbig\n+ξ2\n3/braceleftBig/bracketleftbig\n(Dx∆σσ,x)(Dy∆σσ,y)∗+c.c./bracketrightbig\n+/bracketleftbig\n(Dx∆σσ,y)(Dy∆σσ,x)∗+c.c./bracketrightbig/bracerightBig\n.(7)\nWe adopt the conventional notation for the covariant deriva -\ntiveDj=∇j+(2πi/Φ0)AjandΦ0=hc/2|e|. Other nota-\ntions have been explained in Ref. 45. We omitted the label r\nto simplifythedescriptionofEq.(7).\nAs we have discussed, the spin-orbit coupling plays a cru-\ncial role in the spin-triplet superconductors in the tetrag onal\nlattice even if it is small. Our GL model takes into account\ntwo spin-orbitcouplingterms,\nfSO1=ǫ/summationdisplay\nσσ(i∆σσ,x∆∗\nσσ,y+c.c.), (8)\nfSO2=δ/bracketleftbig\n(∆↑↑,x∆∗\n↓↓,x−∆↑↑,y∆∗\n↓↓,y)+c.c./bracketrightbig\n.(9)\nThe coupling constants ǫandδare related with the spin-\norbit coupling in Cooper pairs as 2ǫ=ηA1uB1u=ηA2uB2u\nand2δ=ηA1uA2u=ηB1uB2u. According to the selection\nrulesshownin §3.3,thefSO1termis givenbythe quasi-one-\ndimensional ( α,β)-bands. The leading order term has been\nobtained as ǫ=O/parenleftbig\nλ/EF·t5/EF·|∆αβ/∆γ|2/parenrightbig\n,14where\n∆αβand∆γare the magnitude of SC gap in the ( α,β)-\nandγ-bands, respectively. For |∆αβ/∆γ| ∼0.3, we find\n|ǫ|= 0.001∼0.01. On the other hand, the fSO2term\noriginates from the coupling between the γ-band and (α,β)-\nbands. The magnitude has been numerically estimated to be\n|δ|= 0.001∼0.01inFig. 3(b).Thus,themagnitudesof two6 J.Phys.Soc.Jpn. FULLPAPER Author Name\nspin-orbitcouplingsarein thesameorder.\nWedeterminethepairingstatefortemperatures Tandmag-\nnetic fields/vectorH=Hˆcby minimizing the GL free energywith\nuseofthevariationalmethod.Werewritetheorderparamete rs\nusing the chirality basis, as ∆σσ,1≡(∆σσ,x−i∆σσ,y)/√\n2\nand∆σσ,2≡(∆σσ,x+i∆σσ,y)/√\n2. Theyareassumedto be\na linearcombinationofthebasisfunctions,\n/parenleftbigg∆↑↑,1(r)\n∆↑↑,2(r)/parenrightbigg\n=C1/parenleftbiggψ1+(r,0)\nψ2+(r,0)/parenrightbigg\n+C2/parenleftbiggψ1−(r,δ2)\nψ2−(r,δ2)/parenrightbigg\n,\n(10)\n/parenleftbigg∆↓↓,1(r)\n∆↓↓,2(r)/parenrightbigg\n=C3/parenleftbiggψ1+(r,δ3)\nψ2+(r,δ3)/parenrightbigg\n+C4/parenleftbiggψ1−(r,δ4)\nψ2−(r,δ4)/parenrightbigg\n,\n(11)\nwhereδ2,δ3,δ4denote the positions of vortex cores in each\nbasis function. We optimize the free energy with respect to\nthese vectors. The basis functions ψj±(r,δ)are obtained by\nsolvingthelinearizedGL equation,\n2πH\nΦ0/parenleftbigg\nκ(1+2Π +Π−)−ρ−Π2\n+−ρ+Π2\n−\n−ρ+Π2\n+−ρ−Π2\n−κ(1+2Π +Π−)/parenrightbigg/parenleftbigg\nψ1±\nψ2±/parenrightbigg\n= 2λ±\nmin/parenleftbiggψ1±\nψ2±/parenrightbigg\n. (12)\nWith use of the Landau level expansion, the basis func-\ntions are described as ψ1+(r,δ) =/summationtext\nn≥0a4nϕ4n(r,δ),\nψ2+(r,δ) =/summationtext\nn≥0a4n+2ϕ4n+2(r,δ)andψ1−(r,δ) =/summationtext\nn≥0b4n+2ϕ4n+2(r,δ),ψ2−(r,δ) =/summationtext\nn≥0b4nϕ4n(r,δ),\nrespectively. The n-th Landau level wave functions are de-\nnoted asϕn(r,0)andϕn(r,δ) =e−iδx(y−δy)ϕn(r−δ,0).\nλ+\nmin(λ−\nmin) is the minimum eigenvalue of Eq. (12) in the\npositive (negative) chirality channel. Variational param eters\n(C1,C2,C3,C4,δ2,δ3,δ4)areoptimizedtominimizetheGL\nfree-energy.\nH/H 0\nc2\nT/T 0\ncTc\nTc2\nTf.o.\nTcr\n 0 0.2 0.4 0.6 0.8 1\n0 0.2 0.4 0.6 0.8 1Chiral-II\nphase\nHelical phaseNon-unitary\nphase\nH/H 0\nc2\nT/T 0\ncTc\nTc2\nTf.o.\nTcr\n 0 0.2 0.4 0.6 0.8 1\n0 0.2 0.4 0.6 0.8 1Helical phaseChiral-II phase(a) (b)\nFig. 4. (Color online) Phase diagram of the four-component G L model\n[Eq. (6)] for temperatures Tand magnetic fields H//[001].45We as-\nsume(β2/β1,β3/β1,ξ1/ξ,ξ2/ξ,ξ3/ξ) = (0.5,0.5,1.2,0.83,0.3).\nSpin-orbit couplings are chosen to be (a) (ǫ,δ) = (−0.01,0.005)and\n(b)(ǫ,δ) = (−0.002,0.001). The unit of magnetic field is H0\nc2=\nΦ0/2πξ2.Thered solid line shows the SC transition temperature Tc(H).\nThe green dot-dashed line, blue double-dot-dashed line, an d black dashed\nline show the second order transition, first-order transiti on, and crossover\nin theSC state, respectively.\nWe show the H-Tphase diagramfor small spin-orbitcou-\nplings(ǫ,δ) = (−0.01,0.005)and that for tiny spin-orbit\ncouplings (ǫ,δ) = (−0.002,0.001)inFig.4(a)andFig.4(b),respectively.Weseethatnotonlythehelicalstatebutalso the\nchiral-II state and the non-unitary state appear in the phas e\ndiagram. The helical state is characterized by the variatio nal\nparameters |C1| ∼ |C4|andC2=C3= 0, and crossoversto\nthe non-unitary state ( |C1| ≫ |C4|) nearT=Tc(H). In the\nhigh magnetic field region, the chiral-II state is stabilize d by\nthe coupling of magnetic field and chirality. Then, we obtain\nthe variationalparameters, |C1| ∼ |C3| ≫ |C4| ∼ |C2|>0,\nwhich is described in the d-vector form as d= (px+ipy)ˆx\nord= (px+ipy)ˆy. This state is distinguished fromthe chi-\nral state [ d= (px±ipy)ˆz] because of the difference in the\ndirectionof d-vector.\nAlthough the H-Tphase diagram is independent of the\nsignofspin-orbitcouplings,the d-vectordependsonthesign\nofǫandδ. We here assume ǫ<0andδ >0so that the spin-\norbitcouplingfavorsthe A2ustate amongthefourhelicalSC\nstates. When we change the sign of ǫandδ, the d-vector in\nthe helical state changes as summarized in Tables III. The d-\nvector in the chiral-II and non-unitary states also depend o n\nthesignofξ1−ξ2.Weassume ξ1−ξ2>0inFig.4asusually\nexpected,46,47buttheperturbationanalysisofthethree-orbital\nHubbard model shows ξ1−ξ2<0.48The parameter depen-\ndenceofthe d-vectorissummarizedin TableIV.\nTheH-Tphase diagram for H/ba∇dbl[001]is basically de-\ntermined by the competition between the spin-orbit couplin g\nand the magnetic-field-chirality coupling. Generally spea k-\ning, the spin-orbit coupling stabilizes one of the irreduci ble\nrepresentationsin Table I. Indeed, the helical state belon ging\nto theA1u,A2u,B1u, orB2urepresentation is stabilized at\nlow magnetic fields as shown in Fig. 4. On the other hand,\nthe coupling of magnetic field and chirality favors the chira l\nSC state. Although the chirality is canceled out in the helic al\nstate, the chiral-II state, which belongs to a mixed represe n-\ntationofthe D4hpointgroupsymmetry,hasa finitechirality.\nThis is the reason why the chiral-II state is stabilized at hi gh\nmagneticfields.Thus,thechiral-IIphaseisstableinthela rge\npart ofH-Tphase diagram when the spin-orbit coupling is\ndecreased(see Fig.4(b)).\nUnfortunately, double SC transitions have not been ob-\nserved in Sr 2RuO4in the magnetic field along the c-axis up\nto now.2,3This experimental status is consistent with our re-\nsult for tiny spin-orbit couplings (Fig. 4(b)). We again str ess\nthat such a tiny spin-orbit coupling is compatible with the\nmicroscopic estimations based on the three-orbital Hubbar d\nmodel. Then, the helical phase at low magnetic fields may\nbe masked by the Meissner phase, which is not negligible in\nSr2RuO4havinga moderateGinzburgparameter κ∼2.6.3If\nso, thevortexstate inthe c-axismagneticfieldis thechiral-II\nphase. Thisphasehas anintriguingproperty,that is, the fr ac-\ntional vortex lattice. We showed that the fractional vortic es\naccompanied by the Majorana zero mode49form the lattice\nowingtothespin-orbitcoupling.45Thisisinsharpcontrastto\nthe fact that the fractional vortex is destabilized by the sp in-\norbitcouplinginnon-chiralspin-tripletsuperconductor s.50In\nother words, the chiral spin-triplet superconductor will b e a\ngood platform of the fractional vortex. Although the vortex\nlattice at low magnetic fields has been clarified by the small\nangle neutron scattering measurement,51future experimental\nsearches for the exotic vortex lattice structure at high mag -\nnetic fieldsare desired.Anotherintriguingfeatureisits t opo-\nlogically non-trivial property of the chiral II state. Rece ntly,J.Phys.Soc.Jpn. FULLPAPER Author Name 7\nthechiral-IIstate hasbeenidentifiedto bea topologicalcr ys-\ntallinesuperconductor.52\n5. Superconductingphasesin Sr 2RuO4forH/bardbl[100]\nNow we turn to the SC phases in the magnetic field along\nthe crystallographic a- orb-axis. TheEustate is robust\nagainst the paramagnetic depairing effect for this field dir ec-\ntion. Since the Eustate has two orbital components, the SC\ndouble transition occurs; the chiral state [ d= (px±ipy)ˆz]\nchanges to the non-chiral state [ d=pxˆzord=pyˆz] at a\nmoderate magnetic field. This chiral to non-chiral transiti on\nwaspredictedbyAgterbergusingtheGLtheory.46\n 0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1H/Hc2\nT/Tcky\nkx+iky\nApprox.\nFig. 5. (Coloronline) Phasediagram oftwocomponent (px,py)-wavesu-\nperconductor obtained by the quasi-classical Eilenberger equation.53Red\nmarks+show the high field state d=pyˆz, and green marks ×show the\nlow field state d= (δpx+ipy)ˆz. Dashed line shows the phase bound-\nary determined by the approximate analytical solution of th e Eilenberger\nequation (Pesch’s approximation).\nWe investigated the chiral to non-chiral transition us-\ning the quantitatively appropriate calculation based on th e\nquasi-classical theory.53We numerically solve the Eilen-\nberger equation for two component (px,py)-wave super-\nconductors with use of the Riccati equation and self-\nconsistently determine the order parameter ˆ∆(r,kF) =\nˆσx(∆x(r)φx(kF)+∆y(r)φy(kF))and the vector potential\nA(r). The chiral to non-chiral transition occurs as predicted\nby Agterberg (see Figure 5). We find that the px-wave com-\nponent∆x(r)vanishesathighfieldswhenwechoose φx(kF)\nandφy(kF)in accordance with the analysis of three-orbital\nHubbard model.48Thus, the non-chiral state ( d=pyˆz) is\nstabilized in the high magnetic field region. The chiral stat e\nd= (δpx+ipy)ˆz(0≤δ≤1)inthelowmagneticfieldregion\nadiabaticallychangestothezero-fieldstate d= (px+ipy)ˆz.\nLet us discuss the experimentaldata of Sr 2RuO4. Figure 5\nshows that the chiral to non-chiral transition occurs aroun d\nH∼0.6Hc2at low temperatures.54This is in agreement\nwith the experimental observation of the magnetization kin k\natH= 8∼9kOe (∼0.6Hc2) forT <0.6K.56Thus, the\nmagnetization kink may a fingerprint of the double SC tran-\nsition. On the other hand, specific heat measurements have\nnot observed any indication for the double SC transition at\nmoderatemagnetic fields.2,3Accordingto the quasi-classical\ntheory, the specific heat jump is too small to be observed at\nlow temperatures ( T <0.5Tc).53However, our calculationshowed a sizable jump of the specific heat around T=Tc,\nwhich has not been observed in experiments. This inconsis-\ntencymayimplythatthezero-fieldstateis notthechiralstate\n(Eustate). When the zero-field state is a helical state ( A1u,\nA2u,B1u, orB2ustate), the chiral to non-chiral transition\ndoesnotoccur.However,thisisnota strongevidenceagains t\ntheEustate, as the B-C phase transition of UPt 3was missed\nin the specific heat measurements.6Recent calculation based\non the quasi-classical theory also pointed out that the sign a-\nture of double SC transition disappears when the magnetic\nfield isslightlytiltedlessthan 1◦fromtheab-plane.57\n 0.4 0.5 0.6 0.7 0.8 0.9\n 0.4 0.5 0.6 0.7 0.8H/H0\nT/Tc\"triple point\"double\npeak\nsingle\npeakHmin\nHc2\n 0 0.3 0.6 0.9 1.2\n-0.2 -0.1 0 0.1C−Cn\n∆C(H)\nT−Tc(H)\nTcH/H0=0.4\n0.7\n0.8\n 0.9 1 1.1\n-0.05 0Tmin(a) (b)\nFig. 6. (Color online) (a) Phase diagram near the upper criti cal field for\nH/bardbl[100].53The green dashed line shows the upper critical field, and\nthe red solid line shows the crossover from the non-unitary s tate [d=\npy(ˆz−iˆy)] to the unitary state [ d=pyˆz]. (b) The specific heat shows a\ndouble peak (single peak) near Tcin the high (low) magnetic field region.\nWeobtained these results on the basis of the quasi-classica l theory of two\ncomponent order parameters, pyˆxandpyˆz. The Pesch’s approximation\nwas used to solve Elenberger equation.\nFinally,webrieflydiscussanotherphasetransitionnearth e\nupper critical field.48That is the unitary to non-unitary tran-\nsition from d=pyˆztod=py(ˆz−iˆy).58Strictly speaking,\nthis transition is a crossover in the presence of the spin-or bit\ncoupling. However, the specific heat shows a peak, when the\nspin-orbitcouplingis tiny η∼0.001.48Indeed,bothGL the-\nory48and quasi-classical theory53(see Fig. 6) reproducesthe\nfeatures of specific heat data indicating the double SC tran-\nsition.59However, recently observed first order transition at\nH=Hc260is not reproduced. Because any weak coupling\ntheoryforthespin-tripletsuperconductivityisnotcompa tible\nwith the first order SC transition with a sizable latent heat, it\nisdesiredtoexaminethestrongcouplingeffect,whichsome -\ntimeschangesthethermodynamicpropertiesofsuperconduc -\ntors.Inordertoexplainthefirst orderSCtransition,thesp in-\nsinglet superconductivityhasbeen consideredforSr 2RuO461\nin spiteofseveralcontradictoryexperiments.3\n6. Summary andDiscussion\nIn this article we reviewed the spin-orbit coupling in spin-\ntriplet Cooper pairs and multiple SC phases in Sr 2RuO4. We\ndemonstrated that the spin-orbit coupling arises from the L S\ncoupling of electrons. Interestingly, not only the magnitu de\nbutalsotherolesofthespin-orbitcouplingaredetermined by\ntheselectionrules whichoriginatefromthesymmetriesofthe\nlocalelectronorbital,crystalstructure,andsupercondu ctivity.\nTherefore,theanisotropyandtheeasyaxisofthe d-vectorare\ndeterminedwithoutrelyingonthemicroscopiccalculation .\nBasedontheselectionrulesandmicroscopicanalysisofthe\nthree-orbital Hubbard model for Sr 2RuO4we found that the8 J.Phys.Soc.Jpn. FULLPAPER Author Name\nchiral SC state ( Eustate) can be stabilized when the quasi-\ntwo-dimensional γ-band is mainly superconducting. On the\nother hand, one of the helical states ( A1u,A2u,B1u, orB2u\nstate) is stable when the quasi-one-dimensional( α,β)-bands\nareresponsibleforthe superconductivity.Thespin-orbit cou-\npling of Cooper pairs is tiny in the former case, althoughthe\nLScouplingismuchlargerthantheenergyscaleofthesuper-\nconductivity. On the other hand, a moderate spin-orbit cou-\nplingappearsinthelatter case.\nIt has been claimed that the superconductivityin Sr 2RuO4\nis likely caused by the γ-band, as it is indicated by the ther-\nmodynamic and transport properties.2,3The features of the\nspin-orbit coupling also point to this case according to the\ncomparison with NMR measurements28–30and the observa-\ntion of the half-quantum vortex32and TRSB.10,11However,\nsome controversialdata remainsto be resolved.For instanc e,\nthe chiral edge mode has not been detected,27and the O-site\nNQR measurement62implies the helical spin-triplet pairing\nstate.63These seemingly controversial data may be under-\nstood by consideringthe small spin-orbitcouplingin Coope r\npairs which allows various textures near the edges, domain\nwalls, andimpurities.\nWhen the spin-orbit coupling in spin-triplet Cooper pairs\nis small as estimated by the microscopic calculation, mul-\ntiple SC phase transitions occur in the magnetic field. We\nelucidated the H-Tphase diagram for both field directions\nH/ba∇dbl[001]andH/ba∇dbl[100]on the basis of the GL theory\nandquasi-classicaltheorytakingaccountofasmallspin-o rbit\ncoupling.For H/ba∇dbl[100],the unitaryto non-unitarytransition\noccurs near the upper critical field, and chiral to non-chira l\ntransition occursat moderatemagnetic fields. For H/ba∇dbl[001],\nthehighfieldSCphaseisthechiral-IIstate[ d= (px+ipy)ˆx\nord= (px+ipy)ˆy]whichaccompaniesthefractionalvortex\nlattice.TheseSCphasesarefingerprintofthesmallspin-or bit\ncoupling.We discussedexperimentalindicationsforthemu l-\ntipleSCphasesinSr 2RuO4,butconvincingevidenceforthem\nis still on the hunt. As the observation of multiple phases in\nUPt3has been an convincingevidencefor the spin-tripletsu-\nperconductivity,5,6it is desirable to elucidate whether multi-\npleSC phasesappearinSr 2RuO4ornot.Ourstudiesprovide\na basisforthefutureexperimentaltest.\nAcknowledgements\nA part of this work was carried out in collaboration with\nM. Mochizuki and M. Ogata. The authors are grateful to D.\nF.Agterberg,K.Deguchi,K.Ishida,S.Kittaka,Y.Maeno,K.\nMiyake,T.Nomura,M. Sigrist, K. Tenya,K. Yamada,andS.\nYonezawa for fruitful discussions. This work was supported\nby KAKENHI (Nos. 24740221, 24740230, and 25103711).\nPartofnumericalcomputationinthisworkwascarriedoutat\ntheYukawaInstituteComputerFacility.\n1) Y.Maeno,H.Hashimoto,K.Yoshida,S.Nishizaki, T.Fujit a, J.G.Bed-\nnorz, and F.Lichtenberg: Nature 372(1994) 532.\n2) A.P.Mackenzie and Y.Maeno: Rev. Mod. Phys. 75(2003) 657.\n3) Y. Maeno, S.Kittaka, T.Nomura, S. Yonezawa, and K. Ishida : J.Phys.\nSoc. Jpn. 81(2012) 011009.\n4) A.J.Leggett: Rev. Mod. Phys. 47(1975) 331.\n5) J.A.Sauls: Adv. Phys. 43(1994) 153.\n6) R. Joyntand L.Taillefer: Rev. Mod. Phys. 74(2002) 235.\n7) H. Tou, Y. Kitaoka, K. Asayama, N. Kimura, Y. Onuki, E. Yama moto,and K. Maezawa: Phys.Rev. Lett. 77(1996) 1374; H.Tou, Y.Kitaoka,\nK. Ishida, K. Asayama, N. Kimura, Y. Onuki, E. Yamamoto, Y. Ha ga,\nand K.Maezawa: Phys.Rev. Lett. 80(1998) 3129.\n8) For a review, D. Aoki and J. Flouquet: J. Phys. Soc. Jpn. 81(2012)\n011003.\n9) M. Sigrist and K.Ueda: Rev. Mod.Phys. 63(1991) 239.\n10) G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merr in, B.\nNachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamur a,\nand M.Sigrist: Nature 374(1998) 558.\n11) J.Xia,Y.Maeno,P.T.Beyersdorf,M.M.Fejer,andA.Kapi tulnik:Phys.\nRev. Lett. 97(2006) 167002.\n12) Y. Yanase and H.Harima: Kotai Butsuri 46(2011) 229 (in Japanese).\n13) K.Yosida: “Theory ofMagnetism” (Springer-Verlag, 1996).\n14) Y. Yanase and M.Ogata: J. Phys.Soc.Jpn. 72(2003) 673.\n15) Y. Yanase, M, Mochizuki and M. Ogata: J. Phys. Soc. Jpn. 74(2005)\n2568.\n16) T.Oguchi: Phys.Rev. B 51(1995) 1385.\n17) D.J. Singh: Phys.Rev. B 52(1995) 1358.\n18) Y.Yanase,T.Jujo,T.Nomura,H.Ikeda,T.Hotta,andK.Ya mada:Phys.\nRep.387(2003) 1.\n19) T.Nomuraand K.Yamada: J.Phys.Soc. Jpn. 71(2002) 1993.\n20) K.Hoshihara and K.Miyake: J.Phys.Soc. Jpn. 74(2005) 2679.\n21) Q.-H.Wang, C. Platt, Y. Yang, C. Honerkamp, F.C. Zhang, W .Hanke,\nT.M.Rice, and R.Thomale: Europhys.Lett. 104(2013) 17013.\n22) T.Takimoto: Phys.Rev. B 62(2000) R14641.\n23) S.Raghu,A.Kapitulnik,andS.A.Kivelson:Phys.Rev.Le tt.105(2010)\n136401.\n24) D. F.Agterberg, T.M. Rice, and M. Sigrist: Phys. Rev. Let t.78(1997)\n3374.\n25) K.K. Ngand M.Sigrist: Europhys.Lett. 49(2000) 473.\n26) Y. Yoshioka and K.Miyake: J.Phys.Soc. Jpn. 78(2009) 074701.\n27) For areview, C.Kallin: Rep. Prog. Phys. 75(2012) 042501.\n28) K.Ishida,H.Mukuda,Y.Kitaoka,K.Asayama,Z.Q.Mao,Y. Mori,and\nY. Maeno: Nature 396(1998) 658.\n29) H.Murakawa, K.Ishida,K.Kitagawa, Z.Q.Mao,and Y.Maen o: Phys.\nRev. Lett. 93(2004) 167004.\n30) H. Murakawa, K. Ishida, K. Kitagawa, H. Ikeda, Z. Q. Mao, a nd Y.\nMaeno: J.Phys.Soc. Jpn. 76(2007) 024716.\n31) K. Kitagawa, K. Ishida, R. S. Perry, H. Murakawa, K. Yoshi mura, and\nY. Maeno: Phys.Rev. B 75(2007) 024421.\n32) J. Jang, D.G. Ferguson, V.Vakaryuk, R.Budakian, S.B. Ch ung, P.M.\nGoldbard, and Y. Maeno: Science 331(2011) 186.\n33) G.E.Volovik and V.P.Mineev: JETPLett. 24(1976) 561.\n34) H.-Y. Keeand M.Sigrist: arXiv:1307.5859.\n35) S.B.Chung, H.Bluhm,E.-A.Kim,Phys.Rev.Lett. 99(2007) 197002.\n36) V. Vakaryuk and A.J.Leggett, Phys.Rev. Lett. 103(2009) 057003.\n37) T.Nomuraand K.Yamada: J.Phys.Soc. Jpn. 71(2002) 404.\n38) T.Nomura: J.Phys.Soc. Jpn. 74(2005) 1818.\n39)Non-Centrosymmetric Superconductors: Introduction and O verview,\ned. by E.Bauer and M.Sigrist (Springer-Verlag, 2012).\n40) P.A.Frigeri, D.F.Agterberg, A.Koga,and M.Sigrist: Ph ys.Rev.Lett.\n92(2004) 097001.\n41) Y. Yanase and M.Sigrist: J.Phys.Soc. Jpn. 77(2008) 124711.\n42) Y. Yanase: J.Phys.Soc. Jpn. 82(2013) 044711.\n43) Y. Yanase: J.Phys.Soc. Jpn. 79(2010) 084701.\n44) S. Kittaka, S. Fusanobori, S. Yonezawa, H.Yaguchi, Y. Ma eno, R. Fit-\ntipaldi, and A.Vecchione: Phys.Rev. B 77(2008) 214511.\n45) S.Takamatsu and Y.Yanase: J.Phys.Soc. Jpn. 82(2013) 063706.\n46) D.F.Agterberg: Phys. Rev. Lett. 80(1998) 5184.\n47) D.F.Agterberg: Phys. Rev. B 58(1998) 14484.\n48) M. Udagawa, Y. Yanase, and M. Ogata: J. Phys. Soc. Jpn. 74(2005)\n2905.\n49) D.A. Ivanov: Phys.Rev. Lett. 86(2001) 268.\n50) S. B. Chung, D. F. Agterberg, and E.-A. Kim: New J. Phys. 11(2009)\n085004.\n51) T.M.Riseman,P.G.Kealy,E.M.Forgan,A.P.Mackenzie,L .M.Galvin,\nA.W.Tyler,S.L.Lee,C.Ager,D.McK.Paul,C.M.Aegerter,R. Cubitt,\nZ.Q.Mao,T.Akima,andY.Maeno:Nature 396(1998)19; 404(2000)\n629.\n52) Y. Ueno, A. Yamakage, Y. Tanaka, and M. Sato: Phys. Rev. Le tt.111\n(2013) 087002.\n53) M. Udagawa: Doctor Thesis in University of Tokyo (2007).\n54) The second order chiral to non-chiral phase transition o ccurs at mod-J.Phys.Soc.Jpn. FULLPAPER Author Name 9\nerate magnetic fields unless we precisely adjust the paramet ers so that\nξ1≈ξ2.55\n55) R. P. Kaur, D. F. Agterberg, and H. Kusunose: Phys. Rev. B 72(2005)\n144528.\n56) K. Tenya, S. Yasuda, M. Yokoyama, H. Amitsuka, K. Deguchi , and Y.\nMaeno: J.Phys.Soc. Jpn. 75(2006) 023702.\n57) M. Ishihara, Y. Amano, M. Ichioka, and K. Machida: Phys. R ev. B87\n(2013) 224509.\n58) Theunitary to non-unitary transition is similar to the A1-A2transition\nof superfluid3He.459) K.Deguchi, M.A.Tanatar, Z.Mao,T.Ishiguro, andY.Maen o: J.Phys.\nSoc. Jpn. 71(2002) 2839.\n60) S. Yonezawa, T. Kajikawa, and Y. Maeno: Phys. Rev. Lett. 110(2013)\n077003.\n61) K.Machida and M.Ichioka: Phys.Rev. B 77(2008) 184515.\n62) H.Mukuda, K.Ishida, Y.Kitaoka, K.Miyake, Z.Q.Mao, Y.M ori,and\nY. Maeno: Phys.Rev. B 65(2002) 132507.\n63) K.Miyake: J.Phys.Soc. Jpn. 79(2010) 024714.10 J.Phys.Soc. Jpn. FULLPAPER Author Name\nCrystal symmetry Tetragonal Hexagonal\nLocal electron orbital dxy dyz,dzx A1g Eg\nOrbital symmetry of SC P-wave P-or F-wave P-wave F-wave\nEasy axis of d-vector (d/bardblc)d/bardblab both d/bardblab both\nAnisotropy [η= 1−TΓ1c/TΓ0c]O/parenleftbig\nλ2/E2\nF/parenrightbig\nO(λ/EF)O/parenleftbig\nλ2/E2\nF/parenrightbig\nO(λ/EF)O/parenleftbig\nλ2/E2\nF/parenrightbig\nTable II. Tableofspin-tripletsuperconductivity in t2gelectron systems.14,15Weshowtheselection ruleswhichdeterminetheeasyaxisand theanisotropyof\nspin-triplet Cooper pairs onthebasis ofthesymmetries ofc rystal, local electron orbital, andCooper pairs’ orbital. Although theeasy axis isnotdetermined\nbytheselection ruleforthed xy-orbitalinthetetragonallattice, weshowtheresultobtai ned bytheperturbation theoryassumingtheparametersforS r2RuO4\n(Fig. 3(a)).\nǫ >0 ǫ <0\nδ >0 δ <0 δ >0 δ <0\npxˆx+pyˆypyˆx−pxˆypxˆx−pyˆypyˆx+pxˆy\nTable III. Thed-vector in the helical state (last row) when w eassumethe sign of couplings ǫandδas in thefirstand second row,respectively.\nξ1> ξ2 ξ1< ξ2\nǫ >0 ǫ <0 ǫ >0 ǫ <0\nChiral II state (px+ipy)ˆx,(px+ipy)ˆy (px−ipy)ˆx,(px−ipy)ˆy\nNon-unitary state (ˆx−iˆy)(px+ipy)(ˆx+iˆy)(px+ipy)(ˆx+iˆy)(px−ipy)(ˆx−iˆy)(px−ipy)\nTable IV. The d-vector in the chiral II state (third row) and i n the non-unitary state (fourth row). The chirality in these states depends on the magnitude\nrelation between ξ1andξ2(first row). The spin component in the non-unitary state also depends on the sign of ǫ(second row). The two-fold degeneracy\nbetweend/bardblˆxandd/bardblˆyis not lifted in the chiral-II state." }, { "title": "2010.02289v1.Terrestrial_Orbit_Spin_Coupling_Torque_Episodes_in_Late_2020.pdf", "content": "Terrestrial Orbit -Spin Coupling To rque E pisodes in Late 2020 \n \nJames H. Shirley \nJet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA \n5 October 2020 \n \nAbstract: Orbit -spin coupling torques on the Earth in Nove mber 2020 are larger than at any \nother time between 2000 and 2050. This affords an opportunity to observe the terrestrial \natmospheric response to the putative torque in near real time. \nFigure 1 illustrates the variability with time of orbit -spin coupling torque s on the Earth in \n2020. Fig. 1 is annotated with a number of key dates. T he peak dates of three future torque \nepisodes are indicated . The torqu es peak near 12 October, 8 November, and 7 December 2020. \nThe dates of the minim a separating the episodes are also of interest . Earth recently passed \nthrough a near-zero-torque interval at the beginning of October. Subsequent minima are found on \ndays 296, 326, and 355 of 2020, corresponding to 23 October, 21 November, and 21 December, \nrespectively. \n \nFigure 1 . Diurnal and seasonal v ariability of the magnitude of the orbit -spin coupling torque for \na location on Earth ’s equator (time step 6 hours). See Methods (below) for calculations. \n The orbit -spin coupling hypothesis [1] predicts that an intensific ation, or powering up, of \nthe large -scale circulation of the atmosphere will occur at times when the torque s are large. We \nthus expect that m omentum will be deposited , within the terrestrial atmosphere, as a \nconsequence of the torque, during each of the upcoming torque episodes of Fig. 1. Experience \ngained through ext ensive simulation of the Mars atmosphere suggests that a strengthening of \nmeridi onal overturning circulations may be expected, together with a strengthening of pressure \ngradien ts for mid -latitudes c yclones and anticyclones. Stronger variability of local weather \nconditions is an expected outcome with a more vigorous large -scale circulation . In the absence of \nnume rical modeling results for this subject body and this time period, we cannot at present more \naccurat ely specify the changes in atmospheric conditions and behaviors that may accompany t he \nanticipated spinning -up of the atmospheric circulation during these three episodes . \n The year 2020 has been one of remarkable extremes of weather and climate. It is beyond \nthe scope of this short note to attempt to correlate past weather events with all of the oscillations \ndisplayed in Fig. 1. However, we have added three additional key dates to Fig. 1 that identify \ntwo recent torque episode peak s, together with one relaxation -phase date. Consideration of these \nepisodes yields added perspective on possible outcomes during the three future torque episodes \nof Fig. 1. \n The peak of the mid-August torque episode of Fig. 1 corresponds in time to an outbreak \nof >12,000 lightning strikes in Northern California. For est fires from that episode are still \nburning. (The excess energy of forced atmospheric motions is largely dissipated by frictional \nprocesses [ 2]). \n The peak of the early September torque episode corresponds in time to the occurrence of \nthe early -season snowstorm that brought extreme temperature changes in B oulder and Denver. \n Each of the se examples describe s phenomena that may result from or be associated with a \npowering -up of the large -scale circulation on short timescales. The third example relates to the \nopposite condition, when the torques are much reduce d. Experience at Mars has shown that the \nspinning -down of a previously intensifi ed circulation may likewise give rise to distinctive \nweather events a nd conditions [2]. The arrow symbol at 22 September 2020 c orresponds to the \nstalling of tropical storm Beta over the U. S. G ulf co ast. This storm dumped ~10 i nches of rain in \nSoutheast Texas and produced flooding in Texas, Arkansas, and Louisiana. The relaxation phase \nof the torque “cycle ” is characterized by much reduced deposi tion of momentum within the \natmosphere. At such times, the spun -up atmosphere may “take a breather, ” plausibly accounting \nfor phenomena such as the stalling -out of tro pical storm Beta. \n We will assess the applicability and accuracy of the above predictions after the \nhighlighted three -month period is completed. Future work will more fully character ize the nature \nof the terrestrial atmospheric response to the orbit -spin coupling torques. Possible lags in the \nresponse are of interest. Quantitative analyses may be formulated and are encouraged. \nBackground: \n The orbit -spin coupling hypot hesis [1] predicts driven cycles of intensification and \nrelaxation of planetary atmospheric circulations . The coupling, given by expression (1), takes the \nform of a reversing torque [1-4] with axis lying in the equatorial plane of the subject body ; \n - c (L̇ ωα) r. (1) \n \nHere L̇ (or dL/dt) represents the time rate of change of the orbital angular momentum of the \nsubject body with respect to the solar system center of mass (or barycenter ), while the axial \nrotation of the subject body (with respect to the same inertial coordinate system ) is represented \nby the angular velocity vector ωα. r denotes a p osition vector in a rotating Cartesian body -fixed \ncoordinate system, while c is a scalar coupling efficiency coefficient , which is constrained by \nobservations of planetary motions to be quite small [ 1, 3-5]. Expression (1) describe s a global \nacceleration field , with units of m s-2, where in the accelerations everywhere lie in directions \ntangential to a spherical surface. \n The orbit -spin coupling hypothesis has been remarkably successful in explaining the \nintermittent occurrence of global dust storms on Mars [ 3-5]. Atmospheric general circulation \nmodel simulations [ 4, 5] reveal that a n intermittent strengthening and weak ening of meridional \noverturning circulations is a characteri stic feature of the mechanism . The predicted \nintensification was observed by spacecraft observations at the sta rt of the Martian global dust \nstorm of 2018 [6]. Adding orbi t-spin coupling accelerations to Mars GCMs significantly \nimproves the agreement between atmospheric simulations and observations [4-5, 7]. \n Recent work [7] indicate s that the orbit -spin coupling torques on the Earth are \nsignificantly larger than those on Mars . The terrestrial torques , in addition , cycle much more \nrapidly, due to the presence of Earth ’s Moon . (The nearby Moon gives rise to significant time \nvariability of the Earth ’s orbital angular momentum with respect to the solar system barycenter ). \nWe have thus begun to investigate the question of the possible response of the Earth atmosphere \nto the deterministic orbit -spin coupling torques given by expression (1). \n \nMethods \n In prio r work we have presented curves representing the time rate of change of the orbital \nangular momentum (with respect to the solar system barycenter) (dL/dt) in order to characteriz e \nthe orbit -spin coupling “forcing function. ” No information regarding the quasi -diurnal cycle of \nthe acceleration at any specific location is provided by this approach. In F ig. 1 of this note , we \nhave gone one s tep farther, for ming the cross product (L̇ ωα) as a functi on of time. This \nidentifies a vector lying in the equatorial plane of the subject body. While the direction in inertial \nspace of this vec tor does not greatly chan ge over short periods, its direction within the \nconventional body -fixed Cartesian coo rdinate frame does cycle (approximately over the sidereal \nperiod ) as a consequence of Earth ’s rotational motion. The curve in Fig. 1 is obtained by \nsampling the quasi -sinusoidal variation of the x or y component of the (L̇ ωα) vector at 6 hour \nintervals. We emphasiz e that this approach serves mainly to reveal the variability of the \nmagnitude of the torque. Fo cused studies will require a more sophisticated approach. \n Methods for obtaining the angular momentum of E arth (or any other solar system body) \nwith respect to th e solar system barycenter are described in [1 -3]. To obtain units of acceleration, \nthe values on the y -axis of Fig. 1 must be multiplied by two factors: the va lue (in meters) of r, the 3-component position vector for a given location; and c, the coupling efficiency coefficient. \nThe coefficient value has not yet been determined for the case of the terrestrial atmosphere. An \noptimized c value obtaine d for the Mars a tmosphere [ 4] was c=5.0 x10-13. \nAcknowledgem ents \n Portions of t his work w ere performed at the Jet Propulsion Laboratory, California \nInstitute of Technology, u nder a contract from NASA. Copyright 2020 , California Institute of \nTechnolog y. Gov ernment sponsorship acknowledged. \n \n \nReferenc es \n[1] Shirley, J.H. (2017). Orbit -spin coupling and the circulation of the Martian atmosphere, \nPlanetary and Space Science 141, 1 –16. doi: 10.1016/j.pss .2017.04.006 \n[2] Shirley, J. H., McKim, R. J., Battalio, J. M., & Kass, D. M. (2020). Orbit -spin coupling and \nthe triggering of the Martian planet -encircling dust storm of 2018. Journal of \nGeophysical Research: Planets , 125, e2019JE006077. \nhttps://doi.org/10.1029/2019JE006077 \n[3] Shirley, J. H. & Mischna, M.A. (2017). Orbit -spin coupling and the interannual variability of \nglobal -scale dust storm occurrence on Mars, Planetary and Space Science 139, 37 –50. \ndoi: 10.1016/j.pss.2017.01.001 \n[4] Mischna, M. A., & Shirley, J. H. (2017). Numerical Modeling of Orbit -Spin Coupling \nAccelerations in a Mars General Circulation Model: Implications for Global Dust Storm \nActivity, Planetary and Space Science 141, 45 –72. doi: 10.1016/j.pss.2017.04.003 \n[5] Shirley, J. H., Newman, C. E., Mischna, M. A., & Richardson, M. I. (2019). Replica tion of \nthe historic record of M artian global dust storm occurrence in an atmospheric general \ncirculation model, Icarus 317, 192 -208. https://doi.org/10.1016/j.icarus.2018.07.024 \n[6] Shirley, J. H., Kleinbӧhl, A., Kass, D. M., Steele, L. J., Heavens, N. G., Suzuki, S., Piqueux, \nS., Schofield, J. T., & McCleese, D. J. (2019b). Rapid expansion and evolution of a \nreginal dust storm in the Acidalia Corridor during the in itial growth phase of the Martian \nGlobal dust storm of 2018, Geophysical Research Letters 46. \nhttps://doi.org/10.1029/2019GL084317 \n[7] Shirley, J. H., 2020. Nontidal coupling of the orbital and rotati onal motions of extended \nbodies, in preparation . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n " }, { "title": "1006.1202v3.Duality_of_the_spin_and_density_dynamics_for_two_dimensional_electrons_with_a_spin_orbit_coupling.pdf", "content": "arXiv:1006.1202v3 [cond-mat.mes-hall] 19 Oct 2010Duality of the spin and density dynamics for two-dimensiona l electrons with a\nspin-orbit coupling\nI. V. Tokatly1,3and E. Ya. Sherman2,3\n1Nano-bio Spectroscopy group and ETSF Scientific Developmen t Centre,\nDpto. F´ ısica de Materiales, Universidad del Pa´ ıs Vasco,\nCentro de F´ ısica de Materiale CSIC-UPV/EHU-MPC, E-20018 S an Sebastian, Spain\n2Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080 Bilbao, Spain\n3Basque Foundation for Science IKERBASQUE, 48011, Bilbao, S pain\nWe study spin dynamics in a two-dimensional electron gas wit h a pure gauge non-Abelian spin-\norbit field, for which systems with balanced Rashba and Dress elhaus spin-orbit couplings, and the\n(110)-axis grown GaAs quantum wells are typical examples. W e demonstrate the duality of the\nspin evolution and the electron density dynamics in a system without spin-orbit coupling, which\nconsiderably simplifies and deepens the analysis of spin-de pendent processes. This duality opens\na venue for the understanding of this class of systems, highl y interesting for their applications in\nspintronics, through known properties of the systems witho ut spin-orbit coupling.\nPACS numbers: 72.25.-b\nThe understanding of spin dynamics in a two-\ndimensional (2D) electron gas with spin-orbit (SO) in-\nteraction is highly important both for theoretical and\napplied spintronics, [1–4] including the design of devices\nwith controlled spin transport. In many physically in-\nteresting situations the SO coupling can be elegantly de-\nscribed as an effective non-Abelian vector potential. [5–\n17]Thereexistsaclassofsystems, wheretheSOcoupling\ncorrespondsto apure gaugenon-Abelian field. Therefore\nit can be gauged away and the behavior of a physical\nsystem should map to that of a system without SO cou-\npling. In practice, there are at least two systems of this\nsort widely investigated from the applied point of view.\nThese are 2D electrons with balanced Rashba and Dres-\nselhaus couplings, and the electron gas in the (110)-axis\ngrown GaAs quantum wells.\nThe coupled spin-charge dynamics is commonly de-\nscribed using the diffusion approximation, where the rate\nof the spin precession is much less than the momentum\nscatteringrate.[18,19]Currently,high-mobility2Dstruc-\ntures, where the time scale of momentum relaxation is\nlonger than the spin rotation time,[20, 21] became avail-\nable. Since here spins can make several turns between\ncollisions with impurities, the conventional Dyakonov-\nPerel mechanism [18] is not applicable,[22] and another\ntype of analysisis required. In the present paper we solve\nthis problem for systems with a pure gauge SO coupling,\nincluding quantum effects due to the weak localization.\nWe show for this general class of systems the existence\nof a duality of observables allowing the spin dynamics\nto be fully mapped to the density dynamics in systems\nwithout SO coupling. As a result, several regimes, in-\ncluding magnetic field dependence of the spin dynamics,\ncan easily be explored using a single formula.\nWerepresenttheHamiltonianofa2Delectrongaswith\nSO coupling as follows [16, 17] (the system of units with¯h= 1 is employed)\nH=1\n2m/integraldisplay\nd2ρΨ+(i∂i+Ai)2Ψ+W/bracketleftbig\nΨ+,Ψ/bracketrightbig\n,(1)\nwhere Ψ( ρ) is a spinor field operator, and the func-\ntionalW[Ψ+,Ψ] contains all spin-independent contribu-\ntions, including the external potential, electron-electron\ninteractions, and, possibly, the electron-phononcoupling.\nHeremis the electron effective mass, and Aiare 2×2\nmatrix-valuedcomponentsofanon-Abelian SU(2) gauge\nfield describing the SO coupling. In the broad class of\nsystems of interest, Aiis a pure gauge, that is it can be\nremoved by a local SU(2) transformation. The general\nform of a pure gauge vector potential,\nAi=mαi(h·σ), (2)\ncorresponds to the following SO Hamiltonian\nHso=α(h·σ)(k·ν), (3)\nwhereσis a vector of Pauli matrices, αis the SO\ncoupling constant, his a three-component unit vec-\ntor for the SO field direction, νis a 2D vector in the\n(x,y) plane, and αi=ανi.The two practically impor-\ntant systems described by the Hamiltonian of Eq. (3)\nare: (i) the balanced Rashba-Dresselhaus system [23, 24]\nwithh=(±1,±1,0)/√\n2,ν= (±1,±1)/√\n2,and (ii)\nthe (110)-axis grown GaAs quantum well,[25–28] where\nh= (0,0,1),ν= (1,0) with the well axes chosen with\nrespect to the crystal axes as x/bardbl[110], y/bardbl[001],andz/bardbl\n[110]. Both systems are expected to demonstrate highly\nanisotropic spin relaxation times with the spin compo-\nnent along the h-axis having a very low relaxation rate,\narising only due to a spin-dependent disorder.[29] Spin\ncurrents in the thermodynamical equilibrium state,[30]\nbeing common for 2Delectron systems with SO coupling,\nare absent [16] in the structures described by the Hamil-\ntonian in Eq. (3).2\nA localSU(2) transformation, which gauges away the\nabove type of SO coupling is ˜Ψ(ρ) =UAΨ(ρ), where\nUA= exp[imα(h·σ)(ρ·ν)]. (4)\nThe transformation (4) renders invariant all spin-\nindependent quantities, such as the charge and current\ndensities, while the spin density transforms covariantly:\n/tildewideS=1\n2tr{σU−1\nA(S·σ)UA}. (5)\nWhen the SO coupling is gauged away the dynamics\nof the transformed spin density /tildewideS(ρ,t) reduces to the\nspin dynamics in the electron gas without SO interac-\ntion, which significantly simplifies the analysis. Then,\nthe physical spin density S(ρ,t) is restored by\nS=1\n2tr{σUA(/tildewideS·σ)U−1\nA}, (6)\nto obtain measurable results. Here we follow this guide-\nline and show that this approach allows to describe all\nregimes of spin dynamics on the same footing.\nWe consider a 2D electron gas with a SO coupling of\nEq. (1) and, initially, a uniform spin density Sproduced,\nfor example, by a static magnetic field B. Att= 0\nthe magnetic field is released and the spin relaxes due to\na disorder potential and other interactions. To describe\nthis process we first eliminate the SO by the gauge trans-\nformation of Eq. (4). Using Eq. (5) we find that the ini-\ntial uniform physical spin density is mapped to the spin\ntexture/tildewideS(ρ,0) =/tildewideS/bardbl(ρ,0)+/tildewideS⊥(ρ,0),where the term\n/tildewideS/bardbl(ρ,0) =h(S·h), (7)\nbeing parallel to h, is untouched by the transformation\nand remains uniform, while the orthogonal to hpart\ntransforms into the helix structure [31, 32]\n/tildewideS⊥(ρ,0) = [S−h(S·h)]cos(Qhx·ρ)−(S×h)sin(Qhx·ρ),\n(8)\nwhereQhx= 2mανis the helix wave vector.\nSince in the transformed system there is no SO cou-\npling, the uniform part of the initial spin distribution,\nEq. (7), is constant in time. A nontrivial dynamics oc-\ncurs in the orthogonal channel due to a diffusional decay\nof the initial helix spin texture described by Eq. (8):\n/tildewideSβ1\n⊥(ρ,t) =/integraldisplay\nDβ1β2(ρ−ρ′,t)/tildewideSβ2\n⊥(ρ′,0)d2ρ′,(9)\nwhereDβ1β2(ρ−ρ′,t) is the exact spin diffusion Green’s\nfunction of a 2D electron gas, which takes into ac-\ncountthedisorder,electron-electronandelectron-phonon\ninteractions. To proceed further we note that in a\nnonmagnetic system without SO coupling the spin dif-\nfusion Green’s function is diagonal in spin subspace\nDβ1β2(ρ,t) =δβ1β2D(ρ,t). Hence Eq. (9) simplifies as\n/tildewideS⊥(ρ,t) =/tildewideS⊥(ρ,0)D(Qhx,t), (10)whereD(q,t) is a Fourier component of the spin diffusion\nGreen’s function\nD(q,t) =/integraldisplay\nd2ρe−i(q·ρ)D(ρ,t), (11)\nand we have taken into account that only the Fourier\ncomponents of D(ρ,t) with the modulus of the wave vec-\ntorq=Qhxcontribute to the dynamics of the helix in\nEq. (8). Since the time-dependent factor in Eq. (10)\nis scalar, the transformation back to the physical spin,\nEq. (6), simply reduces to removing tildas and the coor-\ndinatedependenceinEq.(10). Thus,wegetthefollowing\nexactresult for the observable spin evolution\nS⊥(t) =S⊥(0)/integraldisplaydω\n2πD(Qhx,ω)e−iωt.(12)\nIn Eq. (12) we represented D(q,t) via the Fourier inte-\ngral because in the ω-domain there is a simple expression\nof the spin diffusion Green’s function D(q,ω) in terms\nof the spin-spin correlator (the spin response function)\nχ[S]\nββ(q,ω), where β= (x,y,z) is the Cartesian index cor-\nresponding to the spin component\nD(Qhx,ω) =1\niω/bracketleftBigg\nχ[S]\nββ(Qhx,ω)\nχ[S]\nββ(Qhx,0)−1/bracketrightBigg\n.(13)\nThis equation can be derived by considering a linear re-\nsponse on a time-dependent magnetic field that is adi-\nabatically switched on at t=−∞, and then suddenly\nswitched off at t= 0, i. e. B(t) =eδtθ(−t)Bwithδ→0\n(see, e. g., Ref. [33] for similar calculations). Usually the\nSO coupling is weak on the Fermi energy scale, which\nimpliesQhx≪kF, wherekFis the Fermi momentum.\nTherefore in most situations one can safely replace the\nstaticω= 0 response function in Eq. (13) by the macro-\nscopic Pauli spin susceptibility, χ[S]\nββ(Qhx,0) =χP.\nEquations (12) and (13) are the result of the spin-\ndensity dynamics duality and give the exactevolution\nof the uniform spin density. The problem is solved\nby mapping the spin relaxation in the physical system\nto the “washing out” an inhomogeneous spin texture\nin a dual system without SO coupling. The real spin\nrelaxes because of SO-induced precession and random-\nness introduced by disorder, phonons, and interelectron\ninteractions.[34] For the transformed spin, it is the evolu-\ntionofthenonuniformspindensitydistributions, againin\nthe presence of the disorder and interactions between the\ncarriers. An interesting exact feature of the pure gauge\nSO coupling (in addition to the well known anisotropy)\nis the absence of the spin precession – the vector S⊥(t)\nis always collinear to its initial direction. In the trans-\nformed picture this is related to the diagonal structure of\nthe spin response in a nonmagnetic electron gas. For the\nreal system this translates to the fact that spins of elec-\ntrons with opposite momenta precess around the h-axis\nin the opposite directions with the same rate.3\nAt the level of the random phase approximation the\nspin response function χ[S]\nββ(q,ω) is equal to the den-\nsity response function χ(q,ω) of a noninteracting, but\npossibly disordered and/or coupled to phonons electron\ngas, while the Pauli susceptibility χPis proportional\nto the compressibility ∂n/∂µ, withnandµbeing the\nelectron concentration and chemical potential, respec-\ntively. Hence the spin diffusion Green’s function entering\nEq. (12) reduces to\nD(Qhx,ω) =1\niω/bracketleftbiggχ(Qhx,ω)\n∂n/∂µ−1/bracketrightbigg\n,(14)\nwhich is exactly the density diffusion Green’s function.\nTherefore in this physically important case the spin re-\nlaxation is mapped to the ordinary density diffusion.\nNow we apply Eqs. (14) and (12) to a noninteracting\ndisordered 2D electron gas with a momentum relaxation\ntimeτ, andstudypossibleregimesofspindynamics. The\ndensity-density correlator χ(Qhx,ω) can be obtained ei-\nther diagrammaticallyor by solving the kinetic equation.\nIn the semiclassicalregime, correspondingto the summa-\ntion of ladder diagrams, one obtains:\nD(Qhx,ω) =K(Qhx,ω)\n1−K(Qhx,ω), (15)\nK(Qhx,ω) =1\n2π/integraldisplaydθ\n1−iωτ+iΩsoτcosθ,(16)\nwhere the only SO-dependent parameter in the problem\nΩso≡QhxvF(vFis the Fermi velocity) is the maximum\nspin precession rate, and Ω soτ=ℓQhx(electron mean\nfree path ℓ=vFτ) characterizes the relaxation regime.\nWe begin with Ω soτ≪1 regime, which coresponds to\na pure diffusion, studied in the coordinate representation\ninRef. [17]. HerethediffusionGreen’sfunction, Eq.(15),\nreduces to\nD(Qhx,ω) =1\nDQ2\nhx−iω, (17)\nwhereD=v2\nFτ/2 is the diffusion coefficient. Inserting\nD(Qhx,ω) of Eq. (17) into Eq. (12) we obtain:\nS⊥(t) =S⊥(0)exp/parenleftbig\n−DQ2\nhxt/parenrightbig\n, (18)\nwhich exactly corresponds to the Dyakonov-Perel’ mech-\nanism with the spin relaxation rate Γ s=DQ2\nhx. More-\nover, the factor 1 /2 in the definition of Dacquires an\ninteresting physical meaning in terms of the spin pre-\ncession: it corresponds to the angular averaging of the\nprecession rate/angbracketleftbig\nΩ2\nso(k)/angbracketrightbig\n= Ω2\nso/2.\nThe opposite, clean limit Ω soτ≫1, in terms of the\ndual (transformed) system corresponds to a reversible,\npurely ballistic washing out the helix texture. In this\nregimeD(Qhx,ω)≈ K(Qhx,ω) (see Ref.[35]) and the\nintegration in Eqs. (12) and (16) yields:\nS⊥(t) =S⊥(0)J0(Ωsot) =S⊥(0)J0(QhxvFt),(19)\nFIG. 1: (Color online.) Time dependence of the spin for\ndifferent parameters of SO coupling, shown near the plots,\nwiths(t) defined as S⊥(t)≡s(t)S⊥(0)/S⊥(0).\nwhereJ0(Ωsot) is the Bessel function. The same result\ncan be derived directly from the microscopic spin preces-\nsion with a k-dependent rate Ω so(k) = Ωsocosφ, where\nφis the angle between kandν. Indeed, the net result of\nthe inhomogeneous precession reproduces Eq. (19),\nS⊥(t) =S⊥(0)/integraldisplay\ncos(Ωsotcosφ)dφ\n2π=S⊥(0)J0(Ωsot).\n(20)\nIn contrast, in the systems with pure Rashba or Dressel-\nhaus coupling, spin precession rate does not depend on\nthe direction of momentum. As a result, in the ballistic\nregime, the z-component of the total spin demonstrates\ncosine-like rather than the Bessel function time depen-\ndence.\nAn intermediate regime of Ω soτ∼1, can be investi-\ngated numerically. The results are presented in Fig.1 for\ndifferentparameters. Onecanclearlyseeacrossoverfrom\nthe oscillating Bessel function-like behavior to the expo-\nnential Dyakonov-Perel’ decay. At short times, the be-\nhavior of spin is universal: S⊥(t) =S⊥(0)/parenleftbig\n1−Ω2\nsot2/2/parenrightbig\ndue to the unperturbed precession of the spins. For the\ndensity dynamics the universal short-time behavior is a\ndirect consequence of the f-sum rule,[36] as can be seen\nby expanding Eq. (12) at t→0.\nAnalysis of Eqs. (12),(15), and (16) shows that Ω soτ=\n1 is a critical point. With the decrease in Ω soτin the\nrange Ω soτ >1,the first zero of S⊥(t) rapidly shifts to\nlarger times, with negative value regions becoming very\nshallow. At Ω soτ <1 zeroes of S⊥(t) disappear and the\ndynamics is a pure decay.\nThe gauge transformation approach allows to analyze\nsystems, where a direct treatment of the SO coupling\nwould cause difficulties. The first effect we consider is\nthe influence of the orbital motion in a nonquantizing\nmagnetic field Balong the z-axis on the spin dynamics.\nWe assume that due to a small g−factor of electron, the\nZeeman coupling to the magnetic field does not cause a\nrelevant spin precession. If Ω soτ≪1, the density evolu-\ntion atB= 0 is diffusive, and the electron mobility de-4\ncreases with Bdue to the Lorentz force as/parenleftbig\n1+ω2\ncτ2/parenrightbig−1,\nwhereωc=|e|B/mcis the cyclotron frequency. By\nthe Einstein relation, the diffusion coefficient is renor-\nmalized by the same factor D(B) =D(0)//parenleftbig\n1+ω2\ncτ2/parenrightbig\n.\nHence the spin relaxation rate in Eq. (18) decreases as\nΓs(B) = Γs(0)//parenleftbig\n1+ω2\ncτ2/parenrightbig\n, which reproduces the results\nof the direct quantum kinetic theory.[39] For illustration,\nwe consider the limit ωcτ≫1 at short times t≪τin\na more detail. Here the trajectories of electrons are very\ncloseto circlesand the spin-independent kernelin Eq.(9)\ncan be represented as (cf. Ref.[35])\nD(ρ′−ρ,t) =1\n2πd(t)δ(|ρ′−ρ|−d(t)),(21)\nwith the displacement d(t) = 2Rc|sin(ωct/2)|,where\nRc=vF/ωcis the cyclotron radius, and RcQhx=\nΩso/ωc. Straightforward integration in Eq. (9) yields (cf.\nEq.(19))\nS⊥(t) =S⊥(0)J0(Qhxd(t)), (22)\nwhereQhxd(t) = 2Ω so|sin(ωct/2)|/ωc. For a very weak\nfield (in a very clean system) ωc≪Ωso, the result re-\nproduces Eq. (19), as expected. In the opposite limit\nωc≫Ωso, no relaxation occurs. In terms of spin pre-\ncession this can be understood[22, 37, 38] as very fast\nchanges at the frequency ωcin the direction of the SO\nfield, keeping the total spin out of relaxation. In terms\nof the nonuniform density dynamics, this implies that\nthe electrons, forced to circulate around small radius cy-\nclotron orbits, can not spread out to destroy large scale\n∼1/Qhx≫Rcdensity variations. At long times, the dif-\nfusion behavior takes over, and the relaxation becomes\nexponential.\nAs a second example we briefly discuss the effect of\nweak localization on the spin relaxation [40, 41] by con-\nsideringω-dependent renormalization of the momentum\nrelaxation rate with the correction:[42–44]\nδτ−1\nwl(ω) =2\nτ1\n∂n/∂µ/integraldisplayd2q\n(2π)21\nDq2−iω.(23)\nThis correction arises due to the enhanced return prob-\nability, which slows down of the density dynamics and\neventually leads to the algebraic tail in spin relaxation at\nlong time as S⊥(t)∼1/t.[41] In terms of spin preces-\nsion, the enhanced backscattering slows down the spin\nrelaxation because upon the return to the initial point\nthe direction of the electron spin remains the same.\nTo conclude, we have shown that in a wide class of\nsystems the non-Abelian gauge field description of SO\ncoupling reveals the duality of experimental observables\nand ensures the exact mapping of the spin dynamics\nto the density evolution. The evolution is described in\nterms of the responce to an external perturbation with\nthe wavevector equal to the spin helix wavevector Qhx.We presentedexplicit resultsforaweakSOcouplingwith\nQhx≪kFvalidforallsystemsofinterest(thisrestriction\nis not required in general). This exact mapping opens a\nvenueforunderstandingthe wholeclassofpracticallyim-\nportant systems through better studied, and more simple\nproperties of the systems without SO coupling.\nIVT acknowledges funding by the Spanish MEC\n(FIS2007-65702-C02-01), ”Grupos Consolidados\nUPV/EHU del Gobierno Vasco” (IT-319-07), and\nthe European Community through e-I3 ETSF project\n(Grant Agreement: 211956). This work of EYS was sup-\nported by the University of Basque Country UPV/EHU\ngrant GIU07/40, MCI of Spain grant FIS2009-12773-\nC02-01, and ”Grupos Consolidados UPV/EHU del\nGobierno Vasco” grant IT-472-10.\n[1] R. Winkler, Spin-orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems , Springer Tracts\nin Modern Physics (2003).\n[2] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004); J. Fabian, A. Matos-Abiague, C. Ertler,\nP. Stano, and I. Zutic, Acta Physica Slovaca 57, 565\n(2007).\n[3]Spin Physics in Semiconductors , Springer Series in Solid-\nState Sciences, Ed. by M.I. Dyakonov, Springer (2008)\n[4] M.W. Wu, J.H. Jiang, and M.Q. Weng, Physics Reports\n493, 61 (2010).\n[5] V. P. Mineev and G. E. Volovik, Journal of Low Temper-\nature Physics 89, 823 (1992).\n[6] J. Fr¨ ohlich and U. M. Studer, Rev. Mod Phys. 65, 733\n(1993).\n[7] I. L. Aleiner and V. I. Fal’ko, Phys. Rev. Lett. 87, 256801\n(2001).\n[8] L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324\n(2003).\n[9] Y. Lyanda-Geller, Phys. Rev. Lett. 80, 4273 (1998), Y.\nLyanda-GellerandA.D.Mirlin, Phys.Rev.Lett. 72.1894\n(1994), J. B. Miller, D. M. Zumb¨ uhl, C. M. Marcus, Y.\nB. Lyanda-Geller, D. Goldhaber-Gordon, K. Campman,\nand A. C. Gossard, Phys. Rev. Lett. 90, 076807 (2003).\n[10] S.-R. Eric Yang and N. Y. Hwang, Phys. Rev. B 73,\n125330 (2006).\n[11] Q. Liu, T. Ma, and S.-C. Zhang, Phys. Rev. B 76, 233409\n(2007).\n[12] N. Hatano, R. Shirasaki, and H. Nakamura, Phys. Rev.\nA75, 032107 (2007).\n[13] J.-S. Yang, X.-G. He, S.-H. Chen, and C.-R. Chang,\nPhys. Rev. B 78, 085312 (2008).\n[14] B.W.A. Leurs, Z. Nazario, D.I. Santiago, and J. Zaanen,\nAnnals of Physics 323, 907 (2008).\n[15] R. Raimondi and P. Schwab, Europhys. Lett. 87, 37008\n(2009); M. Milletar, R. Raimondi, and P. Schwab, Euro-\nphys. Lett. 82, 67005 (2008).\n[16] I. V. Tokatly, Phys. Rev. Lett. 101, 106601 (2008).\n[17] I. V. Tokatly and E.Ya. Sherman, Annals of Physics 325,\n1104 (2010).\n[18] M.I. Dyakonovand V.I. Perel’, Sov. Phys. Solid State 13,\n3023 (1972).5\n[19] P. Kleinert and V. V. Bryksin, Phys. Rev. B 79, 045317\n(2009); T. D. Stanescu and V. Galitski, Phys. Rev. B 75,\n125307 (2007).\n[20] M. Griesbeck, M. M. Glazov, T. Korn, E. Ya. Sherman,\nD. Waller, C. Reichl, D. Schuh, W. Wegscheider, and C.\nSch¨ uller, Phys. Rev. B 80, 241314 (2009)\n[21] T. Korn, Physics Reports 494415 (2010) and references\ntherein.\n[22] See e.g., D. Culcer and R. Winkler, Phys. Rev. B 76,\n195204 (2007).\n[23] N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582\n(1999).\n[24] J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett.\n90, 146801 (2003).\n[25] M.I. Dyakonov and V.Yu. Kachorovskii, Fiz. Tekh.\nPoluprovodn. (St.-Petersburg) 20, 178 (1986), [Sov.\nPhys. Semicond. 20, 110 (1986)].\n[26] G. M. M¨ uller, M. R¨ omer, D. Schuh, W. Wegscheider, J.\nH¨ ubner, and M. Oestreich, Phys. Rev. Lett. 101, 206601\n(2008).\n[27] V. V. Bel’kov, P. Olbrich, S. A. Tarasenko, D. Schuh, W.\nWegscheider, T. Korn, C. Sch¨ uller, D. Weiss, W. Prettl,\nandS.D. Ganichev, Phys.Rev.Lett. 100, 176806 (2008).\n[28] S. A. Tarasenko, Phys. Rev. B 80, 165317 (2009); Y.\nZhou and M. Wu, EPL 89, 57001 (2010).\n[29] M. M. Glazov and E. Ya. Sherman, Phys. Rev. B 71,\n241312(R) (2005); V. K. Dugaev, E. Ya. Sherman, V. I.\nIvanov, and J. Barna´ s, Phys. Rev. B 80, 081301 (2009);\nM. M. Glazov, M. A. Semina, and E. Ya. Sherman Phys.\nRev. B81, 115332 (2010); M. M. Glazov, E. Ya. Sher-\nman, and V. K. Dugaev, Physica E (2010) in print.\n[30] E. I. Rashba, Phys. Rev. B 68, 241315 (2003).\n[31] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys.\nRev. Lett. 97, 236601 (2006).\n[32] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,Shou-Cheng Zhang, S. Mack, and D. D. Awschalom, Na-\nture458, 610 (2009).\n[33] D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66,\n261 (1994).\n[34] M. M. Glazov and E. L. Ivchenko, Pis’ma Zh. Eksp.\nTeor. Fiz. 75476 (2002) [JETP Lett. 75, 403 (2002)];\nM. M. Glazov and E. L. Ivchenko, JETP 991279 (2004);\nD. Stich, J. Zhou, T. Korn, R. Schulz, D. Schuh, W.\nWegscheider, M. W. Wu, and C. Sch¨ uller, Phys. Rev.\nLett.98, 176401 (2007).\n[35] In the time-coordinate representation in this limit D(ρ′−\nρ,t) =δ(|ρ′−ρ|−vFt)/(2πvFt).\n[36] G. Giuliani and G. Vignale, Quantum Theory of the\nElectron Liquid , Cambridge University Press (2005), 798\npages.\n[37] M. M. Glazov, Solid State Commun. 142, 531 (2007).\n[38] C.H. Chang, J. Tsai, H.-F. Lo, and A. G. Malshukov,\nPhys. Rev. B 79, 125310, (2009).\n[39] E.L. Ivchenko, Fiz. Tverd. Tela 15, (1973) 1566 [Sov.\nPhys. Solid State 15, (1973) 1048]. The Green’s function\nformalism: A. A. Burkov and L. Balents, Phys. Rev. B\n69, 245312 (2004), in the presence of SO coupling leads\nto the same effect.\n[40] A. G. Mal’shukov, K. A. Chao, and M. Willander, Phys.\nRev. Lett. 76, 3794 (1996).\n[41] I. S. Lyubinskiy and V. Yu. Kachorovskii, Phys. Rev. B\n70, 205335 (2004).\n[42] S. Hershfield and V. Ambegaokar, Phys. Rev. B 34, 2147\n(1986).\n[43] G. Strinati, C. Castellani, and C. Di Castro, Phys. Rev.\nB40, 12237 (1989).\n[44] V.K. Dugaev and D.E. Khmel’nitskii, Zh. Eksp. Teor.\nFiz.901871 (1986) [JETP Lett. 63, 1097 (1986)]." }, { "title": "1312.2292v1.The_Pairing_of_Spin_orbit_Coupled_Fermi_Gas_in_Optical_Lattice.pdf", "content": "The Pairing of Spin-orbit Coupled Fermi Gas in Optical Lattice\nHo-Kin Tang,1, 2Xiaosen Yang,1,\u0003Jinhua Sun,1and Hai-Qing Lin1,y\n1Beijing Computational Science Research Center,Beijing, 100084, China\n2Department of Physics, The Chinese University of Hong Kong, Hong Kong, China\n(Dated: January 25, 2021)\nWe investigate Rashba spin-orbit coupled Fermi gases in square optical lattice by using the de-\nterminant quantum Monte Carlo (DQMC) simulations which is free of the sign-problem. We show\nthat the Berezinskii-Kosterlitz-Thoules phase transition temperature is \frstly enhanced and then\nsuppressed by spin-orbit coupling in the strong attraction region. In the intermediate attraction\nregion, spin-orbit coupling always suppresses the transition temperature. We also show that the\nspin susceptibility becomes anisotropic and retains \fnite values at zero temperature.\nPACS numbers: 03.75.Ss, 71.10.Fd, 02.70.Uu\nIntroduction: Spin-orbit coupling (SOC), breaking the\ninversion symmetry, has attracted extensive attentions\nin condensed matter[1, 2]. Recently, SOC in both the\nbosonic[3, 4] and fermionic[5, 6] systems has been real-\nized in ultracold atomic experiments. These milestone\nbreakthroughs have opened up an exciting route to study\nthe novel phases [7{15] induced by SOC in these systems.\nBy introducing SOC, two dimensional (2D) fermionic\nsystems exhibit much more rich phenomena[16{20]. SOC\ncan stabilize the topological nontrivial super\ruid states\n[21{24]. Majorana zero mode exists in the vortices of\nthese topological nontrivial phases and plays a crucial\nrole in topological quantum computation[25]. It was\nfound that SOC has nontrivial e\u000bect on pairing and\nsuper\ruidity[22, 26] in homogeneous systems. SOC en-\nhances the pairing but suppresses the super\ruidity. On\nlattice, SOC exhibits opposite \flling-dependent behav-\niors for the super\ruidity[27]. These interesting physics\ninduced by SOC are all investigated by the Bogoliubov-\nde Gennes (BdG) approach. Moreover, the study of the\nspin-orbit coupled Fermi gases in lattice at \fnite temper-\nature is still waiting to be explored.\nTwo e\u000bects are resulted by applying SOC in the Fermi\nHubbard model. First, SOC enhances the e\u000bective hop-\nping amplitude and enlarges the bandwidth. The other\nis that SOC \rips the spin of the fermion which breaks the\nrotational symmetry of the spin and signi\fcantly changes\nthe properties of the Fermi surface. When the system\nonly contains the SOC, the ground state is semimetal\nnear half-\flling[27] with vanishingly small density of\nstate(DOS)( \u001a(E)\u0018jEj). In the strong attractive limit,\nthe fermions are strongly bounded and the super\ruid\ntransition temperature is determined by the center-of-\nmass motion which is proportional to the inverse of the\nattraction. Therefore, our major concern here is to in-\nvestigate what e\u000bects can be induced by the SOC on the\npairing at \fnite temperature beyond the BdG approach.\nIn this Letter we investigate the pairing of the at-\ntractive Fermi gases in 2D square optical lattice with\nSOC using both DQMC simulations[28{32] and mean\n\feld theory. To our knowledge, this is the \frst unbi-ased numeric simulation of the spin-orbit coupled Fermi\ngases. Our results give us a detailed description about\nthe pairing behavior and the super\ruid phase transition\nof this spin-orbit coupled system at \fnite temperature.\nThe main results are summarized as following: (1) With\nSOC, there exists Berezinskii-Kosterlitz-Thoules (BKT)\nphase transition even in the absence of the hopping term.\nThe super\ruid phase transition temperature is enhanced\nby SOC in strong attraction region. In intermediate re-\ngion, the super\ruid transition temperature is always sup-\npressed by SOC. The peak of the transition temperature\nis approximately proportional to the bandwidth, which\nis enlarged signi\fcantly by large SOC. (2) SOC always\nsuppresses the pairing temperature at strong attraction\nregion. Thus, SOC has an opposite e\u000bect on the pairing\nand the super\ruidity in this region. These are qualita-\ntively di\u000berent from the continuous case[19, 22]. (3) Due\nto the emergence of spin-triplet pairing and the breaking\nof the rotational symmetry of spin, the spin susceptibility\nbecomes anisotropic. When the temperature decreases to\nzero, spin susceptibility retains \fnite values.\nModel and Method: We start with the 2D Rashba spin-\norbit coupled fermionic Hubbard model on a square lat-\ntice which can be written as following.\nH=\u0000tX\nhi;jicy\ni;scj;s+i\u0015X\nhi;jicy\ni;s(ei;j\u0002\u001b)s;s0\nzcj;s0\n\u0000UX\nini;\"ni;#\u0000\u0016X\nini; (1)\nwherecy\ni;s(ci;s) denotes the creation (annihilation) op-\nerators for fermionic atoms with spin s\u0011(\";#) at\nsitei.niis the fermionic density operator at site i:\nni= \u0006sni;s= \u0006scy\ni;sci;s.\u001bis the Pauli matrices, ^ ei;j\nis the vector connecting sites iandj.hi;jidenotes the\nsummation over the nearest neighbors. t,\u0015,U(U > 0)\nand\u0016stand for the hopping amplitude, Rashba SOC\nstrength, on-site attractive interaction, and chemical po-\ntential, respectively.\nSOC lifts the spin degeneracy and gives rise to two\nsplited helical branches for noninteracting case. ThearXiv:1312.2292v1 [cond-mat.quant-gas] 9 Dec 20132\ntwo helical branches have four contact Dirac cones at\n[(0;0);(0;\u0019);(\u0019;0) and (\u0019;\u0019)]. The splitting between two\nbranches increases with SOC. The bandwidth is enlarged\nand the half bandwidth is W(t;\u0015) = 4tq\n2\n(2+\u00152=t2)+\n2\u0015q\n2\u00152=t2\n2+\u00152=t2. The DOS diverges at four van Hove sin-\ngularities!=\u00062t\u00062tp\n1 +\u00152=t2instead at != 0\n[33]. At half \flling, the Hamiltonian has a particle-\nhole symmetry with ci;s!dy\ni;s= (\u00001)ix+iyci;sand\ncy\ni;s!di;s= (\u00001)ix+iycy\ni;s. The Fermi surface is per-\nfectly nested with a nesting vector Q= (\u0019;\u0019). Through-\nout this paper, we use the hopping amplitude tas the\nunit energy and assume t= 1.\nSince the SOC is a complex spin-\rip term, the BSS\nalgorithm of DQMC should be modi\fed to updates the\nup-spin and down-spin simultaneously instead of updat-\ning them separately. By this modi\fcation, the notorious\nsign-problem becomes more troublesome. Fortunately,\nour model is free of sign-problem in the DQMC simula-\ntions. This guarantees our DQMC simulations to achieve\na good numerical precision at large size and low tem-\nperature. Typical system in our DQMC simulations is\n10\u000210 and periodic boundary condition, the Suzuki-\nTrotter decomposition (the step is \u0001 \u001c=\f=M = 0:125\nwith\f= 1=T) is used and then a discrete Hubbard-\nStratonovich transformation is introduced to decouple\nthe on-site attractive interaction into a bilinear form.\nThe systematic error of our DQMC simulations on the\norder of (\u0001 \u001c)2.\nBerezinskii-Kosterlitz-Thoules phase transition: In\ntwo dimensions, although pairs can be formed, there is no\nlong-range super\ruid order at \fnite temperature because\nof the spatially-dependent phase \ructuation, so there is\nno condensation. At \fnite temperature, BKT phase tran-\nsition is possible for the emergence of quasi-long-range\n(algebraic long-range) super\ruid order. When temper-\nature drops below a critical temperature ( TBKT), the\nsystem undergoes a phase transition from the pseudogap\nphase to the super\ruid phase. On the two sides of TBKT,\nthe super\ruid density has a universal jump. The TBKT\ncan be precisely determined by this jump[33, 34].\nTBKT =\u0019\n2Ds(\u0015;U;TBKT) (2)\nwhereDs(\u0015;U;TBKT), which can be determined by the\ncurrent-current correlation function[35], is the super\ruid\ndensity at the super\ruid side of TBKT.\nIn Fig.1, we show TBKT as a function of \u0015for di\u000ber-\nenthniwithU= 4;6;8. We also have performed the\nDQMC simulations on 12 \u000212 lattice size for U= 6 with\nhni= 0:7 case.TBKT curve of 12\u000212 lattice size al-\nmost coincides with the curve of 10 \u000210 size as shown\nin Fig.1. Thus, our simulations are credible for 10 \u000210\nlattice size. For U= 6;8 cases,TBKT is \frstly enhanced\nand then suppressed by SOC, whereas TBKT is always\nsuppressed by SOC for U= 4 case. These are resulted\n01 2 0.00.10.20.30.4 10X10 12X12 \n=0.7 U=6 QMC =0.7 U=6 QMC \n=0.9 U=6 QMC \n=0.1 U=6 QMC \n=0.7 U=4 QMC \n=0.7 U=8 QMC \n=0.7 U=6 MF \n=0.1 U=6 MFTBKTλ\nFIG. 1.TBKT VS\u0015for varioushniandU. The solid curves\nrepresent the results of DQMC, while the dashed curves is\nthe results of mean \feld theory. TBKT is enhanced \frstly\nand then suppressed by SOC at strong attraction( U= 6;8).\nWhereasTBKT is always suppressed by SOC at intermediate\nattraction(U= 4). TheTBKT curves at the size 12 \u000212 and\n10\u000210 almost coincide for hni= 0:7 withU= 6 case. The\nresults of mean \feld and DQMC are consistent quantitatively\nfor small \flling and only qualitatively for large \flling.\nby the competition between the pair breaking and the\ncenter-of-mass motion. In strong attraction case( U >z ,\nzis coordinate number), the fermions form tight cooper\npairs andTBKT is controlled by the center-of-mass mo-\ntion. When SOC increases, the center-of-mass motion\nis enhanced due to the enlargement of the bandwidth.\nTherefore,TBKT will be enhanced. When SOC becomes\nlarger than a critical value \u0015c, the pair breaking would be\ndominant comparing to the enhancement of the center-\nof-mass motion, then SOC would suppress TBKT. When\nUincreases, the cooper pairs will become tighter, and\nthus\u0015cwill increases. For U= 4 case, this is an inter-\nmediate region between the strong and weak attraction\nregion. The cooper pairs are more loosely formed. There-\nfore, theTBKT will be suppressed by increasing SOC. For\nweak attraction case( U 0). Super\ruid instability is signaled by\n\u0000\r\u0001eP\r!\u00001.\nFig.3 shows the pairing vertex of spin-singlet(up row)\nand spin-triplet(down row) with \u0015= 0;0:5;1;2, and\nU= 6 versus temperature. Fig.3(a) and (b) show that\nspin-singlet pairing vertex converge to \u00001 asT!0 and\ncontribute to super\ruid. While the triplet pairing vertex\nconverge to 0 in the absence of SOC and converge to \u00001\nin the presence of SOC (except Fig.3(d) \u0015= 0:5 case).\nThis indicates that spin-triplet pairing can emerges and\ncontributes to super\ruid in the presence of SOC. The\npairing of the super\ruid is a mixture of spin-singlet and\nspin-triplet. They compete with each other in the system\nas the SOC increases. In Fig.3(d), pairing vertex does not\nconverge to\u00001 for\u0015= 0:5. This indicates that there ex-\nists a critical SOC strength( \u0015c) above which spin-triplet\npairing has contribution to super\ruid. The convergence\nis decelerated by SOC for spin-singlet pairing while is\naccelerated for spin-triplet pairing.\nSpin susceptibilities: Because the symmetry of the\npairing has been changed by SOC, the spin response4\n0.11 1 00.000.050.100\n.11 1 00.000.050.100.15 χz U=4 \nχy U=4 \nχz U=6 \nχy U=6(b)χαT\n(c)T\n(d)χα(a)\nFIG. 4. The spin susceptibilities vs temperature with\n\u0015=(a)0,(b)0.5,(c)1,(d)2 and hni= 0:7. When the temper-\nature decreases to zero, spin susceptibilities tend to \fnite val-\nues for\u00156= 0 and 0 for \u0015= 0.\nwill be very di\u000berent from the case without SOC, es-\npecially the spin susceptibility. Without the SOC, the\npairing is only spin-singlet and the spin susceptibility is\nisotropic. When the temperature decreases, thermody-\nnamic \ructuation will be suppressed and this will render\nthe enhancement of spin susceptibility. When the tem-\nperature decreases to a critical value, spin-singlet pairs\nare formed, so the spin susceptibility will be suppressed.\nWhen the temperature decreases to zero, all the fermions\nare paired. Thus, spin susceptibility would decrease to\nzero [31, 32]. In the presence of SOC, the spin susceptibil-\nity becomes anisotropic and can be written as following:\n\u001f\u000b=1\nNX\ni;je\u0000i~ q\u0001(~ ri\u0000~ rj)Z\f\n0d\u001c j~ q!0;\n(6)\nwheres\u000bis the spin with \u000b= (x;y;z ).\nFig.4 shows spin susceptibilities as functions of temper-\nature with \u0015= 0;0:5;1;2,U= 4;6 andhni= 0:7. The\ncurves of spin susceptibilities are smooth. The spin sus-\nceptibilities remain unchanged across TBKT. For\u0015= 0\nour result agrees with the Ref[31] as shown in Fig.4(a).\nIn the presence of SOC, the anisotropic spin suscepti-\nbilities as shown in Fig.4(b)-(d) for di\u000berent \u0015. When\nthe temperature decreases, spin susceptibilities increase\n\frstly and then gradually decrease. Signi\fcantly di\u000ber-\nent from the \u0015= 0 case, the spin susceptibilities does not\ndrop to zero but remains \fnite even when temperature\ndecreases to zero. This can be understood by the forma-\ntion of spin-triplet pairing. Spin-singlet pairing has zero\ntotal spin and has no contribution to the spin susceptibil-\nities unless being broken by thermodynamic \ructuation.\nQuite the contrary, the spin-triplet pairing possesses total\nspin and contributes to spin susceptibilities even at zero\ntemperature. Thus, the spin susceptibilities retain \fnite\nvalues when temperature approaches to zero. The \fnite\n01 2 3 0.000.040.080.12 χz =0.1 \nχy =0.1 \nχz =0.7 \nχx =0.7 \nχz =0.9 \nχx =0.9χαλ\nFIG. 5. Spin susceptibilities \u001f\u000bvs\u0015withU= 6 andhni=\n0:1;0:7;0:9 for\f= 1=T= 10. At small SOC limit, spin\nsusceptibilities are the quadratic functions of \u0015.\nvalues of spin susceptibilities reveal the weight of spin-\ntriplet pairing. The spin susceptibilities are also sup-\npressed by attraction Ufor the on-site attraction favors\nthe spin-singlet pairing. Certainly, we can also estimate\nthe pairing temperature Tpairfrom Fig.4 by the location\nof the peak of \u001f.Tpairis approximately equal to 1 which\nis much larger than TBKT. Therefore, there is a large\npseudogap region in \fnite temperature phase diagram\nwhich con\frms the validity of the mean \feld phase dia-\ngram in Fig.2. As for the spin-triplet pairing, here Tpair\nis underestimated.\nFig.5 shows the spin susceptibilities as functions of \u0015\nfor di\u000berenthniat\f= 10 with U= 6. Spin suscep-\ntibilities increase \frstly with \u0015for the increasing of the\nspin-triplet pairing. In small SOC limit, spin susceptibil-\nities are the quadratic functions of \u0015which is in accord\nwith the continuous case[37]. At large SOC, the spin sus-\nceptibilities are suppressed for the reason that SOC sup-\npresses the pairing of both spin-singlet and spin-triplet\nas discussed above.\nDiscussion and Conclusion: Obviously, our DQMC\nsimulations and the results could be applicable to the La\nAlO 3/SrTiO 3interface[38{40] and noncentrosymmetric\nsuperconductors such as CePt 3Si, Li 2(Pt1\u0000xPdx)3B[41,\n42], because strong SOC exists in these materials.\nThe behavior of spin susceptibilities can be determined\nby Knight shift in nuclear magnetic resonance(NMR)\nmeasurements[43].\nWe have performed simulations for the attractive\nfermionic Hubbard model with Rashba SOC in 2D square\noptical lattice using DQMC and mean \feld theory. There\nexists a \fnite temperature super\ruid phase transition.\nThe transition temperature is suppressed by SOC in in-\ntermediate attraction. With the strong attraction, the\nsuper\ruid transition temperature is enhanced \frstly and\nthen suppressed by SOC. The spin susceptibility becomes\nanisotropic and retains \fnite values when the tempera-5\nture approach to zero. This nontrivial behavior of spin\nsusceptibilities can be con\frmed by speckle imaging[44]\nin experiments. We also check the anisotropic SOC case\nwhich can be consider as a mixture of Rashba and Dresel-\nhaus SOC. We \fnd that the behavior of super\ruid transi-\ntion temperature resemble to the Rashba SOC case while\nthe isotropic of spin susceptibility in x\u0000yplane will be\nfurther destroyed.\nAcknowledges: We would like to thank Prof. W. Yi,\nYoujin Deng, Hui Zhai, Tianxing Ma and Q. Sun for\nhelpful discussions. H.-K. Tang would like to thank the\nsupports from Prof. ShiJian Gu. This work is supported\nby NSFC 91230203, CAEP, and China Postdoctoral Sci-\nence Foundation (No. 2012M520147).\n\u0003yangxs@csrc.ac.cn\nyhaiqing0@csrc.ac.cn\n[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n[3] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,\nJ. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102,\n130401 (2009).\n[4] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V.\nPorto, and I. B. Spielman, Nature 462, 628 (2009).\n[5] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai,\nH. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301\n(2012).\n[6] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[7] S.-L. Zhu, H. Fu, C.-J. Wu, S.-C. Zhang, and L.-M.\nDuan, Phys. Rev. Lett. 97, 240401 (2006).\n[8] N. Goldman, A. Kubasiak, A. Bermudez, P. Gaspard,\nM. Lewenstein, and M. A. Martin-Delgado, Phys. Rev.\nLett. 103, 035301 (2009).\n[9] W. Yi and G.-C. Guo, Phys. Rev. A 84, 031608 (2011).\n[10] T. Ozawa, L. P. Pitaevskii, and S. Stringari, Phys. Rev.\nA87, 063610 (2013).\n[11] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi,\nPhys. Rev. Lett. 109, 085302 (2012).\n[12] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett.\n108, 225301 (2012).\n[13] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev.\nLett. 105, 160403 (2010).\n[14] Z.-Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305\n(2011).\n[15] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012).\n[16] J. Radi\u0013 c, A. Di Ciolo, K. Sun, and V. Galitski, Phys.\nRev. Lett. 109, 085303 (2012).\n[17] W. Yi and W. Zhang, Phys. Rev. Lett. 109, 140402(2012).\n[18] X. Yang and S. Wan, Phys. Rev. A 85, 023633 (2012).\n[19] K. Zhou and Z. Zhang, Phys. Rev. Lett. 108, 025301\n(2012).\n[20] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett.\n107, 195304 (2011).\n[21] M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev.\nLett. 103, 020401 (2009).\n[22] M. Gong, G. Chen, S. Jia, and C. Zhang, Phys. Rev.\nLett. 109, 105302 (2012).\n[23] C. Qu, Z. Zheng, M. Gong, Y. Xu, L. Mao, X. Zou,\nG. Guo, and C. Zhang, Nature communications 4, 2710\n(2013).\n[24] W. Zhang and W. Yi, Nature communications 4, 2711\n(2013).\n[25] S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, and\nP. Zoller, Phys. Rev. Lett. 98, 010506 (2007).\n[26] L. He and X.-G. Huang, Phys. Rev. Lett. 108, 145302\n(2012).\n[27] Q. Sun, G.-B. Zhu, W.-M. Liu, and A.-C. Ji,\narXiv:1304.4511 (2013).\n[28] J. E. Hirsch, Phys. Rev. B 38, 12023 (1988).\n[29] J. E. Hirsch and H. Q. Lin, Phys. Rev. B 37, 5070 (1988).\n[30] R. T. Scalettar, E. Y. Loh, J. E. Gubernatis, A. Moreo,\nS. R. White, D. J. Scalapino, R. L. Sugar, and\nE. Dagotto, Phys. Rev. Lett. 62, 1407 (1989).\n[31] T. Paiva, R. Scalettar, M. Randeria, and N. Trivedi,\nPhys. Rev. Lett. 104, 066406 (2010).\n[32] M. Randeria, N. Trivedi, A. Moreo, and R. T. Scalettar,\nPhys. Rev. Lett. 69, 2001 (1992).\n[33] See Supplemental Material for details.\n[34] D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39,\n1201 (1977).\n[35] D. J. Scalapino, S. R. White, and S. C. Zhang, Phys.\nRev. Lett. 68, 2830 (1992).\n[36] S. R. White, D. J. Scalapino, R. L. Sugar, N. E. Bickers,\nand R. T. Scalettar, Phys. Rev. B 39, 839 (1989).\n[37] L. P. Gor'kov and E. I. Rashba, Phys. Rev. Lett. 87,\n037004 (2001).\n[38] D. A. Dikin, M. Mehta, C. W. Bark, C. M. Folkman,\nC. B. Eom, and V. Chandrasekhar, Phys. Rev. Lett.\n107, 056802 (2011).\n[39] J. A. Bert, B. Kalisky, C. Bell, M. Kim, Y. Hikita, H. Y.\nHwang, and K. A. Moler, Nature physics 7, 767 (2011).\n[40] S. Banerjee, O. Erten, and M. Randeria, Nature physics\n9, 626 (2013).\n[41] H. Q. Yuan, D. F. Agterberg, N. Hayashi, P. Badica,\nD. Vandervelde, K. Togano, M. Sigrist, and M. B. Sala-\nmon, Phys. Rev. Lett. 97, 017006 (2006).\n[42] G. Eguchi, D. C. Peets, M. Kriener, S. Yonezawa, G. Bao,\nS. Harada, Y. Inada, G.-q. Zheng, and Y. Maeno, Phys.\nRev. B 87, 161203 (2013).\n[43] M. Nishiyama, Y. Inada, and G.-q. Zheng, Phys. Rev.\nLett. 98, 047002 (2007).\n[44] C. Sanner, E. J. Su, A. Keshet, W. Huang, J. Gillen,\nR. Gommers, and W. Ketterle, Phys. Rev. Lett. 106,\n010402 (2011).6\nSUPPLEMENTARY MATERIAL\nIn this supplementary material, we present some details of the calculations.\nvan Hove singularity\nThe single-particle Hamiltonian H0=\u0000tP\nhi;jicy\ni;scj;s+i\u0015P\nhi;jicy\ni;s(ei;j\u0002\u001b)s;s0\nzcj;s0. The dispersion of the two\nhelical branches are \u000fk;\u0017=\u0006=\u00002t(coskx+ cosky) + 2\u0015\u0017q\nsin2kx+ sin2ky. The van Hove singularity is\njr\u000fk;\u0017j= 0\n)8\n<\n:sinkc\nx= 0orcoskc\nx=\u0000\u0017tq\nsin2kcx+ sin2kcy=\u0015\nsinkc\ny= 0orcoskc\ny=\u0000\u0017tq\nsin2kcx+ sin2kcy=\u0015\nThere are three types of van Hove singularities: (I) sin kc\nx= sinkc\ny= 0 with\u000fkc=\u00064t;0; (II) sinkc\nx= 0;coskc\ny=\n\u0000\u0017tq\nsin2kcx+ sin2kcy=\u0015(or sinkc\ny= 0;coskc\nx=\u0000\u0017tq\nsin2kcx+ sin2kcy=\u0015) with\u000fkc=\u00062t\u00062tp\n1 + (\u0015=t)2; (III)\ncoskc\nx= coskc\ny=\u0000\u0017tq\nsin2kcx+ sin2kcy=\u0015with\u000fkc=\u00064tq\n2\n2+(\u0015=t)2\u00062tq\n2(\u0015=t)2\n2+(\u0015=t)2.\nThe half bandwidth is W(t;\u0015) = 4tq\n2\n2+(\u0015=t)2+ 2\u0015q\n2(\u0015=t)2\n2+(\u0015=t)2which increases with SOC.\nThe divergence of DOS only comes from the narrow region which contains the kc. Thus, we dive the integral into\ntwo parts: alabels the narrow region which contains kcandblabels the other region of the integral. The DOS at\nvan Hove singularities is\n\u001a(\u000fc\nk) =1\nNX\nk;\u0017\u000e[\u000fkc\u0000\u000fk;\u0017]\n=X\n\u0017Z\u0019\n0dkxdky\n\u00192\u000e[\u000fkc\u0000\u000fk;\u0017]\n=X\n\u0017Z\na+bdkxdky\n\u00192\u000e[\u000fkc\u0000\u000fk;\u0017]\nWe only consider the integral in aregion that contributes the divergence of the DOS. At this narrow region, the\ndispersion can be expanded as \u000fk;\u0017=\u000fkc+\u000fk0;\u0017with k=kc+k0(k0\nx;y= [0;\u0003];\u0003<<\u0019 ).\n\u001a(\u000fkc) =X\n\u0017Z\u0003\n0dk0\nxdk0\ny\n\u00192\u000e[\u000fk0;\u0017]\nwith\n\u000fk0;\u0017=8\n><\n>:c1q\nk02x+k02y for I case\nc2;xk02\nx\u0000c2;yk02\nyfor II case\nc3k02\nx+c3k02\ny for III case\nwhere,c1= 2\u0017\u0015,c2;x= (tcoskc\nx+\u0017tp\n1 + (\u0015=t)2),c2;y=\u0017(\u00152=t+t)=p\n1 + (\u0015=t)2,c3=\u0000\u0017p\n2(t2+\u00152)p\n2t2+\u00152. Here\nsgn(c2;x) = sgn(c2;y). Therefore, DOS logarithmical diverges for II case and converges for I and III cases.\nAt the bottom(III case) of the dispersion, the e\u000bective mass is1\n[m\u0003]ij=@\u000fk;\u0017\n@ki@kjjkcwithi;j= (x;y).\n1\nm\u0003xx=1\nm\u0003yy=tcoskc\nx+p\n2\u0015sinkc\nx\n1\nm\u0003xy=1\nm\u0003yx=tcoskc\nx\nThen,2\nm\u0003xx+2\nm\u0003xy=W(t;\u0015). Therefore, the e\u000bective mass is suppressed by increasing SOC.7\n0.00 .20 .40 .60 .81 .00246λ<\nn>00.15000.30000.45000.60000.75000.90000\n2 4 0246810ωλ\n00.15000.30000.45000.60000.75000.9000\nFIG. 6. Left is the DOS at Fermi surface on hni-\u0015plane. Right is the DOS on \u0015-!plane.\nIf the system only contains the SOC term, the two helical branches dispersion of the single-particle are \u000fk;\u0017=\u0006=\n2\u0015\u0017q\nsin2kx+ sin2ky. Near half \flling, the dispersion can be expanded at (0 ;\u0019) and (\u0019;0) points. Then the Fermi\nsurface DOS is \u001a(E) =P\nk;\u0017\u000e(E\u00002\u0017\u0015q\nk2x+k2y)\u0018jEj=\u0015.\nBKT transition temperature\nAt \fnite temperature, the spatially-dependent phase \ructuation will always breaks the long-rang order in two\ndimensions. The vortex like phase \ructuation can induce a phase transition between the algebraic long-rang order\n(quasi-long-rang order) and the short-rang order. This is the BKT phase transition. The super\ruid density has a\nuniversal jump and can be determined by current-current correlation. Here, we give the derivation of current formula\nby linear response. The current formula can also be directly derived by ~J=i[H;~P] with the polarization operator\n~P=P\ni~Rini. In the presence of a small vector potential Ax(i), the hopping and the SOC term are modi\fed by a\nPeierls phase\nHA\n0=\u0000tX\ni;sh\ncy\ni+x;sci;seieAx(i)+cy\ni;sci+x;se\u0000ieAx(i)+cy\ni+y;sci;s+cy\ni;sci+y;si\n\u0000\u0015X\nih\n(cy\ni\u0000x;#ci;\"e\u0000ieAx(i)\u0000cy\ni+x;#ci;\"eieAx(i)) +i(cy\ni\u0000y;#ci;\"\u0000cy\ni+y;#ci;\") +H:c:i\n: (7)\nThe Hamiltonian can be expanded in the order of the small vector potential.\nHA\n0=H0+\u0016HA\n0; (8)\nwhere\n\u0016HA\n0=\u0000X\ni\u0014\neJP\nxAx(i) +e2A2\nx(i)\n2Kx(i)\u0015\nwith\nJP\nx=itX\ni;s(cy\ni+x;sci;s\u0000cy\ni\u0000x;sci;s) +i\u0015X\ni(cy\ni\u0000x;#ci;\"+cy\ni+x;#ci;\")\u0000i\u0015X\ni(cy\ni;\"ci\u0000x;#+cy\ni;\"ci+x;#)\nKx(i) =\u0000tX\ni;s(cy\ni+x;sci;s+cy\ni;sci+x;s)\u0000\u0015X\ni(cy\ni\u0000x;#ci;\"\u0000cy\ni+x;#ci;\")\u0000\u0015X\ni(cy\ni;\"ci\u0000x;#\u0000cy\ni;\"ci+x;#)8\n0.00 .51 .00.000.050.100.150.200.252\n46810120.080.120.160.200.24T\nBKT<\nn>(b)TBKTU\n(a)\nFIG. 7.TBKT as a function of (a) Uwith\u0015= 1 andhni= 0:7, (b)hniwithU= 6 and\u0015= 1.TBKT drop rapidly ashni\napproach to half \flling.\nThe current-current correlation is\n\u0003xx(q;i!m) =Z\f\n0d\u001cei!m\u001chJP\nx(q;\u001c)JP\nx(\u0000q;0)i: (9)\nThen, the super\ruid density is:\nDs(T) =1\n4h\n<\u0000Kx>\u0000\u0003xx(qx= 0;qy!0;i!m= 0)i\n: (10)\nThe BKT transition temperature satis\fes\nTBKT =\u0019\n2Ds(TBKT): (11)\nMean \feld framework\nIn mean \feld framework, the partition function of our system can be written as following by introducing the basis\n i= (ci;\";ci;#;cy\ni;\";cy\ni;#)T.\nZ=Z\nD[\u0016 ; ]e\u0000S[\u0016 ; ]; (12)\nwhere the action is\nS[\u0016 ; ] =Z\f\n0d\u001c\"X\ns\u0016 @\u001c +H(\u0016 ; )#\n: (13)\nWith the Hubbard-Stratonovich transformation \u0001 i=\u0000Uhci;#ci;\"iand integrating out the fermion degrees of the\nfreedom, we have the partition function Z=R\nD[\u0016\u0001;\u0001]e\u0000Seff[\u0016\u0001;\u0001]with the e\u000bective action\nSeff[\u0016\u0001;\u0001] =Z\f\n0d\u001c\u0010\nj\u0001j2=U+\"k\u0011\n\u00001\n2Tr[lnG\u00001]: (14)\nHere, the inverse Green function is\nG\u00001=\u0012@\u001c+\"k+gk\u0000i\u0001\u001by\ni\u0016\u0001\u001by@\u001c\u0000\"k+gT\nk\u0013\n; (15)9\n0.20.40.60.81.00246λ<\nn>00.050000.10000.15000.2000(\na)0\n.20.40.60.81.00246λ<\nn>00.25000.50000.7500(\nb)\nFIG. 8.Tpairas a function ofhniand\u0015for (a)U= 2tand (b)U= 4t.Tpairis always suppressed by SOC for U= 4 and is\ndominated by the DOS at Fermi surface for U= 2.\nwith\"k=\u000fk\u0000\u0016,\u000fk=\u00002t(coskx+ cosky) andgk= 2\u0015(sinky\u001bx\u0000sinkx\u001by).\nIf we ignore the spatial-dependent phase \ructuation \u0001 i= \u0001, we have the gap and the number equations as following:\n1\nU=X\nk;\u0017=\u00061\n4Ek;\u0017[1\u00002f(Ek;\u0017)];\nn=1\n2X\nk;\u0017=\u0006\u0014\n1\u0000\"k;\u0017\nEk;\u0017(1\u00002f(Ek;\u0017))\u0015\n:\nHere, the excitation spectrum is Ek;\u0017=p\n\"k;\u0017+ \u00012with\"k;\u0017=\"k+\u0017jgkj. The pairing temperature is determined\nby \u0001 = 0.\nThe spin susceptibility is \u001fi;j=\u0000P\nk;!nTrf\u001biG(k;!n)\u001bjG(k;!n)\u0000\u001biF(k;!n)\u001bjFy(k;!n)g.GandFcan be\nsolved by Eq.15. Here, we show the result of \u001fzz.\n\u001fzz=\u00001\n\fX\nk;!n2(i!n+\u000fk)2\u00002jgkj2+ 2\u00012\n(!2n+E2\nk;+)(!2n+E2\nk;\u0000);\n=X\nk;\u0017ntanh(\fEk;\u0017=2)\n2Ek;\u0017\u00004(\u000f2\nk+ \u00012) tanh(\fEk;\u0017=2)\n2Ek;\u0017(E2\nk;\u0017\u0000E2\nk;\u0000\u0017)o\n: (16)\nAtT= 0,\u001fzz=P\nkn\n(Ek;++Ek;\u0000)2\u00004(\u000f2\nk+\u00012)\n2Ek;+Ek;\u0000(Ek;++Ek;\u0000)o\n. When\u0015\u001cft;\u0016;\u0001g, we can expand the spin susceptibility by \u0015.\n\u001fzz:=X\nk\u00012\nE5\nkjgkj2\u0018\u00152: (17)\nAt zero temperature, spin susceptibility is a quadratic function of \u0015.\nTo investigate the phase \ructuation in mean \feld framework, we can impose a phase twist on the pairing potential\n\u0001i= \u0001eir\u0012\u0001rj+i@\u001c\u0012\u0001\u001c. The partition function can be expanded by @i\u0012. The partition function has a symmetry to\n\u0012!\u0000\u0012. Therefore, the leading order is ( @i\u0012)2.\nSeff=1\n2Z\nd2r[P(@\u001c\u0012)2+Ds(r\u0012)2];10\n07 1 40.00.10.20\n7 1 401230\n1 02 00.00.20.40\n1 02 0051015 n=1 \nn=0.7 \nn=0.1χzzS\nOCU=6 n=1 \nn=0.7 \nn=0.1ΔS\nOCU=6 \nn=1 \nn=0.7 \nn=0.1χzzU\nSOC=1 n=1 \nn=0.7 \nn=0.1ΔU\nSOC=1\nFIG. 9.\u001fzzVSUand\u0015atT= 0. The behavior of \u001fzz\nqualitatively matches with results of DQMC.\nwith\nP=X\nk;\u0017=\u0006(\n\u00012\n8E3\nk;\u0017tanh\u0012\fEk;\u0017\n2\u0013\n+\"2\nk;\u0017\n16E2\nk;\u0017sech2\u0012\fEk;\u0017\n2\u0013)\n;\nDs=X\nk;\u00178\n<\n:\u00002tcos(kx)\"k;\u0017\nEk;\u0017h\n1\u00002f(Ek;\u0017)i\n+ 2\u0015\u0017\"k;\u0017sin2kx\n2Ek;\u0017q\nsin2kx+ sin2kytanh\u0012\fEk;\u0017\n2\u0013\n\u00002\u0015\u0017\u000f2\nk+ 2\u0017\u0015\u000fkq\nsin2kx+ sin2ky+ \u00012\n2\u000fkEk;\u0017sin2kycos2kx\n(sin2kx+ sin2ky)3=2tanh\u0012\fEk;\u0017\n2\u00139\n=\n;\n+f0(Ek;\u0017) sin2kx0\n@2t+\u00172\u0015coskxq\nsin2kx+ sin2ky1\nA2\n;\nwheref(x) = (1 +e\fx)\u00001is the Fermi-Dirac distribution." }, { "title": "1812.07979v1.Global_phase_diagram_of_a_spin_orbit_coupled_Kondo_lattice_model_on_the_honeycomb_lattice.pdf", "content": "Global phase diagram of a spin-orbit-coupled Kondo lattice model on the honeycomb\nlattice\nXin Li,1, 2Rong Yu,3,\u0003and Qimiao Si4,y\n1Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,\nChinese Academy of Sciences, Beijing 100190, China\n2University of Chinese Academy of Sciences, Beijing 100049, China\n3Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices,\nRenmin University of China, Beijing 100872, China\n4Department of Physics & Astronomy, Rice Center for Quantum Materials, Rice University, Houston, Texas 77005,USA\n(Dated: December 20, 2018)\nMotivated by the growing interest in the novel quantum phases in materials with strong electron\ncorrelations and spin-orbit coupling, we study the interplay between the spin-orbit coupling, Kondo\ninteraction, and magnetic frustration of a Kondo lattice model on a two-dimensional honeycomb\nlattice. We calculate the renormalized electronic structure and correlation functions at the saddle\npoint based on a fermionic representation of the spin operators. We \fnd a global phase diagram of\nthe model at half-\flling, which contains a variety of phases due to the competing interactions. In\naddition to a Kondo insulator, there is a topological insulator with valence bond solid correlations\nin the spin sector, and two antiferromagnetic phases. Due to a competition between the spin-orbit\ncoupling and Kondo interaction, the direction of the magnetic moments in the antiferromagnetic\nphases can be either within or perpendicular to the lattice plane. The latter antiferromagnetic state\nis topologically nontrivial for moderate and strong spin-orbit couplings.\nI. INTRODUCTION\nExploring novel quantum phases and the associated\nphase transitions in systems with strong electron corre-\nlations is a major subject of contemporary condensed\nmatter physics.1{3In this context, heavy fermion (HF)\ncompounds play a crucial role.3{6In these materials,\nthe coexisted itinerant electrons and local magnetic\nmoments (from localized felectrons) interact via the\nantiferromagnetic exchange coupling, resulting in the\nKondo e\u000bect.7Meanwhile, the Ruderman-Kittel-Kasuya-\nYosida (RKKY) interaction, namely the exchange cou-\npling among the local moments mediated by the itin-\nerant electrons, competes with the Kondo e\u000bect.8This\ncompetition gives rise to a rich phase diagram with an\nantiferromagnetic (AFM) quantum critical point (QCP)\nand various emergent phases nearby.3,9\nIn the HF metals, experiments10,11have provide strong\nevidence for local quantum criticality,12,13which is char-\nacterized by the beyond-Landau physics of Kondo de-\nstruction at the AFM QCP. Across this local QCP, the\nFermi surface jumps from large in the paramagnetic HF\nliquid phase to small in the AFM phase with Kondo de-\nstruction. A natural question is how this local QCP con-\nnects to the conventional spin density wave (SDW) QCP,\ndescribed by the Hertz-Millis theory14,15. A proposed\nglobal phase diagram16{19makes this connection via the\ntuning of the quantum \ructuations in the local-moment\nmagnetism. Besides the HF metals, it is also interesting\nto know whether a similar global phase diagram can be\nrealized in Kondo insulators (KIs), where the chemical\npotential is inside the Kondo hybridization gap when the\nelectron \flling is commensurate. The KIs are nontriv-\nial band insulators because the band gap originates from\nstrong electron-correlation e\u000bects. A Kondo-destructiontransition is expected to accompany the closure of the\nband gap. The question that remains open is whether the\nlocal moments immediately order or form a di\u000berent type\nof magnetic states, such as spin liquid or valence bond\nsolid (VBS), when the Kondo destruction takes place.\nRecent years have seen extensive studies about the ef-\nfect of a \fne spin-orbit coupling (SOC) on the electronic\nbands. In topological insulators (TIs), the bulk band gap\nopens due to a nonzero SOC, and there exist gapless sur-\nface states. The nontrivial topology of the bandstructure\nis protected by the time reversal symmetry (TRS). Even\nfor a system with broken TRS, the conservation of com-\nbination of TRS and translational symmetry can give rise\nto a topological antiferromagnetic insulator (T-AFMI).20\nIn general, these TIs and TAFIs can be tuned to topo-\nlogically trivial insulators via topological quantum phase\ntransitions. But how the strong electron correlations in-\n\ruence the properties of these symmetry dictated topo-\nlogical phases and related phase transitions is still under\nactive discussion.\nThe SOC also has important e\u000bects in HF materi-\nals19. For example, the SOC can produce a topologi-\ncally nontrivial bandstructure and induce exotic Kondo\nphysics.21,22it may give rise to a topological Kondo in-\nsulator (TKI),23which has been invoked to understand\nthe resistivity plateau of the heavy-fermion SmB 6at low\ntemperatures.24.\nFrom a more general perspective, SOC provides an ad-\nditional tuning parameter enriching the global phase dia-\ngram of HF systems19,25. Whether and how the topolog-\nical nontrivial quantum phases can emerge in this phase\ndiagram is a timely issue. Recent studies have advanced\na Weyl-Kondo semimetal phase26. Experimental evi-\ndence has come from the new heavy fermion compound\nCe3Bi4Pd3, which display thermodynamic27and zero-arXiv:1812.07979v1 [cond-mat.str-el] 19 Dec 20182\n\feld Hall transport28properties that provide evidence for\nthe salient features of the Weyl-Kondo semimetal. These\nmeasurements respectively probe a linearly dispersing\nelectronic excitations with a velocity that is renormal-\nized by several orders of magnitude and singularities in\nthe Berry-curvature distribution.\nThis type of theoretical studies are also of interest for\na Kondo lattice model de\fned on a honeycomb lattice,29\nwhich readily accommodates the SOC30. In the dilute-\ncarrier limit, this model supports a nontrivial Dirac-\nKondo semimetal (DKSM) phase, which can be tuned\nto a TKI by increasing SOC.31In Ref. 29, it was shown\nthat, at half-\flling, increasing the Kondo coupling in-\nduces a direct transition from a TI to a KI. A related\nmodel, with the conduction-electron part of the Hamilto-\nnian described by a Haldane model32on the honeycomb\nlattice, was subsequently studied.33\nHere we investigate the global phase diagram of a spin-\norbit-coupled Kondo lattice model on the honeycomb lat-\ntice at half-\flling. We show that the competing interac-\ntions in this model give rise to a very rich phase diagram\ncontaining a TI, a KI, and two AFM phases. We focus\non discussing the in\ruence of magnetic frustration on the\nphase diagram. In the TI, the local moments develop a\nVBS order. In the two AFM phases, the moments are\nordered, respectively, in the plane of the honeycomb lat-\ntice (denoted as AFM xy) and perpendicular to the plane\n(AFMz). Particularly in the AFM zphase, the conduc-\ntion electrons may have a topologically nontrivial band-\nstructure, although the TRS is explicitly broken. This\nT-AFMzstate connects to the trivial AFM zphase via a\ntopological phase transition as the SOC is reduced.\nThe remainder of the paper is organized as follows.\nWe start by introducing the model and our theoretical\nprocedure in Sec.II. In Sec.III we discuss the magnetic\nphase diagram of the Heisenberg model for the local mo-\nments. Next we obtain the global phase diagram of the\nfull model in Sec. IV. In Sec V we examine the nature\nof the conduction-electron bandstructures in the AFM\nstates, with a focus on their topological characters. We\ndiscuss the implications of our results in Sec. VI.\nII. MODEL AND METHOD\nThe model we considere here is de\fned on an e\u000bective\ndouble-layer honeycomb lattice. The top layer contains\nconduction electrons realizing the Kane-Mele Hamilto-\nnian30. The conduction electrons are Kondo coupled to\n(i.e., experiencing an AF exchange coupling JKwith) the\nlocalized magnetic moments in the bottom layer. The lo-\ncal moments interact among themselves through direct\nexchange interaction as well as the conduction electron\nmediated RKKY interaction; this interaction is described\nby a simple J1-J2model. Both the conduction bands and\nthe localized bands are half-\flled. This Kondo-lattice\nHamiltonian takes the following form on the honeycomblattice:\nH=tX\nhiji\u001bcy\ni\u001bcj\u001b+i\u0015soX\n\u001cij\u001d\u001b\u001b0vijcy\ni\u001b\u001bz\n\u001b\u001b0cj\u001b0\n+JKX\ni~ si\u0001~Si+J1X\nhiji~Si\u0001~Sj+J2X\nhhijii~Si\u0001~Sj;(1)\nwherecy\ni\u001bcreates a conduction electron at site iwith spin\nindex\u001b.tis the hopping parameter between the near-\nest neighboring (NN) sites, and \u0015sois the strength of\nthe SOC between next-nearest neighboring (NNN) sites.\nvij=\u00061, depending on the direction of the NNN hop-\nping.~ si=cy\ni\u001b~ \u001b\u001b\u001b0ci\u001b0, is the spin operator of the con-\nduction electrons at site iwith~ \u001b=\u001bx;\u001by;\u001bzbeing the\npauli matrices. ~Sirefers to the spin operator of the lo-\ncal moments with spin size S= 1=2. In the model we\nconsidered here, JK,J1, andJ2are all AF. By incor-\nporating the Heisenberg interactions, the Kondo-lattice\nmodel we study readily captures the e\u000bect of geometrical\nfrustration. In addition, instead of treating the Kondo\nscreening and magnetic order in terms of the longitudi-\nnal and transverse components of the Kondo-exchange\ninteractions33,34,36, we will treat both e\u000bects in terms of\ninteractions that are spin-rotationally invariant; this will\nturn out to be important in mapping out the global phase\ndiagram.\nFIG. 1. Top panels: De\fnition of nearest neighboring and\nnext nearest neighboring valence bond \felds Qij. Filled and\nempty circles denote the two sublattices A and B, respec-\ntively. Di\u000berent bond directions are labeled by di\u000berent col-\nors. Bottom panel: First Brillouin zone corresponding to the\ntwo-sublattice unit cell.\nWe use the spinon representation for ~Si,i.e., by\nrewriting~Si=fy\ni\u001b~ \u001b\u001b\u001b0fi\u001b0along with the constraintP\n\u001bfy\ni\u001bfi\u001b= 1, where fy\ni\u001bis the spinon operator. The\nconstraint is enforced by introducing the Lagrange mul-\ntiplier termP\ni\u0015i(P\n\u001bfy\ni\u001bfi\u001b\u00001) in the Hamiltonian.\nIn order to study both the non-magnetic and magnetic\nphases, we decouple the Heisenberg Hamiltonian into two3\nchannels:\nJSi\u0001Sj\n=xJSi\u0001Sj+ (1\u0000x)JSi\u0001Sj\n'x\u0012J\n2jQijj2\u0000J\n2Q\u0003\nijfy\ni\u000bfj\u000b\u0000J\n2Qijfy\nj\u000bfi\u000b\u0013\n+ (1\u0000x) (\u0000JMi\u0001Mj+JMj\u0001Si+JMi\u0001Sj) (2)\nHerexis a parameter that is introduced in keeping with\nthe generalized procedure of Hubbard-Stratonovich de-\ncouplings and will be \fxed to conveniently describe the\ne\u000bect of quantum \ructuations. The corresponding va-\nlence bond (VB) parameter Qijand sublattice magne-\ntization MiareQij=hP\n\u000bfy\ni\u000bfj\u000biandMi=hSii,\nrespectively. Throughout this paper, we consider the\ntwo-site unit cell thus excluding any states that breaks\nlattice translation symmetry. Under this construction,\nthere are 3 independent VB mean \felds Qi,i= 1;2;3,\nfor the NN bonds and 6 independent VB mean \felds Qi,\ni= 4;5;:::;9, for the NNN bonds. They are illustrated\nin Fig. 1. We consider only AF exchange interactions,\nJ1>0 andJ2>0, and will thus only take into account\nAF order with M=Mi2A=\u0000Mi2B.\nx\n00.20.40.60.81\n \nJ\n2\n/J\n100.10.20.30.4\nAFM\nVBS\nFIG. 2. Ground-state phase diagram of the J1-J2Hamiltonian\nfor the local moments in the x-J2=J1plane. A NN VBS and\nan AFM state are stabilized in the parameter regime shown.\nTo take into account the Kondo hybridization and the\npossible magnetic order on an equal footing, we follow\nthe treatment of the Heisenberg interaction as outlined\nin Eq. 2 and decouple the Kondo interaction as follows:\nJKS\u0001s\n'y\u0012JK\n2jbj2\u0000JK\n2bfy\ni\u000bci\u000b\u0000JK\n2b\u0003cy\ni\u000bfi\u000b\u0013\n+ (1\u0000y) (\u0000JKMi\u0001mi+JKSi\u0001mi+JKsi\u0001Mi):(3)\nHere we have introduced the mean-\feld parameter for\nthe Kondo hybridization, b=hP\n\u000bcy\ni\u000bfi\u000bi, and the con-\nduction electron magnetization: mi=hsii. For nonzero\nb, the conduction band will Kondo hybridize with the lo-\ncal moments and the system at half-\flling is a KI. On\nthe other hand, when bis zero and Mis nozero, mag-\nnetization ( m6= 0) on the conduction electron band willbe induced by the Kondo coupling, and various AF or-\nders can be stabilized depending on the strength of the\nSOC. Just like the parameter xof Eq. 2 is chosen so that\na saddle-point treatment captures the quantum \ructua-\ntions in the form of spin-singlet bond parameters18, the\nparameterywill be speci\fed according to the criterion\nthat the treatment at the same level describes the quan-\ntum \ructuations in the form of Kondo-insulator state\n(see below).\n|\nM\n|\n00.20.40.6\nJ\n2\n/J\n100.10.20.30.40.50.6\n(b)\nQ\n00.51\n x=0.3\n x=0.4\n x=0.5\n(a)\nFIG. 3. Evolution of the VBS order parameter Q[in (a)] and\nthe AFM order parameter M[in (b)] as a function of J2=J1\nforx= 0:3;0:4;0:5.\nIII. PHASE DIAGRAM OF THE HEISENBERG\nMODEL FOR THE LOCAL MOMENTS\nBecause of the complexity of the full Hamiltonian, we\nstart by setting JK= 0 and discuss the possible ground-\nstate phases of the J1-J2Heisenberg model for the local\nmoments. By treating the problem at the saddle-point\nlevel in Eq. (2), we obtain the phase diagram in the\nx-J2=J1plane shown in Fig.2. Here the x-dependence\nis studied in the same spirit as that of Ref. 18 for the\nShastry-Sutherland lattice. In the parameter regime ex-\nplored, an AF ordered phase (labeled as \\AFM\" in the\n\fgure) and a valence bond solid (VBS) phase are stabi-\nlized. The AF order stabilized is the two-sublattice N\u0013 eel\norder on the honeycomb lattice, and the VBS order refers\nto covering of dimer singlets with jQij=Q6= 0 for one\nout of the three NN bonds (e.g. Q16= 0;Q2=Q3= 0)\nandjQij= 0 for all the NNN bonds. This VBS state\nspontaneously breaks the C 3rotational symmetry of the\nlattice. We thus de\fne the order parameter for VBS state\nto beQ=jP\nj=1;2;3Qjei(2\u0019j=3)j.4\nIn Fig. 3 we plot the evolution of VBS and AF order\nparameters QandMas a function of J2=J1. A direct\n\frst-order transition (signaled by the mid-point of the\njump of the order parameters) between these two phases\nis observed for x.0:6. For the sake of understanding\nthe global phase diagram of the full Kondo-Heisenberg\nmodel, we limit our discussion to J2=J1<1, where only\nthe NN VBS is relevant. A di\u000berent decoupling scheme\napproach was used to study this model35found results\nthat are, in the parameter regime of overlap, consistent\nwith ours. To \fx the parameter x, we compare our results\nwith those about the J1\u0000J2model derived from previ-\nous numerical studies. DMRG studies37found that the\nAFM state is stabilized for J2=J1<0:22, and VBS ex-\nists forJ2=J1>0:35, while in between the nature of the\nground states are still under debate. In this parameter\nregime, the DMRG calculations suggest a plaquette res-\nonating valence bond (RVB) state,37while other meth-\nods implicate possibly spin liquids.38In light of these\nnumerical results, we take x= 0:4 in our calculations.\nThis leads to a direct transition from AFM to VBS at\nJ2=J1'0:27, close to the values of phase boundaries of\nthese two phases determined by other numerical meth-\nods.\nIV. GLOBAL PHASE DIAGRAM OF THE\nKONDO-LATTICE MODEL\nWe now turn to the global phase diagram of the full\nmodel by turning on the Kondo coupling. For de\fnite-\nness, we set J1= 1 and consider t= 1 and\u0015so= 0:4. As\nprescribed in the previous section, we take x= 0:4. Sim-\nilar considerations for yrequire that its value allows for\nquantum \ructuations in the form of Kondo-singlet for-\nmation. This has guided us to take y= 0:7 (see below).\nThe corresponding phase diagram as a function of JK\nand the frustration parameter J2=J1is shown in Fig. 4.\nFIG. 4. Global phase diagram at T= 0 from the saddle-point\ncalculations with x= 0:4,y= 0:7. The ground states include\nthe valence-bond solid (VBS) and Kondo insulator (KI), as\nwell as two antiferromagnetic orders, T-AFM zand AFM xy,\nas described in Sec. V.\nIn our calculation, the phase boundaries are deter-\nmined by sweeping JKwhile along multiple horizontal\nb\n00.51\nJ\n2\n/J\n1\n=0.1 \nJ\n2\n/J\n1\n=0.3 \nJ\n2\n/J\n1\n=0.5\nQ\n00.51\n \nM\nx\n00.20.40.6\n \nM\nz\n00.20.40.6\n \nJ\nK012345FIG. 5. Evolution of the parameters b,Q,MxandMzas a\nfunction of JKfor di\u000berent ratio of J2=J1.\ncuts for several \fxed J2=J1values, as shown in Fig. 5.\nFor smallJKand largeJ2=J1, the local moments and the\nconduction electrons are still e\u000bectively decoupled. The\nconduction electrons form a TI for \fnite SOC, and the\nlocal moments are in the VBS ground state as discussed\nin the previous section. When both JKandJ2=J1are\nsmall, the ground state is AFM. Due to the Kondo cou-\npling, \fnite magnetization mis induced for the conduc-\ntion electrons. This opens a spin density wave (SDW)\ngap in the conduction band, and therefore the ground\nstate of the system is an AFM insulator. The SOC cou-\nples the rotational symmetry in the spin space to the one\nin real space. As a consequence, the ordered moments\nin the AFM phase can be either along the zdirection\n(AFMz) or in the x-yplane (AFM xy). For \fnite SOC,\nthese two AFM states have di\u000berent energies, which can\nbe tuned by JK. As shown in the phase diagram, the\nAFM phase contains two ordered states, the AFM zand\nAFMxy. They are separated by a spin reorientation tran-\nsition atJK=J1\u00190:8. For the value of SOC taken, the\nAFM state is topologically nontrivial, and is hence de-\nnoted as T-AFM zstate. The nature of this state and\nthe associated topological phase transition is discussed\nin detail in the next section.5\nFor su\u000eciently large JK, the Kondo hybridization bis\nnonzero (see Fig.5(a)), and the ground state is a KI. Note\nthat for \fnite SOC, this KI does not have a topological\nnontrivial edge state, as a consequence of the topological\nno-go theorem29,39,40. In our calculation at the saddle-\npoint level, the KI exists for y\u00150:6; this provides the\nbasis for taking y= 0:7, as noted earlier. Going beyond\nthe saddle-point level, the dynamical e\u000bects of the Kondo\ncoupling will appear, and we will expect the KI phase to\narise for other choices of y.\nSeveral remarks are in order. The phase diagram,\nFig. 4, has a similar pro\fle of the global phase diagram\nfor the Kondo insulating systems25,41. However, the pres-\nence of SOC has enriched the phase diagram. In the AF\nstate, the ordered moment may lie either within the plane\nor be perpendicular to it. These two states have very dif-\nferent topological properties. We now turn to a detailed\ndiscussion of this last point.\nV. TOPOLOGICAL PROPERTIES OF THE\nAFM STATES\n|\nm\n|, |\nM\n|\n00.10.20.30.40.50.6\nJ\nK00.511.522.53\n|\nm\n| AFM\nxy \n|\nm\n| AFM\nz \n|\nM\n| AFM\nxy \n|\nM\n| AFM\nz\nFIG. 6. The conduction electron magnetization for the\nAFM xyand AFM zstates at\u0015so= 0:1.\nIn this section we discuss the properties of the AFM xyand AFM zstates, in particular to address their topolog-\nical nature. For a clear discussion, we \fx t= 1,J1= 1,\nandJ2=0. Since the Kondo hybridization is not essen-\ntial to the nature of the AFM states, in this section we\nsimply the discussion by setting y= 0.\nWe start by de\fning the order parameters of the two\nstates:\nMx=hSx\nf;Ai=\u0000hSx\nf;Bi; (4)\nMz=hSz\nf;Ai=\u0000hSz\nf;Bi; (5)\nmx=\u0000hsx\nc;Ai=hsx\nc;Bi; (6)\nmz=\u0000hsz\nc;Ai=hsz\nc;Bi: (7)\nNote that for AFM xystate we set Mx=my= 0 with-\nout losing generality. In Fig.(6) we plot the evolution of\nthese AFM order parameters with JKfor a representa-\ntive value of SOC \u0015so= 0:1. Due to the large J1value we\ntake, the sublattice magnetizations of the local moments\nare already saturated to 0 :5. Therefore, at the saddle-\npoint level, they serve as e\u000bective (staggered) magnetic\n\felds to the conduction electrons. The Kondo coupling\nthen induces \fnite sublattice magnetizations for the con-\nduction electrons, and they increase linearly with JKfor\nsmallJKvalues. But mxis generically di\u000berent from mz.\nThis is important for the stabilization of the states.\nWe then discuss the energy competition between the\nAFMxyand AFM zstates. The conduction electron part\nof the mean-\feld Hamiltonian reads:\nHc=\u0010\ncy\nA\"cy\nA#cy\nB\"cy\nB#\u0011T\nhMF0\nB@cA\"\ncA#\ncB\"\ncB#1\nCA (8)\nwith\nhMF=0\nB@\u0003(k)JKMx=2\u000f(k)\nJKMx=2\u0000\u0003(k) \u000f(k)\n\u000f\u0003(k)\u0000\u0003(k)\u0000JKMx=2\n\u000f\u0003(k)\u0000JKMx=2 \u0003(k)1\nCA\n(9)\nfor the AFM xystate and\nhMF=0\nB@\u0003(k) +JKMz=2 \u000f(k)\n\u0000\u0003(k)\u0000JKMz=2 \u000f(k)\n\u000f\u0003(k) \u0000\u0003(k)\u0000JKMz=2\n\u000f\u0003(k) \u0003( k) +JKMz=21\nCA (10)\nfor the AFM zstate. Here \u0003( k) =\n2\u0015so(sin(k\u0001a1)\u0000sin(k\u0001a2)\u0000sin(k\u0001(a1\u0000a2))),\n\u000f(k) =t1(1 +e\u0000ik\u0001a1+e\u0000ik\u0001a2),\u000f\u0003(k) is the complex con-\njugate of\u000f(k), anda1= (p\n3=2;1=2),a2= (p\n3=2;\u00001=2)\nare the primitive vectors. For both states the eigenvaluesare doubly degenerate.\nEc\n\u0006;xy(k) =\u0006p\n\u0003(k)2+ (JKMx=2)2+j\u000f(k)j2(11)\nEc\n\u0006;z(k) =\u0006p\n(\u0003(k) +JKMz=2)2+j\u000f(k)j2(12)\nThe eigenenergies of the spinon band can be obtained6\nin a similar way:\nEf\n\u0006;xy(k) =\u00061\n2(3J1Mx+JKmx); (13)\nEf\n\u0006;z(k) =\u00061\n2(3J1Mz+JKmz): (14)\nThe expression of total energy for either state is then\nEtot= 21\nNkX\nkEc\n\u0000(k) + 21\nNkX\nkEf\n\u0000(k)\n+ 3J1jMj2+ 2JK(M\u0001m): (15)\nThe \frst line of the above expression comes from \flling\nthe bands up to the Fermi energy (which is \fxed to be\nzero here). The second line is the constant term in the\nmean-\feld decomposition. The factor of 2 in the ksum-\nmation is to take into account the double degeneracy of\nthe energies. Nkrefers to the number of kpoints in the\n\frst Brillouin zone.\nBy comparing the expressions of Ec\n\u0000(k) in Eqns. (11)\nand (12), we \fnd that adding a small Mxis to increase\nthe size of the gap at both of the two (inequivalent) Dirac\npoints, thereby pushing the states further away from the\nFermi-energy. While adding a small Mzis to enlarge the\ngap at one Dirac point but reduce the gap size at the\nother one. Therefore, an AFM xystate is more favorable\nthan the AFM zstate in lowering the energy of conduction\nelectronsP\nkEc\n\u0000(k).\nOn the other hand, from Eqns.(13)-(15), we see that\nthe overall e\u000bect of adding a magnetization of the con-\nduction band, m, is to increase the total energy Etot(the\nmain energy increase comes from the 2 JK(M\u0001m) term).\nBecausejmzj 0, and the ground state is an AFM zstate. With\nincreasingJK, the contribution from the 2 JK(M\u0001m)\nterm is more important. \u0001 Ecrosses zero to be negative,\nand the AFM xystate is eventually energetically favorable\nfor largeJK.\nNext we discuss the topological nature of the AFM z\nand AFM xystate. In the absence of Kondo coupling JK,\nthe conduction electrons form a TI, which is protected\nby the TRS. Their the left- and right-moving edge states\nconnecting the conduction and valence bands are respec-\ntively coupled to up and down spin \ravors (eigenstates of\ntheSzoperator) as the consequence of SOC, and these\ntwo spin polarized edge states do not mix.\nOnce the TRS is broken by the AFM order, generically,\ntopologically nontrivial edge states are no longer guaran-\nteed. However, in the AFM zstate, the structure of the\nHamiltonian for the conduction electrons is as same as\nthat in a TI. This is clearly shown in Eq. (10): the e\u000bect\nof magnetic order is only to shift \u0003( k) to \u0003(k)+JKMz=2.\nIn particular, the spin-up and spin-down sectors still do\nnot mix each other. Therefore, the two spin polarized\nedge states are still well de\fned as in the TI, and the sys-\ntem is topologically nontrivial though without the pro-\ntection of TRS. Note that the above analysis is based\non assuming JKMz\u001c\u0003(k), where the bulk gap be-\ntween the conduction and valence bands is \fnite. For\nJKMz>6p\n3\u0015so=(1\u0000y), the bulk gap closes at one of\nthe inequivalent Dirac points and the system is driven\nto a topologically trivial phase via a topological phase\ntransition.29. We also note that a similar AFM zstate\narises in a Kondo lattice model without SOC but with a\nHaldane coupling, as analyzed in Ref. 33.\nFor the AFM xystate, we can examine the Hamiltonian\nfor the conduction electrons in a similar way. As shown\nin Eq. (9), the transverse magnetic order Mxmixes the\nspin-up and spin-down sectors. As a result, a \fnite hy-\nbridization gap opens between the two edge states mak-\ning the system topologically trivial.\nTo support the above analysis, we perform calculations\nof the energy spectrum of the conduction electrons in\nthe AFM zand AFM xystates, as shown in Eq.(9) and\nEq.(10), on a \fnite slab of size Lx\u0002Ly, withLx= 200\nandLy= 40. The boundary condition is chosen to be\nperiodic along the xdirection and open and zig-zag-type\nalong theydirection. In Fig. 8 we show the plots of the\nenergy spectra with three di\u000berent set of parameters: (a)\n\u0015so= 0:01,JK= 0:4,Mz= 0:5, (b)\u0015so= 0:0,JK= 0:4,\nMz= 0:5, and (c)\u0015so= 0:0,JK= 0:8,Mx= 0:5, which\nrespectively correspond to the topologically trivial AFM z\nstate, topological AFM zinsulator, and AFM xystate. As\nclearly seen, the gapless edge states only exist for pa-\nrameter set (b), where the system is in the topological\nAFMzstate. Note that in this state, the spectrum is\nasymmetric with respect to the Brilluion zone boundary7\n(kx=\u0019), re\recting the explicit breaking of TRS. Based\non our analysis and numerical calculations, we construct\na phase diagram, shown in Fig. 9, to illustrate the com-\npetition of these AFM states. As expected, the AFM z\nstate is stabilized for JK.0:7, and is topological for\nJK<12p\n3\u0015so(above the red line).\nFIG. 8. Energy spectra of the trivial AFM zstate [in (a)], the\ntopological AFM zinsulator [in (b)], and the AFM xystate\n[in (c)] from \fnite slab calculations. Black lines denote the\nbulk states and red lines denote edge states. The topological\nAFM zstate is characterized by the gapless edge states. See\ntext for detailed information on the parameters.\nVI. DISCUSSION AND CONCLUSION\nWe have discussed the properties of various phases in\nthe ground-state phase diagram of the spin-orbit-coupled\nKondo lattice model on the honeycomb lattice at half\n\flling. We have shown how the competition of SOC,\nλ\nso\n00.050.10.150.2\nJ\nK00.20.40.60.81\nAFM\nz\nTopological Insulator\nAFM\nxy\nTrivial\nInsulator\nAFM\nz\nTrivial InsulatorFIG. 9. Phase diagram in the \u0015so-JKplane showing the com-\npetition of various AFM states. The red line denotes a topo-\nlogical phase transition between the topological trivial and\ntopological nontrivial AFM zstates, and the black curve gives\nthe boundary between the AFM zand AFM xystates. These\ntwo states become equivalent in the limit of \u0015so!0.\nKondo interaction, and magnetic frustration stabilizes\nthese phases. For example, in the AFM phase the mo-\nments can order either along the z-direction or within\nthex-yplane. In our model, the AFM order is driven by\nthe RKKY interaction, and the competition of SOC and\nKondo interaction dictates the direction of the ordered\nmagnetic moments.\nThroughout this work, we have discussed the phase\ndiagram of the model at half \flling. The phase dia-\ngram away from half-\flling is also an interesting problem.\nWe expect that the competition between the AFM zand\nAFMxystates persist at generic \fllings, but the topolog-\nical feature will not. Another interesting \flling would be\nthe dilute-carrier limit, where a DKSM exists, and can\nbe tuned to a TKI by increasing SOC.31\nIn this work we have considered a particular type of\nSOC, which is inherent in the bandstructure of the itin-\nerant electrons. In real materials, there are also SOC\nterms that involve the magnetic ions. Such couplings\nwill lead to models beyond the current work, and may\nfurther enrich the global phase diagram.\nIn conclusion, we have investigated the ground-\nstate phase diagram of a spin-orbit coupled Kondo-\nlattice model at half-\flling. The combination of SOC,\nKondo and RKKY interactions produces various quan-\ntum phases, including a Kondo insulator, a topologi-\ncal insulator with VBS spin correlations, and two AFM\nphases. Depending on the strength of SOC, the magnetic\nmoments in the AFM phase can be either ordered per-\npendicular to or in the x-yplane. We further show that\nthez-AFM state is topologically nontrivial for strong and\nmoderate SOC, and can be tuned to a topologically triv-\nial one via a topological phase transition by varying ei-\nther the SOC or the Kondo coupling. Our results shed\nnew light on the global phase diagram of heavy fermion\nmaterials.8\nACKNOWLEDGEMENTS\nWe thank W. Ding, P. Goswami, S. E. Grefe, H.-H.\nLai, Y. Liu, S. Paschen, J. H. Pixley, T. Xiang, and\nG. M. Zhang for useful discussions. Work at Renmin\nUniversity was supported by the Ministry of Science and\nTechnology of China, National Program on Key Research\nProject Grant number 2016YFA0300504, the NationalScience Foundation of China Grant number 11674392\nand the Research Funds of Remnin University of China\nGrant number 18XNLG24. Work at Rice was in part sup-\nported by the NSF Grant DMR-1611392 and the Robert\nA. Welch Foundation Grant C-1411. Q.S. acknowledges\nthe hospitality and support by a Ulam Scholarship from\nthe Center for Nonlinear Studies at Los Alamos National\nLaboratory.\n\u0003rong.yu@ruc.edu.cn\nyqmsi@rice.edu\n1Special issue on Quantum Phase Transitions, J. Low Temp.\nPhys. 161, 1 (2010).\n2S. Sachdev, Quantum Phase Transitions (Cambridge Uni-\nversity Press, Cambridge, 2011), 2nd ed.\n3Q. Si and F. Steglich, Science 329, 1161C1166 (2010).\n4P. Gegenwart, Q. Si, and F. Steglich Nat. Phys. 4, 186 -\n197 (2008).\n5H. von L ohneysen, A. Rosch, M. Vojta, and P.Wol\re, Rev.\nMod. Phys. 79, 1015 (2007).\n6H. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys.\n69, 809 (1997).\n7A. C. Hewson, The Kondo Problem to Heavy Fermions ,\nCambridge Univ. Press, Cambridge, England (1993).\n8S. Doniach, Physica B+C 91, 231-234 (1977).\n9J. Custers, et al. , Nature 424, 524-527 (2003).\n10A. Schr oder et al. , Nature 407, 351 (2000).\n11S. Paschen, T. Luhmann, S. Wirth, P. Gegenwart, O.\nTrovarelli, C. Geibel, F. Steglich, P. Coleman, and Q. Si,\nNature 432, 881 (2004).\n12Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Nature\n413, 804 (2001).\n13P. Coleman, C. P\u0013 epin, Q. Si, and R. Ramazashvili, J.\nPhys.: Condens. Matt. 13, R723-R738 (2001).\n14J. A. Hertz, Phys. Rev. B 14, 1165-1184 (1976).\n15A. J. Millis, Phys. Rev. B 48, 7183-7196 (1993).\n16Q. Si, Physica B 378-380 , 23-27 (2006).\n17Q. Si, Phys. Stat. Solid. B 247, 476-484 (2010).\n18J. H. Pixley, R. Yu, and Q. Si, Phys. Rev. Lett. 113,\n176402 (2014).\n19Q. Si and S. Paschen, Phys. Stat. Solid. B 250, 425-438\n(2013).\n20R. S. K. Mong, A. M. Essin, and J. E. Moore, Phys. Rev.\nB81, 245209 (2010).\n21S. Nakatsuji et al. , Phys. Rev. Lett. 96, 087204 (2006).22G. Chen, Phys. Rev. B 94, 205107 (2016).\n23M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys. Rev.\nLett.104, 106408 (2010).\n24A. Barla et al. , Phys. Rev. Lett. 94, 166401 (2005).\n25S. Yamamoto and Q. Si, J. Low Temp. Phys. 161, 233-262\n(2010).\n26H.-H. Lai, S. E. Grefe, S. Paschen, and Q. Si, PNAS 115,\n93 (2018).\n27S. Dzsaber et al. , Phys. Rev. Lett. 118, 246601 (2017).\n28S. Dzsaber et al. arXiv:1811.02819.\n29X.-Y. Feng, C.-H. Chung, J. Dai, and Q. Si, Phys. Rev.\nLett.111, 016402 (2013).\n30C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n31X.-Y. Feng, H. Zhong, J. Dai, and Q. Si, arXiv:1605.02380\n(2016).\n32F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).\n33Y. Zhong, Y.-F. Wang, Y.-Q. Wang, and H.-G. Luo, Phys.\nRev. B 87, 035128 (2013).\n34C. Lacroix and M. Cyrot, Phys. Rev. B 20, 1969 (1979).\n35H. Li, H.-F. Song, and Y. Liu, EuroPhys. Lett. 116, 37005\n(2016).\n36H. Li, Y. Liu, G.-M. Zhang, and L. Yu, J. Phys.: Condens.\nMatter 27, 425601 (2015).\n37R. Ganesh, J. van den Brink, and S. Nishimoto, Phys. Rev.\nLett.110, 127203 (2013).\n38B. K. Clark , D. A. Abanin, S. L. Sondhi, Phys. Rev. Lett.\n107, 087204(2011).\n39M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n40X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057C1110\n(2011).\n41J. H. Pixley, R. Yu, S. Paschen, and Q. Si, Phys. Rev. B\n98, 085110 (2018)." }, { "title": "2307.14673v1.Spin_orbit_torque_emerging_from_orbital_textures_in_centrosymmetric_materials.pdf", "content": "Spin-orbit torque emerging from orbital textures in centrosymmetric materials\nLuis M. Canonico,1Jose H. Garcia,1,∗and Stephan Roche1,2\n1Catalan Institute of Nanoscience and Nanotechnology (ICN2),\nCSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain\n2ICREA, Institució Catalana de Recerca i Estudis Avançats, 08070 Barcelona, Spain\n(Dated: July 28, 2023)\nWe unveil a hitherto concealed spin-orbit torque mechanism driven by orbital degrees of freedom\nin centrosymmetric two-dimensional transition metal dichalcogenides (focusing on PtSe 2). Using\nfirst-principles simulations, tight-binding models and large-scale quantum transport calculations,\nwe show that such a mechanism fundamentally stems from a spatial localization of orbital textures\nat opposite sides of the material, which imprints their symmetries onto spin-orbit coupling effects,\nfurther producing efficient and tunable spin-orbit torque. Our study suggests that orbital-spin\nentanglement at play in centrosymmetric materials can be harnessed as a resource for outperforming\nconventional spin-orbit torques generated by the Rashba-type effects.\nSpin-orbit torque (SOT) is a central mechanism me-\ndiated by charge-to-spin conversion that enables efficient\nmanipulation of magnetization in spintronic devices1–3.\nSOT is usually generated by nonequilibrium manifesta-\ntions of the spin-orbit coupling (SOC) such the spin-Hall\nand Rashba-Edelstein effects which accumulate intrin-\nsic angular momentum source that exerts some torque\non nearby magnets. These mechanisms are well de-\nscribed in noncentrosymmetric systems4and are funda-\nmentally related to spin-momentum locking in reciprocal\nspace. Other mechanisms have been discussed for low-\ndimensional systems, including the contribution of the\nBerry curvature5or skew scattering6effects, but they re-\nquire inversion symmetry breaking, so to date, all known\nSOT mechanisms exclude the large family of centrosym-\nmetricmaterials.\nRecently, the possibility of unconventional spatially-\ndependent spin-momentum locking properties has been\ndiscussed for a particular family of centrosymmetric ma-\nterials, namely the 1T-phase transition metal dichalco-\ngenides (TMDs). In these materials, the emergence of\nhidden spin textures is dictated by the presence of dipo-\nlar fields within the crystal structure7,8, as confirmed by\nARPES experiments9,10. On the other hand, the elec-\ntrical manipulation of the orbital degrees of freedom can\ngive rise to the orbital-Hall effect, where a longitudinal\ncurrent drives a sizable transverse flow of orbital angu-\nlar momentum even in systems with negligible SOC11–19.\nSuch orbital effects have been shown to also signal the\npresence of high-order topological phases20, pointing to\nnovel paradigms in “orbitronics\". Importantly, the role\nof the orbital degrees of freedom and intertwined spin\nresponse was recognized in a more general description\nof the Rashba SOC4,21–23and the driving or enhance-\nmentofSOT24–26. However, afundamentalorbital-based\ndescription of complex spin phenomena such as the hid-\nden spin textures and their ultimate nature in generating\nnovel SOT effects, remains largely unexplored.\nIn this letter, we predict a novel SOT mechanism in\ncentrosymmetric materials (such as monolayer PtSe 2)\nwhose origin stems from intrinsic orbital textures of\nthe electronic states entangled with their spin features.The hybridization between orbitals with different sym-\nmetries is the mechanism behind the emergence of this\nunconventional SOT, which can be further tailored us-\ning external electric fields. The theoretical results are\nobtained using realistic tight-binding models elaborated\nwith first-principles simulations and implemented in\nlarge-scale (real-space) quantum transport simulations.\nThesediscoverednewSOTcomponentsin centrosymmet-\nricstructures enlarge the portfolio of available materi-\nals for designing ultracompact 2D materials-based spin-\ntronicarchitecturesinthecontextofnon-volatilememory\ntechnologies27.\nMethods. We performed Density Functional The-\nory (DFT) calculations using the plane-wave-based code\nQuantum Espresso28. The correlation and exchange\nfunctionals are treated within the generalized gradient\napproximation (GGA)29. We described the ionic cores\nusing fully relativistic projected augmented wave poten-\ntials (PAW)30and set an energy cutoff of 120Ry and 800\nRy for computing the wave functions and charge densi-\nties, respectively. For the self-consistent run, we used a\nk-mesh of 24×24×1points, and for subsequent calcula-\ntions, we set a 32×32×1mesh. To avoid spurious inter-\nactions due to periodic boundary conditions, we added\na vacuum spacing of 17.57and set the forces and the\nself-consistent cutoffs to 5.142meV/ and 4.03×10−9Ry,\nrespectively. We obtained an energy gap of 1.17 eV and\na lattice constant of 3.75 in good agreement with pre-\nvious first-principles calculations. We then constructed\nan effective tight-binding Hamiltonians from our DFT\ncalculations using the pseudo-atomic orbital projection\n(PAO) method31–34implemented in PAOFLOW35,36.\nThis approach consists in projecting the Kohn-Sham en-\nergy states into the compact subspace spanned by the\npseudo-atomic orbitals built in the PAW potentials. We\nused PAW potentials for the Pt and Se constructed with\nanspdandspbasis, respectively. Thecomputationofthe\nspin-orbit torque is achieved using T=P\niˆm∆xc×⟨Si⟩,\nwhere ˆmis the magnetization direction, ∆xcis the ex-\nchange coupling strength and ⟨Si⟩is the nonequilibrium\nspin density at the i-th atomic plane calculated with the\nKubo-Bastin formula37:arXiv:2307.14673v1 [cond-mat.mes-hall] 27 Jul 20232\nFIG. 1. (a) Bandstructure of free-standing monolayer PtSe 2computed using fully relativistic band structure DFT (red dotted\nlines) and PAO Hamiltonian (blue solid lines). Inset: Side view of PtSe 2monolayer (left) where its height is chosen to match\nthey-axis of the dipolar electric field as a function of the z−direction computed from DFT (right). (b) Orbital texture of\nPtSe 2projected in the top Se atoms, computed in the energy window shown in (a) (grey area), where Lx(k)andLy(k)are\nrepresented as the xandycomponents of each arrow and Lz(k)as the background color. (c) Spin texture of PtSe 2represented\nfollowing the convention in (b) but replacing the orbital angular momentum with the spin operator.\n⟨Si(ε)⟩=−2ImZε\n−∞dε′f(ε′)×\n\u0010\nTr⟨Siδ(ˆH−ε′)∂ε′Gε′+(ˆJ·E)⟩\u0011\n,(1)\nwhere Siis the vector of spin density operators at the\ni-th atomic plane, Eis the electric field, ˆJis the cur-\nrent density operator ˆJ=−ei\n¯hh\nˆH,Ri\nwith ˆRthe po-\nsition operator and ˆHthe Hamiltonian of the system,\nandG+\nεandδ(H−ε)the retarded Green’s and spectral\nfunctions are approximated using the kernel-polynomial\nmethod38,39asimplementedinthe LSQUANT toolkit40.\nWe used 1024Chebyshev moments with the Jackson ker-\nnel to obtain an energy resolution of δε= 33meV and\nfully exploit the symmetries of the system, which allows\nconsidering sizes of 256×256unit cells with 34orbitals\neach, amountingtoover 2millionsofatomicorbitalswith\n∼3.067×109hopping terms per unit cell achieving DFT-\nlevel precision (See SM41).\nOrbital and spin-momentum locking. FIG. 1 (a) shows\na perfect agreement between the DFT energy bands (red\ndots) against those computed using the PAO Hamilto-\nnian(bluesolidlines)withintheconsideredenergyrange.\nIn the inset of FIG. 1 (a), we show the crystalline struc-\nture of PtSe 2(left) together with the electric dipole field\n(right) computed directly from the DFT as the partial\nderivative of the planar average of the potential energy\nversus the vertical distance within the simulation box.\nWe found a finite and sizable field located at both Se\natomic planes (which are inversion partners) with oppo-\nsite values. Moreover, the fields vanish on the Pt ion as\na consequence of the inversion symmetry of the system.\nPanels FIG. 1 (b-c) present the orbital and spin texturesnearby the Fermi contour defined by the energy EF=\n−0.05eV,respectively. Theorbital(spin)texturesarede-\nfinedastheFermisurfaceaverageoftheorbital(spin)op-\nerators( ⟨On(k)⟩=−4KBTP\nnf′\nnk⟨ψkn|O|ψkn⟩), where\nKBis the Boltzmann constant, Tis the temperature,\nf′\nnkis the derivative of the Fermi-Dirac distribution and\n|ψn\nk⟩is the eigenstate of the n-th band evaluated at k.\nThe arrows in these figures indicate the xandycompo-\nnents of the orbital (spin) textures while the color rep-\nresents their out-of-plane components as quantified by\nthe color bars. FIG. 1 (b) shows that within our en-\nergy range, the bands display orbital-momentum locking.\nThe orbital textures are reversed at opposite Se planes\n(see SM41), which is related to the internal symmetries\nof the system. Indeed, 1Tmonolayers of PtSe 2belong\nto the D3dpoint group so that the chalcogen sublayers\nare rotated 180◦to each other, yielding an equal dis-\ntribution of charges across the Se-Pt-Se atomic layers.\nThus, this charge distribution and the absence of a hor-\nizontal mirror plane allow the existence of the vertical\ndipole fields (inset of FIG. 1 (a)), which are similar to\nthe ones observed in layers of bulk HfS 210. These fields\ninduce a local out-of-the-plane asymmetry and separate\nthe manifold of porbitals of the chalcogen atoms in two\nirreducible representations (Irrep). One of these repre-\nsentations is one-dimensional, and it is related to the pz\norbitals, which are moved to lower energies, whereas the\nother is a 2×2Irrep spanned by linear combinations of\nthepxandpyorbitals and are the main constituents of\nthe top valence bands at Γ. Since the latter Irrep is 2×2,\nin close analogy with spin, the top valence bands near the\nBrillouin zone are mappable into orbital angular momen-\ntum (OAM) pseudospinors. In this picture, it becomes\nclear that the hybridizations with the pzorbitals occur-\nring far from Γare translated as an effective coupling3\nFIG. 2. (a) Layer-projected nonequilibrium Sydensities as a function of the energy for the cases with ∆0= 0(dotted line)\nand∆0= 100meV/¯h(solid lines) with β→ ∞. The colors indicate the i=top (red) and i=bottom (blue) contributions for\nthe case with ∆0̸= 0. Inset: Layer-projected nonequilibrium spin densities for a wider energy range. (b) Torque efficiency ξx\nfor∆0= 100meV/¯has a function of the energy. The colors signal the speed of the decay of the effective exchange interaction\nwith the vertical distance.\nbetween these OAM pseudospinors, which consequently\nleads to the emergence of the orbital Rashba effect22,42.\nThismechanismexplainingtheappearanceoforbitaltex-\ntures is similar to the one presented in Ref.[43]. However,\nhere the global inversion symmetry in the system im-\nposes that the orbital (spin) textures on the two atomic\nchalcogen planes are oppositely aligned and degenerate\nin reciprocal space, leading to the vanishing of the ab-\nsolute spin and OAM. Such type of orbital textures and\ntheir real-space configuration can therefore form even in\nthe absence of spin-orbit coupling (See SM41), reinforc-\ning their crystal field origin. Finally, FIG. 1 (c) shows\nthat when the spin-orbit coupling is present, such orbital\ntextures are imprinted into the spin features, inducing\nspin-momentum locking, which has the particularity to\nbe opposite at each chalcogen atomic plane. This is in\nagreement with recent experimental findings by Refs. [9]\nand[10], whichhaverevealedthepresenceofhiddenspin-\ntextures at the surfaces of PtSe 2and HfS 2.\nAtomic-plane localized spin-orbit torque . The domi-\nnant mechanism for SOT in conventional 2D materials is\ntheREE.ItreliesontheREthatemergesinsystemswith\nbroken inversion symmetry, typically due to the presence\nof an interface, which splits the energy bands of oppo-\nsite chirality and generates a nonequilibrium spin den-\nsity due to the differences in the Fermi contour. Here\nfor PtSe 2, we demonstrate that spin-momentum locking\nemerges naturally from the interplay between the SOC\nand the crystal fields. However, momentum-space degen-\neracy enforced by the inversion symmetry of the system\nprevents the formation of a net nonequilibrium spin den-\nsities. Nonetheless, practical2DSOTdeviceswillbetyp-\nically based on PtSe 2/Ferromagnetic (FM) heterostruc-\ntures. Such stacking produces two effects at the origin of\nSOT, namely (I) a built-in electric field induced by the\npresenceoftheFMandthesubstrate, andresponsiblefor\nthe spin-splitting (Stark effect); and (II) an asymmetric\nelectronic coupling between the closest and farthest Se-planes with respect to the FM. Generally, (I) is expected\nto dominate most of the conventional SOT phenomenol-\nogy in centrosymmetric systems. However, the complex\ncrystalline structure and the real-space localization of or-\nbital and spin textures observed in 1T PtSe 2and other\n1T TMDs suggest that the asymmetric coupling between\nthe chalcogen sublayers and the FM will disrupt the com-\npensation of the angular momentum textures, resulting\nin sizable-spin-to-charge conversion and favoring the ap-\npearance of new SOT components. To fully quantify the\nresulting contribution of (II), driven by such localized\nnature of the interaction between the exchange coupling\nfield and the orbital and spin textures in this system,\nwe included some phenomenological position-dependent\nexchange term in our Hamiltonian as:\nHxc(β) =X\ni∆0exp \n−β\u0012zi−ztop\nzbot\u00132!\n·Si\nz,(2)\nwhere Si\nzis the zcomponent of the spin operator of\nthei-atomic plane, ∆0= 100meV/¯his the intensity of\nthe exchange field, βis a phenomenological decay factor\nof the proximity-induced exchange interaction, ztopand\nzbotare the positions of the top and bottom Se atomic\nplanes, respectively.\nWe first analyze the effect associated with the contri-\nbution (II) on the spin-to-charge conversion and on the\ntorque generation in the system. FIG. 2 (a) compares the\nenergy-resolvednonequilibrium Syspindensityprojected\nat both chalcogen planes in the cases with (considering\nβ→ ∞) and without exchange coupling. From the fig-\nure, it is clear that the coupling term breaks the inversion\nsymmetry and disrupts the compensation of the nonequi-\nlibrium spin densities of the top and bottom Se planes.\nFollowingthis, alayer-dependentenergyshiftoccursnear\nthe band gap and signals the real-space decoupling of the\nenergy states of both layers in this energy region. FIG. 24\nFIG. 3. Total and extrinsic torque efficiencies ξxandξxext, as a function of the energy for the case with ∆0= 100meV/¯hand\nβ→ ∞for negative and positive fields. Inset: Comparison between the torque efficiencies in the absence of external electric\nfields and with E=±300mV/, the labels refer to the dominant contribution for SOT. The colors indicate the value of the\nout-of-plane electric field.\n(a) inset, present the overall layer-dependent nonequilib-\nrium Syspin densities for both cases. Qualitatively, the\nspin-to-charge conversion in this system is not altered by\nthe inclusion of the exchange term at the top Se-plane,\nindicating that the resilience of these layer-localized spin-\norbital textures is due to the dominating energy scale set\nby the crystal field. Due to the concealed nature of these\nspin textures and their Fermi-surface dependence, we de-\nfine a specific figure of merit; the torque efficiency as\nξ(β) =P\ni∆xc(β)×Si(β)/∆0P\ni|Si(β)|, which quan-\ntifies the fraction of the total spin angular momentum\nthat participates in the torque generation irrespective of\nthe specific details associated to the bandstructure of the\nsystem.\nFig. 2(b) shows ξxassociated with the torque compo-\nnentalongthex-direction. Oneobservesthat ξxincreases\nwith increasing β, reaching values as large as ξx≃0.5\nover most of the considered energy range. This enhance-\nment of ξxis related to the suppression of the coupling\nbetween the bottom Se-plane and the FM, which leaves\nthe top Se-plane as the sole contributor to the torque in-\ntensity. In the vicinity of the charge neutrality point, ξx\nexceeds 0.6, indicating the predominance of the contribu-\ntion coming from the top Se plane to the total current-\ninducednonequilibriumspindensities. Additionally,near\nthe charge neutrality point, ξxexhibits a dip that occurs\nfor all the values of β. This reduction is related to the\npredominance of contributions to the spin-to-charge con-\nversion coming from the bottom Se and Pt atoms. In\ncontrast, the y-component of the ξy(shown in SM41) is\nonly sizable near the band gap edge and inverts its value\nfor large enough β, owing to the suppression of the SOT\ncontributions from the bottom Se and Pt atoms.\nElectrical tailoring of spin-orbit torque. We further in-\nclude the effect of an external electric field in our simula-\ntions to deepen the analysis of the interplay between the\ncontributions (I) and (II) to the spin-orbit torque compo-\nnents. FIG. 3 (a) shows the ξxtorque efficiency for var-\nious negative and positive out-of-the-plane electric fields\nfor large β. To facilitate the comparison with the efficien-cies portrayed in Fig. 2, we compute the torque efficiency\nusing the total spin angular momentum in the system for\nthe case without external electric fields as a reference to\nquantify the additional contributions to the torque com-\ning from (I). Upon inspection, one can identify two re-\ngionswheretheeffectsoperatedifferently. Awayfromthe\nenergy gap (shown as (I) in the inset), both field config-\nurations produce similar changes in the torque efficiency\nbut with opposite signs. This sign dependence shows\nthat contributions (I) dominate the torque generation in\nthis region. Conversely, near the charge neutrality point\n(shaded area labeled as (II) on the inset), the combined\neffect of (I) and (II) modulates the total torque efficiency.\nFIG. 3 (b) illustrates this more clearly. Here we defined\nthe extrinsic efficiency as ξxext=ξx(E)−ξx(E= 0),\nwhich captures solely the effects associated with the ex-\nternalfields. Thisfigureshowsthatfornegativefields, (I)\nand (II) have opposite signs, in the limit of large negative\nfields, (I) dominates and defines the sign of the torques\nin the system. In contrast, for positive fields, (I) and\n(II) work cooperatively, and for large field amplitudes,\nthe torque efficiency reaches approximately ∼ ×3the ef-\nficiency without an external field. Thus, this electrical\ntunability creates opportunities for electrical control of\nSOT in compact devices.\nIn conclusion, by combining first-principles calcula-\ntions, tight-binding models, and large-scale quantum\ntransport simulations, we uncovered the orbital origin of\nthe hidden spin textures in inversion symmetric TMD\nand analyzed their real-space characteristics. Using\nPtSe 2as a prototypical member of the 1T TMD fam-\nily, we demonstrated that the overall efficiency in gen-\nerating spin-orbit torques in these systems is substan-\ntially increased by the real-space proximity interactions\nbreaking inversion symmetry between the two chalcogen\natoms (which are inversion partners). Our results sug-\ngest that other members of the 1T TMD family, such as\nHfS2, PtTe 2and PdSe 2could be good candidates for ex-\nploiting hidden spin-orbital textures and their real-space\nlocalization in torque applications. This discovered phe-5\nnomenon, evidencing the importance of the cooperative\ninterplay between the orbital and spin degrees of free-\ndom, paves the way to more systematic real-space con-\ntrol of spin-orbit torques for developing novel spin-based\ntechnologies.\nACKNOWLEDGMENTS\nWe acknowledge discussions with A.I. Figueroa, J.F.\nSierra, S.O. Valenzuela, and T.G. Rappoport. L.M.C.\nacknowledges funding from Ministerio de Ciencia e Inno-\nvación de España under grant No. PID2019-106684GB-\nI00 / AEI / 10.13039/501100011033, FJC2021-047300-I, financiado por MCIN/AEI/10.13039/501100011033\nand the European Union “NextGenerationEU”/PRTR.\".\nJ.H.G. acknowledge funding from the European Union\n(ERC, AI4SPIN, 101078370). S.R and J.H.G, acknowl-\nedge funding from MCIN/AEI /10.13039/501100011033\nand European Union \"NextGenerationEU/PRTR” under\ngrant PCI2021-122035-2A-2a and funding from the Eu-\nropean Union’s Horizon 2020 research and innovation\nprogramme under grant agreement No 881603. ICN2\nis funded by the CERCA Programme/Generalitat de\nCatalunya and supported by the Severo Ochoa Cen-\ntresofExcellenceprogramme, GrantCEX2021-001214-S,\nfunded by MCIN/AEI/10.13039.501100011033.\n∗josehugo.garcia@icn2.cat\n1P. Gambardella and I. M. Miron, Philosophical Transac-\ntions of the Royal Society A: Mathematical, Physical and\nEngineering Sciences 369, 3175 (2011).\n2A. Manchon, J. Železn` y, I. M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nReviews of Modern Physics 91, 035004 (2019).\n3Q. Shao, P. Li, L. Liu, H. Yang, S. Fukami, A. Razavi,\nH. Wu, K. Wang, F. Freimuth, Y. Mokrousov, et al., IEEE\nTransactions on Magnetics 57, 1 (2021).\n4G. Bihlmayer, P. Noël, D. V. Vyalikh, E. V. Chulkov, and\nA. Manchon, Nature Reviews Physics 4, 642 (2022).\n5H. Kurebayashi, J. Sinova, D. Fang, A. Irvine, T. Skin-\nner, J. Wunderlich, V. Novák, R. Campion, B. Gallagher,\nE. Vehstedt, et al., Nature nanotechnology 9, 211 (2014).\n6F. Sousa, G. Tatara, and A. Ferreira, Physical Review\nResearch 2, 043401 (2020).\n7X. Zhang, Q. Liu, J.-W. Luo, A. J. Freeman, and\nA. Zunger, Nature Physics 10, 387 (2014).\n8L. Yuan, Q. Liu, X. Zhang, J.-W. Luo, S.-S. Li, and\nA. Zunger, Nature communications 10, 906 (2019).\n9W. Yao, E. Wang, H. Huang, K. Deng, M. Yan, K. Zhang,\nK. Miyamoto, T. Okuda, L. Li, Y. Wang, H. Gao, C. Liu,\nW. Duan, and S. Zhou, Nature Communications 8, 14216\n(2017).\n10O. J. Clark, O. Dowinton, M. S. Bahramy, and J. Sánchez-\nBarriga, Nature Communications 13, 4147 (2022).\n11B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Phys.\nRev. Lett. 95, 066601 (2005).\n12D. Go, D. Jo, C. Kim, and H.-W. Lee, Phys. Rev. Lett.\n121, 086602 (2018).\n13D. Jo, D. Go, and H.-W. Lee, Phys. Rev. B 98, 214405\n(2018).\n14S. Bhowal and S. Satpathy, Phys. Rev. B 102, 035409\n(2020).\n15S. Bhowal and G. Vignale, Phys. Rev. B 103, 195309\n(2021).\n16L. M. Canonico, T. P. Cysne, A. Molina-Sanchez, R. B.\nMuniz, and T. G. Rappoport, Phys. Rev. B 101, 161409\n(2020).\n17T. P. Cysne, M. Costa, L. M. Canonico, M. B. Nardelli,\nR. B. Muniz, and T. G. Rappoport, Phys. Rev. Lett. 126,\n056601 (2021).\n18A. Pezo, D. G. Ovalle, and A. Manchon, Physical ReviewB106, 104414 (2022).\n19Y.-G. Choi, D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H. G.\nPark, C. Kim, B.-C. Min, G.-M. Choi, and H.-W. Lee,\nNature 619, 52 (2023).\n20M. Costa, B. Focassio, L. M. Canonico, T. P. Cysne, G. R.\nSchleder, R. B. Muniz, A. Fazzio, and T. G. Rappoport,\nPhys. Rev. Lett. 130, 116204 (2023).\n21S. R. Park, C. H. Kim, J. Yu, J. H. Han, and C. Kim,\nPhys. Rev. Lett. 107, 156803 (2011).\n22J.-H. Park, C. H. Kim, J.-W. Rhim, and J. H. Han, Phys.\nRev. B 85, 195401 (2012).\n23J. Kim and Y. Otani, Journal of Magnetism and Magnetic\nMaterials 563, 169974 (2022).\n24D. Go, F. Freimuth, J.-P. Hanke, F. Xue, O. Gomonay,\nK.-J. Lee, S. Blügel, P. M. Haney, H.-W. Lee, and\nY. Mokrousov, Phys. Rev. Research 2, 033401 (2020).\n25S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth,\nS. Becker, F. Kammerbauer, J. Yang, G. Jakob,\nY. Mokrousov, and M. Kläui, Phys. Rev. Lett. 125,\n177201 (2020).\n26D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun,\nD. Jo, S. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park,\nB.-C. Min, H. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee,\nNature Communications 12, 6710 (2021).\n27H. Yang, S. O.Valenzuela, M.Chshiev, S. Couet, B. Dieny,\nB. Dlubak, A. Fert, K. Garello, M. Jamet, D.-E. Jeong,\net al., Nature 606, 663 (2022).\n28P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau,\nM. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavaz-\nzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo,\nA. D. Corso, S. de Gironcoli, P. Delugas, R. A. D. Jr,\nA. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer,\nU. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawa-\nmura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri,\nM. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V.\nNguyen, A.O.de-laRoza, L.Paulatto, S.Poncé, D.Rocca,\nR.Sabatini,B.Santra,M.Schlipf,A.P.Seitsonen,A.Smo-\ngunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast,\nX. Wu, and S. Baroni, Journal of Physics: Condensed\nMatter 29, 465901 (2017).\n29J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.77, 3865 (1996).\n30G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n31L. A. Agapito, A. Ferretti, A. Calzolari, S. Curtarolo, and6\nM. Buongiorno Nardelli, Phys. Rev. B 88, 165127 (2013).\n32L. A. Agapito, S. Curtarolo, and M. Buongiorno Nardelli,\nPhys. Rev. X 5, 011006 (2015).\n33L. A. Agapito, M. Fornari, D. Ceresoli, A. Ferretti, S. Cur-\ntarolo, and M. Buongiorno Nardelli, Phys. Rev. B 93,\n125137 (2016).\n34L. A. Agapito, S. Ismail-Beigi, S. Curtarolo, M. Fornari,\nand M. Buongiorno Nardelli, Phys. Rev. B 93, 035104\n(2016).\n35M. Buongiorno Nardelli, F. T. Cerasoli, M. Costa, S. Cur-\ntarolo, R. D. Gennaro, M. Fornari, L. Liyanage, A. R.\nSupka, and H. Wang, Computational Materials Science\n143, 462 (2018).\n36F. T. Cerasoli, A. R. Supka, A. Jayaraj, M. Costa, I. Siloi,\nJ. Sławińska, S. Curtarolo, M. Fornari, D. Ceresoli, and\nM. Buongiorno Nardelli, Computational Materials Science200, 110828 (2021).\n37A. Bastin, C. Lewiner, O. Betbeder-Matibet, and\nP. Nozieres, Journal of Physics and Chemistry of Solids\n32, 1811 (1971).\n38S. M. João, M. Anđelković, L. Covaci, T. G. Rappoport,\nJ. M. V. P. Lopes, and A. Ferreira, Royal Society Open\nScience 7, 191809 (2020).\n39Z. Fan, J. H. Garcia, A. W. Cummings, J. E. Barrios-\nVargas, M.Panhans, A.Harju, F.Ortmann, andS.Roche,\nPhysics Reports 903, 1 (2021).\n40“LSQUANT,” www.lsquant.org .\n41Supplementary Material.\n42D. Go, D. Jo, T. Gao, K. Ando, S. Blügel, H.-W. Lee, and\nY. Mokrousov, Phys. Rev. B 103, L121113 (2021).\n43S. Han, H.-W. Lee, and K.-W. Kim, Phys. Rev. Lett. 128,\n176601 (2022)." }, { "title": "0811.4104v1.Spin_orbital_frustrations_and_anomalous_metallic_state_in_iron_pnictide_superconductors.pdf", "content": "arXiv:0811.4104v1 [cond-mat.supr-con] 25 Nov 2008Spin-orbital frustrations and anomalous metallic state in iron-pnictide\nsuperconductors\nFrank Kr¨ uger1, Sanjeev Kumar2,3, Jan Zaanen2, and Jeroen van den Brink2,4\n1Department of Physics, University of Illinois, 1110 W. Gree n St., Urbana, IL 61801\n2Instituut-Lorentz, Universiteit Leiden, P. O. Box 9506, 23 00 RA Leiden, The Netherlands\n3Faculty of Science and Technology, University of Twente,\nP. O. Box 217, 7500 AE Enschede, The Netherlands\n4Institute for Molecules and Materials, Radboud Universite it Nijmegen,\nP. O. Box 9010, 6500 GL Nijmegen, The Netherlands\n(Dated: June 27, 2021)\nWe develop an understanding of the anomalous metal state of t he parent compounds of recently\ndiscovered iron based superconductors starting from a stro ng coupling viewpoint, including orbital\ndegrees of freedom. On the basis of an intermediate-spin ( S=1) state for the Fe2+ions, we derive a\nKugel-Khomskii spin-orbital Hamiltonian for the active t2gorbitals. It turns out to be a highly com-\nplex model with frustrated spin and orbital interactions. W e compute its classical phase diagrams\nand provide an understanding for the stability of the variou s phases by investigating its spin-only\nand orbital-only limits. The experimentally observed spin -stripe state is found to be stable over\na wide regime of physical parameters and can be accompanied b y three different types of orbital\norders. Of these the orbital-ferro and orbital-stripe orde rs are particularly interesting since they\nbreak the in-plane lattice symmetry – a robust feature of the undoped compounds. We compute\nthe magnetic excitation spectra for the effective spin Hamil tonian, observing a strong reduction of\nthe ordered moment, and point out that the proposed orbital o rdering pattern can be measured in\nresonant X-ray diffraction.\nPACS numbers: 74.25.Jb, 74.70.-b, 74.20.Mn, 74.25.Ha\nI. INTRODUCTION\nThe begining of this year marked the discovery of a\nnew and very unusual family of high temperature super-\nconductors: the iron pnictides. Superconductivity at 26\nK was discovered in fluorine doped rare-earth iron oxyp-\nnictide LaOFeAs1,2. In subsequent experimental studies\ninvolving different rare earth elements a superconduct-\ningTclarger than 50 K was reported3,4,5. Since then a\nlargenumberofexperimentalandtheoreticalpapershave\nbeen published, making evident the immense interest of\nthe condensed matter community in this subject6.\nIthasbecomeclearthattheironpnictidesuperconduc-\ntors have, besides a number of substantial differences,\nat least one striking similarity with the copper oxides:\nthe superconductivity emerges by doping an antiferro-\nmagnetic, non-superconducting parent compound. This\nantiferromagetism is however of a very unusual kind. In-\nstead of the simple staggered ’( π,π)’ antiferromagnetism\nof the undoped cuprates, this ’stripe’ or ’( π,0)’ spin or-\nder involves rows of parallel spins on the square Fe-ion\nlattice that are mutually staggered7. In fact, before this\norder sets in a structural phase transition occurs where\nthe two in-plane lattice constants become inequivalent.\nThis structural distortion is very small, but it appears\nthat the electron system undergoes a major reorganiza-\ntion at this transition. This is manifested by resistivity\nanomalies, drastic changes in the Hall- and Seebeck co-\nefficients, and so forth8. Although the magnetic- and\nstructural distortion appear to be coincident in the 122\nfamily7,9, in the 1111 compounds they are clearly sepa-rated7, andthereitisobviousthatthelargescalechanges\nin the electron system occur at the structural transition\nwhile barely anything is seen at the magnetic transition.\nGiven that the structural deformation is minute, this\nis an apparent paradox. Assuming that only the spins\nmatter one could envisage that the spin ordering would\nlead to a drastic nesting type reorganizationof the Fermi\nsurfaces, causing a strong change in the electronic prop-\nerties. But why is so little happening at the magnetic\ntransition? One could speculate that the spins are fluc-\ntuating in fanciful ways, and that these fluctuations re-\nact strongly to the structural change10,11,12. Such pos-\nsibilities cannot be excluded on theoretical grounds but\nwhichever way one wants to proceed invoking only spins\nand itinerant carriers: one is facing a problem of princi-\nple.\nThis paper is dedicated to the cause that valuable\nlessons can be learned from the experiences with man-\nganites when dealing with the pnictides. A crucial lesson\nlearned over a decade ago, when dealing with the colos-\nsal magneto resistance (CMR) physics of the mangan-\nites, was the demonstration by Millis, Littlewood, and\nShraiman13that the coupling between fluctuating spins\nand charge carriers can only cause relatively weak trans-\nport anomalies. In the pnictides one finds that the re-\nsistivity drops by a couple of milliohm centimeters, that\nthe Hall mobility increases by 2-3 orders of magnitude,\nand most significantly the Seebeck coefficient drops by\nan order of magnitude from a high temperature limit or-\nder value of 40 µV/K in crossing the transition. It is\nvery questionable if spin-carrier coupling of any kind, be\nit itinerant or strongly coupled, can explain such large2\nchanges in the transport properties.\nA. Role of Electron-Electron Interactions\nComparing the pnictides with the cuprate supercon-\nductors there is now a consensus that in two regards\nthese systems are clearly different: (i) in the pnictide\nsystem no Mott insulator has been identified indicating\nthat they are ’less strongly correlated’ than the cuprates\nin the sense ofthe Hubbard type local interactions; (ii) in\nthe pnictide onehas toaccountforthe presenceofseveral\n3dorbitals playing a role in the low energy physics, con-\ntrasting with the single 3 dx2−y2orbital that is relevant\nin the cuprates.\nAs a consequence, the prevailing viewpoint is to regard\nthe pnictides as LDA metals, where the multi-orbital\nnature of the electronic structure gives rise to a multi-\nsheeted Fermi surface, while the ’correlation effects’ are\njust perturbative corrections, causing moderate mass en-\nhancements and so on.\nAlthough there is evidence that the system eventually\ndiscoversthis ’Fermi-liquid fixed point’ at sufficiently low\ntemperatures, it is hard to see how this can explain the\nproperties of the metallic state at higher temperatures.\nThe data alluded to in the above indicate pronounced\n’bad metal’ behavior, and these bad metal characteris-\ntics do not disappear with doping. In fact, one can ar-\ngue that the term ’bad metal’ actually refers to a state\nof ignorance: it implies that the electron system cannot\npossibly be a simple, coherent Fermi-liquid.\nB. Spin-Charge-Orbital Correlations\nAnother important lesson from the manganites is that\nthe presence of multiple orbitals can mean much more\nthan just the presence of multiple LDA bands at the\nFermi-energy. Manganite metalshave a degree of itin-\neracy in common with the pnictides, but they still ex-\nhibit correlated electron physics tied to orbital degen-\neracy which is far beyond the reach of standard band\nstructure theory.\nTheseminal workbyKugelandKhomskiiin the1970’s\nmade clear that in Mott insulators orbital degrees of\nfreedom turn into dynamical spin like entities that are\ncapable of spin-like ordering phenomena under the con-\ndition that in the local limit one has a Jahn-Teller (or-\nbital) degeneracy14. The resulting orbitaldegrees offree-\ndom can have in dynamical regards a ’life of their own’.\nThis manifestsitself typicallyin transitionscharacterized\nby small changes in the lattice accompanied by drastic\nchanges in the electronic properties.\nIn the manganites there are numerous vivid examples\nof the workings of orbital ordering15,16,17. Under the\nright circumstances one can find a transition from a high\ntemperature cubic phase to a low temperature tetrago-\nnalphaseaccompaniedbyaquite moderatechangeinthelattice, but with a change in the electron system that is\nas drastic as a ’dimensional transmutation’: this system\nchanges from an isotropic 3D metal at high temperature\nto a quasi 2D electron system at low temperatures where\nthe in-plane resistivityis ordersofmagntitude lowerthan\nthec-axis resistivity18,19,20.\nTheexplanationisthatoneisdealinginthecubicman-\nganite with a Mn3+ion with an egJahn-Teller degener-\nacy involving 3 dx2−y2and 3d3z2−1orbitals. In the low\ntemperature ’A-phase’ one finds a ’ferro’ orbital order\nwherecooperativelythe x2-y2orbitalsareoccupied. This\ngreatlyfacilitatesthe hoppinginthe planeswhile forsim-\nple orthogonality reasons coherent transport along the\nc-axis is blocked. Since the d-electrons only contribute\nmodestly to the cohesive energy of the crystal, this large\nscale change in the low energy degrees of freedom of the\nelectronic system reflect only barely in the properties of\nthe lattice. On the other hand, this orbital order is a\nnecessary condition for the spin system to order, and at\na lower temperature one finds a transition to a simple\nstaggered antiferromagnet, in tune with the observation\nthat in the A-phase the effective microscopic electronic\nstructure is quite similar to the ones found in cuprate\nplanes.\nThe ruthenates are another class of materials in which\ntheorbitaldegreesoffreedomplayadecisiverole,in both\nthe metallic and insulating phases. Bilayer Ca 3Ru2O7,\nfor instance, has attracted considerable interest because\nthe observed CMR-effect is possibly driven by orbital\nscatteringprocessesamongthe conduction electrons21,22.\nAnother example is Tl 2Ru2O7, in which below 120 K its\n3Dmetallicstateshowsadramaticdimensionalreduction\nand freezes into a quasi-1D spin system, accompanied by\na fundamental orbital reorganization23,24.\nIt is very remarkable that the groundstate of all\niron pnictides is characterized by a very similar spatial\nanisotropy of the magnetic exchange interactions: along\none direction in the plane the Fe-Fe bonds are strong and\nantiferromagnetic, whereas in the orthogonal direction\nthey are very weak and possibly even ferromagnetic25.\nWith all the others, also this observation is consistent\nwith our hypothesis that the undoped iron pnictides are\ncontrolled by ’spin-charge-orbital’ physics, very similar\nin spirit to the ruthenates and manganites.\nC. Organization of this Paper\nIn Sec. II of this paper we derive the spin-orbital\nHamiltonianstartingwithathree-orbitalHubbardmodel\nfortheironsquarelatticeoftheironpnictides. Thephase\ndiagrams in the classical limit of this Hamiltonian are\ndiscussed in Sec. III. We analyze the various phase tran-\nsitions by also considering the corresponding spin-only\nand orbital-only models. Sec. IV deals with the results\non magnetic excitation spectra, which provide a possi-\nble explanation for the reduction of magnetic moment, a\ncentral puzzle in the iron superconductors. We conclude3\nby commenting briefly on how the itineracy may go hand\nin hand with the orbital ’tweed’ order that we put for-\nward in the present study, and point out that the ’tweed’\norbital ordered state can, in principle, be observed in\nresonant X-ray diffraction experiments.\nII. SPIN-ORBITAL MODEL FOR IRON PLANES\nAs stated above, the superconducting iron-pnictides\nare not strongly coupled doped Mott insulators. Staying\nwithin the realm of Hubbard-model language they are\nlikely to be in the intermediate coupling regime where\nthe Hubbard U’s are of order of the bandwidth. To at\nleastdevelop qualitativeinsightin the underlyingphysics\nit is usually a good idea to approach this regime from\nstrong coupling for the simple reason that more is going\non in strong coupling than in the weak coupling band\nstructure limit. As the experience with for instance the\nmanganites and ruthenates shows, this is even more true\nwhen we are dealing with the physics associated with or-\nbital degeneracy. The orbital ordering phenomena that\nwe have already alluded to, take place in itinerant sys-\ntems but their logicis quite comprehensiblestartingfrom\nthe strongly coupled side.\nThus as a first step we will derive the spin-orbital\nmodel of pnictides starting from a localized electron\nframework. A condition for orbital phenomena to oc-\ncur is then that the crystal fields conspire to stabilize\nan intermediate spin ( S= 1) ionic states. These crystal\nfields come in two natural varieties: one associated with\nthe tetrahedral coordination of Fe by the As atoms, and\na tetragonalfield associatedwith the fact that the overall\ncrystal structure consists of layers. When these crystal\nfields would be both very large the Fe 3 d6ions would\nform a low spin singlet state. This is excluded by the\nobservation of magnetism, and moreover band structure\ncalculations indicate that the crystal fields are relatively\nsmall.\nThe other extreme would be the total domination of\nHund’s rule couplingsandthis would resultin a highspin\nS= 2 state, which appears to be the outcome of spin po-\nlarized LDA and LDA+U calculations26. However, given\nthat for elementary chemistry reasons one expects that\nthe tetrahedral splitting is much larger than the tetrago-\nnal splitting there is the possibility that the Hunds rule\noverwhelms the latter but looses from the former, result-\ning in an ’intermediate’ S= 1 state. Although the issue\nis difficult to decide on microscopic grounds, for orbital\nphysics to be relevant we need an intermediate spin state\nas in the present crystal field scheme this is the only ionic\nd6state that exhibits a Jahn-Teller groundstate degen-\neracy (see Fig. 1).\nIn this situation the starting Hubbard model involves\na non-degenerate |xy/angbracketrightand two doubly-degenerate |xz/angbracketright\nand|yz/angbracketrightorbitals, as will be defined in subsection A. The\ndetails of the derivation of the model are given in sub-\nsection B. The derivation does not assume any specificFIG. 1: (Color online) (a) Fe square lattice (black circles)\nand relative positions of the As ions. The latter are located\nin adjacent layers above (filled red squares) and below (empt y\nred squares) the Fe plaquettes. (b) Schematic illustration of a\nground-state d6configuration of the Fe ions corresponding to\nan intermediate S= 1 spin state. (c) Multiplet structure of\nthed6\nid6\nj⇋d7\nid5\njcharge excitations for localized egelectrons.\nstructure for the hopping parameters and hence, is com-\npletely general. The algebra involved in the derivation\nis tedious but straightforward and a general reader may\nwish to skip subsection B and jump directly to subsec-\ntion C where we discuss the relevant hopping processes\nfor the Fe-As plane. Incorporating these hopping param-\neters leads to the model relevant to the iron plane.\nA. Hubbard model for pnictide planes for the\nintermediate-spin d6state\nTheironionsareina d6configurationwhereweassume\nthe low lying egorbitals to be fully occupied due to a\nlarge crystal-field splitting between the egandt2gstates.\nThetworemainingelectronsoccupythe three t2gorbitals\n|a/angbracketright:=|xz/angbracketright,|b/angbracketright:=|yz/angbracketright, and|c/angbracketright:=|xy/angbracketrightwithxandy\npointing along the bonds of the iron square lattice. Due\nto the Hund’s coupling JHbetween the t2gelectrons,\nsuch a configuration leads to an S= 1 intermediate spin\nstate of the d6Fe ions. Further, we incorporate a small\ntetragonal splitting ∆ between the |xy/angbracketrightstate and the\n|xz/angbracketright,|yz/angbracketrightdoublet (see Fig. 1).\nAssuming the egelectrons to be localized, the phys-\nical situation is very similar to almost cubic vanadates\nlike YVO 3or LaVO 3where the two d-electrons of the\nV3+ions occupy nearly degenerate t2gorbitals. Interest-\ningly, in theses systems orbital ordering in the presence4\nof a small crystal-field splitting ∆ can lead to C-type\nantiferromagnetism27,28,29characterized by an ordering\nwavevector Q= (π,π,0). The effective Hubbard model\nfor thet2gelectrons consists of a kinetic energy part Ht,\na crystal field splitting Hcf, and of the on-site electron-\nelectron interactions Hint,\nH=Ht+Hcf+Hint, (1)\nwith a kinetic energy contribution that is much richer\nthan in the vanadates. For the nearest neighbor bonds\nthe effective hoppings between the Fe t2gorbitals have\ncontributions from both direct d−dandd−p−dpro-\ncesses via As p-orbitals. These As ions are located in\nadjacent layersaboveorbelow the Fe ion plaquettes as il-\nlustrated in Fig. 1a. Because of this particular geometry,\nthe indirect As mediated hoppings should be of similar\nstrength for nearest and next-nearest neighbor Fe ions.\nAt this point, we do not specify the effective hopping ma-\ntrix elements t(i,j)\nα,βbetween orbitals α,β=a,b,calong a\nparticular bond ( i,j) and write the kinetic energy oper-\nator in the most general form,\nHt=−/summationdisplay\n(i,j)/summationdisplay\nαβ,σt(i,j)\nαβ(d†\niασdjβσ+h.c.),(2)\nwhered†\niασ(diασ)creates(annihilates) anelectrononsite\niinorbital αwith spin σ=↑,↓. Thecrystal-fieldsplitting\nbetween the t2gorbitals is simply given by\nHcf=/summationdisplay\niαǫαˆniα, (3)\nwith ˆniα=/summationtext\nσˆniασand ˆniασ=d†\niασdiασ. In our case\nthe electron energies are given by ǫc= 0 for the xyand\nǫa=ǫb= ∆ for the xzandyzorbitals. The electron-\nelectroninteractionsaredescribedbythe on-siteterms,30\nHint=U/summationdisplay\niαˆniα↑ˆniα↓+1\n2/parenleftbigg\nU−5\n2JH/parenrightbiggα/negationslash=β/summationdisplay\niαβˆniαˆniβ\n+JHα/negationslash=β/summationdisplay\niαβd†\niα↑d†\niα↓diβ↓diβ↑−JHα/negationslash=β/summationdisplay\niαβˆSiαˆSiβ,(4)\nwith the Coulomb element Uand a Hund’s exchange el-\nementJH.\nB. Superexchange model\nIn the limit of strong Coulomb repulsion, t≪U,\ncharge fluctuations d6\nid6\nj⇋d7\nid5\njare suppressed and on\neach site the two t2gelectrons have to form a state be-\nlonging to the ground-state manifold of Hint+Hcfin thetwo-electron sector. For sufficiently small crystal-field\nsplitting, ∆2<8J2\nH, these states are given by two S= 1\ntriplets in which on each site either the xzoryzis unoc-\ncupied. This orbital degree of freedom can be viewed as\naT=1\n2pseudospin. From Eqs. (3), (4) we easily obtain\nE0=U−3JH+∆ as the ground-state energy of the t2\n2g\nsector.\nA general spin-orbital superexchange model can be de-\nrived by second order perturbation theory controlled by\nthekinetic energycontribution Ht, wherewehavetocon-\nsider all virtual processes t2\n2gt2\n2g→t1\n2gt3\n2g→t2\n2gt2\n2gacting\non theS= 1,T= 1/2 ground-state manifold. The most\ngeneral superexchange Hamiltonian in the sense of Kugel\nand Khomskii for a given bond ( i,j) takes the form\nH(i,j)\nKK=−/summationdisplay\nτi,τj/summationdisplay\nsi,sjJ(i,j)\nτi,τj,si,sjA(i,j)\nτi,τj(ˆTi,ˆTj)\n×Bsi,sj(ˆSi,ˆSj), (5)\nwhereˆSandˆTdenoteS= 1 spin and T=1\n2pseudospin\noperators. The functional form of Bonly depends on\ntotal spins siandsjon the two sites in the intermedi-\natet1\n2gt3\n2gstates. Whereas the single occupied site has\nnecessarily s= 1/2 the other site can be in a high-spin\n(s= 3/2)or low-spin ( s= 1/2) state. Likewise, the func-\ntionsA(i,j)are determined by the pseudospins τi,τjof\nthe involved intermediate states.\nTo derive the effective spin-orbital superexchange\nmodel we first have to find the multiplet structure of the\nvirtual intermediate t3\n2gconfigurations. It is straightfor-\nward to diagonalize Hcf+Hint(3,4) in the three-particle\nsector. The lowest energy we find for the4A2quar-\ntet ofs= 3/2 high-spin intermediate states |4A2,3\n2,sz/angbracketright,\nwith|sz/angbracketright=|3\n2/angbracketright=d†\na↑d†\nb↑d†\nc↑|0/angbracketright,|1\n2/angbracketright=1√\n3(d†\na↑d†\nb↑d†\nc↓+\nd†\na↑d†\nb↓d†\nc↑+d†\na↓d†\nb↑d†\nc↑)|0/angbracketright,| −1\n2/angbracketright=1√\n3(d†\na↓d†\nb↓d†\nc↑+\nd†\na↓d†\nb↑d†\nc↓+d†\na↑d†\nb↓d†\nc↓)|0/angbracketrightand|−3\n2/angbracketright=d†\na↓d†\nb↓d†\nc↓|0/angbracketright. Their\nenergy is ǫ(4A2) =E(4A2)−2E0=U−3JH, where\nE0=U−3JH+∆ is the groundstate energy in the t2\n2g\nsector. In order for the approach to be valid we have\nto assume that the system has a charge-transfer gap,\nU−3JH>0 and that the hopping matrix elements\nare sufficiently small compared to the charge-transfer\ngap. All the other multiplets consist of intermediate\ns= 1/2 doublets. The2Emultiplet with excitation en-\nergyǫ(2E) =Uconsist of the two spin-1\n2doublets\n|2E,1\n2,σ/angbracketright1=1√\n6(2d†\naσd†\nbσd†\nc,−σ−d†\naσd†\nb,−σd†\nc,σ\n−d†\na,−σd†\nbσd†\ncσ)|0/angbracketright (6)\n|2E,1\n2,σ/angbracketright2=1√\n2(d†\na,−σd†\nbσd†\ncσ−d†\naσd†\nb,−σd†\ncσ)|0/angbracketright.(7)\nFinally, we have multiplets2T(∆)\n1,2T(∆)\n2which consist\nof spin-1\n2doublets and invoke doubly occupied orbitals,\n|2T1/2,1\n2,σ/angbracketright=1√\n2d†\ncσ(d†\na↑d†\na↓∓d†\nb↑d†\nb↓)|0/angbracketright(8)5\nwithexcitationenergies ǫ(2T1) =Uandǫ(2T2) =U+2JH\nand\n|2T∆\n1/2,1\n2,σ/angbracketright1=d†\naσ(/radicalBig\n1−v2\n∓d†\nc↑d†\nc↓∓v∓d†\na↑d†\na↓)|0/angbracketright(9)\n|2T∆\n1/2,1\n2,σ/angbracketright2=d†\nbσ(/radicalBig\n1−v2\n∓d†\nc↑d†\nc↓∓v∓d†\nb↑d†\nb↓)|0/angbracketright(10)\nwithv∓=JH//radicalBig\nJ2\nH+(∆±/radicalbig\n∆2+J2\nH)2and excitation\nenergies ǫ(2T∆\n1/2) =U+JH∓/radicalbig\n∆2+J2\nH.\nThe resulting charge-excitation spectrum is shown\nschematically in Fig. 1c. Although the single occupied\nt1\n2gsite of a virtual t1\n2gt3\n2gintermediate state gives no\ncontribution to the on-site electron-electron interaction\nit can lead to an additional crystal-field energy ∆ if the\nelectron is in the aorborbital.\nLet us first focus on the purely magnetic parts\nBsi,sj(ˆSi,ˆSj) of the superexchange Hamiltonian, which\ncan be determined entirely by group theoretical meth-\nods. To be precise, we consider a two-ion system in the\nstate|SA,MA/angbracketright⊗|SB,MB/angbracketrightwhich can be classified by the\ntotal spin Stand the z-component Mt. Applying a hop-\nping operator of the form Ht=−t/summationtext\nσ(c†\nAσcBσ+ h.c),\nwhich preserves the quantum numbers StandMtbe-\ncause of the spin-rotation invariance we obtain an inter-\nmediate state |sA,mA/angbracketright ⊗ |sB,mB/angbracketrightwithsA=SA±1/2\nandsB=SB±1/2. The effective superexchange involv-\ningintermediatespins sA,sBisgivenbythe secondorder\nprocess\nE(St,sA,sB) =−/summationdisplay\nma,mB|/angbracketleftsAmA,sBmB|Ht|StMt/angbracketright|2\n∆E.\nUsing Clebsch-Gordan coefficients Cj1j2j\nm1m2m=\n/angbracketleftj1j2m1m2|jm/angbracketright, we can express the total spin states as\n|StMt/angbracketright=/summationdisplay\nMA,MBCSASBSt\nmAmBMt|SA,MA/angbracketright⊗|SB,MB/angbracketright.\nSince the operators c†\nσandcσare irreducible tensor oper-\nators of rank 1/2 we can use the Wigner-Eckart theorem\nto obtain\n/angbracketleftsAmA|c†\nAσ|SAMA/angbracketright=/bardblc†\nA/bardblCSA1\n2sA\nMAσmA\n/angbracketleftsBmB|cBσ|SBMB/angbracketright=/bardblcB/bardblCSB1\n2sB\nMB(−σ)mB(−1)1\n2−σ,\nwhere we have used /bardbl · /bardblas a short-hand notation for\nthe reduced matrix elements. Using these expressions\nwe can rewrite the exchange energy as E(St,sA,sB) =\nt2\n∆E(/bardblc†\nA/bardbl·/bardblcB/bardbl)2B(St,sA,sB),where we can express the\nfunction Bin terms of a Wigner 6 j-symbol as\nB(St,sA,sB) =−(2sA+1)(2sB+1)/braceleftbigg\nSAsA1\n2\nsBSBSt/bracerightbigg2\n,\nwhich by using the relation St(St+1) =SA(SA+ 1) +SB(SB+1)+2 ˆSAˆSBcan be simplified further to\nBsA,sB=−2\n(2SA+1)(2SB+1)×/braceleftbigg\n(sA+1\n2)(sB+1\n2)\n−sign[(sA−SA)(sB−SB)]ˆSAˆSB/bracerightbigg\n.\nThis expression we can evaluate for SA=SB=S= 1\nfor thes= 3/2 high- and s= 1/2 low-spin intermediate\nstates to obtain the (normalized) spin-projection opera-\ntors\nB3\n2,1\n2(ˆSi,ˆSj) =−1\n3(ˆSiˆSj+2) (11)\nB1\n2,1\n2(ˆSi,ˆSj) =1\n3(ˆSiˆSj−1) (12)\nin agreement with Refs. 27,28. Hence, the Kugel-\nKomskii superexchange Hamiltonian for a given bond\n(i,j) can be written as,\nH(i,j)\nKK=−1\n3(ˆSiˆSj+2)Q(1)(ˆTi,ˆTj)\n+1\n3(ˆSiˆSj−1)Q(2)(ˆTi,ˆTj),(13)\nwhereQ(n)are functions of orbital pseudospin operators.\nTheir functional form can be obtained by tracking the\norbital occupancies in the initial and final states during\na virtual hopping process. In terms of spinless Fermi\noperators a+\ni,b+\niincreasing the occupancy of the aor\nborbital on site ithe pseudospin-1 /2 operators acting\non the ground-state manifold can be expressed as ˆTz\ni=\n(ˆnia−ˆnib)/2,ˆT+\ni=b+\niai, andˆT−\ni=a+\nibi, where ˆnia=\na+\niaiand ˆnib=b+\nibiwith the constraint ˆ nia+ ˆnib=\n1. Whereas it is straightforward to see that the general\nfunctional form is given by\nQ(n)(ˆTi,ˆTj) =f(n)\nzzˆTz\niˆTz\nj+1\n2f(n)\n+−(ˆT+\niˆT−\nj+ˆT−\niˆT+\nj)\n+1\n2f(n)\n++(ˆT+\niˆT+\nj+ˆT−\niˆT−\nj)+f(n)\nzx(ˆTz\niˆTx\nj+ˆTx\niˆTz\nj)\n+f(n)\nz(ˆTz\ni+ˆTz\nj)+f(n)\nx(ˆTx\ni+ˆTx\nj)+f(n)\n0,(14)\nit is quite tedious to determine the coefficients by acting\nwith the hopping operator Ht(2) on all states in the\nground state sector and calculating the overlap of the\nresulting states projected on the different intermediate\nstates listed above. The resulting explicit expressions\nare given in Appendix A.\nC. Hopping and Resulting Hamiltonian\nIn the previoussection wehavederived the generalKK\nsuperexchange Hamiltonian only assuming the effective\nhopping matrices to be symmetric, tαβ=tβα. In or-\nder to write down the spin-orbital model specific to the6\nFIG. 2: Illustration of the effective hopping parameters tαβ,\n(a) between the dxzanddyzorbitals, and (b) those involving\nthedxyorbitals. Theprojectionsofthe dxzanddyzorbitalson\nthe Fe-plane are depicted in white and light grey, respectiv ely,\nand the dxyorbitals are shown in dark grey\npnictide planes we have to use the corresponding hop-\nping parameters. We use the Slater-Koster integrals31\nalong with the geometry of the Fe-As planes to deter-\nmine all the hopping parameters involving the three t2g\norbitals on the nearest neighbor and next nearest neigh-\nbor Fe sites. This considerably reduces the number of\nindependent hopping parameters that enter the Hamil-\ntonian. The direct d−dhoppings are considered to be\nmuch smaller therefore we use hoppings via the As- por-\nbitals only which are given in Appendix B and depend\non the direction cosines l,m,nof the As-Fe bond, as well\nas on the ratio γ= (pdπ)/(pdσ)32,33,34. These result-\ning effective hopping matrix elements between the t2gFe\norbitals are shown schematically in Fig. 2 and can be\nparametrized by the lattice parameter λ=|n/l|andγ.\nIn Fig. 3 the dependence of the hopping matrix el-\nements on the ratio γ= (pdπ)/(pdσ) is shown for a\nlattice parameter λ= 0.7 which is slightly below the\nvalue resulting from the Fe-Fe spacing and the distance\nof the As ions to the Fe planes. Over a realistic range\n−0.2≤γ≤0.2 we find a very strong dependence of the\nhopping amplitudes on γand therefore expect the sta-\nbility of possible phases to depend crucially on γ. This\nparameter cannot be obtained by geometrical consider-\nations but depends for instance on how strongly the or-\nbitals delocalize.\nHaving specified the effective hopping parameters\nαi:=ti/tbetween the Fe orbitals for nearest and next-\nnearest neighbors (see Fig. 2) which are parametrized\nentirely by the ratio γ= (pdπ)/(pdσ) and the lattice pa-\nrameter λ=|n/l|we can now write down the effective\nKK model for the Fe planes. For convenience, we rewrite\nthe Hamiltonian in the form\nHKK=J/summationdisplay\n(i,j)/parenleftbigg1\n2(ˆSiˆSj+1)ˆΩ(i,j)+ˆΓ(i,j)/parenrightbigg\n(15)\nintroducing an overall energy scale J= 4t2/U. The or-\nbital bond operators are defined as ˆΩ =U\n6t2(Q(2)−Q(1))\nandˆΓ =−U\n12t2(Q(1)+2Q(2)) and depend on the effective-0.2 -0.1 0 0.1 0.2γ-0.2-0.100.10.20.3\nαii=1\ni=2\ni=3i=6\ni=5i=4\ni=7\nFIG. 3: (Color online) Various hopping parameters αi:=\nti/tas illustrated in Fig. 2 as a function of the ratio γ=\n(pdπ)/(pdσ) for the lattice parameter λ= 0.7.\ncouplings αi, the relative strength of the Hunds coupling\nη=JH/Uand the crystal-field splitting δ= ∆/U. For\nthe nearest neighbor bonds along ˆ xand ˆyalong the ˆ x±ˆy\ndiagonals the operators are given in Appendix C.\nIII. CLASSICAL PHASE DIAGRAMS\nIn this section we discuss the phase diagrams of the\nspin-orbital Hamiltonian in the classical limit. We have\nfour parameters that enter the model: λandγdeter-\nmine the relative strength of various hopping parameters\nandηandδenter via the energy denominators. Zero\ntemperature phase transitions are discussed in subsec-\ntion A, subsection B is devoted to the understanding of\nfinite temperature transitions and subsection C analyses\nthe phases in terms of the corresponding spin-only and\norbital-only models.\nThe results that we discuss below demonstrate that\nthe Hamiltonian is highly frustrated in the spin and or-\nbital variables. While the spin frustration is largely due\nto the competing interactions between nearest-and next-\nnearestneighbors,thefrustrationinorbitalsectorismore\nintrinsic and exists within a single bond in the Hamilto-\nnian. The spin ( π,0) state is found to be stable over\na wide range of parameter space due to the strong nnn\nAF coupling. However, depending on the parameters,\nthere are three possible orderings of the orbitals that ac-\ncompany the spin ’stripe’ order. Two out of these three\norbital ordering patterns break the in-plane symmetry of\nthe lattice and hence are likely candidates for explain-\ning the orthorhombic transition observed in the parent\ncompounds.\nA. Zero temperature\nSince the effective KK Hamiltonian derived in section\nIIcontainsalargenumberofcompetingtermsitisalmost7\nFIG. 4: (Color online) η-γphase diagram for λ= 0.7 and\nδ= 0.01.η=JH/U,γ= (pdπ)/(pdσ). The phases are\ndenoted by their ordering wavevectors in the spin and orbita l\nvariables. TzorTxrefers to the component of the orbital\npseudospin that is saturated in the ordered state.\nimpossible to anticipate what kind of spin-orbital order-\nings are realized for different parameter values, in partic-\nular since the signs and relative strengths of the effective\nhoppings αibetween nearest and next-nearest neighbor\nFe orbitalscruciallydepend onthe ratio γ= (pdπ)/(pdσ)\nas pictured in Fig. 3. While the parameters α1,α4and\nα7do not show large relative changes over the range of\nγshown in the figure, there are very clear crossings be-\ntweenα2andα3andα5andα6.\nRecallthat α5andα6arethehoppingsbetweennearest\nand next-nearest neighbors involving orbital |c/angbracketright:=|xy/angbracketright.\nIf we infer the spin order arising purely from the non-\ndegenerate |c/angbracketrightorbital, it suggests that the spin state\nshould be ( π,0)-ordered for α2\n5<2α2\n6and (π,π)-ordered\notherwise. Therefore, this would imply that as γ→ −0.2\nthe magnetic superexchange resulting from the |c/angbracketrightor-\nbitals only favors ( π,π) antiferromagnetism, whereas the\n(π,0) stripe AF becomes favorable for γ→0.2.\nA similar spin-only analysis for the degenerate orbitals\n|a/angbracketright,|b/angbracketrightis not possible and one has to treat the full spin-\norbital Hamiltonian in order to find the groundstates.\nNevertheless, the complicated variations in the hopping\nparametersalreadysuggestthatwecanexpectaveryrich\nand complex phase diagram for the groundstate of the\nspin-orbital Hamiltonian. In particular in the region of\nintermediate γwhere the magnetic superexchange model\nresulting form the |c/angbracketrightorbitals only becomes highly frus-\ntrated we expect the magnetic ordering to depend cru-\ncially on the orbital degrees of freedom.\nWe first look at the classical groundstates of this\nmodel. We make use of classical Monte-Carlo method\nin order to anneal the spin and orbital variables simul-\nFIG. 5: (Color online) Schematic pictures of the three\nground-state orbital ordering patterns that accompany the\nspin-stripe phase. (a) Orbital-ferro, (b) orbital-stripe , and\n(c) orbital-antiferro\ntaneously, starting with a completely random high tem-\nperature configuration. Using this method we identify\nthe various groundstates that exist for a combination\nof model parameters. In order to obtain a groundstate\nphase diagram, we minimize the total energy for a set of\nvariational states which also include all the Monte-Carlo\ngroundstates obtained for different choice of parameters.\nFig. 4 shows the resulting T= 0 phase diagram for\nvaryingη=JH/Uandγ= (pdπ)/(pdσ). The lattice pa-\nrameterλis fixed to 0 .7, which is close to the experimen-\ntal value for the oxypnictides. The crystal-field splitting\nbetween the |c/angbracketrightand the |a/angbracketright,|b/angbracketrightorbitals is considered to\nbe very small, δ= ∆/U= 0.01. As expected, a large\nnumber of phases are present in the phase diagram.\nWith increasing γwe indeed find a transition from a\n(π,π) to a (π,0) antiferromagnet as suggested from the\nanalysis of the frustrated magnetic superexchange model\ninvolvingonlythe |c/angbracketrightorbitals. Thisisnotsurprisingsince\nthe correspondingcouplings α2\n5and/orα2\n6are sufficiently\nstrong and as γ→0.2 the biggest hopping element is\nin fact given by α6between next-nearest neighbor |c/angbracketright\norbitals(seeFig. 3). Whereasthe( π,0) stripe magnetfor\nlargeγis accompanied by an antiferro-orbital ordering\nof theTzcomponents corresponding to a checkerboard\narrangement of the |a/angbracketrightand|b/angbracketrightorbitals (see Fig. 5c) for\nintermediate, small γwe find two ( π,0) magnetic phases\npossessing orbital orderings which are likely to break the\ninplane symmetry of the lattice structure.\nFor small ηwe find a ferro-orbital arrangement of the\nTzcomponents corresponding to the formation of chains\nalong the ferromagnetically coupled spin directions (see\nFig. 5a). The existence of this orbital order crucially de-\npends on the pre-existence of a spin stripe state, which\ngenerates magnetic-field-like terms for the orbital pseu-\ndospins. This will be discussed in detail when we try to8\nFIG. 6: (Color online) η-γphase diagram for λ= 0.8 and\nδ= 0.01. Note that the orbital ordered states that break the\northorhombic symmetry do not exist for this choice of λ.\nunderstand the thermal phase transitions. For larger η\nthe orbital order changes to an orbital-( π,0) ’tweed’ pat-\nternwithacondensationofthe Txcomponents. Thiscor-\nresponds to the formation of orbital zig-zag chains along\nthe antiferromagnetically coupled spin direction as pic-\nturedinFig.5b. Interestingly,thestripesinthemagnetic\nand orbital sectors have the same orientation, contrary\nto the conventional Goodenough-Kanamori rules. How-\never, since we are dealing with a highly frustrated spin-\norbital model involving nearest and next-nearest neigh-\nborbondsthesenaiverulesarenotexpectedtohold. The\n’tweed’ orbital order is expected to lead to a displace-\nment pattern of the As ions, which can in principle be\nobserved in X-ray differaction experiments. The ’tweed’\norbital pattern should show up as a higher order struc-\ntural Bragg peak at ( π,0). The orbital order might also\nbe directly visible resonant X-ray diffraction at the iron\nK-edge, a technique that was pioneered in the mangan-\nites35,36,37, and is nowadays available for all transition\nmetal K-edges, in particular the iron one38. Polariza-\ntion analysis and azimuthal angle dependence can distin-\nguish between charge, spin and orbital contributions to\nthe resonant signal35which gives the possibility in the\niron pnictides to single out the ’tweed’ orbital pattern.\nThe orbital stripe order persists to the regime of larger\nnegative γwherethemagneticorderchangestothe( π,π)\nantiferromagnet. This shows that the orbital ’tweed’\nstate does not have spin-( π,0) order as a pre-requisite\nand therefore this orbital order can, in principle, exist\nat temperatures higher than the spin transition tempera-\ntures. In the regimeoflargeHund’s coupling, η≥0.3the\nsystem becomes ferromagnetic. This tendency is easy to\nunderstand since in the limit η→1/3the charge-transfer\nFIG. 7: (Color online) η-δ/ηphase diagram for λ= 0.7 and\nγ= (pdπ)/(pdσ) =−0.05. This phase diagram illustrates the\npoint that δis not a crucial parameter in the Hamiltonian.\ngap closes and the KK model is dominated by processes\ninvolving the low-lying4A2high-spin multiplet favoring\na ferromagnetic superexchange.\nLet us further explore how the groundstate phase di-\nagram changes as we vary the lattice parameter λand\nthe crystal-field splitting δ. Fig. 6 shows the same phase\ndiagramas in Fig. 4 but for a slightly largerseparationof\ntheAsionstotheFe-planes, λ= 0.8. Thetwointeresting\nphaseswith magnetic( π,0) andorbital-stripeandorbital\nferro orderings do not appear in this phase diagram indi-\ncatingthat the stability ofthese phases cruciallydepends\non the relative strength of nearest and next-nereast hop-\npingswhichcanbe tunedby λ. Presenceofatetracritical\npoint is an interesting feature in this phase diagram.\nFinally, we analyze the dependence on the crystal field\nsplitting δ= ∆/Uwhich so far we assumed to be tiny.\nWe do not find any qualitative change of the groundstate\nphase diagram with increasing δ. In particular, there\nare no new phases that appear and therefore the crystal-\nfield splitting does not seem to be a crucial parameter.\nFor example, the phase diagram in the η-δ/η-plane for\nλ= 0.7 andγ=−0.05 shown in Fig. 7 indicates that\na change in δonly leads to a small shift of the phase\nboundaries.\nB. Finite temperature\nTo obtain the transition temperatures for the various\nphase transitions, we track different order parameters as\na function of temperature during Monte-Carlo annealing\nwhere we measure the temperature in units of the energy\nscaleJ. For example, the spin structure factor is defined9\nas\nS(q) =1\nN2/summationdisplay\ni,j/angbracketleftSi·Sj/angbracketrightaveiq·(ri−rj),(16)\nwhere/angbracketleft.../angbracketrightavdenotes thermal averaging and Nis the to-\ntal number of lattice sites. The orbital structure factor\nO(q) is defined analogously by replacing the spin vari-\nables by the orbital variables in the above expression.\nDepending on the groundstate, different components of\nthese structure factors show a characteristic rise upon\nreducing temperature.\nWe fixδ= 0.01,λ= 0.7 andγ=−0.05 and track the\ntemperature dependence of the system for varying η. For\nT= 0 this choice of parameters corresponds to a cut of\nthe phase diagram shown in Fig. 4 through four different\nphases including the two ( π,0) stripe AFs with orbital\norderings breaking the in-plane lattice symmetry.\nIn Fig. 8 the temperature dependence of the corre-\nsponding structure factors is shown for representative\nvalues of the Hund’s rule coupling η. For small values\nofηthe groundstate corresponds to the orbital-ferro and\nspin-stripe state as shown in the phase diagram in Fig. 4.\nFig. 8a shows the temperature dependence of S(π,0) and\nO(0,0) which arethe orderparametersforthe spin-stripe\nand orbital-ferro state, respectively. While the S(π,0)\nleads to a characteristic curve with the steepest rise at\nT∼0.5, the rise in O(0,0) is qualitatively different. In\nfact there is no transition at any finite Tin the orbital\nsector. We can still mark a temperature below which a\nsignificant orbital-ferro ordering is present. The origin of\nthis behavior lies in the presence of a Zeeman-like term\nfor the orbital pseudospin.\nForη= 0.15, the phase diagram of Fig. 4 suggests a\nstate with stripe ordering in both spin and orbital vari-\nables. We show the temperature dependence of S(π,0)\nandO(π,0) in Fig. 8b. In this case both the spin and\norbital variables show a spontaneous ordering, with the\nspins ordering at a much higher temperature. An inter-\nesting sequence of transitions is observed upon reducing\ntemperature for η= 0.18 (see Fig. 8c). This point lies\nclosetothephaseboundarybetweenspin-stripeandspin-\nferro state with the orbital-stripe ordering. The spin-\nstripe order parameter S(π,0) shows a strong rise near\nT= 0.4. The orbital stripe order sets in at T∼0.15.\nThe onset of this orbital order kills the spin-stripe or-\nder. Instead, we find that the ( π,π) components of the\nspin structure factor shows a strong rise along with the\n(π,0) component of the orbital structure factor. Finally\nforη= 0.24 the orbital stripe ordering is accompanied\nby the spin antiferro ordering, with the orbital ordering\nsetting in at slightly higher temperatures (see Fig. 8d).\nThe results shown in Fig. 8 are summarized in the\nT−ηphase diagram shown in Fig. 9. For small η,\nthe groundstate is spin-stripe and orbital-ferro ordered.\nWhilethespinorderoccursathighertemperatures, there\nis no genuine transition to the orbital-ferro state. The00.51\n00.51\n0 0.2 0.4 0.6\nT00.51\n0 0.2 0.4 0.6\nT00.51S(π,0)O(0,0)(a) (b)\n(c) (d)S(π,0)O(π,0)η=0.01 η=0.14\nη=0.18 η=0.24\nO(π,0)O(π,0) S(π,0)\nS(π,π)S(π,π)\nFIG. 8: (Color online)Relevantstructurefactors as afunct ion\nof temperature for different values of η. The lattice parameter\nand the relative strength of σandπhopping are fixed as\nλ= 0.7 andγ=−0.05, respectively.\norbital-ferrostateisdrivenbythepresenceofamagnetic-\nfield-like term for the orbital pseudospin in the Kugel-\nKhomskii Hamiltonian. The stability of the orbital-ferro\nstate crucially depends on the presence of the spin-stripe\norder. The dotted line joining the black circles in the\nsmall-ηrange is only to indicate the temperature below\nwhich the orbital-ferro order is significant. This typical\ntemperature scale reduces with increasing η, until the\nsystem finds a different groundstate for the orbital vari-\nables. Note that the temperature scales involvedarevery\nsmall owing to the highly frustrated nature of the orbital\nmodel, nevertheless there is no zero-temperature transi-\ntion in this purely classical limit.\nThespin-stripestateremainsstablewiththetransition\ntemperature reducing slightly. The transition tempera-\nture for the orbital stripe state increases upon further\nincreasing η. For 0.15< η <0.2, multiple thermal tran-\nsitions are found for the magnetic state. The spin-stripe\norder which sets in nicely at T∼0.35 is spoiled by the\nonset of orbital-stripe state, which instead stabilizes the\nspin (π,π) state. Beyond η= 0.2, The orbital stripe\nstate occurs together with the spin antiferro state, with\nthe spin ordering temperatures slightly lower than those\nfor the orbital ordering. For η >0.3, the spin state be-\ncomes ferromagnetic.10\n0 0.1 0.2η00.20.40.6\nTS(π,0)\nS(π,π)\nO(π,0) S(π,0)\nO(0,0)S(π,0)\nO(π,0)O(π,0)\nFIG. 9: (Color online) T-ηphase diagram for δ= 0.01,λ=\n0.7, andγ=−0.05\nC. Corresponding Orbital-only and Spin-only\nmodels\nIn an attempt to provide a clear understanding of the\nspin and orbital ordered phases, we derive the orbital\n(spin) model that emerges by freezing the spin (orbital)\nstates. For fixed spin correlations, the orbital model can\nbe written as\nHO=/summationdisplay\nµKµµ\nx/summationdisplay\n/angbracketlefti,j/angbracketright/bardblxTµ\niTµ\nj+/summationdisplay\nµKµµ\ny/summationdisplay\n/angbracketlefti,j/angbracketright/bardblyTµ\niTµ\nj\n+/summationdisplay\nµKµµ\nd/summationdisplay\n/angbracketleft /angbracketlefti,j/angbracketright /angbracketrightTµ\niTµ\nj+Kz/summationdisplay\niTz\ni. (17)\nHere and below /angbracketleft·,·/angbracketrightand/angbracketleft /angbracketleft·,·/angbracketright /angbracketrightdenote bonds between\nnearest and next-nearest neighbor pseudospins on the\nsquare lattice, respectively. µdenotes the component\nof the orbital pseudospin. The effective exchange cou-\nplings for this orbital-only model are shown in Fig. 10\nas a function of the Hund’s coupling η=JH/Uwith\nthe other parameters fixed as δ= 0.01,γ=−0.05, and\nλ= 0.7, as before. The solid lines are obtained by fixing\nthe spin degrees of freedom by the classical ground-state\nconfigurationsof the corresponding phases. For compari-\nson, the effectivecouplingsfordisorderedspinsareshown\nby dashed lines.\nSimilarly, we can freeze the orbital degrees of freedom\nto obtain an effective Heisenberg model for spins,\nHS=Jx/summationdisplay\n/angbracketlefti,j/angbracketright/bardblxSiSj+Jy/summationdisplay\n/angbracketlefti,j/angbracketright/bardblySiSj+Jd/summationdisplay\n/angbracketleft /angbracketlefti,j/angbracketright /angbracketrightSiSj.(18)\nThe coupling constants Jx,Jy, andJdfor spins are plot-\nted in Fig. 11.-0.4-0.200.20.4\nKxµµ\n-0.4-0.200.20.4\nKyµµ\n0 0.1 0.2 0.3η-0.500.511.5\nKdµµzz\nxx\nxx\nzz xxzz(a)\nyy\nyyKz(b)\n(c)\nyyxx,yy\nS(π,0) S(π,π)\nFIG. 10: (Color online) The coupling constants as a function\nofη=JH/Ufor the orbital-only model with frozen spin cor-\nrelations for δ= 0.01,γ=−0.05, andλ= 0.7. The couplings\nalong x, y and diagonal directions are plotted in panels (a),\n(b) and (c) respectively. The single site term is plotted in ( b)\nto indicate that this term arises due to a ferromagnetic bond\nalong y-direction. The solid lines correspond to the ground\nstate spin order and the dashed lines are for a paramagnetic\nspin state. The vertical dashed line indicates the location inη\nof the phase transition from spin-stripe to spin-antiferro state\nas seen in Fig. 9\nLet us try to understand the phase diagram of Fig. 9\nin terms of these coupling constants. We begin with\nthe small- ηregime where the groundstate is spin-stripe\nand orbital-ferro. Approaching from the high tempera-\nture limit, we should look at the spin (orbital) couplings\nwith disordered orbitals (spins). The strongest constants\nturn out to be Jd, which is slightly larger than Jxand\nJy, all three being antiferromagnetic. This suggests that\nthe system should undergo a transition to a spin-stripe\nstate consistent with the phase diagram. The coupling\nconstants of the orbital model are much weaker in the\nsmall-ηregime. The largest constant is Kxx\ndsuggesting\nan orbital-stripe order. However, since the spin- stripe\nstate sets in at higher temperatures, in order to deter-\nmine the orbital order one should look at the coupling\nconstants corresponding to the spin-stripe state. There11\n0 0.1 0.2 0.3η-0.500.51coupling constantsorbital disordered\nT=0 orbital order\nJx Jy\nJd\nS(π,0)\nO(0,0)S(π,0)\nO(π,0)JyJx\nJd\nS(0,0)\nS(π,π)\nO(π,0)\nFIG. 11: (Color online) Effective exchange couplings Jx,Jy\nfor nearest-neighbor and Jdfor next-nearest-neighbor spins as\na function of η=JH/Uforδ= 0.01,γ=−0.05, andλ= 0.7.\nThe solid lines correspond to the couplings resulting for th e\ncorresponding orbital ground states whereas the dashed lin es\ncorrespond to the orbitally disordered case. Note that for t he\norbitally disordered case Jx=Jyfor all values of η. The\nvertical dashed line indicates the location in ηof the various\nphase transition as seen in Fig. 9.\narethreemaineffects(comparethesolidanddashedlines\nin the low ηregime in Fig. 10), (i) x- and y-directions be-\ncome inequivalent in the sense that the couplings along\nx are suppressed while those along y are enhanced, (ii)\nthe diagonal couplings are reduced strongly, and (iii) a\nsingle-site term is generated which acts as magnetic field\nfor the orbital pseudospins. It is in fact this single site\ntermthat controlsthe orderingofthe orbitalsatlowtem-\nperatures. This also explains the qualitatively different\nbehavior of the orbital-ferro order parameter observed\nin Fig. 8a. Within the spin-stripe order, the single-site\nterm becomes weaker with increasing ηwhereas the di-\nagonal term increases. This leads to a transition in the\norbital sector from an orbital-ferro to an orbital-stripe\nphase near η= 0.11. The region between 0 .14 and 0 .2\nin eta is very interesting. Approaching from the high\ntemperature the spins order into the ’stripe’ state but\nas soon as the orbitals order into stripe state at lower\ntemperature the diagonal couplings Jdare strongly re-\nduced and become smaller than Jy/2. This destabilizes\nthe spin-stripe state and leads to an spin antiferro order-\ning. For larger ηthe orbital ordering occurs at higher\ntemperature. There is another transition slightly below\nη= 0.3 where spins order into a ferro state. This is sim-\nply understood as Jx=−Jyfrom the coupling constants\nof the Heisenberg model.IV. MAGNETIC EXCITATION SPECTRA\nWe now set out to compute the magnetic excitation\nspectra, treating the orbital pseudospins as classical and\nstatic variables. Fixing the orbital degrees of freedom for\na given set of parameters by the corresponding ground-\nstate configuration we are left with an S= 1 Heisen-\nberg model written in (18). The exchange couplings are\nplotted in Fig. 11. Assuming the presence of local mo-\nments, such J1-J2models with a sufficiently large next-\nnearest neighborexchangehave been motivated and used\nto rationalizethe ( π,0) magnetism in the ironpnictides39\nand been used subsequently to calculate the magnetic\nexcitation spectra40,41, where the incorporation of a rel-\natively strong anisotropy between the nearest-neighbor\ncouplings turned out to be necessary to understand the\nlow energy spin-wave excitations41.\nIn the presence of orbital ordering such an anisotropy\nofthe effective magnetic exchangecouplings appearsnat-\nurally. Both, the orbitalferroand the orbitalstripe order\nlead to a seizable anisotropy in the nearest-neighbor cou-\nplingsJx,Jy, where the anisotropy is much stronger for\ntheorbitalferroorder(seeFig.11). Anevenmoredrastic\neffect is the huge suppression of Jdin the orbital-stripe\nregime.\nOn a classical level, the magnetic transitions are easily\nunderstood in the spin-only model (18) as discussed be-\nfore. The transition from the stripe-AF to the ( π,π) AF\natη≈0.14 occurs exactly at the point where Jy= 2Jd\nwhereas the transition from ( π,π) to ferromagnetic order\natη≈0.3 corresponds to the point Jx=−Jy.\nWe proceed to calculate the magnetic excitation spec-\ntra in the Q= (π,0) and (π,π) phases within a linear\nspin-waveapproximation. Theclassicalgroundstatesare\ngiven given by Sr=S(0,0,σr) withσr= exp(iQr) =\n±1. After performing a simple spin rotation, Sx=\n˜Sx\nr,Sy\nr=σr˜Sy\nr, andSz\nr=σr˜Sz\nr, we express the ro-\ntated spin operatorsby Holstein-Primakoffbosons, ˜S+=√\n2S−ˆnb,˜S−=b†√\n2S−ˆn,and˜Sz=S−ˆnwith ˆn=b†b\nto obtain the spin-wave Hamiltonian\nH=S/integraldisplay\nq/braceleftBig\nAq(b†\nqbq+b−qb†\n−q)+Bq(b†\nqb†\n−q+b−qbq)/bracerightBig\n,\nwith\nAq=/bracketleftbigg\n−JxcosQx+Jx1+cosQx\n2cosqx\n−JdcosQxcosQy\n+Jd\n2(1+cosQxcosQy)cosqxcosqy/bracketrightbigg\n+x↔y\nBq=Jx1−cosQx\n2cosqx+Jy1−cosQy\n2cosqy\n+Jd(1−cosQxcosQy)cosqxcosqy,\nyielding the spin-wave dispersion ωq=S/radicalBig\nA2q−B2qand12\nFIG. 12: (Color online) Spin-wave excitation spectra for di f-\nferent values of η. Top: (π,0) magnet for disordered orbitals.\nThe spectral weights are coded by line thickness and color,\nhigh intensity corresponds to red, low intensity to blue. Mi d-\ndle: (π,0) magnet for orbital-ferro ( η= 0.03,0.07,0.11) and\norbital-stripe order ( η= 0.12,0.13,0.14). Bottom: ( π,π)\nmagnet with orbital-stripe order ( η= 0.15,0.17,...).\nthe inelastic structure factor at zero temperature42\nSinel(q,ω) =/radicalBigg\n1−γq\n1+γqδ(ω−ωq) (19)\nwithγq=Bq/Aq. The resulting excitation spectra\nare shown in Fig. 12 for different values of η. In the\ncase of disordered orbitals, the ( π,0) antiferromagnet\norder is stable up to η≈0.25. Since Jx=Jythe\nspectrum is gapless not only at the ordering wave vec-\ntor (π,0) but also at the antiferromagnetic wave vector\n(π,π). However, the spectral weight is centered close\nto the ordering wave vector and goes strictly to zero\nat the antiferromagnetic wave vector. In the presence\nof orbital ordering the next-nearest neighbor couplings\nare anisotropic Jx> Jywhich in the case of the ( π,0)-\nAF leads to a gap at the antiferromagnetic wave vector,\n∆(π,π)= 2/radicalbig\n(2Jd−Jy)(Jx−Jy). Since the anisotropyand the diagonal exchange are large in the orbital ferro\nstate we find a very big gap at ( π,π). This gap reduces\ndrastically for bigger ηwhere the orbital stripe state be-\ncomes favorable. Due to the large reduction of Jdand\nalso of the anisotropy, the gap is considerably smaller\nand continuously goes to zero as we approach the tran-\nsition to the ( π,π)-AF atη≈0.14 where 2 Jd−Jy= 0.\nThis of course also leads to a strong anisotropy of the\nspin-wave velocities, vy/vx=/radicalbig\n(2Jd−Jy)/(2Jd+Jx).\nOn approaching the magnetic transition we find a signif-\nicantsoftening ofmodesalongthe ( π,0)−(π,π) direction\nwhich leads to a considerable reduction of magnetic mo-\nments close to the transition.\nV. DISCUSSION AND CONCLUSIONS\nIn the preceding we have derived and studied a spin-\norbital Kugel-Khomskii Hamiltonian relevant to the Fe-\nAs planes of the parent compound of the iron supercon-\nductors. A variety of interesting spin and orbital or-\ndered phases exist over a physical regime in parameter\nspace. Due to the peculiarities of the pnictide lattice\nand this particular crystal field state we show that the\nrelevant Kugel-Khomskii model is of a particularly inter-\nesting kind.\nThe essence of the ’spin-charge-orbital’ physics is dy-\nnamical frustration . With so many ’wheels in the equa-\ntion’ it tends to be difficult to find solutions that satisfy\nsimultaneously the desires of the various types of degrees\nof freedom in the problem. This principle underlies the\nquite complex phase diagrams of for instance mangan-\nites. But this dynamical frustration is also a generic\nproperty of the spin-orbital models describing the Jahn-\nTeller degenerate Mott-insulators. In the ’classic’ Kugel-\nKhomskii model43describing egdegenerate S= 1/2 3d9\nsystems of cubic 3d systems, Feiner et al.44,45discovered\na point in parameter space where on the classical level\nthis frustration becomes perfect. In the present context\nof pnictides this appears as particularly relevant since\nthis opens up the possibility that quantum fluctuations\ncan become quite important.\nWe proposetwospecificorbitalorderedphasesthat ex-\nplain the orthorhombic transition observed in the exper-\niments. These are orbital-ferro and orbital-stripe states.\nThe orbital-stripeorderis particularlyinterestingsince it\nleads to a spin model that provides possible explanation\nfor the reduction of magnetic moment. It is our main\nfinding that in the idealized pnictide spin-orbital model\nthe conditions appear optimal for the frustration physics\nto take over. We find large areas in parameter space\nwhere frustration is near perfect. The cause turns out\nto be a mix of intrinsic frustration associated with hav-\ningt2gtype orbital degeneracy, and the frustration of a\ngeometrical origin coming from the pnictide lattice with\nits competing ” J1−J2” superexchange pathways. The\nsignificance of this finding is that this generic frustration\nwill render the spin-orbital degrees of freedom to be ex-13\ntremely soft, opening up the possibility that they turn\ninto strongly fluctuating degrees of freedom – a desired\nproperty when one considers pnictide physics.\nWe argued that the orthorhombic transition in half\nfilledpnicitidesandtheassociatedanomaliesintransport\nproperties can be related to orbital order. When the pa-\nrameters are tuned away from the frustration regime the\nmaintendency ofthe systemistoanti-ferroorbitalorder-\ning, whichisthe usualsituationforantiferromagnets. An\nimportant result is that in the regime of relevance to the\npnictides where the frustrations dominate we find phases\nthat are at the same time ( π,0) magnets and forms of or-\nbital orderthat are compatible with orthorhombiclattice\ndistortions (Fig.’s 4,5). Besides the literal ferro-orbital\nordered state (Fig. 5a), we find also a ( π,0) or ’tweed’\norbital order (Fig. 5b). This appears to be the more\nnatural possibility in the insulating limit and if the weak\nsuperlattice reflections associated with this state would\nbe observed this could be considered as a strong support\nforthe literalnessofthe strongcouplinglimit. Surely, the\neffects of itinerancy are expected to modify the picture\nsubstantially. Propagating fermions are expected to sta-\nbilize ferro-orbitalorders46,47, whichenhances the spatial\nanisotropy of the spin-spin interactions further25.\nAmong the observable consequences of this orbital\nphysics is its impact of the spin fluctuations. We con-\nclude the paper with an analysis of the spin waves in\nthe orbital ordered phases, coming to the conclusion that\nalsothespinsectorisquitefrustrated, indicatingthatthe\nquantum spin fluctuations shouldbe quite strongoffering\na rational for a strong reduction of the order parameter.\nThus we have forwarded the hypothesis that the un-\ndoped iron pnictides are controlled by a very similar\n’spin-charge-orbital’ physics as found in ruthenates and\nmanganites. To develop a more quantitative theoretical\nexpectation is less straightforward and as it is certainly\nbeyond standard LDA and LDA+U approaches will re-\nquire investigations of correlated electron models such as\nwe have derived here48,49, taking note of the fact that\nthe pnictides most likely belong to the border line cases\nwheretheHubbard Uisneither smallnorlargecompared\nto the bandwidth50.\nAcknowledgments\nThe authors would like to acknowledge useful discus-\nsions with G. Giovannetti, J. Moore and G.A. Sawatzky.\nThis work is financially supported by Nanoned , a nan-\notechnology programme of the Dutch Ministry of Eco-\nnomic Affairs and by the Nederlandse Organisatie voor\nWetenschappelijk Onderzoek (NWO), and the Stichting\nvoor Fundamenteel Onderzoek der Materie (FOM) .APPENDIX A: EFFECTIVE INTERACTION\nAMPLITUDES\nBy acting with the hopping operator Ht(2) on all\nstatesin the groundstate sectorandcalculatingthe over-\nlap of the resulting states projected on the different in-\ntermediate states we find the effective interaction ampli-\ntudes. For the high-spin intermediate state ( n= 1) we\nfind by projecting on the intermediate4A2multiplet,\nf(1)\nzz=4t2\nab−2(t2\naa+t2\nbb)\nǫ(4A2)\nf(1)\n+−=−4taatbb\nǫ(4A2)\nf(1)\n++=−4t2\nab\nǫ(4A2)\nf(1)\nzx=4tab(tbb−taa)\nǫ(4A2)\nf(1)\nz=t2\nbc−t2\nac\nǫ(4A2)+∆\nf(1)\nx=−2tactbc\nǫ(4A2)+∆\nf(1)\n0=1\n22t2\nab+t2\naa+t2\nbb\nǫ(4A2)+t2\nac+t2\nbc\nǫ(4A2)+∆,(A1)\nwhere the hopping matrix elements have to be specified\nfor a particular bond. Likewise, we find by projections\non the intermediate low-spin states ( n= 2)14\nf(2)\nzz=1\n2(2t2\nab−(t2\naa+t2\nbb))\n×/parenleftbigg4\nǫ(2E)−3\nǫ(2T1)−3\nǫ(2T2)/parenrightbigg\nf(2)\n+−=2taatbb\nǫ(2E)+3t2\nab/parenleftbigg1\nǫ(2T1)−1\nǫ(2T2)/parenrightbigg\nf(2)\n++=2t2\nab\nǫ(2E)+3taatbb/parenleftbigg1\nǫ(2T1)−1\nǫ(2T2)/parenrightbigg\nf(2)\nzx=tab(tbb−taa)/parenleftbigg1\nǫ(2E)+3\nǫ(2T2)/parenrightbigg\nf(2)\nz=1\n2(t2\nbc−t2\nac)/parenleftbigg4\nǫ(2E)+∆+3\nǫ(2T1)+∆\n+3\nǫ(2T2)+∆−3(1−v2\n−)\nǫ(2T∆\n1)−3(1−v2\n+)\nǫ(2T∆\n2)/parenrightbigg\nf(2)\nx=3\n2tab(taa+tbb)/parenleftbigg1\nǫ(2E)+1\nǫ(2T1)/parenrightbigg\n+tactbc/parenleftbigg2\nǫ(2E)+∆+3\nǫ(2T1)+∆\n−3\nǫ(2T2)+∆+3(1−v2\n−)\nǫ(2T∆\n1)+3(1−v2\n+)\nǫ(2T∆\n2)/parenrightbigg\nf(2)\n0=1\n8(2t2\nab+t2\naa+t2\nbb)\n×/parenleftbigg4\nǫ(2E)+3\nǫ(2T1)+3\nǫ(2T2)/parenrightbigg\n+1\n2(t2\nac+t2\nbc)/parenleftbigg4\nǫ(2E)+∆+3\nǫ(2T1)+∆\n+3\nǫ(2T2)+∆+3(1−v2\n−)\nǫ(2T∆\n1)+3(1−v2\n+)\nǫ(2T∆\n2)/parenrightbigg\n+3t2\ncc/parenleftbigg1−v2\n−\nǫ(2T∆\n1)+∆+1−v2\n+\nǫ(2T∆\n2)+∆/parenrightbigg\n.(A2)\nThe terms bilinear in the pseudospin operators result\nsolely from hopping processes involving the |a/angbracketrightand|b/angbracketright\norbitals only. The hoppings between the |c/angbracketright:=|xy/angbracketrightor-\nbitals enter only as a positive constant in Q(2)leading\nto a conventional antiferromagnetic superexchange con-\ntribution. Interestingly, the coupling between the |c/angbracketrightand\n|a/angbracketright,|b/angbracketrightorbitals results in magnetic field terms for the or-\nbital pseudospins.\nAPPENDIX B: HOPPING MATRIX ELEMENTS\nFor a given As-Fe bond with direction cosines l,m,n,\ntheptot2ghoppings are given by31tx,zx=n[√\n3l2(pdσ)+(1−2l2)(pdπ)]\ntx,yz=lmn[√\n3(pdσ)−2(pdπ)]\ntx,xy=m[√\n3l2(pdσ)+(1−2l2)(pdπ)]\nty,zx=tx,yz=tz,xy\nty,yz=n[√\n3m2(pdσ)+(1−2m2)(pdπ)]\nty,xy=l[√\n3m2(pdσ)+(1−2m2)(pdπ)]\ntz,zx=l[√\n3n2(pdσ)+(1−2n2)(pdπ)]\ntz,yz=m[√\n3n2(pdσ)+(1−2n2)(pdπ)].(B1)\nUsing direction cosines( l,m,n) (l2+m2+n2= 1) with\n|l|=|m|resulting from the orthorhombic symmetry, we\nfind that only the following hopping-matrix elements are\nnon-zero,\ntx\naa=ty\nbb=:t1,\ntx\nbb=ty\naa=:t2\ntd\naa=td\nbb=:t3\ntd−\nab=−td+\nab=:t4\ntx\ncc=ty\ncc=:t5\ntd\ncc=:t6\ntx\nac=ty\nbc=:t7. (B2)\nThese hopping matrix elements which are shown\nschematicallyin Fig. 2 can be parametrizedby the lattice\nparameter λ=|n/l|and the ratio γ= (pdπ)/(pdσ) as\nt1/t=−2(B2−A2−C2)\nt2/t=−2(B2−A2+C2)\nt3/t=−(B2+A2−C2)\nt4/t= 2AB−C2\nt5/t= 2A2\nt6/t= 2(B/λ)2−A2\nt7/t= 2(AC+AB/λ−B2/λ),(B3)\nwhere we have introduced the overall energy scale t=\n(pdσ)2/∆pdand defined for abbreviation\nA=λ(√\n3−2γ)\n√\n2+λ23\nB=λ(√\n3+λ2γ)\n√\n2+λ23\nC=√\n3λ2+(2−λ2)γ\n√\n2+λ23. (B4)15\nAPPENDIX C: ORBITAL PART OF THE\nHAMILTONIAN\nFor the nearest neighbor bonds along ˆ xand ˆythe or-\nbital operators in the spin-orbital Hamiltonian are given\nby\nˆΩx,y=1\n2(α2\n1+α2\n2)(1+2ηr1−ηr3)ˆTz\niˆTz\nj\n+α1α2(1+2ηr1+ηr3)ˆTx\niˆTx\nj\n+α1α2(1+2ηr1−ηr3)ˆTy\niˆTy\nj\n∓1\n12α2\n7(7˜r2+3˜r3−2˜r1−3g1)(ˆTz\ni+ˆTz\nj)\n+1\n8(α2\n1+α2\n2)(1−2ηr1−ηr3)\n+1\n12α2\n7(7˜r2+3˜r3−2˜r1+3g1)+1\n2α2\n5g2(C1)\nˆΓx,y=1\n2(α2\n1+α2\n2)η(r1+r3)ˆTz\niˆTz\nj\n+α1α2η(r1−r3)ˆTx\niˆTx\nj+α1α2η(r1+r3)ˆTy\niˆTy\nj\n−1\n8(α2\n1+α2\n2)(2+ηr1−ηr3)\n−1\n12α2\n7(˜r1+7˜r2+3˜r3+3g1)−1\n2α2\n5g2,(C2)\nwhere we have defined ˜ r1= 1/(1−3η+δ), ˜r2= ˜r1|η=0,\n˜r3= 1/(1+2η+δ), andri= ˜ri|δ=0and introduced the\nfunctionsg1=1+η+δ\n1+2η−δ2(C3)\ng2=1+η+2δ\n1+2η(1+δ)+2δ. (C4)\nLikewise, for the bonds along the ˆ x±ˆydiagonals we\nobtain\nˆΩd±= (α2\n3−α2\n4)(1+2ηr1−ηr3)ˆTz\niˆTz\nj\n+(α2\n3+α2\n4)(1+2ηr1+ηr3)ˆTx\niˆTx\nj\n+(α2\n3−α2\n4)(1+2ηr1−ηr3)ˆTy\niˆTy\nj\n∓α3α4(ˆTx\ni+ˆTx\nj)\n+1\n4(α2\n3+α2\n4)(1−2ηr1−ηr3)\n+1\n2α2\n6g2 (C5)\nˆΓd±= (α2\n3−α2\n4)η(r1+r3)ˆTz\niˆTz\nj\n+(α2\n3+α2\n4)η(r1−r3)ˆTx\niˆTx\nj\n+(α2\n3−α2\n4)η(r1+r3)ˆTy\niˆTy\nj\n−1\n4(α2\n3+α2\n4)(2+ηr1−ηr3)\n−1\n2α2\n6g2. (C6)\n1Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J.\nAm. Chem. Soc. 130, 3296 (2008).\n2H. Takahashi, K. Igawa, A. Kazunobu, Y. Kamihara,\nM. Hirano, and H. Hosono, Nature (London) 453, 376\n(2008).\n3Z.-A. Ren, J. Yang, W. Lu, W. Yi, G.-C. Che, X.-L. Dong,\nL.-L. Sun, and Z.-X. Zhao, arXiv: p. 08034283 (2008).\n4X. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and D. F.\nFang, Nature (London) 453, 761 (2008).\n5G. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong,\nP. Zheng, J. L. Luo, and N. L. Wang, Phys. Rev. Lett.\n100, 247002 (2008).\n6M. R. Norman, Physics 1, 21 (2008).\n7C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. Ratcliff,\nJ. L. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. L.\nWang, et al., Nature (London) 453, 899 (2008).\n8M. A. McGuire, A. D. Christianson, A. S. Sefat, B. C.\nSales, M. D. Lumsden, R. Jin, E. A. Payzant, D. Mandrus,\nY. Luan, V. Keppens, et al., Phys. Rev. B 78, 094517\n(2008).\n9A. I. Goldman, D. N. Argyriou, B. Ouladdiaf, T. Chatterji,\nA. Kreyssig, S. Nandi, N. Ni, S. L. Bud ´ko, P. C. Canfield,\nand R. J. McQueeney, Phys. Rev. B 78, 100506 (2008).\n10C. Fang, H. Yao, W-FTsai, J. Hu, and S. A. Kivelson,\nPhys. Rev. B 77, 224509 (2008).\n11C. Xu, M. Mueller, and S. Sachdev, Phys. Rev. B 78,020501 (2008).\n12I. I.Mazin andM. D. Johannes, arXiv: p. 08073737 (2008).\n13A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys.\nRev. Lett. 74, 5144 (1995).\n14K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 136, 621\n(1982).\n15J. van den Brink and D. Khomskii, Phys. Rev. B 63,\n140416 (2001).\n16J. van den Brink and D. Khomskii, Phys. Rev. Lett. 82,\n1016 (1999).\n17J. van den Brink, G. Khaliullin, and D. Khomskii, Phys.\nRev. Lett. 83, 5118 (1999).\n18J. Dho, E. O. Chi, W. S. Kim, N. H. Hur, and Y. N. Choi,\nPhysical Review B 65, 132414 (2002).\n19T. Akimoto, Y. Maruyama, Y. Moritomo, A. Nakamura,\nK. Hirota, K. Ohoyama, and M. Ohashi, Physical Review\nB57, 5594 (1998).\n20H. Kawano, R. Kajimoto, H. Yoshizawa, Y. Tomioka,\nH. Kuwahara, and Y. Tokura, Phys. Rev. Lett. 78, 4253\n(1997).\n21X. N. Lin, Z. X. Zhou, V. Durairaj, P. Schlottmann, and\nG. Cao, Physical Review Letters 95, 017203 (2005).\n22M. Cuoco, F. Forte, and C. Noce, Physical Review B 73,\n094428 (2006).\n23S.Lee, J.-G. Park, D.T.Adroja, D.Khomskii, S.Streltsov,\nK. A. McEwen, H. Sakai, K. Yoshimura, V. I. Anisimov,16\nD. Mori, et al., Nature Materials 5, 471 (2005).\n24J. van den Brink, Nature Materials 5, 427 (2005).\n25M. J. Han, Q. Yin, W. E. Pickett, and S. Y. Savrasov,\narXiv: p. 08110034 (2008).\n26G. Giovannetti, S. Kumar, and J. van den Brink, Physica\nB403, 3653 (2008).\n27A. M. Ole´ s, G. Khaliullin, P. Horsch, and L. F. Feiner,\nPhys. Rev. B 72, 214431 (2005).\n28A. M. Ole´ s, P. Horsch, and G. Khaliullin, Phys. Rev. B\n75, 184434 (2007).\n29P. Horsch, A. M. Ole´ s, L. F. Feiner, and G. Khaliullin,\nPhys. Rev. Lett. 100, 167205 (2008).\n30A. M. Ole´ s, Phys. Rev. B 28, 327 (1983).\n31J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).\n32S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C.\nZhang, Phys. Rev. B 77, 220503 (2008).\n33C. Cao, P. J. Hirschfeld, and H.-P. Cheng, Phys. Rev. B\n77, 220506 (2008).\n34M. Daghofer, A. Moreo, J. A. Riera, E. Arrigoni, D. J.\nScalapino, and E. Dagotto, arXiv: p. 08050148 (2008).\n35Y. Murakami, H. Kawada, H. Kawata, M. Tanaka,\nT. Arima, Y. Moritomo, and Y. Tokura, Phys. Rev. Lett.\n80, 1932 (1998).\n36I. S. Elfimov, V. I. Anisimov, and G. A. Sawatzky, Phys.\nRev. Lett. 82, 4264 (1999).\n37P. Benedetti, J. van den Brink, E. Pavarini, A. Vigliante,\nand P. Wochner, Phys. Rev. B 63, 060408 (2001).\n38Y. Joly, J. E. Lorenzo, E. Nazarenko, J.-L. Hodeau,D. Mannix, and C. Marin, Physical Review B 78, 134110\n(2008).\n39Q. Si and E. Abrahams, Phys. Rev. Lett. 101, 076401\n(2008).\n40D.-X. Yao and E. W. Carlson, Phys. Rev. B 78, 052507\n(2008).\n41J. Zhao, D.-X. Yao, S. Li, T. Hong, Y. Chen, S. Chang,\nW. R. II, J. W. Lynn, H. A. Mook, G. F. Chen, et al.,\nPhys. Rev. Lett. 101, 167203 (2008).\n42F. Kr¨ uger and S. Scheidl, Phys. Rev. B 67, 134512 (2003).\n43K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231\n(1982).\n44L. F. Feiner, A. M. Ole´ s, and J. Zaanen, Phys. Rev. Lett.\n78, 2799 (1997).\n45A. M. Ole´ s, L. F. Feiner, and J. Zaanen, Phys. Rev. B 61,\n6257 (2000).\n46A. M. Ole´ s, L. F. Feiner, and J. Zaanen, Phys. Rev. B 78,\n155113 (2008).\n47S. Kumar, A. P. Kampf, and P. Majumdar, Phys. Rev.\nLett.97, 176403 (2006).\n48V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev.\nB44, 943 (1991).\n49A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.\nRev. B52, 5467 (1995).\n50G. A. Sawatzky, I. S. Elfimov, J. van den Brink, and J. Za-\nanen, arXiv: p. 08081390 (2008)." }, { "title": "2106.11377v1.Spin_structure_factors_of_doped_monolayer_Germanene_in_the_presence_of_spin_orbit_coupling.pdf", "content": "arXiv:2106.11377v1 [cond-mat.str-el] 21 Jun 2021Spin structure factors of doped monolayer\nGermanene in the presence of spin-orbit\ncoupling\nFarshad Azizi and Hamed Rezania∗\nDepartment of Physics, Razi University, Kermanshah, Iran\nCorresponding author’s email: rezania.hamed@gmail.com\nAbstract\nIn this paper, we present a Kane-Mele model in the presence of magnetic field and\nnext nearest neighbors hopping amplitudes for investigati ons of the spin susceptibilities of\nGermanene layer. Green’s function approach has been implem ented to find the behavior\nof dynamical spin susceptibilities of Germanene layer with in linear response theoryand in\nthe presence of magnetic field and spin-orbit coupling at fini te temperature. Our results\nshow the magnetic excitation mode for both longitudinal and transverse components of\nspintends tohigherfrequencies withspin-orbit couplings trength. Moreover thefrequency\npositions of sharp peaks in longitudinal dynamical spin sus ceptibility are not affected by\nvariation of magnetic field while the peaks in transverse dyn amical susceptibility moves\n∗Corresponding author. Tel./fax: +98 831 427 4569., Tel: +98 831 42 7 4569\n1to lower frequencies with magnetic field. The effects of electr on doping on frequency\nbehaviors of spin susceptibilities have been addressed in d etails. Finally the tempera-\nture dependence of static spin structure factors due to the e ffects of spin-orbit coupling,\nmagnetic field and chemical potential has been studied.\nKeywords: Germanene; Green’s function; Optical absorption\n1 Introduction\nA lot of theoretical and experimental studies have been performe d on Graphene as a one-\natom-thick layer of graphite since it’s fabrication[1]. The low energy lin ear dispersion and\nchiral property of carbon structure leads to map the nearest ne ighbor hopping tight binding\nhamiltonian which at low energy to a relativistic Dirac Hamiltonian for mas sless fermions with\nFermi velocity vF. Novel electronic properties have been exhibited by Graphene laye r with a\nzero band gap which compared to materials with a non-zero energy g ap. These materials have\nintriguing physical properties and numerous potential practical a pplications in optoelectronics\nand sensors[2].\nRecently, the hybrid systems consisting of Graphene and various t wo-dimensional materials\nhave been studied extensively both experimentally and theoretically [3, 4, 5]. Also, 2D materi-\nals could be used for a extensive applications in nanotechnology [6, 7 ] and memory technology\n[8]. While the research interest in Graphene-based superlattices is g rowing rapidly, people have\nstarted to question whether the Graphene could be replaced by its close relatives, such as 2 di-\nmensional hexagonal crystal of Germanene. This material shows a zero gap semiconductor with\n2massless fermion charge carriers since their πandπ∗bands are also linear at the Fermi level[9].\nGermanene as counterpart of Graphene, is predicted to have a ge ometry with low-buckled hon-\neycomb structure for its most stable structures in contrast to t he Graphene monolayer [9, 10].\nSuch small buckling as vertical distance between two planes of atom s for Germanene comes\nfrom the mixing of sp2andsp3hybridization[11, 12]. The behavior of Germanene electronic\nstructure shows a linear dispersion close to K and K’ points of the fir st Brillouin zone. However\nab initio calculations indicated that spin-orbit coupling in Germanene causes t o small band gap\nopening at the Dirac point and thus the Germanene has massive Dirac fermions[10, 13]. Also\nthe band gap due to the spin orbit coupling in Germanene is more remar kable rather than that\nin Graphene[14]. The intrinsic carrier mobility of Germanene is higher th an Graphene[15]. The\ndifferent dopants within the Germanene layer gives arise to the sizab le band gap opening at\nthe Dirac point an the electronic properties of this material are affe cted by that[16, 17]. In a\ntheoretical work, the structural and electronic properties of s uperlattices made with alternate\nstacking ofGermanenelayer aresystematically investigated byusin gadensity functionaltheory\nwith the van der Waals correction[18]. It was predicted that spin orb it coupling and exchange\nfield together open a nontrivial bulk gap in Graphene like structures leading to the quantum\nspin hall effect [19, 20]. The topological phase transitions in the 2D cr ystals can be understood\nbased on intrinsic spin orbit coupling which arises due to perpendicular electric field or interac-\ntion with a substrate. Kane and Mele[21] applied a model Hamiltonian to describe topological\ninsulators. Such model consists of a hopping and an intrinsic spin-or bit term on the Graphene\nlike structures. The Kane-Mele model essentially includes two copies with different sign for\nup and down spins of a model introduced earlier by Haldane[22]. Such m icroscopic model was\n3originally proposed to describe the quantum spin Hall effect in Graphe ne[21]. Subsequent band\nstructure calculations showed, however, that the spin orbit gap in Graphene is so small[23, 24]\nthat the quantum spin Hall effect in Graphene like structures is beyo nd experimental relevance.\nIn-plane magnetic field affects the magneto conductivity of honeyc omb structures so that\nthe results show the negative for intrinsic gapless Graphene. Howe ver the magneto-resistance\nof gapless Graphene presents a positive value for fields lower than t he critical magnetic field\nand negative above the critical magnetic[25]. Moreover, microwave magneto transport in doped\nGraphene is an open problem[26].\nThe many body effects such as Coulomb interaction and its dynamical screening present the\nnovel features for any electronic material. The collective spectru m and quasiparticle properties\nofelectronicsystems aredeterminedformdynamicalspinstructu refactors. Theseareappliedto\nimplytheopticalpropertiesofthesystem. Moreoveralotofstudie shavebeendoneoncollective\nmodes of monolayer Graphene both theoretically [27] and experimen tally[28]. However there\nare no the extensive theoretical studies on doped bilayer systems .\nThe frequency dependence of dynamical spin susceptibility has bee n studied and the re-\nsults causes to find the collective magnetic excitation spectrum of m any body system. It is\nworthwhile to explain the experimental interpretation of imaginary p art of dynamical spin sus-\nceptibilities. Slow neutrons scatter from solids via magnetic dipole inte raction in which the\nmagnetic moment of the neutron interacts with the spin magnetic mo ment of electrons in the\nsolid[29]. We can readily express the inelastic cross-section of scatt ering of neutron beam from\na magnetic system based on correlation functions between spin den sity operators. In other\nwords the differential inelastic cross section d2σ/dΩdωcorresponds to imaginary part of spin\n4susceptibilities. ωdescribes the energy loss of neutron beam which is defined as the diff erence\nbetween incident and scattered neutron energies. Ω introduces s olid angle of scattered neu-\ntrons. The spin excitation modes of the magnetic system have been found via the frequency\nposition of peaks in d2σ/dΩdω. Depending on component of spin magnetic moment of electrons\nthat interacts with spin of neutrons, transverse and longitudinal spin susceptibility behaviors\nhave been investigated. The imaginary and real part of non-intera cting change susceptibilities\nof Graphene within an analytical approach have been calculated A th eoretical work has been\nperformedforcalculating both[30]. Theresults ofthisstudy show s thereisno remarkableangle\ndependence for imaginary part of polarizability around the van Hove singularity, i.e, ¯ hω/t= 2.0\nwheretimplies nearest neighbor hopping integral.\nIt is worthwhile to add few comments regarding the comparison Germ anene and Graphene\nlike structure. As we have mentioned, the most important differenc e between Germanene struc-\nture ad Graphene one arises from the nonzero overlap function be tween nearest neighbor atoms\nin Germanene structure. Response functions of Graphene in the p resence of spin-orbit cou-\npling have been studied recently[31, 32]. In this references, the op tical absorption of Graphene\nstructure in the presence of spin-orbit coupling and magnetic field h as been theoretically stud-\nied. Optical absorption rate corresponds to the charge transitio n rate of electrons between\nenergy levels. However in our work, we have investigated the dynam ical spin susceptibility of\nGermanene structure due to spin-orbit coupling. In other words w e have specially studied the\ntransition rate of magnetic degrees freedom of electrons. Such s tudy is a novelty of our work so\nthat there is no the theoretical work on the study of magnetic exc itation modes in Germanene\nstructure due to spin-orbit coupling. These studies have been per formed for Graphene like\n5structures and the most important difference between our result s and the spin susceptibilities\nof Graphene structure is the effects of overlap function in German ene structure compared to\nGraphene lattice. In fact overlap function has considerable impact on the frequency position of\nexcitation modeandalso onthe intensity ofscattered neutron bea mfromGermanene structure.\nThe purpose of this paper is to provide a Kane Mele model including intr insic spin-orbit\ninteraction for studying frequency behavior of dynamical spin sus ceptibility of Germanene layer\ninthepresence ofmagnetic fieldperpendicular totheplane. Using th esuitable hopping integral\nand on site parameter values, the band dispersion of electrons has been calculated. Full band\ncalculation beyond Dirac approximation has been implemented to deriv e both transverse and\nlongitudinal dynamical spin susceptibilities. We have exploited Green’s function approach to\ncalculate the spin susceptibility, i.e. the time ordered spin operator c orrelation. The effects of\nelectron doping, magnetic field and spin-orbit coupling on the spin str ucture factors have been\nstudied. Also we discuss and analyze to show how spin-orbit coupling a ffects the frequency\nbehavior of the longitudinal and transverse spin susceptibilities. Als o we study the frequency\nbehavior of dynamical spin susceptibility of Germanene due to variat ion of chemical potential\nand magnetic field. Also the effects of spin-orbit coupling constant a nd magnetic field on\ntemperature dependence of both transverse and longitudinal st atic spin susceptibilities have\nbeen investigated in details.\n2 Model Hamiltonian and formalism\nThe crystal structure of Germanene has been shown in Fig.(1). Th e unit cell of Germanene\nstructure is similar to Graphene layer and this honeycomb lattice dep icted in Fig.(2). The\n6primitive unit cell vectors of honeycomb lattice have been shown by a1anda2. In the presence\nof longitudinal magnetic field, the Kane-Mele model[21] ( H) for Germanene structure includes\nthe tight binding model ( HTB), the intrinsic spin-orbit coupling ( HISOC) and the Zeeman term\n(HZeeman) due to the coupling of spin degrees of freedom of electrons with ex ternal longitudinal\nmagnetic field B\nH=HTB+HISOC+HZeeman. (1)\nThe tight binding part of model Hamiltonian consists of three parts; nearest neighbor hop-\nping, next nearest neighbor (2NN) hopping and next next nearest neighbor (3NN) hopping\nterms. The tight binding part, the spin orbit coupling term and the Ze eman part of the model\nHamiltonian on the honeycomb lattice are given by\nHTB=−t/summationdisplay\ni,∆,σ/parenleftig\naσ†\nj=i+∆bσ\ni+h.c./parenrightig\n−t′/summationdisplay\ni,∆′,σ/parenleftig\naσ†\nj=i+∆′aσ\ni+bσ†\ni+∆′bσ\ni/parenrightig\n−t”/summationdisplay\ni,∆”,σ/parenleftig\naσ†\nj=i+∆”bσ\ni+h.c./parenrightig\n−/summationdisplay\ni,σµ/parenleftig\na†σ\niaσ\ni+b†σ\nibσ\ni/parenrightig\n,\nHISOC=iλ/summationdisplay\ni,∆′,σ/summationdisplay\nα=A,B/parenleftig\nνa\ni+∆′,ia†σ\ni+∆′σz\nσσ′aσ′\ni+νb\ni+∆′,ib†σ\ni+∆′σz\nσσ′bσ′\ni/parenrightig\n,\nHZeeman=−/summationdisplay\ni,σσgµBB/parenleftig\na†σ\niaσ\ni+b†σ\nibσ\ni/parenrightig\n. (2)\nHereaσ\ni(bσ\ni) is an annihilation operator of electron with spin σon sublattice A(B) in unit cell\nindexi. The operators fulfill the fermionic standard anti commutation re lations{aσ\ni,aσ′†\nj}=\nδijδσσ′. As usual t,t′,t′′denote the nearest neighbor, next nearest neighbor and next ne xt\nnearest neighbor hopping integral amplitudes, respectively. The p arameterλintroduces the\nspin-orbit coupling strength. Also Brefers to strength of applied magnetic field. gandµB\nintroduce the gyromagnetic and Bohr magneton constants, resp ectively.σzis the third Pauli\nmatrix, and νa(b)\nji=±1 as discussed below. Based on Fig.(2), a1anda2are the primitive\n7vectors of unit cell and the length of them is assumed to be unit. The symbol∆=0,∆1,∆2\nimplies the indexes of lattice vectors connecting the unit cells including nearest neighbor lattice\nsites. The translational vectors ∆1,∆2connecting neighbor unit cells are given by\n∆1=i√\n3\n2+j1\n2,∆2=i√\n3\n2−j1\n2. (3)\nAlso index ∆′=∆1,∆2,−∆1,−∆2,j,−jimplies the characters of lattice vectors connecting\nthe unit cells including next nearest neighbor lattice sites. Moreover index∆” =√\n3i,j,−j\ndenotes the characters of lattice vectors connecting the unit ce lls including next next nearest\nneighbor lattice sites. We consider the intrinsic spin-orbit term[21] of the KM Hamiltonian in\nEq.(2). The expression νa(b)\njigives±1 depending on the orientation of the sites. A standard\ndefinition for να\njiin each sublattice α=A,Bisνα\nji=/parenleftigdα\nj×dα\ni\n|dα\nj×dα\ni|/parenrightig\n.ez=±1 where dα\njanddα\ni\nare the two unit vectors along the nearest neighbor bonds connec ting siteito its next-nearest\nneighborj. Moreover ezimplies the unit vector perpendicular to the plane. Because of two\nsublattice atoms, the band wave function ψσ\nn(k,r) can be expanded in terms of Bloch functions\nΦσ\nα(k,r). The index αimplies two inequivalent sublattice atoms A,Bin the unit cell, rdenotes\nthe position vector of electron, kis the wave function belonging in the first Brillouin zone of\nhoneycomb structure. Such band wave function can be written as\nψσ\nn(k,r) =/summationdisplay\nα=A,BCσ\nnα(k)Φσ\nα(k,r), (4)\nwhereCσ\nnα(k) is the expansion coefficients and n=c,vrefers to condition and valence bands.\nAlso we expand the Bloch wave function in terms of Wannier wave func tion as\nΦσ\nα(k,r) =1√\nN/summationdisplay\nRieik.Riφσ\nα(r−Ri), (5)\n8so thatRiimplies the position vector of ith unit cell in the crystal and φαis the Wannier wave\nfunction of electron in the vicinity of atom in ith unit cell on sublattice index α. By inverting\nthe expansion Eq.(4), we can expand the Bloch wave functions in ter ms of band wave function\nas following relation\nΦσ\nα(k,r) =/summationdisplay\nn=c,vDσ\nαn(k)ψσ\nn(k,r), (6)\nwhereDσ\nαn(k) is the expansion coefficients and we explain these coefficients in the f ollowing.\nThesmallBucklinginGermanenecausestotheconsiderablevaluefor 2NNand3NNhopping\namplitude. Moreover we have considerable values for overlap param eters of electron wave\nfunctions between 2NN and 3NN atoms. The band structures of ele ctrons with spin σof\nGermanene described by model Hamiltonian in Eq.(2) are obtained by u sing the matrix form\nof Schrodinger as follows\nHσ(k)Cσ(k) =Eσ\nn(k)Sσ(k)Cσ(k),\nHσ(k) =\nHσ\nAA(k)Hσ\nAB(k)\nHσ\nBA(k)Hσ\nBB(k)\n,Cσ(k) =\nCσ\nnA(k)\nCσ\nnB(k)\n,\nSσ(k) =\nSσ\nAA(k)Sσ\nAB(k)\nSσ\nBA(k)Sσ\nBB(k)\n. (7)\nUsing the Bloch wave functions, i.e. Φ α(k), the matrix elements of HandSare given by\nHσ\nαβ(k) =/an}bracketle{tΦσ\nα(k)|H|Φσ\nβ(k)/an}bracketri}ht, Sσ\nαβ(k) =/an}bracketle{tΦσ\nα(k)|Φσ\nβ(k)/an}bracketri}ht. (8)\nThe matrix elements of Hσ\nαβandSσ\nαβare expressed based on hopping amplitude and spin-orbit\ncoupling between two neighbor atoms on lattice sites and can be expa nded in terms of hopping\namplitudes t,t′,t”, spin orbit coupling λand overlap parameters. The diagonal elements of\n9matrixes Hin Eq.(7) arise from hopping amplitude of electrons between next nea rest neighbor\natoms on the same sublattice and spin-orbit coupling. Also the off diag onal matrix elements\nwith spin channel σ, i.e.Hσ\nAB,Hσ\nBA, raise from hopping amplitude of electrons between nearest\nneighbor atomsandnext next nearest neighboratomsonthediffer ent sublattices. These matrix\nelements are obtained as\nHσ\nAB(k) =t/parenleftig\n1+eik.∆1+eik.∆2/parenrightig\n+t”/parenleftig\n2cos(ky)+e−i√\n3kx/parenrightig\n=t/parenleftig\n1+2cos(ky/2)e−i√\n3kx/2/parenrightig\n+t”/parenleftig\n2cos(ky)+e−i√\n3kx/parenrightig\n,\nHσ\nAA(k) = 2t′/parenleftig\ncos(√\n3kx/2+ky/2)+cos(√\n3kx/2+ky/2)+cos(ky/2)/parenrightig\n−2λ/parenleftig\nsin/parenleftig1\n2ky/parenrightig\n−sin/parenleftig√\n3\n2kx+1\n2ky/parenrightig\n−sin/parenleftig√\n3\n2kx−1\n2ky/parenrightig/parenrightig\n−µ−σgµBB,\nHσ\nBB(k) =−2t′/parenleftig\ncos(√\n3kx/2+ky/2)+cos(√\n3kx/2+ky/2)+cos(ky/2)/parenrightig\n+ 2λ/parenleftig\nsin/parenleftig1\n2ky/parenrightig\n−sin/parenleftig√\n3\n2kx+1\n2ky/parenrightig\n−sin/parenleftig√\n3\n2kx−1\n2ky/parenrightig/parenrightig\n−µ−σgµBB,\nHσ\nBA(k) =H∗\nAB(k). (9)\nBased on matrix elements Hσ\nαβ(k), the model Hamiltonian in Eq.(2) is written in terms of\nFourier transformation of creation and annihilation fermionic opera tors as\nH=/summationdisplay\nk,σ/bracketleftig\nHσ\nAA(k)aσ†\nkaσ\nk+HAB(k)aσ†\nkbσ\nk+HBA(k)bσ†\nkaσ\nk+Hσ\nBB(k)bσ†\nkbσ\nk/bracketrightig\n, (10)\nso that the operator aσ\nk(bσ\nk) annihilates an electron at wave vector kwith spin index σon\nsublattice A(B) and has the following relation as\naσ\nk=1√\nN/summationdisplay\nkeik·Riaσ\ni, bσ\nk=1√\nN/summationdisplay\nkeik·Ribσ\ni. (11)\nThe matrix elements of S(k), i.e.SAA(k) ,SAB(k),SBA(k) andSBB(k) are expressed as\nSAB(k) =s/parenleftig\n1+eik.∆1+eik.∆2/parenrightig\n+s”/parenleftig\n2cos(ky)+e−i√\n3kx/parenrightig\n10=s/parenleftig\n1+2cos(ky/2)e−i√\n3kx/2/parenrightig\n+s”/parenleftig\n2cos(ky)+e−i√\n3kx/parenrightig\nSAA(k) = 1+2 s′/parenleftig\ncos(√\n3kx/2+ky/2)+cos(√\n3kx/2+ky/2)+cos(ky/2)/parenrightig\nSBB(k) =SAA(k), SBA(k) =S∗\nAB(k), (12)\nso thatsis theoverlap between orbital wave functionof electron respect t o the nearest neighbor\natoms,s′denotes the overlap between orbital wave function of electron re spect to the next\nnearest neighbor atoms and s” implies the overlap between orbital wave function of electron\nrespect to the next next nearest neighbor atoms. The density fu nctional theory and ab initio\ncalculations has been determined the hopping amplitudes and overlap valuess,s′,s” as[18]t=\n−1.163,t′=−0.055,t” =−0.0836,s= 0.01207,s′= 0.0128,s” = 0.048. Using the Hamiltonian\nand overlap matrix forms in Eqs.(9,12), the band structure of elect rons, i.e.Eσ\nη(k) has been\nfound by solving equation det/parenleftig\nH(k)−E(k)S(k)/parenrightig\n= 0. Moreover the matrix elements of C(k)\ncanbefoundbasedoneigenvalueequationinEq.(7). Eq.(7)canbere writtenasmatrixequation\nas follows\n\nψc(k,r)\nψv(k,r)\n=\nCσ\ncA(k)Cσ\ncB(k)\nCσ\nvA(k)Cσ\nvB(k)\n\nΦσ\nA(k,r)\nΦσ\nB(k,r)\n. (13)\nIn a similar way, we can express the matrix from for Eq.(6)\n\nΦσ\nA(k,r)\nΦσ\nB(k,r)\n=\nDσ\nAc(k)Dσ\nAv(k)\nDσ\nBc(k)Dσ\nBv(k)\n\nψσ\nc(k,r)\nψσ\nv(k,r)\n,\n\nDσ\nAc(k)Dσ\nAv(k)\nDσ\nBc(k)Dσ\nBv(k)\n=\nCσ\ncA(k)Cσ\ncB(k)\nCσ\nvA(k)Cσ\nvB(k)\n−1\n(14)\nThe final results for band structure and expansion coefficients, i.e .Cσ\nnαandDσ\nαn, are lengthy\nand are not given here. The valence and condition bands of electron s have been presented by\n11Eσ\nv(k) andEσ\nc(k) respectively. In the second quantization representation, we ca n rewrite the\nEq.(14) as\n\naσ†\nk\nbσ†\nk\n=\nDσ\nAc(k)Dσ\nAv(k)\nDσ\nBc(k)Dσ\nBv(k)\n\ncσ†\nc,k\ncσ†\nv,k\n, (15)\nUsing band energy spectrum, the Hamiltonian in Eq.(2) can be rewritt en by\nH=/summationdisplay\nk,σ,η=c,vEσ\nη(k)cσ†\nη,kcσ\nη,k, (16)\nwherecσ\nη,kdefines the creation operator of electron with spin σin band index ηat wave vector\nk. Since longitudinal magnetic field has been applied perpendicular to th e Germanene layer,\nthe electronic Green’s function depends on the spin index σ=↑,↓. According to the model\nHamiltonian introduced in Eq.(2), the elements of spin resolved Matsu bara Green’s function\nare introduced as the following forms\nGσ\nAA(k,τ) =−/an}bracketle{tT(ak,σ(τ)a†\nk,σ(0))/an}bracketri}ht, Gσ\nAB(k,τ) =−/an}bracketle{tT(ak,σ(τ)b†\nk,σ(0))/an}bracketri}ht,\nGσ\nBA(k,τ) =−/an}bracketle{tT(bk,σ(τ)a†\nk,σ(0))/an}bracketri}ht, Gσ\nBB(k,τ) =−/an}bracketle{tT(bk,σ(τ)b†\nk,σ(0))/an}bracketri}ht.(17)\nTintroduces the time ordering operator and arranges the creation and annihilation operators\nin terms of time of them without attention to the their algebra. The F ourier transformation of\neach Green’s function element is obtained by\nGσ\nαβ(k,iωm) =/integraldisplay1/kBT\n0dτeiωmτGσ\nαβ(k,τ), α,β=A,B, (18)\nwhereωm= (2m+1)πkBTdenotes the Fermionic Matsubara frequency. After some algebra ic\ncalculation, the following expression is obtained for Green’s function s in Fourier presentation\nGσ\nαβ(k,iωm) =/summationdisplay\nη=c,vDσ∗\nαη(k)Dσ\nβη(k)\niωm−Eσ\nη(k), (19)\n12whereα,βrefer to the each atomic basis of honeycomb lattice and Eσ\nη(k) is the band structure\nof Germanene layer in the presence of magnetic field and spin-orbit c oupling. For determining\nthe chemical potential, µσ, we use the relation between concentration of electrons ( ne) and\nchemical potential. This relation is given by\nne=1\n4N/summationdisplay\nk,η,σ1\neEση(k)/kBT+1. (20)\nInfactthediagonalmatrixelements oftheHamiltonianinEq.(9) depe ndsonchemical potential\nµ. Thus eigenvalues, i.e. Eσ\nη(k) includes the factor µ. Therefore the right hand of Eq.(20)\ndepends on chemical potential µ. With an initial guess for chemical potential µ, we can solve\nthe algebraic equation 20 so that we can find the chemical potential value for each amount\nfor electronic concentration ne. These statements have been added to the manuscript after\nEq.(20). Based on the values of electronic concentration ne, the chemical potential, µ, can be\nobtained by means Eq.(20). In order to obtain the magnetic excitat ion spectrum of Germanene\nstructure both transverse and longitudinal dynamical spin susce ptibilities have been presented\nusing Green’s function method in the following section.\n3 Dynamical and static spin structure factors\nThe correlation function between spin components of itinerant elec trons in Germanene layer\nat different times can be expressed in terms of one particle Green’s f unctions. The frequency\nFourier transformation of this correlation function produces the dynamical spin susceptibility.\nThe frequency position of peaks in dynamical spin susceptibility are a ssociated with collective\nexcitation of electronic gas described by Kane-Mele model Hamiltonia n in the presence of\n13magnetic field. These excitations are related to the spin excitation s pectrum of electrons on\nHoneycomb structure. In the view point of experimental interpre tation, the dynamical spin\nsusceptibility of the localized electrons of the system is proportiona l to inelastic cross-section\nfor magnetic neutron scattering from a magnetic system that can be expressed in terms of spin\ndensity correlationfunctions ofthe system. Inother words thed ifferential inelastic crosssection\nd2σ/dΩdωis proportionalto imaginary partof spin susceptibilities. ωdenotes the energy loss of\nneutronbeamwhichisdefinedasthedifferencebetweenincidentand scatteredneutronenergies.\nThe solid angle Ω implies the orientation of wave vector of scattered n eutrons from the localized\nelectronsofthesample. Wecanassumethewave vectorofincident neutronsisalong zdirection.\nThe solidangle Ωdepends onthe polaranglebetween wave vector of s cattered neutrons andthe\nwave vector of the incident neutrons. The frequency position of p eaks ind2σ/dΩdωdetermines\nthe spin excitation spectrum of the magnetic system[29]. In order t o study the general spin\nexcitation spectrum of the localized electron of the systems, both transverse and longitudinal\ndynamical spin-spin correlation functions have been calculated. Lin ear response theory gives us\nthe dynamical spin response functions based on the correlation fu nction between components\nof spin operators. We introduce χ+−as transverse spin susceptibility and its relation is given\nby\nχ+−(q,ω) =i/integraldisplay+∞\n−∞dteiωt/an}bracketle{t[S+(q,t),S−(−q,0)]/an}bracketri}ht\n= lim\niΩn−→ω+i0+/integraldisplay1/(kBT)\n0dτeiΩnτ/an}bracketle{tTS+(q,τ)S−(−q,0)/an}bracketri}ht\n=χ+−(q,iΩn−→ω+i0+), (21)\nin which Ω n= 2nπkBTis the bosonic Matsubara frequency and Timplies the time order op-\nerator. Also the wave vector qin Eq.(21) implies the difference between incident and scattered\n14neutron wave vectors. The Fourier transformations of transve rse components of spin density\noperators, ( S+(−)), in terms of fermionic operators is given by\nS+(q) =/summationdisplay\nk/parenleftig\na†↑\nk+qa↓\nk+b†↑\nk+qb↓\nk/parenrightig\n, S−(q) =/summationdisplay\nk/parenleftig\na†↓\nk+qa↑\nk+b†↓\nk+qb↑\nk/parenrightig\n. (22)\nSubstituting the operator form of S+andS−into definition of transverse spin susceptibil-\nity in Eq.(21), we arrive the following expression for transverse dyn amical spin susceptibility\n(χ+−(q,iΩn))\nχ+−(q,iΩn) =/integraldisplay1/(kBT)\n0dτeiΩnτ1\nN2/summationdisplay\nk,k′/angbracketleftig\nT/parenleftig\na†↑\nk+q(τ)a↓\nk(τ)+b†↑\nk+q(τ)b↓\nk(τ)/parenrightig\n×/parenleftig\na†↓\nk+q(0)a↑\nk(0)+b†↓\nk+q(0)b↑\nk(0)/parenrightig/angbracketrightig\n, (23)\nthatNis the number of unit cells in Germanene structure. In order to calcu late the correlation\nfunction in Eq.(23), one particle spin dependent Green’s function ma trix elements presented in\nEq.(19) should be exploited. After applying Wick’s theorem andtaking Fourier transformation,\nwe can transverse susceptibility in terms of one particle spin depend ent Green’s function\nχ+−(q,iΩn) =−kBT\nN/summationdisplay\nk/summationdisplay\nα,β/summationdisplay\nmG↑\nβα(k,iωm)G↓\nαβ(k+q,iΩn+iωm). (24)\nThe applied magnetic field to the Germanene layer causes to the spin d ependent property for\none particle Green’s function. In order to perform summation over Matsubara frequency ωm\nin Eq.(24), the Matsubara Green’s function elements should be writt en in terms of imaginary\npart of retarded Green’s function matrix elements using Lehman eq uation[33] as\nGσ\nαβ(k,iωm) =/integraldisplay+∞\n−∞dǫ\n2π−2ImGσ\nαβ(k,iωm−→ǫ+i0+)\niωm−ǫ, (25)\nwhereGσ\nαβ(k,iωm−→ǫ+i0+) denotes the retarded Green’s function matrix element. By\nreplacing Lehman representation for Matsubara Green’s function matrix elements into Eq.(24)\n15andtaking summation over fermionic Matsubara frequency ωm, we obtaindynamical transverse\nspin susceptibility of electrons on Germanene structure as following form\nχ+−(q,iΩn) =−1\nN/summationdisplay\nk/summationdisplay\nα,β/integraldisplay+∞\n−∞/integraldisplay+∞\n−∞dǫdǫ′\nπ2ImG↑\nβα(k,ǫ+i0+)ImG↓\nαβ(k+q,ǫ′+i0+)\n×nF(ǫ)−nF(ǫ′)\niΩn+ǫ−ǫ′, (26)\nwherenF(x) =1\nex/kBT+1implies well known Fermi-Dirac distortion function. The Fourier\ntransformation of longitudinal component of the spin, i.e. Sz(q), is given in terms of fermionic\noperators as\nSz(q) =/summationdisplay\nk,σσ/parenleftig\na†σ\nk+qaσ\nk+b†σ\nk+qbσ\nk/parenrightig\n(27)\nAlsoχzzis introduced as longitudinal spin susceptibility and its relation can be e xpressed in\nterms of correlation function between zcomponent of spin operators as\nχzz(q,ω) =i/integraldisplay+∞\n−∞dteiωt/an}bracketle{t[Sz(q,t),Sz(−q,0)]/an}bracketri}ht\n= lim\niΩn−→ω+i0+/integraldisplay1/(kBT)\n0dτeiΩnτ/an}bracketle{tTSz(q,τ)Sz(−q,0)/an}bracketri}ht\n=χzz(q,iΩn−→ω+i0+). (28)\nAfter some algebraic calculations similar to transverse spin suscept ibility case, we arrive te final\nresults for Matsubara representation of longitudinal dynamical s pin susceptibility as\nχzz(q,iΩn) =−1\nN/summationdisplay\nk/summationdisplay\nα,β,σ/integraldisplay+∞\n−∞/integraldisplay+∞\n−∞dǫdǫ′\nπ2ImGσ\nβα(k,ǫ+i0+)ImGσ\nαβ(k+q,ǫ′+i0+)\n×nF(ǫ)−nF(ǫ′)\niΩn+ǫ−ǫ′, (29)\nThe dynamical spin structure factor for both longitudinal and tra nsverse spin directions are\nobtained based on retarded presentation of susceptibilities as\nχzz(q,ω) =χzz(q,iΩn−→ω+i0+), χ+−(q,ω) =χ+−(q,iΩn−→ω+i0+).(30)\n16so thatχzzandχ+−are retarded dynamical spin structure factors for longitudinal a nd trans-\nverse components of spins, respectively. The imaginary part of re tarded dynamical spin struc-\nture factor, i.e. Im χ+−(q,ω),Imχzz(q,ω), is proportional to the contribution of localized spins\nin the neutron differential cross-section. For each qthe dynamical structure factor has peaks at\ncertain energies which represent collective excitations spectrum o f the system. The imaginary\nparts of both retarded transverse and longitudinal spin structu re factors are given by\nImχ+−(q,ω) =1\nN/summationdisplay\nk/summationdisplay\nα,β/integraldisplay+∞\n−∞dǫ\nπImG↑\nβα(k,ǫ+i0+)ImG↓\nαβ(k+q,ǫ+ω+i0+)\n×/parenleftig\nnF(ǫ)−nF(ǫ+ω)/parenrightig\n,\nImχzz(q,ω) =1\nN/summationdisplay\nk/summationdisplay\nα,β,σ/integraldisplay+∞\n−∞dǫ\nπImGσ\nβα(k,ǫ+i0+)ImGσ\nαβ(k+q,ǫ+ω+i0+)\n×/parenleftig\nnF(ǫ)−nF(ǫ+ω)/parenrightig\n. (31)\nThe frequencies of collective magnetic excitation modes are determ ined via finding the position\nof peaks in imaginary part of of imaginary part of dynamical spin susc eptibilities.\nStatic transverse spin structure factor ( s+−(q)) which is a measure of magnetic long range\nordering for spin components along the plane, i.e. transverse direc tion, can be related to\nimaginary part of retarded dynamical spin susceptibility using followin g relation\ns+−(q) =/angbracketleftig\nS+(q)S−(−q)/angbracketrightig\n=kBT/summationdisplay\nn1\n2π/integraldisplay∞\n−∞dω−2Imχ+−(q,iωn−→ω+i0+)\niωn−ω\n=/integraldisplay+∞\n−∞dωnB(ω)\nπImχ+−(q,iωn−→ω+i0+). (32)\nMoreover canfindthe staticlongitudinal spin structure factor, i.e .s+−(q), by using Im χzz(q,ω)\nas\nszz(q) =/an}bracketle{tSz(q)Sz(−q)/an}bracketri}ht=kBT/summationdisplay\nn1\n2π/integraldisplay∞\n−∞dω−2Imχzz(q,iωn−→ω+i0+)\niωn−ω\n=/integraldisplay+∞\n−∞dωnB(ω)\nπImχzz(q,iωn−→ω+i0+). (33)\n17In the next section, the numerical results of dynamical spin struc ture and static spin structures\nof Germanene layer have been presented for various magnetic field and spin-orbit coupling\nstrength.\n4 Numerical Results and Discussions\nWeturntoapresentationofourmainnumericalresultsofimaginary partofdynamicalstructure\nfactorsof Germanene layer atfinite temperature inthe presence of magnetic field andspin-orbit\ncoupling. Alsothetemperaturedependence ofstaticstructuref actorshasbeenaddressedinthis\nsection. Using the electronic band structure, the matrix elements of Fourier transformations\nof spin dependent Green’s function are calculated according to Eq.( 19). The imaginary part\nof both transverse and longitudinal dynamical spin susceptibilities is made by substituting the\nGreen’s function matrix elements into Eq.(31). In the following, the f requency behavior of\nimaginary part of dynamical spin susceptibilities is studied at fixed wav e number q= (0,4π\n3)\nin the Brillouin zone where the length of unit cell vector of honeycomb lattice is taken to be\nunit. Furthermore the static structure factors have been obta ined by using Eqs.(32,33).\nThe frequency behaviors of both the transverse and longitudinal dynamical spin suscepti-\nbilities have been addressed in this present study. Also the spin stru cture factors behaviors\nhave been investigated for Germanene structure.\nThe optimized atomic structure of the Germanene with primitive unit c ell vector length\na= 1 is shown in Fig.(1). The primitive unit cell include two Ge atoms.\nIn Fig.(3), we depict the frequency dependence of imaginary part o f longitudinal dynamical\nspin susceptibility, Imχzz(q0,ω), of undoped Germanene layer for different values of spin-\n18orbit coupling, namely λ/t= 0.08,0.12,0.16,0.2, in the absence of magnetic field by setting\nkBT/t= 0.05. In fact the effects of spin-orbit coupling strength on frequen cy dependence of\nImχzz(q,ω) have been studied in this figure. As shown in Fig.(3), the frequency positions\nof sharp peaks in Imχzz(q0,ω), that imply spin excitation mode for longitudinal components\nof spins, moves to higher frequencies with increase of spin-orbit co upling. This fact can be\nunderstood from this point that the increase of spin-orbit coupling leads to enhance band gap\nin density of states and consequently the excitation mode appears in higher frequency. Note\nthat this figure shows the inelastic cross section neutron particles from itinerant electrons of\nthe system due to longitudinal component along zdirection of magnetic moment of electrons\nand neutron beam. Another novel feature in Fig.(3) is the increase of intensity of sharp peak\nwithλ. For frequencies above normalized value 1.5, there is no collective ma gnetic excitation\nmode for longitudinal components of electron spins as shown in Fig.(3 ).\nThe frequency dependence of imaginary part of longitudinal dynam ical spin structure factor\nof undoped Germanene layer in the presence of spin polarization for different magnetic field\nvaluesgµBB/thas been shown in Fig.(4) at fixed λ/t= 0.6 by setting kBT/t= 0.05. A\nnovel feature has been pronounced in this figure. It is clearly obse rved that all curves for\ndifferent magnetic field indicates two clear magnetic excitation collect ive modes at frequencies\nω/t≈0.45 andω/t≈2.1. The frequency positions of magnetic excitation mode is independe nt\nof magnetic field. The intensity of sharp peaks in ω/t≈2.1 in imaginary part of longitudinal\nsusceptibility, i.e. Imχzz(q0,ω), decreases with magnetic field. However the height of sharp\npeaks in frequency position ω/t≈0.45 enhances with increase of spin-orbit coupling as shown\nin Fig.(4).\n19In Fig.(5), the imaginary part of longitudinal dynamical spin structu re factor of Germanene\nlayer has been plotted for different values of chemical potential, na melyµ/t= 0.2,0.3,0.4,0.5,\nat fixed spin-orbit coupling λ/t= 0.3 by setting kBT/t= 0.05 in the absence of magnetic\nfield. This figure implies the frequency position and intensity of collect ive excitation mode in\nω/t≈1.5 has no dependence on chemical potential. Although the intensity o f sharp peak in\nImχzz(q0,ω) at frequency position ω/t≈0.4 increases with chemical potential. There is no\nconsiderable change forfrequency positionin Imχzz(q0,ω)atω/t≈0.4 withchemical potential\naccording to Fig.(5). Also the intensity of low energy magnetic excita tion mode reduces with\ndecreasing chemical potential value µ.\nThe behavior of longitudinal static spin structure factor szz(q0) of undoped Germanene\nlayer in terms of normalized temperature kBT/tfor different values of λ/thas been presented\nin Fig.(6). The applied magnetic field is assumed to be zero. This functio n is a measure for\nthe tendency to magnetic long range ordering for the longitudinal c omponents of spins in the\nitinerant electrons. This figure implies static spin structure for spin components perpendicular\nto the Germanene plane includes a peak for each value of λ/t. The temperature position of\npeak in longitudinal spin static structure factor is the same for all s pin-orbit coupling strengths.\nAlthough the height of peak increases with λ/twhich justifies the long range ordering for\nzcomponents of spins improves with spin-orbit coupling. Another nov el feature is the non\nzero value for static structure factor szz(q0) at zero temperature. However the increase of\ntemperature up to peak position leads to raise magnetic long range o rdering. Upon more\nincreasing temperature above normalized value 0.25, the thermal fl uctuations causes to reduce\nszz(q0) so that magnetic long range ordering of the electrons decays as s hown in Fig.(6).\n20Thetemperaturedependenceofstaticlongitudinalspinstructur efactorofdopedGermanene\nlayer for various chemical potential has been studied in Fig.(7) for gµBB/t= 0.0 by setting\nλ/t= 0.4. In contrast to the undoped case in Fig.(6), it is clearly observed t he longitudinal\nspin structure factor for each value finite chemical potential get s zero value at zero temperature\naccording to Fig.(7). In fact the quantum fluctuations at zero tem perature for doped case leads\nto destroy any magnetic long range ordering. Moreover the peak in temperature dependence of\nszz(q0) tends to higher temperature upon increasing electron doping. Als o the height of peak\n,as a measure of magnetic long range ordering for longitudinal compo nents of spins, reduces\nwith chemical potential according to Fig.(7). Upon increasing tempe rature above normalized\nvalue 1.5,szz(q0) increases with µ/tand consequently the increase of electron doping improves\nthe long range ordering in temperature region above normalized valu e 1.5.\nThe effect of longitudinal magnetic field on the behavior of szz(q0) in terms of normalized\ntemperature kBT/tin undoped case for different values of magnetic field, namely gµBB/t=\n0.3,0.4,0.5,0.6,0.7 has been plotted in Fig.(8). The static structure is considerably aff ected by\nmagnetic field at low temperatures below normalized amount 0.5 where the quantum effects\nare more remarkable. In addition, at fixed values of temperatures above normalized value 0.5,\nszz(q0) is independent of magnetic field and all curves fall on each other on the whole range\nof temperature in this temperature region. Also temperature pos ition of peak in longitudinal\nstatic spin structure factormoves to lower temperature with incr easing magnetic field according\nto Fig.(8). Moreover the height of peak enhances with magnetic field . It can be understood\nfrom the fact that applying magnetic field along zdirection perpendicular to the plane causes\nlong range ordering of zcomponents of spins of electrons which increases the longitudinal s tatic\n21structure factor at low temperatures below 0.5.\nWe have also studied the dynamical and static transverse spin stru cture factors of Ger-\nmanene layer in the presence of magnetic field and spin-orbit coupling . Our main results for\nimaginary part of dynamical transverse spin susceptibility of undop ed Germanene layer for\ndifferent spin-orbit coupling strengths at fixed temperature kBT/t= 0.05 in the absence of\nmagnetic field are summarized in Fig.(9). The collective magnetic excita tion modes for spin\ncomponents parallel to the plane tends to higher values with λ/taccording to Fig.(9). This\nfeature arises from the increase of band gap with spin-orbit couplin g so that collective mode ap-\npears at higher frequencies as shown in Fig.(9). Also the height of pe ak in Imχ+−(q0,ω), which\nis proportional to intensity of scattered neutron beam from itiner ant electrons of Germanene\nstructure, reduces with decreasing spin-orbit coupling strength .\nIn Fig.(10) we plot the numerical results of Im χ+−(q,ω) of undoped Germanene layer as a\nfunctionofnormalizedfrequency ω/tforvariousmagneticfield,namely gµBB/t= 0.0,0.2,0.4,0.6\nby settingkBT/t= 0.05. It is clearly observed that the frequency position of collective m ag-\nnetic excitation mode moves to lower values with magnetic field. It can be understood from the\nfact that the applying magnetic field gives rise to reduce the band ga p and consequently the\nexcitation mode takes place at lower frequencies. Moreover the int ensity of collective excita-\ntion mode for transverse components of spins is clearly independen t of magnetic field strength\naccording to Fig.(10).\nFig.(11) presents the effect of electron doping on the frequency d ependence of Im χ+−(q0,ω)\nofGermanenelayerbysetting λ/t= 0.3atfixedvalueoftemperatureintheabsenceofmagnetic\nfield. There are two collective magnetic excitation mode for each valu e chemical potential. The\n22frequency positions of excitation mode are the same for all values o f chemical potential. The\nintensity of low frequency peak increases with chemical potential h owever the intensity of high\nfrequency peak reduces with electron doping as shown in Fig.(11). T he increase of electron\ndoping leads to decrease transition rate of electron from valence b and to conduction one and\nconsequently the intensity of excitation mode decreases.\nThe behavior of transverse static spin structure factor s+−(q0) of undoped Germanene\nlayer as a function of normalized temperature kBT/tfor different values of λ/tin the absence\nof magnetic field has been presented in Fig.(12). This function is a mea sure for the tendency to\nmagnetic long range ordering for the transverse components of s pins in the itinerant electrons.\nA peak ins+−(q0) is clearly observed for each value of λ/t. The peak is located at 0.25 for all\nvalues of spin-orbit coupling strengths. Although the height of pea k increases with λ/twhich\njustifies the long range ordering for transverse components of s pins improves with spin-orbit\ncoupling. Another novel feature is the non zero value for static st ructure factor s+−(q0) at zero\ntemperature. In temperature region below peak position, the incr ease of temperature leads to\nraise magnetic long range ordering. Upon more increasing temperat ure above normalized value\n0.25, the thermal fluctuations causes to reduce szz(q0) and magnetic long range ordering of the\nelectrons as shown in fig.(12).\nThe temperature dependence of static transverse spin structu re factor of doped Germanene\nlayer for various chemical potential in the absence of magnetic field has been studied in Fig.(13)\nby settingλ/t= 0.3. In contrast to the undoped case in Fig.(12), it is clearly observed the\ntransverse spin structure factor for each value finite chemical p otential gets zero value at zero\ntemperature according to Fig.(13). In fact the quantum fluctuat ions at zero temperature for\n23doped case leads to destroy any magnetic long range ordering. Mor eover the temperature\nposition of peak in s+−(q0) appears in kBT/t= 0.5 for all amounts of chemical potential. Also\nthe height of peak ,as a measure of magnetic long range ordering for transverse components\nof spins, reduces with chemical potential according to Fig.(13). In other words the increase of\nelectron doping leads to decrease magnetic long range ordering for transverse components of\nspins.\nThe effect of longitudinal magnetic field on the behavior of s+−(q0) in terms of normalized\ntemperature kBT/tin undoped case for different values of magnetic field, namely gµBB/t=\n0.4,0.45,0.5,0.55,0.6 has been plotted in Fig.(14). The static transverse spin structur e factor\nis considerably affected by magnetic field at low temperatures below n ormalized amount 0.25\nwhere the quantum effects are more remarkable. In addition, at fix ed values of temperatures\nabove normalized value 0.25, s+−(q0) is independent of magnetic field and all curves fall on\neach other on the whole range of temperature in this temperature region. Also temperature\nposition of peak in longitudinal static spin structure factor moves t o lower temperature with\nincreasing magnetic field. Moreover the height of peak enhances wit h magnetic field. It can be\nunderstood from the fact that applying magnetic field along zdirection perpendicular to the\nplane causes long range ordering of components of spins parallel to the plane of electrons which\nincreases the longitudinal static structure factor at low tempera tures below 0.25.\n5 Conflict Of interest Statement\nThere is no conflict of interest statement in this manuscript.\n24Figure 1: Crystal structure of Germanene\nFigure 2: The structure of honeycomb structure is shown. The ligh t dashed lines denote the\nBravais lattice unit cell. Each cell includes two nonequivalent sites, wh ich are indicated by A\nand B.a1anda2are the primitive vectors of unit cell.\n250 0.5 1 1.5 2 2.5 300.050.10.150.2\n0.08\n0.12\n0.16\n0.2\nω/tλ/t\nχ ( ,ω)q\n0zzIm\nFigure 3: The imaginary part of dynamical longitudinal spin susceptib ility Imχzz(q0,ω) of\nundoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nspin-orbit coupling λ/tat fixed temperature kBT/t= 0.05. The magnetic field is considered to\nbe zero.\n260 0.5 1 1.5 2 2.5 300.511.52\n0.6\n0.7\n0.8\n0.9\nω/tB/tµBg\nImχ ( ,ω)q0zz\nFigure 4: The imaginary part of dynamical longitudinal spin susceptib ility Imχzz(q0,ω) of\nundoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nmagnetic field gµBB/tat fixed spin-orbit coupling λ/t= 0.6. The normalized temperature is\nconsidered to be kBT/t= 0.05.\n270 0.5 1 1.5 2 2.5 300.10.20.30.40.5\n0.2\n0.3\n0.4\n0.5µ/\ntω/t\nImχ ( ,ω)0qzz\nFigure 5: The imaginary part of dynamical longitudinal spin susceptib ility Imχzz(q0,ω) of\ndoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nchemical potential µ/tat fixed spin-orbit coupling λ/t= 0.3. The magnetic field is assumed to\nbe zero. The normalized temperature is considered to be kBT/t= 0.05.\n280 1 2 3 4 5012345\n0.2\n0.3\n0.4\n0.5\n0.6\ns (q , T)zz0λ/t\nk T/tB\nFigure 6: The longitudinal static spin structure factor szz(q0,T) of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of spin-orbit coupling strength λ/tin\nthe absence of magnetic field.\n290 1 2 3 4 500.511.52\n1.0\n1.2\n1.4\n1.6\n1.8\n2.0\ns ( ,T)q0µ/t\nk T/tBzz\nFigure 7: The longitudinal static spin structure factor szz(q0,T)of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of chemical potential µ/tin the ab-\nsence of magnetic field. Spin-orbit coupling strength has been fixed atλ/t= 0.4.\n300 1 2 3 4 502468 0.3\n0.4\n0.5\n0.6\n0.7\ns ( ,T)q0zz\nk T/tBB/tµ\nBg\nFigure 8: The longitudinal static spin structure factor szz(q0,T) of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of magnetic field gµBB/tat fixed\nspin-orbit coupling strength λ/t= 0.4.\n310 1 2 3 4 501234\n0.3\n0.4\n0.5\n0.6λ/t\nImχ ( ,ω)q0+-\nω/t\nFigure 9: The imaginary part of dynamical transverse spin suscept ibility Imχ+−(q0,ω) of\nundoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nspin-orbit coupling λ/tat fixed temperature kBT/t= 0.05. The magnetic field is considered to\nbe zero.\n320 1 2 3 4 501234\n0.0\n0.2\n0.4\n0.6\nImχ ( ,ω)+-B/tµg\nB\nω/tq0\nFigure 10: The imaginary part of dynamical transverse spin suscep tibility Imχ+−(q0,ω) of\nundoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nmagnetic field gµBB/tat fixed spin-orbit coupling λ/t= 0.6. The normalized temperature is\nconsidered to be kBT/t= 0.05.\n330 1 2 3 4 5012345\n0.7\n0.75\n0.8\n0.85\n0.9µ/t\nImχ ( ,ω)+-q\n0\nω/t\nFigure 11: The imaginary part of dynamical transverse spin suscep tibility Imχ+−(q0,ω) of\ndoped Germanene layer versus normalized frequency ω/tfor different values of normalized\nchemical potential µ/tat fixed spin-orbit coupling λ/t= 0.3. The magnetic field is assumed to\nbe zero. The normalized temperature is considered to be kBT/t= 0.05.\n340 1 2 3 4 501234\n0.1\n0.3\n0.5\n0.7\n0.9\ns (q )+-λ/t\nT/t kB\nFigure 12: The transverse static spin structure factor s+−(q0,T) of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of spin-orbit coupling strength λ/tin\nthe absence of magnetic field.\n350 1 2 3 4 500.20.40.60.8 0.1\n0.2\n0.3\n0.4\ns (q )+-0µ/\nT/tBkt\nFigure 13: The transverse static spin structure factor s+−(q0,T) of doped Germanene layer\nversusnormalizedtemperature kBT/tfordifferentvaluesofchemicalpotential µ/tintheabsence\nof magnetic field with spin-orbit coupling λ/t= 0.3.\n360 0.5 1 1.5 2 2.5 3012345\n0.4\n0.45\n0.5\n0.55\n0.6\ns (q )+-\nT/tkBgµBB/t\nFigure 14: The transverse static spin structure factor s+−(q0,T)of undoped Germanene layer\nversus normalized temperature kBT/tfor different values of magnetic field gµBB/t. Spin-orbit\ncoupling strength has been fixed at λ/t= 0.4.\n37References\n[1] K. S. Novoselov, A. K. Geim, S. V. Morosov, D. Jiang, Y. Zhang, S . V. Dubonos, I. V.\nGrigorieva and A. A. Firsov, Electric field effect in atomically thin carbo n films, Science\n306, 666 (2004)\n[2] X. -L. Wang, X. S. Dou and C. Zhang, zero gap materials for futu re spintronics, NPG Asia\nMaterials 2, 31 (2010)\n[3] Y.Xu, Y.Liu, H.Chen, X.Lin, B.Yu, J.Luo, Anabinitiostudyofen ergy-bandmodulation\nin Graphene-based two dimensional layered superlattices, J. Mat. Chem 22, 23821 (2012)\n[4] K. Chang and W. X. Chen, In situ synthesis of MoS 2/Graphene nanosheet composites\nwith extraordinarily high electrochemical performance for lithium ion batteries, Chem.\nCommunication 47, 4252 (2011)\n[5] K. Chang and W. X. Chen, I-Cysteine-Assisted synthersis of La yered MoS 2/ Graphene\ncomposites with excellent electrochemical performance for lithium io n batteries, Acs Nano\n5, 4720 (2011)\n[6] C. R. Dean, etal, boron nitride substrates for high-quality Graphene electonics, N at. Nan-\notechnol 5, 722 (2010)\n[7] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V Khotkev ich, S. V. Mosorov, A.\nK. Geim, Proc, Natl, Two dimensional atomic crystals, Acad. Sci. USA 102, 10541 (2005)\n[8] S. Bertolazzi, D. Krasnozhon and A. Kis, Nonvolatile memory cells b ased on\nMos2/Graphene heterostructures, Acs Nano 7, 3246 (2013)\n38[9] S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin and S. Ciraci, tw o-and one-dimensional\nhoneycomb structures of silicon and Germanene, Phys. Rev. Lett 102, 236804 (2009)\n[10] C. C. Liu, W. Feng and Y. Yao, Quantum spin hall effect in Silicene an d two-dimensional\nGermanium, Phys. Rev. Lett 107, 076802 (2011)\n[11] J. E. Padilha and R. B. Pontes, free-standding bilayer silicene: T he effect of stacking order\non structural, electronic, and transport properties, The Journ al of Chemistry C 119, 3818\n(2015)\n[12] S. Chowdhury and D. Jana, A theoretical review on electonic, m agnetic and optical prop-\nerties of silicene, Reports on Progress in Physics 79, 126501 (2016 )\n[13] T. P. Kaloni, Tuning the structural, electronic, and magnetic pr operties of Germanene by\nthe adsorption of 3d transition metal atoms, The Journal of Phys ical Chemistry C 118,\n25200 (2014)\n[14] C. -C. Liu, H. Jiang and Y. Yao, Low energy effective Hamiltonian in volving spin-orbit\ncoupling in silicene and two dimensional germanium and Tin, Phys. Rev .B 8 4, 195430\n(2011)\n[15] X. -S. Ye, etal, intrinsic carrier mobility of Germanene is larger than Graphene’s: fir st\nprinciple calculations, Rsc Advances 4, 21216 (2016)\n[16] M. M. Monshi, S. M. Aghaei and I. Calizo, DFT study of adsorptio n behavior NO, Co,\nNO2 nad NH3 molecules on Graphene like BC3: A search for highly sensit ive molecular\nsensor, Surface Science 665, 96 (2017)\n39[17] M. Sun, Magnetism in transition-metal-doped Germanene: a firs t principles study, Com-\nputational Materials Science 118, 112 (2016)\n[18] X. Li, S. Wu, S. Zhouand Z. Zhu, structural and electronic pro perties of Germanene/MoS 2\nmonolayer and silicene/MoS 2monolayer superlattices, Nanoscale Research Letters 9, 110\n(2014)\n[19] Z. Qiao, etal, Quantum Anomalous hall effact in Graphene from Rashba and excha nge\neffects, Phys. Rev. B 82, 161414 (R) (2010)\n[20] W. -K. Tse, etal, Quantum anomalous hall effect in single-layer and bilayer Graphene,\nPhys. Rev. B 83, 155447 (2011)\n[21] C. L. Kane and E. J. Mele, Z2 Topological order and the quantum spin hall effect, Phys.\nRev. Lett 95, 146802 (2005)\n[22] F. D. M. Haldane, Model for a quantum hall effect without landau levels: condensed\nmatter realization of the ”parity anomaly” Phys. Rev. Lett 61, 201 5 (1988)\n[23] H. Min, J. E. Hill, N.A. Sinitsyn, B. R.Sahu, L. Kleinman, andA. H.M a Donald, Intrinsic\nand rashba spin-orbit interactions in Graphene sheets, Phys. Rev . B 74, 165310 (2006)\n[24] Y. Yao, F. Ye, X. -L. Qi, S. -C. Zhang, and Z. Fang, Spin orbit ga p of Graphene: First\nprinciples calculations, Phys. Rev. B 75, 041401 (2007)\n[25] E. H. Hwang and S. Das Sarma, Graphene magnetoresistance in a parallel magnetic field\nspin polarization effect, Phys. Rev. B 80, 075417 (2009)\n40[26] R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. J ohnson and V. Uman-\nsky, Zero-resistance staes induced by electromagnetic-wave ex citation in GaAs/AlGaAs,\nNature (London) 420, 646 (2002)\n[27] E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007) ; B. Wunsch etalNew J.\nPhy 8, 318 (2006)\n[28] Y. Liu, R. F. Willis, K. V. Emtsev, and T. Seyller, Phy. Rev. B 78, 20 1403 (2008)\n[29] S. Doniach and E. H. Sondheimer, Green’s functions for solid sta te physicists (Imperial\nCollege Press, London, 1999)\n[30] T. Stauber, J. Schliemann and N M Peres, Phys. Rev. B 81, 0854 09 (2010)\n[31] M. Inglot, V. K. Dugaev, E. Ya. Sherman and J. Barnas, Phys. Rev. B 91, 195428 (2015)\n[32] M. Inglot, V. K. Dugaev, E. Ya. Sherman and J. Barnas, Phys. Rev. B 89, 155411 (2014)\n[33] G. D. Mahan, Many Particle Physics, Plenumn Press, New York, 1 993\n41" }, { "title": "2003.10090v2.Spin_orbit_coupling_assisted_roton_softening_and_superstripes_in_a_Rydberg_dressed_Bose_Einstein_Condensate.pdf", "content": "arXiv:2003.10090v2 [cond-mat.quant-gas] 1 Sep 2020Spin-orbit-coupling-assisted roton softening and supers tripes in a Rydberg-dressed\nBose-Einstein Condensate\nHao Lyu and Yongping Zhang∗\nInternational Center of Quantum Artificial Intelligence fo r Science and Technology\n(QuArtist) and Department of Physics, Shanghai University , Shanghai 200444, China\nRotons can exist in ultracold atomic gases either with long- range interactions or with spin-orbit-\ncoupled dispersions. We find that two different kinds of roton s coexist in a joint system combining\nlong-range interactions and spin-orbit coupling. One roto n originates from spin-orbit coupling and\ntwo others come from long-range interactions. Their soften ing can be controlled separately. The\ninteresting new phenomenon which we find is that spin-orbit- coupled roton can push down the\nenergy of one long-range-interaction roton. The spin-orbi t coupling accelerates the softening of this\nroton. The post phase of spin-orbit-coupling-assisted rot on softening and instability is identified as\na superstripe.\nI. INTRODUCTION\nIn quantum many-body systems, roton is a particular\nkind ofcollectiveexcitations andfeatures a parabolic-like\ndispersion relation at a finite momentum. Supersolid [ 1–\n5] is a unique phase of ground states and is characterized\nby the coexistence of crystalline densities due to sponta-\nneousbreakingofcontinuoustranslationalsymmetryand\nsuperfluidity associated with gauge symmetry breaking.\nThese two different phenomena are connected by the so-\ncalled roton softening and instability. Varying relevant\nparametersgivesrisetosofteningoftherotonenergy,and\nvanishingofthe rotongappredictsthatthe homogeneous\nstate hosting roton excitations is energetically unstable.\nSuch a roton instability demands that the ground state\npossibly is a supersolid due to the simultaneous occu-\npation of the original momentum and roton momentum\nwhere roton instability happens. Therefore, the roton\nsoftening and instability provides an accessible route to\nsearching for supersolid phases [ 6,7].\nRecently, there is a big advance for the study of these\ntwin phenomena in ultracold atomic gases, produced by\ntheir experimental implementations. In ultracold atoms,\nthere are two different “arenas” to accommodate rotons\nand supersolids. One is with long-range interactions and\nthe other is with special single-particle dispersion rela-\ntions featuring multiple energy minima. Long-range in-\nteractions can form self-attractions around a certain mo-\nmentum in momentum domain which generate a local\nminimum with infinite density of state, i.e., a roton [ 8–\n11]. Dominant long-rangeinteractionscanbe experimen-\ntally realized with the assistance of optical cavities. Ro-\nton softening [ 12] and supersolid phases [ 13] have been\nobserved in pioneering optical cavity experiments. Very\nrecently, experimental achievements of Bose-Einstein\ncondensation in strongly magnetic lanthanide atoms pro-\nvide another promising perspective to measure the dipo-\nlar roton softening [ 7] and identify supersolids [ 14–17],\n∗yongping11@t.shu.edu.cnwith the assistance of dominant long-range dipolar in-\nteractions. Furthermore, dipolar supersolids have been\ncharacterized by observations of their collective excita-\ntions from different aspects [ 18–22]. Rydberg-dressed\nBose-Einsteincondensates(BECs)offeranotherplatform\nto realize long-range interactions [ 23–26]. They can be\nachieved by off-resonantly coupling BEC atoms to a Ry-\ndberg state [ 10,23]. Unlike anisotropic dipolar inter-\nactions, nonlocal Rydberg-dressed interactions can be\nisotropic. Supersolid phases have been identified theo-\nretically in Rydberg-dressed BECs [ 10,27].\nThe other mechanism to introduce rotons and super-\nsolids relates to single-particle dispersion relations. If\nsingle-particle dispersions possess multiple energy min-\nima, the homogeneous ground state chooses to occupy\none of them, anotherunoccupied minima are modified by\nrepulsiveatomicinteractionstogenerateroton-likestruc-\ntures in collective excitations, no matter the interactions\nare short-range or long-range [ 28–30]. There already ex-\nists two experimental approaches to prepare such spe-\ncial single-particle dispersions. One utilizes periodically\nshaking optical lattices [ 31] and the other makes use of\nspin-orbit coupling induced by Raman coupling between\nlasers and atoms [ 32–36]. Roton softening has been ex-\namined in both shaking lattices and spin-orbit coupled\ngases [30,31,37]. Spin-orbit-coupled supersolids have\nalso been observed experimentally in Ref. [ 38]. Because\nof their one-dimensional nature, they are called super-\nstripes [39].\nThe combination of these two different arenas together\nendows rotons and supersolids with more interesting\nproperties. Supersolids and superstripes have been in-\nvestigated in spin-orbit-coupled systems with long-range\ninteractionsindifferentspatialdimensions[ 40–44]. Much\nattention has focused on supersolids. However, whether\ntherotonsgeneratedbythetwodifferentmechanismscan\ncoexist and how the rotons in the combined arenassoften\nare not known yet.\nIn this paper, we address these questions by study-\ning a spin-orbit-coupled BEC with long-range Rydberg-\ndressed interactions. In atomic BECs, the pseudospin\nstates of spin-orbit coupling could be hyperfine ground2\nstates. The coupling of hyperfine states by Raman lasers\nrequires particular atoms [ 45]. Currently, all spin-orbit-\ncoupled BEC experiments with hyperfine states are im-\nplemented in87Rb BECs [ 32,37]. While, the possible re-\nalization of Rydberg-dressed interactions does not need\nto choose specific atoms [ 23], which provides a hope to\nimplement both in87Rb atoms. Until now, it is still a\nchallenge to realize Rydberg-dressed BECs [ 46]. How-\never, Rydberg-dressed BECs attract lots of current the-\noretical interest as an outstanding platform to explore\nphenomena of long-range interactions [ 47,48].\nWe find that there exists two different kinds of rotons\nseparately originating from single-particle spin-orbit-\ncoupled dispersion and long-rangeinteractions. The soft-\neningoftheserotonscanbeadjustedindependently. This\ntunability introduces the system into a new stage where\nthe spin-orbit-coupled roton can assist the softening of\nlong-range interaction-rotons. The consequence of such\nspin-orbit-coupling-assisted roton softening and instabil-\nity is superstripe ground states. We characterize the su-\nperstripes by their collective excitation spectrum. Our\nstudy is organized as follows. In Sec. II, we present\nour theoretical frame for the analysis of roton and su-\nperstripe. In Sec. III, we uncover that two kinds of ro-\ntons with different origination mechanism can coexist,\nand spin-orbit-coupling-assisted roton softening will be\ndemonstrated. The post-roton-softening phase, i.e., su-\nperstripes, will be discussed in Sec. IV, and the conclu-\nsion follows in Sec. V.\nII. MEAN-FIELD THEORY\nWe start from a three dimensional atomicBEC and as-\nsume that harmonic traps along the transverse direction\nare strong enough so that the transversemotion of atoms\nis completely confined into the ground state of harmonic\ntraps. After integrating over the transverse motion, we\nget a quasi-one-dimension (1D) spin-orbit-coupled BEC\nwith Rydberg-dressed interactions. The mean-field en-\nergy functional of the system is,\nE=/integraldisplay\ndxψ†(x)Hsocψ(x)\n+1\n2/summationdisplay\ni,j=1,2/integraldisplay\ndxgij|ψi(x)|2|ψj(x)|2(1)\n+1\n2/summationdisplay\ni,j=1,2/integraldisplay\ndxdx′Vij(x−x′)|ψi(x)|2|ψj(x′)|2.\nHereψ= (ψ1,ψ2)Tis the two-component wave function.\ngij(i,j= 1,2) characterize contact interactions between\nintra- and inter-species, which are proportionalto s-wave\nscattering lengths and the atom number. Hsocis the\nsingle-particle spin-orbit-coupled Hamiltonian,\nHsoc=−1\n2∂2\n∂x2−iγ∂\n∂xσz+Ω\n2σx. (2)In experiments, the spin-orbit coupling term −iγ∂/∂xσ z\ncan be artificially introduced into atoms by momentum\nexchanges between Raman lasers and atoms [ 30,32,37].\nThe spin-orbit coupling strength γ=/planckover2pi1kRam/mrelates\nto the wavelength λRamof Raman lasers with kRam=\n2π/λRam, wheremis the atom mass. Ω is the Raman\ncoupling and is proportional to the laser intensity, so\nthat it can be easily tuned in experiments. In all our\ndimensionless equations, we choose the unit of momen-\ntum, length, andenergyas kRam, 1/kRamand/planckover2pi12k2\nRam/m,\nrespectively. Therefore, under these units, the dimen-\nsionless parameter γ= 1. However, the value of γcan\nbe varied by the periodic modulation of Ω according to\nthe proposal in Ref. [ 49] and the experimental realiza-\ntion in [50]. The effective Rydberg-dressing potentials\nbetween intra- and inter-components are [ 10,23,51],\nVij(x) =˜Cij\n6\nx6+R6c, (3)\nwith˜Cij\n6being the interaction strengths of Rydberg-\ndressing and Rcbeing the blockade radius.\nBy minimizing the free energy F=E−µNwithµ\nbeing the chemical potential and Nbeing the total atom\nnumber, we get stationary Gross-Pitaevskii (GP) equa-\ntions [5],\nµψ= (Hsoc+Hs[ψ]+HRyd[ψ])ψ. (4)\nHereHsdenotes the contact interactions,\nHs[ψ] =/parenleftbigg\ng11|ψ1|2+g12|ψ2|20\n0 g12|ψ1|2+g22|ψ2|2/parenrightbigg\n.\nHRydrepresents the long-rangeRydberg-dressed interac-\ntions,\nHRyd[ψ] =\n/summationdisplay\ni=1,2/integraldisplay\ndx′/parenleftbigg\nV1i(x′−x)|ψi(x′)|20\n0 V2i(x′−x)|ψi(x′)|2/parenrightbigg\n.\nOnce we know the ground-statewave function ψand cor-\nresponding µ, we can study their collective excitationsby\nadding perturbations into the ground state. Therefore\ntotal wave functions are,\nΨ1,2=e−iµt/bracketleftbig\nψ1,2(x)+u1,2(x)e−iωt+v∗\n1,2(x)eiωt/bracketrightbig\n,(5)\nwhereωis the excitation energy, and u1,2(x),v1,2(x)\nare perturbation amplitudes, satisfying the normaliza-\ntion condition,/summationtext\nl=1,2/integraltext\ndx/parenleftbig\n|ul(x)|2−|vl(x)|2/parenrightbig\n= 1. Af-\nter substituting Ψ 1,2into the time-dependent version of\nEq. (4) and keeping linear terms of uandv, we get\nBogoliubov-de Gennes (BdG) equations [ 52],\nL1φ+L2[φ] =ωφ, (6)\nwithφ= (u1(x),u2(x),v1(x),v2(x))T. The matrix L1is\nL1=/parenleftbigg\nHsoc+A−µ B\nB∗−Hsoc−A∗−µ/parenrightbigg\n,(7)3\nwith\nA=Hs[ψ]+HRyd[ψ]+/parenleftbigg\ng11|ψ1|2g12ψ∗\n1ψ2\ng12ψ1ψ∗\n2g22|ψ2|2/parenrightbigg\n,\nand\nB=/parenleftbigg\ng11ψ2\n1g12ψ1ψ2\ng12ψ1ψ2g22ψ2\n2/parenrightbigg\n.\nThe matrix L2[φ] is given by\nL2[φ] =/integraldisplay\ndx′/parenleftbigg\nM1(x,x′)M2(x,x′)\nM∗\n2(x,x′)M∗\n1(x,x′)/parenrightbigg\nφ(x′),(8)\nwith\nM1(x′,x) =/parenleftbigg\nV11(x′−x)ψ∗\n1(x′)ψ1(x)V12(x′−x)ψ∗\n2(x′)ψ1(x)\nV12(x′−x)ψ∗\n1(x′)ψ2(x)V22(x′−x)ψ∗\n2(x′)ψ2(x)/parenrightbigg\n,\nM2(x′,x) =/parenleftbigg\nV11(x′−x)ψ1(x′)ψ1(x)V12(x′−x)ψ2(x′)ψ1(x)\nV12(x′−x)ψ1(x′)ψ2(x)V22(x′−x)ψ2(x′)ψ2(x)/parenrightbigg\n.\nIn the following, we solve GP equations ( 4) to search\nfor ground states and solve BdG equations ( 6) to ana-\nlyze the collective excitation spectrum of corresponding\nground states. We set g11=g22=g12=g, since they\nare approximately the same in experiments [ 32]. The\ntwo-component Rydberg-dressed BEC may be realized\nby coupling two ground hyperfine states to different Ry-\ndberg states [ 44,51]. The strength of long-range inter-\nactions˜Cij\n6and the blockade radius Rcdepend on the\ntwo-photon detuning and Rabi frequency of excitation\nlasers, which are tunable in experiments [ 23]. In this\nwork, we choose V11(x) =V22(x) =V12(x) =V(x) for\nsimplicity, so that ˜Cij\n6=˜C6. For further convenience, we\ntransformV(x) into the momentum space, i.e., ˜V(k) =/integraltextdxV(x)exp(ikx) =˜V0f(k), with ˜V0= 2π˜C6/3R5\nc\nandf(k) =1\n2e−|k|Rc/2/bracketleftbig\ne−|k|Rc/2+cos/parenleftbig√\n3|k|Rc/2/parenrightbig\n+√\n3sin/parenleftbig√\n3|k|Rc/2/parenrightbig/bracketrightbig\n. We use parameters ˜V0andRcto\ncharacterize the Rydberg-dressed interactions. The va-\nlidity of theoretical mean-field frame in our quasi-1Dsys-\ntem is examined by the calculation of quantum depletion\ndemonstrated the appendix.\nIII. SPIN-ORBIT-COUPLING-ASSISTED\nROTON SOFTENING\nWithout the spin-orbit coupling ( γ= 0, Ω = 0), it\nis known that the collective excitation spectrum ω(q) of\na BEC with long-range interactions can be analytically\ncalculated from the BdG equations [ 53]. In collective\nexcitations, there are two branches [ 54,55]; one is spin-\ndensity excitations ω=q2/2, and the other is density/s45/s52 /s45/s50 /s48 /s50 /s52/s48/s50/s52/s54\n/s45/s50 /s48 /s50/s48/s49/s50\n/s32/s69/s110/s101/s114/s103/s121/s32/s82\n/s99/s32/s61/s32/s49\n/s32/s82\n/s99/s32/s61/s32/s49/s46/s53\n/s32/s82\n/s99/s32/s61/s32/s50/s40/s97/s41\n/s113/s69/s110/s101/s114/s103/s121\n/s113/s32 /s32/s61/s32/s48/s46/s52\n/s32 /s61/s32/s48/s46/s54\n/s32 /s32/s61/s32/s48/s46/s56/s40/s98/s41\nFIG. 1. (a) The density excitation of a Rydberg-dressed BEC\nfor various Rcwith a fixed ˜V0= 10. (b) The lower branch of\nthe collective excitation of the plane-wave ground state in a\nspin-orbit coupled BEC without long-range interactions. T he\nspin-orbit coupling strength is set as γ= 0.6. In both (a) and\n(b),s-wave interaction strength is fixed as g= 0.5.\nexcitations,\nω=1\n2/radicalbigg\nq4+4q2/bracketleftBig\ng+˜V(q)/bracketrightBig\n. (9)\nThe density branch is consistent with the collective ex-\ncitation spectrum of a single component with long-range\ninteractions [ 10]. In Fig. 1(a), the density branch as a\nfunctionofRcisshownforafixed ˜V0. Wecheckthat ˜V(q)\nbecomes attractive around |q| ∼4.3/Rc. It predicts [ 6]\nthat the roton momentum qrotwhere minimum of roton\nenergy happens can be characterized by the blockade ra-\ndius, which is consistent with the finding in Ref. [ 10].\nOne of the interesting features of Rydberg-dressed ro-\ntons is that rotons appear in a pair and symmetrically\ndistribute at qrot∼ ±4.3/Rc. Increasing Rcleads to the\nshrinkingofrotonminimumandmeanwhilethereduction\nof roton gap. Thus, roton softening can be introduced\nby increasing Rc[see Fig. 1(a)]. Once the roton gap is\nclosed, roton instability happens and further increasing\nRcleads to the roton energy becoming complex-valued.\nAs mentioned above, we unravel the mechanism of the\nexistence of roton softening in a Rydberg-dressed BEC.\nIn the following, we show that roton softening can exist\nin a spin-orbit-coupled BEC. The dispersion relation of\nthe single-particle spin-orbit-coupled Hamiltonian Hsoc\npossesses two bands and the lowest band has two degen-\nerate energy minima. These two minima give rise to the4\nroton spectrum. In order to show this, we assume that\nthe ground state is a plane-wave, so that the wave func-\ntions areψ(x) =eikx(ϕ1,ϕ2)T[28,56] withkbeing the\nground-state quasimomentum. ϕ1,2are independent of\nspatial coordinates and satisfy the renormalization con-\ndition|ϕ1|2+|ϕ2|2= 1. With this ansatz, the energy\nfunctional becomes\nE=1\n2k2+γk(|ϕ1|2−|ϕ2|2)+Ω\n2(ϕ∗\n1ϕ2+ϕ1ϕ∗\n2)\n+1\n2(g+˜V0).\nFrom the above expression, it is clear that long-range\nand contact interactions contribute only an overall shift\nof the energy since all their coefficients are the same.\nThis leads to the fact that the properties of the ground\nstate should be similar to that of the single-particle case.\nFor Ω<2γ2, minimization of the energy functional re-\nsults in two energy minima laying at quasimomentum\nk±=±γ/radicalbig\n1−Ω2/4γ2, and the plane-wave ground state\nspontaneously chooses one of them to occupy. By us-\ning the results of the minimization with the calculated\nchemical potential and assuming u1,2(x) =u1,2ei(k+q)x\nandv1,2(x) =v1,2e−i(k−q)x[28] withqbeing the pertur-\nbation quasimomentum, we calculate the BdG equations\nto obtain the collective excitation spectrum ω(q) of the\nplane-wavegroundstate. Figure 1(b)demonstratesroton\nsofteningbyvaryingthe RamancouplingΩwithoutlong-\nrange interactions ( ˜V0= 0). The spin-orbit-coupled ro-\nton only exists in one side of q. This is because the plane-\nwave ground state that we use to plot Fig. 1(b) sponta-\nneously occupies k−. Since the contact interactions g\ncontributes positive energy to the system, they displace\nsingle-particle dispersion upwards and meanwhile shape\nthe dispersion around k−(corresponding to q= 0 in the\nperturbation momentum frame) into a linear form. A\nphonon mode can be formed in this mechanism. There-\nfore, the otherunoccupied single-particleminimum at k+\nappears as a roton-like structure in collective excitations.\nIn the perturbation quasimomentum frame, a roton sits\natqrot≈k+−k−= 2γ/radicalbig\n1−Ω2/4γ2. Decrease of Ω\ncauses roton softening, as well as the increase of roton\nmomentum [see Fig. 1(b)]. Further decrease of Ω can\nnot result in roton instability. This is because there is no\nphase transition from the plane-wave ground state to su-\nperstripes, with all equal interaction coefficients [ 56]. If\nwe make the interaction coefficients unequal, roton insta-\nbility occurs after the softening and it features negative\nroton energy [ 30].\nWe conclude that the momentum of Rydberg-dressed\nroton is governed by the blockade radius Rc, while\nthespin-orbit-coupled-roton’smomentumrelatestospin-\norbit coupling parameters γand Ω. They have com-\npletely different originations, which intuitively makes\ntheir coexistence possible. We perform numerical cal-\nculations to confirm this expectation. The collective\nexcitation of a BEC in the presence of both spin-orbit\ncoupling and long-range Rydberg-dressed interactions/s45/s50 /s48 /s50/s48/s49/s50\n/s45/s50 /s48 /s50/s48/s49/s50\n/s32/s32/s69/s110/s101/s114/s103/s121/s32 /s61/s32/s48/s46/s52\n/s32 /s32/s61/s32/s48/s46/s54\n/s32 /s61/s32/s48/s46/s56/s40\n/s32/s113/s69/s110/s101/s114/s103/s121\n/s113/s32/s82\n/s99/s32/s61/s32/s49/s46/s56\n/s32/s82\n/s99/s32/s61/s32/s50\n/s32/s82\n/s99/s32/s61/s32/s50/s46/s51\n/s32/s82\n/s99/s32/s61/s32/s50/s46/s53/s40/s97/s41\n/s40/s98/s41\nFIG. 2. Collective excitation spectrum of the Rydberg-\ndressed BEC with spin-orbit coupling, in which we choose\nγ= 0.6,g= 0.5, and˜V0= 10. (a) Evolution of the spectrum\nfor various Ω with a fixed Rc= 1.8. (b) Evolution of the\nspectrum for various Rcwith a fixed Ω = 0 .5. For the line\nwithRc= 2.5, the right-hand Rydberg-dressed roton become\nunstable with complex-valued energy which is not shown in\nthe plot.\nare demonstrated in Fig. 2(a). ForRc= 1.8, we see\nthat there are only Rydberg-dressed rotons sitting at\nqrot∼ ±4.3/Rc∼ ±2.2 symmetrically when Ω is large\n[see the line with Ω = 0 .8 in Fig. 2(a)]. This is expected\nsince for a larger Ω there is no spin-orbit-coupled roton\neven without long-range interactions [see the line with\nΩ = 0.8 in Fig. 1(b)]. Decreasing Ω leads to the ap-\npearance of the spin-orbit-coupled roton sitting approx-\nimately at 2 γ/radicalbig\n1−Ω2/4γ2. The newborn roton pushes\ndown the spectrum at right-hand side, and pushes up\nthe left-hand side spectrum. However, it does not af-\nfect the location of Rydberg-dressed rotons [see the lines\nwith Ω = 0 .6 and 0.4 in Fig. 2(a)]. From the line with\nΩ = 0.4 in Fig. 2(a), where the spin-orbit-coupled roton\ndominates, the Rydberg-dressed roton at right side fades\naway while the left side roton survives.\nConsequently, the two different kinds of rotons can\nexist independently. Meanwhile, the spin-orbit-coupled\nroton dramatically pushes spectrum on its side down,\nwhich provides a new phenomenon for the softening of\nthe Rydberg-dressed roton. In Fig. 2(b) we show the\nRydberg-dressedrotonsoftening byvarying Rc. We start\nfrom the coexistence case with Rc= 1.8 (the red line).\nDuetothespin-orbit-coupledroton,thespectrumiscom-\npletelyasymmetricwithrespectto q= 0. Theright-hand5\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s49 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s49 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s32/s110 /s40/s120 /s41\n/s120/s40/s97/s41\n/s32/s69/s110/s101/s114/s103/s121\n/s113/s47/s75/s40/s98/s41/s32/s32/s110 /s40/s120 /s41\n/s120/s40/s99/s41\n/s32/s32/s69/s110/s101/s114/s103/s121\n/s113/s47/s75/s40/s100/s41\nFIG. 3. Density profiles (a,c) and excitation spectrums (b,d ) of the superstripe phase. We choose Ω = 0 .4 in (a,b) and 0.8 in\n(c,d). The other parameters are γ= 0.6,g= 0.5,˜V0= 10, and Rc= 3.\nside Rydberg-dressed roton is obviously lower than the\nleft-hand one. Surprisingly, the increase of Rcquickly\nbrings the right-hand roton into the unstable situation,\nwhile the left-hand one softens but still has a large ro-\nton gap [see the line with Rc= 2.5 in Fig. 2(b)]. Very\ninterestingly, the change of Rcdoes not affect the spin-\norbit-coupled roton.\nTherefore, we can have parameter regimes in which\nonly one Rydberg-dressed roton becomes unstable. This\nis fully distinguishable with the case without spin-orbit\ncouplingwherepairedRydberg-dressedrotonssoften and\nare unstable in the same time. The reason of one unsta-\nble roton is that it is the spin-orbit-coupled roton that\nhelps to lower its energy to prepare for the occurring of\ninstability. Thus, the spin-orbit coupling plays a role\nof an assistance for one Rydberg-dressed roton softening\nand instability. It can enhance the roton softening and\naccelerate instability to happen. We emphasize that for\nour parameters the spin-orbit-coupled roton can not be-\ncome unstable and the instability always happens in the\nRydberg-dressed rotons.\nIV. POST-ROTON-INSTABILITY PHASE:\nSUPERSTRIPE\nNow we are in the position to ask: what are prop-\nerties of post-roton-instability phase induced by such a\nnew mechanism of the spin-orbit-coupling-assisted roton\nsoftening. To address this question, we numerically findthe existence of superstripe ground state. Considering\nthe periodic density distributions of the superstripe, we\nassume that the wave functions are periodic and can be\nexpressed in a plane-wave basis [ 39,57],\nψ(x) =L/summationdisplay\nn=−L/parenleftBigg\nψ(n)\n1\nψ(n)\n2/parenrightBigg\neinKx, (10)\nwhereψ(n)\n1andψ(n)\nnsatisfy the renormalization condi-\ntion/summationtext\nn(|ψ(n)\n1|2+|ψ(n)\n2|2) = 1, and Lis the cutoff of\nthe plane-wave modes. This ansatz indicates that the\nperiod of wave functions is relevant to K, whose value\nshall be determined by minimizing corresponding en-\nergy functional obtained by substituting Eq. ( 10) into\nEq. (1). Minimization procedure demonstrates that only\nthe coefficients of odd-numbered plane-wave modes (i.e.,\ne±iKx,e±i3Kx, etc.) are nonzero. Therefore, the period\nofwavefunctions isnumericallyexact2 π/Kandthe den-\nsity period π/K. These results are similar to that in\nspin-orbit-coupled BECs without Rydberg-dressed inter-\nactions [ 39,57]. Two typical density distributions are\nshown in Fig. 3(a,c) for Ω = 0 .4 and 0.8 respectively.\nWe see that Ω has little impact on density periodicity.\nThe density period π/Kis∼4.6, which is consistent\nwith the predicted value from the roton instability. For\nRc= 3 which is used in Fig. 3, we check the roton\ninstability centered around qrot∼1.36, which predicts\nthat the period of post-roton-instability phase should be\n2π/qrot∼4.6. ThedensityperiodforvariousΩisdemon-\nstrated in Fig. 4(a). It confirms that density period is6\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s52/s46/s50/s52/s46/s52/s52/s46/s54/s52/s46/s56/s53/s46/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s48/s49/s50/s51/s32/s83/s116/s114/s105/s112/s101/s32/s112/s101/s114/s105/s111/s100/s40/s97/s41\n/s32/s67/s111/s110/s116/s114/s97/s115/s116/s40/s98/s41\n/s83/s111/s117/s110/s100/s32/s118/s101/s108/s111/s99/s105/s116/s121/s40/s99/s41\nFIG. 4. The period (a) and contrast (b) of the superstripes as a function of the Raman coupling strength Ω. (c) Sound veloci ty\nof the upper (circles) and lower (squares) phonon branches a s a function of the Raman coupling strength Ω. The parameters\nareγ= 0.6,g= 0.5,Rc= 3, and ˜V0= 10.\ninsensitive to Ω.\nWe further characterize the superstripe ground states\nby their collective excitation spectrum. It is calculated\nby substituting the superstripe wave functions into BdG\nequations ( 6) and using the following perturbation am-\nplitudes,\nu1,2(x) =eiqxL/summationdisplay\nn=−Lu(n)\n1,2ei(2n+1)Kx,\nv1,2(x) =eiqxL/summationdisplay\nn=−Lv(n)\n1,2ei(2n+1)Kx.(11)\nThe reason for the choice of above amplitudes is that the\nBdG equations are periodic with the period being π/K,\nas a result of the periodic density of the superstripes.\nTherefore, the amplitudes should be Bloch states with q\nbeing perturbation quasimomentum and we expand their\nperiodic parts by a plane-wave basis. The results ob-\ntained by diagonalizing the BdG equations are shown in\nFig.3(b,d), corresponding to the superstripes plotted in\nFig.3(a,c) respectively. There is a slight difference be-\ntween the two superstripes in Fig. 3(a,c). But the collec-\ntive excitation spectrum are obviously different, which\nfeatures a Bloch band-gap structure. The gap sizes in\nFig.3(d) is clearly larger than these in Fig. 3(b). This\nis because the density of superstripe in Fig. 3(c) have a\nhigher amplitude than that in Fig. 3(a). A high density\nwill open a large gap in collective excitations. We la-\nbel the density amplitude by the contrast of superstripe,\nwhich is defined as\nC=nmax−nmin\nnmax+nmin, (12)\nwithnmaxandnminbeing the maximum and minimum\nof density respectively. The contrast as a function of Ω is\nplotted in Fig. 4(b). It increases with the increase of the\nΩ. Thus, we expect that a larger Raman coupling gives\nrise to larger gap sizes in collective excitations.\nIn the long wavelength regime, the lowest two bands\nof collective excitation spectrum are phonon modes withdifferentsoundvelocities. Theexistenceofthesetwogap-\nless Goldstone modes [ 39,58] is guaranteed by two spon-\ntaneous symmetry breaking of superstripes, one of which\nis continuous translational symmetry breaking and the\nother is gauge symmetry breaking. Sound velocities re-\nlate to slopes ofphonon modes areanalyzed as a function\nof Ω, and the result is shown in Fig. 4(c). The velocity of\nthe upper phonon branch decreases slowly with increas-\ningΩ[seecirclesin Fig 4(c)]. However,the velocityofthe\nlower phonon branch slowly increases and then decreases\nasafunctionofΩ[seesquaresinFig 4(c)]. Soundvelocity\nshows a jump around phase transition from plane-wave\nto superstripe phases. Therefore, the phase transition is\nfirst order.\nFinally, we emphasize that the superstripes stud-\nied above are ground states supported by spin-orbit-\ncoupling-assistedsofteningandinstability. Withoutspin-\norbit coupling, the ground states of relevant parameters\nare density homogeneous with roton excitations.\nV. CONCLUSION\nLong-range Rydberg-dressed interactions and spin-\norbit coupling can separately generate collective exci-\ntations with roton structures. In a system with both\nRydberg-dressed interactions and spin-orbit coupling,\ntwo different kinds of rotons from different originations\ncan coexist. The location and softening of these ro-\ntons are adjustable independently. The interplay of them\nleads to an interesting phenomenon that the spin-orbit-\ncoupled roton can assist Rydberg-dressed roton soften-\ning. The post-roton-instability phase is a superstripe,\nwhich is identified by analyzing their collective excita-\ntions. We conclude that spin-orbit coupling provides a\npossible means to accelerate roton softening.\nACKNOWLEDGEMENT\nWe thank Zhaoxin Liang and Zhu Chen for the useful\nhelp. ThisworkissupportedbytheNSF ofChina(Grant7\n/s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s49/s50/s51/s52\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s49/s50/s51/s52/s53/s110\n/s101/s120/s32/s47/s32/s110/s32 /s40/s37/s41\n/s32\n/s82\n/s99/s40/s97/s41\n/s32/s32/s110\n/s101/s120/s32/s47/s32/s110/s32 /s40/s37/s41/s40/s98/s41\nFIG. 5. Quantum depletion nex/nof the Rydberg-dressed\nBEC with spin-orbit coupling. (a) Quantum depletion as a\nfunction of Rcwith a fixed Ω = 0 .5. (b) Quantum depletion\nas a function of Ω with a fixed Rc= 3. Other parameters are\nγ= 0.6,g= 0.5, Ω = 0 .5,˜V0= 10, and n= 200kRam. A\ncutoffqc= 0.01kRamis introduced to prevent divergence.\nNos. 1174219 and 11974235), the Thousand Young Tal-\nents Program of China, and the Eastern Scholar and\nShuguang (Program No. 17SG39) Program. H. L. ac-\nknowledges the support by China Postdoctoral Science\nFoundation (Grant No. 2019M661457).Appendix A: Quantum depletion\nIn atomic BECs, quantum fluctuation induces a frac-\ntion of the condensate depleted even at zero tempera-\nture. The mean-field theory is valid when the number\nof depleted atoms is much smaller than the total atom\nnumber. Quantum depletion has been calculated in spin-\norbit-coupled BECs [ 57,59–62]. In the following, we\nshow that quantum depletion of our system is small and\nthe mean-field description is reasonable.\nWe start from three-dimensional GP equations and\nconsidertheatomsarestronglyconfinedinthetransverse\ndirection. The transverse wave functions can be assumed\nto have a Gaussian shape. By integrating the transverse\nmotion, we can obtain the quasi-1D GP equations. At\nzero temperature, the density of depleted atoms can be\ncalculated as [ 57]\nnex=/summationdisplay\nj/summationdisplay\nl=1,2/summationdisplay\nq/negationslash=0/integraldisplay\ndx/vextendsingle/vextendsingle/vextendsinglev(j)\nq,l(x)/vextendsingle/vextendsingle/vextendsingle2\n.(A1)\nHere, the superscript jlabels thejth Bogoliubov band.\nTheperturbationamplitude vj\nq,l(x)dependsonthequasi-\nmomentum q, which is given in the main text. Here, the\ntransverse excitations are neglected due to the strong\nconfinement. In Fig. 5(a), we show that by tuning\nthe Rydberg blockade radius, the quantum depletion\nnex/njumps around the phase transition from plane-\nwave (Rc<2.5) to superstripe ( Rc>2.5) phases. n\nis the BEC density, here we use a value n= 200kRam\nthat is typical in usual BEC experiments. In the calcu-\nlation, a cutoff of qmust be introduced to avoid diver-\ngence at very close to q= 0 [63], here the cutoff we used\nisqc= 0.01kRam. For the superstripe phase, the quan-\ntum depletion increase with the increase of the Rabi fre-\nquency, as shown in Fig. 5(b). In our parameter regimes,\nnex/nis less than 5%, which indicates the validity of our\nmean-field theory.\n[1] O. Penrose and L. Onsager, Bose-Einstein condensation\nand liquid Helium, Phys. Rev. 104, 576 (1956).\n[2] E. P. Gross, Unified theory of interacting Bosons,\nPhys. Rev. 106, 161 (1957).\n[3] G. Chester, Speculations on Bose-Einstein condensatio n\nand quantum crystals, Phys. Rev. A 2, 256 (1970).\n[4] A. J. Leggett, Can a solid be “Superfluid”?\nPhys. Rev. Lett. 25, 1543 (1970).\n[5] Y. Pomeau and S. Rica, Dynamics of a model of super-\nsolid, Phys. Rev. Lett. 72, 2426 (1994).\n[6] F. Ancilotto, M. Rossi, and F. Toigo, Supersolid struc-\nture and excitation spectrum of soft-core bosons in three\ndimensions, Phys. Rev. A 88, 033618 (2013).\n[7] D. Petter, G. Natale, R. M. W. van Bijnen, A. Patschei-\nder, M. J. Mark, L. Chomaz, and F. Ferlaino, Probing\nthe roton excitation spectrum of a stable dipolar Bose\ngas, Phys. Rev. Lett. 122, 183401 (2019).[8] L. Santos, G. V. Shlyapnikov, and M. Lewenstein,\nRoton-Maxon spectrum and stability of trapped dipolar\nBose-Einstein Condensates, Phys. Rev. Lett. 90, 250403\n(2003).\n[9] D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, Rotons in\ngaseous Bose-Einstein condensates irradiated by a laser,\nPhys. Rev. Lett. 90, 110402 (2003).\n[10] N. Henkel, R. Nath, and T. Pohl, Three-dimensional\nRoton excitations and supersolid formation in\nRydberg-excited Bose-Einstein Condensates,\nPhys. Rev. Lett. 104, 195302 (2010).\n[11] Y. Kora and M. Boninsegni, Patterned supersolids in\ndipolar Bose systems, J. Low Temp. Phys. 197, 337\n(2019).\n[12] R.Mottl, F.Brennecke, K.Baumann, R.Landig, T. Don-\nner, and T. Esslinger, Roton-type mode softening in a\nquantum gas with cavity-mediated long-range interac-8\ntions, Science 336, 1570 (2012).\n[13] J. L´ eonard, A. Morales, P. Zupancic, T. Esslinger, and\nT. Donner, Supersolid formation in aquantumgas break-\ning a continuous translational symmetry, Nature (Lon-\ndon)543, 87 (2017).\n[14] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni,\nS. Baier, J. H. Becher, M. J. Mark, F. W¨ achtler, L. San-\ntos, and F. Ferlaino, Observation of roton mode popula-\ntion in a dipolar quantumgas, Nat. Phys. 14, 442 (2018).\n[15] F. B¨ ottcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn,\nM. Guo, T. Langen, and T. Pfau, Transient super-\nsolid properties in an array of dipolar quantum droplets,\nPhys. Rev. X 9, 011051 (2019).\n[16] L. Chomaz, D. Petter, P. Ilzh¨ ofer, G. Natale, A. Traut-\nmann, C. Politi, G. Durastante, R. M. W. van Bijnen,\nA. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino,\nLong-lived and transient supersolid behaviors in dipolar\nquantum gases, Phys. Rev. X 9, 021012 (2019).\n[17] L. Tanzi, E. Lucioni, F. Fam´ a, J. Catani, A. Fioretti,\nC. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno,\nObservation of a dipolar quantum gas with metastable\nsupersolid properties, Phys. Rev. Lett. 122, 130405\n(2019).\n[18] J. L´ eonard, A. Morales, P. Zupancic, T. Donner, and\nT. Esslinger, Monitoring and manipulating Higgs and\nGoldstone modes in a supersolid quantum gas, Sci-\nence358, 1415 (2017).\n[19] G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Pet-\nter, M. J. Mark, L. Chomaz, and F. Ferlaino, Excitation\nspectrum of a trapped dipolar supersolid and its experi-\nmental evidence, Phys. Rev. Lett. 123, 050402 (2019).\n[20] L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Fam` a,\nA. Fioretti, C. Gabbanini, G. Modugno, A. Recati, and\nS. Stringari, Supersolid symmetry breaking from com-\npressional oscillations in a dipolar quantum gas, Nature\n(London) 574, 382 (2019).\n[21] M. Guo, F. B¨ ottcher, J. Hertkorn, J.-N. Schmidt,\nM. Wenzel, H. P. B¨ uchler, T. Langen, and T. Pfau, The\nlow-energy Goldstone mode in a trapped dipolar super-\nsolid, Nature (London) 574, 386 (2019).\n[22] J. Hertkorn, F. B¨ ottcher, M. Guo, J.-N. Schmidt,\nT. Langen, H. P. B¨ uchler, and T. Pfau, Fate of\nthe amplitude mode in a trapped dipolar supersolid,\nPhys. Rev. Lett. 123, 193002 (2019).\n[23] M. Saffman, T. G. Walker, and K. M¨ olmer, Quantum\ninformation with Rydberg atoms, Rev. Mod. Phys. 82,\n2313 (2010).\n[24] P. Schauß, J. Zeiher, T. Fukuhara, S. Hild, M. Cheneau,\nT. Macr` ı, T. Pohl, I. Bloch, and C. Gross, Crystallization\nin Ising quantum magnet, Science 347, 1455 (2015).\n[25] O. Firstenberg, C. S. Adams, and S. Hofferberth, Non-\nlinear quantum optics mediated by Rydberginteractions,\nJ. Phys. B: At. Mol. Opt. Phys. 49, 152003 (2016).\n[26] A. Browaeys and T. Lahaye, Many-body physics with\nindividually controlled Rydberg atoms, Nat. Phys. 16,\n132 (2020).\n[27] N. Henkel, F. Cinti, P. Jain, G. Pupillo, and T. Pohl,\nSupersolid vortex crystals in Rydberg-dressed Bose-\nEinstein condensates, Phys. Rev. Lett. 108, 265301\n(2012).\n[28] G. I. Martone, Y. Li, L. P. Pitaevskii, and S. Stringari,\nAnisotropic dynamics of a spin-orbit-coupled Bose-\nEinstein condensate, Phys. Rev. A 86, 063621 (2012).\n[29] W. Zheng, Z.-Q. Yu, X. Cui, and H. Zhai, Properties ofBose gases with the Raman-induced spin-orbit coupling,\nJ. Phys. B: At. Mol. Opt. Phys. 46, 134007 (2013).\n[30] M. A. Khamehchi, Y. Zhang, C. Hamner, T. Busch,\nand P. Engels, Measurement of collective excita-\ntions in a spin-orbit-coupled Bose-Einstein condensate,\nPhys. Rev. A 90, 063624 (2014).\n[31] L.-C.Ha, L.W.Clark, C.V.Parker, B.M.Anderson,and\nC. Chin, Roton-maxon excitation spectrum of Bose con-\ndensates in ashakenoptical lattice, Phys. Rev.Lett. 114,\n055301 (2015).\n[32] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spi n-\norbit-coupled Bose-Einstein condensates, Nature (Lon-\ndon)471, 83 (2011).\n[33] V. Galitski and I. B. Spielman, Spin-orbit coupling in\nquantum gases, Nature (London) 494, 49 (2013).\n[34] N. Goldman, G. Juzeli¯ unas, P. ¨Ohberg, and I. B. Spiel-\nman, Light-induced gauge fields for ultracold atoms,\nRep. Prog. Phys. 77, 126401 (2014).\n[35] H. Zhai, Degenerate quantum gases with spin-orbit cou-\npling: a review, Rep. Prog. Phys. 78, 026001 (2015).\n[36] Y. Zhang, M. E. Mossman, T. Busch, P. Engels, and\nC. Zhang, Properties of spin–orbit-coupled Bose-Einstein\ncondensates, Front. Phys. 11, 118103 (2016).\n[37] S.-C. Ji, L. Zhang, X.-T. Xu, Z. Wu, Y. Deng, S. Chen,\nand J.-W. Pan, Softening of roton and phonon modes\nin a Bose-Einstein condensate with spin-orbit coupling,\nPhys. Rev. Lett. 114, 105301 (2015).\n[38] J. -R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas,\nF. C ¸. Top, A. O. Jamison, and W. Ketterle, A stripe\nphase with supersolid properties in spin-orbit-coupled\nBose-Einstein condensates, Nature (London) 543, 91\n(2017).\n[39] Y. Li, G. I. Martone, L. P. Pitaevskii, and S. Stringari,\nSuperstripes and the excitation spectrum of a spin-orbit-\ncoupled Bose-Einstein condensate, Phys. Rev. Lett. 110,\n235302 (2013).\n[40] Y. Deng, J. Cheng, H. Jing, C.-P. Sun, and S. Yi,\nSpin-orbit-coupled dipolar Bose-Einstein condensates,\nPhys. Rev. Lett. 108, 125301 (2012).\n[41] R. M. Wilson, B. M. Anderson, and C. W. Clark, Meron\nground state of Rashbaspin-orbit-coupled dipolar Boson,\nPhys. Rev. Lett. 111, 185303 (2013).\n[42] S. Gopalakrishnan, I. Martin, and E. A. Demler, Quan-\ntum quasicrystals of spin-orbit-coupled dipolar Bosons,\nPhys. Rev. Lett. 111, 185304 (2013).\n[43] H. L¨ u, S.-B. Zhu, J. Qian, and Y.-Z. Wang, Spin-\norbit coupled Bose-Einstein condensates with Rydberg-\ndressing interactions, Chin. Phys. B 24, 090308 (2015).\n[44] W. Han, X.-F. Zhang, D.-S. Wang, H.-F. Jiang,\nW. Zhang, and S.-G. Zhang, Chiral supersolid in spin-\norbit-coupled Bose gases with soft-core long-range inter-\nactions, Phys. Rev. Lett. 121, 030404 (2018).\n[45] I. B. Spielman, Raman processes and effective gauge po-\ntentials, Phys. Rev. A 79, 063613 (20109).\n[46] J. B. Balewski, A. T. Krupp, A. Gaj, S. Hofferberth,\nR. L¨ ow and T. Pfau, Rydberg dressing: understanding\nof collective many-body effects and implications for ex-\nperiments, New J. Phys. 16, 063012 (2014).\n[47] I. Seydi, S. H. Abedinpour, R. E. Zillich, R. Asgari, and\nB. Tanatar, Rotons and Bose condensation in Rydberg-\ndressed Bose gases, Phys. Rev. A 101, 013628 (2020).\n[48] Y. Zhou, Y. Li, R. Nath, and W. Li, Quench dynamics of\nRydberg-dressed bosons on two-dimensional square lat-9\ntices, Phys. Rev. A 101, 013427 ( 2020).\n[49] Y. Zhang, G. Chen, and C. Zhang, Tunable spin-orbit\ncoupling and quantum phase transition in a trapped\nBose-Einstein condensate, Sci. Rep. 3, 1937 (2013).\n[50] K. Jim´ enez-Garc´ ıa, L. J. LeBlanc, R. A. Williams,\nM. C. Beeler, C. Qu, M. Gong, C. Zhang, and I. B. Spiel-\nman, Tunable spin-orbit coupling via strong driving in\nultracold-atom systems, Phys. Rev. Lett. 114, 125301\n(2015).\n[51] C.-H. Hsueh, Y.-C. Tsai, K.-S. Wu, M.-S. Chang, and\nW. C. Wu, Pseudospin orders in the supersolid phases\nin binary Rydberg-dressed Bose-Einstein condensates,\nPhys. Rev. A 88, 043646 (2013).\n[52] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. String ari,\nTheory of Bose-Einstein condensation in trapped gases,\nRev. Mod. Phys. 71, 463 (1999).\n[53] R. M. Wilson, C. Ticknor, J. L.Bohn, and E. Timmer-\nmans, Phys. Rev. A 86, 033606 (2012).\n[54] M. Abad and A. Recati, A study of coherently\ncoupled two-component Bose-Einstein condensates,\nEur. Phys. J. D 67, 148 (2013).\n[55] R. Wu and Z. Liang, Beliaev damping of a spin-orbit-\ncoupled Bose-Einstein condensate, Phys. Rev. Lett. 121,\n180401 (2018).\n[56] Y. Li, L. P. Pitaevskii, and S. Stringari, Quantum tri-criticality and phase transitions in spin-orbit coupled\nBose-Einstein condensates, Phys. Rev Lett. 108, 225301\n(2012).\n[57] X.-L. Chen, J. Wang, Y. Li, X.-J. Liu, and H. Hu, Quan-\ntum depletion and superfluid density of a supersolid in\nRaman spin-orbit-coupled Bose gases, Phys. Rev. A 98,\n013614 (2018).\n[58] S. Saccani, S. Moroni, and M. Boninsegni, Excitation\nspectrum of a supersolid, Phys. Rev Lett. 108, 175301\n(2012).\n[59] T. Ozawa and G. Baym, Stability of ultracold\natomic Bose condensates with Rashba spin-orbit cou-\npling against quantum and thermal fluctuations,\nPhys. Rev. Lett. 109, 025301 (2012).\n[60] X. Cui and Q. Zhou, Enhancement of condensate de-\npletion due to spin-orbit coupling, Phys. Rev. A 87,\n031604(R) (2013).\n[61] Z. Chen and Z. Liang, Ground-state phase diagram of\na spin-orbit-coupled bosonic superfluid in an optical lat-\ntice, Phys. Rev. A 93, 013601 (2016).\n[62] L. Liang and P. T¨ orm¨ a, Quantum corrections\nto a spin-orbit-coupled Bose-Einstein condensate,\nPhys. Rev. A 100, 023619 (2019).\n[63] C. A. M¨ uller and C. Gaul, Condensate deformation and\nquantum depletion of BoseEinstein condensates in exter-\nnal potentials, New J. Phys. 14, 075025 (2012) ." }, { "title": "1903.12379v2.Spin_Orbital_Hallmarks_of_Unconventional_Superconductors_Without_Inversion_Symmetry.pdf", "content": "arXiv:1903.12379v2 [cond-mat.supr-con] 24 Sep 2019Spin-Orbital Hallmarks of Unconventional Superconductor s\nWithout Inversion Symmetry\nYuri Fukaya,1Shun Tamura,1Keiji Yada,1Yukio Tanaka,1Paola Gentile,2and Mario Cuoco2\n1Department of Applied Physics, Nagoya University, Nagoya 4 64-8603, Japan\n2CNR-SPIN, I-84084 Fisciano (Salerno), Italy, c/o Universi t´ a di Salerno, I-84084 Fisciano (Salerno), Italy\nThe spin-orbital polarization of superconducting excitat ions in momentum space is shown to\nprovide distinctive marks of unconventional pairing in the presence of inversion symmetry breaking.\nTaking the prototypical example of an electronic system wit h atomic spin-orbit and orbital-Rashba\ncouplings, we provide a general description of the spin-orb ital textures and their most striking\nchangeover moving from the normal to the superconducting st ate. We find that the variation\nof the spin-texture is strongly imprinted by the combinatio n of the misalignment of spin-triplet\nd-vector with the inversion asymmetry g-vector coupling and the occurrence of superconducting\nnodal excitations. Remarkably, the multi-orbital charact er of the superconducting state allows to\nunveil a unique type of topological transition for the spin- winding around the nodal points. This\nfinding indicates the fundamental topological relation bet ween chiral and spin-winding in nodal\nsuperconductors. By analogy between spin- and orbital-tri plet pairing we point out how orbital\npolarization patterns can also be employed to assess the cha racter of the superconducting state.\nI. INTRODUCTION\nThe Rashba spin-orbit (SO) coupling1,2is the mani-\nfestation of a fundamental relativistic effect due to struc-\ntural inversion symmetry breaking (ISB) that leads to\nspin-momentum locking with lifting of spin degeneracy\nand remarkable phenomena such as non-standard mag-\nnetic textures3,4, spin Hall5and topological spin Hall6,\nEdelstein effects7, etc.8.\nRecently, it has been realized that spin-momentum\nlocking can also occur from the ISB driven orbital po-\nlarization of electrons in solids which is, then, linked\nwith the spin-sector by the atomic SO coupling. The role\nof spin and orbital polarization in materials has built a\ndifferent view of the manifestation of ISB with respect\nto the conventional spin-Rashba effect, leading to the\nso-called orbital-driven Rashba coupling9. The orbital\nRashba (OR) effect can yield chiral orbital textures and\norbital dependent spin-vector via the SO coupling9–15.\nEvidences of anomalous energy splitting and of a key\nrole played by the orbital degree of freedom have been\ndemonstrated on a large variety of surfaces, i.e. Au(111),\nPb/Ag(111)16, Bi/Ag(111)17, etc. as well as in transition\nmetal oxides based interfaces, i.e., LaAlO 3-SrTiO 318,19.\nIn superconductors without inversion symmetry20,21\nthe presence of non-degenerate spin- and orbital polar-\nized electronic states is generally expected to lead to un-\nconventional pairing, with the occurrence of spin-triplet\norder parameters and singlet-triplet spin mixing22–24,\nnon-standard surface states25,26, as well as topological\nphases27–35.\nExperimental direct probes by using angle- and spin-\norbital resolved photoemission spectroscopy in the nor-\nmal36–39and superconducting (SC) phase40can be ex-\ntremely useful for establishing the nature of the SC state\nandtheunderlyingdegreeofspin-orbitalentanglementor\nthe occurrence of competing orders. A successful photoe-\nmission observation of Dirac-cone with spin-helical sur-face states at Fermi level and their modification below\nthe superconducting critical temperature due to the gap\nopeninghasbeenrecentlydemonstratedintheiron-based\nsuperconductor FeTe 1−xSex41. Along this line it would\nbe highly desirable to have distinctive detectable signa-\nturesassociatedwiththespin-orbitalpolarizationstosin-\ngleoutthe natureofthe SCphase. Symmetryplaysarel-\nevantroleinsuchidentification. Forinstance, skyrmionic\npatterns in the Brillouin zone (BZ) have been suggested\nas marks to make the topological order more accessible\nin ferromagnetic semiconductor/ s-wave superconductor\nheterostructure assuming that both time-reversal (TR)\nand inversion symmetry is broken42. On the other hand,\nthe fundamental interrelation between chiral spin-orbital\ntextures in reciprocal space and unconventional pairing\nsolely due to ISB has not been yet fully established.\nIn this paper we focus on the class of low-dimensional\nsuperconductors with TR and broken inversion symme-\ntry. The aim is to assess how the spin-orbital texture of\nthe SC excitations can unveil the nature of the SC state\nand, eventually, its topological character.\nWe show that the spin-polarization pattern is gener-\nally imprinted by the relative alignment of spin-triplet\nd-vector with the inversion asymmetry g-vector coupling\n(Sect. II). A fundamental issue emergesin nodal topolog-\nical superconductors when considering the occurrence of\nspin-winding around the nodal points. To face this prob-\nlem on a general ground we employ a prototypical elec-\ntronic system with atomic SO and orbital-Rashba cou-\npling,whosespin-orbitaltexturescanmanifestdeviations\nfrom the typical ones due to the spin-Rashba coupling\n(Sect. III) and can exhibit topological SC phases with\norbital-driven pairing (Sect. IV).\nFinally, we find that at the nodal points topological\ntransitions for the spin-winding can occur due to the\nemergence of vanishing spin amplitude lines connecting\nthe nodal points (Sect. IV). This outcome sets the fun-\ndamental interplay between chiral and spin- or orbital\nwinding in nodal superconductors with ISB.2\nII. TOPOLOGICAL SPIN-TEXTURE: SINGLE\nORBITAL MODEL DESCRIPTION\nA. Model Hamiltonian and spin-texture\nWe start by introducing a minimal model that can de-\nscribe the spin-texture of the SC state due to the inter-\nplayofinversionasymmetricSOcouplingandspin-triplet\npairing. Due to ISB the pairing has mixture of spin-\ntriplet and singlet components. Since the spin-singlet\npairing does not affect the spin-texture, the central focus\nis on the consequences of the spin-triplet pair potential.\nInthe superconductingstateweconsiderthe Bogoliubov-\nde Gennes (BdG) Hamiltonian constructed from ˆh(k)\nand including both spin-singlet and triplet pairings as\nfollows\nˆHBdG(k) =/parenleftbigg\n−µˆσ0+ˆh(k)ˆ∆(k)\nˆ∆†(k)µˆσ0−ˆht(−k)/parenrightbigg\n,(1)\nwhereµandˆ∆(k) =iˆσy[|∆S|ψ+|∆T|ˆσ·d(k)] denote the\nchemical potential, and the singlet ( ψ) and triplet order\nparameters ( d), and the corresponding gap amplitudes\n|∆S|and|∆T|, andˆh(k) is the normal state term\nˆh(k) =ε(k)ˆσ0+Λg(k)·ˆσ, (2)\ng(k) = (gx(k),gy(k),gz(k)), (3)\nwithε(k) andg(k) being the kinetic energy and inver-\nsion asymmetry coupling, while Λ denotes the strength\nof the ISB potential, and ˆ σi(i= 0,x,y,z) are the Pauli\nmatrices in spin space. Here, the d-vector has the usual\nmatrix form in terms of the components associated with\nthe spin-triplet configurations as ∆ ↑,↑−∆↓,↓=−2dx(k),\n∆↑,↑+∆↓,↓= 2idy(k), and ∆ ↑,↓+∆↓,↑= 2dz(k).\nWe determine the spin polarization components by\nevaluating the expectation values of the related spin op-\nerators. In the normal state, we assume that g-vector\nlies onxy-plane and gz(k) = 0. Then, the eigenvalues\nand the correspondingeigenstates ofthe Hamiltonian are\ngiven by\nE±=ε(k)±Λ/radicalig\ng2x(k)+g2y(k), (4)\n|+/angbracketright=/parenleftbigg\ncosθ\n2\neiφsinθ\n2/parenrightbigg\n,|−/angbracketright=/parenleftbigg\n−e−iφsinθ\n2\ncosθ\n2/parenrightbigg\n,\nwithθ=π/2, cosφ=gx(k)//radicalig\ng2x(k)+g2y(k), and\nsinφ=gy(k)//radicalig\ng2x(k)+g2y(k). It is immediate to ver-\nify that the expectation values of the spin operators are\ngiven by\n/angbracketleft±|ˆSx|±/angbracketright=±gx(k)/radicalig\ng2x(k)+g2y(k), (5)\n/angbracketleft±|ˆSy|±/angbracketright=±gy(k)/radicalig\ng2x(k)+g2y(k), (6)\n/angbracketleft±|ˆSz|±/angbracketright= 0, (7)whereˆSi=x,y,zare the spin operators expressed in terms\nof the Pauli matrices. Thus, the z-component of the\nspin operator is zero (except that at the high symmetry\npoints) and the in-plane xandy-components are gener-\nally non-vanishing.\nThe planar structure of the spin polarization is a gen-\neral consequence of the symmetry property of the model\nHamiltonian. If the transformation ˆSz→ −ˆSzis a sym-\nmetry for the quantum system upon examination, then,\ndue to the absence of degeneracy at any ( kx,ky) different\nfrom the time reversal invariant momenta, the expecta-\ntion value of the z-component of the spin operator is\nidentically zero. Thus, one can focus the analysis only\non the spin orientation in the xy-plane.\nFor convenience and clarity of computation, starting\nfrom the BdG Hamiltonian, one can introduce the elec-\ntron component of the spin polarization within the xy-\nplane for the m-th excited state of the superconducting\nspectrum by means of the following relation\nθSCm\nS= arg[/angbracketleftΨm|˜Se\nx|Ψm/angbracketright+i/angbracketleftΨm|˜Se\ny|Ψm/angbracketright],(8)\nwhere|Ψm/angbracketrightis them-th eigenstate of the spectrum of\nthe BdG Hamiltonian and ˜Se\ni=x,y,zare the spin operators\nprojected onto the electron space:\n˜Se\ni=1\n2[1+ ˆτ3]⊗ˆSi, (9)\nˆτ3=/parenleftbigg\n1 0\n0−1/parenrightbigg\n, (10)\nwith the Pauli matrix ˆ τ3in Nambu space.\nBefore considering the full diagonalization of the BdG\nexcited states, it is much instructive to consider an effec-\ntive perturbation approach which allows to extract the\nmain issues of the general behavior of the spin polar-\nization of the superconducting excited state. Hence, we\nconsider the BdG Hamiltonian by taking the first order\nperturbation in the pairing term,\nˆH=ˆH0+ˆH′, (11)\nˆH|Ψn/angbracketright=En|Ψn/angbracketright, (12)\nˆH0|Ψ(0)\nn/angbracketright=εn|Ψ(0)\nn/angbracketright. (13)\nHere,ˆH,ˆH0, andˆH′correspond to the total, the un-\nperturbed, and the perturbing Hamiltonian, respectively.\nEnand|Ψn/angbracketright(εnand|Ψ(0)\nn/angbracketright) are the eigenvalue and the\ncorrespondingeigenstate of the total Hamiltonian ˆH(the\nunperturbed Hamiltonian ˆH0). Here, Fig. 1 indicates the\nrelation between the eigenstates |Ψn/angbracketrightand BdG bands.\nWe assume for convenience of computation that the g-\nvector is parallel to the z-axis (g(k) = (0,0,gz(k))) and\nconsider only the spin-triplet pairing ( ψ= 0). Then, the\nunperturbed and perturbed terms of the Hamiltonian at3\na givenkare written by\nˆH0=−µˆσ0⊗ˆτ3+/parenleftbiggˆh(k) 0\n0−ˆht(−k)/parenrightbigg\n,(14)\nˆh(k) =/parenleftbigg\nε(k)+Λgz(k) 0\n0ε(k)−Λgz(k)/parenrightbigg\n,(15)\nˆH′=/parenleftbigg\n0ˆ∆(k)\nˆ∆†(k) 0/parenrightbigg\n, (16)\nˆ∆(k) =/parenleftbigg\n∆↑,↑(k) ∆↑,↓(k)\n∆↓,↑(k) ∆↓,↓(k)/parenrightbigg\n. (17)\nFor the spin-triplet pairing the d-vector can be further\nexpressed in terms of the polar angles ( θd,φd) that iden-\ntify its direction in the spin space (see Fig. 2(a)) as\nd(k) = (dx(k),dy(k),dz(k))\n=ˆn(k)|d(k)|\n= (sinθdcosφd,sinθdsinφd,cosθd)|d(k)|.(18)\nThe eigenstate |Ψ(0)\na/angbracketright(|Ψ(0)\nc/angbracketright) corresponds to |e,↑/angbracketright(|h,↑\n/angbracketright), and|Ψ(0)\nb/angbracketright(|Ψ(0)\nd/angbracketright) is related to |e,↓/angbracketright(|h,↓/angbracketright) where e\nand h are electron and hole, respectively. The eigenval-\nuesand the correspondingeigenstatesofthe unperturbed\nHamiltonian ˆH0are given by\nεa(k) =−εd(k) =ε(k)+Λgz(k), (19)\nεb(k) =−εc(k) =ε(k)−Λgz(k), (20)\n|Ψ(0)\na/angbracketright=/parenleftbiggˆα+\nˆ0/parenrightbigg\n,|Ψ(0)\nb/angbracketright=/parenleftbiggˆα−\nˆ0/parenrightbigg\n,\n|Ψ(0)\nc/angbracketright=/parenleftbiggˆ0\nˆβ+/parenrightbigg\n,|Ψ(0)\nd/angbracketright=/parenleftbiggˆ0\nˆβ−/parenrightbigg\n.\nHere, ˆα±andˆβ±denote the eigenstates of ˆh(k) and\n−ˆht(−k),\nˆα+=ˆβ+=/parenleftbigg\n1\n0/parenrightbigg\n,ˆα−=ˆβ−=/parenleftbigg\n0\n1/parenrightbigg\n.\nFor the electron-like branch, the perturbation within the\nfirst order is zero. It means that the spin polarization for\nthe electron-like branch is not modified within the first\norder perturbation in |∆T|, with|∆T|being the ampli-\ntude of the spin-triplet order parameter. On the other\nhand, since the eigenvalues and the corresponding eigen-\nstates for the hole-like branch change within the first or-\ndercorrection,thespinorientationoftheexcitedstatefor\nthe hole-like branch acquiresa non-trivial pattern. Thus,\nwe focus on the hole-like branch of the excited state to\ninvestigate the spin-texture and we extract the electron\ncomponent of the spin-polarization for the first excited\nstate of the spectrum.\nAt this stage, by the benefit of the analytical expres-\nsion of the first order eigenstates, we can calculate the\nspin-texture for the hole-likebranchawayfrom the Fermi\nlevel,thatis, |−µ+ε(k)| ≫ |Λgz(k)|. Fromtheperformed\nFIG. 1. Schematic illustration of the relation between the\neigenstates |Ψn/angbracketrightandBdGbandsinthecase withnodal points.\nRed (green) line is the first (second) excited band and white\ncircle is thenodal point. Thesame eigenstates are alsoplot ted\nin Fig. 2 (b) with the following correspondence: |ψ+/angbracketright=|Ψa/angbracketright\nand|ψ−/angbracketright=|Ψd/angbracketright.\nanalysis, we can approximate the expectation values in\nthe Appendix A as\n/angbracketleftΨc|˜Se\nx|Ψc/angbracketright ∼ −ascosφdsin2θd,\n/angbracketleftΨc|˜Se\ny|Ψc/angbracketright ∼ −assinφdsin2θd,\n/angbracketleftΨc|˜Se\nz|Ψc/angbracketright ∼ −ascos2θd, (21)\n/angbracketleftΨd|˜Se\nx|Ψd/angbracketright ∼ascosφdsin2θd,\n/angbracketleftΨd|˜Se\ny|Ψd/angbracketright ∼assinφdsin2θd,\n/angbracketleftΨd|˜Se\nz|Ψd/angbracketright ∼ascos2θd, (22)\nwhereasis an amplitude depending on the energy dis-\ntance of the excited state from the Fermi level and the\nstrength of the superconducting pairing,\nas=|∆T|2|d(k)|2\n8[−µ+ε(k)]2. (23)\nHence, one can evaluate the characterof the spin-texture\nfrom these expectation values of the spin operators.\nThen, we focus on the electron- and hole-like branch for\n|Ψa/angbracketrightand|Ψd/angbracketrightas shown in Fig 1.\nOn the basis of the above observations, if d- andg-\nvectorsaremisalignedbyanangle θd[Fig. 2(a)], then the\nelectronspin orientationcorrespondingto the excitations\nclose to the Fermi level ( kF) will manifest a distinctive\npattern.\nThis result is confirmed by full numerical determina-\ntion of the BdG excited states and of the corresponding\nspin polarization [Fig. 2(b)]. Indeed, one can show that\nfork≥kF(electron-like branch |Ψa/angbracketright) the spin orienta-\ntion is collinear to the g-vector while it gets rotated by\nan angle 2θdfork 0is the elemen-\ntary positive charge. The effect of SOC is included into the\nHamiltonian as Hsoc=\u0015soc^L\u0001^S, where\u0015socis the spin-orbit\nstrength, ^Sis the operator of spin, and\n^L=P\nR^LR;\n^LR=P\nnmj\u001enRih\u001enRj(r\u0000R)\u0002pj\u001emRih\u001emRj(3)\nis the representation of the atomic contribution of the orbital\nangular momentum operator [41]. Here, randpdenote the\ncanonical position and momentum operators, respectively. Al-\nthough the general conclusions that we draw from the tight-\nbinding model do not depend on the exact choice of the hop-\nping parameters, they can be easily tuned such that the tight-\nbinding band structure closely resembles the first principles\none of the BiAg2monolayer (Fig. 1).\nOrbital Rashba effect. Neglecting for the moment the effect\nof SOC and assuming jHp(k)\u0000Hs(k)j\u001djh(k)jas is the\ncase for BiAg2in the long-wavelength limit of k!0, we can\nperturbatively downfold the h(k)term to arrive at an effective\nHamiltonian (see Supplementary Information):\nHeff(k) =\u0012\nHp;eff(k) 0\n0Hs;eff(k)\u0013\n; (4)\nwhereHp;eff(k) =Hp(k) +HOR(k). The expression\nHOR(k) =\u000bOR\n\u0016h^L\u0001(^z\u0002k) (5)is known as the orbital Rashba Hamiltonian since it resem-\nbles the conventional spin Rashba Hamiltonian with the or-\nbital angular momentum operator replacing that of spin. Re-\nmarkably, the combined effect of sphybridization and surface\npotential gradient is concisely described by the orbital Rashba\nHamiltonian within the subspace spanned by porbitals. Thus,\nthe orbital Rashba physics arises not from the electron’s spin\nbut from the orbital degrees of freedom even without SOC.\nIn analogy to the Rashba constant of the conventional spin\nRashba model, the parameter\n\u000bOR=\u0011a\rspVz(k)\n\u0001Esp(k)\f\f\f\f\nk=0(6)\nis called the orbital Rashba constant , with \u0001Esp(k)denoting\nthe energy gap between s- andp-derived bands, and \u0011\u00181be-\ning a parameter dependent on the lattice structure. Using the\nspecific tight-binding model parameters for BiAg2(see Sup-\nplementary Information), we estimate the orbital Rashba con-\nstant to be about 1eV\u0001˚A. From Eq. (6) it is clear that the ORE\nroots in the sporbital hybridization, and that the strength of\nthe orbital Rashba effect is directly proportional to the value\nof the surface potential gradient associated with the buckling\nof Bi atoms in BiAg2. This implies that the desired properties\nof the ORE can be designed by controlling the band hybridiza-\ntion via chemical and structural engineering.\nCrystal field splitting. In contrast to the conventional Rashba\neffect, the ORE is very sensitive to the crystal field split-\nting (CFS), which quenches the OM. In the absence of the\nCFS, that is, when \u0001CFS=Epx(y)\u0000Epz= 0withEpx(y);Epz\nas corresponding energy eigenvalues of px(y)- andpz-derived\nstates at k= 0, the expectation value of the OM is given by\nmACA\nl(k) =\u0000\u0016B\n\u0016hX\nRh plkj^LRj plki=\u0016Bl^k\u0002^z;\n(7)3\nwithin the so-called atom-centered approximation for the\nOM, which takes into account only intra-atomic contributions.\nHere, ^zand^kare unit vectors along the z-axis and and vector\nk, respectively, plkis thepl-derived eigenstate of Hp;eff(k),\nandl=f\u00001;0;+1gis the angular momentum quantum num-\nber with respect to the quantization axis ^z\u0002^k. In the vicinity\nof the Fermi energy, the p-derived bands can thus be denoted\nin terms of their dominant orbital character as p\u00001,p0, and\np+1in the order of increasing energy (Fig. 1 c-d). The corre-\nsponding k-dependent eigenenergies are E\u00001(k),E0(k), and\nE+1(k). The associated k-linear orbital-dependent energy\nsplitting arising due to the orbital Rashba term in Hp;eff(k)\namounts to \u0001EOR=l\u000bORjkjin the long-wavelength limit.\nHowever, in the presence of the CFS, although its direction\nremains intact, the expectation value of the OM is reduced by\na factor of 2\u000bORjkj=\u0001CFS, when assumingj\u0001CFSj\u001dj\u000bORkj.\nFor this reason, orbital-dependent energy splitting appears\nfrom the second order in kif\u0001CFS6= 0, which can be seen\nfrom Eq. (5).\nRelation to electric polarization. Consider an orbital-\ncoherent state plk, that is, an eigenstate of Hhop(k)with\n\u0001CFS= 0 which exhibits a quantized value of OM (e.g., due\nto an applied in-plane magnetic field or the ORE):\nmACA\nl(k) = \u0016hl^nk; (8)\nwherel=f\u00001;0;+1gis the orbital angular momentum\nquantum number with respect to the direction of some in-\nplane unit vector ^nk. While in previous studies the electric\npolarization within the ORE was introduced phenomenologi-\ncally [17, 19], one can show from perturbation theory argu-\nments that such an orbital-coherent state naturally exhibits\nelectric polarization perpendicular to the surface plane:\nPz\nl(k) =\u0000eh pmkjzj pmki=\u001fOR\n\u0016BmACA\nl(k)\u0001(k\u0002^z);\n(9)\nwhere the parameter\n\u001fOR=\u0011ae\rsph'pzkjzj'ski\n\u0001Esp(k)\f\f\f\f\nk=0(10)\nis what we call the orbital Rashba susceptibility . The structure\nof\u001fORreflects that the electric polarization arises from the sp\nhybridization (Fig. 2 a). From Eq. (6) it is clear that \u000bOR=\nEz\u001fOR, and that the orbital Rashba Hamiltonian as given by\nEq. (5) can be understood as a dipole coupling between an\nelectric field at the surface with the operator of electric polar-\nization,HOR=\u0000Ez^Pz(k), where ^Pz(k) =\u0000(\u001fOR=\u0016h)k\u0002^L\n(Fig. 2 b). Physically, it means that within the ORE the non-\ntrivial orbital texture of states in k-space arises so as to gain\nmaximal energy by the interaction of the states’ polarization\nwith the surface electric field. Thus, the ORE can be seen as a\nconsequence of orbital magnetoelectric coupling at surfaces.\nAtom-centered approximation from first principles. In\norder to verify our predictions in a realistic system, we\nevaluate the k- and band-resolved value of the OM in\na\nb\nelectron\nek\nEzˆzˆP\nsphybridization\nkˆP\nˆLˆLFIG. 2. Orbital Rashba effect in sp-alloys. (a) After preparing the\nsystem in a state with non-vanishing orbital angular momentum ^L,\nfor example, by applying an in-plane magnetic field, the hybridiza-\ntion between sandporbitals generates an electric polarization ^P\naccording to Eq. (9). (b)The electric dipole coupling of this elec-\ntric polarization to the surface potential gradient Ez^z[Eq. (5)] leads\nto the formation of an orbital texture in reciprocal space, which is\nknown as orbital Rashba effect.\nBiAg2from first principles, which is given by mACA\nn(k) =\n\u0000(\u0016B=\u0016h)P\n\u0016h nkjr\u0016\u0002pj nki\u0016in ACA. Here, nkis an\neigenstate, r\u0016is the position operator relative to atom \u0016, the\nsummation is performed over all atoms in the lattice, and the\nreal-space integration is restricted locally to atom-centered\nmuffin-tin spheres . Computed in such a way OM is a direct\ngeneralization to the first principles framework of the OM\ncomputed in ACA from tight-binding. In Fig. 3 a-c, we show\nthe distribution of the OM in k-space for the p-derived bands\np\u00001,p0, andp+1in absence of SOC, and observe clock-\nwise (\u00001), zero, and counter-clockwise ( +1) type of chiral\nbehavior of this distribution around the \u0000-point, respectively.\nThe magnitude of the OM without SOC along different high-\nsymmetry lines is shown in Fig. 3 d. While the OM reaches\nas much as 0:27\u0016Bfor thep+1band in the middle of the Bril-\nlouin zone, we can use its behaviour with kin the vicinity of\nthe\u0000-point to estimate the magnitude of the orbital Rashba\nconstant\u000bOR. First, we note that a small but finite value of\nthe slope in the OM of p0as well as visible differences in the\nbehavior of the OM of p\u00001andp+1bands can be observed, in\ncontrast to our prediction Eq. (8). The reason for the discrep-\nancy lies in a non-zero crystal field splitting, which we can\nestimate from Fig. 1 dto be \u0001CFS\u00190:76eV. Taking this into\naccount, the orbital Rashba constant of p\u00001andp+1bands\namounts to 0:96eV\u0001˚A and 1:38eV\u0001˚A, respectively. Similarly,\nwe also find orbital chiralities near K-point. Investigating fur-\nther the influence of SOC, we found no qualitative changes in\nthe distribution of the OM in the Brillouin zone. However, we4\nda b c\np-1p+1 p0\n00.050.10.150.20.25\n|k|( Å−1)0.050.100.150.200.250.30\np-1\np0p+1ΓMΓK|m(k)| (µ B) ( ≡1.0 )µB\nFIG. 3. Orbital texture of p-derived bands in BiAg2without spin-\norbit coupling. (a) -(c)Arrows indicate the first principles values for\nthek-resolved in-plane orbital moment (OM) of the p-derived bands\nwithin the atom-centered approximation. The hexagonal Brillouin\nzone of BiAg2is indicated by thin black lines. In the vicinity of the\n\u0000-point, the chirality of the OM texture is +1,0, and\u00001forp\u00001,\np0, andp+1, respectively. (d)The magnitude of the in-plane OM\njmn(k)jof thep-derived bands along the high-symmetry lines \u0000M\nand\u0000K.\nclearly see that the resulting “Rashba” spin texture in recip-\nrocal space, emerging upon including SOC, is strictly bound\nto the local direction of the OM. Details on these results are\nprovided in the Supplementary Information. Finally, we dis-\nplay in Figs. 4 fand 4 gthe chirality of the OM at the Fermi\nsurface both with and without SOC taken into consideration,\nrespectively.\nBerry phase theory in films. The primary manifestation of\nthe ORE in solids is the k- and band-dependent generation\nof finite OM. For understanding this fundamental effect and\npredicting its magnitude, it is crucial to evaluate the magni-\ntude of the OM properly without assuming any approxima-\ntions such as the ACA. Recently, it was shown that a rigorous\ntreatment of OM in solids within the complete Berry phase de-\nscription [30–33] naturally accounts for non-local effects [35].\nThereby, theoretical estimations of OM in magnetic materials\nof various nature have significantly improved [24, 25, 36].\nSince the ORE manifests in the generation of in-plane OM,\nfollowing the procedure of Ref. [33], we extended the previ-\nous formulation to the case of in-plane components of OM of\na film finite along the z-axis. The expression for the in-planecomponents of the OM is given by\nmmod\nn(k) =\ne\n\u0016hRe[hunkj(z^z)\u0002fH(k) +En(k)\u00002EFgj@kunki];\n(11)\nwhereunkis an eigenstate of the lattice-periodic Hamil-\ntonianH(k)with the eigenvalue En(k), andEFis\nthe Fermi energy. Equation (11) can be further de-\ncomposed into the so-called local circulation term\nmLC\nn(k) = (e=\u0016h)Re[hunkj(z^z)\u0002fH(k)\u0000EFgj@kunki],\nand the itinerant circulation term mIC\nn(k) =\n(e=\u0016h)fH(k)\u0000EFgRe[hunkj(z^z)\u0002j@kunki]. The\nlatter expression is connected to the projected Berry curvature\n\nn\nx(y)(k) =Re\u0002\nh@kx(y)unkjzjunki\u0003\n; (12)\nwhich is closely related to the well-known Berry curvature in\nbulk systems by formally replacing zwithi@kz.\nInserting the model orbital Rashba Hamiltonian, Eq. (5),\ndirectly into Eq. (11), we find that\nmmod\nl(k) =\u0000\u001fOR(Epx(y)+Epz\u00002EF)\n2\u0016B\u0016h\u0001mACA\nl(k)(13)\nin the long-wavelength limit, where lis the angular-\nmomentum quantum number defined in Eq. (7). Assuming\nthe typical valuesEz\u00180:1 eV=˚Aand(Epx(y)+Epz\u00002EF)\u0018\n1 eV , we find not only that ACA and modern OM exhibit the\nsame chirality of the distribution, but also that mmod\nl(k)\u0018\nmACA\nl(k)in the vicinity of the \u0000-point within the model anal-\nysis.\nBerry phase theory from first principles. To investigate\nwhether significant differences between the ACA and mod-\nern treatment of OM arise in a realistic situation, we evaluate\nfrom first principles mACA\nn(k)andmmod\nn(k). In Fig. 4, we\nshow the OM distribution evaluated from the modern theory\n(Figs. 4 a, 4c) and ACA (Figs. 4 b, 4d), where the individual\ncontributions of all occupied bands were summed up for each\nkpoint. In this figure, the distribution of the OM within the\nmodern approach and ACA is similar around the \u0000-point both\nin magnitude and overall distribution, in accordance with our\nmodel considerations. Near the K-point, however, the distri-\nbution of the modern OM without SOC deviates significantly\nfrom the ACA result. If SOC is taken into account, the dif-\nference between the two approaches is even more drastic as\nit amounts to one order of magnitude. In Fig. 4, the visible\ndiscontinuity of the OM distribution occurs along the Fermi\nsurface lines (Figs. 4 e-4h). Directly at the Fermi surface, the\nitinerant contribution to the OM in the modern theory van-\nishes and we may restrict ourselves to visualizing in Figs. 4 e\nand 4 gthe local circulation from the modern theory, without\nand with SOC, respectively. In contrast, within ACA there\nis no such decomposition of the Fermi-surface contribution\n(Figs. 4 f, 4h). The OM of Fermi-surface states plays a crucial\nrole in various orbital magnetoelectric phenomena [26, 27].\nWithin the Berry phase theory, one of the most remarkable\nfeatures of the OM in bulk is its correlation with the Berry cur-\nvature in k-space, which often exhibits a spiky behavior in the5\nFIG. 4. In-plane orbital moment (OM) in BiAg2from first principles. (a) -(d)Summing up the individual contributions of all occupied\nbands below the Fermi energy, we obtain the distribution of the in-plane OM m(k)ink-space. (e)-(h)The in-plane OM directly at the\nFermi surface, which is important to orbital magnetoelectric response. In all cases, colors represent the magnitude jm(k)j, arrows indicate\nthe in-plane direction of the OM with the arrow size proportional to jm(k)j, and thin lines mark the Brillouin zone. The results of both Berry\nphase theory (modern) and atom-centered approximation (ACA) are shown with and without taking into account spin-orbit coupling (SOC).\nThe in-plane OM within the two approaches deviates drastically around the K-point, and the modern theory predicts overall a richer structure.\nvicinity of band crossings [34]. In our formalism for the in-\nplane OM in thin films as expressed by Eq. (11), the projected\nBerry curvature, Eq. (12), is a key ingredient. It behaves sim-\nilarly to the conventional Berry curvature in that it can exhibit\nsingular behavior at band crossings as a consequence of the\nrapid variation of the wave functions with k. Following this\nspirit, we also seek for such a spiky behavior in the local circu-\nlation mLC(k)by studying the OM in the vicinity of the band\ncrossing in the electronic structure of BiAg2which is close\nto the M-point and about 0.2 eV below the Fermi level. Set-\ntingEFin Eq. (11) to the energy of the crossing and treating\nall bands below this energy as occupied, we observe a singu-\nlar behavior of the OM magnitude within the modern theory\nin the vicinity of the crossing point (Fig. 5 a). This behavior\ncan be directly correlated with sizable OM contributions of\nthose states which constitute the band crossing. At the point of\nsingularity, the modern-theory OM is purely due to the local\ncirculation and it reaches as much as 1:01\u0016Bin magnitude,\nwhile the ACA predicts a tiny value of 0:03\u0016B. Remarkably,\nboth ACA and modern theory agree qualitatively in their pre-\ndiction of the behavior of the OM away from the band crossing\n(see Supplementary Information).\nTwo-band model near the band crossing. To study the\naforementioned behavior of the OM near the band crossing\nfrom the model point-of-view, we consider the following two-band Hamiltonian:\nH(k) =d(k)\u0001\u001b; (14)\nwhere \u001bis the vector of Pauli matrices reflecting the or-\nbital degree of freedom of two chosen basis states: j's;ki,\nandj'p;ki= (1=p\n2)j'px;ki\u0000(i=p\n2)j'pz;ki. We assume\nthatdx(k) = 0 ,dy(k) =\rspkxa+Vz, anddz(k) =\u000e,\nwhereais the lattice constant, \u000eis the energy gap of the\nmassive Dirac-like dispersion along kx,\rspis the nearest-\nneighbor hopping amplitude between sandpxorbitals, and\nVz=\u0000(eEz=p\n2)h's;kjzj'pz;kiexpresses the breaking of\ninversion symmetry. The band structure of this Hamiltonian\nis shown in Fig. 5 a, where we assumed a= 5 ˚A,Vz= 0:1eV,\n\rsp= 1eV, and\u000e= 5meV. We represent the position oper-\natorzin the specified basis as\nz=\u0012\nh's;kjzj's;ki h's;kjzj'p;ki\nh'p;kjzj's;ki h'p;kjzj'p;ki\u0013\n=p(k)\u0001\u001b; (15)\nwherepx=pz= 0andpy=\u0000(1=p\n2)h's;kjzj'pz;ki. Tak-\ning into account the representation (15) and applying Eq. (11),\nwe obtain an expression for the in-plane OM from the modern\ntheory for the general two-band Hamiltonian (14):\nmmod\n\u0006(k) =\u0006e\n\u0016hEFh\n(^z\u0002@k)^d(k)i\n\u0001p(k); (16)\n\u0019\u0000EF\n\u0016h(^z\u0002@k)P\u0006\nz(k) (17)6\na b\n22.822.923.0\nM↔Γ(%)0.00.51.01.5ACA (total)\nmodern (total)\nmodern\n−0.2−0.10.0\nkx(Å−1)0.01.02.03.0ACA (total)\nmodern (total)\nmodern (local)\n22.822.923.0\nM↔Γ(%)-205-204-203-202-201En(k)−EF(meV )\n−0.2−0.10.0\nkx(Å−1)−100−50050100En(k) (meV)|m(k)| (µ B)\n|m(k)| (µ B)(lower)\nFIG. 5. In-plane orbital moment (OM) near band crossings in BiAg2. (a) First principles results for the total in-plane OM near the band\ncrossing, which is shown as an inset. All states below the energy indicated by the black horizontal line were considered as occupied in the\nband summation of the OM. While the ACA value is insensitive to the presence of band crossings, the Berry phase theory predicts a gigantic\nenhancement of the local circulation of the OM in the vicinity of such points. In particular, the band-resolved contributions (shown here for the\nlower band) display singular behavior due to the rapid variation of the wave function near the crossing. (b)Similar conclusions can be drawn\nfrom tight-binding model calculations assuming a Dirac-like dispersion (inset). Whereas the ACA prediction is strictly bound by \u0016Bwithin\nthe band gap, pronounced values of the in-plane OM jm(k)jcan be reached in the modern theory. Likewise, the corresponding Fermi-surface\ncontribution is strongly enhanced in the gap region.\nwhere ^d(k)is the direction of d(k)and the “ +” (“\u0000”) sign\nstands for upper (lower) band. In the second line, owing to the\nfact that in our model (^z\u0002@k)p(k)\u00190, we related the mod-\nern OM to the derivative of P\u0006\nz(k) =\u0006(\u0000e)[^d(k)\u0001p(k)],\nthat is, thez-component of the electric polarization of the up-\nper (lower) band as defined in Eq. (9).\nFrom this generic expression we clearly observe that the\norigin of the non-vanishing OM within the modern theory lies\nin the gradient of the electric polarization in reciprocal space.\nUsing the parameters stated above and setting EF= 3 meV\nandpy= 2 ˚A, we compute the OM of all occupied states in\nthe vicinity of the band crossing from both modern theory, Eq.\n(16), and ACA, Eq. (3). By replacing further EFwithjd(k)j\nin Eq. (16), we obtain the Fermi-surface contribution due to\nthe self-rotation of the wave packet [30], which determines\nthe orbital magnetoelectric response [26, 27]. Based on the\nresults shown in Fig. 5 b, we conclude that while the values\nofmACA\n\u0006(k)are strictly bound by \u0016Beverywhere as can be\nconfirmed analytically, the pronounced singular behavior of\nmmod\n\u0006(k)with large values within the gap can be attributed\nsolely to the rapid variation of the electric polarization in the\nvicinity of the crossing.\nDiscussion\nWe have shown that sporbital hybridization is the main mech-\nanism for the ORE at surfaces of sp-alloys, which is manifest\nalready without SOC. Just like the spin Rashba effect follows\nfrom the ORE via SOC, we can expect other orbital-dependent\nphenomena to be formulated and discovered, from which SOC\nrecovers their spin analogues. One remarkable example is or-\nbital analogue of the quantum anomalous Hall effect, where\nthe quantized edge state is orbital-polarized [42]. Another ex-ample is the recent formulation of the orbital version of the\nDzyaloshinskii-Moriya interaction governing the formation of\nchiral structures such as domain walls and skyrmions [43].\nOur simulations reveal the complexity of the orbital tex-\ntures driven by ORE, and their sensitivity to the electronic\nstructure of realistic materials. This means that the desired\nproperties of the ORE, and phenomena it gives rise to, can be\ndesigned by proper electronic-structure engineering. More-\nover, by making use of both spin and orbital degrees of free-\ndom, one can generate and manipulate arising spin and orbital\ntextures in k-space. While for the case of BiAg2considered\nhere the spin aligns collinearly to the OM in the presence of\nSOC, a Rashba effect with respect to the total angular momen-\ntum emerges in each of the j= 3=2andj= 1=2branches\nin the regime where SOC is dominant over the ORE [17, 19].\nBased on a similar idea, an orbital version of the Chern insu-\nlator state in a situation of very large SOC has been recently\nproposed [44].\nWe predict that within the ORE, in contrast to the spin\nRashba effect, the magnitude of the OM in the vicinity of\nband crossings can reach gigantic values. This observation\nhas very far-reaching consequences for the magnitude of ef-\nfects which are directly associated with the OM at the Fermi\nsurface. This particularly concerns the orbital magnetoelectric\neffect, as recently discussed in the context of orbital Edelstein\neffect [27] and gyrotropic magnetic effect [26]. Within the\norbital Edelstein effect, a finite OM at the Fermi surface is\ngenerated by an asymmetric change in the distribution func-\ntion created by an electric field. In the gyrotropic magnetic\neffect, discussed intensively these days with respect to topo-\nlogical metals [45, 46], an external magnetic field gives rise\nto an electrical current [26]. Both phenomena rely crucially7\non the magnitude of the local orbital moments at the Fermi\nsurface of materials, and we predict here that they can be\ndrastically enhanced by tuning the electronic structure such\nthat the singularities in the OM, which we disclose in our\nwork, are positioned at the Fermi level. The generally en-\nhanced predicted magnitude of the Berry-phase OM of the\noccupied states, as compared to the OM computed from the\ncommonly used approximation, can also manifest in the re-\nevaluation of the magnitude of other effects such as the orbital\nHall effect [22, 23].\nAn additional flavor to the ORE is the intrinsic valley de-\ngree of freedom inherent to systems of the type studied here:\nbecause of time-reversal symmetry the points of singularity in\nthe OM always come in pairs. Since the OM is opposite for\nopposite valleys, the overall OM integrated over the Brillouin\nzone is zero in the ground state, and orbital magnetoelectric\nresponse becomes strongly valley-dependent. Exploiting the\nORE for the purpose of generating sizable ground state net\nOM at surfaces has to be done in combination with generat-\ning a non-vanishing exchange field and magnetization in the\nsystem. This can give rise to a plethora of effects relying on\nintertwined spin and orbital degrees of freedom in complex\nmagnetic materials.\nMethods\nFirst principles calculation. We performed self-consistent\ndensity-functional theory calculations of the electronic struc-\nture of BiAg2using the film mode of the FLEUR code [47],\nwhich implements the FLAPW method [48, 49]. Exchange\nand correlation effects were treated within the generalized gra-\ndient approximation [50]. We assumed a (p\n3\u0002p\n3)R30\u000e\nunit cell (see Fig. 2 a) with the in-plane lattice constant\na= 9:47a0, wherea0is Bohr’s radius. The surface re-\nlaxation of Bi was set to d= 1:61a0, and the muffin-tinradii of Bi and Ag were chosen as 2:80a0and2:59a0, re-\nspectively. We used Kmax= 4:0a\u00001\n0as plane-wave cutoff\nand sampled the irreducible Brillouin zone using 110points.\nSpin-orbit coupling was included self-consistently within the\nsecond-variation scheme [51].\nBased on the converged charge density, maximally-\nlocalized Wannier functions (MLWFs) were obtained in a\npost-processing step employing an equidistant 16\u000216k-\nmesh. Starting from sp2andpztrial orbitals on Bi as well as\ns,p, anddtrial functions on Ag, we constructed 44 MLWFs\nout of 120 energy bands using the WANNIER90 program [52].\nThe frozen window was set 2:78eV above the Fermi en-\nergy. Subsequently, we calculated the OM in (i) the ACA, and\n(ii) the Berry phase theory according to the scheme proposed\nby Lopez et al. [25].\nAcknowledgment\nD.G. acknowledges the financial support from the\nGlobal Ph.D. Fellowship Program funded by NRF\n(2014H1A2A1019219). H.W.L is supported by the Na-\ntional Research Foundation of Korea (NRF) grant (No.\n2011-0030046). We thank Juba Bouaziz, Manuel dos Santos\nDias, Philipp R ¨ußmann, and Samir Lounis for stimulating\ndiscussions.\nAuthor contributions\nD.G. uncovered the mechanism of the orbital Rashba effect by\ntight-binding model consideration. D.G., J.-P.H., and P.M.B.\nperformed first principles calculations. D.G., J.-P.H., and\nY .M. wrote the manuscript. All authors discussed the data\nand contributed to the paper.\nCompeting financial interests\nThe authors declare no competing financial interests.\n[1] E. I. Rashba, Cyclotron and combinational resonance in a mag-\nnetic field perpendicular to the plane of the loop , Sov. Phys.\nSolid State 2, 1109 (1960).\n[2] J. Nitta, T Akazaki, H. Takayanagi, T. Enoki, Gate control\nof spin-orbit interaction in an inverted InGaAs/InAlAs het-\nerostructure , Phys. Rev. Lett. 78, 1335 (1997).\n[3] S. LaShell, B. A. McDougall, and E. Jensen, Spin Splitting of\nan Au(111) Surface State Band Observed with Angle Resolved\nPhotoelectron Spectroscopy , Phys. Rev. Lett. 77, 3419 (1996).\n[4] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov R. A. Duine,\nNew perspectives for Rashba spinorbit coupling , Nat. Mat. 14,\n871 (2015).\n[5] I. E. Dzyaloshinskii, Thermodynamic theory of weak ferromag-\nnetism in antiferromagnetic substances . Sov. Phys. JETP 5,\n1259 (1957).\n[6] T. Moriya, Anisotropic superexchange interaction and weak fer-\nromagnetism , Phys. Rev. 120, 91 (1960).\n[7] S. Lounis, A. Bringer, and S. Bl ¨ugel, Magnetic Adatom Induced\nSkyrmion-Like Spin Texture in Surface Electron Waves , Phys.\nRev. Lett. 108, 207202 (2012).\n[8] K.-W. Kim, H.-W. Lee, K.-J. Lee, and M. D. Stiles, Chirality\nfrom Interfacial Spin-Orbit Coupling Effects in Magnetic Bilay-ers, Phys. Rev. Lett. 111, 216601 (2013).\n[9] T. Kikuchi, T. Koretsune, R. Arita, and G. Tatara,\nDzyaloshinskii-Moriya Interaction as a Consequence of a\nDoppler Shift due to Spin-Orbit-Induced Intrinsic Spin Current ,\nPhys. Rev. Lett. 116, 247201 (2016).\n[10] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and\nA. H. MacDonald, Universal Intrinsic Spin Hall Effect , Phys.\nRev. Lett. 92, 126603 (2004).\n[11] V . M. Edelstein, Spin polarization of conduction electrons in-\nduced by electric current in two-dimensional asymmetric elec-\ntron systems , Solid State Commun., 73, 233 (1990).\n[12] F. Freimuth, S. Bl ¨ugel, and Y . Mokrousov, Direct and inverse\nspin-orbit torques , Phys. Rev. B 92, 064415 (2015).\n[13] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo,\nK. Li, Y . Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y . Feng, S. Ji, X.\nChen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y . Wang,\nL. Lu, X.-C. Ma, and Q.-K. Xue, Experimental Observation of\nthe Quantum Anomalous Hall Effect in a Magnetic Topological\nInsulator , Science 340, 167 (2013).\n[14] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V .\nCostache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and\nP. Gambadella, Perpendicular switching of a single ferromag-8\nnetic layer induced by in-plane current injection , Nature 476,\n189 (2011).\n[15] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y . Mokrousov,\nS. Bl ¨ugel, S. Auffret, O. Boulle, G. Gaudin, P. Gambardella,\nSymmetry and magnitude of spinorbit torques in ferromagnetic\nheterostructures , Nat. Nanotechnol. 8, 587 (2013).\n[16] H. Kurebayashi, J. Sinova, D. Fang, A. C. Irvine, T. D. Skinner,\nJ. Wunderlich, V . Nov ´ak, R. P. Campion, B. L. Gallagher, E.\nK. Vehstedt, L. P. Z ˆarbo, K. V ´yborn ´y, A. J. Ferguson, and T.\nJungwirth, An antidamping spinorbit torque originating from\nthe Berry curvature , Nat. Nanotechnol. 9, 211 (2014).\n[17] S. R. Park, C. H. Kim, J. Yu, J. H. Han, and C. Kim,\nOrbital-Angular-Momentum Based Origin of Rashba-Type Sur-\nface Band Splitting , Phys. Rev. Lett. 107, 156803 (2011).\n[18] J.-H. Park, C. H. Kim, H.-W. Lee, and J. H. Han, Orbital chiral-\nity and Rashba interaction in magnetic bands , Phys. Rev. B(R)\n87, 041301 (2013).\n[19] J. Hong, J.-W. Rhim, C. Kim, S. R. Park, and J.-H. Shim, Quan-\ntitative analysis on electric dipole energy in Rashba band split-\nting, Sci. Rep. 5, 13448 (2015).\n[20] B. Kim, C. H. Kim, P. Kim, W. Jung, Y . Kim, Y . Koh, M. Arita,\nK. Shimada, H. Namatame, M. Taniguchi, J. Yu, and C. Kim,\nSpin and orbital angular momentum structure of Cu(111) and\nAu(111) surface states , Phys. Rev. B 85, 195402 (2012).\n[21] S. R. Park, J. Han, C. Kim, Y . Y . Koh, C. Kim, H. Lee, H. J.\nChoi, J. H. Han, K. D. Lee, N. J. Hur, M. Arita, K. Shimada,\nH. Namatame, and M. Taniguchi, Chiral Orbital-Angular Mo-\nmentum in the Surface States of Bi2Se3, Phys. Rev. Lett. 108,\n046805 (2012).\n[22] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Orbitronics:\nThe Intrinsic Orbital Current in p-Doped Silicon , Phys. Rev.\nLett. 95, 066601 (2005).\n[23] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. In-\noue, Giant Orbital Hall Effect in Transition Metals: Origin of\nLarge Spin and Anomalous Hall Effects , Phys. Rev. Lett. 102,\n016601 (2009).\n[24] D. Ceresoli, U. Gerstmann, A. P. Seitsonen, and F. Mauri, First-\nprinciples theory of orbital magnetization , Phys. Rev. B 81,\n060409(R) (2010).\n[25] M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza,\nWannier-based calculation of the orbital magnetization in crys-\ntals, Phys. Rev. B 85, 014435 (2012).\n[26] S. Zhong, J. E. Moore, and I. Souza, Gyrotropic Magnetic Effect\nand the Magnetic Moment on the Fermi Surface , Phys. Rev.\nLett. 116, 077201 (2016).\n[27] T. Yoda, T. Yokoyama, and S. Murakami, Current-induced Or-\nbital and Spin Magnetizations in Crystals with Helical Struc-\nture, Sci. Rep. 5, 12024, (2015).\n[28] L. El-Kareh, G. Bihlmayer, A. Buchter, H. Bentmann, S.\nBl¨ugel, F. Reinert, and M. Bode, A combined experimental\nand theoretical study of Rashba-split surface states on the\n(p\n3\u0002p\n3)Pb/Ag (111)R 30\u000esurface alloy , New J. of Phys.\n16045017 (2014).\n[29] S. Schirone, E. E. Krasovskii, G. Bihlmayer, R. Piquerel,\nP. Gambardella, and A. Mugarza, Spin-Flip and Element-\nSensitive Electron Scattering in the BiAg 2Surface Alloy , Phys.\nRev. Lett. 114, 166801 (2015).\n[30] D. Xiao, J. Shi, and Q. Niu, Berry Phase Correction to Electron\nDensity of States in Solids , Phys. Rev. Lett. 95, 137204 (2005).\n[31] T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Or-\nbital Magnetization in Periodic Insulators , Phys. Rev. Lett. 95,\n137205 (2005).\n[32] D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Or-\nbital magnetization in crystalline solids: Multi-band insulators,Chern insulators, and metals , Phys. Rev. B 74, 024408 (2006).\n[33] J. Shi, G. Vignale, D. Xiao, and Q. Niu, Quantum Theory of Or-\nbital Magnetization and Its Generalization to Interacting Sys-\ntems, Phys. Rev. Lett. 99, 197202 (2007).\n[34] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on elec-\ntronic properties , Rev. Mod. Phys. 82, 1959 (2010).\n[35] S. A. Nikolaev and I. V . Solovjev, Orbital magnetization of\ninsulating perovskite transition-metal oxides with a net ferro-\nmagnetic moment in the ground state , Phys. Rev. B 89, 064428\n(2014).\n[36] J.-P. Hanke, F. Freimuth, A. K. Nandy, H. Zhang, S. Blgel, Y .\nMokrousov, Role of Berry phase theory for describing orbital\nmagnetism: From magnetic heterostructures to topological or-\nbital ferromagnets , Phys. Rev. B 94121114(R) (2016).\n[37] L. Petersen and P. Hedeg ˚ard, A simple tight-binding model of\nspin-orbit splitting of sp-derived surface states , Surf. Sci. 459,\n49 (2000).\n[38] C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D.\nPacil, P. Bruno, K. Kern, and M. Grioni, Giant Spin Splitting\nthrough Surface Alloying , Phys. Rev. Lett. 98, 186807 (2007).\n[39] G. Bian, X. Wang, T. Miller, and T.-C. Chiang, Origin of giant\nRashba spin splitting in Bi/Ag surface alloys , Phys. Rev. B 88,\n085427 (2013).\n[40]http://xcrysden.org/ .\n[41] G. Bihlmayer, Yu. M. Koroteev, P. M. Echenique, E. V .\nChulkov, and S. Bl ¨ugel, The Rashba-effect at metallic surfaces ,\nSurf. Sci. 600, 3888 (2006).\n[42] C. Wu, Orbital Analogue of the Quantum Anomalous Hall Ef-\nfect in p-Band Systems , Phys. Rev. Lett. 101, 186807 (2008).\n[43] P. Kim and J. H. Han, Orbital Dzyaloshinskii-Moriya exchange\ninteraction , Phys. Rev. B 87, 205119 (2013).\n[44] H. Zhang, F. Freimuth, G. Bihlmayer, M. Le ˘zai´c , S. Bl ¨ugel, and\nY . Mokrousov, Engineering quantum anomalous Hall phases\nwith orbital and spin degrees of freedom , Phys. Rev. B 87,\n205132 (2013).\n[45] S.-Y . Xu, I Belopolski, N. Alidoust, M. Neupane, G. Bian, C.\nZhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang,\nH. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou,\nP. P. Shibayev, H. Lin, S. Jia, M. Z. Hasan, Discovery of a Weyl\nfermion semimetal and topological Fermi arcs , Science 349,\n613 (2015).\n[46] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P.\nRichard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X.\nDai, T. Qian, and H. Ding, Experimental Discovery of Weyl\nSemimetal TaAs , Phys. Rev. X 5, 031013 (2015).\n[47]http://www.flapw.de .\n[48] E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Full-\npotential self-consistent linearized-augmented-plane-wave\nmethod for calculating the electronic structure of molecules\nand surfaces: O2molecule , Phys. Rev. B 24, 864 (1981).\n[49] H. Krakauer, M. Posternak, and A. J. Freeman, Linearized aug-\nmented plane-wave method for the electronic band structure of\nthin films , Phys. Rev. B 19, 1706 (1979).\n[50] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient\nApproximation Made Simple , Phys. Rev. Lett. 77, 3865 (1996).\n[51] C. Li, A. J. Freeman, H. J. F. Jansen, and C. L. Fu, Mag-\nnetic anisotropy in low-dimensional ferromagnetic systems:\nFe monolayers on Ag(001), Au(001), and Pd(001) substrates ,\nPhys. Rev. B 42, 5433 (1990).\n[52] A. A. Mostofi, J. R. Yates, G. Pizzi, Y . S. Lee, I. Souza, D. Van-\nderbilt, N. Marzari, An updated version of wannier90: A tool\nfor obtaining maximally-localised Wannier functions , Comput.\nPhys. Commun. 185, 2309 (2014)." }, { "title": "1409.8090v1.Absence_of_a_transport_signature_of_spin_orbit_coupling_in_graphene_with_indium_adatoms.pdf", "content": "arXiv:1409.8090v1 [cond-mat.mes-hall] 29 Sep 2014Absence of a transport signature of spin-orbit coupling in g raphene with indium\nadatoms\nZhenzhao Jia, Baoming Yan, Jingjing Niu, Qi Han, Rui Zhu, Xiaosong W u,∗and Dapeng Yu\nState Key Laboratory for Artificial Microstructure and Meso scopic Physics, Peking University, Beijing 100871, China\nCollaborative Innovation Center of Quantum Matter, Beijin g 100871, China\nEnhancement of the spin-orbit coupling in graphene may lead to various topological phenomena\nand also find applications in spintronics. Adatom absorptio n has been proposed as an effective\nway to achieve the goal. In particular, great hope has been he ld for indium in strengthening\nthe spin-orbit coupling and realizing the quantum spin Hall effect. To search for evidence of the\nspin-orbit coupling in graphene absorbed with indium adato ms, we carry out extensive transport\nmeasurements, i.e., weak localization magnetoresistance, quantum Hall effect and non-local spin\nHall effect. No signature of the spin-orbit coupling is found . Possible explanations are discussed.\nPACS numbers: 72.25.Rb 72.80.Vp 73.43.-f 81.05.ue\nThe intrinsic spin orbit coupling (SOC) in graphene is\nextremely weak[1–3]. Enhancement of the coupling may\ngive rise to a variety of topological phenomena, such as\nthe quantum spin Hall effect (two dimensional topolog-\nical insulators)[4–9], quantum anomalous Hall effect[10–\n15] and Chern half metals[16]. These phenomena are\namong the hottest topics in condensed matter physics.\nMoreover, graphene endowed with strong SOC can have\npotential use in spintronics, as SOC provides a means\nto control the spin electrically, which is at the heart of\nspintronics.\nAbsorptionofadatomshasbeentheoreticallyproposed\nas an effective way to enhance SOC in graphene[6–24].\nBy distorting the carbon sp2bond[17, 19], breaking the\ninversion symmetry[6, 10, 19], or mediating the hopping\nbetween the second-nearest-neighbours[4, 6], intrinsic or\nRashba SOC can be enhanced or induced. The intrinsic\nSOC is required for the predicted quantum spin Hall ef-\nfect, whereasRashbaSOCdestroysit[4]. It hasbeen pro-\nposed that if the outer shell electrons of adatoms derive\nfromporbitals, the induced intrinsic SOC always dom-\ninates over the induced Rashba interaction. Under this\ncondition, it is possible to realize two dimensional(2D)\ntopological insulators in graphene[6]. The most promis-\ning candidates are indium and thallium, which can open\nup a significanttopologicallynontrivialgap. Furtherthe-\noretical work has confirmed that the two systems are in-\ndeed stable topological insulators[8, 9].\nTwo experimental groups have reported angle-resolved\nphotoemission studies on the spin-orbit splitting in a\nrelated system, graphene on metal substrates[25–28].\nGraphene on gold displays a very strong Rashba effect.\nOn the other hand, it has been found that the spin re-\nlaxation rate measured by non-local spin valves is not\nenhanced by gold adatoms[29], suggesting SOC is neg-\nligible. Recently, a strong SOC has been observed in\nhydrogenated graphene and chemical vapor deposited\ngraphenebythe spin Halleffect (SHE)[30, 31]. Neverthe-\nless, in sharp contrast to numerous theoretical work on\nthis topic, relevantexperimental results, especiallytrans-port experiments, are scarce. This is in part due to two\nissues. One is related to the low diffusion barrier for\nmetal adatoms[32], which causes clustering of adatoms\nat room temperature. The other is oxidation of adatoms.\nIn thiswork, weemployanultralowtemperaturemag-\nnetotransport measurement system, with in situthermal\ndeposition capability, to circumvent the two aforemen-\ntioned issues. We choose indium, as it is reckoned by a\nfew theoretical work as an ideal candidate[6, 8, 9]. Weak\nlocalization (WL), quantum Hall effect (QHE) and non-\nlocal SHE measurements have been carried out for dif-\nferent indium coverages with the aim of searching for\nevidence of SOC. Comparison with relevant theories has\nbeen made and no signature ofSOC has been found. The\nimplications have been discussed.\nGraphene flakes were exfoliated from Kish graphite\nonto 285 nm SiO 2/Si substrates. Standard e-beam\nlithography and metallization processes were used to\nmake Hall bar structures. Electrodes are made of 5 nm\nPd/ 80 nm Au. Samples were annealed in Ar/H 2atmo-\nsphere at 260◦C for 2 hours to remove photoresist and\nother chemical residues and then transferred into our di-\nlution refrigerator. The systemis amodified Oxforddilu-\ntion refrigerator, in which in situthermal deposition can\nbe performed[33]. Before the first deposition, current an-\nnealing was done to clean the surface. During deposition\nand measurements, the sample temperature was main-\ntained below 5 K. Thus, the thermal diffusion of adatoms\nwas strongly suppressed. Electrical measurements were\ndone by a standard low frequency lock-in technique.\nThe sample geometry can be seen in the scanning elec-\ntron microscopy image in Fig. 1(a). The half integer\nQHE is well developed, which confirms that the sample\nis monolayer graphene. The low field magnetoresistivity\nis plotted in Fig. 1(b). The narrow negative magnetore-\nsistivity peak at B= 0 is WL, while the noise-like but\nreproducible fluctuations are universal conductance fluc-\ntuations. In graphene, electrons are chiral and have a\nBerryphase π, whichinvertstheconstructiveinterference\nto a destructive one. Thus, intrinsic graphene should dis-2\nPSfrag replacements\nn(1012cm−2)l(nm)∆σ(e2/h)\nB(mT)\nρxx(Ω)\nB(mT)ρxx(kΩ)\nVg(V)\nlφ(µm)σxy(e2/h)\n(d) (c)(b) (a)\n0 0.5 1 1 .5 2 2 .5 05 10 15 20 25−100 −50 0 50 100 −50−30−10 10 30\n0306090120150180\n00.20.40.60.811.2\n186187188189190191\n051015202530\n00.511.52−14−10−6−2261014\nFIG. 1. Magnetotransport of a graphene Hall bar device\nbefore indium deposition. (a) Longitudinal resistivity an d\nthe transverse conductivity versus the gate voltage in 9 T\nat 150 mK. The inset is a scanning electron microscopy pic-\nture of the device. (b) Low field magnetoresistivity exhibit s\ntwo features, the weak localization peak at B= 0 and the\nuniversal conductance fluctuations. (c) Fits to Eq. (1) for\nthe low field magnetoresistivity at different carrier densit ies,\nn= 2.15,1.43,0.77,0.50×1012cm−2. (d) The mean free path\nland the phase coherence length lφas a function of ns.\nplay weak anti-localization (WAL). But, in the presence\nof intervalley scattering, resulting from short range po-\ntential, there will be a crossover from suppressed WL to\nWAL as the field increases[34–36]. When the intervalley\nscattering rate exceeds the phase coherence rate, the low\nfield WL correction to the conductivity can be expressed\nas[37]:\n∆σ=−e2\n4π2/planckover2pi1/bracketleftbigg\nF/parenleftbiggB\nBφ/parenrightbigg\n−F/parenleftbiggB\nBφ+2Basy/parenrightbigg\n−2F/parenleftbiggB\nBφ+Basy+Bsym/parenrightbigg/bracketrightbigg\n,\nF(z) = lnz+ψ/parenleftbigg1\n2+1\nz/parenrightbigg\n,Bφ,asy,sym=/planckover2pi1\n4Deτφ,asy,sym.\n(1)\nwhereψis the digamma function, ethe elementary\ncharge and /planckover2pi1the reduced Planck constant. Dis the dif-\nfusion constant and τφis the phase coherence time. τasy,\nτsymare thez→ −zasymmetric and symmetric spin-\norbitscatteringtime, respectively. Forpristine graphene,\nSOCis negligible. Toestablishthe baselineforlatercom-\nparison, we have measured WL at different gate voltages\n(carrier densities), shown in Fig. 1(c). Data are fitted\nto Eq. (1) with only one parameter τφ. To meet the low\nfieldrequirementofEq.(1)andalsoavoidtheinfluenceof\nthe universal conductance fluctuations, only the low field\npositive magnetoconductance are fitted. A good agree-ment with the theory is found. The mean free path lis\ncalculated from the resistivity and carrier density. Con-\nsidering the charge puddles in graphene, the carrier den-\nsity at the Dirac point is taken as 0 .5×1012cm−2[38].l\nandthephasecoherencelength lφareplottedinFig.1(d).\nlφdecreases as one approaches the Dirac point. This is\nbecauseτφingrapheneisdeterminedbyelectron-electron\ninteraction, which is enhanced when screening is weak-\nened. The suppression of τφis further enhanced by the\nreduction of the mean free time τ[39].\n PSfrag replacements\n−∆VD(V)4th depos3rd depos2nd depos1st deposPristine\nl(nm)σ(e2/h)\nVg(V)\nµ(m2V−1s−1)(b) (a)\n0 20 40 60 80 −80−60−40−20 0 200100200300400\n01020304050607080\n00.511.5\nFIG. 2. Deposition of indium. (a) The conductivity σversus\ngate voltage Vgcurves for the device after each deposition.\nThe solid lines are linear fits, from which the field effect mo-\nbility is obtained. (b) The dependence of the mean free path\nland mobility µon the shift of the Dirac point ∆ VD.\nIndium deposition wasperformed in situat a veryslow\nrate for severaltimes, each lasting 22 - 300 seconds. Dur-\ning deposition, the sample resistance was monitored so\nthat a desired shift of the Dirac point could be obtained.\nAfter each deposition, electrical measurements were car-\nried out. The conductivity σasa function ofgate voltage\nis plotted in Fig. 2(a). The Dirac point gradually shifts\ntonegativegatevoltageasthe indium coverageincreases,\nindicating electron doping. At the same time, the con-\nductivity turns from sublinear to linear in Vg. The lin-\neardependence is attributed to chargedimpurity scatter-\ning being dominant[38]. So, the transition to the linear\ndependence suggests that indium adatoms mainly intro-\nduce charged impurities (long range potential), rather\nthan short range potential. The minimum conductiv-\nityσminat the Dirac point remains relatively constant,\nabout 6e2/h, while a closer look shows a slight decrease\nwith increasing adatom density. A similar dependence\nofσminhas also been observed in potassium absorbed\ngraphene[40]. According to a self-consistent theory pro-\nposed by Adams et al.[38],σminis a consequence of two\ncompeting effects ofchargedimpurities. One is to scatter\nelectrons. The other is to generate a residue carrier den-\nsity at the Dirac point by doping. The result is a weak\nnegative dependence of σminon the impurity density. At\nthe same, the width of the σminplateau increases, which\nis observedin our experiment. So, all features in the den-\nsity dependence of the conductivity are consistent with\ncharged impurity scattering. Its implication on SOC will\nbe discussed later.3\nWe now estimate the area density of indium adatoms\nnIn. Assume that each indium adatom transfers Zelec-\ntrons to graphene. If adatoms are dilute, Zshould be a\nconstant[38]. Then, the doped carrier density n=ZnIn.\nncanbe estimatedfromthe shift ofthe Diracpoint ∆ VD,\nasn=cg∆VD/e. Herecgis the gate capacitance for 285\nnm SiO 2dielectric. The only uncertainty is the value of\nZ. According to first-principles calculations, Zfor in-\ndium on graphene is 0 .8∼1[6, 32, 41]. To get an idea of\nthe coverage, we adopt Z= 1 to obtain its lower bound.\nConsequently, the area density after the third deposition\nis 3.1×1012cm−2, corresponding to a coverageof 0.25%.\nFromthe gatedependence ofthe conductivity, the field\neffect mobility µis obtained. Its dependence on ∆ VD,\nwhich is proportional to nIn, is plotted in Fig. 2(b),\nas well the mean free path lat a carrier density of\n3.8×1012cm−2. As the mobility is substantially re-\nduced after deposition, it is evident that adatom scat-\ntering dominates. We now look for signature of SOC\ninduced by adatoms. WAL has been employed as a sen-\nsitive probe for SOC[42, 43]. In conventional 2D electron\ngases with absence of SOC, the magnetoconductance is\npositive, the Hallmark of WL, stemming from construc-\ntive interference of electrons along time reversal paths.\nWhen SOC is turned on, it rotates the electron spin and\nproduces destructive interference, giving rise to WAL, a\nnegative magnetoconductance. W(A)L can be seen as\na time-of-flight experiment. Specifically, interactions of\na longer time scale manifest themselves in a lower mag-\nnetic field[44]. So, as SOC increases, WAL first emerges\nfrom zero field and eventually dictates the whole field\nregime. The conductance correction is given by the HLN\nequation[42]:\nσ(B) =−gsgve2\n2π2/planckover2pi1/bracketleftbigg\nψ/parenleftbigg1\n2+B1\nB/parenrightbigg\n−ψ/parenleftbigg1\n2+B2\nB/parenrightbigg\n+1\n2ψ/parenleftbigg1\n2+B3\nB/parenrightbigg\n−1\n2ψ/parenleftbigg1\n2+B4\nB/parenrightbigg/bracketrightbigg\n.(2)\nwhere\nB1=B0+Bso+Bs\nB2=4\n3Bso+2\n3Bs+Bφ\nB3= 2Bs+Bφ\nB4=2\n3Bs+4\n3Bso+Bφ\nHereB0=/planckover2pi1/4Deτ,Bso,s,φ=/planckover2pi1/4Deτso,s,φ.τso,τs, rep-\nresent spin-orbit scattering time and magnetic scattering\ntime, respectively. Sincetherearenomagneticimpurities\nin our system, we neglect magnetic scattering.\nIn graphene, the expected evolution of magnetocon-\nductance with increasing SOC is qualitatively similar.\nThe reason is that, although intrinsic graphene display\nWAL, opposite to conventional 2D electron gases, typicalgraphenefilmsshowWLduetopresenceofdefects. From\nthe theory in Ref. [37], the magnetoresistance is given by\nEq. (1)[37]. Compared with conventional 2D electron\ngases, the effect of SOC on WL depends on symmetry.\nForz→ −zasymmetricSOC, normalcrossoverfromWL\nto WAL occurs, while for z→ −zsymmetric SOC, WL\nwill be suppressed. For adatom absorbed graphene, if\nany induced SOC, the z→ −zasymmetric component\nshould be substantial[37]. It is anticipated that the mag-\nnetoconductance goes from negative to positive as the\nmagnetic field increases. Therefore, both conventional\n2D electron gases and graphene are predicted to show\nsimilar non-monotonic magnetoconductance. This is the\nfeature that we are particularly interested in.\n PSfrag replacements\n27 V7 V-3 V-13 V-23 V∆σ(e2/h)\nB(mT)∆σ(e2/h)\nB(mT)(b) (a)\n0 5 10 15 20 −20−10 0 10 2000.050.10.150.20.250.30.350.4\n−0.100.10.20.30.4\nFIG. 3. Low field magnetoconductivity after the third de-\nposition. (a) Fits of the low field magnetoconductivity to\nEq. (1) and Eq. (2). The circles are experimental data. The\nsolid lines are the best fits to the equations, red for Eq. (1)\nand blue for Eq. (2). The dotted lines are the plot of two\nequations, assuming a spin-orbit scattering time τso=τφ.\n(b) Magnetoconductivity data and fits to Eq. (1) at different\ngate voltages relative to the Dirac point ( Vg−VD).\nFig.3(a) showsthelowfield magnetoconductanceafter\nthe third deposition. The magnetoconductance mono-\ntonically increases with field, except for universal con-\nductance fluctuations. No trace of WAL near B= 0 has\nbeen found. Fitting of the data to Eq. (1) yields τφ= 8.6\nps, whileτasyandτsymare an order of magnitude larger\nthanτφwith significant standard deviations, which es-\nsentially suggests inappreciable SOC. We have also per-\nformed fitting to Eq. (2). The obtained τφis similar,\n≈10.9 ps. Again, τsois much larger than τφ, consis-\ntent with Eq. (1). To illustrate the expected influence\nof SOC, both equations are plotted with τφobtained by\nfitting and all spin-orbit scattering time being equal to\nτφ. The resultant non-monotonic magnetoconductance\nis distinct from the experiment data. In fact, extensive\nmeasurementsofthe magnetoconductanceatvariouscar-\nrier densities and after each deposition have been carried\nout and none of them shows WAL around B= 0 (See\nsupplementarymaterials). Asacomparison,wehavealso\nperformed the same experiments on deposition ofmagne-\nsium, which is too light to induce appreciable SOC(See\nsupplementary materials). Qualitatively similar results\nhave been obtained, which confirms absence of induced4\nSOC by In.\nSinceτφcan be seen as a cut-off time for the quan-\ntum interference, it is reasonable to estimate that τso\nis longer than τφat least. For Elliott-Yafet spin-orbit\nscattering, which is most likely the case for adatoms,\nτso= (EF/∆so)2τ. The upper-bound of the spin-orbit\ncoupling strength ∆ sois then estimated as 12 meV at a\ncarrier density of 1 .67×1012cm−2. We emphasize that\nthis is the local SOC strength at an adatom, but not the\noverall spin-orbit gap of 7 meV at a 6% coverage calcu-\nlated in Ref. [6]. Because the gap approximately linearly\ndiminishes with the coverage, the upper bound obtained\nin our experiment is actually much smaller than the pre-\ndiction.\n PSfrag replacements\n3rd depospristineρxx(Ω)\nVg−VD(V)ρxx(kΩ)\nVg(V)\nσxy(e2/h)(b) (a)\n−20−10 0 10 20 −60−50−40−30−20−10 0−0.8−0.6−0.4−0.200.20.40.6\n012345\n−10−6−22610\nFIG. 4. Quantum Hall effect and SHE measurements.\n(a)Quantum Hall effect at B= 14 T after the second de-\nposition. (b) Non-local SHE.\nHaving not been able to observe SOC through WL, we\nturn to the quantum Hall effect. The famous half integer\nHall effect in graphene stems from the Berry phase of\nπ. In the presence of Rashba SOC, the electron spin is\nlocked to its momentum[45, 46]. This spin texture adds\nanotherπto the Berry phase[46], as in topological insu-\nlators. The additional phase is expected to modify the\nquantum Hall effect. A theory proposes that the spin\ndegeneracy in all Landau levels will be lifted[45]. Since\nthere are two zero modes, one from the n= 0 level and\nthe other from n= 1, the first quantum Hall plateaus\nstay at±2e2/h(a spacing of 4 e2/h), while the rest are\nspaced by 2 e2/h. We have measured the quantum Hall\neffect after the second deposition, shown in Fig. 4(a).\nDespite a reduced mobility, the quantum Hall effect is\nevident and n= 0,±1 quantum Hall plateaus remain at\ntheir original positions. The predicted lift of the degen-\neracy forn=±1 levels is not observed.\nAnother effect that may arise because of SOC is SHE.\nIn a spin-orbit coupled system, a charge current gener-\nates a spin transport in the transverse direction, called\nSHE,andviceversa,calledreverseSHE.Thecooperation\nof two effects leads to a non-local resistance[47], Rnl=\n1\n2/parenleftBig\nβs\nσ/parenrightBig2W\nσlse−L/ls, whereβsis the spin Hall conductivity,\nlsis the spin diffusion length, LandWare the length\nand width of the sample, respectively. This effect can\nbe used to detect SOC. A large SOC in hydrogenatedgraphene and chemical vapor deposited graphene has\nbeen experimentally confirmed by this method[30, 31].\nHere, we have measured the non-local resistance by in-\njecting current through one pair of Hall probes of the\nHall bar while monitoring the voltage signal across the\nother pair of Hall probes. The non-local resistance as a\nfunction of the gate voltage before and after deposition\nis plotted in Fig. 4(b). There is 0.6 Ohm non-local re-\nsistance before deposition. This resistance is caused by\nOhmic contribution which decays as e−πL/W. After de-\nposition, no substantial change has been observed, indi-\ncationofnoappreciableinducedSOC.Taking0.6Ohmas\nthe upper-bound of the non-local resistance due to SHE\nandβs/σ≈0.45 at a carrier density of 1 ×1012cm2/Vs\nfrom Ref. [30] for hydrogenated graphene, we estimate\nτso= 37 ps, i.e.∆so= 1.3 meV. This is an order of mag-\nnitude smaller than the 12 meV upper-bound estimated\nby WL. It should be pointed out that the estimation here\nis crude in that βs/σis apparently a function of the SOC\nand unlikely the same as hydrogenated graphene.\nWhereas the theories have listed indium as an impor-\ntant candidate for enhancing SOC in graphene and real-\nizinga2DTopologicalinsulator, wefail tofind anysigna-\nture of SOC by transport measurements. It is notewor-\nthy that the potential of adatoms has been theoretically\ntreated as a short range one, as SOC is induced by me-\ndiating the hopping between the first, second and third\nnearest neighbours[24]. However, the carrier density de-\npendence of the conductivity doesn’t support consider-\nable increase of short range scattering. Similar observa-\ntions have been made for magnesium (see supplementary\nmaterials) and potassium[40]. The absence of induced\nSOC may be associated with the lack of short range scat-\ntering. The reason for the potential being short range\ncan be accounted for by the Coulomb potential of ionized\nadatoms, which is long range. We notice that a recent\nstudy have shown that titanium particles dope graphene\nand give rise to long range scattering. But, when these\nparticles are oxidized, accompanied by diminishing dop-\ning, significant short range scattering appears[48]. This\nimplies that long range potential could “screen” short\nrange potential. In the presence of a strong long range\npotential, electrons will have less chance to get close\nenough to experience the local SOC, which will reduce\nits averagestrength. Another possibility is that the bond\nbetween indium adatoms and graphene is Van der Waals\nin nature. The interaction is too weak to modify the\nhopping between neighbours. Further study may focus\non elements that induce less charge transfer, such as Fe\nor can form a stronger bond to graphene.\nThis work was supported by National Key Ba-\nsic Research Program of China (No. 2012CB933404,\n2013CBA01603) and NSFC (project No. 11074007,\n11222436, 11234001). X. W. thanks P. Xiong for pro-\nviding details of his in situdeposition design.5\n∗xswu@pku.edu.cn\n[1] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B74, 155426 (2006).\n[2] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Klein-\nman, and A. H. MacDonald, Phys. Rev. B 74, 165310\n(2006).\n[3] Y. G. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang,\nPhys. Rev. B 75, 041401 (2007).\n[4] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[5] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802\n(2005).\n[6] C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu, Phys.\nRev. X1, 021001 (2011).\n[7] J. Hu, J. Alicea, R. Wu, and M. Franz, Phys. Rev. Lett.\n109, 266801 (2012).\n[8] H. Jiang, Z. Qiao, H. Liu, J. Shi, and Q. Niu, Phys. Rev.\nLett.109, 116803 (2012).\n[9] O. Shevtsov, P. Carmier, C. Groth, X. Waintal, and\nD. Carpentier, Phys. Rev. B 85, 245441 (2012).\n[10] Z. H. Qiao, S. Y. A. Yang, W. X. Feng, W. K. Tse,\nJ. Ding, Y. G. Yao, J. Wang, and Q. Niu, Phys. Rev. B\n82, 161414 (2010).\n[11] J. Ding, Z. Qiao, W. Feng, Y. Yao, and Q. Niu, Phys.\nRev. B84, 195444 (2011).\n[12] W.-K. Tse, Z. Qiao, Y. Yao, A. H. MacDonald, and\nQ. Niu, Phys. Rev. B 83, 155447 (2011).\n[13] Z. Qiao, H. Jiang, X. Li, Y. Yao, and Q. Niu, Phys. Rev.\nB85, 115439 (2012).\n[14] H. Zhang, C. Lazo, S. Bl¨ ugel, S. Heinze, and\nY. Mokrousov, Phys. Rev. Lett. 108, 056802 (2012).\n[15] Z. Qiao, X. Li, W.-K. Tse, H. Jiang, Y. Yao, and Q. Niu,\nPhys. Rev. B 87, 125405 (2013).\n[16] J. Hu, Z. Zhu, and R. Wu, arXiv:1401.5453 (2014).\n[17] A. H. Castro Neto and F. Guinea, Phys. Rev. Lett. 103,\n026804 (2009).\n[18] K.-H. Ding, Z.-G. Zhu, and J. Berakdar, Europhys. Lett.\n88, 58001 (2009).\n[19] S. Abdelouahed, A. Ernst, J. Henk, I. V. Maznichenko,\nand I. Mertig, Phys. Rev. B 82, 125424 (2010).\n[20] A. Dyrda/suppress l and J. Barna´ s, Phys. Rev. B 86, 161401\n(2012).\n[21] D. Ma, Z. Li, and Z. Yang, Carbon 50, 297 (2012).\n[22] M. Gmitra, D. Kochan, and J. Fabian, Phys. Rev. Lett.\n110, 246602 (2013).\n[23] A. Ferreira, T. G. Rappoport, M. A. Cazalilla, and A. H.\nCastro Neto, Phys. Rev. Lett. 112, 066601 (2014).\n[24] A. Pachoud, A. Ferreira, B. ¨Ozyilmaz, and A. H. Cas-tro Neto, Phys. Rev. B 90, 035444 (2014).\n[25] A. Varykhalov, J. S´ anchez-Barriga, A. M. Shikin,\nC. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and\nO. Rader, Phys. Rev. Lett. 101, 157601 (2008).\n[26] Y. S. Dedkov, M. Fonin, U. R¨ udiger, and C. Laubschat,\nPhys. Rev. Lett. 100, 107602 (2008).\n[27] O. Rader, A. Varykhalov, J. S´ anchez-Barriga,\nD. Marchenko, A. Rybkin, and A. M. Shikin, Phys. Rev.\nLett.102, 057602 (2009).\n[28] D. Marchenko, A. Varykhalov, M. Scholz, G. Bihlmayer,\nE. Rashba, A. Rybkin, A. Shikin, and O. Rader, Nat\nCommun 3, 1232 (2012).\n[29] K. Pi, W. Han, K. M. McCreary, A. G. Swartz, Y. Li, and\nR. K. Kawakami, Phys. Rev. Lett. 104, 187201 (2010).\n[30] J. Balakrishnan, G. Kok Wai Koon, M. Jaiswal, A. H.\nCastro Neto, and B. Ozyilmaz, Nat. Phys. 9, 284 (2013).\n[31] J. Balakrishnan, G. K. W. Koon, A. Avsar, Y. Ho, J. H.\nLee, M. Jaiswal, S.-J. Baeck, J.-H. Ahn, A. Ferreira,\nM. A. Cazalilla, et al., Nat. Commun. 5, 4748 (2014).\n[32] K. T. Chan, J. B. Neaton, and M. L. Cohen, Phys. Rev.\nB77, 235430 (2008).\n[33] J. S. Parker, D. E. Read, A. Kumar, and P. Xiong, Eu-\nrophys. Lett. 75, 950 (2006).\n[34] E. McCann, K. Kechedzhi, V. I. Fal’ko, H. Suzuura,\nT. Ando, and B. L. Altshuler, Phys. Rev. Lett. 97,\n146805 (2006).\n[35] X. S. Wu, X. B. Li, Z. M. Song, C. Berger, and W. A.\nde Heer, Phys. Rev. Lett. 98, 136801 (2007).\n[36] F. V. Tikhonenko, D. W. Horsell, R. V. Gorbachev, and\nA. K. Savchenko, Phys. Rev. Lett. 100, 056802 (2008).\n[37] E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 108,\n166606 (2012).\n[38] S. Adam, E. H. Hwang, V. M. Galitski, and\nS. Das Sarma, Proc. Natl. Acad. Sci. USA 104, 18392\n(2007).\n[39] E. Abrahams, P. W. Anderson, P. A. Lee, and T. V.\nRamakrishnan, Phys. Rev. B 24, 6783 (1981).\n[40] J. H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D.\nWilliams, and M. Ishigami, Nat. Phys. 4, 377 (2008).\n[41] F. J. Ribeiro, J. B. Neaton, S. G. Louie, and M. L. Cohen,\nPhys. Rev. B 72, 075302 (2005).\n[42] S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor.\nPhys.63, 707 (1980).\n[43] G. Bergmann, Phys. Rev. Lett. 48, 1046 (1982).\n[44] G. Bergmann, Physics Reports 107, 1 (1984).\n[45] E. I. Rashba, Phys. Rev. B 79, 161409 (2009).\n[46] X. Zhai and G. Jin, Phys. Rev. B 89, 085430 (2014).\n[47] D. A. Abanin, R. V. Gorbachev, K. S. Novoselov, A. K.\nGeim, and L. S. Levitov, Phys. Rev. Lett. 107, 096601\n(2011).\n[48] K. M. McCreary, K. Pi, and R. K. Kawakami, Appl.\nPhys. Lett. 98, 192101 (2011)." }, { "title": "1204.1887v1.Spin_Orbit_Coupled_Degenerate_Fermi_Gases.pdf", "content": "arXiv:1204.1887v1 [cond-mat.quant-gas] 9 Apr 2012Spin-Orbit Coupled Degenerate Fermi Gases\nPengjun Wang∗,1Zeng-Qiang Yu†,2Zhengkun Fu,1Jiao Miao,2\nLianghui Huang,1Shijie Chai,1Hui Zhai,2,‡and Jing Zhang1,§\n1State Key Laboratory of Quantum Optics and Quantum Optics De vices,\nInstitute of Opto-Electronics, Shanxi University, Taiyua n 030006, P.R.China\n2Institute for Advanced Study, Tsinghua University, Beijin g, 100084, P.R.China\nSpin-orbit coupling plays an increasingly important role i n the modern condensed matter physics.\nFor instance, it gives birth to topological insulators and t opological superconductors. Quantum\nsimulation of spin-orbit coupling using ultracold Fermi ga ses will offer opportunities to study these\nnew phenomenain a more controllable setting. Here we report the first experimental study of a spin-\norbit coupled Fermi gas. We observe spin dephasing in spin dy namics and momentum distribution\nasymmetry in the equilibrium state as hallmarks of spin-orb it coupling. We also observe evidences\nof Lifshitz transition where the topology of Fermi surfaces change. This serves as an important first\nstep toward finding Majorana fermions in this system.\nInthepastdecade,quantumsimulationswithultracold\natoms have investigated many fascinating quantum phe-\nnomenain ahighlycontrollableandtunable way. Studies\nof ultracold Fermi gases with resonant interaction have\nshedlightsonunderstandingstronglyinteractingfermion\nsystems in nature, including neutron stars and quark-\ngluon plasma; simulating Fermi Hubbard model with\noptical lattices helps to understand the physical mech-\nanism of unconventional high-Tc superconductors. How-\never,until veryrecently, animportantinteractionhasnot\nbeen explored in cold atom systems, that is, the spin-\norbit (SO) coupling. Recently using two-photon Raman\nprocess, SO coupled Bose-Einstein condensate has been\nrealized in the laboratory [1], which gives rise to new\nquantum phases such as stripe superfluid [2, 3]. In real\nmaterials, SO coupling plays an important role in many\nphysical systems over a wide range of energy scale, from\ndetermining the nuclear structure inside a nuclei, and\nthe electronic structure inside an atom, to giving birth\nto topological insulators in solid state materials [4, 5].\nSince all these systems are fermionic, from the viewpoint\nof quantum simulation it is desirable to experimentally\nrealize SO coupled degenerate Fermi gases. The physical\neffects of SO coupling in a degenerate Fermi gas are quite\ndifferent from those in a Bose system. In this work we\nreport the first experimental study of a SO coupled de-\ngenerate Fermi gas. Evidences of SO coupling have been\nobtained from both the Raman induced quantum spin\ndynamics and the spin-resolved momentum distribution.\nWith SO coupling, we also observe evidences for Lifshitz\ntransition where the Fermi surface changes its topology\nas the density of fermion increases. This progresswill en-\nable us to study stronger pairing and higher Tcenhanced\nby SO coupling in resonant interacting Fermi gases [6–9]\nand topological insulator and topological superfluid in a\nmore flexible setup [10, 11] in the near future.\nIn ultracold atom systems, SO coupling is generated\nthrough atom-light interaction induced artificial gauge\nfields, and therefore it has two advantages. First, both\nthe strength and the configuration of SO coupling aretunableviacontrollingtheatom-lightcoupling; Secondly,\nthe coefficient ofSO coupling is naturally on the sameor-\nder of the inverse of laser wavelength, and in a gaseous\nsystem it gives rise to a coupling strength as strong as\nthe Fermi energy. In this regime SO coupling dramati-\ncallychangesthedensity-of-stateatFermienergyandthe\ntopology of Fermi surface, which gives rise to many in-\ntriguing phenomena in many-body systems [6–12], while\nsuch a regime is not easy to access in conventional solid\nstate materials.\nIn our experiment, a degenerate Fermi gas of 2 ×106\n40K atoms in the lowest hyperfine state |F= 9/2,mF=\n9/2/angbracketrightstate is first prepared in an optical dipole trap.\nThe optical dipole trap is composed of two horizontal\ncrossed beams of 1064 nm at 90oalong the ˆ x±ˆydirec-\ntion overlapped at the focus, as shown in Fig. 1(a). The\ntemperature of the Fermi gas is about 0 .3-0.4TF(TF\nthe Fermi temperature) when the trap frequency reaches\n2π×(116,116,164)Hz along(ˆ x,ˆy,ˆz) direction (see Meth-\nodsfor details). We can change the trap frequency adi-\nabatically to achieve different fermion density. A pair\nof Helmholtz coils (green ones in Fig. 1(a)) provides a\nhomogeneous bias magnetic field along ˆ y(quantization\naxis), whichispreciselycontrolledbyacarefullydesigned\nschemedescribedinRef. [13] toreducethe magneticfield\ndrift and the magnetic noise.\nThe method we used to generate SO coupling is the\nsame as reported by the NIST group for the87Rb Bose\ncondensate[1]. Inthe40Ksystem, twospin-1 /2statesare\nchosen as two magnetic sublevels |↑/angbracketright=|F= 9/2,mF=\n9/2/angbracketrightand|↓/angbracketright=|F= 9/2,mF= 7/2/angbracketrightof theF= 9/2 hy-\nperfinelevel. TheyarecoupledbyapairofRamanbeams\nwith coupling strength Ω. Two Raman lasers with the\nwavelength λ= 773 nm and the frequency difference ω\ncounter-propagate along ˆ xaxis and are linearly polar-\nized along ˆ yand ˆzdirections, respectively, corresponding\ntoπandσpolarization relative to quantization axis ˆ y\n(as shown in Fig. 1(a)). The recoil momentum kr=\nk0sin(θ/2), and recoil energy Er=k2\nr/2m=h×8.34\nkHz are taken as natural momentum and energy units,2\nwherek0= 2¯hπ/λandθ= 180ois the angle between\ntwo Raman beams. A Zeeman shift ωZ/2π= 10.4 MHz\nbetween these two magnetic sublevels is produced by the\nhomogeneous bias magnetic field at 31 G. When the Ra-\nman coupling is at resonance (at ω/2π= 10.4 MHz and\ntwo-photon Raman detuning δ= ¯h(ωZ−ω)≈0), the\ndetuning between |F= 9/2,mF= 7/2/angbracketrightand other mag-\nnetic sublevels like |F= 9/2,mF= 5/2/angbracketrightis about h×170\nkHz, which is one order of magnitude larger than the\nFermi energy. Thus we can safely disregard other levels\nand treat this system as a spin-1 /2 system. Same as in\nthe boson experiment, this scheme generates an effective\nsingle particle Hamiltonian as [1]\nH=/parenleftbigg1\n2m(p−krˆex)2−δ\n2Ω\n2Ω\n21\n2m(p+krˆ ex)2+δ\n2/parenrightbigg\n(1)\nHere,pdenotes the quasi-momentum of atoms, which\nrelates to the real momentum kask=p∓krˆexwith∓\nforspin-upanddown,respectively. ThisHamiltoniancan\nbe interpretedasanequalweightcombinationofRashba-\ntype and Dresselhaus-type SO coupling [1]. Finally, be-\nfore time-of-flight measurement, the Raman beams, opti-\ncal dipole trap and the homogeneous bias magnetic field\nare turned off abruptly at the same time, and a magnetic\nfield gradient along ˆ ydirection provided by Ioffe coil is\nturned on. Two spin states are separated along ˆ ydirec-\ntion during the time-of-flight due to the Stern-Gerlach\neffect, and imaging of atoms along ˆ zdirection after 12\nms expansion gives the momentum distribution for each\nspin component.\nThe main part of this paper is to study manifesta-\ntion SO coupling in a Fermi gas system. We first study\nthe Rabi oscillation between the two spin states induced\nby the Raman coupling. This experiment, on one hand,\nmeasuresthecouplingstrengthΩ; andontheotherhand,\nclearlymanifestsdifferenteffectofSOcouplingcompared\nto boson system. All atoms are initially prepared in the\n|↑/angbracketrightstate. Thehomogeneousbiasmagneticfieldisramped\nto a certain value so that δ=−4Er, that is, the k= 0\nstate of |↑/angbracketrightcomponent is at resonance with k= 2krˆex\nstate of |↓/angbracketrightcomponent, as shown in Fig. 1(b). Then\nwe apply a Raman pulse to the system, and measure the\nspin population for different duration time of the Raman\npulse. Similar experiment in boson system yields an un-\ndamped and completely periodic oscillation, which can\nbe well described by a sinusoidal function with frequency\nΩ [14]. This is because for bosons, macroscopic number\nof atoms occupy the resonant k= 0 mode, and therefore\nthere is a single Rabi frequency determined by the Ra-\nman coupling only. While for fermions, due to the Pauli\nexclusion principle, atoms occupy different momentum\nstates. Precisely due to the effect of SO coupling, the\ncoupling between the two spin states and the resulting\nenergy splitting are momentum dependent, and atoms\nin different momentum states oscillate with different fre-\nquencies (as shown explicitly in Eq. (2) later). Hence,- 6 -4 -2 0 2 4 6\n-2 0 20510\n \n - 4 r E δ=\nQuasimomentum px[units of kr]\nMom entum kx[units of kr]\nImaging Beam\nBias field coils\nIoffe coilOptical dipole\ntrap laser\nRam an laser1Raman laser2\nQuadrupole\ncoilsxyz(a)\n(b) (c)\nFraction of |9/2,7/2>\nRaman pulse time [ m s ]| 9/2,9/2> | 9/2,7/2>\n0.00 0.05 0.10 0.15 0.20 0.25 0.300.00.10.20.30.40.5\n | 9/2,9/2>\n| 9/2,7/2>δ\nEnergy [units of Er ]\n(d)1 6 µ s4 µ s \n3 2 µ s\n6 8 µ s\n1 2 8 µ s\n1 6 0 µ s\nFIG. 1: Experimental setup and Raman-induced quantum\nspin dynamics: (a) Schematic of experimental setup and the\nRaman coupling of two hyperfine levels of40K. (b) The en-\nergy dispersion with δ=−4Er. The system is initially pre-\npared with all atoms in |9/2,9/2/angbracketrightstate. (c) The population\nin|9/2,7/2/angbracketrightas a function of duration time of Raman pulse.\nkF= 1.9krandT/TF= 0.30 for red circles, kF= 1.35krand\nT/TF= 0.35 for blue squares, kF= 1.1krandT/TF= 0.29\nfor green triangles. The solid lines are theory curves with\nΩ = 1.52Er. (d) Time-of-flight image (left) and integrated\ntime-of-flight image (integrated along ˆ y) at different duration\ntime for | ↑/angbracketright(blue) and | ↓/angbracketright(red). The parameters are the\nsame as blue ones in (c)\ndephasing naturally occurs and the oscillation will be in-\nevitably damped after several oscillation periods. This\nprocess is very similar to the spin dynamics of a spin po-\nlarized current ejected into a semiconductor. For semi-\nconductor spintronics, one would like to have a polarized\nspin current, however, because different electron has dif-\nferent velocity and thus their spins precess in different\nway, the current will be unpolarized. Such spin dynam-\nics has been extensively studied during recent decades\n[15]. In our case, we can observe momentum-resolved\nspin dynamics with Stern-Gerlach technique and there-\nfore we can clearly reveal this physical process. In the\nFig. 1(c), we show the momentum distribution for both\nspin components at several different moments. One can\nsee the multiple-peaks feature in momentum distribution\nin Fig. 1(d), which clearly shows the out-of-phase oscil-\nlation for different momentum states.3\nUnlike bosons, the Raman coupling Ω cannot be read\nout directly from the period of oscillation. In fact, the\nfrequency depends on both atoms density and tempera-\nture, and it is generally smaller than Ω due to the effect\nof averaging over different momentum states. To deter-\nmine the value of Ω from the measurements, we fix Ra-\nman coupling and vary atoms density by changing the\ntotal number of fermions or the trapping frequency, and\nwe obtain several different oscillation curve, as shown in\nFig. 1(d). Then we fit them to the theory with a single\nfitting parameter Ω. Theoretically, for a non-interacting\nsystem, the population of |↓/angbracketrightcomponent is given by\nn↓(k+2krˆex,r,t) =n↑(k,r,0)1\n1+/parenleftbig2kxkr\nΩm/parenrightbig2×\nsin2/radicalbig\n(kxkr/m)2+Ω2/4t,(2)\nwheretis the duration time of Raman pulse, n↑(k,r,0)\nis the equilibrium distribution of the initial state in local\ndensity approximation, and temperature of initial cloud\nis determined by fitting the time-of-flight image to mo-\nmentum distribution of free fermions in a harmonic trap.\nFrom Eq. (2) one can see that the momentum distribu-\ntion along ˆ xdirection of |↓/angbracketrightcomponent is always sym-\nmetric respect to 2 krat any time, and the experimental\ndata is indeed the case, as shown in Fig. 1(c). The the-\noretical expectation of the total population in |↓/angbracketrightcom-\nponent is given by N↓(t) =/integraltextd3kd3rn↓(k,r,t), and in\nFig. 1(d), one can see there is an excellent agreement\nbetween the experiment data and theory, from which we\ndetermine Ω = 1 .52(5)Er. Since our current experiment\nisperformedintheweaklyinteractingregimewith s-wave\nscattering length as= 169a0, we have verified that the\ninteraction effect is negligible (see Methodfor details).\nNext, we focus on the case with δ= 0, and study\nthe momentum distribution in the equilibrium state. We\nfirst transfer half of40K atoms from |↓/angbracketrightto|↑/angbracketrightusing radio\nfrequency sweep within 100 ms. Then the Raman cou-\npling strength is ramped up adiabatically in 100 ms from\nzero to its final value and the system is held for another\n50 ms before time-of-flight measurement. We have also\nvaried the holding time and find the momentum distri-\nbution does not change, and thus we conclude that the\nsystem has reached equilibrium in the presence of SO\ncoupling. Since SO coupling breaks spatial reflectional\nsymmetry ( x→ −xandkx→ −kx), the momentum\ndistribution for each spin component will be asymmet-\nric, i.e. nσ(k)/negationslash=nσ(−k), withσ=↑,↓; While on the\nother hand, when δ= 0 the system still preserves time-\nreversal symmetry, which requires n↑(k) =n↓(−k). The\nasymmetry can be clearly seen in the spin-resolved time-\nof-flight images and integrated distributions displayed in\nFig. 2(a) and (b), where the fermion density is relatively\nlow. While it becomes less significant when the fermion\ndensity becomes higher, as shown in Fig. 2(c), because\nthe strength of SO coupling is relatively weaker com-/s45/s50 /s48 /s50/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s45/s50 /s48 /s50 /s45/s50 /s48 /s50/s45/s50 /s48 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s50 /s48 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s50 /s48 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s100/s41\n/s32/s32/s110 /s40/s107\n/s120/s41/s32/s45/s32 /s110 /s40/s45 /s107\n/s120/s41\n/s107\n/s120/s32/s47/s32 /s107\n/s114/s124/s57/s47/s50\n/s124/s55/s47/s50\n/s40/s101/s41\n/s32/s32/s73\n/s107\n/s120/s32/s47/s32 /s107\n/s114/s40/s102/s41\n/s32/s32/s78\n/s107\n/s120/s32/s47/s32 /s107\n/s114/s32\n/s107\n/s120/s32/s47/s32 /s107\n/s114\n/s32/s32/s32\n/s32\n/s32/s32\n/s40/s97/s41\n/s32\n/s32/s107\n/s120/s32/s47/s32 /s107\n/s114/s32/s32\n/s32/s40/s98/s41\n/s32/s32\n/s32/s107\n/s120/s32/s47/s32 /s107\n/s114\n/s32 /s32/s32/s32\n/s32/s40/s99/s41\n/s32/s32/s32\n/s32\n/s32/s32\n/s32\n/s32\n/s32/s32\n/s32\n/s32\n/s32/s32\nFIG. 2: Momentum distribution asymmetry as a hallmark of\nSO coupling: (a-c) time-of-flight measurement of momentum\ndistribution for both | ↑/angbracketright(blue) and | ↓/angbracketright(red). Solid lines are\ntheory curves. (a) kF= 0.9krandT/TF= 0.8 (b)kF= 1.6kr\nandT/TF= 0.63; (c)kF= 1.8krandT/TF= 0.57. (d-f): plot\nof integrated momentum distribution nσ(k)−nσ(−k) for the\ncase of (a-c).\npared to the Fermi energy. In Fig. 2(a-c) the integrated\nmomentum distribution is fitted by the theoretical calcu-\nlationtodeterminethetemperatureandthe chemicalpo-\ntential at the center of the trap. (See Methodfor detail).\nWe find that the Raman lasers indeed cause additional\nheating to the cloud. Nevertheless, the temperature we\nfind is within the range of 0 .5−0.8TF[16], which is still\nbelow degenerate temperature. In Fig. 2(d-f), we also\nshownσ(kx)−nσ(−kx) to reveal the distribution asym-\nmetry more clearly. These are smoking gun evidences of\na degenerate Fermi gas with SO coupling.\nWith SO coupling, the single particle spectra of Eq. 1\nare dramatically changed from two parabolic dispersions\ninto two helicity branches as shown in the inset of Fig.\n3(a). Here, two different branches are eigenstates of “he-\nlicity” ˆsand the “helicity” operator describes whether\nspinσpis parallel or anti-parallel to the “effective Zee-\nman field” hp= (−Ω,0,krpx/m+δ) at each momentum,\ni.e. ˆs=σp·hp/|σp·hp|.s= 1 for the upper branch and\ns=−1 for the lower branch. The topology of Fermi sur-\nface exhibits two transitions as the atoms density varies.\nAt sufficient low density, it contains two disjointed Fermi\nsurfaces with s=−1, and they gradually merge into\na single Fermi surface as the density increases to nc1.\nFinally a new small Fermi surface appears at the cen-\nter of large Fermi surface when density further increases\nand fermions begin to occupy s= 1 helicity branch at\nnc2. A theoretical ground state phase diagram for the\nuniform system is shown in Fig. 3(a), and an illustra-\ntion of the Fermi surfaces at different density are shown\nin Fig. 3(b). Across the phase boundaries, the system\nexperiences Lifshitz transitions as density increases [17],\nwhich is a unique property in a Fermi gas due to Pauli4\nprinciple. At sufficiently low temperature, the derivative\nof the thermodynamic quantities like the compressibility\nwill exhibit singularity in the critical regime around the\ntransition point.\nRigorously speaking Lifshitz transition only exists at\nzero temperature and at finite temperature it becomes\na crossover. However, we can still obtain several consis-\ntent features that supports the existence of such a tran-\nsition at zero temperature. We fix the Raman coupling\nand vary the atoms density at the center of the trap by\ncontrolling total fermion number or trap frequency, as\nindicated by the red arrow in Fig. 3(a). The quasi-\nmomentum distribution in the helicity bases can be ob-\ntained from a transformation of momentum distribution\nin spin bases as follows (See Methodfor the definition of\nupandvp):\nn+(p) =u2\npn↑(p−¯hkrˆ ex)−v2\npn↓(p+¯hkrˆ ex)\nu2p−v2p(3)\nn−(p) =v2\npn↑(p−¯hkrˆ ex)−u2\npn↓(p+¯hkrˆ ex)\nv2p−u2p(4)\nIn Fig. 3(c1-c5), we plot the quasi-momentum distribu-\ntion in the helicity bases for different atoms density. At\nthe lowest density, the s= 1 helicity branch is nearly un-\noccupied, which is consistent with that the Fermi surface\nis belows= 1helicitybranch. The quasi-momentumdis-\ntribution of the s=−1 helicity branch exhibits clearly\na double-peak structure, which reveals the emergence of\ntwo disjointed Fermi surfaces at s=−1 helicity branch.\nAs density increases, the double-peak feature gradually\ndisappears, indicating the Fermi surface of s=−1 helic-\nity branch finally becomes a single elongated one, as the\ntop one in Fig. 3(b). Here we define a quality of visibil-\nityv= (nA−nB)/(nA+nB), where nAis the density of\ns=−1 branch at the peak and nBis the density at the\ndip between two peaks. Theoretically one expects vap-\nproachesunityat lowdensityregimeandapproacheszero\nathighdensityregime. InFig3(d) weshowthat ourdata\ndecreases as density increases and agrees very well with\na theoretical curve with fixed temperature T/TF= 0.65.\nMoreover, across the phase boundary between SFS and\nDFS-1, there will be a significant increase of population\nons= 1 helicity branch. In Fig. 3 (e), the fraction of\natom number population at s= 1 helicity branch is plot-\nted as a function of Fermi momentum kF, which grows\nup rapidly nearby the critical point predicted in zero-\ntemperature phase diagram. The blue solid line is a the-\noretical calculation for N+/NwithT/TF= 0.65, and\nthe small deviation between the data and this line is due\nto the temperature variation between different measure-\nments. Both two features are consistent with a Lifishitz\ntransition. Recently, topological change of Fermi surface\nand Lifshitz transition have also been studied in Fermi\ngas in honeycomb optical lattices, where single particle\nspectrum exhibits Dirac point behavior [18]. A moreaccurate experimental determination of Lifshitz transi-\ntion point requires more well control of temperature in\npresence of Raman lasers and the information of local\nequation-of-state. In the near future, we also plan to\nbring the system close to a Feshbach resonance where\nthes-wave interaction becomes strongly attractive, and\nwe will further cool the system below the superfluid tran-\nsition temperature. There we expect to find Majorana\nfermion modes at the phase boundaries when Fermi sur-\nface topology changes [19–21].\nInsummary,wehaveforthefirsttimestudiedeffectsof\nspin-orbit coupling in a ultracold atomic Fermi gas. We\nstudy Raman induced spin dynamics and reveal clearly\nthe physical process of spin dephasing due to SO cou-\npling. We measure the momentum distribution in the\nequilibrium state and find the momentum distribution\nasymmetry due to the broken of the reflection symme-\ntry caused by SO coupling. We also find evidences for\nthe change of Fermi surface topology and Lifshitz transi-\ntion from the quasi-momentum distribution in the helic-\nity bases. All these features are unique manifestations of\nSOcouplingin fermionicsystem, which areabsentin pre-\nviously realized SO coupled boson condensate [1]. This\nresearch opens up an avenue toward rich physics of SO\ncoupled Fermi gases.\nMethods.\nExperimental setup. The experimental setup (see Fig.\n1) is the same as discussed in [22–24], which employs a\nmixture of87Rb and40K atoms. In short, a mixture of\n1×107 87Rb atoms at the spin state |F= 2,mF= 2/angbracketrightand\n4×106 40K atoms at |F= 9/2,mF= 9/2/angbracketrightare precooled\nto 1.5µKby radio-frequency evaporative cooling in a\nquadrupole-Ioffeconfiguration(QUIC)trap,andthenare\ntransported to the center of the glass cell [25] in favor of\noptical access. Both species are loaded into the optical\ndipole trap and further evaporatively cooled in the op-\ntical dipole trap by evaporative cooling. At the end of\nabove process, the 2 ×106 40K atoms reaches quantum\ndegeneracy at ∼0.3TF. We use a resonant laser beam\nfor 0.03 ms to remove the Rb atoms without heating40K\natoms.\nTwo laser beams for generating Raman coupling are\nextracted from a Ti-sapphirelaser operatingat the wave-\nlength of 773 nm with the narrow linewidth single-\nfrequency and focused at the position of the atomic\nclouds with 1 /e2radii of 200 µm. Two Raman beams\nare frequency-shifted with -100 MHz and -110.4 MHz by\ntwo single-pass acousto-optic modulators (AOM) respec-\ntively, and then are coupled into two polarization main-\ntaining single-mode fibers in order to increase stability of\nthe beam pointing and obtain better beam-profile qual-\nity. The frequency difference ωof two signal generators\nfor two AOMs is phase-locked by a source locking CW\nmicrowave frequency counters. To enhance the intensity\nstability of the two Raman beams, a small fraction of\nthe light is sent into a photodiode and the regulator is5\n/s32/s32/s40/s98/s41\n/s68/s83/s70/s45/s49/s32/s32/s32\n/s83/s70/s83/s32/s32/s68/s70/s83/s45/s50\n/s32 /s32/s32\n/s32/s32\nFIG. 3: Topological change of Fermi surface and Lifshitz\ntransition: (a) Theoretical phase diagram at T= 0.k0\nF=\n¯h(3π2n)1/3. “SFS”means single Fermi surface. “DFS”means\ndouble Fermi surface. (b) Illustration of different topolog y of\nFermi surfaces as Fermi energy increases. The single parti-\ncle energy dispersion is drawn for small Ω, in which red and\ngreen lines are for s=−1 ands= 1 helicity branches, respec-\ntively. Dashed blue line is the chemical potential. Red and\ngreen surfaces are Fermi surfaces for s=−1 helicity branch\nands= 1 helicity branch, respectively. (c) Quasi-momentum\ndistribution in the helicity bases. Red and green points are\ndistributions for s=−1 ands= 1 helicity branches, re-\nspectively. kF= 0.9kr,T/TF= 0.80 for (c1); kF= 1.2kr,\nT/TF= 0.69 for (c2); kF= 1.4kr,T/TF= 0.61 for (c3);\nkF= 1.6kr,T/TF= 0.63 for (c4); kF= 1.8kr,T/TF= 0.57\nfor (c5). All these points are marked on phase diagram in (a).\n(d) Visibility v= (nA−nB)/(nA+nB) decreases as kF/krin-\ncreases (AandBpointsaremarkedin(c1)). (e)Atomnumber\npopulation in s= 1 helicity branch N+/Nincreases as kF/kr\nincreases increases. In both (d) and (e), the blue solid line\nis a theoretical curve with T/TF= 0.65, and the background\ncolor indicates three different phases in the phase diagram.\nused for comparing the intensity measured by the pho-\ntodiode to a set voltage value from the computer. The\nnon-zero error signal is compensated by adjusting the\nradio-frequency power in the AOM in front of the optical\nfiber.\nCalculating Momentum Distribution in a Trap. The\nenergy eigenstates for the single particle Hamiltonian in\nEq. (1) are the dressed states usually denoted as helicity\nbases with s=±1, with|+,p/angbracketright=up| ↑,p/angbracketright+vp| ↓,p/angbracketrightand|−,p/angbracketright=vp| ↑,p/angbracketright−up| ↓,p/angbracketright, where the coefficients\nu2\np= 1−v2\np=1\n2/bracketleftbig\n1+pxvr−δ/2√\n(pxvr−δ/2)2+Ω2/4/bracketrightbig\nareonlyfunctions\nofpxwithvr=kr/m. The energy dispersion for each\nbranch of these dressed states are given by E±,p= (p2+\nk2\nr)/(2m)±/radicalbig\n(pxvr−δ/2)2+Ω2/4.\nThe trapping potential V(r) is taken into account by\nlocal density approximation. Even in presence of Raman\ncoupling, what can be measured in the time-of-flight im-\nagewithStern-Gerlachtechniquearestillmomentumdis-\ntributions of original hyperfine spin states. Theoretically\nthey are given by\nn↑(p−krˆ ex) =/integraldisplayd3r\n(2π¯h)3/bracketleftBig\nu2\npf(Ep,+;r)+v2\npf(Ep,−;r)/bracketrightBig\nn↓(p+krˆ ex) =/integraldisplayd3r\n(2π¯h)3/bracketleftBig\nv2\npf(Ep,+;r)+u2\npf(Ep,−;r)/bracketrightBig\nwheref(E;r) is the Fermi-Dirac distribution function\nwith a local chemical potential µ(r) =µ−V(r), andµis\ndetermined by the equation for total particle number\nN=/integraldisplay\nd3p/bracketleftBig\nn↑(p)+n↓(p)/bracketrightBig\n(5)\nBy fitting the measured data to the theoretical\ncurve of integrated momentum profile n1d,σ(px) =/integraltext\ndpydpznσ(p) withσ=↑or↓, we obtain the temper-\nature of atoms in the experiment.\nInteraction Induced Change of Rabi Frequency. We\nshall also compare this experiment to the clock experi-\nment in Fermi gases[26], wherethe couplingbetween two\ncomponents is momentum independent if the light inten-\nsity is assumed to be uniform. In that case, all atoms\nundergo Rabi oscillation with exact same frequency, and\nthereforeremain as identical particles during the process.\nHence, the s-wave interaction plays no role except when\nthe spatial inhomogeneity is taken into account [26]. In\ncontrast, in our case different atoms oscillate with dif-\nferent frequencies, thus they immediately become distin-\nguishable particles as the oscillation starts, and can in-\nteractwith each othervia s-wavecollisions. Since our ex-\nperiment is performed in the weakly interacting regime,\nwe include the interaction effect with mean-field theory.\nFor a non-interacting system, the equation of motion\nfor a spin Spwith quasi-momentum pis given by\n∂\n∂tSp=Sp×hp (6)\nwherehp= (−Ω,0,krpx/m+δ) is the momentum de-\npendent “effective magnetic field”. The frequency of the\nspin rotation is given by |hp|=/radicalbig\n(pxkr/m+δ)2+Ω2,\nand for δ=−4Er, it is just the frequency of popu-\nlation oscillation as shown in Eq. (2). For a weakly-\ninteracting system, the interaction can be approximated\nbyHint=g\nVS·Satmean-fieldlevel, where g= 4π¯h2as/m\nis the coupling constant, asis thes-wave scattering6\nlength, and S=/summationtext\npSpis the total spin. Thus, an addi-\ntional mean-field term appearsin the equationof motion,\n∂\n∂tSp=Sp×/parenleftbig\nhp+2g\nVS/parenrightbig\n. (7)\nwhere the solution of Spmust be determined self-\nconsistently. We numerically solve this equation of mo-\ntion to determine the spin dynamics. Using the exper-\nimental parameters, we find the frequency shift is only\na few percent of Ω, which is beyond the measurement\nresolution of current experiment.\nAcknowledgements. We would like to thank Zhen-\nhua Yu, Cheng Chin, Tin-Lun Ho and Sandy Fetter\nfor helpful discussions. This research is supported by\nNational Basic Research Program of China (Grant No.\n2011CB921601, 2010CB923103, 2011CB921500), NSFC\n(Grant No. 10725416, 61121064, 11004118, 11174176),\nDPFMEC (Approval No. 20111401130001) and Ts-\ninghua University Initiative Scientific Research Program.\n‡Electronic address: hzhai@mail.tsinghua.edu.cn\n§Electronicaddress: jzhang74@sxu.edu.cn; jzhang74@yaho o.com\n[1] Lin, Y.-J., Jim´ enez-Garc´ ıa, K., & Spielman, I. B. Spin -\norbit-coupled Bose-Einstein condensates. Nature 471, 83\n(2011).\n[2] Wang, C., Gao, C., Jian, C.-M., & Zhai, H. Spin-orbit\ncoupled spinor Bose-Einstein condensates. Phys. Rev.\nLett.105, 160403 (2010).\n[3] Ho, T.-L. & Zhang, S. Bose-einstein condensates with\nspin-orbit interaction. Phys. Rev. Lett. 107, 150403\n(2011).\n[4] Forareview, see Hasan, M. Z.&Kane, C.L. Colloquium:\nTopological insulators. Rev. Mod. Phys. 82, 3045 (2010).\n[5] For a review, see also Qi, X. L. & Zhang, S. C. Topologi-\ncal insulators and superconductors. Rev. Mod. Phys. 83,\n1057 (2011).\n[6] Vyasanakere, J. P. & Shenoy, V. B. Bound state of two\nspin-1/2 fermions in a synthetic non-Abelian gauge field.\nPhys. Rev. B 83, 094515 (2011).\n[7] Vyasanakere, J. P., Zhang, S., & Shenoy,V. B. BCS-BEC\ncrossover induced by a synthetic nob-Abelian gague field.\nPhys. Rev. B 84, 014512 (2011).\n[8] Yu, Z.-Q. & Zhai, H. Spin-orbit coupled Fermi gases\nacross aFeshbach resonance. Phys Rev Lett. 107, 195305\n(2011).\n[9] Hu, H., Jiang, L., Liu, X.-J., & Pu, H. Probing\nanisotropic superfluidity in atomic Fermi gases with\nRashbaspin-orbitcoupling. Phys. Rev.Lett. 107, 195304\n(2011).\n[10] Gong, M., Tewari, S., & Zhang, C. BCS-BEC crossover\nand topological phase transition in 3D spin-orbit coupled\ndegenerate Fermi gases. Phys. Rev. Lett. 107, 195303\n(2011).\n[11] Goldman, N., Satija, I., Nikolic, P., Bermudez, A.,\nMartin-Delgado, M. A., Lewenstein, M., & Spielman, I.B. Realistic time-reversal invariant topological insulat ors\nwith neutral atoms. Phys. Rev. Lett. 105, 255302 (2010).\n[12] Zhai, H. Spin-orbit coupled quantum gases. Int. J. Mod.\nPhys. B 26, 1230001 (2012)\n[13] Fu, Z., Wang, P., Chai, S., Huang, L., & Zhang, J. Bose-\nEinstein condensate in a light-induced vector gauge po-\ntential usingthe 1064 nmoptical dipole trap lasers. Phys.\nRev. A84, 043609 (2011).\n[14] Lin, Y.-J., Compton, R. L., Perry, A. R., Phillips, W. D. ,\nPorto, J. V., & Spielman, I. B. Bose-Einstein condensate\nin a uniform light-induced vector potential. Phys. Rev.\nLett.102, 130401 (2009).\n[15] Wu, M. W., Jiang, J. H., & Weng, M. Q. Spin dynamics\nin semiconductors. Phys. Rep. 493, 61 (2010).\n[16] Here, we define Fermi energy and Fermi temperature in\nthe conventional way as the trapped Fermi gas with-\nout SO coupling, EF=kBTF=k2\nF/(2m), withkF=\n(24N)1/6√¯hmωhothe Fermi momentum and ωhothe av-\nerage trap frequency.\n[17] Lifshitz, I. M. Anomalies of electron characteristics of a\nmetal in the high pressure region. Sov. Phys. JETP 11,\n1130 (1960).\n[18] Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G., &\nEsslinger, T. Creating, moving and merging Dirac points\nwith a Fermi gas in a tunable honeycomb lattice. arXiv:\n1111.5020, accepted by Nature.\n[19] Lutchyn, R. M., Sau, J. D., & Das Sarma, S. Ma-\njorana Fermions and a topological phase transition\ninsemiconductor-superconductorheterostructures.Phys .\nRev. Lett. 105, 077001 (2010).\n[20] Oreg, Y., Refael, G., & von Oppen, F. Helical liquids\nandMajorana boundstates inquantumwires. Phys.Rev.\nLett.105, 177002 (2010).\n[21] Jiang, L., Kitagawa, T., Alicea, J., Akhmerov, A. R.,\nPekker, D., Refael, G., Cirac, J. I., Demler, E., Lukin, M.\nD., & Zoller, P. Majorana Fermions in equilibrium and in\ndriven cold-atom quantum wires. Phys. Rev. Lett. 106,\n220402 (2011).\n[22] Wei, D., Xiong, D., Chen, H., Wang, P., Guo, L.,\n& Zhang, J. Simultaneous magneto-optical trapping of\nfermionic40K and bosonic87Rb atoms. Chin. Phys. Lett.\n24, 1541 (2007).\n[23] Xiong, D., Chen, H., Wang, P., Yu, X., Gao, F., &\nZhang, J. Quantum degenerate Fermi-Bose mixtures of\n40K and87Rb atoms in a quadrupole-Ioffe configuration\ntrap. Chin. Phys. Lett. 25, 843 (2008).\n[24] Wang, P., Deng, L., Hagley, E. W., Fu, Z., Chai, S.,\n& Zhang, J. Observation of collective atomic recoil mo-\ntion in a degenerate Fermion gas. Phys. Rev. Lett. 106,\n210401 (2011).\n[25] Xiong, D., Wang, P., Fu, Z., & Zhang, J. Transport of\nBose-Einstein condensate in QUIC trap and separation\nof trapping spin states. Opt. Exppress. 18, 1649 (2010).\n[26] Campbell, G. K., Boyd, M. M., Thomsen, J. W., Martin,\nM. J., Blatt, S., Swallows, M., Nicholson, T. L., Fortier,\nT., Oates, C. W., Diddams, S. A., Lemke, N. D., Naidon,\nP., Julienne, P., Ye, J., & Ludlow, A. D. Probing in-\nteractions between ultracold fermions. Science 324, 360\n(2009)." }, { "title": "2312.06054v2.Dzyaloshinskii_Moriya_interaction_from_unquenched_orbital_angular_momentum.pdf", "content": "Dzyaloshinskii-Moriya interaction from unquenched orbital angular momentum\nRunze Chen,1, 2Dongwook Go,2, 3Stefan Bl ¨ugel,2Weisheng Zhao,1,∗and Yuriy Mokrousov2, 3,†\n1Fert Beijing Institute, School of Integrated Circuit Science and Engineering,\nBeijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, China\n2Peter Gr ¨unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ¨ulich and JARA, 52425 J ¨ulich, Germany\n3Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany\n(Dated: December 13, 2023)\nOrbitronics is an emerging and fascinating field that explores the utilization of the orbital degree of freedom\nof electrons for information processing. An increasing number of orbital phenomena are being currently discov-\nered, with spin-orbit coupling mediating the interplay between orbital and spin effects, thus providing a wealth\nof control mechanisms and device applications. In this context, the orbital analog of spin Dzyaloshinskii-Moriya\ninteraction (DMI), i.e. orbital DMI , deserves to be explored in depth, since it is believed to be capable of in-\nducing chiral orbital structures. Here, we unveil the main features and microscopic mechanisms of the orbital\nDMI in a two-dimensional square lattice using a tight-binding model of t2gorbitals in combination with the\nBerry phase theory. This approach allows us to investigate and transparently disentangle the role of inversion\nsymmetry breaking, strength of orbital exchange interaction and spin-orbit coupling in shaping the properties\nof the orbital DMI. By scrutinizing the band-resolved contributions we are able to understand the microscopic\nmechanisms and guiding principles behind the orbital DMI and its anisotropy in two dimensional magnetic ma-\nterials, and uncover a fundamental relation between the orbital DMI and its spin counterpart, which is currently\nexplored very intensively. The insights gained from our work contribute to advancing our knowledge of orbital-\nrelated effects and their potential applications in spintronics, providing a path for future research in the field of\nchiral orbitronics.\nI. INTRODUCTION\nIn condensed matter physics, a new and promising area of\nresearch known as orbitronics has emerged in recent years.\nOrbitronics focuses on utilizing the orbital degree of free-\ndom as a means of transmitting and manipulating information\nin solid-state devices [1–3]. Recent investigations have re-\nvealed intriguing possibilities within this realm, demonstrat-\ning that an orbital current, resulting from the movement of\nelectrons possessing finite orbital angular momentum, can be\ngenerated in diverse range of materials, despite the presence\nof orbital quenching in the ground state [4–11]. The fun-\ndamental principle behind orbitronics lies in the precise con-\ntrol of orbital dynamics and information transport and subse-\nquent manipulation of spin transport and magnetization dy-\nnamics through spin-orbit coupling (SOC). This demonstrates\nthe potential of utilizing the orbital current as an alternative to\nconventional spin current in the field of spintronics [12–19].\nMoreover, the orbital degree of freedom serves as an interme-\ndiary, facilitating the appearance of diverse spin phenomena\nby mediating the coupling between magnetic moments and\nthe lattice, thus triggering e.g. the k-dependent spin splitting\nthrough structural inversion symmetry breaking and magneti-\nzation dynamics in an applied external electric field [4, 20].\nCorrespondingly, in the past years the community witnessed\na considerable interest in exploring the fundamental relation\nbetween various spin phenomena and their orbital analogs,\nsuch as orbital Rashba effect, orbital Hall effect and orbital\ntorque [1, 4, 5, 12, 13, 20–38].\n∗weisheng.zhao@buaa.edu.cn\n†y.mokrousov@fz-juelich.deIn the realm of chiral spin magnetism, so-called the\nDzyaloshinskii-Moriya interaction (DMI) has attracted signif-\nicant attention due to its pivotal role in the formation and sta-\nbilization of topological magnetic textures, including chiral\ndomain walls and magnetic skyrmions [39–42], which hold\ngreat promise as information carries in emerging memory-\nstorage and neuromorphic computing [43, 44]. The DMI is an\nantisymmetric spin exchange interaction that originates from\nthe combination of SOC with structural inversion asymme-\ntry [45–47]. Recently, a growing body of theoretical and ex-\nperimental research has illuminated the pivotal role played by\norbital degrees of freedom in the emergence and manipulation\nof the DMI, thereby igniting a surge of interest in exploring\norbital facets in the DMI.\nImportantly, Yamamoto et al. theoretically investigated the\nintricate connection between the DMI and orbital moments\nin heavy metal/ferromagnetic metal structures, revealing a\nclose correlation between the sign of the DMI and the in-\nduced orbital moments of heavy metal elements [48]. Mean-\nwhile, density functional theory calculations by Belabbes et\nal.showed a correlation of the DMI with an electric dipole\nmoment at the oxide/ferromagnetic metal interface [49]. The\nelectric dipole moment arises from the charge transfer trig-\ngered by orbital hybridization at the interface between an ox-\nide and a ferromagnet, tightly correlated with the interfacial\norbital moment. It established a strong relationship among\nthe DMI, interfacial orbital hybridization, and the interfacial\norbital moment. Subsequently, Nembach et al. conducted ex-\nperiments to substantiate the calculations by Belabbes et al. ,\nrevealing that the DMI and the spectroscopic splitting fac-\ntor, which measures the orbital moment, are indeed corre-\nlated [50]. Furthermore, Zhu et al. experimentally investi-\ngated the correlation between interfacial orbital hybridizationarXiv:2312.06054v2 [cond-mat.mtrl-sci] 12 Dec 20232\nand DMI in heavy metal/ferromagnetic metal structures, pro-\nviding conclusive evidence of the pivotal role played by or-\nbital hybridization of magnetic interfaces in the determination\nand regulation of the DMI [51]. It has also been found that\nthe DMI appears to exhibit a close correlation with the or-\nbital moment anisotropy. Kim et al. experimentally unveiled\nthat the DMI is governed by the orbital anisotropy, attribut-\ning its microscopic origin to the asymmetric charge distribu-\ntion at the interface due to electron hoppings driven by the\ninversion symmetry breaking (ISB) [52]. Moreover, the asym-\nmetric charge distribution results in chiral orbital angular mo-\nmentum, which is then converted into spin canting via SOC,\nthereby explaining the emergence of the DMI.\nAt the same time, while it is clear that the various aspects of\norbital nature and orbital contributions to the spin DMI gov-\nerning the energetics of spin canting deserve further efforts,\nthe physics and properties of the orbital DMI, which reflects\nthe energy changes due to chiral canting of orbital moments,\nhave been practically unexplored so far. Katsnelson et al. has\napplied the method of infinitesimal rotations to compute the\nvalues of orbital part of the DMI in magnetic La 2CuO 4from\nLDA+ U+SOC calculations, finding that the orbital part, while\nbeing generally small, is by far larger than the spin contribu-\ntion [53]. On the other hand, Kim and Han have considered\nthe superexchange interaction in a multi-orbital tight-binding\nHubbard model, finding a contribution which is proportional\nto a chiral product between neighboring orbital moments [54].\nIdentifying strong on-site correltaions and ISB as the origin of\norbital version of DMI, they have demonstrated that the latter\ncan dictate the formation of complex orbital textures [54].\nThe orbital DMI and the resulting orbital chiral structures\npresent a novel avenue for creating and manipulating chiral\ntopological textures or angular momentum by controlling or-\nbital properties, marking the orbital DMI as an object deserv-\ning further in-depth investigation. Therefore, a comprehensive\nunderstanding of orbital DMI is imperative. However, a clear\nand simple physical picture of the orbital DMI, reaching into\nthe realm of electronic structure models not necessarily rooted\nin strong correlations, is still elusive, which necessitates fur-\nther clarification of the influence of various orbital parameters\non the orbital DMI.\nHere, we evaluate and analyze in depth the orbital DMI be-\nhavior in a two-dimensional square lattice based on a simple\ntight-binding model of t2gorbitals. This model satisfies the\nrequirement that the crystal field splitting in combination with\nISB promote rich unquenched orbital magnetism, making it\nan optimal choice to extract the orbital-related effects [28].\nWe resort to the Berry phase theory of orbital DMI, which is\nobtained as a direct extension of the successful methodology\nto compute the spin DMI, developed in the past [55]. As one\nof the key variables in this method, we introduce the orbital\nexchange coupling term in the Hamiltonian, discussing its na-\nture and physical origins, assessing the orbital DMI as the en-\nergy response of the model to its spiralization, and relating it\nto the mixed orbital Berry curvature. We demonstrate that the\norbital hybridization induced by ISB not only leads to chiral\norbital angular momentum (OAM) structures in momentum\nspace but also gives rise to prominent orbital DMI localizedin specific regions of the Brillouin zone. Furthermore, we ex-\npore in which way the orbital exchange and ISB strengths im-\npact the orbital DMI. We find that the orbital DMI exhibits\na strong anisotropy with respect to the orbital magnetization\ndirection. We also show that even in the absence of SOC,\nthe orbital DMI persists while the spin DMI vanishes, finding\nthat non-relativistic by nature orbital DMI can dictate the be-\nhavior of spin DMI, consistent with the behavior of other or-\nbital effects. With our work we thus contribute significantly to\nputting the orbital DMI physics onto the rails of material engi-\nneering in wider classes of two-dimensional metallic surfaces,\ninterfaces and heterostructures, opening new perspectives for\nexperimental realization of chiral orbital textures.\nII. METHODS\nA. Motivation and physical setups\nBefore proceeding further with a detailed description of our\napproach, it is necessary to elucidate the motivation behind\nour treatment of OAM contribution to the DMI. Our starting\npoint is the non-relativistic non-magnetic Hamiltonian of an\norbitally complex system, represented by a t2gorbital model\non a square lattice with broken inversion symmetry. Starting\nwith this model, our next step is to break the time-reversal\nsymmetry by allowing unquenched local orbital moments,\nwhich serve as the primary objects whose canting energet-\nics we are to study. The conventional way to achieve this\nwould be to allow for spin exchange interaction which, when\ncombined with an effect of SOC, results in orbital magnetism.\nHowever, here, we choose a different path and explicitly in-\nclude the effect of the orbital exchange field into considera-\ntion, which couples directly to the OAM, see Eq. (3) below,\nand gives rise to unquenched orbital magnetism without re-\nsorting to spin magnetism and SOC. In turn, one can perceive\nthe effect of the orbital exchange field as a combined effect\nof the two latter phenomena, projected onto the orbital sub-\nspace only, which allows for a transparent way to perturb the\ndirections of orbital moments by tracking the energetics of the\norbital sub-system.\nWhen thinking of a physical situation in which the effect of\nthe orbital exchange field can be most easily singled out, we\ncan imagine a weakly coupled bilayer type of system, where\na nonmagnetic layer develops orbital magnetism via the SOC\nand the nonlocal exchange coupling with the second layer that\nis strongly magnetic, e.g. 3 doverlayer separated by a spacer\nfrom a 5 dsubstrate [56–58]. In the latter case, considering\nthat we can freely impose magnetic order in the 3 dmetal, this\nwill be translated into corresponding orbital order via the ef-\nfect of SOC and effective orbital field, the energetics of which\nwe can study separately assuming that it would be possible to\nneglect the spin contributions arising in the 5 dlayer itself due\nto smallness of corresponding induced spin moments.\nSecondly, we can consider a situation in which the shell\nof orbitals which drive spin magnetism is separated in energy\nfrom a nominally spin-degenerate but orbitally active shell of\nstates residing around the Fermi level, such as the case e.g. for3\nmaterials incorporating 4f-ions [59]. Here, again, the SOC-\nimposed contribution to (e.g. p-d-f) hybridization between\nthe two shells would results in the presence of an effective\norbital field. In the latter two cases, for clear identification\nof the orbital DMI, it seems to be particularly promising to\nconsider antiferromagnetic bilayers and f-based compounds,\nwhere the net effect of spin splitting on the orbital states can\nbe vanishingly small, and they can be considered practically\nspin degenerate [59].\nFinally, we can turn to a situation of a non-magnetic thin\nfilm exposed to a spatially varying external magnetic field\nwhich drives a direct interaction with orbital moments in the\nsubstrate of the Zeeman type −i.e. orbital Zeeman effect −\nas given by Eq. (3). The latter case has been considered re-\ncently in a situation of orbital pumping, where magnetic field\ndynamics drives a prominent orbital response [60, 61].\nIn the following, we provide the details of the t2gorbital\nmodel and the derivations of the expression of orbital DMI.\nB.t2gorbitals on a two-dimensional square lattice\nWe employ a tight-binding model description of a\nsimple two-dimensional square lattice with t2gorbitals\n(dxy, dyz, dzx) on each site. We operate in terms of Bloch\nwaves eik·r|φnk⟩=P\nReik·R|ϕnR⟩as basis, where ϕnR\nis the localized state at the Bravais lattice Rwith the orbital\ncharacter n,n=dxy, dyz, dzx. The spinless tight-binding\nHamiltonian in k-space is written as\nHtot(k) =Hkin(k) +HL\nexc, (1)\nwhere Hkin(k)is the kinetic part of the Hamiltonian arising\nfrom hoppings and on-site energies, and HL\nexcdescribes or-\nbital exchange interaction. Hkin(k)is independent of the spin\nand its nonzero matrix elements are\n\nφdxyk|Hkin|φdxyk\u000b\n=E∥−2tδ[cos (kxa) + cos ( kya)],\n(2a)\nφdyzk|Hkin|φdyzk\u000b\n=E⊥−2tπcos (kxa)−2tδcos (kya),\n(2b)\n⟨φdzxk|Hkin|φdzxk⟩=E⊥−2tπcos (kya)−2tδcos (kxa),\n(2c)\nφdxyk|Hkin|φdyzk\u000b\n= 2iγsin (kxa), (2d)\nφdxyk|Hkin|φdzxk\u000b\n= 2iγsin (kya). (2e)\nHere, E∥andE⊥are on-site energies for the in-plane ( dxy)\nand out-of-plane ( dyzanddzx) orbitals, respectively, and tπ\nandtδare the nearest neighbor hopping amplitudes between\nt2gorbitals via the πandδbondings, respectively. The inver-\nsion symmetry breaking at the surface is equivalent to the po-\ntential gradient along the surface-normal direction ( z), which\npromotes hybridization between dxyanddyz, as well as dxy\nanddzxstates, with hopping integral γcharacterizing the\nstrength of the symmetry breaking. The orbital exchange in-\nteraction is written as\nHL\nexc=JL\n¯hˆML·L, (3)where Lis the atomic OAM operator, ˆMLdenotes the direc-\ntion of the orbital exchange field, and JLdenotes the coupling\nstrength of the orbital exchange interaction. For t2gorbitals,\nL= (Lx, Ly, Lz)becomes\nLx= ¯h\n0 0−i\n0 0 0\ni0 0\n, (4a)\nLy= ¯h\n0i0\n−i0 0\n0 0 0\n, (4b)\nLz= ¯h\n0 0 0\n0 0 i\n0−i0\n, (4c)\nin the matrix representation using the basis states\nφdxyk, φdyzk, and φdzxk.\nA spinfull model Hamiltonian is constructed by adding\nSOC Hsoc, and spin exchange interaction HS\nexc. The SOC\nis given by\nHsoc=2λsoc\n¯h2L·S, (5)\nwhere Sis the spin angular momentum operator and λsocis\nthe SOC strength. The spin exchange interaction is written as\nHS\nexc=JS\n¯hˆMS·S, (6)\nwhere ˆMSdenotes the direction of the spin exchange field,\nandJSdenotes the spin exchange interaction strength. The\nspin angular momentum operator is represented by the vector\nof the Pauli matrices σ= (σx, σy, σz)within each orbital\n⟨φnσk|S|φnσ′k⟩=¯h\n2[σ]σσ′, (7)\nwhere σ, σ′are spin indices (up or down).\nThe values of the model parameters used in the calcula-\ntion are E∥= 1.6, E⊥= 1.0, tπ= 0.5, tσ= 0.1, γ=\n0.02, λsoc= 0.04, JL= 1.0, JS= 1.0, all in unit of eV .\nThe lattice constant is set a= 1 ˚A. All parameters are set as\nabove unless specified otherwise.\nC. Berry phase expression for the orbital DMI\nWe define the orbital DMI as the energy contribution to the\nfree energy functional by the spiralization of the orbital ex-\nchange field ˆML,\nFL\nDMI=Z\nd2rX\nijDL\nijˆri·\u0010\nˆML×∂rjˆML\u0011\n, (8)\nwhere ˆriis the unit vector in the idirection, and ∂rjis the\npartial derivative in the jdirection ( i, j=x, y, z ). We derive\na microscopic expression for the orbital DMI coefficient DL\nij\nby means of the Berry phase theory, which has been applied in4\nthe past to calculate the spin DMI [55, 62, 63]. In this method,\nDL\nijcan be quantified by expanding the free energy in terms\nof small spatial gradients of orbital magnetization direction\nwithin quantum mechanical perturbation theory.\nGiven the orbital exchange interaction in the spinless\nHamiltonian in Eq. (3), we define the torque operator on the\norbital exchange field due to the OAM by\nTL=−1\ni¯h[L, HL\nexc] =JL\n¯hˆML×L. (9)\nFollowing the detailed derivation in Ref. [62], the first-\norder perturbation by the spiralization of ˆMLstarting from\na collinear configuration of ˆMLgives rise the the following\nexpression in terms of electronic states\nDL\nij=Zd2k\n(2π)2X\nnh\nf(εnk)An\nij(k)\n+1\nβln\u0010\n1 +e−β(εnk−µ)\u0011\nBn\nij(k)i\n, (10)\nwhere f(εnk) = 1 /[1 +eβ(εnk−µ)]is the Fermi-Dirac distri-\nbution function for the unperturbed band energy εkn,µis the\nelectrochemical potential, and β= 1/kBTfor the Boltzmann\nconstant kBand the temperature T. The quantities An\nij(k)and\nBn\nij(k)are given by the correlation between the orbital torque\noperator and the velocity operator,\nAn\nij(k) =−¯hX\nm̸=nIm\u0002\nunk\f\fTL\ni\f\fumk\u000b\n⟨umk|vj|unk⟩\u0003\nεnk−εmk,\n(11)\nand\nBn\nij(k) =−2¯hX\nm̸=nIm\u0002\nunk\f\fTL\ni\f\fumk\u000b\n⟨umk|vj|unk⟩\u0003\n(εnk−εmk)2,\n(12)\nwhere unkdenotes the lattice-periodic part of the unperturbed\nBloch wave function of band natkandvj=∂kjHkin/¯his\nthej-th component of the velocity operator.\nAt zero temperature Eq. (10) becomes\nDL\nij=Zd2k\n(2π)2X\nnf(εnk)\u0002\nAn\nij(k)−(εnk−µ)Bn\nij(k)\u0003\n.\n(13)\nBy utilizing TL=ˆML×∂ˆMLHtot, Eq. (13) becomes\nDL\nij=Zd2k\n(2π)2X\nnf(εnk)ˆri· (14)\nImh\nˆML× ⟨∂MLunk|(Htot+εnk−2µ)|∂kjunk⟩i\n.\nNote the similarity of this expression to the mixed Berry cur-\nvature\nΩML\nikj=−2ˆri·Imh\nˆML× ⟨∂MLunk|∂kjunk⟩i\n.(15)\nUp to date, the properties of orbital mixed Berry curvature\nin solid state systems have been not explored. According toEq. (14), the orbital spiralization, in analogy to spin spiral-\nization and modern theory of orbital magnetization [64–66],\ncomprises two terms: one, “itinerant” in the language of or-\nbital magnetization, proportional to the orbital mixed Berry\ncurvature and associated with the Bn\nij-part in Eq. (13), and an-\nother one, which should be treated as the “self-rotation” con-\ntribution, driven by the Ak\nij-part in Eq. (13). While the second\npart of the orbital DMI is arising from the local breaking of\ninversion symmetry, the itinerant mixed Berry curvature part\ncan be attributed to the breaking of inversion symmetry at the\nboundary of a finite sample taken to the thermodynamic limit.\nIn considering the spin, as mentioned above, the Hamil-\ntonian for spinfull systems necessitates the inclusion of both\nspin-orbit coupling and spin exchange interaction. In this\ncase, the spin torque operator can be represented as TS=\n(JS/¯h)ˆMS×S. From this, we can obtain the spin spiraliza-\ntionDS\nijthrough the above equations by replacing the orbital\nexchange field ˆMLby the spin exchange field ˆMS.\nIII. RESULTS\nA. Orbital DMI by orbital Rashba effect in k-space\nFirst, let us unravel the emergence of an intrinsic orbital\ntexture of t2gorbitals without considering the effect of SOC\nand orbital/spin exchange interaction. In Figs. 1(a)-(b) we\npresent the band structure of the model along the high sym-\nmetry directions and near Γ, respectively. We can see clearly\nin Fig. 1(b) the effect of orbital hybridization mediated by in-\nversion symmetry breaking, inducing avoided crossings of the\nbands with different orbital characters. In Figs. 1(c)-(e), we\nshow the distribution of the expectation value of OAM in k-\nspace near the Γpoint for the bands E1,k,E2,k, and E3,k,\nwhich are labeled in the increasing order of energy. We ob-\nserve that the OAM exhibits a chiral texture around the Γ\npoint, specifically, of clockwise, counter-clockwise, and zero\ncharacter around the Γpoint, respectively. The chiral behav-\nior of the t2gorbital textures is driven by the orbital Rashba\neffect of the form ˆz·(k×L), which is consistent with previous\nstudies [28]. Investigating the influence of SOC reveals that\nthe spin Rashba effect plays a crucial role in driving the spin\ntexture’s chirality appearing on top of the orbital distribution.\nAs a result, the spin textures are closely tied to the local orien-\ntation of the OAM in the Brillouin zone, manifesting as either\nparallel or antiparallel alignment with the underlying OAM\n(See supplementary section S1 [67]).\nNext, we consider the effect of the orbital Zeeman ef-\nfectHL\nexcdue to the orbital exchange field ˆML(keeping the\nvalue of JLat 1.0 eV). In the two-dimensional square lat-\ntice with an out-of-plane inversion symmetry breaking, the\norbital DMI spiralization is described by an antisymmetric\ntensor due to the four-fold rotational symmetry around the\naxis normal to the film, which amounts to DL\nxx=DL\nyy= 0\nandDL\nyx=−DL\nxy. We compute the magnitude of DL\nyxand\npresent its magnitude in Fig. 2(a) as a function of band filling\nas given by the position of the Fermi level EF, without SOC5\nFIG. 1. Orbital texture of the spinless model. (a) The bandstruc-\nture of the t2gmodel along the high symmetry directions for two-\ndimensional square lattice calculated without SOC and spin/orbital\nexchange interaction. (b) The band structure near Γcorresponding\nto the region marked with an orange circle in (a). The blue circle\nmarks the region of strong orbital hybridization mediated by inver-\nsion symmetry breaking. (c)-(e) Band-resolved orbital textures in\nthe Brillouin zone, where E1,k,E2,k, and E3,kdenote the different\nbands as indicated in (a). The direction of orbital angular momentum\nis shown with an arrow with a length corresponding to its magnitude\n(in arbitrary units).\nand with the orbital exchange field pointing out of the plane.\nThe computed oscillatory behavior is strongly reminiscent of\nthe Fermi energy dependence of the spin DMI typical for thin\nfilms of transition metals [63]. Remarkably, we observe that\nthe magnitude of orbital DMI spiralization DL\nyxwithin our\nmodel reaches a large value of 16.8meV/ ˚A for EF= 2.7eV ,\nwhich is comparable to the theoretical magnitude of spin DMI\nin typical transition-metal systems, such as e.g. Co/Pt, Co/Ir\nand Co/Au thin films [62, 63]. This finding serves as com-\npelling evidence that significant orbital DMI can manifest in\na system lacking SOC but with inversion symmetry breaking\nwhen this system possesses a strong orbital exchange interac-\ntion.\nMore insight can be gained from a band-projected analy-\nsis of orbital DMI at EF= 2.7eV , as shown in Fig. 2(b),\nwhere the color marks the value of projected orbital DMI onto\neach band. We observe that significant orbital DMI can be\nobserved predominantly in the vicinity of the avoided band\ncrossing between ΓandMpoints in the Brillouin zone. As\nshown in Fig. 2(c), the k-resolved orbital DMI displays a very\nspiky behavior in this region, which corresponds to the region\nof orbital hybridization associated with inversion symmetry\nbreaking. At this Fermi energy, another significant contribu-\ntion to the orbital DMI can be seen between M and X in the\nregion of another band anti-crossing along that path. This un-\nderlines the crucial importance of the band-crossing points for\nachieving large values of the orbital DMI in realistic systems.\nGiven a close relation between the orbital DMI and orbital\nFIG. 2. Orbital DMI without SOC . (a) The orbital DMI variation\nwith respect to the position of Fermi energy without SOC. (b) The\nband-projected orbital DMI along the high symmetry path in the Bril-\nlouin zone at EF= 2.7eV . The color marks the value of orbital DMI\nprojected on each band. The projected orbital DMI is pronounced in\nthe vicinity of the band avoided crossing between ΓandMpoints.\n(c) The k-resolved orbital DMI along the high symmetry path in the\nBrillouin zone at EF= 2.7eV .\nFIG. 3. Orbital mixed Berry curvature . Distribution of the orbital\nmixed Berry curvature ΩMLykx(left) and the orbital Dzyaloshinskii-\nMoriya spiralization DL\nyx(right) of all occupied bands in the Bril-\nlouin zone of the model for two Fermi energy values: (a)-(b) EF=\n1.0eV and (c)-(d) EF= 2.7eV . All calculations are performed\nwithout considering SOC. The data are shown on a logarithmic scale,\nsgn(x) log(1 + |x|).6\nmixed Berry curvature, it is insightful to compare the two\nquantities directly. In Fig. 3 we plot the k-resolved distri-\nbution of the orbital spiralization DL\nyxand of the mixed Berry\ncurvature ΩMLykx, which are summed over the occupied bands\nbelow EF= 1.0eV and EF= 2.7eV , respectively. At\nEF= 1.0eV , both ΩMLykxandDL\nyxexhibit complex patterns\nwith significant background contributions arising from broad\nareas of the Brillouin zone. Moreover, they show relatively\nsmall values throughout the entire Brillouin zone, consistent\nwith the low values of the orbital DMI at this energy as visi-\nble in Fig. 1(a). In contrast, when EF= 2.7eV , both ΩMLykx\nandDL\nyxare sharply peaked in narrow regions of the Brillouin\nzone, corresponding to quasi-nodal lines arising due to near\ndegenerate bands crossing the Fermi level. Clearly, the nodal\nlines serve as prominent sources of both orbital mixed Berry\ncurvature and orbital DMI, with the two quantities being di-\nrectly correlated in sign and magnitude.\nNext, we explore the impact of SOC on orbital DMI.\nNamely, we perform calculations to examine the energy de-\npendence, band dispersion, and k-space distribution of orbital\nDMI while considering the presence of SOC. Remarkably, we\nfind that the qualitative nature of orbital DMI remains unaf-\nfected by SOC, see supplementary sections S2 and S3 [67].\nOur findings align with previous studies on other orbital ef-\nfects, such as the orbital Rashba effect and orbital Hall ef-\nfect [4, 20], indicating that SOC does not play a decisive role\nin the emergence of orbital DMI, although it impacts its mag-\nnitude via the influence on the electronic structure details, see\nalso discussion below.\nB. Anatomy of orbital DMI\nWe analyze in great detail to reveal the anatomy of the or-\nbital DMI. At this point, by keeping the SOC strength at a\nconstant value of 0.04 eV , we study the behavior of the or-\nbital DMI in response to changing the key parameters of the\nsystem −the strength of orbital exchange coupling JL, the\nstrength of inversion symmetry breaking γ, and the angle θ\nthat the orbital exchange field makes with the z-axis−pre-\nsenting the results in Fig. 4. In Fig. 4(a), the orbital DMI DL\nyx\nis shown as a function of the Fermi energy EF, for increas-\ning orbital exchange coupling strength JL. It is noteworthy\nthat with increasing JL, both the amplitude and position of\nthe peak exhibit distinct deviations. Specifically, the height\nof the peak exhibits a positive correlation with JL, whereas\nthe peak position undergoes an upward shift with increasing\nJL. This observation underscores the critical significance of\norbital exchange coupling in determining the magnitude of the\norbital DMI. When JLis sufficiently small, the orbital DMI\nassumes a significantly diminished value, aligning with the\ngeneral scenario in practical materials where the orbital Zee-\nman effect is negligibly small and consequently conceals the\nmanifestation of the orbital DMI.\nIn order to gain a deeper understanding of the trend of\norbital DMI with respect to the orbital exchange coupling\nstrength JL, we conduct band-projected as well as k-resolved\ncalculations of orbital DMI. In Fig. 5 we compare two cases\nFIG. 4. The anatomy of orbital DMI . The orbital DMI\u0000\nDL\nyx\u0001\nas a\nfunction of the Fermi energy (EF)for (a) different orbital exchange\ncoupling strengths JLand (b) inversion symmetry breaking strengths\nγ. The orbital DMI\u0000\nDL\nyx\u0001\nas a function of the Fermi energy (EF)\nfor different angles θof the orbital exchange field with the z-axis in\nthe ranges of (c) 0◦to45◦and (d) 45◦to90◦. All calculations take\ninto account the effect of SOC ( λsoc= 0.04eV).\n−JL= 0.3eV and JL= 0.7eV−with the Fermi energy set\nto the peak position for each case. It is evident from Figs. 5(a)\nand (b) that as JLincreases, the position of the band avoided\ncrossing region moves up in energy in response to band dy-\nnamics. This is directly reflected in a shift of the peak in\norbital DMI as seen in Fig. 4(a). Furthermore, by compar-\ning Figs. 5(c) and (d), we observe an increase in the absolute\nvalue of DL\nyx(k)in the k-region associated with orbital hy-\nbridization, resulting in an enhanced peak height in Fig. 4(a)\nwith increasing JL. The analysis for other values of JLin\npresented in the supplementary section S4 [67].\nIn Fig. 4(b) we show the variation of the orbital DMI DL\nyx\nwith the inversion symmetry breaking strength as given by\nparameter γ. As mentioned above, the strength of inversion\nsymmetry breaking directly influences the “intensity” of or-\nbital hybridization, ultimately resulting in the formation of\nchiral orbital textures. We observe that the orbital DMI mono-\ntonically increases with the gradual increase of γ, and zero γ\nresults in vanishing orbital DMI. This highlights the decisive\nrole of inversion symmetry breaking as a dominant factor for\norbital DMI. The presence of ISB, along with the induced or-\nbital hybridization and chiral orbital textures, governs the ex-\nistence of orbital DMI, consistent with previous studies on the7\nFIG. 5. Orbital DMI with varying JL. The band-projected orbital\nDMI along the high symmetry path in the Brillouin zone at (a) EF=\n0.909eV for JL= 0.3eV and (b) EF= 1.939eV for JL= 0.7eV .\nThe color marks the value of orbital DMI projected on each band.\nClearly, the band avoided crossing region contributes primarily to\nthe orbital DMI. (c-d) The k-resolved orbital DMI along the high\nsymmetry path in the Brillouin zone corresponding to the case in (a)\nand (b), respectively.\norbital Hall effect [4].\nThe analysis of band-projected and k-resolved orbital DMI\nforγ= 0.005eV and γ= 0.015eV (results for other val-\nues of γcan be found in the supplementary section S5 [67]),\nshown in Figs. 6(a) and 6(b), reveals that the position of the or-\nbital hybridization points, which contribute predominantly to\nthe orbital DMI, remains unaffected as γundergoes variation\nwithin the considered range of values. Consequently, there is\nno discernible shift of the peak position in the orbital DMI\ncurve, as evident from Fig. 4(b). As shown in Figs. 6(c) and\n6(d), it is also directly evident that the increase in the degree\nof inversion symmetry breaking results in an amplification of\norbital hybridization at the respective hybridization points, as\nreflected in the degree of orbital mixing [68]. This leads to an\nenhancement of the local orbital DMI, and an increase in the\npeak height of the orbital DMI curve in Fig. 4(b).\nAt last, we present in Figs. 4(c) and 4(d) the dependence\nofDL\nyxon the angle of the orbital exchange field θ. Specif-\nically, the range of θfrom 0◦to45◦is depicted in Fig. 4(c),\nwhile the range from 45◦to90◦is shown in Fig. 4(d). We\ncan clearly observe that for θless than 45◦, DL\nyxgradually de-\ncreases with increasing θat around EF= 2.7eV, while in\nother energy regions the DMI values stay stable. On the other\nhand, for θvalues above 45◦, the situation is reversed, and it is\nthe DMI between 0 and 1 eV which is increasing rapidly with\nincreasing θon the background of relative stability at other en-\nergies. Such behavior is a manifestation of strong anisotropy\nof orbital DMI with respect to the direction of the orbital ex-\nchange field. To understand this effect better, we plot the\nband-projected and k-resolved orbital DMI for two orienta-\ntions of the orbital exchange field: the out-of-plane direction\n(θ= 0◦), and the in-plane direction ( θ= 90◦). As shown\nin Fig. 7, drastic changes in the band structure are observed\nFIG. 6. Orbital DMI with varying γ. The band-projected orbital\nDMI along the high symmetry path in the Brillouin zone at EF=\n2.7eV for (a) γ= 0.005eV and (b) γ= 0.015eV . The color marks\nthe value of orbital DMI projected on each band. Clearly, the band\navoided crossing region contributes primarily to the orbital DMI. (c-\nd) The k-resolved orbital DMI along the high symmetry path in the\nBrillouin zone corresponding to the case in (a) and (b), respectively.\nFIG. 7. Orbital DMI with varying θ. The band-projected orbital\nDMI along the high symmetry path in the Brillouin zone at (a) EF=\n2.7eV for θ= 0◦and (b) EF= 0.2eV for θ= 90◦. The color\nmarks the value of orbital DMI projected on each band. (c-d) The k-\nresolved orbital DMI along the high symmetry path in the Brillouin\nzone corresponding to the case in (a) and (b), respectively.\nas the orbital exchange field direction varies. As a result, the\nposition of the band avoided crossing region that mainly con-\ntributes to the orbital DMI experiences alteration. Simulta-\nneously, the degree of orbital hybridization also exhibits pro-\nnounced variation, manifested as changes in the magnitude\nof orbital DMI at the hybridization positions. These factors\ncombined lead to modifications in both the peak position and\npeak height of the orbital DMI curve as the orbital exchange\nfield direction transitions from the out-of-plane to the in-plane\nconfiguration, showcasing a strong anisotropy, as depicted in\nFigs. 4 (c) and 4(d).8\nC. Relation between orbital DMI and spin DMI\nFinally, we consider the case when in addition to the orbital\nexchange and SOC the spin exchange interaction as given by\nHS\nexcis also present in the Hamiltonian. We keep the values\nofJLandJSat 1 eV , while keeping the out-of-plane direc-\ntions of the spin and orbital exchange fields, and compute both\nspin and orbital DMI by following the Berry phase theory out-\nlined above. We focus specifically on the dependence of the\nspin and orbital DMI on the SOC strength, presenting the re-\nsults in Fig. 8. First, we consider the regime of small λsoc,\nFigs. 8(a-b). We observe that in this regime the orbital DMI\nexhibits minimal variations, whereas the spin DMI consis-\ntently increases with increasing the SOC strength, vanishing\nidentically without spin-orbit interaction. More importantly,\nwe observe that for the energies above +1.5 eV the qualitative\nbehavior of the spin and orbital DMI is quite similar in that\nboth quantities develop large peaks at about 2.2 and 3.3 eV ,\nalbeit of opposite sign in the case of spin.\nTo understand the origin of this correlation, we scrutinize\nthe band-resolved contributions to the spin and orbital DMI\nat the value of λsoc= 0.04eV and EF= 2.2eV , shown\nin Figs. 8(e-f). We observe that in the discussed region of\nenergy both falvors of DMI come from the same anticross-\nings in the electronic structure along ΓM. These anticrossings\nare nothing else but the ISB-driven hybridization points be-\ntween orbitally-different bands −discussed in depth above\nand shifted in energy by spin exchange splitting −which in-\nduce orbital DMI without SOC. We thus come to a fundamen-\ntal conclusion that the orbital DMI is the effect which is parent\nto spin DMI. As in the case of a relation between the orbital\nHall effect and spin Hall effect [4], the spin DMI is “pulled”\nby the orbital DMI via SOC. In the case when DMI-driving\nhybridizations are well-defined in energy and k-space, such\nas e.g. for energies above +1.5eV , the behavior of spin and\norbital DMI can be very similar but not necessarily identical:\nfor example, in the case considered here, the switch in sign\ncorrelation between DL\nyxandDS\nyxat+2.2and+3.3eV can\nbe explained by the opposite sign of the spin-orbit correlation\n⟨LS⟩of participating bands [4] at these energies, similarly to\nthe case of Hall effects.\nAs a result, when at a given energy the DMI contributions\nare small and spread over several bands with different orbital\ncharacter filling larger areas of k-space, as it is the case be-\nlow+1eV , Figs. 8(a-b), the correlation between DL\nyxandDS\nyx\nis much less pronounced, or not at all present. This is ulti-\nmately the reason why it is difficult to observe a direct relation\nbetween DL\nyxandDS\nyxfor the case of larger SOC strength,\nshown in Figs. 8(c-d). In the latter case, the strongly mod-\nified by SOC bands develop larger areas in k-space where\nthe effect of SOC on the DMI is active. This is also con-\nsistent with the saturation in the values of the spin DMI with\nincreasing SOC towards the values comparable to the strength\nof exchange interaction. It is especially worth mentioning that\nwhen the strength of orbital exchange interaction JLis similar\nto the magnitude of spin exchange JS, the values of the orbital\nDMI on average exceed those of the spin DMI by an order of\nmagnitude. This underlines the fact that the orbital angular\nFIG. 8. Variation of spin and orbital DMI with SOC . (a) The or-\nbital DMI\u0000\nDL\nyx\u0001\nand (b) spin DMI\u0000\nDS\nyx\u0001\nas a function of the Fermi\nenergy (EF)for various SOC strengths λin small SOC regime. (c)\nDL\nyxand (d) DS\nyxin large SOC regime. (e-f) The band-resolved con-\ntributions to the orbital (e) and spin (f) DMI for spin-orbit strength\nofλsoc= 0.04eV at EF= 2.2eV .\nmomentum in solids is much more prone to the effects of chi-\nralization when compared to spin, due to its stronger coupling\nto the lattice and qualitatively different energetics of orbital\ndynamics relying on crystal field structure rather than spin-\norbit interaction.\nIV . CONCLUSION\nIn summary, we theoretically demonstrate that the chiral\nexchange interaction among orbital moments, closely resem-\nbling the spin Dzyaloshinskii-Moriya interaction, can arise\nin a two-dimensional square lattice with t2gorbitals. We\nfind that orbital hybridization, induced by inversion symme-\ntry breaking, is crucial for generating orbital DMI. Further-\nmore, the position and strength of orbital hybridization have\na crucial impact on both the magnitude and distribution of or-\nbital DMI. We also argue that in the vicinity of isolated Fermi9\nsurface features the orbital DMI is closely correlated with the\nbehavior of the orbital mixed Berry curvature. Moreover, the\nstrength of orbital exchange interaction exerts a decisive im-\npact on the magnitude of orbital DMI. In situations where the\nstrength of the orbital exchange interaction is comparable to\nthat of the spin exchange interaction, the orbital DMI can ex-\nceed the spin DMI by an order of magnitude. Importantly,\nconsistent with other orbital phenomena, the orbital DMI can\nalso emerge in the absence of spin-orbit interaction, subse-\nquently inducing the spin DMI through the effect of SOC.\nWhile with our work we provide first basic insights into the\nphysics and properties of orbital DMI, further experimental\nverification is required to validate the existence and control of\norbital DMI. On the other hand, more effort is needed to sug-\ngest specific material candidates where the formation of chiralorbital textures could be experimentally observed.\nACKNOWLEDGMENTS\nThis work was supported by the Deutsche Forschungs-\ngemeinschaft (DFG, German Research Foundation) - TRR\n173/2 - 268565370 (project A11), and TRR 288 – 422213477\n(project B06). The authors would also like to thank the sup-\nports by the projects from National Natural Science Foun-\ndation of China (No.61627813, 62204018 and 61571023),\nthe Beijing Municipal Science and Technology Project under\nGrant Z201100004220002, the National Key Technology Pro-\ngram of China 2017ZX01032101, the Program of Introducing\nTalents of Discipline to Universities in China (No. B16001),\nthe VR innovation platform from Qingdao Science and Tech-\nnology Commission.\n[1] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Orbitronics:\nThe intrinsic orbital current in p-doped silicon, Phys. Rev. Lett.\n95, 066601 (2005).\n[2] V . o. T. Phong, Z. Addison, S. Ahn, H. Min, R. Agarwal, and\nE. J. Mele, Optically controlled orbitronics on a triangular lat-\ntice, Phys. Rev. Lett. 123, 236403 (2019).\n[3] D. Go, D. Jo, H.-W. Lee, M. Kl ¨aui, and Y . Mokrousov, Or-\nbitronics: Orbital currents in solids, Europhysics Letters 135,\n37001 (2021).\n[4] D. Go, D. Jo, C. Kim, and H.-W. Lee, Intrinsic spin and orbital\nhall effects from orbital texture, Phys. Rev. Lett. 121, 086602\n(2018).\n[5] D. Jo, D. Go, and H.-W. Lee, Gigantic intrinsic orbital hall\neffects in weakly spin-orbit coupled metals, Phys. Rev. B 98,\n214405 (2018).\n[6] S. Bhowal and S. Satpathy, Intrinsic orbital moment and predic-\ntion of a large orbital hall effect in two-dimensional transition\nmetal dichalcogenides, Phys. Rev. B 101, 121112 (2020).\n[7] S. Bhowal and S. Satpathy, Intrinsic orbital and spin hall ef-\nfects in monolayer transition metal dichalcogenides, Phys. Rev.\nB102, 035409 (2020).\n[8] L. M. Canonico, T. P. Cysne, T. G. Rappoport, and R. B. Mu-\nniz, Two-dimensional orbital hall insulators, Phys. Rev. B 101,\n075429 (2020).\n[9] L. M. Canonico, T. P. Cysne, A. Molina-Sanchez, R. B. Muniz,\nand T. G. Rappoport, Orbital hall insulating phase in transition\nmetal dichalcogenide monolayers, Phys. Rev. B 101, 161409\n(2020).\n[10] T. P. Cysne, M. Costa, L. M. Canonico, M. B. Nardelli, R. B.\nMuniz, and T. G. Rappoport, Disentangling orbital and val-\nley hall effects in bilayers of transition metal dichalcogenides,\nPhys. Rev. Lett. 126, 056601 (2021).\n[11] P. Sahu, S. Bhowal, and S. Satpathy, Effect of the inversion\nsymmetry breaking on the orbital hall effect: A model study,\nPhys. Rev. B 103, 085113 (2021).\n[12] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima,\nK. Yamada, and J. Inoue, Intrinsic spin hall effect and orbital\nhall effect in 4dand5dtransition metals, Phys. Rev. B 77,\n165117 (2008).\n[13] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. In-\noue, Giant orbital hall effect in transition metals: Origin of largespin and anomalous hall effects, Phys. Rev. Lett. 102, 016601\n(2009).\n[14] S. Bhowal and G. Vignale, Orbital hall effect as an alternative to\nvalley hall effect in gapped graphene, Phys. Rev. B 103, 195309\n(2021).\n[15] J. Kim, D. Go, H. Tsai, D. Jo, K. Kondou, H.-W. Lee, and\nY . Otani, Nontrivial torque generation by orbital angular mo-\nmentum injection in ferromagnetic-metal/ Cu/al2o3trilayers,\nPhys. Rev. B 103, L020407 (2021).\n[16] S. Lee, M.-G. Kang, D. Go, D. Kim, J.-H. Kang, T. Lee, G.-H.\nLee, J. Kang, N. J. Lee, Y . Mokrousov, S. Kim, K.-J. Kim, K.-J.\nLee, and B.-G. Park, Efficient conversion of orbital hall current\nto spin current for spin-orbit torque switching, Commun. Phys.\n4, 234 (2021).\n[17] S. Ding, Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu, S. Becker,\nW. Yang, H. Du, C. Wang, Y . Yang, G. Jakob, M. Kl ¨aui,\nY . Mokrousov, and J. Yang, Observation of the orbital rashba-\nedelstein magnetoresistance, Phys. Rev. Lett. 128, 067201\n(2022).\n[18] Y . Tazaki, Y . Kageyama, H. Hayashi, T. Harumoto, T. Gao,\nJ. Shi, and K. Ando, Current-induced torque originating from\norbital current (2020), arXiv:2004.09165 [cond-mat.mtrl-sci].\n[19] S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth,\nS. Becker, F. Kammerbauer, J. Yang, G. Jakob, Y . Mokrousov,\nand M. Kl ¨aui, Harnessing orbital-to-spin conversion of interfa-\ncial orbital currents for efficient spin-orbit torques, Phys. Rev.\nLett. 125, 177201 (2020).\n[20] D. Go, J.-P. Hanke, P. M. Buhl, F. Freimuth, G. Bihlmayer, H.-\nW. Lee, Y . Mokrousov, and S. Bl ¨ugel, Toward surface orbitron-\nics: giant orbital magnetism from the orbital rashba effect at the\nsurface of sp-metals, Sci. Rep. 7, 46742 (2017).\n[21] Y .-G. Choi, D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H. G. Park,\nC. Kim, B.-C. Min, G.-M. Choi, and H.-W. Lee, Observation of\nthe orbital hall effect in a light metal ti, Nature 619, 52 (2023).\n[22] I. V . Tokatly, Orbital momentum hall effect in p-doped\ngraphane, Phys. Rev. B 82, 161404 (2010).\n[23] G. Sala and P. Gambardella, Giant orbital hall effect and orbital-\nto-spin conversion in 3d,5d, and 4fmetallic heterostructures,\nPhys. Rev. Res. 4, 033037 (2022).\n[24] A. Johansson, B. G ¨obel, J. Henk, M. Bibes, and I. Mertig, Spin\nand orbital edelstein effects in a two-dimensional electron gas:10\nTheory and application to srtio 3interfaces, Phys. Rev. Res. 3,\n013275 (2021).\n[25] L. Salemi, M. Berritta, A. K. Nandy, and P. M. Oppeneer, Or-\nbitally dominated rashba-edelstein effect in noncentrosymmet-\nric antiferromagnets, Nat. Commun. 10, 5381 (2019).\n[26] T. Yoda, T. Yokoyama, and S. Murakami, Orbital edelstein ef-\nfect as a condensed-matter analog of solenoids, Nano Lett. 18,\n916 (2018).\n[27] J. Hong, J.-W. Rhim, C. Kim, S. Ryong Park, and J. Hoon Shim,\nQuantitative analysis on electric dipole energy in rashba band\nsplitting, Sci. Rep. 5, 13488 (2015).\n[28] P. Kim, K. T. Kang, G. Go, and J. H. Han, Nature of orbital and\nspin rashba coupling in the surface bands of srtio 3andktao 3,\nPhys. Rev. B 90, 205423 (2014).\n[29] J.-H. Park, C. H. Kim, H.-W. Lee, and J. H. Han, Orbital chiral-\nity and rashba interaction in magnetic bands, Phys. Rev. B 87,\n041301 (2013).\n[30] J.-H. Park, C. H. Kim, J.-W. Rhim, and J. H. Han, Orbital rashba\neffect and its detection by circular dichroism angle-resolved\nphotoemission spectroscopy, Phys. Rev. B 85, 195401 (2012).\n[31] S. R. Park, C. H. Kim, J. Yu, J. H. Han, and C. Kim, Orbital-\nangular-momentum based origin of rashba-type surface band\nsplitting, Phys. Rev. Lett. 107, 156803 (2011).\n[32] D. Go, D. Jo, T. Gao, K. Ando, S. Bl ¨ugel, H.-W. Lee, and\nY . Mokrousov, Orbital rashba effect in a surface-oxidized cu\nfilm, Phys. Rev. B 103, L121113 (2021).\n[33] D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun, D. Jo,\nS. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park, B.-C. Min,\nH. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee, Orbital torque in\nmagnetic bilayers, Nat. Commun. 12, 6710 (2021).\n[34] D. Go, F. Freimuth, J.-P. Hanke, F. Xue, O. Gomonay, K.-J.\nLee, S. Bl ¨ugel, P. M. Haney, H.-W. Lee, and Y . Mokrousov,\nTheory of current-induced angular momentum transfer dynam-\nics in spin-orbit coupled systems, Phys. Rev. Res. 2, 033401\n(2020).\n[35] D. Go and H.-W. Lee, Orbital torque: Torque generation by\norbital current injection, Phys. Rev. Res. 2, 013177 (2020).\n[36] D. Go, D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park,\nS. Bl ¨ugel, H.-W. Lee, and Y . Mokrousov, Long-range orbital\ntorque by momentum-space hotspots, Phys. Rev. Lett. 130,\n246701 (2023).\n[37] A. Bose, F. Kammerbauer, R. Gupta, D. Go, Y . Mokrousov,\nG. Jakob, and M. Kl ¨aui, Detection of long-range orbital-hall\ntorques, Phys. Rev. B 107, 134423 (2023).\n[38] H. Hayashi, D. Jo, D. Go, T. Gao, S. Haku, Y . Mokrousov, H.-\nW. Lee, and K. Ando, Observation of long-range orbital trans-\nport and giant orbital torque, Commun. Phys. 6, 32 (2023).\n[39] M. Robertson, C. J. Agostino, G. Chen, S. P. Kang, A. Mas-\ncaraque, E. Garcia Michel, C. Won, Y . Wu, A. K. Schmid, and\nK. Liu, In-plane n ´eel wall chirality and orientation of interfa-\ncial dzyaloshinskii-moriya vector in magnetic films, Phys. Rev.\nB102, 024417 (2020).\n[40] W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch,\nF. Y . Fradin, J. E. Pearson, Y . Tserkovnyak, K. L. Wang,\nO. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, Blowing\nmagnetic skyrmion bubbles, Science 349, 283 (2015).\n[41] W. Jiang, G. Chen, K. Liu, J. Zang, S. G. te Velthuis, and\nA. Hoffmann, Skyrmions in magnetic multilayers, Physics Re-\nports 704, 1 (2017).\n[42] R. Chen, X. Wang, H. Cheng, K.-J. Lee, D. Xiong, J.-Y .\nKim, S. Li, H. Yang, H. Zhang, K. Cao, M. Kl ¨aui, S. Peng,\nX. Zhang, and W. Zhao, Large dzyaloshinskii-moriya interac-\ntion and room-temperature nanoscale skyrmions in cofeb/mgo\nheterostructures, Cell Reports Phys. Sci. 2, 100618 (2021).[43] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. Ben-\njamin Jungfleisch, J. Pearson, X. Cheng, O. Heinonen, K. L.\nWang, Y . Zhou, A. Hoffmann, and S. te Velthuis, Direct obser-\nvation of the skyrmion hall effect, Nat. Phys. 13, 162 (2017).\n[44] Y . Guan, X. Zhou, T. Ma, R. Bl ¨asing, H. Deniz, S.-H. Yang, and\nS. S. P. Parkin, Increased efficiency of current-induced motion\nof chiral domain walls by interface engineering, Adv. Mater. 33,\n2007991 (2021).\n[45] I. Dzyaloshinsky, A thermodynamic theory of “weak” ferro-\nmagnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241\n(1958).\n[46] T. Moriya, Anisotropic superexchange interaction and weak fer-\nromagnetism, Phys. Rev. 120, 91 (1960).\n[47] A. Fert and P. M. Levy, Role of anisotropic exchange interac-\ntions in determining the properties of spin-glasses, Phys. Rev.\nLett. 44, 1538 (1980).\n[48] K. Yamamoto, A.-M. Pradipto, K. Nawa, T. Akiyama, T. Ito,\nT. Ono, and K. Nakamura, Interfacial Dzyaloshinskii-Moriya\ninteraction and orbital magnetic moments of metallic multilayer\nfilms, AIP Adv. 7, 056302 (2016).\n[49] A. Belabbes, G. Bihlmayer, S. Bl ¨ugel, and A. Manchon,\nOxygen-enabled control of dzyaloshinskii-moriya interaction\nin ultra-thin magnetic films, Sci. Rep. 6, 24634 (2016).\n[50] H. T. Nembach, E. Ju ´e, E. R. Evarts, and J. M. Shaw, Correla-\ntion between dzyaloshinskii-moriya interaction and orbital an-\ngular momentum at an oxide-ferromagnet interface, Phys. Rev.\nB101, 020409 (2020).\n[51] L. Zhu, L. Zhu, X. Ma, X. Li, and R. A. Buhrman, Critical role\nof orbital hybridization in the dzyaloshinskii-moriya interaction\nof magnetic interfaces, Commun. Phys. 5, 151 (2022).\n[52] S. Kim, K. Ueda, G. Go, P.-H. Jang, K.-J. Lee, A. Be-\nlabbes, A. Manchon, M. Suzuki, Y . Kotani, T. Nakamura,\nK. Nakamura, T. Koyama, D. Chiba, K. T. Yamada, D.-H.\nKim, T. Moriyama, K.-J. Kim, and T. Ono, Correlation of the\ndzyaloshinskii–moriya interaction with heisenberg exchange\nand orbital asphericity, Nat. Commun. 9, 1648 (2018).\n[53] M. I. Katsnelson, Y . O. Kvashnin, V . V . Mazurenko, and A. I.\nLichtenstein, Correlated band theory of spin and orbital contri-\nbutions to dzyaloshinskii-moriya interactions, Phys. Rev. B 82,\n100403 (2010).\n[54] P. Kim and J. H. Han, Orbital dzyaloshinskii-moriya exchange\ninteraction, Phys. Rev. B 87, 205119 (2013).\n[55] F. Freimuth, R. Bamler, Y . Mokrousov, and A. Rosch, Phase-\nspace berry phases in chiral magnets: Dzyaloshinskii-moriya\ninteraction and the charge of skyrmions, Phys. Rev. B 88,\n214409 (2013).\n[56] A. Belabbes, G. Bihlmayer, F. Bechstedt, S. Bl ¨ugel, and\nA. Manchon, Hund’s rule-driven dzyaloshinskii-moriya inter-\naction at 3d−5dinterfaces, Phys. Rev. Lett. 117, 247202\n(2016).\n[57] H. Yang, A. Thiaville, S. Rohart, A. Fert, and M. Chshiev,\nAnatomy of dzyaloshinskii-moriya interaction at Co/Ptinter-\nfaces, Phys. Rev. Lett. 115, 267210 (2015).\n[58] V . Kashid, T. Schena, B. Zimmermann, Y . Mokrousov,\nS. Bl ¨ugel, V . Shah, and H. G. Salunke, Dzyaloshinskii-moriya\ninteraction and chiral magnetism in 3d−5dzigzag chains:\nTight-binding model and ab initio calculations, Phys. Rev. B\n90, 054412 (2014).\n[59] M. Zeer, D. Go, P. Schmitz, T. G. Saunderson, H. Wang,\nJ. Ghabboun, S. Bl ¨ugel, W. Wulfhekel, and Y . Mokrousov,\nPromoting p-based hall effects by p-d-fhybridization in\ngd-based dichalcogenides (2023), arXiv:2308.08207 [cond-\nmat.mes-hall].11\n[60] S. Han, H.-W. Ko, J. H. Oh, H.-W. Lee, K.-J. Lee, and K.-\nW. Kim, Theory of orbital pumping (2023), arXiv:2311.00362\n[cond-mat.mes-hall].\n[61] D. Go, K. Ando, A. Pezo, S. Bl ¨ugel, A. Manchon, and\nY . Mokrousov, Orbital pumping by magnetization dynamics in\nferromagnets (2023), arXiv:2309.14817 [cond-mat.mes-hall].\n[62] F. Freimuth, S. Bl ¨ugel, and Y . Mokrousov, Berry phase theory\nof dzyaloshinskii–moriya interaction and spin–orbit torques, J.\nPhys. Condens. Matter 26, 104202 (2014).\n[63] J.-P. Hanke, F. Freimuth, S. Bl ¨ugel, and Y . Mokrousov, Higher-\ndimensional wannier interpolation for the modern theory of the\ndzyaloshinskii–moriya interaction: Application to co-based tri-layers, J. Phys. Soc. Japan 87, 041010 (2018).\n[64] T. THONHAUSER, Theory of orbital magnetization in solids,\nInt. J. Mod. Phys. B 25, 1429 (2011).\n[65] R. Resta, Electrical polarization and orbital magnetization: the\nmodern theories, J. Phys.: Condens. Matter 22, 123201 (2010).\n[66] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on elec-\ntronic properties, Rev. Mod. Phys. 82, 1959 (2010).\n[67] R. Chen, (2023), See Supplemental Material.\n[68] M. Zeer, D. Go, J. P. Carbone, T. G. Saunderson, M. Redies,\nM. Kl ¨aui, J. Ghabboun, W. Wulfhekel, S. Bl ¨ugel, and\nY . Mokrousov, Spin and orbital transport in rare-earth dichalco-\ngenides: The case of eus2, Phys. Rev. Mater. 6, 074004 (2022)." }, { "title": "1905.12539v1.Spin_Orbit_Coupling_and_the_Evolution_of_Transverse_Spin.pdf", "content": "Spin-Orbit Coupling and the Evolution of Transverse Spin\nJ org S. Eismann,1, 2Peter Banzer,1, 2and Martin Neugebauer1, 2,\u0003\n1Max Planck Institute for the Science of Light, Staudtstr. 2, D-91058 Erlangen, Germany\n2Institute of Optics, Information and Photonics,\nUniversity Erlangen-Nuremberg, Staudtstr. 7/B2, D-91058 Erlangen, Germany\nWe investigate the evolution of transverse spin in tightly focused circularly polarized beams of\nlight, where spin-orbit coupling causes a local rotation of the polarization ellipses upon propaga-\ntion through the focal volume. The e\u000bect can be explained as a relative Gouy-phase shift between\nthe circularly polarized transverse \feld and the longitudinal \feld carrying orbital angular momen-\ntum. The corresponding rotation of the electric transverse spin density is observed experimentally\nby utilizing a recently developed reconstruction scheme, which relies on transverse-spin-dependent\ndirectional scattering of a nano-probe.\nI. INTRODUCTION\nThe investigation of spin-orbit interactions of light has\nbecome an integral \feld in modern optics, with a huge va-\nriety of related e\u000bects being relevant for a manifold of ap-\nplications [1]. Spin-orbit coupling plays an important role\nin the design of spin-dependent meta-surfaces [2{4], liq-\nuid crystal mode converters [5, 6], and directional waveg-\nuide and plasmon couplers [7{9], etc. Furthermore, it is\nof relevance in the \feld of super-resolution microscopy in\nthe context of proper depletion beams [10, 11], and in\noptical manipulation experiments [12, 13].\nThe spin-orbit coupling also occurs naturally when a\ncircularly polarized beam is focused [1]. The arising or-\nbital angular momentum can thereby be described by a\ngeometric Berry-phase e\u000bect [14], where the longitudinal\ncomponent of the \feld accumulates a phase of 2 \u0019for one\ntrip around the optical axis [1, 15]. The corresponding fo-\ncal \feld distributions and their properties regarding spin\nand orbital angular momentum have been investigated in\nvarious works over the last decade [1, 16{18].\nIn this Letter, we report on a novel e\u000bect caused by\nspin-orbit interactions, which links the three-dimensional\ndistribution of the spin density (SD) to the orbital an-\ngular momentum of the beam. Again, the e\u000bect occurs\nwhen an initially circularly polarized collimated beam\nof light|for simplicity, we consider a Gaussian beam\npro\fle|is tightly focused. Because of the aforemen-\ntioned spin-orbit coupling, the focused beam carries not\nonly spin, but also orbital angular momentum, which\narises in the form of a phase vortex of the longitudinal\n\feld component [13, 15]. The superposition of the longi-\ntudinal \feld and the circularly polarized transverse \feld\nresults in a tilted polarization ellipse o\u000bside the optical\naxis. Consequently, the corresponding SD features trans-\nverse components with respect to the propagation direc-\ntion of the beam (optical axis). The actual local orienta-\ntion of the spin depends on the relative phase between the\nlongitudinal and transverse \felds, which changes upon\npropagation [19]. As we will show later on, this causes the\n\u0003martin.neugebauer@mpl.mpg.de; http://www.mpl.mpg.de/transverse components of the SD to rotate while travers-\ning the focal region.\nIn the following, we start by describing the aforemen-\ntioned e\u000bect as a Gouy-phase-dependent interaction of\nlongitudinal and transverse \felds [20]. For this we elabo-\nrate on a simpli\fed theoretical model in the framework of\nan extended paraxial approximation considering also lon-\ngitudinal \feld components. To demonstrate the rotation\nof the transverse SD experimentally, we use a recently de-\nveloped scheme for measuring transversely spinning \felds\nin tightly focused light beams [21, 22]. Finally, we com-\npare our results with numerical calculations and discuss\nthe possible implications of the e\u000bect on future works.\nII. THEORETICAL MODEL\nWe begin with a simpli\fed paraxial description of a\ntime-harmonic circularly polarized Gaussian beam. Uti-\nlizing the complex beam parameter q(z) =z\u0000{zR, where\nzRis the Rayleigh range of the beam, the transverse \feld\ncomponents are described by [23]\n\u0012\nEx\nEy\u0013\n=\u0012\n1\n\u0006{\u0013u0\nq(z)eik\u001a2\n2q(z)+ikz=\u001b\u0006u(r) , (1)\nwith radius \u001a=p\nx2+y2,u0a complex amplitude,\nand the wave number k. The sign of the polarization\nvector \u001b\u0006indicates right- or left-handed circular polar-\nization. However, the \feld distribution as described by\nEq. (1) does not ful\fll Gauss's law in vacuum, rE= 0,\nwhich requires an additional longitudinal \feld compo-\nnent. Within the paraxial approximation, the missing\ncomponent can be calculated using [24]\nE\u0006\nz={\nk\u0012@Ex\n@x+@Ey\n@y\u0013\n=\u0000u(r)\nq(z)(x\u0006{y) . (2)\nFor a more intuitive description of the distribution of Ez,\nwe rewrite Eq. (2) using the azimuth \u001e= arg (x+{y) and\nthe Gouy-phase \u0011(z) = tan\u00001(z=zR):\nE\u0006\nz=\u0000\u001a{u(r)p\nz2+z2\nRe\u0006{\u001e\u0000{\u0011(z). (3)arXiv:1905.12539v1 [physics.optics] 29 May 20192\nAs we can see, Ezis represented by a \frst order Laguerre-\nGaussian mode with radial mode index 0 and azimuthal\nmode index\u00061 (the sign depends on the handedness of\nthe incoming circularly polarized beam). Thus, the lon-\ngitudinal \feld exhibits the helical phase-front typically\nassociated with the occurrence of orbital angular mo-\nmentum [25]. Furthermore, in comparison to the trans-\nverse \feld (zero order Laguerre-Gaussian mode), Ezex-\nhibits an additional Gouy-phase factor [25]. Therefore,\nthe relative phase between longitudinal and transverse\n\felds changes upon propagation, which consequently af-\nfects the three-dimensional polarization state [26].\nHere, we take a closer look at the evolution of the SD of\nthe electric \feld sE, which describes the local orientation\nand sense of the spinning axis of the three-dimensional\npolarization ellipse [1, 27, 28]. For our paraxial model,\nwe result in:\ns\u0006\nE=\u000f0Im [E\u0003\u0002E]\n4!=0\nBB@\u0000\u001asin[\u001e\u0007\u0011(z)]p\nz2+z2\nR\n\u001acos[\u001e\u0007\u0011(z)]p\nz2+z2\nR\n\u000611\nCCA\u000f0ju(r)j2\n2!. (4)\nThe longitudinal component of the SD has exactly the\nsame Gaussian distribution as the electric \feld intensity\ndistributionsjExj2=jEyj2/sz\nEwhereby the sign indi-\ncates right- and left-handed circular polarization. Most\nimportantly, it is shape-invariant upon propagation. In\ncontrast, the shapes of the transverse SD components de-\npend on the Gouy phase and, as a consequence, also on z.\nIn particular, the distributions of sx\nEandsy\nErotate upon\npropagation along the z-axis. From z=\u00001 toz=1,\nthe SD vector undergoes one half-twist around the z-axis.\nThe rotation direction depends on the handedness of the\nincoming circular polarization.\nFor a more intuitive understanding, the e\u000bect is visu-\nalized in Fig. 1, where we consider a right-handed cir-\ncularly polarized beam propagating top-down, with the\nlocal spin density marked as blue vectors and the corre-\nsponding orientation and spinning direction of the elec-\ntric \feld indicated by black arrows. In the far-\feld of the\nupper half-space ( z<0), the spin points towards the ge-\nometrical focus. A projection onto the x-y-plane reveals\nthat the transverse SD, s?\nE=sx\nEex+sy\nEey, is pointing\ntowards the optical axis (see top right inset). Upon prop-\nagation, the relative phase between the transverse and\nlongitudinal \feld components changes, which results in a\nrotation of the transverse SD. In the focal plane, s?\nEex-\nhibits a purely azimuthal distribution (see central inset).\nBelow the focal plane ( z > 0), the rotation continues,\n\fnally reaching a radial distribution pointing away from\nthe optical axis in the far-\feld (lowest inset).\nIn order to con\frm the half-twist of the spin density,\nwe elaborate on this phenomenon with an experimental\ndemonstration and numerical calculations.\nxyzFIG. 1. Illustration of the electric spin density sEdistribution\nof a tightly focused right-handed circularly polarized beam.\nThe beam (red) propagates top-down, with the blue vectors\nindicating the local orientation of sEand the black vectors\ncorresponding to the polarization ellipse of the electric \feld\nE. The sketches on the right side represent the transverse\nspin density s?\nEfor di\u000berent planes of observation.\nIII. EXPERIMENTAL OBSERVATION\nFirst, we investigate the rotation of the SD experi-\nmentally. Since the strength of the transverse spin de-\npends on the lateral con\fnement [27], we investigate\na circularly polarized beam (wavelength \u0015= 532 nm)\ntightly focused by a high numerical aperture microscope\nobjective (NA = 0 :9, pupil \flling factor \u00190:8). The\nbeam impinges onto a dipole-like gold nano-sphere (ra-\ndius\u001940 nm) sitting on a glass substrate, which is\nscanned through a focal volume of \u00193\u00023\u00023\u0016m3,\nwhere we use a step size of 30 nm in lateral directions\n(xandy) and steps of 200 nm along the propagation di-\nrection (z). For each position, the light scattered into\nthe glass half space is collected with an index-matched\nimmersion-type objective. The directionality of the scat-\ntered light into the substrate allows for determining the\ntransverse SD of the excitation \feld at the particle posi-\ntion [21, 22]. This is due to the fact that the directional\nscattering is a direct consequence of the spinning electric\ndipole induced in the particle by the locally transverse\ncomponents of the SD [27]. Finally, we assemble the\nscanning results to three-dimensional representations of\nthe transverse SD components sx\nEandsy\nE. The measure-\nment results for right- and left-handed circularly polar-\nized incoming beams are depicted in Figs. 2(a)-2(b) and\nFigs. 2(f)-2(g). As predicted by the paraxial model and\nEq. (4), the distributions of the transverse SD compo-\nnents rotate upon propagation, with the rotation direc-\ntion depending on the handedness of the incoming circu-\nlar polarization. This veri\fes the coupling of the trans-\nverse SD distribution and the longitudinal orbital angular\nmomentum. For a quantitative comparison, we calculate\nthe corresponding transverse SD distributions using vec-\ntorial di\u000braction theory [29, 30], where we use the same3\n01\nx [µm]01\ny [µm]0\n1z [µm]\n-1 -1-1sExsEy\n01\nx [µm]01\ny [µm]\n01\n-1\n-1 -1\n01\nx [µm]01\ny [µm]0\n1z [µm]\n-1 -1-1sExsEy\n01\nx [µm]01\ny [µm]\n01\n-1\n-1 -101\nx [µm]01\ny [µm]0\n1z [µm]\n-1 -1-1sExsEy\n01\nx [µm]01\ny [µm] z [µm]\nz [µm]\n01\n-1\n-1 -1\n01\nx [µm]01\ny [µm]0\n1z [µm]\n-1 -1-1sExsEy\n01\nx [µm]01\ny [µm]\n01\n-1\n-1 -1experiment theoryRCP LCP\n-1 0 1-180-90090180\n-1 0 1-180-90090180Φxmeas\nΦymeas\nΦxtheo\nΦytheo\nΦxmeas\nΦymeas\nΦxtheo\nΦytheoΦ [°] Φ [°]angles\n(b) (d)\n(f)(e) (c) (a)\n(g) (h) (j) (i)\nFIG. 2. Experimental and theoretical results. (a)-(d) The experimentally measured and theoretically calculated transverse SD\ncomponents sx\nEandsy\nEin the focal volume of a tightly focused right-handed circularly polarized (RCP) beam. For each plane of\nobservation along the z-axis, the transverse SD is normalized to its maximum amplitude for better visibility. (e) Experimental\n(green and purple circles) and theoretical (green and purple lines) rotation angles of the transverse SD distributions calculated\nfrom the distributions in (a)-(d). The Gouy-phase factor is \ftted to the experimental data (gray lines). (f)-(j) Similar to\n(a)-(e), but for a left-handed circularly polarized (LCP) beam.\nparameters as in the experiment and consider the beam\nto be in free-space (e\u000bects of the glass substrate on the\n\feld distributions are not taken into account). The theo-\nretical results are shown in Figs. 2(c)-2(d) and Figs. 2(h)-\n2(i). All four theoretical distributions are in very good\nagreement with their corresponding experimental coun-\nterparts. Also the rotation of the transverse spin den-\nsities predicted by the simpli\fed paraxial model is con-\n\frmed by the vectorial di\u000braction theory. As a next step,\nwe determine the rotation angles of the distributions of\nsx\nEandsy\nE. For this purpose, we calculate the centroids\nof the positive and negative parts of the respective SD\ncomponent for each x-y-plane of observation along the\nz-axis, and de\fne the angle between the connection line\nof both centroids and the x-axis as rotation angle \u001e(z).\nThe theoretical (solid lines) and experimental (circles)\nrotation angles are plotted in Figs. 2(e) and 2(j), where\nthe green and purple colors correspond to sx\nEandsy\nE,\nrespectively. For the right-handed circularly polarized\nbeam the experimentally measured rotation angles ex-\nhibit a negative angular o\u000bset with respect to the the-\noretical curves, while for the left-handed circularly po-\nlarized beam the o\u000bset is positive. This spin-dependent\no\u000bset is caused by the interference of the incoming beam\nand the light re\rected by the glass substrate, which is\nnot considered in the theoretical treatment. Still, the ro-\ntation angles follow a modi\fed inverse tangent function,\n\u001e(z) =\u0006tan\u00001[(z+zo)=zR] +\u001eo, withzothe o\u000bset\nalong thez-axis and\u001eothe angular o\u000bset. The \fts, which\noverlap very well with the experimental data, are plotted\nas gray lines. This veri\fes that even in the tight focus-\ning regime, we can use the relative Gouy-phase factorbetween longitudinal and transverse \felds derived from\nthe paraxial model as a qualitative explanation for the\nhalf-twist of the SD.\nIV. DISCUSSION\nIn conclusion, we observed the rotation of the trans-\nverse SD upon propagation in tightly focused circularly\npolarized beams. The e\u000bect can be explained by a di\u000ber-\nence in mode orders and, therefore, a relative Gouy-phase\nbetween the transverse \feld components and the longi-\ntudinal \feld, which carries orbital angular momentum\ndue to spin-orbit coupling. The theoretical description\nof this e\u000bect is conceptually related to similar polariza-\ntion interference e\u000bect, where two orthogonally polarized\nbeams with di\u000berent mode orders interfere resulting in a\nchanging two-dimensional polarization distribution upon\npropagation due to di\u000berent Gouy-phase shifts [6, 31, 32].\nIn our case, the e\u000bect is caused by spin-orbit interaction\nand occurs for the transverse SD components represent-\ning a three-dimensional polarization parameter [33]. In\nthis regard, the measurement of the rotation of the trans-\nverse spin density can also be interpreted as an experi-\nmental demonstration of the non-separability of three-\ndimensional \felds [34, 35], and the generation of orbital\nangular momentum by tight focusing [1, 16, 17].\nThe evolution of three-dimensional polarization states\nupon propagation might \fnd application in novel polar-\nization based metrology approaches, where the local po-\nlarization state entails information on the position of a\nscatterer relative to an excitation \feld [36{38]. By uti-4\nlizing a more complex input \feld distribution, it is pos-\nsible to tailor the rotation of the spin density for a given\naxis in space [19], which might facilitate the practical\nimplementation of such position sensing techniques and\nspin-based directional coupling experiments [8, 9]. Fi-\nnally, the notion of a position dependent orientation of\nthe spin density in tightly focused circularly polarized\nbeams can be relevant for optical manipulation experi-\nments, with the local spin exerting a torque or a lateral\nforce on nano-particles [39, 40].ACKNOWLEDGMENTS\nWe gratefully acknowledge discussions with Sergey\nNechayev and Gerd Leuchs. This project has received\nfunding from the European Union's Horizon 2020 re-\nsearch and innovation programme under the Future and\nEmerging Technologies Open grant agreement Super-\npixels No 829116.\n[1] K. Y. Bliokh, F. J. Rodr\u0013 \u0010guez-Fortu~ no, F. Nori, and\nA. V. Zayats, Nat. Photon. 9, 796 (2015).\n[2] N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, E. Veksler,\nV. Kleiner, and E. Hasman, Science 340, 724 (2013).\n[3] E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Up-\nham, and R. W. Boyd, Light. Sci. Appl. 3, e167 (2014).\n[4] X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo,\nS. Wen, and D. Fan, Light. Sci. Appl. 4, e290 (2015).\n[5] L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett.\n96, 163905 (2006).\n[6] F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and\nE. Santamato, Opt. Express 21, 8815 (2013), 1211.4163.\n[7] J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou,\nX.-C. Yuan, and F. Capasso, Science 340, 331 (2013).\n[8] F. J. Rodr\u0013 \u0010guez-Fortu~ no, G. Marino, P. Ginzburg,\nD. O'Connor, A. Mart\u0013 \u0010nez, G. A. Wurtz, and A. V.\nZayats, Science 340, 328 (2013).\n[9] M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs,\nNano Lett. 14, 2546 (2014).\n[10] E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and\nS. W. Hell, Nat. Photon. 3, 144 (2009).\n[11] X. Hao, C. Kuang, T. Wang, and X. Liu, J. Opt. 12,\n115707 (2010).\n[12] H. Adachi, S. Akahoshi, and K. Miyakawa, Phys. Rev.\nA75, 063409 (2007).\n[13] Y. Zhao, J. S. Edgar, G. D. M. Je\u000bries, D. McGloin, and\nD. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).\n[14] R. Bhandari, Phys. Rep. 281, 1 (1997).\n[15] Z. Bomzon and M. Gu, Opt. Express 32, 3017 (2007).\n[16] T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and\nH. Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 10,\n115005 (2008).\n[17] K. Y. Bliokh and F. Nori, Phys. Rep. 592, 1 (2015).\n[18] D. Sugic and M. R. Dennis, J. Opt. Soc. Am. A 35, 1987\n(2018).\n[19] X. Pang and W. Miao, Opt. Lett. 43, 4831 (2018).\n[20] L. G. Gouy, Acad. Sci. Paris 110, 1251 (1890).[21] M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer,\nPhys. Rev. Lett. 114, 063901 (2015).\n[22] M. Neugebauer, J. S. Eismann, T. Bauer, and P. Banzer,\nPhys. Rev. X 8, 021042 (2018).\n[23] Q. Zhan, Adv. Opt. Photon. 1, 1 (2009).\n[24] W. L. Erikson and S. Singh, Phys. Rev. E 49, 5778\n(1994).\n[25] D. L. Andrews and M. Babiker, The angular momentum\nof light , 1st ed. (Cambridge University Press, Cambridge,\n2013).\n[26] X. Pang and T. D. Visser, Opt. Express 21, 8331 (2013).\n[27] A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs,\nNat. Photon. 9, 789 (2015).\n[28] T. Bauer, M. Neugebauer, G. Leuchs, and P. Banzer,\nPhys. Rev. Lett. 117, 013601 (2016), 1601.06072.\n[29] B. Richards and E. Wolf, Proc. R. Soc. A 253, 358 (1959).\n[30] L. Novotny and B. Hecht, Principles of Nano-Optics , 2nd\ned. (Cambridge University Press, Cambridge, 2006).\n[31] A. M. Beckley, T. G. Brown, and M. A. Alonso, Opt.\nExp.18, 10777 (2010).\n[32] M. Neugebauer, S. Grosche, S. Rothau, G. Leuchs, and\nP. Banzer, Opt. Lett. 41, 3499 (2016).\n[33] T. Set al a, A. Shevchenko, M. Kaivola, and A. T. Friberg,\nPhys. Rev. E 66, 016615 (2002).\n[34] A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and\nG. Leuchs, Opt. Express 19, 9714 (2011).\n[35] X.-F. Qian and J. H. Eberly, Opt. Lett. 36, 4110 (2011).\n[36] S. Berg-Johansen, T. Falk, B. Stiller, P. Banzer,\nM. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and\nC. Marquardt, Optica 2, 864 (2015).\n[37] Z. Xi, L. Wei, A. J. L. Adam, H. P. Urbach, and L. Du,\nPhys. Rev. Lett. 117, 113903 (2016).\n[38] A. Bag, M. Neugebauer, P. Wo\u0013 zniak, G. Leuchs,\nand P. Banzer, Phys. Rev. Lett. 121, 193902 (2018),\n1804.10176.\n[39] A. Canaguier-Durand, A. Cuche, C. Genet, and T. W.\nEbbesen, Phys. Rev. A 88, 033831 (2013).\n[40] F. J. Rodr\u0013 \u0010guez-Fortu~ no, N. Engheta, A. Mart\u0013 \u0010nez, and\nA. V. Zayats, Nat. Commun. 6, 8799 (2015), 1504.03464." }, { "title": "1306.0611v1.The_role_of_the_Rashba_coupling_in_spin_current_of_monolayer_gapped_graphene.pdf", "content": "arXiv:1306.0611v1 [cond-mat.mes-hall] 3 Jun 2013The role of the Rashba coupling in spin current of\nmonolayer gapped graphene.\nK. Hasanirokh, J. Azizi, A. Phirouznia, H. Mohammadpour\nDepartment of Physics, Azarbaijan Shahid Madani University, 537 14-161, Tabriz,\nIran\nAbstract. In the current work we have investigated the influence of the Rash ba\nspin-orbit coupling on spin-current of a single layer gapped graphen e. It was shown\nthat the Rashba coupling has a considerable role in generation of the spin-current of\nvertical spins in mono-layer graphene. The behavior of the spin-cu rrent is determined\nby density of impurities. It was also shown that the spin-current of the system could\nincrease by increasing the Rashba coupling strength and band-gap of the graphene and\nthe sign of the spin-current could be controlled by the direction of t he current-driving\nelectric field.\nPACS numbers: 72.80.Vp, 73.23.-b, 73.22.PrThe role of the Rashba coupling in spin current of monolayer g apped graphene. 2\n1. Introduction\nCarbon nano materials reveal an interesting polymorphism of differe nt allotropes\nexhibiting each possible dimensionality: 1) fullerene molecule (0D), 2) n ano tubes (1D),\n3) graphene and graphite platelets (2D) and 4) diamond (3D) are se lected examples.\nSince graphene was discovered in 2004 by Geim and his team [1], it has co ntinued to\nsurprise scientists. This is due to some spectacular and fantastic t ransport properties\nlike the high carrier mobility and universal minimal conductivity at the D irac point\nwhere the valence and conduction bands cross each other [2, 3].\nIn the presence of spin-orbit couplings, spin polarized states in gra phene is a response to\nan external in-plane electron field. In today’s spintronics, it is a pos sible key ingredient\ntowards electrical spin control with spin-orbit interaction which ha s attracted a great\nattention, such as anomalous Hall effect [4]. Extrinsic spin-orbit inte raction (Rashba)\nthat originates from the structure inversion asymmetry (SIA), a rises from symmetry\nbreaking generated at interface between graphene and substra te or by external fields\n[5, 6, 7].\nSpin-current control is of crucial importance in electronic devices . Some attempts\nhave been made for applying Rashba interaction to control spin-cu rrent. Since Rashba\ncoupling strength in graphene is high relative to other materials, stu dying the role of\nthis interaction in graphene could remove most of the existing obsta cles in spin-current\ncontrol. As mentioned in previous studies, amount of Rashba intera ction in graphene\ncan be at a very high. For example, 0.2 ev which has been reported in [ 8] can be pointed\nout.\nDepending on graphene’s substrate, strength of the Rashba inte raction in graphene\ncan be much higher than its intrinsic spin-orbit interaction. In additio n to the spin\ntransport related aspects generated by applying the Rashba inte raction, this coupling\ncan be utilized in optical devices as well. As an example, by applying this in teraction in\ntwo-dimensional electron gas and graphene, controllable blue shift could be obtained in\nabsorption spectrum [9, 10]. Spin-current is an important tool for studying spin related\nfeatures in graphene and is also important in the development of the graphene quantum\ncomputer.\nBased onthe Tight-Binding model, Soodchomshom has studied the eff ect of theuniaxial\nstrain in the spin transport through a magnetic barrier of the stra ined graphene system\nand shown that graphene has a fantastic potential for application s in nano-mechanical\nspintronic devices. Strain in graphene will induce the pseudo-poten tials at the barrier\nthat can control the spin currents of the junction [11]. Ezawa has considered a nano disk\nconnected with two leads and shown this system acts as a spin filter a nd can generate\na spin-current [12].\nIn this work, we have considered the influence of Rashba spin-orbit coupling on spin-\ncurrent of a monolayer gapped graphene. It has been found that the spin-current\nassociated with normal spins is influenced by the Rashba spin-orbit c oupling and the\nenergy gap of the system.The role of the Rashba coupling in spin current of monolayer g apped graphene. 3\n2. Model and approach\nThe low energy charge carriers in graphene are satisfied in a massles s Dirac equation.\nThis equation have an isotropic linear energy dispersion near the Dira c points [1].\nMeanwhile the Hamiltonian of gapped graphene with Rashba spin-orbit (SO) coupling\nis [13, 14]\nˆH=ˆHG+ˆHR+ˆHgap+ˆVim. (1)\nˆHGis the Dirac Hamiltonian for massless fermions which is given as follows [1 5, 16]\nˆHG=−iγψ†(σxτz∂x+σy∂y)ψ, (2)\nvfis the Fermi velocity in graphene, γ= ¯hvf,σandτrepresent the Pauli matrices\nwhere,σz=±1,τz=±1 denote the Pauli matrice of pseudospin on the A and B\nsublattices and different Dirac points, respectively.\nThe second term in equation (1), HR, is the Rashba (SO) interaction, and can be\nexpressed as [14, 17]\nˆHR=λR\n2ψ†(σysx−σxτzsy)ψ, (3)\nwhereλRis the (SO) coupling constant and sdenotes Pauli matrix representing the spin\nof electron. In an ideal graphene sheet, the Dirac electrons are m assless and the band\nstructure has no energy gap. Experimentally, it is possible to manipu late an energy\ngap (from a few to hundreds of meV) in graphene’s band structure , namely a Dirac gap\n[18, 19, 20]. As a result of the asymmetry in graphene sublattices, A and B, the band\ngap can have a nonzero value. The last term in Hgap=τ∆σzequation (1), referred\nto mass term, arises from the energy gap ∆ in the spectrum of grap hene, that τ= 1\n(τ=−1) corresponds to the K(K′) valley. The last term in equation (1), ˆVim, is\ninduced due to the short range impurities which can be written as\nVim(r) =/summationdisplay\njVδ(/vector r−/vector rj), (4)\nwhere the summation is over the position of impurities. The eigenfunc tions of ˆH0=\nˆHG+ˆHR+ˆHgapare\n|ǫ1(k)>=\n−iΩ1(k)e−2iϕ\nΩ2(k)e−iϕ\niΩ3(k)e−iϕ\n1\n, (5)\n|ǫ2(k)>=\niΩ4(k)e−2iϕ\nΩ5(k)e−iϕ\niΩ3(k)e−iϕ\n1\n, (6)The role of the Rashba coupling in spin current of monolayer g apped graphene. 4\n|ǫ3(k)>=\niΩ6(k)e−2iϕ\nΩ7(k)e−iϕ\n−iΩ8(k)e−iϕ\n1\n, (7)\n|ǫ4(k)>=\n−iΩ9(k)e−2iϕ\nΩ10(k)e−iϕ\n−iΩ8(k)e−iϕ\n1\n, (8)\nand the corresponding eigenvalues are\nǫ1(k) =−/radicaligg\n∆2+γ2k2+λ2\n8−λ\n8Γ(k)\nǫ2(k) =−ǫ1(k) (9)\nǫ3(k) =−/radicaligg\n∆2+γ2k2+λ2\n8+λ\n8Γ(k)\nǫ4(k) =−ǫ3(k)\nwhere we have defined\nΩ1(k) =−Λ+(k)ξ(1)\n+(k) Ω 2(k) =ξ(1)\n+(k)\nΩ3(k) = Λ−(k) Ω 4(k) = Λ+(k)ξ(1)\n−(k) (10)\nΩ5(k) =ξ(1)\n−(k) Ω 6(k) =−Λ−(k)ξ(2)\n+(k)\nΩ7(k) =ξ(2)\n+(k) Ω 8(k) = Λ+(k)\nΩ9(k) = Λ−(k)ξ(2)\n−(k) Ω 10(k) =ξ(2)\n−(k)\ninwhich Γ(k) = (16γ2k2+λ2)1/2, Λ±(k) = (±λ+Γ(k))/(4γk),ξ(1)\n±(k) = (∆±ǫ1(k))/(γk)\nandξ(2)\n±(k) = (∆±ǫ3(k))/(γk).\nThe transition probabilities between the |ǫi(k)>and|ǫj(k′)>states are given by the\nFermi’s golden rule,\nωij(/vectork,/vectork′) = (2π/¯h)ni||2δ(ǫi(/vectork)−ǫj(/vectork′)),(i,j= 1,2,3,4).(11)\nthatδ(ǫi(/vectork)−ǫj(/vectork′))≈δ(/vectork−/vectork′)/γ.\nScattering potential of the impurities is described by the ˆVim. These are assumed to be\ndistributed randomly with real density ni\n¯ωi=K/integraldisplay\nd´ϕ/summationdisplay\njωji(ϕ,ϕ′),(i,j= 1,2,3,4) (12)\nin whichK= (2π/¯h)niandϕ,ϕ′are two angles characterizing the direction of /vectorkand/vectork′\nstates relative to the xaxis respectively. Then the non-equilibrium distribution function\nof a givenǫi(k) energy band can be written as [21]\nδfi=−eviE(−∂ǫf0)[ai(ϕ)cosθ+bi(ϕ)sinθ], (13)The role of the Rashba coupling in spin current of monolayer g apped graphene. 5\nin whichEis the current driven electric field, θis the angle between the xaxis and the\ndirection of the electric field and the unknown functions ai(ϕ) andbi(ϕ) can be written\nas follows\nai(ϕ) =a0+/summationdisplay\nj=1acjicos(jϕ)+/summationdisplay\nj=1asjisin(jϕ) (14)\nbi(ϕ) =b0+/summationdisplay\nj=1bcjicos(jϕ)+/summationdisplay\nj=1bsjisin(jϕ) (15)\nai(ϕ) andbi(ϕ) (i= 1,2,3,4) must satisfy [21]\ncosϕ= ¯ωiai(ϕ)−K/integraldisplay\nd´ϕ/summationdisplay\nj[ωij(ϕ,ϕ′)aj(ϕ)] (16)\nsinϕ= ¯ωibi(ϕ)−K/integraldisplay\nd´ϕ/summationdisplay\nj[ωij(ϕ,ϕ′)bj(ϕ)] (17)\nit can be easily obtained that the non-vanishing parameters are ac1iandbs1iin which\nac1i=bs1i(i= 1−4).This can be inferred from the Dirac point approximation in\nwhich at this level band energies are appeared to be isotropic in k-sp ace. Meanwhile,\nthese parameters can be obtained through the equations (16)-( 17) The non-equilibrium\ndistribution function is then given by\nδfi=−eviE(−∂ǫf0)[ac1icosϕcosθ+bs1isinϕsinθ] (18)\nThe spin current operator is defined as [5]\nJsi\nj={si,ˆvj}, (19)\nwhere ˆvj= ¯h−1(∂ˆH\n∂kj) (i,j=x,y) is the velocity operator. The spin current operator in\nthe basise(i/vectork./vector r)|sσ>is given as follows,\nJsz\nx=\n0 ¯hγ0 0\n¯hγ0 0 0\n0 0 0 −¯hγ\n0 0 ¯hγ0\n, (20)\nJsz\ny=\n0−i¯hγ0 0\ni¯hγ0 0 0\n0 0 0 i¯hγ\n0 0 −i¯hγ0\n. (21)\nThen the average spin currents in x and y directions are given by\n=1\n(2π)2/integraldisplay\nd2k4/summationdisplay\nλ=1f λ(k) (i,j=x,y),(22)\nusing the equations (18) and (22), one can obtain\n=−Js\n0(ǫ2(kf)bs12kf\nγ−γλ\nΓ(kf)[Ω4(kf)Ω5(kf)−Ω3(kf)][Ω3(kf)Ω4(kf)+Ω5(kf)]\n+ǫ4(kf)bs14kf\nγ+γλ\nΓ(kf)[Ω8(kf)−Ω9(kf)Ω10(kf)][Ω8(kf)Ω9(kf)+Ω10(kf)]),\n(23)The role of the Rashba coupling in spin current of monolayer g apped graphene. 6\nwhereJs\n0=¯heE\nπ.\nSimil1ary one can easily obtain other components of the spin-curren t as follows\n=−,\n== 0, (24)\n== 0.\nThis means that the Rashba coupling cannot generate spin current of in-plane spin\ncomponents ( Jsx\niandJsy\ni). This is in agreement with the results of the similar case\nin two-dimensional electron gas in which the Rashba coupling induced s pin current\nidentically vanishes [22, 23].\nElectrical current will be as following\n=1\n(2π)2/integraldisplay\nd2k4/summationdisplay\nλ=1f λ(k) =\n−J0(ǫ2(kf)ac12kf\nγ−γλ\nΓ(kf)[Ω3(kf)Ω4(kf)+Ω5(kf)]2+\nǫ4(kf)ac14kf\nγ+γλ\nΓ(kf)[Ω8(kf)Ω9(kf)+Ω10(kf)]2), (25)\nthatJ0= (¯he2Ekf)/(πγ).\nCalculation will obtain the following result, directly < Jx>=< Jy>which can be\nregarded as a consequence of the Dirac point approximation that c ould eliminate the\nanisotropic effects of the band structure such as trigonal warpin g.\n3. Results\nHere, thespin-currentofamonolayergappedgraphenehasbeen obtainedinthepresence\nof Rashba interaction. In this work, it was shown that the non-equ ilibrium spin-current\nof vertical spins can be effectively controlled by this spin-orbit inter action.\nIt was assumed that the electrical field has been applied along the xaxis and the nu-\nmerical parameters have been chosen as follows ǫf= 1meVis the Fermi energy [24] and\nni= 1010cm−2is the density of impurities.\nDifferent non-equilibrium spin-current components have been depic ted as a function\nof the graphene gap in figures 1-2. These figures clearly show that the longitudinal\nand transverse non-equilibrium spin-currents of normal spins hav e accountable values in\nwhich their signs and magnitudes can be controlled by the graphene’s gap. The absolute\nvalue of non-equilibrium spin-current components, with respect to the gap of graphene,\nare increasing at by increasing the Rashba coupling strength. One o f the important\nfeatures which can be inferred from the figures 1 and 2 is the fact t hat the spin current\ncan be of either sign, depending on the direction of the driving electr ic field.The role of the Rashba coupling in spin current of monolayer g apped graphene. 7\n0 2 4 6 810−14−12−10−8−6−4−202x 1012\n∆/ǫFJszx/Js\n0\nλ/εF=2\nλ/εF=8\nλ/εF=10\nFigure 1. non-equilibrium longitudinal spin current as a function of the graphe ne gap\nat different Rashba couplings.\n0 2 4 6 8 10−202468101214x 1012\n∆/ǫFJszy/Js\n0λ/εF=1\nλ/εF=8\nλ/εF=10\nFigure 2. non-equilibrium transverse spin current as a function of the graph ene gap\nat different Rashba couplings.\nFigure 3 displays the electric current along the x direction as a funct ion of the gap. As\nillustrated inthisfiguretheelectric current ingapedgraphenecanb eeffectively changed\nby the Rashba coupling. The absolute value of electrical current inc reases by increasing\nthe amount of the gap. The deference between the curves inside t his figuredemonstrates\nthe importance of the Rashba coupling. The longitudinal non-equilibr ium spin-currents\nof a monolayer gapped graphene have indicated as a function of the Rashba coupling in\nfigure 4. The Rashba spin-orbit coupling strength can reach high va lues up to 0.2eV in\nmonolayer graphene. As shown in this figure, the absolute value of s pin-current inducedThe role of the Rashba coupling in spin current of monolayer g apped graphene. 8\n0 2 4 6 8 10−4−3−2−101x 1015\n∆/ǫFJx/J0\nλ/εF=10\nλ/εF=8\nλ/εF=2\nFigure 3. Longitudinalelectriccurrentasafunctionofthegapingraphenea tdifferent\nRashba couplings.\nby the Rashba coupling increases by increasing the gap.\nAs can be seen in figures 1-4, absolute value of spin-current would in crease by in-\ncreasing the Rashba coupling strength and also the energy gap bec ause, on the one\nhand, the increased energy gap would reduce the possibility of spin r elaxation and\nspin mixing and on the other hand, increased Rashba coupling streng th would raise\neffective magnetic field of this coupling. This effective magnetic field ca n be regarded\nasBeff=λR/(2µB)(σyˆx−σxˆy). Anisotropy induced by current-driving electric field\nresults in a non-vanishing average effective magnetic field; i.e. if the c urrent-driving\nelectric field is along with x, hopping along with xaxis would be more likely to hap-\npen and< σx> > < σ y>therefore the existing electrons at Fermi level would\nfeel a non-zero effective magnetic field where the spin-current is o riginating from this\nfield. Therefore, external in-plane electric field plays an important role in generating\nthe effective magnetic field on spin carriers. Consequently, it is expe cted that, if bias\nvoltage is applied along the yaxis, the direction of effective magnetic field would also\nchange; thus, type of spin majority carriers would also modify. This phenomenon can be\nclearly seen in figures 1-4 so that spin-current sign changes depen ding on the direction\nof the applied bias voltage. Therefore, the sign of spin current wou ld be controllable\nby external bias. According to the mentioned points, generating s pin-current of vertical\nspins at least in non-equilibrium regime can be expected.\nThe behavior of the spin-current is determined by the impurity dens ity as depicted in\nfigure 5. In this figure, we have taken ∆ /ǫF= 5 and it can be inferred from the data\ndepicted in this figure that increasing the spin-mixing rate (which cou ld take place by\nincreasing the density of impurities) decreases the spin-polarizatio n and spin-current of\nthe system.The role of the Rashba coupling in spin current of monolayer g apped graphene. 9\n0 50 100 150 200−202468x 1018\nλ/ǫFJszx/Js\n0∆/εF=5\n∆/εF=8\n∆/εF=10\nFigure 4. Longitudinal spin current as a function of the Rashba coupling.\n0 50 100 150 200−5−4−3−2−101x 1018\nλ/ǫFJszy/Js\n0\nni=10101/cm−2\nni=8 ×1091/cm−2\nni=5 ×1091/cm−2\nFigure 5. Transverse spin current as a function of the Rashba coupling at ∆ /ǫF= 5.\n4. Conclusion\nIn the present work, the influence of the Rashba coupling on spin-r elated transport\neffects have been studied. Results of the present study show tha t the Rashba interaction\nhas an important role in generation of the non-equilibrium spin-curre nt of vertical spins\nin a monolayer gapped graphene. The absolute value of spin-curren t as a function\nof the gap, increases by increasing the Rashba interaction streng th in non-equilibrium\nregime. Another important point in the results of the present stud y can be describe as\nfollows; Not only the amount of spin-current in grapheme is controlla ble by gate voltage\n(responsible for Rashba interaction) but also its sign is predictable b y the direction ofThe role of the Rashba coupling in spin current of monolayer g apped graphene. 10\nthe applied bias voltage.\nReferences\n[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I , Grigo rieva I V, Dubonos S V\nand Firsov A A 2005 Nature 438197\n[2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V , Grigorieva I V and\nFirsov A A 2004 Science 306666\n[3] Tombros N, Jozsa C, Popinciuc M, Jonkman H T and Wees B J van 200 7 Nature 448571\n[4] Han W, Kawakami R K 2001 Phys. Rev. Lett. 107047207\n[5] Rashba Emmanuel I 2003 Phys. Rev. B 68241315\n[6] Rashba E I 1960 Sov. Phys. Solid State 21109\n[7] Huertas-Hernando D, Guinea F and Brataas A 2006 Phys. Rev. B 74155426\n[8] Dedkov Yu S, Fonin M, Rudiger U, and Laubschat C 2008 Phys. Rev . Lett.100107602\n[9] Phirouznia A, Shateri S Safari, Poursamad Bonab J, and Jamshid i-Ghaleh K 2012 Applied Physics\nLetters101111905\n[10] Jamshidi-Ghaleh K, Phirouznia A and Sharifnia R 2012 J. Opt. 14035601\n[11] Soodchomshom B 2011 Physica B 406614-619\n[12] Ezawa M 2010 Physica E 42703706\n[13] Giavaras G and Nori F 2010 Applied Physics Letters 97243106\n[14] Kane C L and Mele E J 2005 Phys. Rev. Lett 95226801\n[15] Ahmadi S, Esmaeilzade M, Namvar E and Pan Genhua 2012 AIP ADV ANCES 2012130\n[16] Rashba Emmanuel I 2003 Phys. Rev. B 68241315\n[17] Yi K S, Kim D and Park K S 2007 Phys. Rev. B 76115410\n[18] Vitali L, Riedl C, Ohmann R, Brihuega I, Starke U and Kern K, Sci Surf 2008 Lett 127602\n[19] Zhou S Y, Gweon G-H, Fedorov A V, First P N, de Heer W A, Lee D-H , Guinea F, Castro Neto\nA H and Lanzara A 2007 Nature Mater 6770\n[20] Enderlein C, Kim Y S, Bostwick A, Rotenberg E and Horn K 2010 New J. Phys. 12033014\n[21] V´ yborn´ y K, Kovalev Alexey A, Sinova J and Jungwirth T 2009 Ph ys. Rev. B 79045427\n[22] Huang Zhian and Hu Liangbin 2006 Phys. Rev. B 73113312\n[23] Inoue Jun-ichiro, Bauer Gerrit E W and Molenkamp Laurens W 200 3 Phys. Rev. B 67033104\n[24] Liewrian W, Hoonsawat R, Tang I-M 2010 Physica E 421287-1292" }, { "title": "1610.01534v3.Exotic_orbits_due_to_spin_spin_coupling_around_Kerr_black_holes.pdf", "content": "September 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\nInternational Journal of Modern Physics D\nc\rWorld Scienti\fc Publishing Company\nExotic orbits due to spin-spin coupling around\nKerr black holes\nWen-Biao Han* and Shu-Cheng Yang\nShanghai Astronomical Observatory,\nCAS, Shanghai, 200030, P. R. China\nSchool of Astronomy and Space Science,\nUniversity of Chinese Academy of Sciences,\nBeijing 100049, P. R. China\n*wbhan@shao.ac.cn\nReceived 8 June 2017\nRevised 14 August 2017\nAccepted 17 August 2017\nPublished\nWe report exotic orbital phenomena of spinning test particles orbiting around a Kerr\nblack hole, i.e., some orbits of spinning particles are asymmetrical about the equatorial\nplane. When a nonspinning test particle orbits around a Kerr black hole in a strong \feld\nregion, due to relativistic orbital precessions, the pattern of trajectories is symmetrical\nabout the equatorial plane of the Kerr black hole. However, the patterns of the spinning\nparticles' orbit are no longer symmetrical about the equatorial plane for some orbital\ncon\fgurations and large spins. We argue that these asymmetrical patterns come from\nthe spin-spin interactions between spinning particles and Kerr black holes, because the\ndirections of spin-spin forces can be arbitrary, and distribute asymmetrically about the\nequatorial plane.\nKeywords : Mathisson-Papapetrou-Dixon equations; spin-spin coupling; Kerr black hole.\nPACS number(s): 04.20.Cu, 04.25.dg, 04.25.dk, 97.80.-d\n1. Introduction\nThe equations of motion of a nonspinning test particle around a Kerr black hole are\nfully integrable because of the existence of four conserved quantities: rest mass, en-\nergy, angular momentum and the Carter constant.1The axisymmetry of spacetime\ndrives the geodesic orbits to \fll the volume in an axisymmetric manner. The same\nholds for the re\rection symmetry of the background along the equatorial plane.\nAs a result, in the strong gravitational \feld region, due to two relativistic orbital\nprecessions: perihelion and Lense-Thirring precessions which re\rect the spacetime\nsymmetries, the pattern of trajectories of the test particle is symmetrical about\nboth the rotating axis and equatorial plane of the corresponding Kerr black hole,\ni.e., after many laps the Kerr geodesic orbits crudely \fll a volume that is loosely\nsymmetric about the polar axis and equatorial plane. In principle, even for the zone\n1arXiv:1610.01534v3 [gr-qc] 6 Sep 2017September 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\n2W.-B. Han, S.-C. Yang\nfar from black holes, providing a su\u000ecient time scale, all orbital con\fgurations of\ntest particles around Kerr black holes have these two symmetries.\nThough almost all astrophysical bodies have spins, in the region far away from\nthe central massive body, for extreme mass-ratio cases, the motion of a small body\ncan be described accurately enough with the nonspinning test particle approxima-\ntion. For example, the spins of S-stars around the supermassive black hole in our\nGalactic center2{6can be ignored.\nHowever, in the strong \feld region and a large spin, due to the spin-orbit and\nspin-spin interactions, the trajectories of a spinning particle in Kerr spacetime can\ndeviate from geodesic motion. Unlike the nonspinning case, for the spinning parti-\ncles, because of the extra degrees of freedom caused by the spin vector and absence\nof the Carter constant, the equations of motion of spinning particles are no longer\nintegrable. The spin of the particle is important in dynamics and gravitational waves\nfor extreme mass-ratio systems.7{13For the nonspinning case, the orbits around a\nKerr black hole are always regular. However, under some conditions, and for ex-\ntreme spin values the orbital motions of extreme spinning particles can be chaotic\n(see Refs. 14{19 and references inside). Such extreme spin values are actually im-\npossible for compact objects like black holes, neutron stars, white dwarfs etc. For\nnoncompact bodies like planets, the spin magnitude can approach 1 (in our units,\nsee next paragraph), for example, the Jupiter-Sun system. Unfortunately, due to\nthe tidal in\ruence from the central black hole, such a noncompact body will be dis-\nrupted by the black hole in the strong \feld region (see Sec. 2 for details). Therefore,\nfor relativistic large-mass-ratio binary systems, the spin magnitude of the small\nobject should be much less than 1. However, the phenomena and characteristics\nof extreme spinning particles orbiting near a black hole are very interesting for re-\nsearchers.14{21In this paper, we focus on an exotic orbital con\fguration whose orbit\npattern is asymmetrical about the equatorial plane of the Kerr black hole. We try\nto study this interesting phenomenon in details and reveal its physical reasons.\nThrough this paper, we use units where G=c= 1 and sign conventions\n(\u0000;+;+;+). The time and space scale is measured by the mass of black hole M,\nand energy of particle is measured by it's mass \u0016, the angular momentum and spin\nby\u0016M, and linear momentum by \u0016. We also assume that \u0016=M\u001c1.\n2. Mathisson-Papapetrou-Dixon equations and repulsive e\u000bect\nfrom spin-spin coupling\nThe popular equations for describing the motion of a spinning particle in curved\nspace-time are Mathisson-Papapetrou-Dixon (MPD) equations,22{25\nDp\u0016\nD\u001c=\u00001\n2R\u0016\n\u0017\u001a\u001b\u001d\u0017S\u001a\u001b; (1)\nDS\u0016\u0017\nD\u001c=p\u0016\u001d\u0017\u0000\u001d\u0017p\u0016; (2)September 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\nExotic orbits due to spin-spin coupling around Kerr black holes 3\nwhere\u001d\u0017\u0011dx\u0017=d\u001cis the four-velocity if \u001cis the proper time of the spinning\nparticle,p\u0016the linear momentum, S\u0016\u0017the anti-symmetrical spin tensor, and R\u0016\n\u0017\u001a\u001b\nthe Riemann tensor of the background. Alternative approaches to the spinning\nparticle equations can be found in Refs. 26 and 27. The spin tensor is then related\nwith the spin vector by\nS\u0016\u0017=\u000f\u0016\u0017\u000b\fu\u000bS\f; (3)\nwhereu\u000b\u0011p\u000b=\u0016,\u000f\u0016\u0017\u000b\f=\"\u0016\u0017\u000b\f=p\u0000gis a tensor and \"\u0016\u0017\u000b\fthe Levi-Civita alter-\nnating symbol ( \"0123\u00111; \"0123\u0011\u00001). Following Tulczyjew,28we choose\np\u0016S\u0016\u0017= 0 =)p\u0016S\u0016= 0 (4)\nas a spin supplementary condition which de\fnes a unique worldline identi\fed with\nthe center of mass. Condition (4) leads to the velocity-momentum relation (see e.g.\nRef. 20)\n\u001d\u0016=m\n\u0016\u0012\nu\u0016+2S\u0016\u0017R\u0017\u001b\u0014\u0015u\u001bS\u0014\u0015\n4\u0016+R\u000b\f\r\u000eS\u000b\fS\r\u000e\u0013\n; (5)\nwhere\u0016is the \\dynamical\" rest mass of the particle de\fned by p\u0017p\u0017=\u0000\u00162and is a\nconstant here because of the supplementary condition we chose. mis the \\kinemat-\nical\" mass which is not a constant and de\fned by p\u0017\u001d\u0017=\u0000m. In order to obtain\nthe four-velocity through Eq. (5), one normalizes min such way that v\u0016v\u0016=\u00001.\nOne can see Ref. 20 for detailed discussions. Now, the MPD equations become a\nclosed form and can be calculated.\nDue to the lack of enough conserved quantities, there is no analytical solution for\nEqs.(1) and (2), and then numerical integration is used to calculate the motion of\nthe spinning particle. We choose Boyer-Lindquist coordinates for calculations, and\n\frstly we should give a set of initial conditions at time t0:r0;\u00120;\u001e0;u\u0016\n0andS\u0016\n0. It is\nnoted that there are three constraints and two constants of motion: the constraints\nu\u0016u\u0016=\u00001;S\u0016S\u0016=S2(S is the spin magnitude), and the spin supplementary\ncondition (4), as well as the energy and angular momentum constants, because of\nthe two Killing vectors \u0018\u0016\nt; \u0018\u0016\n\u001e. The energy and angular momentum are given as\n(e.g. Refs. 20 and 21),\nE=\u0000pt+1\n2gt\u0016;\u0017S\u0016\u0017; (6)\nLz=p\u001e\u00001\n2g\u001e\u0016;\u0017S\u0016\u0017: (7)\nHereafter, we use the dimensionless quantities u\u0017to replacep\u0017, because we basically\ncan utilize the dimensionless counterparts of the quantities presented up to this\npoint. Note that in numerics the dimensionless and the dimensionful quantities are\nequivalent if one sets \u0016=M= 1.\nAs a result, for setting the initial conditions, three components of u\u0016\n0, two com-\nponents of S\u0016\n0are not arbitrary, they must satisfy the above \fve constraint and\nconservation equations. In this paper, we set a group of initial conditions by hand:September 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\n4W.-B. Han, S.-C. Yang\nr0;\u00120;\u001e0;u\u0012\n0;Sr\n0andS\u0012\n0, and also the energy E, the orbital angular momentum Lz\nand the spin magnitude S. From the \fve mentioned equations, we can solve out the\nleft initial conditions: ut;ur;u\u001eandSt;S\u001e. The relative accuracy of the calculated\ninitial conditions can achieve 10\u000015with double precision codes. With these initial\nconditions, one can immediately get \u001d\u0016with the help of Eq. (5). In every numerical\nstep, we integrate Eqs. (1) and (2) and solve the velocity-momentum relation (5)\nat the same time. At the end of numerical evolution, all these conserved quantities\nmust be checked again to make sure the calculations are accurate enough. During\nour numerical simulations, the relative errors of all these constraints are about 10\u000013\nafter\u001c= 106Mevolution.\nFor simpli\fcation, \frstly, we assume a spinning particle locating at the polar\ndirection of the Kerr black hole (i.e. \u0012= 0). Because of the bad behavior of Boyer-\nLindquist coordinates at \u0012= 0, we transfer the coordinates to Cartesian-Kerr-Schild\nones, then the line element is written in ( t;x;y;z ) as29\nds2=\u0000dt2+dx2+dy2+dz2\n+2Mr2\nr4+a2z2\u0014\ndt+r(xdx+ydy)\na2+r2+a(ydx\u0000xdy)\na2+r2+z\nrdz\u00152\n: (8)\nIn the Cartesian coordinates, the spinning particle is put at (0 ;0; z0) originally,\nthe components of the initial u\u0016are all zero except for ut, and the only nonzero\ncomponent of the spin vector is Sz. From Eq. (5), the only nonzero component of\nthe four-velocity is \u001dt. The schematic diagram of an aligned spin con\fguration is\nshown in Fig. 1.\nFig. 1. A spinning particle with aligned spin to a Kerr black hole locates at the z-axis.September 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\nExotic orbits due to spin-spin coupling around Kerr black holes 5\nBased on these assumptions, Eq. (1) is reduced to\nduz\nd\u001c=\u0000ut\u001dt\u0012\n\u0000z\ntt+1\n2SRz\ntxygttpgzz\u0013\n\u0011ut\u001dt(Fm+Fss); (9)\nwhereFmmeans the gravitational interaction due to the curvature (mass) and Fss\nthe spin-spin interaction. The second term is de\fnitely zero when S= 0 ora= 0.\nFor clarity, we write down the expressions of them\nFm=\u0000M(z2\u0000a2)(z2\u00002Mz+a2)\n(z2+a2)3; (10)\nFss= +MaSr\nz2\u00002Mz+a2\nz2+a2[z3(3z\u00006M) + 2a2z(z+M)]\u0000a4\n(z2+a2)4: (11)\nThe behaviors of these two functions near horizon are plotted in Fig. 2. Outside\nof the horizon, the value of Fmis always negative to o\u000ber \\regular\" gravity (here\nwe assume zis positive, for negative z, vice versa). However, we can clearly \fnd\nthat the spin-spin coupling force Fssis positive with aligned spin. If we change\nthe direction of spin, the direction of spin-spin coupling is also changed (See the\nanti-aligned case in Fig. 2). In this way, the spin-spin coupling can be thought as\na kind of phenomenological counter-gravity. However, when the spin value \u00141, the\nspin-spin force is not as large as the interaction induced by the mass so that it\ncannot fully counteract the latter one.\nActually, the physically allowed value of spin of the particle in the extreme-\nmass-ratio system should be much less than 1. For compact objects like black holes,\nneutron stars or white dwarfs, the magnitudes of spins are \u0018\u00162=\u0016M =\u0016=M\u001c1.16\nHowever, for a noncompact body like Jupiter, the spin value of it in the Jupiter-\nstellar mass black hole system can be as large as 1. For this case, in the ultra-\nrelativistic region, the tidal in\ruence from the black hole cannot be ignored. The\ntidal radius rt\u0018Rp(M=\u0016 )1=3\u001dRs(Rpis the radius of planet and Rsthe\nSchwarzschild radius of the black hole, see Eq. (6.1) in Ref. 30), then the planet will\nbe disrupted by the black hole in the strong \feld region.\nFig. 2. (Color online) Left panel: Fm(red solid line), Fss(aligned and anti-aligned cases); Right\npanel:Fm+Fssfor the aligned spin case (directions of the spins of particle and black hole are the\nsame). The parameters used for plotting are a= 1;jSj= 1.September 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\n6W.-B. Han, S.-C. Yang\nEven though the astrophysically relevant dimensionless values of the spin should\nbe much less than 1, in the study we are interested in the dynamical aspects of\nthe MPD equations and these aspects S\u00181. For example, in Fig. 2, for the spin\nmagnitudeS= 1 and an extreme Kerr black hole, we \fnd that Fssis always less than\nFm. Mathematically, equilibrium points between the two forces do not exist until the\nspin reaches the value 2.4925. This spin value is impossible for an extreme-mass-ratio\nsystem involving stellar compact objects, so there is no equilibrium point for such\nsystems. On the other hand, there is no equilibrium point for noncompact objects,\ntoo. As analyzed by Wald, the MPD equations will be invalid in this extremely large\nspin cases, because this spin magnitude asks for a body whose size is greater than\nthe back ground curvature (see Ref. 31).\nGenerally, the direction of the spin-spin coupling can be arbitrary due to the\norientation of the spin vector, unlike the mass part, which always points to the\nmass center. Along the z axis, only the spin-spin interaction is present; a spin-\norbit interaction will appear if the small body is moved o\u000b the axis. It should be\nmentioned here that several papers7,31,32have already discussed this situation, and\nthe results in this paper coincide with theirs.\nNow, we analyze the \\acceleration\" along the \u0012direction (du\u0012=d\u001c) when a spin-\nning particles lies on the equatorial plane. For convenience, we are back to the\nBoyer-Lindquist coordinates. The contribution of the curvature part on du\u0012=d\u001cis\n\u0000u\u0012vr=r, which is re\rectively symmetrical about the equatorial plane. When the\nsign ofu\u0012is changed , the sign of du\u0012=d\u001cis also changed but the magnitude remains\nunchanged.\nFor simpli\fcation, we set the initial vr= 0, then the contribution of the curva-\nture part disappears. If we allow the spin direction of particle to be arbitrary (no\nlonger aligned with the rotational axis of Kerr black hole), the contribution from\nspin-curvature coupling on du\u0012=d\u001c(i.e. the right hand of Eq. (1)) is nonzero:\ndu\u0012\nd\u001c=MSr\nr7fv\u001e[(Lz\u0000aE)(3a3\u00002aMr ) +ar2(3Lz\u00005aE)\u00002r4E]\n\u0000vt[(Lz\u0000aE)(3a2\u00002Mr) +r2(Lz\u00003aE)]g: (12)\nWe notice that the above equation does not include u\u0012. In the \frst-order approx-\nimation of S, vt;\u001e\u0019+ut;\u001e+O(S2), Eq. (12) is independent from u\u0012. Actually,\nunder our assumption vr= 0;\u0012=\u0019=2, the right-hand side of (12) is independent\nfromu\u0012exactly, though the mathematical proof is complicated (solve vt,v\u001efrom\nEq.(5) and take into Eq. (12). We will see this point in the following numerical\nexperiments. This means when u\u0012changes sign, the \\acceleration\" contributed by\nthe spindu\u0012=d\u001c(s) does not change its sign and at the same time the magnitude\nremains. In this sense, the re\rection symmetry is destroyed due to the spin of the\nparticle.\nOne can see an example demonstrated in Fig. 3. The direction angle of spin\nis \fxed as ^ \u000bs= 83:1\u000e;j^\fsj= 52:9\u000e(see the next section for the details of our\nde\fnition of spin direction), and ur=vr= 0 at this moment. When u\u0012changesSeptember 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\nExotic orbits due to spin-spin coupling around Kerr black holes 7\nFig. 3. Spinning particles locates at the equatorial plane of a Kerr black hole.\nits value from 3 :8\u000210\u00002to\u00003:8\u000210\u00002,du\u0012=d\u001cdoes not change its sign (always\nequals 3:0\u000210\u00005). This property may give a clue to the exotic orbits studied in\nthis paper.\n3. Exotically asymmetrical orbits\nAs we have mentioned before, due to the axisymmetry of the spacetime, and the\nre\rection symmetry of the background along the equatorial plane, for nonspinning\ntest particles orbiting the Kerr black hole, the perihelion advance makes the pericen-\nter to precess in the orbital plane, and the frame-dragging e\u000bect causes the orbital\nplane to precess around the rotation axis of the Kerr black hole at the same time.\nBecause of these two precessions, the patterns of the particles' trajectories distribute\nsymmetrically about the equatorial plane and the rotation axis of the black hole.\nFor the spinning particles, even for the highly spinning ones, in most cases, the pat-\nterns still have these two symmetries (sometimes approximately). Figure 4 shows\nthe orbit of a spinning particle with S= 0:8, energyE= 0:9 and total angular\nmomentum Lz= 2:5 around an extreme Kerr black hole with a= 1, and clearly\ndemonstrates the symmetry.\nFor the numerical calculation of the orbits, \frst we need to input the initial\nconditions. The free parameters inputted by hand are the initial coordinates t0=\n0;r0;\u00120;\u001e0, one initial component of the four velocity u\u0012\n0, two initial components of\nthe spin vector Sr\n0; S\u0012\n0, and the values of E; LzandS. The remaining \fve initial\nconditionsut\n0;ur\n0;u\u001e\n0andSt\n0; S\u001e\n0are subsidiary quantities which are calculated from\nthe \fve constraint equations. We also compute the initial direction of the spin from\nthe initial conditions for understanding better the spin vector. For describing the\ndirection of spin, we introduce a local hypersurface-orthogonal observer (HOO).\nIn a Kerr space-time, the HOO is represented by an observer with zero angular\nmomentum with respect to the symmetry axis, ZAMO, havingSeptember 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\n8W.-B. Han, S.-C. Yang\n-10\n0\n y1010 x0-103\n2\n1\n0\n-1\n-2\n-3 z\n x-10 -5 0 5 10 y\n-6-4-202468\n ;0 2 4 6 810 z\n-3-2-10123\n x or y-5 0 5 z\n-6-4-20246\ny-z plane\nx-z plane\nFig. 4. Orbits of a spinning particle with spin parameter S= 0:8, energyE= 0:9 and total\nangular momentum Lz= 2:5 around a Kerr black hole with a= 1. The initial conditions are\nr0= 4; \u00120=\u0019=2;\u001e0= 0,u\u0012\n0= 0 andSr;\u0012\n0= (\u00000:32;\u00000:16). The subsidiary data are 1.623, 0.285,\n0.156, -0.390 and -0.096 for ut\n0; ur,u\u001e,StandS\u001e, respectively. The corresponding initial angles\n^\u000bs\n0;^\fs\n0are 120:2\u000e;29:1\u000e. The top-left panel shows the 3D trajectories, the top-right and bottom-\nleft ones show the projection orbits on x\u0000yand\u001a\u0000zplanes respectively, where \u001a=p\nx2+y2.\nThe bottom-right panel shows the projection points when the trajectories pass through the y\u0000z\nandx\u0000zplanes.\nu\u0016\nZAMO =r\nA\n\u0001\u0006(1;0;0;2Mar\nA); (13)\nwhere \u0001 = r2\u00002Mr+a2, \u0006 =r2+a2cos2\u0012andA= (r2+a2)2\u0000\u0001a2sin2\u0012. The\nrelative spin with respect to the HOO is given as\n^S\u0016=^\u0000\u00001(\u000e\u0016\n\u0017+u\u0016\nZAMOuZAMO\u0017)S\u0017; (14)\nwhere ^\u0000 =\u0000u\u0016u\u0016\nZAMO is the relative boost factor. Now, one can project the relativeSeptember 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\nExotic orbits due to spin-spin coupling around Kerr black holes 9\nspin vector to observer's local Cartesian triad with basis vectors\ne^r\n\u0016= (0;pgrr;0;0); (15)\ne^\u0012\n\u0016= (0;0;pg\u0012\u0012;0); (16)\ne^\u001e\n\u0016= (gt\u001epg\u001e\u001e;0;0;pg\u001e\u001e); (17)\nto get the spin components with respect to this local orthonormal space triad\n^S^i=^S(cos ^\u000bs;sin ^\u000bscos^\fs;sin ^\u000bssin^\fs): (18)\nThe angles ^ \u000bs;^\fsrepresent the orientation of the spin. For a detailed description,\nplease see Ref. 20.\nThe orbit con\fguration demonstrated in Fig. 4 represents the normal behavior\nof spinning particles, i.e., having equatorial symmetric patterns like the nonspinning\ncases. If one calculates the average value of z-coordinates when the particle passes\nthrough the x-z or y-z plane (i.e. the y= 0 plane or x = 0 plane), the average\nvalue will go to 0 after su\u000ecient orbital evolution (see the bottom-right panel of\nFig. 4). In this symmetric case, we get in x-z plane \u0016 zy=0= 6\u000210\u00005and in y-z\nplane \u0016zx=0= 5\u000210\u00005. Another criterion is the di\u000berence of the maximum jzjvalue\nachieved by the particle above and below the equatorial plane, i.e., z++z\u0000. For the\nsymmetric pattern, z+equals\u0000z\u0000approximately, then z++z\u0000\u00190.\nHowever, from the simple analysis in Sec. 2, we know that the spin-spin coupling\nwill supply a kind of \\force\" with di\u000berent direction from the gravity of the mass.\nThe direction of spin-spin interaction depends on the direction of spin. We also\n\fnd that the spin-curvature coupling can destroy the re\rection symmetry about\nthe equatorial plane. For the generic orbits, the spin vector precesses along the tra-\njectory in a very complicated way. In general, the spin directions are not re\rectively\nsymmetric about the equatorial plane, and then the total \\force\" is no longer sym-\nmetrical about the equatorial plane. In some situations, this asymmetry is enough\nobviously to be seen (as shown in Fig. 5 and 6).\nThese orbits show a kind of exotic con\fguration, i.e., an asymmetry pattern\nappears about the equatorial plane of a Kerr black hole. It seems that the orbits\nhave \\polarized\" directions. For example, we just change the initial velocity and\nthe spin direction in the case of Fig. 4, and as a result we get an asymmetrical\n\\upward\" orbit (Fig. 5). In this case, the initial angles of the spin vector with\nrespect to the local orthonormal space triad are ^ \u000bs\nup= 61:9\u000e;^\fs\nup= 293:6\u000e. It\nmeans that spin points downward respect to the equatorial plane. Obviously, the\npattern is asymmetric about the equatorial plane in the \fgure. From the bottom-\nleft panel of Fig. 5, we can \fnd that z++z\u0000\u00192, and the average zvalues across\ny-z and x-z plane are 0.144 and 0.136 respectively (from the bottom-right panel).\nThese two numbers deviate from 0 obviously comparing with the symmetric case.\nWe keep all parameters except for the directions of momentum and spin (^ \u000bs\ndown =\n118:1\u000e;^\fs\ndown = 66:4\u000e), then we get an asymmetrical \\downward polarized\" orbit\nin Fig. 6. We notice that ^ \u000bs\nup+ ^\u000bs\ndown = 180\u000eand^\fs\nup+^\fs\ndown = 360\u000e. It means thatSeptember 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\n10W.-B. Han, S.-C. Yang\nthe initial direction of the spin in the downwards pattern points to upwards from the\nequatorial plane. Notice that this asymmetry is only about the equatorial plane, the\norbital con\fguration is still axis-symmetric with su\u000ecient evolution time. It looks\nlike a force with one direction (along or anti-along with z axis) to push the particle\n\roating above or sink down about the equatorial plane. This exotic asymmetrical\nphenomenon was found in Ref. 18. In this paper, we study this phenomenon more\nthoroughly.\n-10\n y0101050\n x-5-100\n-1\n-21234 z\n x-5 0 5 y\n-6-4-20246\n ;2 4 6 8 z\n-10123\n x or y-6-4-20246 z\n-4-20246\ny-z plane\nx-z plane\nFig. 5. Orbits of spinning particles with spin parameter S= 0:8, energyE= 0:9 and total\nangular momentum Lz= 2:5 around a Kerr black hole with a= 1. These panels show a kind\nof \\upward\" orbit, the particle is put at r0= 4; \u00120=\u0019=2;\u001e0= 0 at beginning, and u\u0012\n0= 0.\nThe initial spin vector for the \"upward\" orbit is Sr; \u0012\n0= (0:32;\u00000:08). The subsidiary data are\n1.615, 0.187, 0.172, 0.594 and 0.192 for ut\n0; ur,u\u001e,StandS\u001e, respectively. The corresponding\ninitial angles ^ \u000bs;^\fsare 61:9\u000e;293:6\u000e. The top-left panel shows the 3D trajectories, the top-right\nand bottom-left ones show the projection orbits on x\u0000yand\u001a\u0000zplanes respectively, where\n\u001a=p\nx2+y2. The bottom-right panel shows the projection points when the trajectories pass\nthrough the y\u0000zandx\u0000zplanes.\nFor revealing the relation between the orbital polarization orientation and the\ninitial spin direction, in Fig. 7, we plot the contour of z++z\u0000with variable angles\n^\u000bs;^\fs.z+andz\u0000mean the maximum and minimum of zreached by a spinning\nparticle with a set of given parameters after enough orbital evolution. We use varied\ncolor for di\u000berent values of z++z\u0000. Green points denote z++z\u0000\u00190, and dark\nred or blue ones denote z++z\u0000deviates from zero obviously. So red or blue points\nrepresent the asymmetry patterns. Obviously, z++z\u0000>0 implies an upwards orbit,September 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\nExotic orbits due to spin-spin coupling around Kerr black holes 11\n-10\n0\n y1010 x0-102\n1\n0\n-1\n-2\n-3\n-4 z\n x-5 0 5 y\n-6-4-20246\n ;2345678 z\n-4-3-2-1012\n x or y-6-4-20246 z\n-6-4-2024\ny-z plane\nx-z plane\nFig. 6. Orbits of spinning particles with spin parameter S= 0:8, energyE= 0:9 and total\nangular momentum Lz= 2:5 around a Kerr black hole with a= 1. These panels show a kind of\n\\downward\" orbit, the particle is put at r0= 4; \u00120=\u0019=2;\u001e0= 0 at beginning, and and u\u0012\n0= 0.\nThe initial spin vector for the \"upward\" orbit is S\u0016\n0= (\u00000:32;\u00000:08). The subsidiary data are\n1.615, -0.187, 0.172, -0.594 and -0.192 for ut\n0; ur,u\u001e,StandS\u001e, respectively. The corresponding\ninitial angles ^ \u000bs;^\fsare 118:1\u000e;66:4\u000e. The top-left panel shows the 3D trajectories, the top-right\nand bottom-left ones show the projection orbits on x\u0000yand\u001a\u0000zplanes respectively, where\n\u001a=p\nx2+y2. The bottom-right panel shows the projection points when the trajectories pass\nthrough the y\u0000zandx\u0000zplanes.\nand vice versa. It is clear that the orbital polarization direction is decided by the\ninitial direction of the spin. Furthermore, we also give the results for a smaller spin\nvalueS= 0:4, and do not \fnd obvious asymmetric orbits (all points are green).\nUnfortunately we do not \fnd a general quantitative criterion to determine which\nkind of initial conditions will produce asymmetric patterns. By a lot of scans in\nthe parameter space, we can de\fnitely conclude that the exotic orbits found by\nus can only happen with arti\fcially large spin (i.e. S\u00181). Actually, we did not\n\fnd any obviously exotic orbits when S= 0:4 for an extreme Kerr black hole.\nWe can speculate carefully that the asymmetric phenomena cannot happen when\nS <0:1. If we \fx all parameters except for the spin components SrandS\u0012, which\nare equivalent to the angles \u000bsand\fs, we may determine the range of the angles\nthat the exotic behavior occurs. From the top-left panel of Fig. 7, one can \fnd\nthat whenj\fsj>50\u000e(e.g. we can take 310\u000eas the same as\u000050\u000e) the asymmetricSeptember 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\n12W.-B. Han, S.-C. Yang\n ,s0 50 100 150 -s\n050100150200250300350\n-2-1.5-1-0.500.511.52\n ,s0 50 100 150 -s\n050100150200250300350\n-2-1.5-1-0.500.511.52\n ,s0 50 100 150 -s\n050100150200250300350\n-2-1.5-1-0.500.511.52\n ,s0 50 100 150 -s\n050100150200250300350\n-2-1.5-1-0.500.511.52\nFig. 7. (Color online) The values of z++z\u0000with variable initial spin directions (^ \u000bs;^\fs). The\ncolor of point represents the values of z++z\u0000. Top-left:E= 0:9; Lz= 2:5; a= 1 andS= 0:8.\nTop-right:E= 0:9237; Lz= 2:8; a= 1 andS= 0:8. Bottom panels: all parameters are the same\nbutS= 0:4.\nbehavior happens. For the cases of initial angle j\fsj<50\u000e, patterns of orbits are\napproximately symmetric. Be careful, this criterion is only correct for the special\ncase ofE= 0:9; Lz= 2:5; a= 1 andS= 0:8. If changing any one of these four\nparameters, the range of angles for exotic behaviors is di\u000berent.\nWe emphasize that there is no direct connection of the nonre\rection symmetric\norbits with chaotic behaviors. Some nonre\rection symmetric orbits can be regular\n(for example a case with energy 0.9237, angular momentum 2.8 and spin magnitude\n0.8), and some may be chaotic. Such kind of exotic phenomenon is restored for\nvery large evolution time (we evolve the orbits up to 107M), we believe that it\nis not a transient phenomenon. Notice that in Figs. 5-7, we choose the extreme\nKerr black hole just for demonstrating the most prominent e\u000bects. However, there\nis no connection between the asymmetry and the naked singularity ( a= 1). An\nimmediate example is in the case of E= 0:9237;Lz= 2:8 andS= 0:8, instead of\na= 1 witha= 0:998, the asymmetric pattern happens too.\nAdditionally, we do not \fnd any obviously asymmetric pattern if the black hole\nis a Schwarzschild one in the parameter space we scanned (energy from 0.8 to 0.95,\nangular momentum from 2.0 to 5.0 and r0from 8 to 10). This may imply thatSeptember 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\nExotic orbits due to spin-spin coupling around Kerr black holes 13\nonly the spin-spin interaction but not spin-orbit one causes the exotic phenomena.\nHowever, our scanning do not cover all the parameter space. This statement may\nbe only valid for the parameter space we scanned.\nFor the Kerr spacetime, the spin-spin coupling is a necessary condition but\nnot a su\u000ecient one for the appearance of asymmetric patterns. Without this spin-\nspin e\u000bect between the spinning particle and fast rotating black hole, asymmetric\npatterns may not happen. However, the spin-spin interaction does not guarantee\nthe appearance of asymmetric orbits. A fast rotating Kerr black hole can be easily\nfound in the universe, but the spin magnitude of particle should be treated carefully.\nAs we discussed in the Sec. 2, the physical value of spin must be \u001c1 for an extreme-\nmass-ratio system. That means it is di\u000ecult to \fnd such exotic orbits in the realistic\nastrophysics. Therefore, our \fnding may have no in\ruence on the gravitational-wave\ndetection of LISA, Taiji, and Tianqin.\n4. Conclusions\nIt is a well-known fact that the gravitational force is an attractive force. However,\nas already revealed by a few researchers, we know that the spin-spin interaction\nbetween the spinning particle and Kerr black hole can have arbitrary action direc-\ntions (e.g. Ref. 31). Phenomenologically, the spin-spin coupling can actually o\u000ber a\nkind of \\counter-gravity\". The exotically asymmetrical orbit con\fgurations about\nthe equatorial plane demonstrated in Figs. 5 and 6 should come from the spin-spin\ncoupling, because we have not found this asymmetry either for nonspinning parti-\ncles or for the Schwarzschild black hole. However, the orbits in Figs. 5 and 6 are\nquite complicated, and in this paper we do not plan to analyze the quantitative re-\nlation between the asymmetry and spin-spin interaction. As analyzed in Sec. 2, the\nexistence of spin can destroy the re\rection symmetry about the equatorial plane.\nThis may give a clue to the physical origin of the asymmetrical phenomena.\nHowever, not all the spinning particles demonstrate such asymmetrical orbit\npatterns, the asymmetry appears only for cases with special physical parameters.\nWe still have not found a criterion to determine if a spinning particle with a certain\nset of parameters will have an asymmetrical orbit shape, but the numerical results\nshow that it may easier to appear for large eccentricities. Actually, we do not give a\ncritical spin value for the appearance of asymmetry because it depends on too many\nparameters. There is, however, no evidence that asymmetrical phenomena happen\nwhen dimensionless spin magnitude S\u001c1. We conclude that the asymmetry can\nonly happen for astrophysically irrelevant large spin values. On the other hand,\nit is interesting to study the complicated behavior and dynamical nature of these\nextreme spinning particles.\nFor comparable mass-ratio binary systems, the spin of both components can be\n\u00181. Until now, there is no report on the analogous asymmetrical orbits for compa-\nrable mass-ratio binaries. It is very interesting to investigate if there are asymmet-\nrical orbits in the comparable mass-ratio binary systems or not. The phenomenaSeptember 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\n14W.-B. Han, S.-C. Yang\nrevealed in this paper should be interesting for the study of dynamical properties\nof the spinning particles in strong gravitational \feld. The gravitational waves from\nthe asymmetrical orbits should have some obvious properties which distinguish from\nthe normal orbits. However, considering the asymmetry can only appear in the as-\ntrophysically unrealistic cases, it should have no in\ruence on the gravitational wave\ndetections. More detailed studies on this asymmetry should be done in the future\nworks.\nAcknowledgements\nWe appreciate the anonymous Referee for pointing an error in Eq. (11). This work\nis supported by NSFC No. U1431120, QYZDB-SSW-SYS016 of CAS; W.-B. Han is\nalso supported by Youth Innovation Promotion Association CAS.\nReferences\n1. B. Carter, Phys. Rev. 174(1968) 1559.\n2. Z.-Q. Shen, K. Y. Lo, M.-C. Liang, P. T. P., Ho & J.-H. Zhao, Nature 438(2005) 62.\n3. A. M. Ghez et al., Astrophys. J. 689(2008) 1044.\n4. S. Gillessen et al., Astrophys. J. 692(2009) 1075.\n5. R. Genzel, F. Eisenhauer and S. Gillessen, Rev. Mod. Phys. 82(2010) 3121.\n6. W.-B. Han, Res. Astron. Astrophys. 14(2014) 1415.\n7. T. Tanaka, Y. Mino, M. Sasaki and M. Shibata, Phys. Rev. D 54(1996) 3762.\n8. D. Bini, F. de Felice and A. Geralico, Class. Quantum Grav. 21(2004) 5441.\n9. W.-B. Han, Phys. Rev. D 82(2010) 084013.\n10. E. A. Huerta and J. R. Gair, Phys. Rev. D 84(2011) 064023.\n11. E. Hackmann, C. L ammerzahl, Y. N. Obukhov, D. Puetzfeld and I. Scha\u000ber, Phys.\nRev. D 90(2014) 064035.\n12. U. Ruangsri, S. J. Vigeland and S. A. Hughes, arXiv:1512.00376.\n13. E. Harms, G. Lukes-Gerakopoulos, S. Bernuzzi and A. Nagar, Phys. Rev. D 93(2016)\n044015.\n14. S. Suzuki and K. Maeda, Phys. Rev. D 55(1997) 4848.\n15. S. Suzuki and K. Maeda, Phys. Rev. D 58(1998) 023005.\n16. M.D. Hartl, Phys. Rev. D 67(2003) 024005.\n17. M.D. Hartl, Phys. Rev. D 67(2004) 104023.\n18. W. Han, Gen. Relativ. Gravit., 40(2008) 1831.\n19. G. Lukes-Gerakopoulos, M. Katsanikas, P. A. Patsis and J. Seyrich, Phys. Rev. D 94\n(2016) 024024.\n20. O. Semer\u0013 ak, Mon. Not. R. Astron. Soc. 308(1999) 863.\n21. K. Kyrian and O. Semer\u0013 ak, Mon. Not. R. Astron. Soc. 382(2007) 1922.\n22. M. Mathisson, Acta Phys. Pol. 6(1937) 163.\n23. A. Papapetrou, Proc. R. Soc. Lond. A 209(1951) 248.\n24. W. G. Dixon, Nuovo Cimento 34(1964) 317.\n25. W. G. Dixon, Philos. Trans. R. Soc. Lond. A 277(1974) 59.\n26. A. A. Deriglazov and W. G. Ram\u0013 \u0010rez, Int. J. Mod. Phys. D 26(2017) 1750047.\n27. A. A. Deriglazov and W. G. Ram\u0013 \u0010rez, Adv. High Energy Phys. 2016 (2016) 1376016.\n28. W. Tulczyjew, Acta Phys. Pol. 18(1959) 393.September 7, 2017 0:44 WSPC/INSTRUCTION FILE 20170901ws-ijmpd-\n2017\nExotic orbits due to spin-spin coupling around Kerr black holes 15\n29. R. Kerr, in the First Texas Symp. Relativistic Astrophysics, eds. I. Robinson et.al.\n(University of Chicago Press, Chicago, 1965), pp. 99{102.\n30. T. Alexander, Phys. Rep. 419(2005) 65.\n31. R. Wald, Phys. Rev. D 6(1972) 406.\n32. L. F. O. Costa, J. Nat\u0013 ario, and M. Zilh~ ao, Phys. Rev. D 93(2016) 104006." }, { "title": "1406.2715v2.Weak_Localization__Spin_Relaxation__and_Spin_Diffusion__The_Crossover_Between_Weak_and_Strong_Rashba_Coupling_Limits.pdf", "content": "arXiv:1406.2715v2 [cond-mat.mes-hall] 12 Jun 2014Weak Localization, Spin Relaxation, and Spin-Diffusion:\nThe Crossover Between Weak and Strong Rashba Coupling Limit s\nYasufumi Araki,1Guru Khalsa,1,2and Allan H. MacDonald1\n1Department of Physics, University of Texas at Austin, Austi n, Texas 78712, USA\n2Center for Nanoscale Science and Technology, National Inst itute of Standards and Technology, Gaithersburg, MD 20899, USA\nDisorderscatteringandspin-orbitcouplingaretogetherr esponsible forthediffusionandrelaxation\nof spin-density in time-reversal invariant systems. We stu dy spin-relaxation and diffusion in a two-\ndimensional electron gas with Rashba spin-orbit coupling a nd spin-independent disorder, focusing\non the role of Rashba spin-orbit coupling in transport. Spin -orbit coupling contributes to spin\nrelaxation, transforming the quantuminterference contri bution toconductivityfrom anegative weak\nlocalization (WL) correction to a positive weak anti-local ization (WAL) correction. The importance\nof spin channel mixing in transport is largest in the regime w here the Bloch state energy uncertainty\n/planckover2pi1/τand the Rashba spin-orbit splitting ∆ SOare comparable. We find that as a consequence of this\nspin channel mixing, the WL-WAL crossover is non-monotonic in this intermediate regime, and use\nour results to address recent experimental studies of trans port at two-dimensional oxide interfaces.\nPACS numbers: 73.20.Fz,71.70.Ej,72.25.Rb,71.10.Ca\nI. INTRODUCTION\nSpin-orbit coupling, present whenever electrons move\nin a strongelectricfield, hasrecentlybeen playingamore\nprominent role in electronics. When spin-orbit coupling\nis present, broken inversion symmetry lifts the two-fold\nspin-degeneracy of Bloch states in a crystal. For ex-\nample, as pointed out by Rashba1, spin-orbit coupling\nproduces spin-splitting at surfaces and at interfaces be-\ntween different materials. In spintronics, Rashba spin-\norbit coupling can provide a handle for electrical con-\ntrol of spin since its strength and character depends not\nonly on atomic structural asymmetry but also on exter-\nnal gate voltages2, allowing for the possibility of a spin-\nbased field-effect transistor.3Alternately, spin-splitting\ndue to Rashba spin-orbit coupling in proximity coupled\nnanowires can lead to topological superconductivity and\nMajorana edge states4, which can provide an attractive\nHilbert space for quantum state manipulation for the\npurpose of quantum information processing.\nBecause spin-orbit coupling does not conserve spin,\none of its most important consequences in spintronics is\nits role in providing a mechanism for relaxation of non-\nequilibrium spin densities. In the absence of spin-orbit\ncoupling, total charge and all three components of total\nspin are conserved. Spin relaxation mechanisms due to\nspin-orbit coupling can be classified into two types: the\nElliott–Yafet (EY) mechanism,5,6where skew-scattering\ndue to spin-orbit interactions with scattering centers\nis the the most obvious spin-relaxation process, and\nthe more subtle but equally important Dyakonov–Perel\n(DP) mechanism,7in which the momentum-dependent\nspin-orbit effective magnetic fields responsible for spin-\nsplitting of the Bloch states cause spin-precession be-\ntween collisions. The DP spin relaxation mechanism is\noften dominant in spintronics, and can cause subtle in-\nterplays between charge and spin transport.8\nSpin relaxation has an important indirect effect on theFIG. 1: (Color online) A schematic illustration of the the\nquantum correction to conductivity. Quantum interference\nbetween a closed electron path (red) and a nearly time-\nreversed counterpart (blue), alters the backscattering ra te\nwhenq, the sum of the two incoming momenta, is close to\nzero. The interference is constructive in the absence of spi n-\norbit coupling, enhancing back scattering and suppressing the\ndiffusion constant and the conductivity, but can be destruc-\ntive and enhance the conductivity when spin-orbit is presen t.\nquantum contribution to the conductivity. In weakly\ndisordered metallic systems with no spin-orbit coupling,\nbackscattering is enhanced by constructive interference\nbetween time-reversed paths (see Fig.1) yielding a nega-\ntive quantum correction to the classical conductivity cal-\nculated from Drude’s formula. This effect is referred to\nas weak localization (WL).9–12In two-dimensional sys-\ntems, WL acts as a precursor to the transition into the\nAnderson insulator state in which disorder is sufficiently\nstrong to localize electrons. When the spin degree-of-\nfreedom is accounted for in the absence of spin-orbit cou-\npling, spin-degeneracymultipliesthe conductivitycorrec-\ntion by a factor of two. In general there are four two-\nparticle spin states which contribute to the interference.\nWhen parsed in terms of total spin eigenstates, inter-\nference between time-reversed paths is constructive for2\nthe three triplet channels, but destructive for the sin-\nglet channel because of the Berry phase contributed by\nrotation of the spin wave function along the path, re-\ncovering the factor of two enhancement. Spin relaxation\nchanges this situation. Because the spin-density present\nin the triplet channels relaxes their contribution to the\nconductivity, the correction is reduced when spin-orbit\ncoupling is present, whereas the singlet channel is unaf-\nfected due to charge conservation. When this effect is\nstrong, either due to strong spin-orbit coupling or due\nto long phase coherence times, the quantum contribu-\ntion becomes positive. In this case the quantum correc-\ntion is referred to as weak antilocalization (WAL). WL\nand WAL can be identified by studying the temperature\nand magnetic field dependence of the conductivity, since\nthese parameters limit the phase coherence length L, the\ncharacteristic length within which electrons can propa-\ngate without losing their phase coherence. The theory\nof WAL onset was developed microscopically by Hikami,\nLarkin and Nagaoka (HLN) using a model with EY spin-\nrelaxation,13and later macroscopically using a nonlinear\nσ-model approach14which demonstrated that the effect\ndepends mainly on global symmetries and not on micro-\nscopicdetails. Iordanskii,Lyanda-GellerandPikus(ILP)\nlater were the first to point out that DP spin relaxation\nalso leads to WAL, and that the triplet channels contri-\nbution is modified compared to that implied by the EY\nmechanism.15WAL induced by DP spin relaxation has\nbeen identified as responsible for negative magnetocon-\nductivity in quantum wells16,17and in topological insu-\nlator surface states.18–20A recent experiment on trans-\nport at the interface between LaAlO 3(LAO) and SrTiO 3\n(STO) has demonstrated that a WL-WAL crossover can\nbe induced by gate voltage modulation.21.\nMotivatedpartlybytheexperimentsinLAO/STOhet-\nerostructures, we attempt to investigate in detail the\ndependence of WL and WAL transport contributions\non spin-orbit coupling strength across the crossover be-\ntween resolved spin-splitting (where disorder broaden-\ning is much smaller that spin-orbit coupling) and spin-\nsplitting obscured by disorder, by tuning the Rashba\nspin-orbit coupling strength in a two-dimensional elec-\ntrongases(2DEG).This crossoveriscontrolledbyacom-\npetition between two energy scales: the Bloch state spin-\nsplitting ∆ SOinduced by spin-orbit coupling in systems\nwithout inversion symmetry, and the Bloch state energy\nuncertainty ηdue to the finite lifetime of Bloch states\nin a disordered system. The asymptotic behavior in the\nextreme cases is obvious from the arguments we have\nsummarized briefly above. In the band-unresolved limit\n(∆SO≪η) we can apply ILP’s analysis15by taking spin-\norbit coupling into account perterbatively. In this limit\nWALemergesfromWLbehaviorinthe long-phasecoher-\nence limit. On the other hand, in the band-resolved limit\n(∆SO≫η), only the spin singlet channel contributes to\nquantum interference and we obtain perfect WAL behav-\nior. What we intend to investigatehere is the behaviorin\nthe intermediate regime (∆ SO≈η). For this purpose, wehave developed tools which enable us to investigate spin-\nrelaxationanddiffusion, andtoevaluatequantumcorrec-\ntions to Boltzmann transport at any value of ∆ SO/η. We\nhavefound that fora two-dimensionalelectron-gasmodel\nwith Rashba spin-orbit interactions, spin relaxation in\nthe intermediate regime cannot be simply described by\nILP’s picture. Spin relaxation is partially suppressed by\ninterference between channels, leading to a new plateau\non which the WL/WAL behavior is relatively insensitive\nto spin-orbit coupling strength. We suggest that such a\nbehavior can be confirmed experimentally by tuning the\nspin-orbit coupling with gate voltageand fixing other pa-\nrameters.\nThis paper is organized as follows. In Section II, we\nexplain how we evaluate the low-energy long-wavelength\nlimit of the electron-pair(Cooperon) propagatortreating\nspin-orbit coupling and spin-relaxation it producees non-\nperturbatively. In Section III, we discuss our numerical\nresults for the Cooperon of the Rashba model, and cal-\nculate the spin relaxation lengths for each triplet channel\nin order to characterize spin relaxation behavior across\nthe crossover between the spin-resolved and unresolved\nlimits. Using the spin-relaxation characteristics we have\ncalculated, we summarize the WL-WAL crossoverin Sec-\ntionIV,constructingaphasediagraminthe ∆ SO-ηplane\nwhich identifiesthree regimes: perfect WL, perfect WAL,\nand an intermediate plateau regime. Finally, in Section\nV, we briefly summarize our findings and present our\nconclusions.\nII. MICROSCOPIC THEORY OF WL AND WAL\nAs a typical example of a system with broken inversion\nsymmetry, we consider an isotropic 2DEG band Hamil-\ntonian with a Rashba spin-orbit interaction term,\nˆH(k) =k2+ ˆσihi(k), (1)\nwhere we have set /planckover2pi1= 1 so that wave vector can be\nidentified as momentum and for simplicity rescaled mo-\nmentum to set 2 m= 1. We distinguish 2 ×2 matri-\nces in the spin-up/down representation by a hataccent.\nThe second term in Eq. 1 allows for arbitrary spin-orbit\ncoupling given a model with a single spin-split band.\n(ˆσi(i=x,y,z) are Pauli matrices.) For the Rashba\nmodel, the effective magnetic field is perpendicular to\nmomentum k. and has a coupling strength characterized\nby the parameter α:h(k) =α(ky,−kx,0). Rashba cou-\npling is symmetry-allowed in systems in which inversion\nsymmetry is broken because the two-dimensional system\nis not a mirror plane. For example Rashba coupling can\nbe induced by a gate-induced electric-field perpendicu-\nlar to the two-dimensional electron gas plane. It leads\nto spin-splitting 2 αkat momentum k, where the band\nenergies are\nEn(k) =k2+nαk, (2)3\nwith band index n=±1. Limiting the Fermi energy\nǫFto be positive, the two bands have Fermi surfaces\nwith different Fermi radii kFn= (vF−nα)/2, but equal\nFermi velocities vF=√α2+4ǫF. In this article we de-\nfine ∆ SO= 2α¯kFand use this number to characterize\nthe strength of spin-orbit coupling at the Fermi energy.\nHere¯kF= (kF++kF−)/2 =vF/2 is the typical value of\nthe Fermi momentum, independent of n.\nWe assume a disorder model with randomly-\ndistributed, spin-independent, δ-function scatterers:\nˆHdis(r) =VN/summationdisplay\ni=1δ(r−ri), (3)\nwhereNis the total number of impurities. After disor-\nder averaging disorder vertices are linked in pairs with\nfour-point vertex amplitude NV2/Ω2≡γ/Ω, where\nΩ is the volume of the system. Thus, the disor-\nder unaveraged one-particle Green’s function ˆG±\n0=/bracketleftig\nǫF−ˆH−ˆHdis±i0/bracketrightig−1\nreduces to a translationally in-\nvariant one,\nˆG±(k) =/angbracketleftig\nˆG±\n0(k,k)/angbracketrightig\ndis=1\nǫF−ˆH(k)±iη,(4)\nin the Born approximation, where ±distinguishes re-\ntarded and advanced Green’s functions, ∝an}bracketle{t·∝an}bracketri}htdisrepresents\nthe average over disorder configuration, and\nη=1\n2τ=−γ\nΩIm/summationdisplay\nkG+(k) =γ\n4. (5)\nThe spectral weight of the Green’s function is spread\nover the energy interval η, corresponding to the finite-\nlifetime energy uncertainty of the Bloch states. When\n∆SO≪η, the two bands are degenerate to within en-\nergy resolution and the role of spin-orbit interactions is\nsimply to cause spin-precessionbetween collisions. When\n∆SO≫η, on the other hand, the two-band energies are\nwell resolved and coherence between bands is negligible.\nIn our analysis we assume that ∆ SO,η≪ǫF, the nor-\nmal experimental situation, but allow the ratio ∆ SO/η\nto vary. In our discussion section, we comment briefly\non the ∆ SO,η≫ǫFcase, which corresponds closely to\nthecircumstanceachievedintopologicalinsulatorsurface\nstates.\nIn general, the longitudinal conductivity at zero tem-\nperature is given by the Kubo–Streda formula,\nσ=1\n2πΩRe/summationdisplay\nk,k′Tr/angbracketleftig\nˆjx(k)ˆG+\n0(k,k′)ˆjx(k′)ˆG−\n0(k′,k)/angbracketrightig\ndis,\n(6)\nwhere the current matrix is defined by ˆjx(k) =eˆvx(k) =\ne[∂ˆH(k)/∂kx]. Forδ-function scatterers the semi-\nclassical Boltzmann theory result for the conductiv-(a)\n(b)\nFIG. 2: (Color online) Feynman diagrams for the dominant\nquantum correction to the conductivity. (a) The upper line\nand lower lines represent the retarded and advanced Green’s\nfunctions ˆG±, respectively. The maximally crossed diagrams\ncan be reorganized into a particle-particle ladder-diagra m\nsum. (b) Graphical representation of the ladder diagram sum\nfor the Cooperon which can be performed by solving a Bethe–\nSalpeter equation.\nity, namely Drude’s formula, is recovered by disorder-\naveraging the two Green’s functions separately:\nσ0=1\n2πΩRe/summationdisplay\nkTr/bracketleftig\nˆjx(k)ˆG+(k)ˆjx(k)ˆG−(k)/bracketrightig\n.(7)\nσ0is proportional to the density of states at the Fermi\nenergy. Since the total density of states at fixed ǫFis\nindependent of α, the classical conductivity σ0is inde-\npendent of αprovided that ∆ SOis small compared to\nthe Fermi energy ǫF.\nThe leading quantum correction to the conductivity\ncomes from the interference between a closed multiple-\nscattering path and its time-reversed counterpart, as il-\nlustrated in Fig.1. In its diagrammatic representation,\nthe sum of this interference over all classical paths is\ncaptured by summing the diagrams in which disorder\ninteraction lines connecting the retarded and advanced\nGreen’s functions are maximally crossed, as illustrated\nin Fig. 2(a). The particle-particle ladder diagram sum is\nreferred to as the Cooperon ˇΓ(q), withq=k+k′the to-\ntalmomentum flowingintothe Cooperon,orequivalently\nthe deviation from the perfect backscattering which oc-\ncurs for q= 0. In the following we distinguish matri-\nces in the 4 ×4 tensor product space, with the basis\n{| ↑↑∝an}bracketri}ht,| ↑↓∝an}bracketri}ht,| ↓↑∝an}bracketri}ht,| ↓↓∝an}bracketri}ht}, by acheckaccent over the let-\nters as in ˇO. The contribution of the Cooperon to the\nconductivity is\n∆σ=Re\n2πΩ/summationdisplay\nk,k′(ˆG−ˆjxˆG+)ν′µ(k)ˇΓµµ′\nνν′(q)(ˆG+ˆjxˆG−)µ′ν(k′)\n=e2\n2πReTr/bracketleftigg\nˇW/summationdisplay\nqˇΓ(q)/bracketrightigg\n, (8)\nwhereµ,µ′,ν,ν′take the spin indices ↑or↓. We will as-\nsume that ˇΓ(q) has a peak at backscattering q= 0, and4\nthat it is large only for small total momentum q=k+k′.\nThe area of summation by qis limited by the character-\nistic length scales ofthe system. The lowercutoff is given\nby the inverse of a large length scale L, within which the\nelectron can move without losing its phase coherence. L\nacts like the (effective) size of a phase coherent system.\nThe lengthscale Ldecreaseswith increasingtemperature\ndue to increased inelastic scattering by phonons or other\nelectrons, or with an increase of magnetic field due to\nthe cyclotron motion of the Cooper pair center of mass.\nIn our analysis, we represent both effects by the length\nL. The upper wave vector cutoff is given by the inverse\nof the elastic mean free path l=√\n2Dτ, above which\nelectron dynamics is ballistic rather than diffusive. Here\nD=v2\nFτ/2 is the diffusion coefficient.\nTheweight factor (ˇW) in Eq. 8 specifies how each\nCooperon channel contributes to the conductivity, and\nis defined by\nˇWµ′µ\nν′ν=1\nΩ/summationdisplay\nk(ˆG−ˆvxˆG+)ν′µ(k)(ˆG+ˆvxˆG−)µ′ν(−k).(9)\nSince the original Hamiltonian is isotropic, the matrix\nstructure of ˇWis not changed by coordinate rotations\nwhich replace ˆ vxby velocity in some other direction.\nThe Cooperon factor ( ˇΓ(q)) in Eq. 8 is defined as an\ninfinite sum of ladder diagrams,\nˇΓ(q) =γ\nΩ+γ\nΩΩˇP(q)γ\nΩ+γ\nΩΩˇP(q)γ\nΩΩˇP(q)γ\nΩ+···.\n(10)\nThe structure factor ˇPassociated with a single rung of\nthe ladder, is given by the tensor product\nˇP(q) =1\nΩ/summationdisplay\nkˆG+(k+q\n2)⊗ˆG−(−k+q\n2).(11)\nEq.(10) can be summed analytically by solving an alge-\nbraic equation as illustrated in Fig. 2(b), to obtain\nˇΓ(q) =1\nΩ/bracketleftbig\nγ−1−ˇP(q)/bracketrightbig−1. (12)\nThus the matrix structure of ˇP(q) determines which\nCooperon channel contribute to the conductivity correc-\ntion. Eigenvalues of ˇP(q) that are close to γ−1lead to\nlarge contributions to the conductivity. In the next sec-\ntion, we investigate the matrix structure of Cooperon\nin detail, and calculate the conductivity correction as a\nfunction of the Rashba coupling constant αboth analyt-\nically and numerically.\nIII. EVALUATION OF THE CHARACTERISTIC\nFACTORS\nA. Cooperon\nSince we expect the Cooperon to be large only in the\nvicinity of backscattering, we set q=q(cosθ,sinθ) andexpandˇP(q) in powers of qup to order O(q2):\nˇP(q) =ˇP(0)+qˇP(1)\nθ+q2ˇP(2)\nθ+O(q3).(13)\nWe obtain the following expressions for the expansion\ncoefficients:\nˇP(0)=1\nΩ/summationdisplay\nkˆG+⊗ˆG−(14)\nˇP(1)\nθ=1\n2Ω/summationdisplay\nk/bracketleftig\n(ˆG+ˆvθˆG+)⊗ˆG−+ˆG+⊗(ˆG−ˆvθˆG−)/bracketrightig\nˇP(2)\nθ=1\n2Ω/summationdisplay\nk/bracketleftig\n(ˆG+ˆvθˆG+)⊗(ˆG−ˆvθˆG−)/bracketrightig\n,\nwhere the underlined matrices are evaluated at momen-\ntum−k, and other matrices are evaluated at k. ˆvθ\ndenotes the velocity projected onto the direction of q:\nˆvθ= ˆvxcosθ+ ˆvysinθ. In the spinless (or α= 0) case\nthe expansion simplifies to P(q) =γ−1[1−Dτq2]. The\nCooperon therefore has a pole at q= 0 and this leads to\nthe well-known WL correction to the conductivity. Our\ngoalhereistoinvestigatethedeviationfromconventional\nCooperon structure due to Rashba spin-orbit coupling.\nThe matrix structure of ˇP(q) can be understood\nthrough the symmetries of the Hamiltonian. Consider\na spin rotation by πaround the in-plane axis perpendic-\nular to the q-direction generated by the Pauli matrix\nˆσθ≡ˆσycosθ−ˆσxsinθ=/parenleftbigg\n0e−i(θ−π/2)\nei(θ−π/2)0/parenrightbigg\n.\n(15)\nThe unitary transformation ˆ σθtransforms the Rashba\nHamiltonian ˆ σihi(k) to ˆσihi(k′), where k′is the mirror\nreflection of kin a plane perpendicular to q. Thus, the\nGreen’s function ˆG±(±k+q\n2) gets rotated to ˆG±(±k′+\nq\n2). By replacing the summation over kby one over k′\nwe can conclude that ˇP(q) in Eq. (11) is invariant under\nˇΣθ≡ˆσθ⊗ˆσθ. It follows that ˇP(q) andˇΣθcan be diag-\nonalized simultaneously. Since the eigenstates of ˇΣθare\ntwofold degenerate, with the eigenvalues ±1 respectively,\nˇP(q) is at least block diagonal in this basis.\nNext consider spin rotation by πaround the z-axis\nwhich is generated by ˆ σz. Since this operation trans-\nformsthe Rashbaterm ˆ σihi(k) to ˆσihi(−k),ˇP(q) goesto\nˇP(−q) under the unitary transformation ˇΣz≡ˆσz⊗ˆσz.\nAlthough ˇP(q) is not invariant under this transforma-\ntion, even-ordered expansion terms like ˇP(0)andˇP(2)\nare invariant. It follows that these terms are diagonal in\nthe representation formed by the mutual eigenstates of\nˇΣθandˇΣz:\n\n|χ1∝an}bracketri}ht\n|χ2∝an}bracketri}ht\n|χ3∝an}bracketri}ht\n|χ4∝an}bracketri}ht\n=1√\n2\neiθ−e−iθ\n1 1\n1−1\neiθe−iθ\n\n| ↑↑∝an}bracketri}ht\n| ↑↓∝an}bracketri}ht\n| ↓↑∝an}bracketri}ht\n| ↓↓∝an}bracketri}ht\n.(16)\nThis argument does not rule out off-diagonal elements in\neach block of ˇP(1). In fact as we emphasize later these5\nFIG. 3: (Color online) Matrix elements of ˇΓ−1(q) =γ−1−\nˇP(q) at zeroth, first, and second order in a wave vector mag-\nnitude ( q) expansion. The illustrated calculation was for\nη/ǫF= 0.01. The horizontal axis is the Rashba band split-\nting ∆ SO= 2αkFnormalized by η. The plotted quantities\nare defined in Eq.(18).\nterms do appear in ˇP(1)and are responsible for anoma-\nlous spin relaxation behavior. We will refer to this rep-\nresentation as the “singlet-triplet basis”, since |χ3∝an}bracketri}htcor-\nresponds to the spin singlet state. and the other three\nstates span the three triplet states. Note that this two-\nparticle basis depends on the direction of q.\nWe now discuss the numerical evaluation of the ˇP(q)\nwave-vector-magnitude expansion coefficients defined in\nEqs. (14) as a function of the Rashba coupling strength\nα. We make all physical quantities dimensionless by\ninvoking scale transformations which reduce the Fermi\nenergyǫFand the mass 2 mto unity. As explained in\nmore detail in Appendix A, the momentum integrations\ncan be performed analytically by using a gradient ex-\npansion around the Fermi level and extending the inte-\ngration contour to a closed path in the complex plane.\nWe calculate the structure factor matrix ˇP(q) in the\nsinglet-triplet basis motivated above. The Cooperon,\nˇΓ(q) = Ω−1/bracketleftbig\nγ−1−ˇP(q)/bracketrightbig−1, has the block-diagonal ma-trix structure\nˇΓ(q) =1\nΩ/parenleftbiggˆΓ12(q) 0\n0ˆΓ34(q)/parenrightbigg\n, (17)\nwith\nˆΓ−1\n12(q) =/parenleftigg\nA(0)\n1+q2A(2)\n1iqA(1)\n12\n−iqA(1)\n12A(0)\n2+q2A(2)\n2/parenrightigg\n(18)\nˆΓ−1\n34(q) =/parenleftigg\nq2A(2)\n3qA(1)\n34\n−qA(1)\n34A(0)\n4+q2A(2)\n4/parenrightigg\n.\nThe definitions of the coefficients A(0),A(1)andA(2)are\ngiven in Appendix A. In Fig.3 all distinct expansioncoef-\nficients are plotted as a function of the Rashba coupling\nstrengthα. (SinceA(1)is zero forα= 0, we normalize\nit byγ−1√\nDτ, which is comparable to√\nA(0)A(2).)\nFor orientation we first comment on the characteristic\nbehavior of these coefficients under some extreme condi-\ntions:\n•When the spin-orbit coupling is switched off ( α=\n0), all the constant A(0)and linearA(1)coefficients\nvanish, and the quadratic coefficients A(2)reduce\ntoDτ/γ. This result for the Cooperon leads to\nthe conventionalWL expression for the maximally-\ncrossed diagram correction to the conductivity of a\n2DEG that is free of spin-orbit coupling.\n•When ∆ SO≪η,A(0)andA(2)depart from their\ndegenerate values by O(α2), whileA(1)isO(α1).\nThesefindings agreewith resultsobtainedbyILP15\nby treating Rashba spin-orbit coupling as a pertur-\nbation.\n•In the strong (∆ SO≫η) spin-orbit coupling limit,\nA(0)andA(2)reach asymptotic values with the ra-\ntios\nA(0)\n1:A(0)\n2:A(0)\n3:A(0)\n4= 1 : 2 : 0 : 1 (19)\nA(2)\n1:A(2)\n2:A(2)\n3:A(2)\n4= 1 : 0 : 4 : 3 ,\nwhich coincides with the behavior of the Cooperon\ncoefficients of the massless Dirac Hamiltonian. The\nRashba and massless Dirac models agree in this\nlimitbecausetheeigenstatesofthetwomodelshave\nthe same structure. The agreement occurs even\nthough the Rashba model normally has two Fermi\nsurfaces, whereas the massless Dirac model always\nhas a single Fermi surface.\nFig.3 describes the crossover behavior of the Cooperon\nfrom the weak spin-orbit coupling regime captured by\nILP’s analysis,15to the strongly spin-orbit coupled limit\nwith partial equivalence between massless Dirac and\nRashba models. Since the spin singlet channel is un-\naffected by the Rashba internal magnetic field h(k), the\ncoefficients A(0)\n3andA(2)\n3for the singlet channel are in-\ndependent of α.6\nIt is important here to note the behavior of the O(q1)\nterm, which is absent in the spinless model. Its off-\ndiagonal components give rise to mixing between differ-\nent spin channels. Although the mixing between the sin-\nglet channel and one of the triplet channels, specified by\nA(1)\n34, is weak provided only that ∆ SO≪ǫF, the mix-\ning between two triplet channels, specified by A(1)\n12shows\nquite a nontrivial behavior. It vanishes in both strong\nand weakspin-orbit coupling limits: A(1)\n12∝ǫ1/2\nF∆SO/8η3\nin the band unresolved limit, and A(1)\n12∝2ǫ1/2\nFη/∆3\nSO\nin the resolved spin-splitting limit. In the intermedi-\nate regime (∆ SO≈η), on the other hand, it is ∼\nO(√\nA(0)A(2)). Due to this effect, spin relaxation is no\nlonger described by simple exponential decay, but rather\nlike a damped oscillation in which spins precess as they\nrelax. We elaborate on this point in the next subsection.\nB. Spin relaxation\nUsing the symmetry-dictated block-diagonal structure\nof theˇP(q) matrix in the singlet-triplet basis, we can\nexpress the Cooperon in terms of its non-zero matrix el-\nements,\nˇΓ(q) =1\nΩ\nX11X120 0\nX21X220 0\n0 0X33X34\n0 0X43X44.\n,(20)\nThe elements Xij(q) are determined by inverting\nEqs.(18). Since the singlet-triplet basis depends on the\ndirection of the momentum q, we need to change the ba-\nsis back to a momentum independent form before taking\nthe sum over qin Eq. (8). Going backto the tensorprod-\nuct basis and integrating out the angular dependence we\nobtain\n/summationdisplay\nqˇΓ(q) =/integraldisplayl−1\nL−1dq q\n2π\n˜X1(q)\n˜X2(q)˜X3(q)\n˜X3(q)˜X2(q)\n˜X1(q)\n,\n(21)\nwhere\n˜X1=X11+X44\n2,˜X2=X22+X33\n2,˜X3=X22−X33\n2.\n(22)\nThus we need to calculate only the diagonal elements\nof the Cooperon matrix in Eq.(20). When there is no\nlinearterm in q, asin the spin-orbitdecoupled limit, each\ndiagonalelement has a diffusion peak; for example X11=\n[A(0)\n1+q2A(2)\n1]−1. TheO(q) terms in the inverse matrix\nmix contributions from the two channels in each block.\nThe diagonalelements arethen convenientlyexpressedinFIG. 4: (Color online) Upper panel: Inverse relaxation leng th\n|λi|for each channel, normalized by the mean free path l. The\ntriplet channels 1,2 and 4 belong to the eigenstates of Γ( q) in\nEq. (17), which are linear combinations of the triplet basis in\nEq.(16). Lower panel: The argument of λ−2\n1normalized by π.\nThe values plotted in this figure were calculated for η/ǫF=\n0.01. It should be noted that λ1,2is complex for ∆ SO/lessorsimilar0.8η,\nwhere the O(q1) channel mixing effect is dominant.\nterms of partial fraction decompositions with the form:\nX11=c11\nλ−2\n1+q2+c12\nλ−2\n2+q2, X22=c21\nλ−2\n1+q2+c22\nλ−2\n2+q2,\nX33=c33\nq2+c34\nλ−2\n4+q2, X44=c44\nλ−2\n4+q2. (23)\nThe wavelengths λin the denominators, whose depen-\ndence on spin-orbit coupling strength is plotted in Fig. 4,\ndetermine the characteristic length scales within which\nparticle-hole pairs in different channels can propagate\nwithout loss. λ1,2,4are the “relaxation lengths” for\ntriplet channels, which correspond to q-dependent lin-\near combinations of |χ1∝an}bracketri}ht,|χ2∝an}bracketri}htand|χ4∝an}bracketri}ht. Recall that λis\ninfinite in the spinless case, i.e. λ−2\n3= 0.\nThe maximally crossed diagram contribution to the\nconductivity is proportional to an integral over wave vec-\ntor magnitude of the diagonal elements of the Cooperon\nmatrix. When corrections to the conductivity are sub-\nstantial this integral is dominated by contributions from\nsmallqwhere our wave vector expansion is valid. These\nconsiderationslead to a sum overchannels of the familiar\nlogarithmic integral:\n/integraldisplayl−1\nL−1dq q\n2πc\nλ−2+q2=c\n4πln1+(λ/l)2\n1+(λ/L)2.(24)\nThe quantum correction to the conductivity is deter-\nmined by three length scales: the mean free path l, the\nphase length L, which has simple power law dependences\non temperature and magnetic field, and the spin relax-\nation length λ. We note that\n•(i) When the spin relaxation length λis long com-\npared toL,i.e.forlλ, sothat arelativelystrongmagneticfield\nmight be necessary to observe unconventional magneto-\nresistance and this might give rise to other effects, suchas classical magnetoresistance or Shubnikov–de Haas os-\ncillation. Since the classical conductivity is insensitive to\nthe spin-orbit coupling strength as long as ∆ SO≪ǫF,\nmeasurements of the α-dependence at fixed temperature\nand magnetic field might be able to distinguish the two\nplateaus and might be possible if αis tuned by vary-\ning the electric field at fixed carrier density in a two-\ndimensional sample with both front and back gates.\nV. CONCLUSION\nIn this paper, we have examined the crossover behav-\nior between WL and WAL in a two-dimensional electron\ngasthat is triggeredbyvariationofthe Rashbaspin-orbit\ncoupling strength. We have used a numerical approach\nto evaluate the Cooperon contribution to the conductiv-\nity, assuming only that the energy-uncertainty of Bloch\nstatesηdue to disorder scattering is small compared to\nthe Fermi energy and treating spin-orbit coupling in a\nnon-perturbative fashion. For this reason we are able\nto evaluate quantum corrections to the conductivity for\nany value of the ratio of the Rashba spin-splitting to dis-\norder broadening η; an approach applicable beyond the\nband-unresolved limit when ∆ SO≪η. When ∆ SO≫η,\nthere is no trace of the double degeneracy and each band\ncontributes independently to the conductivity. In this\nlimit, the system exhibits perfect WAL behavior, where\nthe quantum interference for spin triplet combinations\nquickly vanishes within the order of the mean free path.\nIn the strong spin-orbit coupling limit, the Cooperon has\nthe same structure as that for the 2D Dirac Hamilto-\nnian, since the band eigenstates of the two models are\nidentical. On the other hand, when ∆ SO/lessorsimilarη, a mixing\nbetween two spin triplet particle-hole channels at first\norder in a long-wavelength expansion is present, which\nhas been previously identified as an important feature of\nspin diffusion8. Here we show that this coupling has pro-\nfound effect on quantum corrections to conductivity. In\nthis regimethe spin relaxationlength becomes imaginary\nwhen different channels are strongly coupled, suppress-\ning the damping of quantum interference corrections to\nconductivity. As a result, a new plateau-like region ap-\npears near the ∆ SO=ηline when the maximally crossed\ndiagrams are evaluated as a function of spin-orbit cou-\npling strength at fixed phase coherence length. Although\nit seems difficult to identify these two plateaus by the\ninvestigation of the differential behavior by L, like the\nmagneto-resistance/conductance measurement under a\nfinite magnetic field, we suggest that they can be dis-\ntinguished by tuning the spin-orbit coupling strength by\nan external gate voltage and fixing the coherence length\n(temperature and magnetic field).\nAlthough we have limited our attention here to a sim-\nple model with spin-independent disorder scattering and\na single spin-split band that has circularly symmetric\nFermi surfaces, the numerical approach we have taken\nis readily generalized to an arbitrary band model and to10\nmodels with spin-dependent disorder scattering. Deal-\ning with anisotropy requires only that an angular aver-\nage over the Fermi surface be added to sums over band\nstate labels. Qualitative aspects of the Rashba model\nresults reported on here apply to other two-dimensional\nelectron systems with broken inversion symmetry. For\ntwo-dimensional electron systems, inversion symmetry\ncan usually be varied in situby tuning gate voltages.\nFor any two-dimensional electron system without inver-\nsion symmetry, the double spin-degeneracy of the Bloch\nbands is lifted by spin-orbit coupling. When the spin-\nsplitting ∆ SOis larger than the Bloch state energy\nuncertainty η, the spin-split bands can be viewed as\ndistinct independent bands with momentum-dependent\nspin-orientations. It follows that in this regime, the spin-\nrelaxation length is on the order the mean-free path,\ni.e.spin-memory is lost at every collision. Once this\noccurs we do not expect to see a crossover from WAL to\nWL when the phase length Lis decreased by increasing\nthe magnetic field or decreasing temperature. At weaker\nspin-orbit coupling strengths we do expect to see WL at\nsome temperatures and fields. However, our study shows\nthat the way in which a WL regime emerges at weaker\nspin-orbit coupling can be nontrivial and is determined\nby specific features of the band structure of a particular\nsystem.\nOnepotentiallyinterestingapplicationofourapproach\nis to two-dimensional electron gases formed at oxide het-\nerojunctions, for instance to the t2gelectron-gas systems\nat the interface between LaAlO 3and SrTiO 3. The pres-\nence in this case of three different orbitals, each of which\ncan have Fermi surfaces, might make the spin relaxation\nscenario rich22. It is known that Rashba spin-orbit split-\nting in these systems23,24is strongest near the avoided\ncrossing between two higher energy (lower density) el-\nlipticalxz,yzsubbands and a lower energy (higher den-\nsity)xysubband. Thereareindeed indicationsthatmag-\nnetoresistive transport anomalies occur when the Fermi\nlevel is near these weakly avoided crossings25. Another\npotentiallyinterestingsystemis twodimensionalelectron\ngases formed in the layers of transition metal dichalco-\ngenide two-dimensional materials. Spin-orbit coupling\nand band spin-splitting is particularly strong in the va-\nlence bands of this class of materials. Coupling between\nspin and other degrees of freedom MoS 226, may give rise\nto interesting complex behavior27, although we note that\nstudies of transport in these materials are at a very early\nstage28.\nAcknowledgments\nYA is supported by a Japan Society for the Promotion\nof Science Postdoctoral Fellowship for Research Abroad\n(No.25-56). Work at the University of Texas was sup-\nported by the Welch Foundation under grant TBF1473\nand by the DOE under Division of Materials Science and\nEngineering grant DE-FG03-02ER45958.Appendix A: Calculating the Cooperon and the\nweight factor by contour integration\nIn this section, we discuss in detail the procedure we\nuse to obtain Cooperon and weight factor matrices. The\nkey ingredient here is to split the Green’s function into a\nsum of contributions from each band:\nˆG±(k) =/summationdisplay\nnψn(k)ψ†\nn(k)\ng±n(k), (A1)\nwhereψn(k) is the eigenfunction for band n(=±1) in\nmomentum and spin representation, and g±\nn(k) =ǫF−\nEn(k)±iη. In our 2DEG model, the Green’s function\nsimplifies to,\nˆG±(k) =/summationdisplay\nn1\n2g±n(k)/parenleftbigg\n1−ine−iφ\nineiφ1/parenrightbigg\n.(A2)\nSince the band structure is isotropic, the denominator\nis independent of φ, the direction of the wave vector k.\nSimilarly, the velocity matrix ˆ vθ= ˆvxcosθ+ˆvysinθcan\nbe written as\nˆvθ(k) =/summationdisplay\nnvn(k)\n2/parenleftbigg\ncos(θ−φ)ine−iθ\n−ineiθcos(θ−φ)/parenrightbigg\n.(A3)\nUsing these expressions, the sum over kin Eq.(14) can\nbe separated into integrations over the orientation φand\nthe modulus kof band wave vector. The phase inte-\ngration eliminates elements which vary like exp( imφ) for\nsome non-zero value of mand hence determines the ma-\ntrix structure. The integrations over khave the general\nform\n/integraldisplay∞\n0dk kfn1···nj\nm1···mj(k)\ng+n1(k)···g+nj(k)g−m1(k)···g−mj(k).(A4)\nThey can be completed by extending the path of integra-\ntion into a large contour in the complex plane and using\nCauchy’s theorem. If we choose to close the contour in\nthe upper half complex plane, the poles ¯knare given by\nthe solutions of the equations g+\nn(¯kn) = 0. As long as\nwe limit the disorder strength to lie within the diffusive\nregimeη≪ǫF, we can solve this equation by using a\ngradient expansion around the Fermi surface,\ng+\nn(¯kn)≃vF(¯kn−kFn)+iη. (A5)\nBy summing over the band indices, we determine the\nvalues of the matrix elements. Because the resulting ex-\npressions are extremely cumbersome, we have evaluated\nthe residues and summed over band indices numerically.\nIn the following subsections, we show how the matrix el-\nements can be constructed at each order in q-expansion.11\n1.O(q0)\nToleadingorderinthe q-expansion,weobtaintheform\nˇP(0)=p(0)\n1\n1\n1\n1\n1\n−p(0)\n2\n0\n1\n1\n0\n(A6)\nin the tensor product basis, where\np(0)\n1=/summationdisplay\nn,mI(0)\nn,m, p(0)\n2=/summationdisplay\nn,mI(0)\nn,mnm, (A7)\nI(0)\nn,m=1\n2π/integraldisplay\ndkk\n4g+ng−m. (A8)\nThe coefficients in the singlet-triplet basis are\nA(0)\n1=A(0)\n4=γ−1−p(0)\n1, A(0)\n2=γ−1−p(0)\n1+p(0)\n2.\n(A9)\nWe can show analytically that A(0)\n3=γ−1−p(0)\n1−p(0)\n2\nvanishes at any value of α: Sincep(0)\n1+p(0)\n2=/summationtext\nn,m(1+\nnm)I(0)\nn,mvanishes when nm=−1, only particle-hole\npairswith band indices n=mcontributeto A(0)\n3. Taking\nthe residual value, we obtain\nI(0)\nn,n=i¯kn\n−4vFg−n(¯kn)∼kFn\n8vFη=1\n4γ,(A10)\nwhere we have used g−\nn(¯kn) =g+\nn(¯kn)−2iη=−2iη.\nTherefore,p(0)\n1+p(0)\n2=γ−1, which leads to A(0)\n3= 0.\n2.O(q1)\nAt linear and the quadratic orders in the q-expansion,\nwe should note that ˆG±ˆvθˆG±in Eq.(14) can be decom-\nposed as\n(ˆG±ˆvθˆG±)(k) =/summationdisplay\nn1n2n3vn2\n8g±n1g±n3/parenleftbigg\nn2−n1+n3\n2/parenrightbigg\n×/bracketleftigg/parenleftbigg\n0ie−iθ\n−ieiθ0/parenrightbigg\n−(n1+n3)cos(θ−φ)/parenleftbigg\n1 0\n0 1/parenrightbigg\n+n1n3/parenleftbigg\n0iei(θ−2φ)\n−iei(2φ−θ)0/parenrightbigg/bracketrightigg\n. (A11)\nSubstituting this decomposition to Eq.(14) and integrat-\ning out the phase φ, we obtain the matrix decomposition\nofˇP(1)\nθin the tensor product basis,\nˇP(1)\nθ= (A12)\np(1)\nΘ∗\nΘ∗\nΘ\nΘ\n−(p(1))∗\n−Θ∗\n−Θ\nΘ\nΘ∗\n,with the shorthand notation Θ = −ieiθ. Here the factor\np(1)is defined by\np(1)=1\n2/summationdisplay\nnm1m2m3/bracketleftig/parenleftig\nI(1)\nnm1m2m3/parenrightig∗\n−nm1I(1)\nnm1m2m3/bracketrightig\n,\n(A13)\nI(1)\nnm1m2m3=1\n2π/integraldisplay\ndkkvm2\n16g+ng−m1g−m3/parenleftbigg\nm2−m1+m3\n2/parenrightbigg\n.\n(A14)\nApplyingtheunitarytransformationby ˇTθ, weobtainthe\ncorrespondence to the coefficients in the singlet-triplet\nbasis,\nA(1)\n12= Rep(1), A(1)\n34=−Imp(1). (A15)\nIt should be noted that the linear term in qis not diago-\nnal even in the singlet-triplet basis, and accounts for the\ncoupling between different channels at finite momentum.\n3.O(q2)\nSubstituting the decomposition in Eq.(A11) to\nEq.(14), we obtain the matrix decomposition of ˇP(2)\nθin\nthe tensor product basis,\nˇP(2)\nθ=−p(2)\n1\n−e−2iθ\n1\n1\n−e2iθ\n+p(2)\n2\n1\n1\n1\n1\n\n−p(2)\n3\n0\n1\n1\n0\n, (A16)\nwith the coefficients\np(2)\n1=/summationdisplay\n{n,m}I(2)\n{n,m}, p(2)\n2=/summationdisplay\n{n,m}2n1m1I(2)\n{n,m},(A17)\np(2)\n3=/summationdisplay\n{n,m}n1n3m1m3I(2)\n{n,m},\nI(2)\n{n,m}=/integraldisplaydk\n2πkvn2vm2(2n2−n1−n3)(2m2−m1−m3)\n512g+n1g+n3g−m1g−m3,\n(A18)\nwhere{n,m}={n1,n2,n3,m1,m2,m3}. This can be di-\nagonalizedby the unitary transformation ˇTθ, which leads\nto the following connection to the singlet-triplet basis:\nA(2)\n1=−p(2)\n1+p(2)\n2, A(2)\n2=−p(2)\n1+p(2)\n2−p(2)\n3,(A19)\nA(2)\n4=p(2)\n1+p(2)\n2, A(2)\n3=p(2)\n1+p(2)\n2−p(2)\n3.\n4. Weight factor\nThe decomposition in Eq.(A11) can also be applied to\nthe calculation of the weight factor matrix. Substituting12\nthe decomposition to the definition of weight factor ma-\ntrix in Eq.(9) and integrating out the phase φ, we obtain\nthe form in Eq.(25), with\nR1=/summationdisplay\n{n,m}J{n,m}, R2=/summationdisplay\n{n,m}(n1m3+n3m1)J{n,m},\n(A20)\nR′\n2=/summationdisplay\n{n,m}(n1m1+n3m3)J{n,m},\nR3=/summationdisplay\n{n,m}n1n3m1m3J{n,m},\nJ{n,m}=/integraldisplaydk\n2πkvn2vm2\n64g+m1g+n3g−n1g−m3/parenleftbig\nn2−n1+n3\n2/parenrightbig/parenleftbig\nm2−m1+m3\n2/parenrightbig\n.The definition of J{n,m}looks similar to I(2)\n{n,m}, while\nthe difference appears in the retarded/advanced indices\nin the denominator. We should note that ( n1,n3) and\n(m1,m3) cannot be exchanged here, respectively.\n1Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6093\n(1984).\n2J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki,\nPhys. Rev. Lett. 78, 1335 (1997); T. Koga, J. Nitta,\nH. Takayanagi, and S. Datta, Phys. Rev. Lett. 88, 126601\n(2002).\n3S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n4J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).\n5R. J. Elliott, Phys. Rev. 96, 266 (1954).\n6Y. Yafet, Solid State Phys. 14, 1 (1963).\n7M. I. D’yakonov and V. I. Perel’, Sov. Phys. Solid State\n13, 3023 (1972).\n8A. A. Burkov, A. S. N´ u�� nez, and A. H. MacDonald,\nPhys. Rev. B 70, 155308 (2004).\n9E. Abrahams, P. W. Anderson, D. C. Licciardello, and\nT. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).\n10L. G. Gorkov, A. I. Larkin, and D. E. Khmel’nitzkii, JETP\nLett.30, 228 (1979); B. L. Altshuler, D. Khmel’nitzkii,\nA. I. Larkin, and P. A. Lee, Phys. Rev. B 22, 5142 (1980).\n11G. Bergmann, Phys. Rep. 107, 1 (1984).\n12E. Akkermans and G. Montambaux, Mesoscopic\nPhysics of Electrons and Photons , (Cambridge University\nPress,2007).\n13S. Hikami, A. I. Larkin, and Y. Nagaoka,\nProg. Theor. Phys. 63, 707 (1980).\n14S. Hikami, Phys. Rev. B 24, 2671 (1981).\n15S. V. Iordanskii, Yu. B. Lyanda-Geller, and G. E. Pikus,\nJETP Lett. 60, 206 (1994); W. Knap et al., Phys. Rev. B53, 3912 (1996).\n16T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi,\nPhys. Rev. Lett. 89, 046801 (2002).\n17E. B. Olshanetsky et al., JETP Lett. 91, 347 (2010).\n18J. Chen et al., Phys. Rev. Lett. 105, 176602 (2010).\n19H.-T. He et al., Phys. Rev. Lett. 106, 166805 (2011).\n20J. G. Checkelsky, Y. S. Hor, R. J. Cava, and N. P. Ong,\nPhys. Rev. Lett. 106, 196801 (2011).\n21A. D. Caviglia et al., Phys. Rev. Lett. 104, 126803 (2010).\n22J. A. Sulpizio, S. Ilani, P. Irvin, and J. Levy, Annual Re-\nview of Materials Research 44, (2014); S. Stemmer and\nS. J. Allen, Annual Review of Materials Research 44,\n(2014); H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer,\nN. Nagaosa, and Y. Tokura, Nature Materials 11, (2012).\n23G. Khalsa, B. Lee, and A. H. MacDonald, Phys. Rev. B\n88, 041302 (2013).\n24Z. Zhong, A. T´ oth, and K. Held, Phys. Rev. B 87, 161102\n(2013).\n25A. Joshua, S. Pecker, J. Ruhman, E. Altman, and S. Ilani,\nNat. Commun. 3, 1129 (2012).\n26D. Xiao, G.-B. Liu, W. Feng, X. Xu, and Wa. Yao,\nPhys. Rev. Lett. 108, 196802 (2012).\n27H.-Z. Lu, W. Yao, D. Xiao, and S.-Q. Shen,\nPhys. Rev. Lett. 110, 016806 (2013).\n28A. T. Neal, H. Liu, J. Gu , and P. D.Ye, ACS Nano 7, 7077\n(2013)." }, { "title": "0910.3050v1.Current_induced_spin_polarization_for_a_general_two_dimensional_electron_system.pdf", "content": "arXiv:0910.3050v1 [cond-mat.mes-hall] 16 Oct 2009physica statussolidi, 2 November2018\nCurrent-induced spinpolarizationfor\na general two-dimensional electron\nsystem\nC.M. Wang*,1, H. T. Cui1, andQ. Lin2\n1Department ofPhysics, Anyang Normal University, Anyang 45 5000, China\n2Shanghai Universityof Engineering Science, 333Longteng R oad, Songjiang, Shanghai 201620, China\nPACS71.70.Ej, 72.10.Bg, 72.25.Dc\n∗Corresponding author: e-mail cmwangsjtu@gmail.com\nIn this paper, current-inducedspin polarization for two-\ndimensional electron gas with a general spin-orbit in-\nteraction is investigated. For isotropic energy spectrum,\nthe in-plane current-induced spin polarization is found\nto be dependent on the electron density for non-linear\nspin-orbit interaction and increases with the increment\nof sheet density, in contrast to the case for k-linear\nspin-orbit coupling model. The numerical evaluation is\nperformed for InAs/InSb heterojunction with spin-orbit\ncoupling of both linear and cubic spin-orbit coupling\ntypes.For δ-typeshort-rangeelectron-impurityscatteri-ng, it is found that the current-induced spin polariza-\ntion increases with increasing the density when cubic\nspin-orbitcouplingsareconsidered.However,forremote\ndisorders, a rapid enhancement of current-induced spin\npolarization is always observed at high electron den-\nsity, even in the case without cubic spin-orbit coupling.\nThis result demonstrates the collision-related feature of\ncurrent-induced spin polarization. The effects of differ-\nent high order spin-orbit couplings on spin polarization\ncanbecomparable.\nCopyrightlinewillbe provided by the publisher\n1 introduction Spintronics, where the spin degree\nof freedom is manipulated to control the electronic de-\nvices,hasbeenbecomingarapidfieldofcondensedmatter\nphysics[1]. Current-inducedspin polarization (CISP) dis -\ncloses the possibility of the spin polarization generated\nin semiconductor directly by the electric field. It refers to\na spatially homogeneous spin polarization in two dimen-\nsional systems due to an in-plane charge current. CISP\ndue to spin-dependent scattering was first reported by\nD’yakonov and Perel’ in 1971 [2]. Later it was realized\nthat such transport phenomenon exists in semiconductor-\nbasedtwo-dimensionalelectrongas(2DEG)withoutstruc-\nture or bulk inversion symmetry [3,4,5]. Experimentally,\nCISPwasfirstmeasuredbySilov etal.intwodimensional\nhole system with the help of polarizedphotoluminescence\ntechnology [6]. Observations of CISP in strained semi-\nconductors have been reported by Kato [7,8]. And later\nSihet al.demonstrated the existence of CISP in AlGaAs\nquantumwell [9].\nSofar,thetheoreticalinvestigationsabouttheCISPare\nmainly focused on the 2DEG with k-linear SOC, such asRashba SOC due to structure inversion asymmetry HR=\nα(kyσx−kxσy)[3,5,10], linear Dresselhaus SOC due\nto bulk inversion asymmetry H(1)\nD=β(kxσx−kyσy)\n[11], and the combination of linear Rashba and Dressel-\nhaus coupling types [12,13,14]. Here σ= (σx,σy,σz)\nrepresents the set of Pauli matrices, and k= (kx,ky) =\nk(cosθ,sinθ)is the two-dimensional momentum, αand\nβare linear Rashba and Dresselhaus SOC factors, respec-\ntively. CISP is found to be proportional to the SOC con-\nstant and independent of the electron density for 2DEG\nwithk-linearRashbaor DresselhausSOC whena dc elec-\ntric fieldisapplied[3,5,11].\nHowever situations may become different for 2DEG\nwith non-linear SOC, just like the spin Hall effect [15,\n16,17,18,19,20,21,22], the investigation of CISP on this\nsystem is desirable. Liu et al.studied the CISP in hole-\ndoped two dimensionalsystem lacking structure inversion\nsymmetry, and they found that CISP is dependent on the\nFermi energy, in vivid contrast against the case in 2DEG\nwithk-linear SOC [23]. While it was pointed by Liu et\nal.that the “spin” for hole system is actually the total an-\nCopyrightlinewillbe provided by the publisher2 C.M. Wang et al.: Current-induced spin polarization forag eneral two-dimensional electron system\ngular momentum, where the spin of hole system is not a\nconserved physical quantity [23]. Hence, we expect the\nbehavior of the spin polarization of conduction electron\nin the presence of non-linear SOC. We know that at high\nelectronsheetdensity,thecubictermofDresselhausSOC,\nH(3)\nD=η(kxk2\nyσx−k2\nxkyσy)[24], (ηis the cubic Dres-\nselhaus SOC constant), has to be taken into account [25].\nRecently, by using the double-grouprepresentations, Car-\ntoix` aet al.[26] found that there is another cubic term for\nheterojunctiondue to bulk inversion asymmetry, H(3)\nBIA=\nζ(k3\nyσy−k3\nxσx), (i. e.the last term of Eq. (3) in Ref.\n[26]). Here ζ=a2β/6is the cubic SOC constant with\naas the well width. Apart from the type of SOC, most\nof the pioneering researches on CISP treat the electron-\nimpurity collision with a simple momentum-independent\nform,described by a relaxationtime τ. However in realis-\ntic2DEG,theelectrondensityisnotlargeenoughtoscreen\nthe charged impurities. The interaction between electron\nanddisorderislongranged.\nIn this paper we consider the CISP in 2DEG with a\ngeneral SOC, which can be applied to describe Rashba,\nlinearandcubicDresselhausSOCs,andmanyotherSOCs.\nFor the simple isotropic energy band form, the analytical\nresult of CISP is obtained. The numerical evaluation for\nthe electron system with both linear and cubic SOCs due\ntobulkinversionasymmetryisalsoperformed,considering\nbothshort-andlong-rangedisorders.\n2 formalism We consider a two-dimensional non-\ninteractingelectron system with a general SOC, described\nbythe followingone-particleHamiltonian:\nˆH=ε0(k)+b(k)·σ. (1)\nForsimplicitywehaveassumedthat ε0(k),theenergydis-\npersionintheabsenceofSOC,isisotropicfunctionofmo-\nmentumk. We set¯h= 1throughoutthis paper. As a fur-\nther simplification of SOC, we shall later specialize to the\nmodel,wherethespin orbitfield b(k)hastheform\nbx(k)+iby(k) = ˜αkM1(sinM2θ)λeiM3θ.(2)\nHerethecomplexnumber ˜α=αr+iαiisthegeneralcou-\npling constant, irrespective of the momentum k, withαr\nandαiastherealandimaginarypart,respectively. λ= 0,1\nandM1,M2,M3areintegernumbers.Itwillbenotedlater\nthat the index λdetermines whether the energy spectrum\nisisotropic.Thenumber M1usuallyispositive.Itisfound\nthat when λ= 0andM1=M3, our model becomes the\noneinRef.[21],wherethemodelisusedtodiscussthespin\nHall effect. Due to time reversal symmetry, the spin orbit\nfieldb(k)satisfiesb(k) =−b(−k).Thereforeforthecase\nλ= 0,i.e.isotropicenergyspectrum, M3mustbeanodd\nintegernumber;while forthe case λ= 1,i. e.anisotropic\nenergy spectrum, M2+M3must be an odd integer num-\nber.NowwelistsomespecialSOCforms:forpureRashba\nSOC,˜α=−iα,λ= 0,M1=M3= 1;while for k-linearDresselhausSOC, ˜α=β,λ= 0,M1= 1,M3=−1;and\nthe case˜α=−1\n2iη,λ= 1,M1= 3,M2= 2,M3= 1\ncorrespondsthecubicDresselhausterm.\nWiththehelpofthelocalunitarymatrix\nUk=1√\n2/parenleftBigg\n1 1\nieiχk−ieiχk/parenrightBigg\n, (3)\nwhereχksatisfies\ntanχk=αisinM3θ−αrcosM3θ\nαicosM3θ+αrsinM3θ,(4)\nthe Hamiltonian (1) can be diagonalized into H=\ndiag[ε1(k),ε2(k)]inthehelicitybasis. Here\nεµ(k) =ε0(k)+(−1)µεM(k), (5)\nwithεM(k) =|˜α|kM1(sinM2θ)λandµ= 1,2as the he-\nlix band index. We note that when λ= 0, the energy dis-\npersionεµ(k)becomes isotropic function of wave vector\nk, whereas for the case λ= 1, the energy spectrum relies\nontheangleofmomentum.\nWhen the system is driven by a weak dc electric field\napplied along the ˆxdirection, E=E0ˆx. Following the\nprocedure of Ref. [27], the kinetic equation of the distri-\nbution function ρ(k)can be derived, with the equilibrium\ndistributionfunction\nρ(0)=/parenleftBigg\nnF[ε1(k)] 0\n0nF[ε2(k)]/parenrightBigg\n. (6)\nHerenF(x)istheFermidistribution.Thedistributionfunc-\ntionρ(k)to first order of electric field comprises two\nterms.Thefirst termiswrittenas\nρ(1)\n12=ρ(1)\n21=−eE0\n4εM(k)∂χk\n∂kx/braceleftBig\nnF[ε1(k)]−nF[ε2(k)]/bracerightBig\n.\n(7)\nAnd the second term ρ(2)(k)is determined by the set of\nequationswith theform\neE0∂nF[εµ(k)]\n∂kx=π/summationdisplay\nqµ′|u(k−q)|2Ωµµ′\n×/bracketleftBig\nρ(2)\nµµ(k)−ρ(2)\nµ′µ′(q)/bracketrightBig\nδ(εµ(k)−εµ′(q)),\n(8)\n4εM(k)Reρ(2)\n12(k) =π/summationdisplay\nqµµ′|u(k−q)|2¯Ωµµ′\n×/bracketleftBig\nρ(2)\nµµ(k)−ρ(2)\nµ′µ′(q)/bracketrightBig\nδ(εµ(k)−εµ′(q)).\n(9)\nHereΩµµ′= 1 + (−1)µ+µ′cos(χk−χq)and¯Ωµµ′=\n(−1)µ′sin(χk−χq).Reρ(2)\n12(k)represents the real part\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3\nof the off-diagonal distribution function ρ(2)\n12(k).u(k)is\ntheelectron-impurityscatteringmatrix.Notethattheabo ve\nequations of distribution function, Eqs. (8) and (9), have\nbeenderivedinRefs. [3,4,13].\nFinally, in helix spin basis, the single-particle opera-\ntorsof spin polarizationare givenby ˆSi=U†\nk1\n2σiUkwith\ni=x,y,z. The corresponding macroscopical quantities\nare obtained by taking the statistical average over them,\nSi=/summationtext\nkTr[ρ(k)ˆSi], andare expressedas\nSx=1\n2/summationdisplay\nkµsinχk[ρ22(k)−ρ11(k)], (10)\nSy=1\n2/summationdisplay\nkµcosχk[ρ22(k)−ρ11(k)], (11)\nSz=/summationdisplay\nkReρ12(k). (12)\n3 spinpolarization\n3.1 analytical result For this angle-dependentSOC,\nthe formulas to be derived will become much less trans-\nparent, and the integrals are more difficult to solve ana-\nlytically. Therefore, we first limit ourselves to isotropic\nenergy spectrum, i. e.λ= 0and the parabolic case\nε0(k) =k2\n2m. The spin polarization is examined in the\npresence of electron-impurity scattering with δ-potential,\n|u(k−q)|2=niu2\n0. Heremis the effective mass of\ntwo-dimensionalelectronsand niistheimpuritydensity.\nKeepingonlythelowest-orderofspin-orbitinteraction,\nthe diagonal elements of distribution function can be ob-\ntainedanalyticallyfromEq.(8). For M3=±1,the diago-\nnalelementsof ρ(2)(k)taketheform\nρ(2)\n11(k) =−eE0τ\nm/bracketleftBig\nk+m|˜α|(1−M1)(2πN)M1−1\n2/bracketrightBig\n×cosθδ(εk1−εF), (13)\nρ(2)\n22(k) =−eE0τ\nm/bracketleftBig\nk−m|˜α|(1−M1)(2πN)M1−1\n2/bracketrightBig\n×cosθδ(εk2−εF). (14)\nAndforthecase M3>1orM3<−1,theyaregivenby\nρ(2)\n11(k) =−eE0τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ε1(k)\n∂k/vextendsingle/vextendsingle/vextendsingle/vextendsinglecosθδ(εk1−εF),(15)\nρ(2)\n22(k) =−eE0τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ε2(k)\n∂k/vextendsingle/vextendsingle/vextendsingle/vextendsinglecosθδ(εk2−εF),(16)\nwithτ= 1/mniu2\n0astherelaxationtime, εFastheFermi\nenergy.With the help of Eq. (9), the off-diagonaldistribu-\ntion function can be obtained directly. Finally, we find thespinpolarizationtakesthe form:\nSx=\n\nemαrτE0\n2π(2πN)M1−1\n2,|M3|= 1\n0, |M3|>1,(17)\nSy=\n\nemαiτE0\n2π(2πN)M1−1\n2,|M3|= 1\n0, |M3|>1,(18)\nSz= 0. (19)\nWe come to the conclusion that the in-plane CISPs exist\nonly when the winding number |M3|= 1, nevertheless\nthe out-of-plane component of CISP is always zero for\nthis 2DEG with isotropic general SOC. As expected, for\nM3=±1andM1= 1, we obtain the CISP for 2DEG\nwith Rashba or k-linear Dresselhaus SOC, in agreement\nwith previous theoretical studies [3,5,11,12,13]. Our re-\nsultsimplythat,incontrasttothe2DEGwithRashbaor k-\nlinearDresselhausSOC, the in-planeCISPs fornon-linear\nSOC system, M1>1, become dependent on the electron\ndensity and enhance for high density semiconductor. The\naboveanalyticalcalculationisvalidintheweakSOCcase,\nεM(kF)≪ε0(kF), but the relationship between εM(kF)\nandτ−1isarbitrary.Here kFistheFermiwavevector.For\nstrong SOC case, the dependence of CISP on the electron\ndensitycanbeevaluatednumerically.\nIt should be noted that the in-plane CISP comes from\nthe interband processes, arising from the SOC, which can\nbe seen from Eqs. (10) and (11). In the absence of spin-\norbit interaction, ρ11=ρ22leads to the vanishing CISP.\nHence, although the result, we obtained here, is for the\nparabolic case, the nonparabolic contribution of ε0(k)to\nCISPmayonlychangeitsvalueslightlythroughtheFermi\nenergy. Further, it can be confirmed below by the numeri-\ncal calculation.Ingeneral,SOCfield will be thecombina-\ntionofEq.(2)withbothlinearandhighorderterms.From\nour analytical result, one can deduce that with increasing\nthehighorderSOCconstant,thedensity-relatedfeatureof\nCISP will becomemoreandmoreevident.\n3.2 numericalresult Nowweperformthenumerical\nevaluation for CISP in InAs/InSb heterojunctions without\ntheadditionallargebiasvoltage,wherethemainSOCcon-\ntribution terms arise owing to the absence of the center of\ninversionin thebulkmaterial.\nFurther, to take account of the nonparabolicity of the\nenergy band of InAs, we use the isotropic Kane band\nmodel:\nε0(k) =1\n2γ/parenleftBigg/radicalbigg\n1+2γk2\nm−1/parenrightBigg\n,(20)\nwhereγ≈1/εgis the nonparabolicparameter,with εgas\ntheenergygapbetweentheconductionandvalencebands.\nNote that Kane energy band becomes the parabolic case\nfor vanishing γ. The nonparabolic factor in the numerical\ncalculation is set to be γ= 2.73eV−1for InAs [28]. The\nCopyrightlinewillbe provided by the publisher4 C.M. Wang et al.: Current-induced spin polarization forag eneral two-dimensional electron system\n0 2 4 6 8 10 0243.9 4.0 4.1 4.2 4.3 \nN\nni S x / E 0 η = 0 \n η = 0.2 η0\n η = 0.4 η0\n η = 0.6 η0\n η = 0.8 η0\n η = η0ni S x / E 0 (10 26 g µB/Vm 3)\nN ( 10 11 cm -2 ) ( b ) \n γ = 0 τ = 10 ps ni S x / E 0 (10 25 g µB/Vm 3)( a ) \n \n \n9 10 4.5 5.0 \nη = η0γ = 2.73 eV -1 \nγ = 0 \n \n \nFigure 1 niSx/E0as functions of electron density for\n(a) short- or (b) long-range impurity scattering for γ=\n2.73eV−1. Hereη0= 1.0×10−28eVm3,gis the effec-\ntive g-factor,and µBis the Bohr magneton.The thin solid\nline in (a) is obtained for γ= 0andη=η0. The thin and\nthick solid lines in inset of (b) is calculated when η=η0\nforγ= 0andγ= 2.73eV−1, respectively. The unit of\nelectrondensity Nis1011cm−2, andtheunitof niSx/E0\nis1026gµB/Vm3.\nδ-form short-range or the remote charged impurity scat-\ntering is considered in the calculation. The scattering ma-\ntrix of remote electron-impurityscattering takes the form :\n|u(q)|2≃nie−2sqI(q)2[29], with I(q)as the form fac-\ntor. We set the electron effective mass at the band bottom\nm= 0.04me(meisthe freeelectronmass),remoteimpu-\nritiesinInSbbarrierarelocatedatadistanceof s= 10nm\nfromtheinterfaceoftheheterojuction[22,28].\n3.2.1H(1)\nD+H(3)\nDFirstweconsiderthesystemwith\nlinear and cubic Dresselhaus SOC. In this case, the spin\norbit field b(k) = (βkx+ηkxk2\ny,−βky−ηkyk2\nx). At\nthe same time, the equations about distribution function,\nfrom Eq. (6) to Eq. (9), can be obtained, by substituting\nthe new form of energy εM(k)andχkwithεM(k) =/radicalbig\nbx(k)2+by(k)2,χk=−tan−1bx(k)\nby(k). It is noted that\ntheenergyspectrumbecomesanisotropiccompletely.\nFrom Eqs. (8) and (10), we find that the xcompo-\nnent of CISP is inverse proportional to the impurity den-\nsityni. Therefore, niSx/E0is plotted in these figures. In\nthe calculation, we set linear Dresselhaus SOC coefficient\nβ= 1.0×10−11eVm.Theshort-rangeimpurityscattering\nis considered with relaxation time τ= 10ps. When the\ncubicDresselhausSOCisconsidered,itisfoundthatCISP\nisstill alongthe xdirection.0.0 0.2 0.4 0.6 0.8 1.0 012343.9 4.0 4.1 4.2 ni S x / E 0 (10 26 g µB/Vm 3)\nη ( 10 -28 ev m 3 ) ( b ) \n N = 1.0x10 11 cm -2 \n 3.0 \n 5.0 \n 7.0 \n 9.0 τ = 10 ps ni S x / E 0 (10 25 g µB/Vm 3)( a ) \n \n \nFigure 2 Dependenciesof niSx/E0oncubicDresselhaus\nSOC parameter for (a) short- or (b) long-range impurity\nscattering.\nIn Fig. 1, the xcomponentof CISP is plotted as func-\ntion of electron density. For short-rangedisorder, the den -\nsity dependence of CISP can be observed when η/negationslash= 0.\nNotethattheresultant Sxisproportionaltorelaxationtime\nτ. With the incrementof the sheet density, CISP increases\nmonotonously, and almost saturates at high density for\nlargeη. From Fig. 1(b), for long-range electron-impurity\nscattering it is evident that, unlike the case for short-ran ge\ndisorder, here CISP always increases with ascending the\ndensity even for the system without cubic SOC. In the\nparameter regime, N <1012cm−2, long-range disorders\nhave strong effect on CISP, where CISP increases rapidly\nwith the rise of density. It can be seen that the role of cu-\nbic term of Dresselhaus SOC on CISP becomesimportant\nat high sheet density for both short- and long-range colli-\nsion.CISP forparabolicenergybandisalso plottedinthis\nfigurewithathinline.Wefindthattheweakeffectofnon-\nparabolicityonCISP appearsathighdensity.\nCISP is shown as a function of the cubic Dresselhaus\nSOC parameter ηin Fig. 2. For momentum independent\npotential, CISP begins with the value emβτE 0/2π, inde-\npendent of the density, and increases with ascending η. In\nFig. 2(b), the calculated CISP for long-range collision is\nalmostlinearproportionaltocubicDresselhausconstant η.\n3.2.2H(1)\nD+H(3)\nD+H(3)\nBIAInthissubsection,the\nadditional high-order contribution H(3)\nBIAdue to bulk in-\nversion asymmetry in Ref. [26] is also considered. Now\nthe spin orbit field becomes b(k) = (βkx+ηkxk2\ny−\nζk3\nx,−βky−ηkyk2\nx+ζk3\ny),andthecorresponding εM(k),\nχkcan be obtained analogously. We take the well width\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 5\n0 2 4 6 8 10 0244.0 4.4 4.8 \nζ = 0 \nN\nni S x / E 0 η = 0 \n η = 0.2 η0\n η = 0.4 η0\n η = 0.6 η0\n η = 0.8 η0\n η = η0ni S x / E 0 (10 26 g µB/Vm 3)\nN ( 10 11 cm -2 ) ( b ) \n ζ = 0 τ = 10 ps ni S x / E 0 (10 25 g µB/Vm 3)\n( a ) \n \n \n8 9 10 45\n \n \nFigure 3 niSx/E0is shown as functions of electron den-\nsity for (a) short- or (b) long-range impurity scattering\nwhen the additional cubic SOC term H(3)\nBIAis considered.\nThethinsolid linesin(a)and(b)are obtainedwhen ζ= 0\nandη=η0for short-range and long-range collisions,\nrespectively. The inset in (b) shows the dependencies of\nniSx/E0onNat high density regime. The other parame-\ntersarethe sameasthoseinFigure1.\na= 5nm. The calculated niSx/E0as functions of elec-\ntrondensity NisshowninFig.3.\nWhen the additional high-order term H(3)\nBIAis in-\ncluded, the magnitude of CISP rises for both short- and\nlong-rangedisorders.For short-rangescattering, niSx/E0\nalways increases with ascending the density. However, at\nhigh density, the magnitude of CISP for large ηmay be\nless than the one for small η. This is due to the interplay\nbetween two cubic terms. It has been seen that CISP satu-\nratesathighdensitywhenonlytheDresselhauscubicterm\nH(3)\nDis included. However, one can see, from the thick\nsolid line in Fig. 3(a), this behavior will not occur when\nwe only include the term H(3)\nBIA. For large η, the effect on\nCISP of cubic Dresselhaus term H(3)\nDexceeds the one of\nthis additional cubic term H(3)\nBIA. CISP saturates again at\nhigh density, hence its magnitude becomes less than the\none with small η. However, such phenomenon can not be\nobservedforthe caseoflong-rangecollision.4 conclusion In summary, the CISP for 2DEG with\na general SOC is investigated. For isotropic energy band,\nwefindthatthein-planeCISP becomesdensity-dependent\nfornon-linearSOC,andincreaseswithenhancingthesheet\ndensity. We have numerically studied the linear and cubic\nSOCcontributionstoCISP,consideringboththeshort-and\nlong-range disorders. For short-range collision, we have\ndemonstrated the dependencies of CISP on density when\nhigh-order SOCs are included. When impurity scattering\nbecomes long ranged, however, CISP increases rapidly\nwith raising the density even for the system without cubic\nSOC. Our investigation indicates that the remote disorder\nhasa stronginfluenceonspinpolarization,andthemagni-\ntude of CISP strongly relies on the scattering matrix. The\ncontributionsofdifferentcubicSOCstoCISPcanbecom-\nparable.\nAcknowledgements WeareverythankfultoN.S.Averkiev\nfor useful information. HTC gratefully acknowledges suppo rt\nfrom the Special Foundation of Theoretical Physics of NSF in\nChina (grant 10747159).\nReferences\n[1] I.ˇZuti´ c, J. Fabian, and S.D. Sarma, Rev. Mod. Phys. 76,\n323 (2004).\n[2] M.I. D’yakonov and V.I. Perel’, Phys. Lett. A 35, 459\n(1971).\n[3] V.M.Edelstein, SolidStateCommun. 73, 233 (1990).\n[4] A. G. Aronov, Y. B. Lyanda-Geller, and G. E. Pikus, Sov.\nPhys. JETP 73, 537 (1991).\n[5] J.I. Inoue, G.E.W. Bauer, and L.W. Molenkamp, Phys.\nRev. B67, 033104 (2003).\n[6] A.Y. Silov, P.A. Blajnov, J.H. Wolter, R. Hey, K.H.\nPloog, and N.S. Averkiev, Appl. Phys. Lett. 85, 5929\n(2004).\n[7] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Nature (London) 427, 50(2004).\n[8] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Phys.Rev. Lett. 93, 176601 (2004).\n[9] V. Sih,R.C. Myers, Y.K. Kato, W.H. Lau, A.C.Gossard,\nand D.D.Awschalom, Nature Physics 1, 31(2005).\n[10] V.V. Bryksin and P. Kleinert, Phys. Rev. B 73, 165313\n(2006).\n[11] M. Trushin and J. Schliemann, Phys. Rev. B 75, 155323\n(2007).\n[12] A.V. Chaplik, M.V. Entin, and L.I. Magarill, Physica E\n13, 744 (2002).\n[13] N.S.Averkiev, andA.Yu.Silov,Semiconductors 39,1323\n(2005).\n[14] V.V. Bryksin and P. Kleinert, Int. J. Mod. Phys. B 20, 1\n(2006).\n[15] J. E.Hirsch,Phys. Rev. Lett. 83, 1834 (1999).\n[16] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301,\n1348 (2003).\n[17] J.Sinova,D.Culcer,Q.Niu,N.Sinitsyn,T.Jungwirth, and\nA.MacDonald, Phys.Rev. Lett. 92, 126603 (2004).\n[18] B.A. Bernevig and S.C. Zhang, Phys. Rev. Lett. 95,\n016801 (2005).\nCopyrightlinewillbe provided by the publisher6 C.M. Wang et al.: Current-induced spin polarization forag eneral two-dimensional electron system\n[19] A.G.Mal’shukovandK.A.Chao,Phys.Rev.B 71,121308\n(2005).\n[20] S.Murakami, Phys.Rev. B 69, 241202 (2004).\n[21] A.V. Shytov, E.G. Mishchenko, H.A. Engel, and B.I.\nHalperin,Phys. Rev. B 73, 075316 (2006).\n[22] Q. Lin, S.Y. Liu, and X.L. Lei, Appl. Phys. Lett. 88,\n122105 (2006).\n[23] C.X.Liu,B.Zhou, S.Q.Shen, and B.fenZhu, Phys.Rev.\nB77, 125345 (2008).\n[24] G.Dresselhaus, Phys. Rev. 100, 580 (1955).\n[25] B. Jusserand, D. Richards, H. Peric, and B. Etienne, Phy s.\nRev. Lett. 69, 848 (1992).\n[26] X.Cartoix` a, L.-W.Wang, D.Z.-Y.Ting, and Y.-C.Chang ,\nPhys.Rev. B 73, 205341 (2006).\n[27] S.Y. Liuand X.L.Lei,Phys.Rev. B 72, 155314 (2005).\n[28] J.C. Cao, X.L. Lei, A.Z. Li, M. Qi, and H.C. Liu, Semi-\ncond. Sci.Technol. 17, 215 (2002).\n[29] X.L. Lei, J.L. Birman, and C.S. Ting, J. Appl. Phys. 58,\n2270 (1985).\nCopyrightlinewillbe provided by the publisher" }, { "title": "1304.3234v1.Large_spin_orbit_coupling_in_carbon_nanotubes.pdf", "content": "Large spin-orbit coupling in carbon nanotubes\nG. A. Steele1,∗, F. Pei1, E. A. Laird1, J. M. Jol1, H. B. Meerwaldt1, L. P. Kouwenhoven1\n1Kavli Institute of NanoScience, Delft University of Technology, PO Box 5046, 2600 GA, Delft, The Nether-\nlands.\n��Correspondence and requests for materials should be address to G.A.S. (email: g.a.steele@tudelft.nl).\nIt has recently been recognized that the strong spin-orbit interaction present in solids can\nlead to new phenomena, such as materials with non-trivial topological order. Although the\natomic spin-orbit coupling in carbon is weak, the spin-orbit coupling in carbon nanotubes can\nbe significant due to their curved surface. Previous works have reported spin-orbit couplings\nin reasonable agreement with theory, and this coupling strength has formed the basis of a large\nnumber of theoretical proposals. Here we report a spin-orbit coupling in three carbon nanotube\ndevices that is an order of magnitude larger than measured before. We find a zero-field spin\nsplitting of up to 3.4 meV, corresponding to a built-in effective magnetic field of 29 T aligned\nalong the nanotube axis. While the origin of the large spin-orbit coupling is not explained\nby existing theories, its strength is promising for applications of the spin-orbit interaction in\ncarbon nanotubes devices.\nIn solids, spin-orbit coupling has recently become a very active topic, in particular in the context of its role\nin a new class of materials with a non-trivial topological order[1, 2, 3], and its use to enable new control\ntechniques in solid-state qubits based on manipulating spins with electric fields[4, 5]. Due to the low atomic\nnumber of the carbon nucleus, the spin-orbit interaction in carbon materials is, in general, weak. An example\nof this is flat graphene, in which intrinsic spin-orbit effects are expected to appear at energy scales of only 1\nµeV (10 mK)[6, 7]. In carbon nanotubes, however, the curvature of the surface breaks a symmetry that is\npresent in graphene. This broken symmetry enhances the intrinsic spin-orbit coupling in carbon nanotubes\ncompared to flat graphene, with theoretical estimates predicting splittings on the order of 100 µeV, an energy\nscale easily accessible in transport measurements at dilution refrigerator temperatures, and recently observed\n1arXiv:1304.3234v1 [cond-mat.mes-hall] 11 Apr 2013in experiments[8, 9, 10, 11]. Experiments so far have reported spin-orbit splittings typically in the range of\nhundreds of µeV, and which were reasonably consistent with theoretical predictions.\nSince its first experimental observation, the spin-orbit interaction in carbon nanotubes has attracted\nsignificant theoretical attention, and has been the basis of a large number of theoretical proposals. Recent\ncalculations predict that it enables fast electrical spin manipulation in carbon nanotube spin qubits [12, 13],\nthat it can couple to the phase of Josephson supercurrents through Andreev bound states in nanotube\nsuperconducting junctions[14, 15], that it allows the spin to couple to the high quality vibrational modes of\nnanotubes[16, 17], and that it could be interesting for the study of topological liquids and Majorana bound\nstates [18, 19, 20, 21, 22]. These many exciting proposed applications could potentially benefit from a stronger\nspin-orbit coupling.\nHere, we present measurements of three carbon nanotube devices which have spin-orbit couplings an\norder of magnitude larger than that predicted by theory. We observe the spin-orbit coupling by measuring the\nmagnetic field dependence of the ground states of clean carbon nanotube quantum dots in the few-electron\nand few-hole regime [23]. We use a Dirac-point crossing at a low magnetic field as a tool for distinguishing\norbital-type coupling[24, 25, 6] from the recently predicted Zeeman-type coupling[26, 27, 28]. While it is not\nunderstood why the spin-orbit coupling we observe is so much larger than that predicted by tight-binding\ncalculations, its large magnitude is attractive for implementing the theoretical proposals for using the carbon\nnanotube spin-orbit coupling for a wide range of new experiments.\nResults\nLarge spin orbit coupling in a few electron nanotube quantum dot. The devices are made using a\nfabrication technique in which the nanotube is deposited in the last step of the fabrication. Figure 1a shows\na schematic of a single quantum dot device with three gates. Figure 1b shows a scanning electron microscope\n(SEM) image of device 1, taken after all measurements were completed. Similar to previous reports[23], we\nare able to tune the device to contain only a single electron (see Supplementary Note 1 and Supplementary\nFigures S1 and S2). An external magnetic field is applied in-plane, perpendicular to the trench. As we do\nnot control the direction of the growth process, this magnetic field often has a misalignment to the nanotube,\n2but still contains a large component parallel to the nanotube axis. All measurements were performed in a\ndilution refrigerator with an electron temperature of 100 mK.\nIn figures 1c–f, we show the magnetic field dependence of the Coulomb peaks of the first four electrons\nin a carbon nanotube quantum dot in device 1. In the few electron regime, we estimate the single-particle\nlevel spacing of the quantum dot to be ∆ ESP= 11 meV (see Supplementary Figure S3). Note that similar\nto recent reports[29], this device exhibits a crossing of the Dirac point at an anomalously low magnetic field,\ncausing a reversal of the orbital magnetic moment of one of the valleys at BDirac = 2.2 T (see figures 2c–f).\nThe lowBDirac indicates a small shift of the k⊥quantization line from the Dirac point (Figure 2a), and would\npredict a small electronic bandgap contribution from the momentum k⊥of the electronic states around the\nnanotube circumference: Ek⊥\ngap= 2¯hvFk⊥= 7 meV. We describe a nanotube with a low Dirac-field crossing\nas “nearly metallic”, as the k⊥quantization line nearly passes through the Dirac point. The bandgap in our\ndevice does not vanish at BDirac , as would be expected, but instead retains a large residual contribution\nEresidual\ngap = 80 meV, similar to previous reports[29]. It has been suggested that this residual energy gap could\narise from a Mott-insulating state, although its exact origin remains a topic of investigation that we will not\naddress here. This low Dirac field crossing does not affect the spin-orbit spectra we observe, and will later\nprovide a unique signature for distinguishing orbital [24, 25, 6] from Zeeman [26, 27, 28] type coupling. We\nfirst focus on the behaviour at magnetic fields below BDirac .\nThe unambiguous signature of the nanotube spin-orbit interaction can be seen by comparing the low\nmagnetic field behaviour in figures 1c and d. Due to the opposite direction of circulation of the electronic\nstates about the nanotube circumference, the bandgap of the KandK/primevalleys both change in the presence\nof a parallel magnetic field[30, 31]. The bandgap in one valley increases and the other decreases, both with a\nrate given by dE/dB = 2µorb, whereµorb=devF⊥/4 (µorb∼220µeV / T for d= 1 nm). In the absence of\nspin-orbit coupling, the first two electrons would both occupy the valley with lower energy, and thus the first\ntwo ground states would both shift down in energy with magnetic field. In figures 1c and d, we observe a\ndifferent behaviour: in particular, at low magnetic fields, the second electron instead occupies the valley that\nis increasing in energy with magnetic field. The occupation of the “wrong” valley by the second electron is a\nresult of the nanotube spin-orbit interaction[8]: The spin-orbit coupling in nanotubes results in an effective\n3magnetic field aligned along the nanotube axis, which points in opposite directions for the KandK/primevalleys\n(Fig. 2d). This magnetic field produces a spin splitting ∆ SOfor the two spin species in the same valley. In\nan external magnetic field, the second electron then enters the “wrong” valley, and persists there until the\nenergy penalty for this exceeds ∆ SO. In device 1, from the extract ground state spectra shown in figure 3(a),\nwe find a ∆ SO= 3.4±0.3 meV. In addition to the ground state measurements, states consistent with such a\nsplitting have been observed in finite bias excited state spectroscopy (see Supplementary Figures S3 and S4).\nWe have also observed a large ∆ SO= 1.5±0.2 meV in a second similar single-dot device (see Supplementary\nFigures S5-S9, and Supplementary Note 2).\nSpin-orbit coupling in nearly metallic carbon nanotubes. In figure 2, we show calculated energy levels\nof a nearly metallic carbon nanotube including the spin-orbit interaction. In carbon nanotubes, there are\ntwo contributions to the spin-orbit coupling, one which we describe as orbital-type coupling, which induces a\nshift in the k⊥quantization line[26, 27, 28] and results in an energy shift proportional to the orbital magnetic\nmoment. The second type, which we describe as Zeeman-type, shifts only the energy of the electron spin\nwith no shift in k⊥. The energy and momentum shifts from these couplings are illustrated in figures 2e and\nf. Combining these two effects, we have the following Hamiltonian for the spin-orbit interaction (equation 71\nin [28]):\nHcv\nSO=αSzσ1+τβSz(1)\nwhereSzis the spin component along the axis of the nanotube, σ1leads to a spin-dependent horizontal shift of\nthe dispersion relation along k⊥that is of opposite sign in different valleys, while τleads to a spin-dependent\nvertical shift that is opposite in the two valleys. The first term represents the orbital-type of coupling, while\nthe second represents the Zeeman-type coupling. The coefficients αandβdetermine the strength of the two\ntypes of coupling, with ∆orb\nSO=α= (−0.08 meV nm) /ratk||= 0, and ∆Zeeman\nSO =β= (−0.31 cos 3θmeV\nnm)/rwhereθis the chiral angle of the nanotube wrapping vector[28], and ris the radius of the nanotube\nin nanometers. Through the cos(3 θ) term, ∆Zeeman\nSO is dependent on the chirality of the nanotube, and\nis maximum for nanotubes with θ= 0, corresponding to the zigzag wrapping vector. Direct experimental\nobservation of the Zeeman-type coupling has been, until now, difficult. There have been two reported\nindications of a Zeeman-type coupling. The first is a different ∆ SOfor holes and electrons[26, 27], which is\n4not present in the orbital-type spin-orbit models[24, 25, 6]. Such an asymmetry was observed in the initial\nexperiments by Kuemmeth et al. , and motivated in part the initial theoretical work predicting the Zeeman-\ntype coupling[26, 27]. The second indication is a scaling of ∆ SOover a large number of electronic shells, as\nseen in recent experiments[11], from which a small Zeeman-type contribution was extracted.\nThe low Dirac field crossing in the nearly-metallic carbon nanotubes studied here provides a unique\nsignature that allows us to identify the type of coupling by looking at the energy spectrum of only a single\nshell. In figure 2g, we show the calculated energy spectrum for a nearly-metallic carbon nanotube with purely\norbital-type coupling (see Supplementary Note 3 for details of the model). Since the orbital-type coupling\nshiftsk⊥, the spin-up and spin-down states cross the Dirac point at significantly different magnetic fields[10].\nFor a purely Zeeman-type coupling, figure 2h, the two spin states cross the Dirac point at the same magnetic\nfield. By comparing the theoretical predictions in figures 2g and 2h to the observed energy spectrum extracted\nfrom the Coulomb peaks in figure 3a, we can clearly identify a Zeeman-type spin-orbit coupling, suggesting\nthat this nanotube has a chiral vector near θ= 0. However, the magnitude of the spin-orbit splitting is much\nlarger than that predicted by theory (see Supplementary Table S1 and Supplementary Note 4 for a summary\nof expected theoretical values and previous experimental observations). One possible origin for the observed\ndiscrepancy is an underestimate of the bare atomic spin-orbit coupling parameter from ab-initio calculations,\nwhich enters the tight-binding calculations as an empirical input parameter.\nIn figure 3, we show the ground state energies of the first 12 electrons as a function of magnetic field,\nextracted from the Coulomb peak positions (Supplementary Figure S9). The ground states energies follow\na four-fold periodic shell-filling pattern, with the spin-orbit split energy spectrum reproduced in the second\nand third electronic shell. In figure 3e, we plot the orbital magnetic moment as a function of shell number,\nincluding a correction for the angle between the magnetic field and the nanotube axis. As reported previously\n[32], the orbital magnetic moment changes with shell number, an effect particularly strong in our device due\nto the small k⊥implied by the low magnetic field Dirac crossing. In figure 3f, we plot the observed ∆ SOas\na function of the orbital magnetic moment, together with the theoretical predictions from equation 1. In the\nplot, we have included the fact that the orbital coupling coefficient αin equation 1 scales with the orbital\nmagnetic moment[11]. The green dashed line shows the prediction from equation 1 for a nanotube with a 3\n5nm diameter, emphasizing the disagreement between measured and the theoretically predicted values. Also\nshown is the same prediction with the coefficients scaled by a factor of 8 in order to obtain the order of\nmagnitude of the observed splitting.\nNote that there are some discrepancies between the energy spectrum extracted from the Coulomb peak\npositions (figure 3a-c) and the theoretical spectra presented in figure 2. The first discrepancy is a small\ncurvature of the extracted ground state energies at B < 0.15 T in figures 3a-c, which we attribute to artifacts\nfrom way in which the magnetic field sweeps were performed (see Supplementary Note 5 and Supplementary\nFigure S10). The second discrepancy is a bending of the extracted energies at B < 1.5 T, particularly\nnoticeable in the upper two states of the second and third shells (blue and purple lines in figures 3b,c), and\na resulting suppressed slope for B < 1.5 T in these states. Correlated with the gate voltages and magnetic\nfields where the suppressed slopes occur, we observed a strong Kondo effect present in the odd valleys (see\nSupplmentary Figure S2). Due to the strong tunnel coupling to the leads, the Kondo current in the valley\ncan persist up to fields of 1.5 T (see Supplementary Figure S9), and is stronger in the higher shells where the\ntunnel coupling to the leads is larger. The model described in figure 2 does not include higher-order effects,\nsuch as Kondo correlations, and it seems that it is no able to correctly predict the position of the Coulomb\npeak in these regions. Qualitatively, the magnetic moments associated with the states appear to be reduced\nby the strong Kondo effect, although the reason for this is not understood. Note that a suppressed magnetic\nmoment will reduce the apparent spin-orbit splitting, and thus the large spin-orbit splittings reported here\nrepresent a lower bound.\nLarge spin-orbit coupling in a nanotube double quantum dot. In figure 4, we present data from\na third device in a p-n double quantum dot configuration that also exhibits an unexpectedly large spin-\norbit coupling (see Supplementary Note 6 and Supplementary Figures S11 and S12 for device details and\ncharacterization). Figures 4c and d show measurements of the ground state energies of the first two electrons\nand first two holes in the device as a function of parallel magnetic field, measured by tracking the position\nof a fixed point on the bias triangle in gate space (coloured circles in 4a) as a function of magnetic field.\nThe signature of the nanotube spin-orbit interaction can be clearly seen by the opposite slope of the first\ntwo electrons (holes) in Figure 4c (4d), and is consistent with the carbon nanotube spin-orbit spectrum far\n6from the Dirac crossing, shown in figure 4b. The difference in the high magnetic field slopes corresponds to\na Zeeman splitting with g∼2, as expected from the spin-orbit spectrum. By calibrating the gate voltage\nshifts into energy using the size and orientation of the finite bias triangles (see Supplementary Note 6), we\nextract an orbital magnetic moment of µorb= 0.8 meV/T, a spin-orbit splitting ∆1e\nSO= 1.7±0.1 meV for\nthe first electron shell, and ∆1h\nSO= 1.3±0.1 meV for the first hole shell. Estimating the diameter from the\norbital magnetic moment, theory would predict a ∆max\nSO∼0.2 meV for this device, an order of magnitude\nbelow the observed values. Note that device 3 exhibits a large spin-orbit coupling without a low BDirac ,\nsuggesting that these two phenomena are not linked.\nFrom the slopes of the ground states, we predict that first two electron levels will cross at a magnetic\nfieldB2= ∆ SO/gµB= 15 T, while the first two hole levels do not cross. The crossing of the first two\nelectron levels instead of the hole states, as was observed by Kuemmeth et al. , implies the opposite sign of\nthe spin-orbit interaction, likely due to a different chirality of our nanotube. The absence of the low Dirac\nfield crossing, however, does not allow us to clearly separate the orbital and Zeeman contributions, as was\npossible for the other two devices.\nDiscussion\nWe have observed strong spin-orbit couplings in carbon nanotubes that are an order of magnitude larger\nthan that predicted by theory, with splittings up to ∆ SO= 3.4 meV. By using a low Dirac field, we are able\nto identify a strong Zeeman-type coupling in two devices. The origin of the large magnitude of the spin-orbit\nsplitting observed remains an open question. Nonetheless, the observed strength of the coupling is promising\nfor many applications of the spin-orbit interaction in carbon nanotube devices.\nMethods\nSample Fabrication The devices are made using a fabrication technique in which the nanotube is deposited\nin the last step of the fabrication. Single quantum dot devices were fabricated by growing the device across\npredefined structures with three gates, using W/Pt electrodes for electrical contacts to the nanotube, and a\ndry-etched doped silicon layer to make gates[23]. Double quantum dot devices were fabricated by growing\n7the nanotube on a separate chip[33].\nMeasurements Measurements were performed with a base electron temperature of 100 mK. For measure-\nments performed with single quantum dot devices, a magnetic field was applied with an orientation in the\nplane of the sample, perpendicular to the trench. In measurements with double quantum dot devices, a 3D\nvector magnet was used to align the direction of the magnetic field along the axis of the nanotube. The\nmeasurement datasets presented in this manuscript are available online, see Supplemenatary Data 1.\nExtraction of the ground state energies In order to convert changes in gate voltage position of the\nCoulomb peak to changes in energy of the ground state, a scaling factor αis required that converts gate\nvoltage shifts into an energy scale. This scaling factor is measured by the lever-arm factor from the Coulomb\ndiamond data, such as that shown in Figure S3. In addition to the scaling of gate voltage to energy, the\nground state magnetic field dependence traces must be offset by an appropriate amount, corresponding to\nsubtracting the Coulomb energy from the addition energy, to produce spectra such as that shown in figure 3\nof the main text. To determine this offset, we use the fact that at B= 0, time-reversal symmetry requires\nthat the electron states are two-fold degenerate. The offset for the 1e/2e curves was thus chosen such that the\nextrapolated states are degenerate at B= 0. This was also used to determine the offset between the 3e/4e\ncurves. For the remaining offset between the 2e and 3e curves, we use the level crossing that occurs at B1.\nAtB1, the levels may exhibit a splitting due to intervalley scattering. This results in a ground state energy\nwhich does not show a sharp kink at B1, but instead becomes rounded. The rounding of this kink in our\ndata, however, is small. We estimate ∆ KK∼0.1 meV, and have offset the 2e/3e curves by this amount at the\ncrossing at B1. The spin-orbit splittings are determined by the zero-field gap in the resulting ground-state\nspectra. The error bars quoted on the spin-orbit splittings are estimates based on the accuracy with which\nthe ground states energy curves can be aligned to produced plots such as those in figure 3 of the main text.\nReferences\n[1] Kane, C. & Moore, J. Topological insulators. Physics World 24, 32 (2011).\n[2] Hasan, M. & Kane, C. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045 (2010).\n8[3] Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–974 (2008).\n[4] Nowack, K., Koppens, F., Nazarov, Y. & Vandersypen, L. Coherent control of a single electron spin\nwith electric fields. Science 318, 1430 (2007).\n[5] Nadj-Perge, S., Frolov, S., Bakkers, E. & Kouwenhoven, L. Spin-orbit qubit in a semiconductor nanowire.\nNature 468, 1084–1087 (2010).\n[6] Huertas-Hernando, D., Guinea, F. & Brataas, A. Spin-orbit coupling in curved graphene, fullerenes,\nnanotubes, and nanotube caps. Phys. Rev. B 74, 155426 (2006).\n[7] Min, H. et al. Intrinsic and Rashba spin-orbit interactions in graphene sheets. Phys. Rev. B 74, 165310\n(2006).\n[8] Kuemmeth, F., Ilani, S., Ralph, D. & McEuen, P. Coupling of spin and orbital motion of electrons in\ncarbon nanotubes. Nature 452, 448–452 (2008).\n[9] Churchill, H. et al. Relaxation and dephasing in a two-electron13C nanotube double quantum dot.\nPhys. Rev. Lett. 102, 166802 (2009).\n[10] Jhang, S. et al. Spin-orbit interaction in chiral carbon nanotubes probed in pulsed magnetic fields. Phys.\nRev. B 82, 041404 (2010).\n[11] Jespersen, T. et al. Gate-dependent spin-orbit coupling in multielectron carbon nanotubes. Nature Phys.\n7, 348–353 (2011).\n[12] Bulaev, D., Trauzettel, B. & Loss, D. Spin-orbit interaction and anomalous spin relaxation in carbon\nnanotube quantum dots. Phys. Rev. B 77, 235301 (2008).\n[13] Flensberg, K. & Marcus, C. Bends in nanotubes allow electric spin control and coupling. Phys. Rev. B\n81, 195418 (2010).\n[14] Zazunov, A., Egger, R., Jonckheere, T. & Martin, T. Anomalous Josephson current through a spin-orbit\ncoupled quantum dot. Phys. Rev. Lett. 103, 147004 (2009).\n9[15] Lim, J., L´ opez, R. & Aguado, R. Josephson current in carbon nanotubes with spin-orbit interaction.\nPhys. Rev. Lett. 107, 196801 (2011).\n[16] P´ alyi, A., Struck, P., Rudner, M., Flensberg, K. & Burkard, G. Spin-orbit-induced strong coupling of a\nsingle spin to a nanomechanical resonator. Physical Review Letters 108, 206811 (2012).\n[17] Ohm, C., Stampfer, C., Splettstoesser, J. & Wegewijs, M. Readout of carbon nanotube vibrations based\non spin-phonon coupling. Applied Physics Letters 100, 143103–143103 (2012).\n[18] Lutchyn, R., Sau, J. & Das Sarma, S. Majorana fermions and a topological phase transition in\nsemiconductor-superconductor heterostructures. Physical review letters 105, 77001 (2010).\n[19] Oreg, Y., Refael, G. & Von Oppen, F. Helical liquids and majorana bound states in quantum wires.\nPhys. Rev. Lett. 105, 177002 (2010).\n[20] Klinovaja, J., Schmidt, M., Braunecker, B. & Loss, D. Helical modes in carbon nanotubes generated by\nstrong electric fields. Phys. Rev. Lett. 106, 156809 (2011).\n[21] Egger, R. & Flensberg, K. Emerging dirac and majorana fermions for carbon nanotubes with proximity-\ninduced pairing and spiral magnetic field. Physical Review B 85, 235462 (2012).\n[22] Sau, J. & Tewari, S. Majorana fermions in carbon nanotubes. Arxiv preprint arXiv:1111.5622 (2011).\n[23] Steele, G., Gotz, G. & Kouwenhoven, L. Tunable few-electron double quantum dots and klein tunnelling\nin ultraclean carbon nanotubes. Nature Nano. 4, 363–367 (2009).\n[24] Ando, T. Spin-orbit interaction in carbon nanotubes. J. Phys. Soc. Jpn. 69, 1757–1763 (2000).\n[25] De Martino, A., Egger, R., Hallberg, K. & Balseiro, C. Spin-orbit coupling and electron spin resonance\ntheory for carbon nanotubes. Physical review letters 88, 206402 (2002).\n[26] Izumida, W., Sato, K. & Saito, R. Spin-orbit interaction in single wall carbon nanotubes: Symmetry\nadapted tight-binding calculation and effective model analysis. J. Phys. Soc. Jpn. 78, 4707 (2009).\n[27] Jeong, J. & Lee, H. Curvature-enhanced spin-orbit coupling in a carbon nanotube. Phys. Rev. B 80,\n075409 (2009).\n10[28] Klinovaja, J., Schmidt, M., Braunecker, B. & Loss, D. Carbon nanotubes in electric and magnetic fields.\nPhys. Rev. B 84, 085452 (2011).\n[29] Deshpande, V. et al. Mott insulating state in ultraclean carbon nanotubes. Science 323, 106 (2009).\n[30] Minot, E., Yaish, Y., Sazonova, V. & McEuen, P. Determination of electron orbital magnetic moments\nin carbon nanotubes. Nature 428, 536–539 (2004).\n[31] Jarillo-Herrero, P. et al. Electronic transport spectroscopy of carbon nanotubes in a magnetic field.\nPhys. Rev. Lett. 94, 156802 (2005).\n[32] Jespersen, T. et al. Gate-dependent orbital magnetic moments in carbon nanotubes. Phys. Rev. Lett.\n107, 186802 (2011).\n[33] Pei, F., Laird, E., Steele, G. & Kouwenhoven, L. Valley-spin blockade and spin resonance in carbon\nnanotubes. Nature Nanotechnology (2012).\nAcknowledgments\nWe thank Daniel Loss, Jelena Klinovaja, and Karsten Flensberg for helpful discussions. This work was\nsupported by the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Orga-\nnization for Scientific Research (NWO), the EU FP7 STREP program (QNEMS).\nAuthor Contributions\nG.A.S., F.P., E.A.L., J.M.J., and H.B.M. performed the experiments; G.A.S., F.P., and H.B.M. fabricated the\nsamples; G.A.S. wrote the manuscript; all authors discussed the results and contributed to the manuscript.\nAdditional Information\nSupplementary information accompanies this paper at www.nature.com/naturecommunictions. Reprints and\npermission information is available online at http://npg.nature.com/reprintsandpermissions/. Correspon-\ndence and requests for materials should be addressed to G.A.S.\nThe authors declare that they have no competing financial interests.\n11Figure 1 :A 29 T spin-orbit magnetic field in a carbon nanotube. a, A schematic of device 1. b,\nA SEM image of device 1. Scale bar, 300 nm. Scale bar, 300 nm. The arrow indicates the direction of the\napplied magnetic field B.c-f,Magnetic field dependence of the Coulomb peak positions of the first four\nelectrons in the device. VSD= 200µV inc,dandVSD= 150µV ine,f. ∆VGcorresponds to a small offset\nin gate voltage used to track the Coulomb peaks as a function of magnetic field. The crossing of the Dirac\npoint reverses the sign of the orbital magnetic moment of the lower energy valley at a field BDirac = 2.2 T.\nWithout spin-orbit coupling, the first two electrons would both occupy the valley with the decreasing orbital\nenergy, and would result in a downwards slope in both canddat fields below BDirac . Here, the second\nelectron, d, instead occupies a valley with increasing orbital energy, a unique signature of the nanotube\nspin-orbit coupling, up to a field B1= 1.6 T. From the ground state energies extracted from the Coulomb\npeak positions, figure 3 a, we obtain a spin-orbit splitting ∆ SO= 3.4±0.3 meV, corresponding to a built-in\nspin-orbit magnetic field BSO= 29 T seen by the electron spin. The sharp kinks at B1indandeimply\nweak valley mixing: we estimate ∆ KK/prime∼0.1 meV.\n12Figure 2 :Spin-orbit coupled states in a nearly metallic nanotube. a The two nanotube valleys\n(K and K/prime) arise from the intersection of the k⊥quantization lines (dashed) with the Dirac cones of the\ngraphene bandstructure. A magnetic field applied parallel to the nanotube axis shifts both quantization lines\nhorizontally, reducing the bandgap in one K point and increasing it for the other, illustrated in b. At a\nsufficiently large magnetic field BDirac , one valley (red line) crosses the Dirac point, after which the orbital\nmagnetic moment changes sign. c,Withk/bardbl= 0, the lowest energy state in the conduction band would\nfollow a v-shape with a sharp kink at BDirac (red line in bandc). A finite k/bardblfrom confinement in the\naxial direction results instead in a hyperbolic shape (orange line in c).d,The spin-orbit interaction in the\nnanotube results in an internal magnetic field aligned along the nanotube axis whose direction depends on the\nvalley the electron occupies. e,In the orbital-type spin-orbit coupling[24, 25, 6], this magnetic field results\nin a spin-dependent shift of k⊥, while the Zeeman-type coupling, f, gives a valley dependent vertical shift in\nenergy[26, 27, 28]. g, h, Calculated energy spectrum of the first shell for a purely orbital-type coupling, g,\nand a purely Zeeman-type coupling, h, with parameters chosen to illustrate the difference between the two\ntypes of spectra. Colours indicate the ground state energies of the four electrons that would fill the shell.\nIng, electrons experience a spin-dependent k⊥shift, resulting in two separate Dirac crossings[10], an effect\nabsent in h.\n13Figure 3 :Spin-orbit coupling in the first three electronic shells. a -c, Observed energy spectra of the\nfirst twelve electrons in device 1. The spectra exhibits a four-fold shell filling, with the spin-orbit electronic\nspectrum visible in all three shells. Extracted ∆ SOare shown in d. Comparing to the spectra for the two\ntypes of nanotube spin-orbit coupling (fig. 2 gand 2h), it is clear the device exhibits a Zeeman-type coupling.\nDeviations from the model are discussed in the main text. e,µorbas a function of the shell number. For\nlarger shells, electrons are confined in an electronic level with a larger value of k/bardbl. The correspondingly larger\nmomentum along the nanotube axis decreases the velocity around the nanotube circumference, reducing the\norbital magnetic moment[11, 32]. f,∆SOas a function of µorb. The green dashed line shows the maximum\nspin-orbit coupling expected from theory with αandβfor a 3 nm nanotube (equation 1 together with the\nscaling ofαwith the magnetic moment). By scaling coefficients αandβby a factor of 8 (blue line), we can\nreproduce the order of magnitude of the spin-orbit coupling in our device.\n14Figure 4 :Large spin-orbit coupling in a (p,n) double quantum dot. a, Colourscale plots of the\ncurrent at a source-drain bias Vsd= 5 mV and B= 0. Black dashed lines indicate the baseline of the triple-\npoint bias triangles. Movement of the tip of the bias triangles (coloured circles) in gate space along line cuts\nin gate space (white dashed lines) with magnetic field is used in canddto track the ground state energies.\nb,Expected energy spectrum for the first shell of electrons and holes including the spin-orbit interaction.\nc,d, Magnetic field dependence of line cuts in gate space (white dashed lines in a) for the first two electrons,\nc, and holes, d. Coloured circles indicate positions on the corresponding bias triangles in a, and the dashed\nlines indicate the observed magnetic field dependence of the ground states, in good agreement with the spin-\norbit spectrum, b. High magnetic field slopes for the ground state energies are indicated in the figures. We\nextract spin-orbit splittings ∆1e\nSO= 1.7±0.1 meV for the first electron shell and ∆1h\nSO= 1.3±0.1 meV for\nthe first hole shell. Excited states inside the bias triangles (colourscale data above dashed lines) exhibit a\nrich structure as a function magnetic field, which we discuss elsewhere [33].\n15Supplementary Figure S1: Characterization of Device 1. a, Schematic of device 1, consisting of a\nclean nanotube grown over a predefined trench. Source and drain contacts to the nanotube are made by a\n5/25 nm W/Pt bilayer (dark grey). Two gates embedded in the oxide (red) are used to induce charges in\nthe nanotube. A backgate (blue) is kept grounded. b,A scanning electron microscope (SEM) image of the\nactual device, taken after all measurements. The nanotube axis lies at an angle of 48 degrees relative to the\nmagnetic field orientation. c,Imeas at aVsd= 1 mV as a function of the two gate voltages. d,Magnetic\nfield dependence of Imeas along a diagonal line cut Vg1=Vg2=Vginc. The device exhibits a minimum\ngap atBDirac = 2.2 T (white arrows), corresponding to a crossing through the Dirac point of the graphene\nbandstructure by the k⊥quantization line. e,Plot ofImeas (logscale) at Vsd= 1 mV showing the gate\nvoltages corresponding to the first electrons in the quantum dot.\n1arXiv:1304.3234v1 [cond-mat.mes-hall] 11 Apr 2013Supplementary Figure S2: Stability diagram showing Coulomb diamonds of Device 1. a, Dif-\nferential conductance of device 1 showing the Coulomb diamonds of the first electrons, the empty device,\nand the threshold for hole conduction, taken at B= 0. For larger gate voltage, a four-fold pattern of Kondo\nresonances is observed, together with strong instabilities in the Coulomb diamonds which we attribute to\nmechanical excitation of mechanical resonances of the suspended nanotube by single-electron tunnelling [33].\nDue to the lack of p-n junction barriers for holes, the hole doped region shows Fabry-Perot type oscillations.\nFor the hole doped device, and for large electron numbers, we estimate the single-particle energy of the\nconfined states to be ∼5 meV. b,The same data as in awith the contrast enhanced in order to clearly show\nthe Coulomb diamond of the first electron.\n2Supplementary Figure S3: Excited states of the 2e charge state in Device 1 Differential conduc-\ntance vs.VGandVSDfor the 1e to 2e transition, in which excited states could be resolved. The dashed lines\nindicates excited states we identify as the single-particle energy splitting ∆ ESP. For the 1e-2e transition,\nwe extract ∆ ESP= 11 meV. From DeltaE = ¯hvF/(2L), we estimate the size of the quantum dot L∼200\nnm. From the measured angle in the SEM image, the total length of the nanotube over the trench is ∼400\nnm. This implies a 100 nm length for the pn depletion region and p doped regions from the work function\ninduced doping for the 1e quantum dot. For higher electron numbers, and similarly for holes, the single\nparticle energy drops to 5 meV (see figure S4), implying a confinement length responding to the full length\nof the suspended nanotube. The white arrow indicates the position of a faint excited state with an energy\n∼3 meV, consistent with the spin-orbit splitting we observe from the ground state measurements.\n3Supplementary Figure S4: Evidence for spin-orbit splitting in magnetic field spectroscopy of\nexcited states in Device 1. A colourscale plot showing dI/dV gas a function of magnetic field for the\n1e/2e transition of device 1 taken at VSD= 5.5 mV. Excited states of the 2e ground state appear as positive\npeaks indI/dV g. The dashed line indicates the magnetic field dependence of a 2e excited state consistent\nwith the expected spectrum from spin-orbit splitting. From the excited state data, we extra a spin-orbit\nsplitting ∆ SO= 2.9 meV, lower than that from the ground state measurements. This difference can arise\nfrom a difference between the excited state energies of the 2e state compared to the ground state energy of 3e\nfrom electron interactions. The origin of the extra excited state running parallel to the ground state for fields\nless thanB1is not understood. The value of ∆ SOfrom the excited state measurement is, similar to that\nfrom the ground state measurements, an magnitude larger than the expected maximum ∆max\nSO= 106µeV\nexpected from theory (see table S1). Excited states in the 0/1e transition did not show clear visibility, and\nthose of higher states were masked by instabilities we attribute to mechanical excitation of the suspended\nnanotube (also visible here above 6T).\n4Supplementary Figure S5: Spin-orbit split states in Device 2. Magnetic field dependence of the first\ntwo electrons in device 2, showing the signature of the nanotube spin-orbit coupling with ∆ SO= 1.5 meV.\n5Supplementary Figure S6: Spin-orbit spectrum of the first shell in Device 2. Extracted ground\nstate energies of the first four electrons in device 2. Device 2 also shows a large spin orbit coupling with a\ndominant Zeeman-type contribution. Note that the flat behaviour of the ground states at zero magnetic field\nis a measurement artifact from the magnetic field controller, see text for discussion.\n6Supplementary Figure S7: Magnetic field dependence of the first four Coulomb peaks in Device\n2.Coulomb blockade current vs. magnetic field and gate voltage for device 2, used to extract the ground\nstate energies of the first shell, Vsd= 1 mV. The data in figure S6 is a zoom of the data here.\n7Supplementary Figure S8: Stability diagram of Device 2 in few-electron regime. a, Differential\nconductance of device 2 showing the Coulomb diamonds of the first electrons, the empty device, and the\nthreshold for hole conduction, taken at B= 0.\n8Supplementary Figure S9: Coulomb peak data for Shells 2 and 3 of Device 1. Measurements of\nImeas vs.VgandBon device 1 that are used to extract the energy spectra shown in figure 3 of the main\ntext, taken at Vsd= 70µV.\n9Supplementary Figure S10: High resolution datasets in Device 1 of spin-orbit states. High\nresolution datasets of the 1e and 2e Coulomb peaks of device 1 with VSD= 1 mV, showing behaviour at\nlow magnetic fields. Here, the low field behaviour is not affected by artifacts in the first gate sweep first\ngate sweep at because of a slow sweep rate of the gate that provided sufficient time for the magnetic field\ncontroller to settle before reaching the position of the Coulomb peak.\n10Supplementary Figure S11: Characterization of Device 3. a, Schematic of device 3. The total device\nlength is 600 nm. The device includes 5 local gates embedded in oxide under the suspended nanotube. In the\nmeasurements, the outermost gates ( VGL,VGR) are used to tune the electron/hole number in a (p,n) type\ndouble quantum dot, while the inner three gates are used to tune the interdot tunnel barrier. b,An overview\nof gate space, indicating the (p,p), (p,n), (n,p), and (n,n) regions of gate space, and the identification of the\n(0,0) configuration. c,A color scale plot of the measured current as a function of VGLandVGRatVSD= 10\nmV. The boxes outlined by dashed lines show the triple points used to track the ground state energies in\nfigure S12 and figure 4 of the main text.\n11Supplementary Figure S12: Energy spectra of the first electron shell and hole shell of Device\n3.Extended datasets from figure 4 of the main text, showing the ground states of the first four electrons and\nthe first four holes, extracted from the motion of the triple points indicated in the dashed boxes in figure S11\nwith magnetic field. a,Double-dot stability diagrams taken at Vsd= 5 mV. b,Expected spectra for the first\nfour electrons (solid lines), as well as spectra from the next higher shell (dashed lines). c,d, Magnetic field\ndependence of gate space cuts (white dashed lines in a) for the first four electrons cand holes d. Similar\nto previous works [8], the magnetic field dependence of the third and fourth electrons/holes does not follow\nexactly the single-shell spin orbit spectrum (solid lines in b), but instead show extra crossings from downward\nmoving levels in higher shells (dashed lines in b).\n12Supplementary Table S1: Summary of previous spin-orbit measurements.\nReference µorb d⋆∆max\nSOtheory ∆SOobserved\nKuemmeth et al. [8] 1.55 meV/T 7.0 nm 110µeV 370µeV (1e)\n210µeV (1h)\nJespersen et al. [8] 0.63 meV/T 2.9 nm 168µeV 150µeV (many electrons)\nJespersen et al. [31] 0.87 meV/T 5.3 nm†146µeV 200µeV (many electrons)\nChurchill et al. [9] 0.33 meV/T 1.5 nm 520µeV 170µeV (1e)\nJhang et al. [10] 0.33 meV/T/diamondmath1.5 nm⋆⋆520µeV 2500µeV††\nDevice 1 1.6 meV/T 7.2 nm 106µeV 3400µeV (1e)\nDevice 1 1.6 meV/T 3 nm‡260µeV 3400µeV (1e)\nDevice 2 1.5 meV/T 6.8 nm 116µeV 1500µeV (1e)\nDevice 3 0.9 meV/T 4.1 nm 190µeV 1700µeV (1e)\nDevice 3 0.8 meV/T 3.7 nm 208µeV 1300µeV (1h)\n⋆Estimated from the observed orbital magnetic moment, ignoring effects of k||, unless otherwise noted\n†The value of the diameter for this entry is based on a detailed analysis of µorbas a function of shell\nnumber performed by the authors.\n‡The diameter for this entry is based on the observed AFM height of the nanotube.\n/diamondmathOrbital moment implied from AFM diameter.\n⋆⋆Diameter from AFM.\n††Implied from bulk bandgap measurements.\n13Supplementary Note 1: Characterization of Device 1\nA schematic of Device 1 is shown in figure S1 a. Similar to previous studies[22], we make a clean suspended\ncarbon nanotube quantum dot by growing the nanotube across a pre-defined structure in the last step of\nthe fabrication. A SEM image of the actual device (taken after all measurements were completed) is shown\nin figure S1 b. As we do not control the direction of the nanotube growth, it often crosses the trench at an\nangle, as can be seen in this device. From AFM measurements, we estimate the nanotube diameter to be 3\nnm.\nWe apply a d.c. voltage across the source and drain of the device and measure the current through the\nnanotube as we sweep the gates, as shown in figure S1 c. In the upper left corner of the plot, the gates dope\nthe center of the nanotube with holes. Near the edge of the device, the gate electric fields are screened by the\nohmic contact metal; here, the doping is set by the work function difference between the metal (Φ Pt∼5.6 eV)\nand the nanotube (Φ CNT∼4.9 eV), resulting in a gate-independent hole doping at the edge of the trench.\nThis, combined with hole doping of the suspended segment from the gates, results in a p/primepp/primeconfiguration\nin the upper left corner of figure S1 c. In this region, we observe only weak modulations of the conductance\nwhich does not vanish between peaks, indicating a highly transparent interface between the Pt metal and\nclean nanotube. In the lower right corner of figure S1 c, the gates induce electrons in the suspended segment,\ngiving a p/primenp/primedoping profile. Electrons occupy a quantum dot with tunnel barriers defined by p-n junctions\n[22], in which we can count the number of carriers starting from zero, shown in figure S1 e.\nFigure S1 dshowsImeas vs.Vgtaken along the dotted line in S1 cas a function of an external magnetic\nfield applied in the plane of the sample, perpendicular to the trench. The distance in gate voltage between\nthe onset of electron and hole current is a measure of the electronic bandgap of the nanotube. In carbon\nnanotubes, a magnetic field component parallel to the nanotube axis shifts the quantization condition of\nthe states circling the circumference ( k⊥) by an Arahonov-Bohm flux, and therefore reduces the nanotube\nbandgap (see figure 2 aof main text). For sufficiently large magnetic fields, the k⊥quantization line will\ncross the Dirac point of the graphene bandstructure and the bandgap begins to increase again. In our\ndevice, this occurs at a magnetic field of BDirac = 2.2 T, indicated by white arrows in figure S1 d. This\nimplies a contribution to the electronic bandgap Ek⊥\ngap= 2¯hvF∆k⊥∼7 meV arising from the shift of\nthek⊥quantization line. In this sense, our nanotube is very close to the metallic condition in which the\nk⊥quantization line passes directly through the center of the Dirac cone. This is a very different regime\ncompared to previous devices where the nanotube spin orbit coupling was studied [8,11], in which no such\nevidence of a low Dirac field was seen. Similar to previous studies where low Dirac fields were reported [28],\nthe bandgap do not vanish at the Dirac point. We observe a residual gap in the transport data at the Dirac\n14point of about 80 meV, measured by subtracting the average of the addition energies from the first electron\nand the first hole from the addition energy of the empty quantum dot.\n15Supplementary Note 2: Spin orbit splitting in Device 2\nIn figures S6-S9, we present the magnetic field dependence of the ground states of the first four electrons in\na second nearly metallic carbon nanotube (device 2). Device 2 is similar in design to device 1, but includes\nonly a backgate. The trench length is 800 nm. In device 2, we observe a Dirac field of 0.8 T, an orbital\nmagnetic moment µorb= 1.5 meV/T, and a spin orbit splitting ∆ SO= 1.5 meV.\n16Supplementary Note 3: Model for a nearly metallic nanotube with spin-orbit coupling\nIn order to calculate the spectra plotted in figures 2 gandhof the main text, we use a model of the nanotube\nbased on the graphene bandstructure with a parallel magnetic field. In a basis of spin and valley eigenstates\nin which the spin direction is defined parallel to the axis of the nanotube, the Hamiltonian consists of a 4x4\nmatrix with only diagonal elements given by:\nE(v,s,B ) =/radicalBig\n(Ek⊥+vs∆orb\nSO+vµorbB)2+E2\nk/bardbl+vs∆Zeeman\nSO +1\n2sgµBB (S1)\nHere,vandstake on values±1 depending on the electron spin and the valley it occupies, Ek(/bardbl,⊥)= ¯hvFk(/bardbl,⊥)\nwherek(/bardbl,⊥)are the momentum of the electron relative to the Dirac points in the directions parallel and\nperpendicular to the axis of the nanotube, and ∆orb\nSOand ∆Zeeman\nSO are the orbital and Zeeman type spin orbit\nsplittings at k||= 0 (αandβ). These diagonal elements correspond to the energies plotted in figure 3. In the\ncalculations, we have chosen to make the total spin orbit coupling either purely orbital or purely Zeeman for\nillustrative purposes, and have used the following parameters: ∆ SO= 2 meV,Ek/bardbl= 1 meV,Ek⊥= 2 meV,\nandµorb= 0.9 meV/T.\nIncluding the observed 48 degree misalignment of the magnetic field to the nanotube axis, the Zeeman\nsplitting Hamiltonian gµB/vectorB·/vectorSis no longer diagonal in this basis, and the eigenstates are mixtures of the\nfour basis states described above. However, because the Bohr magneton is small compared to the orbital\nmagnetic moment, this effect is weak and does not result in qualitative different spectra.\nThe Zeeman-type contribution to the spin-orbit splitting, according to current theoretical estimates, is\nexpected to be larger than the orbital-type contribution by as much as a factor of 4, except for in nanotube\nchiralities where it vanishes or is small due to the cos(3 θ) term (θ= 0 corresponding to a zigzag nanotube).\nIt is an open question, however, why the spin-orbit splitting we observe in devices 1 and 2 is so dominantly\nof the Zeeman-type, with little indication of an orbital contribution.\n17Supplementary Note 4: Discussion of summary table of previous spin-orbit spliting measure-\nments\nIn Supplementary Table S1, we summarize in a table our measurements together with other measurements\nof the spin-orbit coupling reported in literature. As described in the main text, we use the formula for the\nnanotube spin-orbit splitting from [27], given by:\nHcv\nSO=αSzσ1+τβSz(S2)\nwith the orbital contribution given by:\n∆orb\nSO=α=−0.08 meV nm\nr(S3)\nand the Zeeman contribution given by:\n∆Zeeman\nSO =β=−0.31 meV nm\nr(S4)\nwhereris the radius of the nanotube. For the maximum theoretical value, we choose θ= 0, giving:\n∆max\nSO=780µeV\nd(in nm)(S5)\nIn order to provide a consistent comparison, we have estimated the (minimum) diameter using the\nobserved value of the orbital magnetic moment µorb. Assuming a Fermi velocity of 0 .9×106m/s,µorbis\ngiven by:\nµorb=devF\n4= 220µeV / T×d(in nm) (S6)\nwherevFis the Fermi velocity of the graphene bandstructure, which we take here as 0 .9×106m/s. Here,\nwe assume vF⊥=vF, and therefore have not accounted for the reduction of µorbfrom a finite k||[31]. The\nresulting estimates of dfromµorbrepresent a lower bound on the diameter (and thus also an upper bound\non ∆max\nSO).\nWe have also included three entries in which we calculate ∆max\nSObased on a different estimate of the\ndiameter. These three entries correspond to the diameter d= 3 nm we estimate from AFM measurements\non device 1, the diameter d= 5.3 nm estimated by [31] from an extensive analysis of µorbas a function of\ngate voltage, and the diameter d= 1.5 nm measured by Jhang et al. [10]. Note that the tapping-mode\nAFM measurement of the diameter may underestimate the diameter of single-wall carbon nanotubes due to\n18compression forces from the AFM tip.\nThe measurements referred to in the table were performed by tracking the electronic states of individual\nlevels in a quantum dot at low temperatures, except for the measurements of Jhang et al. [10]. The values in\n[10] are based on measurements of the nanotube bandgap implied from device conductance near the bandgap\nas a function of magnetic field at different fixed gate voltages, together with the nanotube diameter as\nmeasured by AFM. The devices measured here and those measured by Kuemmeth et al. [8] were made using\nclean nanotubes grown in the last step of the fabrication, while the other measurements were performed on\nnanotubes which were grown first and subsequently underwent processing in the cleanroom.\nFinally, we also note that when using µorbto estimate the nanotube diameter, we obtain a number\nthat is not only larger than the AFM measurement for device 1, but also larger than the largest diameter\nexpected for single wall carbon nanotubes in, for example, transmission electron microscope studies. This is\nalso the case for many of the devices in Table 1. Such a discrepancy was also noted by earlier authors [31],\nand remains unresolved. One suggestion of the authors of [31] was a renormalization of the Fermi velocity.\nSuch a renormalization could arise from, for example, discrepancies between the experimental tight binding\nparameters of carbon nanotubes and those obtained from ab initio calculations.\n19Supplementary Note 5: Artifacts in extracted ground state energies at B < 0.15T\nNote that there is glitch in the first line of the data set in figure S6. This artifact is also present to a lesser\ndegree figures 1(c)-(f) and the resulting extracted energies in figures 3(a)-(c) of the main text. This glitch\nresults in an artifact in the resulting extracted ground state energies plotted in figure S5 in the form of a flat\nslope forB < 150 mT. The glitch and resulting artifacts arise from the inability of our magnetic controller\nto track the setpoint field during faster magnetic field sweeps. The effects of these artifacts are limited to\nthe first gates sweep (row) of the Coulomb peak magnetic field dependence data. These artifacts have been\naccounted for in the estimation of the error bar on ∆ SO.\nIn order to demonstrate that these artifacts are not obscuring possible other phenomena at very low\nmagnetic fields, we have also included high resolution datasets in figure S10 for the data in figure 1(c) and\n1(d) of the main text. Here, the gate was swept sufficiently slowly that the magnet controller had time to\nsettle before the gate voltage reached the position of the first Coulomb peak, and thus the artifacts are not\npresent.\n20Supplmentary Note 6: Device 3 characterization and analysis\nIn this section, we present a basic characterization of device 3 (figure S11), together with measurements\nthe magnetic field dependence of the ground state energies of the first four electrons and first four holes\nin the device (figure S12), and discuss the extraction of the ground state energies from the magnetic field\ndependence of the gate-space cuts through the triple-point triangles.\nBy tracking the gate voltage position of any fixed point on the triple-point bias triangles as a function\nof magnetic field, we can independently track the ground state energy of the left and right dot in the double\nquantum dot device. This is analogous to the tracking of the ground states of a single quantum dot by\nfollowing the Coulomb peak position with magnetic field. To make this concrete, we illustrate this in the\ncontext of upper left bias triangle in figure 4 aof the main text, corresponding to the (3h,1e) ↔(2h,0e)\ntransition. In the case that there is very small crosstalk capacitance from the left gate to the right dot (as\nis the case in figure 4 aof the main text where the edges of the triple-point bias triangle are nearly vertical),\nvertical shifts of the bias triangle arise from shifts in the 3h ground state, while shifts in the 1e ground state\nshift the bias triangle horizontally. In measuring the shift of the bias triangle, it is equivalent to track any\nfixed point on the triangle. We choose to extract the ground state energies by following a point near the tip\nof the triangle, as the current on the baseline in our device is weak due to weakly tunnel-coupled ground\nstates.\n21Supplementary References\n[34] Steele, G. et al. Strong coupling between single-electron tunneling and nanomechanical motion. Science\n325, 1103 (2009).\n22" }, { "title": "1301.3565v1.Vortex_lattice_solutions_to_the_Gross_Pitaevskii_equation_with_spin_orbit_coupling_in_optical_lattices.pdf", "content": "arXiv:1301.3565v1 [cond-mat.quant-gas] 16 Jan 2013Vortex lattice solutions to the Gross-Pitaevskii equation with spin-orbit coupling in\noptical lattices\nHidetsugu Sakaguchi and Ben Li\nDepartment of Applied Science for Electronics and Material s,\nInterdisciplinary Graduate School of Engineering Science s,\nKyushu University, Kasuga, Fukuoka 816-8580, Japan\nEffective spin-orbit coupling can be created in cold atom sys tems using atom-light interaction. We\nstudy the BECs in an optical lattice using the Gross-Pitaevs kii equation with spin-orbit coupling.\nBlochstatesforthelinearequationarenumericallyobtain ed, andcomparedwithstationarysolutions\nto the Gross-Pitaevskii equation with nonlinear terms. Var ious vortex lattice states are found when\nthe spin-orbit coupling is strong.\nPACS numbers: 03.75.-b, 03.75.Mn, 05.30.Jp, 67.85.Hj\nRecently, Bose-Einstein condensates (BECs) with effective spin-o rbit coupling were created in cold atom systems\nusing atom-light interaction [1]. The spin-orbit-coupled BECs are act ively studied theoretically [2]. Wang et al. found\nthat the mean-field ground state has two different phases: plane- wave and stripe phases depending on the nonlinear\ninteractions [3]. Half vortex states were found in a spin-orbit couple d BECs confined in a harmonic potential [4, 5].\nExotic spin textures were predicted in Bose-Hubbard models corre sponding to spin-orbit coupled BECs in the Mott-\ninsulator phase [6, 7].\nThe GRoss-Pitaevskii (GP) equation is a mean-field approximation fo r the BECs with the spin-orbit coupling.\nThere are some studies for the GP equation with spin-orbit coupling in optical lattices [8, 9]. In this paper, we study\nvortex lattice solutions to the GP equation in a square type optical la ttice. The model equation is expressed as\ni∂ψ+\n∂t=−1\n2∇2ψ++(g|ψ+|2+γ|ψ−|2)ψ+−ǫ{cos(2πx)+cos(2πy)}ψ++λ/parenleftbigg∂ψ−\n∂x−i∂ψ−\n∂y/parenrightbigg\n,\ni∂ψ−\n∂t=−1\n2∇2ψ−+(g|ψ−|2+γ|ψ+|2)ψ−−ǫ{cos(2πx)+cos(2πy)}ψ−+λ/parenleftbigg\n−∂ψ+\n∂x−i∂ψ+\n∂y/parenrightbigg\n,(1)\nwhereψ= (ψ+,ψ−) denotes the wave function of the spinor BECs, ǫis the strength of the optical lattice, gandγ\nexpress the strengths of interactions respectively between the same and the different kinds of atoms, and λdenotes\nthe strength of the Rashba spin-orbit coupling. We have assumed t hat the wavelength of the optical lattice is 1.\nIfgandγare zero, Eq. (1) becomes linear equations with spatially-periodic po tential. The Bloch states are\nstationary solutions to the linear equations, which are expressed a s\nψ+(x,y,t) =φ+(x,y)exp(ikxx+ikyy−iµt), ψ−(x,y,t) =φ−(x,y)exp(ikxx+ikyy−iµt), (2)\nwhereφ+andφ−are periodic functions of wavelength 1. Therefore, φ+andφ−satisfy\nµφ+=−1\n2∇2φ++1\n2(k2\nx+k2\ny)φ+−ikx∂φ+\n∂x−iky∂φ+\n∂y−ǫ{cos(2πx)+cos(2πy)}φ+\n+λ/parenleftbigg∂φ−\n∂x−i∂φ−\n∂y+ikxφ−+kyφ−/parenrightbigg\n,\nµφ−=−1\n2∇2φ−+1\n2(k2\nx+k2\ny)φ−−ikx∂φ−\n∂x−iky∂φ−\n∂y−ǫ{cos(2πx)+cos(2πy)}φ−\n+λ/parenleftbigg\n−∂φ+\n∂x−i∂φ+\n∂y−ikxφ++kyφ+/parenrightbigg\n. (3)\nThe eigenvalue µand the eigen function φ+andφ−can be numerically obtained from the stationary solution of the2\n/j120/j131/j202/j40/j97/j41\n/j131/j202\n/j120(b)\n/j131/j202\n/j120(c)\n-2.8\r-2.5\r-2.2\r-1.9\r-1.6\r\n0\r 1\r 2\r 3\r 4\r 5\r 6\r\nk\r-13.8\r-13.6\r-13.4\r-13.2\r-13\r-12.8\r\n0\r 1\r 2\r 3\r 4\r 5\r 6\rk\r-8\r-7\r-6\r-5\r-4\r\n0\r 1\r 2\r 3\r 4\r 5\r 6\r\nk\r\nFIG. 1: Eigenvalue µvs.kxfor (a)λ=π/2,ky= 0, (b)λ= 3π/2,ky= 0, and (c) λ=π,ky=kx. Dashed curve in Fig.1(a) is\nplotted using Eq. (8) and the dashed curve in Fig. 1(c) is obta ined using Eq. (6).\nlinear equation [10]:\n∂φ+\n∂t=1\n2∇2φ+−1\n2(k2\nx+k2\ny)φ++ikx∂φ+\n∂x+iky∂φ+\n∂y+ǫ{cos(2πx)+cos(2πy)}φ+\n−λ/parenleftbigg∂φ−\n∂x−i∂φ−\n∂y+ikxφ−+kyφ−/parenrightbigg\n+µφ+,\n∂φ−\n∂t=1\n2∇2φ−−1\n2(k2\nx+k2\ny)φ−+ikx∂φ−\n∂x+iky∂φ−\n∂y+ǫ{cos(2πx)+cos(2πy)}φ−\n−λ/parenleftbigg\n−∂φ+\n∂x−i∂φ+\n∂y−ikxφ1+kyφ+/parenrightbigg\n+µφ−,\ndµ\ndt=α(N0−N), (4)\nwhereα>0 is a parameter and fixed to be 5 in our numerical simulation. N=/integraltext1\n0/integraltext1\n0(|φ+|2+|φ−|2)dxdyis the total\nnorm, and N0is fixed to be 1 by the normalization condition. The time evolution of the dissipative equation (4)\nleads to a stationary state and the total norm Napproaches N0= 1. The eigenvalue µin Eq. (3) is obtained as µin\nEq. (4) at the stationary state. In this numerical method, the gr ound state for fixed values of kxandkyis obtained\nat the stationary state, starting from most initial conditions, bec ause the total energy decreases in the time evolution\nof Eq. (4). Excited states are obtained by removing the ground st ate by the method of orthogonalization. Figure\n1(a) shows µ(kx) as a function of kxforky= 0,ǫ= 5, andλ=π/2.µ(kx) is a periodic function of kxwith period\n2π. There are peaks near kx= 0,πand 2πand minima at kx∼π/2 and 3π/2. The peak point at kx=πis a cusp\npoint, where two µ(kx) curves corresponding to the ground state and the excited stat e cross, although the branch\nof the excited state is not shown. For λ= 0,µ(kx) increases monotonously as kxincreases from 0 and reaches the\nmaximum at the edge of the Brillouin zone at kx=π. If there is no optical lattice, i.e., ǫ= 0,µ(k) takes a minimum\natk=λwherek=/radicalBig\nk2x+k2y[2, 3]. The minimum point for λ=π/2 locates near kx=λ, and the peak corresponds\nto the edge of the Brillouin zone. Figure 1(b) shows µ(kx) atλ= (3/2)π. There are a large peak at kx= 0 and\n2πand a small peak at kx=πand minima at kx= 2 andkx= 4.3. Figure 1(c) shows µ(k) as a function of kxfor\nkx=ky,λ=π,ǫ= 5. There is a large peak at kx= 0 and 2π, a small peak at kx=π, and minima at kx∼2.5 and\n3.8. The wavenumber kx∼2.5 is close to λ/√\n2∼2.22 by the simplest approximation k=λbut slightly deviated.\nThe approximation kmin=λfor the minimum point of µ(k) becomes worse for large λ. The small peaks correspond\nto the edge of the Brillouin zone.\nFigure 2(a) shows |φ+(x,y)|and|φ−(x,y)|as a function of yin the section x= 0 atkx= 3π/2 forλ= 3π/2. The\nmodulus |φ+|and|φ−|take maximum at different positions. The minimum value is almost zero, w hich implies the\nexistence of vortices. Figure 2(b) shows a contour plot of |φ+|for the same parameter. The locations of vortices for\nφ+can be calculated from the phase distribution θ+(x,y) = sin−1(Imφ+(x,y)/|φ+(x,y)|). There exist a vortex at a\npoint, if the integral of the phase grandient along an anticlockwise p ath encircling the point is a nontrivial multiple of\n2π. We have counted the path integral by discretizing the ( x,y) space with ∆ x= 1/64. Figure 2(c) shows positions\nof vortices of vorticity ±1 with square and ×marks. The vortex cores locate near (0 ,−0.28) and (0,−0.48) forφ+.\nIn generic cases, there is a vortex of vorticity 1 or -1 at a position s atisfying |φ+|= 0, where a line of Re φ+= 0\nintersects with a line of Im φ+= 0 [11]. We do not show explicitly the positions of vortices later in Fig. 3 a nd Fig. 4,\nhowever, we have checked the existence of vortices of vorticity 1 or -1 at positions satisfying |φ±|= 0 by calculating3\n00.511.5\n-0.25 0 0.25\ny00.020.040.06\n3.6 3.8 4 4.2 4.4\n/j131/j201/j124/j131/j211/j124\n/j109/j124/j131/j211/j124/j40/j97/j41/j40/j98/j41\n/j40/j99/j41/j40/j100/j41\n-0.4\r-0.2\r0\r0.2\r0.4\r\n-0.4\r -0.2\r 0\r 0.2\r 0.4\ry\r\nx\r\nFIG. 2: (a) |φ+|(solid curve) and |φ−|(dashed curve) along the line x= 0 forλ= 3π/2, andkx= 3π/2. (b) Contour plot\nof|φ+|. (c) Square shows a vortex with vorticity 1 and ×shows a vortex with vorticity -1. (d) Minimum values of |φ+|as a\nfunction of λforkx=λandky= 0.\nthe phase distribution. Figure 2(d) shows the minimum value of |φ+|as a function of λforkx=λ,ky= 0 atǫ= 5.\nThe minimum value becomes zero and a vortex-antivortex pair appea rs forλ>λc∼4.2.\nBecauseφ±are periodic functions with wavelength 1, φ±can be expressed as the simplest approximation:\nφ+=C0++C1+e2πix+C2+e−2πix+C3+e2πiy+C4+e−2πiy,\nφ−=C0−+C1−e2πix+C2−e−2πix+C3−e2πiy+C4−e−2πiy. (5)\nSubstitution of this ansatz into Eq. (3) yields\nµC0±= (k2\nx+k2\ny)C0±/2−(ǫ/2)(C1±+C2±+C3±+C4±)+λ(±ikx+ky)C0∓,\nµC1±={(kx+2π)2+k2\ny}C1±/2−(ǫ/2)C0±+λ{±i(kx+2π)+ky}C1∓,\nµC2±={(kx−2π)2+k2\ny}C2±/2−(ǫ/2)C0±+λ{±i(kx+2π)+ky}C2∓,\nµC3±={k2\nx+(ky+2π)2}C3±/2−(ǫ/2)C0±+λ{±ikx+(ky+2π)}C3∓\nµC4±={k2\nx+(ky−2π)2}C4±/2−(ǫ/2)C0±+λ{±ikx+(ky−2π)}C4∓. (6)\nForky= 0,C0−=iC0+,C1−=iC1+,C2−=iC2+are satisfied, and then\nC1+=−(ǫ/2)C0+\nµ−(kx+2π)2/2+λ(kx+2π),\nC2+=−(ǫ/2)C0+\nµ−(kx−2π)2/2+λ(kx−2π),\nC3+=[−(ǫ/2){µ−(k2\nx+4π2)/2)}+(ǫ/2)λ(kx−2πi)]C0+\n{µ−(k2x+4π2)/2}2−λ2(k2x+4π2),\nC4+=[−(ǫ/2){µ−(k2\nx+4π2)/2)}+(ǫ/2)λ(kx+2πi)]C0+\n{µ−(k2x+4π2)/2}2−λ2(k2x+4π2), (7)\nwhereµis given by a solution of the equation\nµ=k2\nx\n2−λkx+ǫ2/4\nµ−(kx+2π)2/2+λ(kx+2π)+ǫ2/4\nµ−(kx−2π)2/2+λ(kx−2π)\n+ǫ2\n42µ−(k2\nx+4π2)−2λkx\n{µ−(k2x+4π2)/2}2−λ2(k2x+4π2). (8)\nFurthermore, C3−=iC∗\n3+,C4−=iC∗\n4+are satisfied. Here,∗denotes the complex conjugate. The dashed curve in\nFig. 1(a) denotes µ(kx) by Eq. (8) at λ=π/2. The approximation is good for λ=π/2 but is not so good for large λ,\nbecause the higher harmonics is necessary for the expansion in Eq. (5). We can assume that C0+,C1+andC2+are\nreal numbers and C4+=C∗\n3+= ReC3+−iImC3+. Then,φ+andφ−are expressed as\nφ+=C0++(C1++C2+)cos(2πx)+i(C1+−C2+)sin(2πx)+2ReC3+cos(2πy)−2ImC3+sin(2πy),\nφ−=iC0++i(C1++C2+)cos(2πx)−(C1+−C2+)sin(2πx)+2iImC3+sin(2πy)+2iReC3+cos(2πy).(9)4\n00.020.040.06\n22.2 2.4 2.6 2.8 33.2 3.4\n/j131/j201/j124/j131/j211/j124/j40/j97/j41/j40/j98/j41/j40/j99/j41\n/j40/j100/j41/j109\nFIG. 3: (a) Contour plot of |ψ+|forλ=π,g= 1,γ= 0.5 andL= 8.kx=kyare evaluated as 3 π/4. (b) Contour plot of |ψ−|.\n(c) Contour plot of |φ+|to the linear equation Eq. (3) for λ=π,kx=ky= 3π/4. (d) Minimum values of |φ+|as a function of\nλforkx=ky=λ/√\n2.\n/j40/j97/j41\n/j40/j98/j41/j40/j99/j41\n/j40/j100/j41\nFIG. 4: (a) Contour plot of |ψ+|forλ=π,g= 1,γ= 2 andL= 8.kx=kyare evaluated as 3 π/4. (b) Contour plot of |ψ−|.\n(c) Contour plot of the superposition of |(φ+++φ+−)/√\n2|to the linear equation Eq. (3) for λ=πandkx=ky=±3π/4. (d)\nContour plot of the superposition of |(φ+++φ+−)/√\n2|to the linear equation Eq. (3) for λ= 1 andkx=ky=±π/4\nImφ+= 0 and Re φ−= 0 are satisfied on the line x= 0. When λis small, the minimum values of Re φ+and Imφ−\nare positive and there are no vortices. When λis increased the minimum values decrease and reach 0, and then a\nvortex pair is created. A vortex core of φ+is located at a point on the line x= 0 where Re φ+(0,y) = 0 is satisfied,\nand similarly a vortex core of φ−is located at a point on the line x= 0 where Im φ−(0,y) = 0 is satisfied.\nEven forgandγis not zero, the Bloch state is a good approximation for the stationa ry state for γ < g. We\nhave performed numerical simulation of Eq. (1) by the imaginary time evolution method similar to Eq. (4) and found\nstationary solutions. The system size is Lx×Ly=L×Land the total norm N=/integraltextL\n0/integraltextL\n0(|ψ+|2+|ψ−|2)dxdyis set\nto beL2in this paper. Periodic boundary conditions are imposed. The potent ial is shifted as U=−ǫ[cos{2π(x−\n1/2)}+cos{2π(y−1/2)}] by (1/2,1/2) to confine the wave pattern in the range of [0 ,L]×[0,L].\nFigure 3(a) and (b) show contour plots of |ψ+|and|ψ−|atg= 1,γ= 0.5,L= 8,λ=π, andǫ= 5. The contour\nplot is drawn in the region [0 ,2]×[0,2], and the contour lines are drawn for |ψ±|= 0.025,0.05,0.075,0.1,1 and 1.5.\nVortex pairs exist in each cell of size 1 for this parameter, and a vor tex lattice is constructed as a whole. Vortex\nlattices were experimentally found first in rotating BECs [12] and re cently in BECs under synthetic magnetic fields\nby atom-light interaction [13]. In our model equation, vortices are s pontaneously created by the spin-orbit coupling.\nThe wavevector ( kx,ky) is evaluated at (3 π/4,3π/4). Positions of vortex cores for ψ+andψ−are mutually deviated.\nFigure 3(c) shows a contour plot of |φ+|for the linear equation corresponding to g= 0,γ= 0 forkx=ky= 3π/4 at\nλ=πandǫ= 5. The eigenvalue µtakes a minimum at ( kx,ky) = (3π/4,3π/4) in the finite size system of L= 8,\nwherekx(ky) takes a discrete value 2 πnx/L(2πny/L) with integer nx(ny). The contour plot is almost the same as\nFig. 3(a). It means that the Bloch wave is a good approximation for t he solution to the GP equation. Figure 3(d)\nshows the minimum values of |φ+|for the linear equation as a function of λforkx=ky=λ/√\n2 atǫ= 5. The\nminimum value becomes zero and vortices appear for λ>2.9. It is related to the existence of vortices at λ=π.\nStripe wave states are expected to appear for γ >g. The superposition of Bloch waves of ( kx,ky) and (−kx,−ky)\nis a simple approximation for γ >g. Figure 4(a) and (b) show contour plots of |ψ+|and|ψ−|atg= 1,γ= 2,L= 8,5\n2\r4\r6\r8\r\n2\r 4\r 6\r 8\rj\r\ni\r2\r4\r6\r8\r\n2\r 4\r 6\r 8\rj\r\ni\r/j40/j97/j41 /j40/j98/j41\n0\r2\r4\r6\r8\r\n0\r 2\r 4\r 6\r 8\rj\r\ni\r/j40/j99/j41\nFIG. 5: (a) Spinconfiguration of ( sx(i,j),sy(i,j)) atλ=π,g= 1,γ= 0.5 andL= 8. (b)Spin configuration of ( sx(i,j),sy(i,j))\natλ=π,g= 1,γ= 2 andL= 8. (c) Spin configuration of sz(i,j) atλ=π,g= 1,γ= 2 andL= 8.\nandλ=π. The wavevector is evaluated as ( kx,ky) = (3π/4,3π/4) in this case, too. The contour lines are drawn\nfor|ψ±|= 0.025,0.05,0.075,0.1,1 and 1.5. Vortex cores exist in dark pointed regions. The vortex lat tice structure is\nrather complicated. The circular contour lines correspond to peak regions of |ψ±|. The peak regions stand in a line\nin the direction of angle −π/4 and the peak lines for ψ+andψ−alternates in the diagonal direction of angle π/4.\nFigure 4(c) shows a contour plot of a linear combination |(φ+++φ+−)/√\n2|of two Bloch waves φ++andφ+−with\n(kx,ky) = (3π/4,3π/4) and (−3π/4,−3π/4) for the + component at λ=π. The superposition of the Bloch waves\nis a good approximation for the stationary solution to the GP equatio n. The superposition of two Bloch waves with\nopposite wavevectors generates a standing wave. For plane wave s, the amplitude becomes zero at the nodal lines.\nThe nodal lines are perturbed by the optical lattice and vortices ar e generated. A vortex lattice structure therefore\nappears even for small λin case ofγ > g. Figure 4(d) shows a vortex lattice pattern with kx=ky=π/4 atλ= 1\nandǫ= 5. For large λ, a vortex pair is created in a single Bloch wave and the superposition o f the two Bloch waves\nmake the vortex lattice structure more complicated as shown in Fig. 4(c).\nThe complicated patterns might be simplified, if a spin representation is used, which was discussed in the Bose-\nHubbard model [6, 7]. The whole system is divided into cell regions of [ i−1,i]×[j−1,j]. The spin variables\nsx(i,j),sy(i,j) andsz(i,j) are defined for each cell labeled by ( i,j) as\nsx(i,j) =/integraldisplayi\ni−1/integraldisplayj\nj−1ψ†σxψdxdy=/integraldisplayi\ni−1/integraldisplayj\nj−1(ψ∗\n+ψ−+ψ∗\n−ψ+)dxdy,\nsy(i,j) =/integraldisplayi\ni−1/integraldisplayj\nj−1ψ†σyψdxdy=/integraldisplayi\ni−1/integraldisplayj\nj−1(−iψ∗\n+ψ−+iψ∗\n−ψ+)dxdy,\nsz(i,j) =/integraldisplayi\ni−1/integraldisplayj\nj−1ψ†σzψdxdy=/integraldisplayi\ni−1/integraldisplayj\nj−1(|ψ+|2−|ψ−|2)dxdy, (10)\nwhereσx,σyandσzare the Pauli matrix, and†denotes the complex conjugate transpose. Figure 5(a) shows ( sx,sy)\ncorresponding to the pattern in Figs. 3(a) and (b) for g= 1,γ= 0.5,λ=πandǫ= 5. The vector ( sx(i,j),sy(i,j)) is\nexpressed as an arrow on each lattice point at ( i−1/2,j−1/2). The pattern is interpreted as a ferromagnetic state\nin the (x,y) plane in this spin representation. The spin szis zero for this pattern. Figures 5(b) and (c) show spin\nconfigurations respectively for ( sx,sy) andszfor the pattern at g= 1 andγ= 2 shown in Figs. 4(a) and (b). The spin\nconfiguration is also rather complicated. The wavelength of the spin configuration is 4 both in the iandjdirections.\nAn anti-ferromagnetic order is seen in the diagonal direction of ang leπ/4 and a ferromagnetic order appears in its\northogonal direction of angle −π/4 both for the ( sx,sy) andszpatterns. The ( sx,sy) component appears at the sites\nwhere theszcomponent vanishes, and the szcomponent appears at the sites where the ( sx,sy) component vanishes.\nTo summarize, we havestudied the Gross-Pitaevskiiequation with s pin-orbit coupling in an optical lattice. We have\nfound that a vortex lattice structure appears for large λin case ofγ g, becausethenodallinesinthestripewavepatternareperturbed bytheopticallattice. Wehave\nfound a complicated spin configuration in a case of γ >g. The complicated patterns can be qualitatively understood\nby the corresponding Bloch waves. The Bloch waves are further ap proximated by a Fourier series expansion with five\nmodes to understand the formation of the vortices. We have obta ined various spin configurations by changing the6\nparameterλ. The detailed phase diagrams by changing various parameters are u nder study.\n[1] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Natu re471, 83 (2011).\n[2] H. Zhai, Int. J. Mod. Phys. 26, 1230001 (2012).\n[3] C. Wang, C. Gao, C. -M. Jian, and H. Zhai, Phys. Rev. Lett. 105, 160403 (2010).\n[4] B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D. Dru mmond, and H. Hu, Phys. Rev. A 85, 023606 (2012).\n[5] Y. Zhang, L. Mao, and C. Zhang, Phys. Rev. Lett. 108, 035302 (2012).\n[6] J. Radic, A. Di Ciolo, K. Sun, and V. Galitski, Phys. Rev. L ett.109, 085303 (2012).\n[7] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi, Phys . Rev. Lett. 109. 085302 (2012).\n[8] J. Larson and E. Sj¨ oqvist, Phys. Rev. A. 79, 043627 (2009).\n[9] Y. Zhang and C. Zhang, arXiv:1203.2389 (2012).\n[10] H. Sakaguchi and H. Takeshita, J. Phys.Soc. Jpn. 77, 054003 (2008).\n[11] A. Ohta, R. Kashiwa, and H. Sakaguchi, Phys. Rev. A 82, 055602 (2010).\n[12] J. E. Williams and M. J. Holland, Nature 401, 568 (1999).\n[13] Y.-L. Lin, R. L. Compton, K. Jim´ enez-Garc´ ıa, J. V. Por to,and I. B. Spielman, Nature 462, 628 (2009)" }, { "title": "1301.4718v2.Spin_orbit_induced_bound_state_and_molecular_signature_of_the_degenerate_Fermi_gas_in_a_narrow_Feshbach_resonance.pdf", "content": "arXiv:1301.4718v2 [cond-mat.quant-gas] 9 Jul 2013Spin-orbit-induced bound state and molecular signature of the degenerate Fermi gas\nin a narrow Feshbach resonance\nKuang Zhang,1Gang Chen,1,∗and Suotang Jia1\n1State Key Laboratory of Quantum Optics and Quantum Optics De vices,\nInstitute of Laser spectroscopy, Shanxi University, Taiyu an 030006, People’s Republic of China\nIn this paper we explore the spin-orbit-induced bound state and molecular signature of the degen-\nerate Fermi gas in a narrow Feshbach resonance based on a gene ralized two-channel model. Without\nthe atom-atom interactions, only one bound state can be foun d even if spin-orbit coupling exists.\nMoreover, the corresponding bound-state energy depends st rongly on the strength of spin-orbit cou-\npling, but is influenced slightly by its type. In addition, we find that when increasing the strength\nof spin-orbit coupling, the critical point at which the mole cular fraction vanishes shifts from zero to\nthe negative detuning. In the weak spin-orbit coupling, thi s shifting is proportional to the square of\nits strength. Finally, we also show that the molecular fract ion can be well controlled by spin-orbit\ncoupling.\nPACS numbers: 03.75. Ss, 05.30. Fk, 67.85. Lm\nI. INTRODUCTION\nRecently, the investigation of spin-orbit (SO) coupling\nin neutral atoms has attracted much attentions [1]. In\nparticular, a one-dimensional (1D) equal Rashba and\nDresselhaus SO coupling has been first realized in the ul-\ntracold87Rb atoms by a couple of Raman lasers [2]. By\napplying the same laser technique, this 1D SO coupling\nhas been also achieved experimentally in the degenerate\nFermi gas with40K [3] and6Li [4]. Theoretical inves-\ntigations have been revealed that in the presence of SO\ncoupling, the degenerate Fermi gas can exhibit the inter-\nesting physics in both three [5–21] and lower dimensions\n[22–29]. For example, by increasing the strength of SO\ncoupling, the density of state at the Fermi surface is in-\ncreased,andtheCooperparinggapcanbe thusenhanced\nsignificantly[6–8]. Moreimportantly, this systemmaybe\nchanged from the Bardeen-Cooper-Schrieffer (BCS) su-\nperfluid to the Bose-Einstein condensate (BEC) with a\nnew two-body bound state called Rashbon [5, 8]. When\nan effective Zeeman field is applied, the 2D degenerate\nFermigaswiththeRashbaSOcouplingexhibitsanexotic\ntopological superfluid supporting the Majorana fermions\n[23], which is the heart for realizing the topologicalquan-\ntum computing [30]. Recently, a universal midgap bound\nstateinthetopologicalsuperfluidhasbeenpredicted[31].\nTo illustrate the SO-driven fundamental physics, a\ngeneralized one-channel model has been introduced in\nall previous considerations [5–29]. In this one-channel\nmodel, only the atoms tuned via the Feshbash-resonant\ntechnique are taken into account. However, it is valid\nfor thebroadFeshbash-resonant regime with Γ ≫1\n[32], where the dimensionless parameter is defined as\nΓ =/radicalBig\n32mµBa2\nbgB2w/(/planckover2pi12πEF),µBis the Bohr magne-\nton,mis the atom mass, abgis the background s-wave\n∗Corresponding author: chengang971@163.comscattering strength, Bwis the resonant width and EFis\nthe Fermi energy. In fact, to get a more realistic and\ncomplete description of the degenerate Fermi gas, espe-\ncially in the narrowFeshbash-resonant limit with Γ ≪1,\nwe must introduce a two-channel model [33–37], which\nincludes both the atoms in the open channel and the\nmolecules in the closed channel. Moreover, in the narrow\nFeshbash-resonant regime, some fundamental properties\ncan be observed experimentally by detecting the striking\nmolecular signature [38], additional to measuring the su-\nperfluid pairing gap applied usually in the one-channel\nmodel [39–41]. More importantly, new quantum phase\ntransitions have been predicted [42, 43], attributed to\nthe existence of extra U(1) symmetry for the molecular\nfield. Onthe experimentalside, the degenerateFermi gas\nin the narrowFeshbach-resonantregime has been also re-\nported successfully in6Li [44] and the Fermi-Fermi mix-\nture of6Li and40K [45]. Thus, it is crucially important\nto explore the SO-induced exotic physics in this regime\n[46].\nIn the present paper we investigate the SO-induced\nbound state and molecular signature of the degenerate\nFermi gas in the narrow Feshbach resonance. The main\nresults are given as follows. (i) Without the atom-atom\ninteractions, only one bound state can be found even if\nSO coupling exists. Moreover, the corresponding bound-\nstate energy depends strongly on the strength of SO cou-\npling, but is influenced slightly on its type. (ii) With\nthe increasing of the strength of SO coupling, the critical\npointat whichthe molecularfractionvanishesshifts from\nzero to the negative detuning. In the weak SO coupling,\nthis shifting is proportional to the square of its strength.\n(iii) Finally, we also show that the molecular fraction can\nbe well controlled by SO coupling. We believe that in ex-\nperiments it is a good signature to detect the SO-induced\nphysics.2\nII. MODEL AND HAMILTONIAN\nFor the SO-driventwo-channelmodel, the total Hamil-\ntonian can be written formally as\nH=HF+HM+HI+HS. (1)\nIn Hamiltonian (1),\nHF=/summationdisplay\nkσǫkC†\nkσCkσ (2)\nis the atom Hamiltonian, where C†\nkσis the creation oper-\nator for a atom with the momentum kandσ=↑,↓, and\nǫk=k2/(2m) is the kinetic energy of the atom.\nHM=/summationdisplay\nq(ǫq+δ0)b†\nqbq (3)\nis the molecular Hamiltonian, where b†\nqis the creation\noperator of a molecule with the momentum q,ǫq=\n/planckover2pi12q2/(2M) withM= 2mis the kinetic energy of the\nmolecule, and δ0is the bare detuning determined by the\nFeshbash-resonant position δvia a relation\nδ0=δ+g2/summationdisplay\nk1\n2ǫk. (4)\nWithout SO coupling, the system has the BCS superfluid\nin the positive detuning ( δ >0), and enters into the\nBEC regime in the negative detuning ( δ <0) [33–37] .\nAt lower energy, the position is given approximately by\nδ≃2µB(B−B0), whereB0is the magnetic field at which\nthe resonance is at zero energy, and Bis the tunable\nmagnetic field [47]. The atom-molecule interconversion\nterm is governed by the following Hamiltonian\nHI=g/summationdisplay\nqkk′b†\nqC−k+q\n2↓Ck+q\n2↑+C†\nk′+q\n2↑C†\n−k′+q\n2↓bq,(5)\nwheregis the coupling constant that measures the am-\nplitude of the decay of the molecule in the closed channel\ninto a pair of the open-channel atoms. Finally, the SO\ncoupling is chosen as a generalized Rashba and Dressel-\nhaus type. The corresponding Hamiltonian is given by\nHS=α/summationdisplay\nk[(ky+iλkx)C†\nk↑Ck↓+(ky−iλkx)C†\nk↓Ck↑] (6)\nwithα= (αR+αD) andλ= (αR−αD)/(αR+αD), where\nαRandαDare the SO coupling strengths for the Rashba\nand Dresselhaus types, respectively. Clearly, αis the\ngeneralized strength of SO coupling and the dimension-\nless parameter λreflects the competition between these\ndifferent types of SO coupling. For example, for λ= 1\n(αD= 0), the 2D Rashba SO coupling can be found.\nWhereas, for λ= 0 (αD=αR), the 1D equal Rashba and\nDresselhaus SO coupling can be generated. Fortunately,\nthis 1D SO coupling has been realized experimentally in\nthe ultracold neutral atoms [2–4].In the absence of SO coupling, the limit q=0can\nbe applied usually to discuss the standard two-channel\nmodel including the effective Zeeman field [32]. However,\nin the presence of SO coupling, the result is quite com-\nplicated. If both SO coupling and the effective Zeeman\nfield are taken into account, the parity and time-reversal\nsymmetries are broken. As a result, the q-dependent or-\nder parameter should be introduced [48]. However, in\nthis paper we do not consider the effect of the effective\nZeeman field and thus may focus on the case of q=0.\nIII. TWO-BODY BOUND STATE\nWe begin to discuss the two-body bound state of the\ngeneralized two-channel model (1) by introducing the\nansatz wavefunction. In the absence of SO coupling\n(α= 0), Hamiltonian (1) reduces to the standard two-\nchannel model H=/summationtext\nkσξkC†\nkσCkσ+/summationtext(δ0−2µ)b†\n0b0+\ng/summationtext\nkk′b†\n0C−k↓Ck↑+C†\nk′↑C†\n−k′↓b0[33–36], in which only\nthe singlet Cooper paring can be formed. However, in\nthe presence of SO coupling, both the singlet and triplet\nCooper parings can coexist [49]. As a result, the ansatz\nwavefunction should be written formally as\n|Ψ/angbracketright= (/summationdisplay\nk′σσ′βk′σσ′C†\nk′σC†\nk′σ′+γb†\n0)|0,0,0,0/angbracketright⊗|0/angbracketright,(7)\nwhereβk′↑↓andβk′↓↑(βk′↑↑andβk′↓↓) represent the\nprobability amplitude of the singlet (triplet) Cooper\nparing,γstands for the probability amplitude of the\nmolecule, and |0,0,0,0/angbracketright⊗|0/angbracketrightis the direct multiple of the\nfermion vacuum with spin flipping and the molecule vac-\nuum. Substituting the wavefunction in Eq. (7) into the\nstationary Schr¨ odinger equation\n(H−E)|Ψ/angbracketright= 0, (8)\nwe find that the six coefficients determining the ansatz\nwavefunction |Ψ/angbracketrightand the energy Eare governed by the\nfollowing equations:\n\n\ngγ+βk′↑↑αk−= Ξkβk↓↑\ngγ+βk′↓↓αk+= Ξkβk↑↓\n(βk′↓↑+βk′↑↓)αk+= Ξkβk′↑↑\n(βk′↓↑+βk′↑↓)αk−= Ξkβk′↓↓\n2(δ0−E)γ=−g/summationtext\nk′(βk′↑↓+βk′↓↑),(9)\nwhere Ξk=E−2ǫkandk±=ky±iλkx. Eq. (9) can not\nbe solved directly because of lack of a coefficient equa-\ntion. If we define the spin symmetry and anti-symmetry\nvectors as\n/braceleftBigg\nψs(k) =1√\n2(βk↓↑−βk↑↓)\nψa(k) =1√\n2(βk↓↑+βk↑↓), (10)\nthe stationary Schr¨ odinger equation is rewritten as\n(H−E)ψk= 0 in the representation of ψk=3\nFIG. 1: (Color online) The bound-state energy for the 2D\nRashbaSOcoupling( λ= 1)asafunctionofthedetuning δfor\nthe different strengths of SO coupling (a) αKF= 0.5EF, (b)\nαKF= 3.6EF, and (c) αKF= 7.2EFwhen Γ = 0 .1. In Fig.\n1(a), the red open symbol corresponds to the analytical resu lt\n(AR) and the black solid line represents the direct numerica l\nsimulation (NS).\n[βk↑↓,βk↓↑,βk↓↓,βk↑↑,γ]T. This leads to another equa-\ntions for the coefficients βk′σ,σ′andγ, that is,\n\n\nγ=−/summationtext\nk′g√\n2ψa(k′)\n(δ0−E)\nβk′↑↑=√\n2ψa(k)αk+\nΞk\nβk′↓↓=√\n2ψa(k)αk−\nΞk. (11)\nSubstituting Eq. (11) into Eq. (9) yields\n/summationdisplay\nk(Ξk\nΞ2\nk−4α2k+k−+1\n2ǫk) =E−δ\ng2.(12)\nEquation (12), which is the main result of this pa-\nper, determines the SO-induced bound-state energy of\nthe generalized two-channel model (1). The procedure is\ngivenasfollows. (i)Wefirstobtaintheenergy EfromEq.\n(12). (ii) Then, we introduce the threshold energy ET,\nwhich is the lowest energy of the free particle (i.e., the\nlowest band), to judge whether this energy is the bound-\nstate energy or not. If E < ET, the bound state exists\nand the correspondingenergy Eis called the bound-state\nenergy, and vice versa [5]. According to its definition,\nthe threshold energy ETof the generalized two-channel\nmodel (1) is evaluated as\nET=−2mα2\n/planckover2pi12. (13)\nIn the absence of SO coupling ( α= 0), this threshold\nenergy becomes ET= 0, as expected. In the following\ndiscussions, we mainly consider two interesting cases, in-\ncludingthe 2DRashbaSOcoupling( λ= 1)and1Dequal\nRashba and Dresselhaus SO coupling ( λ= 0), to reveal\nthe fundamental properties of the bound state.FIG.2: (Coloronline)Thebound-stateenergyasafunctiono f\nthe detuning δfor the different types of SO coupling including\nλ= 1 (the 2D Rashba SO coupling) and λ= 0 (the 1D\nequal Rashba and Dresselhaus SO coupling) when Γ = 0 .1\nandαKF= 4.2EF.\nWefirstaddressthecaseofthe2DRashbaSOcoupling\n(λ= 1). In the absence of SO coupling ( α= 0), the\nanalytical bound-state energy is derived from Eq. (12)\nby\nE=1\n32π2/planckover2pi16(−g4m3+32π2δ/planckover2pi16−g2/radicalbig\ng4m6−64m3π2δ/planckover2pi16).\n(14)\nIt implies that in such a case only one bound state can be\nfound [32]. In the presence of SO coupling ( α/negationslash= 0), the\nexplicit expression for the bound-state energy can not be\nobtained. However, for the weak SO coupling, Eq. (12)\nis simplified as\nm3\n2[2/planckover2pi12E+(1+λ2)mα2]\n8π/planckover2pi15√\n−E=E−δ\ng2(15)\nwith the help of a Taylor expansion with respect to the\nstrength of SO coupling. In this case, Hamiltonian (1)\nalso exhibits one bound state, as shown in Fig. 1(a).\nBy further solving Eq. (15) approximately, we find that\nthe bound-state energy is proportional to −α4. This be-\nhavior agrees well with the numerical simulation, as also\nshown in Fig. 1(a). It implies that the bound-state\nenergy can decrease by increasing the strength of SO\ncoupling. For the strong SO coupling, the perturbation\nmethod is invalid. Here we numerically solve Eq. (12) to\nevaluate the bound-state energy E. Even if the strong\nSO coupling exists, only one bound state can be found,\nas shown in Figs. 1(b) and 1(c)\nIn Fig. 2, we plot the bound-state energy with respect\nto the detuning δfor the different types of SO coupling\nincludingλ= 1 (the 2D Rashba SO coupling) and λ= 0\n(the 1D equal Rashba and Dresselhaus SO coupling). It\ncanbeseenclearlythatthebound-stateenergyisaffected\nslightly by the type of SO coupling.\nWhen the Feshbach-resonant width parameter Γ in-\ncreases, the system changes from the narrow limit (Γ ≪4\nFIG. 3: (Color online) The bound-state energy for the 1D\nequal Rashba and Dresselhaus SO coupling with λ= 0 as a\nfunction of the detuning δfor the different Feshbach-resonant\nwidths (a) Γ = 0 .1, (b) Γ = 30, and (c) Γ = 300 .0 when\nαKF= 0.7EF. In (b) and (c), the red dash line corresponds\nto the numerical solution of Eq. (16). For the 2D Rashba SO\ncoupling, the similar conclusion can be also found.\n1) to the broad limit (Γ ≫1). Especially, for the broad\nlimit (Γ≫1), Eq. (12) becomes\n/summationdisplay\nk(Ξk\nΞ2\nk−4α2k+k−+1\n2ǫk)≃ −δ\ng2=m\n4π/planckover2pi12as,(16)\nwhich is similar to the result of Ref. [7]. In Fig. 3, we\nplot the bound-state energy for the 1D equal Rashba and\nDresselhaus SO coupling as a function of the detuning δ\nfor the different Feshbach-resonant width parameters (a)\nΓ = 0.1, (b) Γ = 30 .0, and (c) Γ = 300 .0. This fig-\nure shows again that for the broad limit our considered\ntwo-channel model reduces to the single-channel model.\nHowever, it should be remarked that the energy Efor\nthe broad Feshbach resonance is not a bound-state en-\nergy, but is a two-body interaction energy approaching\ninfinitely the bound-state energy [50]\nIV. MOLECULAR SIGNATURE\nHaving obtained the bound-state energy in the gen-\neralized two-channel model, it is conveniently to discuss\nthe experimentally-measurable molecular signature. In\nterms of the Hellmann-Feymann theorem, the molecular\nfraction is obtained by\nN0=/angbracketleftΨ|b†\n0b0|Ψ/angbracketright=/angbracketleftΨ|∂H\n∂δ0|Ψ/angbracketright=dE\ndδ,(17)\nwhere the bound-state energy Ecan be derived from Eq.\n(12).\nFigure4isplottedthescaledmolecularfraction2 N0/N\nof both the 2D Rashba SO coupling ( λ= 1) and the 1D\nequal Rashba and Dresselhaus SO coupling ( λ= 0) withFIG. 4: (Color online) The molecular fraction of both (a)\nthe 2D Rashba SO coupling with λ= 1 and (b) the 1D equal\nRashbaand Dresselhaus SO coupling with λ= 0 as a function\nof the detuning δfor the different strengths of SO coupling\nwhen Γ = 0 .1.\nrespect to the detuning δfor the different strengths of\nSO coupling. In the absence of SO coupling ( α= 0),\nthe molecule exists in the negative detuning ( δ <0).\nHowever, for the positive detuning ( δ≥0), the physical\nbound state vanishes [50], i.e., there is no real molecular\nfraction. With the increasing of the strength of SO cou-\npling, the critical point at which the molecular fraction\nvanishes shifts from zero to the negative detuning. The\nphysics can be understood as follows. In the generalized\ntwo-channel model, the molecules play two roles. One is\nthat they interact directly with the atoms via Hamilto-\nnianHI. The other (the most important) is that they\ninduce the indirect atom-atom interactions, which gen-\nerate the Cooper pairing. When the Cooper pairing is\nenhanced by SO coupling [6–8], the molecules are thus\nsuppressed because the system need guarantee a con-\nserved number N= 2b†\n0b0+/summationtext\nkσC†\nkσCkσ. In order to\nsee clearly this behavior induced by SO coupling, we in-\ntroduce a key parameter δm. This parameter describes\nthe maximum detuning at which the molecular fraction\nexists. In terms of the definition, the parameter δmis\ngiven by\nδm=−2mα2\n/planckover2pi12−g2[/summationdisplay\nk(Ξk\nΞ2\nk−4α2k+k−+1\n2ǫk)]E=ET.\n(18)\nIn the case of the weak SO coupling, the parameter δm\nis obtained explicitly by\nδm=−32/planckover2pi12mπα2+√\n2g2m2(λ2−3)α\n16π/planckover2pi14≃ −α2.(19)\nEq. (19) shows clearly that δmdecreases with the in-\ncreasing of the strength of SO coupling.\nInFig. 5, weplotthemolecularfractionofboth the2D\nRashba SO coupling ( λ= 1) and the 1D equal Rashba\nand Dresselhaus SO coupling ( λ= 0) with respect to the5\nFIG. 5: (Color online) The molecular fraction of both (a)\nthe 2D Rashba SO coupling with λ= 1 and (b) the 1D equal\nRashba and Dresselhaus SO coupling with λ= 0 as a function\nof the strength αKFof SOcoupling for the different detunings\nwhen Γ = 0 .1.\nstrength of SO coupling for the different detunings. In\nthe negative detuning ( δ <δm) we shows again that the\nSO coupling suppresses the molecular fraction. With the\nincreasing of the strength of SO coupling, the molecu-\nlar fraction also vanishes. It means that in experiment\nthe molecular fraction can be well controlled by tuning\nthe SO strength. In addition, in the positive detuning,\nno molecular fraction can be found even if SO coupling\nexists.\nV. CONCLUSIONS AND REMARKS\nIn summary, motivated by the recent experimental de-\nvelopments, we have investigated the SO-driven degen-erate Fermi gas in the narrow Feshbash resonance based\non the generalized two-channel model. We have found\nthat in the absence of the atom-atom interactions, only\none bound state can be found even if SO coupling exists.\nIn addition, we have shown that the molecular fraction\ncan be well controlled by SO coupling. We believe that\nin experiments it is a good signature to explore the SO-\ninduced physics.\nVI. ACKNOWLEDGEMENTS\nWe thank Professors Peng Zhang, Wei Yi, Wei Zhang,\nShizhong Zhang and Doctor Zengqiang Yu for their help-\nful discussions. This work was supported partly by\nthe 973 program under Grant No. 2012CB921603; the\nNNSFC under Grants No. 10934004, No. 11074154, and\nNo. 61275211; NNSFC Project for Excellent Research\nTeam under Grant No. 61121064; and International Sci-\nence and Technology Cooperation Program of China un-\nder Grant No.2001DFA12490.\nNote added –During preparing this paper, we noticed\nthat two bound states for the SO-driven two-channel\nmodel with the atom-atom interactions was predicted by\nV. B. Shenoy in terms of a renormalizable quantum field\ntheory [51]. However, that paper does not consider the\nmolecular fraction.\n[1] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg,\nRev. Mod. Phys. 83, 1523 (2011).\n[2] Y.-J.Lin, K.Jimenez-Garcia, andI.B.Spielman, Nature\n(London), 471, 83 (2011).\n[3] P. Wang, Z. -Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.\nZhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).\n[4] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,\nW. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109,\n095302 (2012).\n[5] J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B 83,\n094515 (2011).\n[6] M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. 107,\n195303 (2011).\n[7] Z. -Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305\n(2011).\n[8] H. Hu, L. Jiang, X. -J. Liu, and H. Pu, Phys. Rev. Lett.\n107, 195304 (2011).\n[9] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys.\nRev. B84, 014512 (2011).\n[10] M. Iskin, and A. L. Subasi, Phys. Rev. Lett. 107, 050402(2011); Phys. Rev. A 84, 043621 (2011).\n[11] L. Jiang, X. -J. Liu, H. Hu, and H. Pu, Phys. Rev. A 84,\n063618 (2011).\n[12] W. Yi and G. -C. Guo, Phys. Rev. A 84, 031608 (2011).\n[13] L. Dell’Anna, G. Mazzarella, and L. Salasnich, Phys.\nRev. A 84, 033633 (2011); Phys. Rev. A 86, 053632\n(2012).\n[14] L. Han andC. A.R. S´ adeMelo, Phys.Rev. A 85, 011606\n(2012).\n[15] K. Seo, L. Han, andC. A.R.S´ adeMelo, Phys.Rev.Lett.\n109, 105303 (2012); Phys. Rev. A 85, 033601 (2012);\narXiv:1301.1353 (2013).\n[16] K. Zhou and Z. Zhang, Phys. Rev. Lett. 108, 025301\n(2012).\n[17] R. Liao, Y. -X. Yu, and W. -M. Liu, Phys. Rev. Lett.\n108, 080406 (2012).\n[18] P. Zhang, L. Zhang, and Y. Deng, Phys. Rev. A 86,\n053608 (2012); P. Zhang, L. Zhang, and W. Zhang, Phys.\nRev. A86, 042707 (2012).\n[19] S. -G. Peng, X. -J. Liu, H. Hu, and K. Jiang, Phys. Rev.6\nA86, 063610 (2012).\n[20] L. He andX. -G. Huang, Phys. Rev.B 86, 014511 (2012).\n[21] L. Dong, L. Jiang, H. Hu, and H. Pu, Phys. Rev. A 87,\n043616 (2013).\n[22] G. Chen, M. Gong, and C. Zhang, Phys. Rev. A 85,\n013601 (2012).\n[23] M. Gong, G. Chen, S. Jia, and C. Zhang, Phys. Rev.\nLett.109, 105302 (2012).\n[24] L. He and X. -G. Huang, Phys. Rev. Lett. 108, 145302\n(2012); Phys. Rev. A 86, 043618 (2012).\n[25] J. Zhou, W. Zhang, and W. Yi, Phys. Rev. A 84, 063603\n(2012).\n[26] W. Yi and W. Zhang, Phys. Rev. Lett. 109, 140402\n(2012).\n[27] X. Yang and S. Wan, Phys. Rev. A 85, 023633 (2012).\n[28] J. -N. Zhang, Y. -H. Chan, and L. -M. Duan,\narXiv:1110.2241 (2011).\n[29] F. Wu, G. -C. Guo, W. Zhang, and W. Yi, Phys. Rev.\nLett.110, 110401 (2013).\n[30] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S.\nDas Sarma, Rev. Mod. Phys. 80, 1083 (2008).\n[31] H. Hu, L. Jiang, H. Pu, Y. Chen, and X. -J. Liu, Phys.\nRev. Lett. 110, 020401 (2013).\n[32] D. E. Sheehy and L. Radzihovsky, Phys. Rev. Lett. 96,\n060401 (2006); Ann. Phys. 322, 1790 (2007).\n[33] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo,\nand R. Walser, Phys. Rev. Lett. 87, 120406 (2001).\n[34] E. Timmermans, K. Furuya, P. W. Milonni, and A. K.\nKerman, Phys. Lett. A 285, 228 (2001).\n[35] Y. Ohashi and A. Griffin, Phys. Rev. Lett. 89, 130402\n(2002).\n[36] R. Duine and H. Stoof, Phys. Rep. 396, 115 (2004).[37] S. -G. Peng, H. Hu, X. -J. Liu, and K. Jiang, Phys. Rev.\nA86, 033601 (2012).\n[38] G. B. Partridge, K.E. Strecker, R. I.Kamar, M. W.Jack,\nand R. G. Hulet, Phys. Rev. Lett. 95, 020404 (2005).\n[39] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S.\nJochim, J. H. Denschlag, and R. Grimm, Science 305,\n1128 (2004).\n[40] Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle,\nPhys. Rev. Lett. 99, 090403 (2007).\n[41] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mo d.\nPhys.80, 1215 (2008).\n[42] Y. Nishida, Phys. Rev. Lett. 109, 240401 (2012).\n[43] Z. Shen, L. Radzihovsky, and V. Gurarie, Phys. Rev.\nLett.109, 245302 (2012).\n[44] E. L. Hazlett, Y. Zhang, R.W. Stites, and K. M. O’Hara,\nPhys. Rev. Lett. 108, 045304 (2012).\n[45] L. Costa, J. Brachmann, A. -C. Voigt, C. Hahn, M.\nTaglieber, T. W. H¨ ansch, and K. Dieckmann, Phys. Rev.\nLett.105, 123201 (2010).\n[46] J. -X. Cui, X. -J. Liu, G. L. Long, and H. Hu, Phys. Rev.\nA.86, 053628 (2012).\n[47] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.\nMod. Phys. 82, 1225 (2010).\n[48] D. F. Agterberg, Physica C 387, 13 (2003); D. F. Agter-\nberg and R. P. Kaur, Phys. Rev. B 75, 064511 (2007).\n[49] L. P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. 87,\n037004 (2001).\n[50] L. D. Landau and E. M. Lifshitz, Quantum Mechanics:\nNon-relativistic Theory (Permagon, New York, 1977).\n[51] V. B. Shenoy, arXiv: 1212. 2858 (2012)." }, { "title": "2008.01182v1.Interfacial_spin_orbit_torques.pdf", "content": "Interfacial spin-orbit torques\nV. P. Amin,1, 2,a)P. M. Haney,2,b)and M. D. Stiles2,c)\n1)Department of Chemistry & Biochemistry, University of Maryland, College Park,\nMD 20742\n2)National Institute of Standards and Technology, Gaithersburg, Maryland 20899,\nUSA\n(Dated: 5 August 2020)\nSpin-orbit torques o\u000ber a promising mechanism for electrically controlling magnetization dynamics in\nnanoscale heterostructures. While spin-orbit torques occur predominately at interfaces, the physical mecha-\nnisms underlying these torques can originate in both the bulk layers and at interfaces. Classifying spin-orbit\ntorques based on the region that they originate in provides clues as to how to optimize the e\u000bect. While\nmost bulk spin-orbit torque contributions are well studied, many of the interfacial contributions allowed\nby symmetry have yet to be fully explored theoretically and experimentally. To facilitate progress, we re-\nview interfacial spin-orbit torques from a semiclassical viewpoint and relate these contributions to recent\nexperimental results. Within the same model, we show the relationship between di\u000berent interface transport\nparameters. For charges and spins \rowing perpendicular to the interface, interfacial spin-orbit coupling both\nmodi\fes the mixing conductance of magnetoelectronic circuit theory and gives rise to spin memory loss. For\nin-plane electric \felds, interfacial spin-orbit coupling gives rise to torques described by spin-orbit \fltering,\nspin swapping and precession. In addition, these same interfacial processes generate spin currents that \row\ninto the non-magnetic layer. For in-plane electric \felds in trilayer structures, the spin currents generated at\nthe interface between one ferromagnetic layer and the non-magnetic spacer layer can propagate through the\nnon-magnetic layer to produce novel torques on the other ferromagnetic layer.\nI. INTRODUCTION\nSpintronic devices can augment modern integrated\ncircuits with novel functionality, as exempli\fed by\nmagnetoresistive random access memories. However,\nwidespread adoption of additional spintronic devices\ndepends on reducing the energy these devices require\nto control their magnetization dynamics via electri-\ncal currents.1,2Most commercial uses and many an-\nticipated applications of spintronic devices are based\non magnetic tunnel junctions because their large\nmagnetoresistance3{5makes it easy to measure their con-\n\fguration. In most cases, the magnetization of one layer\nis \fxed and the magnetization of the other layer is ma-\nnipulated electrically. For manipulating the magnetiza-\ntion direction, all-electrical methods are preferred due to\ntheir compatibility with conventional electronic devices.\nIn most devices, the control current typically \rows across\nthe tunnel junctions along the same path as the read cur-\nrent, see Fig. 1(a). Such devices have challenging fabrica-\ntion margins because the current that \rows through the\ntunnel barrier must be much smaller than the current\nthat can cause breakdown of the barrier.\nAn alternative geometry was proposed about a decade\nago in which the read currents also \row out-of-plane,\nbut in which the control currents \row in-plane through a\nnon-magnetic layer, usually a heavy metal, grown under-\nneath the tunnel junction, see Fig. 1(b). The torques in\na)Electronic mail: vpamin@iu.edu\nb)Electronic mail: paul.haney@nist.gov\nc)Electronic mail: mark.stiles@nist.govthis geometry are called spin-orbit torques because spin-\norbit coupling either in the interior of the layers or at the\ninterfaces between them plays an essential role. By using\nthese torques, such structures reduce the maximum cur-\nrent \row through the barrier and all but eliminate the\nproblem of breakdown, while increasing the design space\nof possible devices.6\nOptimizing the electrical control of magnetization\ncould allow for a variety of new commercial applications\nof magnetic tunnel junctions. For applications in mag-\nnetic random access memory, the alternate geometry has\na disadvantage compared to the original geometry be-\ncause as a three terminal device it takes up more space\non the chip. On the other hand, there are indications\nthat it switches faster. Due to this tradeo\u000b between\nfootprint and speed, the traditional and alternative ge-\nometries may be better suited for di\u000berent applications,\nFIG. 1. Magnetic tunnel junctions (dark red arrows repre-\nsent magnetization direction). (a) Standard magnetic tunnel\njunction with \fxed and free layers and the control current fol-\nlowing the same path as the read current. (b) magnetic tunnel\njunction grown on heavy metal layer with separate read and\ncontrol current paths.arXiv:2008.01182v1 [cond-mat.mes-hall] 3 Aug 20202\nsuch as di\u000berent levels of cache memory.6Another do-\nmain of potential applications is in neuromorphic com-\nputing, where magnetic tunnel junctions can be used as\nlocal memory, superparamagnetic tunnel junctions, and\nspin torque nano-oscillators.7,8Since one of the main\ndriving forces for neuromorphic computing is reducing\nthe energy consumption for di\u000berent cognitive computing\ntasks, reducing the control current by optimizing spin-\norbit torques becomes a key goal for spintronics-based\napproaches.\nHere, we focus on controlling the free layer magneti-\nzation in a magnetic tunnel junction by passing current\nthrough an adjacent non-magnetic metal. We ignore the\n\fxed layer and the tunnel barrier of the magnetic tun-\nnel junction and focus on the magnetic free layer and\nthe adjacent non-magnetic layer, referring to this pair as\na bilayer structure. In addition, we consider a trilayer\nstructure, in which an additional magnetic layer, not\npart of the magnetic tunnel junction, is added below the\nnon-magnetic layer. This trilayer structure, sometimes\ncalled a spin valve, allows for non-zero torques on the\nfree layer magnetization when symmetry requires that\nthese torques be zero in bilayer structures.\nThe interfaces between layers play a fundamental role\nin spin-orbit torques. They break inversion symmetry, as\nis necessary to generate a net torque on the magnetiza-\ntion. In addition, the reduced symmetry at the interface\ncan enhance the role of spin-orbit coupling there, giving\nrise to interfacial coupling between the electric current\nand the spins. The goal of the paper is to provide under-\nstanding of the interfacial contributions to the spin-orbit\ntorques in these bilayer and trilayer structures. Hope-\nfully, this understanding will help lead to a reduction of\nthe energy consumption for a variety of applications.\nSpin-orbit torques have two classes of mechanisms,\nthose due to spin-orbit coupling in the interior of the lay-\ners, called bulk mechanisms, and those due to spin-orbit\ncoupling at the interfaces between layers, called inter-\nfacial mechanisms. The \frst reported observation of a\nspin-orbit torque was an observation of modi\fed damp-\ning in a bilayer composed of a ferromagnet and a heavy\nmetal.9The authors interpreted the mechanism as the\nheavy metal layer generating an out-of-plane spin cur-\nrent under the applied in-plane electric \feld from the spin\nHall e\u000bect.10{12That spin current exerts a spin trans-\nfer torque13{16upon \rowing into the ferromagnetic layer.\nThe mechanism was the motivation for an early exper-\niment demonstrating the excitation of precessional dy-\nnamics through a spin-orbit torque.17\nThe prediction of an interfacial mechanism18for spin-\norbit torques was based on the Rashba-Edelstein e\u000bect.19\nIn this model, the two thin \flms are viewed as a\ntwo-dimensional electron gas. Electrons in this two-\ndimensional gas become spin-polarized under the applied\nin-plane electric \feld; these spin polarized electrons then\nexert torques on the magnetization of the ferromagnetic\nlayer via the exchange interaction. For the \frst obser-\nvation of switching due to spin-orbit torques,20the au-thors invoked this prediction to explain their results. In\nboth the bulk and interfacial mechanisms, the applied\nin-plane electric \feld results in a torque on the magneti-\nzation, but the physical mechanism and the qualitative\nnature of the torque di\u000ber signi\fcantly. For a comprehen-\nsive review of theoretical and experimental progress on\nspin-orbit torques since then, see Ref. 21. In the present\nreview, we focus on a pedagogical description interfacial\ncontributions to spin-orbit torques.\nThere has only been limited research addressing the\nrole of interfacial spin-orbit coupling. Experimentally,\nit is di\u000ecult to distinguish between bulk and interfacial\nmechanisms of spin-orbit torques because there is no dif-\nference in the symmetry of the resulting torques. One can\nonly hope to di\u000berentiate them through indirect measure-\nments like thickness dependence or material variations.\nUnfortunately, doing so through such measurements re-\nquires that other properties of the sample do not change\nas the thickness or materials are varied, which is almost\nnever the case. In addition, as we discuss below, the im-\nportance of multiple length scales can make it di\u000ecult to\ninterpret the experiments.\nFirst principles calculations of spin-orbit torques22{30\nnaturally include the processes that contribute to both\nbulk and interfacial mechanisms. Unfortunately, they are\nnot at a state where they can de\fnitively identify the\norigin of the torques. These calculations are numerically\nintensive, so that few systematic thickness and material\nstudies have been done.23,24,29,30Of those, some but not\nall suggest interfacial contributions. Most experimental\nsystems are quite disordered and disorder is di\u000ecult to\ntreat in \frst principles calculations. Furthermore, the\ntypes of disorder that can be treated do not necessarily\nre\rect the relevant experimental systems. Including on-\nsite disorder29,31allows calculated systems to have the\nhigh resistivities measured experimentally, but it is un-\nclear how e\u000bectively such calculations capture the role\nof structural disorder, including amorphous structures,\npolycrystallinity, and grain boundaries that may be im-\nportant in these systems.\nIn this paper we adopt a semiclassical approach,32{34\nwhich despite of some disadvantages compared to a \frst\nprinciples approach o\u000bers signi\fcant advantages for ped-\nagogy. Semiclassical calculations are based on assump-\ntions that are seldom justi\fed in these systems. They as-\nsume that system sizes and scattering lengths are much\nlarger than the electron wavelengths. However, layer\nthicknesses in experimentally-relevant systems tend to\napproach that length scale. Semiclassical approximations\nleave out quantum interference e\u000bects, though these ef-\nfects have not been observed experimentally in connec-\ntion with spin-orbit torque. An additional drawback of\nsemiclassical approximations is an explosion of parame-\nters that are not all constrained by experiment. On the\nother hand, semiclassical calculations are be easier to in-\nterpret than \frst principles calculations and o\u000ber a clear\nseparation between bulk and interface e\u000bects.\nThe most common semiclassical approach is the drift-3\ndi\u000busion approximation in which the system is described\nin terms of densities and currents. While this approach\nis often considered the most natural way to describe ex-\nperimental results, there are at least two reasons to use\na description based on the Boltzmann equation. The\n\frst is that since the early theories of current-in-plane\ngiant magnetoresistance35it is known that the appropri-\nate length scale for in-plane transport is the mean free\npath rather than the spin di\u000busion length. Variations\non the length scale of the mean free path are captured\nby the Boltzmann equation but not drift di\u000busion ap-\nproaches. More importantly, since spin-orbit coupling\ncouples the electron spin to its motion, a wave-vector-\ndependent approach is needed to capture its e\u000bects. In\nthis work, we start with simple model described by the\nBoltzmann equation and show how that model connects\nto the parameters that might enter a description based\non the drift-di\u000busion equation.\nGiven the experimental and theoretical di\u000eculties in\ndistinguishing bulk and interfacial mechanisms for spin-\norbit torques, what is the rationale for studying inter-\nfacial mechanisms? The main reason is to develop a\nclear picture of what system properties lead to opti-\nmal behavior. For example, an analysis of interfacial\nspin-orbit torques could help determine whether to min-\nimize or maximize interfacial spin-orbit coupling or to\nminimize or maximize the bulk spin di\u000busion length.\nAnother important reason to study interfacial mecha-\nnisms lies in recent experiments on trilayer structures\ndriven by in-plane currents. In these systems, spin cur-\nrents generated at the interfaces and/or the ferromag-\nnetic layers enable additional functionality compared to\nbilayers, such as \feld-free switching of perpendicularly-\nmagnetized layers.36Determining exactly what drives\nmagnetization dynamics in these systems will o\u000ber new\ninsights into the nature of spin-orbit torque. Both\ntheory27,37,38and experiment36,39,40suggest that bulk\nand interfacial mechanisms could play a role in these\nsystems, but here the bulk mechanisms originate in the\nferromagnetic layers rather than a heavy metal. Thus,\ndisentangling bulk and interfacial contributions remains\nan important challenge, even as new device geometries\nare explored.\nThe goal of this paper is to provide a pedagogical ex-\nplanation of interfacial contributions to spin-orbit torque,\nusing a semiclassical approach. In Section II, we give\nbackground for subsequent discussions. This background\nincludes a discussion of the \row of angular momen-\ntum between reservoirs (Sec. II A), the role that inter-\nfaces play in perpendicular transport (Sec. II B) and in-\nplane transport both for bilayers (Sec. II C) and trilayers\n(Sec. II D), the distinctions between extrinsic and intrin-\nsic mechanisms (Sec. II E), the angular dependence of\nthe torques that are allowed by symmetry (Sec. II F),\nand complications associated with distinguishing bulk\nand interface contributions from the thickness depen-\ndence (Sec. II G). With that background, in Sec. III, we\nuse a highly simpli\fed model to describe the di\u000berentmechanisms that can generate interfacial contributions\nto spin-orbit torques.\nII. BACKGROUND\nA. Angular Momentum\nTracking the \row of angular momentum in the system\nprovides a useful framework for understanding spin-orbit\ntorques. The total angular momentum of the system in-\ncludes contributions from the ions comprising the lat-\ntice and the electrons, which possess an orbital angular\nmomentum and an intrinsic angular momentum derived\nfrom their spin degree of freedom. It is useful to further\npartition the electrons' spin angular momentum into a\ncomponent from the magnetic order parameter ( e.g., the\nelectrons comprising the magnetization condensate) and\na component from non-equilibrium states participating in\ntransport. Each of these components represent a reser-\nvoir of angular momentum, and our interest is in tracking\nthe \row of angular momentum from these reservoirs to\nthe magnetization upon the application of an electric \feld\nas shown in Fig. 2.\nThe transfer of angular momentum between reservoirs\nis mediated by interactions, which are described in the\nFIG. 2. Schematic of di\u000berent angular momentum reservoirs\nand the interactions coupling them. In ferromagnetic metals,\nthe net magnetization is the sum of the magnetic moments of\nelectrons carrying both orbital and spin angular momentum,\nwith the latter dominating in transition metal ferromagnets.\nThe magnetic exchange potential couples the spin angular\nmomentum of the magnetization to the spin angular momen-\ntum of the carriers. The spin-orbit interaction couples the\nspin angular momentum of the carriers to their orbital angu-\nlar momentum. The crystal \feld potential couples the orbital\nangular momentum of carriers to the angular momentum of\nthe atomic lattice. Spin-orbit torques arise when an applied\nelectric \feld promotes angular momentum transfer from the\natomic lattice to the magnetization using carriers as media-\ntors for the transfer.4\nHamiltonian for the electrons:\nH=~2r2\n2me+V0(r) + \u0001(r) (^m\u0001^\u001b) +Vso(L\u0001^\u001b):(1)\nThe \frst term is the kinetic energy. The second term\nV0(r) is the crystal \feld potential, which breaks rota-\ntional symmetry for the electrons, so that the angular\nmomentum of electrons is not conserved. This term en-\nables the \row of angular momentum from the electronic\nsystem to the lattice. Note that the Hamiltonian for the\nwhole system, including that of the lattice, is rotationally\ninvariant, so that total angular momentum is conserved.\nThe third term is the exchange interaction between elec-\ntron spin\u001band the magnetization, which is oriented in\nthe^mdirection. Its magnitude \u0001( r) is position depen-\ndent, and can be determined self-consistently in a mean-\n\feld theory approach, or taken as a constant in simpler\nmodels, such as the Stoner model. The fourth term is the\nspin-orbit coupling, where we only include contributions\nfrom the onsite, atomic-like form L\u0001\u001b, and parameter-\nize its strength with \u000b. This is the dominant source of\nspin-orbit coupling in most materials, owing to the rapid\norbital motion (compared to linear motion) of electrons\nand the strong electric \felds near the nucleus.\nThe degrees of freedom in Eq. 1 represent the di\u000ber-\nent reservoirs of angular momentum, while the coupling\nbetween degrees of freedom mediate the transfer of an-\ngular momentum between reservoirs, as shown schemat-\nically in Fig. 2. Spin transfer torque, which we discuss\nin the following section, is a transfer of angular momen-\ntum between the magnetization and the electron spin of\ncurrent-carrying electrons. In systems with strong spin-\norbit coupling, the magnetization is also coupled to the\norbital angular momentum of the electrons and to the\nlattice, opening up a wider array of mechanisms for ex-\nerting torques on the magnetization. This framework of\ntracking angular momentum \row is quite general and de-\nscribed in more details in Refs. 41 and 42.\nB. Perpendicular Transport\nThe study of perpendicular transport in mag-\nnetic multilayers (see Fig. 3) began with measure-\nments of the current-perpendicular-to-the-plane gi-\nant magnetoresistance.43,44Following that, two inter-\ntwined phenomena dominated the \feld, spin transfer\ntorques13,14,45{47and tunneling magnetoresistance.3{5,48\nIn all of these, interfaces play a crucial role. For\ngiant magnetoresistance, spin-dependent scattering at\nthe interface leads to a spin-dependent interface\nresistance,44,49{51which can dominate the resistance for\nthin enough layers. This same spin-dependent scattering\nleads to a spin-transfer torque.\nIn magnetic multilayers or tunnel junctions, spin trans-\nfer torques are the torques on the magnetizations exerted\nby the spins of non-equilibrium, current-carrying elec-\ntrons for currents \rowing perpendicular to the plane of\nFIG. 3. Magnetic trilayer and perpendicular transport. The\ntop and bottom ferromagnetic layers are separated by a non-\nmagnetic layer. In each layer, the charge current \rows along\nthe electric \feld, where \row directions are given as block ar-\nrows. In each ferromagnetic layer, the spin current \rows along\nthe charge current with spins aligned with the magnetization\n(red arrows). For spin currents, block arrows indicate elec-\ntron \row direction and blue arrows indicate spin direction.\nEquivalently, block arrows could also indicate charge \row di-\nrection with blue arrows indicating magnetic moment. In the\nnon-magnetic layer, the spins in the spin current are a combi-\nnation of spins aligned with the lower layer magnetization and\nanti-aligned with the upper layer magnetization (here given\nby^x\u0000^z). In the absence of spin-orbit coupling, the spin\ncurrent with spin direction longitudinal to the magnetization\nis conserved across the interfaces. Note that spin currents are\nunchanged by \ripping both the \row and spin directions. The\ndiscontinuity in the spin current at the interfaces, given by\nthe spin direction transverse to the magnetization, is the spin\ntransfer torque, indicated for the top and bottom layers by\nthe green arrows. Typically, one layer will be able to respond\nto the torques and the other layer will be essentially \fxed\nthough one of several mechanisms.\nthe layers. These torques are generically present when an\nelectric current is applied to a system where the magne-\ntization is oriented di\u000berently in di\u000berent layers. When\nelectrons with spin-polarization aligned with one mag-\nnetic layer interact with a subsequent magnetic layer,\ntwo processes contribute to the torque.16,52The \frst is\nthat the electron spins precess around the magnetiza-\ntion at the interface and exert a reaction torque on the\nmagnetization. The second is that the spin current that\npropagates into the ferromagnetic layer rapidly dephases\nand becomes aligned with the magnetization. These pro-\ncesses are discussed in more detail in Sec. III.\nThe physics of spin transfer torque is most easily un-\nderstood in the limit where spin-orbit coupling is small\ncompared to the magnetic exchange energy. In this case,\nan equation of continuity for total spin (magnetization\nplus conduction electron spin) relates the torque on a\nvolume of magnetization to the net \rux of transverse5\nspin current into the volume. Magnetoelectronic circuit\ntheory53,54provides a description of perpendicular trans-\nport when interfacial spin-orbit coupling is weak.\nThe \frst indication of the importance of interfacial\nspin-orbit coupling was the determination that some\ncurrent-perpendicular-to-the-plane giant magnetoresis-\ntance measurements could not be adequately \ft unless\nthey allowed for \fnite spin relaxation at interfaces55{57\nrather than simply spread out through the layers. Re-\ncent \frst-principles calculations58,59support this phe-\nnomenology. This is most dramatically illustrated\nby spin memory loss at interfaces between to normal\nmetals.58,60,61Since inversion symmetry is broken at such\ninterfaces, special forms of spin-orbit coupling are al-\nlowed. These cause wave-vector-dependent precession in\nthe spin-orbit e\u000bective \feld and a reduction of spin cur-\nrent crossing the interface.\nC. In-plane Transport in Bilayers\nFor bilayer systems composed of nonmagnetic and fer-\nromagnetic layers, in-plane transport leads to torques\non the magnetization from several distinct sources. Al-\nthough the bilayer geometry is simpler than that of trilay-\ners, the materials are chosen to utilize spin-orbit coupling\nfor generating torques. This enlarges the set of reservoirs\nand interactions which contribute to the torque, so that\nidentifying the di\u000berent sources of torque is a more dif-\n\fcult task. In this section we review the mechanisms\nof spin-orbit torque in this geometry. We \frst brie\ry de-\nscribe the spin Hall and orbital Hall contributions, which\narise from transporting angular momentum from the non-\nmagnetic layer to the ferromagnet. We then discuss the\nrecently discovered anomalous torque, and conclude with\na longer discussion on interfacial torques.\nThe spin Hall e\u000bect plus spin transfer torque mech-\nanism was proposed to explain one of the early exper-\niments on spin-orbit torques.17For many of the sys-\ntems studied to date, this mechanism is considered to\nprovide the primary contribution to the dampinglike\ntorque. It is based on the spin Hall e\u000bect in the non-\nmagnetic layer, which results in a spin current which\n\rows in all directions perpendicular to the electric \feld,\nwith the spin directions perpendicular to both the spin\n\row and electric \feld directions. This e\u000bect was \frst\npredicted by D'yakonov and Perel10using a semiclas-\nsical approach and later explained using several other\nmechanisms,11,12,62,63eventually resulting in a mostly\nuni\fed picture.64We refer interested readers to more in-\ndepth reviews on the spin Hall e\u000bect.64{68The spin cur-\nrent generated in the nonmagnetic layer is injected into\nthe ferromagnet. If the spin-orbit coupling at the inter-\nface and in the ferromagnet is much smaller than the\nexchange splitting, then the torque on the magnetization\nequals the incoming spin current due to the spin transfer\ntorque mechanism.\nThe orbital Hall e\u000bect plus spin transfer torque is a\nFIG. 4. Magnetic bilayer and in-plane transport. In both\nthe top ferromagnetic layer and bottom non-magnetic layer,\ncharge currents \row along the electric \feld, where \row direc-\ntions are given as block arrows. In the ferromagnetic layer,\nthe spin current \rows along the charge current with spins\naligned with the magnetization (red arrow), where for spin\ncurrents block arrows give \row direction and blue arrows give\nspin direction. Green arrows indicate the two components of\nthe torque on the magnetization. In the bottom nonmagnetic\nlayer, the spin Hall e\u000bect generates a spin current with \row\nalong ^zand spin direction along ^y. In the absence of spin-\norbit coupling at the interface, the discontinuity of the spin\nHall current across the interface gives the interfacial contri-\nbution to spin-orbit torque on the magnetization. However,\nwith nonvanishing interfacial spin-orbit coupling, the Rashba-\nEdelstein e\u000bect generates a spin accumulation at the interface\nthat exerts an exchange torque on the magnetization. As will\nbe discussed throughout this review article, additional torques\narise from spin-orbit scattering at the interface (not shown\nhere), possibly contributing to torques measured in experi-\nments.\nmore recently proposed mechanism of spin-orbit torque.\nIn this case, the applied electric \feld induces orbital an-\ngular momentum \row in the nonmagnet, with similar\nsymmetry properties to the spin Hall e\u000bect: the \row\ndirection is perpendicular to the applied electric \feld,\nand the angular momentum direction is perpendicular\nto the \feld and \row directions.69{73This orbital angu-\nlar momentum is injected into the adjacent ferromagnet,\nwhere spin-orbit coupling in the ferromagnet transduces\nthe orbital current to a spin accumulation, which exerts\na torque on the magnetization.42,74Experimentally dis-\ntinguishing orbital Hall from spin Hall contributions is\nchallenging, and is discussed in Ref. 42. Note that or-\nbital angular momentum also plays a crucial role in the\nspin Hall e\u000bect. The electric \feld does not couple directly\nto the electrons' spins but rather couples to their orbital\nmoments. These in turn couple to the spins through\nspin-orbit coupling.\nThe anomalous torque is an e\u000bect in which the appli-\ncation of an electric \feld to a single ferromagnetic layer\nleads to torques in the ferromagnet where inversion sym-6\nmetry is broken, for example, at interfaces. Recent ex-\nperimental work has con\frmed that single layer ferro-\nmagnets experience spin torques at their layer bound-\naries under applied electric \felds.75,76These anomalous\ntorques may arise from spin currents generated in the\nbulk with spin direction transverse to the magnetization.\nTheoretical studies38show that such spin currents, which\nare allowed by symmetry and not subject to dephasing,\nare comparable in strength to spin Hall currents in Pt.\nWhen these spin currents \row to the layer boundaries,\nwhere inversion symmetry is broken, they can exert spin\ntransfer torques. The same e\u000bect was studied in a di\u000ber-\nent context in Ref. 77, which proposed dampinglike spin-\norbit torques in ferromagnets with broken bulk inversion\nsymmetry, and experimentally observed these torques in\na strained ferromagnetic semiconductor, GaMnAs.\nIn addition to these spin-orbit-torque mechanisms in\nwhich angular momentum is supplied from the interior\nof the layers, there are also contributions in which the\nangular momentum is supplied at the interfaces between\nlayers. These interfacial contributions to the spin-orbit\ntorque are the focus of this review. In the initial model\nfor spin-orbit torque18, the interface plays a direct role\nin generating a magnetic torque through interfacial spin-\norbit coupling. It's useful to study interfacial contribu-\ntions by examining the Rashba model, which provides\na minimal description of spin-orbit coupling in systems\nwith broken structural inversion symmetry. For broken\ninversion symmetry along the z-direction, the Rashba\ninteraction is given by \u001b\u0001(k\u0002z). For nonmagnetic sys-\ntems, the Rashba interaction lifts the spin-degeneracy of\nstates with nonzero Bloch wave vector k. Electron states\nare still doubly degenerate (Kramer's doublet) but now\nthe two degenerate states exist at kand\u0000k, with time\nreversal symmetry ensuring s(k) =\u0000s(\u0000k). This de-\ngeneracy implies that the net spin density of the Rashba\nmodel without ferromagnetism or a magnetic \feld van-\nishes in equilibrium. However, under an applied elec-\ntric \feld, the nonequilibrium occupation of carriers with\nwavevectors\u0006kdi\u000bers in general, so a net nonequilib-\nrium spin density (or spin accumulation) forms at the\ninterface, as shown in Fig. 6(b).\nIn ferromagnetic systems that lack structural inversion\nsymmetry, a spin accumulation still forms in response to\nan in-plane electric \feld and this spin accumulation ex-\nerts an exchange torque on the magnetization. For bro-\nken inversion symmetry along the ^z-direction, the mini-\nmal Hamiltonian is\nH=~2r2\n2me+V0+ \u0001 ( ^m\u0001^\u001b) +\u000bR\u001b\u0001(k\u0002z):(2)\nThis Hamiltonian di\u000bers from that in Eq. 1 in that the\ncrystal \feld potential and the atomic spin-orbit coupling\nhave been replaced by the Rashba form of spin-orbit cou-\npling, which is wave-vector dependent. This transforma-\ntion is based on the assumption that the wave vector\nperpendicular to the symmetry-breaking direction ^zis\na good quantum number and that all of the e\u000bects ofthe crystal-\feld potential can be absorbed into V0,\u000bR,\nand possibly an e\u000bective mass (see Ref. 78). In a multi-\nlayer, these parameters vary from layer to layer and \u000bR\nbecomes large at interfaces where symmetry breaking is\nstrongest.\nSeveral studies22,79have addressed the relevance of\nsimpli\fed models, like the Rashba model, using density\nfunctional theory to describe bilayers in a slab geometry,\nin which several atomic layers are included away from the\ninterfaces. These calculations22,79show that the Rashba\nspin-orbit interaction, as measured by the misalignment\nbetween the spin and the local exchange \feld, is highly\nlocalized and dominant on the interfacial atoms. These\nresults provide a motivation for the models of interfacial\nRashba spin-orbit coupling described below in Sec. III.\nWhile Rashba spin-orbit coupling is highly localized on\nthe interfacial atoms, electronic transport is not con\fned\nto the interface, as assumed in the typical description\nof the Rashba-Edelstein e\u000bect. Although experiments\nand \frst principles calculations have con\frmed that lo-\ncalized interface electronic states form at various mate-\nrial interfaces80{83, the remaining electronic states in the\nbilayer are not con\fned to the interface plane. There-\nfore, it is useful to extend the Rashba-Edelstein model\nfrom two dimensions to three dimensions by treating the\nspin-dependent scattering of electrons o\u000b a localized in-\nterfacial potential32{34. Section III discusses such a cal-\nculation in detail.\nTo motivate the more complete discussion in Sec. III,\nwe start with a simple extension of the Rashba model\nfrom two to three dimensions27,36. In this model, we\nomit an interfacial exchange interaction and assume the\ncrystal \feld potential and Rashba potential are localized\nat the interface ( z= 0):\nH=~2r2\n2me+t\u000e(z)\u0002\nV0+\u000bR\u001b\u0001(k\u0002z)\u0003\n(3)\nHerezis the out-of-plane direction and tis the rele-\nvant interfacial length scale. In what follows, we discuss\nwhat happens when free electrons from the bulk layers\nscatter o\u000b the interface, modeled by the delta function\npotential given in Eq. 3. Even under in-plane electric\n\felds, carrier motion is largely isotropic, so the electron\ndistribution functions within an average elastic scatter-\ning length (mean free path) of the interface are modi\fed\nby interfacial scattering despite the formation of net in-\nplane currents in the bulk layers. In response to in-plane\nelectric \felds, the interfacial scattering leads to out-of-\nplane spin currents that can exert spin torques on the\nferromagnetic layer, as depicted in Fig. 6(c)-(f).\nUnpolarized free electrons from the bulk layers be-\ncome spin polarized (for nonvanishing V0and\u000bR) af-\nter scattering o\u000b the interface. This \fltering e\u000bect oc-\ncurs because the Rashba potential acts as a spin- and\nmomentum-dependent potential barrier. In particular,\nthe Rashba potential preferentially re\rects or transmits\nelectrons based on their spin, so an unpolarized stream\nof electrons becomes spin-polarized after scattering, as7\nFIG. 5. Real and reciprocal space depictions of the Rashba-Edelstein e\u000bect. Carriers are restricted to an idealized two-\ndimensional interface. (a) The carriers feel an e\u000bective magnetic \feld along u(k) =k\u0002^z(green arrow) due to spin-orbit\ncoupling. (b) The band structure obtained from the Rashba Hamiltonian in Eq. 2 for vanishing exchange interaction (\u0001 = 0).\nThe spin expectation values (arrows) are shown at the Fermi energy EF, whereEF>E(k=0). (c) The Fermi surface forms\ntwo circular sheets distinguished by their spin expectation values being parallel (outer circle) or antiparallel (inner circle) to\nu(k). An electric \feld biases carrier occupations, where blue arrows indicate increased occupation and red decreased, leading\nto a net spin polarization along E\u0002^z.\nseen in Fig. 6(c). However, in equilibrium, the net spin\ncurrent vanishes after summing over all k-states, much\nlike the vanishing equilibrium spin density under the con-\nventional Rashba-Edelstein e\u000bect. However, the Rashba\npotential also depends on the momentum of incident elec-\ntrons. In the presence of an in-plane, applied electric \feld\nE, the occupation of carriers becomes anisotropic, so the\nnet spin current carried by the scattered electrons does\nnotvanish after summation over all k-states (Fig. 6(d)).\nThe spin current carried by the scattered electrons \rows\nout-of-plane with net spin direction along ^z\u0002E. Fol-\nlowing Ref. 27, we refer to this mechanism of generating\nspin currents as spin-orbit \fltering , because the Rashba\nspin-orbit potential \flters the unpolarized, incident spins\nas they scatter o\u000b the interface, yielding an out-of-plane\nspin current. Note that while the spin Hall e\u000bect occurs\nin bulk materials and spin-orbit \fltering occurs only at\ninterfaces, both e\u000bects can generate spin currents with\nthe same spin and \row orientation, making them di\u000e-\ncult to distinguish in experiments. Unlike the spin Hall\ne\u000bect, which mostly depends on material properties from\nthe single originating layer, spin-orbit \fltering depends\nstrongly on the momentum relaxation times and elec-\ntronic structure of the two adjacent layers and requires\ninversion symmetry to be broken by the interface.\nIf one of the layers is ferromagnetic, there is another\ninterfacial mechanism that generates spin currents. As-\nsume that in one layer the in-plane charge current is spin-\npolarized along p. This occurs in ferromagnetic layers,\nwhereppoints along the magnetization ^m. In this case,\nthe spin polarized carriers will rotate about the spin-\norbit \feld while scattering o\u000b the interface, as seen in\nFig. 6(e). This phenomenon occurs in addition to the\n\fltering e\u000bect described above. After summing over all\nk-states, the net out-of-plane spin current has a compo-\nnent with the spin direction along p\u0002(^z\u0002E) (Fig. 6(f)).\nWe refer to this phenomena as spin-orbit precession , be-cause it describes spins precessing about the spin-orbit\n\feld while they scatter o\u000b the interface. The spin swap-\nping e\u000bect, \frst predicted in Ref.84, has a similar phe-\nnomenogical form to spin-orbit precession when it occurs\nnear interfaces85,86. Like the spin Hall e\u000bect and spin-\norbit \fltering e\u000bects, spin swapping and spin-orbit pre-\ncession di\u000ber in that the latter depends more intimately\non the relaxation times and electronic structure of both\nmaterial layers. Another key di\u000berence is that the \row\nand spin orientations described by spin swapping repre-\nsent a subset of those allowed by the spin-orbit precession\nmechanism, as discussed in section III.\nThe spin-orbit \fltering and precession e\u000bects discussed\nhere represent the simplest generalization of the Rashba-\nEdelstein model18that sparked this \feld of study. How-\never, several other important spin-orbit torque mecha-\nnisms have been predicted. For instance, theory pre-\ndicts spin-orbit torques that are directly generated at\ninterfaces77that share a common origin with the spin\nHall e\u000bect but are not caused by that e\u000bect. The spin\nHall e\u000bect arises due to both intrinsic and extrinsic mech-\nanisms, which we discuss below. The intrinsic mecha-\nnism can be interpreted as capturing the perturbation\nof the electronic wavefunctions under an applied electric\n\feld, creating nonequilibrium electronic states that carry\nspin currents. The same perturbation to electronic wave-\nfunctions occurs for carriers in regions that break inver-\nsion symmetry (like at interfaces), yielding the additional\ntorques that were proposed in Ref. 77.\nD. In-plane Transport in Trilayers\nTrilayers have a more complex geometry than bilay-\ners, and therefore have more degrees of freedom and\nexperimentally-controllable (and uncontrollable) param-8\nFIG. 6. Real and reciprocal space depictions of the role of\ninterfacial spin-orbit coupling. Green arrows indicate the ef-\nfective magnetic \feld along u(k) =k\u0002^zdue to spin-orbit\ncoupling. In panels (a) and (c), red and blue arrows indicate\nspin directions. In panels (b) and (d), blue arrows indicate\nincreased occupation due to the in-plane electric \feld and\nred decreased. (a) Unpolarized carriers scattering from the\ninterfacial spin-orbit \feld become spin-polarized (spin-orbit\n\fltering) because the \feld creates a spin-dependent poten-\ntial barrier. (b) Spin polarization after transmission through\nthe interface for unpolarized incoming carriers on a circular\nslice (constant kz) of one sheet of the Fermi surface. The\nnon-equilibrium occupation due to the electric \feld leads to\na net \row of transmitted electrons along ^zwith a net spin\npolarization along ^z\u0002E. (c) In ferromagnetic layers, car-\nriers are spin-polarized along the magnetization. Scattering\nfrom the interface, these spins precess around u(k) (spin-orbit\nprecession). (d) In-plane spin polarization after transmission\nthrough the interface for incoming carriers polarized along ^z\non a circular slice (constant kz) of one sheet of the Fermi sur-\nface. The non-equilibrium occupation due to the electric \feld\nleads to the transmitted spins carrying a net spin \row along\n^zwith net spin polarization along ^m\u0002(^z\u0002E).\neters. For instance, a spin valve is a trilayer system\nwhere the magnetization directions of each ferromagnetic\nlayer can be di\u000berent. Based on symmetry considerations\nalone, the magnetization dependencies of the torques in\ntrilayers are more complex than in bilayers. This addi-\ntional complexity occurs in part because spin currents\noriginating in one ferromagnetic layer or at the adja-\ncent interface can \row through the spacer layer and ex-\nert torques on the other ferromagnetic layer. By vary-\ning each ferromagnetic layer's magnetization direction\nand/or selectively inserting additional layers, one can ob-\ntain information about spin-orbit torques not available in\nbilayers that could help parse spin-orbit torque mecha-\nnisms. As a result, trilayers present unique structures to\nFIG. 7. Magnetic trilayer and in-plane transport. The top\nand bottom ferromagnetic layers are separated by a non-\nmagnetic layer. In each layer, the current \rows along the\nelectric \feld, where \row directions are given as block arrows.\nIn the ferromagnetic layers, the spin currents shown here \row\nalong the charge currents with spins aligned with the mag-\nnetization (indicated by red arrows). Note that for spin cur-\nrents, block arrows give \row direction and blue arrows give\nspin direction. Green arrows indicate the two components of\nthe torque on the magnetization. In the non-magnetic layer,\nspin currents originating in the lower ferromagnetic layers and\n\rowing out-of-plane ( ^z) have spin directions along both ^yand\n^z(other contributions to the spin current are not shown). The\nspin currents with y-spin direction can arise from the spin Hall\ne\u000bect in any layer or through the spin-orbit \fltering e\u000bect at\ninterfaces. The spin currents with z-spin direction are not al-\nlowed by symmetry in bulk nonmagnets; these spin currents\ncan only arise in the ferromagnetic layers through various pro-\ncesses or at the interfaces through the spin-orbit precession\ne\u000bect. Spin transfer torques arising from the spin currents\nwithz-spin direction could switch ferromagnetic layers with\nperpendicular magnetic anisotropy, suggesting applications of\npossible technological interest.\nfurther investigate the spin-orbit torques \frst proposed\nin bilayers.\nMeasurements of spin-orbit torque in trilayers are\nnearly as old as measurements in bilayers. Trilayers were\n\frst investigated as means to suppress interfacial spin-\norbit torque contributions.87{89In these experiments, a\nlight spacer material (typically Cu) with a spin di\u000busion\nlength far exceeding layer thicknesses was placed in be-\ntween the heavy metal and the ferromagnet, resulting in\na trilayer. Since the heavy metal is no longer adjacent to\nthe ferromagnet, no interface exists in the trilayer that\nhas both strong spin-orbit coupling and ferromagnetic\nexchange. The lack of such an interface was thought\nto suppress interfacial spin-orbit torques, such that all\ntorques could be attributed to spin current generation in\nthe heavy metal.\nTwo problems exist with this interpretation. First,\nthose trilayers still contain a nomagnetic interface with9\nstrong spin-orbit coupling (formed by the heavy metal\nand the spacer layer). As previously discussed, under an\nin-plane electric \feld, theory predicts that these inter-\nfaces can generate spin currents of comparable strength\nto spin Hall currents in Pt. If such interface-generated\nspin currents occur in the system, the measured torque\nis no longer solely due to the spin Hall e\u000bect in the\nheavy metal. Second, despite lower spin-orbit coupling\nstrength at the interface between the ferromagnet and\nspacer layer, both interfacial and anomalous spin-orbit\ntorques could still contribute to the measured torques.\nThe trilayers discussed so far have a single ferromag-\nnetic layer. Investigations of trilayers with two ferro-\nmagnetic layers and a nonmagnetic spacer layer (spin\nvalves) have expanded the reach of spin-orbit torque\nmeasurements36,39. In these experiments, the spin cur-\nrents generated in ferromagnets or at adjacent interfaces\ncan be measured through their e\u000bect on the other ferro-\nmagnetic layer, creating an independent measurement of\nspin-current driven torques. In the following, we discuss\nthe rami\fcations of spin currents generated both at in-\nterfaces and in bulk ferromagnetic layers on spin torques\nin trilayers.\nInterfacial spin current generation in trilayers |The\nexperimental results reported in Ref. 36 demonstrated\nthat if one ferromagnetic layer has an out-of-plane mag-\nnetization, current-induced torques could switch that\nmagnetization without external magnetic \felds if the\nother ferromagnetic layer's magnetization was in-plane.\nThis behavior is allowed by symmetry, and one possi-\nble mechanism explaining these results involves interface-\ngenerated spin currents. The spin-orbit precession cur-\nrent generated at the interface between the in-plane mag-\nnetized layer and the spacer layer has spin direction\n^mIP\u0002(^z\u0002E), where ^mIPis the in-plane magnetization\ndirection. If the electric \feld and in-plane magnetization\nare parallel, the resulting out-of-plane spin current has an\nout-of-plane spin direction. This spin current can then\n\row through the spacer layer into the out-of-plane ferro-\nmagnetic layer and exert a spin transfer torque with the\nright orientation to enable switching. The authors pre-\nsented evidence of spin currents with out-of-plane spin\ndirection in the form of this \feld-free switching, and\nthrough current-induced shifting of the hysteresis loops\nof the out-of-plane layer.\nRecent experiments have expanded these \fndings by\nconsidering di\u000berent magnetization con\fgurations and\nusing other experimental techniques. Hibino et al.40in-\nvestigated spin-orbit torques in Py/Pt/Co trilayers using\nharmonic Hall analysis. They found two distinct damp-\ninglike torques through the angular dependence of the\nharmonic Hall signal, which damped the magnetization\ntowards thepand^z\u0002pdirections, where p=^z\u0002E.\nSpin transfer torques originating from the spin Hall e\u000bect\ndamp the magnetization towards pand can incite magne-\ntization precession about pthrough spin-dependent scat-\ntering at the interface (parameterized by the spin mixing\nconductance). However, torques that damp the magneti-zation towards the ^z\u0002pcannot be explained by the spin\nHall e\u000bect since its spin direction (which points along\np) is tightly constrained by symmetry. The authors at-\ntributed the unconventional dampinglike torque to the\nspin-orbit precession e\u000bect at the Pt/Co interface and\nextracted an associated spin torque e\u000eciency in reason-\nable agreement with \frst principles calculations.27The\nauthors further showed that the spin torque strength de-\npended greatly on the material composition of the inter-\nfaces, further suggesting an interfacial origin. In another\nwork,90Hibino et al. \fnd further experimental evidence\nof spin currents with both pand^z\u0002pspin directions\ngenerated in FeB/Cu/CoNi multilayers, this time using\nspin torque ferromagnetic resonance techniques to mea-\nsure the angular dependence of the associated torques.\nBulk ferromagnetic spin current generation in trilay-\ners|Many experiments have measured spin currents\noriginating in ferromagnetic layers with a magnetization-\naligned spin direction.91{94Since charge currents in\nferromagnets are spin-polarized, both the planar and\nanomalous Hall e\u000bects are expected to generate spin-\npolarized currents with spin directions aligned with the\nmagnetization37,95, which could explain some of these re-\ncent experiments. Other experiments measured contri-\nbutions from both transverse and magnetization-aligned\nspin directions in Py,96{98further supporting the claim\nthat ferromagnets are robust generators of spin current.\nWhile magnetization-aligned spin currents cannot exert\nspin torques in single layer ferromagnets, they can exert\ntorques on other ferromagnetic layers within trilayers, as\nlong as the magnetization direction of the other ferro-\nmagnetic layer is noncollinear to the magnetization of\nthe generating layer.\nE. Extrinsic vs Intrinsic E\u000bects\nThe electrical control of magnetization through spin-\norbit torques can be understood by examining the re-\nsponse of the system, Eq. 1, to an applied electric \feld.\nHeavy metal-ferromagnet thin \flm bilayers typically op-\nerate in the linear response regime. In this case, the\nelectric \feld impacts the system in two ways: \frst by\nchanging the electrons' distribution function, and second\nby changing the electrons' wave functions. Often, each of\nthese two aspects of the electric \feld perturbation result\nin the same observable (e.g. anomalous Hall current).\nThe pre\fx \\extrinsic\" or \\intrinsic\" indicates the physi-\ncal mechanism under consideration ( e.g., extrinsic versus\nintrinsic anomalous Hall current). While there is not a\nuniversally agreed upon usage of intrinsic and extrinsic,\nwe \fnd it useful to use \\extrinsic\" to describe the con-\ntributions where the perturbing electric \feld changes the\noccupation of the states and \\intrinsic\" to describe the\ncontributions where the perturbing electric \feld changes\nthe states themselves. This distinction is straightforward\nto make in calculations based on the Kubo formula but\nmay not be so straightforward in other approaches.10\nWhen the electric \feld perturbs the distribution\nfunction, the nonequilibrium distribution function can\nbe obtained by solving the Boltzmann equation, and\nhas so far been studied in the relaxation time\napproximation.18,23,32,99Scattering plays a central role\nin determining the nonequilibrium distribution function\nand all subsequent observables (e.g. charge and spin cur-\nrent, magnetization torques). The magnitude of these\ne\u000bects typically scale linearly with the scattering time \u001c,\nin the limit where ~=\u001c < \" , where\"is an energy scale\nthat is characteristic of the typical band splitting near\nthe Fermi energy. Interestingly, this scaling implies that\nwhen scattering is very weak ( e.g.,\u001cis very large) the\nscattering-based contribution to the spin Hall conductiv-\nity, for example, dominates over other contributions.100A\nmore common regime for transition metals is the clean to\ndirty metal limit, in which the intrinsic mechanism dom-\ninates. In this case, the intrinsic response is independent\nof\u001cfor~=\u001c <\" ,70and varies as \u001cmfor~=\u001c <\" , where\nm= 2 for simple models of scattering,70but whose spe-\nci\fc value generally depends on the observable and the\nmicroscopic model.101\nThe extrinsic and intrinsic contributions have been\nstudied extensively for the two-dimensional Rashba\nmodel, Eq. 2. The extrinsic response was analyzed in\nRefs. 102 and 103. The application of an electric \feld\nperturbs the distribution function, introducing asymme-\ntry in the occupation of states with Bloch wave vector\nalongE. This naturally leads to a spin accumulation\naligned in the E\u0002zdirection, as illustrated in Fig. 6(b).\nIn the remainder of this review, we focus on extrinsic\ncontributions to the interfacial spin-orbit torque.\nF. Symmetry\nIn this section, we demonstrate how symmetry con-\nstrains spin-orbit torques in various material systems fol-\nlowing earlier discussions.29,104We \frst show that in\nnonmagnet/ferromagnet bilayers, symmetry forces the\ntorque to vanish along the axis ^z\u0002E. Next, we de-\nrive the general form of the response tensor that relates\nthe torque and the electric \feld as a function of magne-\ntization direction, \frst considering only continuous sym-\nmetries and later including discrete crystal symmetries.\nFinally, we discuss how unique crystal symmetries a\u000bect\nspin-orbit torque and lead to novel phenomena.\nIgnoring crystal structure, only two types of spatial\nsymmetries exist in a nonmagnetic bilayer: 1) contin-\nuous rotational symmetry about ^zand 2) mirror-plane\nsymmetry with respect to planes whose normal vector ^n\nlies within the interface plane. Fig. 8a illustrates these\nsymmetries, where \u001erdenotes the angle of rotation about\n^zand\u001empdenotes the angle of the mirror-plane normal\nvector ^n. As we show, \u001erand\u001empparameterize every\nspatial symmetry transformation for the bilayer system,\nwhere\u001er2[0;2\u0019] and\u001emp2[0;\u0019].\nAn applied, in-plane electric \feld Ebreaks all of thesesymmetries except the single mirror-plane that contains\nE(i.e. when ^n?E). To see this, note that Eis a\npolar vector, so when Elies within the mirror-plane it is\ninvariant upon re\rection (Fig. 8b). All rotations about\n^zwill change the orientation of Esince it lies in-plane,\nso those transformations are no longer symmetries of the\nsystem. Assuming now that one layer is ferromagnetic,\nits magnetization direction ^mmust also be invariant to\nany allowed symmetry transformation. Since magnetiza-\ntion is a pseudovector, it is invariant to a mirror-plane re-\n\rection only when it is normal to the mirror-plane. Thus,\nas depicted in Fig. 8b, the only remaining symmetry is\na mirror plane containing Eand^zwhen ^m=\u0006^z\u0002E.\nOnly in this scenario is Ewithin the mirror-plane and\n^mnormal to the mirror-plane.\nUnder the assumption that only the magnetization di-\nFIG. 8. Depiction of symmetries and their consequences in a\npolycrystalline bilayer. (a) For nonmagnetic bilayers, any ro-\ntation\u001eRabout the out-of-plane direction ( z-axis) leaves the\nsystem unchanged. Likewise, any mirror-plane transforma-\ntion where the mirror-plane normal lies in-plane (parameter-\nized by the angle \u001eMP) also leaves the system invariant. (b)\nUnder an applied, in-plane electric \feld E, all symmetries are\nbroken except the mirror-plane that lies parallel to the electric\n\feld, since the electric \feld is a polar vector. If one layer is\nferromagnetic, this symmetry is broken unless the magnetiza-\ntion^mpoints normal to the mirror-plane, since magnetization\nis a pseudovector. In this con\fguration, the torque \u001cmust\nvanish, because a nonvanishing torque re\rected through the\nmirror-plane will reverse, violating the system's symmetry.11\nrection is variable (i.e. the magnetization is saturated),\nthe net torque acting on the magnetization must be or-\nthogonal to ^m. Thus, for the case when ^mpoints normal\nto the mirror-plane as described above, the torques ( \u001c)\nmust be parallel to the mirror-plane. Since torque is a\npseudovector, and since any pseudovector parallel to a\nmirror-plane will \rip sign upon re\rection (see Fig. 8b),\nthe torque must vanish to preserve the mirror-plane sym-\nmetry. Thus, in nonmagnet/ferromagnet bilayers under\nan applied, in-plane electric \feld, all spin-orbit torques\nmust vanish when ^m=^z\u0002E.\nSo far, we have considered two constraints on spin-\norbit torques in nonmagnet/ferromagnet bilayers: 1)\nthey point orthogonally to ^mand 2) they vanish when\n^m=^z\u0002E. Given these constraints, spin-orbit torques\ncould be written as a linear combination of the following\nterms,\n\u001cD=cD(^m)^m\u0002\u0000\np\u0002^m\u0001\n(4)\n\u001cF=cF(^m)p\u0002^m; (5)\nwhereD/Frefers to the dampinglike/\feldlike compo-\nnent (visualized in Fig. 9), cD=F(^m) are magnetization-\ndependent scalar functions, and p=^z\u0002^E. The damp-\ninglike and \feldlike vectors span the plane perpendicular\nto the magnetization and vanish when ^m=p, satisfy-\ning the two constraints above. Eqs. 4 and 5 describe two\ntypes of behavior: damping towards ^z\u0002^E(Eq. 4) and\nprecession about ^z\u0002^E(Eq. 5).\nIt is important to note that, in general, the coe\u000ecients\ncD=F(^m) cannot be given by all functions of ^m. In other\nwords, Eqs. 4 and 5 are under-constrained. The general\nexpression104for spin-orbit torque in a bilayer subject to\nthe continuous symmetries described above is given by\n\u001c=1X\nl=0(mz)2l\u0000\nalp\u0002^m+bl^m\u0002(p\u0002^m)\n+cl(^m\u0001E)^z\u0002^m+dl(^m\u0001E)^m\u0002(^z\u0002^m)\u0001\n(6)\nwhereal,bl,cl, anddlare the coe\u000ecients of expansion.\nNote that the \frst four terms in the expansion (zeroth or-\nder inmz) are the traditional \feldlike torque and damp-\ninglike torque plus two additional terms. The additional\nterms behave like \feldlike and dampinglike torques de-\n\fned relative to ^zinstead ofp=^z\u0002^E, but also carry\na factor ^m\u0001^E, which ensures the torque vanishes when\n^m=pas required.\nThe vector forms in Eq. 6 are shown in Fig. 9(c-f).\nEach of these forms can additionally be multiplied by\n(mz)2l, each power of which suppresses the torque at\n\u0012= 0;\u0019. An important point is that measuring the\ntorque at the poles, \u0012= 0;\u0019, or the equator, \u0012=\u0019=2 does\nnot necessarily predict the behavior at the other set of\npoints. The di\u000berence, if large, can be important for con-\nnecting measurements of the torque at speci\fc magneti-\nzation directions with magnetic dynamics, which depend\non the values of the torques at many points.104There areindications in both model calculations105and \frst prin-\nciples calculations29that the angular dependence can be\nmore complicated than just a sum of the simple \feld-\nlike and dampinglike torques. The expansion in Eq. 6\nis complete and it is easy to envision what each of the\nterms looks like. Unfortunately, the di\u000berent terms are\nnot orthogonal to each other. That means that if a \fnite\ntruncation of the series is used to \ft experimental (or cal-\nculated) data, the \ft parameters, aletc., will depend on\nthe order at which the series is truncated. Ref. 30 gives an\northogonal expansion in terms of modi\fed vector spher-\nical harmonics. That expansion is more appropriate for\n\ftting data although the higher order terms have a less\ntransparent form.\nWe \fnally note that, quite generally, reducing the\nsystem symmetry relaxes the constraints on the sys-\ntem response and enables more nonzero components of\nthe torque. Recent work has utilized substrates which\nlack in-plane mirror symmetry, such as transition metal\ndichalcogenides WTe 2, MoTe 2, and others.106{111For\nthese materials, applying a current in the single mirror-\nplane results in an out-of-plane torque. Such a torque\nmay enable switching of perpendicular magnetic layers,\nwhich possess technological advantages relative to in-\nplane magnetic layers.112,113\nG. Distinguishing Bulk from Interface E\u000bects\nFor spin transfer torques and current-perpendicular-to-\nthe-plane giant magnetoresistance,44the most important\nlength scale is the spin di\u000busion length, the length scale\nover which spin currents and spin accumulations decay.\nIt is de\fned as the distance a spin di\u000buses before it under-\ngoes spin-\rip scattering and is given by `2\nsf=\u0015vF\u001csf=6,\nwhere\u0015is the elastic mean free path, which is the aver-\nage distance between elastic scattering events, vFis the\nFermi velocity, and \u001csfis the spin-\rip scattering time. On\nthe other hand, for current-in-the-plane giant magnetore-\nsistance, the important length scale is the mean free path.\nSpin-orbit torques combine both in-plane charge trans-\nport with out-of-plane spin transport making it likely\nthat both length scales are important. The importance\nof multiple length scales can complicate the interpreta-\ntion of calculations and experiments.\nOther length scales may be important as well. For\nexample, the spin current associated with the spin Hall\ne\u000bect di\u000bers qualitatively from di\u000busive spin currents. If\nthe spin Hall spin current is intrinsic (see Sec. II E), it\ncan be described as arising from an anomalous velocity\nof electrons at special points on the Fermi surface where\nspin-orbit coupling leads to large band splittings. It is not\nclear whether such spin currents vary with the spin dif-\nfusion length or with yet a di\u000berent length scale. Studies\nof the thickness dependence57,114,115are frequently inter-\npreted by assigning the observed length scale to the spin\ndi\u000busion length, but that length scale could be entirely\ndi\u000berent.12\nFIG. 9. (a) Depiction of a bilayer consisting of a heavy metal layer (blue region) and a ferromagnetic layer (red region)\nunder an applied, in-plane electric \feld E(here along ^x). The high symmetry direction p=^z\u0002Eis normal to the x=z\nmirror-plane, where ^zpoints out-of-plane. (b) Spin-orbit torque is conveniently de\fned using two basis vectors: dampinglike\n(^m\u0002(p\u0002^m)) and \feldlike ( p\u0002^m), which are de\fned relative to the high-symmetry direction p. These basis vectors span\nthe plane perpendicular to the magnetization and vanish when ^mjjp, satisfying the bilayer's symmetry constraints. However,\nthe dampinglike and \feldlike basis vectors are not su\u000ecient to describe the magnetization-dependence of spin-orbit torque\nunless they have magnetization-dependent coe\u000ecients. The full expansion of spin-orbit torque using constant coe\u000ecients is\nmore complicated, and is given by Eq. 6. This expansion consists of four-vector functions of the magnetization, depicted in\n(c)-(f). Each vector function can be additionally multiplied by any power of m2\nz. The in-plane and out-of-plane torques are\nprojected below and above the unit sphere respectively. The full expansion suggests that if measurements of in-plane and\nout-of-plane torques are interpreted as arising from only dampinglike or \feldlike torques, the full magnetization dependence\nmay be misrepresented. For example, when ^mjj^z, measuring a small torque component pointing along pindicates a small\ndampinglike torque, but this measurement could be incorrectly interpreted as weak potential for magnetization switching,\nbecause the torque component ( ^m\u0001E)^z\u0002^mshown in (e) is zero at ^mjj^zbut contributes to the switching process for other\nmagnetization directions.\nA \fnal unknown length scale that complicates the in-\nterpretation of experiment is the length scale over which\nstructural details of the layers vary. There are many pro-\ncesses that can contribute to structural variations, in-\ncluding relaxation of strain due to lattice mismatch and\ngrain growth. Without detailed structural characteriza-\ntion and related calculations it is di\u000ecult to know how\nmuch of a measured variation with layer thickness could\nbe due to structural changes.\nIII. PHENOMENOLOGICAL MECHANISMS\nBuilding on the idea that the overlapping spin-orbit\ncoupling and exchange interaction at an interface can\nmake a substantial contribution to the spin-orbit torque,we introduce an extremely simple model that displays all\nof the important interfacial e\u000bects.\nFirst, we describe a spin torque on a strong magnetic\nimpurity, where the absorbed spin current equals the spin\ntorque in the absence of spin-orbit coupling. We then\nshow that in the presence of spin-orbit coupling this is\nno longer the case, because spin-orbit coupling opens an-\nother channel of angular momentum transfer to and from\nthe atomic lattice.\nNext, we introduce a model based on embedding a two-\ndimensional Hamiltonian describing the interface within\na structureless, three-dimensional bulk material, yield-\ning a bilayer. The two-dimensional Hamiltonian describ-\ning the interface breaks structural inversion symmetry\nby design, enabling a nonvanishing spin-orbit torque. In\nthis approach, the important interactions are due to elec-13\ntrons re\recting from or transmitting through the inter-\nface. As they do, they interact with the interfacial spin-\norbit coupling and exchange interaction, which modify\nthe transmission and re\rection amplitudes. These ampli-\ntudes combine to give all of the currents and spin currents\nat the interface as well as the torques on the magnetiza-\ntion.\nThe simple models presented here are not realistic since\ndetails of the electronic structure and disorder are ab-\nsent. However, such models accomplish three goals: 1)\nthey give a qualitative understanding of interfacial spin-\norbit e\u000bects, 2) they present a quasi-analytical way of\nseparating contributions from spin-orbit coupling and the\nexchange interaction, and 3) they provide a template to\ncompare to ab-initio calculations so that some physical\nintuition may be extracted.\nA. Spin torque on a strong magnetic impurity in one\ndimension\nIn this section we provide a simple example of a spin\ntorque on a strong magnetic impurity. Later, we show\nthat the qualitative behavior in this example is analogous\nto the role that interfaces play on spin torques in bilayers.\nImagine an electron scattering o\u000b of a magnetic im-\npurity in one dimension, described by the coordinate z,\n(Fig. 10). We assume the magnetic impurity (located at\nz= 0) is captured by a delta function potential\nV\"=#(z)/\u000e(z)u\"=#; (7)\nwhereu\"=#is the barrier strength for spins parallel ( \") or\nantiparallel (#) to the magnetic \feld of the impurity ( B).\nIn what follows, we use the coordinate system ( x0;y0;z0)\nfor spin space, such that the z0direction points along B.\nFor simplicity, let u#= 0 andu\"!1 . The impu-\nrity thus behaves as a perfect re\rector for \"spins and\na perfect transmitter for #spins, though each spin state\ncan acquire a phase factor upon scattering. Now assume\nan electron with spin transverse to Bscatters o\u000b of this\nimpurity. The incoming transverse spin state could be\ndescribed by the following spinor (assuming spin along\nx0):\n\u001fI=1p\n2\u0000\nj\"i+j#i\u0001\n: (8)\nThe re\rected and transmitted states are then\n\u001fR=\u000bp\n2j\"i; \u001f T=\fp\n2j#i; (9)\nwhere\u000band\fare the phase factors acquired upon scat-\ntering. Thus, while the incoming spin is transverse to B\n(alongx0), the re\rected and transmitted spins are parallel\nand antiparallel to B(along\u0006z0). To ensure continuity\nof the wavefunction at z= 0, such that \u001fI+\u001fR=\u001fT,\nwe may choose \u000b=\u00001 and\f= 1.\nHaL\nx\nz yDQ=t\nB\nsy'x'\nz'\ny'Real space Torque Spin space\nHbL\nB cIeikzz\ncRe-ikzzcTeikzz\nHcL\nz=0dHzLu\ndHzLu¯Qzx'=DQ=t\nQzz'FIG. 10. Schematic of an electron scattering o\u000b of an in-\n\fnitely strong magnetic impurity. (a) Coordinate systems for\nreal space and spin space, where transport occurs along zand\nthe impurity's magnetic moment points along z0. The basis\nstatesj\"iandj#icorrespond to spins along \u0006z0. (b) In the\nlimit thatB! 1 , the impurity perfectly re\rects \"spins\nand perfectly transmits #spins. Thus, for an incoming spin\nstate\u001fI/j\"i +j#ialongx0(transverse to the impurity's mag-\nnetic moment), the re\rected and transmitted spin states point\nalong\u0006z0(parallel or antiparallel to the impurity's magnetic\nmoment) (c) Plot of the spin current as a function of position.\nThe incoming state carries spin current Qzx0; the indices in-\ndicate \row along zand spin direction along x0. The re\rected\nand transmitted states each carry the same spin current Qzz0;\nthe indices indicate \row along zand spin direction along z0.\nThus, the net change in spin current across the impurity is\nQzx0, indicating that the incoming spin angular momentum is\ncompletely absorbed. The absorption of spin current results\nin a torque on the impurity's magnetic moment.\nThe incident, re\rected, and transmitted states each\ncarry a spin current that \rows along z(Fig. 10b/c).\nSince the re\rected and transmitted states have equal\nand opposite spin andequal and opposite velocities, they\nboth carry the same spin current ( Qzz0), which is contin-\nuous across the impurity and implies no angular momen-\ntum transfer. However, the incoming spin current ( Qzx0)\nis completely absorbed by the impurity, since this spin\ncurrent exists on one side of the interface but not on the\nother side. The absorption of the incoming spin current\nis a simple example of a spin transfer torque ( \u001c), where\nin this case\u001c=Qzx0^x0.\nFor there to be a spin transfer torque along x0on the\nimpurity, the spin density satz= 0 must have a non-\nvanishingy0component and a vanishing x0component.\nThis is because the spin torque is given in general by:\n\u001c/(u\"\u0000u#)s\u0002B!u\"s\u0002B; (10)14\nwhere the last form holds for u#= 0. Since\u001cjj^x0and\nBjj^z0, the spin accumulation smust lie in the y=zplane\nto satisfy Eq. 10. Here we run into an apparent problem.\nIf the wavefunction at z= 0 equals \u001f0=\u001fI+\u001fR=\n\u001fT=j#i, then the spin density spoints solely along\n\u0000z0and has no y0component, resulting in a vanishing\ntorque. Where then did the angular momentum from\nthe absorbed, incoming spin current go?\nThe inconsistency described above is corrected by care-\nfully taking the limit as u\"!1 . As we show in Ap-\npendix A, the wavefunction at z= 0 is actually a super-\nposition of spin states given by\n\u001f0=\u001fT/\u0000(ia=u\")j\"i+j#i; (11)\nbecause when u\"is large but not in\fnite, a tiny amount\nofj\"imust be transmitted (here we have omitted the nor-\nmalization factor). Note that ais a dimensionless con-\nstant determined by details of the scattering potential.\nThen, asu\"!1 ,\u001f0approaches the previous solution,\nbut now carries a component of spin along y0as required\ns=\u001fy\n0\u001b\u001f0=0\n@0\n2a=u\"\n\u00001 +a=u2\n\"1\nA!0\n@0\n0\n\u000011\nA (12)\nwhere\u001bis the vector of Pauli matrices. Even though\nsy0vanishes as u\"!1 , the torque it exerts does not,\nbecause the prefactor u\"in Eq. 10 exactly cancels the\nfactor of 1=u\"insy0. Because the torque equals the in-\ncoming spin current, which does not depend on u\", the\ntorque cannot depend on u\"either. The cancellation of\nu\"re\rects this fact.\nB. Role of spin-orbit coupling on the torque\nWe now consider the case in which the impurity's to-\ntal magnetic \feld is given by B=Bex+Bsoc, where\nBexdenotes the exchange \feld while Bsocis the e\u000bective\nmagnetic \feld from the spin-orbit interaction. Let us as-\nsume again that the total magnetic \feld Bpoints along\n^z0whileBexandBsocindividually do not. As before,\n\u001cpoints along x0ands0lies in the y0=z0plane. The\ntorque on the impurity's magnetic moment is due to the\nmisalignment of the spin and the exchange \feld:41\n\u001c/s0\u0002Bex: (13)\nSinceBexis not required to point along z0as before, there\nis no guarantee that the absorbed spin current equals the\nexchange torque, as can be seen in Fig. 11). In other\nwords, while the absorbed spin current must equal the\ntorque on the total e\u000bective \feld B, it need not equal\nthe torque on only part of that e\u000bective \feld (i.e. Bex).\nIn this case, spin-orbit coupling has introduced another\nchannel of angular momentum transfer between the scat-\ntered electron and the lattice at the magnetic impurity.\nBex\nBexBSOCDQ=t\nB\nsy'DQt\nB\nsy'x'\nz'\ny'NoSOC With SOC Spin spaceHaL HbLFIG. 11. Result of adding spin-orbit coupling to the mag-\nnetic impurity. (a) Without spin-orbit coupling, the magnetic\n\feld of the impurity Bequals the exchange \feld Bex. The\nabsorbed spin current \u0001 Qequals the torque \u001c/s\u0002Bex.\n(b) With spin-orbit coupling, Bis the sum of the exchange\n\feldBexand the spin-orbit \feld Bsoc. The absorbed spin\ncurrent is perpendicular to the total \feld, not the exchange\n\feld. However, only the part of the absorbed spin current\nthat is perpendicular to the exchange \feld contributes to the\ntorque on the magnetization. The rest of the absorbed spin\ncurrent exerts a torque on the lattice through the spin-orbit\ncoupling.\nWithout spin-orbit coupling at the interface, the lon-\ngitudinal spin current (spins aligned along the magneti-\nzation) is conserved, but the transverse spin current is\nnot, as discussed in Sec. III A. This no longer holds at\ninterfaces with spin-orbit coupling. It does hold for indi-\nvidual states with respect to the total e\u000bective \feld, but\nnot with respect to the exchange \feld (aligned with mag-\nnetization) alone. This di\u000berence becomes important in\nSec. III C 4.\nC. Spin currents and spin torques in a bilayer\nIn the previous section, we showed that an electron\ntransfers angular momentum to a magnetic impurity af-\nter scattering o\u000b of it. The change in spin \rux always\nequals the total torque, but in the presence of spin-orbit\ncoupling, the torque on the impurity's magnetic moment\nis only part of the total torque. Now, we consider a bi-\nlayer system in which the material interface plays the role\nof the magnetic impurity.\nOur primary goal is to relate the accumulations and\ncurrents that drive transport with the resulting spin cur-\nrents and spin torques at the interface. In equilibrium,\nwe model both layers as free electron gasses with identi-\ncal, spin-independent, spherical Fermi surfaces. We do so\nmainly for simplicity, but also to focus on the important\nqualitative behavior arising in nonequilibrium. To distin-\nguish between layers, we assume each layer has di\u000berent\nmomentum relaxation times ( \u001c), and to model ferromag-\nnetic layers, we assume \u001cis spin-dependent.\nOur description of electron transport and the result-\ning spin currents and torques comes from the Boltzmann\nequation. A full solution of the Boltzmann equation116\ninvolves taking a general form for the solution of the bulk\nBoltzmann equation in each layer, computing match-\ning conditions at their common interface, and applying\nboundary conditions based on the behavior of the so-15\nlution at in\fnity. While this approach is straightfor-\nward, it precludes simple analytical models for the re-\nsults. Here, we approximate the non-equilibrium distri-\nbution of electrons by neglecting the e\u000bect of scattering\nnear the interface on the distribution functions, focusing\non the matching conditions across the interface. This ap-\nproach enables analytical solutions that are impossible if\nwe consider the full solution. While these approximate\nsolutions di\u000ber quantitatively from the full solution, they\nare qualitatively the same and allow full consideration of\nwhat processes are possible at interfaces.\nAt the interface, we adopt a basic quantum mechan-\nical picture where electrons are plane waves with two\nspin degrees of freedom. E\u000bective magnetic \felds cap-\nture the exchange interaction and spin-orbit coupling at\nthe interface, similar to the previous section. These e\u000bec-\ntive magnetic \felds behave like spin-dependent potential\nbarriers, leading to spin-dependent scattering. From this\nscattering, spin currents and spin torques arise.\nIn the following, we formally de\fne this model, de-\nscribe the crucial approximations, and present the results\nwithout derivation, which can be found in Appendix B.\n1. Boltzmann model with quantum mechanical interfacial\nscattering\nFirst, we formally de\fne the model, beginning with a\ndescription of the interface. The e\u000bective magnetic \feld\nBe\u000bseen by carriers at the interface is given by\nBe\u000b=Bex+Bsoc/ue\u000b (14)\nwhereue\u000bis a unitless quantity proportional to the e\u000bec-\ntive magnetic \feld (note that ue\u000bis not a unit vector).\nAssuming Rashba-type spin-orbit coupling, ue\u000bis given\nby:\nue\u000b=uex^m+uR^z\u0002k=kF; (15)\nwhere ^mis the unit vector pointing along the interfacial\nmagnetization, kis the crystal momentum of the electron\ntraveling towards the interface, kFis the Fermi wave vec-\ntor, anduexanduRare unitless parameters describing\nthe relative strengths of the exchange and spin-orbit in-\nteractions respectively. Throughout this section, we use\nthe following parameterization:\nuex=jue\u000bjcos(\u000b); u R=jue\u000bjsin(\u000b): (16)\nThis parameterization is useful because the results have\na simple dependence on \u000b(despite having a complicated\ndependence onjue\u000bj). This simple \u000bdependence allows\nus to probe the limits of vanishing exchange interaction\nor vanishing spin-orbit coupling.\nIncluding a spin-independent potential barrier (param-\neterized by u0), this system is described by the following\n2\u00022 Hamiltonian\nH(^r) =~2k2\n2mI2\u00022+~2kF\nm\u000e(z)\u0000\nu0I2\u00022+\u001b\u0001ue\u000b\u0001\n;(17)whereI2\u00022is the identity matrix in spin space, and\nthe interface is located at z= 0. Note that the factor\n~2kF\u000e(z)=mconverts the unitless vector ue\u000bto units of\nenergy. Wavefunction matching at the interface yields re-\n\rection and transmission amplitudes, which are derived\nin Appendix A. To determine the resulting spin currents\nand spin torques, we use these re\rection and transmis-\nsion amplitudes as boundary conditions for the Boltz-\nmann equation.\nThe statistics of carriers in each layer of the system\nare captured by the Boltzmann distribution function. In\nthe model we consider here, with full coherence between\nall spin states at each point in reciprocal space but no\ncoherence between di\u000berent points in reciprocal space,\nthe full distribution function can be captured by four\nfunctions,f\u000b(r;k) where\u000b2[x;y;z;c ], representing the\nexpectation value of spin along each axis and the number\noperator. Since we are interested in the linear transport\nregime, we can linearize the distribution function around\nits equilibrium form\nf\u000b(z;k) =feq(\u000fk)\u000e\u000bc+@feq\n@\u000fkg\u000b(z;k) (18)\nwhere\u000fkis thek-dependent energy, feqis the spin-\nindependent equilibrium distribution function, g\u000bis the\nnonequilibrium perturbation of the distribution function,\nand\u000b2[x;y;z;c ]. In the simple model for the electronic\nstructure that we consider here, the equilibrium distri-\nbution is independent of spin, as re\rected in the \u000e\u000bcin\nthe \frst term.\nTo evaluate the accumulations and currents at the\ninterfaces, we must know the distribution functions\ng\u000b(z;k) atz= 0\u0006. We could solve the Boltzmann equa-\ntion for the entire bilayer using the quantum mechanical\nscattering matrices as boundary conditions. This pro-\ncess requires numerical solutions that are cumbersome to\ncompare with experiments. In the next section, we out-\nline a simple but e\u000bective approximation that bypasses a\nfull solution of the Boltzmann equation for the bilayer.\n2. Obtaining qualitative results without solving the\nBoltzmann equation for the entire heterostructure\nTo obtain a useful, quasi-analytical description of spin\ncurrents and spin torques at interfaces without solving\nthe Boltzmann equation, we focus on two cases:\n1. Perpendicular transport in which spin and charge\naccumulations at z= 0\u0006drive out-of-plane cur-\nrents\n2. In-plane transport that drives out-of-plane spin\ncurrents\nFor each case, we assume a reasonable form for the distri-\nbution function of incoming electrons, de\fned by kz>0\nforz<0 andkz<0 forz>0. The outgoing (scattered)16\nelectrons are then determined by the quantum mechan-\nical scattering matrices. To simplify our notation, we\nrewrite the distribution function as a four-vector labeled\nby components \u000b2[x;y;z;c ], such that g\u000b!g. For\nperpendicular transport, , we then approximate gfor in-\ncoming carriers as\ng(0\u0006;k) =eq\u0006; (19)\nwhere q\u0006is a constant four-vector de\fned separately on\nboth sides of the interface z= 0\u0006. Such distributions\nare shown in the blue regions of panels (a) and (b) of\nFig. 12 for the cases of charge accumulation and spin\naccumulation respectively. For in-plane electric \felds,\nthe incoming distribution is de\fned by\ng(0\u0006;k) =e~kxq\u0006; (20)\nwhere the constant q\u000b\u0006=EvF\u001cl\u000bP\u000b\u0006,Eis the magni-\ntude of the in-plane electric \feld, vFthe Fermi velocity,\n\u001c\u000b\u0006the momentum relaxation times, and P\u000b\u0006the di-\nmensionless spin or charge polarization with \u0006indicat-\ning either side of the interface. These distributions are\nshown in the blue regions of panels (c) and (d) of Fig. 12\nfor the cases of charge accumulation and spin accumu-\nlation respectively. The panels in Fig. 12 also give, in\nred, the outgoing distribution functions resulting from\ninterfacial scattering for each of the di\u000berent incoming\ndistributions. When they scatter from the interface, in-\ncoming unpolarized carriers become spin polarized and\nincoming spin-polarized carriers rotate their spin polar-\nization. These phenomena arise from the in\ruence of the\ninterfacial exchange and spin-orbit interactions, and lead\nto the modi\fcation of spin accumulations and spin cur-\nrents at the interface, which we discuss in detail in the\nnext subsection.\n3. Calculating the spin/charge accumulations and\nspin/charge currents at interfaces\nUsing the incoming distributions as de\fned in Eq. 19\nor Eq. 20, we use the matching conditions at the in-\nterface (discussed below) to determine the full distribu-\ntions shown in Fig. 12. From these distribution, we can\ncompute the non-equilibrium spin and charge densities\n(also called spin and charge accumulations) and the non-\nequilibrium spin and charge currents. At the two sides of\nthe interface, z= 0\u0006, these quantities are given by the\nfollowing integrals over the spherical Fermi surfaces\n\u0016(0\u0006) =c\u0016Z\nFSd2kg(0\u0006;k) (21)\njz(0\u0006) =cjZ\nFSd2kkzg(0\u0006;k); (22)\nwherecsandcjare constants. The components of the\nfour-vector \u0016represent the spin ( \u000b=x;y;z ) and charge\n(\u000b=c) accumulations at z= 0. The components of jzsimilarly represent the spin and charge current \rowing\nout-of-plane at z= 0\u0006. The conductance matrix would\nbe constructed by solving for q\u0006in terms of the accumu-\nlations\u0016(0\u0006) and inserting those into the expression for\njz(0\u0006), see Refs. 33 and 34. Here, we do not reproduce\nthe derivation of that matrix but focus on the di\u000berent\nphysical processes that contribute to it.\nThe dimensionless vector ue\u000bcharacterizing the poten-\ntial given in Eq. 16 describes the interaction between the\nexchange potential and the spin-orbit potential. If we\nchoose the magnetization to be out-of-plane, ^m=^z, the\nexchange \feld and the spin-orbit \feld are always perpen-\ndicular to each other, greatly simplifying the form of the\nresults. We present results for this case because inter-\nmediate results can be cast in a physically transparent\nform that allows for a clear understanding of the role\nplayed by the exchange interaction and spin-orbit cou-\npling in the processes that occur at the interface. The\n\fnal result, obtained after integrating over the full Fermi\nsurface obscures these simple roles. For in-plane or gen-\neral direction magnetizations, the physics is the same but\neven the intermediate forms are complicated enough to\nobscure the physical interpretation. The full results can\nbe found numerically as is done in Refs. 33 and 34.\nFor a magnetization ^m=^zthe vectorue\u000bdepends on\neach electrons' wave vector in two ways, most easily seen\nin spherical coordinates\nkx=kFsin(\u0012) cos(\u001e) (23)\nky=kFsin(\u0012) sin(\u001e) (24)\nkz=kFcos(\u0012): (25)\nThe relative strength of the spin-orbit interaction de-\npends on the polar angle \u0012, going to zero as \u0012goes to\nzero and its direction depends on the azimuthal angle \u001e.\nIt turns out we can analytically evaluate the integrals in\nEqs. 21 and 22 over azimuthal angle \u001eusing the de\fnition\nforggiven by Eq. 19 or Eq. 20. We refrain from evalu-\nating the remaining integral over polar angle \u0012because\nit is cumbersome and not necessary to obtain physical\ninsight. Even though the azimuthal average of the spin-\norbit potential is zero, it still makes substantial contri-\nbutions to the transport when the average is weighted by\nthe distribution functions with either a spin-dependence\nor an angular dependence. Carrying out these azimuthal\nintegrations highlights the e\u000bects that remain.\nIn the following, we write the results in terms of the\naverage and di\u000berence in values of jzand qacross the\ninterface:\n\u0001jz=1\n2\u0000\njz(0\u0000)\u0000jz(0+)\u0001\n\u0001q=1\n2(q\u0000\u0000q+) (26)\n\u0016jz=1\n2\u0000\njz(0\u0000) +jz(0+)\u0001\u0016q=1\n2(q\u0000+q+) (27)\nUsing this notation, the accumulations and out-of-plane17\nFIG. 12. Plots depicting the nonequilibrium distribution functions g\u000b(0\u0006;k) in the presence of interfacial spin-orbit scattering.\nEach picture illustrates the physics captured by a single matrix element in the matrices de\fned in Eq. 33 or Eq. 38. The\ninterfacial exchange \feld (red arrow) points out-of-plane (along z). The gray sphere represents the equilibrium Fermi surface.\nThe colored surfaces represent the nonequilibrium perturbation to the Fermi surface, given by the charge distribution gc(0\u0006;k).\nThe arrows depict the spin distribution function gi(0\u0006;k) fori2[x;y;z ]. Blue and red regions represent the incoming and\noutgoing (scattered) carriers respectively. The net spin current at z= 0\u0006is shown below the distribution functions, where the\nblock arrows denote spin \row (always out-of-plane) and the tubular arrows denote spin direction. Note that we use transverse\nandlongitudinal to denote spin directions relative to the interfacial exchange \feld. (a) Scenario where the incident carriers have\ntwo di\u000berent charge accumulations but no spin accumulation. Regardless, the scattered carriers are spin polarized from their\ninteraction with the interfacial exchange and spin-orbit \felds. The net spin currents after scattering have longitudinal spin\ndirections and are conserved across the interface. (b) Scenario where the incident carriers have two di\u000berent transverse spin\naccumulations. The net spin currents after scattering also have transverse spin directions but are rotated relative to the spin\naccumulation and not conserved across the interface. (c) Scenario where two di\u000berent in-plane charge currents \row at z= 0\u0006,\nindicated by di\u000bering shifts the Fermi surface. The scattered carriers become spin polarized and the net out-of-plane spin\ncurrents have transverse spin direction. (d) Scenario where two di\u000berent in-plane, longitudinal spin currents \row at z= 0\u0006.\nThe net out-of-plane spin currents have transverse spin direction and are not conserved across the interface.\ncurrents are given by\n\u0016=Z\u0019=2\n0d\u0012w(\u0012)St(\u0012)\u0016q (28)\n\u0016jz=Z\u0019=2\n0d\u0012v(\u0012)\u0016S(\u0012)\u0001q; (29)\n\u0001jz=Z\u0019=2\n0d\u0012v(\u0012)\u0001S(\u0012)\u0016q; (30)\nwherew(\u0012) =cutan(\u0012),v(\u0012) =cjek2\nFsin(\u0012), andSt,\u0016S,\nand \u0001Sare 4\u00024 matrices. The constants c\u0016andcj\nare de\fned in the appendix. These quantities capture\nthe matching conditions for the distribution functions at\nthe interface based on transmission and re\rection prob-\nabilities and the form of the incident distribution, accu-mulations versus in-plane electric \feld. The magnitudes\nof the incident distributions are contained in q\u0006The in-\ndex\u0017=fi-r;tgindicates whether the matrix refers to\ntheincident plus re\rected side or the transmitted side.\nSince we have made the approximation that the elec-\ntronic structure is the same on both sides of the inter-\nface, the transmission and re\rection probabilities are the\nsame for electrons incident from the right and from the\nleft. These matrices are given by the integration over\nazimuthal angle of the appropriate transmission and re-\n\rection coe\u000ecients. They are related as follows:\n\u0016S(\u0012) =Si-r(\u0012) +St(\u0012) (31)\n\u0001S(\u0012) =Si-r(\u0012)\u0000St(\u0012): (32)\nNote that the spin and charge accumulations de\fned in18\nEq. 28 are at the interface and di\u000ber from those, de\fned\nin Eq. 22, on either side of the interface.\nEquations 28-32 relate the important physical quanti-\nties in terms of the boundary conditions. They show that\nthe symmetric response matrix \u0016Sdetermines the average\nspin and charge currents at the interface ( \u0016jz) while the\nantisymmetric response matrix \u0001 Sdetermines the dif-\nference in spin and charge currents across the interface\n(\u0001jz). The form of w,v,Si-r, andStdepends greatly on\nchoice of g, i.e whether accumulations or in-plane cur-\nrents drive the system. In the next two subsections, we\npresent the Si-r, andStmatrices, and discuss how theycapture the e\u000bect of interfacial spin-orbit coupling for\nboth scenarios.\n4. Spin or charge accumulations drive the system\nThe choice of g(0\u0006;k) =\u0000eq\u0006corresponds to a spin\nand/or charge accumulation at z= 0\u0006as might be driven\nby a perpendicular voltage or by the spin Hall e\u000bect for\nin-plane transport. In this case, the Si-r, andStmatrices\nare:\nS\u0017=0\nB@0 0 0 0\n0 0 0 0\n0 0 0 0\n0 0 0 2c1\nCA+ 2fc(\u000b)0\nB@fc(\u000b)a\u0017\u0000b\u0017 0 0\nb\u0017 fc(\u000b)a\u0017 0 0\n0 0 fc( \u000b)c d\n0 0 d 01\nCA+ fs(\u000b)20\nB@a\u0017+c 0 0 0\n0a\u0017+c0 0\n0 0 2 a\u00170\n0 0 0 01\nCA: (33)\nwhere\u00172[i-r;t]. The parameters a\u0017,b\u0017,c, andddepend\nonly on the polar angle \u0012, the magnitude of the e\u000bective\n\feldjue\u000bj, and the spin-independent barrier strength u0.\nAs a reminder, the indices in order are [ x;y;z;c ]. The\nimportance of the spin-orbit interaction to the scattering\ndepends on the wave vector. For normal incidence it is\nzero and is maximal for grazing incidence. Recall that the\nangle\u000bde\fned in Eq. 16 re\rects the relative importance\nof the exchange interaction and the spin-orbit coupling.\nFor a particular angle of incidence \u0012, the dependence of\nS\u0017on angle\u000bis given by the functions fs( \u000b) and fc(\u000b):\nfs(\u000b) =sin(\u000b)q\nsin2(\u0012) cos2(\u000b) + sin2(\u000b)(34)\nfc(\u000b) =sin(\u0012) cos(\u000b)q\nsin2(\u0012) cos2(\u000b) + sin2(\u000b); (35)\nThese obey the following limits:\nfs(0) = 0;fs(\u0019=2) = 1; (36)\nfc(0) = 1;fc(\u0019=2) = 0; (37)\nTherefore, in the limit of vanishing spin-orbit coupling\n(\u000b= 0) or vanishing exchange interaction ( \u000b=\u0019=2),\nonly one of these functions is nonzero. The parameters\ncanddderive from the components of the spin longitu-\ndinal to the e\u000bective \feld, so they are conserved across\nthe interface and hence are equal for the incident plus re-\n\rected and transmitted sides. They do enter some of the\ncoe\u000ecients for the components transverse to the magne-\ntization, because for each incident wave vector, the e\u000bec-\ntive \feld is not along the magnetization. For this spe-\ncial case with the magnetization always perpendicular to\nthe spin-orbit coupling \feld, all but a few of the possi-\nble contributions of this type get integrated away. Theparameters a\u0017,b\u0017are di\u000berent on the incident plus re-\n\rected and transmitted sides as they are associated with\nthe components of the spin current with spins transverse\nto the e\u000bective \feld. For this particular magnetic \feld,\nthey only make a limited but important contribution to\nthe transport for spins longitudinal to the \feld.\nEq. 33 shows how spin/charge accumulations drive\ntransport and clari\fes the role of the exchange and spin-\norbit interactions. Since ^m=^z, we refer to spin polar-\nizations along xandyastransverse and those along z\naslongitudinal . Fig. 13 illustrates the contributions from\neach element of the Si-randStmatrices and can be used\nas a companion to the main text below.\nCharge current conservation |The \frst matrix in\nEq. 33 describes charge current conservation across the\ninterface. The sole nonzero parameter 2 crelates the drop\nin charge accumulation across the interface to the total\ncharge current \rowing across the interface.\nGeneralized Magnetoelectronic Circuit Theory |The\nsecond matrix in Eq. 33 describes a generalization of mag-\nnetoelectronic circuit theory.53,54Because this matrix is\nmultiplied by 2fc( \u000b), it vanishes for zero exchange inter-\naction (\u000b= 0). For a nonmagnet/ferromagnet bilayer,\nthe real and imaginary parts of the spin mixing conduc-\ntance are given by integrating ai-randbi-rusing Eq. 35.\nThe mixing conductance is also generalized to include a\ntransmitted spin mixing conductance given by integrat-\ningatandbtusing Eq. 35. The concept of a transmitted\nmixing conductance has been discussed before117and de-\nscribes the part of the transverse spin current transmit-\nted through the interface. The factor fc( \u000b) generalizes\nthe real part of the mixing conductance to capture fea-\ntures of interfacial spin-orbit coupling.\nThe top left 2\u00022 block relates the transverse spin\naccumulations at z= 0\u0006to the transverse spin currents\natz= 0\u0006and the transverse spin accumulation at z= 0.19\nFIG. 13. Breakdown of the S\u0017matrices (\u00172[i-r;t]) when spin or charge accumulations drive transport at interfaces. The\nmatrixStdetermines the spin and charge accumulation \u0016at the interface (see Eq. 28). The symmetric response \u0016S=Si-r+St\ndetermines the average spin current \u0016jzat the interface (see Eq. 29). The antisymmetric response \u0001 S=Si-r\u0000Stdetermines the\ndi\u000berence in spin current \u0001 jzacross the interface (see Eq. 30). The matrix column speci\fes the spin and charge accumulations\natz= 0\u0006while the row gives the components of \u0016,\u0016jz, or \u0001 jz, depending on whether Eq. 28, Eq. 29, or Eq. 30 is used. The\nimages depict the charge accumulations (gold spheres) or the spin accumulations (gold spheres with arrows) that drive the\nsystem and the resulting spin currents at z= 0\u0006, where block arrows denote \row direction and tubular arrows denote spin\ndirection.\nThe transverse spin currents are not conserved across the\ninterface since ai-r6=atandbi-r6=bt. For vanishing spin-\norbit coupling, the transverse spin current at z= 0\u0000\ngives the total spin transfer torque.\nThe bottom right 2 \u00022 block relates the longitudinal\nspin accumulation and charge accumulation at z= 0\u0006\nto the longitudinal spin current and charge current at\nz= 0\u0006. Because this block depends only on candd,\nthe longitudinal spin currents governed by this block are\nconserved across the interface.\nSpin Memory Loss |Finally, the third matrix in Eq. 33\ncaptures spin memory loss. Because this matrix is mul-\ntiplied by fs( \u000b)2, it vanishes for zero spin-orbit coupling\n(\u000b=\u0019=2). The three nonzero matrix elements parame-\nterize spin memory loss, which we describe as a magni-\ntude di\u000berence in the spin current driven by spin/charge\naccumulations at z= 0\u0006. While all spin components ex-\nperience spin memory loss (since ai-r6=at), the degree of\nspin memory loss di\u000bers for transverse and longitudinal\nspin currents.\nWhen the magnetization is oriented in any other di-\nrection than normal to the interface, the form of these\nresults becomes much more complicated. The four by\nfour matrix loses the diagonal two by two simpli\fcation.\nHowever, the same processes described above still take\nplace, though their e\u000bects change quantitatively and getspread out throughout the matrix.\n5. In-plane spin or charge currents drive the system\nThe choice of g=~kxq\u0006corresponds to an in-plane\ndriving current at z= 0\u0006(here \rowing along x). The\ncomponents qx,qy, andqzdescribe the spin polarization\nof thex-\rowing spin current driving the system while qc\ndescribes the x-\rowing charge current driving the system.\nFor this system, the S\u0017matrices are given by:\nS\u0017= fs(\u000b)0\nB@0 0 \u0000b\u0017 0\n0 0 fc( \u000b)(a\u0017\u0000c)\u0000d\nb\u0017fc(\u000b)(a\u0017\u0000c) 0 0\n0\u0000d 0 01\nCA:\n(38)\nThe parameters a\u0017,b\u0017,c, anddare the same as those\nused in Eq. 33. Since S\u0017is proportional to fs( \u000b), we\nsee that a nonzero interfacial spin-orbit interaction is re-\nquired to couple an in-plane driving current with out-of-\nplane spin currents or a spin torque. As a reminder, the\nindices in order are [ x;y;z;c ]. Below, we discuss the im-\nportant features of Eq. 38, which are also illustrated in\nFig. 14.20\nFIG. 14. Breakdown of the S\u0017matrices when in-plane spin/charge currents drive transport at interfaces. As in Fig. 13,\nthe matrix Stdetermines the spin/charge accumulation \u0016at the interface (see Eq. 28), the symmetric response \u0016S=Si-r+St\ndetermines the average spin current \u0016jzat the interface (see Eq. 29), and the antisymmetric response \u0001 S=Si-r\u0000Stdetermines\nthe di\u000berence in spin current \u0001 jzacross the interface (see Eq. 30). The column speci\fes the in-plane spin/charge currents at\nz= 0\u0006while the row gives the components of \u0016,\u0016jz, or \u0001 jz, depending on whether Eq. 28, Eq. 29, or Eq. 30 is used. The images\ndepict both the in-plane and out-of-plane spin currents at z= 0\u0006using block arrows for \row direction and tubular arrows for\nspin direction.\nGeneralized Rashba-Edelstein e\u000bect |First, we show\nthat in-plane currents create interfacial spin accumula-\ntions. The spin/charge accumulations at z= 0 are given\nby\n\u0016(0) =Z\u0019=2\n0d\u0012w(\u0012)St(\u0012)\u0016q (39)\nand thus governed by the Stmatrix. As seen from Eq. 38,\nthe parameter\u0000din the second row, fourth column of\nStrelates an in-plane charge current to the spin accu-\nmulation along y. This describes the Rashba-Edelstein\ne\u000bect. Spin accumulations in other directions only arise\nwhen in-plane spin currents drive the system. In ferro-\nmagnet/nonmagnet bilayers, an in-plane charge current\nbecomes spin polarized along the magnetization ^min the\nferromagnetic layer (here ^m=^z). According to the \frst\nrow, third column in Eq. 38, an in-plane spin current\npolarized along zcreates a spin accumulation along x.\nTo describe these additional spin accumulations arising\nfrom in-plane spin currents, we use the term generalized\nRashba-Edelstein e\u000bect .\nSpin-orbit \fltering |Second, we show that an in-plane\ncharge current (here along x) generates an out-of-plane\nspin current (here along z) with spin direction along y.\nThis spin current shares the same orientation as the spin\nHall current. According to our model, when in-plane\ncurrents di\u000ber at z= 0\u0000andz= 0+, an out-of-plane\nspin/charge current develops across the interface. This\nout-of-plane current is given by\n\u0016jz=Z\u0019=2\n0d\u0012v(\u0012)\u0016S(\u0012)\u0001q (40)where \u0016S=Si-r+St. According to Eq. 38, in-plane charge\ncurrents create out-of-plane spin currents with spin di-\nrection along y. Although not strictly an inverse e\u000bect,\nEq. 38 suggests that in-plane spin currents with spin di-\nrectionyalso result in out-of-plane charge currents. Both\nof these e\u000bects are proportional to d. We call both ef-\nfects spin-orbit \fltering , because they result from elec-\ntron spins being \fltered by the spin-orbit \feld while scat-\ntering o\u000b the interface.\nSpin-orbit precession |Finally, we show that in-plane\nspin currents generate out-of-plane spin currents at in-\nterfaces. To see this, note that the parameter b\u0017(which\nappears twice in Eq. 38) describes the following two cases:\n1) anx-\rowing spin current with z-spin direction creates\naz-\rowing spin current with x-spin direction and 2) an\nx-\rowing spin current with x-spin direction creates a z-\n\rowing spin current with z-spin direction. Both of these\ncases are phenomenologically identical to spin swapping ,\nwhere nonmagnets convert spin currents into other spin\ncurrents by swapping their \row and spin directions. How-\never, the terms in Eq. 38 proportional to fc( \u000b)(a\u0017\u0000c)\ndo not follow the spin swapping mechanism, but never-\ntheless convert in-plane spin currents into out-of-plane\nspin currents. To unify these concepts, we refer to this\nfamily of e\u000bects at interfaces as spin-orbit precession , be-\ncause they result from electron spins rotating about the\nspin-orbit \feld while scattering o\u000b the interface.\nThe spin currents generated at interfaces are not neces-\nsarily identical at z= 0\u0000andz= 0+. This discontinuity21\nin spin current across the interface is given by\n\u0001jz=Z\u0019=2\n0d\u0012v(\u0012)\u0001S(\u0012)\u0016q; (41)\nwhere nonvanishing terms in the antisymmetric response\n\u0001S=Si-r\u0000Stcontribute to the discontinuities. Inspec-\ntion of Eq. 38 reveals that spin-orbit precession currents\nare discontinuous at the interface. In general, both spin-\norbit \fltering and spin-orbit precession currents can be\ndiscontinuous at the interface. Here, the continuity of\nspin-orbit \fltering currents is a result of the simplicity of\nthis model.\nTogether, Eq. 33 and Eq. 38 capture the processes that\ncontribute to spin-orbit torques when the magnetization\nis perpendicular to the interface. Eq. 38 captures the\ndirect processes due to an in-plane electric \feld at the\ninterface between two di\u000berent materials and Eq. 33 cap-\ntures the processes that are initiated in the interior of\nthe layers through e\u000bects like the spin Hall e\u000bect that\ngives rise to a spin current scattering from the interface.\nThe relevant parts of the incoming distribution functions\nare combined with the relevant S\u0017matrices to give the\ninterfacial torques through the interfacial spin accumula-\ntion. The same matrices give the outgoing spin currents.\nThose directed into the ferromagnet typically dephase\nand contribute to the torque on that layer. Those di-\nrected into the non-magnetic layer can traverse that layer\nand in trilayers contribute to the torque on the other fer-\nromagnetic layer.\nIV. OUTLOOK\nIn the previous section, we introduced a quasi-\nanalytical model that captures how spin-orbit scattering\nat interfaces generates out-of-plane spin and charge cur-\nrents and spin torques. These currents and torques were\nstudied for two driving mechanisms: 1) spin/charge ac-\ncumulations form on each side of the interface and 2)\nin-plane spin/charge currents \row on each side of the in-\nterface. The system could be nonmagnetic or contain\na ferromagnetic layer. In the latter case, magnetism at\nthe interface came from an interfacial exchange interac-\ntion while magnetism in the bulk layers was omitted in\nthe electronic structure; however, the spin-polarized cur-\nrent in the ferromagnetic layer was captured via spin-\ndependent momentum relaxation times. When in-plane\nspin currents drive the system, we allow their spin direc-\ntion to be longitudinal or transverse to the ferromagnetic\nlayer's magnetization, capturing symmetry-allowed spin\ncurrents that are typically not considered in such sys-\ntems.\nAlthough the bulk layers in the model have a triv-\nial electronic structure, the driving mechanisms we con-\nsider are fairly general, allowing exploration of many\nscenarios, albeit qualitatively. For instance, in nonmag-\nnet/ferromagnet bilayers, the spin Hall e\u000bect generates a\nspin accumulation at the interface which exerts a torqueon the ferromagnetic layer. The spin Hall e\u000bect arises\nfrom an in-plane charge current, and we \fnd that this in-\nplane charge current also generates an out-of-plane spin\ncurrent at the interface. This interface-generated spin\ncurrent can have a di\u000berent spin direction than the spin\nHall current, thus enabling di\u000berent torques.\nThe model also describes the role of in-plane spin cur-\nrents in the ferromagnetic layer when generating spin cur-\nrents and spin torques. For example, in-plane charge cur-\nrents are spin-polarized in ferromagnets along the mag-\nnetization direction. Near the interface, the electrons\ncarrying this spin-polarized current interact with inter-\nfacial spin-orbit \felds, which rotate their spin polariza-\ntion and generate spin accumulations not captured by\nthe two-dimensional inverse galvanic e\u000bect (or Rashba-\nEdelstein e\u000bect). We also consider in-plane spin cur-\nrents with spin direction transverse to the magnetization,\nwhich are allowed by symmetry but not well studied in\nthe context of spin-orbit torque. These in-plane spin cur-\nrents generate out-of-plane spin currents that also exert\ntorques not predicted by traditional models that omit\nthree-dimensional spin-orbit scattering. We show that\nthis family of e\u000bects, which we called spin-orbit preces-\nsion, includes phenomena like spin swapping that was\n\frst predicted in nonmagnets84and later studied in fer-\nromagnetic systems.85,86\nMoving away from bilayers, we can also consider spin\ncurrents created in other layers not adjacent to the in-\nterface, as in spin valves. Such spin currents can eventu-\nally \row across the interface and undergo spin memory\nloss. If one of the layers adjacent to the interface is ferro-\nmagnetic, the degree of spin memory loss di\u000bers for spin\ncurrents with transverse and longitudinal spin directions\n(where transverse and longitudinal are de\fned relative to\nthe magnetization).\nThe phenomena discussed here only scratch the sur-\nface of what is allowed at interfaces with spin-orbit cou-\npling. Various magnetoresistance e\u000bects (like the spin\nHall magnetoresistance) should be a\u000bected by spin-orbit\nscattering at interfaces. Following the methods in this\npaper, one may extend our model to describe how in-\nplane electric \felds generate in-plane spin and charge\ncurrents near interfaces that are modulated by magne-\ntization direction. Thus, simple extensions to this model\nshould capture the e\u000bect of interfacial spin-orbit scatter-\ning on current-in-plane magnetoresistance e\u000bects.\nExperiments have yet to verify many of these theo-\nretical predictions. Part of the di\u000eculty comes from\nthe lack of reliable experimental techniques to indepen-\ndently quantify bulk and interfacial contributions to spin\ntorques. We do not o\u000ber a solution to this problem. How-\never, some of the di\u000eculty also arises from bilayer sys-\ntems, where the sum of several e\u000bects are lumped into a\nsingle measurement. Experiments in ferromagnetic mul-\ntilayers have already shown the existence of competing\ntorques that each damp the magnetization towards two\nseparate axes;36,39,40this phenomena could be explained\nby the spin-orbit precession e\u000bects discussed earlier. By22\ngiving a clear, qualitative picture of what interfacial spin-\norbit scattering enables, we hope to guide new experi-\nments that can probe these e\u000bects (perhaps in uncon-\nventional heterostructures), and motivate new methods\nto electrically control magnetization dynamics.\nACKNOWLEDGMENTS\nWork by V.P.A. was supported by Quantum Materi-\nals for Energy E\u000ecient Neuromorphic Computing, an\nEnergy Frontier Research Center funded by the U.S.\nDepartment of Energy (DOE), O\u000ece of Science, Basic\nEnergy Sciences (BES), under Award #DE-SC0019273.\nThe authors appreciate useful comments from Robert\nMcMichael, Jabez McClelland, Hans Nembach, Ivan\nSchuller, Andrew Kent, and Axel Ho\u000bmann.\nAppendix A: Spin torques in bilayers\nFirst, we derive the quantum mechanical scattering\namplitudes relevant to the phenomenological model. In\nthis model, only the interface between layers has mag-\nnetism, which is captured by an e\u000bective magnetic \feld\nB. A free electron gas describes the bulk of each layer\nwhile a delta function potential describes the interface.\nAlthough this model is three-dimensional, it reduces to\nthe one-dimensional model derived earlier for each in-\ncoming electron, except now we relax the condition that\nu\"!1 andu#= 0.\nThe 2\u00022 Hamiltonian for the system is,\nH(^r) =~2k2\n2mI2\u00022+\u000e(z)\u0000\nV0I2\u00022+Jex\u001b\u0001^B\u0001\n(A1)\nwhere the spin-independent potential V0and interfacial\nexchange energy Jexcan be written as:\nV0=~2kF(u\"+u#)=2m (A2)\nJex=~2kF(u\"\u0000u#)=2m (A3)\nHerekFis the Fermi momentum (which is the same for\nboth layers) and u\"=#is the unitless spin-dependent bar-\nrier strength at the interface.\nAlternatively, we can write this Hamiltonian explicitly\nin the spin basis aligned with the e\u000bective magnetic \feld\nB:\nH(^r) =~2\nm\u0012\nk2=2 +\u000e(z)kFu\" 0\n0k2=2 +\u000e(z)kFu#\u0013\n(A4)\nIn this form, the problem reduces to two independent\nchannels for spins parallel or antiparallel with B.\nConsider an electron scattering o\u000b the interface. The\nelectron arrives at the interface in one layer (layer 1) and\nis either re\rected back into this layer or transmitted into\nthe other layer (layer 2). Assuming that during scat-\ntering, the electron's in-plane momentum is conserved(specular scattering), the wavefunctions in layers 1 and\n2 are given by\n 1(r) =eik?\u0001r?\u0000\n\u001fIeikzz+\u001fRe\u0000ikzz\u0001\n; (A5)\n 2(r) =eik?\u0001r?\u001fTeikzz; (A6)\nwherezis the out-of-plane direction, kzis the out-of-\nplane component of momentum, and r?andk?are the\nin-plane position and momentum vectors, such that k=\n(k?;kz) andr= (r?;z). The spinors \u001fI,\u001fR, and\u001fT\ndescribe the incoming, re\rected, and transmitted states\nrespectively.\nThe re\rected and transmitted wavefunctions are re-\nlated to the incoming wavefunction through the scatter-\ning matrices. Thus, we may assume, for some 2 \u00022 ma-\ntricesrandtthat\u001fR=r\u001fIand\u001fT=t\u001fI. Assuming\nthe interface lies at z= 0, we have:\n 1= (1 +r)\u001fI;\n@z 1= ikz(1\u0000r)\u001fI;\u001b\nz= 0\u0000(A7)\n 2=t\u001fI;\n@z 2= ikzt\u001fI;\u001b\nz= 0+(A8)\nDue to in-plane momentum conservation, the scatter-\ning problem has now been reduced to a one-dimensional\nproblem de\fned along z.\nBoundary conditions dictate that the wavefunction\nand particle current match at z= 0\u0000andz= 0+, which\ngives:\n1 +r=t; (A9)\n1\u0000ryr=tyt: (A10)\nThe latter condition arises from matching the probability\ncurrent\nj=~\n2mi\u0010\n y(@z )\u0000(@z y) \u0011\n; (A11)\natz= 0\u0000andz= 0+, and can be checked using Eqs. A7\nand A8. The spin density ( si) and out-of-plane \rowing\nspin current ( Qzi) are\nsi= y\u001bi (A12)\nQzi=~\n2mi\u0010\n y\u001bi(@z )\u0000(@z y)\u001bi \u0011\n; (A13)\nwhere\u001biare the Pauli matrices corresponding to direc-\ntionsi2[x0;y0;z0] in spin space, where as before z0jjB.\nIn this notation, Qzz0describes the spin current \row-\ning out-of-plane ( z) with spin direction aligned with B\n(i.e.z0), whileQzx0andQzy0describe the spin currents\n\rowing out-of-plane with spin direction transverse to B.\nUsing Eqs. A7 and A8 and Eqs. A12 and A13, the spin\ndensity and spin currents near the impurity are:\ns0\ni=\u001fy\nI\u0000\nty\u001bit\u0001\n\u001fI (A14)\nQ0\u0000\nzi=~kz\nm\u001fy\nI\u0000\n\u001bi\u0000ry\u001bir\u0001\n\u001fI (A15)\nQ0+\nzi=~kz\nm\u001fy\nI\u0000\nty\u001bit\u0001\n\u001fI (A16)23\nα↑ α↓kz*\nu↑ u↓\nFIG. 15. Visual representation of the relationship between\nthe angles\u000b\"=#, the barrier strengths u\"=#, and the scaled z-\nvelocityk\u0003\nz=kz=kF. In the limit that u#= 0 andu\"!1 ,\n\u000b#=\u0019=2 and\u000b\"!0.\nThe re\rection ( r) and transmission ( t) matrices are di-\nagonal in spin space,\nr=\u0012\nr\"0\n0r#\u0013\n; t=\u0012\nt\"0\n0t#\u0013\n; (A17)\nwhere as before the \"=#labels denotes the spin aligned\nor opposite to the interfacial magnetic \feld B.\nBased on the Hamiltonian given by Eq. A4, the spin-\ndependent re\rection and transmission amplitudes are:\nr\"=#=u\"=#\nik\u0003z\u0000u\"=#t\"=#=ik\u0003\nz\nik\u0003z\u0000u\"=#(A18)\nwherek\u0003\nz=kz=kFis the out-of-plane component of the\nincident crystal momentum (along ^z) scaled by the Fermi\nmomentum. We can further simplify this notation by\nintroducing an angle \u000b\"=#(de\fned geometrically in Fig.\n15) such that:\ncos(\u000b\"=#) =u\"=#p\n(k\u0003z)2+ (u\"=#)2; (A19)\nsin(\u000b\"=#) =k\u0003\nzp\n(k\u0003z)2+ (u\"=#)2: (A20)\nWithout loss of generality, we may assume that k\u0003\nzand\nu\"=#are either zero or positive de\fnite, so that \u000b\"=#2\n[0;\u0019=2]. The scattering amplitudes then become:\nr\"=#=\u0000ei\u000b\"=#cos(\u000b\"=#) (A21)\nt\"=#=\u0000iei\u000b\"=#sin(\u000b\"=#) (A22)\nUsing these scattering amplitudes, we may determine\nthe fate of an incident electron spin oriented transverse\nto the e\u000bective magnetic \feld at the interface. Say the\nincident electron spin points along x0, which corresponds\nto:\n\u001fI=1p\n2\u0012\n1\n1\u0013\n: (A23)\nThe re\rected and transmitted spinors are then:\n\u001fR=\u00001p\n2\u0012\nei\u000b\"cos(\u000b\")\nei\u000b#cos(\u000b#)\u0013\n; (A24)\n\u001fT=\u0000ip\n2\u0012\nei\u000b\"sin(\u000b\")\nei\u000b#sin(\u000b#)\u0013\n: (A25)Let us pause to connect back to the main text, in which\nu\"!1 andu#= 0. In this limit, \u000b\"!0 and\u000b#=\u0019=2,\nwhich gives\n\u001fR!\u0012\n\u00001\u0000i\u000b\"\n0\u0013\n=\u0012\n\u00001\u0000ik\u0003\nz=u\"\n0\u0013\n; (A26)\n\u001fT!\u0012\n\u0000i\u000b\"\n1\u0013\n=\u0012\n\u0000ik\u0003\nz=u\"\n1\u0013\n; (A27)\nto \frst order in \u000b\". Note that we have dropped the nor-\nmalization constant here. In the previous section, the\nimaginary part of \u001fTgives rise to the transverse spin\ndensity required to have a spin torque. Here, we show\nquantitatively that the absorbed spin current equals the\nspin torque.\nTo verify that the absorbed spin current, given by the\ndiscontinuity in spin current across the interface,\n\u0001Qz=Q0\u0000\nz\u0000Q0+\nz (A28)\nequals the spin torque\n\u001c=~kF\nm(u\"\u0000u#)s0\u0002^B; (A29)\nwe evaluate the expression \u001c= \u0001Qzusing Eqs. A14-A16\nand Eqs. A21 and A22. To prove these two quantities are\nequal, it is easier to divide both by k\u0003\nz, yielding:\n\u001c=k\u0003\nz= \u0001Qz=k\u0003\nz=~kF\nm0\n@sin2(\u0001\u000b)\nsin(\u0001\u000b) cos(\u0001\u000b)\n01\nA (A30)\nThe \fnal expression for the torque (scaled by k\u0003\nz) de-\npends only on the di\u000berence in angles \u0001 \u000band the Fermi\nmomentum.\nFrom Eq. A30 we see that the spin torque at the inter-\nface equals the drop in spin current across the interface.\nThe lost spin current was absorbed by the magnetic part\nof the interface, which resulted in the torque. Further-\nmore, we see that the spin current component Qzz0is\ncontinuous across the interface (i.e. \u0001 Qzz0= 0).\nAppendix B: Phenomenological Theory of Spin Transport\nat Interfaces with Spin-Orbit Coupling\nThe presence of spin-orbit coupling at interfaces\ngreatly complicates spin transport because spin-orbit\ncoupling opens a channel for angular momentum transfer\nto and from the atomic lattice. Since nothing in prin-\nciple restricts the direction of angular momentum \row\nbetween conduction electrons and the atomic lattice, in-\nterfacial spin-orbit coupling has two consequences: 1)\nspin currents may give some angular momentum to the\natomic lattice when \rowing across the interface and 2)\nthe atomic lattice may generate spin currents at the in-\nterface. The former is called spin memory loss and the\nlatter is called interface-generated spin currents .24\nOur goal is to develop a simple-enough model that\nqualitatively describes spin memory loss and interface-\ngenerated spin currents, as well as other features of spin\ntransport at interfaces with spin-orbit coupling. While\nquantitative estimates of these phenomena have been ob-\ntained from \frst principles calculations, a simple model\nhelps to introduce the wide variety of phenomena driven\nand/or in\ruenced by interfacial spin-orbit coupling.\nBy assuming various boundary conditions, we can\nqualitatively describe the spin currents and spin torques\nresulting from both in-plane and out-of-plane electric\n\felds, as well as from spin currents generated elsewhere\nin the system. Here, boundary conditions refer to our\nchoice of the spin and occupation probability of carri-\ners incident to the interface. Such freedom in bound-\nary conditions enables a description of several important\nphenomena within the same model, including spin mem-\nory loss, interface-generated spin currents, the e\u000bect of\nspin-orbit coupling on the spin mixing conductance, spin\ntransfer torques, and spin-orbit torques.\nFirst, how do we describe the occupation of carriers\nin a given state? Here, we use a semiclassical descrip-\ntion based on the spin-dependent Boltzmann equation.\nThe simplest relevant description, introduced by Cam-\nley and Barnas,35assumes carrier spins are parallel or\nantiparallel to a given axis, and that carriers of each\nspin species are described by a separate occupation func-\ntionf\"=#(r;k). The occupation function f\"=#(r;k) is the\nprobability to \fnd a carrier with spin \"or#at position\nrwith momentum k. However, to describe spins along\nmultiple (non-collinear) axes, a more general formalism\nis required. We could, for instance, assign an occupation\nfunction to parallel and antiparallel spins along all three\nCartesian axes, giving six occupation functions fis(r;k)\nfori2[x;y;z ] ands2[\";#]. However, two of these\noccupation functions are redundant if we only wish to\nkeep track of the spin polarization and the total charge\ndensity, in which case we may write four occupation func-\ntions instead:\nfi(r;k) =fi\"\u0000fi#fori2[x;y;z ] (B1)\nfc(r;k) =X\ni(fi\"+fi#): (B2)\nFor simplicity, let us assume that the Boltzmann distri-\nbution varies along zbut is isotropic along xandy. For\nsystems just out of equilibrium, we describe the pertur-\nbation of the distribution function as follows\nf\u000b(z;k) =feq(\u000fk)\u000e\u000bc+@feq\n@\u000fkg\u000b(z;k) (B3)\nwhere\u000fkis thek-dependent energy, feqis the (spin-\nindependent) equilibrium distribution function, and g\u000b\nis the nonequilibrium perturbation of the distribution\nfunction. Note that the distribution functions can be\narranged as four-vectors (i.e. f\u000b!f) with components\ndenoted by \u000b2[x;y;z;c ]. In the four-vector notation,we have\nf(z;k) = feq(\u000fk) +@feq\n@\u000fkg(z;k) (B4)\nwhere\nfeq(\u000fk) =0\nB@0\n0\n0\nfeq(\u000fk)1\nCA;g(z;k) =0\nB@gx(z;k)\ngy(z;k)\ngz(z;k)\ngc(z;k)1\nCA:(B5)\nThe Boltzmann equation is an integro-di\u000berential equa-\ntion that can be used to solve for gas a function of po-\nsition and momentum. We omit details of solving the\nBoltzmann equation here, and instead refer the reader\nto Ref. 116. However, it is important to note that, when\nsolving the Boltzmann equation, boundary conditions are\nneeded at interfaces and these can be supplied by quan-\ntum mechanical scattering amplitudes. For instance, at\nan interface ( z= 0), the nonequilibrium distribution of\nincident states is related to the re\rected and transmitted\nstates like so\ng(0\u0000;kx;ky;\u0000kz) =R(k)g(0\u0000;kx;ky;kz)\n+T(k)g(0+;kx;ky;\u0000kz) (B6)\ng(0+;kx;ky;kz) =T(k)g(0\u0000;kx;ky;kz);\n+R(k)g(0+;kx;ky;\u0000kz) (B7)\nwhereR(k) andT(k) are 4\u00024 matrices describing re-\n\rection and transmission respectively. The RandTma-\ntrices used in Eqs. B6 and B7 are the same regardless\nof what layer the carriers are incident from because we\nassume the layers are identical in equilibrium. We re-\nmind the reader that in this model, the nonequilibrium\ndistribution function gcaptures the di\u000berences in each\nlayer.\nWe can simplify this notation for spherical Fermi sur-\nfaces, where the incident, re\rected, and transmitted dis-\ntribution functions are de\fned on hemispheres speci\fed\nby the sign of kz. Thus, we may write\ngR(0\u0000;kjj) =R(kjj)gI(0\u0000;kjj) (B8)\ngT(0+;kjj) =T(kjj)gI(0\u0000;kjj) (B9)\ngR(0+;kjj) =R(kjj)gI(0+;kjj) (B10)\ngT(0\u0000;kjj) =T(kjj)gI(0+;kjj); (B11)\nwherekjj= (kx;ky) is the in-plane crystal momentum of\nthe incoming electrons and the superscripts I,RandT\ndenote the incident, re\rected and transmitted distribu-\ntion functions respectively.\nThe last step in setting up the calculation is to relate\nthe 4\u00024 Boltzmann interface scattering matrices Rand\nTto the 2\u00022 quantum mechanical scattering matrices\nr(kjj) andt(kjj) that were derived in earlier sections:\n[R(kjj)]\u000b\f=1\n2tr[ry(kjj)\u001b\u000br(kjj)\u001b\f] (B12)\n[T(kjj)]\u000b\f=1\n2tr[ty(kjj)\u001b\u000bt(kjj)\u001b\f] (B13)25\nWe omit the derivation of these equations here, which\ncan be found in Ref. 34. The expression for the charge\nand spin currents \rowing in direction i(i2[x;y;z ]) are\nji(z) =e\n~(2\u0019)3Z\nFSdkki\nkFgc(z;k) (B14)\nQis(z) =1\n2(2\u0019)3Z\nFSdkki\nkFgs(z;k) (B15)\nin units of A =m2(charge current density) and J =m2(an-\ngular momentum current density) respectively. We can\ncombine these de\fnitions into a single de\fnition\nji\u000b(z) =cjZ\nFSdkki\nkFg\u000b(z;k) (B16)\nwherecj=e=~(2\u0019)3and\u000b2[x;y;z;c ] as before. Note\nthat the spin current tensor elements ( \u000b=x;y;z for any\ni) are given in units of charge current density and can be\nconverted back to an angular momentum current density\nby multiplying by ~=2e. In the main text, we also de\fne\nthe spin/charge accumulation at z= 0 using the constant\nc\u0016=\u00001=4\u0019ek2\nF.\nIn what follows, we are only interested in the out-of-\nplane \rowing charge and spin currents (i.e. along ^z). We\ncan then rewrite the above expression in four-vector no-\ntation at the interface as:\njz(0\u0006) =cjZ\nFSdkkz\nkFg(0\u0006;k): (B17)\nWe know that the incident distribution functions (de-\n\fned on one hemisphere of the Fermi surface) at z= 0\u0006\nare related to the re\rected and transmitted distribution\nfunctions (de\fned on the other hemisphere) by Eqs. B8-\nB11. Thus we may write the total spin/charge currents\nin terms of the incident, re\rected, and transmitted con-\ntributions as follows,\njz(0\u0006) =\u0007\u0000\njI\nz(0\u0006)\u0000jR\nz(0\u0006)\u0000jT\nz(0\u0006)\u0001\n(B18)\n=\u0007cjZ\n2DBZdkjj(I\u0000R)gI(0\u0006)\u0000TgI(0\u0007)\n(B19)\nwhere the last line is rewritten as an integral over the\ntwo-dimensional Brillouin zone (2DBZ) spanned by kx\nandky. Note that the kjj-dependence of R,T, and gI\nhas been omitted for simplicity.\nAs before, we assume that carriers see an e\u000bective mag-\nnetic \feldB(k) =Bex+Bsoc(k) at the interface, where\nBexis the exchange \feld and Bsoc(k) is the momentum-\ndependent spin-orbit \feld. A free electron gas describes\nthe bulk of each layer while a delta function potential\ndescribes the interface. If B(k) points along ^z0(which\ncorresponds to the spin reference frame), then the Rand\nTmatrices computed using Eqs. B12 and B13 are givenby:\n\u0016R=0\nB@1\u0000ai-rbi-r 0 0\n\u0000bi-r1\u0000ai-r 0 0\n0 0 1\u0000c\u0000d\n0 0\u0000d1\u0000c1\nCA; (B20)\n\u0016T=0\nB@tat\u0000bt0 0\nbtat0 0\n0 0c d\n0 0d c1\nCA (B21)\nwhere all tensor elements are real-valued and depend on\nkxandky. The parameters canddare identical in both\ntensors and are a consequence of particle conservation\nduring scattering. Note that the matrices I\u0000RandT\nboth have the following form\n0\nB@a\u0017\u0000b\u00170 0\nb\u0017a\u00170 0\n0 0c d\n0 0d c1\nCA; (B22)\nwhere\u00172[i-r;t]. IfB(k) points along some general\ndirection, the scattering matrices become\nR(k) =O(k)\u0016R(k)O(k)y(B23)\nT(k) =O(k)\u0016T(k)O(k)y(B24)\nwhereOis any orthogonal transformation rotating the\nvector ^zto the direction parallel to B(k). By switching\nto spherical coordinates\nkx=kFsin(\u0012) cos(\u001e) (B25)\nky=kFsin(\u0012) sin(\u001e) (B26)\nkz=kFcos(\u0012); (B27)\nit becomes apparent that the \u0016Rand \u0016Tmatrices only de-\npend on\u0012while the orthogonal transformations Oencode\nthe\u001e-dependence. This is because the 2 \u00022 re\rection and\ntransmission matrices de\fned in Eq. A18 that are used\nto calculate \u0016Rand \u0016Tdepend only on kz, or alternatively,\nonly on\u0012. Thus, we may write:\nR(\u0012;\u001e) =O(\u0012;\u001e)\u0016R(\u0012)O(\u0012;\u001e)y(B28)\nT(\u0012;\u001e) =O(\u0012;\u001e)\u0016T(\u0012)O(\u0012;\u001e)y: (B29)\nIn spherical coordinates we can more easily write the ex-\nplict form of the Omatrices. For an out-of-plane mag-\nnetization, these matrices are given by\nO(r;\u001e) =0\nB@1 0 0 0\n0 fs(\u000b)(\u0012;\u000b)\u0000fc(\u000b)(\u0012;\u000b) 0\n0 fc(\u000b)(\u0012;\u000b) fs(\u000b)(\u0012;\u000b) 0\n0 0 0 11\nCA\n\u00020\nB@cos(\u001e) sin(\u001e) 0 0\n0 0 1 0\nsin(\u001e)\u0000cos(\u001e) 0 0\n0 0 0 11\nCA; (B30)26\nwhere the functions fs and fc are given in Eq. 35. We\nremind the reader that \u000bencodes the relative dependence\non the interfacial exchange and spin-orbit interactions,\nwhereuex=jue\u000bjcos(\u000b) anduR=jue\u000bjsin(\u000b).\nRewriting jz(0\u0006) in spherical coordinates gives\njz(0\u0006) =\u0007cjk2\nFZ\nd\u0012sin(\u0012)Z\nd\u001e (B31)\n\u0002(I\u0000R)gI(0\u0006)\u0000TgI(0\u0007):(B32)\nAs seen in the main text, it is convenient to analyze\nthe spin/charge currents and the distribution functions\nin terms of their average values and di\u000berence in values\nacross the interface, de\fned in Eq. 27. Some algebra\ngives:\n\u0001jz=cjk2\nFZ\nd\u0012sin(\u0012)Z\nd\u001e (B33)\n\u0002(I\u0000R\u0000T)\u0016gI(B34)\n\u0016jz=cjk2\nFZ\nd\u0012sin(\u0012)Z\nd\u001e (B35)\n\u0002(I\u0000R+T)\u0001gI(B36)\nPerforming the \u001eintegral is tedious but straightforward,\nwhile performing the \u0012integral is much more di\u000ecult\nand does not change the conceptual understanding of the\nmodel. In this spirit, we de\fne the following matrices:\nSn\ni-r=Z\nd\u001ecosn(\u001e)\u0000\nI\u0000R(\u0012;\u001e)\u0001\n(B37)\n=Z\nd\u001ecosn(\u001e)\u0000\nI\u0000O(\u0012;\u001e)\u0016R(\u0012)O(\u0012;\u001e)y\u0001\n(B38)\nSn\nt=Z\nd\u001ecosn(\u001e)T(\u0012;\u001e) (B39)\n=Z\nd\u001ecosn(\u001e)O(\u0012;\u001e)\u0016T(\u0012)O(\u0012;\u001e)y(B40)\nThe result of evaluating these integrals yields the expres-\nsions in Eq. 33 (for n= 0) and Eq. 38 (for n= 1) in the\nmain text. Using the de\fnition v(\u0012) =cjek2\nFsin(\u0012), we\nmay then write:\n\u0001jz=Z\nd\u0012v(\u0012)(S0\ni-r\u0000S0\nt)\u0016q (B41)\n\u0016jz=Z\nd\u0012v(\u0012)(S0\ni-r+S0\nt)\u0001q (B42)\nwhen g(0\u0006;k) =eq\u0006as de\fned in Eq. 19 and\n\u0001jz=Z\nd\u0012v(\u0012)(S1\ni-r\u0000S1\nt)\u0016q (B43)\n\u0016jz=Z\nd\u0012v(\u0012)(S1\ni-r+S1\nt)\u0001q (B44)\nwhen g(0\u0006;k) =e~kxq\u0006=ecos(\u001e)q\u0006as de\fned in Eq. 20.\nThe forms of both Eqs. B42 and B44 are quite similar,so following the main text, we write\n\u0001jz=Z\nd\u0012v(\u0012)\u0001S(\u0012)\u0016q (B45)\n\u0016jz=Z\nd\u0012v(\u0012)\u0016S(\u0012)\u0001q (B46)\nfor both choices of g, where the symmetric and antisym-\nmetric matrices are de\fned as\n\u0001S=Si-r\u0000St (B47)\n\u0016S=Si-r+St (B48)\nand the explicit form of Si-randStdepends on the choice\nofgas seen above.\n1D. Apalkov, B. Dieny, and J. Slaughter, Proceedings of the\nIEEE 104, 1796 (2016).\n2T. Hanyu, T. Endoh, D. Suzuki, H. Koike, Y. Ma, N. Onizawa,\nM. Natsui, S. Ikeda, and H. Ohno, Proceedings of the IEEE\n104, 1844 (2016).\n3W. Butler, X.-G. Zhang, T. Schulthess, and J. MacLaren, Phys-\nical Review B 63, 054416 (2001).\n4S. S. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes,\nM. Samant, and S.-H. Yang, Nature materials 3, 862 (2004).\n5S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and\nK. Ando, Nature materials 3, 868 (2004).\n6S.-W. Lee and K.-J. Lee, Proceedings of the IEEE 104, 1831\n(2016).\n7J. Grollier, D. Querlioz, and M. D. Stiles, Proceedings of the\nIEEE 104, 2024 (2016).\n8J. Grollier, D. Querlioz, K. Camsari, K. Everschor-Sitte,\nS. Fukami, and M. Stiles, Nature Electronics 3, 360 (2020).\n9K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa,\nand E. Saitoh, Physical review letters 101, 036601 (2008).\n10M. I. D'yakonov and V. I. Perel, 13, 467 (1971).\n11J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n12S. Zhang, Phys. Rev. Lett. 85, 393 (2000).\n13J. Slonczewski, Journal of Magnetism and Magnetic Materials\n159, L1 (1996).\n14L. Berger, Phys. Rev. B 54, 9353 (1996).\n15M. D. Stiles and J. Miltat, in Spin dynamics in con\fned mag-\nnetic structures III (Springer, 2006) pp. 225{308.\n16D. Ralph and M. Stiles, Journal of Magnetism and Magnetic\nMaterials 320, 1190 (2008).\n17L. Liu, O. Lee, T. Gudmundsen, D. Ralph, and R. Buhrman,\nPhysical Review Letters 109, 096602 (2012).\n18A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008).\n19V. Edelstein, Solid State Communications 73, 233 (1990).\n20I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V.\nCostache, S. Au\u000bret, S. Bandiera, B. Rodmacq, A. Schuhl, and\nP. Gambardella, Nature 476, 189 (2011).\n21A. Manchon, J. \u0014Zelezn\u0012 y, I. M. Miron, T. Jungwirth, J. Sinova,\nA. Thiaville, K. Garello, and P. Gambardella, Reviews of Mod-\nern Physics 91, 035004 (2019).\n22P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M. D.\nStiles, Phys. Rev. B 88, 214417 (2013).\n23F. Freimuth, S. Bl ugel, and Y. Mokrousov, Phys. Rev. B 90,\n174423 (2014).\n24F. Freimuth, S. Bl ugel, and Y. Mokrousov, Phys. Rev. B 92,\n064415 (2015).\n25G. G\u0013 eranton, F. Freimuth, S. Bl ugel, and Y. Mokrousov, Phys.\nRev. B 91, 014417 (2015).\n26G. G\u0013 eranton, B. Zimmermann, N. H. Long, P. Mavropoulos,\nS. Bl ugel, F. Freimuth, and Y. Mokrousov, Phys. Rev. B 93,\n224420 (2016).\n27V. P. Amin, J. Zemen, and M. D. Stiles, Phys. Rev. Lett. 121,\n136805 (2018).27\n28F. Mahfouzi and N. Kioussis, Physical Review B 97, 224426\n(2018).\n29K. Belashchenko, A. A. Kovalev, and M. Van Schilfgaarde,\nPhysical Review Materials 3, 011401 (2019).\n30K. Belashchenko, A. A. Kovalev, and M. van Schilfgaarde,\nPhysical Review B 101, 020407 (2020).\n31L. Wang, R. J. H. Wesselink, Y. Liu, Z. Yuan, K. Xia, and P. J.\nKelly, Phys. Rev. Lett. 116, 196602 (2016).\n32P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M. D.\nStiles, Phys. Rev. B 87, 174411 (2013).\n33V. P. Amin and M. D. Stiles, Phys. Rev. B 94, 104419 (2016).\n34V. P. Amin and M. D. Stiles, Phys. Rev. B 94, 104420 (2016).\n35R. E. Camley and J. Barna\u0013 s, Phys. Rev. Lett. 63, 664 (1989).\n36S. C. Baek, V. P. Amin, Y. Oh, G. Go, S.-J. Lee, M. D. Stiles,\nB.-G. Park, and K.-J. Lee, Nature Materials 17, 509 (2018).\n37T. Taniguchi, J. Grollier, and M. D. Stiles, Phys. Rev. Applied\n3, 044001 (2015).\n38V. P. Amin, J. Li, M. D. Stiles, and P. M. Haney, Phys. Rev.\nB99, 220405 (2019).\n39A. M. Humphries, T. Wang, E. R. J. Edwards, S. R. Allen, J. M.\nShaw, H. T. Nembach, J. Q. Xiao, T. J. Silva, and X. Fan, Na-\nture Communications 8(2017), doi:10.1038/s41467-017-00967-\nw.\n40Y. Hibino, K. Hasegawa, T. Koyama, and D. Chiba, APL Ma-\nterials 8, 041110 (2020), https://doi.org/10.1063/5.0002326.\n41P. M. Haney and M. D. Stiles, Physical Review Letters 105,\n126602 (2010).\n42D. Go, F. Freimuth, J.-P. Hanke, F. Xue, O. Gomonay, K.-J.\nLee, S. Bl ugel, H.-W. Lee, and Y. Mokrousov, arXiv preprint\narXiv:2004.05945 (2020).\n43W. Pratt Jr, S.-F. Lee, J. Slaughter, R. Loloee, P. Schroeder,\nand J. Bass, Physical Review Letters 66, 3060 (1991).\n44J. Bass and W. Pratt Jr, Journal of magnetism and magnetic\nmaterials 200, 274 (1999).\n45M. Tsoi, A. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi,\nand P. Wyder, Physical Review Letters 80, 4281 (1998).\n46E. Myers, D. Ralph, J. Katine, R. Louie, and R. Buhrman,\nScience 285, 867 (1999).\n47J. Katine, F. Albert, R. Buhrman, E. Myers, and D. Ralph,\nPhysical Review Letters 84, 3149 (2000).\n48J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey,\nPhysical Review Letters 74, 3273 (1995).\n49K. M. Schep, J. B. van Hoof, P. J. Kelly, G. E. Bauer, and J. E.\nIngles\feld, Physical Review B 56, 10805 (1997).\n50M. D. Stiles and D. R. Penn, Physical Review B 61, 3200 (2000).\n51.\n52M. D. Stiles and A. Zangwill, Journal of Applied Physics 91,\n6812 (2002).\n53A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev.\nLett. 84, 2481 (2000).\n54A. Brataas, Y. Nazarov, and G. Bauer, The European Physical\nJournal B - Condensed Matter and Complex Systems 22, 99\n(2001).\n55C. Galinon, K. Tewolde, R. Loloee, W.-C. Chiang, S. Olson,\nH. Kurt, W. Pratt Jr, J. Bass, P. Xu, K. Xia, et al. , Applied\nphysics letters 86, 182502 (2005).\n56J. Bass and W. P. Pratt Jr, Journal of Physics: Condensed\nMatter 19, 183201 (2007).\n57J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W. Savero, J.-\nP. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and\nH. Ja\u000br\u0012 es, Phys. Rev. Lett. 112, 106602 (2014).\n58K. Dolui and B. K. Nikoli\u0013 c, Physical Review B 96, 220403\n(2017).\n59K. Gupta, R. J. Wesselink, R. Liu, Z. Yuan, and P. J. Kelly,\nPhysical Review Letters 124, 087702 (2020).\n60H. Kurt, R. Loloee, K. Eid, W. P. Pratt, and J. Bass, Applied\nPhysics Letters 81, 4787 (2002).\n61H. Nguyen, W. P. Jr., and J. Bass, Journal of Magnetism and\nMagnetic Materials 361, 30 (2014).\n62S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348(2003).\n63J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and\nA. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004).\n64J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and\nT. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).\n65S. Murakami, in Advances in Solid State Physics , Advances in\nSolid State Physics, Vol. 45, edited by B. Kramer (Springer\nBerlin Heidelberg, 2006) pp. 197{209.\n66E. M. Hankiewicz and G. Vignale, Journal of Physics: Con-\ndensed Matter 21, 253202 (2009).\n67T. Jungwirth, J. Wunderlich, and K. Olejnk, Nature Materials\n11, 382 (2012).\n68A. Ho\u000bmann, IEEE transactions on magnetics 49, 5172 (2013).\n69B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Physical Review\nLetters 95, 066601 (2005).\n70T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima,\nK. Yamada, and J. Inoue, Physical Review B 77, 165117 (2008).\n71H. Kontani, T. Tanaka, D. Hirashima, K. Yamada, and J. In-\noue, Physical Review Letters 102, 016601 (2009).\n72D. Go, D. Jo, C. Kim, and H.-W. Lee, Physical Review Letters\n121, 086602 (2018).\n73D. Jo, D. Go, and H.-W. Lee, Physical Review B 98, 214405\n(2018).\n74D. Go and H.-W. Lee, Physical Review Research 2, 013177\n(2020).\n75W. Wang, T. Wang, V. P. Amin, Y. Wang, A. Radhakrishnan,\nA. Davidson, S. R. Allen, T. J. Silva, H. Ohldag, D. Balzar,\nB. L. Zink, P. M. Haney, J. Q. Xiao, D. G. Cahill, V. O. Lorenz,\nand X. Fan, arXiv:1902.05490 (2019).\n76C. Safranski, E. A. Montoya, and I. N. Krivorotov, Nature\nNanotechnology 14, 27 (2019).\n77H. Kurebayashi, J. Sinova, D. Fang, A. Irvine, T. Skinner,\nJ. Wunderlich, V. Nov\u0013 ak, R. Campion, B. Gallagher, E. Vehst-\nedt, et al. , Nature nanotechnology 9, 211 (2014).\n78S. R. Park, C. H. Kim, J. Yu, J. H. Han, and C. Kim, Phys.\nRev. Lett. 107, 156803 (2011).\n79S. Grytsyuk, A. Belabbes, P. M. Haney, H.-W. Lee, K.-J. Lee,\nM. D. Stiles, U. Schwingenschl ogl, and A. Manchon, Physical\nReview B 93, 174421 (2016).\n80S. LaShell, B. McDougall, and E. Jensen, Physical Review Let-\nters77, 3419 (1996).\n81G. Nicolay, F. Reinert, S. H ufner, and P. Blaha, Physical Re-\nview B 65, 033407 (2001).\n82H. Cercellier, Y. Fagot-Revurat, B. Kierren, F. Reinert,\nD. Popovi\u0013 c, and D. Malterre, Physical Review B 70, 193412\n(2004).\n83A. Bendounan, K. A \u0010t-Mansour, J. Braun, J. Min\u0013 ar, S. Borne-\nmann, R. Fasel, O. Gr oning, F. Sirotti, and H. Ebert, Physical\nReview B 83, 195427 (2011).\n84M. B. Lifshits and M. I. Dyakonov, Phys. Rev. Lett. 103, 186601\n(2009).\n85H. B. M. Saidaoui and A. Manchon, Phys. Rev. Lett. 117,\n036601 (2016).\n86C. O. Pauyac, M. Chshiev, A. Manchon, and S. A. Nikolaev,\nPhys. Rev. Lett. 120, 176802 (2018).\n87X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, and J. Q. Xiao,\nNature communications 4, 1 (2013).\n88X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V. O. Lorenz, and\nJ. Q. Xiao, Nature communications 5, 1 (2014).\n89T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm, J. G.\nJones, B. M. Howe, G. J. Brown, and N. X. Sun, Physical\nReview B 91, 214416 (2015).\n90Y. Hibino, K. Yakushiji, A. Fukushima, H. Kubota, and\nS. Yuasa, Phys. Rev. B 101, 174441 (2020).\n91H. Wu, X. Wang, L. Huang, J. Qin, C. Fang, X. Zhang, C. Wan,\nand X. Han, Journal of Magnetism and Magnetic Materials 441,\n149 (2017).\n92J. D. Gibbons, D. MacNeill, R. A. Buhrman, and D. C. Ralph,\nPhys. Rev. Applied 9, 064033 (2018).\n93A. Bose, D. D. Lam, S. Bhuktare, S. Dutta, H. Singh, Y. Jibiki,28\nM. Goto, S. Miwa, and A. A. Tulapurkar, Phys. Rev. Applied\n9, 064026 (2018).\n94Y. Omori, E. Sagasta, Y. Niimi, M. Gradhand, L. E. Hueso,\nF. Casanova, and Y. Otani, Phys. Rev. B 99, 014403 (2019).\n95C. Safranski, J. Z. Sun, J.-W. Xu, and A. D. Kent, Physical\nReview Letters 124, 197204 (2020).\n96B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys. Rev.\nLett. 111, 066602 (2013).\n97D. Tian, Y. Li, D. Qu, S. Y. Huang, X. Jin, and C. L. Chien,\nPhys. Rev. B 94, 020403 (2016).\n98K. S. Das, W. Y. Schoemaker, B. J. van Wees, and I. J. Vera-\nMarun, Phys. Rev. B 96, 220408 (2017).\n99I. Garate and A. H. MacDonald, Physical Review B 80, 134403\n(2009).\n100L. Zhu, L. Zhu, M. Sui, D. C. Ralph, and R. A. Buhrman,\nScience advances 5, eaav8025 (2019).\n101N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P.\nOng, Rev. Mod. Phys. 82, 1539 (2010).\n102A. Manchon and S. Zhang, Physical Review B 78, 212405\n(2008).\n103A. Manchon and S. Zhang, Physical Review B 79, 094422\n(2009).\n104K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov,\nS. Bl ugel, S. Au\u000bret, O. Boulle, G. Gaudin, and P. Gam-\nbardella, Nature nanotechnology 8, 587 (2013).\n105K.-S. Lee, D. Go, A. Manchon, P. M. Haney, M. D. Stiles, H.-W.\nLee, and K.-J. Lee, Physical Review B 91, 144401 (2015).\n106D. MacNeill, G. Stiehl, M. Guimaraes, R. Buhrman, J. Park,and D. Ralph, Nature Physics 13, 300 (2017).\n107D. MacNeill, G. M. Stiehl, M. H. Guimaraes, N. D. Reynolds,\nR. A. Buhrman, and D. C. Ralph, Physical Review B 96, 054450\n(2017).\n108M. H. Guimaraes, G. M. Stiehl, D. MacNeill, N. D. Reynolds,\nand D. C. Ralph, Nano letters 18, 1311 (2018).\n109G. M. Stiehl, R. Li, V. Gupta, I. El Baggari, S. Jiang, H. Xie,\nL. F. Kourkoutis, K. F. Mak, J. Shan, R. A. Buhrman, et al. ,\nPhysical Review B 100, 184402 (2019).\n110S. Shi, S. Liang, Z. Zhu, K. Cai, S. D. Pollard, Y. Wang,\nJ. Wang, Q. Wang, P. He, J. Yu, et al. , Nature nanotechnol-\nogy14, 945 (2019).\n111Q. Shao, G. Yu, Y.-W. Lan, Y. Shi, M.-Y. Li, C. Zheng, X. Zhu,\nL.-J. Li, P. K. Amiri, and K. L. Wang, Nano letters 16, 7514\n(2016).\n112K. Wang, J. Alzate, and P. K. Amiri, Journal of Physics D:\nApplied Physics 46, 074003 (2013).\n113K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner, A. Ghosh,\nS. Au\u000bret, O. Boulle, G. Gaudin, and P. Gambardella, Applied\nPhysics Letters 105, 212402 (2014).\n114J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami,\nT. Suzuki, S. Mitani, and H. Ohno, Nature materials 12, 240\n(2013).\n115M.-H. Nguyen, D. Ralph, and R. Buhrman, Physical Review\nLetters 116, 126601 (2016).\n116J. Xiao, A. Zangwill, and M. D. Stiles, Eur. Phys. J. B 59, 415\n(2007).\n117A. Brataas, G. Zar\u0013 and, Y. Tserkovnyak, and G. E. Bauer, Phys-\nical Review Letters 91, 166601 (2003)." }, { "title": "1703.05986v2.Spin_Degenerate_Regimes_for_Single_Quantum_Dots_in_Transition_Metal_Dichalcogenide_Monolayers.pdf", "content": "Spin-Degenerate Regimes for Single Quantum Dots in Transition Metal\nDichalcogenide Monolayers\nMatthew Brooks\u0003and Guido Burkard\nDepartment of Physics, University of Konstanz, D-78464, Germany\nStrong spin-orbit coupling in transition metal dichalcogenide (TMDC) monolayers results in spin\nresolvable band structures about the KandK0valleys such that the eigenbasis of a 2D quantum\ndot (QD) in a TMDC monolayer in zero \feld is described by the Kramers pairs j0i\u0000=jK0\"i,\nj1i\u0000=jK#iandj0i+=jK\"i,j1i+=jK0#i. The strong spin-orbit coupling limits the usefulness\nof single TMDC QDs as spin qubits. Possible regimes of spin-degenerate states, overcoming the\nspin-orbit coupling in monolayer TMDC QDs are investigated in both zero \feld, where the spin and\nvalley degrees of freedom become fourfold degenerate, and twofold degeneracy in some magnetic\n\feld, localised to a given valley. Such regimes are shown to be achievable in MoS 2, where the spin\norbit coupling is su\u000eciently low and of the right sign such that the spin resolved conduction bands\nintersect at points about the KandK0valleys and as such may be exploited by selecting suitable\ncritical dot radii.\nI. INTRODUCTION\nTransition metal dichalcogenide (TMDC) monolayers\nare atomically thin crystal layers exfoliated down from\nbulk weakly cohesive stacks. Similarly to graphene, a\nhexagonal lattice of alternating lattice sites results in\ntwo inequivalent, time-reversal symmetric valleys ( Kand\nK0), see Fig. 1 (b)1{4. Unlike graphene, the monolayer\ncrystals posses broken inversion symmetry, see Fig. 1 (a),\ninducing direct band gaps in the visible range about the\ntwo valleys5{7. Furthermore, strong spin-orbit coupling\nfrom the transition metal atoms introduces a strong cou-\npling between the spin and valley degrees of freedom, see\nFig. 1 (a)8{10. TMDCs are characterised by the chemical\ncomposition MX 2, where M denotes the transition metal\n(Mo or W) and X denotes the chalcogenide (S or Se). The\npresence of a direct band gap and spin-valley coupling in\na two-dimensional material allows for a number of inter-\nesting electronic, spintronic and valleytronic applications\nincluding room temperature quantum spin Hall insula-\ntors, optically pumped valley polarisation, long lived ex-\nciton spin polarisation and 2D quantum dots (QDs)11{16.\nWhile the strong spin-valley coupling of TMDC mono-\nlayers o\u000bers numerous interesting physical phenomena,\nit presents a di\u000eculty for qubit implementation in gated\nQDs. Kramers pairs of the spin and valley degrees of\nfreedom result from this coupling13,14,17. At low en-\nergy thej0i\u0000=jK0\"iandj1i\u0000=jK#istates are\ndegenerate in zero \feld and are energetically separate\nfrom thej0i+=jK\"iandj1i+=jK0#istates14,18,19.\nThis e\u000bect can be observed in the spin resolvable struc-\nture of the conduction band (CB) about the K(K0)\npoints20,21as shown in Fig. 1 (c). The obvious choice\nfor the computational basis of a qubit is therefore a spin-\nvalley qubit consisting of the two states of the lowest\nlying Kramers pair, j0i\u0000(j1i\u0000) in Mo X2andj0i+(j1i+)\nin WX2, where the required energy di\u000berence may be\nachieved by spin-valley Zeeman splitting induced by a\nperpendicular magnetic \feld22{25. However, such qubits\nare inherently limited by a necessity for coupling of the\nxzy\n(a) (b)\n(c)\nkEMoX2 WX2\nxy\nFIG. 1. (a) 3D view of a TMDC unit cell (red denoting M\natoms, blue denoting X atoms) showing the three sub layers\nof a TMDC monolayer and the broken inversion of the crystal\nlattice. (b) Planar (X-Y) view of a TMDC lattice. (c) Spin\nresolved conduction band (red: j0i\u0000=jK0\"iandj1i\u0000=\njK#i, blue:j0i+=jK\"iandj1i+=jK0#i) around the\nKvalley in the BZ of Mo and W based TMDC monolayers\ndemonstrating the spin crossings present in Mo TMDCs and\nnot in W, the K0valley may be visualised simply by the time-\nreversal of the given band structure.\nvalley states. Methods of doing so have been proposed in\ncarbon nanotubes by means of short range disorder in the\ndots22,26, requiring atomic level engineering, or by opti-\ncal manipulation27. Additionally, the valley coherence of\nWSe 2excitons has been measured28, demonstrating an\norder of magnitude lower coherence times than spin in\nother TMDC monolayer crystals11. If qubits in TMDC\nmonolayers could operate similarly to semiconductor spin\nqubits then the broad theoretical and experimental \fnd-arXiv:1703.05986v2 [cond-mat.mes-hall] 24 Apr 20182\nings of the \feld29{31may be directly utilised. In so doing,\na novel breed of 2D, optically active, direct band gap, and\nrelatively nuclear spin free15semiconductor spin qubits\nare gained without the need for an arti\fcially induced\nband gap, as is needed in graphene32. This requires a\nmethod of manipulating the dots such that the spin-orbit\ncoupling may be suppressed and regimes of pure spin\nqubits may be accessed.\nThere is a noticeable and useful di\u000berence between the\nlow energy band structures of Mo based and W based\nmonolayers as demonstrated in Fig. 1 (c): the band\ncrossings observed in the spin resolved CB structures\nin Mo monolayers which are absent in W monolayers\nwhich suggest that it is possible to achieve spin degener-\nacy localised within a given valley. Such spin-degenerate\nregimes o\u000ber the possibility of implementing the desired\npure spin qubits in TMDCs. Additionally, by placing a\nTMDC material in a perpendicular magnetic \feld, break-\ning time reversal symmetry, valley Zeeman splitting may\nbe introduced to the system. Previous work14has sug-\ngested that it may be possible to access regimes of spin\ndegeneracy within the same valley by introducing a large\nmagnetic \feld. In this work, we build upon previous anal-\nyses of TMDC QDs in a e\u000bective low energy regime by\nsolving for various conditions in which a spin qubit may\nbe viable, demonstrating a dot size tuneable spin-orbit\nsplitting and investigating the e\u000bects of a \fnite poten-\ntial well model as opposed to previous assumptions of an\nin\fnite potential.\nHere, we present methods of achieving spin degeneracy\nwithin a given valley of a QD in a TMDC monolayer at\nzero or moderate \felds. Firstly in Sec. II a zero exter-\nnal \feld model is discussed, demonstrating the Kramers\npairing of states as to derive an expression for a criti-\ncal radius at which fourfold spin-valley degeneracy may\nbe expected. Also we discuss the best candidate mono-\nlayer for a pure spin qubit. Then in Sec. III an exter-\nnal magnetic \feld perpendicular to the dot is considered\nand numerical solutions to the necessary external \feld\nstrengths at a given dot radius are shown at which a\nspin-degenerate state within a given valley is expected.\nNext the e\u000bects of \fnite con\fnement potential are shown\non the two previously discussed regimes is given in Sec.\nIV. Finally, an e\u000bective implementation regime for the\nvarious methods of achieving valley independent spin de-\ngeneracy is discussed in Sec. V before a summary is given\nin VI.\nII. ZERO FIELD\nTo describe a QD in monolayer TMDC the following\ne\u000bective low energy Hamiltonian about the KandK0\npoint in the CB is employed14\nHdot=H\u001c;s\nel+Hintr\nso+V=~2q+q\u0000\n2m\u001c;s\ne\u000b+\u001c\u0001cbsz+V:(1)Here,\u001c= 1(\u00001) refers to the KandK0valley,szgives\nthe spin Pauli-z matrix with eigenvalues s= 1(\u00001) for\nspin\"(#), wave number operators q\u0006=qx\u0006iqywhere\nqk=\u0000i@k, \u0001cbis the energy spliting in the CB due\nto the strong intrinsic spin-orbit coupling of the TMDC\nmonolayer and the spin-valley dependant e\u000bective elec-\ntron mass is de\fned as 1 =m\u001c;s\ne\u000b= 1=m0\nel\u0000\u001cs=\u000em e\u000bwhere\n\u000eme\u000bis material dependant. Initially, it is assumed that\nthe QD potential Vis su\u000eciently deep such that it may\nbe described by an in\fnite hard walled potential\nV=(\n0r\u0014RD\n1r>RD(2)\nwhereris the radial coordinate and RDis the radius of\nthe dot. This may be assumed in lieu of a harmonic po-\ntential, as is often used in bulk semiconductor QD mod-\nels, since the 2D nature of a TMDC allows for a more\ndirect interface between the gates and the plane in which\nan electron will be con\fned. Additionally, such an as-\nsumption allows for edge e\u000bects at the boundary of the\ndot to be neglected. In 2D polar coordinates, the wave\nnumber operators may be de\fned as\nq\u0006=\u0006ie\u0006i\u001e(\u0007@r\u0000i\nr@\u001e): (3)\nwhere\u001eis the angular coordinate. Assuming the dot\nto be circular, rotational symmetry about the z-axis dic-\ntates that the dot's Hamiltonian will commute and share\neigenstates with the z-component of the angular mo-\nmentum operator ( lz). This allows for the normalised\nsolution of the angular component of the wavefunction\n\t(r;\u001e) =R(r)\b(\u001e) to be given as\n\b(\u001e) =eil\u001e\np\n2\u0019: (4)\nSince the radial component of the wavefunction ob-\nserves the boundary condition R(RD) = 0, the fol-\nlowing expression is derived where jn;lis thenthzero\n(n= 1;2;3;:::) of thelthBessel function of the \frst\nkindJl(l= 0;\u00061;\u00062;:::)\nRn;l(r) =(\u00001)jlj\u0000l\n2p\n2Jjlj\u0010jn;jlj\nRDr\u0011\nRDjn;jlj+1: (5)\nAs such, the full normalised solutions of a hard wall\nTMDC quantum dot in zero external \feld are given in\nthe spinor form as\n\t\"\nn;l(r;\u001e) =eil\u001e\np\n2\u0019\u0012\n1\n0\u0013\nRn;l(r); (6a)\n\t#\nn;l(r;\u001e) =eil\u001e\np\n2\u0019\u0012\n0\n1\u0013\nRn;l(r); (6b)3\nand the spin, valley and dot radius dependant energy\neigenvalues are given as\nEn;l\n\u001c;s(RD) =~2j2\nn;jlj\n2m\u001c;s\ne\u000bR2\nD+\u001cs\u0001cb: (7)\nFrom the four realisations of spin and valley, only\ntwo separate energy solutions in zero \feld emerge, i.e.\nEn;l\nK;\"=En;l\nK0;#=En;l\n+andEn;l\nK0;\"=En;l\nK;#=En;l\n\u0000.\nThese two possible solutions describe the j0i+(j1i+) and\nj0i\u0000(j1i\u0000) Kramers pairs respectively. If the two solu-\ntions are assumed to be equivalent, then Eq. (7) may be\nused to describe the radius at which fourfold degeneracy\nin the valley-spin Hilbert space is achieved. As such, a\ncritical radius Rn;l\ncat whichEn;l\n+=En;l\n\u0000is given by\nRn;l\nc=~jn;jlj\n2p\u0001cbs\n1\nm\u0000\ne\u000b\u00001\nm+\ne\u000b(8)\nwherem\u0000\ne\u000b=mK#=K0\"\ne\u000bandm+\ne\u000b=mK\"=K0#\ne\u000b. Therefore,\nthere are real solutions to the critical radius at which\nfourfold valley-spin degeneracy may exist for dots with\nintrinsic spin-orbit coupling such that \u0001 cb>0 andm+\ne\u000b>\nm\u0000\ne\u000b. The latter condition is given for all possible TMDC\nmonolayers while the former is only satis\fed by Mo based\nTMDCs (\u0001 cb= 1:5 meV for MoS 2and \u0001cb= 11:5 meV\nfor MoSe 2)4,14(see Fig. 2). Alternatively, real solutions\nofRcmay be found in materials where both \u0001 cb<0\nandm+\ne\u000b0\nnot being satis\fed by W based TMDCs.\ng-factor,gvlis the valley g-factor and \u000b\u0006denote the\nmodi\fed wavenumber operators \u000b\u0006=\u0007ilBq\u0006=p\n2 where\nlB=p\n~=eBzis the magnetic length. After appropri-\nate gauge selection wavefunctions in terms of the di-\nmensionless length parameter \u001a=r2=2l2\nBare given as\nPn;l(\u001a) =\u001ajlj=2e\u0000\u001a=2M(an;l;jlj+1;\u001a) wherean;ldescribes\nthenthsolution of the following bound state identity\nM(an;l;jlj+ 1;\u001aD) = 0, where \u001aD=\u001a[r=RD] and\nM(a;b;c ) is the con\ruent hypergeometric function of the\n\frst kind. The addition of an out of plane magnetic \feld\ndoes not break the rotational symmetry of the dot, hence\nthe angular component of the wavefuntion is not a\u000bected\nby this change. The eigenenergies are therefore given as\nE\u001c;s\nn;l=~!\u001c;s\nc\u00121 +\u001c\n2Bz\njBzj+jlj+l\n2\u0000an;l\u0013\n+\u001c\u0001cbsz+1\n2(\u001cgvl+sgsp)\u0016BBz:(10)\nFrom Eq. (10), spectra demonstrating the e\u000bect of an4\n� � �� �� �� �� �������\nFIG. 3. Energy spectra of the n= 1,l= 0 state in a QD of\n20 nm radius on a MoS 2monolayer with under a perpendic-\nular magnetic \feld. Here the critical \feld strength at which\nEn=1;l=0\nK0;#=En=1;l=0\nK0;\"is observed at the high magnetic \feld\nstrength of\u001823 T. Blue solid (dashed) line: jK0\"i(jK#i)\nand red solid (dashed) line: jK\"i(jK0#i).\nout of plane magnetic \feld for QDs in MoS 2monolayers\nmay be calculated numerically. The splitting of the spin\nand valley states due to the external magnetic \feld allows\nfor spin-degenerate crossings for a given radius within the\nK0valley, i.e. at some external magnetic \feld strength\nEn;l\nK0;\"=En;l\nK0;#, see Fig. 3. These critical magnetic \feld\nstrengths (Bc) for given dot radii may be determined for\na range of radii to give the spin-degenerate regime spectra\nshown in Fig. 4.\nThese spectra show separate plateaus in the critical\n\feld strength at relatively high dot radii ( R> 20 nm) for\nthel\u00150 andl <0 angular states, di\u000bering by up to\n\u00185 T, but with both still at high \feld strengths. This is\nthe limit at which the maximum Kramers pair energy dif-\nference at zero \feld is observed and valley Zeeman split-\nting alone is used to achieve spin degeneracy. On the\nother end of the spectra, at low external \feld strengths\nthe gradient of the regime curves increases compromising\nthe fabrication error robustness of single dot spin qubits,\ni.e. small errors ( \u00181 nm) in QD radii would make the\ndi\u000berence between operating the qubit at 1 T and 6 T ex-\nternal \feld. Thus operating a spin qubit with a single\nelectron regime in the groundstate is not easily imple-\nmented. The possibility of operation at excited states\nand alternative enhancment methods are considered and\ndiscussed in Sec. V.\nIV. FINITE WELL\nUp to this point, all models used assume QDs with an\nin\fnite hard wall potential. Here the e\u000bects of transi-\n� �� �� �� ������������FIG. 4. Spin degeneracy curves of critical out of plane mag-\nnetic \feld strength Bcwith the radius of QD on MoS 2mono-\nlayer for the \frst few states, black solid (dashed): n= 1 (2),\nl= 0, red solid (dashed): n= 1, l= 1 (\u00001), blue solid\n(dashed): n= 1, l= 2 (\u00002), purple solid (dashed): n= 1,\nl= 2 (\u00002).\ntioning to a \fnite hard wall potential\nV=(\n0r\u0014RD\nV0r\u0015RD;(11)\non the spin-degenerate regimes discussed are shown.\nThus, for both the zero \feld and perpendicular magnetic\n\feld regimes, the \t( r=RD;\u001e) = 0 boundary condition\nis replaced by the continuity condition at the potential\ninterface@rln[\tr\u0015RD\nn;l(r=RD;\u001e)] =@rln[\tr\u0014RD\nn;l(r=\nRD;\u001e)]33.\nIn zero \feld the unnormalised radial portions of the\nwavefunction within and outside of the potential barrier\nare described as follows\nRn;l(r) =(\nJjlj(\u000fin\nn;lr)r\u0014RD\neil\u0019\n2Kjlj(\u000fout\nn;lr)r\u0015RD: (12)\nHereKlis thelthmodi\fed Bessel function of the sec-\nond kind,\u000fin\nn;l=p\n2m\u001c;s\ne\u000b[En;l\u0000\u001c\u0001cbsz]=~and\u000fout\nn;l=p\n2m\u001c;s\ne\u000b[V0\u0000En;l+\u001c\u0001cbsz]=~. Eigenenergies as a func-\ntion of potential height may then be numerically calcu-\nlated by applying the continuity condition to Eq. (12),\n\u000fin\nn;lJjlj+1(\u000fin\nn;lRD)\nJjlj(\u000fin\nn;lRD)=\u000fout\nn;lKjlj+1(\u000fout\nn;lRD)\nKjlj(\u000fout\nn;lRD): (13)\nFrom this, the fourfold degenerate critical radii as a func-\ntion of potential height may be calculated, leading to the5\n���������������������������������\nFIG. 5. Spin-degenerate critical radii Rcof QD of \fnite po-\ntential height in MoS 2monolayers at the ground and \frst few\nexcited states, red: n= 1,l= 0, blue: n= 1,jlj= 1, purple:\nn= 1,jlj= 2.\nresult shown in Fig. 5. The e\u000bect of a \fnite potential\nis only noticeable at low potential heights <100 meV,\nwhereafter a sharp drop in the critical radii is observed.\nSimilarly, when a \fnite potential is considered with an\nexternal magnetic \feld over the QD, the unnormalised\nradial component of the wavefunction is described as\nPn;l(\u001a) =\u001ajlj=2e\u0000\u001a=2(\nM(~ain\nn;l;jlj+ 1;\u001a)r\u0014RD\nU(~aout\nn;l;jlj+ 1;\u001a)r\u0015RD\n(14)\nwhereU(~aout\nn;l;jlj+1;\u001a) is Tricomi's hypergeometric func-\ntion and ~ain\nn;lis thenthnumerical solution to the con-\ntinuity equation at the potential barrier and ~ aout\nn;l=\n~ain\nn;l+V0=~!\u001c;s\nc. The continuity condition may then be\napplied to achieve the following characteristic equation\n(1 +jlj)~aout\nn;lM(~ain\nn;l;jlj+ 1;\u001aD)U(1 + ~aout\nn;l;jlj+ 2;\u001aD)\n+ ~ain\nn;lM(1 + ~ain\nn;l;jlj+ 2;\u001aD)U(~aout\nn;l;jlj+ 1;\u001aD) = 0\n(15)\nfrom which ~ ain\nn;lmay be numerically extracted and ap-\nplied to Eq. (10) in lieu of an;l. The e\u000bect of a \fnite\npotential height model on the spin-degenerate regimes of\nMoS 2is shown in Fig. 6.\nA similar e\u000bect on the spin degeneracy regimes in\nshown in both FIGs. 5 and 6. At shallow potential\nheights the required critical radius of the dot decreases by\n\u00181\u00002 nm. However at high magnetic \feld, there is no\ndiscernible di\u000berence between the \fnite and in\fnite po-\ntential solutions. This result will pose little threat to the\noperation of dots with a single electron charged into the\ngroundstate as the potential height may be selected to\nbe su\u000eciently high such that little to no di\u000berence in the\ncritical radii will be observed. Although, as is discussed\n� � �� �� �����������FIG. 6. Spin-degenerate critical magnetic \feld Bcof QD of\n\fnite potential heights in MoS 2monolayers at the ground of\nheights 1 eV (red), 0 :5 eV (blue), 0 :25 eV (purple) and in\fnite\npotential (black dashed) for reference\nin Sec. V, this e\u000bect must be considered when switching\nto an excited operational electron state by charging.\nV. SINGLE QUANTUM DOTS AS QUBITS\nTo achieve a pure spin qubit in a single MoS 2QD, some\nconsidered parameter selection is required to gain a cer-\ntain robustness of the operational regime. As previously\nstated in Sec. III, a regime with a single electron in the\nlowest spin-degenerate state either requires a very large\nexternal \feld ( >20 T) or extreme precision in the QD's\nradius. This is not ideal, however these problems may\nbe mitigated by charging the dot to operate at higher\ndegenerate states. As can be seen in Fig. 4, at reason-\nable external \felds ( \u001410 T), for each increasing excited\nstate the necessary QD radius increases in accordance\nwith Eq. (8). These regimes allowing for larger dot radii\nare more reliably achieved by gated monolayer QD fab-\nrication methods. Moreover, the ( jlj+l)=2 term of Eq.\n(10) splits the plateaus of the regime curves shown in\nFig. 4 into the higher plateaus of the l\u00140 and lower\nl= 1;2;::: plateaus. Therefore, if a charged excited\nstate is chosen as the operational state, the ideal choice\nwould be an l>0 angular-state.\nEven in the lowest spin-degenerate state, some charg-\ning may be required. The operational electron con\fned\nto theK0valley is at a higher energy than the two other\npossible states in the Kvalley (see Fig. 3). Although\nvalley lifetime is expectedly long11,34, eventually the elec-\ntron will decay out of the higher operational state to these\nempty states. Also, since each excitation state may be\nsplit into four di\u000berent con\fgurations of spin and valley,\nthe total number of electrons needed to charge the dot\nup to the desired operational regime is 3 + 4 Nwhere\nNis an integer describing the excitation level of the op-\nerational state, i.e. N= 0 corresponds to the ground-6\nstaten= 1l= 0,N= 1 corresponds to the \frst ex-\ncited state n= 1l=\u00001 etc. The direct band gap of\nmonolayer MoS 2is\u00181:8 eV6, and current advances in\ngated QD nanostructures in MoS 2give a charging en-\nergy of 2 meV at a dot radius of 70 nm35. This result was\nsaid to align well with the self capacitance model29,35,36,\ntherefore, using said model, the charging energy at de-\nsired radii for spin-degenerate regimes ( \u001810 nm) may be\napproximately shown to increase to \u001814 meV. This is\nhowever a broad approximation, therefore further study\nof the perturbation of the energy levels due to Coulomb\ninteraction mediated by the Keldysh potential37is war-\nranted, however such e\u000bects are spin and valley indepen-\ndant and should only serve as a renormalisation of the\ne\u000bects studied here. These considerations do however\nlimit the choice of excited operational states, as is evi-\ndent in Fig. 5, at highly charged states relative to the\npotential height and band gap, the critical radii will be\ncompromised.\nAdditionally, ferromagnetic substrates may be em-\nployed to enhance the valley splitting due to an external\nmagnetic \feld. Recent experiments demonstrate an e\u000bec-\ntive\u00182 T addition to the magnetic \feld inducing valley\nZeeman splitting in WSe 2monolayers on EuS ferromag-\nnetic substrate38. Such techniques may be employed to\nreduce the necessary external \feld strength to reasonable\nquantities.\nAn alternative quantum con\fnement method with\nTMDC monolayers has been proposed by way of het-\nerostructures consisting of islands of one form of Mo\nbased TMDC within a sea of a the corresponding W\nbased monolayer15,39, or by su\u000eciently small free stand-\ning \rakes40. While such methods o\u000ber quantum con\fne-\nment on the desired scale, high inter-vally coupling terms\nare introduced at small dot radii due to edge e\u000bects, of-\nfering a decoherence channel to the system. Addition-\nally, such structures o\u000ber scalability challenges such as\nthe lack of a method of adjusting the exchange coupling\nif the proposed model is extended to a double QD sys-\ntem. However, such studies of quantum con\fnement in\nTMDCs pay close attention to the e\u000bect of dot shape, a\nconsideration omitted here for simple symmetry consid-\nerations, but could yet warrant consideration in further\nresearch.\nWith a suitable operational regime selected, operation\nof the spin qubit is relatively straightforward. The en-\nergy gap between the up and down spin computational\nbasis is tuneable by the external magnetic \feld, whileBychkov-Rashba spin orbit coupling induced by an ex-\nternal electric \feld perpendicular to the device may be\nused to provide o\u000b diagonal spin coupling terms in the\nspin Hilbert space14.\nVI. SUMMARY\nOverall, given selection of a proper operational regime\nand reasonable accuracy in QD fabrication at low radii,\nMoS 2monolayer QDs do o\u000ber novel pure spin qubits\nin 2D semiconductors. Overcoming the Kramers pairs\nof gated QDs on TMDC monolayers is explored, as to\nachieve operational regimes of pure spin qubits, thus\navoiding the problem of achieving valley state mixing\nand low valley coherence times. Zero \feld fourfold spin-\nvalley degeneracy was demonstrated to be achievable\nin Mo based TMDC monolayers, unlike their W based\ncounterparts, at low QD radii whilst spin degeneracy\nsolely within a given valley was shown be achieved by\napplication of a su\u000eciently high external magnetic \feld\nperpendicular to the dot. Regime restrictions for spin-\ndegenerate MoS 2QDs have been shown, demonstrating\nradially sensitive low external \feld regimes which may\nbe made to be more robust when charged into higher\noperational states and enhanced valley-Zeeman splitting\nsubstrates. Switching from an in\fnite to a \fnite poten-\ntial barrier model did demonstrate a drop in the expected\nvalues of spin-degenerate critical radii, but only at par-\nticularly low barrier heights. In addition to the moder-\nate expected charging energy this somewhat limits the\nusefulness of highly charged operational states, but will\nnot substantially e\u000bect operation at the \frst few excited\nstates. To conclude, a theoretical demonstration of QD\nradius dependant spin-orbit e\u000bects in TMDC monolay-\ners is given along with descriptions of possible methods of\nimplementing novel pure spin qubits on two-dimensional\nsemiconductor crystals.\nVII. ACKNOWLEDGEMENTS\nWe acknowledge helpful discussions with A.\nKorm\u0013 anyos, A. Pearce, M. Ran\u0014 ci\u0013 c and M. Russ\nand funding through both the European Union by way\nof the Marie Curie ITN Spin-Nano and the DFG through\nSFB 767.\n\u0003matthew.brooks@uni-konstanz.de\n1Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,\nand M. S. Strano, Nat. Nanotechnol. 7, 699 (2012).\n2R. Suzuki, M. Sakano, Y. Zhang, R. Akashi, D. Morikawa,\nA. Harasawa, K. Yaji, K. Kuroda, K. Miyamoto,\nT. Okuda, et al. , Nat. Nano. 9, 611 (2014).3T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu,\nP. Tan, E. Wang, B. Liu, et al. , Nat. Comms. 3, 887 (2012).\n4A. Korm\u0013 anyos, G. Burkard, M. Gmitra, J. Fabian,\nV. Z\u0013 olyomi, N. D. Drummond, and V. Fal'ko, 2D Mater.\n2, 022001 (2015).\n5A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y.\nChim, G. Galli, and F. Wang, Nano lett. 10, 1271 (2010).7\n6K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz,\nPhys. Rev. Lett. 105, 136805 (2010).\n7H.-Z. Lu, W. Yao, D. Xiao, and S.-Q. Shen, Phys. Rev.\nLett.110, 016806 (2013).\n8D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys.\nRev. Lett. 108, 196802 (2012).\n9W.-Y. Shan, H.-Z. Lu, and D. Xiao, Phys. Rev. B 88,\n125301 (2013).\n10X. Xu, W. Yao, D. Xiao, and T. F. Heinz, Nat. Phys. 10,\n343 (2014).\n11L. Yang, N. A. Sinitsyn, W. Chen, J. Yuan, J. Zhang,\nJ. Lou, and S. A. Crooker, Nat. Phys. 11, 830 (2015).\n12Y. Ma, L. Kou, X. Li, Y. Dai, and T. Heine, Phys. Rev.\nB93, 035442 (2016).\n13M. Cazalilla, H. Ochoa, and F. Guinea, Phys. Rev. Lett.\n113, 077201 (2014).\n14A. Korm\u0013 anyos, V. Z\u0013 olyomi, N. D. Drummond, and\nG. Burkard, Phys. Rev. X 4, 011034 (2014).\n15Y. Wu, Q. Tong, G.-B. Liu, H. Yu, and W. Yao, Phys.\nRev. B 93, 045313 (2016).\n16H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nat. Nan-\notechnol. 7, 490 (2012).\n17J. Klinovaja and D. Loss, Phys. Rev. B 88, 075404 (2013).\n18Y. Song and H. Dery, Phys. Rev. Lett. 111, 026601 (2013).\n19A. Korm\u0013 anyos, V. Z\u0013 olyomi, N. D. Drummond, P. Rakyta,\nG. Burkard, and V. I. Fal'ko, Phys. Rev. B 88, 045416\n(2013).\n20K. Ko\u0013 smider, J. W. Gonz\u0013 alez, and J. Fern\u0013 andez-Rossier,\nPhys. Rev. B 88, 245436 (2013).\n21Z. Zhu, Y. Cheng, and U. Schwingenschl ogl, Phys. Rev.\nB84, 153402 (2011).\n22K. Flensberg and C. M. Marcus, Phys. Rev. B 81, 195418\n(2010).\n23G. Aivazian, Z. Gong, A. M. Jones, R.-L. Chu, J. Yan,\nD. G. Mandrus, C. Zhang, D. Cobden, W. Yao, and X. Xu,\nNat. Phys. 11, 148 (2015).24A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke,\nA. Kis, and A. Imamo\u0015 glu, Nat. Phys. 11, 141 (2015).\n25H. Rostami and R. Asgari, Phys. Rev. B 91, 075433 (2015).\n26A. P\u0013 alyi and G. Burkard, Phys. Rev. Lett. 106, 086801\n(2011).\n27Z. Ye, D. Sun, and T. F. Heinz, Nat. Phys. 13, 26 (2016).\n28K. Hao, G. Moody, F. Wu, C. K. Dass, L. Xu, C.-H. Chen,\nL. Sun, M.-Y. Li, L.-J. Li, A. H. MacDonald, et al. , Nat.\nPhys. 12, 667 (2016).\n29R. Hanson, L. Kouwenhoven, J. Petta, S. Tarucha, and\nL. Vandersypen, Rev. Mod. Phys. 79, 1217 (2007).\n30F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Sim-\nmons, L. C. Hollenberg, G. Klimeck, S. Rogge, S. N. Cop-\npersmith, and M. A. Eriksson, Rev. Mod. Phys. 85, 961\n(2013).\n31J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science 309, 2180 (2005).\n32S. Y. Zhou, G.-H. Gweon, A. Fedorov, P. First,\nW. De Heer, D.-H. Lee, F. Guinea, A. C. Neto, and\nA. Lanzara, Nat. Mater. 6, 770 (2007).\n33P. Recher, J. Nilsson, G. Burkard, and B. Trauzettel,\nPhys. Rev. B 79, 085407 (2009).\n34G. Sallen, L. Bouet, X. Marie, G. Wang, C. Zhu, W. Han,\nY. Lu, P. Tan, T. Amand, B. Liu, et al. , Phys. Rev. B 86,\n081301 (2012).\n35K. Wang, T. Taniguchi, K. Watanabe, and P. Kim, arXiv\npreprint arXiv:1610.02929 (2016).\n36L. P. Kouwenhoven, D. Austing, and S. Tarucha, Rep.\nProg. Phys. 64, 701 (2001).\n37A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi,\nY. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen,\nand T. F. Heinz, Phys. Rev. Lett. 113, 076802 (2014).\n38C. Zhao, T. Norden, P. Zhang, P. Zhao, Y. Cheng, F. Sun,\nJ. P. Parry, P. Taheri, J. Wang, Y. Yang, et al. , Nat. Nano.\n12, 757 (2017).\n39G.-B. Liu, H. Pang, Y. Yao, and W. Yao, New J. Phys.\n16, 105011 (2014).\n40S. Pavlovi\u0013 c and F. Peeters, Phys. Rev. B 91, 155410 (2015)." }, { "title": "1512.01661v1.Kinetic_theory_of_spin_polarized_systems_in_electric_and_magnetic_fields_with_spin_orbit_coupling__II__RPA_response_functions_and_collective_modes.pdf", "content": "Kinetic theory of spin-polarized systems in electric and magnetic \felds with spin-orbit\ncoupling: II. RPA response functions and collective modes\nK. Morawetz1;2;3\n1M unster University of Applied Sciences, Stegerwaldstrasse 39, 48565 Steinfurt, Germany\n2International Institute of Physics (IIP) Federal University of Rio Grande\ndo Norte Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil and\n3Max-Planck-Institute for the Physics of Complex Systems, 01187 Dresden, Germany\nThe spin and density response functions in the random phase approximation (RPA) are derived\nby linearizing the kinetic equation including a magnetic \feld, the spin-orbit coupling, and mean\n\felds with respect to an external electric \feld. Di\u000berent polarization functions appear describing\nvarious precession motions showing Rabi satellites due to an e\u000bective Zeeman \feld. The latter\nturns out to consist of the mean-\feld magnetization, the magnetic \feld, and the spin-orbit vector.\nThe collective modes for charged and neutral systems are derived and a threefold splitting of the\nspin waves dependent on the polarization and spin-orbit coupling is shown. The dielectric function\nincluding spin-orbit coupling, polarization and magnetic \felds is presented analytically for long\nwave lengths and in the static limit. The dynamical screening length as well as the long-wavelength\ndielectric function shows an instability in charge modes, which are interpreted as spin segregation\nand domain formation. The spin response describes a crossover from damped oscillatory behavior to\nexponentially damped behavior dependent on the polarization and collision frequency. The magnetic\n\feld causes ellipsoidal trajectories of the spin response to an external electric \feld and the spin-orbit\ncoupling causes a rotation of the spin axes. The spin-dephasing times are extracted and discussed\nin dependence on the polarization, magnetic \feld, spin-orbit coupling and single-particle relaxation\ntimes.\nPACS numbers: 75.30.Fv, 71.70.Ej, 85.75.Ss, 77.22.Ch\nI. INTRODUCTION\nThe development of spintronic devices is largely based\non the understanding of collective spin waves. Spin\nwaves, besides density waves, are one of the fundamen-\ntal collective excitations in strongly interacting Fermi\nsystems, e.g., in ferromagnetic materials1,2, graphene3,4,\nor isospin excitations in nuclear matter5. In the past,\nthis had motivated people to develop Green functions\ntechniques for the quantum transport theory of spin\nresonance6,7.\nIf the range of interaction is shorter than the DeBroglie\nwavelength, such excitations are also predicted8{13and\nobserved14,15in dilute spin-polarized gases. The trans-\nverse spin-wave dynamics has been the subject of a series\nof theoretical investigations12,16. In ultracold gases, spin\nwaves have been observed, even spatially resolved17,18,\nand were described by longitudinal spin waves19. The\nspin di\u000busion in trapped Bose gases shows an anisotropy\nin modes20and the collisionless damping has been seen to\ndeviate for quadrupole modes from experiments21indi-\ncating the role of collisional correlations. The spin-wave\ndamping has been measured in a polarized Fermi-liquid-\nlike3He-4He mixture even at zero temperature22and as\nan 'identical spin-rotation e\u000bect'23.\nThe in\ruence of the magnetic \feld on such spin waves\nis of special interest, e.g., as magneto-transport e\u000bects in\nparamagnetic gases24. The Landau levels in\ruence the\nspin relaxation25,26, which has been measured with the\nhelp of spin coherence times27. The in\ruence of magnetic\n\felds is treated in various systems ranging from plasma28,solid-state plasmas29, and semiconductors30to spin-orbit\ncoupled systems31and graphene32. The feedback of mag-\nnetization dynamics due to spins on the spin dynamics\nitself is reviewed in Ref.33. The Zeeman \feld is reported\nto trigger a transition from a charge density wave to a\nspin density wave34. Quite promising for technological\napplications turns out the possibility to create magnetic\nnanooscillations by pure spin currents35. The spin cur-\nrent can be converted into a terahertz electromagnetic\npulse due to the inverse spin Hall e\u000bect36.\nQuite recently the spin-orbit coupling has moved to\nthe center of interest37,38since this coupling allows to\nconvert spin waves into spin currents, which is impor-\ntant for spintronic devices39. There has been observed a\nspin-orbit-driven ferromagnetic resonance40which shows\nthat an e\u000bective magnetic \feld is created in the magnetic\nmaterial by oscillating electric currents. This is also the\nbasis of microwave spectroscopy41. Earlier this has been\nidenti\fed as a magneto-electric e\u000bect where a charge cur-\nrent induces a spin polarization known as the Edelstein\ne\u000bect42,43. Spin-polarized longitudinal currents can be\ninduced due to spin-orbit interaction in certain crystal\nsymmetries44. Experimentally even a planar Hall e\u000bect\nhas been reported using spin waves45as well as spin po-\nlarization oscillations without spin precession46.\nCoulomb interactions are known to reduce the e\u000bect of\nspin-orbit coupling in the spin-Hall e\u000bect47. The phonon-\nmodulated spin-orbit interaction has been investigated\nto show that the screening is in\ruenced by the spin-orbit\ncoupling48. Screening e\u000bects play a crucial role for the\ntemperature dependence of conductivity in quasi two-arXiv:1512.01661v1 [cond-mat.str-el] 5 Dec 20152\ndimensional systems49, monolayer graphene50and multi-\nlayer graphene51. The Coulomb correction to the conduc-\ntivity in graphene had covered an involved debate52{55.\nWith this respect the extraction of correct spin relaxation\nor dephasing times has been in the center of interest56{58\nsince it is most promising for new storage devices.\nDuring the last few years, many researchers have\nshown an appreciable interest in the dielectric function\nand the properties of screening in two-dimensional gases\nwith spin-orbit coupling59{61. Similar results appear\nif the pseudospin response function in doped graphene\nis calculated62{65. The random phase approximation\n(RPA) is calculated in this respect for single and mul-\ntilayer graphene66,67. These approaches calculate the\nLindhard dielectric function with form factors arising\nfrom chirality subbands. Additional energy denomina-\ntors appear if four-band approximations are considered65\nwhere a band gap appears68. The comparison of pristine\ngraphene, Dirac cones, and gaped graphene with an an-\ntidot lattice can be found in Ref.69. These responses are\nneeded if one wants to understand the optical properties\nof graphene irradiated by an external electric \feld70.\nAll of these approaches consider the spin degree of free-\ndom as an inner property of particles leading to form fac-\ntors in the Lindhard dielectric function. Here, the spin\ndegree of freedom is considered on equal statistical foot-\ning with the particle distribution leading to more forms\nof the response function due to the spin-orbit coupling,\nsatellites, and Zeeman splitting by magnetic \felds and\nself-energy e\u000bects that cannot be cast into a Lindhard\nform with form factors. This has an impact on the col-\nlective density and spin modes. The will calculate analyt-\nically the threefold splitting of spin modes71as a function\nof the spin-orbit coupling and the e\u000bective Zeeman \feld.\nIn this paper, we want to present a unifying treatment\nof density and spin waves in the random phase approxi-\nmation including the spin-orbit coupling, magnetic \felds,\nand an arbitrary magnetization for systems with charged\nand neutral scattering. This will allow us to investi-\ngate the in\ruence of spin-orbit coupling on the screen-\ning properties of the Coulomb interaction as well as the\ncollective modes in systems with neutral scattering. For\nthis purpose, we linearize the kinetic equation derived in\nthe \frst part of this paper. Linearizing the mean-\feld\nkinetic equation yields the RPA response since a lower-\nlevel kinetic equation provides a response of higher-order\nmany-body correlations72.\nFollowing a short summary of the basic kinetic equa-\ntion derived in the \frst part of the paper, the linear re-\nsponse to an external electric \feld is presented in the\nsecond section. This results into coupled equations for\nthe spin and density response with a variety of dynamical\npolarization functions describing di\u000berent precessions. In\nSec. III, the charge and spin density response functions\nare analyzed with respect to their collective modes and\nthe spin waves are discussed for neutral and charged scat-\ntering. The polarization causes a splitting of spin modes.\nFor certain polarizations and spin-orbit coupling, an in-stability occurs, which is interpreted as spin-domain sep-\naration. This is underlined by the in\ruence of spin-orbit\ncoupling and polarization on the screening properties\nin charged systems where the instability occurs in spa-\ntial domain. The dielectric function including the spin-\norbit coupling and an e\u000bective medium-dependent Zee-\nman \feld is derived analytically in the long-wavelength\nand static limits respectively. The dynamic and static\nscreening lengths are discussed there. The spin response\ndue to an applied electric \feld is then extracted and the\nspin-dephasing times are discussed. As in the Edelstein\ne\u000bect, the applied electric \feld causes a charge current\nand a spin response that shows oscillations dependent on\nthe spin-orbit coupling, magnetic \feld, and relaxation\ntime. In Sec. V, we present the linear response including\narbitrary magnetic \felds and show how the normal Hall\nand quantum Hall e\u000bects appear from the kinetic theory.\nAs an important point, a subtlety in retardations due\nto the magnetic \feld is presented. A summary \fnishes\nthis second part of the paper. In Appendix, some use-\nful expressions for solving involved vector equations are\npresented.\nLet us now shortly summarize the quantum kinetic\nequation derived in the \frst part of the paper. We de-\nscribe the density and polarization density by their cor-\nresponding Wigner functions\nX\npf(~ x;~ p;t ) =n(~ x;t);X\np~ g(~ x;~ p;t ) =~ s(~ x;t) (1)\nwhereP\np=R\ndDp=(2\u0019~)DforDdimensions. As a result\nof the \frst part of this paper, the four Wigner functions\n^\u001a=f+~ \u001b\u0001~ g=\u0012\nf+gzgx\u0000igy\ngx+igyf\u0000gz\u0013\n(2)\nhave been shown to obey coupled kinetic equations\nDtf+~A\u0001~ g= 0\nDt~ g+~Af= 2(~\u0006\u0002~ g) (3)\nwhereDt= (@t+~F~@p+~ v~@x) describes the drift and force\nof the scalar and vector part with the velocity\nv=p\nme+@p\u00060 (4)\nand the e\u000bective Lorentz force\n~F= (e~E+e~ v\u0002~B\u0000~@x\u00060): (5)\nThis e\u000bective Lorentz force as well as the velocity both\nbecome modi\fed due to the scalar mean\feld selfenergy\n\u00060(~ q;t) =n(~ q;t)V0(~ q) +~ s(~ q;t)\u0001~V(~ q) (6)\nas a spatial convolution between the density and spin po-\nlarization with the Fourier transformed scalar and vector\npotentials, respectively. The latter ones originate from3\nmagnetic impurities and/or e\u000bective magnetizations in\nthe material. Here, we concentrate on the intrinsic spin-\norbit coupling. The mean\felds with extrinsic spin-orbit\ncoupling are given in III.C of the \frst part of the paper.\nThe second parts on the left side of (3) represent the\ncoupling between the spin parts of the Wigner distribu-\ntion given by the vector drift\nAi= (~@p\u0006i~@x\u0000~@x\u0006i~@p+e(~@p\u0006i\u0002~B)~@p): (7)\nWe subsumed in the vector selfenergy\n~\u0006 =~\u0006H(~ x;t) +~b(~ x;~ p;t ) +\u0016B~B(~ x;t) (8)\nthe magnetic impurity mean\feld\n~\u0006H=n(~ q;t)~V(q) +~ s(~ q;t)V0(q); (9)\nand the spin-orbit coupling vector ~b, as well as the Zee-\nman term\u0016B~Bsuch that the e\u000bective Hamiltonian pos-\nsesses the Pauli structure\nHe\u000b=H+~ \u001b\u0001~\u0006 (10)\nwith the e\u000bective scalar Hamiltonian\nH=k2\n2me+ \u0006 0(~ x;~k;T) +e\b(~ x;T) (11)\nwhere~k=~ p\u0000e~A(~ x;t) ensures gauge invariance. Any\nspin-orbit coupling found in the literature can be recast\ninto the form ~ \u001b\u0001~b(p) as illustrated in Table I of the \frst\npart of this paper. The vector part of (3) \fnally contains\nadditionally the spin-rotation term on the right-hand side\nresponsible for spin precession.\nII. LINEAR RESPONSE\nA. Without magnetic \feld but conserving\nrelaxation time\nLet us consider the linearization of the kinetic equation\n(3) with respect to an external electric \feld, no mag-\nnetic \feld and in a homogeneous situation. We Fourier\ntransform the time @t!\u0000i!and the spatial coordinates\n~@x!i~ q. The Wigner functions are linearized according\nto ^\u001a(~ x;~ p;t ) =f(~ p) +\u000ef(~ x;~ p;t ) +~ \u001b\u0001[~ g(~ p) +\u000e~ g(~ x;~ p;t )]\ndue to the external electric \feld perturbation e\u000e~E=\ne~E(~ x;t) =\u0000r\b. The density and spin-density variation\nreads\n\u000en(~ q;!) =X\np\u000ef(~ q;~ p;! )\n\u000e~ s(~ q;!) =X\np\u000e~ g(~ q;~ p;! ) (12)\nand the density and spin-density linear response func-\ntions are given by\n\u000en(~ q;!) =\u001f(~ q;!)\b\n\u000e~ s(~ q;!) =~ \u001fs(~ q;!)\b: (13)Further, we assume a collision integral of the relaxation\ntime approximation73\n\u00001\n2[^\u001c\u00001;\u000e^\u001al]+ (14)\nwith the vector and scalar parts of the relaxation times\n^\u001c\u00001=\u001c\u00001+~ \u001b\u0001~ \u001c\u00001where\n\u001c=\u001c\u00001\n\u001c\u00002\u0000j~ \u001c\u00001j2; ~ \u001c =\u0000~ \u001c\u00001\n\u001c\u00002\u0000j~ \u001c\u00001j2: (15)\nIn the \frst part of the paper, the relaxation of the\nkinetic equations (3) has been shown towards the two-\nband distribution f=f++f\u0000\n2and~ g=~ ef+\u0000f\u0000\n2with the\nFermi-Dirac distribution\nf0(\u000fp(r)\u0006j~\u0006(p;r)j) (16)\nand the precession direction ~ e=~\u0006=\u0006. Now we as-\nsume a relaxation towards a local distribution fl=\nf0(\u000f\u0006j\u0006j\u0000\u0016\u0000\u000e\u0016) such that the density conservation\ncan be enforced74,75,\n\u000en=X\np(f\u0000f0) =X\np(f\u0000fl+fl\u0000f0)\n=X\np(fl\u0000f0) =@\u0016n\u000e\u0016; (17)\nas expressed by the second line. Therefore the scalar\nrelaxation term becomes\n\u0000\u000e^\u001al\n\u001c=\u0000\u000e^\u001a\n\u001c+\u000en\n\u001c@\u0016n@\u0016^\u001a0: (18)\nIn this way the density is conserved in the response func-\ntion which could be extended to include more conserva-\ntion laws76,77. If we consider only the density conserva-\ntion but not the polarization conservation, we can restrict\nourselves to the @\u0016f0term. Please note that we neglect\nin this way the interference e\u000bects of disorder78.\nAbbreviating now \u0000i!+i~ p\u0001~ q=m+\u001c\u00001=aandiq@p~\u0006+\n~ \u001c\u00001=~B, the coupled kinetic equations (3) take then the\nform\na\u000ef+~B\u000e~ g=S0\na\u000e~ g+~B\u000ef\u00002~\u0006\u0002\u000e~ g=~S (19)\nwithe~E=\u0000i~ q\b and\nS0=iq@pf(\b +\u000e\u00060) +iq@p~ g\u0001\u000e~\u0006 +\u000en\n\u001c@\u0016n@\u0016f0\n~S=iq@p~ g(\b +\u000e\u00060) +iq@pf\u000e~\u0006 + 2(\u000e~\u0006\u0002~ g) +\u000en@\u0016~ g\n\u001c@\u0016n:\n(20)\nIn order to facilitate the vector notation we want to un-\nderstandq@p=~ q\u0001~@pin the following.4\nB. Collective modes from balance equations\nMultiplying the linearized kinetic equations (19) with\npowers of momentum and integrating one obtains cou-\npled hierarchies of moments. A large variety of treat-\nments neglect certain Landau-liquid parameters79based\non the work of Ref.80in order to close such system. A\nmore advanced closing procedure was provided by Ref.81\nwhere the energy dependence of \u000e~ swas assumed to be\nfactorized from space and direction ~ pdependencies.\nWe will not follow these approximations here but solve\nthe linearized equation exactly to provide the solution\nof the balance equations and the dispersion. Amazingly,\nthis yields a quite involved and extensive structure with\nmuch more terms than usually presented in the literature.\nNevertheless, it is instructive to have a \frst look at the\nbalance equation for the densities\n@t\u000en+@xi~Ji+~ \u001c\u00001\u0001\u000e~ s= 0\n@t\u000e~ s+@xi~Si+~ \u001c\u00001\u000en\u00002X\np~\u0006\u0002\u000e~ g= 2\u000e~\u0006\u0002~ s(21)\nwhere we Fourier transformed the wavevector qback to\nspatial coordinates x. Then the density currents and\nmagnetization currents\n^Jj=1\n2X\np[^\u001a;vj]+=X\nph\nfvj+~ g\u0001@pj~b+~ \u001b\u0001(vj~ g+f@pj~b)i\n=Jj+~ \u001b\u0001~Sj: (22)\nappear exactly as expected from the elementary de\fni-\ntions, see Sec. III.G of part I.\nWe are now interested in the long wavelength limit\nq!0, which means we neglect any spatial derivative in\n(21). Alternatively, we might consider this as the spa-\ntially integrated values providing the change of number\nof particles and magnetization\nN=Z\nd3xn(x) =nq=0; ~ m =Z\nd3x~ s(x) =~ sq=0:(23)\nThe \frst equation of (21) gives in frequency space\n\u0000i!\u000eN +~ \u001c\u00001\u0001\u000e~ m= 0 (24)\nwith the help of which we get the closed equation for the\nmagnetization from (21)\n\u0000i!\u000e~ m\u0000i\n!(~ \u001c\u00001+2~ m\u0002~V)~ \u001c\u00001\u0001\u000e~ m\u00002(N~V+\u0016B~B)\u0002\u000e~ m\n= 2X\np~b(p)\u0002\u000e~ gq=0: (25)\nIn the right-hand side, all terms that are coming from the\nexplicit knowledge of the solution \u000e~ gneeded to evaluate\nthis sum over the momentum-dependent spin-orbit term\n~bare collected. The separation of the balance equation\nin this form has the merit to see already the collectivespin mode structure. The \fne details are then worked\nout when we know the explicit solution in Sec. III.\nSince we have ~ m=m~ eZand~V=V~ eZthe equation\nfor the magnetization becomes\n0\n@\u0000i! 2(NV+\u0016BB) 0\n\u00002(NV+\u0016BB)\u0000i! 0\n0 0 \u0000i!1\nA\u000e~ m=2X\np~b\u0002\u000e~ gq=0\n(26)\nneglecting the quadratic terms of the vector relaxation\ntimes~ \u001c\u00001. The latter would add a term \u0000i(~ \u001c\u00001+2~ m\u0002\nV)~ \u001c\u00001\u0001\u000e~ m=! to the left-hand side.\nInverting (26) provides the solution of the magneti-\nzation change provided we know the solution of \u000e~ gon\nthe right hand side. Interestingly, this inversion is only\npossible for a nonzero determinant. The vanishing deter-\nminant provides therefore the eigenmodes of spin waves\nwith some possible modi\fcations due to the spin-orbit\ncoupling term.\nThe dispersion relation from the condition of vanishing\ndeterminant in (26) yields the two spin waves\n!spin=\u00062jNV+\u0016BBj (27)\nwhich shows the linear splitting due to the driving ex-\nternal magnetic \feld and the permanent magnetization\n~V=V~ ez.\nC. Solution of linearized coupled kinetic equations\nAs we have seen, even the balance equations for the lin-\nearized kinetic equations (19) are not closed if we do not\nknow the solution for \u000e~ g, which comes from the momen-\ntum dependence of ~\u0006. In the following we will present\ntwo ways to solve (19). First the elementary direct way\nand secondly with the help of operator algebra. The lat-\nter is then applicable directly to solve the kinetic equa-\ntion with magnetic \felds in the next section.\n1. Solution with the help of vector equation\nSolving the \frst equation of (19) for\n\u000ef=\u00001\na\u0010\n~B\u0001\u000e~ g\u0000S0\u0011\n(28)\nand introducing the result into the second equation leads\nto a vector equation of type (A6) and with the help of\nthe abbreviations\n~ c=\u0000~B\na; ~ z =~\u0006\na; ~ o =1\na\u0010\n~ cS0+~S\u0011\n(29)\nand setting B=~ o,A= 2~ z, andQ=\u0000V=~ cit can be\nreadily solved [ see Eq. (A8)], which becomes\n\u000e~ g=~ o+~ c\u0002(~ c\u0002~ o)+2(~ z\u0002~ o)+4~ z(~ z\u0001~ o)\u00002(~ z\u0001~ c)(~ c\u0002~ o)\n1\u0000c2+ 4z2\u00004(~ z\u0001~ c)2\n\u000ef=~ c\u0001~\u000eg+S0\na: (30)5\n2. Solution with the help of operator algebra\nAs a second possibility, we rewrite equations (19) with\nthe help of the identity\n~ c\u0001~ g+ (~ \u001b\u0001~ c)f\u00002~ \u001b\u0001(~ \u001b\u0002~ g) =\n~ \u001b\u0001~ c+ 2i~ \u001b\n2^\u001a+ ^\u001a~ \u001b\u0001~ c\u00002i~ \u001b\n2(31)\ninto one operator equation for \u000e^F=\u000ef+~ \u001b\u0001~\u000egand\ntransform the frequency back in time \u0000i!!@t:\n^Sp(t) =S0(t)+~ \u001c\u0001~S(t) =\u0010\n@t+ipq\nm+\u001c\u00001\u0011\n\u000e^F(t)\n+ ~b\n2+i~\u0006!\n\u0001~ \u001b\u000e^F+\u000e^F ~b\n2\u0000i~\u0006!\n\u0001~ \u001b: (32)\nThis equation is easily solved\n\u000e^F(t) =tZ\n\u00001d\u0016tei(pq\nm\u0000i\n\u001c)(\u0016t\u0000t)e\u0010~b\n2+i~\u0006\u0011\n\u0001~ \u001b(\u0016t\u0000t)^Sp(\u0016t)e\u0010~b\n2\u0000i~\u0006\u0011\n\u0001~ \u001b(\u0016t\u0000t)\n(33)\nand transformed back in frequency space to obtain\n\u000e^F(!) =0Z\n\u00001dxei(pq\nm\u0000!\u0000i\n\u001c)xe\u0010~b\n2+i~\u0006\u0011\n\u0001~ \u001bx^Sp(!)e\u0010~b\n2\u0000i~\u0006\u0011\n\u0001~ \u001bx:\n(34)\nFurther evaluation is presented in Appendix B. With the\nhelp of (B11) and (B12) we evaluate the corresponding\nintegrals and all scalar terms determine \u000ef, and all terms\nproportional to \u001bdetermine~\u000eg. We obtain again the\nresult (30)98\n\u000e~ g=~ o+~ c\u0002(~ c\u0002~ o)+2(~ z\u0002~ o)+4~ z\u0001(~ z\u0001~ o)\u00002(~ z\u0001~ c)(~ c\u0002~ o)\n(1\u0000c2)(1 + 4z2):\n(35)\nTo see the known limits, we inspect some further approx-\nimations.\n3. Long-wavelength limit\nWe assume only a scalar relaxation time and neglect\ntherefore the vector part describing skew scattering and\nside jump e\u000bects. Further we use the long wavelength\nlimit and expand in \frst orders of q@p~\u0006 which translates\ninto~ c2\u00190 in (35). Here, we have abbreviated the form\nof the mean-\feld self-energy\nU= \b +\u000e\u00060\n~U=\u000e~\u0006 (36)and we used e~E=\u0000i~ q\b, \u0016!=ia=!\u0000~ p\u0001~ q=m +i\u001c\u00001\nand the mean\felds\n\u000e\u00060=V0\u000en+~V\u0001\u000e~ s; \u000e~\u0006 =~V\u000en +V0\u000e~ s: (37)\nThen the solution (28) and (35) reads [ ~ g=g~ e]\n\u000ef=\u00001\n\u0016!\u001a\nUq@pf+~Uq@p~ g\u0000i\u000en\n\u001c@\u0016n@\u0016f\u0000q@p~\u0006\u0001~\u000eg\u001b\n\u000e~ g=\u000e~ gasy+\u000e~ gsym\nRabi+\u000e~ gsym\nn (38)\nEach of the terms corresponds to one of the three possible\nprecession motions:\n1\n4j\u0006j2\u0000!20\n@\u0000i!\n4ij\u0006j2\n!\n2j\u0006j1\nA=1Z\n0ei!t0\n@cos 2j\u0006jt\n1\u0000cos 2j\u0006jt\nsin 2j\u0006tj1\nA:(39)\nThe sin 2j\u0006jtmotion is responsible for the anomalous\nHall e\u000bect and their terms are collected in \u000e~ gasy. Let\nus write out the explicit forms. The symmetric solution\nconsists of an frequency denominator with Rabi-satellites\n\u000e~ gsym\nRabi=1\n2\u0010\n1\n\u0016!\u00002\u0006+1\n\u0016!\u00002\u0006\u0011h\n\u0000Ugq@p~ e\u00002i~ g\u0002\u000e~\u0006\n+q@pf~ e\u0002(~ e\u0002~U) +i\u000en\n\u001c@\u0016ng@\u0016~ ei\n(40)\nand a part with a normal denominator \u0016 !=!+i=\u001c\u0000\n~ p~ q=m :\n\u000e~ gsym\nn=1\n\u0016!\u0014\n\u0000U~ eq@pg\u0000q@pf~ e\u0002(~ e\u0001~U) +i\u000en\n\u001c@\u0016n@\u0016g~ e\u0015\n(41)\nwhich can be combined together\n\u000e~ gsym=1\n\u0016!(4j~\u0006j2\u0000\u0016!2)\u001a\n[\u0016!2~U\u00004j~\u0006j2~ e(~ e\u0001~U)]q@pf\n+ \u0016!2(Uq@p~ g\u0000i\u000en\n\u001c@\u0016n@\u0016~ g)\n\u00004\u00062(Uq@pg\u0000i\u000en\n\u001c@\u0016n@\u0016g)~ e\u00002i\u0016!2~U\u0002~ g\u001b\n:(42)\nThe term responsible for the anomalous Hall e\u000bect\nreads\n\u000e~ gasy=i\n2\u00121\n\u0016!+ 2\u0006\u00001\n\u0016!\u00002\u0006\u0013\u001a\n~ e\u0002~Uq@pf\n+\u0014\nU~ e\u0002q@p~ e\u00002i~ e\u0002(~U\u0002~ e)\u0000i\u000en\n\u001c@\u0016n~ e\u0002@\u0016~ e\u0015\ng\u001b\n(43)\nIn order to compare with the homogeneous solution\npresented in Eq. (143) of part I\n\u000e~ \u001a(!;k) =i!\n4j\u0006j2\u0000!2eE@k~ g\n\u00004i1\n!(4j\u0006j2\u0000!2)~\u0006(~\u0006\u0001eE@k~ g)\n\u000021\n4j\u0006j2\u0000!2~\u0006\u0002eE@k~ g: (44)6\nwe take the q!0 limit of (38) with ~ qU=ie~E+\no(q);q~U=o(q) and obtain\n\u000ef=\u0000i\n\u0016!\u0012\neE@pf\u0000\u000en\n\u001c@\u0016n@\u0016f\u0013\n\u000e~ gsym=\u0000i\u0016!\n\u0016!2\u00004j\u0006j2\u0012\neE@p~ g\u0000\u000en\n\u001c@\u0016n@\u0016~ g\u00002~U\u0002~ g\u0013\n+4i\u00062\n\u0016!(\u0016!2\u00004j\u0006j2)\u0012\neE@pg\u0000\u000en\n\u001c@\u0016n@\u0016g\u0013\n~ e\n\u000e~ gasy=2j\u0006jg\n\u0016!2\u00004j\u0006j2h\n~ e\u0002eE@p~ e\u00002~ e\u0002(~U\u0002~ e)\n\u0000\u000en\n\u001c@\u0016n~ e\u0002@\u0016~ e\u0015\n: (45)\nWithout vector mean\feld variation ~U\u00190 and relaxation\ntime, the last term responsible for the anomalous Hall\ne\u000bect corresponds directly to the third one in Eq. (44).\nThe \frst two terms correspond to the \frst two ones in Eq.\n(44) as simple algebra shows observing that ~\u0006\u0001@~ e= 0\nsince~ e=~\u0006=j~\u0006j,~ e(~ e\u0001@~ g) =~ e@gand~ e\u0002(~ e\u0002@~ g) =\n\u0000g@~ e. The term ~ e\u0002eE@p~ eof Eq. (45) corresponds\nto the precession term found in Ref.31as an additional\nrotation of the magnetization.\nD. Response functions\nWe want now to integrate the linearized solution (40),\n(41) and (43) over the momentum to obtain the density\nand spin response functions including the mean\feld and\nspin-orbit coupling e\u000bects. This will lead to a selfconsis-\ntent equation. We can design the dimensionality of the\nconsidered problem as done above after Eq. (1).\nThe \fnal result for the particle and spin density re-\nsponse using only intrinsic mean\felds \u000e\u00060=V0\u000en+~V\u0001\u000e~ s\nand\u000e~\u0006 =~V\u000en+V0\u000e~ sleads to the following linear system\n(1\u0000\u00050V0\u0000~\u0005\u0001~V+i\u00050\u0016\n\u001c@\u0016n)\u000en= \u0005 0\b + (\u0005 0~V+~\u0005V0)\u0001\u000e~ s\n(1\u0000\u00050V0\u0000 !\u0005V0)\u000e~ s=~\u00053\b +~\u00053(~V\u0001\u000e~ s) +V0~\u00052\u0002\u000e~ s\n+ (V0~\u00053+ \u0005 0~V+~\u00052\u0002~V+ !\u0005\u0001~V\u0000i\n\u001c@\u0016n~\u0005@)\u000en\n(46)\nwith the abbreviations for the polarizations\n~\u00052=~\u0005g+~\u0005xf\n~\u00053=~\u0005 +~\u0005xg+~\u0005e\n~\u0005@=~\u0005x\u0016+~\u0005\u0016+~\u0005\u0016e\n !\u0005 = !\u0005fe+ !\u0005xe: (47)\nThe 1=\u001cterms come from the Mermin conserving\nrelaxation-time approximation which means a relaxation\ntowards a local equilibrium speci\fed such that local con-\nservation laws are obbeyed74,75,77.It was helpful here to de\fne the polarization functions\naccording to the three precessions expressed by the parts\n(40), (41) and (43). The standard polarization functions\nfor scalar and vector distribution coming from (40) read\nwith \u0016!=!\u0000~ p\u0001~ q=m +i=\u001c\n\u00050(q!) =\u0000X\npq@pf1\n\u0016!\n\u00050\u0016(q!) =\u0000X\np@\u0016f1\n\u0016!\n~\u0005(q!) =\u0000X\npq@p~ g1\n\u0016!\n~\u0005\u0016(q!) =\u0000X\np@\u0016~ g1\n\u0016!: (48)\nThe remaining parts of (40) and (41) combine into the\nforms of\n4\u00062\n\u0016!(4\u00062\u0000\u0016!2)=1\n\u0016!\u00001\n2\u00121\n\u0016!+ 2\u0006+1\n\u0016!\u00002\u0006\u0013\n(49)\nwhich vanish quadratically with the vector mean\feld\n~\u0005e(q!) =X\npgq@p~ e4\u00062\n\u0016!(4\u00062\u0000\u0016!2)\n~\u0005\u0016e(q!) =X\npg@\u0016~ e4\u00062\n\u0016!(4\u00062\u0000\u0016!2)\n !\u0005fe(q!) =X\npq@pf(1\u0000~ e\u000e~ e)4\u00062\n\u0016!(4\u00062\u0000\u0016!2)(50)\nand a rotation part\n~\u0005g(q!) =\u0000iX\np~ g\u00121\n\u0016!+ 2\u0006+1\n\u0016!\u00002\u0006\u0013\n:(51)\nThe responses from the asymmetric part (43) lead to\n~\u0005xf(q!) =iX\np~ eq@pf1\n2\u00121\n\u0016!+ 2\u0006\u00001\n\u0016!\u00002\u0006\u0013\n~\u0005xg(q!) =iX\np~ e\u0002q@p~ g1\n2\u00121\n\u0016!+ 2\u0006\u00001\n\u0016!\u00002\u0006\u0013\n~\u0005x\u0016(q!) =iX\np~ e\u0002@\u0016~ g1\n2\u00121\n\u0016!+ 2\u0006\u00001\n\u0016!\u00002\u0006\u0013\n !\u0005xe(q!) = 2X\npg(1\u0000~ e\u000e~ e)1\n2\u00121\n\u0016!+ 2\u0006\u00001\n\u0016!\u00002\u0006\u0013\n(52)\nwhich vanish linearly in orders of the self-energy. In the\ncase of vanishing spin-orbit coupling, i.e. no momentum\ndependence of ~ e, we have~\u0005xg=~\u0005x\u0016=~\u0005e=~\u0005\u0016e= 0.\nIt is known that vertex corrections lead to addi-\ntional structure factors in the RPA polarization func-\ntions, which would extend the expressions here. Here, it7\nis shown that the spin-orbit coupling causes a zoo of ad-\nditional RPA polarization forms even on the level of the\nsingle-loop approximation, which is the highest level to\nbe obtained by the linearization of the mean-\feld equa-\ntions.\nIII. SPIN AND DENSITY WAVES\nThe system of density and spin responses (46) allows\nus to determine the spin and density waves that might be\nexcited in the system. It is convenient to continue to work\nin the wave number space r$q. For the magnetization\nand total particle number (23), we consider only dipole\nmodes that are characterized by the \frst-order moments\n\tj=Z\nd3rxj\u000en(r) =\u0000i@qj\u000enqjq=0\n~\tj=Z\nd3rxj\u000e~ s(r) =\u0000i@qj\u000e~ sqjq=0: (53)\nThe linear response for dipole modes is assumed to be\ncharacterized by linear deviations \u000ef=~\f1\u0001~@rf0+~\f2\u0001~@pf0\nand\u000eg=~ \u000b1\u0001~@rg0+~ \u000b2\u0001~@pg0such that we have\n\u000enjq=0=i~\f1\u0001~ qnjq=0= 0\n\u000e~ sjq=0=i~ \u000b1\u0001~ q~ sjq=0= 0 (54)\nfor the long-wavelength limit of the deviations them-\nselves. In order to determine the eigenmodes of (46),\nwe do not need the actual values of \u000band\f. Due to the\nproperties (53) and (54), we can apply the q-derivative\ndirectly to (46) and obtain an equation system where\n\u000enand\u000e~ sare replaced by \t jand~\tjin (46) and all re-\nsponse functions have to be taken in the q!0 limit. Any\nderivative of the latter vanishes since they are connected\nwith terms (54). This simpli\fes the analysis appreciably\nand shows the strength of the response system (46). For\nquadrupole modes where the second derivative is needed,\none has to calculate also the q!0 limit of the derivatives\nof the polarization functions.\nFurther analysis relays on the expansion of di\u000berent\npolarization functions as presented in Appendix C. Then,\nthe momentum integration still cannot be performed an-\nalytically if the momentum-dependent spin-orbit term\nb(p) in~\u0006 is present. Therefore we treat them linearly\nin~b(p), which allows to give an explicit form. In fact,\nthese terms are only present if the spin-orbit coupling\ncreates an explicit p-dependence in ~\u0006 =~\u0006n+~b(p), where\nwe denote the momentum-independent selfenergy with\n~\u0006n=n~V+V0~ s+\u0016B~B. Let us de\fne the z-direction by\nthis last term ~ ez=~\u0006n=j~\u0006njand expand all directions in\n\frst order of ~b(p).\nThe direction of e\u000bective polarization becomes\n~ e=~\u0006\nj\u0006j=~ ez\u0012\n1\u0000b2\n?\n2\u0013\n+~b?(1\u0000b3) (55)where we use the short-hand notation\n~bp\n\u0006n=~b?+~ ezb3: (56)\nSince the distribution functions in equilibrium are func-\ntions ofj~\u0006jaccording to (16), i.e. functions of b2\n?andb3,\nand since the latter ones are even in momentum direc-\ntion, the distributions are even in momentum direction\nas well. Therefore the polarization becomes\n~ s=X\npg~ e=~ ezX\npg\u0012\n1\u0000b2\n?\n2\u0013\n=~ ez \ns0\u0000B2\ng\n2!\n(57)\nwith\ns0=X\npg;B2\ng=X\npb2\n?g;m=sq=0:(58)\nTheq!0 limit is of course dependent on the q-\ndependence of the potential. Therefore let us analyze the\nneutral scattering V0and charged scattering V0=e2=\u000fq2\nseparately. As we will see, only the latter provides den-\nsity waves as a collective plasma oscillation.\nA. Excitation with neutral scattering\nTo study the excitation modes for scattering with neu-\ntral impurities, we can restrict ourselves to the lowest-\norder expansion in q, which simpli\fes the results of the\nlast section again. Especially \u0005 0=~\u0005 =~\u0005xf=~\u0005xg=\n~\u0005e= !\u0005fe= 0. According to (53) and (54) we obtain\nfor the density excitation from (46)\n\u0012\n1\u0000i\n!+\u001c+o(q2)\u0013\n\tj= 0 (59)\nwith!+=!+i=\u001cwhich means that we have either\nthe zero mode != 0 or \t j= 0, i.e., no density dipole\nwave excitations. This will be di\u000berent when we consider\ncharged scattering in the next section.\nWith the help of this result the equation for the spin\nexcitation becomes from (46) in long-wavelength expan-\nsion simply\nD~\tj\u0011(1\u0000V0 !\u0005 )~\tj\u0000V0~\u00052\u0002~\tj= 0 (60)\nwith\n~\u00052=i~ ez\"\n\u0000s0+B2\ng\n2(1\u0000\u0006n@\u0006n)\u0000q2Eg\nDme@2\n!#\n2!\n!2\u00004\u00062n\n !\u0005 =0\nB@s0+Bg11+B2\ng\u0006n\n2@\u0006n 0 0\n0 s0+Bg22+B2\ng\u0006n\n2@\u0006n0\n0 0 \u0000B2\ng1\nCA\n\u00024\u0006n\n4\u00062n\u0000!2\n+: (61)8\nFIG. 1: The three frequencies of spin excitation mode (63) as\na function of selfenergy \u0006 and thermally averaged spin-orbit\ncoupling (58) and magnetization m=sq=0.\nThe vanishing determinant of Din (60) yields the collec-\ntive spin excitation wave.\nNeglecting \frst the thermally averaged spin-orbit\nterms completely we obtain the modes\n!spin=\u0000i\n\u001c\u00062jsV0\u0000\u0006nj (62)\nwhich shows that two spin modes are excited. Provided\nwe have an interaction V0the mode is shifted simply by\nthe mean\feld selfenergy \u0006 n=nV+V0s+\u0016BBand this\nmode!\u00182\u0016BBe\u000bis exclusively dependent on the e\u000bec-\ntive Zeeman shift \u0016BBe\u000b=nV+\u0016BB. This result we\nhad already obtained as zero order in spin-orbit coupling\n(27) in agreement with the recent report of a transition\nfrom charge to spin density waves only appearing at a\n\fnite Zeeman \feld34.\nAs discussed in Sec. III.H of part I of this paper [], the\nselfconsistency would result into the replacement \u0006 n=\nnV+V0s+\u0016BB!(nV+\u0016BB)=(1 +me\n2\u0019~2V0). In order\nto facilitate the following notation we write shortly \u0006 for\nthis selfconsistent \u0006 n.\nThe dispersion including the spin-orbit coupling is de-\npendent on the parameter B2\ng=B2\ng11+B2\ng22given in\n(58). The spin modes as zeros of jDjof (60) appear to be\n\u0012\n!+i\n\u001c\u00132\n\u00004(\u0006\u0000sV0)2= 2V0\u0002\nB2\ng\u0006\n+(sV0\u00002\u0006)\u0012\ns\u0006r\ns2\u00002\nV0\u0006B2g\u0013\u0015\n(63)\ntogether with a third mode as the sum of the right side.\nThe result is plotted in Fig. 1. One sees that the two\nmodes (62) appear, which di\u000ber with increasing \u0006. The\nthreefold splitting of spin modes was reported in Ref.71.\nA closer inspection shows that one mode can become\nimaginary for small selfenergies \u0006. In fact, expanding up\n0.00.51.01.52.001234\nBg2\nm/CapSigma\nm/LBracket1V0/RBracket1\n0.51.01.5/CapSigma\nm/LBracket1V0/RBracket1\n/Minus4/Minus224Ω2\nm2/LBracket1V02/RBracket1\nBg2/Equal1mFIG. 2: Left: The maximal range of spin-wave instability for\nvanishing collisional damping. The outer area (blue) designs\nthe Im!26= 0 modes and the inner (yellow) area the !2<0\nmodes. Right: The real (black solid) and imaginary (dashed\nred) part of !2versus \u0006\nto +o(B2\n?;) one gets\n\u0012\n!spin+i\n\u001c\u00132\n= 48\n<\n:\u00062\u0010\n1\u0000B2\ng\ns\u0011\n(\u0006\u0000sV0)2+B2\ng\u0006(\u0006\u0000sV0)\ns(64)\nwhere the square of the \frst mode can become negative\nwhich means an imaginary mode. Due to the six-order\npolynomial in !, for each dispersion, also the complex\nconjugated one is a solution. A \fnite imaginary part\nmeans instability when it overcomes the damping by col-\nlisions 1=\u001c.\nThe maximal range of such possible spin-wave insta-\nbility (without collisional damping 1 =\u001c!0) is shown\nin Fig. 2. Here, we distinguish the region of !2<0 ap-\npearing as the inner region (yellow) and the region where\nIm!26= 0 as the outer (blue) one. The twofold regions\nof complex frequencies are seen in the cut in Fig. 2.\nThe physics of these instabilities can be seen more\nexplicitly in the zero temperature limit where in quasi-\ntwo-dimensions and in the presence of linear Dresselhaus\n\f=\fDor Rashba\f=\fRspin-orbit coupling the density\nand the polarization become\nn=X\npf=me\n2\u0019~2(\u000ff+\u000f\f)\ns=X\npg=\u0000me\n2\u0019~2q\n\u000f\f(\u000f\f+ 2\u000ff) + \u00062n (65)\nwith the spin-orbit energy \u000f\f=me\f2. Further we have\nB2\ng=4\u0019~2\nmepn2\u000f\f\n\u00062n(66)\nwith the polarization p=s=n.\nIn Fig. 3 we plot the non-self-consistent and self-\nconsistent modes in dependence on the density. Using\nthe polarization and the scaling with the Fermi energy\n\u000ffallows us to get rid of the spin-orbit energy. One sees\nthat the selfconsistency leads to smaller modes at smaller9\n01234n/LBracket1ΕF/Slash1V0/RBracket112345Ω/LBracket1ΕF/RBracket1\n0.00.51.01.52.0n/LBracket1ΕF/Slash1V0/RBracket10.51.01.52.0Ω/LBracket1ΕF/RBracket1\nFIG. 3: The real (solid) and imaginary (dashed) part of !\nas a function of the density for the e\u000bective magnetic \feld\n\u0016BBe\u000b=nV+\u0016BB= 1\u000ff, the polarization p= 0:5 and\nthe dimensionless potential v0=meV0=2\u0019~2= 1 at zero tem-\nperature. Left \fgure shows the selfconsistent result and right\n\fgure the non-selfconsistent one. Di\u000berent branches of the\nsingle mode are distinguished additionally by di\u000berent colors.\n0.00.51.01.52.0n/LBracket1ΕF/Slash1V0/RBracket10.51.01.52.0Ω/LBracket1ΕF/RBracket1\n0.00.51.01.52.0n/LBracket1ΕF/Slash1V0/RBracket10.51.01.52.0Ω/LBracket1ΕF/RBracket1\nFIG. 4: The selfconsistent modes of \fgure 3 for e\u000bective\nmagnetic \felds \u0016BBe\u000b=nV+\u0016BB= 0:5\u000ff(left) and\n\u0016BBe\u000b=nV+\u0016BB= 0:1\u000ff(right).\ndensities. Up to a critical density, we do not have any\nimaginary part. As soon as the two spin modes vanish, a\ndamping occurs that is symmetric in sign such that it de-\nnotes an instability. The third mode of (63) is vanishing\nat higher densities.\nThe dependence on the e\u000bective magnetic \feld is seen\nin Fig. 4, which shows that the two spin modes become\nnontrivial and vanish at the same density as the third\nmode for vanishing magnetic \feld.\nThe expansion of (63) in small spin-orbit coupling\nreads\n\u0012\n!spin+i\n\u001c\u00132\n=\u001a\n4\u00062\u00008\u000ff\u000f\f\n(4\u00062+ 8\u000ff\u000f\f)(1 +me\n2\u0019~2V0)2+o(\u000f2\n\f)\n(67)\nand the third mode 4\u00062\u00008meV0\n2\u0019~2\u000ff\u000f\f. This shows that\nthe spin modes becomes\n!+i=\u001c=8\n<\n:i\u0010\n2\fpf\u0000\u00062\n\fpf\u0011\n\u0010\n2\fpf+\u00062\n\fpf\u0011\u0000\n1 +meV0\n2\u0019~2\u0001+o(\u000f2\n\f;\u00064) (68)\nfor small e\u000bective Zeeman \felds providing a linear de-\npendence of the energy and damping on \f. This is in\ncontrast to the in\ruence of the Landau levels which pro-\nvides quadratic dependencies25,26.\nB. Excitation with charged scattering\nNow we consider the charged scattering with an im-\npurity Coulomb potential V0=e2=\u000foq2or the scatteringbetween charged particles as mean\felds and in the re-\nlaxation time approximation. Using the long-wavelength\nexpansions of Appendix C, we obtain the following equa-\ntion system from (46)\n\u0012!2\n+\n!2p+ 1\u0000i!+\n!2p\u001c\u0013\n\tj=~ m\u0001~\tj\n2X\npg(1\u0000~ e\u000e~ e)~\tj+i!~ m\u0002~\tj= 0 (69)\nwith~ m=~ sq=0=~ ezs. We used only the most divergent\nterms\u00181=q2in the equation for \u000e~ sand have introduced\nthe plasma frequency !2\np=e2n=\u000f0me. The eigenmodes\nof (69) can be seen to decouple for density and spin modes\nsince the equation provides the eigenmode as a vanishing\ndeterminant of the matrix in front of ~\tj. The density\neigenmodes are given by the vanishing left side of the\n\frst equation which implies ~ m?~\tjwhich means that\nwe have only transverse spin modes with respect to the\ne\u000bective magnetization axes ~ mof (57). The frequency of\nthe density modes is just the damped plasma oscillation,\n!n=\u0000i\n2\u001c\u0006r\n!2p\u00001\n4\u001c2(70)\nand the spin oscillation becomes\n!spin=\u0000i\n\u001c\u00062[\u0006 +o(b2)] (71)\nwhere we have used the linear expansion in bof the last\nparagraph. Compared with the spin mode of the neutral\nexcitation (62), we see that the term sV0is absent.\n1. Dielectric function\nThe response function (13) describes the density\nchange with respect to the external potential \u000en=\n\u001f\b while the polarization function \u000en= \u0005\bindis the\ndensity variation with respect to the induced potential\n\bindVq\u000en+ \b. Therefore one has \u001f= \u0005=(1\u0000Vq\u0005), and\nthe dielectric function as a ratio of the induced to the\nexternal potential is\n1\n\u000f= 1 +Vq\u001f: (72)\nIf we expand the response function up to quadratic orders\nof the wave vector, as performed in appendix C, the result\nfor the dielectric function can be compactly written as\n\u000f(!;q) = 1\u00001\n1\n1\u0000\u000f(!;0)\u0000q2\n\u0014eff(!)(73)\nwhere the long-wavelength dielectric function can be ex-\npressed as\n\u000f(!;0) =\u000f!\n+p2\u0012\n1\u00001\n\u000f!\u0013\"\n1\u0000\u000f!\u0000B2\nf\n\u000f! \n(1\u0000\u000f!)2\u0000!2\np\n!2(1+p2)!#\n(74)10\nFIG. 5: The long-wave excitation function (74) for a collision\nfrequency\u001c\u00001= 0:3!p(left) and\u001c\u00001= 1!p(right).\n0.51.01.52.0Ω/Slash1Ωp0.51.01.52.02.53.0ImΕ/Minus1\np/Equal1p/Equal0.5p/Equal0.3p/Equal0\nFIG. 6: The long-wave excitation function (74) for di\u000berent\npolarizations versus frequency as cuts of \fgure 5 for \u001c\u00001=\n0:3!p.\nin terms of the Drude's expression\n\u000f!= 1\u0000!2\np\n!(!+i\n\u001c)(75)\nand the e\u000bective polarization\np=s\nn=n\"\u0000n#\nn\u0000B2\ng\n2n: (76)\nHere we use the zero temperature result for linear spin-\norbit coupling\nB2\ng=2p\n1 +p2B2\nf (77)\nwith (58). Let us discuss the long-wavelength limit of\nthe dielectric function \frst without spin-orbit coupling\nBf= 0 but \fnite polarization (76). This corresponds to\nthe treatments of two-\ruid models, e.g. one-dimensional\nquantum wires82, with \fnite polarization. In \fgure 5 we\nplot the excitation function which yields the weight of the\ncollective modes as a function of frequency and polariza-\ntion. There we plotted a larger range of polarizations.\nSince the latter one is an e\u000bective one according to (76)\nwe can have values smaller than \u00001. A smaller relaxation\ntime leads to a higher damping of modes, of course.\nWe see the appearance of two collective modes with\nincreasing polarization. The plasma mode becomes split\nin a fast decaying mode with increasing polarization and\n0.00.51.01.52.02.53.0/Minus0.20.00.20.4\n/VertBar1p/VertBar1ImΕ/Minus10.4Ωp0.1Ωp0.01ΩpFIG. 7: The long-wave excitation function (74) for three\nsmall frequencies versus e\u000bective polarization (76).\n0.51.01.52.0Ω/Slash1Ωp\n/Minus224ImΕ/Minus1\nBf/Equal5Bf/Equal3Bf/Equal1Bf/Equal0\nFIG. 8: The long-wave excitation function (74) for di\u000berent\nspin-orbit couplings Bfversus frequency with a \fxed polar-\nizationp= 0:3 of \fgure 6.\na mode which becomes sharper again with increasing po-\nlarization. This is illustrated in \fgure 6 as cuts for special\npolarizations.\nNear the point of vanishing \frst mode around p= 1\nthe excitation function becomes negative indicating an\ninstability. This is illustrated in the next \fgure 7. The\ne\u000bective polarization as the sum of polarization and spin-\norbit coupling term Bgcan lead to negative excitation\nfunctions indicating an instability. We will interpreted\nthis instability as a de-mixing of spin states later when\ndiscussing the screening length.\nNext we consider the in\ruence of the spin-orbit cou-\nplingBfwhich is given in \fgure 8 for di\u000berent values and\na \fxed polarization. We see that the spin-orbit coupling\nhas basically the same e\u000bect as an additional polariza-\ntion. Above a certain spin-orbit coupling the excitation\nfunction becomes negative indicating an instability. This\nis also visible in the \fgure 9.\nThe range where the excitation function becomes neg-\native indicating an instability is plotted in the next \fgure\n10. This range indicates a spin domain separation and\nbecomes large for increasing spin-orbit coupling parame-\nterB2\nfof (C6). We interpret this as spin segregation as11\n0.00.51.01.52.0/Minus0.20.00.20.4\n/VertBar1p/VertBar1ImΕ/Minus1 Bf/Equal0.6Bf/Equal0.4Bf/Equal0.2Bf/Equal0\nFIG. 9: The long-wave excitation function (74) for di\u000berent\nspin-orbit couplings Bfversus e\u000bective polarization (76) with\na \fxed polarization jpj= 1 and frequency != 0:1!pof \fgure\n7.\n0.00.20.40.60.81.00.81.01.21.41.61.82.0\nΩ/Slash1Ωp/VertBar1p/VertBar1\n0.00.51.01.52.00.00.51.01.52.0\nΩ/Slash1Ωp/VertBar1p/VertBar1\nFIG. 10: The region where the excitation function becomes\nnegative versus frequency and polarization for Bf= 0 (left)\nandBf= 2 (right) . The upper range (yellow) is for 1 =\u001c=\n1!p, the middle (red) for 1 =\u001c= 0:3!pand the bottom (blue)\none for 1=\u001c= 0:001!p.\nobserved in17and described in19.\na. Screening length Next we discuss the e\u000bective dy-\nnamical screening length \u001fe\u000b(!) of (73) which can be\nexpressed shortly in terms of (75) as\n\u0012\u0014\n\u0014e\u000b(!)\u00132\n=1\n1\u0000i!\u001c\u0000p\u00142\nV\u000f!\u00001\n\u000f!\u001a\n1+1\n(1\u0000\u000f!)(1\u0000q!)\n+Bf\u000f!\u00001\n\u000f!(1\u0000q!)\u0014\n1+\u0012\n1+2(1\u0000i!\u001c)\n!2p\u001c2(\u000f!(1 +q!)\u00001)\u0013\n\u0002\u00121\n(1\u0000\u000f!)(1\u0000q!)\u0000q!\u0013\u0015\u001b\n(78)\nand the short-hand notation q!=p2(\u000f!\u00001)=\u000f!. The\nstatic limit where ( \u000f!\u00001)=\u000f!!1 and\u000f!!1 reads\ntherefore\n\u0012\u0014\n\u0014e\u000b(0)\u00132\n= 1\u0000p\u00142\nV\u0014\n1 +Bf\u0012\n1\u00002p2\n!2p\u001c2(1\u0000p4)\u0013\u0015\n:\n(79)\nWe have abbreviated \u00142\nV= (V+\u0016BB=n)@\u0016n=\n\u0016BBe\u000b@nn=n. This result is explicitly an analytic long-\nwavelength expression of the in\ruence of polarization on\n01234/Minus10/Minus50510\n/VertBar1p/VertBar1/LParen1Κeff/LParen10/RParen1/Slash1Κ/RParen12ΚV/Equal1.5ΚV/Equal0FIG. 11: The static e\u000bective screening length (79) versus\npolarization for di\u000berent \u0014VandBf= 0 (solid) and Bf= 1\n(dashed).\nFIG. 12: (Color online) The range where the static e\u000bective\nscreening length (79) becomes negative for 1 =\u001c= 1!p(left)\nand 1=\u001c= 0!p(right).\nthe screening length which was treated other wise by ex-\ntensive numerics28,83.\nThe static screening length (79) changes only for \f-\nnite\u00142\nVwhich means a \fnite magnetic \feld or ferromag-\nnetic impurity polarization V. In other words we need a\npreferred direction of motion in order to see a change of\nstatic screening length. In the latter case it is then depen-\ndent on the spin-orbit coupling Bfas illustrated in \fgure\n11. One sees that the screening length increases with in-\ncreasing polarization for \u00142\nV6= 0 and diverges at the zero\nof (79) which provides the critical \u00142\nVin terms of the\npolarization pand the material parameter !p\u001c. This in-\nstability appears here in the spatial screening length and\nwe can interpret it as a spatial domain separation of spin-\npolarized electrons known as domain wall formation84,85.\nInterestingly, for \fnite spin-orbit coupling there appears\nan upper singularity at p= 1.\nThe range where the real part becomes negative is plot-\nted in \fgure 12. With increasing collision frequency the\nrange of instability becomes smaller.\nComparing the case with vanishing collision frequency\nin \fgure 12 we see that only one range at positive polar-12\n12345Ω/Slash1Ωp\n/Minus4/Minus2246/LParen1Κeff/Slash1Κ/RParen12\np/Equal0.3p/Equal0.1p/Equal0\nFIG. 13: The dynamical e\u000bective screening length (78) ver-\nsus frequency for di\u000berent polarizations, \u0014V= 1, 1=\u001c= 0:3!p\nandBf= 0.\nization appears. In other words there appears an asym-\nmetric second range in the instability due to the collisions\nfor positive and negative polarizations.\nIt is interesting to discuss the dynamical screening\nlength as well. First one notes that the correct static\nlimit (79) only appears if we have a relaxation damping\n1=\u001cwhich drops out of the result. In contrast if we \frst\nset 1=\u001c!0 before the static limit we would obtain\nlim\n\u001c!1\u0012\u0014e\u000b(!)\n\u0014\u00132\n=(!2\u00001)(!2+p2\u00001)\n\u00142\nVp(!4+p2\u00001)(80)\nleading to the wrong static limit \u00001=p\u00142\nVwhich is clearly\nunphysical. Therefore an even in\fnitesimal friction is\nnecessary in order to ensure the correct static screening\nlength. One can see this also from the limit of vanishing\npolarization p!0 which yields\nlim\np!0\u0012\u0014e\u000b(!)\n\u0014\u00132\n= 1\u0000i!\u001c: (81)\nThe dynamical screening length is plotted in \fgures 13\nand 14 for di\u000berent cuts. Like in the static limit above,\nat certain\u0014Vthe dynamical screening length becomes\nnegative indicating a domain-wall formation.\nThe range where the real part becomes negative is\ngiven in \fgure 15. With increasing collision frequency\nthe range of instability becomes smaller. If we addition-\nally demand that the imaginary part of the dynamical\nscreening length should be positive which means spatially\nunstable modes, we get a smaller region. Above a certain\ncollisional damping there is no such region.\n12345Ω/Slash1Ωp\n/Minus4/Minus2246/LParen1Κeff/Slash1Κ/RParen12\nΚV2/Equal1ΚV2/Equal0.5ΚV2/Equal0FIG. 14: The dynamical e\u000bective screening length (78)\nversus frequency for di\u000berent \u0014V, a polarization p= 0:1,\n1=\u001c= 0:3!p, andBf= 0.\n0.00.51.01.52.00.00.20.40.60.81.0\nΩ/Slash1Ωpp\n0.00.51.01.52.00.00.20.40.60.81.0\nΩ/Slash1Ωpp\nFIG. 15: (Color online) Left: The range where the dynam-\nical e\u000bective screening length (78) becomes negative versus\nfrequency and polarization for \u0014V= 1. The inner range (yel-\nlow) is for 1 =\u001c= 0:5!p, the middle (red) for 1 =\u001c= 0:3!p\nand the outer (blue) one for 1 =\u001c= 0:1!p. Right: additional\ndemand that Im \u0014eff>0\nC. Spin response\nThe spin response \u000e~ s=~ \u001fs\b can be calculated as well\nand we obtain with ~ q\b =ie~Eq\n\u000e~ sq\nE=ie\nq~ \u001fs=en\n!p0\nB@s1(!)b1(q)\nq+Bcs2(!)b2(q)\nq\ns1(!)b2(q)\nq\u0000Bcs2(!)b1(q)\nq\niq\u000f0!p\nne2s3(!)1\nCA(82)\nwhich means we have an induced spin due to an ap-\nplied electric \feld as used in microwave spectroscopy41.\nThis is purely transverse to the z-direction of the ef-\nfective magnetic and ferromagnetization \feld for long-\nwavelength. We consider this as the response of spin-\nHall e\u000bect, described by the spin-orbit coupling ~b(q) =\n(B?1;B?2;B?3) according to (56) and (C6). Since ~ qjj~E\nthis q-dependent spin-orbit coupling describes the exci-\ntation due an external electric \feld. Oscillating electric\ncurrents are used experimentally to create an e\u000bective\nmagnetic \feld and ferromagnetic resonances40.\nThe external magnetic \feld enters (82) by the dimen-13\nsionless quantity\nBc=!c\u000ff\nnD~!2p=\u000f0B\nnDe~\u000ff\nn: (83)\nThe frequency-dependent functions (82) can be recast\ninto the form\ns1+ 1 =\u00001 +p\n2I(p)\u00001\u0000p\n2I(\u0000p)\ns2=2I1(p)\n1\u0000p2\u0000I(p)\u0000I(\u0000p)\np+I2(p)\np(1 +p)\u0000I2(\u0000p)\np(1\u0000p)\ns3=\u00001 +p\n2I(p) +1\u0000p\n2I(\u0000p) (84)\nwith\nI(p) =1\n\u00001\u0000p+i\n!2p\u001c!+!2\n!2p!\u0000!psin\rt\n\re\u0000t\n2\u001c\nI1(p) =1\n1\u0000i!\u001c!1\n\u001ce\u0000t\n\u001c\nI2(p) =i!\n\u001c!2pI1(p)!1\n\u001c@\n@tsin\rt\n\re\u0000t\n2\u001c(85)\nand\n\r=r\n(1 +p)!2p\u00001\n4\u001c2: (86)\nSince we apply a frequency-constant electric \feld it\nmeans we have an instant disturbance of the system at\ntimet= 0 in the form E(t) =E\u000e(t). This \feld itself has\nto be subtracted from the response which is represented\nby the constant s1(!) + 1. Further we present the lin-\nearized result with respect to the spin-orbit coupling. A\nnonlinear analytic result with some more drastic simpli-\n\fcations can be found in86.\nThe collisions are responsible for the damping of this\noscillatory motion. Dependent on the temperature we\nwill have a transition from collision-dominated damped\nmotion towards an oscillatory regime as observed in87.\nThis transition is here explicitly seen in the expression\nfor\rin (86) which turns the oscillatory behavior into an\nexponential one if\n(1 +p)!2\np<1\n4\u001c2(87)\nwhich provides density, polarization, and (due to the re-\nlaxation time) temperature-dependent criteria for such a\ntransition.\nIt is now interesting to inspect the spin response for\nlinear Dresselhaus and Rashba spin-orbit coupling. If\nthe electric \feld is excited in x-direction we have for\nDresselhaus b1=\fDqx=\u0006n;b2= 0 and for Rashba\nb1= 0;b2=\u0000\fRqx=\u0006n. This translates into the spin\nresponse\n\u000e~ sq\nE\f\f\f\f\nD=\fRqxen\nq\u0006n!p(s1;\u0000s2Bc;0)\n\u000e~ sq\nE\f\f\f\f\nR=\fDqxen\nq\u0006n!p(\u0000s2Bc;\u0000s1;0) (88)\nFIG. 16: (Movie online) The time-dependent trajectories\nof the induced spin with the disturbance of the electric \feld\n~E\u000e(t) inx-direction for Dresselhaus spin-orbit coupling and\n1=\u001c= 0:1!p. The external magnetic \feld was chosen Bc= 1\naccording to (83).\nIf the electric \feld is excited in y-direction we have\nfor Dresselhaus b2=\u0000\fDqy=q;b 1= 0 and for Rashba\nb2= 0;b1=\fRqy=qand the response in (88) are inter-\nchanged between Dresselhaus and Rashba. Therefore it\nis su\u000ecient to discuss one of the cases, say excitation in\nx-direction. Let us concentrate on the Dresselhaus relax-\nation. From (88) we see that the external magnetic \feld\nBccauses ellipsoid trajectories. If it is absent, we have a\nmere linear-polarized damped oscillation in x-direction.\nIn the next \fgure 16 we plot the trajectories for di\u000ber-\nent polarizations.\nOne recognizes that with increasing polarization the\nspin response turns to the perpendicular direction of the\napplied electric \feld which is a spin-Hall e\u000bect. Here we\ncan see how the evolution of trajectories changes with\nincreasing polarization. The Rashba spin-orbit coupling\nwill lead to the same curves but with 90oclockwise rota-\ntion as one sees from (88) too.\nSince the change of spins (82) is di\u000berent in each spa-\ntial direction triggered by spin-orbit coupling and split\nfurther by the magnetic \feld one can predict that this\nwill lead to an anomalous spin segregation as was ob-14\n51015202530t/LBracket11/Slash1Ωp/RBracket1\n/Minus2/Minus112\ns3/LParen1t/RParen1s2/LParen1t/RParen1s1/LParen1t/RParen1\n51015202530t/LBracket11/Slash1Ωp/RBracket1\n/Minus2/Minus112\ns3/LParen1t/RParen1s2/LParen1t/RParen1s1/LParen1t/RParen1\n51015202530t/LBracket11/Slash1Ωp/RBracket1\n/Minus2/Minus112\ns3/LParen1t/RParen1s2/LParen1t/RParen1s1/LParen1t/RParen1\n51015202530t/LBracket11/Slash1Ωp/RBracket1\n/Minus2/Minus112\ns3/LParen1t/RParen1s2/LParen1t/RParen1s1/LParen1t/RParen1\nFIG. 17: The time dependent spin response function (82)\nfor 1=\u001c= 0:3!pand polarizations p= 0:01;0:3;0:6;0:9 from\nupper left to lower right.\nserved in88and investigated in one-dimensional systems\nin89.\nThe spin dephasing time is of special interest90{92\nwhere one has found discrepancies between the experi-\nmental values and earlier treatments. It is now quite dif-\n\fcult to extract dephasing times since the envelope of the\noscillation in each direction shows maxima and a quite\nnonlinear behavior as illustrated in \fgure 17 by the con-\nstituent time-dependent functions of (82). One sees that\nbesides oscillations with the frequency (86) the s1(t) pos-\nsesses a maximum and all functions become quite non-\nlinear for higher polarizations. These components mix\nadditionally due to the spin-orbit coupling and the mag-\nnetic \feld. If we, nevertheless, \ft these time dependence\nto a damped exponential oscillator, we can extract the\nspin-dephasing time \u001csanalogously to92.\nThe results are given in \fgure 18. The overall observa-\ntion is that the spin dephasing time is an order of magni-\ntude larger than the relaxation time. One sees that the\ns1component, which corresponds to the x-component for\nthe Dresselhaus and ycomponent for the Rashba cou-\npling has a minimum at a polarization which increases\nwith increasing relaxation time. The minima in s2and\ns3are not so pronounced and shift to larger polariza-\ntions as well with increasing relaxation time. This result\nis di\u000berent from92where only an increasing spin dephas-\ning time in dependence on the polarization has been re-\nported. The combined e\u000bect of the Rashba and the Dres-\nselhaus coupling as well as the magnetic \feld mixes these\nresults according to (82).\nHere, we extracted the spin-dephasing time as an enve-\nlope of the precessional motion of the spins after a sudden\ndistortion by an electric \feld. This is the Dyakonov-Perel\nmechanism of relaxation, e.g. investigated experimen-\ntally and theoretically in56{58\n0.00.20.40.60.81.0010203040\npΤs/LBracket11/Slash1Ωp/RBracket1\ns3s2s1\n0.00.20.40.60.81.0024681012\npΤs/LBracket11/Slash1Ωp/RBracket1s3s2s1\n0.00.20.40.60.81.002468\npΤs/LBracket11/Slash1Ωp/RBracket1s3s2s1\n0.00.20.40.60.81.0024681012\npΤs/LBracket11/Slash1Ωp/RBracket1\nΤ/Equal1/Slash1ΩpΤ/Equal2/Slash1ΩpΤ/Equal3/Slash1ΩpΤ/Equal4/Slash1ΩpΤ/Equal5/Slash1ΩpFIG. 18: The spin dephasing times (82) from exponential\nenvelops for the di\u000berent directions versus polarization with\n1=\u001c= 0:2;0:5;0:8 from upper left to lower left and s1for\ndi\u000berent relaxation times (lower right).\nIV. RESPONSE WITH MAGNETIC FIELDS\nA. Linearizing kinetic equation with magnetic \feld\nWe want to consider the spin and density response to\nan external perturbing electric \feld now under a constant\nbias of magnetic \feld. The magnetic \feld consists of a\nconstant and an induced part B(x;t) =B+\u000eB(x;t).\nSince the external electric \feld perturbation is produced\ndue to an external potential Uextone sees from the\nMaxwell equation that \u000e_~B=\u0000r\u0002\u000e~E= 0, which means\nthat all terms linear in the induced magnetic \feld vanish.\nIt is convenient to work in velocity variables instead of\nmomentum de\fned according to (4) and we use for the\nquasiparticle energy \u000fp=p2=2me+\u00060. We obtain \fnally\nfrom (3) with the Larmor frequency !c=eB\nme\n\u0014\n\u0000i!+iqv+1\n\u001c+ (v\u0002!c)@v\u0015\n\u000ef\n+\u0014iq\nme@v\u0006i+ (~ \u001c\u00001)i\u0000\u0012@v\u0006i\nme\u0002!c\u0013\n@v\u0015\n\u000egi=S0\n(89)\nand\n\u0014\n\u0000i!+iqv+1\n\u001c+ (v\u0002!c)@v\u0015\n\u000egi\u00002me(\u0006\u0002\u000eg)i\n+\u0014iq\nme@v\u0006i+ (~ \u001c\u00001)i\u0000\u0012@v\u0006i\nme\u0002!c\u0013\n@v\u0015\n\u000ef=Si(90)15\nwith the source terms arising from the external \feld ~ q\b =\nie~Eand the induced mean\feld variations (37)\nS0=iq@vf\nme\b +\u000en\n\u001c@\u0016n@\u0016f0\n+\u0012\niq\nme+!c\u0002@\u0006\nv\nme\u0013\n\u000e\u00060@vf+\u0012\niq\nme\u0000!c\u0002@\u0006\nv\nme\u0013\n\u000e\u0006i@vgi\nSi=iq@vgi\nme\b + 2(\u000e\u0006\u0002g)i+\u000en@\u0016g\n\u001c@\u0016nei\n+\u0012\niq\nme+!c\u0002@\u0006\nv\nme\u0013\n\u000e\u00060@vgi+\u0012\niq\nme\u0000!c\u0002@\u0006\nv\nme\u0013\n\u000e\u0006i@vf:\n(91)\nCompared with the result without magnetic \feld we\nsee that the source terms (20) get additional rotation\nterms coupled to the momentum-dependent derivation of\nmean\felds (37) which is present only with extrinsic spin-\norbit coupling. Further the drift side gets an explicit\nderivative with respect to the velocity which we will take\ninto account in the following.\nB. Solution of linearized equations\nIn order to solve (89) and (90) we use the same coordi-\nnate system as Bernstein93. The magnetic \feld ~Bpoints\nin thevzdirection and the qvector is in the vz\u0000vxplane\nwith an angle \u0002 between vxandq\n~ q=qsin \u0002~ ex+qcos \u0002~ ez: (92)\nFor the velocity vwe use polar coordinates around ~B\nwith an azimuthal angle \u001e\n~ v(\u001e) =wcos\u001e~ ex+wsin\u001e~ ey+u~ ez (93)\nand one gets\n1\n\u001c\u0000i!+i~ q~ v+ \n~ v\u0002e~B\nm!\n~@v=1\n\u001c\u0000i!+i~ q\u0001~ v(\u001e)\u0000!c@\u001e\n=1\n\u001c\u0000i!+i~ q\u0001~ v(!ct\u001e)\u0000@t\u0011\u0000i\nt\u001e\u0000@t\u001e (94)\nwith the orbiting time\nt\u001e=\u001e=!c: (95)\nWe can write the equations (89) and (90) as\n\u0000\n\u0000i\nt\u001e\u0000@t\u001e\u0001\n\u000ef+ \niq@v~\u0006\nme+~ \u001c\u00001!\n\u0001~\u000eg=S0\n\u0000\n\u0000i\nt\u001e\u0000@t\u001e\u0001\n\u000egi+\u0012iq@v\u0006i\nme+(~ \u001c\u00001)i\u0013\n\u000ef\u00002(~\u0006\u0002~\u000eg)i=Si:\n(96)\nwhere the corresponding right hand sides are given by\n(91). Now we employ the identity\n~B\u0001\u000e~ g+ (~ \u001b\u0001~B)\u000ef\u00002~ \u001b\u0001(~\u0006\u0002\u000e~ g) =\n~ \u001b\u0001~B+ 2i~\u0006\n2\u000e^F+\u000e^F~ \u001b\u0001~B\u00002i~\u0006\n2(97)with~B=iq@v~\u0006=me+~ \u001c\u00001and\u000e^F=\u000ef+~ \u001b\u0001\u000e~ gwhich\none proves with the help of ( \u001ca)(\u001cb) =a\u0001b+i\u001c(a\u0002b).\nThis allows to rewrite (96) into\n\u0000@t\u001e^\u000eF\u0000i\nt\u001e^\u000eF\n+~ \u001b\u0001 ~B\n2+i~\u0006!\n\u000e^F+\u000e^F~ \u001b\u0001 ~B\n2\u0000i~\u0006!\n=^Sp\u001e(!)\n(98)\nwhere ^Sp\u001e=S0+~ \u001b\u0001~S.\nPlease note that due to (95) the integration over the\nazimuthal angle is translated into the time integration\nabout orbiting intervals. Therefore Eq. (98) has a great\nsimilarity to the time-dependent Eq. (32).\nEquation (98) is easily solved as\n\u000eF=\u0000tZ\n1d\u0016tei\u0016tR\nt\n+\nt0dt0\ne~ \u001btR\n\u0016t\u0012~Bt0\n2+i~\u0006t0\u0013\ndt0\n^Sp!c\u0016te~ \u001btR\n\u0016t\u0012~Bt0\n2\u0000i~\u0006t0\u0013\ndt0\n=0Z\n\u00001dxe\u0000ixR\n0\n+\nt\u0000ydy\n\u0002e~ \u001bxR\n0(1\n2~Bt\u0000y+i~\u0006t\u0000y)dy^Sp!c(t\u0000x)(!)e~ \u001bxR\n0(1\n2~Bt\u0000y\u0000i~\u0006t\u0000y)dy\n(99)\nwhere we used !+=!+i\u001c\u00001as before. The \frst expo-\nnent can be calculated explicitly with the de\fnitions of\n(93) and (94)\nixZ\n0\n+\nt\u0000ydy=i!+x\u0000i~ q\u0001Rx\u0001~ v(t) (100)\nwith the matrix94\nRx=1\n!c0\n@sin!cx 1\u0000cos!cx0\ncos!cx\u00001 sin!cx 0\n0 0 !cx1\nA (101)\nhaving the property R\u0000x=\u0000RT\nx. Neglecting the\nmagnetic-\feld dependence in the phase qRv=qv+o(B)\nwe obtain with (99) exactly again the solution (34) but\nwith an additional retardation in the momentum Sp!c(t\u0000x)\ninstead ofSp!ct=Spin (34).\nWe employ the long-wavelength approximation ~B\u0019 0\nneglecting the vector relaxation. The integration over\nan azimuthal angle x=\u001e=!cis coupled to the momen-\ntum (velocity) arguments. The spin-orbit coupling pro-\nvides a momentum-dependent ~\u0006 which couples basically\ntoq@p~\u0006. Since\nmeq@p=qsin \u0002\u0012\ncos\u001e@w\u0000sin\u001e\nw@\u001e\u0013\n+qcos \u0002@u\n(102)16\nin the coordinates (92) and (93), it means we neglect\nhigher than \frst order derivatives in \u001eand@p~\u0006 when\napproximating ~\u0006t\u0000y\u0019~\u0006tin the exponent. We obtain\n\u000e^F=\u000ef+~ \u001b\u0001\u000e~ g=0Z\n\u00001dxei(~ qRx~ vt\u0000!+x)eix~ \u001b~\u0006t^Sp!c(t\u0000x)e\u0000ix~ \u001b~\u0006t:\n(103)\nTo work it out further we use ei\u001c\u0001a= cosjaj+i\u001c\u0001a\njajsinjaj\nto see that\nei~ \u001b\u0001~\u0006x(S0+~ \u001b\u0001~S)e\u0000i~ \u001b\u0001~\u0006x=S0+ (~ \u001b\u0001~S) cos(2xj\u0006j)\n+~ \u001b(~S\u0002~ e) sin(2xj\u0006j) + (~ \u001b\u0001~ e)(~S\u0001~ e)(1\u0000cos(2xj\u0006j))\n(104)\nwith the direction ~ e=~\u0006=j\u0006jand (8).\nThe e\u000bect of a magnetic \feld is basically condensed at\ntwo places. First the phase term ~ q\u0001Rx\u0001~ p=~ q\u0001~ p+o(B)\nand we have \u0016 !=!\u0000~ p\u0001~ q=m +i=\u001c\n0Z\n\u00001e\u0000i(!x\u0000~ q\u0001Rx\u0001~ p\nm)0\n@cos 2j\u0006jx\n1\u0000cos 2j\u0006jx\n\u0000sin 2j\u0006jx1\nA\n=1\n20\n@i\n\u0016!+ 2\u0006+i\n\u0016!\u00002\u00062i\n\u0016!\u0000i\n\u0016!+ 2\u0006\u0000i\n\u0016!\u00002\u00061\n\u0016!\u00002\u0006\u00001\n\u0016!+ 2\u00061\nA+o(q2;B) =0\n@i\n\u0016!\n0\n2\u0006\n!21\nA+o(B;\u00062):\n(105)\nThe magnetic-\feld-dependent phase factor Rxdoes play\na role only in inhomogeneous systems with \fnite wave-\nlength. In the limit of large wavelength this e\u000bect can be\nignored.\nThesinandcosterms are results of the precession of\nspins around the e\u000bective direction ~ e=~\u0006=\u0006 and can be\nconsidered as Rabi oscillations. For the limit of small\n\u0006 we can expand the cosandsinterms in \frst order\n\u0019S0+~ \u001b\u0001~S\u00002~ \u001b\u0001(~S\u0002~\u0006)xas was analyzed in95. The\nsecond e\u000bect is the retardation in t=\u001e=!cwhich means\nthat the precession time in the arguments S(t\u0000x) con-\ntains important magnetic \feld e\u000bects. In fact, this retar-\ndation represents all kinds of normal Hall e\u000bects as we\nwill convince ourselves now.\nC. Retardation subtleties by magnetic \feld\nThe magnetic \feld causes a retarding integral in the\nlast section over the precession time t=\u001e=!ccoupled to\nany momentum by the representation in Bernstein coor-\ndinates\n~ p\u001e\nm= (wcos\u001e;wsin\u001e;u): (106)\nThis retardation is crucial for any kind of Hall e\u000bect. In\norder to get a handle on such expressions we concentrate\frst on the mean values of the scalar part \u000ef. The general\n\feld-dependent solution provides a form\nhAi=X\np\u001e0Z\n\u00001dxe\u0000i(!+x\u0000~ qRx~ p\u001e\nm)A(~ p\u001e)S0(p\u001e\u0000x!c)(107)\nwhereS0is the scalar source term. The trick is to per-\nform \frst a shift \u001e!\u001e+!cxand integrate then about\np=p\u001e. This has the e\u000bect that the retardation is only\ncondensed in the momentum of variable A\nP(x) =~ p\u001e+!cx\n=~ p\u001ecos(!cx)+~ ez(~ ez\u0001~ p\u001e)[1\u0000cos(!cx)]+~ ez\u0002~ p\u001esin(!cx)\n=~ p\u001e+~ ez\u0002~ p\u001e!cx+o(!2\nc): (108)\nand the exponent\nRx~ p\u001e+!cx\nm=~ p\u001e\nmsin!cx\n!c+\u0012\n~ ez\u0002~ p\u001e\nm\u00131\u0000cos!cx\n!c\n=~ p\u001e\nmx+!cx2\n2~ ez\u0002~ p\u001e\nm+o(!2\nc): (109)\nThe phase e\u000bect leads to the \frst order corrections in !c\nor alternatively in wave length q\nhAi=X\npS0(p)h\n1\u0000!c\n2m~ q\u0001(~ ez\u0002~ p)@2\n!+o(!2\nc)i\n\u00020Z\n\u00001dxe\u0000i(!+x\u0000~ q\u0001~ p\nm)A[P(x)]\n=X\npS0(p)A[P(i@!)]\n\u0002h\n1\u0000!c\n2m~ q\u0001(~ ez\u0002~ p)@2\n!+o(!2\nc)ii\n!+\u0000~ q\u0001~ p\nm:(110)\nwhere the integration variable xin the momentum (108)\ncan be transformed into derivatives of !if needed.\nCompletely analogously we can perform any mean\nvalue over the vector part of the distribution ~\u000eg. We\nhave\nh~Ai=X\npA(p)\u000e~ g=X\nph\n1\u0000!c\n2m~ q\u0001(~ ez\u0002~ p)@2\n!+o(!2\nc)i\n\u00020Z\n\u00001dxei(~ q~ p\nmx\u0000!+x+o(!c;q2))A[P(x)]\n\u0002h\n~Scos(2\u0006x)+~ e\u0002~Ssin(2\u0006x)+~ e(~ e\u0001~S)(1\u0000cos(2\u0006x))i\n(111)\nwhere the arguments of ~S, \u0006 and~ eare the momentum p\nand no retardation anymore. The exponent can be writ-\nten in complete B-dependence with Rxof course. Then\nthe x-integration over the cosandsinterms has to be\nperformed numerically. Analytically we can proceed if17\nwe expand the phase e\u000bect in orders of ~ q. We obtain\nwith the help of (105)\nh~Ai=X\npA[P(i@!)]h\n1\u0000!c\n2m~ q\u0001(~ ez\u0002~ p)@2\n!+o(!2\nc)i\n\u0002\u0014\n[~ e\u0002(~S\u0002~ e)]i\n2\u00121\n\u0016!+ 2\u0006+1\n\u0016!\u00002\u0006\u0013\n+~ e\u0002~S1\n2\u00121\n\u0016!+ 2\u0006\u00001\n\u0016!\u00002\u0006\u0013\n+~ e(~ e\u0001~S)i\n\u0016!\u0015\n(112)\nwith \u0016!=!+i\n\u001c\u0000~ p~ q\nmand (108).\nThe formula (110) and (112) establish the rules for cal-\nculating mean values with magnetic \felds. The useful-\nness of these rules can be demonstrated since it simpli\fes\nthe way to obtain the linearized solutions (40), (41) and\n(43) tremendously. In fact integrating with A= 1 we\nobtain straightforwardly the response functions and the\nequation system (46). This shows that up to linear or-\nder in wave-vector the magnetic \feld enters only via the\nZeeman term in ~\u0006.\nD. Classical Hall e\u000bect\nNow, we are in a position to see how the Hall e\u000bect is\nburied in the theory. Therefore we neglect any mean\feld\nand spin-orbit coupling for the moment such that the f\nand g distributions decouple and use the q!0 limit, i.e.\nhomogeneous situation. We obtain from (99) with (104)\nand (91)\n\u000ef=\u00000Z\n\u00001dxe\u0000i!+xe~E\u0001~@p\u001ef(p\u001e\u0000!cx) (113)\nwhere we now pay special care to the retardation since\nthis provides the Hall e\u000bect which was overseen in many\ntreatments of magnetized plasmas.\nAfter the shift of coordinates in azimuthal angle \u001e\nas outlined in the last section, we can carry out the x-\nintegration with the help of (108), (110) and (105):\n~J=eX\np\u001e~ p\u001e\nme\u000ef\n=\u0000e2X\np\u001e~E\u0001~@p\u001ef(p\u001e)0Z\n\u00001dxe\u0000i!+x~ p\u001e+!cx\nm\n=\u001b01\u0000i!\u001c\n(1\u0000i!\u001c)2+ (!c\u001c)2\u0014\n~E+(!c\u001c)2\n(1\u0000i!\u001c)2(~E\u0001~ ez)~ ez\n+!c\u001c\n1\u0000i!\u001c~E\u0002~ ez\u0015\n(114)\nwhich agrees of course with the elementary solution of\nme_~ v=e(~ v\u0002~B) +e~E\u0000me~ v\n\u001c: (115)\nIn order to obtain all three precession terms we have\nused the complete form (108) and no expansion in !c.E. Quantum Hall e\u000bect\nIf we consider low temperatures such that the mo-\ntion of electrons become quantized in Landau levels we\nhave to use the quantum kinetic equation and not the\nquasi-classical one. However, we can establish a sim-\nplere-quantization rule which allows us to translate the\nabove discussed quasi-classical results into the quantum\nexpressions. Therefore we recall the linearization of the\nquantum-Vlasov equation, which is the quantum kinetic\nequation with only the mea-\feld in operator form:\n_\u001a\u0000i\n~[\u001a;H] = 0: (116)\nThe perturbing Hamiltonian due to external electric\n\felds is\u000eH=e~E\u0001~^xsuch that the linearization in eigen-\nstatesEnof the unperturbed Hamiltonian reads\n\u000e\u001ann0=\u0000e~E\u0001~ xnn0\u001an\u0000\u001an0\n~!\u0000En+En0: (117)\nOne obtains the same result in the vector gauge since\n[\u001a;\u000eH ] =e~Et\nme[\u001a;~ p] =e~E[\u001a;~ v] and the same matrix el-\nements appear. Now, we investigate the quasi-classical\nlimit where the momentum states are proper representa-\ntions. We chose\nhnj=hp1j=hp+q\n2j;jni0=jp2i=j\u0000p+q\n2i(118)\nand we have in the quasi-classical q!0 approximation\n~ xnn0(\u001an\u0000\u001an0) =~\ni~@q\u000e(~ q)(\u001ap+q\n2\u0000\u001ap\u0000q\n2)\n\u0019~\ni~@q\u000e(~ q)~ q\u0001~@p\u001ap=\u0000~\ni\u000e(~ q)~@p\u001ap(119)\nfrom which follows\n\u000e\u001a\u0019\u0000i~e~E\u0001~@p\u001a\n~!\u0000~ p\u0001~ q\nme: (120)\nThis is precisely the quasi-classical result we obtain from\nquasi-classical kinetic equations. Turning the argument\naround we see that we can re-quantize our quasi-classical\nresults by applying the rule\n~E\u0001~@pf!~E\u0001~ vnn0fn\u0000fn0\nEn0\u0000En: (121)\nLet us apply it to the normal Hall conductivity. We\nuse the area density 1 =Aand re-normalize the level dis-\ntributionP\nnfn= 1 to obtain for the static conductivity\n!= 0\n\u001b\u000b\f=e2~i\nAX\nnn0fn(1\u0000fn0)1\u0000e\f(En\u0000En0)\n(En\u0000En0)2v\u000b\nnn0v\f\nn0n(122)\nwhich is nothing but the Kubo formula . Further\nevaluation for Landau levels has been performed by18\nVasilopoulos96,97. Therefore one chose the gauge ~A=\n(0;Bx; 0) and the corresponding energy levels are\nEn=\u0012\nn+1\n2\u0013\n~!c+p2\nz\n2me(123)\nwhere the last term is only in 3D. The wave functions\nread\njni=1p\nA\u001en(x+x0)eipyy=~eipzz=~(124)\nwith the harmonic oscillator functions \u001en,x0=l2py=~,\nl2=~=eB, andA=LyLzwhere the corresponding z\nparts are absent in 2D. The calculation in 3D can be\nfound in97. Here, we represent the 2D calculation. One\neasily obtains\nvnn0vn0n=i~!c\n2m[n\u000en0;n\u00001\u0000(n+1)\u000en0;n+ 1]\u000e(py\u0000p0\ny):(125)\nIntroducing this into (122) and using\nX\npy=Ly\n2\u0019~~Lx\n2l2Z\n\u0000~Lx\n2l2dpy=A\n2\u0019l2(126)\none arrives at\ne2\nhX\nn0(n0+ 1)fn0(1\u0000fn0+1)\u0000\n1\u0000e\u0000\f~!c\u0001\n!e2\nh(\u0016n+ 1)\n(127)\nwith \u0016n!c\u0014\u000ff\u0014(\u0016n+ 1)!c. This is von Klitzing's result\nforT!0.\nF. Polarization functions\nIntegrating (99) over the momentum p=vmeand solv-\ning algebraically for \u000enand\u000esone gets the response func-\ntions (46) with the B-\feld modi\fcations. This concerns\nthe precession-time integration instead of the energy de-\nnominator coupled to the tensor qRpand retardations\nin the momentum integration as described above. Espe-\ncially with the help of (112) the discussed polarizations\n(47)-(52) can be easily translated with (105) such that\nthe e\u000bect of the magnetic \feld in the phase can be con-\nsidered. The retardations does not play any role since for\nthe density and spin response we do not have moments\nof momentum that would be retarded. The numerical\nresults of these phase e\u000bects for small \u0006 have been dis-\ncussed in95leading to a staircase structure of the response\nfunctions with respect to the frequency at Landau levels.\nThe excitation shows a splitting of the collective mode\ninto Bernstein modes. Since it was presented in95the\nrepetition of results is avoided here.V. SUMMARY\nWe have solved the linearized coupled kinetic equa-\ntions for the density and spin Wigner distributions in\narbitrary magnetic \felds, vector and scalar mean\felds,\nspin-orbit coupling and relaxation time approximations\nobeying the conservation of density. The response func-\ntions for the density and spin polarization with respect\nto an external electric \feld are derived. Various forms of\npolarization functions appear re\recting the complicated\nnature of di\u000berent precessions and including Rabi shifts\ndue to the e\u000bective Zeeman \feld. The latter consists of\nthe magnetic \feld, the magnetization due to impurities,\nthe spin polarization, and the spin-orbit coupling.\nThe long wavelength expansions are presented and the\ndensity and spin collective modes are determined for neu-\ntral and charged scattering separately. For a neutral\nsystem, no optical charge mode appears but there are\nthree optical spin modes. These are dependent on the\nspin-orbit coupling and the e\u000bective Zeeman \feld. The\nenergy and damping of these modes are found to be lin-\nearly dependent on the spin-orbit coupling. A spin-wave\ninstability is reported and the range where such spin seg-\nregation can appear are calculated.\nThe charge and spin waves for charged Coulomb scat-\ntering show that only transverse spin modes can exist\nwith respect to an e\u000bective magnetization axis. The\ncharge density waves are damped plasma oscillations and\nthe spin waves are splitting into two modes dependent\non the polarization. One mode decreases in energy and\nbecomes damped with increasing polarization while the\nsecond mode increases and becomes sharper again with\nincreasing polarization. This analysis was possible with\nthe help of the polarization, magnetic \feld and spin-orbit\ndependent dielectric function which was presented here as\na new result. The range of instability with respect to fre-\nquency, polarization and collisional damping is presented\nwhich is again interpreted as spin segregation. The lat-\nter view is supported by the discussion of the statical and\ndynamical screening length whose dependence on the po-\nlarization and spin-orbit coupling is derived.\nFinally, the spin response shows an interesting damped\noscillation behavior di\u000berent in each direction originat-\ning from the o\u000b-diagonal responses. The magnetic \feld\ncauses an ellipsoidal relaxation which shows a rotation\nof the polarization axes depending on the spin-orbit cou-\npling. We \fnd a cross-over from damped oscillation to\nexponentially decay dependent on the polarization and\ncollision frequency. Spin segregation as a consequence is\ndiscussed and the dephasing times are extracted.\nThe response with an external magnetic \feld shows\nsome subtleties in retardations when observables of the\nWigner functions are calculated. In fact, the Hall e\u000bect is\npossible to obtain only when these retardations are taken\ninto account. The quantum version of the quasi-classical\nkinetic equations is shown to provide the quantum-Hall\ne\u000bect. Explicit calculations of the response function show\na staircase behavior with respect to the frequency at the19\nLandau levels. At these frequencies the Rabi satellite\nresponse functions become large leading to out-of-plane\nresonances95.\nAppendix A: Solving vector equations\nIn the following all symbols are vectors and we search\nfor solutions yin terms of capital symbols. We start with\nthe simplest vector equation\ny1=B\u0000Q(V\u0001y1) (A1)\nwhich is easily solved by iteration and the geometrical\nsum\ny1=B\u0000Q(V\u0001B)1\n1 +Q\u0001V=B+V\u0002(B\u0002Q)\n1 +Q\u0001V:(A2)\nNext we consider the equation of the type\ny2=B\u0000A\u0002y2 (A3)\nwhich by iterating once leads to\n(1 +A2)y2=B\u0000A\u0002B+A(A\u0001y2) (A4)\nwhich is again of the type (A1) with B!(B\u0000A\u0002\nB)=(1 +A2),V!\u0000A, andQ!A=(1 +A2) such that\nthe solution reads\ny2=B\u0000A\u0002B+A(A\u0001B)\n1 +A2: (A5)\nThe combined type reads\ny3+A\u0002y3=B\u0000Q(V\u0001y3) (A6)\nwhere in a \frst step we consider the right hand side as\naBof the problem (A3) and get the solution according\nto (A5). This leads to the problem (A1) with B!(B\u0000\nA\u0002B+A(A\u0001B))=(1 +A2) andQ!(Q\u0000A\u0002Q+\nA(A\u0001Q))=(1 +A2) such that the solution can be written\naccording to (A2)\ny3=B\u0000A\u0002B+A(A\u0001B) +V\u0002H\n1 +A2+Q\u0001V\u0000V(A\u0002Q) + (A\u0001V)(A\u0001Q)\nH= (B\u0000A\u0002B+A(A\u0001B))\u0002Q\u0000A\u0002Q+A(A\u0001Q)\n1 +A2\n=B\u0002Q+A\u0002(Q\u0002B) (A7)\nwhere the last equality is a matter of algebra. The \fnal\nsolution reads therefore\ny3=\nB\u0000A\u0002B+A(A\u0001B) +V\u0002[B\u0002Q+A\u0002(Q\u0002B)]\n1 +A2+Q\u0001V\u0000V\u0001(A\u0002Q) + (A\u0001V)(A\u0001Q)\n\u0011y3z\ny3n: (A8)As a next complication we consider the vector equation\nwhere the scalar products appear with respect to two\nvectors\ny4+A\u0002y4+Q(V\u0001y4) =B\u0000P(T\u0001y4) (A9)\nwhich is recast into the problem (A6) with B!B\u0000\nP(T\u0001y4) such that we obtain\ny4=y3\u0000(T\u0001y4)Q1\nQ1=\nP+P\u0002A+V\u0002(A\u0002(Q\u0002P)\u0000Q\u0002P) +A(A\u0001P)\ny3n\n(A10)\nwhich is the problem (A1) with V!T,B!y3and\nQ!Q1such that we obtain\ny4=y3\u0000T\u0002(Q1\u0002y3)\n1 +T\u0001Q1: (A11)\nThe cross product in the numerator can be shown by\nsomewhat lengthy calculation to be\nQ1\u0002y3=P\u0002B+A\u0002(B\u0002P) +V(B\u0001P\u0002Q)\ny3n\n(A12)\nsuch that the \fnal solution reads\ny4=\ny3z\u0000T\u0002[A\u0002(B\u0002P)\u0000B\u0002P+VP\u0001(B\u0002Q)]\ny3n+T\u0001y4h\ny4h=\nP+P\u0002A\u0000V\u0002[A\u0002(P\u0002Q) +P\u0002Q]+A(A\u0001P) (A13)\nAppendix B: Evaluation of operator forms\nIn this appendix we evaluate the operator form\n1Z\n0dxei\u0016!xe(~b+~ a)\u0001~ \u001bx(S0+~ \u001b\u0001~S)e(~b\u0000~ a)\u0001~ \u001bx\n(B1)\nwith \u0016!=!+i=\u001c. First we rewrite the exponential of\nPauli matrices to obtain\ne(~b\u0006~ a)\u0001~ \u001bx= cosc\u0006x+i~ e\u0006\u0001~ \u001bsinc\u0006x (B2)\nwhere we use\nc\u0006=jb\u0006aj;~ e\u0006=~b\u0006~ a\nj~b\u0006~ aj: (B3)\nTo evaluate the occurring products it is useful to deduce\nfrom (~ a\u0001~ \u001b)(~b\u0001~ \u001b) =~ a\u0001~b+i~ \u001b\u0001(~ a\u0002~b) the relations\n~ \u001b\u0001(~ a\u0001~ \u001b) =~ a+i(~ a\u0002~ \u001b);(~ a\u0001~ \u001b)\u0001~ \u001b=~ a\u0000i(~ a\u0002~ \u001b) (B4)20\nwith the help of which we \fnd\n(\u001b\u0001~ a)(~b\u0002~ \u001b) =\u0000((\u001b\u0001~ a)~ \u001b\u0002~b) =\u0000(~ a\u0000i(~ a\u0002~ \u001b))\u0002~b\n=\u0000~ a\u0002~b+i~b\u0002(~ \u001b\u0002~ a) =\u0000~ a\u0002~b+i~ \u001b(~ a\u0001~b)\u0000i~ a(~b\u0001~ \u001b):\n(B5)\nOne obtains from (B2)\ne(~b+~ a)\u0001~ \u001bxe(~b\u0000~ a)\u0001~ \u001bx= cosc+xcosc\u0000x\n+i(~ e+\u0001~ \u001b) sinc+xcosc\u0000x+i(~ e\u0000\u0001~ \u001b) sinc\u0000xcosc+x\n\u0000[~ e+\u0001~ e\u0000+i~ \u001b\u0001(~ e+\u0002~ e\u0000)] sinc+xsinc\u0000x (B6)\nand\ne(~b+~ a)\u0001~ \u001bx~ \u001be(~b\u0000~ a)\u0001~ \u001bx=~ \u001bcosc+xcosc\u0000x\n+i(~ e+\u0000~ \u001b\u0002~ e+) sinc+xcosc\u0000x\n+i(~ e\u0000+~ \u001b\u0002~ e+) sinc\u0000xcosc+x\n+[~ \u001b(~ e+\u0001~ e\u0000)\u0000(~ \u001b\u0001~ e+)~ e\u0000\u0000(~ \u001b\u0001~ e\u0000)~ e++i(~ e+\u0002~ e\u0000)]\n\u0002sinc+xsinc\u0000x: (B7)\nNow we evaluate the integrals over the cos and sin func-\ntions. Due to the positive imaginary part of !the integral\nvanishes at the upper in\fnite limit and one has\n1Z\n0dxei!xcos(cx) =i!\n!2\u0000c2\n1Z\n0dxei!xsin(cx) =\u0000c\n!2\u0000c2: (B8)\nWe will need\n(c+\u0006c\u0000)2= 2(a2+b2\u0006ja2\u0000b2j): (B9)\nwhich leads to either to 4 a2or 4b2dependent whether\na2?b2and the\u0006sign respectively. Therefore one ob-\ntains\n1Z\n0dxei!xcos(c+x) cos(c\u0000x) =i!\n2\u00121\n!2\u00004b2+1\n!2\u00004a2\u0013\n1Z\n0dxei!xsin(c\u0006x) sin(c\u0007x)\nja2\u0000b2j=\u00002i!\n(!2\u00004a2)(!2\u00004b2)\n1Z\n0dxei!x \nsin(c+x) cos(c\u0000x)\nj~ a+~bj+sin(c\u0000x) cos(c+x)\nj~ a\u0000~bj!\n=\u00002!2\n(!2\u00004a2)(!2\u00004b2)\n1Z\n0dxei!x \nsin(c+x) cos(c\u0000x)\nj~ a+~bj\u0000sin(c\u0000x) cos(c+x)\nj~ a\u0000~bj!\n=8~ a\u0001~b\n(!2\u00004a2)(!2\u00004b2): (B10)This allows to calculate the di\u000berent occurring inte-\ngrals in (B1) with (B6) and (B7) as\n1Z\n0dxei!xe(~b+~ a)\u0001~ \u001bxe(~b\u0000~ a)\u0001~ \u001bx\n=4!~ \u001b\u0001(~b\u0002~ a)+i!(!2\u00004a2)+8i(~ \u001b\u0001~ a)(~ a\u0001~b)\u00002i!2(~ \u001b\u0001~b)\n(!2\u00004a2)(!2\u00004b2)\n(B11)\nand\n1Z\n0dxei!xe(~b+~ a)\u0001~ \u001bx~ \u001be(~b\u0000~ a)\u0001~ \u001bx=\u001a\n4!(~ a\u0002~b)+8i~ a(~ a\u0001~b)\n+i![~ \u001b(!2\u00004b2)+4~b(~ \u001b\u0001~b)\u00004~ a(~ \u001b\u0001~ a)\u00002~b!]\n+8(~ a\u0001~b)(~b\u0002~ \u001b)+2!2(~ \u001b\u0002~ a)\u001b1\n(!2\u00004a2)(!2\u00004b2):\n(B12)\nAppendix C: Long wavelength expansion\nIn order to discuss dispersion relations and collective\nmodes we need the expansion of all polarization functions\nup to second order in wavelength appearing in terms of\n\u0005\u0012\n!+i\n\u001c\u0000~ p\u0001~ q\nme\u0013\n=\u0012\n1\u0000~ p\u0001~ q\nme@!+(~ p\u0001~ q)2\nm2e@2\n!\u0013\n\u0005(!+)\n(C1)\nwhere we use !+=!+i=\u001c. A further wave-length term\ncomes from the magnetic \feld dependence of the polar-\nization function discussed in chapter IV.C which leads to\na term linear in the magnetic \feld\n1\u0000!c\n2me~ q\u0001(~ ez\u0002~ p)@2\n!: (C2)\nAny function of \u0006 we can expand therefore as\n\u0005(\u0006)=\u0014\n1+\u0012b2\n?\n2+b3\u0013\n\u0006n@\u0006n+b2\n3\n2\u00062\nn@2\n\u0006n\u0015\n\u0005(\u0006n):(C3)\nSummarizing we have to apply (C1), (C2) and (C3)\nto all polarization function and calculate the momentum\nintegration. Here we give the \fnal results which may\nbe obtained after some lengthy calculation. Since b?(~ p)\nis uneven and b3(p) is even in momentum, from various\nmean values with the momentum-even distributions only\nthe following terms remain nonzero\n!c\n2meX\np\u0012\nf\ng\u0013\n~ q\u0001(~ ez\u0002~ p)~b?(p) =!c\nD~b?(q)\u0002~ ez\u0012\nEf\nEg\u0013\nX\np\u0012\nf\ng\u0013(~ p\u0001~ q)2\n2m2e=q2\nDme\u0012\nEf\nEg\u0013\n(C4)21\nwhereDdenotes the dimension and the mean (polariza-\ntion) kinetic energy is denoted as\n\u0012\nEf\nEg\u0013\n=X\np\u0012\nf\ng\u0013p2\n2me: (C5)\nHere and in the following we use ~ q~@pb(p)\u0019b(q) strictly\nvalid only for linear spin-orbit coupling and neglect\nhigher-order moments than o(p2b(p)). Besides (58) we\nwill use further shorthand notations\nBg3=X\npb3g; Bgii=X\npb2\n?ig;~B?=~b?(q) (C6)\nand analogously for g$f.\nThe equation for the density dipole-excitation is given\nby the \frst line of (46) and one needs the expansion of\nthe polarizations\n\u00050=nq2\nme!2\n++o(q3) (C7)\nand analogously\n~\u0005 =~ mq2\nme!2\n++o(q3): (C8)\nThis means for neutral scattering the combinations V0\u00050\nandV0~\u0005 vanish.\nAccording to (47) we need the expansion of (51)\n~\u0005g=\u0000i(\n~ ez\"\ns0\u0000B2\ng\n2(1\u0000\u0006n@\u0006n) +Bg3\u0006n@\u0006n\n+Bg33\n2\u00062\nn@2\n\u0006n+q2Eg\nDm@2\n!\u0015\n\u0000~B?\u0002~ ez!cEg\nD@2\n!\u001b2!\n!2\u00004\u00062n\n(C9)\nand also +o(q3)\n~\u0005xf=i(\n~ ez\"\nq2\nm(n\u0000B2\nf\n2(1\u0000\u0006n@\u0006n) +Bf3\u0006n@\u0006n\n+Bf33\n2\u00062\nn@2\n\u0006n)@!\u0000nB?3\u0006n@\u0006n\u0000B?3Bf3\u00062\nn@2\n\u0006n\u0015\n\u0000~B?(n\u0000Bf3(1\u0000\u0006n@\u0006n))\n\u0000~B?\u0002~ ezB?3!cEg\nD(1\u0000\u0006n@\u0006n@!\u001b2\u0006n\n4\u00062n\u0000!2(C10)\nsuch that we have the precession term ~\u00052=~\u0005xf+~\u0005g.\nFor \u0005 3we need besides (C8) according to (47)\n~\u0005xg=in\n~ ez\u0002~B?(s0+Bg3\u0006n@\u0006n)\n+h\n~B?B?3\u0000~ ezB2\n?i!cEg\nD@2\n!\u001b2\u0006\n4\u00062\u0000!2(C11)and\n~\u0005e=n\n~B?(s0\u0000Bg3(1\u0000\u0006n@\u0006n)\u0000Bg33\u0006n@\u0006n)\n+~b?(q)\u0002~ ez!cEg\nD@omega2\u001b4\u00062\nn\n!(4\u00062n\u0000!2)(C12)\nsuch that~\u00053=o(q):The in-plane terms are a little bit\nmore lengthy\n !\u0005xe= 28\n<\n:0\n@1 0 0\n0 1 0\n0 0 01\nA \ns0+q2Eg\nmD@2\n!+ (Bg3+B2\ng\n2)\u0006n@\u0006n\n+Bg33\n2\u00062\nn@2\n\u0006n\u0013\n+0\n@0 0 B?2\n0 0\u0000B?1\nB?2\u0000B?1 01\nA!cEg\nD@2\n!\n\u00000\n@Bg11 0 0\n0Bg22 0\n0 0\u0000B2\ng1\nA9\n=\n;2\u0006n\n4\u00062n\u0000!2\n+(C13)\nand\n !\u0005fe=8\n<\n:\u0000B?30\n@1 0 0\n0 1 0\n0 0 01\nA\u0000\nn\u0006n@\u0006n+Bf3\u00062\nn@2\n\u0006n\u0001\n+0\n@2B?1B?2B2\n?2\u0000B2\n?1B?2\nB2\n?2\u0000B2\n?1\u00002B?1B?2\u0000B?1\nB?2\u0000B?1 01\nA!cEf\nD@2\n!\n+2\n40\n@1 0 0\n0 1 0\n0 0 01\nA\" \nBf3+B2\nf\n2!\n\u0006n@\u0006n+Bf33\n2\u00062\nn@2\n\u0006n#\n+0\n@n\u0000Bf11 0 0\n0n\u0000Bf220\n0 0 B2\nf1\nA3\n5q2\nm@!9\n=\n;4\u00062\nn\n!(4\u00062n\u0000!2\n+)\n(C14)\nThe terms coming from the Mermin relaxation time\nbecome\n\u00050\u0016=\u0000@\u0016n\n!+\u0000nq2\nme!3\n++o(q3;b3;\u001bnb(q)2)\n~\u0005\u0016=\u0000@\u0016~ s\n!+\u0000sq2\nme!3\n+~ ez+o(q3;b3;\u001bnb(q)2) (C15)\nwhere we use (57). The terms ~\u0005\u0016eand~\u0005x\u0016vanish at\nthis level of expansion.\n1. Long wave length expansion in quasi 2D systems\nIn the cases discussed in this paper we are not inter-\nested in 3D spin-orbit coupling such that we can neglect\nthe termsb3. Summarizing the results of the last chapter\nwe obtain for the coupled dispersion (46) the terms\n\u00050=nq2\nme!2\n+;~\u0005 =~ sq2\nme!2\n+; s =s0\u0000B2\ng\n2;(C16)22\n~\u00052=i~ ez(\"\n\u0000s0+B2\ng\n2(1\u0000\u0006n@\u0006n)\u0000q2Eg\nDme@2\n!#\n2!\n!2\u00004\u00062n\n\u0000~B?\u0002~ ez!cEg\nD@2\n!+q2\nm(n\u0000B2\nf\n2(1\u0000\u0006n@\u0006n))2\u0006n\n4\u00062n\u0000!2)\n+i~B\u0002~ ez!cEg\nD@2\n!2!\n!2\u00004\u00062n\u0000~B?n2\u0006n\n4\u00062n\u0000!2;(C17)\n~\u00053=~ sq2\nme!2\n+\u0000i\u0014\n~ ez\u0002~B?s0\u0000~ ezB2\n?!cEg\nD@2\n!\u00152\u0006\n4\u00062\u0000!2\n+~B?s04\u00062\nn\n!(4\u00062n\u0000!2); (C18)\nand\n !\u0005 =8\n<\n:0\n@1 0 0\n0 1 0\n0 0 01\nA \ns0+q2Eg\nmD@2\n!+B2\ng\n2\u0006n@\u0006n!\n+0\nB@\u0000Bg11 0B?2!cEg\nD@2\n!\n0\u0000Bg22\u0000B?1!cEg\nD@2\n!\nB?2!cEg\nD@2\n!\u0000B?1!cEg\nD@2\n!\u0000B2\ng1\nCA9\n>=\n>;\n\u00024\u0006n\n4\u00062n\u0000!2\n+\n+8\n<\n:0\n@2B?1B?2B2\n?2\u0000B2\n?1B?2\nB2\n?2\u0000B2\n?1\u00002B?1B?2\u0000B?1\nB?2\u0000B?1 01\nA!cEf\nD@2\n!\n+0\nB@n+B2\nf\n2\u0006n@\u0006n\u0000Bf11 0 0\n0 n+B2\nf\n2\u0006n@\u0006n\u0000Bf220\n0 0 B2\nf1\nCA\n\u0002q2\nme@!\u001b4\u00062\nn\n!(4\u00062n\u0000!2\n+)(C19)together with (C15).\n1N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys.\nRev. B 72, 174406 (2005).\n2I. J. Vera-Marun, B. J. van Wees, and R. Jansen, Phys.\nRev. Lett. 112, 056602 (2014).\n3S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev.\nMod. Phys. 83, 407 (2011).\n4H. Tomita and J. Nakamura, J. Vac. Sci. Technol. B 31,\n04D105 (2013).\n5E. P. Bashkin, C. da Provid^ encia, and J. da Provid^ encia,\nPhys. Rev. C 50, 2800 (1994).\n6D. C. Langreth and J. W. Wilkins, Phys. Rev. B 6, 3189\n(1972).\n7J. Sweer, D. C. Langreth, and J. W. Wilkins, Phys. Rev.\nB13, 192 (1976).\n8A. J. Leggett and M. J. Rice, Phys. Rev. Lett. 20, 586\n(1968), erratum Phys. Rev. Lett. 21 506 (1968).\n9E. P. Bashkin, JETP Lett. 33, 8 (1981).\n10C. Lhuilier and F. Lalo e, J. Phys. (Paris) 43, 197 (1982).\n11W. J. Gully and W. J. Mullin, Phys. Rev. Lett. 52, 1810(1984).\n12E. P. Bashkin, Sov. Phys. Usp. 29, 238 (1986).\n13A. E. Ruckenstein and L. P. L\u0013 evy, Phys. Rev. B 39, 183\n(1989).\n14B. R. Johnson, J. S. Denker, N. Bigelow, L. P. L\u0013 evy, J. H.\nFreed, and D. M. Lee, Phys. Rev. Lett. 52, 1508 (1984).\n15N. P. Bigelow, J. H. Freed, and D. M. Lee, Phys. Rev. Lett.\n63, 1609 (1989).\n16W. J. Mullin and R. J. Ragan, J. Low Temp. Phys. 138,\n73 (2005).\n17H. J. Lewandowski, D. M. Harber, D. L. Whitaker, and\nE. A. Cornell, Phys. Rev. Lett. 88, 070403 (2002).\n18J. M. McGuirk, H. J. Lewandowski, D. M. Harber,\nT. Nikuni, J. E. Williams, and E. A. Cornell, Phys. Rev.\nLett. 89, 090402 (2002).\n19J. E. Williams, T. Nikuni, and C. W. Clark, Phys. Rev.\nLett. 88, 230405 (2002).\n20W. J. Mullin and R. J. Ragan, Phys. Rev. A 74, 043607\n(2006).23\n21T. Nikuni, J. E. Williams, and C. W. Clark, Phys. Rev. A\n66, 043411 (2002).\n22S. Perisanu and G. Vermeulen, Phys. Rev. B 73, 214519\n(2006).\n23P. J. Nacher, G. Tastevin, S. B. Crampton, and F. Lalo e,\nJ. Physique Lett. 45, L (1984).\n24E. P. Bashkin, Phys. Rev. B 44, 12440 (1991).\n25A. A. Burkov and L. Balents, Phys. Rev. B 69, 245312\n(2004).\n26V. M. Edelstein, Phys. Rev. B 74, 193310 (2006).\n27V. Sih, W. H. Lau, R. C. Myers, A. C. Gossard, M. E.\nFlatt\u0013 e, and D. D. Awschalom, Phys. Rev. B 70, 161313\n(2004).\n28R. A. Cover, G. Kalman, and P. Bakshi, Phys. Rev. D 20,\n3015 (1979).\n29N. J. M. Horing and M. M. Yildiz, Phys. Rev. B 33, 3895\n(1986).\n30V. L\u0013 opez-Richard, G. E. Marques, and C. Trallero-Giner,\nJ. Appl. Phys. 89, 6400 (2001).\n31O. Bleibaum, Phys. Rev. B 71, 235318 (2005).\n32K. Shizuya, Phys. Rev. B 75, 245417 (2007).\n33K.-J. Lee, M. Stiles, H.-W. Lee, J.-H. Moon, K.-W. Kim,\nand S.-W. Lee, Phys. Rep. 531, 89 (2013).\n34A. Aperis, M. Georgiou, R. Roumpos, S. Tsonis, G. Varel-\nogiannis, and P. B. Littlewood, Europhys. Lett. 83, 67008\n(2008).\n35V. E. Demidov, H. Ulrichs, S. V. Gurevich, S. O. Demokri-\ntov, V. S. Tiberkevich, A. N. Slavin, A. Zholud, and\nS. Urazhdin, Nature comm. 5, 3179 (2014).\n36T. Kampfrath, M. Battiato, P. Maldonado, G. Eil-\ners, J. N otzold, S. M ahrlein, V. Zbarsky, F. Freimuth,\nY. Mokrousov, S. Bl ugel, et al., Nature Nanotechnology\n8, 256 (2013).\n37R. Winkler, Spin-Orbit Coupling E\u000bects in Two-\nDimensional Electron and Hole Systems (Springer - Verlag,\nBerlin Heidelberg, 2003).\n38M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Rep. 493,\n61 (2010).\n39C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak,\nY. Tserkovnyak, H. Kurebayashi, A. Brataas, and A. Fer-\nguson, Nature Nanotechnology 10, 50 (2015).\n40D. Fang, H. Kurebayashi, J. Wunderlich, K. V\u0012 yborn\u0012 y,\nL. P. Z^ arbo, R. P. Campion, A. Casiraghi, B. L. Gallagher,\nT. Jungwirth, and A. J. Ferguson, Nature Nanotechnology\n6, 413 (2015).\n41R. H. Liu, W. L. Lim, and S. Urazhdin, Phys. Rev. Lett.\n110, 147601 (2013).\n42V. M. Edelstein, Solid State Comm. 73, 233 (1990).\n43J. Borge, C. Gorini, G. Vignale, and R. Raimondi, Phys.\nRev. B 89, 245443 (2014).\n44S. Wimmer, M. Seemann, K. Chadova, D. K odderitzsch,\nand H. Ebert, Phys. Rev. B 92, 041101 (2015).\n45Y. V. Kobljanskyj, G. A. Melkov, A. A. Serga, A. N. Slavin,\nand B. Hillebrands (2015), arXiv:1503.04638.\n46D. H. Berman, M. Khodas, and M. E. Flatt\u0013 e, Phys. Rev.\nX4, 011048 (2014).\n47L. B. Hu, J. Gao, and S. Q. Shen, Phys. Rev. B 68, 153303\n(2003).\n48C. Grimaldi and P. Fulde, Phys. Rev. B 55, 15523 (1997).\n49K. Morawetz, Europhys. Lett. 67, 77 (2004).\n50T. Ando, J. Phys. Soc. Japan 75, 074716 (2006).\n51F. Guinea, Phys. Rev. B 75, 235433 (2007).\n52G. Gazzola, A. L. Cherchiglia, L. A. Cabral, M. C. Nemes,\nand M. Sampaio, Europ. Phys. Lett. 104, 27002 (2013).53V. Juri\u0014 ci\u0013 c, O. Vafek, and I. F. Herbut, Phys. Rev. B 82,\n235402 (2010).\n54I. F. Herbut, V. Juri\u0014 ci\u0013 c, and O. Vafek, Phys. Rev. Lett.\n100, 046403 (2008).\n55E. G. Mishchenko, Europ. Phys. Lett. 83, 17005 (2008).\n56W. J. H. Leyland, G. H. John, R. T. Harley, M. M. Glazov,\nE. L. Ivchenko, D. A. Ritchie, I. Farrer, A. J. Shields, and\nM. Henini, Phys. Rev. B 75, 165309 (2007).\n57W. J. H. Leyland, R. T. Harley, M. Henini, A. J. Shields,\nI. Farrer, and D. A. Ritchie, Phys. Rev. B 76, 195305\n(2007).\n58W. J. H. Leyland, R. T. Harley, M. Henini, A. J. Shields,\nI. Farrer, and D. A. Ritchie, Phys. Rev. B 77, 205321\n(2008).\n59M. Pletyukhov and V. Gritsev, Phys. Rev. B 74, 045307\n(2006).\n60S. M. Badalyan, A. Matos-Abiague, G. Vignale, and\nJ. Fabian, Phys. Rev. B 79, 205305 (2009).\n61A. Scholz, T. Stauber, and J. Schliemann, Phys. Rev. B\n86, 195424 (2012).\n62X.-F. Wang and T. Chakraborty, Phys. Rev. B 75, 033408\n(2007).\n63G. Borghi, M. Polini, R. Asgari, and A. H. MacDonald,\nPhys. Rev. B 80, 241402 (2009).\n64A. Principi, M. Polini, and G. Vignale, Phys. Rev. B 80,\n075418 (2009).\n65O. V. Gamayun, Phys. Rev. B 84, 085112 (2011).\n66S. Yuan, R. Rold\u0013 an, and M. I. Katsnelson, Phys. Rev. B\n84, 035439 (2011).\n67R. E. V. Profumo, R. Asgari, M. Polini, and A. H. Mac-\nDonald, Phys. Rev. B 85, 085443 (2012).\n68C. Triola and E. Rossi, Phys. Rev. B 86, 161408 (2012).\n69M. H. Schultz, A. P. Jauho, and T. G. Pedersen, Phys.\nRev. B 84, 045428 (2011).\n70M. Busl, G. Platero, and A.-P. Jauho, Phys. Rev. B 85,\n155449 (2012).\n71C. A. Ullrich and M. E. Flatt\u0013 e, Phys. Rev. B 68, 235310\n(2003).\n72K. Morawetz, Phys. Rev. B 67, 115125 (2003).\n73R. Hakim, L. Mornas, P. Peter, and H. D. Sivak, Phys.\nRev. D 46, 4603 (1992).\n74N. Mermin, Phys. Rev. B 1, 2362 (1970).\n75A. K. Das, J. Phys. F 5, 2035 (1975).\n76K. Morawetz, Phys. Rev. B 66, 075125 (2002), errata:\nPhys. Rev. B 88, 039905(E).\n77K. Morawetz, Phys. Rev. E 88, 022148 (2013).\n78G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B\n64, 214204 (2001).\n79V. P. Mineev, Phys. Rev. B 69, 144429 (2004).\n80A. J. Leggett, J. Phys. C 3, 448 (1970).\n81V. P. Mineev, Phys. Rev. B 72, 144418 (2005).\n82R. Bala, R. Moudgil, S. Srivastava, and K. Pathak, Euro-\npean Phys. J. B 87, 5 (2014).\n83S. DasSarma, Solid State Commun. 36, 357 (1980).\n84W. Nicolazzi and S. Pillet, Phys. Rev. B 85, 094101 (2012).\n85A. L. Chernyshev, A. H. Castro Neto, and A. R. Bishop,\nPhys. Rev. Lett. 84, 4922 (2000).\n86P. Schwab, M. Dzierzawa, C. Gorini, and R. Raimondi,\nPhys. Rev. B 74, 155316 (2006).\n87M. A. Brand, A. Malinowski, O. Z. Karimov, P. A. Mars-\nden, R. T. Harley, A. J. Shields, D. Sanvitto, D. A. Ritchie,\nand M. Y. Simmons, Phys. Rev. Lett. 89, 236601 (2002).\n88X. Du, L. Luo, B. Clancy, and J. E. Thomas, Phys. Rev.\nLett. 101, 150401 (2008).24\n89U. Ebling, A. Eckardt, and M. Lewenstein, Phys. Rev. A\n84, 063607 (2011).\n90A. Malinowski, R. S. Britton, T. Grevatt, R. T. Harley,\nD. A. Ritchie, and M. Y. Simmons, Phys. Rev. B 62, 13034\n(2000).\n91P. H. Song and K. W. Kim, Phys. Rev. B 66, 035207\n(2002).\n92M. Q. Weng and M. W. Wu, Phys. Rev. B 68, 075312\n(2003).\n93I. B. Bernstein, Phys. Rev. 109, 10 (1958).\n94M. Walter, G. Zwicknagel, and C. Toep\u000ber, Eur. Phys. J.D35, 527 (2005).\n95Morawetz, K., EPL 104, 27005 (2013).\n96P. Vasilopoulos, Phys. Rev. B 32, 771 (1985).\n97C. M. V. Vliet and P. Vasilopoulos, Journal of Physics and\nChemistry of Solids 49, 639 (1988).\n98There is a puzzling di\u000berence in the denominator. We have\ninstead of the square of scalar product ( ~ c\u0001~ z)2nowc2z2\nwhich di\u000berence is just the squared cross product. Since\nwe restrict later to linear orders in q@p, i.e. linear orders in\n~ cwe can neglect this di\u000berence." }, { "title": "1312.6458v3.Long_Range_Spin_Triplet_Helix_in_Proximity_Induced_Superconductivity_in_Spin_Orbit_Coupled_Systems.pdf", "content": "Long-Range Spin-Triplet Helix in Proximity Induced Superconductivity in\nSpin-Orbit-Coupled Systems\nXin Liu, J. K. Jain, and Chao-Xing Liu\nDepartment of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802-6300\n(Dated: April 6, 2022)\nWe study proximity induced triplet superconductivity in a spin-orbit-coupled system, and show\nthat the dvector of the induced triplet superconductivity undergoes precession that can be controlled\nby varying the relative strengths of Rashba and Dresselhaus spin-orbit couplings. In particular, a\nlong-range spin-triplet helix is predicted when these two spin-orbit couplings have equal strengths.\nWe also study the Josephson junction geometry and show that a transition between 0 and \u0019junctions\ncan be induced by controlling the spin-orbit coupling with a gate voltage. An experimental setup\nis proposed to verify these e\u000bects. Conversely, the observation of these e\u000bects can serve as a direct\ncon\frmation of triplet superconductivity.\nPACS numbers: 74.45.+c, 75.70.Tj, 85.25.Cp\nIntroduction - Crucial to the success of spintronics [1]\nare injection of spin, its long decay length and its manip-\nulation. The study of spin transport in a superconductor\nhas given rise to the sub\feld known as superconducting\nspintronics [2{4]. One may wonder if the spin-1 of Cooper\npairs in a triplet superconductor can play a similar role\nas the electron spin in spintronics. The observation of\nsurprisingly long-range proximity e\u000bect in a supercon-\nductor (SC)/ferromagnet (FM) junction [5{11] has been\ninterpreted in terms of an injection into the FM of triplet\nCooper pairs with a long decay length [12{17]. However,\nit is unclear how to manipulate the long-range part of\nthe induced triplet pair.\nWe propose here a geometry in which the triplet pairs\nare injected into a material with spin-orbit coupling\n(SOC) and show, theoretically, that they can be manip-\nulated by varying the relative strengths of the Rashba\nand Dresselhaus SOCs. In particular, we predict a long-\nrange spin-triplet helix, which can be veri\fed by observ-\ning a 0\u0000\u0019transition in Josephson junctions as a function\nof the SOC strengths. We show that the e\u000bect is robust\nagainst any spin independent scattering. Proximity e\u000bect\nin SOC materials has been considered previously,[18, 19]\nbut with only Rashba SOC, which does not produce long-\nrange e\u000bects discussed below.\nBefore presenting the detailed microscopic theory, we\n\frst illustrate the underlying physics, shown in Fig 1.\nIn the absence of magnetization and SOC, four kinds\nof Cooper pairs (singlet j\"#i\u0000j#\"i and triplet pairs\nj\"#i +j#\"i,j\"\"i\u0006j##i ) are allowed with a zero center-\nof-mass momentum. The magnetization breaks the de-\ngeneracy between jk;\"iandj\u0000k;#i. It will lead to a spa-\ntially modulated oscillation e\u0000iqxj\"#i\u0006eiqxj#\"i [20, 21]\nfor the Cooper pairs with opposite spins but leave the\npairsj\"\"i\u0006j##i unchanged, as shown in Fig 1(b). (Here\nwe assume that the system is uniform along y and z direc-\ntions so that the center-of-mass momentum of pairs is al-\nways zero along these directions.) On the contrary, SOC\nbreaks the degeneracy between jk;\"(#)iandj\u0000k;\"(#)i,as shown in Fig. 1(c,d). Thus, the Cooper pairs with par-\nallel spins will oscillate spatially as e\u0000iqxj\"\"i +eiqxj##i,\nwhile the pairsj\"#i\u0006j\"#i remain unchanged. Here we\nemphasize that the spin quantization axis aligns along\ndi\u000berent directions for di\u000berent momenta, determined by\nthe form of SOC in Fig. 1(c). The spatially oscillatory\npairs will decay after taking into account all possible\nwave vectors of q[15] in the case of Fig. 1(b). Simi-\nlarly, the triplet pairs j\"\"i andj##i in Fig. 1(c) will also\ngenerally decay rapidly in the SOC region. Therefore,\nin the presence of magnetization and generic SOC, only\nthe pairs with zero center-of-mass momenta exhibit long-\nrange proximity e\u000bect. However, there is an exception for\na system with equal strengths of Rashba and Dresselhaus\nSOCs. In this case, the Fermi surfaces for two spin bands\nshifted in opposite directions by Q= 4m\u000b, shown in Fig.\n1(d). Here mbeing the electron e\u000bective mass and \u000b\nbeing the Rashba SOC strength. Thus, all of spatially\noscillatory pairs have the same wave vector Qand will\nnot decay even in the presence of spin independent scat-\ntering. We show below that these oscillatory triplet pairs\nresult in a long-range helical mode, dubbed \"long-range\nspin-triplet helix\", in analogy to the persistent spin helix\nobserved in two dimensional electron gases (2DEGs)[22{\n26].\nHamiltonian and pairing functions - We study\nthe SC/normal-conductor structure whose Hamiltonian\ntakes the form\n^H=\u0012H0^\u0001\n^\u0001y\u0000H\u0003\n0\u0013\n;^\u0001 = \u0001(x)i\u001by;\nH0=\u0012p2\n2m\u0000\u0016\u0013\n\u001b0+ (M(x) +h(x;k))\u0001\u001b;\nin the basis [ c\";c#;cy\n\";cy\n#]T, wherec\";#andcy\n\";#are elec-\ntron annihilation and creation operators for di\u000berent\nspins,mis the electron mass, \u0016is the chemical poten-\ntial, ^\u0001 is the spin-singlet s-wave superconducting gap, M\nis the magnetization, his the e\u000bective magnetic \feld of\nSOC and\u001bdenotes the spin operators. The gap strengtharXiv:1312.6458v3 [cond-mat.mes-hall] 29 Nov 20142\n(a)(b)\n(c) (d)\nFIG. 1. Energy dispersion and Fermi surfaces are shown\nfor (a) normal metals, (b) ferromagnets, (c) a 2DEG with\nRashba SOC and (d) a 2DEG with equal strengths of Rashba\nand Dresselhaus SOCs. The possible forms of spin states of\nCooper pairs, including singlet and triplet pairs, are also il-\nlustrated in the \fgures. For ky6= 0;kx= 0, the gap between\ntwo spin bands in (c) is j2\u000bkyj.\n\u0001(x) is zero in the proximity region and has a constant\nvalue \u0001 in the superconducting region. The magnetiza-\ntionM(x) and e\u000bective magnetic \feld of SOC h(x;k) are\nonly present in the normal-conductor and depend on the\nspatial coordinate xshown in Fig. 2 and Fig. 3(a,b).\nCooper pairs in spin space can be described microscop-\nically by a pairing function fR(E;r) = (d0\u001b0+d\u0001\u001b)i\u001by\n[12, 27], which is the o\u000b diagonal block of the retarded\nGreen's function\nGR(E;r;r0)\f\f\nr=r0= \ngR(E;r)fR(E;r)\nfR(E;r)gR(E;r)!\n:(1)\nHered0anddare the expectation value of singlet and\ntriplet pairs respectively, Eis the energy, randr'are the\nspatial coordinates; we have fR\nij(E;r) =\u0000(fR\nij(\u0000E;r))y;\nandgR(gR) is the electron (hole) Green's function. Both\nfRandgRare 2\u00022 matrices in spin space. The su-\nperconducting gap is related to the pairing function by\nthe equality ^\u0001 = (1=2\u0019)R\ndE\u0015fEImfRwhere\u0015is the\nattractive interaction strength and fEis the Fermi dis-\ntribution. In the proximity region, the superconducting\ngap is zero because of \u0015= 0, but the pairing function fR\ncan be nonzero. Below, we will calculate, in the presence\nof either magnetization or SOC, the spatial evolution of\nFIG. 2. A schematic plot of a SC/FM/SOC junction. Energy\ndispersions for di\u000berent regions are shown above the junc-\ntion structure. The colors in the dispersion relation represent\ndi\u000berent spin indices and the solid lines (dashed lines) de-\nnote electron (hole) bands. k1(2);fandk3(4);fare the Fermi\nmomenta of di\u000berent spin bands for SOC and FM regions,\nrespectively. Di\u000berent propagation or re\rection processes are\ndenoted by Tin(out)\nfm(soc)orRad.\nthe pairing function fR(E;r) in the proximity region and\nshow its consistence to the physical picture in Fig. 1.\ndvector in a one-dimensional (1D) SC/FM/SOC\njunction - In the ferromagnetic region ( x2(\u0000a;0)) the\nSOC is zero, while in the SOC region ( x >0) the mag-\nnetization is zero shown in Fig. 2. In the SOC (FM)\nregion, the Fermi wave vectors of the spin split bands,\nk1f;k2f(k3f;k4f) in Fig 2, satisfy\nk2f\u0000k1f=2jh(kf)j\n~vf; k4f\u0000k3f=2M\n~vf; (2)\nwith~vf=~kf=m=p\n2\u0016=m, assuming h;M\u001c\u0016.\nThe Green's functions GRcan be related to the re-\n\rection matrix Rby the Fisher-Lee relation[28] which\nhas been applied to the superconducting proximity e\u000bect\n[29, 30]. For 1D case, Fisher-Lee relation in the basis\n[c\";c#;cy\n\";cy\n#]Ttakes the form [29{31]\nRij(E;r) =\u0000\u000eij+i~pvivjGR\nij(E;r); (3)\nwherei;j= 1;:::; 4 andvi(j)is the velocity of the par-\nticle at energy Eini(j) channels. Therefore, we will\ncalculate the re\rection matrix to extract pairing func-\ntions in 1D case. For simplicity, we consider the clean\nlimit with perfect transmission at FM/SOC boundary\nand ideal Andreev re\rection at the FM/SC boundary.\nThe re\rection matrix R(x) in the SOC region can be de-\ncomposed into \fve matrices representing \fve steps shown\nin Fig. 2: an electron \frst propagates from x=rto the\ninterface at x= 0 (Tin\nsoc); it then propagates to the inter-\nface atx=\u0000a(Tin\nfm); ideal Andreev re\rection occurs at\nthe SC/FM interface of x=\u0000a(Rad), where the elec-\ntron is completely re\rected as a hole; the re\rected hole\ntransmits back to x= 0 (Trf\nfm), and \fnally to the SOC\nregion atx=r(Trf\nsoc) [32]. Consequently, the scattering\nmatrixR(r) takes the form\nR(r) =Trf\nsocTrf\nfmRadTin\nfmTin\nsoc: (4)3\nWhen there is no SOC (i.e., r= 0), the re\rection ma-\ntrix at FM/SOC boundary takes the form [32]\nR(r= 0) =Trf\nfmRadTin\nfm=\u0000~vf(d0\u001b0+d\u0001\u001b)i\u001by\n\u001cy;(5)\nwhere\n(d0;d) =\u0000ie\u0000i\u000b\n~vf\u0012\ncos\u00122Ma\n~vf\u0013\n;isin\u00122Ma\n~vf\u0013\nm\u0013\n;\n(6)\nm=M=M,\u000b= arccos(E=\u0001) and\u001cz= +1(\u00001) for the\nelectron (hole) in the Nambu space. In the limit M\u001c\u0016,\nwe takepvivj\u0019vf. Eqs. (S9) and (S10) show oscillation\nbetween singlet and triplet pairs as a function of a, the\ndistance from the SC/FM interface. Thus, by choosing\nan appropriate length aof the FM region, one can use\nthe SC/FM junction to inject singlet or triplet pairs into\nthe SOC region.\nWhen there is no FM ( a= 0), the re\rection matrix\nreduces to R(r) =Trf\nsocRadTin\nsoc=Rad[32] in the SOC\nregion. This is because SOC does not lift the degeneracy\nof time reversed pairs, as shown in Fig 1 (c) and (d). For\nan FM of length asatisfying 2Ma=~vf=\u0019=2, only triplet\npairs with dvector along Mare injected into the SOC\nregion. When the e\u000bective magnetic \feld of SOC is par-\nallel to the magnetization, say h(k)kM, the re\rection\nmatrix in the SOC region can be written as\nR(r) =\u0000e\u0000i\u000bm\u0001\u001bi\u001by\n\u001cy: (7)\nWhen h(k)?M, the re\rection matrix in the SOC regime\nhas the form\nR(r) =\u0000~vf(d1m\u0001\u001b+d2m\u0002n\u0001\u001b)i\u001by\n\u001cy;(8)\nwhere\n(d1;d2) =e\u0000i\u000b\n~vf(cos(k2f\u0000k1f)r;sin(k2f\u0000k1f)r);(9)\nandnis the unit direction of h(kf). Hered1andd2give\nthe decomposition of the d-vector along the direction m\nandm\u0002n, respectively. Eq. (7) implies that dvector\nkeeps its original direction in the case of dkh(k). In\ncontrast, Eq. (8) shows that in the case of d?h(k),d\nvector precesses in the plane perpendicular to h(k) when\npropagating along 1D SOC region. The above conclu-\nsions are consistent with our physical picture shown in\nFig 1(c,d). Especially, based on Eq (9), the precession\nofdvector leads to a helical structure, which is dubbed\ndhelix or spin-triplet helix and schematically shown by\nred arrows in the SOC region of Fig 3 (b).\n0 and\u0019Josephson junction transition - To con\frm the\npredicted dhelix, we propose an experimental setup of\na SC/FM/SOC/FM/SC junction (Fig. 3(a,b)) and show\nthat the dhelix can lead to a 0 \u0000\u0019transition in Josephson\njunctions [15]. The magnetizations of two ferromagnetic\nlayers point along xand\u0000xdirection (Fig. 3(a,b)), to\nensure a trivial 0-Josephson junction in the absence of\n0 2−101\n0−junction\nπ−junction\n(a) (b)\n(c) (d)\n(f)\n0 1 2−101\n0.1123\n0 1 2−101\n0.1123FIG. 3. The magnetization direction (the green arrows) and\nthe e\u000bective magnetic \feld direction of SOC (the purple ar-\nrow) are shown (a) for a 0-junction and (b) a \u0019-junction. The\nred arrows reveals the spatial distribution of d-vector. The\nphases of SCs at two sides are taken to be \u001e=2 and\u0000\u001e=2.\nThe color in (c) and (d) shows the spectral function of the\nSC/FM/SOC/FM/SC junction (logarithmic plot) as a junc-\ntion of the relative phase \u001efor a 0-junction and \u0019-junction,\nrespectively. The black lines are the Andreev levels from an-\nalytical calculations. (f) shows the current-phase relation for\nthe 0- and\u0019-junction.\nthe SOC region. The lengths of FMs are chosen to satisfy\n2Ma=~vf=\u0019=2, so only triplet pairs with dvector along\nx direction are injected into the SOC region.\nWe consider two cases with the SOC h(k) =\u000bkx^exk\nMin Fig. 3(a) and h(k) =\u000bkx^ey?Min Fig. 3(b). The\nlength of the SOC wire satisfy ( k2f\u0000k1f)L=\u0019. To study\nthe current-phase relation in this setup, we \frst calculate\nthe Andreev levels numerically by evaluating the spectral\nfunction, Tr[P\nngR(E;xn)]=N, in a tight-binding model.\nHeregRis the electron retarded Green's function de\fned\nin Eq. (1), xnrepresents the nth site and Nis the total\nnumber of sites in the proximity region. The spectral\nfunctions are plot as a function of the relative phase \u001e\nbetween two SCs in Fig. 3(c,d). The peaks shown by\nthe red color indicate Andreev levels. We also obtain\nAndreev levels analytically using the standard scattering\nmatrix method [32{34]. The analytical results are shown\nby two black lines in Fig 3(c,d), which are consistent\nwith the numerical results. It is noted that the crossings\nof the black curves at \u001e=\u0019, in Fig. 3(c) and at \u001e= 0;\n2\u0019in Fig. 3(d) turn into anti-crossings in numerical re-\nsults. This is because we impose a barrier potential at\nthe SC/FM interfaces and include the Fermi velocity mis-\nmatch among di\u000berent regions in numerical calculations,\nwhich remove all degeneracies in analytical results. The\nanti-crossing changes the period of the Josephson current\nat zero temperature, Is=2e\n~P\nn@En=@\u001ewith the sum-4\nmation of negative Andreev levels, from 4 \u0019(black curves)\nto 2\u0019[33, 35]. The Josephson current for the Fig. 3(c)\ngives the form of Is\u0018sin(\u001e) (the blue line in Fig. 3(f)),\nwhich corresponds to a 0-junction. In contrast, for the\nFig. 3(d) we have Is\u0018sin(\u001e+\u0019) (the red line in Fig.\n3(f)), indicating a \u0019-junction. This 0-pi junction transi-\ntion is consistent with the physical picture of the d-vector\nprecession, shown by red arrows in Fig. 3(a,b). Further\ncalculations show that the \u0019junction is obtained for L\nsatisfying\u0019=2<(k1f\u0000k2f)L<3\u0019=2 [36].\ndhelix in a 2D system - Having clari\fed the physics\nin a 1D model, we next ask if dhelix also exists in a 2D\nsystem. For a 2DEG, the SOC has the form (assuming\nx-axis along [110] direction)\nHso= (\u000b+\f)kx\u001by+ (\f\u0000\u000b)ky\u001bx;\nwhere\u000band\fare the Rashba and Dresselhaus SOC\nstrengths. When \u000b=\f, the Fermi surface with the\nspin parallel (anti-parallel) to the y axis is shifted along\n\u0000x(x) direction by Q=2, as shown in Fig 1(d). As\na result, the eigenenergies of two spin states satisfy\n\u000f1(k) =\u000f2(k+Q), where Q= 4m\f^ex, and 1 (2) de-\nnotes the spin parallel (anti-parallel) to the y axis. As\nshown in Refs. [22{26], one can construct spin helix op-\nerators, which commute with the Hamiltonian and lead\nto a persistent spin helix mode.\nIn our model with superconductivity, we can de\fne\ntriplet pairing operators\n^dx=1\n20\n@X\nfk;ig\u000e(\u000fk;i\u0000\u0016)cy\nk;icy\n\u0000k\u0000(\u00001)iQ;i+ h:c:1\nA;(10)\n^dz=1\n2i0\n@X\nfk;ig\u000e(\u000fk;i\u0000\u0016)cy\nk;icy\n\u0000k\u0000(\u00001)iQ;i\u0000h:c:1\nA(11)\nwhere the summation is performed in the interval\nfk;ig=fkx<(\u00001)iQ=2;kygat the Fermi surface to\navoid double counting. These two operators represent a\ndhelix of triplet pairs with center-of-mass Qin x-z plane.\nSince the operators ^dx;zcommute with the Hamiltonian\nH0+Hso[32], a persistent dhelix also exists in the triplet\nsuperconducting proximity region. It is also noted that in\nthe case of\u000b=\f, the Hamiltonian even with a spin inde-\npendent scattering potential, H=H0+Hso+V(r)\u001b0, can\nbe transformed to a Hamiltonian without SOC through\nthe unitary matrix U= exp(\u0000iQx= 2)\u001by. This is be-\ncauseUis independent of momenta and commutes with\nV(r)\u001b0. At the same time, the triplet pairs with center-\nof-mass momentum Qas de\fned in Eq. (10, 11) is trans-\nformed to those with zero center-of-mass momentum as\nshown in Fig 1(a). Therefore, we expect that this spin-\ntriplet helix is immune to any spin-independent scatter-\ning and its decay length should be as long as the Cooper\npairs coherence length [33] in the normal region. This can\nbe further con\frmed by solving Usadel equations [37, 38]\nwith SOCs [32].\n(a) (b)\n(c)\nFIG. 4. The spatial dependence of dvector of triplet pairs\n(the red arrows) and the corresponding e\u000bective magnetic\n\feld (the purple arrows) of the SOC are shown for (a) \u000b=\f\n(\u0019-junction) and (b) \u000b=\u0000\f(0-junction). TSC means triplet\nsuperconductor. (c) The proposed 2D SC/FM/SOC/FM/SC\nstructure for an electronic-tunable Josephson junction.\nIn experiments, the Dresselhaus parameter \fis \fxed\nwhile Rashba parameter \u000bcan be tuned by a gate volt-\nage. Therefore, the following geometry can be used to\ncon\frm the oscillatory triplet pairs by observing an elec-\ntrically tunable 0- \u0019transition. The length Lof the SOC\nregion is chosen to satisfy the condition QL=\u0019. From\nthe above discussion, when \u000b=\f, thedvector of triplet\npairs changes its sign after propagating from x=x2\ntox=x3(Fig. S2(a)), leading to a \u0019-junction. If we\ntune the Rashba parameter to \u000b=\u0000\f, the e\u000bective\nmagnetic \feld of SOC h= 2\fky^exis along the x di-\nrection, parallel to dvector. Based on our theory, d\nvector keeps its direction in the SOC region (Fig. S2(b))\nand we will have a 0-junction. The proximity e\u000bect in\nthe 2D Josephson junction for these two cases should be\nlong-range according to our arguments. For realistic ex-\nperiments, InAs quantum wells provide a potential can-\ndidate (Fig S2(c)), because they show strong proximity\ne\u000bect due to their low Schottky barrier [39]. If the two\nFM layer are Ni, 1 nm thickness [9] is enough to convert\nsinglet pairs in SC to triplet pairs on FM/InAs inter-\nface. For the e\u000bective mass me\u000b= 0:04m eand typical\n\u000b= 0:2eV\u0017Ain InAs quantum wells, we \fnd Q\u001940\u0016m\u00001,\nwhich corresponds to the length of \u001880nmof the SOC\nregion to realize the Josephson 0 \u0000\u0019junction transi-\ntion. This length is much smaller than the coherence\nlength,\u0018N=~2p\n2\u0019n=m e\u000b2\u0019kBTc\u00194\u0016m [33], where\nn= 1012cm\u00002is the typical electron density in the InAs\nquantum well and Tc= 1:2Kis the critical temperature\nof Al.\nWe acknowledge Yinghai Wu, Jimmy A. Hutasoit and\nShou-Cheng Zhang for very helpful discussion. X.L. ac-\nknowledges partial support by the DOE under Grant No.\nDE-SC0005042.5\n[1] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[2] K. Ohnishi, T. Kimura, and Y. Otani, Applied Physics\nLetters 96, 192509 (2010).\n[3] C. Quay, D. Chevallier, C. Bena, and M. Aprili, Nature\nPhysics 9, 84 (2013).\n[4] T. Wakamura, N. Hasegawa, K. Ohnishi, Y. Niimi, and\nY. Otani, Phys. Rev. Lett. 112, 036602 (2014).\n[5] R. Keizer, S. Goennenwein, T. Klapwijk, G. Miao,\nG. Xiao, and A. Gupta, NATURE 439, 825 (2006).\n[6] J. Wang, M. Singh, M. Tian, N. Kumar, B. Liu, C. Shi,\nJ. K. Jain, N. Samarth, T. E. Mallouk, and M. H. W.\nChan, NATURE PHYSICS 6, 389 (2010).\n[7] J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire,\nScience (2010), 10.1126/science.1189246.\n[8] T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O.\nBirge, Phys. Rev. Lett. 104, 137002 (2010).\n[9] C. Klose, T. S. Khaire, Y. Wang, W. P. Pratt, N. O. Birge,\nB. J. McMorran, T. P. Ginley, J. A. Borchers, B. J. Kirby,\nB. B. Maranville, and J. Unguris, Phys. Rev. Lett. 108,\n127002 (2012).\n[10] J. W. A. Robinson, F. Chiodi, M. Egilmez, G. B. Ha-\nlasz, and M. G. Blamire, SCIENTIFIC REPORTS 2, 699\n(2012), 10.1038/srep00699.\n[11] P. V. Leksin, N. N. Garif'yanov, I. A. Garifullin, Y. V.\nFominov, J. Schumann, Y. Krupskaya, V. Kataev, O. G.\nSchmidt, and B. B uchner, Phys. Rev. Lett. 109, 057005\n(2012).\n[12] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys.\nRev. Lett. 86, 4096 (2001).\n[13] M. Eschrig and T. Loefwander, NATURE PHYSICS 4,\n138 (2008).\n[14] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev.\nMod. Phys. 77, 1321 (2005).\n[15] A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n[16] S. Takei and V. Galitski, Phys. Rev. B 86, 054521 (2012).\n[17] F. S. Bergeret and I. V. Tokatly, Phys. Rev. Lett. 110,\n117003 (2013).\n[18] Z. Yang, J. Wang, and K. Chan, Superconductor Science\nand Technology 22, 055012 (2009).\n[19] Z. Yang, J. Wang, and K. Chan, Journal of Physics:\nCondensed Matter 22, 045302 (2010).\n[20] E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys.\nRev. B 55, 15174 (1997).\n[21] M. Eschrig, Physics Today 64, No. 1, 43 (2011).\n[22] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys.\nRev. Lett. 97, 236601 (2006).\n[23] T. D. Stanescu and V. Galitski, Phys. Rev. B 75, 125307\n(2007).\n[24] X. Liu and J. Sinova, Phys. Rev. B 86, 174301 (2012).\n[25] C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang,\nJ. Stephens, and D. D. Awschalom, Phys. Rev. Lett. 98,\n076604 (2007).\n[26] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig,\nS.-C. Zhang, S. Mack, and D. D. Awschalom, Nature 458,\n610 (2009).\n[27] T. Champel, T. L ofwander, and M. Eschrig, Phys. Rev.\nLett. 100, 077003 (2008).\n[28] D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).\n[29] C. J. Lambert, V. C. Hui, and S. J. Robinson, Journal\nof Physics: Condensed Matter 5, 4187 (1993).[30] J. Wang, Y. Wei, B. Wang, and H. Guo, Applied Physics\nLetters 79, 3977 (2001).\n[31] S. Datta, Electronic Transport in Mesoscopic Systems\n(Cambridge Studies in Semiconductor Physics and Mi-\ncroelectronic Engineering) (Cambridge University Press,\n1997).\n[32] See Supplemental Material for the derivation of transmis-\nsion matrices Tin\nfm,Tin\nsoc,Trf\nfm,Trf\nsocand Andreev re\rection\nmatrixRadin 1D SC/FM/SOC junction, the property of\nthe triplet pairing operators in 2D and the spatial of the\nevolution of the spin-triplet pairs in the SOC region.\n[33] T. Sch apers, Superconductor/semiconductor junctions ,\nVol. 174 (Springer, 2001).\n[34] C. Beenakker, in Transport Phenomena in Mesoscopic\nSystems , edited by H. Fukuyama and T. Ando, Vol. 109\n(Springer, 1992).\n[35] H. Tang, Z. Wang, and Y. Zhang, Zeitschrift fur Physik\nB Condensed Matter 101, 359 (1997).\n[36] Xin Liu, J. K. Jain and C.-X. Liu, To be published.\n[37] K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n[38] J. Rammer, Quantum Field Theory of Non-equilibrium\nStates (Cambridge University Press, 2007).\n[39] Y.-J. Doh, J. a. van Dam, A. L. Roest, E. P. a. M.\nBakkers, L. P. Kouwenhoven, and S. De Franceschi, Sci-\nence309, 272 (2005).6\nSUPPLEMENTARY MATERIAL\nIn the Supplementary Material, we provide details for the calculation of the propagation matrix and the Andreev\nlevels for various geometries mentioned in the main text in the 1D clean limit. We also present the details for the\nspatial evolution of triplet pairs in the 2D system with general spin-orbit couplings (SOCs). Section I will derive\nthe previously known results for superconductor/ferromagnet (SC/FM) junction [S1, S2] from the scattering matrix\nmethod, and Section II will consider SC/FM/SOC geometry. Section III will show how to calculate the Andreev\nlevels in SC/FM/SOC/FM/SC junctions based on the scattering matrix method. The de\fnition of the triplet pairing\noperators given in Eqs. (10, 11) of the main text and their properties are given in Section IV. Section V describes the\nspatial evolution of the triplet pairs based on the Usadel equation.\n1D SC/FM junction\nWe \frst consider how magnetization mixes di\u000berent pairing functions in a one-dimensional (1D) ferromagnetic\nregion of a SC/FM junction, schematically shown in Fig S1.a. The e\u000bective Hamiltonian for this junction is given by\nHSC=FM=0\n@\u0010^P2\n2m\u0000\u0016\u0011\n\u001b0 0\n0\u0000\u0010^P2\n2m\u0000\u0016\u0011\n\u001b01\nA+ \u0002(x)\u0012M\u0001\u001b 0\n0\u0000M\u0001\u001b\u0003\u0013\n+ \u0002(\u0000x)\u00120 \u0001i\u001by\n\u0000\u0001i\u001by0\u0013\n;(S1)\nwhere \u0002 is the Heaviside step function, Mdenotes magnetization of FM, and \u0001 is the superconducting gap in the SC\nregion. The SC/FM interface re\rects incoming electrons (holes) into outgoing holes (electrons) and thereby induces a\nnon-zero pairing function in the ferromagnetic region. To explore the spatial evolution of pairing function in a clean\nferromagnetic wire, we formulate the re\rection process by a matrix Rfm(a), given by\nRfm(a) =Trf\nfmRadTin\nfm (S2)\nwhich is decomposed into three steps shown in Fig. S1.a. An incoming electron (hole) is transmitted from x=ato\nthe SC/FM interface at x= 0 (Tin\nfm); then an ideal Andreev re\rection occurs at SC/FM interface where the incoming\nelectron (hole) is completely re\rected to the outgoing hole (electron) ( Rad); the re\rected outgoing hole (electron)\npropagates back to x=a(Trf\nfm). We now calculate each factor separately.\nIn the \frst step, the wave functions of two incoming electrons and holes with opposite spins take the form\n\tin\n+e=\u0012 +\n^0\u0013\ne\u0000ik4fx;\tin\n+h=\u0012^0\n \u0003\n+\u0013\ne+ik4fx;\n\tin\n\u0000e=\u0012 \u0000\n^0\u0013\ne\u0000ik3fx;\tin\n\u0000h=\u0012^0\n \u0003\n\u0000\u0013\ne+ik3fx; (S3)\nwhere ^0 = (0;0)T,k3fandk4f(in Fig. S1(a)) are the Fermi wave vectors of the minority and majority spin bands,\nrespectively, and M\u0001\u001b \u0006=\u0006jMj \u0006. In the clean limit, the transmission matrix Tin\nfmdescribes the propagation of\nan incoming electron or hole from x=a>0 to the SC/FM interface and is given by\nTin\nfm=j\tin\n+e(0)ih\tin\n+e(a)j+j\tin\n+h(0)ih\tin\n+h(a)j+j\tin\n\u0000e(0)ih\tin\n\u0000e(a)j+j\tin\n\u0000h(0)ih\tin\n\u0000h(a)j\n=1\n2\u0012eik4fa(\u001b0+m\u0001\u001b) 0\n0e\u0000ik4fa(\u001b0+m\u0001\u001b\u0003)\u0013\n+1\n2\u0012eik3fa(\u001b0\u0000m\u0001\u001b) 0\n0e\u0000ik3fa(\u001b0\u0000m\u0001\u001b\u0003)\u0013\n;\n=\u0012ei\fUi 0\n0e\u0000i\fU\u0003\ni\u0013\n: (S4)\nHere\f= (k4f+k3f)a=2 and\nUfm= exp(i(k4f\u0000k3f)a\n2m\u0001\u001b) = cos((k4f\u0000k3f)\n2a)\u001b0+isin((k4f\u0000k3f)\n2a)m\u0001\u001b;\nUy\nfm= exp(\u0000i(k4f\u0000k3f)a\n2m\u0001\u001b) = cos((k4f\u0000k3f)\n2a)\u001b0\u0000isin((k4f\u0000k3f)\n2a)m\u0001\u001b;\nU\u0003\nfm= exp(\u0000i(k4f\u0000k3f)a\n2m\u0001\u001b\u0003) = cos((k4f\u0000k3f)\n2a)\u001b0\u0000isin((k4f\u0000k3f)\n2a)m\u0001\u001b\u0003;\nUT\nfm= exp(i(k4f\u0000k3f)a\n2m\u0001\u001b\u0003) = cos((k4f\u0000k3f)\n2a)\u001b0+isin((k4f\u0000k3f)\n2a)m\u0001\u001b\u0003:\n(S5)7\n(a)\n(b)\nFIG. S1. Panels (a) and (b) show SC/FM and SC/FM/SOC junctions, respectively. The dispersions relations in various regions\nare shown; the colors represent the di\u000berent spin indices and the solid lines (dashed lines) denote the electron (hole). Di\u000berent\npropagation processes de\fned as Tin\nfm,Rad,Trf\nfm,Tin\nsocandTrf\nsocare shown on the \fgure. The quantities k1f,k2f,k3fandk4fare\nthe Fermi momenta for di\u000berent spin bands.\n.\nIn the second step, the ideal Andreev re\rection matrix has the form [S3]\nRad=e\u0000i\u000b\u00120i\u001by\n\u0000i\u001by0\u0013\n; (S6)\nwhere\u000b= arccos(E=\u0001) with the energy EsatisfyingjEj<\u0001. In the third step, there are four outgoing particles in\nthe ferromagnetic region, with wave functions given by\n\trf\n+e=\u0012 +\n^0\u0013\neik4fx;\trf\n+h=\u0012^0\n \u0003\n+\u0013\ne\u0000ik4fx;\n\trf\n\u0000e=\u0012 \u0000\n^0\u0013\neik3fx;\trf\n\u0000h=\u0012^0\n \u0003\n\u0000\u0013\ne\u0000ik3fx: (S7)\nThe transmission matrix Trf\nfmdescribes the outgoing waves moving back from the SC/FM interface at x= 0 tox=a,\ngiven by\nTrf\nfm=j\trf\n+e(a)ih\trf\n+e(0)j+j\trf\n+h(a)ih\trf\n+h(0)j+j\trf\n\u0000e(a)ih\trf\n\u0000e(0)j+j\trf\n\u0000h(a)ih\trf\n\u0000h(0)j\n=1\n2\u0012eik4fa(\u001b0+m\u0001\u001b) 0\n0e\u0000ik4fa(\u001b0+m\u0001\u001b\u0003)\u0013\n+1\n2\u0012eik3fa(\u001b0\u0000m\u0001\u001b) 0\n0e\u0000ik3fa(\u001b0\u0000m\u0001\u001b\u0003)\u0013\n=\u0012ei\fUi 0\n0e\u0000i\fU\u0003\ni\u0013\n; (S8)8\nIt is noted that Tin\nfmhas the same form to Trf\nfm, which is consistent to the fact that the magnetization respects the\ninversion symmetry. The total re\rection matrix at x=ain the ferromagnetic region is then given by\nRfm(a) =Trf\nfmRadTin\nfm\n=\u0012\nei\fUi 0\n0e\u0000i\fU\u0003\ni\u0013\ne\u0000i\u000b\u0012\n0i\u001by\n\u0000i\u001by0\u0013\u0012\nei\fUi 0\n0e\u0000i\fU\u0003\ni\u0013\n=e\u0000i\u000b\u00120Uii\u001byU\u0003\ni\n\u0000U\u0003\nii\u001byUi 0\u0013\n=\u0012\n0 cos(( k4f\u0000k3f)a)i\u001by+isin((k4f\u0000k3f)a)m\u0001\u001bi\u001by\ncos((k4f\u0000k3f)a)(\u0000i\u001by) +isin((k4f\u0000k3f)a)m\u0001\u001b\u0003i\u001by 0\u0013\n=e\u0000i\u000b\u0014\ncos(k4f\u0000k3f)a\u00120i\u001by\n\u0000i\u001by0\u0013\n+isin(k4f\u0000k3f)a\u00120m\u0001\u001bi\u001by\n(m\u0001\u001bi\u001by)y0\u0013\u0015\n(S9)\nThe pairing function can now be obtained by the Fisher-Lee relation [S4] shown in the main text, and is given by\nfR(E;x) = (d0\u001b0+d\u0001\u001b)i\u001by;\nd0=\u0000ie\u0000i\u000b\n~vfcos((k4f\u0000k3f)a);d=e\u0000i\u000b\n~vfsin((k4f\u0000k3f)a)m: (S10)\nThe above equations display the spatial oscillation between singlet and triplet pairs in the FM region. We note that\nthed-vector is along the direction of magnetization M.\n1D SC/FM/SOC junction\nWe next consider a 1D SC/FM/SOC junction. The calculation is conceptually similar to that given above, although\nthe details are more complicated. The re\rection matrix is now given by\nRsoc(L) =Trf\nsocRfm(a)Tin\nsoc (S11)\nwhereRfm(a) has already been calculated above. The SC/FM junction is utilized as a source of singlet and triplet\npairs, and the relative strengths of the two can be tuned by varying a, the length of the FM region.\nThe Hamiltonian in the SOC wire has the form\nHSOC=0\n@\u0010\n^p2\n2m\u0000\u0016\u0011\n\u001b0 0\n0\u0000\u0010\n^p2\n2m\u0000\u0016\u0011\n\u001b01\nA+\u0012h(^p)\u0001\u001b 0\n0\u0000h\u0003(^p)\u0001\u001b\u0003\u0013\n(S12)\nwhere ^pis the momentum operator, mis the electron mass, h(^p) is the e\u000bective magnetic \feld due to SOC and \u0016\nis the chemical potential. In the SOC region, the incoming electrons and holes propagate to the FM/SOC interface\nwith the wave functions\n\tin\n+e=\u0012 +\n^0\u0013\ne\u0000ik1fx;\tin\n+h=\u0012^0\n \u0003\n+\u0013\neik2fx;\n\tin\n\u0000e=\u0012 \u0000\n^0\u0013\ne\u0000ik2fx;\tin\n\u0000h=\u0012^0\n \u0003\n\u0000\u0013\neik1fx; (S13)\nwherek2fandk1f(in Fig S1(b)) are the Fermi wave vectors of the majority and minority spin bands respectively and\nn\u0001\u001b \u0006=\u0006 \u0006withn=h=jhj. The wave functions of outgoing electrons and holes are given by\n\trf\n+e=\u0012 +\n^0\u0013\neik2fx;\trf\n+h=\u0012^0\n \u0003\n+\u0013\ne\u0000ik1fx;\n\trf\n\u0000e=\u0012 \u0000\n^0\u0013\neik1fx;\trf\n\u0000h=\u0012^0\n \u0003\n\u0000\u0013\ne\u0000ik2fx; (S14)\nTo obtain the pairing function in the SOC wire, we consider a perfect contact at FM/SOC interface. The transmission\nfromx=Lto the SC/FM interface at x= 0 is represented by the matrix\nTin\nsoc=1\n2\u0012eik2fL(\u001b0+n\u0001\u001b) 0\n0e\u0000ik2fL(\u001b0+n\u0001\u001b\u0003)\u0013\n+1\n2\u0012eik1fL(\u001b0\u0000n\u0001\u001b) 0\n0e\u0000ik1fL(\u001b0\u0000n\u0001\u001b\u0003)\u0013\n=\u0012ei\fUi 0\n0e\u0000i\fU\u0003\ni\u0013\n(S15)9\nwhere\f= (k2f+k1f)L=2 and\nUsoc= exp(i(k2f\u0000k1f)L\n2n\u0001\u001b) = cos((k2f\u0000k1f)\n2L)\u001b0+isin((k2f\u0000k1f)\n2L)n\u0001\u001b;\nUy\nsoc= exp(\u0000i(k2f\u0000k1f)L\n2n\u0001\u001b) = cos((k2f\u0000k1f)\n2L)\u001b0\u0000isin((k2f\u0000k1f)\n2L)n\u0001\u001b;\nU\u0003\nsoc= exp(\u0000i(k2f\u0000k1f)L\n2n\u0001\u001b\u0003) = cos((k2f\u0000k1f)\n2L)\u001b0\u0000isin((k2f\u0000k1f)\n2L)n\u0001\u001b\u0003;\nUT\nsoc= exp(i(k2f\u0000k1f)L\n2n\u0001\u001b\u0003) = cos((k2f\u0000k1f)\n2L)\u001b0+isin((k2f\u0000k1f)\n2L)n\u0001\u001b\u0003:\n(S16)\nThe re\rected hole (electron) moves back from the interface to x=L, which is represented by the matrix\nTrf\nsoc=1\n2\u0012eik1fL(\u001b0+n\u0001\u001b) 0\n0e\u0000ik1fL(\u001b0+n\u0001\u001b\u0003)\u0013\n+1\n2\u0012eik2fL(\u001b0\u0000n\u0001\u001b) 0\n0e\u0000ik2fL(\u001b0\u0000n\u0001\u001b\u0003)\u0013\n=\u0012\nei\fUy\ni 0\n0e\u0000i\fUT\ni\u0013\n: (S17)\nIt is noted that Trf\nsoc, describing the propagation away from the F/SOC interface, is di\u000berent from Tin\nsoc, describing the\npropagation towards the F/SOC interface. This indicates the fact that SOC breaks inversion symmetry.\nWe now have all the information needed to evaluate Rsoc(L), and hence the pairing function. We specialize below\nto the case ( k4f\u0000k3f)a=\u0019=2, for which, according to Eq. S9, the SC/FM junction behaves as a reservoir of only\ntriplet pairs whose d-vector is along the magnetization direction. When the magnetization Min the ferromagnetic\nregion is parallel to hin the SOC region, the re\rection matrix at x=Lin the SOC region is the same as that in the\nferromagnetic region\nRsoc(L) =Trf\nsocRfm(a)Tin\nsoc=Rfm(a); (S18)\nwhich implies that the SOC will not a\u000bect the triplet pair whose d-vector is parallel to the e\u000bective magnetic \feld of\nSOC. When Mis perpendicular to h, the refection matrix shows an oscillating behavior\nRsoc(L) =ie\u0000i\u000b\u0014\ncos(k2f\u0000k1f)L\u00120m\u0001\u001bi\u001by\n(m\u0001\u001bi\u001by)y0\u0013\n+ sin(k2f\u0000k1f)L\u00120m\u0002n\u0001\u001bi\u001by\n(m\u0002n\u0001\u001bi\u001by)y0\u0013\u0015\nwhich is identical to rotate the triplet pair in the plane perpendicular to h.\nScattering matrix method in SC/FM/SOC/FM/SC junction\nWe now show the scattering matrix method in the SC/FM/SOC/FM/SC junction. The magnetizations of two\nferromagnetic layers point along xand\u0000xdirection (Fig.3(a,b) in the main text), to ensure a trivial 0-Josephson\njunction in the absence of the SOC region. The lengths of FMs are chosen to satisfy 2 Ma=~vf=\u0019=2, so only triplet\npairs withd-vector along x direction are injected into the SOC region based on Eq. (S10). From Eq. (S9,S10), the\nassociated re\rection matrix at the interface of x2(x3) takes the form\nR\u0000(+)\nfm=ie\u0000i\u000b\u0012\n0e\u0000(+)i\u001e=2\u001bxi\u001by\n(e+(\u0000)i\u001e\u001bxi\u001by)y0\u0013\n: (S19)\nwhere\u0000\u001e=2 (\u001e=2) is the phase of the left (right) superconductor. The discrete Andreev levels in the Josephson\njunction can be obtained from the condition [S3, S5]\nDet\u0000\nI4\u00024\u0000Tin\nsocR\u0000\nfmTrf\nsocR+\nfm\u0001\n= 0; (S20)\nwhereI4\u00024is a 4 by 4 identity matrix. When the e\u000bective magnetic \feld of SOC is along y direction, substituting\nEq. (S15,S17) into Eq. (S20) and taking k2f\u0000k1f=\u0019, we have\nDet(I4\u00024\u0000Tin\nsocR\u0000\nfmTrf\nsocR+\nfm) = Det\u0012(1 +e\u00002i\u000b\u0000i\u001e)\u001b0 0\n0 (1 + e\u00002i\u000b+i\u001e)\u001b0\u0013\n= 0;10\nwhich gives the two-fold degenerate Andreev levels E=\u0006\u0001 cos(\u001e+\u0019\n2). When the e\u000bective magnetic \feld of SOC is\nalong x or -x direction, we have\nDet(I4\u00024\u0000Tin\nsocR\u0000\nfmTrf\nsocR+\nfm) = Det\u0012\n(1\u0000e\u00002i\u000b\u0000i\u001e)\u001b0 0\n0 (1\u0000e\u00002i\u000b+i\u001e)\u001b0\u0013\n= 0;\nwhich gives E=\u0006\u0001 cos(\u001e\n2).\nPersistent triplet helix\nIn the two dimensional case, SOC in general induces a destructive interference of di\u000berent transverse modes shown\nin Fig 1(c) in the main text. This will lead to a rapid decay of triplet pairing function. However, for some particular\nforms of SOC, triplet pairs can precess in a coherent way, resulting in a long range proximity e\u000bect. Below, we will\nshow how to achieve a long range proximity e\u000bect of triplet pairing functions in a 2DEG system. 2DEGs usually\npossess two kinds of SOCs, namely the Rashba and Dresselhaus terms, given by\nHR&D =HRashba +HDresselhaus =\u000b(kx\u001by\u0000ky\u001bx) +\f(kx\u001bx\u0000ky\u001by); (S21)\nwhere\u000band\fare the coe\u000ecients of Rashba and Dresselhaus SOCs, respectively. In the case of \u000b=\f, the SOC\nHamiltonian takes the form\nHR&D =\u000b(kx\u0000ky)(\u001bx+\u001by): (S22)\nThe form of the Hamiltonian is simpli\fed if we re-de\fne ( kx\u0000ky)=p\n2!kxand (\u001bx+\u001by)=p\n2!\u001bztoHR&D = 2\u000bkx\u001bz.\nThe two spin bands with opposite spins are shifted in the opposite directions. The Hamiltonian in the spin and Nambu\nspace has the form\nH=1\n2X\nk;i\u0012\ncy\nk;i\nc\u0000k;i\u0013T\u0012k2=2m\u0000(\u00001)iQkx\u0000\u0016 0\n0\u0000(k2=2m\u0000(\u00001)iQkx\u0000\u0016)\u0013\u0012ck;i\ncy\n\u0000k;i\u0013\n(S23)\nwherei= 1(2) is for the spin parallel (anti-parallel) to zdirection, we de\fne Q= 4m\u000b, andmis the e\u000bective mass\nof 2DEGs. We construct the triplet helix operators as\n^d\u0000=X\nfk;ig\u000e(\u000fk;i\u0000\u0016)c\u0000k\u0000(\u00001)iQ;ick;i; (S24)\n^d+=X\nfk;ig\u000e(\u000fk;i\u0000\u0016)cy\nk;icy\n\u0000k\u0000(\u00001)iQ;i; (S25)\nwhere\n\u000fk;i=k2\u0000(\u00001)iQkx\n2m(S26)\nand the summation is performed over the Fermi surface, where we choose the interval fk;ig=fkx<(\u00001)iQ=2;kyg\nto avoid double counting. Due to the dispersion relation in Eq. S26, ^d\u0006de\fned in Eq. (S24,S25) commute with the\nHamiltonian in Eq. S23\n[H;^d\u0000] =X\nfk;ig\u000e(\u000fk;i\u0000\u0016)\u0000\n\u000f\u0000k\u0000(\u00001)iQ;i\u0000\u000fk;i\u0001\nc\u0000k\u0000(\u00001)iQ;ick= 0; (S27)\n[H;^d+] =X\nfk;ig\u000e(\u000fk;i\u0000\u0016)\u0000\n\u000f\u0000k\u0000(\u00001)iQ;i\u0000\u000fk;i\u0001\ncy\nkcy\n\u0000k\u0000(\u00001)iQ;i= 0: (S28)\nwhich is the reason why SOC with \u000b=\fdoes not cause a decay of the triplet dhelix for the center-of-mass momentum\nQ.\nTo further con\frm the long range triplet order in the dirty limit, we derive the Usadel equation in the proximity\nregion with isotropic spin-independent scattering time \r. First, we derive the dynamic equation of the annihilation\noperator\n \u0016(t;x) =ei^Ht^ \u0016(x)e\u0000i^Ht=ei^Ht X\nk^ \u0016(k)eik\u0001x!\ne\u0000i^Ht(S29)11\nwhere ^H=P\n\u0016;\u0017;k y\n\u0016(k)H\u0016\u0017(k) \u0017(k) and\nH(k) = \nk2\n2m0\n0k2\n2m!\n+M\u0001\u001b+h(k)\u0001\u001b; (S30)\nwhereMis the magnetization and h(k) is the SOC \feld. Therefore we have\ni@t^ \u0016(t;x) =ei^Ht\"X\nk^ \u0016(k)eik\u0001x;^H#\ne\u0000i^Ht\n=ei^HtX\nk;k0;\u0015;\u0017h\n^ \u0016(k)eik\u0001x; y\n\u0015(k0)H\u0015\u0017(k0) \u0017(k0)i\ne\u0000i^Ht\n=ei^HtX\nk;k0;\u0015;\u0017\u0010\n^ \u0016(k) y\n\u0015(k0) \u0017(k0)\u0000 y\n\u0015(k0) \u0017(k0)^ \u0016(k)\u0011\nH\u0015\u0017(k0)eik\u0001xe\u0000i^Ht\n=ei^HtX\nk;k0;\u0015;\u0017\u0010\n^ \u0016(k) y\n\u0015(k0) \u0017(k0) + y\n\u0015(k0)^ \u0016(k) \u0017(k0)\u0011\nH\u0015\u0017(k0)eik\u0001xe\u0000i^Ht\n=ei^HtX\nk;k0;\u0015;\u0017\u000ekk0\u000e\u0016\u0015 \u0017(k0)H\u0015\u0017(k0)eik\u0001xe\u0000i^Ht\n=X\nk;\u0017H\u0016\u0017(k)ei^Ht^ \u0017eik\u0001xe\u0000i^Ht\n=X\n\u0017 \nH\u0016\u0017(\u0000ir)X\nkei^Ht^ \u0017eik\u0001xe\u0000i^Ht!\n=X\n\u0017H\u0016\u0017(\u0000ir)^ \u0017(t;x): (S31)\nSimilar, for the creation operator\n y\n\u0016(t;x) =ei^Ht^ y\n\u0016(x)e\u0000i^Ht=ei^Ht X\nk^ y\n\u0016(k)e\u0000ik\u0001x!\ne\u0000i^Ht; (S32)\nwe have\ni@t^ y\n\u0016(t;x) =ei^Ht\"X\nk^ y\n\u0016(k)eik\u0001x;^H#\ne\u0000i^Ht\n=ei^HtX\nk;k0;\u0015;\u0017h\n^ y\n\u0016(k)e\u0000ik\u0001x; y\n\u0015(k0)H\u0015\u0017(k0) \u0017(k0)i\ne\u0000i^Ht\n=ei^HtX\nk;k0;\u0015;\u0017\u0010\n^ y\n\u0016(k) y\n\u0015(k0) \u0017(k0)\u0000 y\n\u0015(k0) \u0017(k0)^ y\n\u0016(k)\u0011\nH\u0015\u0017(k0)e\u0000ik\u0001xe\u0000i^Ht\n=ei^HtX\nk;k0;\u0015;\u0017\u0000\u0010\n y\n\u0015(k0)^ y\n\u0016(k) \u0017(k0) + y\n\u0015(k0) \u0017(k0)^ y\n\u0016(k)\u0011\nH\u0015\u0017(k0)e\u0000ik\u0001xe\u0000i^Ht\n=ei^HtX\nk;k0;\u0015;\u0017\u0000\u000ekk0\u000e\u0016\u0017 y\n\u0015(k0)H\u0015\u0017(k0)e\u0000ik\u0001xe\u0000i^Ht\n=\u0000X\nk;\u0015H\u0015\u0016(k)ei^Ht^ y\n\u0015e\u0000ik\u0001xe\u0000i^Ht\n=\u0000X\n\u0015 \nH\u0015\u0016(ir)X\nkei^Ht^ \u0015e\u0000ik\u0001xe\u0000i^Ht!\n=\u0000X\n\u0015H\u0015\u0016(ir)^ y\n\u0015(t;x): (S33)\nThe triplet pairs can be described by the G-lessor Green's function f<\n\u0016\u0017(t;x;t0;x0) =h^ \u0017(t0;x0)^ \u0016(t;x)iwhich12\nsatis\fes\ni@tf<\n\u0016\u0017=X\n\u0015H\u0016\u0015(\u0000irx)f<\n\u0015\u0017(t;x;t0;x0) (S34)\ni@t0f<\n\u0016\u0017=X\n\u0015H\u0017\u0015(\u0000irx0)f<\n\u0016\u0015(t;x;t0;x0): (S35)\nBecause\nH(\u0000ir) = \n\u0000r2\n2m0\n0\u0000r2\n2m!\n+M\u0001\u001b+h(\u0000ir)\u0001\u001b; (S36)\nwe have\nHT(\u0000ir) = \n\u0000r2\n2m0\n0\u0000r2\n2m!\n+M\u0001\u001b\u0003+h(\u0000ir)\u0001\u001b\u0003: (S37)\nCombining Eq. (S34,S35,S36,S37), we have\ni@tf<(t0;x0;t;x) =H(\u0000irx)f<(t;x;t0;x0) (S38)\ni@t0f<(t0;x0;t;x) =f<\n\u0016\u0015(t;x;t0;x0)HT(\u0000irx0): (S39)\nWe de\fneR= (x+x0)=2,r=x\u0000x0,T= (t+t0)=2,\u001c=t\u0000t0,rR=rx+rx0,rr= (rx\u0000rx0)=2,@T=@t+@t0\nand@\u001c= (@t\u0000@t0)=2. Therefore, Eq. (S38,S39) can be written in the ( T;\u001c;R;r) coordinates as\nEq:(S38) + Eq:(S39) = i@Tf<=\u0012\n\u0000r2\nr\nm\u0000r2\nR\n4m\u0013\nf<+M\u0001\u001bf<+f