Datasets:

ZWG817

/

Full_Content

like 0

1412.5634v2.Energy_spectra_of_two_interacting_fermions_with_spin_orbit_coupling_in_a_harmonic_trap.pdf

arXiv:1412.5634v2 [cond-mat.quant-gas] 7 Apr 2015Energy spectra of two interacting fermions with spin-orbit coupling in a harmonic trap Cory D. Schillaci∗ Department of Physics, University of California, Berkeley , California 94720, USA Thomas C. Luu† Institute for Advanced Simulation, Institut f¨ ur Kernphysik and J¨ ulich Center for Hadron Phys ics, Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany (Dated: December 17, 2014) We explore the two-body spectra of spin-1 /2 fermions in isotropic harmonic traps with external spin-orbit potentials and short range two-body interactio ns. Using a truncated basis of total angular momentum eigenstates, nonperturbative results are presen ted for experimentally realistic forms of the spin-orbit coupling: a pure Rashba coupling, Rashba and Dresselhaus couplings in equal parts, and a Weyl-type coupling. The technique is easily adapted to bosonic systems and other forms of spin-orbit coupling. PACS numbers: 71.70.Ej, 67.85.-d, 03.75.Mn, 03.65.Ge I. INTRODUCTION Cold atomic gases with spin-orbit coupling (SOC) have recently been a n area of intense interest because of the potential to simulate interesting physical systems with precisely tu nable interactions [1]. In condensed matter physics, spin-orbit couplings are essential for many exotic systems such as topological insulators [2, 3], the quantum spin Hall effect [4], and spintronics [5]. The experimental setup which induces s pin-orbit coupling is intimately related to simulation of synthetic gauge fields [6–9]. Because these couplings ar e parity violating, they potentially play similar roles within nuclear systems that undergo parity-violating transitio ns due to the nuclear weak force. Atomic gases provide an excellent testing ground both to explore universal beha vior of these real life systems and to create new types of spin-orbit coupling which are not yet known to exist (or hav e no solid-state analog) in other materials but are interesting in their own right. Further, these experiments can be performed in an environment with few or no defects and impurities. Spin-orbit coupling was first realized in a Bose condensate of87Rb [10] and extended shortly after to Fermi gases of40K [11] and6Li [12]. These spin-orbit interactions are ‘synthetic’ in the sense th at a subset of the hyperfine states stand in as virtual spin states. A particularlyinteresting conseque nce ofthis is the possibility ofstudying systems with synthetic spin-1 /2 spin-orbit interactions but bosonic statistics [10, 13]. From anoth er point of view, the couplings are equivalent to applying external electromagnetic forces via syn thetic gauge couplings on the physically uncharged particles in the gas [14, 15]. It has alsobeen conjectured that thes e systems could be used to physicallysimulate lattice gauge theories [16, 17]. Spin-orbit couplings in solid-state systems a rise in two-dimensional (2D) systems (Rashba and Dresselhaus types, described in Sec. II), but recently an exp erimental setup has been proposed that can simulate the Weyl-type SOC which is fundamentally three dimensional [18]. Spin-orbit couplings are also of interest from the perspective of fe w-body physics where they arise in a variety of fields, e.g., theweaknuclearinteractionsgoverningproton-proto nscattering[19,20]. Becausethespin-orbitcouplingis long range, it can significantly modify both the threshold scattering behavior and the spectrum of two-body systems [21]. For low-energy scattering, Duan et al.[22] showed analytically that parity-violating SOC leads to the the spontaneous emergence of handedness in outgoing states, a find ing later confirmed in [23]. Even in the presence of a repulsive two-body interaction, an arbitrarily weak SOC has been s hown to bind dimers [24]. For three-particle systems, a new type of universality is conjectured to occur for bo und trimers with negative scattering length [25]. Few-atom systems undergoing SOC within trapping potentials have a lso been explored. For example, the spectrum of particles within a trap with an external SOC of the Weyl type (but no relative interaction) has been theoretically determined [26]. The Rashba SOC with two-particle systems interact ing via short-ranged interactions was investi- gated perturbatively in [27], where it was shown that the leading orde r corrections due to the SOC and short-range interaction are independent when the scattering length is equal fo r all channels. In one dimension, the spectrum for ∗schillaci@berkeley.edu †t.luu@fz-juelich.de2 this type of system has been calculated when the SOC consists of eq ual parts Rashba and Dresselhaus interactions [28]. Information learned from trapped systems augments that fr om scattering experiments while also being relevant to interesting phenomena in trapped many-body systems with SOC s uch as solitons [29, 30] or novel phase diagrams [31]. In all these calculations, the emergent spectrum is rich and complex , offering new insights into few-body behavior. Our objective is to provide some additional insight into two-body phy sics of Fermi gases with spin-orbit interactions in the presence of both three-dimensional trapping potentials and short-ranged two-body interactions, which are necessarily present in dilute cold-atom experiments. Our approach is to numerically diagonalize the Hamiltonian within a suitably truncated basis, and is thus nonperturbative in nat ure. Eigenstates of the interacting Hamiltonian without SOC are used for the basis. Section II introduces the spec ific forms of spin-orbit coupling and two-body interactions which we consider. The general method is detailed in Sec . III for the simplest SOC. In the remaining Secs. IV-V we study the spectra of additional spin-orbit couplings in order of increasing computational complexity. II. HAMILTONIAN FOR SPIN-ORBIT COUPLINGS WITH CONTACT INTE RACTIONS In this paper we simply refer to our systems by their ‘spin’ degrees o f freedom and use the standard notation for spin quantum numbers. We consider three different types of spin-o rbit coupling. The form of spin-orbit coupling realized in experiments is a linear combination of the Rashba [32] and line ar Dresselhaus [33] types, VR≡αR(σxky−σykx), (1) VD≡αD(σxky+σykx), (2) which were originally recognized in two-dimensional solid-state syste ms. In a 2D system, these form a complete basis for spin-orbit couplings linear in momentum. Note that some ref erences use the alternate definitions VR∝ (σxkx+σyky) andVD∝(σxkx−σyky) which are equivalent up to a pseudospin rotation. For solids, these parity- violating interactions are allowed only in the absence of inversion symm etries. Rashba-type SOC typically arises in the presence of applied electric fields or in 2D subspaces such as the surfaces of materials where the boundary breaks the symmetry. Dresselhaus couplings were first studied in the cont ext of bulk inversion asymmetry, when the internal structure leads to gradients in the microscopic electric field. To date, experiments have produced only SOC potentials in which the Rashba and Dresselhaus terms appear with equal strength (also known as the “persistent spin-helix symmetr y point” [34]), VR=D≡αR=Dσxky. (3) After a pseudospin rotation, this potential can be seen as a unidire ctional coupling of the pseudospin and momentum along a single axis. A proposal for tuning the ratio αR/αDhas been given in [35]. An experimental setup which gives the simple three-dimensional Weyl coupling, VW≡αW/vectork·/vector σ, (4) has also been proposed in [18] and [36]. In the following sections we calculate the spectra of two particles wit h a short-range two-body interaction, an isotropic harmonic trapping potential and spin-orbit coupling. The s ingle particle Hamiltonian is H1=/planckover2pi12k2 2m+1 2mω2r2+VSO. (5) For the spin-orbit term VSO, we consider equal Rashba and Dresselhaus (3), pure Rashba (1) , and Weyl (4) spin-orbit couplings because these are generally considered to be experiment ally feasible. We assume that the range of interaction between particles is small c ompared to the size of the oscillator well. The relative interaction between the particles can then be approximate d as a regulated s-wave contact interaction, which in momentum space (as a function of relative momentum) is given by 4π/planckover2pi12 ma(Λ). (6) Here the argument Λ refers to some cutoff scale and a(Λ) is some function of the cutoff and physical scattering length aphys. The exact form of this function depends on the type of regulator used and is not relevant for this work; the only constraint is that a(Λ) reproduce the physical scattering length given by the scatte ringTmatrix at threshold,3 FIG. 1. (Color online) Spectrum of the two-body contact inte raction Hamiltonian as a function of ˜ a. The horizontal lines indicate the dimensionless energy eigenvalues in the unita ry limit|˜a| → ∞. T(E= 0) = 4π/planckover2pi12aphys/m[37]. In the limit Λ → ∞the spectrum of two particles in an oscillator well (without external spin-orbit interaction) was solved by Busch et al.[38] using the method of pseudopotentials. In Ref. [39] the solution for general Λ was given using a Gaussian regulator, whic h in the limit Λ → ∞recovered the Busch et al. solution. For our work below we use the eigenstates and eigenvalues of this two-particle system given in Ref. [38]. III. WEYL COUPLING We tackle the Weyl form first because of its mathematical and nume rical simplicity. In the absence of the two-body interaction, this problem was treated in Ref. [26]. Our approach is to determine the matrix elements of the SOC in an appropriate basis. The eigenvalue is then solved numerically at the desired precision by choosing an appropriately large truncated basis of harmonic oscillator (HO) eigenstates. As usual, the two-body problem is best approached in the dimensionle ss Jacobi coordinates R=r1+r2√ 2b, r=r1−r2√ 2b(7) andthecorrespondingconjugatemomenta q,Qrepresentingthe relativeandtotalmomenta. Foranisotropichar monic oscillator, distances can be expressed in terms of the ground-sta te length scale b=/radicalbig /planckover2pi1/mωand energies will be similarly measured in units of E0=/planckover2pi1ω. We also define the spin operators /vector σ≡/vector σ1−/vector σ2,/vectorΣ≡/vector σ1+/vector σ2. (8) With thesedefinitions, the two-bodyHamiltoniancanbe nondimension alizedandseparatedintorelativeandcenter- of-mass (c.m.) parts, 1 /planckover2pi1ωH=/parenleftbigg h0,rel+˜αW√ 2/vector q·/vector σ+√ 2π˜a(Λ)δ(3)(r)/parenrightbigg +/parenleftbigg h0,c.m.+˜αW√ 2/vectorQ·/vectorΣ/parenrightbigg , (9) whereh0,rel=r2/2 andh0,c.m.=R2/2. Notably, the spin-orbit coupling appears in both terms. The tilde o ver the coupling constants indicates that they are dimensionless, related t o the original coupling constants by dividing out the oscillator length (e.g., ˜ α=α/b). Throughout the remainder of this paper we will refer to dimension less eigenvalues ofH//planckover2pi1ωas the energies of the system. Eigenstates of two particles with a short-range interaction in a har monic oscillator trapping potential form a convenient basis for these calculations. These basis functions wer e first derived in [38] for the isotropic case considered here, and the more general case of an anisotropic trap has been e xplored in [40]. The dependence of the energy spectrum on the scattering length ais shown in Fig. 1 for reference. Qualitatively, the effect of the shor t-range interaction is to shift the harmonic oscillator energies by ±/planckover2pi1ωas the scattering length goes to ±∞. For positive scattering length, there is also an additional negative-energy dime r state. We choose the particular coupling scheme of angular momentum eigen states, |n(ls)j;NL;(jL)J∝angb∇acket∇ight, (10)4 which simplify the matrix elements for the relative-coordinate opera tors. Here nandlrefer to the principal and orbital angular-momentum quantum numbers of the two-particle s ystem in the relative coordinates. NandLrefer to the analogous numbers in the center-of-mass frame. The tota l spin of the two spin-1 /2 particles is denoted by s=s1+s2and may be either 0 or 1. First sandlto make angularmomentum j, which is then recoupled with the c.m. angular momentum Lto make the state’s total angular momentum J. Because all terms in the Hamiltonian (9) are scalars, the interaction is independent of Jzand so we omit this quantum number for clarity. Due to Pauli exclusion , l+smust be even to enforce antisymmetry under exchange of the par ticles. Forl∝negationslash= 0 the states (10) are identical to the well known harmonic oscillato r, withnandl(NandL) indicating the relative (center-of-mass) HO quantum numbers. We use the c onvention that n,N= 0,1,2,..., and therefore E= 2n+l+2N+L+3. The short range interaction (5) modifies the l= 0 states and their spectrum. The principal relative quantum number nfor these states is obtained by solving the transcendental equat ion √ 2Γ(−n) Γ(−n−1/2)=1 a(11) and is no longer integer valued. For the relative-coordinate part of thel= 0 wave function, φ(r) =1 2π3/2A(n)Γ(−n)U(−n,3/2,r2)e−r2/2, (12) A(n) =/parenleftbiggΓ(−n)[ψ0(−n)−ψ0(−n−1/2)] 8π2Γ(−n−1/2)/parenrightbigg−1/2 , (13) whereU(a,b,x) is Kummer’s confluent hypergeometric function and ψ0(x) = Γ′(x)/Γ(x) is the digamma function. A derivation of the normalization factor A(n) is given in the Appendix. Standard angular momentum algebra can be used to determine the m atrix elements of the two spin-orbit coupling terms; we follow the conventions of [41]. For Weyl SOC of two spin-1 /2 fermions, the matrix elements of the coupling in the relative momentum are ∝angb∇acketleftn′(l′s′)j′;N′L′;(j′L′)J′|/vector q·/vector σ|n(ls)j;NL;(jL)J∝angb∇acket∇ight =δN,N′δL,L′δj,j′δJ,J′(−1)l+s′+j3√ 2/braceleftbigg j s′l′ 1l s/bracerightbigg (s′−s)∝angb∇acketleftn′l′||q||nl∝angb∇acket∇ight.(14) To preserve anti-symmetry of the two-particle system, the relat ive momentum term in the Weyl SOC must couple states with relative angular momentum ltol±1, leavingl+seven but changing the parity. For basis states with both l,l′∝negationslash= 0, reduced matrix elements of the momentum operator are calcula ted between pure harmonic oscillator states, ∝angb∇acketleftn′l′||q||nl∝angb∇acket∇ight=(−1)l′(−1)l+l′+1 2/radicalBigg 2(2l+1)(2l′+1) (l+l′+1)∝angb∇acketleftn′l′0|(−i∇0)|nl0∝angb∇acket∇ight (15) =i(−1)l/radicalbigg l+l′+1 2/radicalbig n!n′!Γ(n+l+3/2)Γ(n′+l′+3/2) ×n,n′/summationdisplay m,m′=0 (−1)m+m′/bracketleftBig 2mΓ/parenleftBig m+m′+1+l+l′ 2/parenrightBig −Γ/parenleftBig m+m′+1+l+l′ 2/parenrightBig/bracketrightBig m!m′!(n−m)!(n′−m′)!Γ(m+l+3/2)Γ(m′+l′+3/2)ifl′=l−1 (−1)m+m′+1/bracketleftBig (2m+2l+1)Γ/parenleftBig m+m′+1+l+l′ 2/parenrightBig −Γ/parenleftBig m+m′+1+l+l′ 2/parenrightBig/bracketrightBig m!m′!(n−m)!(n′−m′)!Γ(m+l+3/2)Γ(m′+l′+3/2)ifl′=l+1 0 otherwise(16) Ifl= 1 andl′= 0 or vice versa, reduced matrix elements between one modified wav e function of the form (12) and one pure harmonic oscillator state are needed. These are given by ∝angb∇acketleftnl= 0||q||n′l′= 1∝angb∇acket∇ight=−iA(n)/radicalbigg Γ(n′+5/2) 2π3n′!2n−2n′−1 2(n′−n)(1+n′−n)(17) and its Hermitian conjugate. Our choice of basis makes the relative matrix elements (14) simple at t he cost of complicating the center-of-mass term. We take the approach of expanding the states (10) in the alt ernate coupling scheme, |n(ls)j;NL;(jL)J∝angb∇acket∇ight= (−1)l+s+L+J/radicalbig 2j+1/summationdisplay J√ 2J+1/braceleftbigg l s j L JJ/bracerightbigg |nl;N(Ls)J;(lJ)J∝angb∇acket∇ight. (18)5 FIG. 2. (Color online) Absolute value of the matrix elements |/angbracketleftn′(11)0;00;(00)0 |/vector σ·/vector q|n(00)0;00;(00)0 /angbracketright|between the ground state andl= 1excited states. The horizontal axis is the principal quan tumnumber of the ground state obtained bysolving (11). From left to right, the vertical lines on the negative axis in dicate the values obtained for ˜ a= 1/4, ˜a= 1, ˜a=±∞, and ˜a=−1, respectively. FIG. 3. (Color online) A convergence plot giving the change i n energy eigenvalue, ∆ E, for the lowest eight energy levels when a shell is added as a function of Emax. The left figure shows convergence for ˜ a=−1 and ˜αW= 0.5. In the right panel we show ˜a= 1 and ˜αW= 0.5, demonstrating that convergence of the states with large n egativenis poor. Using this notation, the matrix elements can be written ∝angb∇acketleftn′(l′s′)j′;N′L′;(j′L′)J′|/vectorQ·/vectorΣ|n(ls)j;NL;(jL)J∝angb∇acket∇ight=δn,n′δl,l′δJ,J′δs,1δs1,16(−1)L ×∝angb∇acketleftN′L′||/vectorQ||NL∝angb∇acket∇ight/summationdisplay J(−1)J(2J+1)/braceleftbigg l1j′ L′JJ/bracerightbigg/braceleftbigg l1j L JJ/bracerightbigg/braceleftbigg J1L′ 1L1/bracerightbigg .(19) Again, the reduced matrix element of the center-of-mass moment um changes the parity by connecting states with ∆L=±1. Matrix elements are nonzero only for ∆ s= 0 because the antisymmetry of the spatial wave function depends only on l, which does not change. We also note that the c.m. term does not aff ect states with singlet spin wave functions ( s= 0). Using these matrix elements, we calculated the spectrum of the two interacting particles with Weyl spin-orbit coupling. Our calculations are performed by numerically diagonalizing in a truncated basis of the harmonic oscillator states (10), where a cutoff 2 N+L+2n+l+3≤Emaxis set high enough that the eigenvalues of the matrix have converged to the desired accuracy. This approach converges well only when the ground-state energy is not too low. In particular, for apositive but very small the principal quantum number of the ground state is incr easing from negative infinity as seen in Fig. 1.6 FIG. 4. (Color online) Spectrum of states with total angular momentum J= 0 for the dimensionless Hamiltonian (9). The bottom left figure shows the ground-state energy for ˜ a=−1 as a function of ˜ αW; above are the first few excitation energies. The right figure shows the results in the unitary limit of the t wo-body interaction, |˜a| → ∞. The spectrum is symmetric about ˜αW= 0. FIG. 5. (Color online) For different values of the two-body co upling strength ˜ a, we show the magnitude of the ground state projected onto even parity basis states as a function of the S OC strength. This is given by/vextendsingle/vextendsingleP+|ψGS/angbracketright/vextendsingle/vextendsingle2=/vextendsingle/vextendsingle(1−P−)|ψGS/angbracketright/vextendsingle/vextendsingle2, whereP+(P−) is the projection operator onto the positive- (negative-) parity basis states. The left figure shows negative ˜ a, while the right shows positive ˜ a. Note that the limits ˜ a→ ±∞are physically identical. From Fig. 2, we can see that as nbecomes more negative, the principal quantum number of the domin ant matrix element is also increasing. Because convergenceof any energyleve l requires a cutoff much largerthan the energyof the most strongly coupled states, a sufficiently high Emaxto ensure an accurate ground-state energy becomes infeasible for small positive a. For excited states, nis always positive and matrix elements with similar nalways dominate. The strength of the matrix elements follows a similar qualitative behav ior for the spin-orbit couplings treated in the following sections where the same issues recur. As a result, convergence of the ground state is actually slower tha n that for nearby excited states. Furthermore, our approach gives the fastest convergence when ais not small and positive. We compare the rate of convergence of the ˜a=−1 and ˜a= 1 spectra in Fig. 3 to demonstrate the dependence of convergen ce on the matrix truncation. The actual energy spectrum is shown in Fig. 4. One consequence of parity violation in this system is that the eigenst ates are mixtures of the even- and odd-parity basis states described by Eq. (10). In Fig. 5 we visualize how these s ubspaces are mixed in the ground state as the7 SOC strength increases. For the noninteracting system, ˜ a= 0, more than half of the ground state projects onto negative-parity states even at fairly small values of ˜ αW. However, we see that the short-range interaction reduces this effect. With negative ˜ a, the mixing of the negative-parity states is suppressed as the str ength of the two-body interaction increases. When ˜ ais positive the effect is more striking. Mixing with negative-parity stat es is most strongly suppressed for small positive values of ˜ a, while the projection onto these states increases for larger posit ive values. The admixture is qualitatively the same when considering othe r forms of SOC as described in the following sections. IV. THE PURE RASHBA COUPLING In order to find the matrix elements of the pure Rashba coupling give n in (1), we first note that it can be written as a spherical tensor VR=i√ 2αR[k⊗σ]10. (20) We therefore have the two-body Hamiltonian 1 /planckover2pi1ωH=/parenleftBig h0,rel+i˜αR[/vector q⊗/vector σ]10+√ 2π˜a(Λ)δ(3)(r)/parenrightBig +/parenleftBig h0,c.m.+i˜αR[/vectorQ⊗/vectorΣ]10/parenrightBig . (21) Because the spin-orbit coupling is now a k= 1 tensor rather than a scalar operator, the total angular mome ntumJ isnolongerconserved. Additionally, the matrixelements nowdepend onthe quantumnumber Jz(whichis conserved). For the relative-coordinate part of the SOC, some algebra gives ∝angb∇acketleftn′(l′s′)j′;N′L′;(j′L′)J′J′ z|[/vector q⊗/vector σ]10|n(ls)j;NL;(jL)JJz∝angb∇acket∇ight= 6i(−1)J+J′−J′ z+j′+L+1δN,N′δL,L′δJz,J′z ×/radicalbig (2J+1)(2J′+1)(2j+1)(2j′+1)/parenleftbigg J′1J −Jz0Jz/parenrightbigg/braceleftbigg j′J′L J j1/bracerightbigg/braceleftBiggl′l1 s′s1 j′j1/bracerightBigg (s′−s)∝angb∇acketleftn′l′||q||nl∝angb∇acket∇ight.(22) For the center-of-mass part of the Hamiltonian we again expand th e basis states in the alternate coupling scheme (18) to obtain the matrix elements ∝angb∇acketleftn′(l′s′)j′;N′L′;(j′L′)J′J′ z|[/vectorQ⊗/vectorΣ]10|n(ls)j;NL;(jL)JJz∝angb∇acket∇ight=δn,n′δl,l′δJz,J′zδs,1δs′,1 ×6i√ 2(−1)J+J′−J′ z+l/radicalbig (2J+1)(2J′+1)(2j+1)(2j′+1)/parenleftbigg J′1J −Jz0Jz/parenrightbigg ∝angb∇acketleftN′L′||Q||NL∝angb∇acket∇ight ×/summationdisplay J,J′(−1)J(2J+1)(2J′+1)/braceleftbigg l1j′ L′J′J′/bracerightbigg/braceleftbigg l1j L JJ/bracerightbigg/braceleftbigg J′J′l JJ1/bracerightbigg/braceleftBiggL′L1 1 1 1 J′J1/bracerightBigg .(23) Our results for the Rashba SOC are shown in Fig. 6. Because the Ras hba spin-orbit coupling is a vector operator, states of all possible Jmust be included in any calculation and the size of the basis scales much more quickly with Emax. These spectra were computed with an Emaxof 24/planckover2pi1ω, for which there are approximately 36000 basis states. All displayed eigenvalues of the Hamiltonian shift by less than 10−2/planckover2pi1ωif an additional shell of states is included. This interaction was also studied perturbatively for small αRin [27], including the possibility of a spin-dependent two-body interaction, under the assumption that center-of-ma ss excitations are unimportant. For the specific case of identical fermions with spin-independent scattering length conside red here, they found that the first correction to the energies occurs at order α2 Rand is independent of the scattering length a. We compare their perturbative predictions, which are derived from the non-degenerate theory, with our nume rical results in Fig. 7. By setting all matrix elements with N,L >0 in the bra or ket to zero, we also explored the approximation of ignoring center-of-mass excitations. Fig. 8 shows that this is very accurate for the ground state, but less accurate for excited states. Suppression of the c.m. coordinate has a similar effe ct for the SOCs considered in Secs. III and V. We also note that in the case of small positive a, the landscape of low-lying excited states is dominated by center-o f-mass excitations. When a→0+in the absence of spin-orbit coupling, there are an infinite number of states with nonzero c.m. quantum numbers whose energies lie between the ground state and the first relative-coordinate excitation. V. EQUAL-WEIGHT RASHBA-DRESSELHAUS SPIN-ORBIT COUPLING Experiments have thus far realized only the effective Hamiltonian with equal strength Rashba and Dresselhaus couplings in the form (3). Energy levels of the two-body system in th e one-dimensional equivalent of this Hamiltonian8 FIG. 6. (Color online) Spectrum of states with total angular momentum quantum number Jz= 0 for the Hamiltonian (21). The left figure shows the energies with negative scattering l ength ˜a=−1. The right figure shows the results in the unitary limit|˜a| → ∞. The spectrum is symmetric about ˜ αR= 0. FIG. 7. (Color online) Comparison of selected spectral line s (dashed black) with the perturbative predictions from [27 ] (solid red) when ˜a=∞. with the additional magnetic field couplings present in experimental r ealizations have been calculated in [28]. Here we treat the problem in three dimensions. This is also the most computationally difficult of the three cases. When decomposed into spherical tensors, the interaction (2) becomes VD=iαD/parenleftBig [k⊗σ]2,−2−[k⊗σ]2,2/parenrightBig , (24) and the two-particle Hamiltonian in the presence of equal strength Rashba and Dresselhaus SOC is given by (21) withαR→αR=Dplus the additional spin-orbit terms ∆H=i˜αR=D√ 2/parenleftBig [/vector q⊗/vector σ]2,−2−[/vector q⊗/vector σ]2,2+[/vectorQ⊗/vectorΣ]2,−2−[/vectorQ⊗/vectorΣ]2,2/parenrightBig . (25) Yet again the number of basis states with nonzero matrix elements h as increased; no angular momentum quantum numbers are conserved. The only remaining selection rule will be that the interaction does not change the total magnetic quantum number Jzbetween even and odd.9 FIG. 8. (Color online) A comparison of the energy levels with (dashed black) and without (solid red) the inclusion of exci tations in the c.m. coordinate for ˜ a=−1. The approximation of ignoring c.m. excitations provides very accurate results for the ground state, but not for excited states. Using the same approach as in the previous sections, the matrix elem ents of the relative Dresselhaus term are ∝angb∇acketleftn′(l′s′)j′;N′L′;(j′L′)J′J′ z|i˜αR=D√ 2/parenleftBig [/vector q⊗/vector σ]2,−2−[/vector q⊗/vector σ]2,2/parenrightBig |n(ls)j;NL;(jL)JJz∝angb∇acket∇ight =i√ 30(−1)J+J′−J′ z+j′+LδN,N′δL,L′/radicalbig (2J+1)(2J′+1)(2j+1)(2j′+1)∝angb∇acketleftn′l′||q||nl∝angb∇acket∇ight ×(s′−s)/bracketleftbigg/parenleftbigg J′2J −J′ z−2Jz/parenrightbigg −/parenleftbigg J′2J −J′ z2Jz/parenrightbigg/bracketrightbigg/braceleftbigg j′J′L J j2/bracerightbigg/braceleftBiggl′l1 s′s1 j′j2/bracerightBigg ,(26) while the center-of-mass part is ∝angb∇acketleftn′(l′s′)j′;N′L′;(j′L′)J′J′ z|i˜αR=D√ 2/parenleftbigg/bracketleftBig /vectorQ⊗/vectorΣ/bracketrightBig 2,−2−/bracketleftBig /vectorQ⊗/vectorΣ/bracketrightBig 2,2/parenrightbigg |n(ls)j;NL;(jL)JJz∝angb∇acket∇ight = 2i√ 15(−1)J+J′−J′ z+l+1δn,n′δl,l′δs,1δs′,1 ×/radicalbig (2J+1)(2J′+1)(2j+1)(2j′+1)/bracketleftbigg/parenleftbigg J′2J −J′ z−2Jz/parenrightbigg −/parenleftbigg J′2J −J′ z2Jz/parenrightbigg/bracketrightbigg ∝angb∇acketleftN′L′||Q||NL∝angb∇acket∇ight ×/summationdisplay J,J′(−1)J(2J+1)(2J′+1)/braceleftbigg l1j′ L′J′J′/bracerightbigg/braceleftbigg l1j L JJ/bracerightbigg/braceleftbigg J′J′l JJ2/bracerightbigg/braceleftBiggL′L1 1 1 1 J′J2/bracerightBigg .(27) The richly structured excitation spectrum of low-lying states is sho wn in Fig. 9 for a cutoff of Emax= 17. All displayed energies shift by less than .02 /planckover2pi1ωwhen the final shell is added, giving a slightly faster convergence th an in the pure Rashba case. VI. CONCLUSIONS In this paper we have nonperturbatively calculated the spectrum o f interacting two-particle systems with realistic spin-orbit couplings when the trapping potential cannot be ignored . Matrix elements of a short-range pseudopotential and three types of spin-orbit coupling were determined analytically in a basis of the total angular momentum eigen- states of the interacting two-body problem without SOC. With the a nalytic matrix elements, exact diagonalization of the Hamiltonian within a finite basis was possible. Our energy calculations were performed in a basis truncated in a con sistent way by including all states below an energy cutoff. The resulting spectra show good convergence e xcept in the case where the two-body interaction generates a small positive scattering length. In this regime coupling of the ground state to higher relative-coordinate excited states dominates and convergence in the cutoff paramete rEmaxwas numerically intractable. We are currently investigating alternative methods to deal with this issue. In the limit o f weak SOC we have compared our results to10 FIG. 9. (Color online) Spectrum of states with even total ang ular momentum magnetic quantum number Jz= 0,2,...for the equal-weight Rashba-Dresselhaus SOC (3). The left figure sh ows the energies with negative scattering length ˜ a=−1. The right figure shows the results in the unitary limit |˜a| → ∞. The spectrum is symmetric about ˜ αR=D= 0. the perturbative calculations of [27] and found good agreement. W e also observed that although the ground state does not couple strongly to center-of-mass excitations, their inc lusion is crucial for the excited state spectrum. The relatively weak center-of-mass coupling of the ground state, how ever, suggests that cold atoms with SOC can be used as a surrogate system to probe properties of two-body spin-orb it couplings, e.g., the parity-violating weak interaction in nuclear systems. We provided plots of a variety of spectra calculated with Weyl, Rashb a, and equal weight Rashba-Dresselhaus couplings. Although in this paper we show spectra only within certain s ubspaces of conserved angular momentum quantum numbers, the approach presented is fully capable of gene rating results for all possible states. Larger SO- coupling constants are also accessible with larger basis sizes. The ge neral method can easily be adapted to calculate energies for bosonic systems, or to new forms of SOC such as the r ecently proposed spin-orbital angular momentum coupling [42]. Using the eigenvectors of the truncated basis Hamiltonian, we also e xplored the effect of parity violation on the system. In particular we show how the SOC induces mixing of the posit ive- and negative-parity subspaces for the ground state. Without a two-body interaction, the ground state preferentially projects onto negative parity basis states even for modest SOC strength. The short-range interac tion was seen to suppress this mixing, especially when the scattering length is positive. A natural extension of this work is to consider three particles within a trap. Because of the complex spectrum that is associated with three-body physics at the unitary limit (e.g., E fimov states, limit cycles, etc.), the spectrum under the influence of an external SOC is expected to be quite rich. Couplings between the center-of-mass and relative motion due to the SOC present a potential challenge to trad itional few-body techniques, such as the Faddeev equations, which work only within the relative coordinates. However , in our two-body calculations we found that the coupling of the ground state to the c.m. motion is weak. If this is also t rue in the three-body case, then to a good approximation we can ignore the c.m. motion and utilize existing few-bo dy techniques with little or no modification. ACKNOWLEDGEMENTS We thank Paulo Bedaque, Jordy de Vries and Timo L¨ ahde for their inp ut and discussions related to this work. This paper is based in part on work supported by the U.S. Departmen t of Energy, Office of Science, Office of Nuclear11 Physics, under Award No. de-sc00046548. [1] V. Galitski and I. B. Spielman, Nature 494, 49 (2013). [2] T. Das and A. V. Balatsky, Nature Communications 4(2013). [3] N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Mart in-Delgado, M. Lewenstein, and I. B. Spielman, Phys. Rev. Lett. 105, 255302 (2010). [4] M. C. Beeler, R. A. Williams, K. Jim´ enez-Garcia, L. J. Le Blanc, A. R. Perry, and I. B. Spielman, Nature 498, 201 (2013). [5] I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [6] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg, Rev. Mod. Phys. 83, 1523 (2011). [7] C. Hamner, C. Qu, Y. Zhang, J. Chang, M. Gong, C. Zhang, and P. Engels, Nature Communications 5, 4023 (2014). [8] Y.-J. Lin, R. Compton, A. Perry, W. Phillips, J. Porto, et al., Phys.Rev.Lett. 102, 130401 (2009), arXiv:0809.2976 [cond-mat.other]. [9] A. Bermudez, T. Schaetz, and D. Porras, Phys.Rev.Lett. 107, 150501 (2011), arXiv:1104.4734 [quant-ph]. [10] Y.-J. Lin, K. Jim´ enez-Garcia, and I. B. Spielman, Natu re471, 83 (2011). [11] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zha i, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [12] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [13] S. Ashhab and A. J. Leggett, Phys. Rev. A 68, 063612 (2003). [14] Y. J. Lin, R. L. Compton, K. Jim´ enez-Garcia, J. V. Porto , and I. B. Spielman, Nature 462, 628 (2009). [15] M. Aidelsburger, M. Atala, S. Nascimb` ene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011). [16] A. Bermudez, L. Mazza, M. Rizzi, N. Goldman, M. Lewenste in, and M. A. Martin-Delgado, Phys.Rev.Lett. 105, 190404 (2010), arXiv:1004.5101 [cond-mat.quant-gas]. [17] L. Mazza, A. Bermudez, N. Goldman, M. Rizzi, M. A. Martin -Delgado, and M. Lewenstein, New J.Phys. 14, 015007 (2012), arXiv:1105.0932 [cond-mat.quant-gas]. [18] B. M. Anderson, G. Juzeli¯ unas, V. M. Galitski, and I. B. Spielman, Phys. Rev. Lett. 108, 235301 (2012). [19] W. Haxton and B. Holstein, Prog.Part.Nucl.Phys. 71, 185 (2013), arXiv:1303.4132 [nucl-th]. [20] J. de Vries, N. Li, U.-G. Meißner, N. Kaiser, X. H. Liu, et al., Eur.Phys.J. A50, 108 (2014), arXiv:1404.1576 [nucl-th]. [21] P. Zhang, L. Zhang, and W. Zhang, Phys. Rev. A 86, 042707 (2012). [22] H. Duan, L. You, and B. Gao, Phys. Rev. A 87, 052708 (2013). [23] S.-J. Wang and C. H. Greene, Phys. Rev. A 91, 022706 (2015). [24] J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B 83, 094515 (2011). [25] Z.-Y. Shi, X. Cui, and H. Zhai, Phys. Rev. Lett. 112, 013201 (2014). [26] B. M. Anderson and C. W. Clark, Journal of Physics B: Atom ic, Molecular and Optical Physics 46, 134003 (2013). [27] X. Y. Yin, S. Gopalakrishnan, and D. Blume, Phys. Rev. A 89, 033606 (2014). [28] Q.Guan, X.Y.Yin, S.E.Gharashi, andD.Blume,Journal o f Physics B: Atomic, Molecular and Optical Physics 47, 161001 (2014), arXiv:1406.7177 [cond-mat.quant-gas]. [29] V. Achilleos, J. Stockhofe, P. G. Kevrekidis, D. J. Fran tzeskakis, and P. Schmelcher, EPL (Europhysics Letters) 103, 20002 (2013). [30] Y. Xu, Y. Zhang, and B. Wu, Phys. Rev. A 87, 013614 (2013). [31] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401 (2011). [32] Y. A. Bychkov and E. I. Rashba, Journal of Physics C: Soli d State Physics 17, 6039 (1984). [33] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [34] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev . Lett.97, 236601 (2006). [35] D. L. Campbell, G. Juzeli¯ unas, and I. B. Spielman, Phys ical Review A 84, 025602 (2011). [36] B. M. Anderson, I. B. Spielman, and G. Juzeli¯ unas, Phys . Rev. Lett. 111, 125301 (2013). [37] J. R. Taylor, Scattering Theory , 2nd ed. (Dover, 2000). [38] T. Busch, B.-G. Englert, K. Rza˙ zewski, and M. Wilkens, Foundations of Physics 28, 549 (1998). [39] T. Luu and A. Schwenk, Phys.Rev.Lett. 98, 103202 (2007), arXiv:cond-mat/0606069 [cond-mat]. [40] Z. Idziaszek and T. Calarco, Phys. Rev. A 74, 022712 (2006). [41] A. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, 1996). [42] K. Sun, C. Qu, and C. Zhang, ArXiv e-prints (2014), arXiv :1411.1737 [cond-mat.quant-gas]. [43] K. M. Daily, X. Y. Yin, and D. Blume, Phys. Rev. A 85, 053614 (2012). [44] “NIST Digital Library of Mathematical Functions,” htt p://dlmf.nist.gov/, Release 1.0.9 of 2014-08-29 (2014). [45] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products , 5th ed., edited by A. Jeffrey (Academic Press, Boston, 1994).12 Appendix: Derivation of the normalization factor for Busch wave functions In the original paper by Busch et al.[38], the normalization factor of the wave functions is not given. The closed form expression for this normalization does not seem to be widely kno wn. It was originally presented in [43] without derivation, which we provide here. To find the norm of the wave func tion (12), one must integrate (using a change of variables to z=r2) A−2=Γ(−n)2 8π3/integraldisplay∞ 01 z/bracketleftBig U(−n,3/2,z)e−z/2z3/4/bracketrightBig2 dz. (A.1) The term in brackets is equal to a Whittaker function [44] and so th is can be rewritten, A−2=Γ(−n)2 8π3/integraldisplay∞ 01 z/bracketleftbig Wn+3/4,1/4(z)/bracketrightbig2dz. (A.2) This integral can be found in [45] /integraldisplay∞ 01 z[Wκ,µ(z)]2dz=π sin(2πµ)ψ0(1 2+µ−κ)−ψ0(1 2+µ−κ) Γ(1 2+µ−κ)Γ(1 2−µ−κ). (A.3) Applying this to (A.1) with κ=n+3/4 andµ= 1/4 gives the desired result, A−2=1 8π3Γ(−n) Γ(−n−1/2)[ψ0(−n)−ψ0(−n−1/2)]. (A.4)

1607.08724v1.Twisted_spin_vortices_in_a_spinor_dipolar_Bose_Einstein_condensate_with_Rashba_spin_orbit_coupling.pdf

arXiv:1607.08724v1 [cond-mat.quant-gas] 29 Jul 2016Twisted spin vortices in a spinor-dipolar Bose-Einstein co ndensate with Rashba spin-orbit coupling Masaya Kato,1Xiao-Fei Zhang,1,2Daichi Sasaki,1and Hiroki Saito1 1Department of Engineering Science, University of Electro- Communications, Tokyo 182-8585, Japan 2Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China (Dated: September 20, 2018) We consider a spin-1 Bose-Einstein condensate with Rashba s pin-orbit coupling and dipole-dipole interaction confined in a cigar-shaped trap. Due to the combi ned effects of spin-orbit coupling, dipole-dipole interaction, and trap geometry, the system e xhibits a rich variety of ground-state spin structures, including twisted spin vortices. The ground-s tate phase diagram is determined with respect to the strengths of the spin-orbit coupling and dipo le-dipole interaction. PACS numbers: 03.75.Mn, 03.75.Lm, 67.85.Bc, 67.85.Fg I. INTRODUCTION Spin-orbit coupling is of fundamental importance in many branches of physics, such as quantum spin-Hall ef- fect, topological insulators, and superconductivity [1–4]. Recently, the NIST group has realized the light-induced vector potentials and the synthetic electric and magnetic fields in Bose-Einstein condensates (BECs) of neutral atoms using Raman processes [5–8]. Remarkably, they also created a two-component spin-orbit coupled con- densate of Rb atoms [9]. Artificial spin-orbit coupling (SOC) offers us a tremendous opportunity to study ex- otic quantum phenomena in many-body systems, which exhibit various symmetry-broken and topological con- densate phases in pseudospin-1 /2 systems [10–18]. For spin-1 and -2 condensates, more exotic patterns form due to the competition between the SOC and spin-dependent interactions [19–23]. On the other hand, recent experimental realization of BECs of atomic species with large magnetic moments boosts interest in the field of quantum gases with dipole- dipole interaction (DDI) [24–27]. Previous studies on spinor-dipolar BECs have shown that the interplay be- tween spin-dependent interaction and DDI leads to rich topological defects and spin structures [28–34]. Conse- quently, it is of particular interest to explore the effects of long-rang and anisotropic DDI on such a spin-orbit coupled system, which has recently drawn considerable attentions. More specifically, Deng et al. [35] proposed anexperimentalschemetocreateSOCinspin-3Cratoms using Raman processes. Wilson et al. have investigated the effects of DDI on a pseudospin-1 /2 spin-orbit cou- pled condensate, and predicted the emergence of a ther- modynamically stable ground state having a spin con- figuration called meron [36]. Furthermore, a number of quantum crystalline and quasicrystalline ground states were found in two-dimensional (2D) dipolar bosons with Rashba SOC [37]. In this work, we consider a BEC of spin-1 bosons con- fined in a cigar-shaped trap potential, subject to both 2D SOC and DDI. The 2D SOC tends to create spintextures in the x-yplane, while the DDI can generate z-dependent spin textures in an elongated system. As a result, 3D spin structures emerge in this system. We elu- cidate the ground-state spin textures as functions of the strengths of the SOC and DDI by numerically minimiz- ing the energy functional. We will show a rich variety of ground-state spin textures, such as twisted spin vortices, in which spin vortices twist around each other along the zdirection. The paper is organized as follows. In Sec. II, we for- mulate the theoretical model and briefly introduce the numerical method. In Sec. III, the ground-state phase diagram of the system is determined, and a detailed de- scription of each phase is given. In Sec. IV, the main results of the paper are summarized. II. FORMULATION OF THE PROBLEM We consider a BEC of spin-1 atoms with mass Mcon- fined in a harmonic potential, which are subject to the 2D SOC. We employ the mean-field approximation and the state of the system is described by the spinor order parameter Ψ(r) = (ψ1(r),ψ0(r),ψ−1(r))T.The single- particle energy is given by E0=/integraldisplay drΨ†/bracketleftbigg −/planckover2pi12 2M∇2+V(r)+gsoc/planckover2pi1 i∇⊥·f⊥/bracketrightbigg Ψ, (1) where gsocparametrizes the SOC strength, ∇⊥= (∂x,∂y), andf⊥= (fx,fy) are the 3 ×3 spin-1 ma- trices. The trap potential is axisymmetric, V(r) = Mω2 ⊥(x2+y2+λ2z2)/2, whereω⊥is the radial trap fre- quency and λ=ωz/ω⊥is the aspect ratio between the axial and radial trap frequencies. The s-wave contact interaction energy is written as Es=1 2/integraldisplay dr/bracketleftbig g0ρ(r)+g1F2(r)/bracketrightbig , (2) where g0= 4π/planckover2pi12(a0+ 2a2)/(3M) and g1= 4π/planckover2pi12(a0− a2)/(3M) withas(s= 0,2) being the s-wave scattering2 length for the scattering channel with total spin s. The totalatomicdensity ρ(r) =|ψ1(r)|2+|ψ0(r)|2+|ψ−1(r)|2 satisfies/integraltext ρ(r)dr=N, whereNis the total number of atoms. The spin density has the form F(r) =Ψ† fx fy fz Ψ= √ 2Re[ψ∗ 1ψ0+ψ∗ 0ψ−1]√ 2Im[ψ∗ 1ψ0+ψ∗ 0ψ−1] |ψ1|2−|ψ−1|2 .(3) The DDI energy is given by Eddi=gdd 2/integraldisplay drdr′ˆF(r)·ˆF(r′)−3(ˆF(r)·e)(ˆF(r′)·e) |r−r′|3, (4) where gdd=µ0µ2 d/(4π),µ0is the magnetic permeability of the vacuum, µdis the magnetic dipole moment of the atom, and e= (r−r′)/|r−r′|. The total energy of the system is thus given by E=E0+Es+Eddi. The ground state is numerically obtained by minimiz- ing the totalenergy Eusingthe imaginary-timepropaga- tionmethod. Fortheimaginary-timeevolution, thepseu- dospectral method with the fourth-order Runge-Kutta scheme is used. In the following numerical simulations, we work in dimensionless unit. The energy and length are normalized by /planckover2pi1ω⊥anda⊥=/radicalbig /planckover2pi1/(Mω⊥). In this unit, the wave function, the SOC coefficient gsoc, and the interaction coefficients g0, g1, and gddare normalized by N1/2/a3/2 ⊥,a⊥ω⊥, and/planckover2pi1ω⊥a3 ⊥/N, respectively. III. GROUND-STATE SPIN STRUCTURES The richness of the present system lies in the large number of free parameters, including the strength and sign of the contact interactions, DDI, SOC, aspect ratio, and so on. To highlight the effects of the SOC and DDI, we fixλ= 0.2, g0= 4000, and g1= 0, implicitly assum- ing that the ground-state spin texture is dominated by the SOC and DDI. Our main results are summarized in Fig. 1, which shows the ground-state phase diagram of a spin-orbit coupled dipolar condensate with respect to gsocand gdd. There are eight different phases marked by A-H, which differ in density profiles, spin texture and angular mo- mentum. In the following discussion, we will give a de- tailed description of each phase. In the white region of Fig. 1, the condensatecollapsesdue tothe attractivepart of the DDI [38], where no stable mean-field solution ex- ists [39]. The critical value of gddfor the collapse seems almost independent of gsoc. We start from the case where both the SOC and DDI are sufficiently weak, indicated by the gray region F in Fig. 1. In this phase, the central region of the poten- tial is occupied by mf= 0 component and the system is condensed to such component, leading to vanishing magnetization of the system. We note that this phase disappears with increasing either the SOC or DDI. In the limit of strong SOC but weak DDI, the system exhibits a spin-stripe pattern, indicated by A-phase incollapse HGFA EBC D 1000 500 2.0 1.0 0 03.0 FIG. 1: (color online) Ground-state phase diagram of the spin-orbit coupled dipolar BEC with respect to gsocand gdd for g0= 4000, g1= 0, andλ= 0.2. There are eight different phasesmarkedbyA-H.Thewhiteregionrepresentsinstabili ty against dipolar collapse. Fig. 1. Typical density and spin distributions of such phase are shown in Fig. 2(a). In this phase, the spin texture on the x-yplane shows typical spin stripe struc- ture, which is almost unchanged along the z-axis. Actu- ally,previousstudiesontrappedspin-orbitcoupledBECs haveshown that the spin stripe structure is known as one of the ground states at strong SOC in harmonic poten- tial [15]. In the present system, this state also exists for a strong SOC, but with a weak DDI. With an increase in the strength of the DDI, B-phase emerges as the ground state, as shown in Fig. 1. Typical density and spin distributions of such phase are shown in Fig. 2(b). This phase is characterized by the checker- board lattice of spin vortices on the x-yplane, in which spin vortices with Fz>0 andFz<0 are alternately aligned. Such a pattern may be understood by the long- range nature of the DDI, which leads to a regular density distribution of each component. Similar to A-phase, the spin texture is almost independent of z. Increasing the DDI further, the spin vortex structures begin to have a zdependence, and C-phase emerges as the ground-state of the system, as the yellow-green re- gion in Fig. 1. Typical density and spin distributions of such phase are shown in Fig. 3(a). This phase has a spin-vortex train structure on the x-yplane, where com- ponents 1 and −1 are surrounded by component 0. The numbers of spin vortices with Fz>0 andFz<0 are equal to each other, which increase with SOC. Three- dimensional (3D) isodensity surfaces of the state are shown in Fig. 4, which indicates that the spin vortex structure depends on zdue to the DDI. WithafurtherincreaseintheDDI,C-phasetransforms to D-phase, as shown in Fig. 1. Its density and spin dis- tributions and 3D structures are shown in Figs. 3(b) and 5, respectively. Interestingly, the spin structure signifi-3 0.000 0.002 -0.002 0.002 0total spin (b) B-phase -2 0 2 -2 0 2 -2 0 2total spin (a) A-phase -2 0 2 -2 0 2 -2 0 2 FIG. 2: (color online) Typical density distribution and spi n texture of the system for (a) gdd= 0.0 and gsoc= 2.4 and (b) gdd= 100 and gsoc= 2.4, corresponding to the states represented in A and B phases in Fig. 1, respectively. The arrows in the spin texture represent the transverse spin vec tor (Fx,Fy) with background color representing Fz. cantly depends on zand form a helical structure, leading to twisted spin vortices. We note that this spin struc- ture, as well as C-phase, reflects the features of the SOC and DDI: multiple spin vortices are created by the SOC and they are twisted along the z-axis by the DDI. In the region D of Fig. 1, we also observe three twisted spin vortices for a larger SOC. Our numerical results show that the degree of torsion increases with DDI, while the separation between the spin-vortices decreases. In the limit of strong DDI, the separation almost disappears and E-phase emerges as the ground state, as shown in blue region in Fig. 1. E-phase is characterized by its axisymmetric density distribution of each component, where the central region is occupied by component 1 and outer regions by com- ponents 0 and −1, as shown in Fig. 6(a). Components 0 and−1 have vorticities ±1 and±2, respectively. The spin texture in the x-yplane has a single spin vortex at the center, which is similar to the chiral spin-vortex state [29, 30]. This phase exists for strong DDI or weak SOC, and occupies the largest phase in the ground-state phase diagram in Fig. 1. 0.000 0.002 -0.002 0.002 0total spin (b) D-phase -2 0 2 -2 0 2 -2 0 2total spin (a) C-phase -2 0 2 -2 0 2 -2 0 2 FIG. 3: (color online) Typical density distribution and spi n texture of the system for (a) gdd= 250 and gsoc= 2.0 and (b) gdd= 300 and gsoc= 1.5, corresponding to C and D phases in Fig. 1, respectively. The arrows in the spin textur e represent the transverse spin vector ( Fx,Fy) with background color representing Fz. Finally, we move to another limit of weak SOC and strong DDI. In this region, there are two phases marked G and H in Fig. 1. The G-phase is shown in Figs. 6(b) and 7. This phase has a helical structure along the z- axis, in which component 0 are twined by the other two components, resulting in a double helix of Fz>0 and Fz<0. The state shown in Figs. 6(b) and 7 has not only the spin angular momentum but also the orbital angular momentum in the zdirection. For a small gsoc, +1 and−1 components are balanced and the spin and orbital angular momenta disappear. In the present case of g1= 0, this state is not the ground state for gsoc= 0, while it can be a stationary state. It is found in Ref. [33] that this state can be the ground state for g1<0 even without SOC. The density and spin distributions of H-phase are shown in Fig. 6(c). The spin texture of this state is sim- ilar to that of the polar-core vortex, i.e., |F|= 0 on the z-axis and the transverse spin vectors rotate around the core. However, this state is different from the polar-core vortex, in that the axisymmetry is broken in |ψ±1|2and Fz/negationslash= 0.4 02 4 FIG. 4: (color online) Isodensity surfaces of the three com- ponents |ψ1|2= 0.001 (red), |ψ0|2= 0.0007 (green), and |ψ−1|2= 0.001 (blue), corresponding to C-phase shown in Fig. 3(a). See the Supplemental Material for a movie showing the three-dimensional (3D) structure [40]. 02 4 FIG. 5: (color online) Isodensity surfaces of the three com- ponents |ψ1|2= 0.001 (red), |ψ0|2= 0.0007 (green), and |ψ−1|2= 0.0004 (blue), corresponding to D-phase shown in Fig. 3(b). See the Supplemental Material for a movie showing the 3D structure [40]. In all the phases demonstrated above, space- and time- reversed states of the ground states are also the ground states because of the symmetry of the Hamiltonian, that is, ifψm(r) is a ground state, ( −1)mψ∗ −m(−r) is also a ground state. The rotation about the z-axis also does not change the energy. A-, C-, F-, and H-phases have the space-time reversal symmetry and E-phase has the rotation symmetry about the z-axis. Thez-component of the orbital angular momentum, /angb∇acketleftLz/angb∇acket∇ight=/integraltext rΨ†(r)(xpy−ypx)Ψ(r)isnotableinconsidering our system. Figure 8 shows /angb∇acketleftLz/angb∇acket∇ightas a function of gsocfor gdd= 700 being fixed. The C- and H-phases scarcely have angular momentum, since there is the space-time reversal symmetry and /angb∇acketleftLz/angb∇acket∇ightis canceled between ψ1and ψ−1. The first rapid increase in /angb∇acketleftLz/angb∇acket∇ightoccurs in the G- phase(0.06<∼gsoc<∼0.17). TheE-phasealsohasnonzero /angb∇acketleftLz/angb∇acket∇ight, sinceψ0andψ−1have singly and doubly quantized vortices, respectively. In the D-phase, /angb∇acketleftLz/angb∇acket∇ightchanges at gsoc≃2.0, sincethenumberofspinvorticeschangesfrom 0.000 0.002 -0.002 0.002 0total spin (c) H-phase -2 0 2 -2 0 2 -2 0 2total spin (b) G-phase -2 0 2 -2 0 2 -2 0 2 total spin (a) E-phase -2 0 2 -2 0 2 -2 0 2 0.000 0.004 total FIG. 6: (color online) Typical density distribution and spi n texture of the system for (a) gdd= 300 and gsoc= 0.6, (b) gdd= 700 and gsoc= 0.1, and (c) gdd= 700 and gsoc= 0, corresponding to the states represented in E, G and H phases in Fig. 1, respectively. The arrows in the spin textur e represent the transverse spin vector ( Fx,Fy) with background color representing Fz. two to three. The maximum of /angb∇acketleftLz/angb∇acket∇ightis attained in the D-phase, and at gsoc≃2.4,/angb∇acketleftLz/angb∇acket∇ightdramatically decreases and the ground state transforms to the C-phase with an increase in SOC. We have also examined the cases of g1/negationslash= 0, and found that the B-E phases remained almost unchanged for|g1| ∼0.1g0; the phaseboundariesareslightlyshifted. Our main results are thus unchanged for finite g1.5 02 4 FIG. 7: (color online) Isodensity surfaces of the three com- ponents |ψ1|2= 0.001 (red), |ψ0|2= 0.001 (green), and |ψ−1|2= 0.001 (blue), corresponding to G-phase shown in Fig. 6. See the Supplemental Material for a movie showing the 3D structure [40]. 0 0.5 1.0 1.5 2.0 2.5 3.0 00.5 1.0 1.5 2.0 2.5 3.0 H G E D C FIG. 8: (color online) Orbital angular momentum /angbracketleftLz/angbracketrightas a function of gsocfor gdd= 700. The vertical lines separate the phases and the solid curve is guide to the eyes.IV. CONCLUSIONS We have investigated the ground-state structures of a spin-1 Bose-Einstein condensate with the 2D Rashba SOC and DDI, confined in a cigar-shaped trap potential. Due to the interplay between the 2D-like pattern for- mation by the Rashba SOC and the zdependence aris- ing from the long-range DDI, we found a rich variety of ground-state phases, including the twisted spin vor- tices. We systematically explored the parameter space and obtained the ground-state phase diagram as a func- tion of the strength of the SOC and DDI, which con- sists of eight different phases. For strong SOC and weak DDI, the stripe or plane-wavephase is obtained. Increas- ing the DDI, the spin-vortex lattice emerges (B-phase), which form square pattern due to the long-range nature of the DDI. In the opposite limit, i.e., for strong DDI and weak SOC, we have two symmetry broken states (G- and H-phases). The chiral spin-vortex state (E-phase) is the ground state for a wide parameter region. Between B- and E-phases, we found novel spin structures having both SOC and DDI features, which we call C-and D- phases. In the C-phase, bunches of spin vortices with opposite directions are twisted along the z-axis, and in the D-phase, a few spin vortices form helical structures. Both SOC and DDI couple the internal and external degrees of freedom in a BEC. Combining such effects, a wide variety of spin textures will be realized. Acknowledgments This work was supported by JSPS KAKENHI Grant Numbers JP16K05505,JP26400414,and JP25103007,by the key project fund of the CAS for the “Western Light” Talent Cultivation Plan under Grant No. 2012ZD02,and by the Youth Innovation Promotion Association of CAS under Grant No. 2015334. [1] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett.48, 1559 (1982). [2] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [3] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [4] X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). [5] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102, 130401 (2009). [6] Y.-J. Lin, R. L. Compton, K. Jim´ enez-Garc´ ıa, J. V. Porto, and I. B. Spielman, Nature (London) 462, 628 (2009). [7] Y.-J. Lin, R. L. Compton, K. Jim´ enez-Garc´ ıa, W. D. Phillips, J. V. Porto, and I. B. Spielman, Nat. Phys. 7,531 (2011). [8] J. Dalibard, F. Gerbier, Juzeli¯ unas, and P. ¨Ohberg, Rev. Mod. Phys. 83, 1523 (2011). [9] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Natu re (London) 471, 83 (2011). [10] C. J. Wu, I. Mondragon-Shem, and X. F. Zhou, Chin. Phys. Lett. 28, 097102 (2011). [11] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012); H. Zhai, Rep. Prog. Phys. 78, 026001 (2015). [12] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett.105, 160403 (2010) [13] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011). [14] X.-Q. Xu and J. H. Han, Phys. Rev. Lett. 107, 200401 (2011). [15] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107,6 270401 (2011). [16] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys. Rev. Lett. 108, 010402 (2012). [17] B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D. Drummond, and H. Hu, Phys. Rev. A 85, 023606 (2012). [18] T. Kawakami, T. Mizushima, M. Nitta, and K. Machida, Phys. Rev. Lett. 109, 015301 (2012). [19] Z. F. Xu, R. Lu, and L. You, Phys. Rev. A 83, 053602 (2011); Z. F. Xu, Y. Kawaguchi, L. You, and M. Ueda, Phys. Rev. A 86, 033628 (2012). [20] T. Kawakami, T. Mizushima, and K. Machida, Phys. Rev. A84, 011607 (2011). [21] S.-W. Su, I.-K. Liu, Y.-C. Tsai, W. M. Liu, and S.-C. Gou, Phys. Rev. A 86, 023601 (2012). [22] C.-F. Liu and W. M. Liu, Phys. Rev. A 86, 033602 (2012). [23] Z.Chen andH.Zhai, Phys.Rev.A 86, 041604(R) (2012). [24] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett. 94, 160401 (2005). [25] T. Lahaye, T. Koch, B. Frohlich, M. Fattori, J. Metz, A. Griesmaier, S.Giovanazzi, andT. Pfau, Nature(London) 448, 672 (2007). [26] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Phys. Rev. Lett. 107, 190401 (2011). [27] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F. Ferlaino, Phys. Rev. Lett. 108, 210401 (2012).[28] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett. 96, 080405 (2006). [29] S. Yi and H. Pu, Phys. Rev. Lett. 97, 020401 (2006). [30] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett. 97, 130404 (2006). [31] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett. 98, 110406 (2007). [32] K. Gawryluk, M. Brewczyk, K. Bongs, and M. Gajda, Phys. Rev. Lett. 99, 130401 (2007). [33] J. A. M. Huhtamaki and P. Kuopanportti, Phys. Rev. A 82, 053616 (2010) [34] Y. Eto, H. Saito, and T. Hirano, Phys. Rev. Lett. 112, 185301 (2014). [35] Y. Deng, J. Cheng, H. Jing, C.-P. Sun, and S. Yi, Phys. Rev. Lett. 108, 125301 (2012). [36] R. M. Wilson, B. M. Anderson, and C. W. Clark, Phys. Rev. Lett. 111, 185303 (2013). [37] S. Gopalakrishnan, I. Martin, and E. A. Demler, Phys. Rev. Lett. 111, 185304 (2013). [38] T. Lahaye, J. Metz, B. Fr¨ ohlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. Lett. 101, 080401 (2008). [39] T. Koch, T. Lahaye, J. Metz, B. Frohlich, A. Griesmaier, and T. Pfau, Nat. Phys. 4, 218 (2008). [40] See Supplemental Material at http:... for movies showi ng the three-dimensional structures of the ground states.

1907.00974v2.Spin_dynamics_of_moving_bodies_in_rotating_black_hole_spacetimes.pdf

Spin dynamics of moving bodies in rotating black hole spacetimes Balázs Mikóczi1;yand Zoltán Keresztes2;z 1Research Institute for Particle and Nuclear Physics, Wigner RCP H-1525 Budapest 114, P.O. Box 49, Hungary 2Department of Theoretical Physics, University of Szeged, Tisza Lajos krt. 84-86, Szeged 6720, Hungary yE-mail: mikoczi.balazs@wigner.huzE-mail: zkeresztes@titan.physx.u-szeged.hu The dynamics of spinning test bodies, moving in rotating black hole (Kerr, Bardeen-like and Hayward-like) spacetimes, are investigated. In Kerr spacetime, all the spherical, zoom-whirl and unbound orbits are considered numerically. Along spherical orbits and for high spin, an amplitude modulation is found in the harmonic evolution of the spin precessional angular velocity, caused by the spin-curvature coupling. Along the discussed zoom-whirl and unbound orbits, the test body approachesthecentersomuchthatitpassesthroughtheergosphere. Nearandinsidetheergosphere, the variation of the spin direction can be very rapid. The effects of the spin-curvature coupling is also investigated. The initial values are chosen such a way, that the body and its spin move in the equatorial plane of the coordinate space and of the comoving frame, respectively. Hence, a clear effect of the spin-curvature coupling is observed as the orbit and the spin vector leave the equatorial plane. Additional effects in the spin precessional angular velocity and in the evolution of the Boyer-Lindquist coordinate components of the spin vector is also considered. Finally, in case of different regular black holes, the spin-curvature coupling influences differently the orbit and the spin evolutions. Keywords: black hole physics, spinning test particles, Mathisson-Papapetrou-Dixon equations, Kerr space- time I. INTRODUCTION Both the orbital and the spin dynamics of compact bi- nary systems have a renewed interest. All observed grav- itational waves originated from compact binary systems composed of black holes or neutron stars ([1–8]). In two cases the spin of the merging black holes was identified with high significance [2, 8, 9]. In addition, in a binary system the dominant supermassive black hole spin pre- cession was identified from VLBI radio data spanning over 18 years [10]. In the post-Newtonian (PN) approximation the lowest order spin contributions to the dynamics come from the spin-orbit, spin-spin and quadrupole-monopole interac- tions [11–14]. The spin effects on the orbit leaded to set up generalized Kepler equations [15–18]. The analytical description of the secular spin dynamics for black holes is given in Refs. [19] and [20]. Based on the PN description several interesting spin related behaviours were identified in compact binary systems, like transitional precession [21], equilibrium configurations [22], spin-flip [23], spin flip-flop [24] and wide precession [25]. The Mathisson–Papapetrou–Dixon (MPD) equations [26–30] describe the dynamics of binaries with signif- icantly different masses more accurately than the PN approximation in the strong gravitational field regime, where the PN parameter is not small. The black hole binary systems with small mass ratio are among the most promising sources for gravitational waves in the fre- quency sensitivity range of the planned LISA - Laser In- terferometer Space Antenna [31, 32]. In addition, near the central supermassive black holes in the galaxies many stellar black holes are expected to exist [33–35]. The MPD equations are not closed, a spin supplemen-tary condition1(SSC) is necessary to choose [28, 37–44], which defines the point at which the four-momentum and the spin are evaluated. The Hamiltonian formulations in different SSCs are discussed in Refs. [45–49]. A non- spinning body follows a geodesic trajectory, while a spin- ning one does not [50, 51]. Spinning bodies governed by the MPD equations were already studied on Kerr back- ground. Circular orbits in the equatorial plane can be unstable not only in the radial direction but also in the perpendiculardirectiontotheequatorialplaneduetothe spin [52]. The spin-curvature effect strengthens with spin and with non-homogeneity of the background field [51]. The MPD equations admit many chaotic solutions, how- ever, these do not occur in the case of extreme mass ra- tiobinaryblackholesystems[53–56]. Analyticstudieson the deviation of the orbits from geodesics due to the pres- enceofasmallspinarepresentedinRefs. [57–59]. Highly relativistic circular orbits in the equatorial plane occur in much wider space region for a spinning body than for a non-spinning one [60]. Spin-flip-effects may occur when the magnetic type components of quadrupole tensor are non-negligible [61]. Corrections due to the electric type components of quadrupole tensor to the location of in- nermost stable circular orbit in the equatorial plane and to the associated motion’s frequency were determined in Ref. [62]. An exact expression for the periastron shift of a spinning test body moving in the equatorial plane is derived in Ref. [63]. The influences of the affine param- eter choice on the constants of motion in different SSC 1Both the PN dynamics with spin-orbit coupling and the gravita- tional multipole moments depend on the SSC [13, 36].arXiv:1907.00974v2 [gr-qc] 9 Feb 20222 were also considered [64]. Frequency domain analysis of motion and spin precession was presented in Ref. [65]. The evolution of spinning test particles were investigated in the space-time [66], in non-asymptotically flat space- times [67], and in wormholes [68]. Considering geodesic trajectories, the periastron ad- vance can become such significant in the strong gravita- tional field regime that the test particle follows a zoom- whirl orbit[69–71]. For non-spinningparticles, thetopol- ogy of these orbits was encoded by a rational number [72, 73]. Numerical relativity confirmed the existence of zoom-whirl orbits [74–79], and they also occur in the 3 PN dynamics with spin-orbit interaction [80, 81]. Here we will present zoom-whirl orbits occurring in the MPD dynamics for the first time. In addition, these orbits pass over the ergosphere where the PN approximation fails.2 Hyperbolic orbits of spinning bodies were analytically studied in both the PN [82] and the MPD [83] dynamics. Analytic computations in Ref. [83] were carried out for small spin magnitudes, when the spin is parallel to the central black hole rotation axis and the body moves in the equatorial plane. In this configuration both the spin magnitude and direction are conserved, but they have non-negligible influences on the orbit. In our numerical consideration the spin is not parallel to the black hole rotation axis. As a consequence, the body’s orbit is not confined to the equatorial plane and the spin direction evolves. In addition, the closest approach distance is in- side the ergosphere where the PN approximation cannot be used. Our investigations are not only applied in the Kerr spacetime but also in regular black hole backgrounds. The first spacetime containing a nonrotating regular black hole was suggested by Bardeen [84]. This metric was interpreted as the spacetime surrounding a magnetic monopole occurring in a nonlinear electrodynamics [85]. Another nonrotating regular black hole was introduced by Hayward [86] having similar interpretation [87]. The spacetime family containing the Bardeen and Hayward cases was generalized for rotating black holes [88] which we will use here3. In this paper, we investigate the orbit and spin evo- lutions of bodies moving in Kerr, Bardeen-like and Hayward-like spacetimes and governed by the MPD equations with Frenkel–Mathisson–Pirani (FMP) and Tulczyjew–Dixon (TD) SSCs. When the covariant derivatives of the spin tensor and the four-momentum alongtheintegralcurveofthecentroiddeterminedbythe 2At the ergosphere the value of the PN expansion parameter is typically about 1=2. 3There are discussions (see Refs. [89–91]) on that the rotating regular black hole spacetimes given in Ref. [88] are not exact solutions of the field equations. However the spacetime family given analytically differs only perturbatively from the exact so- lution [91], therefore it is suitable for consideration of spinning bodies evolutions.SSC are small, this system reduces to a geodesic equation with parallel transported spin discussed in Ref. [92]. In thissensethepresentarticlecanbeconsideredasthegen- eralizationofRef. [92]withnon-negligiblespin-curvature correctionscausingthatthecentroidorbitisnon-geodesic andthespinisnotparalleltransported. AsBini,Geralico and Jantzen pointed out that the spin dynamics can be described suitably in the comoving Cartesian-like frame obtained by boosting the Cartesian-like frame associated to the family of static observers (SOs). This is because SOsdonotmovewithrespecttothedistantstars. Hence, the Cartesian-like axes locked to SOs define good refer- ence directions to which the variation of the spin vector can be compared. Here, we derive the spin evolution equation in the comoving Cartesian-like frame based on the MPD system. However, SO does not exist inside the ergosphere of the rotating black hole, and thus its frame cannot be used for description of the dynamics when the spinning body passes over this region. Therefore, the spin dynamics in a Cartesian-like frame obtained by an instantaneous Lorentz-boost from the frame associated to the zero angular momentum observer (ZAMO) is also presented, which can be used inside the ergosphere. The boosted SO and ZAMO frames relate to each other by a spatial rotation outside the ergosphere. The rotation angle between these boosted frames is unsignificant far from the rotating black hole. In Section II, the MPD equations, the spin supple- mentary conditions, the rotating (Kerr, Bardeen-like and Hayward-like) black hole spacetimes and the frames as- sociated with the families of SOs and ZAMOs are intro- duced. In Section III the representations of spin evo- lution are given. For this purpose, we introduce two framesbyinstantaneousLorentzboostsofSOandZAMO frames, which comoves with an observer having an arbi- trary four velocity U. The relation between the boosted frames is discussed (additional expressions are given in Appendix VII). We use the TD SSC, and Umeans ei- ther the centroid or the zero 3-momentum observer four velocity. The spin evolution equation is derived in these U-frames. First, the spin precession is described with respect to the boosted spherical coordinate triad associ- ated with either the SOs or ZAMOs. Then, we intro- duce Cartesian-like triads in the rest spaces of SOs and ZAMOs. The spin precession with respect to the corre- sponding boosted Cartesian-like frames is also derived. The relations between the spin angular velocities in the boosted SO and ZAMO frames are discussed. In Section IV, we apply the derived spin equations for numeric in- vestigations when the body moves along spherical-like, zoom-whirl and unbound orbits. In Subsection IVA the background is the Kerr spacetime, while in Subsection IVB, it is one of the rotating regular black hole space- times. In Appendix VIII, the avoidance of paradoxical behaviour of the MPD equations is checked. Finally, Sec- tion V contains the conclusions. We use the signature + ++, and units where c= G= 1, with speed of light cand gravitational constant3 G. The bold small Greek indices with or without prime take values 1,2and3, while the bold capital and the small Latin indices 0,1,2and3. In addition, the fol- lowing small bold Latin indices i,j,kandi0,k0take values fromfx;y;zg. Finally, the bold indices are frame indices, while the non-bold indices are spacetime coordi- nate indices. II. EQUATIONS OF MOTION FOR SPINNING BODIES IN ROTATING BLACK HOLE SPACETIMES A. MPD equations and SSC In the pole-dipole approximation, the motion of an ex- tended spinning body in curved spacetime is governed by the MPD equations [26–30] which read as Dpa ducrcpa=Fa; (1) DSab ducrcSab=paubuapb; (2) with Fa=1 2Ra bcdubScd: (3) Herercis the covariant derivative, paandSabare the four-momentum and the spin tensor of the moving body, respectively, and Ra bcdis the Riemann tensor. Finally, ua=dxa=dis the four-velocity of the representative point for the extended body at spacetime coordinate xa()with an affine parameter . Note that higher mul- tipoles of the body should occur in the MDP equations when they are nonvanishing. Here they are taken to be zero. Choosing the affine parameter as the proper time [48, 93]uaua=14, Equation (2) can be written as pa=muaubDSab d; (4) wherem=uapais the mass in the rest frame of the observer moving with velocity ua. Equation (4) shows that the momentum paand the kinematic four velocity uaare not proportional to each other for a spinning body in general. We note that if the covariant derivatives of the spin tensor and the four-momentum along the integral curve ofuaare small, i.e. the right hand sides of Equations (1) and (2) are negligible, pabecomes proportional to ua 4Below we derive a condition for the spin magnitude in TD SSC when the proper time parametrization has a sense.which satisfies the geodesic equation because mis a con- stant. Then introducing a spin four-vector perpendicular toua(see Equation (2.5) of Ref. [94]), it will be parallel transported along the trajectory. The geodesic equations with parallel transported spin vector was investigated in Ref. [92]. Ingeneral, inordertoclosetheMPDequationsanSSC is necessary to choose, which defines the representative point of the extended body referred as the center of mass or the centroid. There are some proposed SSC, namely the Frenkel-Mathisson-Pirani [26, 37, 38], the Newton- Wigner-Pryce [40, 41], the Corinaldesi-Papapetrou [27, 39], and the Tulczyjew-Dixon [28, 42]. We will apply the Tulczyjew-Dixon SSC imposing that paSab= 0: (5) This SSC yields two constants of motion, the spin mag- nitudeS2=SabSab=2and the dynamical mass M=ppapa(see Ref. [51]). In addition, the TD SSC to- gether with the MPD equations results in the following velocity-momentum relation [51, 95, 96]: ub=m M2 pb+4S2 vb ; (6) with vb=SbaRaecdpeScd 2S2; (7) and = 4M2+ 2RS2; (8) whereR=RaecdSaeScd=2S2. Sincepbandubare not parallels,ubmay become spacelike from timelike along an integral curve. Where the causal character of a curve is changed, it is known as superluminal bound and has been discussed in different cases (e.g. Refs.: [51, 66–68]). The superluminal motion has no physical meaning, and the timelike condition for ubyields a bound for the spin magnitude as S2<2M3 2vRM; (9) wherev=pvava5. When the spin magnitude obeys this constraint, the proper time parametrization has sense and the normalization ubub=1gives a relation m2=m2 pa;Sbc
as m2=M4  M216S4 2v2: (10) 5Note thatvandRdo not carry information on the spin mag- nitude since Sab=p 2Shas unit norm.4 Since the relation (6) can be inverted [97], both initial data sets xa;pa;Sab jinand xa;m;ua;Sab jinpro- vide a unique solution of the MPD equations with TD SSC. Thespinvectorbeingperpendicularto paisintroduced as Sa=1 2MabcdpbScd: (11) Since SaSab= 0 =Sapa= 0; (12) the contraction of Equation (6) with Sbresults inSbub= 0. Finally, the covariant derivative of Saalong the world- line of the centroid is DSa d=SbFb M2pa: (13) IfFais negligible, Sais parallel transported along the worldline of the centroid, and the centroid moves along a geodesic curve. The latter can be shown from the MPD equations together with (4) and (13). Finally, we mention that the MPD equations are valid only for test particles whose backreaction to the back- ground spacetime curvature are negligible. Hence, when the spinning body is moving in a spacetime around a black hole with a mass parameter , the dimensionless spin magnitude S=Mmust be small [53, 98]. This is consistent with the constraint (9), which becomes for the dimensionless spin magnitude as S M2 <2 2vRMM ; (14) where the mass ratio M=gives a small factor. B. Rotating black hole spacetimes The line element squared describing the considered ro- tating black hole spacetimes in Boyer-Lindquist coordi- nates reads as [88, 99] ds2=
a2sin2 dt22aBsin2 dtd +
dr2+ d2+A sin2d2;(15) with  =r2+a2cos2 ;
=r2+a22 [+(r)]r ; B=r2+a2
; A= r2+a2
2
a2sin2: (16) In the Kerr spacetime (r)vanishes and andade- note the mass and rotation parameters, respectively. Thefunction(r)occurs when a non-linear electromagnetic field is present. It is given by (r) =emr (r+qm) =; (17) whereem=q3 m=is the electromagnetically induced ADM mass. Here controls the strength of nonlinear electrodynamic field and carries the dimension of length squared,qmis related to the magnetic charge (see Ref. [87]), and the powers are ( = 3,= 2) for the Bardeen- like and ( = 3,= 3) for the Hayward-like spacetimes. The stationary limit surfaces and the event horizon (if they exist) are determined by the solutions of equations gtt=
a2sin2= 0andgrr=
= 0 , respectively. The structure of the spacetime depends on the number of real, positive solutions of these equations. For the Kerr spacetime em= 0, then there are two stationary limit surfaces and event horizons for a= < 1. The re- gion which is located outside the outer event horizon but inside the outer stationary limit surface is called ergo- sphere. The spacetime is free from the singularity for = 0and 3. The first and the second panels of Figure 3 in Ref. [88] indicate the regions in the param- eter space of aandq=qm=emfor the Bardeen and the Hayward subcases, respectively, where the above line element squared describes a regular black hole. In the spacetimes having symmetries, constants of mo- tion associated to each Killing vector a(which obeys the Killing equation r(ab)= 0) emerge [29]. Since the rotating black hole spacetimes have a timelike @tand a spatial@Killing vectors due to the staticity and axial symmetry, there are two constants of motion [53]: E=pt1 2Sab@agbt; Jz=p+1 2Sab@agb: (18) At spatial infinity Emeans the energy of the spinning body andJzis the projection of the total momentum to thesymmetryaxis. Theseconstantsareusedforchecking the numerical accuracy. 1. Static and zero angular momentum observers The worldlines of static observers are the integral curves of the Killing vector field @t. This family of ob- servers exists outside the ergosphere, where their frame is given by e0=u(SO)=1pgtt@t; e1=r
@r; e2=@p ; e3=1p
aBsin pgtt@tpgtt sin@ : (19) The dual basis is obtained as eA a=gabABeb B, where AB=diag(1;1;1;1).5 The orbit of a zero angular momentum observer is or- thogonal to the t=const. hypersurfaces [100, 101]. The four velocity along this orbit is u(ZAMO )=r A 
 @t+aB A@ ;(20) which corresponds to the 1-form: dt=p gtt. In con- trasttotheSOs, thisfamilyofobserversalsoexistsinside the ergosphere but outside the outer event horizon. The frame of the ZAMOs is given by f0=u(ZAMO ); f1=r
@r; f2=@p ; f3=r  A@ sin; (21) with dual basis: fA a=gabABfb B. III. REPRESENTATIONS OF SPIN EVOLUTION The spin vector (11) will be considered in both co- moving and zero 3-momentum frames. The definitions of comoving and zero 3-momentum observers will be in- troduced in the next subsection. Then the spin evolution equations will be derived using the boosted spatial spher- ical and Cartesian-like triads. A. Comoving and zero 3-momentum frames In the TD SSC, the center of mass is unique and mea- sured in the zero 3-momentum frame with four velocity pa=M. Ontheotherhandthefourvelocityofthecentroid isua. The comoving indicative will refer to that observer which moves along the centroid worldline. The spin dy- namics will be described in both the zero 3-momentum and the comoving observer’s frames. The velocity of the chosen observer will be denoted by U. The comoving and zero 3-momentum observers’ frames will be set up from the frames of the static and the zero angular momentum observers by an instantaneous Lorentz-boost knowing U numerically. The comoving and zero 3-momentum frames (hereafter unanimously referred as U-frame) obtained from the SO frame are given by E0(e;U)U= (S) e0+v(S)
; E(e;U) =e+U
e 1 + (S) U+u(SO)
:(22) Here=f1;2;3g,v(S)= 1 (S)Uu(SO)is the rela- tive spatial velocity of either the comoving or the zero 3-momentum observer with respect to the SO frame,which is perpendicular to e0, and the Lorentz factor is (S)=U
u(SO). The dot denotes the inner product with respect to the metric gab. The inverse transforma- tion is given by e0= (S) E0(e;U) +w(S)
; e=E(e;U) +u(SO)
E(e;U) 1 + (S) U+u(SO)
;(23) where w(S)=w (S)E(e;U) = 1 (S)u(SO)U;(24) is the relative spatial velocity of the static observer with respect to the U-frame. The corresponding Lorentz-boost from the ZAMO frame reads as E0(f;U)U= (Z) f0+v(Z)
; E(f;U) =f+U
f 1 + (Z) U+u(ZAMO )
;(25) with relative spatial velocity v(Z)= 1 (Z)Uu(ZAMO ) of theU-frame with respect to the ZAMO frame, and Lorentz factor: (Z)=U
u(ZAMO ). The inverse boost transformation is given by f0= (Z) E0(f;U) +w(Z)
; f=E(f;U) +u(ZAMO )
E(f;U) 1 + (Z) U+u(ZAMO )
; (26) where w(Z)=w (Z)E(f;U) = 1 (Z)u(ZAMO )U;(27) is the relative spatial velocity of the ZAMO with respect to either the comoving or the zero 3-momentum frame. SinceE0(e;U) =U=E0(f;U), the transformation between the frames EA(e;U)andEA(f;U)is a rota- tion in the rest space of either the comoving or the zero 3-momentum observer. The rotation axis has the follow- ing non-zero components in both the EA(e;U)and the EA(f;U)frames: n1=w2 (Z)q w1 (Z)
2+ w2 (Z)
2 =w2 (S)q w1 (S)
2+ w2 (S)
2; (28) and n2=w1 (Z)q w1 (Z)
2+ w2 (Z)
2 =w1 (S)q w1 (S)
2+ w2 (S)
2: (29)6 The rotation angle is determined by sin  =" 1s 
gttA! (Z)w3 (Z) 1 + (Z)+aBsinpgttA# (Z)r w1 (Z)2 + w2 (Z)2 1 + (S) =" 1s 
gttA! (S)w3 (S) 1 + (S)aBsinpgttA# (S)r w1 (S)2 + w2 (S)2 1 + (Z); (30) and cos 1 1q 
gttA=2 (Z) w1 (Z)2 + w2 (Z)2 1 + (S)
1 + (Z)
=2 (S) w1 (S)2 + w2 (S)2 1 + (S)
1 + (Z)
:(31) The frame E(e;U)(E(f;U)) is obtained from E(f;U)(E(e;U))byarotationwiththeangle () about the axis n. The rotation angle exists outside the ergosphere where the terms under the square roots in Equations(30)and(31)arepositive. Thetransformation betweenE(e;U)andE(f;U)in another form is given in Appendix VII. The above transformation is a special caseoftheWigner-rotation[102], whichwasdiscussedre- cently in Ref. [103]. However explicit expressions for the rotation between the frames which we denote E(f;U) andE(e;U)were not presented in [103]. B. MPD spin equations in comoving and zero 3-momentum frames Weinvestigatetwocasesrelatedtothechosen U-frame: i)Ua=pa=Mwhen we work in the zero 3-momentum frame; and ii)Ua=uawhich is the four velocity of the center of mass measured in the zero 3-momentum frame. In all cases the spin vector can be expanded as S=SE; (32) sinceS0= 0. Here, the spatial frame vector Ein the U-frame denotes either E(e;U)orE(f;U), which are obtained by boosting the SO and ZAMO frames, respec- tively. The covariant derivative of the spin vector along the integral curve of uis DS d=dS dE+SDE d: (33)Since the frame vectors are perpendicular to each other, we have EA
DEB d=EB
DEA d; (34) forA6=B, and because of the normalization: EA
DEA d= 0: (35) Therefore the covariant derivatives of the spatial frame vectors along the integral curve of ucan be expressed as DE d= E0
DE d E0"  E ;(36) where" Levi-Civita symbol in the 3-dimensional Eu- clidean space, whose frame indices are raised and lowered by the 3-dimensional Kronecker , and the frame compo- nents of the angular velocity are =1 2" E
DE d: (37) Due to Equation (34), the first term6in (36) can be writ- ten as E0
DE d=E
a; (38) where adenotes the acceleration a=DE0=d. Now the spin Equation (33) becomes as DS d=dS d+"  S  E+ (S
a)E0:(39) Finally, we take into account the spin Equation (13). When considering the spin evolution in the zero 3- momentum frame, we find the following equation for the spin: dS d+"  S = 0: (40) The second case when considering the evolution of S in the comoving frame, requires somewhat longer com- putation. Equations (13) and (39) results in dS d+"  S  E+= 0;(41) with = (S
a)uA(S
F)pA M2 EA:(42) 6Notethatthistermvanisheswhenthefirstorderspincorrections, i.e. the right hand sides of Equations (1) and (2), are neglected (see Ref. [92]).7 Using Equations (6) and (12), a straightforward compu- tation shows that u
= 0. Therefore, can be ex- panded as = E. On the other hand is perpen- dicular toS, hence, we can introduce a vector !, whose frame components obey the relation "  !S =: (43) The vector !is determined ambiguously since its frame component parallel with Svanishes in the cross product. As a natural choice, we choose !to be perpendicular to S. Using the definition (43), Equation (41) reads as dS d+"  +!
S = 0: (44) The Equations (40) and (44) can be considered in ei- ther theE(e;U)or theE(f;U)frame. Introducing the notations k=fe;fg; = (S);(Z) ; (45) the angular velocity components (k;U)can be deter- mined by using E(k;U)
DE(k;U) d =k
Dk d +1 1 + [(U
k)k(U
k)k]
Dk0 d +1 1 + [(U
k)k(U
k)k]
DU d;(46) where6=. This can be computed once Uis deter- mined.7 When both SO and ZAMO frames exist, a rotation about the axis ndefined by Equations (28) and (29) [see also Appendix VII for the explicit expressions] relates E(f;U)toE0(e;U)which can be written as E(e;U) =R0 E0(f;U): (47) HereR0 denotes the components of the corresponding rotation matrix. From the definitions of (e;U)and 0(f;U), the following relation between them can be derived: R0  (e;U) = 0(f;U) +R0   (R):(48) Here we have introduced (R)as R1
 0dR0  d="  (R); (49) 7We note that when the right hand sides of Equations (1) and (2) are neglected, the centroid moves along a geodesic, thus !and the last term in (46) vanish. The four-velocity Uis determined from the geodesic equation, and for k=e, we obtain the same system which was investigated in Ref. [92].which is the angular velocity of rotation between the frame bases along the body’s trajectory. 1. Cartesian-like triads and the characterization of spin evolution The evolution of the spin vector can be illustrated suitably by comparison its direction with Cartesian axes which are fixed with respect to the distant stars. The static observers are those fiducial observers, whose frame doesnotmovewithrespecttotheblackhole’sasymptotic frame [104]. A static observer sees the same “nonrotat- ing” sky during the evolution. In this sense the static observers are preferred fiducial observers in the investiga- tion of spin dynamics. Following Ref. [92], we introduce a spatial Cartesian-like triad ex,eyandezin the local rest space of the static observer as e=Ri ei, where =f1;2;3g,i=fx;y;zgandRis the same rotation matrix, which relates the Cartesian and spherical coordi- nates in the 3-dimensional Euclidean space (see Equation (85) of [92]). Since the rotation Rand the boost can be interchanged, we have E(e;U) =Ri Ei(e;U). The family of static observers only determines a frame outside the ergosphere. Therefore, we introduce another Cartesian-like triad fx,fyandfzin the local rest space of ZAMO for representation of the spin evolution inside the ergosphere8asf=Ri fi. Then the boost transfor- mation results in E(f;U) =Ri Ei(f;U). The Cartesian-like triad components of the spin vector in both the boosted SO and ZAMO frames are obtained as Si=Ri S; (50) which obeys the following equation of motion: dSi d=Ri "   (prec)S : (51) Here the precession angular velocity is9  (prec)=  (p)+!; (52) with  (p)=  (orb)+ ; (53) where  (orb)defined [see also Ref. [92]] as R1
 jdRj  d="  (orb); (54) 8Note that the frame associated to the ZAMO moves with respect to the distant stars. 9Note that that the expression of  (prec )reduces to -1 times that of Ref. [92] for = 0.8 and= 0in the zero 3-momentum frame, while = 1 in the comoving frame. The angular velocity  (prec)de- scribes the spin precession in the Cartesian-like frame. The Cartesian-like triad components of  (prec)are ob- tained from Equation (50) with notation change S! (prec). The quantity (p)can also be expressed in terms of the inner product of the Cartesian-like triad vectors Ei and their derivatives along the considered worldline as i (p)Ri   (p)=1 2"ijkEj
DEk d:(55) This expression is analogous with Equation (37). The angular velocities i (p)(e;U)and i0 (p)(f;U)defined in terms ofEi(e;U)andEi0(f;U), respectively, are related by Tk0 i i (p)(e;U) +Sk0   (orb)(e;U) = k0 (p)(f;U) +Rk0 (f)0 0 (orb)(f;U) +Sk0   (R);(56) with Tk0 i R1 (e) iR0 Rk0 (f)0; Sk0 Rk0 (f)0R0 :(57) Noting that !in Equation (43) transforms as a vec- tor for real rotation R0 . The transformation rules for  (prec)and i (prec)follow from the definitions (52) and (56). IV. NUMERICAL INVESTIGATIONS The orbit of the spinning body will be represented in the coordinate space: x=rcossin; y=rsinsin; z=rcos:(58) We characterize the instantaneous plane of the motion in the (x,y,z)-space by the unit vector: l=RV jRVj; (59) whereis the cross product in Euclidean 3-space, R is the position vector with components Rx=x,Ry= y,Rz=z, and Vis a spatial velocity vector with10 Vx=dx d; Vy=dy d; Vz=dz d: (60)The absolute value in the denominator denotes the “Eu- clidean length” of the numerator. Since the considered spacetimes are asymptotically flat, the quantity licoin- cides with the direction of the orbital angular momen- tum11at spatial infinity. The initial data for the spin vector will be character- ized by its magnitude and two angles in the boosted SO Cartesian-like frame as S=SiEi(e;u); (61) with Si=jSj cos(S)sin(S);sin(S)sin(S);cos(S) : (62) Since we use dimensionless quantities during the nu- mericalinvestigation, theparameters ,a,mandMonly appear through the ratios a=andm=M. We choose the initial data set in the TD SSC as pa (TD)=MandSa=M (by Equation (62)), then the initial spin tensor is derived from the inverse of (11), while m(0)=M(0)and the four velocityua (TD)of the centroid from (6). The SSC choice determining the representative world- line of a spinning test body corresponds to a gauge choice in an action approach [113]. In the following we will con- sider the evolution of the spin precessional angular ve- locity and will check its dependence on the SSC choice. For the numerical comparison, we will use the Frenkel– Mathisson–Pirani (FMP) SSC which imposes uaSab= 0. The definition of the spin vector is sa=abcdubScd=2, which is Fermi-Walker transported along the worldline of the centroid making the FMP SSC preferred from mathematical point of view [114–116]. Its frame com- ponents obey the same precessional equation in the co- moving frame like the TD spin vector Sain the zero 3-momentum frame (13). In the FMP SSC, there ex- ists also a velocity-momentum relation, Equation (19) of Ref. [97], like in the TD SSC. Hence the initial data set xa;pa;Sab jinprovides a unique solution of the MPD equation with FMP SSC. However, we must men- tion that, this velocity-momentum relation does not au- tomatically ensure that uaSab= 0for arbitrary paand Sab. In order to ensure this, we have a constraint be- tween the four momentum and the spin tensor emerging from the contraction of this equation with Sab. In ad- dition, the data set xa;pa;Sab jincannot be inverted for the set xa;m;ua;Sab jinlike in the TD SSC. One needs the data set xa;m;ua;Sab;aa jinto fix the tra- jectory. For a set xa;m;ua;Sab jin, we can obtain a non-helical and infinite number of helical trajectories for9 Ω1(prec)(e,u) Ω2(prec)(e,u) Ω3(prec)(e,u) 50 100 150 200τ/μ -0.010-0.0050.0050.0100.0150.020[1/μ] Ω1(prec)(e,u) Ω2(prec)(e,u) Ω3(prec)(e,u) 50 100 150 200τ/μ -0.010-0.0050.0050.0100.0150.020[1/μ] Ω1(prec)(e,u) Ω2(prec)(e,u) Ω3(prec)(e,u) 50 100 150 200τ/μ -0.010-0.0050.0050.0100.0150.020[1/μ] Ω1(prec)(e,u) Ω2(prec)(e,u) Ω3(prec)(e,u) 5001000 1500 2000 2500τ/μ -0.010-0.0050.0050.0100.0150.020[1/μ] Ω1(prec)(e,u) Ω2(prec)(e,u) Ω3(prec)(e,u) 5001000 1500 2000 2500τ/μ -0.010-0.0050.0050.0100.0150.020[1/μ] Ω1(prec)(e,u) Ω2(prec)(e,u) Ω3(prec)(e,u) 5001000 1500 2000 2500τ/μ -0.010-0.0050.0050.0100.0150.020[1/μ] 0500 1000 1500 2000 2500τ /μ0.0010.0020.0030.004Sin (Θ) 0500 1000 1500 2000 2500τ /μ0.0010.0020.0030.004Sin (Θ) 0500 1000 1500 2000 2500τ /μ0.0010.0020.0030.004Sin (Θ) Figure 1: (color online). The evolution of spinning body moving on spherical-like orbits around the Kerr black hole with a= 0:5. From left to right the magnitude of the body’s spin increases as jSj=M = 0:01,0:1and0:9. The rows represent the following: 1. the orbit in coordinate space (x=,y=,z=) (the ergosphere of the central black hole is marked by blue and the initial and the final positions of the spinning body are denoted by green and red dots, respectively,), 2. the instantaneous orbital plane orientation li(initial and final directions are marked by purple and black arrows, respectively), 3. unit spin vector in the boosted SO comoving Cartesian-like frame Ei(e;u)(initial and final spin directions are marked by green and blue arrows, respectively), 4. and 5.  (prec )(e;u)on shorter and longer timescales, respectively, 6. sin . The initial place of the body isr(0) = 8,(0) ==2and(0) = 0. The direction of the initial spin vector is given by (S)(0) ==2and(S)(0) = 0in the boosted SO frame (resulting in Sr(0)=jSj= 0:8682,S(0)=jSj= 0andS(0)=jSj= 0in Boyer-Lindquist coordinates). The four momentum pa (TD)=M is chosen for the TD SSC as pr (TD)(0)=M(0) = 0,p (TD)(0)=M(0) = 0:0442andp (TD)(0)=M(0) =0:0316. The initial centroid four velocity ua (TD)is determined from Equation (6).10 differentaa. In principle all worldlines where the condi- tionsuaua=1,uaSa= 0andpaSa= 0are satisfied can be used for representation of the moving body. Since the tangent vector of the centroid orbit occurs in the spin precessional equation through EAand their derivatives, the spin axis may describe very complicated motion in such observer’s frame, which follows a helical trajectory. In order to characterize the self rotation of the body in the easiest way possible, the helical trajectories should be avoided. However, there is no generic rule for deter- mination of the non-helical trajectory. According to the Authors’ knowledge, the best ansatz is suggested in Ref. [115] as taking pa=mua+SabFb m: (63) In this case aa/Fa=mat leading order in spin, which is plausible for a non-helical trajectory since aa/O S1
for the helical ones. However, the ansatz (63) cannot be imposed as a constraint for the dynamics with signifi- cant spin magnitude in the consideration. We require the ansatz (63) for setting initial conditions in the numeri- cal investigations. This is not forbidden because (63) is consistent with the algebraic velocity-momentum equa- tion. The corresponding initial data set in the FMP SSC are chosen by identifying the initial centroid four velocity and spin vector as ua (FMP )=ua (TD)andsa=m=Sa=M. Then the initial spin tensor and pa (FMP )=mare computed fromSab=ab cducsdand(63),respectively. Bringingfor- ward the result of the SSC dependence, we have found that the evolutions of the spin vectors defined in the TD and the FMP SSCs are barely distinguishable from each other in all cases. The differences in the evolutions of the different considered quantities considered in the sub- sequent subsections remains below 1%. This is in agree- ment with result of Ref. [117], where the evolution of test bodies moving on circular equatorial orbits around a Schwarzschild black hole were investigated. A. Spinning bodies moving in the Kerr spacetime In this subsection, we set em= 0anda= < 1, i.e. the background is a Kerr black hole’s spacetime. Figure 1 shows spherical-like orbits. The initial values are listed in the caption. The orbits, the black curves in the upper row, are shown in the coordinate space ( x=,y=,z=) defined in Equation (58). The initial and the final po- sitions of the body are marked by green and red dots, respectively. The initial position is in the equatorial plane(0) ==2atr(0) = 8and(0) = 0. The blue surface at the center depicts the outer bound of the Kerr black hole’s ergosphere. In the columns from left to right, the spin magnitude jSj=Mvariates as 0:01, 0:1and0:9, respectively, while the other initial values are fixed. For small spin, the orbit is spherical ( _r= 0) and reproduces Figure 3 of [92]. For higher spins (secondand third columns) the orbit becomes less and less spher- ical, but because of _r1, it is spherical-like. On the purplish spheres in the second row, the evolutions of the kinematical quantity defined in Equation (59) are shown under the corresponding orbits. Their initial and final di- rections are marked by purple and black arrows, respec- tively. The evolution of this vector clearly shows that the increasing spin magnitude due to the nonvanishing spin- curvaturecoupling(i.e. thenon-vanishingrighthandside of Equation (1)) in the spin precession, which was not in- cluded in the investigation of Ref. [92]) affects the orbit. On the greenish spheres in the third row, the evolutions of the spin direction are represented in the boosted SO frameEi(e;u). The initial and final spin directions are marked by green and blue arrows, respectively. In Boyer- Lindquist coordinates, the initial spin four vector Sahas only non-vanishing component Sr. The fourth and fifth rows image the evolutions of spin precessional angular velocity  (prec)(e;u)on shorter and longer timescales, respectively. For jSj=M = 0:01, the frame compo- nents of this angular velocity oscillates (see also Figure 3 of Ref. [92] and remembering for that the definition of  (prec)carries an extra sign). For jSj=M = 0:1 and0:9, an amplitude modulation occurs. This is also a clear sign of the spin-curvature effect. We mention that, the evolution of  (prec)(f;u)differs less than 1% from that of  (prec)(e;u). This is because the boosted SO and ZAMO frames are almost the same, i.e. the ro- tation angle between them is small as shown in the last row. We also mention that, all precessional angular velocities  (prec)(e;p=M ),  (prec)(e;u),  (prec)(f;p=M ) and  (prec)(f;u)in the frames E(e;p=M ),E(e;u), E(f;p=M )andE(f;u), respectively, describe the same evolutions within 1%. The Boyer-Lindquist com- ponents of the spin vector are frame independent quan- tities. Their evolutions are presented on Figure 2. The blue, the green and the red curves belong to the different spin magnitude cases jSj=M = 0:01,jSj=M = 0:1and jSj=M = 0:9, respectively. An amplitude modulation due to the spin-curvature coupling occurs in the oscil- lation around a harmonic evolution of the -component, which can be mostly seen along the red curve. In the following, we will consider zoom-whirl and un- bound orbits passing over the ergosphere. In all cases we choose such initial conditions that the body moves in theequatorialplanefornegligiblespinmagnitude. Hence the deviation of the trajectory from this plane is a clear sign of the spin-curvature effect. Zoom-whirl orbits of a nonspinning test body around a spinning black hole were already investigated in Refs. [69–73]. Those orbits did not passed through the ergosphere for which we will fo- cus. In addition, when the test body is spinning, some effects from the spin-curvature coupling are waited which we will consider. In addition, zoom-whirl orbits of com- parable mass black holes, when only one of them is spin- ning, were analyzed in Ref. [80] within the framework of PN approximation. Hyperbolic orbits of spinning11 200 400 600 800 1000 1200 1400τ/μ -0.4-0.20.20.4St/|S| 200 400 600 800 1000 1200 1400τ/μ -0.50.5Sr/|S| 200 400 600 800 1000 1200 1400τ/μ -0.10-0.050.050.10μSθ/|S| 200 400 600 800 1000 1200 1400τ/μ -0.2-0.10.10.2μSϕ/|S| Figure 2: (color online). The evolutions of the Boyer-Lindquist coordinate components of the unit spin vector are shown. The blue and the green curves belonging to the spin magnitude jSj=M = 0:01andjSj=M = 0:1, respectively, almost cover each other. The red curve represents the high spin magnitude case jSj=M = 0:9. An amplitude modulation occurs in the oscillation around a harmonic evolution of the -component which can be mostly seen along the red curve. 5 10 15ρμ -3-2-1123z/μ 5 10 15ρμ -3-2-1123z/μ Figure 3: (color online). The evolution of spinning body moving on zoom-whirl orbits around the Kerr black hole with a= 0:99. The magnitudes of the body’s spin are jSj=M = 0:01(left panel) and 0:1(right panel). The rows represent the following: 1. the orbit in coordinate space (x=,y=,z=) (outer and inner bounds of the ergosphere of the central black hole is marked by blue and red surfaces, respectively, and initial and final positions of the spinning body are denoted by green and red dots, respectively), 2. the orbit in the coordinate space ==rsin=and z==rcos=with marked initial and final positions and bounds of the ergosphere, 3. the unit spin vector in the boosted SO Cartesian-like comoving frame Ei(e;u)on a shorter timescale including the first whirling period, and 4. the unit spin vector in the boosted ZAMO Cartesian-like comoving frame Ei(f;u)on the total timescale (initial and final spin directions are marked by green and blue arrows, respectively). The initial data set:t(0) = 0,r(0) = 14:05,(0) ==2,(0) = 0,pr(0)=M=0:03,p(0)=M= 0,p(0)=M= 0:012,(S)(0) ==2and(S)(0) = 0 (the spatial Boyer-Lindquist coordinate components are Sr(0)=jSj= 0:9293,S(0)=jSj= 0andS(0)=jSj=0:0002).12 0 100 200 300 400τ /μ0.0050.010.015Sin (Θ) 0 100 200 300 400τ /μ0.0050.010.015Sin (Θ) Ω1(prec)(f,u) Ω1(prec)(e,u) 100 200 300 400τ /μ -0.008-0.006-0.004-0.0020.0020.004[1/μ] Ω1(prec)(f,u) Ω1(prec)(e,u) 100 200 300 400τ/μ -0.08-0.06-0.04-0.020.020.04[1/μ] Ω2(prec)(f,u) Ω2(prec)(e,u) 100 200 300 400τ/μ0.20.40.60.8[1/μ] Ω2(prec)(f,u) Ω2(prec)(e,u) 100 200 300 400τ/μ0.20.40.60.8[1/μ] Ω3(prec)(f,u) Ω3(prec)(e,u) 100 200 300 400τ/μ -0.006-0.004-0.0020.0020.004[1/μ] Ω3(prec)(f,u) Ω3(prec)(e,u) 100 200 300 400τ/μ -0.06-0.04-0.020.020.04[1/μ] Figure 4: (color online). The left and right columns belong to the same evolution which are shown on Figure 3. The first row shows sin . The next three rows present the evolutions of the spherical triad components of the spin precessional angular velocities  (prec )(e;u)and  (prec )(f;u). test bodies based on the MPD equations were analyti- cally studied in Ref. [83]. The perturbations caused by the spin-curvature coupling in the equatorial orbits were considered in the case when the spin is parallel to the central black hole rotation axis. In this configuration the spin vector is conserved. Here, in order to discuss non- trivial spin evolution and spin-curvature coupling effects, we choose the initial spin direction to be perpendicular to the central black hole rotation axis. ThefirstrowofFigure3showstheorbitsinthe( x,y,z)- space for increasing spin magnitude jSj=M = 0:01(left panel) and 0:1(right panel). The other initial values listed in the caption are the same. The blue and red sur- faces at the center depict the outer and interior bounds of the ergosphere, respectively (i.e. the outer stationarylimit surface and the outer event horizon). The initial and final positions of the body are marked by green and red dots, respectively. The initial position is in the equa- torial plane (0) ==2atr(0) = 14:05and(0) = 0, and both the initial four momentum and centroid four velocity have vanishing -component. With this initial location and four velocity a non-spinning particle moves in the equatorial plane. However, since the spin direction is not parallel with the rotation axis of the central black hole, the body’s centroid leaves the equatorial plane due to the effect of spin-curvature coupling. This is high- lighted in the second row representing the orbits in coor- dinates==rsin=andz==rcos=. The bounds of the ergosphere are drawn by blue and red curves. The body is inside the ergosphere when it whirls around the13 100 200 300 400τ/μ -2-1123St/|S| 100 200 300 400τ/μ -1.0-0.50.51.0Sr/|S| 100 200 300 400τ/μ -0.10-0.05μSθ/|S| 100 200 300 400τ/μ -1.0-0.50.51.0μSϕ/|S| μ SdSt dτ 100 200 300 400τ/μ -0.50.5 μ SdSr dτ 100 200 300 400τ /μ -0.2-0.10.10.2 μ2 SdSθ dτ 100 200 300 400τ/μ -0.04-0.020.020.04 μ2 SdSϕ dτ 100 200 300 400τ/μ -0.4-0.20.20.4 St S μ SdSt dτ 60 65 70 75 80τ /μ -0.50.51.01.52.02.5 Sr S μ SdSr dτ 60 65 70 75 80τ/μ -0.6-0.4-0.20.20.40.60.8 μSθ S μ2 SdSθ dτ60 65 70 75 80τ/μ -0.10-0.08-0.06-0.04-0.020.02 μSϕ S μ2 SdSϕ dτ 60 65 70 75 80τ/μ0.51.0 Figure 5: (color online). The evolution of the Boyer-Lindquist coordinate components of the unit spin vector and their derivatives rescaled to dimensionless variables is presented for that case which is shown on the right hand sides of Figures 3 and 4. The first and the second rows show the evolution on a timescale which includes the first three whirling period when the body is inside the ergosphere. The third row zooms in on that evolution period where the body is first in the ergosphere which is indicated by the purplish shadow on all panels. In the first row, the black and the red curves represent the evolutions without and with spin-curvature coupling, respectively. central Kerr black hole. This happens during all whirling period. The unit spin vector evolutions in the boosted SO Cartesian-like comoving frame ( Ei(e;u)) during a timescale including the first whirling period are shown in the third row. The initial and final directions are marked by green and blue arrows, respectively. The rotation of the projection of spin vector in the plane (Ex(e;u),Ey(e;u)) is counterclockwise in both cases. In the boosted SO frame the evolution is not contin- uous due to the motion through the ergosphere. The black dots denote the spin directions when the body first enters and leaves the ergosphere. The magnitude of the jump shows that the spin direction changed sig- nificantly inside the ergosphere. The fourth row shows the unit spin vector evolution on the total timescale in the boosted ZAMO Cartesian-like comoving frame (Ei(f;u)). This frame can be used for the spin repre- sentation inside the ergosphere, hence the evolution is continuous. For higher spin, when the spin-curvature coupling is stronger the deflection of the spin direction moves more out of the equatorial plane of ZAMO frame. The first row in Figure 4 shows the rotation angle  between the boosted SO and ZAMO frames. Here and in the following pictures the purplish shadow indicates the time interval where the body moves inside the ergo- sphere during the first whirling period. The next three rows in Figure 4 depict the evolutions of  (prec)(e;u) and  (prec)(f;u). Each row shows one component of these angular velocities. The red and blue curves rep- resent the precessional angular velocities in the boosted ZAMO and SO frames, respectively. The blue curves diverge at the ergosphere where the description in theboosted SO frame fails. The magnitude of the preces- sional angular velocities rapidly increases near and in- side the ergosphere and becomes higher for higher spin magnitude. Finally, we note that the precessional veloc- ities  (prec)(e;p=M )(  (prec)(f;p=M )) and  (prec)(e;u) (  (prec)(f;u)) describe the same evolutions within 1%. From the consideration of the moving body near and inside the ergosphere, we have found that the spin pre- cession was highly increased. Since the presented inves- tigation was based on the introduction of ZAMO, and the precessional angular velocity  (prec)(f;u)described the spin evolution with respect to the boosted ZAMO frame, the highly increased precession effect could be an observer dependent statement. However, this effect is supported in another way. The static observers play fun- damentalroleincomparingthevariationofspindirection with respect to the distant stars. The third row of Figure 3showsintheboostedSOframethatthejumpofthespin direction (between the black dots) happening during that period when the body is staying first in the ergosphere. This jump happens during relatively short period indi- cated by the purplish shadow on Figure 4. More exactly, the evolution period presented in the third row of Fig- ure 3 is given by = [0;94:7]from which the body is inside the ergosphere in the interval = [68:6;73:4]. The evolution of the spin together with these timescales result in the same conclusion that the precession angular velocity is highly increased in the ergosphere. In addi- tion, this effect can also be supported without using any particular reference frame. For the case presented on the right hand side of Figure 4, we show the evolution of the Boyer-Lindquist coordinate components of the unit14 5 10 15ρμ -3-2-1123z/μ 5 10 15ρμ -3-2-1123z/μ Ω1(prec)(f,u) Ω1(prec)(e,u) 100 200 300 400τ /μ -0.06-0.04-0.020.020.040.06[1/μ] Ω1(prec)(f,u) Ω1(prec)(e,u) 100 200 300 400τ /μ -0.06-0.04-0.020.020.040.06[1/μ] Ω2(prec)(f,u) Ω2(prec)(e,u) 100 200 300 400τ/μ0.20.40.60.8[1/μ] Ω2(prec)(f,u) Ω2(prec)(e,u) 100 200 300 400τ/μ0.20.40.60.8[1/μ] Ω3(prec)(f,u) Ω3(prec)(e,u) 100 200 300 400τ /μ -0.050.05[1/μ] Ω3(prec)(f,u) Ω3(prec)(e,u) 100 200 300 400τ /μ -0.050.05[1/μ] Figure 6: (color online). The same as on the right column of Fig- ures 3 and 4 (apart from sin ), but the initial direction of the spin vector is rotated by =2(left column) and by =2(right column). (The spatial Boyer-Lindquist coordinate components of the spin vec- tor areSr(0)=jSj=0:0025(left column), 0:0025(right column), S(0)=jSj= 0(both left and right columns) and S(0)=jSj= 0:0720(left column),0:0720(right column). spin vector and their derivatives rescaled to dimen- sionless variables on Figure 5. The first and the second rows represent the evolution on a timescale which in- cludes the first three whirling period when the body isinside the ergosphere. The third row zooms in on that evolution period where the body is first inside the ergo- sphere. As mentioned, this period is indicated by the purplish shadow. All panels of Figure 5 confirm that the rate of change in the direction of the spin vector is highly increased near and inside the ergosphere. As a reference, the black curves in the first row represent the evolution of the unit spin coordinate components when the spin- curvature coupling is turned off. While the red curves show the evolutions when the spin-curvature coupling is taken into account. Significant differences in the evolu- tions can be seen in the case of the randcoordinate components. The coordinate component identically vanishes when the spin-curvature coupling is neglected. Finally, we mention that since paanduaare not parallel with each other it may happen that uaua= 0[51, 66– 68] orpaua= 0[118–120]. The first case was discussed previously in Section IIA. The second case is equivalent with becoming the momentum light-like papa= 0, which can be seen from the contraction Equation (6) with pa. We have checked in the Appendix VIII that the MPD equations are applied only in that domain where such pathological behaviours do not occur. When the initial direction of the spin vector is oppo- site with respect to the case presented in the Figures 3 and 4, while all other initial conditions are the same, we have found the following. The centroid trajectory be- comes the reflection of the orbit presented on Figure 3 through the equatorial plane. The instantaneous direc- tions of the spin vector in the boosted SO (ZAMO) frame can be obtained from the corresponding picture of Fig- ure 3 by a rotation with an angle about the axis zand Ez(e;u)(Ez(f;u)), respectively. The angle describes thesameevolution. Finally, thecomponents 2 (prec)(e;u) and 2 (prec)(f;u)remain unchanged, while 1 (prec)(e;u), 1 (prec)(f;u), 3 (prec)(e;u)and 3 (prec)(f;u)get an extra sign. On Figure 6, the initial spin direction is rotated by =2(left column) and =2(right column) in the plane (Ex(e;u),Ey(e;u)) with respect to the case presented on Figure 3. These two cases have opposite initial spin directions leading to the following differences in the orbit and spin evolutions. The zoom-whirl orbit on the right hand side is the reflection of the trajectory on the left hand side through the equatorial plane, which are shown in the first two rows. The spin the evolutions presented on the left and the right hand sides in the third and fourth rows are related to each other by a rotation with an angleabout the axis connecting the south and north poles. The evolution of 2 (prec)are the same on the left and right hand sides, while 1 (prec)and 3 (prec)have a sign difference, as it can be seen in the last three rows. For the consideration of evolutions of spinning bodies which follow unbound orbits crossing through the ergo- sphere, the spin magnitude is chosen as jSj=M = 0:1. The initial spin directions on the left (right) hand side of Figure 7 are the same as on Figure 3 (on the left hand15 123456ρμ -1.5-1.0-0.50.51.01.5z/μ 123456ρμ -1.5-1.0-0.50.51.01.5z/μ Figure 7: (color online). The evolutions of spinning body moving on unbound orbits around Kerr black hole with a= 0:99. The spin magnitude chosen asjSj=M = 0:1. The considered unbound orbits are shown in the first row. The near black hole parts of these orbits are represented in the second and third rows in ( x=,y=,z=) and (=,z=) coordinates, respectively. The fourth and fifth rows present the evolutions of the spin vector in the boosted SO and ZAMO frames, respectively. The initial spin direction is determined by (S)(0) = 0 (left col.),=2 (right col.) and (S)(0) ==2(both cols.). The spatial Boyer-Lindquist coordinate components [ Sr(0),S(0),S(0)]=jSjof the spin vector are [0:0134,0,3:1109] and [0:000006,0,0:000005] in the left and right columns, respectively. Additional initial data set is t(0) = 0, r(0) = 2000,(0) ==2,(0) = 0,pr(0)=M=0:9,p(0)=M= 8107andp(0)=M= 0. The final locations [ t()=,r()=,(),()] at= 4433are [6033:2,1999:7,1:527,14:24] (left col.) and [ 6033:3,1999:7,1:671,14:25] (right col.). The final values of the spatial Boyer- Lindquist coordinate components [ pr(),p(),p()]=Mof the four momentum are [ 0:900002,7:65108,8:00107] (left col.) and [0:900003,3:48108,8:01107] (right col.). The final values of the spatial Boyer-Lindquist components [ Sr(),S(),S()]=jSjof the spinvector are [ 0:86,2:9105,3:8104] (left col.) and [1:11,8:6105,2:7104] (right col.). The final spin directions [ (S)(),(S)()] in the boosted SO frame are determined by [ 1:48,0:59] (left col.) and [ 1:31,1:09] (right col.). The angles [ (l)(),(l)()] characterizing the final orbital plane orientations in coordinate space ( x=,y=,z=) are [ 0:11,0:33] (left col.) and [ 0:11,1:27] (right col.).16 side of Figure 6). The first row depicts the unbound or- bits in the ( x,y,z)-space. The initial data set is chosen atr(= 0) = 2000 where the body is in the equatorial plane ((= 0) ==2and(= 0) = 0 ) and the cen- troid four velocity has vanishing -component. We nu- merically checked that r!1as!1. Second and third rows represent the orbits near the black hole in the (x,y,z)andthe(,z)spaces, respectively. Theintervalfor is determined by 5before and +5after the body crossed the outer stationary limit surface. As the body penetrates the ergosphere, it makes two turns around the black hole, then it leaves the ergosphere going to the spatial infinity. These evolutions describe such scatter- ing processes where the center is extremely approached. The deviation of the trajectory from the equatorial plane is an effect of the spin-curvature coupling. The fourth and fifth rows image the evolutions of the unit spin vec- tor represented in the boosted SO and ZAMO frames, respectively. The deviation of the spin vector direction from the equatorial plane also occurs because of the spin- curvaturecoupling. Thejumpintheevolutionofthespin vector in the boosted SO frame (marked by black dots) shows that the variation of spin direction takes place mainly inside the ergosphere. The large part of the vari- ation of spin direction happens during that period when the body is inside the ergosphere. This time interval is 2[2214:8;2218:6]which is short with respect to the considered total evolution period = [0;4433]. The fi- nal value of the proper time = 4433was chosen in such a way, that for  >the spin angles undergo only unsignificant changes. Figure 8 presents the evolutions of  (prec)(e;u)and  (prec)(f;u)for that time interval which is determined by 25before and +25after the body crossed the outer stationary limit surface. The pur- plish shadow denotes that period when the body is inside the ergosphere where the spin precessional angular veloc- ity components increases. The spin-curvature coupling mainly influences the smaller components of the precessional angular velocity 1 (prec)(f;u)and 3 (prec)(f;u), as it can be seen in the first row of Figure 9. The black curve represents the evo- lutions without the spin-curvature coupling. In the case of the red curves, the spin-curvature coupling is taken into account, and they are the same as in the second col- umn of Figure 8. The spin-curvature coupling increases the amplitude of the precessional angular velocity com- ponents. The reparametrization invariance of the representative worldline also implies a gauge freedom [121]. Usually, the following choices for this timelike parameter are ap- plied in the literature: i)the proper time ( uaua=1) [51, 92] also used in this paper; ii)the parameter deter- mined by the normalization uapa=M=1[122, 123]; iii)the coordinate time t[96, 124]. Employing the TD SSC, considerable differences were not found in both the orbit and the spin dynamics when using the parameters eitheri)orii)[64]. The orbit and the spin evolutions are unaffected when using the coordinate time tinstead of the proper time . However, the precessional angularvelocity is changed for  (prec)=u0which is shown in the second row of Figure 9 as a function of t. The black and the red curves represent the evolution without and with the spin-curvature coupling. We can conclude the same effects when we have considered the spin evolution with respect to the proper time. The relatively rapid change in the direction of the spin vector can also be confirmed without using any partic- ular reference frame. In the first row of Figure 10, we present the evolutions of the Boyer-Lindquist coordinate components of the unit spin vector for the case imaged on the right hand sides of Figures 7 and 8. The spin- curvature coupling is included in the evolutions depicted by the red curves. The black curves represent the corre- spondingevolutionswhenthiscouplingisturnedoff. The effect of the spin-curvature coupling can be seen in the evolution of St,SrandScomponents. The latter van- ishes identically in the absence of the spin-curvature cou- pling. However, if the spin-curvature coupling is included in the analysis, the Scomponent deviates significantly fromzerowhenthebodyisclosetothecentralblackhole. In addition, the effect of the spin-curvature coupling re- mains in the StandSrcomponents far from the cen- tral black hole. They approach another constant values when the spin-curvature coupling is taken into account. The evolutions of the components of the unit spin vector and their derivatives rescaled to dimensionless variables on a smaller timescale, when the spinning test body is close the central black hole are represented in the second row. All panels supports a relatively rapid change of the spin vector near and inside the ergosphere. Finally, we mention that the MPD equations were applied only in its validity domain, this check is given in the Appendix VIII. In a wider range of initial conditions, the final values of the polar(S)and azimuthal (S)spin angles (the scat- tering angles) are represented on Figure 11 as functions of gauge invariant, dimensionless energy ^E=E=Mand angular momentum ^Jz=Jz=M. The small black dots in the plane of the initial spin angles ( (S)(0) ==2and (S)(0) = 0) indicate the region, where the body crosses the event horizon of the Kerr black hole. Then, instead of a scattering process, the body falls into the black hole. Close to the left corner, i.e. at smaller ^Eand higher ^Jz values, the body approaches the central black hole less than for higher ^Eand/or for smaller ^Jzvalues. As a con- sequence, the precession and hence the variation of the spin angles are both small. However, close to the diago- nal in the ^E,^Jzplane indicated by the edge of the black dots region, the body enters into the ergosphere, and due to the high precession there, the spin angles undergo a relatively large change. In all case, the initial values are chosen such that, if the spin-curvature coupling is neglected, the polar angle (S)remains=2during the whole evolution, and the spin precession influences only (S). Hence, the variation of (S)shown on the left panel is a clear effect of the spin-curvature coupling. We17 Ω1(prec)(f,u) Ω1(prec)(e,u) 2200 2210 2220 2230 2240τ/μ -0.10-0.050.05[1/μ] Ω1(prec)(f,u) Ω1(prec)(e,u) 2200 2210 2220 2230 2240τ/μ -0.10-0.050.05[1/μ] Ω2(prec)(f,u) Ω2(prec)(e,u) 2200 2210 2220 2230 2240τ/μ0.51.01.52.02.5[1/μ] Ω2(prec)(f,u) Ω2(prec)(e,u) 2200 2210 2220 2230 2240τ/μ0.51.01.52.02.5[1/μ] Ω3(prec)(f,u) Ω3(prec)(e,u) 2200 2210 2220 2230 2240τ /μ -0.050.05[1/μ] Ω3(prec)(f,u) Ω3(prec)(e,u) 2200 2210 2220 2230 2240τ /μ -0.050.05[1/μ] Figure 8: (color online). On the left and right columns the evolutions of the spherical triad components of the spin precessional angular velocities are presented along those orbits which are shown in the left and right columns of Figure 7, respectively. 2200 2210 2220 2230 2240τ/μ -0.10-0.050.05Ω1 (prec) (f,u)[1/μ] 2200 2210 2220 2230 2240τ/μ0.51.01.52.02.5Ω2 (prec) (f,u)[1/μ] 2200 2210 2220 2230 2240τ/μ -0.050.05Ω3 (prec) (f,u)[1/μ] 2990 3000 3010 3020 3030 3040 3050t/μ -0.010-0.0050.005Ω1 (prec) (f,u)/u0 2990 3000 3010 3020 3030 3040 3050t/μ0.050.100.150.200.250.30Ω2 (prec) (f,u)/u0 2990 3000 3010 3020 3030 3040 3050t/μ -0.010-0.0050.0050.010Ω3 (prec) (f,u)/u0 Figure 9: (color online). In the first line, the black and the red curves show the precessional angular velocity spherical frame components without and with spin-curvature coupling, respectively. The second line shows them when the spin evolution is considered as a function of the coordinate timet. These evolutions belong to the case which is presented in the right hand sides of Figures 7 and 8. mention that, both functions (S)and(S)steeply in- crease as approaching the edge of the black dots region. Those maxima, which can be seen on the panels, belong to the chosen grid in the ^E,^Jzplane. B. Spinning bodies moving on zoom-whirl orbits in rotating regular black hole spacetimes In this subsection, we set = 0, = 3anda= 0:99em. The background is either a regular, rotatingBardeen-like ( = 2) or Hayward-like ( = 3) black hole spacetime. For = 2and= 3, the spacetime contains a black hole for q0:081andq0:216, respectively.18 2000 2200 2400 2600τ/μ -6-5-4-3-2-1St/|S| 2400 2600-0.74-0.72-0.70 2000 2200 2400 2600τ/μ -1.0-0.50.5Sr/|S| 2000 2200 2400 2600τ/μ -0.20-0.15-0.10-0.05μSθ/|S| 2000 2200 2400 2600τ/μ -2.5-2.0-1.5-1.0-0.5μSϕ/|S| 2210 2215 2220 2225τ/μ -6-4-22St/|S| 2210 2215 2220 2225τ/μ -1.0-0.50.5Sr/|S| 2210 2215 2220 2225τ/μ -0.20-0.15-0.10-0.050.050.10μSθ/|S| 2210 2215 2220 2225τ/μ -2-11μSϕ/|S| Figure 10: (color online). In the first line, the black and the red curves present the Boyer-Lindquist coordinate components of the unit spin vector without and with spin-curvature coupling. In case of St, the relatively small deviation of the curves when the test body is moving away from the central black hole is shown in a small box. The second line presents the evolutions of the unit spin vector and their derivatives rescaled to dimensionless variables when the spinning test body is close to the central black hole. These evolutions belong to the case, which is presented in the right hand sides of Figures 7 and 8 and also in Figure 9. The time interval, when the body is inside the ergosphere, is indicated by a purplish shadow on all panels. Figure 11: (color online). The left and right panels present the final value of the spin angles (S)()and(S)(), respectively, as functions of the dimensionless energy ^E=E=Mand angular momentum ^Jz=Jz=M. The final values were computed at = 4433. We have checked that the spin angles undergo only unsignificant changes for  > . The initial spin is given by jSj=M = 0:1,(S)(0) ==2and(S)(0) = 0. The initial momentum has vanishing component: p(0)=M= 0, and its additional components were determined from ^E,^Jzandpapa=M2=1. The small black dots in the plane of the initial spin angles represent the region, where the body crosses the event horizon of the Kerr black hole, hence, unbound orbits do not develop. We consider three cases: ( = 2,q= 0:081), (= 3,q= 0:081) and (= 3,q= 0:216). For these parameters the regular black holes have two stationary limit surfaces and event horizons. In addition, the spin magnitude for the moving body is chosen as jSj=M = 0:1. On Figure 12, zoom-whirl orbits in different regular ro- tating black hole spacetimes are presented. The columns from left to right correspond to ( = 2,q= 0:081), (= 3,q= 0:081) and (= 3,q= 0:216). With the notation change !em, the initial values are chosen the same as in the second column of Figure 3. Each row represents the same quantity which was shown on Fig- ure 6. The first two columns show that both the orbit and the spin evolutions are significantly different in the cases of the Bardeen-like and Hayward-like black holes for the same emandqvalues. In addition, the second and the third columns show in the case of Hayward-like backgroundthattheseevolutionsarealsosensitiveforthe value ofq. The way of deviation of the orbit from the equatorial plane, which is the effect of the spin-curvaturecoupling, is also very sensitive for the parameters of the regular black holes. The spin vector evolutions includ- ing the first whirling period in the boosted SO frame is presented in the third row. The black dots represent a jump in the evolution. The part of the evolution which is not shown takes place inside the ergosphere. The amount of the jumps is somewhat different for each cases. The fourth row shows the total evolution of the spin vec- tor in the boosted ZAMO frame. The final directions (blue arrows) of the spin direction are significantly differ- ent. The evolutions of the spherical frame components of the precessional angular velocity including the first three whirling period are shown in the last three rows. These are perturbatively different for the different regular black holes. However, the effects of these small differences add up over the evolution. Finally, we mention that a consideration of unbound orbits about regular black holes can be found in Ref. [125].19 5 10 15ρμem -3-2-1123z/μem 5 10 15ρμem -3-2-1123z/μem 5 10 15ρμem -3-2-1123z/μem Ω1(prec)(f,u) Ω1(prec)(e,u) 100 200 300 400τ/μem -0.08-0.06-0.04-0.020.020.04[1/μem] Ω1(prec)(f,u) Ω1(prec)(e,u) 100 200 300 400τ/μem -0.08-0.06-0.04-0.020.020.04[1/μem] Ω1(prec)(f,u) Ω1(prec)(e,u) 100 200 300 400τ/μem -0.08-0.06-0.04-0.020.020.04[1/μem] Ω2(prec)(f,u) Ω2(prec)(e,u) 100 200 300 400τ /μem0.20.40.60.8[1/μem] Ω2(prec)(f,u) Ω2(prec)(e,u) 100 200 300 400τ /μem0.20.40.60.8[1/μem] Ω2(prec)(f,u) Ω2(prec)(e,u) 100 200 300 400τ /μem0.20.40.60.8[1/μem] Ω3(prec)(f,u) Ω3(prec)(e,u) 100 200 300 400τ/μem -0.06-0.04-0.020.020.040.06[1/μem] Ω3(prec)(f,u) Ω3(prec)(e,u) 100 200 300 400τ/μem -0.06-0.04-0.020.020.040.06[1/μem] Ω3(prec)(f,u) Ω3(prec)(e,u) 100 200 300 400τ/μem -0.06-0.04-0.020.020.040.06[1/μem] Figure 12: (color online). Zoom-whirl orbits are represented around regular, rotating black holes with = 3anda= 0:99em. The first column shows the orbit around a Bardeen-like black hole ( = 2) while the middle and the last around a Hayward-like black hole ( = 3). The parameter qis0:081in the first two columns while 0:216in third one. Applying the notation change !em, the initial values are chosen the same as in the second column of Figure 3. The quantities in each line are the same which are presented in Figure 6.20 V. CONCLUSIONS Wehaveconsiderednumericallytheevolutionofaspin- ning test body governed by the MPD equations, mov- ing along spherical-like, zoom-whirl and unbound orbits around a Kerr black hole. When the spacetime curva- ture and the spin contributions on the right hand sides of the MPD equations can be neglected, we recovered the corresponding results of Ref. [92] for a spherical orbit. However, for higher spin, an amplitude modulation oc- cured in the harmonic evolution of the spin precessional angular velocity caused by the spin-curvature coupling. This amplitude modulation also occured in the Boyer- lindquist coordinate component of the spin vector. Theexistenceofzoom-whirlorbitsareconfirmedbyus- ing the MPD dynamics. The considered zoom-whirl and unbound orbits of spinning body passed over the ergo- sphere, where the PN approximation cannot be applied. In all cases the numerical investigations showed that the spin precessional angular velocity highly increased near and inside the ergosphere. Thus the direction of the spin vector is significantly variated during the evolutionary phase inside the ergosphere. The initial values were cho- sen such that the test body moved in the equatorial plane when the spin-curvature coupling is neglected. Hence, the effect of this coupling occured as a deviation of the orbit from the equatorial plane. In order to investigate non-trivial spin evolution, the initial spin direction was chosen to be perpendicular to the rotation axis of the central black hole. Then, the spin vector evolved in the equatorial plane of the boosted SO and ZAMO frames when the spin-curvature coupling is neglected. The de- viation of the spin vector from this equatorial plane was also the effect of the spin-curvature coupling. Additional effects of the spin-curvature coupling was observed in the evolutions ofthespin precessionalangularvelocityandof the Boyer-Lindquist coordinate components of the spin vector.Zoom-whirl orbits and spin precession including the spin-curvature coupling were also considered in regular spacetimes containing a central rotating black hole. Sig- nificant differences were observed in the way of deviation oftheorbitfromtheequatorialplanewhichweresensitive for the parameters of the regular black hole. Small devi- ations were found in the spin precession angular velocity, which add up over the evolutions. Hence, the direction of the final spin vector can be very different for different parameters of the regular black hole. Finally, wementionthatthenumericinvestigationpre- sentedherecouldbegeneralizedinthefollowingway. Be- sides the spin-curvature coupling another effects would occur if the backreaction of the body to the metric was not neglected. This backreaction appears as a self-force in the equation of motion [126–129], and also causes a deviation from the geodesic orbit like the spin-curvature coupling. Acknowledgements The work of B. M. was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sci- ences. The work of Z. K. was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the UNKP-18-4 New National Excellence Program of the Ministry of Human Capacities and by the Hungarian National Research Development and Innova- tion Office (NKFI) in the form of the grant 123996. VI. CONFLICT OF INTEREST The authors declare no conflict of interest. VII. APPENDIX A: THE RELATION BETWEEN THE FRAMES E(e; U )AND E(f; U ) The frame vectors E(e;U)derived from the SO’s frame are the following linear combination of E(f;U): E1(e;U) =E1(f;U) +(Z)w1 (Z) 1 + (S)" aBsinpgttAE3(f;U) + 1s 
gttA! (Z)w(Z) 1 + (Z)# ; E2(e;U) =E2(f;U) +(Z)w2 (Z) 1 + (S)" aBsinpgttAE3(f;U) + 1s 
gttA! (Z)w(Z) 1 + (Z)# ; E3(e;U)= s 
gttA+ (Z)! E3(f;U) 1 + (S)(Z)w(Z) 1 + (S)" 1s 
gttA! (Z)w3 (Z) 1 + (Z)+aBsinpgttA# :(64) The inverse relations are E1(f;U) =E1(e;U)(S)w1 (S) 1 + (Z)" aBsinpgttAE3(e;U) 1s 
gttA! (S)w(S) 1 + (S)# ;21 E2(f;U) =E2(e;U)(S)w2 (S) 1 + (Z)" aBsinpgttAE3(e;U) 1s 
gttA! (S)w(S) 1 + (S)# ; E3(f;U)= s 
gttA+ (S)! E3(e;U) 1 + (Z)(S)w(S) 1 + (Z)" 1s 
gttA! (S)w3 (S) 1 + (S)aBsinpgttA# :(65) The frame components of any vector field V=(e) VE(e) =(f) VE(f); (66) obey the following transformation rule (e) V1=(f) V1+" 1s 
gttA! (Z)w(Z)
V 1 + (Z)+aBsinpgttA(f) V3# (Z)w1 (Z) 1 + (S); (e) V2=(f) V2+" 1s 
gttA! (Z)w(Z)
V 1 + (Z)+aBsinpgttA(f) V3# (Z)w2 (Z) 1 + (S); (e) V3= s 
gttA+ (Z)! (f) V3 1 + (S)(Z)w(Z)
V 1 + (S)" aBsinpgttA+ 1s 
gttA! (Z)w3 (Z) 1 + (Z)# ;(67) withw(Z)introduced in Equation (27). The inverse relations are (f) V1=(e) V1+" 1s 
gttA! (S)w(S)
V 1 + (S)aBsinpgttA(e) V3# (S)w1 (S) 1 + (Z); (f) V2=(e) V2+" 1s 
gttA! (S)w(S)
V 1 + (S)aBsinpgttA(e) V3# (S)w2 (S) 1 + (Z); (f) V3= s 
gttA+ (S)! (e) V3 1 + (Z)+(S)w(S)
V 1 + (Z)" aBsinpgttA 1s 
gttA! (S)w3 (S) 1 + (S)# ;(68) withw(S)introduced in Equation (24). 200 400 600 800 1000 1200 1400τ/μ -2.5×10-6-2.×10-6-1.5×10-6-1.×10-6-5.×10-7u2+1 65 70 75 80τ/μ -3.×10-7-2.×10-7-1.×10-71.×10-72.×10-7u2+1 200 400 600 800 1000 1200 1400τ/μ -0.000020-0.000015-0.000010-5.×10-6g+1 65 70 75 80τ/μ -0.000020-0.000015-0.000010-5.×10-6g+1 Figure 13: (color online). The evolutions of u2=uauaandgon longer and shorter timescales for a zoom-whirl orbit presented on the left hand sides of Figures 3 and 4.22 1000 2000 3000 4000τ/μ -3.5×10-6-3.×10-6-2.5×10-6-2.×10-6-1.5×10-6-1.×10-6-5.×10-7u2+1 2212 2214 2216 2218 2220 2222τ/μ-3.×10-6-2.8×10-6-2.6×10-6-2.4×10-6u2+1 1000 2000 3000 4000τ/μ -0.00008-0.00006-0.00004-0.00002g+1 2212 2214 2216 2218 2220 2222τ/μ -0.00014-0.00012-0.00010-0.00008-0.00006-0.00004-0.00002g+1 Figure 14: (color online). The evolutions of u2=uauaandgon longer and shorter timescales are shown for an unbound orbit presented on the left hand sides of Figures 7 and 8. VIII. APPENDIX B: CHECKING THE VALIDITY OF THE MPD EQUATIONS The contraction of the inverse of the velocity- momentum relation (6) with uagives that the sign of pauais determined by the quantity: g=uaua1 2M2ubRebcdScdSaeua;(69)which corresponds to _x~T_x= _xG_xin Ref. [119]. Both the functionsganduauaare shown on Figures 13 and 14 for twocaseswhenthetestbodyfollowsazoom-whirlandan unbound orbit, respectively. In both cases gtakes values close to -1 during the whole evolution and uaua=1. Hence, the MPD equations are applied where they are valid. Similar is hold along the all trajectories presented in the article. [1] LIGO Scientific Collaboration and Virgo Collaboration, Phys. Rev. Lett. 2016,116, 061102. [2] LIGO Scientific Collaboration and Virgo Collaboration, Phys. Rev. Lett. 2016, 116, 241103. [3] LIGO Scientific Collaboration and Virgo Collaboration, Phys. Rev. Lett. 2017,118, 221101. [4] LIGO Scientific Collaboration and Virgo Collaboratio, Phys. Rev. Lett. 2017,119, 141101. [5] LIGO Scientific Collaboration and Virgo Collaboratio, Phys. Rev. Lett. 2017,119, 161101. [6] LIGO Scientific Collaboration and Virgo Collaboration, Astrophys. J. Lett. 2017,848, L13. [7] LIGO Scientific Collaboration and Virgo Collaboration, Astrophys. J. 2017,851, L35. [8] LIGO Scientific Collaboration and Virgo Collaboration, Phys. Rev. X 2019,9, 031040. [9] C. Kimball, C. P. L. Berry, V. Kalogera, Res. Notes AAS 2020,4, 2. [10] E. Kun, K. É. Gabányi, M. Karouzos, S. Britzen, L. Á. Gergely, Mon. Not. Royal Astron. Soc. 2014,445(2), 1370. [11] B. M. Barker, R. F. O’Connell, Phys. Rev. D 1970,2, 1428. [12] B. M. Barker, R. F. O’Connell, Phys. Rev. D 1975,12, 329. [13] L. E. Kidder, Phys. Rev. D 1995,52, 821. [14] E. Poisson, Phys. Rev. D 1998,57, 5287. [15] N. Wex, Class. Quantum Grav. 1995 12, 983. [16] C. Königsdörffer, A. Gopakumar, Phys. Rev. D 2005, 71, 024039. [17] Z. Keresztes, B. Mikóczi, L. Á. Gergely, Phys. Rev. D 2005,71, 124043. [18] C. Königsdörffer, A. Gopakumar, Phys. Rev. D 2006, 73, 044011. [19] É. Racine, Phys. Rev. D 2008,78, 044021. [20] M. Kesden, D. Gerosa, R. O’Shaughnessy, E. Berti, U.Sperhake, Phys. Rev. Lett. 2015,114, 081103. [21] T. A. Apostolatos, C. Cutler, G. J. Sussman, K. S. Thorne, Phys. Rev. D 1994,49, 6274. [22] J. D. Schnittman, Phys. Rev. D 2004,70,124020 (2004). [23] L. Á. Gergely, P. L. Biermann, Astrophys. J. 2009 697, 1621. [24] C. O. Lousto, J. Healy, Phys. Rev. Lett. 2015,114, 141101. [arXiv:1410.3830] [25] D. Gerosa, A. Lima, E. Berti, U. Sperhake, M. Kesden, R. O’Shaughnessy, Class. Quantum Grav. 2019 36, 10, 105003. [26] M. Mathisson, Acta. Phys. Polon. 1937,6, 163. [27] A. Papapetrou, Proc. Phys. Soc. 1951 64, 57. [28] W. Dixon, Nuovo Cim. 1964 34, 317. [29] W. G. Dixon, Proc. R. Soc. London A 1970,314, 499. [30] W.G.Dixon, in Isolated Gravitating Systems in General Relativity , Proceedings of the International School of Physics, Course LXVII 1979, ed. by J. Ehlers. [31] P. Amaro-Seoane et al., Laser Interferometer Space An- tenna 2017, arXiv:1702.00786. [32] C. Huwyler, E. K. Porter, P. Jetzer, Phys. Rev. D 2015, 91, 024037. [33] J. N. Bahcall, R. A. Wolf, Astrophys. J. 1977 216, 883. [34] I. Bartos, Z. Haiman, B. Kocsis, Sz. Márka, Phys. Rev. Lett.2013,110, 221102. [35] C. J. Hailey, K. Mori, F. E. Bauer, M. E. Berkowitz, J. Hong, B. J. Hord, Nature 2018,556, 70. [36] B. Mikóczi, Phys. Rev. D 2017,95, 064023. [37] J. Frenkel, Z. Phys. 1926 37, 243. [38] F. A. E. Pirani, Acta Phys. Polon. 1956,15, 389. [39] E. Corinaldesi, A. Papapetrou, Proc. Roy. Soc. A 1951, 209, 259. [40] T. D. Newton, E. P. Wigner, Rev. Mod. Phys. 1949,21, 400. [41] M. H. L. Pryce, Proc. Roy. Soc. Lond. A 1948,195, 62.23 [42] W. M. Tulczyjew, Acta Phys. Polon. 1959,18, 393. [43] A. Ohashi, Phys. Rev. D 2003,68, 044009. [44] K. Kyrian, O. Semerák, Mon. Not. Roy. Astron. Soc. 2007,382, 1922. [45] V.Witzany, J.Steinhoff, G.Lukes-Gerakopoulos, Class. Quantum Grav. 2019,36, 075003. [46] E. Barausse, E. Racine, A. Buonanno, Phys. Rev. D 2009,80, 104025; Erratum: Phys. Rev. D 2012 85, 069904(E). [47] A. A. Deriglazov, W. G. Ramirez, Phys. Rev. D 2015, 92, 124017. [48] G. Lukes-Gerakopoulos, J. Seyrich, D. Kunst, Phys. Rev. D 2014,90, 1040109. [49] D. Kunst, T. Ledvinka, G. Lukes-Gerakopoulos, J. Seyrich, Phys. Rev. D 2016,93, 044004. [50] B. M. Barker, R. F. O’Connell, Gen. Rel. Grav. 1974, 5, 539. [51] O. Semerák, Mon. Not. Roy. Astron. Soc. 1999,308, 863. [52] S. Suzuki, K. Maeda, Phys. Rev. D 1998,58, 023005. [53] M. D. Hartl, Phys. Rev. D 2003,67, 104023. [54] M. D. Hartl, Phys. Rev. D 2003,67, 024005. [55] W.-B. Han, Gen. Relativ. Gravit. 2008,40, 1831. [56] W.-B. Han, R. Cheng, Gen. Relativ. Gravit. 2017,49, 48. [57] B.Mashhoon, D.Singh, Phys. Rev. D 2006,74, 124006. [58] D.Bini,P.Fortini,A.Geralico,A.Ortolan, Class. Quan- tum Grav. 2008,25, 125007. [59] D. Bini, A. Geralico, Phys. Rev. D 2011,84, 104012. [60] R. Plyatsko, M. Fenyk, Phys. Rev. D 2013,87, 044019. [61] D. Bini, A. Geralico, Class. Quantum Grav. 2014,31, 075024. [62] D. Bini, G. Faye, A. Geralico, Phys. Rev. D 2015 92, 104003. [63] E. Hackmann, C. Lämmerzahl, Y. N. Obukhov, D. Puetzfeld, I. Schaffer, Phys. Rev. D 2014,90, 064035. [64] G.Lukes-Gerakopoulos, Phys. Rev. D 2017,96,104023. [65] U. Ruangsri, S. J. Vigeland, S. A. Hughes, Phys. Rev. D2016,94, 044008. [66] B. Toshmatov, D. Malafarina, Phys. Rev. D 2019,100, 104052. [67] B. Toshmatov, O. Rahimov, B. Ahmedov, D. Malafa- rina, Eur. Phys. J. C 2020,80, 675. [68] C. A. Benavides-Gallego, W-B. Han, D. Malafarina, Phys. Rev. D 2021,104, 084024. [69] K. Glampedakis, D. Kennefick, Phys. Rev. D 2002,66, 044002. [70] K. Glampedakis, S. A. Hughes, D. Kennefick, Phys. Rev. D 2002,66, 064005. [71] K.Glampedakis, Class. Quantum Grav. 2005,22, S605. [72] J. Levin, G. Perez-Giz, Phys. Rev. D 2008,77, 103005. [73] R. Grossman, J. Levin, G. Perez-Giz, Phys. Rev. D 2012,85, 023012. [74] F. Pretorius, D. Khurana, Class. Quantum Grav. 2007, 24, S83. [75] J. Healy, J. Levin, D. Shoemaker, Phys Rev. Let. 2009, 103, 131101. [76] U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, T. Hin- derer, N. Yunes Phys. Rev. Lett. 2009,103, 131102. [77] R. Gold, B. Brugmann, Class. Quantum Grav. 2010, 27, 084035. [78] R. Gold, B. Brugmann, Phys. Rev. D 2013,88, 064051. [79] W. E. East, S. T. McWilliams, J. Levin, F. Pretorius, Phys. Rev. D 2013,87, 043004.[80] J. Levin, R. Grossman, Phys. Rev. D 2009,79, 043016. [81] R. Grossman, J. Levin, Phys. Rev. D 2009,79, 043017. [82] L. De Vittori, A. Gopakumar, A. Gupta, P. Jetzer, Phys. Rev. D 2014,90, 124066. [83] D. Bini, A. Geralico, J. Vines, Phys. Rev. D 2017,96, 084044. [84] J. M. Bardeen, Proc. GR5, Tbilisi USSR 1968,174. [85] E. Ayón-Beato, A. García, Phys. Lett. B 2000,493, 149. [86] S. A. Hayward, Phys. Rev. Lett. 2006,96, 031103. [87] Z-Y Fan, X. Wang, Phys. Rev. D 2016,94, 124027. [88] B. Toshmatov, Z. Stuchlík, B. Ahmedov, Phys. Rev. D 2017,95, 084037. [89] K. A. Bronnikov, Phys. Rev. D 2017,96, 128501. [90] M. E. Rodrigues, E. L. B. Junior, Phys. Rev. D 2017, 96, 128502. [91] B. Toshmatov, Z. Stuchlík, B. Ahmedov, Note on the character of the generic rotating charged regular black holes in general relativity coupled to nonlinear electro- dynamics ,2017[arXiv:1712.04763]. [92] D. Bini, A. Geralico, R. T. Jantzen, Phys. Rev. D 2017, 95, 124022. [93] E. Hackmann, C. Lämmerzahl, Y. N. Obukhov, D. Puetzfeld, I. Schaffer Phys. Rev. D 2014,90, 064035. [94] D. Bini, A. Geralico, Phys. Rev. D 2018,98, 084021. [95] K. P. Tod, F. de Felice, M. Calvani, Il Nouvo Cimento 1976,34, 365. [96] R. Hojman, S. Hojman, Phys. Rev. D 1977,15, 2724. [97] L. F. O. Costa, G. Lukes-Gerakopoulos, O. Semerák, Phys. Rev. D 2018,97, 084023. [98] S. Suzuki, K. Maeda, Phys. Rev. D 1997,55, 4848. [99] R. P. Kerr, Phys. Rev. Lett. 1963,11, 237. [100] J. M. Bardeen, P. W. H. Teukolsky, Astrophys J. 1972, 178, 347. [101] O. Semerák, Gen. Rel. Grav. 1993,25, 1041. [102] E. P. Wigner, Annals. Math. 1939,40, 149 [ Nucl. Phys. Proc. Suppl. 1989,6, 9]. [103] D. Bini, A. Geralico, R. T. Jantzen, Phys. Rev. D 2019, 99, 064041. [104] C.W.Misner, K.S.Thorne, J.A.Wheeler, Gravitation , W. H. Freeman and Company, San Francisco 1973. [105] J. L. Synge, Relativity of the general theory , North- Holland Publishing Company, Amsterdam 1960. [106] H. S. Ruse, Quart. J. Math. 1930 1, 146. [107] P. Gunther, Arch. Rat. Mech. Analys. 1965,18, 103. [108] H. A. Buchdahl, Gen. Rel. Grav. 1972,3, 35. [109] H. A. Buchdahl, N. P. Warner, J. Phys. A 1980 13, 509. [110] R. W. John, Ann. Phys. Leipzig 1984,41, 67. [111] R. W. John, Trans. Inst. Phys. Estonian Acad Sci. 1989,65, 58. [112] M. D. Roberts, Astrophys. Lett. Comm. 1993,28, 349. [113] J. Steinhoff, Spin gauge symmetry in the action principle for classical relativistic particles ,2015 [arXiv:1501.04951]. [114] L.F.O.Costa, C.Herdeiro, J.Natário, M.Zilhăo, Phys. Rev. D 2012,85, 024001. [115] L. F. O. Costa, J. Natário, Fund. Theor. Phys. 2015, 179, 215. [116] L. F. O. Costa, J. Natário, M. Zilhăo, Phys. Rev. D 2016,93, 104006. [117] I. Timogiannis, G. Lukes-Gerakopoulos, T. A. Aposto- latos, Phys. Rev. D 2021, 104, 024042. [118] A. A. Deriglazov, W. G. Ramirez, Int. J. Mod. Phys. D 2017,26, 1750047.24 [119] A. A. Deriglazov, W. G. Ramirez, Adv. Math. Phys. 2017,7397159. [120] A. A. Deriglazov, W. G. Ramirez, Phys. Lett. B 2018, 779, 210. [121] A. J. Hanson, T. Regge, Ann. Phys. 1974, 87, 498. [122] J. Ehlers, E. Rudolph, Gen. Relativ. Gravit. 1977, 8, 197. [123] T. Tanaka, Y. Mino, M. Sasaki, M. Shibata, Phys. Rev. D1996, 54, 3762. [124] S. A. Hojman, F. A. Asenjo, Class. Quantum. Grav.2012, 30, 025008. [125] Z. Keresztes, B. Mikóczi, Roman. Astron. J. 2020, 30, 59. [126] A. Pound, Phys. Rev. Lett. 2012, 109, 051101. [127] M. van de Meent, Phys. Rev. D 2018, 97, 104033. [128] L. Barack, A. Pound, Rep. Prog. Phys. 2018, 82, 016904. [129] L. Barack, M. Colleoni, T. Damour, S. Isoyama, N. Sago, Phys. Rev. D 2019, 100, 124015.

1403.0378v1.Chaos_in_two_black_holes_with_next_to_leading_order_spin_spin_interactions.pdf

arXiv:1403.0378v1 [gr-qc] 3 Mar 2014Eur. Phys.J.C manuscriptNo. (will beinsertedbytheeditor) Chaos in two black holes with next-to-leading order spin-sp in interactions GuoqingHuang1, XiaotingNi1,Xin Wua,1 1Department ofPhysics, Nanchang University, Nanchang 3300 31, China Received: date /Accepted: date Abstract We take intoaccountthe dynamicsofa complete thirdpost-NewtonianconservativeHamiltonianoftwospin - ning black holes, where the orbital part arrives at the third post-Newtonian precision level and the spin-spin part with the spin-orbit part includes the leading-order and next-to - leading-order contributions. It is shown through numerica l simulationsthatthenext-to-leadingorderspin-spincoup lings play an important role in chaos. A dynamical sensitivity to thevariationofsingleparameterisalsoinvestigated.Inp ar- ticular,thereareanumberof observable orbitswhoseinitial radiiarelargeenoughandwhichbecomechaoticbeforeco- alescence. Keywords black hole ·post-Newtonian approximations · chaos·Lyapunovexponents PACS04.25.dg·04.25.Nx ·05.45.-a·95.10.Fh 1 Introduction Massivebinaryblack-holesystemsarelikelythemostpromi s- ingsourcesforfuturegravitationalwavedetectors.Thesu c- cessful detection of the waveforms means using matched- filtering techniques to best separate a faint signal from the noise and requiresa very precise modelling of the expected waveforms. Post-Newtonian (PN) approximations can sat- isfy this requirement. Up to now, high-precision PN tem- plates have already been known for the non-spin part up to 3.5PN order ( i.e.the order 1 /c7in the formal expansion in powers of 1 /c2withcbeing the speed of light) [1,2], the spin-orbitpartupto3.5PNorderincludingtheleading-ord er (LO,1.5PN),next-to-leading-order(NLO,2.5PN)andnext- to-next-to-leading-order(NNLO, 3.5PN) interactions[3- 5], and the spin-spinpart up to 4PN orderconsisting of the LO (2PN),NLO (3PN)andNNLO(4PN)couplings[6-9]. ae-mail: xwu@ncu.edu.cnHowever, an extremely sensitive dependence on initial conditions as the basic feature of chaotic systems would pose a challenge to the implementation of such matched filters, since the number of filters required to detect these waveforms is exponentially large with increasing detectio n sensitivity. This has led to some authors focusing on re- search of chaos in the orbits of two spinning black holes. Chaoswasfirstlyfoundandconfirmedinthe2PNLagrangian approximation of comparable mass binaries with the LO spin-orbit and LO spin-spin effects [10]. Moreover, it was reported in [11] that the presence of chaos should be ruled out in these systems because no positive Lyapunov expo- nents could be found. As an answer to this claim, Refs. [12,13]obtainedsomepositiveLyapunovexponentsandpoin ted out these zero Lyapunov exponents of [11] due to the less rigorouscalculationoftheLyapunovexponentsoftwonearb y orbits with unapt rescaling. In fact, the conflicting result s on Lyapunovexponentsare because the two papers [11,12] used different methods to compute their Lyapunov expo- nents, as was mentioned in [14]. Ref. [11] computed the stabilizing limit values of Lyapunov exponents, and Ref. [12] worked out the slopes of fit lines. This is the so-called doubt regarding different chaos indicators causing two dis - tinct claims on the chaotic behavior.Besides this, there wa s second doubt on different dynamical approximations mak- ing the same physical system have distinct dynamical be- haviors.The 2PN harmonic-coordinatesLagrangianformu- lation of the two-black hole system with the LO spin-orbit couplings of one spinning body allows chaos [15], but the 2PN ADM (Arnowitt-Deser-Misner) -coordinates Hamilto- nian does not [16,17]. Levin [18] thought that there is no formal conflict between them since the two approaches are not exactly but approximately equal, and different dynam- ical behaviors between the two approximately related sys- tems are permitted according to dynamical systems theory. Seen from the canonical, conjugate spin coordinates [19],2 the former non-integrability and the latter integrability are clearer. As extensions, both any PN conservative Hamilto- nian binary system with one spinning body and a conser- vativeHamiltonianof two spinningbodieswithoutthe con- straint of equal mass or with the spin-orbit couplings not restricted to the leading order are still integrable. Recen tly, [20,21]arguedtheintegrabilityofthe2PNHamiltonianwit h- outthespin-spincouplingsandwiththeNLOand/orNNLO spin-orbitcontributionsincluded.Onthecontrary,theco rre- spondingLagrangiancounterpartwithspineffectslimited to the spin-orbit interactionsup to the NLO terms exhibitsthe stronger chaoticity [22].Third doubt relates to different de- pendence of chaos on single dynamical parameter or initial condition.Thedescriptionofthechaoticregionsandchaot ic parameter spaces in [15] are inconsistent with that in [23]. The differentclaims are regardedto be correct accordingto thestatementof[24]thatchaosdoesnotdependonlyonsin- glephysicalparameterorinitialconditionbutacomplicat ed combinationofallparametersandinitial conditions. It isworthemphasizingthatthe spin-spineffectsarethe most important source for causing chaos in spinning com- pact binaries, but they were only restricted to the LO term in the published paperson research of the chaotic behavior. It should be significant to discuss the NLO spin-spin cou- plings included to a contribution of chaos. For the sake of this, we shall considera complete3PN conservativeHamil- tonian of two spinning black holes, where the orbital part is up to the 3PN order and the spin-spin part as well as the spin-orbitpartincludestheLOandNLOinteractions.Inthi s way, we want to know whether the inclusion of the NLO spin-spin couplings have an effect on chaos, and whether thereischaosbeforecoalescenceofthe binaries. 2 Third post-NewtonianorderHamiltonianapproach Itistoodifficulttostrictlydescribethedynamicsofasyst em oftwomasscomparablespinningblackholesingeneralrel- ativity.Instead,thePNapproximationmethodisoftenused . Suppose that the two bodies have masses m1andm2with m1≤m2. Other mass parameters are the total mass M= m1+m2,thereducedmass µ=m1m2/M,themassratio β= m1/m2andthemassparameter η=µ/M=β/(1+β)2.As tootherspecifiednotations,a 3-dimensionalvector rrepre- sentstherelativepositionofbody1tobody2,itsunitradia l vectorisn=r/rwiththeradius r=|r|,andpstandsforthe momentaofbody1relativetothecentre.Themomenta,dis- tancesandtime tarerespectivelymeasuredintermsof µ,M andM.Additionally,geometricunits c=G=1areadopted. The two spin vectors are Si=SiˆSi(i=1,2) with unit vec- torsˆSiandthespinmagnitudes Si=χim2 i/M2(0≤χi≤1). InADMcoordinates,thesystemcanbeexpressedasthedi-mensionlessconservative3PN Hamiltonian H(r,p,S1,S2) =Ho(r,p)+Hso(r,p,S1,S2) +Hss(r,p,S1,S2). (1) In the following, we write its detailed expressionsalthoug h theyaretoolong. Fortheconservativecase,theorbitalpart Hodoesnotin- cludethedissipative2.5PNterm(whichistheleadingorder radiationdampinglevel)buttheNewtonianterm HNandthe PNcontributions H1PN,H2PNandH3PN, thatis, Ho=HN+H1PN+H2PN+H3PN. (2) Asgivenin [25],theyare HN=p2 2−1 r, (3) H1PN=1 8(3η−1)p4−1 2[(3+η)p2+η(n·p)2]1 r +1 2r2, (4) H2PN=1 16(1−5η+5η2)p6+1 8[(5−20η−3η2)p4 −2η2(n·p)2p2−3η2(n·p)4]1 r+1 2[(5+8η)p2 +3η(n·p)2]1 r2−1 4(1+3η)1 r3, (5) H3PN=1 128(−5+35η−70η2+35η3)p8+1 16[(−7 +42η−53η2−5η3)p6+(2−3η)η2(n·p)2 ×p4+3(1−η)η2(n·p)4p2−5η3(n·p)6]1 r +[1 16(−27+136η+109η2)p4+1 16(17 +30η)η(n·p)2p2+1 12(5+43η)η(n·p)4]1 r2 +{[−25 8+(1 64π2−335 48)η−23 8η2]p2 +(−85 16−3 64π2−7 4η)η(n·p)2}1 r3 +[1 8+(109 12−21 32π2)η]1 r4. (6) The spin-orbit part Hsois linear functions of the two spins. It is the sum of the LO spin-orbit term HLO soand the NLOspin-orbitterm HNLO so,i.e. Hso(r,p,S1,S2)=HLO so(r,p,S1,S2)+HNLO so(r,p,S1,S2). (7) Ref.[5]gavetheirexpressions Hso=1 r3[g(r,p)S+g∗(r,p)S∗]·L, (8)3 wheretherelatednotationsare S=S1+S2,S∗=1 βS1+βS2, g(r,p) =2+[19 8ηp2+3 2η(n·p)2−(6+2η)1 r], g∗(r,p) =3 2+[−(5 8+2η)p2+3 4η(n·p)2 −(5+2η)1 r], and the Newtonian-lookingorbital angularmomentumvec- toris L=r×p. (9) The constant terms in gandg∗correspond to the LO part, andtheothers,theNLO part. Similarly,thespin-spinHamiltonian Hssalsoconsistsof theLOspin-spincouplingterm HLO ssandtheNLOspin-spin couplingterm HNLO ss, namely, Hss(r,p,S1,S2)=HLO ss(r,S1,S2)+HNLO ss(r,p,S1,S2).(10) Thefirst sub-Hamiltonianreads[25] HLO ss=1 2r3[3(S0·n)2−S2 0] (11) withS0=S+S∗. The second sub-Hamiltonian is made of threeparts, HNLO ss=Hs2 1p2+Hs2 2p2+Hs1s2p2. (12) Theyarewrittenas[7,8] Hs2 1p2=η2 β2r3[1 4(p1·S1)2+3 8(p1·n)2S2 1 −3 8p2 1(S1·n)2−3 4(p1·n)(S1·n)(p1·S1)] −η2 r3[3 4p2 2S2 1−9 4p2 2(S1·n)2] +η2 r3β[3 4(p1·p2)S2 1−9 4(p1·p2)(S1·n)2 −3 2(p1·n)(p2·S1)(S1·n) +3(p2·n)(p1·S1)(S1·n) +3 4(p1·n)(p2·n)S2 1 −15 4(p1·n)(p2·n)(S1·n)2], (13) Hs2 2p2=HS2 1p2(1↔2), (14)Hs1s2p2=η2 2r3{3 2{[(p1×S1)·n][(p2×S2)·n] +6[(p2×S1)·n][(p1×S2)·n] −15(S1·n)(S2·n)(p1·n)(p2·n) −3(S1·n)(S2·n)(p1·p2) +3(S1·p2)(S2·n)(p1·n) +3(S2·p1)(S1·n)(p2·n) +3(S1·p1)(S2·n)(p2·n) +3(S2·p2)(S1·n)(p1·n) −1 2(S1·p2)(S2·p1)+(S1·p1)(S2·p2) −3(S1·S2)(p1·n)(p2·n) +1 2(S1·S2)(p1·p2)} +3η2 2r3β{−[(p1×S1)·n][(p1×S2)·n] +(S1·S2)(p1·n)2 −(S1·n)(S2·p1)(p1·n)} +3η2β 2r3{−[(p2×S2)·n][(p2×S1)·n] +(S1·S2)(p2·n)2 −(S2·n)(S1·p2)(p2·n)} +6η r4[(S1·S2)−2(S1·n)(S2·n)]. (15) Here,p1=−p2=p. In a word, the conservative Hamilto- nian (1) up to the 3PN order is not completely given until Eq. (15) appears. Clearly, Hamiltonian (1) does not depend onanymassbutthemassratio. Theevolutionsofposition randmomentum psatisfythe canonicalequationsofthe Hamiltonian(1): dr dt=∂H ∂p,dp dt=−∂H ∂r. (16) Thespinvariablesvarywithtimeaccordingtothefollowing relations dSi dt=∂H ∂Si×Si. (17) Besides the two spin magnitudes, there are four con- served quantities in the Hamiltonian (1), includingthe tot al energyE=Handthreecomponentsofthetotalangularmo- mentumvector J=L+S.Afifthconstantofmotionisab- sent,sotheHamiltonian(1)isnon-integrable.1Itshighnon- linearity seems to imply that it is a richer source for chaos. Next,we shall search forchaos, andparticularlyinvestiga te 1Based on the idea of [19], the Hamiltonian (1) can be expresse d as a completely canonical Hamiltonian with a 10-dimensional ph ase space when the canonical, conjugate spin coordinates are used ins tead ofthe original spin variables. If the system is integrable, at lea st five inde- pendent integrals of motion beyond the constant spin magnit udes are necessary.4 theeffectoftheNLOspin-spininteractionsonthedynamics ofthesystem. 3 Detectionofchaosbeforecoalescence With numerical simulations, we use some chaos indicators to describe dynamical differences between the NLO spin- spincouplingsexcludedandincluded.Theappropriateones of the indicators are selected to study dependence of chaos on single parameter when the NLO spin-spin couplingsare included. Finally, we expect to find chaos before coales- cencebyestimatingtheLyapunovandinspiraldecaytimes. 3.1 Comparisons Numerical methods are convenient to study nonlinear dy- namics of the Hamiltonian (1). Symplectic integrators are efficientnumericaltoolssincetheyhavegoodgeometricand physical properties, such as the symplectic structure con- servedandenergyerrorswithoutsecularchanges.However, they cannot provide high enough accuracies, and the com- putationsare expensivewhenthemixedsymplecticintegra- tionalgorithms[21,26]withacompositeofthesecond-orde r explicitleapfrogsymplecticintegratorandthesecond-or der implicit midpoint rule are chosen. In this sense, we would prefer to adopt an 8(9) order Runge-Kutta-Fehlberg algo- rithm of variable time steps. In fact, it gives such high ac- curacy to the energy error in the magnitude of about order 10−13∼10−12whenintegrationtimereaches106,asshown in Fig. 1. Here, orbit 1 we consider has initial conditions (p(0);r(0))=(0,0.39,0;8.55,0,0),whichcorrespondtothe initial eccentricity e0=0.30 and the initial semi-majoraxis a0=12.2. Other parameters and initial spin angles are re- spectively β=0.79,χ1=χ2=χ=1.0,θi=78.46◦and φi=60◦, where polar angles θiand azimuthal angles φi satisfy the relations ˆSi=(cosφisinθi,sinφisinθi,cosθi), as commonly used in physics. The NLO spin-spin couplings are not included in Fig. 1(a), but in Fig. 1(b). It can be seen clearly that the inclusion of the NLO spin-spin cou- plings with a rather long expression decreases only slightl y thenumericalaccuracy.Therefore,ournumericalresultsa re showntobereliablealthoughtheenergyerrorshavesecular changes. We apply several chaos indicators to compare dynam- ical behaviors of orbit 1 according to the two cases with- out and with the NLO spin-spin couplings. The method of Poincarésurfaceofsectioncanprovideacleardescription of the structureof phasespace toa conservativesystem whose phasespaceis4dimensions.Asapointtonote,itisnotsuit- able for such a higher dimensional system (1). Fortunately, power spectra, Lyapunov exponents and fast Lyapunov in-dicatorswould work well in finding chaos regardlessof the dimensionalityofphasespace. 3.1.1 Powerspectrumanalysis Power spectrum analysis reveals a distribution of various frequencies ωof a signal x(t). It is the Fourier transforma- tion X(ω)=/integraldisplay+∞ −∞x(t)e−iωtdt, (18) whereiis the imaginary unit. In general, the power spec- traX(ω)are discrete for periodic and quasi periodic orbits but continuous for chaotic orbits. That is to say, the classi - fication of orbits can be distinguished in terms of different featuresofthespectra.Onthebasisofthis,weknowthrough Fig.2thattheorbitseemstoberegularwhentheNLOspin- spin couplingsare not included, but chaotic when the NLO spin-spin couplingsare included.Notice that the method of power spectra is only a rough estimation of the regularity and chaoticity of orbits. More reliable chaos indicators ar e stronglydesired. 3.1.2 Lyapunovexponents The maximum Lyapunov exponent is used to measure the averageseparationrateoftwoneighboringorbitsinthepha se spaceandgivesquantitativeanalysistothestrengthofcha os. Its calculations are usually based on the variational metho d andthetwo-particlemethod[27].Theformerneedssolving thevariationalequationsaswellastheequationsofmotion , and the latter needs solving the equations of motion only. Considering the difficulty in deriving the variational equa - tions of a complicated dynamical system, we pay attention to the application of the latter method. In the configuration space,it is definedas[28] λ=lim t→∞1 tln|Δr(t)| |Δr(0)|, (19) where|Δr(0)|and|Δr(t)|are the separations between the two neighboring orbits at times 0 and t, respectively. The initialdistancecannotbetoobigortoosmall,and10−8isre- gardedasto its suitablechoice in the doubleprecision[27] . Forthesakeoftheoverflowavoided,renormalizationsfrom time to time are vital in the tangent space. A bounded orbit is chaotic if its Lyapunov exponent is positive, but regular when its Lyapunov exponent tends to zero. In this way, we canknowfromFig.3thatorbit1isregularforthecasewith- out the NLO spin-spin couplings, but chaotic for the case withthe NLO spin-spincouplings.Of course,it takesmuch computational cost to distinguish between the ordered and chaoticcases.5 3.1.3 FastLyapunovindicators A quicker method to find chaos than the method of Lya- punov exponents is a fast Lyapunov indicator (FLI). This indicator that was originally considered to measure the ex- pansion rate of a tangential vector [29] does not need any renormalization,whileitsmodifiedversiondealingwithth e useofthe two-particlemethod[30]does.Themodifiedver- sionisoftheform FLI(t)=log10|Δr(t)| |Δr(0)|. (20) Itscomputationisbasedonthefollowingexpression: FLI=−k(1+log10|Δr(0)|)+log10|Δr(t)| |Δr(0)|, (21) wherekdenotes the sequential number of renormalization. The FLI of Fig. 4(a) corresponding to Fig. 3(a) increases algebraically with logarithmic time log10t, and that of Fig. 4(b)correspondingtoFig.3(b)doesexponentiallywithlog - arithmic time. The former indicates the character of order, butthelatter,thefeatureofchaos.Onlywhentheintegrati on time addsupto 1 ×105, canthe orderedandchaotic behav- iorsbe identifiedclearly forthe use of FLI unlikethe appli- cation of Lyapunov exponent.There is a threshold value of the FLIs between order and chaos, 5. Orbits whose FLI are largerthan5arechaotic,whereasthosewhoseFLIsareless than5areregular. The above numerical comparisons seem to tell us that chaosbecomeseasierwhentheNLOspin-spintermsarein- cluded. This sounds reasonable. As claimed in [20,21], the system(1)isintegrableandnotatallchaoticwhenthespin- spin couplings are turned off. The occurrence of chaos is completely due to the spin-spin couplings, which include particularly the NLO spin-spin contributions leading to a sharpincreaseinthestrengthofnonlinearity.Infact,wee m- ploy FLIs to find that there are other orbits (such as orbits 2-5in Table1),whichare notchaoticforthe absenceofthe NLO spin-spin couplings but for the presence of the NLO spin-spincouplings.Inaddition,the strengthofthechaot ic- ity of orbits 6-8 increases. As a point to illustrate, the oth er initial conditions beyond Table 1 are those of orbit 1; the starting spin unit vectorsof orbit 2 are those of orbit 1, and those of orbits 3-8 are θ1=84.26◦,φ1=60◦,θ2=84.26◦ andφ2=45◦. Hereafter,onlythe dynamicsof the complete Hamiltonian (1) with the NLO spin-spin effects included is focusedon. 3.2 Lyapunovandinspiraldecaytimes Takingβ=0.5,theinitialconditionsandtheinitialunitspin vectors of orbit 1 as reference, we start with the spin pa- rameterχat the value 0.2 that is increased in increments of 0.01 up to a final value of 1 and draw dependence ofFLI onχin Fig. 5(a). This makes it clear that chaos oc- curs when χ≥0.7. Precisely speaking, the larger the spin magnitudes get, the stronger the chaos gets. Note that this dependence of chaos on χrelies typically on the choice of the initial conditions, the initial unit spin vectors and th e other parameters. As claimed in [24], there is different de- pendenceof chaoson χif the choice changes. On the other hand,takingtheinitialspinanglesoforbit3,fixingthespi n parameter χ=0.90andtheinitialconditions (p(0);r(0))= (0,0.39,0;8.4,0,0), which correspond to the initial eccen- tricitye0=0.28 and the initial semi-major axis a0=11.6, we study the range of the mass ratio βbeginningat 0.5 and ending at 1 in increments of 0.01. At once, dependence of FLIonβcanbedescribedinFig.5(b).Thereischaoswhen β≤0.86 and chaos seems easier for a smaller mass ratio. Asinthepanel(a),thisresultisgivenonlyunderthepresen t initialconditions,initialunitspinvectorsandotherpar ame- ters. Dothe above-mentionedchaoticbehaviorsoccurbefore themergerofthebinaries?Toanswerit,wehavetocompare the Lyapunov time Tλ=1/λ(i.e.the inverse of the Lya- punov exponent)with the inspiral decay time Td, estimated by[31] Td=12 19c4 0 γ/integraldisplaye0 0e29/19[1+(121/304)e2]1181/2299 (1−e2)3/2de,(22) wherethetwo parametersare c0=a0(1−e2 0)e−12/19 0(1+121 304e2 0)−870/2299(23) andγ=64m1m2M/5.WhenTλislessthan Td(orλTd>1), chaoswouldbeobserved.Because Tλ=3.0×103andTd= 1.3×103for orbit 1, the chaoticity can not be seen before the merger. Values of λTdfor orbits 2-8 are listed in Table 1. Clearly, only chaotic orbit 8 is what we expect. Besides these,we plottwo panels(a) and(b)of Fig. 6 regardingde- pendenceofLyapunovexponentonsingleparameter,which correspondrespectively to Figs 5(a) and 5(b). Here are two facts.First,theresultsinFig.6arethesameasthoseinFig . 5. Second, lots of chaotic orbits whose Lyapunovtimes are many times greater than the inspiral times should be ruled out, and there are only a small quantity of desired chaotic orbitsleft. InordertomaketheaccuracyofthePNapproachbetter, weshouldchooseorbitswhoseinitialradiiarelargerenoug h thanroughly10 M.AllchaoticorbitsinTable2areexpected. Noticethattheotherinitialconditionsoftheseorbitsbey ond this table are y=z=px=pz=0, and the starting spin an- gles are still the same as those of orbit 3. Althoughan orbit hasa largeinitialradius,it maystill bechaoticwhenitsin i- tial eccentricity is high enough. This supports the result o f [23]thata higheccentricorbitcaneasilyyieldchaos.6 Table1ValuesofFLIsand λTdfordifferent orbits.FLIacorresponds totheNLOspin-spin couplingsturned off.FLIb, λ,λdandλTdcorrespond tothe NLO spin-spin couplings included. Orbit β χ x p ye0a0FLIa FLIb λ λ dλTd 2 0.90 1.0 8.55 0.390 0.30 12.2 3.6 10.2 1.9E-4 8.1E-4 0.2 3 0.50 0.93 18.6 0.195 0.29 14.4 4.5 9.0 1.7E-4 3.7E-4 0.5 4 0.71 0.95 17.5 0.200 0.30 13.5 3.9 12.4 2.8E-4 5.3E-4 0.5 5 0.65 0.90 35.4 0.100 0.65 21.5 4.2 9.8 2.1E-4 3.8E-4 0.7 6 0.86 0.90 8.40 0.390 0.28 11.6 6.5 20.4 4.2E-4 9.3E-4 0.5 7 0.50 0.83 19.5 0.185 0.33 14.6 7.0 11.5 2.5E-4 3.8E-4 0.7 8 0.50 0.97 18.7 0.190 0.32 14.1 9.4 22.5 5.5E-4 4.3E-4 1.3 Table 2Values of λTdfor chaotic orbitswithbig initialradii when the NLOspin-s pin contributions are included. Orbit χ β x p ye0a0λ λ dλTd 9 0.95 0.50 14.5 0.240 0.16 12.5 6.8E-4 5.2E-4 1.3 10 0.97 0.50 19.5 0.185 0.33 14.6 3.8E-4 3.7E-4 1.0 11 0.93 0.50 20.5 0.175 0.37 14.9 4.4E-4 3.9E-4 1.1 12 0.93 0.50 25.5 0.140 0.50 17.0 4.7E-4 3.8E-4 1.2 13 0.90 0.48 35.4 0.100 0.65 21.5 4.2E-4 3.5E-4 1.2 14 0.90 0.44 35.4 0.100 0.65 21.5 4.3E-4 3.4E-4 1.3 4 Conclusions This paper is devoted to studying the dynamicsof the com- plete3PNconservativeHamiltonianofspinningcompactbi- naries in which the orbital part is accurate to the 3PN order and the spin-spinpart as well asthe spin-orbitpart include s theLO andNLO contributions.Because ofthehighnonlin- earity,theNLOspin-spincouplingsincludedgiverisetoth e occurrenceof strong chaos in contrast with those excluded. By scanning single parameter with the FLIs, we obtained dependence of chaos on the parameter. It was shown suffi- ciently that chaos appears easier for larger spins or smalle r mass ratios under the present considered initial condition s, starting unit spin vectors and other parameters. So does for a smaller initial radius. In spite of this, an orbit with a lar ge initialradiusisstill possiblychaoticifitsinitialecce ntricity ishighenough.Aboveall,therearesome observable chaotic orbits whose initial radii are suitably large and whose Lya- punovtimesarelessthanthecorrespondinginspiraltimes. Acknowledgements This research is supported by the Natural Sci- ence Foundation ofChina under Grant Nos. 11173012 and 11178 002. References 1. P.Jaranowski, G. Schäfer, Phys. Rev. D 55, 4712 (1997) . 2. N.E.Pati and C. M.Will,Phys. Rev. D 65, 104008 (2002). 3. T. Damour, P. Jaranowski, G. Schäfer, Phys. Rev. D 77, 064032 (2008). 4. T.Damour,P.JaranowskiandG.Schäfer,Phys.Rev.D 78,024009 (2008).5. A. Nagar, Phys. Rev. D 84, 084028 (2011). 6. J. Hartung, J.Steinhoff, Ann. Phys. 523, 919 (2011). 7. J. Steinhoff, S. Hergt and G. Schäfer, Phys. Rev. D 77, 081501 (2008). 8. S. Hergt, G.Schäfer, Phys. Rev. D 78, 124004 (2008). 9. M.Levi, Phys. Rev. D 85,064043 (2012). 10. J. Levin, Phys. Rev. Lett. 84, 3515 (2000). 11. J. D. Schnittman and F. A. Rasio, Phys. Rev. Lett. 87, 121101 (2001). 12. N. J.Cornish and J. Levin, Phys. Rev. Lett. 89,179001 (2002). 13. N. J.Cornish and J. Levin, Phys. Rev. D 68, 024004 (2003). 14. X. Wu and Y.Xie,Phys. Rev. D 76, 124004 (2007). 15. J. Levin, Phys. Rev. D 67,044013 (2003). 16. C. Königsdörffer and A. Gopakumar, Phys. Rev. D 71, 024039 (2006). 17. A.Gopakumar andC.Königsdörffer, Phys.Rev. D 72,121501(R) (2005). 18. J. Levin, Phys. Rev. D 74,124027 (2006). 19. X. Wu,Y.Xie, Phys. Rev. D 81, 084045 (2010). 20. X. Wu and S. Y.Zhong, Gen. Relat.Gravit. 43,2185 (2011). 21. L. Mei,M.Ju, X.Wu,and S.Liu, Mon.Not.R. Astron. Soc. 435, 2246 (2013). 22. Y. Wang and X. Wu, Classical Quantum Gravity 28, 025010 (2011). 23. M.D. Hartl,A.Buonanno, Phys. Rev. D 71, 024027 (2005). 24. X. Wu and Y.Xie,Phys. Rev. D 77, 103012 (2008). 25. A. Buonanno, Y. Chen, T. Damour, Phys. Rev. D 74, 104005 (2006). 26. L. Mei,X.Wu, and F. Liu,Eur. Phys. J. C 73, 2413 (2013). 27. G.Tancredi, A.Sánchez and F.Roig,Astorn.J. 121, 1171 (2001). 28. X. Wu and T.Y.Huang, Phys. Lett.A 313, 77 (2003). 29. C. Froeschlé and E. Lega, Celest. Mech. Dyn. Astron. 78, 167 (2000). 30. X. Wu, T. Y. Huang, and H. Zhang, Phys. Rev. D 74, 083001 (2006). 31. P. C.Peters, Phys. Rev. B 136, 1224 (1964).7 /s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53 /s78/s111/s32/s32/s32/s32 /s32/s72 /s115/s115/s78/s76/s79 /s40/s97/s41 /s32/s32/s69/s47/s49/s48/s45/s49/s51 /s116/s32/s47/s49/s48/s53/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52 /s72 /s115/s115/s78/s76/s79 /s40/s98/s41 /s32/s32/s69/s47/s49/s48/s45/s49/s51 /s116/s47/s49/s48/s53 Fig.1Energy errors of orbit1. The NLO spin-spin couplings are not included inpanel (a) butin Panel (b). /s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s78/s111/s32/s32/s32/s32 /s72 /s115/s115/s78/s76/s79 /s40/s97/s41 /s32/s32/s112/s111/s119/s101/s114/s32/s32/s115/s112/s101/s99/s116/s114/s117/s109 /s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s47/s32/s49/s48/s45/s52/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54 /s72 /s115/s115/s78/s76/s79 /s40/s98/s41 /s32/s32/s112/s111/s119/s101/s114/s32/s32/s115/s112/s101/s99/s116/s114/s117/s109 /s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s47/s32/s49/s48/s45/s52 Fig.2Power spectra corresponding toFig. 1. /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s45/s53/s46/s48/s45/s52/s46/s53/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53 /s78/s111/s32/s32/s32/s32 /s72 /s115/s115/s78/s76/s79 /s40/s97/s41 /s32/s32/s108/s111/s103 /s49/s48/s32/s124 /s124 /s108/s111/s103 /s49/s48/s32/s116/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48 /s72 /s115/s115/s78/s76/s79 /s40/s98/s41 /s32/s32/s108/s111/s103 /s49/s48/s32/s124 /s124 /s108/s111/s103 /s49/s48/s32/s116 Fig.3The maximum Lyapunov exponents λcorresponding to Fig.1.8 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53 /s78/s111/s32/s32/s32/s32 /s72 /s115/s115/s78/s76/s79 /s40/s97/s41 /s32/s32/s70/s76/s73 /s108/s111/s103 /s49/s48/s32/s116/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54 /s72 /s115/s115/s78/s76/s79 /s40/s98/s41 /s32/s32/s70/s76/s73 /s108/s111/s103 /s49/s48/s32/s116 Fig.4The fast Lyapunov indicators (FLIs) corresponding to Fig.1 . /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48 /s32/s32/s70/s76/s73/s40/s97/s41 /s48/s46/s53/s48 /s48/s46/s53/s53 /s48/s46/s54/s48 /s48/s46/s54/s53 /s48/s46/s55/s48 /s48/s46/s55/s53 /s48/s46/s56/s48 /s48/s46/s56/s53 /s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48 /s32/s32/s70/s76/s73/s40/s98/s41 Fig.5(color online)The FLIs asa function of χorβwhen the NLO spin-spin interactions are included. AllFLIs l arger than 5 mean chaos. /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54 /s99/s61/s49/s48/s45/s52/s100/s61/s55/s46/s49/s56/s42/s49/s48/s45/s52/s40/s97/s41 /s32/s32/s47/s49/s48/s45/s52 /s48/s46/s53/s48 /s48/s46/s53/s53 /s48/s46/s54/s48 /s48/s46/s54/s53 /s48/s46/s55/s48 /s48/s46/s55/s53 /s48/s46/s56/s48 /s48/s46/s56/s53 /s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56 /s99/s61/s49/s48/s45/s52/s40/s98/s41 /s100 /s32/s32/s47/s49/s48/s45/s52 Fig. 6(color online) The maximum Lyapunov exponents λcorresponding to Fig. 5. Note that λ>λcmeans chaos, and λ>λdwithλd=1/Td indicatesthe occurrence of chaos before coalescence.

2006.08253v1.Control_of_Spin_Relaxation_Anisotropy_by_Spin_Orbit_Coupled_Diffusive_Spin_Motion.pdf

Control of Spin Relaxation Anisotropy by Spin-Orbit-Coupled Diusive Spin Motion Daisuke Iizasa,1Asuka Aoki,1Takahito Saito,1Junsaku Nitta,1, 2, 3Gian Salis,4and Makoto Kohda1, 2, 3, 5 1Department of Materials Science, Tohoku University, Sendai 980{8579, Japan 2Center for Spintronics Research Network, Tohoku University, Sendai 980{8577, Japan 3Center for Science and Innovation in Spintronics (Core Research Cluster), Tohoku University, Sendai 980{8577, Japan 4IBM Research-Zurich, S aumerstrasse 4, 8803 R uschlikon, Switzerland. 5Division for the Establishment of Frontier Sciences, Tohoku University, Sendai 980-8577, Japan (Dated: June 16, 2020) Spatiotemporal spin dynamics under spin-orbit interaction is investigated in a (001) GaAs two- dimensional electron gas using magneto-optical Kerr rotation microscopy. Spin polarized electrons are diused away from the excited position, resulting in spin precession because of the diusion- induced spin-orbit eld. Near the cancellation between spin-orbit eld and external magnetic eld, the induced spin precession frequency depends nonlinearly on the diusion velocity, which is unex- pected from the conventional linear relation between the spin-orbit eld and the electron velocity. This behavior originates from an enhancement of the spin relaxation anisotropy by the electron ve- locity perpendicular to the diused direction. We demonstrate that the spin relaxation anisotropy, which has been regarded as a material constant, can be controlled via diusive electron motion. Precise control of spin motion is a prerequisite from fundamental physics to spintronics and quantum infor- mation technology [1{4]. In a semiconductor quantum well (QW), Rashba [5, 6] and Dresselhaus [7] spin{orbit (SO) interactions act as eective magnetic elds for mov- ing electrons, enabling coherent spin control via preces- sion, whereas spin relaxation occurs simultaneously be- cause of an interplay between the SO eld and the ran- dom motion of electrons [8]. Both spin precession and relaxation processes are closely tied to one another solely by SO interaction [9]. For stationary electrons with mean zero velocity, the correlation between precession and re- laxation triggers a modulation of spin precessional mo- tion, known as spin relaxation anisotropy [10{19]. For spin rotation by external and/or SO elds in a QW, spins along growth and in-plane orientations do not experience identical torques because of the in-plane orientation of the SO elds. This situation induces anisotropic spin relaxation [10{19] and modulates the spin precession fre- quency [13{16, 18, 19]. Because SO elds are well dened for stationary electrons, the spin relaxation anisotropy has been regarded as a material constant. However, for moving electrons with a nite net velocity induced by drift [20{25] and diusion [24{27], the electron trajectory further modulates SO elds and directly aects the spin relaxation anisotropy through the momentum-dependent spin precession. Moreover, the spin relaxation anisotropy is not limited to particular materials such as III{V semi- conductors because the anisotropic SO elds are ubiqui- tous in solid states, with spin-momentum locking in topo- logical insulators [28, 29], Rashba interface in oxides [30], metal interfaces [31], and Zeeman-type SO eld in 2D ma- terials [32]. Consequently, unveiling the eects of moving electrons on the modulation of spin relaxation anisotropy and induced precession frequency are expected to be cru- cially important for future spintronics, topological elec- tronics, and quantum information technologies. Despitethis, earlier studies of spin relaxation anisotropy have re- mained limited only to stationary cases [13{16, 18, 19]. Here, we experimentally manifest control of spin pre- cessional motion via spin relaxation anisotropy by dif- fusive spin motion in a GaAs-based QW. When the SO eld under diusive motion is nearly compensated by a constant external magnetic eld, the spin precession fre- quency is no longer linear to the diusion velocity. This behavior cannot be anticipated from a conventional spin drift/diusion model. It is explained by a modulation of the spin relaxation anisotropy. The evaluated spin re- laxation anisotropy, which exhibits six-fold enhancement from the stationary case, is explained by a tilting of the spin precession axis from the direction of external mag- netic eld caused by the electron diusive motion. We in uence the spin relaxation anisotropy, as reported for the rst time, by precisely controlling the electron mo- tion. The structure examined for this study was an n-doped 20-nm-thick (001) GaAs QW. In this system, we obtain SO elds characterized by the Rashba parameter (<0), the Dresselhaus parameter =13(>0), with linear 1= hk2 ziand cubic term 3= k2 F=4. Here,hk2 zi denotes the expected value of the squared wavenumber in the QW. The bulk Dresselhaus coecient is < 0. The Fermi wavenumber is kF=p2ns. The carrier density and mobility measured using a Hall bar device werens= 1:721011cm2and 11:2104cm2V1s1, respectively, at 4.2 K. To detect the diusive spin dy- namics, spatiotemporal Kerr rotation microscopy is per- formed using a mode-locked Ti:sapphire laser emitting 2-ps-long pulses at a 79.2 MHz repetition rate. Fig- ure 1(a) depicts an experimental conguration for pump and probe beams with Rashba and Dresselhaus SO elds. Therein, Rand Drespectively represent the spin pre- cession frequency vectors. A circularly polarized pump beam with Gaussian sigma-width ppis focused onto thearXiv:2006.08253v1 [cond-mat.mes-hall] 15 Jun 20200 r x|| [110]-y|| [110] Ωex,yΩex,x ΩD ΩRFIG. 1. (a) Sketch of a pump-probe scanning Kerr microscopy setup with Rashba ( R) and Dresselhaus ( D) elds as pre- cession vectors. An external magnetic eld is depicted as ex;xand ex;yas a precession vector for y- andx-scans con- gurations, respectively. A circularly polarized pump beam excites a spin polarization sz. A linearly polarized probe beam detects szat a delay time tand a position ( x;y). (b) Measuredszat dierent xpositions highlighted as colored circles in (a). sample surface to excite spin polarization szalong the growth direction. A linearly polarized probe beam (spot sizepr) detectsszat delay time tand arbitrary posi- tion by a motor-controlled scanning mirror. All optical measurements are taken at 30 K. The spin precession frequency induced by a velocity v= (vx;vy) in an external magnetic eld Bex= (Bx;By) is generally described as x;y(vy;x) =2m ~2(+)vy;x+gB ~Bx;y: (1) Hereg <0 stands for the electron gfactor,Bdenotes the Bohr magneton, ~is the reduced Plancks constant, andm= 0:067m0expresses eective electron mass of GaAs. The diusion velocity vdif, which is controlled by the center-to-center distance rbetween pump and probe spots, is dened as vdif= 2Dsr=(2Dss+2 e); (2) whereDsis the spin diusion constant, srepresents the D'yakonov-Perel' spin relaxation time, and the convo- luted spot size eis dened by 2 e=2 pp+2 pr[26]. Also,sis a result of the replacement of t=sbe- cause our system satises 2 Dss2 eand smalls. By changing the probe position along the x-axis (y-axis) [26], i.e., the distance rin Eq. (2) , one can set the dif- fusion velocity vdif=vx(vdif=vy) and thereby modu- late the spin precession frequency [ y(vx) or x(vy) in Eq. (2)]. Figure 1(b) shows the time evolution of the ex- perimental Kerr signal ( sz) at dierent probe positions (x= 9:7;0:8 and12:5m) in anx-scan (e= 8:1 m andBy= +0:45 T). The spin precession frequency depends strongly on the probe position, re ecting the momentum (velocity) dependent SO eld induced by the nite diusion velocity. We systematically measured Kerr signals with dierent positions on the x- andy-axes with several spot sizes e. We extracted the precession 1 ΩD 0x|| [110] ΩR- 0y|| [110] ΩRΩD10,167,209 FIG. 2. Measured spin precession frequency j measjobtained for dierent eandBexand for scans of the pump-probe separation along x(a) andy(b). Diusing spins experience strong SO elds for x-scan, but weak elds for y-scan. All symbols represent experimental data. All solid lines show linear ts. Dashed lines in (a) correspond to the nonlinear ts based on Eq. (3) with at=0:076 GHz. frequencyj measjby tting the normalized Kerr signal sz= exp (t=s) cos (2j measjt+) with phase shift . Figures 2(a) and 2(b) summarize extracted j measj inx- andy-scans. For the y-scan [Fig. 2(b)], j measj varies linearly with the yposition for all conditions of Bxande, re ecting the linear dependence of vdifon theyposition, as presented in Eq. (2). In addition, when edecreases from 11.6 to 6.8 m, the slope d meas=dy increases gradually, which agrees well with Eq. (2) and which is consistent with earlier reports of the literature [20, 22, 23, 25{27]. For the x-scan [Fig. 2(a)], however, a linear variation of j measjon thexposition is only ob- served fore= 9:8m andBy= +0:45 T (diamond symbols). Reducing eto 8.1 and 5.6 m exhibits a de- viation from a linear variation; notably most pronounced whenj measjapproaches zero. This cannot be explained using the conventional linear relation between electron velocity and SO eld. To understand this eect, we rst evaluate the SO parameters from the linear frequency variation. From linear ts depicted as solid lines in Figs. 2(a) and 2(b), we obtain =2:891013eVm, 1= 1:861013eVm, and 3= 0:221013eVm. Also,g=0:268 is estimated at r= 0 (vdif= 0). We assume g < 0 based on the QW thickness [23]. Also,Ds= 0:0195 m2=s is derived from the measured s= 75 ps at Bex=0T [33]. Using evaluated 1;3, andns, we obtain =8:31 eV A3which is consistent with values reported in the literature [34]. To explain our observation, we introduce in analogy to anisotropic spin relaxation for stationary electrons modied spin preces- 2sion frequencies [13{16, 18, 19],  x=q x(vdif)22 at;  y=q y(vdif)22 at;(3) where the anisotropic term [15, 18] is at() =(xcos2 + ysin2)=2: (4) Here the relaxation rate of spins oriented along x- and y-axes is x;y= (4Dsm2=~4)[(+)2+2 3], respec- tively, and 2[0;2] is the direction of the spin pre- cession axis, dened as in-plane polar angle from + x- to- ward +y-axis. The term at() describes the relaxation anisotropy between the two relevant orthogonal crystal axes and is responsible for a correction of the precession frequency [Eq. (3)]. For the y-scan ( Bex= (Bx;0)) spins precess in the y-zplane, and at( = 0) =x=2 = (yz)=2 denotes half of the dierence of the relax- ation rate between y- andz-axes, where z= x+ y is the relaxation rate along the z-axis. For the x-scan (Bex= (0;By)), at(=2) =y=2. Because at() additionally contributes to the spin precession frequency shown in Eq. (3),  x;yshows a nonlinear dependence on the probe position r, which becomes pronounced when the precession frequency induced by external and SO elds becomes comparable to x;y=2. Based on the ex- perimentally evaluated values for ;1;3, andDs, we calculatey=2 =0:076 andx=2 =0:99 GHz. Fore= 5:6 and 8.1m, the calculated  yare shown as dashed lines in Fig. 2(a). The calculated values only reproduce the experimental data in a linear frequency region. The rapid decrease of  ythat occurs below 0.8 GHz cannot be explained by y=2 =0:076 GHz. According to Eq. (4), the frequency modulation caused by the relaxation anisotropy depends on the direction of the precession axis (). For stationary electrons under Bex, where the SO eld does not contribute to frequency modulation,  is well-dened by the direction of Bex. However, for moving electrons, the precession axis is de- ned by the sum of Bexand the SO eld, implying that the electron trajectories under diusion further modu- late . Because various electron trajectories can lead from the pump spot to the probe spot, the precession axis is no longer well dened by Bexbecause of dierent diusion velocity vectors v= (vx;vy) in times[dier- ent arrows in Fig. 3(a)]. Specically examining one single trajectory with average velocity v, its direction of aver- age precession axis ( vx;vy) = arctan( y(vx)= x(vy)) can be obtained from Eq. (1). Entering ( vx;vy) into Eq. (4) directly reveals the velocity-dependent form of the anisotropic term at(vx;vy) =x x(vy)2+ y y(vx)2 2 ( x(vy)2+ y(vx)2): (5) It is noteworthy that the precession frequencies caused by opposite velocities ( vy;x) do not cancel each other 2 (a) x|| [110]-y (x, 0) vyvx1 2 3 PumpProbe(b) (x, 0) x xy(c)FIG. 3. (a) x-scan conguration where the probe center is separated by a vector ( x;0) from the pump center (0 ;0). Elec- tron trajectories exist with an average velocity that is tilted from thex-axis, contrary to the macroscopic diusion veloc- ity [Eq. (2)]. (b) Mean velocity components of horizontal and vertical directions, v handv v. (c) Calculated v h,v vbased on Eqs. (6) and (7). v vare constant with respect to rhere, whereasv hdepends on the position. because x;yenter as squares. This nding contrasts to the notion of a single (averaged) diusion velocity for given pump and probe spots [described by Eq. (2)], corre- sponding to the mean value of all velocity vectors leading from the pump to the probe spots [Fig. 3(a)]. Actually, Eq. (5) rather suggests that the anisotropic term is de- ned by the microscopic behavior of the electron motion (velocity). Therefore, we sort all diusion velocity vec- tors along the x- andy-axes according to their sign and evaluate the mean values of horizontal and vertical ve- locities at probe positions v handv v, respectively, as indicated by silver bold arrows in Fig. 3(b). Each veloc- ity is described as v h=es  ser2s 2 eff+ rerfc rps e! ;(6) v v=ep =(s); (7) where  = Ds=(2Dss+2 e). The complementary error function is denoted by erfc. The signs inv handv v respectively correspond to the positive/negative velocity components along the horizontal or vertical axes. Fig- ure 3(c) shows calculated v handv v. Thev+ hincreases rapidly in the + xregion, whereas v hhas the opposite tendency because of radial diusion of electrons from the excited pump spot. When the probe spot is displaced along the x-axis [Fig. 3(a)], the average velocity vector points to the x- axis because there, jv hj>jv+ hj. Remarkably, vertical velocitiesv vare independent of probe position and are of similar size as v h. This suggests that the tilted veloc- ity vectors should contribute to atand the precession axis () no longer points along Bex. For the total mean velocityv+ h+v h+v+ v+v vas sketched in Fig. 3(b), the v vcancel out each other and are linear in the position [dashed line in Fig. 3(c)], corresponding to the conven- tional macroscopic diusion velocity [Eq. (2)]. By considering both horizontal and vertical velocities [Eqs. (6) and (7)] depicted in Fig. 3(b), we average over 301530 x ( 7m)0!atx (GHz) 01530 y ( 7m)!aty (GHz)<eff = 5.6 7m By<eff = 8.1 7m By = 0.45 T<eff = 9.8 7m By = 0.45 T <eff = 6.8 7m Bx = 0.4 T <eff = 8.1 7m Bx = 0.55 T<eff = 11.6 7m Bx = 0.65 T(a) (b)FIG. 4. (a) Calculated anisotropy term x;y atforx-scan (a) andy-scan (b) as obtained from using Eq. (8). Both x;y at are modulated by r. Parameters for the calculation are all experimentally determined values. all contributions to the anisotropic term [Eq. (5)] for the x-scan with x at= at(v+ h;v+ v) + at(v+ h;v v) + at(v h;v+ v) + at(v h;v v) =4: (8) For they-scan conguration, y atis obtained from Eq. (8) by ipping vhandvvwith each other. We calculate x;y at in Figs. 4(a) and 4(b) with parameters evaluated from the experimental conditions. The x atexhibits a peak struc- ture corresponding to the cancellation between external and SO elds, i.e. y(vdif) = 0. At this position, x atis enhanced by more than six times from y=2 =0:076 GHz. Such a peak structure is observed consistently for dierent spot sizes and Byvalues. The enhanced x atis the consequence of a tilting of the spin precession axis away from Bexdirection due to the v vcomponents that introduce a precession contribution x(v v)2. As seen from Eq. (5), x x(v v)2is introduced in the numera- tor of the expression for at. Fory-scan, y atis only gently modulated with ybecause, in this case, the ad- ditional contribution proportional to y y(v v) is weak compared to the case of x-scan (because xy). In other words, a small SO eld along the y-axis does not tilt the spin precession axis signicantly. In both x- and y-scans, when the magnitude of Bexbecomes suciently large compared to the SO eld, x;y atconverges respec- tively to the stationary cases of 0:076 GHz and0:99 GHz. Finally, we reproduce the nonlinear behavior of the x-scan quantitatively by considering the calculated x;y at in Fig. 4. Figure 5(a) shows the experimentally obtained results of color-coded szin anx-scan and a frequency analysis (circles) for e= 5:6m withBy=0:4 T. Solid and dashed lines respectively correspond to the cal- culated  ybased on Eq. (3) with our new x at[Eq. (8)] and the conventional y=2 =0:076 GHz. The re- maining parameters of Eq. (3) are all experimentally ob- tained. The  yobtained with the new x atshows excel- lent agreement with the experimental values, including the nonlinear variation. To conrm the enhancement of atfurther, we conducted Monte Carlo (MC) simulation. Figure 5(b) presents the simulated Kerr traces, showing 01 sz (arb. units)00.20.40.60.8 t (ns) 000.20.40.60.8t (ns) 000.511.5 | +meas| (GHz) 00.5100.20.40.60.8 t (ns) 015 x ( 7m)00.20.40.60.8t (ns) 015 x ( 7m)0123 | +meas| (GHz) 00.51 sz (arb. units)00.20.40.60.8 t (ns) 02040 y ( 7m)00.20.40.60.8t (ns) 012 | +meas| (GHz) 01 sz (arb. units)x = -8.7 7m x = -8.6 7m(a) (b) (c)x = 17.2 7m x = 17.2 7m(d) (e) (f)Experiment Monte Carlo Experiment I Monte Carlo Iy = -18.2 7m (g)(h) Monte CarloFIG. 5. Experimental and Monte Carlo simulated times- pace records of szinx-scan for (a) and (b) with e= 5:6 m atBy=0:4 T, and for (d) and (e) with e= 8:1 m atBy= 0:45 T. All solid lines are  ycalculated with new x at, and dashed lines with conventional anisotropic term y=2 =0:076 GHz. The time evolution of szis shown at (c) forx8:6m and in (f) for x=17:2m for ex- perimental data (diamonds) and for Monte Carlo simulation results (red solid lines). (g) Monte Carlo simulated time-space records ofszin ay-scan withe= 5:6m,Bx= 0:4 T and g=0:268: the solid line shows  xwith new y at; the dashed line is obtained with conventional value x=2 =0:99 GHz. The gray diamond is the extracted frequency from the (h) time evolution of szaty=18:2 m with Monte Carlo simu- lation (red solid) and the tted curve (dotted black). good agreement with the experimentally obtained result. Atx8:6m, whereas  ywith conventional y=2 suggests nite precession with 0.19 GHz,  ywith the new x atshows a halt of spin precession. As presented in Fig. 5(c), the time evolution of szfor both the experiment (diamond) and MC simulation (red solid) at x8:6m conrms a simple exponential decay with no oscillatory behavior. Similar results also hold for e= 8:1m with By= 0:45 T in Figs. 5(d){5(f), supporting enhancement of the anisotropic term. We also compare y-scan by MC simulation for parameters e= 5:6m andBx= 0:4 T [Fig. 5(g)]. Solid and dashed lines are calculated values of  xwith new y atand conventional x=2 =0:99 GHz, respectively, where  xis enhanced for negative yvalues for new y at. This point is conrmed further in Fig. 5(h) by plotting the time evolution of szaty=18:2m. The spin precession frequency obtained from MC simula- tion aty=18:2m [grey diamond in Fig. 5(g)] shows good agreement with our new model. The quantitative 4agreement shown above reveals clearly that precession by the relaxation anisotropy is not a material constant pa- rameter, but is rather controlled by diusive spin motion. In conclusion, we have experimentally observed an en- hancement of the spin relaxation anisotropy by diusive spin motion. We measured the precession frequency in an external magnetic eld by changing the relative distance between excited pump and detected probe positions in a spatiotemporal Kerr rotation microscope. Because of various electron trajectories for electrons travelling from the pump to the probe position, the spin precession axis is tilted substantially from the external magnetic eld di- rection when the diusion-induced SO eld nearly com- pensates the magnitude of the external magnetic eld. Such a SO-coupled spin-diusive motion controls the re- laxation anisotropy. It is detected as a nonlinear pre- cession frequency modulation. Whereas the relaxation anisotropy is regarded as a constant parameter for sta- tionary electrons, it becomes controllable for moving elec- trons. This eect also points out a threshold to start spin precession at a certain velocity. Because this eect is not limited only to diusive motion, but also can be con- trolled by drift and ballistic transport, our ndings link the eect of the precise control of spin states to future spintronics and quantum information technology. We acknowledge nancial support from the Japanese Ministry of Education, Culture, Sports, Science, and Technology (MEXT) Grant-in-Aid for Scientic Research (Nos. 15H02099, 15H05854, 25220604, and 15H05699), EPSRC-JSPS Core-to-Core program, and the Swiss Na- tional Science Foundation through the National Center of Competence in Research (NCCR) QSIT. D.I. thanks the Graduate Program of Spintronics at Tohoku University, Japan, for nancial support. [1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln ar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). [2] D. D. Awschalom and M. E. Flatt e, Nat. Phys. 3, 153 (2007). [3] B. Behin-Aein, D. Datta, S. Salahuddin, and S. Datta, Nat. Nanotechnol. 5, 266 (2010). [4] M. M. Waldrop, Nature 530, 144 (2016). [5] E. I. Rashba, Sov. Physics, Solid State 2, 1109 (1960). [6] Y. A. Bychkov and E. I. Rashba, J. Phys. C Solid State Phys. 17, 6039 (1984). [7] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [8] M. I. D'yakonov and V. I. Perel', Eksp. Teor. Fiz. 33, 1954 (1971). [9] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997). [10] N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582 (1999). [11] N. S. Averkiev, L. E. Golub, and M. Willander, J. Phys. Condens. Matter 14, R271 (2002).[12] J. Kainz, U. R ossler, and R. Winkler, Phys. Rev. B 68, 075322 (2003). [13] S. D ohrmann, D. H agele, J. Rudolph, M. Bichler, D. Schuh, and M. Oestreich, Phys. Rev. Lett. 93, 147405 (2004). [14] K. Morita, H. Sanada, S. Matsuzaka, C. Y. Hu, Y. Ohno, and H. Ohno, Appl. Phys. Lett. 87, 171905 (2005). [15] N. S. Averkiev, L. E. Golub, A. S. Gurevich, V. P. Evtikhiev, V. P. Kochereshko, A. V. Platonov, A. S. Shkolnik, and Y. P. Emov, Phys. Rev. B 74, 033305 (2006). [16] L. Schreiber, D. Duda, B. Beschoten, G. G untherodt, H.- P. Sch onherr, and J. Herfort, Phys. Rev. B 75, 193304 (2007). [17] D. Stich, J. H. Jiang, T. Korn, R. Schulz, D. Schuh, W. Wegscheider, M. W. Wu, and C. Sch uller, Phys. Rev. B76, 073309 (2007). [18] A. V. Larionov and L. E. Golub, Phys. Rev. B 78, 033302 (2008). [19] M. Griesbeck, M. M. Glazov, E. Y. Sherman, D. Schuh, W. Wegscheider, C. Sch uller, and T. Korn, Phys. Rev. B85, 085313 (2012). [20] Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Nature 427, 50 (2004). [21] L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch on, and K. Ensslin, Nat. Phys. 3, 650 (2007). [22] M. Studer, M. P. Walser, S. Baer, H. Rusterholz, S. Sch on, D. Schuh, W. Wegscheider, K. Ensslin, and G. Salis, Phys. Rev. B 82, 235320 (2010). [23] M. P. Walser, U. Siegenthaler, V. Lechner, D. Schuh, S. D. Ganichev, W. Wegscheider, and G. Salis, Phys. Rev. B 86, 195309 (2012). [24] P. Altmann, F. G. G. Hernandez, G. J. Ferreira, M. Ko- hda, C. Reichl, W. Wegscheider, and G. Salis, Phys. Rev. Lett. 116, 196802 (2016). [25] T. Saito, A. Aoki, J. Nitta, and M. Kohda, Appl. Phys. Lett. 115, 052402 (2019). [26] M. Kohda, P. Altmann, D. Schuh, S. D. Ganichev, W. Wegscheider, and G. Salis, Appl. Phys. Lett. 107, 172402 (2015). [27] K. Kawaguchi, T. Fukasawa, I. Takazawa, H. Shida, Y. Saito, D. Iizasa, T. Saito, T. Kitada, Y. Ishitani, M. Kohda, and K. Morita, Appl. Phys. Lett. 115, 172406 (2019). [28] D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Science 323, 919 (2009). [29] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [30] A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Can- cellieri, and J.-M. Triscone, Phys. Rev. Lett. 104, 126803 (2010). [31] F. Meier, H. Dil, J. Lobo-Checa, L. Patthey, and J. Os- terwalder, Phys. Rev. B 77, 165431 (2008). [32] H. Yuan, M. S. Bahramy, K. Morimoto, S. Wu, K. No- mura, B.-J. Yang, H. Shimotani, R. Suzuki, M. Toh, C. Kloc, X. Xu, R. Arita, N. Nagaosa, and Y. Iwasa, Nat. Phys. 9, 563 (2013). [33] T. Henn, L. Czornomaz, and G. Salis, Appl. Phys. Lett. 109, 152104 (2016). [34] M. Kohda and G. Salis, Semicond. Sci. Technol. 32, 073002 (2017). 5

2110.07094v3.Thermalization_in_a_Spin_Orbit_coupled_Bose_gas_by_enhanced_spin_Coulomb_drag.pdf

Thermalization in a Spin-Orbit coupled Bose gas by enhanced spin Coulomb drag D. J. Brown1, 2,and M. D. Hoogerland1 1Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand 2Present address: Light-Matter Interactions for Quantum Technologies Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan An important component of the structure of the atom, the eects of spin-orbit coupling are present in many sub-elds of physics. Most of these eects are present continuously. We present a detailed study of the dynamics of changing the spin-orbit coupling in an ultra-cold Bose gas, coupling the motion of the atoms to their spin. We nd that the spin-orbit coupling greatly increases the damping towards equilibrium. We interpret this damping as spin drag, which is enhanced by spin- orbit coupling rate, scaled by a remarkable factor of 8 :9(6) s. We also nd that spin-orbit coupling lowers the nal temperature of the Bose gas after thermalization. INTRODUCTION The understanding of the transport, diusion and damping of spin, in contrast to those of charge, is im- portant to the eld of spintronics [1], where the spin of particles, rather than the quantity of particles (charge) carries information. Spin currents, in contrast to charge currents, are damped due to collisions between particles of opposite spin, as their relative momentum is not con- served. This damping is known as Spin Drag [2, 3]. Analogous to the spin drag in bilayer electron systems, systems of ultracold bosons can also demonstrate spin drag, with the drag enhanced by the familiar bosonic enhancement [4] prominent in ultracold boson systems. There is a detailed collection of work over the years into the presence of spin Coulomb drag in ultracold atomic systems [5{7] with a small selection of the work featuring the inclusion of spin-orbit coupling [8]. The other important eect in two dimensional elec- tron systems is spin-orbit coupling (SOC) [9{11]. How- ever, solid state materials used to investigate the eects of spin-orbit coupling are often challenging due to the limited control of individual experimental parameters. The body of work surrounding surrounding the topic of spin-drag with added spin-orbit coupling is limited with theoretical investigations looking at the impact of weak coupling on the drag in a 2D electron system [12], or the behaviour of impurities in a spinor condensate sys- tem [13]. In depth understanding of this combination could lead to better understanding of systems such as the topo- logical insulators [14, 15] with their famed protected edge states are dependent on the spin-momentum locking caused by the SOC within the material and have been in- vestigated as a potential platform for fault tolerant quan- tum computation [16, 17]. Ultracold atoms provide an ideal environment for test- ing the eects of SOC on the spin coulomb drag in quan- tum systems due to the ability to control many of the crucial parameters accurately.Previous experimental and theoretical work by Li et. al. [8] demonstrated the generation of spin currents using the same technique of a quench of a spin-orbit coupled Bose-Einstein condensate (BEC) and investigated the in- creased damping of the out of equilibrium system. GPE simulations showed good qualitative agreement with the experimental results, and gathering insight into the BEC shape oscillations and the miscible-imiscible phase tran- sition. As stated by Li et. al. the simulations underes- timate the damping of the BEC oscillations, potentially due to the lack of thermal atoms in the simulations. In particular, it was shown in [4] that the spin-Coulomb drag between thermal atoms and the condensate domi- nates over the mean eld eects, rendering the mean eld GPE only partially eective. In this article we present our experiments on investi- gating the thermodynamic behaviour of spin-orbit cou- pled systems within a conservative potential, and at- tempt to explain the enhanced damping of the atomic oscillations in the presence of SOC as Coulomb spin drag [4] by comparing the results to theoretical calculations. We create synthetic SOC using the ground state man- ifold of a Rubidium-87 (87Rb) BEC, following the Ra- man laser scheme rst demonstrated in the experiments of Spielman et. al [18]. A bias magnetic eld induces a Zeeman shift, breaking the degeneracy of the F=1 ground state of the atom sepa- rating them in energy. A quadratic Zeeman shift shifts themF= +1 state further than the mF=1, allow- ing us to eectively decouple the latter from the system. The atoms in the dierent Zeeman sublevels also dier in momentum by ky= 2kR, wherekR= 2= is the recoil momentum gained by the atom due to absorption of a photon with wavelength . The Hamiltonian of the coupledF= 1 state as a function of the atomic quasi- momentum ~ky, is as follows,arXiv:2110.07094v3 [cond-mat.quant-gas] 15 Feb 20222 (a) (b) FIG. 1: Schematic of the experimental setup. (a) Geometry of the laser orientation and polarization, along with the bias magnetic eld. (b) The level scheme of theF= 1 ground state and the Raman transitions induced by the coupling lasers. The Rabi frequencies of the individual transitions are 1and 2. ^Hy(~ky) =0 BB@h2(~ky+2kR)2 2mhh R 20 h R 2h2~k2 y 2mhh R 2 0h R 2h2(~ky2kR)2 2m+ h1 CCA (1) Diagonalizing the Hamiltonian gives the energies of the spin-orbit coupled dressed states which for Raman cou- pling strengths  h R<4ERfeatures a double minimum. Here,ER= h2k2 R=2m,is the two photon detuning be- tween the bare states, and indicates the quadratic shift. Correctly choosing the detuning for a given coupling re- sults in the ground state being an equal superposition of the two pseudospin states j"i,j#i, corresponding to a spin-orbit coupled state. However, when  h R>4ER, there is only a single min- imum at quasimomentum ~ky=kR. In the experiments reported here, we initially prepare a Bose-Einstein Con- densate, trapped in a harmonic trap, in the latter state with h R= 5:5ER. The system is then quenched to a lower h R<2ER, which takes the system out of equi- librium, and allowed to thermalise. We nd the time constant for this thermalisation, and nd that the rate scales with the coupling strength R. EXPERIMENTAL APPARATUS Our experiments begin with an all-optical BEC com- posed of approximately 104 87Rb atoms, optically pumped into the jF= 1;mF= 0ibefore evaporation, as described in our previous work [19]. The BEC is held in a harmonic trap, with aspect ratios !y:!x:!z= 1 : 1:2 : 2, formed by a crossed-beam optical dipole trap. We use two values of !yfor our experiments. The lowesttrapping frequencies correspond to the the experiments performed with the trap held at the nal power 66 mW achieved after evaporation, with !y= 285 s1. For the larger trapping frequencies, we adiabatically increase the power of the dipole trapping laser to 90 mW, cor- responding to !y= 2112 s1. Note that the larger trap frequency corresponds to a larger trap depth, and thus increases the number of atoms retained in the trap during the thermalization process and increases the rate of collisions between the atoms. During the evaporation to BEC a magnetic bias eld Byis ramped up in the last 2 seconds to 8.35 G providing the a measured !B=2= 5:845 MHz Zeeman shift, and a measured quadratic Zeeman shift of =2= 5 kHz. A schematic of the coupling scheme and geometry is shown in Fig. 1. The two-photon coupling strength Ris ex- perimentally determined by observing the Rabi oscilla- tions between the populations of the states j1i,j0iand j+ 1izero detuning, and tting the time evolution with the three state optical Bloch equations. To induce spin-orbit coupling we use two orthogonally polarized laser beams with wavelength = 790:2 nm counter-propagating along ythat are focused to an 150m diameter beam onto the center of the dipole trap. This wavelength of 790 :2 nm was chosen to minimise the scalar AC Stark shift in the atoms, which would have led to undesirable extraneous forces induced by these beams. The two beams are derived from the same laser, but dier in frequency by
!!B+and couple two of the inter- nalmFlevels of the BEC atoms. For sucient quadratic Zeeman shift, that is h > E RthemF= +1 internal state is tuned out of resonance for the two photon Ra- man coupling. The coupled system becomes an eective two level system of spin-momentum states which we label jmF=1;~ky+ 2kRi=j"0iandjmF= 0;~kyi=j#0i. EXPERIMENTAL PROCEDURE The condensate is prepared in the lowest energy dressed band of the Raman coupled system by adia- batically increasing the Raman coupling to 5.5 ERover 50 ms, where there is a single minimum in the dispersion curve as illustrated in Fig.2(a). The adiabatic increase of the coupling prevents unwanted heating and oscillations of the condensate in the trap caused by synthetic electric elds [18]. We hold the Raman coupling on for a further 30 ms at a constant value in order to ensure the system is in the lowest energy dressed band. We conrm that the ramp is adiabatic by measuring the total momentum of the atoms, obtained from the weighted sum of the quasi- momentum of all momentum components, during this 30 ms period and conrming that it is zero at each point in time. If the total momentum is non-zero during this phase, the ramp speed must be adjusted to ensure the atoms remain in the lowest energy dressed state.3 (a) (b) FIG. 2: Dispersion relations for the coupled BEC for two coupling strengths. (a)  h R= 5:5ER, features a single minima where both spins have equal populations and the same quasimomentum. (b)  h R= 1ERfeatures two minima of the dispersion. Each pseudospin occupies one of the minima with the corresponding quasimomentum. To take the system out of equilibrium, a synthetic elec- tric force is imparted on the dressed BEC by abruptly reducing the Raman coupling strength from the initial i= 5:5ER=hto a nal value fin 1 ms. The rapid decrease of Raman coupling constitutes a quench of the system. The condensate separates into the two pseu- dospin statesj"0iandj#0ithrough the synthetic electric force, each accelerating towards one of the new minima (see Fig. 2(b)) of the dispersion relation, where they then oscillate in the harmonic trap with maximal momentum jk";#j=1hkR. To compensate for both the impact of the mF= +1 state and the changing AC stark shift as the laser in- tensity changes, we adjust the laser frequency dierence to maintain equal populations of the two states, by an amount up to  hAC= 1ER. This shift in the two pho- ton resonance condition is extremely sensitive to small changes in experimental parameters, such as the mag- netic elds. Although care is taken to maintain equal populations of the spin components, the nal spin-orbit coupled state after the quench will occasionally have non- equal populations in each component. To group the data we calculate the population imbalance F"#=N"N# N"+N#; (2) (a) (b)FIG. 3: Plots of the momentum dierence between the two components of the system for two coupling strengths (a)  h f= 1:5ERand (b) h f= 2ERt with a decaying cosine function. The insets show the spin-momentum distribution after the system has returned to equilibrium, and clearly demonstrate the return of the system to a spin orbit coupled state with the nal momentum distribution re ecting the non-zero quasimomentum before release from the trap. whereN"is the population of the mF=1 state and N#is the population of the mF= 0 state. In this paper we focus on the case with balanced populations, where jF"#j<0:1, by post-selecting the data. We let the two pseudospin states oscillate in the dipole trap for time tup to 20 ms before switching the trap and Raman coupling o simultaneously, projecting the atoms onto their bare spin-momentum states. The bare states expand for 15 ms in a Stern-Gerlach gradient separat- ing the spin components in the xdimension before being imaged with a resonant absorption method. We measure the rate of thermalization of the system by evaluating the momentum distribution of both spin states as a function of time. We numerically determine the mean momentum of each of the spin ensembles as they oscillate in the trap with a decaying amplitude. We t the decay of the oscillation of the momentum dierence kt=jk"k#jand measure the nal temperature of the thermalized ensembles.4 THERMALIZATION OF A SPIN-ORBIT COUPLED BEC It is important to note that the momentum imparted on the pseudospins during the quench depends of the nal coupling strength, which arises from the quasimo- mentum minima shifting as a function of the coupling strength. The shift in this work on the order of 0:05hk for each spin, accounting for a 5% dierence in the total momentum of the atoms of  h f= 0 and h f= 2ER Once the system has thermalized and the oscillations have completely damped, a small fraction of the conden- sate remains, with the atoms occupying the minima of the new spin-orbit coupled band. For Raman coupling above h R= 1ERthe time-of- ight images show clearly the system has returned to equilibrium in a spin-orbit coupled state with the pseudospin momentum clearly be- ing non-zero. We conrm the non-zero momentum of the atoms comes from the quasimomentum of the spin-orbit coupled state, rather than residual oscillation energy, by noting the momentum remains unchanged over 5 ms of evolution. Fig. 3 demonstrates two cases where the nal pseu- dospins are separated from zero momentum when reach- ing equilibrium. A clear example is shown in the in- set of Fig. 3(b) shows the pseudospins are positioned k";#=0:25hkR, corresponding to quasimomentum be- fore release ~ky=0:75hkR, the locations of the disper- sion minima obtained from exact diagonalization of the Hamiltonian. Even though the trap frequencies are the same for the two situations in the gure, the dispersion relation is dierent for dierent coupling, giving rise to the observed dierence in oscillation frequency. It is also clear that the higher coupling strength (b) gives rise to a stronger damping of the oscillation. As mentioned, some data was also obtained for im- balanced populations. In this case, qualitatively we ob- serve that the smaller population oscillation damps more rapidly while the larger population continues to oscil- late. Accurately controlling the imbalanced populations proves to be dicult and therefore we do not include these results in this paper. SPIN COULOMB DRAG For a situation with no spin-orbit coupling, the damp- ing coecient can be determined theoretically for an ul- tracold Bose gas. The spin drag between two compo- nents can be calculated from two expressions for the non- condensed and the condensed atoms respectively [4], 22=na2 "# h1 621 (n3)2Z1 0dqd!q2 sinh2(!=2) ln exp q2=16+gn 0!=2 +!2=q2 exp [!] exp [q2=16+gn 0!=2 +!2=q2]1! ; (3) and 12=na2 "# h64n0a 3(2)3nZ1 0dp1dp3p1p3 3  1 +1 exp [(p2 1+p2 3)=4+ 2gn 0]1 1 exp [p2 1=4+gn 0]11 exp [p2 3=4+gn 0]1 p1p3 2gn 0 : (4) Here,  is the Heaviside function,  =q 2h2=mk BT is the thermal de Broglie wavelength, = 1=kBTthe inverse thermal energy and g= 4h2a=2mthe interpar- ticle interaction strength. Calculations were performed using a script provided by Jogundas Armaitis [4] with our experimental parameters, returning the total spin drag relaxation rate for a given density of atoms. Due to the fact the atoms are oscillating in the trap and only overlap and only interact periodically, we multiply by a scaling factor calculated based on the interaction time of the two spin clouds overlapping in the trap. At the time of writing we are unable to obtain theo- retical calculations for the eects of the spin-orbit cou- pling on the spin drag, so we compare the experiments for the uncoupled case with the theory. For the ytrapping frequencies !y= 285 s1and!y= 2112 s1, we calculate a spin drag damping rate of s= 12+ 22=13.6 s1. Comparing this to our experimentally observed damping rate of = 72(9) s1, we observe that part of eis caused by collisions of the condensate with thermal atoms, and is dependent on the atomic density and is also present regardless of spin drag. Taking into account the elastic collision rate, it is clear from Fig. 4 that the increase in the spin-orbit coupling corresponds to a signicantly increased damping rate, with a linear dependence over the range measured. We nd that the damping rate also scales with the calculated spin-drag damping rate and e summarize our results for the damping coecient by the expression = e+ R s (5) whereis a constant. The t parameters in Fig. 4, as well as the spin drag constant without spin-orbit coupling are summarised in table I.5 0.0 0.5 1.0 1.5 2.0 Final Coupling Strength (ER)050100150200250Damping rate (S1) FIG. 4: The damping rate of the system as a function of the Raman coupling strength, for low (green circles) and high (red crosses) trapping frequencies. The damping increases linearly over the range of coupling, with the gradient being a combination of both collisional damping and spin drag. The red and green dashed lines indicates the theoretical spin drag damping rates. !y=2(s1) e(s1) s s(s1) 85 18(6) 63(4) 6.4 112 67(14) 107(13) 13.6 TABLE I: Scaling of the t parameters in gure 4 At the time of writing, we have not found a way to nd from theoretical considerations for our experimental conguration, but from table I we nd that = 8:9(6) s, which is remarkably large as spin-orbit coupling aects the condensate fraction much more strongly than the thermal fraction. We envision that nding an accurate theoretical value may take a truncated Wigner type simu- lation [19, 20] to include both spin components, the spin- orbit coupling along with their interactions with both the opposing spin condensate atoms, but also the atoms be- longing to the thermal cloud. Finally, we measure the nal temperature of the system once it has reached equilibrium. We integrate the time- of- ight region for each spin to obtain a 1D density prole and t them with a sum of a Bose enhanced Gaussian and a Thomas-Fermi prole. Integrating the ts we obtain the atom number for the BEC and thermal component, which we use to obtain the fractional temperature T/T c. We plot the measured temperatures in Fig. 5 along with the initial temperature and uncertainty. We t a straight line to the temperature, obtaining T=Tc= 1:08(0:01) 0:11(0:02) R. It is interesting to note that the condensate is o cen- 1.0 1.2 1.4 1.6 1.8 2.0 Final Coupling Strength (ER)0.800.850.900.951.00Fractional Temperature (T/Tc) FIG. 5: Fractional temperature T/T cof the system as a function of the nal coupling strength. The blue line shows the temperature before the quench averaged over many shots, with the shaded region indicating the error. The temperature of the equilibrium state is shown with red circles, demonstrating a clear decrease in the heating for larger coupling strengths. Inset: The integrated 1D prole showing the bimodal t to one of the spin components ter with respect to the thermal cloud, consistent with the quasimomentum of the nal spin orbit coupled state mentioned in an earlier section. The results surprisingly show that for increasing Raman coupling, the damping results in a reduced nal temperature, possibly indicat- ing the spin-orbit coupling plays a signicant role in the relaxation process. The means by which the temperature decreases is not so obvious, however the condensate frac- tion remaining at the end of the experiment is increased for increasing coupling. CONCLUSION We have presented experiments performed to investi- gate the impact of spin-orbit coupling on the thermal- ization processes present in an out of equilibrium system of ultracold bosons. We measure the spin drag damping rate of the atoms and compare the uncoupled case to the- oretical calculations. We show that introducing the spin- orbit coupling into the system strongly increases the rate at which the system returns to equilibrium, while also reducing the temperature increase caused by the excita- tion. Finally, we have shown that the equilibrium state of the system after rethermalization is a spin-orbit cou- pled BEC, with the quasimomentum measurements af- ter reaching equilibrium corresponding to the dispersion6 relation calculated through exact diagonalization of the Hamiltonian. We anticipate that this work will lead to new understanding of thermalization in the presence of spin-orbit coupling. dylan.brown@oist.jp [1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln ar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). [2] I. D'Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000). [3] C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein, J. Stephens, and D. D. Awschalom, Nature 437, 1330 (2005). [4] R. A. Duine and H. T. C. Stoof, Phys. Rev. Lett. 103, 170401 (2009). [5] A. Sommer, M. Ku, G. Roati, and M. W. Zwierlein, Nature 472, 201 (2011). [6] O. Goulko, F. Chevy, and C. Lobo, Phys. Rev. Lett. 111, 190402 (2013). [7] E. Fava, T. Bienaim e, C. Mordini, G. Colzi, C. Qu, S. Stringari, G. Lamporesi, and G. Ferrari, Phys. Rev. Lett.120, 170401 (2018). [8] C.-H. Li, C. Qu, R. J. Nienegger, S.-J. Wang, M. He, D. B. Blasing, A. J. Olson, C. H. Greene, Y. Lyanda-Geller, Q. Zhou, C. Zhang, and Y. P. Chen, Nature Communications 10, 375 (2019). [9] J. P. Heida, B. J. van Wees, J. J. Kuipers, T. M. Klap- wijk, and G. Borghs, Phys. Rev. B 57, 11911 (1998). [10] J. I~ narrea, Scientic Reports 7, 13573 (2017). [11] C. Zhu, L. Dong, and H. Pu, 49, 145301 (2016). [12] W.-K. Tse and S. Das Sarma, Phys. Rev. B 75, 045333 (2007). [13] R. Liao, O. Fialko, J. Brand, and U. Z ulicke, Phys. Rev. A93, 023625 (2016). [14] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). [15] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006). [16] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). [17] R. Raussendorf, J. Harrington, and K. Goyal, New Jour- nal of Physics 9, 199 (2007). [18] Y.-J. Lin, K. Jim enez-Garc a, and I. B. Spielman, Nature 471, 83 (2011). [19] D. J. Brown, A. V. H. McPhail, D. H. White, D. Baillie, S. K. Ruddell, and M. D. Hoogerland, Phys. Rev. A 98, 013606 (2018). [20] P. Blakie, A. Bradley, M. Davis, R. Ballagh, and C. Gar- diner, Advances in Physics 57, 363 (2008).

1606.04426v2.Modulation_instability_in_quasi_two_dimensional_spin_orbit_coupled_Bose_Einstein_condensates.pdf

Modulation instability in quasi two-dimensional spin-orbit coupled Bose-Einstein condensates S. Bhuvaneswari,1K. Nithyanandan,2P. Muruganandam,1and K. Porsezian2 1Department of Physics, Bharathidasan University, Palkalaiperur, Tiruchirappalli 620024, India. 2Department of Physics, Pondicherry University, Puducherry 605014, Puducherry, India. We theoretically investigate the dynamics of modulation instability (MI) in two-dimensional spin- orbit coupled Bose-Einstein condensates (BECs). The analysis is performed for equal densities of pseudo-spin components. Dierent combination of the signs of intra- and inter-component interac- tion strengths are considered, with a particular emphasize on repulsive interactions. We observe that the unstable modulation builds from originally miscible condensates, depending on the com- bination of the signs of the intra- and inter-component interaction strengths. The repulsive intra- and inter-component interactions admit instability and the MI immiscibility condition is no longer signicant. In uence of interaction parameters such as spin-orbit and Rabi coupling on MI are also investigated. The spin-orbit coupling (SOC) inevitably contributes to instability regardless of the nature of the interaction. In the case of attractive interaction, SOC manifest in enhancing the MI. Thus, a comprehensive study of MI in two-dimensional spin-orbit coupled binary BECs of pseudo-spin components is presented. PACS numbers: 05.45.Yv, 03.75.Lm, 03.75.Mn I. INTRODUCTION Study of spin-orbit (SO) coupled Bose-Einstein con- densates (BECs) is one among the important topics of current research in the context of macroscopic quantum phenomena. Spin-orbit coupling (SOC) describes the in- teraction between the particle's spin and orbital momen- tum and plays a crucial role for many physical phenom- ena in condensed matter systems including spin-Hall ef- fect, topological insulators, spintronics and so on [1{6]. The synthetic SOC in BECs was experimentally achieved very recently. In this realization techniques, two Ra- man laser beams were used to couple with two compo- nent BECs [7]. The momentum transfer between laser beams and atoms leads to synthetic SOC [8{11]. SOC has been realized with cold atomic gases by designat- ing the hyperne atomic states as pseudo-spins and cou- pling them with Raman laser beams [12{14]. For in- stance, in the case of87Rb the pseudo-spin states are j"i=jF= 1;mF= 0iandj#i=jF= 1;mF=1i, which are generated using pair of Raman laser beams. SO coupled BECs have been studied extensively in dierent contexts including phase separation, strip phases [15], spotlighting the phase transition [16], vor- tices with or without rotations [17], and so on. In ad- dition, the study of topological excitations, for exam- ple, skyrmions, has also attracted much along these di- rections [18]. Further, matter wave solitons such as bright and dark solitons have been studied in quasi- one-dimensional with attractive and repulsive SO cou- pled BEC [19, 20]. It should be noted that most of the studies on SO coupled BECs were primarily focused on quasi-one-dimensional systems. Only a few studies were devoted on multi-dimensional SO coupled BECs. How- ever, there were few important studies reported in the context of two-dimensional SO coupled BECs, for in- stance, the dynamics of vortices, the existence of vortex-antivortex pair, to mention few [21{25]. Recently, the study on two-dimensional SO coupled BEC of mixed Rashba-Dresselhaus type and Rabi couplings have earned particular interest [26, 27]. Thus, it is more appropriate, realistic and interesting to study SO coupled BECs in two- and three-dimensions systems. Particularly, here we emphasize on the study of instability of plane wave in two-dimensional SO coupled BECs, in the framework of MI analysis [21, 22]. The degree of instability in a BEC can be character- ized by the MI. MI is an instability process and identied as a requisite mechanism to understand various physics eects in nonlinear systems. The phenomenon of MI was rst observed in hydrodynamics by Benjamin and Feir in 1967 [28]. In the same year, Ostrovskii predicted the possibility of MI in optics [29] and later explained in de- tail by Hasegawa et. al. in 1973 in the context of opti- cal bers [30]. The MI is a general phenomenon occur- ring in many nonlinear wave equation and is of particu- lar interest in dispersive nonlinear systems. In conven- tional dispersive nonlinear systems, MI manifest as a re- sult of the constructive interplay between dispersion and nonlinearity. Such that any deviation from the steady state in the form of perturbation leads to an exponential growth of the weak perturbation, resulting in a break- up of the carrier wave into trains of soliton-like pulses [31]. In addition, MI has been widely studied in various branches such as uid dynamics [28], magnetism [32], plasma physics [33] and BEC [34]. In the context of BEC, the MI has been given consid- erable importance over a long period of time, owing to its fundamental and applied interest in various aspects. In particular, MI has been found to be relevant in un- derstanding the formation and propagation of solitonic waves [35], and also apparent in explaining the domain formation [36] and quantum phase transition [37]. MI has been studied extensively in BECs for both single [38] andarXiv:1606.04426v2 [cond-mat.quant-gas] 15 Jun 20162 two-component systems [39], and realized experimentally as well [40]. In the case of single-component BECs, the MI has been found to be feasible only for attractive in- teraction (self-focusing nonlinearity), in such case, the phase uctuations caused by MI leads to the formation of soliton trains. However, the breakthrough work by Goldstein and Meystre opens up the possibility of MI even for the repulsive interactions [41], in similar lines to the case of cross-phase modulation induced instability in nonlinear optics [42, 44]. Thus, the two component BEC nds particular interest in the study of MI, as it helps to achieve instability even in repulsive interactions. In the case of SO coupled BEC, the MI in one-dimensions was recently explored in Ref. [45], and the higher dimensional case is still an open problem. Thus, inspired by the spe- cial features of SO coupling and the physical relevance of two-component BEC system, we intend to study the dynamical behavior of MI in SO coupled BEC in two- dimensions. In this paper, we present a systematic study of MI in quasi-two-dimensional SO coupled BECs with the inclusion of Rashba and Dresselhaus SO couplings. By considering small perturbation approximation, we ob- tain linearized GP equation. Further, the interplay be- tween dispersion and nonlinear eects have been studied in terms of system parameters. We have also summa- rized the growth of MI gain for dierent combinations of intra- and inter-component of interaction strengths in the presence and absence of SO coupling. The organization of the paper is as follows: After a detailed introduction in Sec. I, the Sec. II features the theoretical model for the case of SO coupled BECs. In Sec. III, we present the MI dispersion relation through linear stability analysis, and systematically explained the eect of SOC and Rabi coupling for a dierent combina- tion of inter- and intra-component interactions strength. Sec. IV, features the results and discussion followed by conclusion in Sec. V. II. THEORETICAL MODEL We consider the spin-orbit coupled Bose-Einstein con- densates conned in a harmonic trap with equal Rashba and Dresselhaus couplings described, within the frame- work of mean eld theory by an energy functional of the following form [7] E=Z+1 1"d~xd~y; (1) where, "=1 2 yH0 + ~g11j "j4+ ~g22j #j4+ 2~g12j "j2j #j2
; (2) = ( "; #)Tis the condensate wave function, "and #are associated with the pseudo-spin components. Themodel Hamiltonian H0in Eq. (2) assumes the form, H0=^p2 2m+V(~r) +~ 2x~~kL m^p~xz (3) where, ^p=i~(@~x; @ ~y) is the momentum operator, V(~r) =1 2m[!2 ?(~x2+ ~y2) +!2 z~z2] is a quasi-2D harmonic trapping potential where !z!?,is the frequency of Raman coupling, x;zare Pauli spin matrices and ~kLis the wave number of the Raman laser which couples the two hyperne states. The eective two dimensional cou- pling constant ~ gij= 4~2aij=m, (i;j= 1;2) represents the intra- (~ g11;~g22) and inter- component (~ g12) interac- tion strengths, which are dened by the corresponding s-wave scattering lengths aijand atomic mass m. Mea- suring energy in units of the radial trap frequency ( !?), i.e.,~!?, length in units of harmonic oscillator length, a?=p ~=(m!?), and time in units of !1 ?the follow- ing dimensionless Gross-Pitaevskii (GP) equations can be derived for dierent components of 1;2from Eq. (2) as [43] i@ 1 @t= 1 2r2 ?+V(r) +g11j 1j2+g12j 2j2 1 + ikL@ @x 1+ 2; (4a) i@ 2 @t= 1 2r2 ?+V(r) +g22j 2j2+g12j 1j2 2 ikL@ @x 2+ 1; (4b) where,V(r) = (x2+y2)=2,x= ~x=a?,y= ~y=a?,t= !?~t,kL=~kLa?, ==2!?,gij= 4Naij=a?and 1;2= ";#a3=2 ?=p N. In the following, we shall proceed with the study of modulational instability in the above two-dimensional model Eq. (4) for spin-orbit coupled BECs. III. ANALYSIS OF MODULATION INSTABILITY A. Linear Stability Analysis The fundamental framework of MI analysis relies on the linear stability analysis (LSA), such that the steady state solution is perturbed by a small amplitude/phase, and then study whether the perturbation amplitude grows or decays [31]. For this purpose, we consider a continuous wave (CW) state of the miscible SO coupled BECs with the two-dimensional density nj0=j j0j2of the form j(x;y;t ) =pnj0eit: (5) Then the stability of the SO coupled BECs can be exam- ined by assuming the perturbed wave functions as j(x;y;t ) = (pnj0+j) eit; (6)3 A set of linearized equations for the perturbation can be obtained by using Eq. (6) in Eq. (4) i@(1) @t=1 2@2(1) @x2+@2(1) @y2 + ikL@(1) @x +  2rn20 n10(1) +g11n10(1+ 1) +g12pn10n20(2+ 2); (7a) i@(2) @t=1 2@2(2) @x2+@2(2) @y2 ikL@(2) @x +  1rn10 n20(2) +g22n20(2+ 2) +g12pn10n20(1+ 1); (7b) where the symboldenotes complex conjugate. Assum- ing a general solution of the form j=jcos (kxx+kyy t) + ijsin (kxx+kyy t); (8) wherekxandky, are the wavenumbers and jandj(j= 1;2) are the amplitudes of wavefunction, and is the eigenfrequency. We further assume that two pseudo-spin states of equal density n10=n20=n. A straightforward substitution of Eq. (8) in Eq. (7) yields the following dispersion relation for . 4 21 4(K2) (2K+G1+G2) + 2k2 xk2 L+ 2G12 + 2kxkL(K2)(G1G2) +K k2 L k2 xk2 L+ 2G121 4(K2) (2K+G1+G2) +K 41 4(K+G1)(K+G2)G2 12 1 2k2 Lk2 y2K 2(G1+G24) + 2k2 Lk2 xK2 + 4 (G12+K) = 0; (9) whereK=k2 x+k2 yandG1= 4g11n2,G2= 4g22n2, andG12= 2g12n+ are the modied intra- and inter- components interaction strengths, respectively. For equal strengths of intra-component interactions, i.e., a11=a22 (g11=g22=g), the dispersion relation recast into a simpler form as 2 =1 2 p 2+ 4
 ; (10) with  =1 2(K2) (K+G) + 2k2 xk2 L+ 2G12;(11a)
=1 2k2 Lk2 y (2K)(G2) + 2k2 Lk2 xK2 + 4 (G12+K) K k2 L k2 xk2 L+ 2G121 2(K2) (K+G) +K 41 4(K+G)2G2 12 ; (11b) where,G1=G2=G. The above Eq. (10) is the disper- sion relation corresponding to the stability of the miscible SO coupled BECs. As it is known from the theory of MI, the system exhibit stable conguration for all real val- ues ofkxandky, if 2 is positive ( 2 >0). If > 0, the eigenfrequency +is always real but may be real or imaginary, which is dependent on
. If the eigenfre- quency has an imaginary part, the spatially modu- lated perturbation become exponential with time, as it is obvious from the form of j. On the other hand, for negative value of (<0), 2 need not to be positive. In such case, 2 is characterized by the values of
. For >0 the value of lower branch 2 is negative if,
<0. Similarly for >0 the value of upper branch 2 +is neg- ative when
<0. Regardless of anything 2 is always negative, and therefore, the MI sets in via the exponen- tial growth of the weak perturbations. The MI growth rate is dened as jIm( )j. Following the mathe- matical calculation pertaining to the dispersion relation corresponding to the stability/instability of the system, the subsequent sections are dedicated to the study on the eect of SOC in the MI. B. Eect of Rabi coupling in the MI of SO coupled BECs In order to study the eect of Rabi coupling in the MI, we turn o the SOC by making kL= 0. For better insight, we consider two special cases, (i) one without Rabi coupling ( = 0) and (ii) other in the presence of Rabi coupling ( 6= 0). 1. Zero Rabi coupling In absence of Rabi coupling ( = 0), the eigenfre- quency of the system for kx6=kyassumes the form, 2 =1 2[K(K+ 2n(gg12))] (12) One can infer from the above Eq. (12), that based on the sign/nature of the interaction strength, the may be real or imaginary. It is obvious from the combination of signs of intra and inter-component interactions, +is found to be real in the following cases: (i) both intra- (g) and inter- (g 12) component interac- tions are repulsive,4 (ii) attractive intra-component and repulsive inter- component interactions, and (iii) repulsive intra-component and attractive inter- component interactions. For attractive intra- and inter-component interactions +becomes imaginary and thereby inevitably con- tributes to MI. However, becomes imaginary for all cases. Thus, as far as MI is concern, contribute better to the instability in all means than the +counterpart. It is worth mentioning, at ky= 0, our results completely agree with the Ref. [45], and could reproduce the results of the MI in the conventional two-component system as in Ref. [39]. 2. Non-zero Rabi coupling Next, we study the eect of Rabi coupling on MI by considering any nite value for ( 6= 0). Here the dispersion relation as given by Eq. (10) can be modied as follows 2 +=1 2[K(K+ 2n(gg12))]:; (13) withkx6=ky, and in order to highlight the eect of Rabi coupling, the coecient of SOC is turned o, i.e., kL= 0. It is straightforward to notice that 2 +given by Eq. (13) for = 0 is similar to Eq. (12) for the case of zero Rabi coupling. Therefore, the instability/stability condition as dened by the zero Rabi coupling in the earlier section is completely applicable here as well. Hence, for non-zero Rabi coupling, 2 +is not dierent from that of zero Rabi coupling, which implies that 2 +is independent of . On the other hand, 2 is found to be signicantly in uenced by Rabi coupling and can be expressed as 2 =K2 4+ (Kn4) (gg12) + 2 (2K):(14) The eect of Rabi coupling from Eq. (14) can be better explored for three representative cases of , namely (i) = 0, (ii) >0, and (iii) <0. For = 0, Eq. (14) reverts to the expression for as given by Eq. (12), and therefore, will not be discussed again here. Our par- ticular focus is on = 0 and >0. For <0, is imaginary only for repulsive intra- and attractive inter- component interaction, and for all other cases does not contribute to MI, as it is real. However, the eect of Rabi coupling is more pronounced for >0, as the instability/stability conditions qualitatively dier from the previous cases. It is found that the is unsta- ble for all combination interactions, except the repulsive intra- and attractive inter-component interaction. Per- haps, the magnitude of intra- and inter-component inter- action strengths are rather identied to be deterministic for MI. It is observed that for repulsive intra- and inter- component interactions the instability is possible onlywhenjgj>jg12j. Similarly, for attractive interaction, the condition for MI can be modied as jg12j>jgj. For a better understanding of the eect of Rabi cou- pling, as a representative case, we have shown in Fig. 1, the MI gain corresponding to the repulsive intra- and inter-component interactions with kL= 0, = 1, g= 2, g12= 1 andn= 1. It should be noted, the condition for instability (jgj>jg12j) in repulsive interactions is true for the above choice of parameters. It is evident, from the −3−2−10123kx−3−2−10123 ky00.250.50.751ξ 00.20.40.60.81 −3.0−1.50.0 1 .5 3 .0 kx−3.0−1.50.01.53.0ky 0.00.51.0 (a) (b) FIG. 1. (color online) (a) Three-dimensional (3D) surface plot showing the MI gain, =jIm( )j, and (b) the corresponding two-dimensional (2D) contour plot for the parameters kL= 0, = 1,n= 1,g= 2 andg12= 1. existence of instability region, that the MI is caused by Rabi coupling for repulsive intra- and inter-component interactions. It should be noted that the instability re- gion is symmetric in momentum space on either side of the wave numbers, kxandky. Overall, it is apparent from the above discussion on the in uence of Rabi coupling in the instability that out of the dierent choices of Rabi coupling strengths, the con- dition >1 is found to carry more information about the MI. Therefore, in the subsequent section we shall study the eect of SO coupling by xing the Rabi cou- pling strength as = 1. IV. THE EFFECT OF RABI AND SPIN-ORBIT COUPLING One can draw out a conclusion from the previous sec- tion, that the sign/nature of the interaction signicantly in uences the stability/instability of the system. For bet- ter insight, in the following section, we would like to brie y emphasize the eect of dierent combinations of intra- and inter- component interaction strength with the inclusion of both Rabi ( 6= 0) and SOC ( kL6= 0). We consider following four representative cases to study MI in the SO coupled BEC system. A. Both repulsive intra- and inter- component interac- tions (g>0,g12>0). B. Attractive intra- and repulsive inter- component in- teractions ( g<0,g12>0). C. Repulsive intra- and attractive inter- component in- teractions ( g>0,g12<0).5 D. Both attractive intra- and inter- component inter- actions (g<0,g12<0). A. Repulsive intra- and inter-component interactions Here, we consider self repulsive intra- ( g >0) and re- pulsive inter- ( g12>0) components of modied interac- tionsG1;G2andG12. Our investigation follows from the general dispersion relation for non-zero SO and Rabi cou- pling as given by Eq. (10). It is apparent from Eq. (10), the expression can be real or complex depends on the sign of>0 and2+ 4
. For>0, the upper branch +will be imaginary only for 2+ 4
<0, and there- fore contribute to MI. Fig. 2 shows the MI gain for + −1.5−1.0−0.5 0.0 0.5 1.0 1.5 ky0.00.10.20.30.40.5ξMiscibleImmiscible g= 0.2 g= 0.5 g= 0.6 g= 0.8 g= 1.0 FIG. 2. (color online) Plot of the MI gain, =jIm( +)j, as a function of kyfor dierent intra-component interaction strengths with kL= 1, = 1, n= 1,g12= 1 andkx= 1. as a function of one of the momentum component ( ky) for dierent values of intra-component ( g) at xed inter- component interaction strength ( g12= 1). The choice of parameters are kL= = 1,n= 1,g12= 1 andkx= 1. It is evident from Fig. 2, there exist two symmetrical in- stability region on either side of the zeros of kxandky. As the intra-component interactions strength increases further, the two instability region approaches to the zero wave number and merge into a single coalesced instability region with elevated gain at higher values of g. On the other hand, from Eq. (10) is more in- teresting, since leads to unstable region even for 2+ 4
>0. Fig. 3 shows the MI gain for as a function of momentum component for similar values as used for +. One can straightforwardly notice, that there exist two pairs of instability region for as against, the single pair of instability region observed for the case +. As the strength of the intra- component interaction in- creases, the instability region at the center unies into single band (similar to the case of +), while the other −3−2−1 0 1 2 3 ky0.00.20.40.60.81.0ξMiscibleImmiscible g= 0.2 g= 0.5 g= 0.6 g= 0.8 g= 1.0FIG. 3. (color online) Plot showing the MI gain, =jIm( )j as a function of kyfor dierent gwithkL= 1, = 1, n= 1, g12= 1 andkx= 1. pair of instability region at higher values of kysubstan- tially decreases in gain and width of the instability re- gion. For insight, we plot in Fig. 4 the 3D variation of −4−2024 kx012345 g00.51ξ 00.20.40.60.81 −4−20 2 4 kx012345g 0.00.51.01.2 (a) (b) FIG. 4. (color online) 3D surface plot of the MI gain, =jIm( )jin thekx-gplane and (b) the corresponding 2D contour plot for kL= 1, = 1, n= 0:3,g12= 1 and ky= 1. MI gain for a range of kxandgwith the inter-component interaction xed ( g12). It is obvious that the two insta- bility regions merge into a single instability region with elevated gain. To explore the eect of inter-component −4−2024 kx012345 g1200.511.3ξ 00.40.81.2 −4−20 2 4 kx012345g12 0.00.51.01.3 (a) (b) FIG. 5. (color online) 3D surface plot showing the MI gain, =jIm( )j, in thekxg12plane and (b) the corresponding 2D contour plot for the parameters kL= 1, = 1 for n= 0:3, g= 1 andky= 1. interaction in the instability, in Fig. 5, we depict the MI6 gain for a range of g12with xedg. It is apparent from Fig. 5, for smaller values of g12, the gain in the inner in- stability band decreases gradually to zero, while the gain in the instability region at higher values of kxgrows with increase in g12. In order to explore the eect of the wave numbers,kxandky, we plot the MI gain as a function of kxfor dierent values of kyand vice-versa. Figs. 6 and 0 1 2 3 kx0.00.10.20.30.40.50.6ξMiscibleImmiscible ky= 0.5 ky= 1.0 ky= 1.5 ky= 2.0 FIG. 6. (color online) Plot showing the MI gain, =jIm( )j as a function of kxfor dierent kyvalues with kL= 1, = 1, n= 0:3g= 1 andg12= 2. 0.0 0.5 1.0 1.5 2.0 2.5 ky0.00.10.20.30.40.50.6ξMiscibleImmiscible kx= 0.5 kx= 1.0 kx= 1.5 kx= 2.5 FIG. 7. (color online) Plot of the MI gain, =jIm( )jas a function of kyfor dierent kxvalues with kL= 1, = 1, n= 0:3,g= 1 andg12= 2. 7 show that the MI bands drift towards the center and coalesced into single instability band with the increase in the wave numbers. Thus, there are no changes in the general trend of shifting of MI band for both cases, how- ever the peak gain and the width of instability region substantially diers. One can infer that the maximum gain is observed for kxas evident from Fig. 6 in compar-ison to the plot of MI gain for kyin Fig. 7. Fig. 8 shows −3−1.501.53ky−3−1.501.53 kx00.20.40.6ξ00.20.40.6 −3.0−1.50.0 1 .5 3 .0 kx−3.0−1.50.01.53.0ky 0.00.20.40.6 (a) (b) FIG. 8. (color online) (a) 3D surface plot showing the MI gain,=jIm( )j, and (b) the corresponding 2D contour plot for the parameters kL= 1, = 1, n= 0:3,g= 5 and g12= 3. the instability gain on the momentum space as a func- tion ofkxandkyfor some representative values of intra- and inter-components interaction strength. It is observed that there exist two instability bands corresponding to kx andky. The inner pair of bands corresponds to kxwith slightly higher gain than the outer band as a result of ky. This combination of intra- and inter-components in- teraction is of particular interest, because the instability is generally not feasible, as both interaction components are repulsive and therefore does not contribute to MI. However, the above results suggest that the MI is still possible even in the repulsive two component BEC with the aide of SOC. B. Attractive intra-component and repulsive inter-component interactions This condition corresponds to the binary BEC with at- tractive intra- component and repulsive inter-component interactions, which is subject to the MI even in the ab- sence of the SOC. Although SOC is not fundamental to the occurrence of MI in this particular case, but signif- icantly aects the instability. The MI corresponding to +produces the same number of bands as in the previ- ous case for repulsive interaction. However, the MI corre- −4−2024 kx−5−4−3−2−10 g00.51ξ 00.511.2 −4−20 2 4 kx−5−4−3−2−10g 0.00.51.01.2 (a) (b) FIG. 9. (color online) 3D surface plot of the MI gain, =jIm( )jin thekx-gplane and (b) the corresponding 2D contour plot for xed g12withkL= 1, = 1, n= 0:3, g12= 1 andky= 1. sponding to qualitatively diers, and it can be better explained in the following two combinations, namely, (i)7 MI gain as a function of gfor xedg12, and (ii) vari- ation of MI gain as a function of g12at constant value ofg. Fig. 9 shows the possibility of three pairs of instabil- ity bands for attractive intra-component at constant g12, and the instability bands grow in gain with the increase ing. Fig. 10 depicts the MI gain for a range of repulsive −4−20 2 4 kx0 2 4 6 8 10g12 0123 −4 −2 0 2 4kx0 2 4 6 8 10g120123ξ0 1 2 3 (a) (b) FIG. 10. (color online) 3D surface plot of the MI gain, =jIm( )jin thekx-g12plane and (b) the corresponding 2D contour plot for xed gwithkL= 1, = 1, n= 0:3, g=1 andky= 1. inter-component interaction strength ( g12) at constant g. It is obvious, there exist two pairs of instability bands on either side of ky, the inner one decreases with increase in g12, while the outer own grow with the increase in g12. The instability gain in momentum space for some repre- sentative value of intra- and inter-components interaction strength is shown in Fig. 11. It is observed that there ex- −4−2024ky −4−2024 kx00.511.52ξ 00.511.5 −4−20 2 4 kx−4−2024ky 0.00.51.01.5 (a) (b) FIG. 11. (color online) (a) 3D surface plot showing the MI gain,=jIm( )j, and (b) the corresponding 2D contour plot for the parameters kL= 1, = 1, n= 0:3,g=5 and g12= 1. ist three symmetric instability bands corresponding to kx, while only two for ky. Unlike the previous case, the instability gain is maximum for the bands corresponding tokyas shown in Fig. 11. C. Repulsive intra-component and attractive inter-component interactions Here, we consider the binary BEC with repulsive intra- component ( g>0) and attractive inter-component inter- actiong12<0. It is obvious from our earlier discussion, in the absence of Rabi and SO coupling, the MI (through ) is observed provided the condition jg12j>jgjis sat- ised. But in the presence of SO coupling, the MI is said to occur regardless of the sign of the interaction strength and there are no conditions imposed. In similar lines withthe previous section, we discuss MI in the two particular cases, i.e. constant intra-component interaction strength with varying inter-component strength and vice-versa. Fig. 12 shows the MI gain at constant gas a function of −4−2024 kx−5−4−3−2−10 g1200.511.5ξ 00.511.5 −4−20 2 4 kx−5−4−3−2−10g12 0.00.51.01.5 (a) (b) FIG. 12. (color online) 3D surface plot of the MI gain, =jIm( )jin thekx-g12plane and (b) the corresponding 2D contour plot for xed gwithkL= 1, = 1, n= 0:3,g= 1 andky= 1. g12. As the strength of the inter-component interaction increases, the outer instability band grows and merges with inner instability band of higher gain. The variation −4−2024 kx012345 g00.511.5ξ 00.511.5 −4−20 2 4 kx012345g 0.00.51.01.5 (a) (b) FIG. 13. (color online) 3D surface plot of the MI gain, =jIm( )jin thekx-gplane and (b) the corresponding 2D contour plot for xed g12withkL= 1, = 1, n= 0:3, g12=1 andky= 1. of MI gain for g12at constant gshows a similar trend, ex- cept the changes in the numerical value of gain as shown in Fig. 13. Fig. 14 depicts the MI gain in momentum space forkxandky. Unlike the earlier cases, the gain of −4 −202 4ky −4−2024 kx0123ξ 0123 −4−20 2 4 kx−4−2024ky 0123 3 (a) (b) FIG. 14. (color online) (a) 3D surface plot showing the MI gain,=jIm( )j, and (b) the corresponding 2D contour plot for the parameters kL= 1, = 1, n= 0:3,g= 2 and g12=13. the inner band is quantitatively same for both bands cor- responding to kxandky. However, the instability gain of the outer band corresponding to kyis slightly larger than the outer band of ky.8 D. Attractive intra-and inter-component interactions In this region, both intra- and inter-component interac- tions are attractive, i.e. g<0 andg12<0, and therefore, MI occurs naturally even without the aide of Rabi and SOC. This case has already been discussed thoroughly in the context of MI in two component BEC, and hence, an extensive investigation is needless. However, for the sake of completeness, we focus on the eect of SOC in the in- stability. Fig. 15 shows that the growth of MI gain with −4−2024kx−5−4−3−2−10 g00.511.5ξ0 0 .5 1 1 .5 −4−20 2 4 kx−5−4−3−2−10g 0.00.51.01.5 (a) (b) FIG. 15. (color online) 3D surface plot of the MI gain, =jIm( )jin thekx-gplane and (b) the 2D corresponding contour plot for xed g12withkL= 1, = 1, n= 0:3, g12=1, andky= 1. the variation of g12for constant g=1. The current case completely concur with our earlier discussion, and the MI becomes independent of the gfor the strong g12 interaction. The variation of MI gain in momentum space is shown in Fig. 16. Like in the previous section, the MI −4−20 2 4 kx−4−2024ky 0.00.51.01.5 (a) (b)−4 −2 0 2 4ky −4−2024 kx00.511.5ξ00.511.5 FIG. 16. (color online) (a) 3D surface plot showing the MI gain,=jIm( )j, and (b) the corresponding 2D contour plot for the parameters kL= 1, = 1, n= 0:3,g=1, g12=5. bands are symmetric across the zero wave number, and the maximum gain occurs for the bands corresponding to ky. E. Results and Discussion For the ease of understanding and to make the analysis self-explanatory, we summarize our results of MI in the two-dimensional SO coupled two-component BEC in Ta- ble I. We systematically discussed the presence/absence of SO/Rabi coupling under dierent combination of signs of intra- and inter-component interaction strength. Asit is evident from our extensive investigation that SO coupling inevitably destabilizes the initial steady state for equal densities of binary BEC, and thereby makes the system unstable for all combinations of interaction strength. Also, we have shown the conventional MI im- miscibility condition, g12> g for repulsive two compo- nent BEC system is no longer signicant for MI. Our par- ticular focus is on repulsive intra- and inter-component interaction, as it is proven to be stable against the pertur- bation, and therefore, MI is generally not feasible. How- ever, we have shown that MI can be achieved with the eect of SOC as demonstrated through Figs. 2 - 8. We discussed the MI gain in momentum space as a function ofkxandkyand emphasize the variation of gain over the wave numbers in the two directions. We noted that the MI gain is not identical on kxandky, and signicant changes in MI gain, width of instability region and the number of instability bands are readily observed. In Sec. IV B, we discussed the MI condition for attrac- tive intra- and repulsive inter-component interactions. Figs. 9 - 11 show the instability gain as a function of gandg12. One can straightforwardly observe the emer- gence of new instability bands in the MI gain plot, which is identied to be the consequence of the incorporation of SOC. Following that, we discussed in Sec. IV C, the MI scenario in the case of repulsive intra- and attractive inter-component interactions. Figs. 12 - 14 portray the variation of MI gain for gandg12. Along the similar lines with the earlier cases, the SOC results in new in- stability bands and thereby help in enhancing the MI in such systems. Finally, we studied MI in attractive intra- and inter-component interactions in Sec. IV D. It is very well known from the theory of MI in BEC, that attrac- tive interactions naturally support MI. Although, SOC is not fundamental for the origin of MI, however, SOC signicantly in uences the instability region in terms of peak gain and width as evident from Figs. 15 and 16. Overall, the eect of SOC can be understood as a means to achieve MI in repulsive interactions, and also enhance instability in the system. Last but not least, it is also important to see the impact of SO and Rabi couplings on MI gain for xed wavenum- berskxandky. Fig. 17 depicts the MI gain as a func- −4−2024Γ −4−2024 kL00.20.40.6ξ 00.20.4 −4−20 2 4 Γ−4−2024kL 0 (a) (b) FIG. 17. (color online) (a) 3D surface plot showing the MI gain,=jIm( )j, and (b) the corresponding 2D contour plot for the parameters kL= 1, = 1, g= 1,g12= 1, n= 0:3,kx= 2 andky= 1. tion of SO and Rabi coupling. As it is evident from our9 TABLE I. Summary of MI in SO coupled two dimensional binary BEC SO coupling Rabi coupling MI gain Dierent combinations Inference both interaction are repulsive Always stable g<0,g12>0 Always stable +g>0,g12<0 Always stable both interaction are attractive Always unstable = 0 both interaction are repulsive g<0,g12>0 stable for all cases g>0,g12<0 both interaction are attractive both interaction are repulsive g<0,g12>0 Similar to the case +g>0,g12<0 with = 0 and + both interaction are attractive kL= 0 <0 both interaction are repulsive Always stable g<0,g12>0 Always unstable g>0,g12<0 Always stable both interaction are attractive Always stable both interaction are repulsive g<0,g12>0 Similar to the case +g>0,g12<0 with = 0 and + both interaction are attractive >0 both interaction are repulsive Unstable ifjgj>jg12j g<0,g12>0 Always stable g>0,g12<0 Always unstable both interaction are attractive Unstable ifjg12j>jgj both interaction are repulsive Always stable g<0,g12>0 Unstable ifjg12j>jgj +g>0,g12<0 Always stable both interaction are attractive Unstable ifjgj>jg12j = 0 both interaction are repulsive Unstable ifjg12j>jgj g<0,g12>0 Always unstable g>0,g12<0 Unstable ifjgj>jg12j both interaction are attractive Always unstable both interaction are repulsive g<0,g12>0 stable for all cases +g>0,g12<0 both interaction are attractive kL= 1 <0 both interaction are repulsive Always stable g<0,g12>0 Always unstable g>0,g12<0 Unstable ifjgj>jg12j both interaction are attractive Always unstable both interaction are repulsive Always unstable g<0,g12>0 Always unstable +g>0,g12<0 Unstable ifjgj>jg12j both interaction are attractive Always unstable >0 both interaction are repulsive g<0,g12>0 Always unstable g>0,g12<0 both interaction are attractive choice of parameters, the instability bands are symmetric for both positive and negative values of the Rabi and SO10 coupling. One can also infer, that the MI is possible even for zero SOC, provided the Rabi coupling is >0. V. CONCLUSIONS To summarize, we investigated the dynamics of MI gain in two-dimensional SO coupled binary BEC at an equal density of pseudo-spin components. The disper- sion relation corresponding to the instability of the at CW background against small perturbation was stud- ied using linear stability analysis. For a comprehensive study, we consider all the possible combination of signs of intra- and inter-component interactions, with a partic- ular, emphasize on repulsive interactions. Our analysis illustrates that SOC inevitably contributes to instability, regardless of the nature of the interaction strength. With detailed interpretation, we have shown that the repulsive intra- and inter-component interaction admit instability and the MI immiscibility condition g12> gis no longer essential for MI. We also have shown, for the strong at- tractive inter-component interaction, the nature of the intra- component interaction is immaterial for constantSO and Rabi coupling. We also analyzed the variation of instability domain in momentum space for kxandky. The MI gain is not identical on kxandky, and signicant changes in MI gain and a number of bands are observed. In the case of systems naturally admitting MI (attractive interactions), the SOC and Rabi coupling manifest in the generation of new instability bands, thereby enhances the MI. Thus, we presented a comprehensive analysis with detailed interpretation and graphical illustration of MI in two-dimensional SO coupled binary BEC for equal den- sities. We believe, the aforementioned results could po- tentially provide new ways to generate and manipulate MI and solitons in two-dimensional BECs. ACKNOWLEDGMENTS The work of PM forms a part of Science & Engi- neering Research Board, Department of Science & Tech- nology, Government of India sponsored research project (No. EMR/2014/000644). KP thanks the DST, IFC- PAR, NBHM, and CSIR, Government of India, for the nancial support through major projects. [1] C. Nayak, S.H. Simon, A. Stern, M. Freedman, and S.D. Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008). [2] C.L. Kane and E.J. Mele, Z2Topological Order and the Quantum Spin Hall Eect, Phys. Rev. Lett. 95, 146802 (2005). [3] M.Z. Hasan and C.L. Kane, Colloquium: Topological in- sulators, Rev. Mod. Phys. 82, 3045 (2010). [4] H. Zhai, Spin orbit coupled quantum gases, Int. J. Mod. Phys. 26, 1230001 (2012). [5] M. K onig, S. Wiedmann, C. Br une, A. Roth, H. Buh- mann, L.W. Molenkamp, X.L. Qi, and S.C. Zhang, Quan- tum Spin Hall Insulator State in HgTe Quantum Wells, Science 318, 766 (2007). [6] B.A. Bernevig, T.L. Hughes, and S.-C. Zhang, Quantum spin Hall eect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006). [7] Y.-J. Lin, K. Jim enez-Garc a and I. B. Spielman, Spin- orbit coupled Bose-Einstein condensates, Nature 471, 83 (2011). [8] J. Ruseckas, G. Juzeli unas, P. Ohberg, and M. Fleis- chhauer, Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark States, Phys. Rev. Lett. 95, 010404 (2005). [9] C. Zhang, Spin-orbit coupling and perpendicular Zeeman eld for fermionic cold atoms: Observation of the intrin- sic anomalous Hall eect, Phys. Rev. A 82, 021607(R) (2010). [10] T. D. Stanescu, C. Zhang, and V. Galitski, Nonequilib- rium Spin Dynamics in a Trapped Fermi Gas with Eec- tive Spin-Orbit Interactions, Phys. Rev. Lett. 99, 110403 (2007). [11] D.L. Campbell, G. Juzelinas, and I.B. Spielman, Realistic Rashba and Dresselhaus spin-orbit coupling for neutralatoms, Phys. Rev. A 84, 025602 (2011). [12] Y. Zhang, L. Mao and C. Zhang, Mean-eld dynamics of spin-orbit coupled Bose-Einstein condensates, Phys. Rev. Lett.108, 035302 (2012). [13] W. Pengjun, Y.-Q. Zeng, F. Zhengkun, M. Jiao, H. Lianghui, C. Shijie, Z. Hui, and -Z. Jing, Spin-Orbit Coupled Degenerate Fermi Gases Phys. Rev. Lett. 109, 095301 (2012). [14] W.-L. Cheuk, A-.T. Sommer, Z. Hadzibabic, T. Yefsah, W-.S. Bakr, and M-.W. Zwierlein, Spin-Injection Spec- troscopy of a Spin-Orbit Coupled Fermi Gas Phys. Rev. Lett.109, 095302 (2012). [15] C. Wang, C. Gao, C.M. Jian, and H. Zhai, Spin-Orbit Coupled Spinor Bose-Einstein Condensates, Phys. Rev. Lett.105, 160403 (2010). [16] Y. Li, L. P. Pitaevskii, and S. Stringari, Quantum Tri- criticality and Phase Transitions in Spin-Orbit Coupled Bose-Einstein Condensates, Phys. Rev. Lett. 108, 225301 (2012). [17] J. Radic, T. A. Sedrakyan, I. B. Spielman, and V. Galit- ski, Vortices in spin-orbit-coupled Bose-Einstein conden- sates, Phys. Rev. A 84, 063604 (2011). [18] C.-F. Liu and W. M. Liu, Spin-orbit-coupling-induced half-skyrmion excitations in rotating and rapidly quenched spin-1 Bose-Einstein condensates, Phys. Rev. A86, 033602 (2012). [19] V. Achilleos, D. J. Frantzeskakis, P. G. Kevrekidis and D. E. Pelinovsky, Matter-wave bright solitons in spin-orbit coupled Bose-Einstein condensates, Phys. Rev. Lett. 110, 264101 (2013). [20] V. Achilleos, J. Stockhofe, D. J. Frantzeskakis, P. G. Kevrekidis and P. Schmelcher, Matter-wave dark solitons and their excitation spectra in spin-orbit coupled Bose- Einstein condensates, Europhys. Lett. 10320002 (2013).11 [21] P.-S. He, Stability of plane-wave Bose-Einstein conden- sation with spin-orbit coupling, Eur. Phys. J. D 48, 67 (2013). [22] P.-S. He, W.-L. You, and W.-M. Liu, Stability of a two- dimensional homogeneous spin-orbit-coupled boson sys- tem, Phys. Rev. A 87, 063603 (2013). [23] Y. Cheng, G. Tang, and S.K. Adhikari, Localization of a spin-orbit-coupled Bose-Einstein condensate in a bichro- matic optical lattice, Phys. Rev. A 89, 063602 (2014). [24] L. Salasnich, W.B. Cardoso, and B.A. Malomed, Local- ized modes in quasi-two-dimensional Bose-Einstein con- densates with spin-orbit and Rabi couplings, Phys. Rev. A90, 033629 (2014). [25] J. Jin, S. Zhang, and W. Han, Zero-momentum coupling induced transitions of ground states in Rashba spin-orbit coupled Bose-Einstein condensates, J. Phys. B: At. Mol. Opt. Phys. 47, 115302 (2014). [26] G. Dresselhaus, Spin-Orbit Coupling Eects in Zinc Blende Structures, Phys. Rev. 100, 580 (1955). [27] Y. A. Bychkov and E. I. Rashba, Oscillatory eects and the magnetic susceptibility of carriers in inversion layers, J. Phys. C 17, 6039 (1984). [28] T. B. Benjamin and J. E. Feir, disintegration of wave trains on deep water Part 1. Theory, J. Fluid Mech. 27, 417 (1967). [29] L.A. Ostrovsky, Propagation of wave packets and space- time self-focusing in a nonlinear medium., Sov. Phys. JETP 24, 797 (1967). [30] A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric bers. I. Anomalous dispersion, Appl. Phys. Lett. 23, 142 (1973). [31] G.P. Agrawal, Nonlinear Fiber Optics, 5th edition (Aca- demic Press San Diego, 2013). [32] A. Z. Assalauov and V. E. Zakharov, Modulation insta- bility and collapse of plasma waves in a magnetic eld, Sov.J. Plasma Phys. 11, 762 (1985). [33] T. Taniuti and H. Washimi, Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma, Phys. Rev. Lett. 21, 209 (1968). [34] I. Kourakis, P.K. Shukla, M. Marklund and L. Sten- o Modulational instability criteria for two-component Bose-Einstein condensates, Eur. Phys. J. B 46, 381(2005). [35] P.G. Kevrekidis, R. Carretero-Gonzlez, G. Theocharis, D.J. Frantzeskakis, B.A. Malomed, Stability of dark soli- tons in a Bose-Einstein condensate trapped in an optical lattice, Phys. Rev. A 68, 035602 (2003). [36] W. X. Zhang, D.L. Zhou, M.-S. Chang, M.S. Chapman, and L. You, Dynamical Instability and Domain Forma- tion in a Spin-1 Bose-Einstein Condensate, Phys. Rev. Lett.95, 180403 (2005). [37] A. Smerzi, A. Trombettoni, P.G. Kevrekidis, and A.R. Bishop, Dynamical Super uid-Insulator Transition in a Chain of Weakly Coupled Bose-Einstein Condensates, Phys. Rev. Lett. 89, 170402 (2002). [38] G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, Modulational insta- bility of Gross-Pitaevskii-type equations in 1 + 1 dimen- sions, Phys. Rev. A 67, 063610 (2003). [39] K. Kasamatsu and M. Tsubota, Modulation instabil- ity and solitary-wave formation in two-component Bose- Einstein condensates, Phys. Rev. A 74, 013617 (2006). [40] K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Formation and propagation of matter-wave soli- ton trains, Nature 417, 150 (2002). [41] E.V. Goldstein and P. Meystre, Quasiparticle instabilities in multicomponent atomic condensates, Phys. Rev. A 55, 2935 (1997). [42] G. Agrawal, Modulation instability induced by cross- phase modulation in optical bers, Phys. Rev. Lett. 59, 880 (1987). [43] K. Bao and Y. Cai, Ground states and dynamics of spin- orbit coupled Bose-Einstein condensates, SIAM J. App. Math. 95, 492 (2015). [44] K. Nithyanandan, R.V.J. Raja, and K. Porsezian, Mod- ulational instability in a twin-core ber with the eect of saturable nonlinear response and coupling coecient dispersion, Phys. Rev. A 87, 043805 (2013). [45] I.A. Bhat, T. Mithun, B.A. Malomed, and K. Porsezian, Modulational instability in binary spin-orbit-coupled Bose-Einstein condensates, Phys. Rev. A 92, 063606 (2015).

2311.00362v1.Theory_of_Orbital_Pumping.pdf

Theory of Orbital Pumping Seungyun Han∗∗,1Hye-Won Ko∗∗,2Jung Hyun Oh,2Hyun-Woo Lee,1,∗Kyung-Jin Lee,2,†and Kyoung-Whan Kim3,‡ 1Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea 2Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea 3Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea We develop a theory of orbital pumping, which corresponds to the emission of orbital currents from orbital dynamics. This phenomenon exhibits two distinct characteristics compared to spin pumping. Firstly, while spin pumping generates solely spin (angular momentum) currents, orbital pumping yields both orbital angular momentum currents and orbital angular position currents. Secondly, lattice vibrations induce orbital dynamics and associated orbital pumping as the orbital angular position is directly coupled to the lattice. These pumped orbital currents can be detected as transverse electric voltages via the inverse orbital(-torsion) Hall effect, stemming from orbital textures. Our work proposes a new avenue for generating orbital currents and provides a broader understanding of angular momentum dynamics encompassing spin, orbital, and phonon. Introduction.– Orbital transport in solids has recently at- tracted considerable theoretical and experimental interest be- cause nonequilibrium orbital quantities arise from strong crystal field coupling rather than weak spin-orbit coupling (SOC) [1]. It results in intriguing phenomena associated with orbital angular momentum (OAM), including the or- bital Hall effect [2–11] and orbital magnetoresistance [12, 13]. In the presence of SOC, these OAM-related phenomena are intimately connected to spin-related ones; the spin Hall ef- fect [14, 15] and spin magnetoresistance [16]. Moreover, anal- ogous to spin torque, which arises from spin injection into a ferromagnet [17, 18], the injection of OAM into a ferromagnet results in orbital torque [19–24] contributing to the net mag- netic torque. These findings underscore the electron’s orbital as an essential degree of freedom for understanding angular momentum transport in solids and realizing novel orbitronic devices. Despite its importance, identifying orbital transport presents challenges because OAM shares the same symmetry opera- tions with spin. Symmetry-wise, distinguishing OAM from spin is thus impossible. Consequently, previous experi- ments have relied on the quantitative difference between spin Hall and orbital Hall conductivities to identify the orbital physics [11, 13, 21–24]. While the spin Hall conductivity is significant only in a limited number of materials, the orbital Hall conductivity is large in a broad range of materials and often surpasses the maximum value of spin Hall conductiv- ity [8, 25]. Although these quantitative differences are valuable, qual- itative differences between the orbital and spin properties are crucial for understanding orbital physics and its unambigu- ous identification. In this regard, orbital angular position (OAP), which describes different aspects of orbital states from OAM [26], emerges as a crucial element for such qualitative distinctions. It is noted that three OAM operators ( Lx,Ly,Lz) are insufficient to completely describe orbital states and addi- tional OAP operators, composed of even-order symmetrized products of L, are essential. Physically, OAP operators capture real orbital states with zero OAM expectation value, such as pxandpyorbitals, and mediate the orbital-lattice (i.e., crystal field) coupling [26]. Unlike OAM, which bears analogies to NM2NM1NM1xzy c M(t) c Oscillating magnetic fieldOscillating lattice jLyz,j{Lz,Lx}z jx FIG. 1. Schematic illustration of orbital pumping in a model sys- tem, NM1/NM2/NM1 (NM = normal metal), driven by an oscillating magnetic field or vibrating lattice. For the magnetic-field-driven case, NM2 may be considered a ferromagnet. ji zis an orbital current flow- ing along zwith orbital i, and jxis a charge current flowing along x. spin in various aspects, OAP notably lacks a spin counterpart. Adiabatic pumping [27] provides critical insights into the dynamics of physical systems [28]. For spin dynamics, adi- abatic spin pumping has elucidated phenomena such as en- hanced magnetic damping [29] and spin motive force [30– 34], and served as an efficient means for generating pure spin currents [35, 36]. Spin pumping is connected with spin torque through the Onsager reciprocity [37]. Given orbital torque [19–24], the Onsager reciprocity also guarantees the presence of orbital pumping. Recent studies have investi- gated orbital pumping [38–40], focusing on OAM pumping but neglecting the contribution of OAP. Therefore, a complete description of orbital pumping is yet to be established. In this Letter, we present a theory of orbital pumping that incorporates the entire orbital degrees of freedom, encom- passing both OAM and OAP. We consider two distinct types of adiabatic orbital pumping. The first type involves the OAM dynamics induced by an AC magnetic field (Fig. 1), similar to the approach used for spin pumping with ferromagnet/normal metal bilayers. The second type, which does not require a ferromagnet, arises from the lattice dynamics, which is real- ized by applying AC stress to a nonmagnet (Fig. 1). ResultingarXiv:2311.00362v1 [cond-mat.mes-hall] 1 Nov 20232 AC strain gives rise to orbital pumping through strong orbital- lattice coupling mediated by OAP. We show that both methods generate not only OAM pumping but also OAP pumping. The latter pumping has no spin counterpart and causes an even- order harmonic AC transverse voltage, which is absent for spin pumping and thus allows for unambiguous identification of orbital pumping in experiments. Orbital pumping by oscillating magnetic field.– For the first type, we examine orbital pumping arising from the dynamics of the orbital moment (i.e., OAM) driven by an AC magnetic field. We ignore the spin degree of freedom to focus solely on orbital responses and neglect SOC for the same reason. To get a tractable analytic formula, we consider a p-orbital system as a minimal model and assume an ideal case where three p- orbitals are degenerate in equilibrium. This ideal case reveals critical qualitative differences between orbital pumping and spin pumping. However, this ideal case is not realized in real materials since p-orbitals are split due to crystal fields. We consider the crystal field effects on orbital pumping in numer- ical calculations below and demonstrate that the predictions from the ideal case persist in real situations. We construct a model system of NM1/NM2/NM1 structure (Fig. 1; NM = normal metal) and derive orbital pumping cur- rents induced by time-dependent perturbations to NM2. The perturbation Hamiltonian is H(t)=JexL·M(t), where M(t) is the unit vector of time-dependent magnetic field, Jexis the coupling strength, and Lis the (dimensionless) OAM operator, which is a 3×3matrix in p-orbital space. For a degenerate p-orbital system, the orbital is conserved, and the 3×3Green’s function is given by, g=g0I+g1−g−1 2L·M(t)+g1+g−1−2g0 2[L·M(t)]2,(1) where gmis the Green’s function associated with an eigenstate having an eigenvalue m(=−1,0,1)ofL·M(t)andgincludes a term quadratic in L(i.e., OAP). We abbreviate the explicit po- sition dependence of gmfor simplicity. Employing the method developed in Ref. [34], we compute the 3×3matrix pumped current density operator jαwhereαis the flow direction: jα/(−e)=jOAM α·L+/summationdisplay βγjOAP α,βγ{Lβ,Lγ}, (2) where the first and second terms represent the OAM and OAP currents, respectively. After some algebra, we obtain jOAM α=1 4πRe/bracketleftigg/parenleftiggG1,0 α+G0,−1 α 2/parenrightigg/parenleftigg M×dM dt−idM dt/parenrightigg/bracketrightigg ,(3) jOAP α,βγ=1 8πRe/bracketleftigg/parenleftiggG1,0 α−G0,−1 α 2/parenrightigg ×Mβ/parenleftigg M×dM dt/parenrightigg γ−iMβdMγ dt+(β↔γ),(4) Gµ,ν α=Jexℏ2 me/integraldisplay dr′[gR µ(r,r′)↔ ∂αgA ν(r′,r)], (5)where gR/Arepresents retarded/advanced Green’s function of the NM1/NM2/NM1 heterostructure,↔ ∂αis the antisymmetric differential operator, and/integraltext dr′denotes the volume integral. The OAM pumping current [Eq. (3)] has the same form as the spin pumping current [29], except for replacing the spin mixing conductance with the orbital mixing conductance [Eq. (5)]. The spin (orbital) mixing conductance arises from the scattering caused by an abrupt change of the spin (orbital) environment at an interface. Since the orbital space encom- passes additional degrees of freedom (OAP), the orbital mixing conductance has more components than the spin mixing con- ductance. More specifically, 9 conductances are required to completely describe p-orbital scattering whereas 4 conduc- tances ( G↑,G↓,Re[Gmix],Im[Gmix]) are sufficient for a full description of spin scattering [41]. The OAP pumping current derived in Eq. (4) corresponds to the additional components and is of the distinct form (including higher order terms in M) from the OAM pumping current (thus from the spin pumping current as well). This OAP current corresponds to a flow of real-orbital-polarized electrons with zero OAM and lacks a counterpart in spin pumping, marking a qualitative distinction between orbital pumping and spin pumping. As an example, when M(t)rotates in the zxplane (M(t)=ˆzcosωt+ˆxsinωt),jOAM z =jOAM α·Land jOAP z=/summationtext βγjOAP α,βγ{Lβ,Lγ}become jOAM z=ω 4π/braceleftig Re[G+ L]Ly+Im[G+ L](Lxcosωt−Lzsinωt)/bracerightig , (6) jOAP z=ω 4πIm[G− L]/parenleftig {Lz,Lx}cos 2ωt−(L2 z−L2 x) sin 2ωt/parenrightig +ω 4πRe[G− L]/parenleftig/braceleftig Ly,Lz/bracerightig cosωt+/braceleftig Lx,Ly/bracerightig sinωt/parenrightig ,(7) where G± L=(G1,0 z±G0,−1 z)/2. Here, jOAM zandjOAP zare the op- erator expressions of the OAM and OAP currents, respectively, which we use to show the dynamics of each orbital degree of freedom explicitly. Equations (6) and (7) predict that orbital pumping currents in ideal situations (i.e., no crystal field spit- ting) consist of a DC component carrying Ly, first-harmonic (1ω) ones carrying Lx,Lz,{Ly,Lz}, and{Lx,Ly}, and second- harmonic ( 2ω) ones carrying{Lz,Lx}and(L2 z−L2 x). Next, we examine orbital pumping in a more realistic situa- tion, where the degeneracy of orbitals is lifted by crystal fields. We adopt a sp3tight-binding model in a simple cubic lattice. With sphybridization, the orbital nature of eigenstates varies with the crystal momentum. Such variation (called orbital texture) is common in real materials [12, 42]. We consider the same M(t)oscillation as above and calcu- late numerically the pumped orbital current using the linear response theory in the adiabatic limit (see Supplementary Ma- terials (SM) for details [43]). We find that all of the orbital cur- rents predicted by Eqs. (6) and (7) are pumped by the M(t)os- cillation, although additional types of orbital currents are also pumped. Figure 2 presents some of the numerical results: DC OAM current ( jLy,DC z ), DC OAP current ( j{Lz,Lx},DC z ) [Fig. 2(a)],3 -1.0-0.50.00.51.0j(t) wtjLyz(t) j{Lz,Lx} z (t) jx(t) 2p p 0 -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0jDC z(lattice constant, a0)jLy,DC z j{Lz,Lx},DC z jDC x(a) (b) FIG. 2. (a) Spatial profile of DC pumping currents, jLy,DC z (red), j{Lz,Lx},DC z (blue), and jDC x(gray), driven by the time-dependent mag- netic field, L·M(t)forM(t)=ˆzcosωt+ˆxsinωt, on NM2 layer. (b) Temporal dependence of pumping currents j(t)at the right interface (z=16a0). The thick dashed horizontal line in (b) shows the DC component of transverse charge current jx(t). and second-harmonic OAM current ( jLy z(t)), second-harmonic OAP current ( j{Lz,Lx} z (t)) [Fig. 2(b)]. We note that the DC OAP current and the second-harmonic OAM current are unexpected from the analytic theory [Eqs. (6) and (7)]. We attribute these unexpected orbital currents to the fact that the crystal field splitting can convert OAM current to OAP current, and vice versa [26]. Note that the relatively short decay length of orbital currents in NM1 [Fig. 2(a)] arises from the orbital characters of the band structure used in our model. It increases with de- creasing the orbital splitting [44], as demonstrated in SM [43]. The pumped OAM and OAP currents are converted to trans- verse charge currents jxthrough the inverse orbital Hall ef- fect [26, 45, 46] and inverse orbital-torsion Hall effect [26], respectively [Figs. 2(a) and (b)]. A recent orbital pumping experiment [39] reported a DC charge current and attributed it entirely to conversion from pumped DC OAM current [44]. However, the measured DC charge current may contain an ad- ditional contribution due to conversion from pumped DC OAP current. The converted charge current also contains a second- harmonic component since pumped OAM and OAP currents contain second-harmonic components [Fig. 2(b)]. The second- harmonic charge current remains unexplored experimentally. It is worth noting that the generalized pumping equation de- rived in SM [43] indicates that a d-orbital system may also produce a fourth-harmonic component. As a spin pumping current contains only DC and first-harmonic components [29], higher-harmonics pumping signals are a unique feature of or- bital pumping. In addition to orbital pumping, we find that the pumped -1.0-0.50.00.51.0jLy,DCztime-dependent hopping integrals [L×u(t)]2(´102) -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0jDCx z(lattice constant, a0)time-dependent hopping integrals [L×u(t)]2(´102) -1.0-0.50.00.51.0j{Lz,z Lx},DCtime-dependent hopping integrals [L×u(t)]2(´102)(a) (b) (c)FIG. 3. Spatial profile of DC pumping currents (a) jLy,DC z, (b) j{Lz,Lx},DC z , and (c) jDC xdriven by the lattice dynamics, which is imposed by time-dependent variations of tight-binding hopping integrals (solid circles) and [L·u(t)]2foru(t)=ˆzcosωt+ˆxsinωt(open diamonds). orbital current is converted to another orbital current, i.e., the orbital swapping effect. Analogous to the spin swapping effect [47–49], it results in the type I conversion (e.g., jLy,DC z→ jLz,DC y) and the type II conversion (e.g., jLz,DC z→jLx,DC x) , which can be called the OAM swapping effect and is in line with a recent theory [50]. We find that the OAP swapping effect also arises (see SM [43]). Similar to the orbital Hall effect [7], the orbital swapping effect arises even without SOC. But it vanishes in the absence of the orbital texture. Orbital pumping by lattice dynamics–. In the second case, we explore orbital pumping due to lattice dynamics, which can be realized by a time-dependent variation of strain, i.e., a time- dependent deformation of crystal. For a crystal under arbitrary deformations, three effects need to be considered: variations of i) orbital splitting, ii) bonding lengths, and iii) bonding angles. Each factor manifests in the tight-binding model as corrections to on-site energies, magnitudes of hopping inte- grals, and directional cosines, respectively. By assuming the time-dependent hopping integrals to follow a power law of bonding length [51, 52], we integrate the periodic lattice dy- namics driven by biaxial strains into our model and numer- ically calculate orbital pumping due to this lattice dynamics (see SM [43] for details). We consider two biaxial strains in the zxplane with a phase difference, which makes the cubic lattice undergo a circularly rotating strain. It resembles a generation of phonon angu-4 lar momentum (PAM) polarized along the ydirection under surface acoustic wave [53]. Numerical calculations with the time-dependent hopping integrals show that DC OAM cur- rent [ jLy,DC z ; Fig. 3(a)], DC OAP current [ j{Lz,Lx}DC z ; Fig. 3(b)], and associated DC transverse charge current [ jDC z; Fig. 3(c)] are pumped. OAM pumping induced by lattice dynamics is the reverse process of crystal field torque [54], where non- equilibrium OAM, generated by external perturbations such as an electric field, is absorbed by the lattice. The reverse process of the OAP current may offer another mechanism of the crystal field torque, which has not been identified yet. Orbital pumping driven by lattice dynamics can be under- stood through an OAP-type perturbation, which is qualitatively distinct from the previously considered perturbation, L·M(t), for the case of oscillating M(t). To demonstrate this, we con- sider the long-wavelength limit ( k→0) of lattice vibration, where we may neglect the effects of spatial variations and fo- cus on the time-dependent variations of orbital splitting (i.e., time-dependent lifting of orbital degeneracy in the analytic model). If the crystal experiences a strain along the udirec- tion, the energy of the puorbital becomes different from that of the other porbitals perpendicular to it. It is important to note that the eigenvalue of (L·u)2with respect to the former is 0 while those with respect to the latter are 1. Therefore, the time-dependent strain in a p-orbital system can be modeled by a perturbation of [L·u(t)]2. In a general sense, a lattice distor- tion is time-reversal even, and thus the corresponding pertur- bation Hamiltonian should be expressed in terms of even-order products of the OAM operators, which is essentially OAP. The open diamonds in Fig. 3 confirm that the [L·u(t)]2perturba- tion incorporated to the numerical model can reproduce the characteristics of orbital pumping from the lattice dynamics described by time-dependent hopping integrals in reasonable consistency, despite the simplicity of the model. Further insight can be gained through a simplified ana- lytic treatment. Assuming that all equilibrium orbitals are degenerate at the Γpoint for simplicity, the Green’s func- tion for the perturbation of [L·u(t)]2can be expressed as g=g0I+(g1−g0)[L·u(t)]2. Utilizing Eqs. (3) and (4), we derive jOAM α=1 4πRe[G1,0 α]u×du dt, (8) jOAP α,βγ=1 8πIm[G1,0 α]d(uβuγ) dt, (9) which explain that both OAM and OAP currents are pumped by lattice dynamics. Equation (8) indicates the presence of nonzero DC OAM pumping when urotates in time, for exam- ple,u(t)=ˆzcosωt+ˆxsinωt. OAM pumping induced by the rotating strain can be understood as the transfer of PAM to elec- tron OAM [55]. Conversely, Eq. (9) suggests the absence of DC OAP pumping for a periodic u, as demonstrated by its time average (1/T)/integraltextT 0jOAP α,βγdt=(1/8πT) Im[G1,0 α]uβ(T)uγ(T)→0 asTincreases. The emergence of DC OAP pumping in nu- merical calculations (Fig. 3) is attributed to the orbital tex- ture, resulting in an interconversion between OAM and OAP,as generally proven in Ref. [26]. Its detailed analytical de- scription would require going beyond our simplified approach and considering the Green’s function more complicated than g=g0I+(g1−g0)[L·u(t)]2, which we leave for future work. Discussion and outlook.– We demonstrated orbital pump- ing driven by either oscillating M(t)or lattice dynamics. In both pumping methods, not only OAM currents but also OAP currents are pumped. The simultaneous emergence of OAM and OAP pumping highlights the necessity of considering both orbital degrees of freedom when describing orbital dynamics. This is a fundamental requirement since a complete descrip- tion of entire orbital degrees of freedom can be achieved only by incorporating both OAM and OAP operators. As a result, OAP pumping always plays a role in physical phenomena aris- ing from OAM pumping. An important consequence is that OAP current contributes to orbital torque. It has been believed that orbital torque originates solely from the injection of OAM current [19]. However, the emergence of OAP current through magnetization dynamics suggests the existence of its inverse process: the generation of magnetic torque through the injec- tion of OAP current. This previously unrecognized inverse process, which we term ”OAP torque”, introduces a new di- mension to the understanding of orbital torque. Moreover, the Onsager reciprocity between orbital pumping and orbital torque is validated only when one considers both OAM and OAP contributions. Our theory of orbital pumping offers an exploitable method for generating and detecting the OAP degree of freedom. We note that previous suggestions in Ref. [26] rely on twisted heterostructures, which can be challenging to implement, or on low-gap semiconductors with limited availability of the required materials. In contrast, the high-harmonics pumping, driven by the OAP degree of freedom, can be realized with various materials. We also note that this property stems from the distinct behaviors of high-order products of Loperators with respect to rotational transformations. Thus, it is a general property that is not limited to p-orbital systems. Furthermore, orbital pumping is not limited to multilayer structures. Recalling that the spin motive force is a continuum version of spin pumping [56], we anticipate the presence of an orbital motive force when a single-layer system exhibits an inhomogeneous crystal field. Thus, the orbital motive force would encompass OAP contributions, which lack their spin counterparts. Notably, the physical properties of the orbital motive force would differ from those of the spin motive force due to the presence of orbital texture, a factor not accounted for in the theory of the spin motive force. The exploration of this topic remains a subject for future work. Lastly, our work sheds light on the transfer of angular mo- mentum between electrons and phonons. Theoretical estima- tions [57–59] and experimental measurements [53, 60, 61] il- lustrate that PAM can have a considerable magnitude contrary to early assumptions and plays a nontrivial role in various phe- nomena such as magnetization relaxation [62] and ultrafast de- magnetization [63, 64]. Intriguingly, recent many-body treat- ment [65] shows that a complete picture of angular momen-5 tum transfer between electron and phonon subsystems requires OAM as a key ingredient since the electron-phonon coupling is independent of spin. The strain-induced orbital pumping weighs heavily on this connection of orbital and lattice and call for a wider viewpoint on PAM dynamics [55, 66, 67] assisted by the orbital degree of freedom [68]. Note added.– During the preparation of our manuscript, we became aware of a recent theoretical work on orbital pumping that focuses only on OAM pumping induced by magnetization dynamics [40]. This work was supported by the National Research Foun- dation of Korea (NRF) funded by the Ministry of Science and ICT (2020R1A2C3013302, 2022M3I7A2079267) and the KIST Institutional Program. S.H. and H.-W.L. were supported by the Samsung Science and Technology Foundation (BA- 1501-51). ∗∗S.H. and H.-W.K. contributed equally to this work. ∗hwl@postech.ac.kr †kjlee@kaist.ac.kr ‡kwk@kist.re.kr [1] D. Go, D. Jo, H.-W. Lee, M. Kl ¨aui, and Y. Mokrousov, Orbi- tronics: Orbital currents in solids, Europhys. Lett. 135, 37001 (2021). [2] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Orbitronics: The Intrinsic Orbital Current in p-Doped Silicon, Phys. Rev. Lett. 95, 066601 (2005). [3] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Inoue, Intrinsic spin Hall effect and orbital Hall effect in 4dand5dtransition metals, Phys. Rev. B 77, 165117 (2008). [4] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Giant Intrinsic Spin and Orbital Hall Effects in Sr2MO4 (M=Ru, Rh, Mo), Phys. Rev. Lett. 100, 096601 (2008). [5] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Giant Orbital Hall Effect in Transition Metals: Origin of Large Spin and Anomalous Hall Effects, Phys. Rev. Lett. 102, 016601 (2009). [6] I. V. Tokatly, Orbital momentum Hall effect in p-doped graphane, Phys. Rev. B 82, 161404(R) (2010). [7] D. Go, D. Jo, C. Kim, and H.-W. Lee, Intrinsic Spin and Orbital Hall Effects from Orbital Texture, Phys. Rev. Lett. 121, 086602 (2018). [8] D. Jo, D. Go, and H.-W. Lee, Gigantic intrinsic orbital Hall effects in weakly spin-orbit coupled metals, Phys. Rev. B 98, 214405 (2018). [9] S. Bhowal and G. Vignale, Orbital Hall effect as an alternative to valley Hall effect in gapped graphene, Phys. Rev. B 103, 195309 (2021). [10] T. P. Cysne, M. Costa, L. M. Canonico, M. B. Nardelli, R. Muniz, and T. G. Rappoport, Disentangling Orbital and Valley Hall Effects in Bilayers of Transition Metal Dichalcogenides, Phys. Rev. Lett. 126, 056601 (2021). [11] Y.-G. Choi, D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H. G. Park, C. Kim, B.-C. Min, G.-M. Choi, and H.-W. Lee, Observation of the orbital Hall effect in a light metal Ti, Nature 619, 52 (2023). [12] H.-W. Ko, H.-J. Park, G. Go, J. H. Oh, K.-W. Kim, and K.-J. Lee, Role of orbital hybridization in anisotropic magnetoresistance,Phys. Rev. B 101, 184413 (2020). [13] S. Ding, Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu, S. Becker, W. Yang, H. Du, C. Wang, Y. Yang, G. Jakob, M. Kl ¨aui, Y. Mokrousov, and J. Yang, Observation of the Orbital Rashba- Edelstein Magnetoresistance, Phys. Rev. Lett. 128, 067201 (2022). [14] J. Sinova, D. Culcer, Q. Niu, N. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Universal Intrinsic Spin Hall Effect, Phys. Rev. Lett. 92, 126603 (2004). [15] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015). [16] H. Nakayama, Y. Kanno, H. An, T. Tashiro, S. Haku, A. Nomura, and K. Ando, Rashba-Edelstein Magnetoresistance in Metallic Heterostructures, Phys. Rev. Lett. 117, 116602 (2016). [17] J. C. Slonczewski, Current-driven excitation of magnetic multi- layers, J. Magn. Magn. Mater. 159, L1 (1996). [18] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B 54, 9353 (1996). [19] D. Go and H.-W. Lee, Orbital torque: Torque generation by orbital current injection, Phys. Rev. Res. 2, 013177 (2020). [20] Z. C. Zheng, Q. X. Guo, D. Jo, D. Go, L. H. Wang, H. C. Chen, W. Yin, X. M. Wang, G. H. Yu, W. He, H.-W. Lee, J. Teng, and T. Zhu, Magnetization switching driven by current-induced torque from weakly spin-orbit coupled Zr, Phys. Rev. Res. 2, 013127 (2020). [21] Y. Tazaki, Y. Kageyama, H. Hayashi, T. Harumoto, T. Gao, J. Shi, and K. Ando, Current-induced torque originating from orbital current, arXiv:2004.09165. [22] D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun, D. Jo, S. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park, B.-C. Min, H. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee, Orbital torque in magnetic bilayers, Nat. Commun. 12, 6710 (2021). [23] J. Kim, D. Go, H. Tsai, D. Jo, K. Kondou, H.-W. Lee, and Y. Otani, Nontrivial torque generation by orbital angular momen- tum injection in ferromagnetic-metal/ Cu/Al2O3trilayers, Phys. Rev. B 103, L020407 (2021). [24] G. Sala and P. Gambardella, Giant orbital Hall effect and orbital- to-spin conversion in 3 d, 5d, and 4 fmetallic heterostructures, Phys. Rev. Res. 4, 033037 (2022). [25] L. Salemi and P. M. Oppeneer, First-principles theory of intrinsic spin and orbital Hall and Nernst effects in metallic monoatomic crystals, Phys. Rev. Mater. 6, 095001 (2022). [26] S. Han, H.-W. Lee, and K.-W, Kim, Orbital Dynamics in Cen- trosymmetric Systems, Phys. Rev. Lett. 128, 176601 (2022). [27] D. J. Thouless, Quantization of particle transport. Phys. Rev. B 27, 6083 (1983). [28] R. Citro and M. Aidelsburger, Thouless pumping and topology, Nat. Rev. Phys. 5, 87 (2023). [29] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Enhanced Gilbert Damping in Thin Ferromagnetic Films, Phys. Rev. Lett. 88, 117601 (2002). [30] K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee, Prediction of Giant Spin Motive Force due to Rashba Spin-Orbit Coupling, Phys. Rev. Lett 108, 217202 (2012). [31] G. Tatara, N. Nakabayashi, and K.-J. Lee, Spin motive force induced by Rashba interaction in the strong sdcoupling regime, Phys. Rev. B 87, 054403 (2013). [32] W. M. Saslow, Spin pumping of current in non-uniform con- ducting magnets, Phys. Rev. B 76, 184434 (2007). [33] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Spin Pumping and Spin-Transfer Torques in Antiferromagnets, Phys. Rev. Lett. 113, 057601 (2014). [34] K. Chen and S. Zhang, Spin Pumping in the Presence of Spin- Orbit Coupling, Phys. Rev. Lett. 114, 126602 (2015).6 [35] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, Quantifying Spin Hall Angles from Spin Pumping: Experiments and Theory, Phys. Rev. Lett. 104, 046601 (2010). [36] K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa, and E. Saitoh, Inverse spin-Hall effect induced by spin pumping in metallic system. J. Appl. Phys. 109, 103913 (2011). [37] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Nonlocal magnetization dynamics in ferromagnetic heterostruc- tures, Rev. Mod. Phys. 77, 1375 (2005). [38] E. Santos, J.E. Abr ˜ao, D. Go, L.K. de Assis, Y. Mokrousov, J.B.S. Mendes, and A. Azevedo, Inverse Orbital Torque via Spin-Orbital Intertwined States, Phys. Rev. Appl. 19, 014069 (2023). [39] H. Hayashi and K. Ando, Observation of orbital pumping, arXiv:2304.05266 (2023). [40] D. Go, K. Ando, A. Pezo, S. Bl¨ ugel, A. Manchon, and Y. Mokrousov, Orbital Pumping by Magnetization Dynamics in Ferromagnets, arXiv:2309.14817 (2023). [41] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Finite-Element Theory of Transport in Ferromagnet–Normal Metal Systems, Phys. Rev. Lett. 84, 2481 (2000). [42] S. Han, H.-W. Lee, and K.-W. Kim, Microscopic study of orbital textures, Curr. Appl. Phys. 50, 13 (2023). [43] See Supplemental Materials for details. [44] D. Go, D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park, S. Bl¨ ugel, H.-W. Lee, and Y. Mokrousov, Long-Range Orbital Torque by Momentum-Space Hotspots, Phys. Rev. Lett. 130, 246701 (2023). [45] P. Wang, Z. Feng, Y. Yang, D. Zhang, Q. Liu, Z. Xu, Z. Jia, Y. Wu, G. Yu, X. Xu, and Y. Jiang, Inverse orbital Hall effect and orbitronic terahertz emission observed in the materials with weak spin-orbit coupling, npj Quantum Mater. 8, 28 (2023). [46] Y. Xu, F. Zhang, A. Fert, H.-Y. Jaffres, Y. Liu, R. Xu, Y. Jiang, H. Cheng, and W. Zhao, Orbitronics: Light-induced Orbit Currents in Terahertz Emission Experiments, arXiv:2307.03490 (2023). [47] M. B. Lifshits and M. I. Dyakonov, Swapping Spin Currents: Interchanging Spin and Flow Directions, Phys. Rev. Lett. 103, 186601 (2009). [48] S. Sadjina, A. Brataas, and A. G. Mal’shukov, Intrinsic spin swapping, Phys. Rev. B 85, 115306 (2012). [49] H. B. M. Saidaoui and A. Manchon, Spin-Swapping Transport and Torques in Ultrathin Magnetic Bilayers, Phys. Rev. Lett. 117, 036601 (2016). [50] A. Manchon, A. Pezo, K.-W. Kim, and K.-J. Lee, Orbital dif- fusion, polarization and swapping in centrosymmetric metals, arXiv:2310.04763 (2023). [51] S. Froyen and W. A. Harrison, Elementary prediction of linear combination of atomic orbitals matrix elements, Phys. Rev. B 20, 2420 (1979). [52] W. A. Harrison, Electronic Structure and the Properties of Solids(Freeman, San Francisco, 1980). [53] R. Sasaki, Y. Nii, and Y. Onose, Magnetization control by an- gular momentum transfer from surface acoustic wave to ferro- magnetic spin moments, Nat. Commun. 12, 2599 (2021). [54] D. Go, F. Freimuth, J.-P. Hanke, F. Xue, O. Gomonay, K.-J. Lee, S. Bl¨ ugel, P. M. Haney, H.-W. Lee, and Y. Mokrousov, Theory of current-induced angular momentum transfer dynamics in spin- orbit coupled systems, Phys. Rev. Res. 2, 033401 (2020). [55] D. Yao and S. Murakami, Chiral-phonon-induced current in helical crystals, Phys. Rev. B 105, 184412 (2022). [56] S. Zhang and S. S.-L. Zhang, Generalization of the Landau- Lifshitz-Gilbert Equation for Conducting Ferromagnets, Phys. Rev. Lett. 102, 086601 (2009). [57] L. Zhang and Q. Niu, Angular Momentum of Phonons and the Einstein-de Haas Effect, Phys. Rev. Lett. 112, 085503 (2014). [58] L. Zhang and Q. Niu, Chiral Phonons at High-Symmetry Points in Monolayer Hexagonal Lattices, Phys. Rev. Lett. 115, 115502 (2015). [59] Y. Ren, C. Xiao, D. Saparov, and Q. Niu, Phonon Magnetic Moment from Electronic Topological Magnetization, Phys. Rev. Lett. 127, 186403 (2021). [60] H. Zhu, J. Yi, M.-Y. Li, J. Xiao, L. Zhang, C.-W. Yang, R. A. Kaindl, L.-J. Li, Y. Wang, and X. Zhang, Observation of chiral phonons, Science 359, 579 (2018). [61] J. Holanda, D. S. Maior, A. Azevedo, and S. M. Rezende, Detect- ing the phonon spin in magnon-phonon conversion experiments, Nat. Phys. 14, 500 (2018). [62] S. Streib, H. Keshtgar, and G. E. W. Bauer, Damping of Mag- netization Dynamics by Phonon Pumping, Phys. Rev. Lett. 121, 027202 (2018). [63] C. Dornes, Y. Acremann, M. Savoini, M. Kubli, M. J. Neuge- bauer, E. Abreu, L. Huber, G. Lantz, C. A. F. Vaz, H, Lemke, E. M. Bothschafter, M. Porer, V. Esposito, L. Rettig, M. Buzzi, A. Alberca, Y. W. Windsor, P. Beaud, U. Staub, D. Zhu, S. Song, J. M. Glownia, and S. L. Johnson, The ultrafast Einstein–de Haas effect, Nature 565, 209 (2019). [64] S. R. Tauchert, M. Volkov, D. Ehberger, D. Kazenwadel, M. Evers, H. Lange, A. Donges, A. Book, W. Kreuzpaintner, U. Nowak, and P. Baum, Polarized phonons carry angular momen- tum in ultrafast demagnetization, Nature 602, 73 (2022). [65] J. H. Mentink, M. I. Katsnelson, and M. Lemeshko, Quantum many-body dynamics of the Einstein–de Haas effect, Phys. Rev. B99, 064428 (2019). [66] M. Hamada and S. Murakami, Conversion between electron spin and microscopic atomic rotation, Phys. Rev. Res. 2, 023275 (2020). [67] D. Yao and S. Murakami, Conversion of phonon angular momentum into magnons in ferromagnets, arXiv:2304.14000 (2023). [68] C. Xiao, Y. Ren, and B. Xiong, Adiabatically induced orbital magnetization, Phys. Rev. B 103, 115432 (2021).Supplemental Material for ”Theory of Orbital Pumping” I. GENERAL PUMPING FORMULA FOR ARBITRARY ORBITAL SYSTEMS A. Generalized algebra for high- lorbital spaces In the case of spin, one can completely describe the dynamics using the Pauli matrix, enabling us to represent arbitrary traceless operators as vectors. Consequently, the pumping formula can also be expressed in vector form. However, in the case of orbitals with angular momentum quantum number l, it is imperative to define a vector space of (2 l+1)2dimensions, leading to the necessity of generalizing the dot product and cross product. This section will provide these definitions and outline their essential algebraic properties. In the following section, we will employ these concepts to demonstrate the straightforward generalization of the pumping formula. We define a new symbol ” ˙ =” to denote the mapping of an (2 l+1)×(2l+1) matrix Ato an (2 l+1)2-dimensional vector as follows, A˙=A, (S1) where A=AiLi(Einstein convention), Tr[ LiLj]=2δi j, and Ai=Tr[ALi]/2. The orthogonality guarantees the uniqueness and existence of the representation. We can generalize the cross product and dot product that map Rn2×Rn2toRn2as, A⊙B/doteq1 2{A,B}, (S2) A⊗B/doteq1 2i[A,B]. (S3) Below, we present some useful properties of generalized dot and cross products. 1. Property 1: Bilinearity (A+B)⊙(C+D)=A⊙B+B⊙C+A⊙D+B⊙D, (S4) (A+B)⊗(C+D)=A⊗B+B⊗C+A⊗D+B⊗D. (S5) 2. Property 2: (anti)commutativity A⊗B=−B⊗A, (S6) A⊙B=B⊙A. (S7) 3. Property 3: Representation of matrix multiplication AB=A⊙B+iA⊗B. (S8) c.f. (a·σ)(b·σ)=a·b+i(a×b)·σfor the spin case. 4. Property 4: Traces Definition: Tr v[A]/doteqTr[A]. Trv[A⊗B]=0, (S9) Trv[A⊙B]=2A·B. (S10)arXiv:2311.00362v1 [cond-mat.mes-hall] 1 Nov 20232 5. Property 5: Completeness relation. Trv[A⊙B]=1 2Trv[A⊙Li]Trv[Li⊙B]. (S11) 6. Property 6: Trace of matrix multiplication. Tr[AB]=2A·B, (S12) Tr[ABC ]=2A·(B⊙C+iB⊗C), (S13) where A·B=AiBi. 7. Property 7: Jacobi identities A⊗(B⊗C)+B⊗(C⊗A)+C⊗(A⊗B)=0, (S14) A⊗(B⊙C)+B⊗(C⊙A)+C⊗(A⊙B)=0. (S15) 8. Property 8: Non-associativity relations A⊙(B⊙C)−(A⊙B)⊙C=−B⊗(C⊗A), (S16) A⊗(B⊗C)−(A⊗B)⊗C=−B⊗(C⊗A), (S17) A⊗(B⊙C)−(A⊗B)⊙C=−B⊙(C⊗A). (S18) c.f. The second relation is equivalent to the first Jacobi identity. 9. Property 9: Triple scalar products A·(B⊙C)=(A⊙B)·C(=1 4Tr[ABC +ACB ]), (S19) A·(B⊗C)=(A⊗B)·C(=1 4iTr[ABC−ACB ]). (S20) c.f.A·(B×C)=(A×B)·Cfor usual cross product. 10. Property 10: spin limit ForLi=σi(i=0,1,2,3), A⊙B˙=(A·B,A0B1+B0A1,A0B2+A2B0,A0B3+A3B0), (S21) A⊗B˙=(0,A2B3−A3B2,A3B1−A1B3,A1B2−A2B1). (S22)3 11. Property 11: Equation of motion For physical operator Aand Hamiltonian H, dA dt=1 iℏ[H,A] ˙=2 ℏH⊗A. (S23) B. The Green’s function formalism for orbital pumping In this section, we derive a pumping formula applicable to arbitrary orbital systems with angular momentum quantum number l. We extend the formula derived in the reference [1] for spin pumping, utilizing the Green’s function approach, to operate even in states with higher angular momentum ( l≥1/2). For state with total angular momentum l, we need (2 l+1)2-dimensional vector space to completely describe pumping e ffects. We define unit vector in (2 l+1)2space as follows, ˆα˙=Lα. (S24) Thenν-direction total current is given by, jν=ℏ 4πJexℏ2 mei/integraldisplay dr′/bracketleftbigg gR(r,r′)ˆα↔ ∂νgA(r′,r)/bracketrightbiggduα dt, (S25) where gR/A(r,r′) is retarded and advanced Green’s function defined in (2 l+1)2-dimensional vector space and ( duα/dt)ˆαconsti- tutes external perturbations also defined in (2 l+1)2-dimensional vector space. Then, we use orbital algebra introduced in the previous section to simplify above equation. Using generalized products defined in previous section, the orbital current is given by, jν=ℏ 4πJexℏ2 meIm/integraldisplay dr′× /bracketleftbigg/parenleftig gR(r,r′)⊙ˆα/parenrightig ⊙↔ ∂νgA(r′,r)−/parenleftig gR(r,r′)⊗ˆα/parenrightig ⊗↔ ∂νgA(r′,r)+i/parenleftig gR(r,r′)⊙ˆα/parenrightig ⊗↔ ∂νgA(r′,r)+i/parenleftig gR(r,r′)⊗ˆα/parenrightig ⊙↔ ∂νgA(r′,r)/bracketrightbiggduα dt. (S26) By projecting the expression for gR/Aonto unit vectors, we can obtain pumping expressions with a generalized mixing conduc- tance, jν=ℏ 4πJexℏ2 me/integraldisplay dr′× /summationdisplay βγIm[gR β(r,r′)↔ ∂νgA γ(r′,r)][(ˆβ⊙ˆα)⊙ˆγ−(ˆβ⊗ˆα)⊗ˆγ]+Re[gR β(r,r′)↔ ∂νgA γ(r′,r)][(ˆβ⊙ˆα)⊗ˆγ+(ˆβ⊗ˆα)⊙ˆγ]duα dt, (S27) where gR/A m=gR/A·ˆm. II. ORBITAL PUMPING BASED ON THE SCATTERING MATRIX APPROACH In this subsection, we show that the pumped orbital current by the magnetization dynamics using the scattering matrix ap- proach [2] is consistent with that of the Green’s function approach. Following the conventional scattering matrix approach [2], the pumped current density operator jαwhereαis the flow direction is given by, jα/(−e)=∂nα ∂XdX(t) dt, (S28) ∂nα ∂X=1 4πi∂S ∂XS†+h.c., (S29) where X(t) is a time-dependent system parameter, ∂nα/∂Xis 3×3 emissivity matrix along αdirection, and Sis the 3×3 scattering matrix in p-orbital space. We then define the orbital mixing conductance as the orbital counterpart of the spin mixing conductance, Gi,j/doteq1−rir∗ j, (S30)4 where riandrjare the reflection coe fficients. Now we calculate the pumped orbital currents by the magnetization dynamics where the Hamiltonian is given by, H(t)=L·M(t). (S31) For this case, scattering matrix is given by, S=/summationdisplay mrm|m⟩⟨m|, (S32) where|m⟩is the eigenstate of the Eq. (S31). Then, the pumped OAM and OAP currents are given by, jOAM α=1 8πRe/bracketleftigg/parenleftiggG1,0+G0,−1 2/parenrightigg/parenleftigg M×dM dt−idM dt/parenrightigg/bracketrightigg , (S33) jOAP α,βγ=1 16πRe/bracketleftigg/parenleftiggG1,0−G0,−1 2/parenrightigg ×Mβ/parenleftigg M×dM dt/parenrightigg γ−iMβdMγ dt+(β↔γ), (S34) whereαis the surface-normal direction. We can see there is direct mapping between pumped orbital currents calculated by Green’s function approach. The same conclusion can be drawn for the orbital pumping by the lattice dynamics. III. LINEAR RESPONSE CALCULATION We consider currents carrying a physical quantity Q. An increase of Qin the region less than the ( ℓ+1)-th layer is given by, dNQ(ℓ) dt=d dt/angbracketleftiggℓ/summationdisplay i=−∞Q(i)(t)/angbracketrightigg ,Q(i)(t)=/summationdisplay α,α′C† iα(t)Qiα,iα′Ciα′(t). From the Heisenberg equation of motion, the increasing rate of Qis expressed as, dNQ(ℓ) dt=1 iℏ/angbracketleftiggℓ/summationdisplay i=−∞Q(i)(t),/summationdisplay C† i1α1Hi1,i2α1,α2Ci2α2/angbracketrightigg , where His the Hamiltonian matrix. Using the commutation relations of {Ciα,C† jα′}=δi jδαα′, we derive an increase of Qin terms of the generation and transfer terms: dNQ(ℓ) dt=WQ L(ℓ)+IQ L(ℓ). Here, WQ L(ℓ) is the generation rate and IQ L(ℓ) is the current of increasing Qin the region less than ( ℓ+1), IQ L(ℓ)=1 iℏQℓα,ℓα′/angbracketleftig C† ℓαHℓ,ℓ+1 α′,α1Cℓ+1α1−C† ℓ+1α1Hℓ+1,ℓ α1,αCℓα′/angbracketrightig . By defining the lesser Green function by ⟨C† αCβ⟩=−iℏG< βα, we arrive at IQ L(ℓ)=2 Re Tr/bracketleftig Hℓ+1,ℓQℓ,ℓG<(ℓ,ℓ+1)/bracketrightig . In a similar way, we can calculate an increase of Qin the region greater than the ℓ-th layer that is given by, IQ R(ℓ)=2 Re Tr/bracketleftig Hℓ,ℓ+1Qℓ+1,ℓ+1G<(ℓ+1,ℓ)/bracketrightig . In an averaged sense, we define the current carrying Qas, jQ(ℓ)=1 2/bracketleftig IQ R(ℓ−1)−IQ L(ℓ)/bracketrightig =Re Tr/bracketleftig Qℓ,ℓ/parenleftig G<(ℓ,ℓ−1)Hℓ−1,ℓ−G<(ℓ,ℓ+1)Hℓ+1,ℓ/parenrightig/bracketrightig . (S35)5 If a time-dependent term U(t) in the Hamiltonian is slow enough relative to electron dynamics, we can treat its time-derivative ˙U(t) as a perturbation. Then, the lesser Green function is expressed as G<(t,t)=−1 2πℏ/integraldisplay dE f o(E)/bracketleftigg GC(E)+iℏ/parenleftigg∂GR ∂E˙U(t)GC(E)−GC(E)˙U(t)∂GA ∂E/parenrightigg +···/bracketrightigg . Here, fo(E) is the Fermi-Dirac distribution and GC(E)≡GR(E)−GA(E).GR,A(E)=gR,A[E−U(t)] is the retarded and advanced Green function, respectively, and we express them as the adiabatic modulation of the unperturbed Green functions, gR,A(E)= (E−H±iΓ)−1for given level broadening Γ =25 meV. Because the zeroth order term is irrelevant to the spin pumping currents, we omit it and obtain the lesser Green function up to the first order as δG<(t,t)=−1 iℏ/integraldisplay dE/parenleftigg∂fo ∂EδN(E)+fo(E)δD(sea)(E)/parenrightigg . Here,δNandδD(sea)are called Fermi surface and sea contributions, respectively, given by δN(E)=ℏ 4π/bracketleftig GR˙U(t)GC−GC˙U(t)GA/bracketrightig , δD(sea)(E)=ℏ 4π/bracketleftigg GR˙U(t)∂GR ∂E−∂GR ∂E˙U(t)GR+h.c./bracketrightigg . UsuallyδD(sea)is considered to be small and is neglected in this work. By associating the lesser Green function and the currents of Eq. (S35), we obtain jQ(ℓ)=1 ℏImTr/integraldisplay dE∂fo ∂E/braceleftbigg Qℓ,ℓ/parenleftbigg/bracketleftig δN(E)/bracketrightigℓ,ℓ+1Hℓ+1,ℓ−/bracketleftig δN(E)/bracketrightigℓ,ℓ−1Hℓ−1,ℓ/parenrightbigg/bracerightbigg =−1 a0ImTr/braceleftbigg Q/bracketleftig δN(E)iv/bracketrightigℓ,ℓ/bracerightbigg E=µ,T−→ 0 (S36) where a0is a lattice constant and the velocity matrix is given by vℓ,ℓ′=±ia0 ℏHℓ,ℓ′δℓ′,ℓ±1. (S37) IV . TIGHT-BINDING MODEL Consider a three-dimensional NM1 /NM2 /NM1 system of simple cubic structure (Fig. 1 in the main text). The tight-binding Hamiltonian composed of sandporbitals is given as [3] H(0)(k)=... T(0)† llh(0) lT(0) la0 0 0 0 0T(0)† lah(0) aT(0) aa 0 0 0 ... 0 0 0T(0)† aa h(0) aT(0) al0 0 0 0 0 T(0)† alh(0) lT(0) ll ..., (S38) where h(0) i−E(0) i=2V(0) ssσ(cx+cy) 2 iV(0) spσsx 2iV(0) spσsy 0 −2iV(0) spσsx2V(0) ppσcx+2V(0) ppπcy 0 0 −2iV(0) spσsy 0 2 V(0) ppπcx+2V(0) ppσcy 0 0 0 0 2 V(0) ppπ(cx+cy), E(0) i=ϵ(0) s 0 0 0 0ϵ(0) px0 0 0 0ϵ(0) py0 0 0 0 ϵ(0) pz,T(0) i j=V(0) ssσ 0 0 V(0) spσ 0 V(0) ppπ0 0 0 0 V(0) ppπ0 −V(0) spσ0 0 V(0) ppσ,6 with abbreviations cx(y)≡coskx(y)a0andsx(y)≡sinkx(y)a0are used for brevity. The basis of h(0) iandT(0) i jis{|s⟩,|px⟩,|py⟩,|pz⟩} where subscripts stand for the active region ( a), leads ( l), and interfaces ( alandla). The tight-binding parameters are set as ϵ(0) s=0.50,ϵ(0) px=ϵ(0) py=ϵ(0) pz=−0.70,V(0) ssσ=−0.30,V(0) ppσ=0.50,V(0) ppπ=−0.20, and V(0) spσ=0.40 for both NM1 and NM2 layers, all in units of eV . To impose the boundary condition that both NM1 layers are semi-infinite, we utilize the repetitive structure of Hamiltonian [4]. Namely, the Hamiltonian is constructed as H(0)(k)=HllIla 0 IalHaaIar 0IraHrr, whereHandIis the on-site and interaction matrices of each region: the left contact ( l), the right contact ( r), and the active region ( a). We define the retarded Green function in a similar manner, GR=GllGlaGlr GalGaaGar GrlGraGrr, thereby obtain the Green function of the active region as (Haa−Σl−Σr)GR aa=1, with the surface self energies defined as Σl=IalH−1 llIla,Σr=IarH−1 rrIra. A. Time-dependent lattice dynamics Now, suppose that a time-dependent biaxial stress is applied along two axes perpendicular to each other, ˆe1=ˆzcosϕ+ˆxsinϕ andˆe2=−ˆzsinϕ+ˆxcosϕ, with phase di fferenceφ. Then, the position vector of nearest neighbors is modified as δδδ(t)=ˆe1δ(0) 1(1+ε1sinωt)+ˆe2δ(0) 2[1+ε2sin(ωt+φ)], whereε1(2)is the maximum strain along the axis ˆe1(2)andδ(0) 1(2)=limε→0δδδ(t)·ˆe1(2). Due to the deformation of crystal structure, the directional cosines for nearest-neighbor bonds and the magnitude of hopping integrals should be altered. The bond-length dependence of hopping integrals is usually fitted to a power law [5, 6] Vαβτ(d)=Vαβτ(d0)/parenleftiggd0 d/parenrightiggηαβτ , (S39) in whichα,βdenote orbitals under consideration and τis the bonding type σ,π,δ,···. In general, the exponent ηαβτis orbital- and material-specific parameter obtained by comparing tight-binding and ab initio data. For simplicity, however, we arbitrarily choose the exponent to be a constant for all hopping integrals, i.e., ηαβτ=2. Then, the Hamiltonian at instantaneous time tis H(k;t)=... T(0)† llh(0) lT(0) la0 0 0 0 0T(0)† laha(t)Taa(t) 0 0 0 ... 0 0 0T† aa(t)ha(t)T(0) al0 0 0 0 0 T(0)† alh(0) lT(0) ll ...,7 where ⟨s;k|ha(t)|s;k⟩=ϵ(0) s+2V(0) ssσ(˜cx+cy), ⟨px;k|ha(t)|px;k⟩=ϵ(0) px+2(V(0) ppσl2 x+V(0) ppπn2 x)˜cx+2V(0) ppπcy, ⟨py;k|ha(t)|py;k⟩=ϵ(0) py+2V(0) ppπ˜cx+2V(0) ppσcy, ⟨pz;k|ha(t)|pz;k⟩=ϵ(0) pz+2(V(0) ppσn2 x+V(0) ppπl2 x)˜cx+2V(0) ppπcy, ⟨s;k|ha(t)|px;k⟩=2iV(0) spσlx˜sx, ⟨s;k|ha(t)|py;k⟩=2iV(0) spσsy, ⟨s;k|ha(t)|pz;k⟩=2iV(0) spσnx˜sx, ⟨px;k|ha(t)|pz;k⟩=2(V(0) ppσ−V(0) ppπ)lxnx˜cx, and Taa(t)=1 |dz(t)|2V(0) ssσ lzV(0) spσ 0 nzV(0) spσ −lzV(0) spσl2 zV(0) ppσ+n2 zV(0) ppπ 0lznz(V(0) ppσ−V(0) ppπ) 0 0 V(0) ppπ 0 −nzV(0) spσlznz(V(0) ppσ−V(0) ppπ) 0 n2 zV(0) ppσ+l2 zV(0) ppπ. Here, we neglect the time-dependent variation of on-site energies and assume ε1=ε2=ε. Note that the strain is uniformly applied in the active region, i.e., ∂ε/∂ y=0. The position vectors directing nearest neighbors are dz(t)=a0ˆz[1+ε{cos2ϕsinωt+sin2ϕsin(ωt+φ)}]+a0ˆxεcosϕsinϕ[sinωt−sin(ωt+φ)], dx(t)=a0ˆzεcosϕsinϕ[sinωt−sin(ωt+φ)]+a0ˆx[1+ε{sin2ϕsinωt+cos2ϕsin(ωt+φ)}], and corresponding directional cosines are defined as li=ˆx·di(t)/|di(t)|andni=ˆz·di(t)/|di(t)|fori=x,z. The revised abbreviations ˜ cx≡coskxa0/|dx(t)|2and ˜sx≡sinkxa0/|dx(t)|2are used. Up to the linear order of strain ε, ha(t)−h(0) a≈−2ε/parenleftig sin2ϕsinωt+cos2ϕsin(ωt+φ)/parenrightig2V(0) ssσcx2iV(0) spσsx 0 0 −2iV(0) spσsx2V(0) ppσcx 0 0 0 0 2 V(0) ppπcx 0 0 0 0 2 V(0) ppπcx +εcosϕsinϕ(sinωt−sin(ωt+φ))0 0 0 2 iV(0) spσsx 0 0 0 2( V(0) ppσ−V(0) ppπ)cx 0 0 0 0 −2iV(0) spσsx2(V(0) ppσ−V(0) ppπ)cx0 0, and Taa(t)−T(0) aa≈−2ε/parenleftig cos2ϕsinωt+sin2ϕsin(ωt+φ)/parenrightigV(0) ssσ 0 0 V(0) spσ 0 V(0) ppπ0 0 0 0 V(0) ppπ0 −V(0) spσ0 0 V(0) ppσ +εcosϕsinϕ(sinωt−sin(ωt+φ))0 V(0) spσ 0 0 −V(0) spσ 0 0 V(0) ppσ−V(0) ppπ 0 0 0 0 0 V(0) ppσ−V(0) ppπ0 0. We superpose two biaxial strains to generate lattice vibration with finite net angular momentum. One biaxial strain with ϕ1=0 andφ1=πand the other strain with ϕ2=π/4 andφ2=πare overlapped where overall phase of the latter is shifted byπ/2 with respect to the former.Note that the strength of each set of strains is set to ε=0.5%. To obtain an insight on the perturbation we take, consider the tight-binding Hamiltonian of porbitals in bulk simple cubic lattice. Then, the time-dependent perturbation Ubulk(t) in the long-wavelength limit is given as Ubulk(t)≈−4ε(V(0) ppσ−V(0) ppπ)/bracketleftig {Lz,Lx}cosωt−(L2 z−L2 x) sinωt/bracketrightig , (S40) which resembles [ L·u(t)]2with halved frequency for ˆu(t)=ˆzcosωt+ˆxsinωt. Thus, one can conclude that the e ffective perturbation [ L·u(t)]2captures the lattice dynamics characterized by the circular motion of an atom about its equilibrium position, which in turn gives rise to adiabatic transformation of the OAP.8 -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0 jLz,DCz z(lattice constant, a0)Vsps(eV) 0.40 0.20 0.00 FIG. S1. Spatial profile of DC orbital pumping current driven by the time-dependent magnetic field L·M(t) for ˆM(t)=ˆxcosωt+ˆysinωtfor various strengths of sphybridization Vspσ. V . DECAY LENGTH OF ORBITAL CURRENTS The transmission of OAM current into adjacent layers depends on the orbital character [7]. For a given geometry, an electron propagating toward the leads experiences the crystal field which splits |pz⟩from|px⟩and|py⟩. Therefore, one can suppress the spatial oscillation of OAM response by choosing ˆM(t)=ˆxcosωt+ˆysinωtthat conveys the OAM Lz. The calculated OAM current jLz,DC z (Fig. S1) exhibits a monotonically decaying behavior rather than an oscillatory decay near the interface as illustrated in Fig. 2. Still, the OAM penetration length is not strikingly enhanced as that previously reported in ferromagnets when the orbital texture exists. Under consideration of all electrons participating in the propagation, we recognize that degenerate |px⟩ and|py⟩states with nonzero in-plane wave vectors are gapped into |pr⟩≡cosϕk|px⟩+sinϕk|py⟩and|pt⟩≡− sinϕk|px⟩+cosϕk|py⟩ by the orbital texture while the degeneracy at k=±kFˆzare preserved. Note that ϕk=arg(kx+iky) is an azimuthal angle of wave vector k. As the operator ˆLz=i|py⟩⟨px|−i|px⟩⟨py|=i|pt⟩⟨pr|−i|pr⟩⟨pt|, the broken degeneracy between |pr⟩and|pt⟩results in increased oscillation for states carrying Lz, and thus short-ranged orbital transport. Conversely, the overall penetration length can be extended by decreasing the strength of orbital texture as shown in Fig. S1. VI. ORBITAL SWAPPING EFFECT The pumped orbital current serves as a primary input of the orbital swapping e ffect, which is an orbital counterpart of the spin swapping e ffect [8–10]. Similar to the spin swapping e ffect which is categorized into two types, the orbital swapping e ffect can be classified into one that illustrates the conversion of OAM current jOAM α,β→jOAM β,αforα/nequalβ[Fig. S2(a)] and the other which represents the conversion of OAM current jOAM α,α→jOAM β,βforα/nequalβ[Fig. S2(c)]. Furthermore, we observe the swapping of OAP current jOAP α,αβ→jOAP γ,βγ[Fig. S2(b)] and jOAP α,βγ→jOAP β,γα(not shown). Note that the latter process converts AC OAP current to another AC OAP current in our model. All processes are suppressed when the orbital texture vanishes.9 -1.0-0.50.00.51.0jDCj{Lz,Lx},DC z j{Lx,Ly},DC y -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0jDC z(lattice constant, a0)jLz,DC z jLx,DC x -1.0-0.50.00.51.0jDCjLy,DC z jLz,DC y(a) (b) (c) FIG. S2. Spatial profile of DC orbital pumping currents and resultant orbital swapping currents driven by the time-dependent magnetic field L·M(t) for (a,b) ˆM(t)=ˆzcosωt+ˆxsinωtand (c) ˆM(t)=ˆxcosωt+ˆysinωt. [1] K. Chen and S. Zhang, Spin Pumping in the Presence of Spin-Orbit Coupling, Phys. Rev. Lett. 114, 126602 (2015). [2] Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Enhanced Gilbert Damping in Thin Ferromagnetic Films, Phys. Rev. Lett. 88, 117601 (2002). [3] J. C. Slater and G. F. Koster, Simplified LCAO Method for the Periodic Potential Problem, Phys. Rev. 94, 1498 (1954). [4] M. Luisier, A. Schenk, W. Fichtner, and G. Klimeck, Atomistic simulation of nanowires in the sp3d5s∗tight-binding formalism: From boundary conditions to strain calculations, Phys. Rev. B 74, 205323 (2006). [5] S. Froyen and W. A. Harrison, Elementary prediction of linear combination of atomic orbitals matrix elements, Phys. Rev. B 20, 2420 (1979). [6] W. A. Harrison, Electronic Structure and the Properties of Solids (Freeman, San Francisco, 1980). [7] D. Go, D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park, S. Bl ¨ugel, H.-W. Lee, and Y . Mokrousov, Long-Range Orbital Torque by Momentum-Space Hotspots, Phys. Rev. Lett. 130, 246701 (2023). [8] M. B. Lifshits and M. I. Dyakonov, Swapping Spin Currents: Interchanging Spin and Flow Directions, Phys. Rev. Lett. 103, 186601 (2009). [9] S. Sadjina, A. Brataas, and A. G. Mal’shukov, Intrinsic spin swapping, Phys. Rev. B 85, 115306 (2012). [10] H. B. M. Saidaoui and A. Manchon, Spin-Swapping Transport and Torques in Ultrathin Magnetic Bilayers, Phys. Rev. Lett. 117, 036601 (2016).

1512.01660v1.Kinetic_theory_of_spin_polarized_systems_in_electric_and_magnetic_fields_with_spin_orbit_coupling__I__Kinetic_equation_and_anomalous_Hall_and_spin_Hall_effects.pdf

arXiv:1512.01660v1 [cond-mat.str-el] 5 Dec 2015Kinetic theory of spin-polarized systems in electric and ma gnetic fields with spin-orbit coupling: I. Kinetic equation and anomalous Hall and spin-H all effects K. Morawetz1,2,3 1M¨ unster University of Applied Sciences, Stegerwaldstras se 39, 48565 Steinfurt, Germany 2International Institute of Physics (IIP) Federal Universi ty of Rio Grande do Norte Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Braz il and 3Max-Planck-Institute for the Physics of Complex Systems, 0 1187 Dresden, Germany The coupled kinetic equations for density and spin Wigner fu nctions are derived including spin- orbit coupling, electric and magnetic fields, selfconsiste nt Hartree meanfields suited for SU(2) trans- port. The interactions are assumed to be with scalar and magn etic impurities as well as scalar and spin-flip potentials among the particles. The spin-orbit in teraction is used in a form suitable for solid state physics with Rashba or Dresselhaus coupling, gr aphene, extrinsic spin-orbit coupling, and effective nuclear matter coupling. The deficiencies of th e two-fluid model are worked out con- sisting of the appearance of an effective in-medium spin prec ession. The stationary solution of all these systems shows a band splitting controlled by an effecti ve medium-dependent Zeeman field. The selfconsistent precession direction is discussed and a cancellation of linear spin-orbit coupling at zero temperature is reported. The precession of spin arou nd this effective direction caused by spin-orbit coupling leads to anomalous charge and spin curr ents in an electric field. Anomalous Hall conductivity is shown to consists of the known results obtai ned from the Kubo formula or Berry phases and a new symmetric part interpreted as an inverse Hal l effect. Analogously the spin-Hall and inverse spin-Hall effects of spin currents are discussed which are present even without magnetic fields showing a spin accumulation triggered by currents. Th e analytical dynamical expressions for zero temperature are derived and discussed in dependence on the magnetic field and effective mag- netizations. The anomalous Hall and spin-Hall effect change s sign at higher than a critical frequency dependent on the relaxation time. PACS numbers: 72.25.-b, 75.76.+j, 71.70.Ej, 85.75.Ss I. INTRODUCTION A. Motivation and outline The interest in spin transport has regained a renais- sance due to the promising application as next genera- tion information storage. The experiments have reached such high precession that single spin transport and spin wave processes can be resolved and investigated in view of applications to new nanodevices where spin-field tran- sistors are proposed1. Many interesting effects have been reported such as anomalous spin segregation in weakly interacting6Li in a trap2which effect has been described by the meanfield, spin-Hall effects and spin-Hall nano- oscillators3. Different spin transport effects have led to the spin current concept4which tries to summarize these effects with respect to the current. Thespin-orbitcouplinghasleadtotheideaofthe spin- Hall effect5–7which was proposed8,9and first observed in bulkn-type semiconductors10and in 2D heavy-hole systems11. Spin and particle currents are coupled such that one observesan accumulationof transversespin cur- rent near the edges of the sample. The spin conductance has been measured in mesoscopic cavities12to extract the part due to spin-orbit coupling. Within the anoma- lous Hall effect first described in Ref.13the spin polariza- tion takes over the role of a magnetic field and creates a current contribution14–16. This effect occurs when time- symmetry is broken17and is related to the inverse spinHall effect18. For the latter one spin-orbit coupling takes the role of an additional electric field. Anisotropic magnetoresistance together with the anomalous Hall coefficients have been measured and attributed to spin-orbit coupling19,20and treated in quantum wires21and mesoscopic rings22. The lattice structure causes strong dependencies on the transport direction23. One distinguishes between extrinsic and intrinsic spin- Hall effects. The extrinsic is due to spin-dependent scat- teringbymixing ofspin and momentum eigenstates. The intrinsic effect is an effect of the momentum-dependent internal magnetic field due to spin-orbit coupled band structures. This leads to a spin splitting of the en- ergy bands in semiconductors due to lacking of inver- sion symmetry. Most observations are performed with the extrinsic10,24–26and only some for the intrinsic spin- Hall effect11,27. There are different model treatments of intrinsic28,29and extrinsic spin-Hall effects30sometimes usingBerrycurvatures17,31–33,theLandauerformula31or even relativistic treatments34,35. A theoretical compar- ison of the relativistic approach with the Kubo formula is found in36. A detailed discussion of possible occur- ring spin-orbit couplings in semiconductor bulk struc- tures and nanostructures can be found in37and in the book5. For the Rashba coupling and quadratic disper- sion in disordered two-dimensional systems it has been shown that the spin-Hall effect vanishes38–40. This is not the case if magnetic scatters are considered41. The in-2 trinsic anomalous Hall effect is treated also in Ref.42and in disordered band ferromagnets43. The (pseudo)spin-Hall effect in graphene is cur- rently a very heavily investigated field44reporting also the anomalous Hall effect in single-layer and bilayer graphene45,46and which is treated like a spin-orbit cou- pled system47. Recently even spin-orbit coupled Bose- Einstein condensates have been realized48. The main motivation of the present paper is to derive in an unambiguous way a kinetic equation of interacting spin-polarized fermions including magnetic and electric fields with spin-orbit coupling. Normally one finds all four problems treated separately in the literature. First, there exists a vast literature to derive kinetic equations with spin-polarized electrons49–60. Second, other quan- tum kinetic approaches focus exclusively on the spin- orbit coupling61–64. Third, the transport in high and low magnetic fields itself is involved due to precession mo- tionsofchargedparticlesandtreatedapproximately65–71. Fourth, the interaction with scalar and magnetic impu- rities requires a certain spin-coupling which is important for transport effects in ferromagnetic materials72–74. Here we will combine all four difficult problems into a unifying quantum kinetic theory. First we restrict our- selves to approximate the many-body interactions by the mean field and a conserving relaxation time. The out- lined formalism is straight forward to derive proper col- lision integrals as done in the literature51,53,56,75–77. The reasons to consider once again the lowest-order many- body approximation is twofold. On one side during the derivation of the proper kinetic equations it turned out that even on the meanfield level all four effects together create additional terms when considered on a common footing not known so far. On the other side, with a meanfield quantum kinetic equation including all these effects we have the possibility to linearize with respect to an external perturbation and to obtain in this way the response function in the random phase approxima- tion (RPA). As a general rule, when a lower-level kinetic equation is linearized, a response of higher-order many- body correlations is obtained78. Most treatments of the response function use approxi- mations already at the beginning and concentrate only on specific effects, such asthe diffusive regime31,79or currents80. We will explicitly work out these response functions in the second paper of this series. Here in the first paper we want to focus on the derivation of the quantum kinetic equation including all these effects as transparently as possible. With the aim to derive the RPA response we concentrate on the correct mean-field formulation and restrict ourselves to a relaxation-time approximation of the collision integral as a first step. This relaxation time will be understood with respect to a local equilibrium which accounts for local conservation laws81–85. Though it can be derived from Boltzmann col- lision integrals, this relaxation time approximation omits quantum interference effects such as weak localization due to disorder, for an adequate treatment of these ef-fects see, e.g.,86. The outline of this first paper is as follows. After ex- plaining the basic notation we present in the second sec- tion the phenomenological two-fluid model and show the insufficiency due to the missing self-consistent precession direction. We present an educative guess for the proper kinetic equation from the demand of SU(2) symmetry. Interestingly, this leads already to the correct form of the kinetic equation except for four parameters which have to be derived microscopically in Sec. III. We use the nonequilibrium Green’s function technique in the no- tation of Langreth and Wilkins87. The obtained kinetic equations lead to a unique static solution which shows the splitting of the band due to spin-orbit interference. The anomalous current is shown to cancel the normal one in the stationary state pointing to the importance of the anomalous currents when the balance is disturbed like in transport. The selfconsistent precession direction is calculated explicitly for zero temperature and linear spin-orbit coupling. In Sec. IV we compare the derived kineticequationwiththeguessedoneofSec. IIdetermin- ing the remaining open parameters. The anomalous Hall and spin-Hall effects are calculated in Sec. IV and ap- pear in agreement with other approaches using the Kubo formula or helicity basis. We obtain a dynamical sym- metric contribution interpreted as inverse spin-Hall and inverse Hall effects. Analytical expressions are discussed for the dynamical conductivities at zero temperature and linearspin-orbitcoupling. Asummaryconcludesthefirst paper of this series. B. Basic notation The spin of a fermion/planckover2pi1 2/vector σwith the Pauli matrices /vector σ as an internal degree of freedom analogously to circular motion leads to the elementary magnetic moment for the spin in terms of the Bohr magneton µB, ˆ/vector m=ge 2me/planckover2pi1 2/vector σ=g 2µB/vector σ (1) with the anomalous gyromagnetic ration g≈2 for elec- trons. If there are many fermions with densities n±of spins parallel/antiparallel to the magnetic field, the to- tal magnetization is mz=gµBszwith the polarization sz= (n+−n−)/2 We want to access the density and polarization density distributions and use therefore four Wigner functions ˆρ(/vector x,/vector p) =f(/vector x,/vector p)+/vector σ·/vector g(/vector x,/vector p) =/parenleftbigg f+gzgx−igy gx+igyf−gz/parenrightbigg (2) with the help of which the density and polarization den- sity are given by /summationdisplay pf=n(/vector x),/summationdisplay p/vector g=/vector s(/vector x) (3)3 where/summationtext p=/integraltext dDp/(2π/planckover2pi1)DforDdimensions and the magnetization density becomes /vectorM(/vector x) =gµB/vector s(/vector x). The advantage is that we can describe any direction of mag- netization density created by microscopic correlations which will be crucial in this paper. Sometimes one finds the probability distribution of spin up/down in the di- rection/vector eof the mean polarization by the spin projection or a twofold additional spin variable69,70is used. Since thereareinversionformulas88alltheseapproachesshould be equivalent. However, we prefer the presentation of the Wigner function in terms of the scalar and vector parts (2) since the coupling between these functions bear clear physical meaning which is somewhat buried in the sometimes used super distribution. Moreoverthe Wigner functions (3) yield directly the total density andthe spin- polarizations. C. Spin-orbit coupling Any spin-orbit coupling used in different fields, say plasma systems, semiconductors, graphene or nuclear physics can be recast into the general form Hs.o.=A(/vector p)σx−B(/vector p)σy+C(/vector p)σz=/vectorb·/vector σ(4) with a momentum-dependent /vectorbillustrated in table I and which can become space and also time-dependent in nonequilibrium. Also the Zeeman term is of this form. The time reversal invariance of spin current due to spin- orbit coupling requires that the coefficients A(p) and B(p) be odd functions of the momentum kand there- fore such couplings have no spatial inversion symmetry. Also 3D systems of spin-1/2 particles can be recast into the form (4). Let us shortly discuss different realizations since we want to treat all of them within the quantum kinetic theory. 1. Extrinsic spin-orbit coupling First we might think on the direct spin-orbit coupling as it appears due to expansion of the Dirac equation where only the Thomas term is relevant /vector σ·ie/planckover2pi12 8m2c2/parenleftbigg ∂R×/vectorE−2i /planckover2pi1/vectorE×/vector p/parenrightbigg ≈λ2/vector σ·(/vector p /planckover2pi1×∇V)(5) withλ2=/planckover2pi12/4m2 ec2≈3.7×10−6˚A2for electrons. The electric field is not an external one, but e.g. created by the nucleus /vectorE=−/vector∇V, and is called extrinsic spin-orbit coupling. The spin-orbit coupling mixes different mo- mentum states and is coupled to inhomogeneities in the material. The matrix elements ofthe spin-orbit potential reads /an}b∇acketle{t/vector p2|Vs.o.|/vector p1/an}b∇acket∇i}ht=iλ2 /planckover2pi12V(/vector p)(/vector p×/vector σ)·/vector q (6)with the center-of-mass momentum /vector q= (/vector p1+/vector p2)/2 and /vector p=/vector p1−/vector p2. Any such spin-orbit coupling possesses the general structure (4). 2. Intrinsic spin-orbit coupling in semiconductors For direct gap cubic semiconductors such as GaAs the form (4) of spin-orbit coupling arises by coupling of the s-type conductance band to p-type valence bands. With in the 8 ×8 Kane model the third-order perturba- tion theory5yieldsλ=P2/3[1/E2 0−1/8(E0+∆0)2] with the gapE0and the spin-orbit splitting ∆ 0between the J= 3/2 andJ= 1/2 hole bands and a matrix element P. For GaAs one finds λ= 5.3˚A2which shows that in n-type GaAs the spin-orbit coupling is six orders of magnitude stronger than in vacuum and has an opposite sign89. The cubic Dresselhaus spin-orbit corrections are usually neglected since they are small and does not ap- pear in the 8 ×8model. Therefore the spin-orbit coupling is considered to come from the potential of the driving field and the impurity centers. In a GaAs/AlGaAs quantum well there can be two types of spin-orbit couplings that are linear in momen- tum. One considers a narrow quantum well in the /vector n= [001] direction. The linear Dresselhaus spin-orbit cou- pling is due to the bulk inversion asymmetry of the zinc- blende type lattice. It is proportional to the kinetic en- ergy of the electron’s out-of-plane motion and decreases therefore quadratically with increasing well width. In lowest-order momentum one obtains Hs.o. D=βD /planckover2pi1(−pyσy+pxσx) (7) again of the form (4) with /vectorb=βD(−px,py,0)//planckover2pi1. The Rashba spin-orbit coupling (SOC) Hs.o. R=βR /planckover2pi1(−pxσy+pyσx) =βR /planckover2pi1/vector σ·(/vector p×/vector n) (8) is finally due to structure inversion asymmetry and the strength can be tuned by a perpendicular electric field, for example by changing the doping imbalance on both sides of the quantum well. The Rashba coupling is again of the form (4) with /vectorb=βR(py,−px,0)//planckover2pi1. Note that the Rashba SOC has winding number +1 as the momen- tum direction winds around once in momentum space, whereas the linear Dresselhaus has the opposite winding -190. There are further types of spin-orbit expansion schemes for quasi-2D systems such as cubic Rashba and cubic Dresselhaus expansions whose competing inter- play is treated too91. These terms including wurtzite structures92all together can be recast into the form of (4) and seen in table I.4 TABLE I: Selected 2D and 3D systems with the Hamiltonian described by (4) taken from92,93 2D−system A(p) B(p) C(p) Rashba βRpy βRpx Dresselhaus[001] βDpx βDpy Dresselhaus[110] βpx −βpx Rashba−Dresselhaus βRpy−βDpxβRpx−βDpy cubicRashba(hole) iβR 2(p3 −−p3 +)βR 2(p3 −+p3 +) cubicDresselhaus βDpxp2 y βDpyp2 x Wurtzitetype ( α+βp2)py(α+βp2)px single−layergraphene vpx −vpy bilayergraphenep2 −+p2 + 4mep2 −−p2 + 4mei 3D−system A(p) B(p) C(p) bulk Dresselhaus px(p2 y−p2 z)py(p2 x−p2 z)pz(p2 x−p2 y) Cooperpairs ∆ 0p2 2m−ǫF extrinsic β=i /planckover2pi1λ2V(p)qypz−qzpyqzpx−qxpzqxpy−qypx neutrons in nuclei β=iW0(nn+np 2)qzpy−qypzqxpz−qzpxqypx−qxpy 3. Spin-orbit coupling in graphene Solving the tight-binding model on the honeycomb lattice including only nearest neighbor hopping gives an effective two-band Hamiltonian for the Bloch wave function which can be considered as coupled pseudo- spins. The interband coupling of these different bands in graphene leads to the spin-orbit coupling of the form (4). 4. Spin-orbit coupling in nuclear matter In nuclearmatter the spin-orbitinteractionis strongin heavy elements and the reason for magic numbers. Shell structures cannot be described properly without consid- eration of the spin-orbit interaction coming from the ten- sor part of the nuclear forces94. The effective spin-orbit couplinginnuclearmatterwithneutronsandprotonscan be considered as a meanfield expression due to schematic Skyrme forces and is expressed in the Rashba form (8) which reads for neutrons95 Hs.o.=−W0 2/vector σ[/vector p×(/vector∇np+2/vector∇nn)] (9) and interchanging np↔nnfor protons. One sees that in this Hartree-Fock expression the effective direction /vector nof Rashba form (8) is given by the gradient of the densities. Therefore the structure appears as in extrinsic spin-orbit coupling. The coupling constant is a matter of debateand dependent on the used density functional96. Further terms can be considered if more involved Skyrme poten- tials are used leading to additional current coupling97,98. It is expected that the spin-orbit coupling plays an im- portantrolein heavy-ionreactions99and in rareisotopes. II. PHENOMENOLOGICAL KINETIC EQUATION A. Deficiencies of two fluid model Nowweconsiderthewidelyusedtwo-fluidmodel100,101 developed from the two-current conduction in iron102. It consists of a distribution for spin-up f↑and spin- downf↓parts. Until recently it was used to explain even anisotropic magnetic resistance103with its limits observed there. Indeed we will show that an important part is missing in this model. Besides the two distribu- tion functions we have to consider the direction of the mean spin or polarization. As we will see this leads to a third equation which is silently overlooked in these mod- els. From general SU(2) symmetry considerations one can already conclude that this part is missing in the two- fluidmodelandhowitsformshouldappear. Thedetailed derivation will be performed in the next section. Here we repeat briefly the two-fluid model and develop the miss- ing parts from general physical grounds. We start with the linearized coupled kinetic equations for the two components ∂tδf↑+/vector p me/vector∂rδf↑−/vector∂r(Uext+U↑↑δn↑+U↑↓δn↓)/vector∂pf0 ↑ =−δf↑ τ↑↑−δf↑−δf↓ τ↑↓ ∂tδf↓+/vector p me/vector∂rδf↓−/vector∂r(Uext+U↓↓δn↓+U↓↑δn↑)/vector∂pf0 ↓ =−δf↓ τ↓↓−δf↓−δf↑ τ↓↑(10) with the external electric field e/vectorE=−/vector∂rUext. The relaxation due to collisions is considered as relaxation times with respect to the same kind of particle τ↑↑or τ↓↓and with respect to the other sort which is described by the cross relaxation time τ↑↓=τ↓↑due to symmetric collisions. For later use we have added also the spin- dependent meanfields U/angbracketrightand their linearization Uij= ∂Ui/∂njwith respect to the densities n↑,↓=/summationtext pf↑,↓. Multiplying (10) with /vector pand integrating together with Fourier transform ∂t→ −iωand/vector∂r→i/vector qleads to the coupled equations for the currents in lowest-order wavevector /vector q (ρ↑+r↑↓)δJ↑−r↑↓δJ↓=E −r↓↑δJ↑+(ρ↓+r↓↑)δJ↓=E (11) where we have introduced the partial and crossed resis-5 ρ ρ ρρ /2 /2/2 /2r FIG. 1: Resistor scheme of (13) illustrating the spin mixing . tivities [(i,j) =↑,↓] 1 ρi=σi=nie2τii me(1−iωτii), r ij=me nie2τij.(12) The partial currents (11) are easily solved and one ob- tains the total resistivity ρ=E δJ↑+δJ↓=ρ↑ρ↓+ρ↑r↑↓+ρ↓r↑↓ ρ↑+ρ↓+2(r↑↓+r↓↑).(13) Assuming further that r=r↓↑=r↑↓this resistivity al- lows an interpretation as composed resistivity illustrated in figure 1 which shows the role of the cross scattering between different species known as spin mixing. Despite the successful application of this model104it has an important inconsistency which is not easy to rec- ognize. We therefore rewrite the kinetic equations (10) into a form for the total density distribution and total density f=1 2(f↑+f↓);n=1 2(n↑+n↓) (14) and the polarization distribution and total spin g=1 2(f↑−f↓);s=1 2(n↑−n↓) (15) which reads (Fourier transform /vector∂r→i/vector q) ∂tδf+i/vector q/vector p meδf+e/vectorE/vector∂pf0−Mf=−δf τ+−δg τ− ∂tδg+i/vector q/vector p meδg+e/vectorE/vector∂pg0−Mg=−δf τ−−δg τ+−2δg τD (16) with the meanfield parts abbreviated as Mf=i/vector q/vector∂pf0(V1δn+V3δs)+i/vector q/vector∂pg0(V2δn+V4δs) Mg=i/vector q/vector∂pg0(V1δn+V3δs)+i/vector q/vector∂pf0(V2δn+V4δs). (17) Here it was convenient to introduce the relaxation times 1 τ±=1 2/parenleftbigg1 τ↑↑±1 τ↓↓/parenrightbigg ;τD=τ↑↓=τ↓↑(18)and the meanfield potentials V1/2=U↑↑+U↑↓±(U↓↑+U↓↓) V3/4=U↑↑−U↑↓±(U↓↑−U↓↓).(19) The crucial point is now that the polarization or total spin has a direction /vector ewhich means we have to consider thevectorquantity /vector g=g/vector ewhichtranslatesintotwoparts when linearized δ/vector g=gδ/vector e+/vector eδg. Only the second part is obviously covered by the second equation of (16). The equation for δ/vector eremains undetermined so far. B. Educated guess from SU(2) symmetry However from general consideration of SU(2) symme- try we can infer the form in which this missing equation will appear and which we will derive in the next section from microscopic theory. It is convenient to write both scalar distribution fand vector distribution /vector gtogether in spinor form (2). Then any collision integral and therefore any kinetic equation must be possible to write as commutator and anticom- mutator in spin-space where the forms 1 2/bracketleftig δˆρ,a+/vector σ·/vectorb/bracketrightig +=aδf+/vectorb·δ/vector g+/vector σ·/parenleftig aδ/vector g+/vectorbδf/parenrightig 1 2[δˆρ,a+/vector σ·/vector c]−=i/vector σ·(δ/vector g×/vector c) (20) can appear making use of ( /vector a·/vector σ)(/vectorb·/vector σ) =/vector a·/vectorb+i/vector σ·(/vector a×/vectorb). Therefore the expected kinetic equation reads ∂tδˆρ+i/vector p/vector q meδˆρ+e/vectorE/vector∂pˆρ0 −i 2/bracketleftig /vector q/vector∂pˆρ0,a+/vectorb/vector σ/bracketrightig +−i 2/bracketleftig /vector q/vector∂pˆρ0,/vector c/vector σ/bracketrightig −−i 2[ˆρ0,/vectorh·/vector σ]− =−1 2/bracketleftbigg δˆρ,1 τ+/vector τ−1·/vector σ/bracketrightbigg +−1 2/bracketleftig δˆρ,/vectord·/vector σ/bracketrightig −(21) with the yet undetermined constants a,/vectorb,/vector c,/vectord,/vectorh. Here the first line describes the scalar drift which can be triv- ially written in anticommutators. The commutator and anticommutator on the second line are the meanfields including a possible precession in the second and third terms. The third line expresses possible relaxations. Now the compact equation (21) is decomposed into the components δfand/vectorδg=/vector eδg+gδ/vector ewith the help of (20). The aim is to specify the three remaining vectors /vector c,/vectordand/vectorhsuch that the results of the two-fluid model (16) are reproduced. The two values a=V1δn+V3δs and/vectorb= (V2δn+V4δs)/vector ecan be already determined since only this choice creates the meanfield terms in (16). It is convenient to decompose the so far unspecified vectors /vector c=c/vector e+c1δ/vector e+c2/vector e×δ/vector e /vectord=d/vector e+d1δ/vector e+d2/vector e×δ/vector e (22)6 accordingto the three orthogonaldirections since /vector e·∂/vector e= 0 due to |/vector e|2= 1. We obtain from (21) the analogous equations to (16) ∂tδf+i/vector q/vector p meδf+e/vectorE/vector∂pf0−Mf=−δf τ−δ/vector g·/vector τ−1 +ig0/vectorb·q∂pδ/vector e ∂tδg+i/vector q/vector p meδg+e/vectorE/vector∂pg0−Mg=−δf(/vector e·/vector τ−1)−δg τ +c1g0/vector e·(q∂p/vector e×δ/vector e)−c2q∂p/vector e·δ/vector e−ig0d2(δ/vector e)2(23) Comparing the two-fluid model (16) with (23) we find the unique identification c2=d2=c1= 0. Further one sees that we have to set 1 τ=1 τ++2 τD;/vector e·/vector τ−1=1 τ−(24) and the cross relaxation time has to be determined by 2 τDδf=−g0(/vector τ−1·δ/vector e)+ig0/vector e·q∂pδ/vector e(V2δn+V4δs). (25) Only with these settings we obtain exactly the two-fluid model (16). The resulting equation (25) reveals that the cross relaxation time τD, see (18), can be only obtained with the solution of the equation for the direction δ/vector e which, however, is a dynamical (frequency-dependent) one. This shows what one silently approximates when using a constant cross relaxation time τD. Finally from (21) the equation for δ/vector etakes the form ∂tδ/vector e+i/vector q/vector p meδ/vector e+e/vectorE/vector∂p/vector e+/vector e×/bracketleftbigg c/vector q/vector∂p/vector e+/vectorh−i(d−δg gd1)δ/vector e/bracketrightbigg =−δf g/bracketleftbig /vector e×(/vector τ−1×/vector e)/bracketrightbig −δ/vector e τ+i/vector q/vector∂p/vector e(V1δn+V3δs).(26) The drift side has the usual form extended by a preces- sion term around the axes /vector e. This spin-precession term is expected and the values of d,d1,cand/vectorhwill be de- rived in the next section which should include the exter- nal magnetic field as one part of the defining axes. The righthandsidein(26)containstherelaxationmechanism which shows a coupling to the solution δfand mean field contributions which will be obtained from a proper mi- croscopic theory. Summarizingthe resultsofthis sectionwe havestarted with the often used two-fluid model together with mean- field terms. Fromaproperwritingofthe kinetic equation in spinor form (21) required by SU(2) symmetry we have seen that a third equation for the change of the spin di- rection is needed. Further it reveals that the cross relax- ation time canbe consideredonly approximatelyastime- independent and constant. The general possible form of the kinetic equation for the scalar (total density) distri- bution, the polarization(total spin) distribution, and the total spin direction is already settled except three scalars c,d,d1and one vector /vectorhto be derived from a microscopictheory, see later (102). It is remarkable that the demand of SU(2) symmetry leads already to such a far leading determination of the structure of equations. III. QUANTUM KINETIC EQUATION A. Green-functions Let us consider spin-polarized fermions which interact with impurity potential Viand are themselves coveredby the Hamiltonian ˆH=/summationdisplay iΨ+ i/bracketleftigg (/vector p−e/vectorA(/vectorRi,t))2 2me+eΦ(/vectorRi,t) −µB/vector σ·/vectorB+/vector σ·/vectorb(/vector p,/vectorRi,t)+ˆVi(/vectorRi)/bracketrightig Ψi +1 2/summationdisplay ijΨ+ iΨ+ jˆV(/vectorRi−/vectorRj)ΨjΨi (27) with the spinor Ψ i= (ψi↑,ψi↓) such that any of the spin- orbit couplings discussed above and the Zeeman term are included. Weassumeatwo-particleinteractionwhichhas a scalar and a spin-dependent part ˆV=V0+/vector σ·/vectorV (28) wherethe latteris responsibleforspin-flip reactions. The vectorpart ofthe potential describese.g. spin-dependent scattering. The scattering off impurities consists of a vector potential from magnetic impurities and a scalar one from charged or neutral impurities ˆVi=Vi0+/vector σ·/vectorVi. (29) In this way we have included the Kondo model as spe- cific case which was solved exactly in equilibrium105and zero temperature. Here we will consider the nonequilib- rium form of this model in the meanfield approximation including relaxation due to collisions. We use the formalism of the nonequilibrium Green’s function technique in the generalizedKadanoff-Baymno- tation introduced by Langreth and Wilkins87. The two independent real-time correlation functions for spin-1 /2 fermions are defined as G> αβ(1,2)=/an}b∇acketle{tψα(1)ψ† β(2)/an}b∇acket∇i}ht, G< αβ(1,2)=/an}b∇acketle{tψ† β(2)ψα(1)/an}b∇acket∇i}ht(30) whereψ†(ψ) are the creation (annihilation) operators, α andβare spin indices, and numbers are cumulative vari- ables for space and time,1 ≡(/vector r1,t1). Accordingly, all the correlation functions without explicit spin indices, are understood as 2 ×2 matrices in spin space, and they can be written in the form ˆC=C+/vector σ·/vectorC, whereC(/vectorC) is the scalar (vectorial) part. This will result in preservation of the quantum mechanical behavior concerning spin com- mutation relations even after taking the quasi classical limits of the kinetic equation. The kinetic equation is7 obtained from the Kadanoff and Baym (KB) equation106 for the correlation function ˆG< −i(ˆG−1 R◦ˆG<−ˆG<◦ˆG−1 R) =i(ˆGR◦ˆΣ<−ˆΣ<◦ˆGA) (31) whereˆΣ is the self-energy, and retarded and advanced functions are defined as CR,A(1,2) =∓iθ(±t1∓t2)[C>(1,2)+C<(1,2)]+CHF (32) whereCHFdenotes the time-diagonal Hartree-Fock terms discussed later. Products ◦are understood as in- tegrations over intermediate variables, space and time, A◦B=/integraltext d¯t/integraltext d¯/vector rA(/vector r1,t1;¯/vector r,¯t)B(¯/vector r,¯t;/vector r2,t2). The nota- tion of Langreth and Wilkins87used here has the ad- vantage that the correlation functions G≷bear a direct physical meaning of occupation of particles and holes. Alternatively sometimes the Keldysh function is used107 which has to be linked to physical quantities. Moreover in this latter Keldysh-matrix notation a superfluous de- gree of freedom occurs canceling in any diagrammatic expansion108which is absent when directly using corre- lationfunctions andtheretarded/advancedGreen’sfunc- tions. We are interested in the Wigner distribution function (2) which is given by the equal time ( t1=t2=T,t= 0) correlation function ˆG<(/vectork,/vectorR,t= 0,T) ˆρ(/vector p,/vectorR,T) =ˆG<(/vector p,/vectorR,t= 0,T) =/integraldisplaydω 2πˆG<(/vector p,ω,/vectorR,T) =f+/vector σ·/vector g.(33) We use the Wigner mixed representation in terms of the center-of-mass variables /vectorR= (/vector r1+/vector r2)/2 and the Fourier transform of the relative variables /vector r=/vector r1−/vector r2→/vector p which separatesthe fast microscopicvariations from slow macroscopic variations. To derive the kinetic equation we expand the convolu- tion up to second-order gradients. Matrix product terms A◦Bappearing in the KB equation can be written as A◦B→ei 2(∂Ω∂B T−∂A T∂B Ω−∂A p∂B R+∂A R∂B p)AB. (34) The quasi-classical limit is obtained by keeping only the first gradients in space and time of the above gradient expansion A◦B→AB+i 2{A,B}, (35) where curly brackets denote Poisson’s brackets, i.e., {A,B}=∂ΩA∂TB−∂TA∂ΩB−∂pA∂RB+∂RA∂pB. Therefore, in the lowest-order gradient approximation, we have the following rule to evaluate the commutators [A,B]−in the KB equation [A◦,B]−→[A,B]−+i 2({A,B}−{B,A}) = [A,B]−+i 2/parenleftbigg [∂RA,∂pB]+−[∂pA,∂RB]+ +[∂ΩA,∂TB]+−[∂TA,∂ΩB]+/parenrightbigg .(36)Please note that the quantum spin structure remains un- touched even after gradient expansion due to the com- mutators. B. Gauge In order to prevent ambiguous results for different choice of gauges, we need to formulate the theory in a gauge-invariant way. Under U(1) local gauge the wave function and vector potential transform as Φ′= e−ie /planckover2pi1α(x)Φ andA′ µ=Aµ+∂µα(x) such that the Green- function transform itself as G′(12) =/an}b∇acketle{tΦ′ 1Φ′ 2/an}b∇acket∇i}ht=/an}b∇acketle{tΦ1Φ2/an}b∇acket∇i}hte−ie /planckover2pi1α(X+x 2)−ie /planckover2pi1α(X−x 2) =/an}b∇acketle{tΦ1Φ2/an}b∇acket∇i}hte−ie /planckover2pi1xµ1/2/integraltext −1/2dλ∂µα(X+λx) . (37) This shift can be compensated if a corresponding phase is added. This is achieved by using a modified Fourier transform G(kX) =/integraldisplay dxei /planckover2pi1xµ/parenleftBigg kµ+e1/2/integraltext −1/2dλAµ(X+λx)/parenrightBigg ×G/parenleftig X+x 2,X−x 2/parenrightig . (38) whereA= (φ(/vectorR,T),/vectorA(/vectorR,T)) and we used four-vector notationx= (t,/vector r)andX= (T,/vectorR). Itisobviousthatthis gauge-invariant Fourier transform leads to gradients as well. To see this we consider the general gauge-invariant Fourier transform for Aµ= (Φ(/vectorR,T),/vectorA(/vectorR,T)) in gradi- ent expansion /vector p=/vectork+e1 2/integraldisplay −1 2dλ/vectorA(/vectorR+λ/vector r,T+λτ) =/vectork+e1 2/integraldisplay −1 2dλeλ/vector r∂A Reλτ∂A T/vectorA(/vectorR,T) →/vectork+e1 2/integraldisplay −1 2dλe−i/planckover2pi1λ∂p∂A Rei/planckover2pi1λ∂ω∂A T/vectorA(/vectorR,T) =/vectork+esinc/parenleftbigg/planckover2pi1 2∂p∂A R/parenrightbigg sinc/parenleftbigg/planckover2pi1 2∂ω∂A T/parenrightbigg /vectorA(/vectorR,T)(39) with sinc(x) = sinx/x= 1−x2/3!+−...and analogously Ω =ω+esinc/parenleftbigg/planckover2pi1 2∂p∂Φ R/parenrightbigg sinc/parenleftbigg/planckover2pi1 2∂ω∂Φ T/parenrightbigg Φ(/vectorR,T).(40) One sees that up to second order gradients we have correctly the following rules for gauge invariant8 formulation109,110: (1) Fourier transform of the differ- ence variable xto the canonical momentum /vector p. (2) Shift from canonical momentum to the gauge invariant (kine- matical) momentum kµ=pµ−e/integraltext1/2 −1/2dλAµ(X+λx), which becomes kµ=pµ−eAµin the lowest-order gradi- ent expansion. (3) Then the gauge invariant functions ¯G reads ¯G(/vectork,ω,/vectorR,T) =G(/vectork+e/vectorA,ω+φ,/vectorR,T) =G(/vector p,Ω,/vectorR,T). (41) This treatment ensures that one has even included all orders of a constant electric field. C. Meanfield We want to consider now the meanfield selfenergy for impurity interactions as well as spin-orbit couplings. A general four-point potential can be written /an}b∇acketle{tx1x2|ˆV|x′ 1x′ 2/an}b∇acket∇i}ht=/summationdisplay p,p′e−ip(x1−x2)+ip′(x′ 1−x′ 2) ×/an}b∇acketle{tp|ˆV/parenleftigg x1+x2 2−x′ 1+x′ 2 2,x1+x2 2+x′ 1+x′ 2 2 2/parenrightigg |p′/an}b∇acket∇i}ht =ˆV−p,p′δ/parenleftbiggx1+x2 2−x′ 1+x′ 2 2/parenrightbigg (42) with ˆV−p,p′= V0(p′−p) /vector σ·/vectorV(p′−p) iλ2 /planckover2pi1/vector σ·(/vector p×/vector p′)V(p′−p)(43) for scalar, magnetic impurities (29), the two-particle interaction (28), and extrinsic spin-orbit coupling (6). Since the potential is time-local, the Hartree-meanfield is the convolution with the Wigner function (33) and writ- ten with Fourier transform of difference coordinates ˆΣ(p,R,T) =/summationdisplay R′qQeiq(R−R′)ˆρ(Q+p,R′,T)ˆVq−Q 2,q+Q 2.(44) Due to the occurring product of the potentials (28), (29) and the Wigner function (33) one has ˆVˆρ=V0f+/vector g·/vectorV+/vector σ·[f/vectorV+V0/vector g+i(/vectorV×/vector g)].(45) The last term in (45) is absent since we work in sym- metrized products as they appear on the left side of (31) from now on. Consequently the selfenergy possesses a scalar and a vector component ˆΣH(/vector p,/vectorR,T) = Σ0(/vector p,/vectorR,T)+/vector σ·/vectorΣH(/vector p,/vectorR,T).(46) The interaction between a conduction electron and the magnetic impurity /vector σ·/vectorViwhere the direction of /vectorViis the local magnetic field deserves some more discussion. Weassume that this magnetic field is randomly distributed on different sites within an angle θlfrom the/vector ezdirection. The directional average74leads then to /summationdisplay pf/vectorV=|V|sinθl θl/vector ezn=/vectorV(q)n /summationdisplay p/vector g/vectorV=|V|sinθl θl/vector ezs=/vectorV(q)s. (47) The angleθlallows us to describe different models. A completely random local magnetic field θl=πis used for magnetic impurities in a paramagnetic spacer layer and in a ferromagnetic layer one uses θl=π/4. The latter one describes the randomly distributed orientation against the host magnet74. For impurity potentials the spatial convolution with the density and spin polarization reads when Fourier transformed, /vectorR→/vector q, Σimp 0(p,q,T) =n(q)V0(q)+/vector s(q)·/vectorV(q) /vectorΣimp(p,q,T) =/vector s(q)V0(q)+n(q)/vectorV(q).(48) For extrinsic spin-orbit coupling we obtain Σs.o. 0(/vector p,/vector q,T) =iλ2 /planckover2pi12V(q)/bracketleftig me(/vectordj(q)×/vector q)j−/vector s(q)·(/vector p×/vector q)/bracketrightig /vectorΣs.o.(/vector p,/vector q,T) =iλ2 /planckover2pi12V(q)/bracketleftig me(/vectorj(q)×/vector q)−n(q)(/vector p×/vector q)/bracketrightig . (49) The used particle density and current are n=/summationdisplay pg(/vector p,/vector q,T);/vectorj=/summationdisplay p/vector p meg(/vector p,/vector q,T) (50) and the spin polarization and spin current /vector s=/summationdisplay p/vector g(/vector p,/vector q,T);/vectordi=/summationdisplay p/vector p me[/vector g(/vector p,/vector q,T)]i;dij=Sji. (51) Please note the summation over indices in the first line of (49) after cross products. Collecting these results the inverse retarded Green’s function reads ˆG−1 R(/vectork,Ω,/vectorR,T) = Ω−H−/vector σ·/vectorΣ(/vectork,/vectorR,T) (52) with the effective scalar Hamiltonian H=k2 2m+Σ0(/vectork,/vectorR,T)+eΦ(/vectorR,T) (53) and/vectork=/vector p−e/vectorA(/vectorR,t). We have summarized the Zeemann term, the intrinsic spin-orbit coupling, and the vector part of the Hartree-Fockselfenergy due to impurities and extrinsic spin-orbit coupling into an effective selfenergy /vectorΣ =/vectorΣH(/vectork,/vectorR,T)+/vectorb(/vectork,/vectorR,T)+µB/vectorB (54)9 such that the effective Hamiltonian possessesPaulistruc- ture ˆHeff=H+/vector σ·/vectorΣ. (55) Pleasenote thatone canconsider(54) asan effectiveZee- man term where the spin-orbit competes with the mag- netic field leading to additional degeneracies in Landau levels111. D. Commutators Nowwe arereadyto evaluatethe commutatorsaccord- ing to (36). In the following we drop the vector notation where it is obvious. We calculate first the commutator with the scalar parts of (52) where we use the gauge- invariant Green’s function (41) such that one has ∂pG=∂k¯G ∂RG=∂R¯G−e∇Φ∂ω¯G−e(∇Ai)∂ki¯G ∂TG=∂T¯G−e∂TA∂k¯G−e∂TΦ∂ω¯G ∂ΩG=∂ω¯G. (56) Further one calculates ∂TH=e˙Φ−ek me˙A−e˙A∂kΣ0+˙Σ0 ∂pH=k me+∂kΣ0 ∂RH=e∂RΦ+1 me[k×∂R×(p−eA)+(k·∂R)(p−eA)] +∂RΣ0−e∂RAi∂kiΣ0 =e∇Φ−k me×eB−e me(k·∇)A+∂RΣ0−e∂RAi∂kiΣ0 (57) where we have used1 2∇u2=u×∇ ×u+(u· ∇)u. We obtain for the commutator with the scalar part of (52) according to (36) 1 i[(Ω−H)◦,G<]−→/bracketleftbigg ∂T+(˙Σ0+evE)∂ω +v∂R+(eE+ev×B−∂RΣ0)∂k/bracketrightbigg ¯G<(58) where the mean velocity of the particles is given by v=k me+∂kΣ0 (59) andE=−e∂RΦ−e˙AandB=∂R×A. InordertogettheequationfortheWignerdistribution we integrate over frequency and the second term on the right hand side of (58) disappears. This term has the structure of the power supplied to the particles which is composed of the contribution by the electric field and the time change of the scalar field which feeds energy to thesystem. The first and third part of (58) together are the co-movingtime derivativeofaparticlewith velocity(59). The fourth term in front of the momentum derivative of ¯G<represents the forces exercised on the particles which appears as the Lorentz force and the negative gradient of the scalar part of the selfenergy which acts therefore like a potential. Next we calculatethe commutator(36) with the vector components of (52). Therefore we employ the relations [/vector σ·/vectorA,ˆG<]+= 2/vector σ·/vectorAG<+2/vectorA·/vectorG< /bracketleftig /vector σ·/vectorA,ˆG</bracketrightig −= 2i/vector σ·(/vectorA×/vectorG<) (60) whereˆG<=G<+/vector σ·/vectorG<and using (56) we obtain −i[−/vector σ·/vectorΣ◦,ˆG<]−→/bracketleftbigg ∂T+(˙Σ0+evE)∂ω+v∂R +(eE+ev×B−∂RΣ0)∂k/bracketrightbigg /vector¯G< −2(/vectorΣ×/vector¯G< ) +/bracketleftig (˙/vectorΣ+evE)∂ω+∂k/vectorΣ∂R+(e∂k/vectorΣ×B−∂R/vectorΣ)∂k/bracketrightig ¯G<. (61) We recognize the same drift terms as for the scalar self- energy components (58). Additionally, the vector self- energy couples the scalar and spinor part of the Green function by an analogous drift but controlled by the vec- tor selfenergy instead of the scalar one. E. Coupled kinetic equations Integrating (58) over frequency and adding (61) we have the complete kinetic equation as required from the Kadanoff and Baym equation (31). In order to make it more transparent we separate the equation according to the occurring Pauli matrices. This is achieved by once forming the trace and once multiplying with /vector σand form- ing the trace. We obtain finally two coupled equations for the scalar and vector part of the Wigner distribution DTf+/vectorA·/vector g= 0 DT/vector g+/vectorAf= 2(/vectorΣ×/vector g) (62) whereDT= (∂T+/vectorF/vector∂k+/vector v/vector∂R) describes the drift and force of the scalar and vector part with the velocity (59) and the effective Lorentz force /vectorF= (e/vectorE+e/vector v×/vectorB−/vector∂RΣ0). (63) The coupling between spinor parts is given by the vector drift Ai= (/vector∂kΣi/vector∂R−/vector∂RΣi/vector∂k+e(/vector∂kΣi×/vectorB)/vector∂k).(64) Remember that we subsumed in the vector selfenergy (54) the magnetic impurity meanfield, the spin-orbit cou- pling vector, and the Zeeman term.10 The term (64) in the second parts on the left sides of (62) represent the coupling between the spin parts of the Wigner distribution. The vector part contains addition- ally the spin-rotation term on the right hand side. These coupled mean field kinetic equations including the mag- netic and electric field, Zeeman coupling, and spin-orbit coupling are the final result of the section. On the right hand side one has to consider additionally collision inte- grals which can be derived from the KB equation taking the selfenergy beyond the meanfield approximation. In the simplest way we will add a relaxation time. Thesystem(62)isthemainresultofthispaperandthe basisforthe further discussionincluding collectivemodes in the secondpaper ofthe series. Thereforeit is time now to compare with other approaches and lay out the gen- eralizations obtained here. If one neglects the coupling of the scalar distribution fto the vector distribution /vector g in the second equation of (62) one has the Eilenberger equation112extended here by magnetic and electric fields as well as selfenergy effects. Compared to51we write both scalar and vector components and have included meanfield quasiparticle renormalizations and the vector self energy. The coupling of the vector equation to the scalar one has been neglected in113,114too but selfconsis- tent quasiparticle energies and Zeeman fields have been taken into account. In75,115only the transverse compo- nents have been considered75which approximates two of the four degrees of freedom in (62). The same reduction of degrees of freedom at the beginning has been used by the projection technique in56,116since the focus of all these papers had been on the proper collision integral in- stead. Selfconsistent quasiparticle equations have been presented in54,60which have been decoupled by Landau Fermi-liquid assumptions57and variational approaches. The coupled kinetic equations have been derived with- out spin-orbit coupling terms and reduced vector selfen- ergies in117which disentangle the equation of moments. Here we present all these effects without the assump- tions found in different places of the above-mentioned approaches. F. Quasi stationary solution The time-independent stationary solution should obey the stationary mean-field equation (62) since any colli- sion integral is then zero providing the Fermi distribu- tion. However,the argumentsand functional dependence as well as spin structure of the solution are already de- termined from the stationary equation of the mean field equation (62). We write them in formal notation Df+/vectorA·/vector g= 0 D/vector g+/vectorAf= 2(/vectorΣ×/vector g) (65) with D={ǫk,...}, Ai={Σi,...} (66)and the Poisson bracket {a,b}=/vector∂ka·/vector∂Rb−/vector∂Ra·/vector∂kb. The electric field is given by a scalar potential e/vectorE(R) = −/vector∇Φ(R) such that we have the quasiparticle energy ǫk(R) =k2 2me+Σ0(k,R)+Φ(R). (67) The choice of gauge is arbitrary since we have ensured that the kinetic equation is gauge invariant. We rewrite (65) into one equation again by the spinor representation ˆ ρ=f+/vector σ·/vector gusing the identity /vector c·/vector g+(/vector σ·/vector c)f−2/vector σ·(/vector σ×/vector g) = /vector σ·/vector c+2i/vector σ 2ˆρ+ ˆρ/vector σ·/vector c−2i/vector σ 2(68) to arrive at [D+/vector σ·/vectorA,ˆρ]++2i[/vector σ·/vectorΣ,ˆρ]−= 0 (69) which is equivalent to (65). Now we search for a solu- tion which renders both anticommutator and commuta- tor zero separately. For the anticommutator, the equation (D+/vector σ·/vectorA)ˆρ= 0 (70) and correspondingly ˆ ρ(D+/vector σ·/vectorA) = 0 are solved by any function of the argument ˆρ0/bracketleftig ǫk(R)+/vectorΣ(k,R)·/vector σ/bracketrightig (71) due to (66). Employing the relation e/vector σ·/vectorΣ= cosh |/vectorΣ|+/vector σ·/vector esinh|/vectorΣ|=/summationdisplay s=±ˆPses|/vectorΣ|(72) with the projectors ˆP±=1 2(1±/vector e·/vector σ) and/vector e=/vectorΣ/|/vectorΣ|we have to have the stationary solution in the form ˆρ0/bracketleftig ǫk(R)+/vectorΣ(k,R)·/vector σ/bracketrightig =/summationdisplay ±P±ˆρ±(ǫk±|/vectorΣ|).(73) The demand of vanishing commutator in (69) works further down the still general possibility of distribution ˆρ±=¯f±+/vector σ·/vector g±. In fact it demands 0 = [/vector σ·/vectorΣ,ˆρ]−= [/vector σ·/vectorΣ,/vector σ·/vector g]−=i/vector σ·(/vectorΣ×/vector g) (74) which implies that /vector g=/vector egwith the effective direction /vector e=/vectorΣ/|/vectorΣ|. Together with (73) we obtain the stationary solutionof(69) and consequentlyof(65) to havethe form ˆρ(ˆε) =/summationdisplay ±ˆP±f±=f++f− 2+/vector σ·/vector ef+−f− 2 ≡ρ+/vector σ·/vector ρ (75) withf±=¯f±+g±=f0(ǫk(R)± |/vectorΣ(k,R)|) andf0an unknown scalarfunction which is determined by the van- ishing of the collision integral to be the Fermi-Dirac dis- tribution.11 G. Currents Duetothespin-orbitcoupling(4)thecurrentpossesses annormalandanomalypart. Using[ /vectorb(/vector p),/vector x] =−i/planckover2pi1∂/vector p/vectorb(/vector p) from elementary quantum mechanics we have ˆvj=i /planckover2pi1[ˆH,ˆxj] =vj+∂pj/vectorb·/vector σ (76) andthequasiparticlevelocity vj=∂pjǫifthesingleparti- cle Hamiltonian is given by the quasiparticle energy ǫ(p). Together with the Wigner function ˆ ρ=f+/vector g·/vector σone has ˆρˆvj=fvi+/vector g·/vectorβj+/vector σ·(vj/vector g+f∂pj/vectorb+i∂pj/vectorb×/vector g) (77) and the particle current and spin current density reads ˆJj=1 2/summationdisplay p[ˆρ,vj]+=/summationdisplay p/bracketleftig fvj+/vector g·∂pj/vectorb+/vector σ·(vj/vector g+f∂pj/vectorb)/bracketrightig =Jj+/vector σ·/vectorSj. (78) Thescalarpartdescribestheparticlecurrent /vectorJ=/vectorJn+/vectorJa consisting of a normal and anomaly current and the vec- tor part describes the spin current Sijnot to be cinfused with the polarization /vector s. The stationary solution allows one to learn about the seemingly cumbersome structure of the particle current (78) consisting of normal and anomalous parts. In fact both parts are necessary to guaranteethe absence of par- ticlecurrentsinstationaryspin-orbitcoupledsystems. In fact both partsseparatelyarenonzeroand onlytheirsum vanishes. We expand the normal particle current with (75) and/vectorΣ =/vectorΣn+/vectorblinear in the spin-orbit coupling /vectorb to get Jn i=1 2/summationdisplay p∂piǫ[f(ǫp+Σ)+f(ǫp−Σ)] =1 2/summationdisplay p∂piǫ/vectorΣn·/vectorb Σn∂ǫ[f(ǫp+Σn)−f(ǫp−Σn)].(79) The anomalous current reads linear in /vectorb Ja i=1 2/summationdisplay p(∂pi/vectorb)·/vector e[f(ǫp+Σ)−f(ǫp−Σ)] =1 2/summationdisplay p/vectorΣn·∂pi/vectorb Σn[f(ǫp+Σ)−f(ǫp−Σ)].(80) Combining both currents one obtains Ji=Ja i+Jn i =/vectorΣn Σn·1 2/summationdisplay p∂pi/braceleftig /vectorb[f(ǫp+Σ)−f(ǫp−Σ)]/bracerightig = 0 (81) as one should. This demonstrates the importance of the anomalous current. One can consider the spin-orbit cou- pling as a continuous current of normal quasiparticlescompensated by the spin-induced one. Any disturbance and linear response will lead to interesting effects due to this disturbed balance such as the anomalous Hall and spin-Hall effects discussed in Sec. IV. H. Selfconsistent precession direction The meanfield approximation establishes a nonlinear relationforparametersofthe distribution functions. The quasi-stationary distribution (75) is determined by the selfenergy(48), (45) and(54) whichin turn is determined againbythe distribution. Without spinpolarizationusu- ally this leads to the selfconsistent determination of the chemical potential. Now we have to accept that the spin precession direc- tion obeys a similar selfconsistency and has to be deter- mined accordingly. Let us assume that the external magnetic field is in the z-direction as is the mean local magnetic field of the magnetic impurities (47) and write µBBeff=nV+µBB. Since the effective local spin (precession) direction is /vector e= /vectorΣ/Σ the mean polarization reads with (54) /vector s=/summationdisplay p/vectorb(p)+µBBeff/vector ez+V0/vector s |/vectorΣ|g =/summationtext p/vectorbg |/vectorΣ| 1−V0/summationtext pg |/vectorΣ|+µBBeff/vector ez/summationtext pg |/vectorΣ| 1−V0/summationtext pg |/vectorΣ|.(82) Since|/vectorΣ|=|/vectorb(p) +µBBeff/vector ez+V0/vector s|one recognizes the selfconsistent equation for the mean polarization /vector swhich in turn determines the local spin precession direction /vector e. In other words the usual selfconsistency due to the meanfields is extended towards a scalar density and a vector polarization yielding the values and the selfcon- sistent precession direction. The procedure is as follows. One calculates the density and spin-polarization accord- ing to (3) where the distributions (75) are dependent on the vector selfenergy (54) which in turn is again deter- mined by the density and spin polarization. The scalar selfenergy we absorb into an effective chemical potential. This selfconsistent precession direction is solely due to the momentum dependence of the spin-orbit coupling /vectorb. Let us inspect all the equations in first order of spin- orbit coupling. We write /vectorΣ =/vectorΣn+/vectorbpwhere we denote the momentum-independent selfenergy with /vectorΣn=n/vectorV+ V0/vector s+µB/vectorB. We expand all directions in first order of /vectorb. The direction of effective polarization becomes /vector e=/vectorΣ |Σ|=/vector ez/parenleftbigg 1−b2 ⊥ 2/parenrightbigg +/vectorb⊥(1−b3) (83) where we will use the convenient separation in the z- direction and the perpendicular direction /vectorbp Σn=/vectorb⊥+/vector ezb3. (84)12 The first impression of (83) suggests that one has a deviation from the z-direction due to the perpendicular direction/vectorb⊥. Let us calculate the selfconsistency and see what remains from this deviation. Since the distribution functions in equilibrium are functions of |/vectorΣ|according to (75), i.e. a function of b2 ⊥andb3, and since the latter ones are even in the momentum direction, the distribu- tions are even in the momentum direction. Therefore the polarization becomes /vector s=/summationdisplay pg/vector e=/vector ez/summationdisplay pg/parenleftbigg 1−b2 ⊥ 2/parenrightbigg =/vector ez/parenleftigg s0−B2 g 2/parenrightigg (85) with s0=/summationdisplay pg;B2 g=/summationdisplay pb2 ⊥g. (86) Now the effective precession direction /vector e=/vectorΣ/|/vectorΣ|is seen to be in the z-direction up to second order in the spin- orbit coupling in contrast to our first view (83). It is instructive to calculate the selfconsistent preces- sion explicitly up to any order now for zero temperature inquasitwodimensionsandlinearDresselhaus β=βDor Rashbaβ=βRspin-orbit coupling. We have the density and the polarization n=/summationdisplay pf=me 2π/planckover2pi12(ǫf+ǫβ) s=/summationdisplay pg=−me 2π/planckover2pi12/radicalig ǫβ(ǫβ+2ǫf)+Σ2n(87) with the spin-orbit energy ǫβ=meβ2. On sees how the Fermienergy ǫfisshifted bythespin-orbitcoupling. The effective Zeeman term Σ ndetermines the polarization in the absence of spin-orbit coupling as s/n=−Σn/ǫf. Since Σ n=µBBeff+sV0withµBBeff=nV+µBB we might conclude from the quadratic equation for sin (87) that the selfconsistent polarization becomes sself =me 2π/planckover2pi12meV0 2π/planckover2pi12µBBeff±/radicalig (µBBeff)2+ǫβ(ǫβ+2ǫf)/parenleftbigmeV0 2π/planckover2pi12/parenrightbig2 1−/parenleftbigmeV0 2π/planckover2pi12/parenrightbig2 (88) However, this procedure oversees just the selfconsistent precession/vector e=/vectorΣ/|/vectorΣ|. In fact instead of (87) we have to calculate the vector quantity /vector s=/summationdisplay p/vector eg=∞/integraldisplay 0dpp (2π/planckover2pi1)22π/integraldisplay 0/vector eg =/vector ezp2/integraldisplay p1dpp 4π/planckover2pi12µBBeff+V0sz/radicalbig β2p2+(µBBeff+V0sz)2 =−/vector ezme 2π/planckover2pi12/parenleftbig µBBeff+V0sz/parenrightbig (89)with p2 1/2 2me=ǫf+ǫβ±/radicalig ǫβ(2ǫf+ǫβ)+(µBBeff+V0sz)2(90) originating from the zero-temperature Fermi functions. We obtainjust theresult(88)but without spin-orbitcou- plingǫβ→0 /vector sself=−/vector ezme 2π/planckover2pi12nV+µBB 1+meV0 2π/planckover2pi12(91) which is quite astonishing. Though the selfconsistent Fermi functions and the selfconsistent precession both contain an involved spin-orbit coupling separately, they cancel each other in the polarization in quasi two dimen- sions and for Rashba or Dresselhaus coupling. It remains to show that this observation is consistent with the general expression for the linearized result (85). With the help of (87) one obtains in fact just (91) /vector s=/vector ez/parenleftigg/summationdisplay pg−1 2/summationdisplay pgb2 ⊥/parenrightigg =−me/vector ez 2π/planckover2pi12/parenleftbig µBBeff+V0sz/parenrightbig +o(β3) =−me/vector ez 2π/planckover2pi12nV+µBB 1+meV0 2π/planckover2pi12 (92) and the effective magnetic field becomes renormalized µBBeff=nV+µBB→nV+µBB 1+meV0 2π/planckover2pi12(93) due to selfconsistency. We can conclude that the self- consistency will determine an effective precession direc- tiondeviatingfromthedirectionofthe externalmagnetic field due to the spin-orbit coupling. However this effect is of higher than second order in spin-orbit coupling /vectorb and vanishes in quasi two-dimensional systems at zero temperature and linear spin-orbit coupling. IV. BALANCE EQUATION A. Linearization to external electric field We consider now the linearization of kinetic equation (62) with respect to an external electric field, no mag- netic field, and a homogeneous situation. We Fourier transform the time ∂t→ −iωand the spatial coordi- nates/vector∂R→i/vector q. The distribution is linearized accord- ing to ˆρ(pRT) =f(p)+δf(pRT)+/vector σ·[/vector g(p)+δ/vector g(pRT)] due to the external electric field perturbation eδ/vectorE= e/vectorE(R,T) =−∇Φ. Further we assume a collision inte- gral of the relaxation time approximation59 −1 2[ˆτ−1,δˆρl]+ (94)13 with a vector and scalar part of relaxation times ˆ τ−1= τ−1+/vector σ·/vector τ−1and τ=τ−1 τ−2−|/vector τ−1|2, /vector τ=−/vector τ−1 τ−2−|/vector τ−1|2.(95) The scalar relaxation is assumed not towards the ab- solute equilibrium f0(ǫ± |Σ| −µ) characterized by the chemical potential µbut towards a local one fl=f0(ǫ± |Σ|−µ−δµ). The latter one can be specified such δn=/summationdisplay p(f−f0) =/summationdisplay p(f−fl+fl−f0) =/summationdisplay p(fl−f0) =∂µnδµ (96) such that the density is conserved81,82as expressedin the step to the second line. Therefore the relaxation term becomes −δˆρl τ=−δˆρ τ+δn τ∂µn∂µˆρ0. (97) In this way the density is conserved in the response func- tion which could be extended to included more conserva- tion laws85,118. If we consider only density conservation but no polarization conservation, of course, we can re- strict ourselves to the ∂µf0terms. Abbreviating −iω+i/vector p·/vector q/m+τ−1=aandiq∂p/vectorΣ + /vector τ−1=/vectorb, the coupled kinetic equations (62) take the form aδf+/vectorbδ/vector g=S0 aδ/vector g+/vectorbδf−2/vectorΣ×δ/vector g=/vectorS (98) withe/vectorE=−i/vector qΦ and S0=iq∂pf(Φ+δΣ0)+iq∂p/vector g·δ/vectorΣ+δn τ∂µn∂µf0 /vectorS=iq∂p/vector g(Φ+δΣ0)+iq∂pfδ/vectorΣ+2(δ/vectorΣ×/vector g)+δn∂µ/vector g τ∂µn. (99) In order to facilitate the vector notation we want to un- derstandq∂p=/vector q·/vector∂pin the following. B. Density and spin current The linearized kinetic equations (98) allow us to write the balance equations for the magnetization density δ/vector s, the density δn, and the currents by integrating over the corresponding moments of momentum ∂tδn+∂Ri/vectorJi+/vector τ−1·δ/vector s= 0 ∂tδ/vector s+∂Ri/vectorSi+/vector τ−1δn−2/summationdisplay p/vectorΣ×δ/vector g= 2δ/vectorΣ×/vector s(100)where we Fourier transformed the wavevector qback to spatial coordinates R. Exactly the expected density cur- rents and magnetization currents (78) appear /vectorJ=/summationdisplay p/parenleftig ∂pǫpδf+∂p/vectorb·δ/vector g/parenrightig /vectorSi=/summationdisplay p/parenleftig ∂piǫpδ/vector g+∂pi/vectorbδf/parenrightig .(101) The right hand side of (100) can be reshuffled to the left sinceδ/vectorΣ =V0δ/vector s+/vectorVδn. Theonlyproblemmakestheterm/summationtext p/vectorΣ×δ/vector gsince the momentum-dependence of the spin- orbit coupling prevents the balance equations from being closed. We need the complete solution of δ/vector gin order to write the correct balance equation for the magnetization density. This will be given in the second part of this paper series. The method ofmomentsdoes not yield a closedsystem of equations since the density couples to the currents, the balance for the currents to the energy currents and so on. Only with specific approximationsthese equations can be closed. One can find a great variety of methods in the literature. Many treatments neglect certain Landau- liquid parameters119based on the work of120. A more advancedclosingprocedurewasprovidedby114wherethe energy dependence of δ/vector swas assumed to be factorized from space and direction /vector pdependencies. We will not follow these approximations but solve the linearized equation exactly in the second part of this pa- per seriesto providethe solutionofthe balanceequations and the dispersion exactly. Amazingly this yields quite involved and extensive structures with many more terms then usually presented in the literature. C. Comparison with two-fluid model We are now in a position to compare the result of mi- croscopic theory with the two-fluid model (16) and the form for the direction (26) extracted from general sym- metric considerations. The first observation is that the momentum-dependence of the direction /vector edue to, e.g., spin-orbit coupling is not covered by the original two- fluid model. We can, however, redefine certain relaxation times to account for these effects as we did in (25). The priceto pay wasthat one hasto considera third equation forthe precessiondirection. The undetermined constants in (26) are now derived to be /vectorh= 2δ/vectorΣ, d=−2iΣ, d 1=c= 0 (102) which one can see from decomposing (98) and (99) into equations for δf,δgandδ/vector eand compare with (23) and (26). The problem with the two-fluid model becomes appar- ent if we try to extract the values for the meanfields (19). One obtains the unique identification V1δn+V3δs=δΣ0 (103)14 and two different forms from the δfand theδgequation /vector e·q∂p/vectorΣδg+η=q∂pg(V2δn+V4δs−/vector e·δ/vectorΣ) /vector e·q∂p/vectorΣδf=q∂pf(V2δn+V4δs−/vector e·δ/vectorΣ)(104) withη= (Σδ/vector e−δ/vectorΣ)q∂p/vector eg≈0. One sees that we have two determining equations which are consistent only if δf/q∂pf=δg/q∂pgwhich is not ensured in general. Therefore the mapping of the momentum-dependent spin-orbit coupling terms to a two-fluid model is not pos- sible in general. Only if we approximate the momentum- dependence of /vectorΣ by a constant direction /vector e(p)≈/vector edo we have the two unique equations (103) and V2δn+V4δs=/vector e·δ/vectorΣ. (105) For illustration we decompose the change of precession directionintothe componentsproportionaltothe density and polarization change δ/vector e=/vector e1δn+/vector e2δsand obtain in this way V1=V0+/vectorV·/vector e, V 2=/vector e·/vectorV+V0/vector e·/vector e1, V4=V0+V0/vector e·/vector e2, V 3=/vector e·/vectorV+/vectorV·/vector e2(106) which provides indeed four different meanfield potentials (19) which in turn can be translated into the original Uij potentials. In the case that U↑↓=U↓↑we must have V0/vector e·/vector e1=/vectorV·/vector e2. V. SPIN-HALL AND ANOMALOUS HALL EFFECT A. Homogeneous situation without magnetic field It is interesting to note that the coupled kinetic equa- tion (62) allowsforafinite conductivityevenwithout col- lisions and a Hall effect without external magnetic field. This is due to the interference between the two-fold split- ting of the band and will be the reason for the anomalous Hall effect. We haveseenalreadyanexpressionofthis in- terference by the two compensating currents, the normal and anomalous ones, in Sec. IIIG. For a homogeneous system neglecting magnetic fields we have from (62) (∂t+eE∂p)f= 0 (∂t+eE∂p)/vector g= 2(/vectorΣ×/vector g). (107) Due to the spin-precession term a nontrivial solution for the polarization part /vector ρof the distribution appears. We will solve these coupled equations in the following by two ways, once in the helicity basis16and once directly in the spin basis. In order to gain trust in the result we then compare the expressions with the Kubo formula. With the help of an effective Hamiltonian H=ǫ+/vector σ×/vectorΣ (108)withǫ=p2 2m+ Σ0+eΦ and the spin-orbit coupling as well as the meanfield by magnetic impurities summarized in/vectorΣ, one can rewrite both coupled kinetic equation ˆ¯ρ= ρ+/vector σ×/vector ρinto (∂t+eE∂p)ˆ¯ρ+i[H,ˆ¯ρ]−= 0. (109) B. Helicity basis Now we go into the helicity basis which means we use the eigenstates H|±/an}b∇acket∇i}ht=ǫ±|±/an}b∇acket∇i}ht (110) and using the convenient notation Σx−iΣy= Σe−iϕ:|Σ|=/radicalig Σ2x+Σ2y+Σ2z(111) we have ǫ±=ǫ±|Σ| |Σ|2= Σ2+Σ2 z |±/an}b∇acket∇i}ht=1/radicalbig 2|Σ|/parenleftigg −e−iϕ/radicalbig |Σ|±Σz ∓/radicalbig |Σ|∓Σz/parenrightigg .(112) The transformation matrix is U= (|+/an}b∇acket∇i}ht,|−/an}b∇acket∇i}ht) and the Hamiltonian becomes diagonal ¯H=U+HU=/parenleftigg ǫ+0 0ǫ−/parenrightigg (113) and since the transformed spin-projection operators read ¯P+=/parenleftigg 1 0 0 0/parenrightigg ,¯P−=/parenleftigg 0 0 0 1/parenrightigg (114) the equilibrium distribution (75) becomes diagonal ˆ¯ρ=/summationdisplay i=±ˆPifi=/parenleftigg f+0 0f−/parenrightigg (115) or in general ˆ¯ρ= ¯ρ+/vector¯ρ·U+/vector σU. (116) The Pauli matrices transformed in the helicity basis can be written with the notation (111) aslinearcombinations of Pauli matrices ¯σx=1 |Σ|/parenleftigg Σcosϕ −i|Σ|sinϕ−Σzcosϕ i|Σ|sinϕ−Σzcosϕ −Σcosϕ/parenrightigg =−Σz |Σ|cosϕσx+sinϕσy+Σ |Σ|cosϕσz ¯σy=i |Σ|/parenleftigg −iΣsinϕ i Σzsinϕ+|Σ|cosϕ iΣzsinϕ−|Σ|cosϕ i Σsinϕ/parenrightigg =−Σz |Σ|sinϕσx−cosϕσy+Σ |Σ|sinϕσz ¯σz=1 |Σ|/parenleftigg ΣzΣ Σ−Σz/parenrightigg =Σ |Σ|σx+Σz |Σ|σz. (117)15 This allows us to transform the velocity operator in the helicity basis ¯ˆvi=∂iǫ+ /parenleftig Σ2 |Σ|∂iΣz Σ/parenrightig −Σ∂iϕ ∂i|Σ| ·/vector σ (118) where∂idenotes the derivative with respect to the i-th component of the momentum. The current would be Ji=e 2Tr[¯ˆviδ¯ρ] (119) where we need the linearized solution of (109) in the he- licity basis with respect to an external electric field. Em- ploying (∂U+)U=−U+∂Uone transforms U+(∂f)U=∂¯f+[U+∂U,¯f]− (120) such that the kinetic equation (109) reads ∂t¯ρ+[U+∂tU,¯ρ]−+e/vectorE·/vector∂p¯ρ+e/vectorE·[U+/vector∂pU,¯ρ]− +i[¯H,¯ρ]−= 0.(121) The correspondinglinearizedkinetic equation ¯ ρ= ¯ρ0+δ¯ρ takes the same form ∂tδ¯ρ+[U+∂tδU,¯ρ0]−+e/vectorE·/vector∂p¯ρ0+e/vectorE·[U+/vector∂pU,¯ρ0]− +i[¯H,δ¯ρ]−= 0. (122) For further use we neglect the time-dependence ∂tδUdue to selfconsistent mean field in the basis. The selfconsis- tently induced meanfield term [ δ¯H,¯ρ0] can be considered more convenient in the next paragraph in spinor repre- sentation by the solution in the spin basis in the next part of this series. With the help of the diagonal Hamiltonian (113) one has for any matrix A={aij} /bracketleftigg/parenleftigg ǫ+0 0ǫ−/parenrightigg ,A/bracketrightigg = (ǫ+−ǫ−)/parenleftigg 0a12 −a210/parenrightigg =1 2(ǫ+−ǫ−)[σz,A] (123) such that we can write (122) with the equilibrium distri- bution (116) ∂tδ¯ρ+e/vectorE·/vector∂p¯ρ0+1 2(f+−f−)e/vectorE·[U+/vector∂pU,σz]− −i 2(ǫ+−ǫ−)[δ¯ρ,σz]−= 0. (124) Consequently, the first part describes the diagonal re- sponse and the second part the response due to off- diagonal or band interference. The solution reads ex- plicitly δ¯ρ=−ie/vectorE·/parenleftigg/vector∂pf+ ω2g ∆ǫ−ω/an}b∇acketle{t+|/vector∂p|−/an}b∇acket∇i}ht 2g ∆ǫ+ω/an}b∇acketle{t−|/vector∂p|+/an}b∇acket∇i}ht/vector∂pf− ω/parenrightigg (125)with2g=f+−f−and∆ǫ=ǫ+−ǫ−= 2|Σ|. Thediagonal part leads to the standard dynamical Drude conductivity if a scattering with impurities is considered ω→ω+i/τ. The second part is the reason for the anomalous Hall effect which we consider in the following. With the help of (112) one has explicitly /an}b∇acketle{t±|∂|∓/an}b∇acket∇i}ht=−iΣ 2|Σ|∂ϕ∓Σ2 2|Σ|2∂/parenleftbiggΣz Σ/parenrightbigg /an}b∇acketle{t±|∂|±/an}b∇acket∇i}ht=−i|Σ|±Σz 2|Σ|∂ϕ (126) and the off-diagonal parts of (125) can be expressed as δ¯ρAH=Σ |Σ|ge/vectorE· ω2−(∆ǫ)2/bracketleftbigg ∆ǫ/parenleftbigg /vector∂pϕσx+Σ |Σ|/vector∂p/parenleftbiggΣz Σ/parenrightbigg σy/parenrightbigg +iω/parenleftbigg /vector∂pϕσy−Σ |Σ|/vector∂p/parenleftbiggΣz Σ/parenrightbigg σx/parenrightbigg/bracketrightbigg . (127) The current (119) reads then Jα=σαβEβ (128) with the two parts of conductivity σαβ=e2 2/summationdisplay pΣ2 |Σ|2g ω2−4|Σ|2 ×/bracketleftbigg 2Σ/parenleftbigg ∂βϕ∂α/parenleftbiggΣz Σ/parenrightbigg −∂αϕ∂β/parenleftbiggΣz Σ/parenrightbigg/parenrightbigg −iω/parenleftbigg ∂βϕ∂αϕ+Σ2 |Σ|2∂β/parenleftbiggΣz Σ/parenrightbigg ∂α/parenleftbiggΣz Σ/parenrightbigg/parenrightbigg/bracketrightbigg .(129) 1. Dynamical asymmetric part The first part of (129) is the standard anomalous Hall effect since it represents an asymmetric matrix noting ∂αa∂βb−∂βa∂αb=ǫαβγ(/vector∂a×/vector∂b)γ.(130) To simplify this asymmetric part further we perform the derivatives explicitly /vector∂/parenleftbiggΣz Σ/parenrightbigg ×/vector∂ϕ=−1 Σ3ǫijkΣi/vector∂Σj×/vector∂Σk(131) and the first asymmetric part of (129) can be written σas αβ=e2 2/summationdisplay pg 1−ω2 4|Σ|2/vector e·(∂α/vector e×∂β/vector e) (132) with/vector e=/vectorΣ/|Σ|. This describes the dynamical anomalous Hallconductivityaswecanverifybythecomparisonwith the dc Hall conductivity from the Kubo formula in the next section.16 2. Dynamical symmetric part The second symmetric part of (129) is a pure dynami- cal conductivity and can be rewritten in a compact form as well by noting ∂βϕ∂αϕ+Σ2 |Σ|2∂β/parenleftbiggΣz Σ/parenrightbigg ∂α/parenleftbiggΣz Σ/parenrightbigg =/parenleftig ∂α/vectorΣ·∂β/vectorΣ−∂α|Σ|∂β|Σ|/parenrightig1 Σ2 = (∂α/vector e·∂β/vector e)|Σ|2 Σ2(133) with/vector e=/vectorΣ/|/vectorΣ|. Therefore we obtain σsym αβ=ie2 2/summationdisplay pω 2|Σ|g 1−ω2 4|Σ|2∂α/vector e·∂β/vector e. (134) C. Anomalous Hall conductivity from Kubo formula For the reason of comparison we re-derive this results from the Kubo formula and consider the dc limit of the interband conductivity with band energies ǫn(p) and oc- cupationsfn(p). Due to band polarizations one has a finite current. The Kubo-Bastin-Streda formula reads (/summationtextfn=nn) σαβ=e2/planckover2pi1 i/summationdisplay nm/summationdisplay pfm−fn (ǫn−ǫm)2vα nmvβ mn(135) with the velocity in band basis /vector vnm=/an}b∇acketle{tn|ˆ/vector v|m/an}b∇acket∇i}ht=1 i/planckover2pi1/an}b∇acketle{tn|[ˆ/vector x,ˆH]|m/an}b∇acket∇i}ht=/an}b∇acketle{tn|[/vector∂p,ˆH]|m/an}b∇acket∇i}ht =/an}b∇acketle{tn|/vector∂p|m/an}b∇acket∇i}ht(ǫm−ǫn). (136) The conductivity becomes therefore with the notation (/vector∂p)α=∂α σαβ=−e2/planckover2pi1 i/summationdisplay nm/summationdisplay p(fm−fn)/an}b∇acketle{tn|∂α|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂β|n/an}b∇acket∇i}ht =e2/planckover2pi1 i/summationdisplay n/summationdisplay pfn/an}b∇acketle{tn|∂α∂β−∂β∂α|n/an}b∇acket∇i}ht =ǫαβγe2/planckover2pi1 i/summationdisplay n/summationdisplay pfn/an}b∇acketle{tn|(/vector∂p×/vector∂p)γ|n/an}b∇acket∇i}ht =ǫαβγe2/planckover2pi1 i/summationdisplay n/summationdisplay pfn(/vector∂p×/an}b∇acketle{tn|/vector∂p|n/an}b∇acket∇i}ht)γ =−ǫαβγe2/summationdisplay n/summationdisplay pfn(/vector∂p×/vector an)γ (137) whereweintroducedin thelaststep theBerry-phasecon- nection /vector an=i/planckover2pi1/an}b∇acketle{tn|/vector∂p|n/an}b∇acket∇i}ht=/an}b∇acketle{tn|/vector x|n/an}b∇acket∇i}ht (138)and/vector∂×/vector anis the Berry-phase curvature. Now we specify this formula for the two-spin band problem where the Berry phase connection with the help of (126) reads /vector a±=i/planckover2pi1/an}b∇acketle{t±|/vector∂p|±/an}b∇acket∇i}ht=/planckover2pi1Σ±Σz 2Σ/vector∂pϕ (139) and the Berry curvature /vector∂p×/vector a±=∓/planckover2pi1 2Σ3ǫijkΣi/vector∂pΣj×/vector∂pΣk(140) or (/vector∂p×/vector a±)γ=∓/planckover2pi1 2Σ3ǫαβγǫijkΣi∂αΣj∂βΣk =∓/planckover2pi1 2Σ3ǫαβγ/vectorΣ·(∂α/vectorΣ×∂β/vectorΣ).(141) Therefore the dc Hall conductivity reads finally σdc αβ=e2/planckover2pi1 2/summationdisplay pg /vector e·(∂α/vector e×∂β/vector e) (142) which is exactly the dc limit of (132). D. Spin-Hall and anomalous Hall effect in spin basis 1. Anomalous and inverse Hall effect Now we solve the equations (107) once more directly in the spin basis. This has the advantage that the am- biguousterm U+∂tUin the helicitybasisdoesnotappear and in this way we will see that it does not contribute to the final result. Moreover we have the relaxation time in the kinetic equation which means that in the end we can understand ω→ω+i/τ. In ordertokeepthe comparison as near as possible to the above two ways of derivation we keepωand shift in the end. Equations (107) are decoupled since we neglect mean- field effects and magnetic fields. Linearizing and noting that/vectorΣ×/vector g= 0 since/vector g=/vector e(f+−f−)/2 we obtain after the Fourier transform of time δf(ω,p) =−i ωeE∂pf δ/vector g(ω,p) =iω 4|Σ|2−ω2eE∂p/vector g −4i1 ω(4|Σ|2−ω2)/vectorΣ(/vectorΣ·eE∂p/vector g) −21 4|Σ|2−ω2/vectorΣ×eE∂p/vector g. (143) With the help of 1 4|Σ|2−ω2 −iω 4i|Σ|2 ω 2|Σ| =∞/integraldisplay 0eiωt cos2|Σ|t 1−cos2|Σ|t sin2|Σt| ,(144)17 one sees that each of the terms in (143) correspond to a specific precession motion analogously to the one seen in the conductivity of a charge in crossed electric and magnetic fields /vectorJ(t) =σ0t/integraldisplay 0d¯t τe−¯t τ/braceleftig cos(ωc¯t)/vectorE(t−¯t) +sin(ωc¯t)/vectorE(t−¯t)×/vectorB0+[1−cos(ωc¯t)][/vectorE(t−¯t)·/vectorB0]/vectorB0/bracerightig (145) as the solution of the Newton equation of motion me˙/vector v=e(/vector v×/vectorB)+e/vectorE−me/vector v τ. (146) It illustrates the threefold orbiting of the electrons with cyclotron frequency: (i) in the direction of the electric and (ii) magnetic field, and (iii) in the direction perpen- dicular to the magnetic and electric field. The chargecurrent (78) or (101) consists of the normal current as the first part, e/summationdisplay p∂αǫδf=ine2 ω+Eα=ine2τ 1−iωτEα(147) and the anomalous current due to spin-polarization as the second part of (101). The latter one contains the standard anomalous Hall effect as the third term of (143) and their first and second term will combine together to the symmetric part of the anomalous conductivity as we will demonstrate now. Since /vectorΣ·∂/vector e= 0, the third term of (143) leads to σas αβ=−e2/summationdisplay p∂α/vectorΣ(/vectorΣ×∂β/vector e)f+−f− 4|Σ|2−ω2 =−e2 2/summationdisplay kpg 1−ω2 4|Σ|2∂α/vector e(/vector e×∂β/vector e) (148) and one recognizes the anomalous Hall conductivity (132). The first and second term of (143) combine together with the symmetric part σsym αβ=ie2 2ω/summationdisplay p2∂α/vectorΣ 4|Σ|2−ω2/bracketleftbigg ω2(g∂β/vector e+/vector e∂βg)−4/vectorΣ|Σ|∂βg/bracketrightbigg =ie2 2ω/summationdisplay p2 4|Σ|2−ω2/bracketleftbiggω2g |Σ|(∂α/vectorΣ∂β/vectorΣ−∂α|Σ|∂β|Σ|) +ω2∂α|Σ|∂βg−4|Σ|2∂α|Σ|∂βg/bracketrightbigg =iωe2 4/summationdisplay pg 1−ω2 4|Σ|21 |Σ|3/parenleftbigg ∂α/vectorΣ∂β/vectorΣ−∂α|Σ|∂β|Σ|/parenrightbigg +ie2 ω/summationdisplay pg∂α∂β|Σ| (149)where we have used in the second step the relation ∂α/vectorΣ∂β/vectorΣ |Σ|=1 |Σ|/parenleftbigg ∂α/vectorΣ∂β/vectorΣ−∂α|Σ|∂β|Σ|/parenrightbigg =|Σ|∂α/vector e·∂β/vector e. (150) Since the last term in (149) vanishes due to symmetry in pwe obtain exactly (134). Summarizing, the total charge current (101) is given by Jα=σDEα+(σas αβ+σsym αβ)Eβ (151) with the usual Drude conductivity σD=ne2τ/meand the symmetric and asymmetric parts of the anomalous Hall conductivity (132) and (134) σas αβ σsym αβ =e2 2/summationdisplay pg 1−ω2 4|Σ|2 /vector e·(∂α/vector e×∂β/vector e) iω 2|Σ|∂α/vector e·∂β/vector e(152) and/vector e=/vectorΣ/|Σ|. Note that from our kinetic equation with the relaxation time approximation we understand the above formulas as ω→ω+i/τwhich leads in the static limit the modifications of the Kubo expression due to collisions. ForzerotemperatureandlinearRashbaspin-orbitcou- pling we can integrate these expressions analytically. We consider the electric field in x-direction and obtain σas yx=e2 4π/planckover2pi1Σnτωarctan/bracketleftbigg2ǫβτω /planckover2pi12+4(2ǫβǫF+Σ2n)τ2ω/bracketrightbigg →e2 4π/planckover2pi1 ǫβΣn 2ǫβǫf+Σ2nω= 0,τ→ ∞ Σn ωartanh/bracketleftig 2ǫβω /planckover2pi12ω2−4(2ǫβǫF+Σ2n)/bracketrightig ω/ne}ationslash= 0,τ→ ∞ (153) with the Rashba energy ǫβ=mβ2 R//planckover2pi1and the dynamical result is given τω=τ/(1−iωτ) in (153). We see that the anomalous Hall effect vanishes with vanishing effective Zeeman field Σn=|n/vectorV+/vector sV0+µB/vectorB|. (154) We see from figure 2 that the anomalous Hall conduc- tivity is strongly dependent on the frequency showing even a sign change at high frequencies which had been reported before121. This sign change is connected with a collisional damping which means that it is suppressed by collisions. This damping as expressed by the imaginary part of the conductivity vanishes in the static limit. The absolute value is dependent on the effective Zee- man field (154) as one sees from the scaling of the static limit plotted in figure 3. The static anomalous Hall con- ductivity possesses a maximum at certain Zeeman terms which are dependent on the relaxation time.18 02468100.0000.0050.0100.0150.0200.025 Ω/LBracket1ΕF/RBracket1Σyxas/LBracket1e/Slash18Π/HBar/RBracket1 /CaΠSigman/Equal1ΕF/CaΠSigman/Equal0.5ΕF/CaΠSigman/Equal0ΕF FIG. 2: The dynamical anomalous Hall conductivity (153) vs. frequency for different values of the effective Zeeman fiel d (154) and a relaxation time τ= 0.3/planckover2pi1ǫFand a Rashba energy ofmβ2 R= 0.1ǫF. Real parts are thick and imaginary are thin lines. 0.00.51.01.52.02.53.00.000.020.040.060.080.10 /CaΠSigma/LBracket1ΕF/RBracket1Σxxsym,Σyxas/LBracket1e/Slash18Π/HBar/RBracket1 Τ/Equal3/HBar/Slash1ΕFΤ/Equal1/HBar/Slash1ΕF FIG. 3: The static anomalous Hall conductivity of (153) (thick) and inverse Hall conductivity (155) (thin) vs. effec tive Zeeman field for two different relaxation times and a Rashba energy of mβ2 R= 0.1ǫF. The symmetric part of (152) in fact yields σsym yx= 0 and σsym xx=e2 16π/planckover2pi1/braceleftbigg4ǫβΣ2 nτω 2ǫβǫf+Σ2n + (1−4Σ2 nτ2 ω)arctan/bracketleftbigg4ǫβτω /planckover2pi12+4(2ǫβǫF+Σ2n)τ2ω/bracketrightbigg/bracerightbigg →e2 4π/planckover2pi1/braceleftigg 0ω= 0,τ→ ∞ imaginary ω/ne}ationslash= 0,τ→ ∞(155) which shows that it represents a contribution in the di- rection of the applied electric field and is caused by col- lisional correlations. We interpret it as an inverse Hall effect. This dynamical result is different from the spin accumulation found in122basically by the arctanterm and therefore no sharp resonance feature. Expanding, however, in small spin-orbit coupling σsym xx=e2 2π/planckover2pi1ǫβτ 1+4Σ2nτ2+o(ǫ2 β) (156)0 510 150.000.010.020.03 Ω/LBracket1ΕF/RBracket1Σxxsym,Σyxas/LBracket1e/Slash18Π/HBar/RBracket1 ΣxxsymΣyxas FIG. 4: The comparison of the dynamical anomalous Hall conductivity (153) (solid) and the inverse Hall conductivi ty (155) (dashed) for a relaxation time τ= 0.3/planckover2pi1ǫF, a Rashba energy of mβ2 R= 0.1ǫFand an effective Zeeman energy Σ n= 1ǫF. The real parts are thick lines, and the imaginary parts are thin lines. showsthat the static limit agreeswith122,123. Pleasenote that if one sets Σ n→0 before expanding a factor 1/2 appears which illustrates the symmetry breaking by the effective Zeeman term. In figure3we comparethis expressionwith the anoma- lous Hall conductivity for two different relaxation times. While the static anomalous Hall conductivity vanishes with the effective Zeeman field the inverse Hall effect re- mains finite which value is easily seen from (155). Both the anomalous Hall conductivity as well as the inverse Hall conductivity possess a maximum at certain effective Zeeman fields. The comparison of the dynamical anomalous Hall ef- fect and the inverse Hall effect finally can be found in figure 4. In contrast to the anomalous Hall effect the inverse Hall effect does not show a sign change which means the current remains in the direction of the applied electric field as it should. 2. Spin-Hall effect Now we can consider the spin current (101) with the help of the long-wavelength solution (143) without mag- netic fields. The part with δfvanishes after partial inte- grationand symmetry in p. The other terms groupinto a normal spin-current and an (anomalous) part represent- ing the spin-Hall effect /vectorSα=−eτ me(1−iωτ)/vector sEα+/vector σαβEβ.(157) The spin-Hall coefficient consists analogously as the anomalous Hall effect of a symmetric and an asymmetric part (ω→ω+i/τ) /vector σas αβ /vector σsym αβ =e meω/summationdisplay ppαg 1−ω2 4|Σ|2 iω 2|Σ|/vector e×∂β/vector e i∂β/vector e.(158)19 We see that both effects the spin-Hall effect and the anomalous Hall effect appear without magnetic fields and have their origin in the anomalous parts of the cur- rents due to spin-orbit coupling. Though the normal and anomalous currents exactly compensate in the station- ary state, the current due to a disturbing external elec- tric field shows a finite asymmetric and symmetric part (158) not known in the literature. Explicit integration of (158) is possible to carry out in zero temperature and linear Rashba and Dresselhaus coupling. We assume the electric field in the x-axis and we get for the Rashba linear spin-orbit coupling σz yx=e 8π/planckover2pi1/bracketleftbigg 1−1+4Σ2 nτ2 ω 4ǫβτωarctan/parenleftbigg4/planckover2pi1ǫβτω /planckover2pi12+4τ2ω(2ǫβǫF+Σ2n)/parenrightbigg/bracketrightbigg (159) withτω=τ/(1−iωτ). Only the z-axis survives in linear spin-orbit coupling. Neglecting the selfenergy and using the static limit it is just the result of6,124. The so-called universal limit appears if one takes the limit of vanishing collision frequency σz yx=e 8π/planckover2pi12ǫβǫf 2ǫβǫf+Σ2n+o(1/τ). (160) We see how the selfenergy including the Zeeman term (154) modifies this ”universal limit” which already hints at questioning of this notion. For the Dresselhaus linear spin-orbit coupling we ob- tain just (159) with opposite sign. Therefore if we ap- proximate the combined effect by adding the two specific results we see that the constant term vanishes and the difference of the expressions with corresponding Rashba and Dresselhaus energies occur. The correct treatment of both couplings together leads to involved angular in- tegrations and escape analytical work. The universal constant e/8π/planckover2pi1has been first de- scribed by125and raised an intensive discussion. It was shown that the vertex corrections cancel this constant126,127. A suppression of Rashba spin-orbit cou- pling has been obtained due to disorder128, or electron- electron interaction129and found to disappearin the self- consistent Born approximation38. The conclusion was thatthetwodimensionalRashbaspin-orbitcouplingdoes not lead to a spin-Hall effect as soon as there are re- laxation mechanisms present which damp the spins to- wards a constant value. In order to include such effects one has to go beyond meanfield and relaxation-time ap- proximation by including vertex corrections86. The spin- Hall effect does not vanish with magnetic fields or spin- dependent scattering processes6. Please note that the universal constant in (159) is nec- essary to obtain the correct small spin-orbit coupling re- sult σz yx=e π/planckover2pi1ǫfτ2 (1−iωτ)2+4Σ2nτ2ǫβ+o(ǫ2 β).(161) Without the Zeeman term Σ n→0 and for small spin- orbit coupling this agrees with the dynamical result of12202468100.000.020.040.06 Ω/LBracket1ΕF/RBracket1Σyxz/LBracket1e/Slash18Π/HBar/RBracket1 /CaΠSigman/Equal1ΕF/CaΠSigman/Equal0.5ΕF/CaΠSigman/Equal0ΕF FIG. 5: The dynamical spin-Hall coefficient vs frequency for different values of the effective Zeeman field and a relaxation timeτ= 0.3/planckover2pi1ǫFand a Rashba energy of mβ2 R= 0.1ǫF, real parts are shown as thick and imaginary parts as thin lines. where the definition of spin current has been employed in terms of physical argumentation. Again the result here differs from the resonant structure found in122by the arctanterm but the static limit agrees with the result of6,124. The dynamical result (159) describes the influence of anexternalmagneticfieldaswellasmeanmagnetizations due to magnetic impurities. The advantage of the result here is the simplicity in which the frequency dependence enters and the combined effect of external magnetic field, spin polarizations and mean magnetization described by one vector selfenergy (54) called the effective Zeeman field. The z-component of the spin-Hall coupling in the x- direction becomes a modified coefficient (159) σz xx=2 /planckover2pi1Σnτσz xy. (162) The medium effects or magnetic field or magnetization condensed in (54) triggers a second spin-Hall direction in plane with the z-axes and the electric field. This obser- vation is interpreted as the inverse spin-Hall effect. The herepresentedinversespin-Halleffectsaretheunderlying physics in the recently observed terahertz spin signals130 in magnetic heterostructures. In figure 5 we plot the dynamical spin-Hall coefficients for different effective Zeeman fields. The coefficients be- comes suppressed with increasing Zeeman field as seen also in figure 6 and with larger scattering frequency 1 /τ. Interestingly the real part of the spin-Hall conductivity shows a sign change at a specific frequency. Though this effect is interesting this reversed current is strongly damped represented by the imaginary part. The static limit becomes suppressed as seen in Fig. 6 dependent on the relaxation time and the effective Zee- man field. The in-plane component (162) is zero for the absent Zeeman term and shows a characteristic maxi- mum at certain effective Zeeman fields. The dynamical spin Hall coefficient and the inverse20 0.00.51.01.52.00.00.20.40.60.81.01.21.4 /CaΠSigma/LBracket1ΕF/RBracket1Σxxz,Σyxz/LBracket1e/Slash18Π/HBar/RBracket1 Τ/Equal3/HBar/Slash1ΕFΤ/Equal1/HBar/Slash1ΕF FIG. 6: The static spin-Hall coefficients of (159) (thick) and (162) (thin) effective Zeeman field for two different relaxati on times and a Rashba energy of mβ2 R= 0.1ǫF. 0246810/MinuΣ0.010.000.010.020.030.040.05 Ω/LBracket1ΕF/RBracket1Σxxz,Σyxz/LBracket1e/Slash18Π/HBar/RBracket1 ΣxxzΣyxz FIG. 7: The comparison of the dynamical spin-Hall coeffi- cients (159) (solid) and the inverse spin-Hall coefficient (1 62) (dashed) for a relaxation time τ= 0.3/planckover2pi1ǫF, a Rashba energy ofmβ2 R= 0.1ǫFand an effective Zeeman energy Σ n= 1ǫF. spin Hall coefficient are compared in figure 7. We recog- nize a sign change at higher frequencies connected with a strongdampingforbotheffects. Comparedtothe inverse Hall effect the inverse spin-Hall current can be directed both parallel and anti-parallel to the electric field. It is important to note that the here presented lin- earized mean-field kinetic equation with a relaxation- time approximation is not capable to describe correctly the collisional correlationswhich has to be performed be- yond the relaxation-time approximation. Especially the above-mentioned vertex corrections seem to lead to can- cellations of the result. Here we restrict ourselves to a Drude level of conductivity. VI. SUMMARY The coupled kinetic equation for particle and spin po- larization is derived including meanfields, spin-orbit cou- pling and arbitrary magnetic and electric field strength.This is achieved by using a gauge-invariant formulation and keeping the quantum spin structure as commuta- tors/anticommutators through gradient approximations. Bothequationshavetheexpectedstructureofadriftcon- trolled by a mean quasiparticle velocity renormalized by meanfields and the effective Lorentz force which contains besidesthemagneticfieldalsothe vectorpartsoftheself- energy. These equations arecoupled exactly by the latter one. Additionally the polarization distribution exercises a precession around a direction given by this vector self- energy. This latter one together with the magnetic field establishesan effective medium-dependent magnetic field and can be considered as a many-body extension of the Zeeman field. The stationary solution shows a unique splitting into two bands controlled by the vector selfen- ergy. Here the selfconsistent precession direction appears. For linear spin-orbit coupling and zero temperature an exact cancellation of spin-orbit coupling is found for the polarization. We have calculated charge and spin currents and show that besides the regular currents an anomalous part ap- pearsduetospin-orbitcoupling. Theregularandanoma- lous currents compensate exactly in the stationary state. For transport with respect to an external electric field we calculate the spin-Hall coefficient as well as the anoma- lous Hall conductivity. Both consist of an asymmetric part which in the case of the anomalousHall effect agrees with the standard one from the Kubo formula or Berry phases and an additional symmetric part interpreted as the inverse Hall effect with an expression not presented so far. The corresponding spin-Hall effects are described as well and are presented here in their dynamical form dependent on the magnetic field and the mean magneti- zation of the system. A sign change of the Hall conductivity is reported for higher frequencies than a critical one given in terms of the relaxation time. The inverse Hall effect does not show such sign change in accordance with causality. The spin-Hall and inverse spin-Hall effects both show such sign change where the latter one is an expression of the non-conserved spin current. In the second part of this paper series we will present the linear response results including the meanfield and magnetic field which allows us to discuss the dynami- cal response functions. From this we will see the collec- tivemodesandhowthe spin-orbitcouplinginfluences the screening properties. Acknowledgments Discussions with Janik Kailasvuori, Alvaro Ferraz and OzgurBozataregratefullymentioned. Financialsupport by the Brazilian Ministry of Science and Technology is acknowledged.21 1J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003). 2S. S. Natu and E. J. Mueller, Phys. Rev. A 79, 051601 (2009). 3V. E. Demidov, H. Ulrichs, S. V. Gurevich, S. O. Demokri- tov, V. S. Tiberkevich, A. N. Slavin, A. Zholud, and S. Urazhdin, Nature comm. 5, 3179 (2014). 4E. B. Sonin, Advances in Physics 59, 181 (2010). 5R. Winkler, Spin-Orbit Coupling Effects in Two- Dimensional Electron and Hole Systems (Springer - Ver- lag, Berlin Heidelberg, 2003). 6J. Schliemann, Int. J. of Mod. Phys. B 20, 1015 (2006). 7M. I. Dyakonov, Int. J. of Mod. Phys. B 23, 2556 (2009). 8M. I. Dyakonov and V. I. Perel, JETP Lett. 13, 467 (1971). 9M. I. Dyakonov and V. I. Perel, Phys.Lett. A 35, 459 (1971). 10Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004). 11J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005). 12I. Adagideli, J. H. Bardarson, and P. Jacquod, J. Phys.: Cond. Mat. 21, 155503 (2009). 13R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954). 14N. A. Sinitsyn, A. H. MacDonald, T. Jungwirth, V. K. Dugaev, and J. Sinova, Phys. Rev. B 75, 045315 (2007). 15A. A. Kovalev, Y. Tserkovnyak, K. Vyborny, and J. Sinova, Phys. Rev. B 79, 195129 (2009). 16S. Y. Liu and X. L. Lei, Phys. Rev. B 72, 195329 (2005). 17N. Nagaosa, J. Sinova, S. Onoda, A. H. Macdonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010). 18P. Schwab, R. Raimondi, and C. Gorini, EPL 90(2010). 19M. Bibes, V. Laukhin, S. Valencia, B. Martinez, J. Fontcuberta, O. Y. Gorbenko, A. R. Kaul, and J. L. Martinez, J. Phys.: Cond. Mat. 17, 2733 (2005). 20C. Gould, K. Pappert, G. Schmidt, and L. W. Molenkamp, Adv. Mater. 19, 323 (2007). 21K. Hattori, J. Phys. Soc. Japan 79, 104709 (2010). 22S. Souma and B. K. Nikolic, Phys. Rev. Lett. 94, 106602 (2005). 23M. H. Liu, G. Bihlmayer, S. Blugel, and C. R. Chang, Phys. Rev. B 76, 121301 (2007). 24V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gos- sard, and D. D. Awschalom, Nature Physics 1, 31 (2005). 25S. O. Valenzuela and M. Tinkham, Nature (London) 442, 176 (2006). 26T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, and H. I. a, Nature Matter 7, 125 (2008). 27C. Br¨ une, A. Roth, E. G. Novik, M. Koenig, H. Buh- mann, E. M. Hankiewicz, W. Hanke, J. Sinova, and L. W. Molenkamp, Nature Physics 6, 448 (2010). 28T. Fujita, M. B. A. Jalil, and S. G. Tan, J. Phys. Soc. Japan78, 104714 (2009). 29E. M. Hankiewicz, L. W. Molenkamp, T. Jungwirth, and J. Sinova, Phys. Rev. B 70, 241301 (2004). 30M. Gradhand, D. V. Fedorov, P. Zahn, and I. Mertig, Phys. Rev. Lett. 104, 186403 (2010). 31B. K. Nikolic, L. P. Zarbo, and S. Souma, Phys. Rev. B 73, 075303 (2006). 32P. Gosselin, A. Berard, H. Mohrbach, and S. Ghosh, Eur.Phys. J. C 59, 883 (2009). 33M. Gradhand, D. V. Fedorov, F. Pientka, P. Zahn, I. Mer- tig, and B. L. Gy¨ orffy, Phys. Rev. B 84, 075113 (2011). 34W. K. Tse, J. Fabian, I. Zutic, and S. DasSarma, Phys. Rev. B72, 241303 (2005). 35S. Lowitzer, D. Kodderitzsch, and H. Ebert, Phys. Rev. B82, 140402 (2010). 36A. Cr´ epieux and P. Bruno, Phys. Rev. B 64, 014416 (2001). 37M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Rep. 493, 61 (2010). 38S. Y. Liu, X. L. Lei, and N. J. M. Horing, Phys. Rev. B 73, 035323 (2006). 39A. Khaetskii, Phys. Rev. Lett. 96, 056602 (2006). 40A. G. Mal’shukov and K. A. Chao, Phys. Rev. B 71, 121308 (2005). 41J. I. Inoue, T. Kato, Y. Ishikawa, H. Itoh, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. Let. 97, 046604 (2006). 42P. Streda and T. Jonckheere, Phys. Rev. B 82, 113303 (2010). 43P. Wolfle and K. A. Muttalib, Ann. Phys. (Leipzig) 15, 508 (2006). 44C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 45Z. Qiao, S. A. Yang, W. Feng, W.-K. Tse, J. Ding, Y. Yao, J. Wang, and Q. Niu, Phys. Rev. B 82, 161414 (2010). 46W.-K. Tse, Z. Qiao, Y. Yao, A. H. MacDonald, and Q. Niu, Phys. Rev. B 83, 155447 (2011). 47G. Tkachov, Phys. Rev. B 79, 045429 (2009). 48Y. J. Lin, K. Jimenez-Garica, and I. B. Spielman, Nature 471, 83 (2011). 49C. Lhuilier and F. Lalo¨ e, J. Phys. (Paris) 43, 197 (1982). 50C. Lhuilier and F. Lalo¨ e, J. Phys. (Paris) 43, 225 (1982). 51A. E. Meyerovich, Phys. Rev. B 39, 9318 (1989). 52A. E. Meyerovich, Physica B 169, 183 (1991). 53J. W. Jeon and W. J. Mullin, J. Phys. (Paris) 49, 1691 (1988). 54J. W. Jeon and W. J. Mullin, Phys. Rev. Lett. 62, 2691 (1989). 55P. J. Nacher, G. Tastevin, and F. Laloe, J. Phys. (Paris) 50, 1907 (1989). 56P. J. Nacher, G. Tastevin, and F. Lalo¨ e, J. Phys. (Paris) I1, 181 (1991). 57A. E. Meyerovich and K. A. Musaelian, J. Low Temp. Phys.89, 781 (1992). 58E. P. Bashkin, Phys. Rev. B 44, 12440 (1991). 59R. Hakim, L. Mornas, P. Peter, and H. D. Sivak, Phys. Rev. D46, 4603 (1992). 60E. P. Bashkin, C. daProvidencia, and J. daProvidencia, Phys. Rev. B 62, 3968 (2000). 61V. M. Edelstein, Solid State Comm. 73, 233 (1990). 62R. Raimondi, C. Gorini, P. Schwab, and M. Dzierzawa, Phys. Rev. B 74, 035340 (2006). 63R. Raimondi and P. Schwab, Physica E 42, 952 (2010). 64R. V. Shchelushkin and A. Brataas, Phys. Rev. B 71, 045123 (2005). 65I. B. Bernstein, Phys. Rev. 109, 10 (1958). 66M. S. Kushwaha and B. Djafari-Rouhani, Ann. Phys. 265, 1 (1998). 67K. Shizuya, Phys. Rev. B 75, 245417 (2007). 68P. Kleinert and V. V. Bryksin, Phys. Rev. B 79, 04531722 (2009). 69J. Zamanian, M. Marklund, and G. Brodin, New J. Phys. 12, 043019 (2010). 70J. Lundin and G. Brodin, Phys. Rev. E 82, 056407 (2010). 71M. Walter, G. Zwicknagel, and C. Toepffer, Eur. Phys. J. D35, 527 (2005). 72G. Strinati, C. Castellani, and C. DiCastro, Phys. Rev. B 40, 12237 (1989). 73L. A. Openov, Phys. Rev. B 58, 9468 (1998). 74J. Chen and S. Hershfield, Phys. Rev. B 57, 1097 (1998). 75A. E. Meyerovich, S. Stepaniants, and F. Lalo¨ e, Phys. Rev. B52, 6808 (1995). 76J. Kailasvuori, J. Stat. Mech. p. P08004 (2009). 77J. Kailasvuori and M. C. Luffe, J. Stat. Mech. p. P06024 (2010). 78K. Morawetz, Phys. Rev. B 67, 115125 (2003). 79O. Bleibaum, Phys. Rev. B 73, 035322 (2006). 80E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93, 226602 (2004). 81N. Mermin, Phys. Rev. B 1, 2362 (1970). 82A. K. Das, J. Phys. F 5, 2035 (1975). 83H. Heiselberg, C. J. Pethick, and D. G. Ravenhall, Ann. Phys.223, 37 (1993). 84M. D. Barriga-Carrasco, Laser and Particle Beams 28, 307 (2010). 85K. Morawetz, Phys. Rev. E 88, 022148 (2013). 86G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B 64, 214204 (2001). 87D. C. Langreth and J. W. Wilkins, Phys. Rev. B 6, 3189 (1972). 88S. S. Mizrahi, Physica A 150, 541 (1988). 89H. A. Engel, B. I. Halperin, and E. I. Rashba, Phys. Rev. Lett.95, 166605 (2005). 90M. Matuszewski, T. C. H. Liew, Y. G. Rubo, and A. V. Kavokin, Phys. Rev. B 86, 115321 (2012). 91R. S. Chang, C. S. Chu, and A. G. Mal’shukov, Phys. Rev. B79, 195314 (2009). 92T.-W. Chen and G.-Y. Guo, Phys. Rev. B 79, 125301 (2009). 93J. Cserti and G. D´ avid, Phys. Rev. B 74, 172305 (2006). 94M. Bender, K. Bennaceur, T. Duguet, P. H. Heenen, T. Lesinski, and J. Meyer, Phys. Rev. C 80, 064302 (2009). 95Y. Xia, J. Xu, B.-A. Li, and W.-Q. Shen, Phys. Rev. C 89, 064606 (2014). 96M. Zalewski, J. Dobaczewski, W. Satu/suppress la, and T. R. Werner, Phys. Rev. C 77, 024316 (2008). 97J. M. Pearson and M. Farine, Phys. Rev. C 50, 185 (1994). 98T. Lesinski, M. Bender, K. Bennaceur, T. Duguet, and J. Meyer, Phys. Rev. C 76, 014312 (2007). 99J. Xu and B.-A. Li, Physics Letters B 724, 346 (2013), ISSN 0370-2693. 100A. Fert and I. A. Campbell, Phys. Rev. Lett. 21, 1190 (1968). 101R. Bala, R. Moudgil, S. Srivastava, and K. Pathak, Euro- pean Phys. J. B 87, 5 (2014).102I. A. Campbell, A. Fert, and R. Pomeroy, Philosophical Magazine 15, 977 (1967). 103R. M. Rowan-Robinson, A. T. Hindmarch, and D. Atkin- son, Phys. Rev. B 90, 104401 (2014). 104R. Coehoorn, Novel magnetoelectronic materials and de- vices, lecture Notes TU/e 1999-2000, 2001-2002, 2003. 105P. B. Wiegmann, J. Phys. C: Solid State Phys. 14, 1463 (1981). 106L. P. Kadanoff and G. Baym, Quantum Statistical Me- chanics (Benjamin, New York, 1962). 107J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). 108P. Danielewicz, Ann. Phys. (NY) 197, 154 (1990). 109R. Bertoncini and A. P. Jauho, Phys.Rev.B 44, 3655 (1991). 110K. Morawetz, Phys. Rev. E 62, 6135 (2000), errata 69, 029902. 111S. Q. Shen, M. Ma, X. C. Xie, and F. C. Zhang, Phys. Rev. Lett. 92, 256603 (2004). 112P. Schwab, M. Dzierzawa, C. Gorini, and R. Raimondi, Phys. Rev. B 74, 155316 (2006). 113A. J. Leggett and M. J. Rice, Phys. Rev. Lett. 20, 586 (1968), erratum Phys. Rev. Lett. 21 506 (1968). 114V. P. Mineev, Phys. Rev. B 72, 144418 (2005). 115D. I. Golosov and A. E. Ruckenstein, Phys. Rev. Lett. 74, 1613 (1995). 116P. J. Nacher, G. Tastevin, and F. Lalo¨ e, Ann. Phys. (Leipzig) 48, 149 (1991). 117J. E. Williams, T. Nikuni, and C. W. Clark, Phys. Rev. Lett.88, 230405 (2002). 118K. Morawetz, Phys. Rev. B 66, 075125 (2002), errata: Phys. Rev. B 88, 039905(E). 119V. P. Mineev, Phys. Rev. B 69, 144429 (2004). 120A. J. Leggett, J. Phys. C 3, 448 (1970). 121B. Zhou, L. Ren, and S. Q. Shen, Phys. Rev. B 73, 165303 (2006). 122V. V. Bryksin and P. Kleinert, Phys. Rev. B 73, 165313 (2006). 123J.-I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B67, 033104 (2003). 124J. Schliemann and D. Loss, Phys. Rev. B 69, 165315 (2004). 125J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung- wirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004). 126J.-I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B70, 041303 (2004). 127R. Raimondi and P. Schwab, Phys. Rev. B 71, 033311 (2005). 128O. Chalaev and D. Loss, Phys. Rev. B 71, 245318 (2005). 129O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005). 130T. Kampfrath, M. Battiato, P. Maldonado, G. Eil- ers, J. N¨ otzold, S. M¨ ahrlein, V. Zbarsky, F. Freimuth, Y. Mokrousov, S. Bl¨ ugel, et al., Nature Nanotechnology 8, 256 (2013).

1407.3446v1.Orbital_angular_momentum_driven_intrinsic_spin_Hall_effect.pdf

arXiv:1407.3446v1 [cond-mat.mes-hall] 13 Jul 2014Orbital angular momentum driven intrinsic spin Hall effect Wonsig Jung,1Dongwook Go,2Hyun-Woo Lee,2,∗and Changyoung Kim1,† 1Institute of Physics and Applied Physics, Yonsei Universit y, Seoul 120-749, Korea 2Department of Physics, Pohang University of Science and Tec hnology, Pohang 790-784, Korea (Dated: June 19, 2021) We propose a mechanism of intrinsic spin Hall effect (SHE). In this mechanism, local orbital angular momentum (OAM) induces electron position shift and couples with the bias electric field to generate orbital Hall effect (OHE). SHE then emerges as a co ncomitant effect of OHE through the atomic spin-orbit coupling. Spin Hall conductivity due to this mechanism is estimated to be comparable to experimental values for heavy metals. This me chanism predicts the sign change of the spin Hall conductivity as the spin-orbit polarization chan ges its sign, and also correlation between the spin Hall conductivity and the splitting of the Rashba-t ype spin splitting at surfaces. PACS numbers: 74.25.Jb,74.70.Xa,78.70.Dm Spin Hall effect (SHE) [1] is a phenomenon in which electrons with opposite spins are deflected in opposite side ways. Its first experimental confirmation [2] was achieved for n-doped GaAs and very small spin Hall con- ductivity σSH∼1 Ω−1m−1was obtained, which was at- tributed [3] to the extrinsic mechanisms [4] of SHE such as skew scattering and side jump. For some heavy met- als, on the other hand, much larger σSHwas reported [5]. For Pt, for instance, reported values range from 2 .4×104 Ω−1m−1[6] to 5.1×105Ω−1m−1[7]. Such large σSH raises hope for device applications of SHE. The current- induced magnetization switching observed in Ta/CoFeB magnetic bilayer [8] is attributed to the large SHE in Ta, which injects strong spin Hall current into CoFeB to switch its magnetization direction. LargeσSHis often attributed to intrinsic mecha- nisms [9–16] of SHE, which do not resort to impurity scattering. Their exact nature remains unclear however. In one mechanism [12], a small spin-orbit energy gap near the Fermi energy resonantly enhances the momen- tumspaceBerryphaseeffecttoproduceastrongeffective magnetic field in momentum space and σSH= 104∼105 Ω−1m−1is predicted for Pt. In another mechanism [13– 16], the orbital angular momentum (OAM) of atomic orbitals generates the Aharonov-Bohm phase and pro- duces a spin-dependent effective magnetic field in real space. For various heavy metals with strong atomic spin-orbit (SO) coupling, resulting σSHis estimated to 104∼105Ω−1m−1and predicted to exhibit a systematic sign changeamongmaterialswith different spin-orbit po- larization, in qualitative agreement with experiments [5]. We report another intrinsic mechanism of SHE based on a special role of OAM with regard to electron posi- tion, whichwasnotrecognizedinpreviousstudies[13–17] on OAM effect. For illustration, we use for now a two- dimensional (2D) square lattice in the plane z= 0. Later we switch back to 3D. When pz±ipxorbitals ( Ly=±¯h) at different lattice sites are superposed to form a Bloch state with crystal momentum /vectorkalong +xdirection, the resulting electron density is notcentered around thez= 0 plane but instead shifted out-of-plane along±z direction due to the interference between atomic orbitals at neighboring sites (see Fig. 2 and related discussion in Ref. [18]). When /vectorkis small, this shift δ/vector ris given by δ/vector r=αK e/vectork×/vectorL, (1) for general directions of /vectorkand/vectorL, where−eis the elec- tron charge and αKis a proportionality constant, which depends on the relative size of atomic orbitals with re- spect to inter-atomic distance. Here /vectorLdenotes OAM of atomicorbitals instead of /vector rׯh/vectork[19]. It thus commutes with/vectorkand also with the position operator /vector r, which is the canonical pair of /vectorkand measures the lattice position of each atomic orbital. Nonzero δ/vector rimplies that /vector rdoes not properly represent the true position of an electron. At surfaces with broken inversion symmetry, this correction couples with an internal electric field to produce large Rashba-type spin splitting [18, 20]. Pedagogical discussion.— To illustrate effects of δ/vector rfor nonmagnetic systems with inversion symmetry, we use the free electron-like unperturbed band Hamiltonian H0, H0=¯h2/vectork2 2m+HLS, (2) where the atomic SO coupling HLS, HLS=αSO/vectorL·/vectorS, (3) is large in heavy metals. We regard the total angular momentum Jas a good quantum number and illustrate orbital Hall effect (OHE) and SHE for J= 1/2 states of a 2D electron system. Note that H0is two-fold de- generate for all /vectorkand provides a general description of nonmagnetic systems with inversion symmetry for small /vectork. We remark that for this H0, previous theories [12–16] of intrinsic SHE do not work. Ourtheorydeviatesfromprevioustheorieswhenacon- stant external electric field /vectorEis applied. The coupling to /vectorEis commonly given by H′ 1=e/vectorE·/vector r. (4)2 ky k0(d) ω(a) ω kxky(b) ω kxkyOAM SAM O kxky (c) kxkk kykk OAM SSAM O FIG. 1: (Color online) Electron dispersion for J= 1/2 in the presence of the bias field /vectorE. Band structure based on (a) H0 and (b)H0+H′ 2together with the occupation change due to H′ 1./vectorEis applied in the −x-direction. Average spin direction of the split Jz=±1/2 bands is anti-parallel to the average OAM direction. (c) Fermi surfaces of the split bands. Red (blue) area represents occupied states with only down (up) spins. (d) The band dispersion along the dash-dot line in (c) . k0is the shift of each band along the kydirection. However δ/vector rimplies that the correct coupling [18] should beH′ 1+H′ 2, where H′ 2=e/vectorE·δ/vector r=αK/vectorE·(/vectork×/vectorL). (5) Previous analyses [13–16] of OAM based intrinsic SHE didnottakeintoaccount H′ 2. ThusthetotalHamiltonian becomes Htot=H0+H′ 1+H′ 2. (6) Its band, spin angular momentum (SAM), and OAM structures are plotted for /vectorE=−E0/vector xwithH′ 2neglected [Fig.1(a)]andwith H′ 2considered[Fig.1(b)]. Inaddition to the overall band structure shift in the kx-direction as shown in Fig. 1(a) (to be more exact, it is actually a shift in the occupation), the originally degenerate Jz=±1/2 bands get split due to H′ 2with the average OAM polar- ized along the + z- or−z-directions as shown in Fig. 1(b) (exaggerated for a better view). The split Fermi surfaces are shown in Fig. 1(c), where the Fermi surfaces with opposite OAM are shifted along opposite kydirections. Consequently, there are k-space regions (shaded areas) where electrons have net OAM; more electrons with up- OAM in the + kyregion (shaded red) and more electrons xyz L Wd OAM SAM FIG. 2: (Color online) Schematic for OAM driven intrinsic SHE. Electrons flow in the x-direction by /vectorEand are deflected in side ways due to H′ 2. Note that the deflection direction depends on the direction of OAM, amounting to OHE. For J= 1/2 band, HLSsets SAM anti-parallel to OAM. Thus SHE arises a concomitant effect of OHE. with down-OAM (shaded blue) in the −kyregion. This naturally leads to OHE. This mechanism of OHE due to δ/vector rdiffers from other mechanisms [13–17] of OHE. For strong HLS, OHE implies SHE since OAM and SAM are correlated; for J= 1/2 withL= 1, they are anti-parallel. Thus the orbital Hall current implies the spin Hallcurrentofopposite sign. Figure2illustratesthe OAM driven intrinsic SHE for the J= 1/2 case. This mechanism of SHE can be generalized to other situations in a straightforward way. For instance, if we apply Htot to theJ= 3/2 case with L= 1 [21], one again finds both OHE and SHE, the only qualitative difference being that the orbital and spin Hall currents now have the same sign since /vectorL·/vectorS >0. This provides an alternative [13–16] explanation for opposite signs of σSHfor materials with opposite signs of the SO polarization /vectorL·/vectorS. Conventional spin current.— The above discussion is incomplete since it demonstrates only the Fermi surface contribution to SHE and neglects a Fermi sea contribu- tion. From now on, we consider a 3D system described by Eq. (6), and evaluate systematically the conventional spin current density operator ˆjS α,βdefined by ˆjS α,β=1 V−e ¯h/2{Sα,vβ} 2, (7) where{···}is the anti-commutator, Vis the volume of the system, and the factor −e/(¯h/2) is introduced to makeˆjS α,βhave the same dimension as the charge cur- rent density. Here vβis theβ(=x,y,z) component of the velocity operator /vector v, /vector v=[/vector r,Htot] i¯h=¯h/vectork m+αK ¯h(/vectorL×/vectorE) =/vector v(0)+/vector v(1).(8) Note that /vector vcontains two contributions. When the anomalous velocity /vector v(1)= (αK/¯h)/vectorL×/vectorEis neglected and3 (a) kyE kyE Occupation change (b) Anomalous velocity (c) E J = 3/2 J = 1/2State changeE mixing FIG. 3: (Color online) Schematic illustration of the three terms tothe intrinsic SHE, (a) occupation change, (b)anoma - lous velocity, and (c) state change. Figures on the left repr e- sent the situation with H0while on the right with H0+H′ 2. the resulting ˆjS α,βis averaged over the shaded momen- tum space region in Fig. 1(c) [to be precise, 3D coun- terpart of Fig. 1(c)], one obtains what we call the oc- cupation change contribution ( jS α,β)occoming from the Fermi surface, as illustrated in the pedagogical discus- sion. The magnitude of ( jS α,β)occan be estimated eas- ily. The density of electrons that contribute to the net spin current density is proportional to 4 πk2 Fk0, where kFis the Fermi wavevector for the unperturbed Fermi surface and k0∼(m/¯h2)αK/vectorE×/vectorLis the Fermi surface shift caused by H′ 2[see Fig. 1(d)]. Each of such elec- trons contributes ±efor [−e/(¯h/2)]Sα, and±¯hkF/mfor vβ. Combined with a symmetry consideration, which re- quires (jS α,β) to be proportional to ǫαβγEγ, whereǫαβγis the Levi-Civita symbol, one finds (jS α,β)oc= (ηJ)ocǫαβγEγeαK4πk3 F/3 (2π)3,(9) where (ηJ)ocis a dimensionless constant. From the exact evaluation [22] of ( jS α,β)oc, we find ( ηJ=1/2)oc= 4/9 and (ηJ=3/2)oc=−20/9 [21]. Note that the sign of ηJis opposite for the two J’s as expected. The anomalous velocity v(1)generates additional con- tribution, which comes from the Fermi sea. When v(0) is neglected and only v(1)is retained, the average of the resulting ˆjS α,βover the unperturbed Fermi sea of H0re- sults in what we call the anomalous velocity contribution(jS α,β)av. To estimate its magnitude, one first notes that ǫαβγSαv(1) β= (αK/¯h)[/vectorS×(/vectorL×/vectorE)]γ= (αK/¯h)[(/vectorS·/vectorE)/vectorL− (/vectorS·/vectorL)/vectorE]γ. While the first term may fluctuate in sign, the second term ( ∝/vectorS·/vectorL) has a definite sign over the Fermi sea. Thus ( jS α,β)avmay be estimated by multiplying the secondterm with the electrondensity ∼(4πk3 F/3)/(2π)3, which results in (jS α,β)av= (ηJ)avǫαβγEγeαK4πk3 F/3 (2π)3,(10) where (ηJ)avis a dimensionless constant. From the exact evaluation [22] of ( jS α,β)av, we find ( ηJ=1/2)av=−4/3 and (ηJ=3/2)av= +4/3 [21]. The sign of ( ηJ)avis again opposite for the two Jvalues due to the sign difference of/vectorS·/vectorL. Figures 3(a) and (b) illustrate schematically (jS α,β)ocand (jS α,β)av. In addition, there exists a third contribution which is illustrated in Fig. 3(c). When /vectorEis applied, /vectorJ=/vectorL+/vectorSis not a good quantum number any more and H′ 2induces the inter-band mixing between the J= 1/2 andJ= 3/2 bands. This contribution ( jS α,β)sc, which we call the state changecontribution,isinverselyproportionaltotheband separation between the J= 1/2 andJ= 3/2 bands, and becomes smaller as HLSbecomes larger. From the exact evaluation [22] of ( jS α,β)sc, we find that ( jS α,β)scis smaller than (jS α,β)ocand (jS α,β)avby the factor (¯ h2k2 F/2m)/∆E, where ∆E= 3¯h2αSO/2 is the energy separation between theJ= 1/2 andJ= 3/2 bands. Since we are inter- ested in the large HLSlimit, we ignore ( jS α,β)scin the subsequent discussion. Proper spin current.— Next we examine whether the OAM driven spin Hall current generates spin accumula- tion at side surfacesofa system, which is what is actually measuredinSHEdetectionschemessuchasKerrrotation spectroscopy [2, 23–27] and photoluminescence [28, 29]. SinceHLSbreaks the spin conservation, nonzero conven- tional spin current does not guarantee the spin accumu- lation [30]. For transparent connection with the spin ac- cumulation, we evaluate the proper spin current density operator [31] ˆjS,prop α,β=1 V−e ¯h/2d(Sαrβ) dt, (11) which captures the combined effect of the conventional spin current and the spin conservation violation. We evaluate [22] the spin current for Htotby using ˆjS,prop α,β instead of ˆjS α,β[Eq. (7)], and find identical results, con- firming the spin accumulation by the OAM driven SHE. To be more rigorous, however, both ˆjS,prop α,βandˆjS α,β fail to capture the full effect of δ/vector r, since both operators are defined in terms of /vector r, which does not represent the true position of electrons. To remedy this problem, /vector rin the definitions should be replaced by /vectorR≡/vector r+δ/vector r. Af- ter this remedy to ˆjS,prop α,β, we find [22] that the anoma- lous contribution ( jS α,β)avbecomes doubled. Thus in the4 strongHLSlimit, the total spin Hall conductivity σSH [(jS α,β)total=ǫαβγσSHEγ] is given by σSH= (ηJ)totaleαK4πk3 F/3 (2π)3(12) for small /vectork, where the dimensionless constant ( ηJ)total= (ηJ)oc+2(ηJ)avis 4/9−8/3 =−20/9 forJ= 1/2 and −20/9 + 8/3 = 4/9 forJ= 3/2. Note that σSHhas opposite signs for J= 1/2 andJ= 3/2. Discussion.— To understand better the mechanism of the OAM driven SHE, it is useful to examine the equa- tion of motion, d/vectorR/dt=/vector v(0)+2/vector v(1)−(αK/e)/vectork×d/vectorS/dt+ [δ/vector r,e/vectorE·δ/vector r]/i¯h. The factor 2 in the second term explains why (jS α,β)avis doubled after the remedy to ˆjS,prop α,β. The third term vanishes in the steady state and does not con- tribute to σSH[22]. The last term ( ∝k2) is small in the small /vectorklimit but is important conceptually. Fur- ther insights can be gained by regarding δ/vector ras a momen- tum space vector potential /vectorA≡ −δ/vector r. Then/vectorR=/vector r−/vectorA amounts to the “gauge-invariant”position operator. The equations of motion become d/vectorR dt=¯h/vectork m+d/vectork dt×/vectorB,d/vectork dt=−e/vectorE ¯h, (13) wherethemomentum spaceeffectivemagneticfield Bα= (1/2)ǫαβγFβγwith Fβγ=∂kβAγ−∂kγAβ+i[Aβ,Aγ].(14) Note that Eq. (13) has the same form as the wavepacket equations of motion [32] in the presence of the momen- tum space Berry phase. There is however an important difference; the momentum space Berry connection /vectorAis now non-Abelian ([ Aβ,Aγ] = [δrβ,δrγ]∝ne}ationslash= 0). For the non-Abelian case, the commutator in Eq. (14) is crucial to keep the field strength tensor Fβγ“gauge-invariant”. This indicates that δ/vector rinduces the momentum space non- Abelian Berry phase, which is responsible for the Fermi sea contribution 2( ηJ)avtoσSH. The non-Abelian /vectorA also implies the noncommutative space, [ Rα,Rβ]∝ne}ationslash= 0. Such noncommutative geometry arises generically when the true position operatoris projected onto a sub-Hilbert space[33]. A well known exampleis the quantum Hall ef- fect, where the noncommutativity emerges after the pro- jection onto the lowest Landau level [34]. For the present case, the noncommutativity arises since /vectorRamounts to the projection of the true position operator onto the sub- Hilbert space with fixed /vectorL2(L= 1). Difference from other mechanisms of intrinsic SHE is now evident. Unlike previous works on the OAM based SHE [13–16], what OAM generates is the momentum space Berry phase instead of the real space Aharonov- Bohm phase. Unlike previous works [12] based on the momentum space Berry phase, its origin is δ/vector rinstead of small SO gap. Thus this mechanism works even forJ= 1/2, for which SO gap is forbidden in nonmag- netic systems with inversion symmetry. In this sense, this mechanism is quite generic; it applies to all non- s- character orbitals, with the only serious constraint being largeHLS. When HLSis small, bands with opposite signs of/vectorL·/vectorSoverlap and their contributions to σSHtend to cancel each other. Finally we estimate the magnitude of σSH∼eαKn, wherenis the electron density. To estimate αK, we uti- lize the connection between αKand the Rashba-type SO coupling constant αRnear a surface where the structural inversion symmetry is broken. Some of us have demon- strated [18, 20] that the maximum αRin the large HLS limit is roughly given by αK|/vectorEint|¯h, where/vectorEintdenotes the internal electric field near surfaces produced by the inversion symmetry breaking and is of order (work func- tion)/(atomicspacing) ∼1 V/A. For αR∼10−11−10−10 eV·m [35–37], one obtains αK∼10−6−10−5m2V−1s−1. Then for typical metallic electron density n∼(3 A)−3, one obtains σSH∼104−105Ω−1m−1, which is compa- rable to experimental values for heavy metals [5]. We note however that this estimation is crude since Eq. (12) is derived in the small /vectorklimit whereas /vectorkis not small in metallic systems. Moreover it ignores complicated band structures of real materials. In conclusion, we presented a generic mechanism of intrinsic SHE based on OAM, which is applicable to all non-s-characterorbitals in nonmagnetic systems with in- version symmetry. The position shift δ/vector rdue to OAM gives rise to the non-Abelian Berry curvature in the mo- mentumspace, whichproducesbothOHEandSHE.This mechanism implies the sign change of σSHas the SO po- larization /vectorS·/vectorLchanges its sign. The resulting σSHis estimated to 104−105Ω−1m−1whenHLSis large. This OAM based theory also predicts the correlation between σSHand the strength of the Rashba-type spin splitting at surfaces. We acknowledge fruitful discussion with G. S. Jeon, J. H. Han, K. J. Lee and B. C. Min. This research was supported by the Converging Research Center Pro- gram through the Ministry of Science, ICT and Fu- ture Planning, Korea(2013K000312). HWL acknowl- edges the financial support of the NRF (2011-0030784 and 2013R1A2A2A05006237). ∗Electronic address: hwl@postech.ac.kr †Electronic address: changyoung@yonsei.ac.kr [1] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). [2] Y. K. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Science 306, 1910 (2004). [3] H.-A. Engel, B. I. Halperin, and E. I. Rashba, Phys. Rev. Lett.95, 166605 (2005). [4] S. Zhang, Phys. Rev. Lett. 85, 393 (2000). [5] M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka,5 H. Kontani, T. Kimura and Y. Otani, Phys. Rev. B 83, 174405 (2011). [6] T. Kimura, Y. Otani, T. Sato, S. Takahashi and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). [7] K. Ando, S. Takanashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). [8] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). [9] S. Murakami, N. Nagaosa and S.-C. Zhang, Science 301, 1348 (2003). [10] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung- wirth, andA.H.MacDonald, Phys.Rev.Lett. 92, 126603 (2004). [11] G. Y. Guo, Y. Yao, and Q. Niu, Phys. Rev. Lett. 94, 226601 (2005). [12] G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa, Phys. Rev. Lett. 100, 096401 (2008). [13] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Phys. Rev. Lett. 100, 096601 (2008). [14] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hi- rashima, K. Yamada, and J. Inoue, Phys. Rev. B 77, 165117 (2008). [15] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Phys. Rev. Lett. 102, 016601 (2009). [16] T. Tanaka and H. Kontani, Phys. Rev. B 81, 224401 (2010). [17] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Phys. Rev. Lett. 95, 066601 (2005). [18] S. R. Park, C. H. Kim, J. J. Yu, J. H. Han and C. Kim, Phys. Rev. Lett. 107, 156803 (2011) [19] S. Zhang and Z. Yang, Phys. Rev. Lett. 94, 066602 (2005). [20] B. Kim et al., Phys. Rev. B 85, 195402 (2012). [21] For J= 3/2, the four-fold degeneracy of H0is generally lifted into two doublets. When this happens, the mo-mentum space Berry phase mechanism in Refs. [9, 12] is turned on. Here we ignore the lifting to emphasize that the OAM based mechanism works even without the de- generacy lifting. [22] See Supplementary Material for detailed calculation. [23] V. Sih et al., Nature Phys. 1, 31 (2005). [24] V. Sih et al., Phys. Rev. Lett. 97, 096605 (2006). [25] N. P. Stern et al., Phys. Rev. Lett. 97, 126603 (2006). [26] N. P. Stern, D. W. Steuerman, S. Mack, A. C. Gossard, and D. D. Awschalom, Nature Phys. 4, 843 (2008). [27] S. Matsuzaka, Y. Ohno and H. Ohno, Phys. Rev. B 80, 241305 (2009). [28] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth , Phys. Rev. Lett. 94, 047204 (2005). [29] H. J. Chang et al., Phys. Rev. Lett. 98, 136403 (2007). [30] In certain limiting cases, finite spin current may not pr o- duce spin accumulation at all. For an example, see S. Mu- rakami, N. Nagaosa, and S.-C. Zhang, Phys. Rev. Lett. 93, 156804 (2004). [31] J. Shi, P. Zhang, D. Xiao, and Q. Niu, Phys. Rev. Lett. 97, 076604 (2006). [32] G. Sundaramand Q.Niu, Phys.Rev. B 59, 14915 (1999). [33] R. Shindou and K.-I. Imura, Nucl. Phys. B 720, 399 (2005). [34] A. P. Polychronakos, J. High Energy Phys. 04, 011 (2001). [35] A. Varykhalov, J. Sanchez-Barriga, A.M. Shikin, W. Gu- dat, W. Eberhardt, and O. Rader, Phys. Rev. Lett. 101, 256601 (2008) [36] A. Bendounan, K. Ait-Mansour, J. Braun, J. Minar, S. Bornemann, R. Fasel, O. Groning, F. Sirotti, and H. Ebert, Phys. Rev. B 83, 195427 (2011). [37] J. Park, S. W. Jung, M.-C. Jung, H. Yamane, N. Kosugi, and H. W. Yeom, Phys. Rev. Lett. 110, 036801 (2013).arXiv:1407.3446v1 [cond-mat.mes-hall] 13 Jul 2014Supplementary Material: Orbital angular momentum driven i ntrinsic spin Hall effect W. S. Jung,1Dongwook Go,2Hyun-Woo Lee,2,∗and C. Kim1,† 1Institute of Physics and Applied Physics, Yonsei Universit y, Seoul 120-749, Korea 2Department of Physics, Pohang University of Science and Tec hnology, Pohang, Kyungbuk 790-784, Korea (Dated: June 19, 2021) PACS numbers: In Secs. I, II, III of the supplementary material, we present the calculation of the spin current density in 3D. Eventually we calculate in Sec. III the proper spin cur- rent density jS,PROP α,β, which is based on the concept of the “proper” spin current1and formulated in terms of the “proper” position operator /vectorR. However its calcula- tion is rather technical and less illuminating. Thus for pedagogical purpose, we present the calculation of more conventional spin current density first in Secs. I and II. In Sec. I, we presentthe calculationofthe conventional spin current density jS α,βformulated in terms of the con- ventional position operator /vector r, where/vector ris the canonical pair of the Bloch momentum /vectorkandjS α,βis the expecta- tion value of the conventional spin current density oper- atorˆjS α,β, ˆjS α,β=1 V−e ¯h/2{Sα,drβ/dt} 2. (S1) Note that ˆjS α,βis defined to have the same dimension as the charge current density. We demonstrate that jS α,β has three independent contributions, which we call the anomalous velocity contribution, the state change con- tribution, and the occupation change contribution. The physical meaning of each contribution will become clear in Sec. I. In Sec. II, we present the calculation of the proper spin currentdensity jS,prop α,βformulatedin termsofthe conven- tionalpositionoperator /vector r. Theconceptoftheproperspin current was proposed1to take into account the violation of the spin conservation and to facilitate the connection with the spin accumulation. jS,prop α,βis the expectation value of the operator ˆjS,prop α,β, ˆjS,prop α,β=1 V−e ¯h/2d dt{Sα,rβ} 2. (S2) Compared to Eq. (S1), where the time derivative ap- plies torβonly, Eq. (S2) differs since the time deriva- tive now applies to the anti-commutator {Sα,rβ}. We demonstrate that jS,prop α,βis identical to jS α,β. In Sec. III, we finally present the calculation of the proper spin current density jS,PROP α,βformulated in terms of the proper position operator /vectorR, where/vectorRdiffers from /vector ras follows, /vectorR=/vector r+αK e/vectork×/vectorL, (S3)andjS,PROP α,βis the expectation value of the operator ˆjS,PROP α,β, ˆjS,PROP α,β=1 V−e ¯h/2d dt{Sα,Rβ} 2. (S4) Note that Eq. (S4) is identical to Eq.(S2) except that Rβ appears instead of rβ. While the calculation of jS,PROP α,β is more tedious than those of the former two counter- parts, the value of jS,PROP α,βturns out to be almost iden- tical tojS α,βandjS,prop α,β, except that the magnitude ofthe anomalous velocity contribution is now two times bigger. I. CONVENTIONAL SPIN CURRENT DENSITY Here we present the calculation of the conventional spin current density jS α,βformulated in terms of the con- ventional position operator. jS α,βis given by jS α,β= Tr/bracketleftBig ˆjS α,βˆρ/bracketrightBig , (S5) where ˆρis the density matrix and the operator ˆjS α,βis defined in Eq. (S1). For/vectorE= 0, both H′ 1andH′ 2vanish and ˆ ρbecomes its equilibrium form ˆ ρ(0), where ˆρ(0)=/summationdisplay nf(0)/parenleftBig E(0) n/parenrightBig |n/an}bracketri}ht(0) (0)/an}bracketle{tn|,(S6) Here|n/an}bracketri}ht(0)denotesaneigenstateof H0withenergyeigen- valueE(0) n, andf(0)(E) is the equilibrium Fermi occupa- tion function. It is straightforward to verify that jS α,β vanishes in equilibrium. When a nonzero /vectorEis applied, we evaluate jS α,βup to the first order in /vectorE. Up to this order, effects of H′ 1and H′ 2may be considered separately. H′ 1alone does not con- tribute to jS α,βat all since as far as H0+H′ 1is concerned, the dynamicsof /vector rinH′ 1is decoupledfrom thatof /vectorS. This is evident from the facts that H′ 1commutes with both /vectorL and/vectorSand that there is no coupling in H0+H′ 1linking/vector r (or/vectork) with/vectorS(or/vectorL). Below we thus ignore effects of H′ 1 and consider effects of H′ 2only. In Secs. IA and IB, we evaluate two contributions to jS α,βunder the assumption that impurity scattering is completely absent. In Sec. IC, we consider the effect of the impurity scattering on jS α,βin the limit of vanishingly weak scatterers.2 A. Anomalous velocity contribution One effect of H′ 2is to modify the velocity operator /vector v. For the total Hamiltonian H0+H′ 2,/vector vis given by /vector v=d/vector r dt=[/vector r,H0+H′ 2] i¯h=¯h/vectork m+αK ¯h/vectorL×/vectorE=/vector v(0)+/vector v(1), (S7) where/vector v(0)and/vector v(1)refer to the terms independent of and linear in /vectorE. Here we call /vector v(1)the anomalous velocity since it denotes the extra contribution to the velocity generated by /vectorE. /vector v(1)generateswhatwecallthe anomalousvelocitycon- tribution ( jS α,β)avto the spin current, (jS α,β)av=1 V−e ¯h/2Tr/bracketleftBigg {Sα,v(1) β} 2ˆρ/bracketrightBigg .(S8) Up to the first order in /vectorE, ˆρin the above equation may be replaced by ˆ ρ(0)sincev(1)is already first order in /vectorE. Then Eq. (S8) reduces to (jS α,β)av=1 V−e ¯h/2/summationdisplay nf(0)/parenleftBig E(0) n/parenrightBig (0)/an}bracketle{tn|Sαv(1) β|n/an}bracketri}ht(0). (S9) Here one used Sαv(1) β=v(1) βSα. Since the eigenstates of H0are completely specified by the three quantum num- bers (/vectork,J,Jz) within the orbital angular momentum L= 1 sector, the state |n/an}bracketri}ht(0)amounts to |/vectork,J,Jz/an}bracketri}ht(0). For a givenJ, the summation over nin Eq. (S9) amounts to the summations over /vectorkandJz. SinceE(0) n=E(0)(/vectork,J) is independent of Jz, the summation over Jzleads to the following partial trace over jz, /summationdisplay Jz(0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingleSαv(1) β/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) .(S10) One then utilizes the relations v(1) β= (αK/¯h)ǫβηγLηEγ and /summationdisplay Jz(0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingleSαLη/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) (S11) =δα,η/summationdisplay Jz(0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingleSzLz/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) =1 3δαη/summationdisplay Jz(0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingle/vectorS·/vectorL/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) . Note that /vectorS·/vectorL= (¯h2/2)/bracketleftbig J(J+1)−1·2−1 2·3 2/bracketrightbig has opposite signs for J= 3/2 andJ= 1/2. As confirmed below, this sign difference leads to the sign difference in (jS α,β)avforJ= 3/2 andJ= 1/2. Subsequent calcula- tion proceeds as follows. One first obtains /summationdisplay Jz(0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingleSαv(1) β/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) (S12) =−ǫαβγEγαK ¯h¯h21 6/bracketleftbigg J(J+1)−11 4/bracketrightbigg (2J+1).Thentheanomalousvelocitycontributiontothespincur- rent density becomes (jS α,β)av (S13) =−e ¯h/2(−ǫαβγ)αK ¯hEγ¯h21 6/bracketleftbigg J(J+1)−11 4/bracketrightbigg (2J+1) ×1 V/summationdisplay /vectorkf(0)/parenleftBig E(0)(/vectork,J)/parenrightBig =ǫαβγEγeαK1 3/bracketleftbigg J(J+1)−11 4/bracketrightbigg (2J+1)4πk3 F/3 (2π)3 =±2 9π2ǫαβγEγeαKk3 F, where the upper and lower signs apply to J= 3/2 and J= 1/2, respectively. Note that ( jS α,β)avindeed has op- positesignsfor J= 3/2andJ= 1/2. Thissigndifference stems from the fact that /vectorSis parallel (antiparallel) to /vectorL forJ= 3/2 (J= 1/2). B. State change contribution Inaddition toEq.(S8), which capturestheeffect ofthe anomalous velocity /vector v(1), the conventional velocity opera- tor/vector v(0)also contributes to the spin current density. We call this contribution the state change contribution for the reason that will become clear below. It is given by (jS α,β)sc=1 V−e ¯h/2Tr/bracketleftBigg {Sα,v(0) β} 2ˆρ/bracketrightBigg .(S14) When ˆρin the above expression is replaced by ˆ ρ(0), the above expression vanishes. Thus ( jS α,β)scarises from the first order correction to ˆ ρdue to/vectorE. Up to this order, one obtains (jS α,β)sc=1 V−e ¯h/2Tr/bracketleftBig Sαv(0) βˆρ(1)/bracketrightBig ,(S15) whereSαv(0) β=v(0) βSαis used. One way to evaluate Eq. (S15) is to use the Kubo formula. Here we evaluate Eq. (S15) in a slightly different way, since this alternative method illustrates better why ( jS α,β)scmay be called the inter-band mixing contribution. It is straightforward to verify that this method and the Kubo formula produce the same result for ( jS α,β)sc. The adiabatic turning-on procedure allows a straight- forward evaluation of ˆ ρ(1). When H′ 2is turned on adi- abatically from the far past t=−∞, ˆρat present time t= 0 is given by ˆρ=/summationdisplay nf(0)/parenleftBig E(0) n/parenrightBig |n/an}bracketri}ht /an}bracketle{tn|. (S16) Here|n/an}bracketri}htdenotes the state at t= 0, to which |n/an}bracketri}ht(0)at t=−∞evolves as H′ 2is adiabatically turned on. Up3 to the first order in /vectorE,|n/an}bracketri}htdiffers from |n/an}bracketri}ht(0)by|n/an}bracketri}ht(1), which is given by |n/an}bracketri}ht(1)=/summationdisplay n′/negationslash=n|n′/an}bracketri}ht(0)(0)/an}bracketle{tn′|H′ 2|n/an}bracketri}ht(0) E(0) n−E(0) n′. (S17) Then ˆρ(1)becomes ˆρ(1)=/summationdisplay nf(0)/parenleftBig E(0) n/parenrightBig/parenleftBig |n/an}bracketri}ht(0) (1)/an}bracketle{tn|+|n/an}bracketri}ht(1) (0)/an}bracketle{tn|/parenrightBig , (S18) and (jS α,β)scin Eq. (S15) becomes (jS α,β)sc=1 V−e ¯h/2/summationdisplay n/summationdisplay n′f(0)/parenleftBig E(0) n/parenrightBig (S19) ×/parenleftBig(0)/an}bracketle{tn′|Sαv(0) β|n/an}bracketri}ht(0) (1)/an}bracketle{tn|n′/an}bracketri}ht(0) +(0)/an}bracketle{tn′|Sαv(0) β|n/an}bracketri}ht(1) (0)/an}bracketle{tn|n′/an}bracketri}ht(0)/parenrightBig . To evaluate this expression, one recalls E(0) nbeing in- dependent of Jzand exploits this energy degeneracy to introduce a new set of quantum numbers ( /vectork,J,J˜z) to specify the state n. HereJ˜zdenotes the component of the total angularmomentum along the direction ˜ z, which points along /vectorE×/vectorkdirection. Note that ˜ zaxis is de- pendent on /vectork. This change of the angular momentum quantization axis from zto ˜zsimplifies the evaluation of Eq. (S17). Considering that H′ 2reduces to αK|/vectorE×/vectork|L˜z, one finds (0)/an}bracketle{tn′|H′ 2|n/an}bracketri}ht(0)(S20) =αK|/vectorE×/vectork|(0)/angbracketleftBig /vectork′,J′,J′ ˜z/vextendsingle/vextendsingle/vextendsingleL˜z/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) =δ/vectork′/vectorkδJ′ ˜zJ˜zαK|/vectorE×/vectork|(0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleL˜z/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) . ThusH′ 2induces the inter-band mixing between |/vectork,J= 1/2,J˜z/an}bracketri}ht(0)and|/vectork,J= 3/2,J˜z/an}bracketri}ht(0). It is now evident that (jS αβ)sccaptures the effect of the state change due to the inter-band mixing caused by H′ 2. The matrix elements that capture this inter-band mixing effect are (0)/angbracketleftbigg /vectork,J=1 2,J˜z=±1 2/vextendsingle/vextendsingle/vextendsingle/vextendsingleH′ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectork,J=3 2,J˜z=±1 2/angbracketrightbigg(0) =(0)/angbracketleftbigg /vectork,J=3 2,J˜z=±1 2/vextendsingle/vextendsingle/vextendsingle/vextendsingleH′ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectork,J=1 2,J˜z=±1 2/angbracketrightbigg(0) =αK|/vectorE×/vectork|/parenleftBigg −√ 2 3¯h/parenrightBigg . (S21)All other matrix elements are zero. Then one obtains /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectork,J=1 2,J˜z=±1 2/angbracketrightbigg(1) (S22) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectork,J=3 2,J˜z=±1 2/angbracketrightbigg(0)αK|/vectorE×/vectork|√ 2 3¯h ∆E, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectork,J=3 2,J˜z=±1 2/angbracketrightbigg(1) (S23) =−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectork,J=1 2,J˜z=±1 2/angbracketrightbigg(0)αK|/vectorE×/vectork|√ 2 3¯h ∆E, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectork,J=3 2,J˜z=±3 2/angbracketrightbigg(1) = 0. (S24) where ∆ E≡E(0)(/vectork,J= 3/2,J˜z)−E(0)(/vectork,J= 1/2,J˜z) = 3αSO¯h2/2 is independent of /vectorkandJ˜z. Then (jS α,β)scin Eq. (S19) reduces to (jS α,β)sc (S25) =∓1 V−e ¯h/2/summationdisplay /vectork/summationdisplay J˜z=±1/2f(0)/parenleftBig E(0)(/vectork,J)/parenrightBig ×αK|/vectorE×/vectork|√ 2 3¯h ∆E/parenleftbigg(0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleSαv(0) β/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) +(0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsingleSαv(0) β/vextendsingle/vextendsingle/vextendsingle/vectork,J′,J˜z/angbracketrightBig(0)/parenrightbigg , where the upper and lower signs apply to J= 3/2 and J= 1/2, respectively. J′= 1/2 (3/2) when J= 3/2 (1/2). Using the relation (0)/angbracketleftbigg /vectork,J=1 2,J˜z/vextendsingle/vextendsingle/vextendsingle/vextendsingleSαv(0) β/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectork,J=3 2,J˜z/angbracketrightbigg(0) (S26) =√ 2¯h 3(/vectorE×/vectork)α |/vectorE×/vectork|¯hkβ m, one obtains (jS α,β)sc (S27) =∓1 V−e ¯h/2/summationdisplay /vectork2f(0)/parenleftBig E(0)(/vectork,J)/parenrightBigαK(/vectorE×/vectork)α ∆E2¯h2 92¯hkβ m. From the relation 1 V/summationdisplay /vectorkf(0)/parenleftBig E(0)(/vectork,J)/parenrightBig (/vectorE×/vectork)αkβ(S28) =1 V/summationdisplay /vectorkf(0)/parenleftBig E(0)(/vectork,J)/parenrightBig/parenleftBig −ǫαβγ 3Eγ/parenrightBig k2 =−ǫαβγ 3Eγ4πk3 F/3 (2π)33k2 F 5, (S29) one finally obtains (jS α,β)sc=∓16 135π2ǫαβγEγeαKk3 F¯h2k2 F/2m ∆E.(S30)4 Note that similarly to ( jS α,β)av, (jS α,β)scalso has opposite signs for J= 3/2 (upper sign) and J= 1/2 (lower sign). C. Occupation change contribution So far we have neglected impurity scattering. Here we consider the scattering effect in the vanishing scattering strength limit. Even in this limit, the scattering is im- portant since it violates the momentum conservation and allows electrons to relax in momentum space. To illus- trateitsimportance, it isusefultoconsiderthe casewhen the scattering is completely absent. Then all throughout the adiabatic turning-on procedure of H′ 2,/vectorkremains a good quantum number and the electron occupation in /vectork space cannot be altered by H′ 2, which is in contrast to what we expect as illustrated in Fig. 2(b). In the Kubo formalism, this effect is often addressed through the ver- tex correction. Here we address this effect by noting that the energy eigenvalues of H0+H′ 2are bounded from be- low. In such a situation, the electron occupation will relax in/vectorkspace to minimize the total energy of the elec- trons. Thus the occupation change contribution ( jS αβ)oc to the spin current density is given by (jS α,β)oc=1 V−e ¯h/2Tr/bracketleftBig Sαv(0) βˆρ(1) oc/bracketrightBig ,(S31) where ˆρ(1) ocdenotes the first order correction to density matrix due to scattering and is given by ˆρ(1) oc=/summationdisplay nf(1) n|n/an}bracketri}ht(0) (0)/an}bracketle{tn|. (S32) Heref(1) n=f(0)(En)−f(0)(E(0) n) denotes the first order correctionto the occupation function and Endenotes the energy eigenvalue of H0+H′ 2. (jS αβ)ocis thus given by (jS α,β)oc=1 V−e ¯h/2/summationdisplay nf(1)(En)(0)/an}bracketle{tn|Sαv(0) β|n/an}bracketri}ht(0), (S33) where Tr[ Sαv(0) βρ(0)] = 0 has been used. To determine En, it is useful to use the quantum num- bers/vectork,J,J˜zinstead of /vectork,J,Jzto specify nsince for given Jsector, the state |/vectork,J,J˜z/an}bracketri}ht(0)diagonalizes H0+H′ 2with the eigenvalue En=E(/vectork,J,J˜z) given by E(/vectork,J,J˜z) =E(0)(/vectork,J)+αK|/vectorE×/vectork|3∓1 3¯hJ˜z,(S34) where the upper and lower signs apply to J= 3/2 and J= 1/2, respectively. To understand the effect of the second term, it is useful to consider one particular case; /vectorE=Ezˆz. Then the second term is proportional to (k2 x+k2 y)1/2. On the other hand, the first term is pro- portional to /vectork2=k2 z+ (k2 x+k2 y). Thus the combined effect of the first and second terms is to expand (shrink)theoriginallysphericalFermisurfacealongthe“equator” direction when the second term is negative (positive). The next step in the evaluation of ( jS αβ)ocis to calcu- late(0)/an}bracketle{tn|Sαv(0) β|n/an}bracketri}ht(0)with|n/an}bracketri}ht(0)replacedby |/vectork,J,J˜z/an}bracketri}ht(0). After straightforward calculation, one obtains (0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsingleSαv(0) β/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) (S35) =(/vectorE×/vectork)α |/vectorE×/vectork|/parenleftbigg ±¯hJ˜z 3/parenrightbigg¯hkβ m, = sgn(Ez)kxδαy−kyδαx/radicalBig k2x+k2y/parenleftbigg ±¯hJ˜z 3/parenrightbigg¯hkβ m, where the upper and lower signs apply to J= 3/2 and J= 1/2, respectively. After the average over the az- imuthal angle in /vectorkspace, the above expression reduces to (0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsingleSαv(0) β/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) (S36) = sgn(Ez)/radicalBig k2x+k2y 2(−ǫαβz)/parenleftbigg ±¯hJ˜z 3/parenrightbigg¯h m, Then (jS αβ)ocbecomes (jS α,β)oc (S37) =1 V−e ¯h/2/summationdisplay /vectork/summationdisplay J˜zf(0)/parenleftBig E/parenleftBig /vectork,J,J˜z/parenrightBig/parenrightBig ×sgn(Ez)/radicalBig k2x+k2y 2(−ǫαβz)/parenleftbigg ±¯hJ˜z 3/parenrightbigg¯h m. After some tedious calculation, and for general direction of/vectorE, one obtains (jS α,β)oc=/braceleftbigg −10 +2/bracerightbigg ×1 27π2ǫαβγEγeαKk3 F.(S38) Here the upper and lower results apply to J= 3/2 and J= 1/2, respectively. Note that ( jS αβ)ochas opposite signs for J= 3/2 andJ= 1/2. D. Summary Finally,jS α,βcan be obtained by summing up all three contributions,( jS αβ)oc, (jS αβ)av, and(jS αβ)sc. ForJ= 3/2, one finds jS αβ=−2 9π2ǫαβγEγeαKk3 F/parenleftbigg5 3−1+8 15¯h2k2 F/2m ∆E/parenrightbigg (S39) and forJ= 1/2, one finds jS αβ=2 9π2ǫαβγEγeαKk3 F/parenleftbigg1 3−1+8 15¯h2k2 F/2m ∆E/parenrightbigg (S40)5 II. PROPER SPIN CURRENT DENSITY FOR /vector r In this section, we calculate the proper spin current density operator jS,prop α,βbased on the conventional posi- tion operator /vector r. The corresponding operator ˆjS,prop α,βin Eq. (S2) may be divided into two pieces as follows, ˆjS,prop α,β=ˆjS α,β+ˆjS,extra α,β, (S41) whereˆjS α,βis the conventional spin current operator as defined in Eq. (S1), and ˆjS,extra α,β=1 V−e ¯h/2/braceleftbigdSα dt,rβ/bracerightbig 2=1 V−e ¯h/2αSOǫαγδLγSδrβ. (S42) Thus the difference ˆjS,extra α,βbetween jS,prop α,βandjS α,β amounts to the expectation value of ˆjS,extra α,β, jS,extra α,β= Tr/bracketleftBig ˆjS,extra α,βˆρ/bracketrightBig , (S43) which will be evaluated below. Among the two perturba- tionsH′ 1andH′ 2,H′ 1cannot generateany contribution to jS,extra α,βsince it does not induce any correlation between /vector r(or/vectork) and/vectorL(or/vectorS). Below we thus consider possible contribution from H′ 2only. A. Anomalous velocity contribution By the “anomalous velocity contribution”, we refer to jS,extra α,βwith ˆρin Eq. (S43) replaced by its equilibrium counterpart ˆ ρ(0)in Eq. (S6). We find jS,extra α,βvanishes identically. Below we demonstrate this for α=z. The generalization to the case with α=xoryis straightfor- ward. For α=z, one obtains jS,extra z,β=1 V−e ¯h/2αSO/summationdisplay /vectork,Jzf(0)/parenleftBig E(0)(/vectork,J)/parenrightBig (S44) ×(0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingle(LxSy−LySx)rβ/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) . To evaluate the expectation value in the above equa- tion, one uses the relations [ LxSy−LySx,/vectork] = [LxSy− LySx,Jz] = [rβ,J] = 0 to obtain (0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingle(LxSy−LySx)rβ/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) (S45) =(0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingle(LxSy−LySx)/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) ×(0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsinglerβ/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) . This expression vanishes since (0)/angbracketleftBig /vectork,J,Jz/vextendsingle/vextendsingle/vextendsingle(LxSy−LySx)/vextendsingle/vextendsingle/vextendsingle/vectork,J,Jz/angbracketrightBig(0) = 0.(S46)This vanishing can be understood as follows. Since LxSy−LySxis hermitian, its expectation value with re- spect to|/vectork,J,Jz/an}bracketri}ht(0)must be real. On the other hand, the conventional representations of LxSyandLySxare pure imaginary. The only way to reconcile these two proper- ties is to make its expectation value zero. Thus one finds ( jS,extra α,β)av= 0 and ( jS,prop α,β)av= (jS α,β)av. This way, one finally obtains /parenleftBig jS,prop α,β/parenrightBig av=±2 9π2ǫαβγEγeαKk3 F.(S47) B. State change contribution The state change contribution is defined as the con- tribution that arises from the deviation of ˆ ρfrom ˆρ(0). Thus the state change contribution from the extra spin current density operator is given by /parenleftBig jS,extra α,β/parenrightBig sc=1 V−e ¯h/2Tr/bracketleftBigg/braceleftbigdSα dt,rβ/bracerightbig 2ˆρ(1)/bracketrightBigg ,(S48) where ˆρ(1)is given in Eq. (S18). Substituting ˆ ρ(1)into the above equation leads to (jS,extra α,β)sc=1 V−e ¯h/2/summationdisplay n/summationdisplay n′f(0)/parenleftBig E(0) n/parenrightBig αSOǫαδγ(S49) ×/parenleftBig(0)/an}bracketle{tn′|LδSγrβ|n/an}bracketri}ht(0) (1)/an}bracketle{tn|n′/an}bracketri}ht(0) +(0)/an}bracketle{tn′|LδSγrβ|n/an}bracketri}ht(1) (0)/an}bracketle{tn|n′/an}bracketri}ht(0)/parenrightBig . Note that this expression has the identical structure as Eq. (S19) except that the conventional spin current den- sity operator ˆjS α,βis replaced by the extra spin current density operator ˆjS,extra α,β. The evaluation ofthis equation proceeds in a similar way. One first adopts the quantum numbers /vectork,J, andJ˜zto specify the state n, whereJ˜z denotes the component of the total angular momentum operator along /vectorE×/vectorkdirection. This allows one to utilize6 Eqs. (S22), (S23), and (S24), and one finds (jS,extra α,β)sc (S50) =∓1 V−e ¯h/2/summationdisplay /vectork/summationdisplay J˜z=±1/2f(0)/parenleftBig E(0)(/vectork,J)/parenrightBig ×αSOǫαδγαK/vextendsingle/vextendsingle/vextendsingle/vectorE×/vectork/vextendsingle/vextendsingle/vextendsingle√ 2 3¯h ∆E ×/parenleftbigg(0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleLδSγrβ/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) +(0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsingleLδSγrβ/vextendsingle/vextendsingle/vextendsingle/vectork,J′,J˜z/angbracketrightBig(0)/parenrightbigg =∓1 V−e ¯h/2/summationdisplay /vectork/summationdisplay J˜z=±1/2f(0)/parenleftBig E(0)(/vectork,J)/parenrightBig ×αSOαK/vextendsingle/vextendsingle/vextendsingle/vectorE×/vectork/vextendsingle/vextendsingle/vextendsingle√ 2 3¯h ∆E ×2Re/bracketleftbigg(0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleǫαδγLδSγrβ/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0)/bracketrightbigg , where the upper and lower signs apply to J= 3/2 and J= 1/2, respectively. J′= 1/2 (3/2) when J′= 3/2 (1/2). Using [ /vector r,J] = [/vector r,J˜z] = [/vectork,Lδ] = [/vectork,Sγ] = 0, the last line of the above equation can be written as 2Re/bracketleftbigg(0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleǫαδγLδSγrβ/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0)/bracketrightbigg (S51) = 2Re/bracketleftbigg(0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleǫαδγLδSγ/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) ×(0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsinglerβ/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0)/bracketrightbigg . Here,(0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsinglerβ/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) is manifestly real since rβis hermitian. It can be also verified that (0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleǫαδγLδSγ/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) is purely imaginary. For this reason, the above equation vanishes identically and one finds ( jS,extra α,β)sc= 0. Therefore ( jS,prop α,β)sc= (jS α,β)sc, and one obtains (jS,prop α,β)sc=∓16 135π2ǫαβγEγeαKk3 F¯h2k2 F/2m ∆E. forJ= 3/2 (upper sign) and J= 1/2 (lower sign), re- spectively. C. Occupation change contribution The occupation change contribution refers to the con- tribution arising from the additional deviation of ˆ ρ from ˆρ(0)due to the impurity scattering of infinitesimal strength. Thus ( jS,extra α,β)ocbecomes (jS,extra α,β)oc= Tr/bracketleftBig ˆjS,extra α,βˆρ(1) oc/bracketrightBig ,(S52)where ˆρ(1) ocdenotes the impurity scattering effect to ˆ ρ. Using its expression in Eq. (S32), one obtains (jS,extra α,β)oc (S53) =1 V−e ¯h/2/summationdisplay nf(1)(En)αSO(0)/an}bracketle{tn|ǫαγδLγSδrβ|n/an}bracketri}ht(0). By following the same analysis as in Sec. IIA, one can verify that(0)/an}bracketle{tn|ǫαγδLγSδrβ|n/an}bracketri}ht(0)= 0. Thus ( jS,extra α,β)oc vanishes identically and ( jS,prop α,β)oc= (jS α,β)oc. Therefore one obtains (jS,prop α,β)oc=/braceleftbigg −10 +2/bracerightbigg ×1 27π2ǫαβγEγeαKk3 F, where the upper and lower numbers apply to J= 3/2 andJ= 1/2, respectively. D. Summary In the preceding subsections, we showed that the extra spin current density operator ˆjS,extra α,βdoes not generate any extra contributions, so the proper spin current den- sityjS,prop α,βis identical to the conventional spin current densityjS α,β. To summarize the result of this section, we obtained jS,prop αβ=−2 9π2ǫαβγEγeαKk3 F/parenleftbigg5 3−1+8 15¯h2k2 F/2m ∆E/parenrightbigg (S54) forJ= 3/2, and jS,prop αβ=2 9π2ǫαβγEγeαKk3 F/parenleftbigg1 3−1+8 15¯h2k2 F/2m ∆E/parenrightbigg (S55) forJ= 1/2. III. PROPER SPIN CURRENT DENSITY FOR /vectorR In this section, we evaluate the proper spin current densityjS,PROP α,β, which is the expectation value of the proper spin current density operator ˆjS,PROP α,β[Eq. (S4)] formulated in terms of the proper position operator /vectorR [Eq. (S3)]. ˆjS,PROP α,βdiffers from the proper spin current density operator ˆjS,prop α,βin that the “proper”position op- erator/vectorRin Eq. (S3) is used instead of the conventional position operator /vector r. One way to understand the differ- ence between the two position operators is to compare the corresponding velocity operators. The “proper” ve-7 locity operator /vectorVbecomes /vectorV=d/vectorR dt=[/vectorR,H0+H′ 1+H′ 2] i¯h(S56) =¯h/vectork m+αK ¯h/vectorL×/vectorE +αK ¯h/vectorL×/vectorE+α2 K e/parenleftBig /vectorE×/vectork/parenrightBig/parenleftBig /vectork·/vectorL/parenrightBig +αKαSO e/vectork×/parenleftBig /vectorS×/vectorL/parenrightBig , where the first two terms amount to /vector v. Compared to the conventional velocity operator /vector vin Eq. (S7), /vectorVdiffers by /vectorV−/vector v=δ/vector va+δ/vector vb+δ/vector vc, (S57) where δ/vector va=αK ¯h/vectorL×/vectorE, (S58) δ/vector vb=α2 K e/parenleftBig /vectorE×/vectork/parenrightBig/parenleftBig /vectork·/vectorL/parenrightBig , (S59) δ/vector vc=αKαSO e/vectork×/parenleftBig /vectorS×/vectorL/parenrightBig . (S60) Among the three terms δ/vector va,δ/vector vb, andδ/vector vc,δ/vector vais identical to the anomalousvelocity /vector v(1)in Eq. (S7). Recallingthat /vector v(1)isresponsiblefortheanomalousvelocitycontribution (jS α,β)av= (jS,prop α,β)avin Eqs. (S13) and (S47), δ/vector vabeing identical to /vector v(1)doubles the anomalous velocity contri- bution, which will be verified explicitly in Sec. IIIA. /vector vbis linear in /vectorEand thus generates a new piece of the anoma- lous velocity. Comparedto /vector va, it is smallerby the dimen- sionless factor αK¯hk2/e, which is much smaller than 1 in the small /vectorklimit. Thus /vector vbis not important in the small /vectork limit, but just for the sake of completeness, we evaluate its contribution to jS,PROP α,βin Sec. IIIA. On the other hand,/vector vcis zeroth order in /vectorE. We examine its possible contribution below. For explicit evaluation of jS,PROP α,β, one needs to deal withˆjS,PROP α,βin Eq. (S4). It is useful to compare ˆjS,PROP α,βwithˆjS α,βandˆjS,prop α,β, ˆjS,PROP α,β=ˆjS,prop α,β+ˆjS,EXTRA α,β(S61) =ˆjS α,β+ˆjS,extra α,β+ˆjS,EXTRA α,β, whereˆjS,extra α,βis defined in Eq. (S42) and ˆjS,EXTRA α,βis given by ˆjS,EXTRA α,β=ˆjS,a α,β+ˆjS,b α,β+ˆjS,c α,β.(S62)Here ˆjS,a α,β=1 V−e ¯h/2αK ¯hSα/parenleftBig /vectorL×/vectorE/parenrightBig β, (S63) ˆjS,b α,β=1 V−e ¯h/2α2 K eSα/parenleftBig /vectorE×/vectork/parenrightBig β/parenleftBig /vectork·/vectorL/parenrightBig ,(S64) ˆjS,c α,β=1 V−e ¯h/2αKαSO 2e/bracketleftbigg¯h2 2/parenleftBig δαβ/vectork·/vectorL−kαLβ/parenrightBig (S65) −/braceleftbigg/parenleftBig /vectorS×/vectorL/parenrightBig α,/parenleftBig /vectork×/vectorL/parenrightBig β/bracerightbigg/bracketrightbigg . It is evident that the expectation value of ˆjS,EXTRA α,βde- terminesthe differencebetween jS,PROP α,βand the twofor- mer spin current densities jS α,βandjS,prop α,β. Simple order counting helps estimate effects of the three terms of ˆjS,EXTRA α,β. SinceˆjS,aandˆjS,bare linear in/vectorE, they can affect only the anomalous velocity contri- bution. They do not affect the state change contribution and the occupation change contribution. Among these termsˆjS,b α,βis smaller than ˆjS,a α,βbyαK¯hk2/e, which ap- proaches zero in the small /vectorklimit. Thus ˆjS,a α,βis expected to be more important. Actually it can be easily verified thatˆjS,a α,βis identical to1 V−e ¯h/2Sαv(1) β[see Eq. (S8)], which is responsible for the anomalous velocity contribution of jS α,β. Thus the presence of ˆjS,a α,βdoubles the anomalous velocity contribution. On the other hand, ˆjS,c α,βdiffers by the factor αKαSO¯hm/efrom1 V−e ¯h/2Sαv(0) β[see Eq. (S31)], which is responsible for the occupation change contribu- tion ofjS α,β. Since this factor may not be small in the strong spin-orbit coupling limit that we consider, it is yet unclear how important ˆjS,c α,βis. Below we demonstrate through explicit calculation thatˆjS,c α,βdoes not generate any important contribution and the only important effect of ˆjS,EXTRA α,βis to double the anomalous velocity contribution. A. Anomalous velocity contribution To calculate the anomalous velocity contribution (jS,PROP α,β)avtojS,PROP α,β, it is sufficient to evaluate (jS,EXTRA α,β)av, which is given by (jS,EXTRA α,β)av= Tr/bracketleftBig ˆjS,EXTRA α,βˆρ(0)/bracketrightBig (S66) = (jS,a α,β)av+(jS,b α,β)av+(jS,c α,β)av, where (jS,a α,β)av= Tr/bracketleftBig ˆjS,a α,βˆρ(0)/bracketrightBig , (S67) (jS,b α,β)av= Tr/bracketleftBig ˆjS,b α,βˆρ(0)/bracketrightBig , (S68) (jS,c α,β)av= Tr/bracketleftBig ˆjS,c α,βˆρ(0)/bracketrightBig . (S69)8 For (jS,c α,β)av, it vanishes simply because ˆjS,c α,βis linear in/vectorkwhereas ˆ ρ(0)puts the same weighting independent of the direction of /vectork. To evaluate ( jS,a α,β)av, one notes ˆjS,a α,β=1 V−e ¯h/2Sαδva β=1 V−e ¯h/2Sαv(1) β.(S70) Thus (jS,a α,β)avis identical to ( jS α,β)avin Eq. (S9), and one obtains /parenleftBig jS,a α,β/parenrightBig av=±2 9π2ǫαβγEγeαKk3 F,(S71) forJ= 3/2 (upper sign) and J= 1/2 (lower sign). To evaluate ( jS,b α,β)av, one notes ˆjS,b α,β=1 V−e ¯h/2Sαδvb β. (S72) We demonstrate the evaluation of ( jS,b α,β)avfor/vectorE=Ezˆz. In this case, it is straightforward to verify that ( jS,b α,β)av is proportional to ǫαβz. It then suffices to evaluate ǫαβz(jS,b α,β)av. One uses the following relation ǫαβzSα/parenleftBig /vectorE×/vectork/parenrightBig β/parenleftBig /vectork·/vectorL/parenrightBig (S73) = (Sxkx+Syky)/parenleftBig /vectork·/vectorL/parenrightBig Ez to obtain ǫαβz/parenleftBig jS,b α,β/parenrightBig av(S74) =1 V−e ¯h/2α2 K eEzTr/bracketleftBig (Sxkx+Syky)/parenleftBig /vectork·/vectorL/parenrightBig ˆρ(0)/bracketrightBig . Sincef(0)(E(0) n) in ˆρ(0)[Eq. (S6)] does not depend on the direction of /vectork, this expression may survive only when the traced expression is even in components of /vectork. It thus reduces to ǫαβz/parenleftBig jS,b α,β/parenrightBig av(S75) =1 V−e ¯h/2α2 K eEzTr/bracketleftBig/parenleftbig SxLxk2 x+SyLyk2 y/parenrightbig ˆρ(0)/bracketrightBig . Since the dependence on /vectorkis decoupled from the depen- dencies on /vectorSand/vectorLas far as ˆ ρ(0)is concerned, k2 xand k2 yin the above expression may be replaced by k2/3, and SxLxandSyLyby/vectorS·/vectorL/3. Thus the above expression reduces further to ǫαβz/parenleftBig jS,b α,β/parenrightBig av(S76) =1 V−e ¯h/2α2 K eEz2 9Tr/bracketleftBig/parenleftBig /vectorS·/vectorL/parenrightBig k2ˆρ(0)/bracketrightBig . Here/vectorS·/vectorLmay be replaced by (¯ h2/2)[J(J+1)−1·2−1 23 2], which is ¯ h2/2 forJ= 3/2 and−¯h2forJ= 1/2. Theremaining calculation is straightforward. Generalizing to general direction of /vectorE, one obtains /parenleftBig jS,b α,β/parenrightBig av=∓1 15π2ǫαβγEγ¯hα2 Kk5 F(S77) forJ= 3/2 (upper sign) and J= 1/2 (lower sign). Note that ( jS,b α,β)avdiffers from ( jS,a α,β)avby the factor −3 5αK¯hk2 F/e, which is smallerthan 1 in the small /vectorklimit. Therefore ( jS,EXTRA α,β)avis given by /parenleftBig jS,EXTRA α,β/parenrightBig av=±2 9π2ǫαβγEγeαKk3 F/parenleftbigg 1−3 10αK¯hk2 F e/parenrightbigg . (S78) Consideringthe relationbetween ˆjS,EXTRA α,βandˆjS,PROP α,β in Eq. (S61), one finally obtains /parenleftBig jS,PROP α,β/parenrightBig av=±4 9π2ǫαβγEγeαKk3 F/parenleftbigg 1−3 20αK¯hk2 F e/parenrightbigg , (S79) forJ= 3/2 (upper sign) and J= 1/2 (lower sign). B. State change contribution The perturbation H′ 1andH′ 2can modify the density matrix ˆρ. To calculate the state change contribution (jS,PROP α,β)scarising from the density matrix change (in the absence of any scattering), it is sufficient to retain onlyˆjS,c α,βout of the three terms for ˆjS,EXTRA α,β[Eq. (S62)] and calculate (jS,c α,β)sc= Tr/bracketleftBig ˆjS,c α,βˆρ(1)/bracketrightBig , (S80) where ˆρ(1)denotesthefirstorder(in /vectorE)changeof ˆ ρ. Here we may ignorethe contributions from ˆjS,a α,βandˆjS,b α,β, since these operators are already first order in /vectorEand generate the second order contribution when combined with ˆ ρ(1). BothH′ 1andH′ 2contribute to ˆ ρ(1). However ˆ ρ(1) due toH′ 1does not contribute to ( jS,c α,β)scsince as far asH0+H′ 1is concerned, there is no coupling between (/vectork,/vector r) and (/vectorL,/vectorS). Combined with the fact that ˆjS,c α,βis odd in/vectorLor/vectorS, this feature prohibits H′ 1from generating any contribution to ( jS,c α,β)sc. Below we confine ourselves to ˆ ρ(1)arising from H′ 2, which has been explicitly constructed in Sec. IB. Fol- lowing the similar calculation procedure in Sec. IB, one obtains (jS,c α,β)sc (S81) =∓/summationdisplay /vectork/summationdisplay J˜z=±1/2f(0)/parenleftBig E(0)(/vectork,J)/parenrightBig ×αK|/vectorE×/vectork|√ 2 3¯h ∆E ×2Re/braceleftbigg(0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleˆjS,c α,β/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0)/bracerightbigg ,9 where the upper and lower signs apply to J= 3/2 and J= 1/2, respectively. J′= 1/2 (3/2) when J= 3/2 (1/2). ThisisthecounterpartofEq.(S25). Fromsymme- try consideration, it can be verified that ( jS,c α,β)scshould be proportional to ǫαβγEγ. Also the expression for ˆjS,c α,β in Eq. (S65) indicates that ( jS,c α,β)scin Eq. (S81) is of the order of /vectorE¯hαKk5 F. Thus one finds (jS,c α,β)sc=ηǫαβγEγ¯hα2 Kk5 F, (S82) whereηis a dimensionless constant. Explicit evalua- tion of Eq. (S81) is necessary to determine η. How- ever even without the explicit evaluation, it is evident that (jS,c α,β)scis smaller than ( jS,PROP α,β)avby the factor αK¯hk2 F/e, which is smaller than 1 in the smaller /vectorklimit. Therefore in the small /vectorklimit, (jS,c α,β)scis not important. Below we demonstrate the explicit evaluate of ( jS,c α,β)sc to determine η. It suffices to assume /vectorE=Ezˆzand evalu- ateǫαβz(jS,c α,β)sc. Forthis, oneutilizes Eq. (S65) to obtain ǫαβzˆjS,c α,β=1 V−e ¯h/2αKαSO 2e/bracketleftbigg −¯h2 2/parenleftBig /vectork×/vectorL/parenrightBig z(S83) −Lz(/vectorS·/vectork×/vectorL)−(/vectork×/vectorL·/vectorS)Lz/bracketrightbigg . After some algebra, one finds ǫαβz2Re/braceleftbigg(0)/angbracketleftBig /vectork,J′,J˜z/vextendsingle/vextendsingle/vextendsingleˆjS,c α,β/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0)/bracerightbigg (S84) =1 V−e ¯h/2αKαSO 2e3√ 2 2¯h3/vextendsingle/vextendsingle/vextendsingleˆz×/vectork/vextendsingle/vextendsingle/vextendsingle (S85) The rest of calculation is straightforward and results in (jS,c α,β)sc=±2 45π2ǫαβγEγ¯hα2 Kk5 F.(S86) Finally by combining with ( jS,prop α,β)sc, one obtains (jS,PROP α,β)sc (S87) =∓2 45π2ǫαβγEγeαKk3 F/parenleftbigg8 3¯h2k2 F/2m ∆E−αK¯hk2 F e/parenrightbigg , forJ= 3/2 (upper sign) and J= 1/2 (lower sign). C. Occupation change contribution The occupation change contribution refers to the con- tribution arising from the additional deviation of ˆ ρ from ˆρ(0)due to the impurity scattering of infinitesimal strength. Due to Eqs. (S61) and (S62), the calculation of (jS,PROP α,β)ocoverlaps a lot with that of ( jS,prop α,β)ocand (jS α,β)oc. The only piece that requires additional calcula- tion is (jS,c α,β)oc, which is given by (jS,c α,β)oc= Tr/bracketleftBig ˆjS,c α,βˆρ(1) oc/bracketrightBig , (S88)where ˆρ(1) ocdenotes the impurity scattering effect to ˆ ρ. The perturbation H′ 1does not make any contribution to (jS,c α,β)ocsince it does not induce any correlation among (/vectork,/vector r) and (/vectorL,/vectorS). Below we thus consider the pertur- bationH′ 2only. Then using the expression for ˆ ρ(1) ocin Eq. (S32), one obtains (jS,c α,β)oc=/summationdisplay nf(1)(En)(0)/an}bracketle{tn|ˆjS,c α,β|n/an}bracketri}ht(0).(S89) From symmetry consideration, one can verify that (jS,c α,β)ocshould be proportional to ǫαβγEγ. It then suf- fices to assume /vectorE=Ezˆzand evaluate ǫαβz(jS,c α,β)oc. For its evaluation, one uses Eq. (S83) and also the relation, /summationdisplay kz/braceleftbigg(0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsingle¯h2 2/parenleftBig /vectork×/vectorL/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) (S90) +(0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsingleLz/parenleftBig /vectorS·/vectork×/vectorL/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0) +(0)/angbracketleftBig /vectork,J,J˜z/vextendsingle/vextendsingle/vextendsingle/parenleftBig /vectork×/vectorL·/vectorS/parenrightBig Lz/vextendsingle/vextendsingle/vextendsingle/vectork,J,J˜z/angbracketrightBig(0)/bracerightbigg = 0, which shows that ( jS,c α,β)oc= 0. Finally by combining with (jS,prop α,β)oc, one obtains (jS,PROP α,β)oc=/braceleftbigg −10 +2/bracerightbigg ×1 27π2ǫαβγEγeαKk3 F, where the upper and lower results apply to J= 3/2 and J= 1/2, respectively. D. Summary To summarize the result of this section, we obtained jS,PROP αβ=−2 9π2ǫαβγEγeαKk3 F (S91) ×/bracketleftbigg5 3−/parenleftbigg 2−3 10αK¯hk2 F e/parenrightbigg +/parenleftbigg8 15¯h2k2 F/2m ∆E−1 5αK¯hk2 F e/parenrightbigg/bracketrightbigg forJ= 3/2 and jS,PROP αβ=2 9π2ǫαβγEγeαKk3 F (S92) ×/bracketleftbigg1 3−/parenleftbigg 2−3 10αK¯hk2 F e/parenrightbigg +/parenleftbigg8 15¯h2k2 F/2m ∆E−1 5αK¯hk2 F e/parenrightbigg/bracketrightbigg forJ= 1/2. Note that jS,PROP αβdiffers from jS αβand jS,prop αβin two ways. One difference is the extra terms, which are of order of /vectorEα2 K¯hk5 Fand thus smaller than10 other leading order terms by the factor αK¯hk2 F/e. Since this factor approaches zero in the small /vectorkregime that we consider, this difference is not important. The other difference is the factor two enhancement of the anoma- lous velocity contribution. Since this enhancement oc-curs at the leading order term of the order of /vectorEeαKk3 F, this enhancement by factor 2 is relevant. Thus the only important deviation of jS,PROP αβfromjS αβandjS,prop αβis the factor two enhancement of the anomalous velocity contribution. ∗Electronic address: hwl@postech.ac.kr †Electronic address: changyoung@yonsei.ac.kr 1J. Shi, P. Zhang, D. Xiao, and Q. Niu, Phys. Rev. Lett. 97,076604 (2006).

1308.6349v1.Spin_orbit_coupling_and_spin_Hall_effect_for_neutral_atoms_without_spin_flips.pdf

Spin-orbit coupling and spin Hall effect for neutral atoms without spin-flips Colin J. Kennedy, Georgios A. Siviloglou, Hirokazu Miyake, William Cody Burton, and Wolfgang Ketterle MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: August 8, 2018) We propose a scheme which realizes spin-orbit coupling and the spin Hall effect for neutral atoms in optical lattices without relying on near resonant laser light to couple different spin states. The spin-orbit coupling is created by modifying the motion of atoms in a spin-dependent way by laser recoil. The spin selectivity is provided by Zeeman shifts created with a magnetic field gradient. Alternatively, a quantum spin Hamiltonian can be created by all-optical means using a period- tripling, spin-dependent superlattice. PACS numbers: 67.85.-d, 03.65.Vf, 03.75.Lm Many recent advances in condensed matter physics are related to the spin degree of freedom. The field of spin- tronics [1], the spin Hall effect [2], and topological insu- lators [3] all rely on the interplay between spin and mo- tionaldegreesoffreedomprovidedbyspin-orbitcoupling. Quantum simulations with neutral atoms have started to implement spin-orbit coupling using Raman transitions between different hyperfine states [4–8]. Since the Ra- man process transfers momentum to the atom, the res- onance frequency is Doppler sensitive, and thus couples motion and spin. The possibility of using spin-flip Raman processes to create interesting gauge fields was first pointed out in [9–11], and extended to non-Abelian gauge fields, which imply spin-orbit coupling, in [12, 13]. With the exception of an atom chip proposal where the spin-flips are induced with localized microwave fields [14], all recently proposed schemes are based on spin-flip Raman processes [8, 15– 20]. The major limitation of these Raman schemes is that spin-flip processes are inevitably connected with heating by spontaneous emission if they rely on spin-orbit cou- pling in the excited state, as in alkali atoms or other atoms with an Sorbital ground state. Since laser beams interact with atoms via the electric dipole interaction, they do not flip the spin. Spin-flips occur only due to intrinsic spin-orbit interactions within the atoms; there- fore, spin-orbit coupling by spin flip Raman processes relies on the spin-orbit coupling withinthe atom. Since the spontaneous emission rate and the two-photon Rabi frequency for Raman spin-flip processes scale in the same way with respect to the ratio of laser power to detuning, for a given atom the coupling strength relative to the spontaneous emission rate is fixed by the fine structure splittingcomparedtothenaturallinewidth. Thishasnot been a limitation for the demonstration of single-particle or mean-field physics [4–8], but will become a severe re- striction for many-body physics where the interactions willintroduceasmallerenergyscaleandthereforerequire longer lifetimes of the atomic sample. Some authors have considered transitions involving metastable states of al-kaline earth atoms to reduce the effects of spontaneous emission [21, 22]. Here we present a spin-orbit coupling scheme that does not involve spin-flips, is diagonal in the spin component, z, and corresponds to an Abelian SU(2)gauge field. This scheme can be implemented with far-off resonant laser beams, thus overcoming the limitationof short sam- ple lifetimes due to spontaneous emission. In the field of cold atoms, many discussions of spin-orbit coupling em- phasize its close relationship to non-Abelian gauge fields [17, 23] which are non-diagonal for any spin component and therefore mix spin and motion in a more compli- cated way. However, a scheme diagonal in the spin com- ponent is sufficient for spin Hall physics and topologi- cal insulators [24, 25], and its implementation has major experimental advantages. In the theoretical proposals [26, 27] and the demonstration [23] of the spin Hall effect for quantum gases, Raman spin-flips are used to create an Abelian gauge field diagonal with respect to one spin component. The physical principle of the spin-orbit coupling scheme presented here is very different from spin-flip schemes. It does not require any kind of spin-orbit cou- pling within the atom. Rather, spin-dependent vector potentials are engineered utilizing the Zeeman effect in a magnetic field – atoms in the spin up and down states interact with different pairs of laser beams, or differently with the same pair, and the photon recoil changes the atom’s motion in a spin-dependent way. This results in spin-orbit coupling which is diagonal in the spin basis. To begin, we summarize the relationship between spin- orbit coupling and spin-dependent vector potentials. For charged particles, the origin of spin-orbit coupling is the relativistic transformation of electromagnetic fields. Whenanelectronmovesthroughanelectricfield E, itex- periences a magnetic field Bin its moving frame which interacts with the spin (described by the Pauli spin matricies). Spin-orbit coupling contributes a term pro- portional to (pE)
in the Hamiltonian. As such, an electric field in the z-direction gives rise to the Rashba spin-orbit coupling (p)z=xpyypx.arXiv:1308.6349v1 [cond-mat.quant-gas] 29 Aug 20132 Assuming a 2D system confined to the x;yplane, and an in-plane electric field, the spin-orbit interac- tion conserves z. Following [25], a radial electric field EE(x;y;0)leads to a spin-orbit coupling term in the Hamiltonian of the form Ez(xpyypx). Such a radial field could be created by a uniformly charged cylinder, or can be induced by applying stress to a semiconductor sample [25]. This spin-coupling term is identical to the A
pterm for the Hamiltonian describing a spin in a magnetic field, zB. Using the symmetric gauge for the vector potential A=zB 2(y;x;0), one obtains a term in the Hamiltonian proportional to zB(xpyypx)or equivalently to B
L, where Lis the orbital angular momentum of the atom. Therefore, this form of spin- orbit coupling is equivalent to a spin-dependent magnetic field which exerts opposite Lorentz forces on spin up and down atoms. This leads to the spin Hall effect which creates a transverse spin current and no charge or mass currents [24, 25]. The A2term constitutes a parabolic spin-independentpotentialwhichisirrelevantforthespin physics discussed here. We now present a scheme which realizes such an Abelian gauge field and manifests itself as a spin- dependent magnetic field. Recently, the MIT group [28,29]andtheMunichgroup[30,31]havesuggestedand implemented a scheme to generate synthetic magnetic fields for neutral atoms in an optical lattice. The scheme is based on the simple Hamiltonian for non-interacting particles in a 2D cubic lattice, H=X m;n Jx^ay m+1;n^am;n+Jy^ay m;n +1^am;n+h:c:
(1) whereJx(y)describestunnelinginthe x-(y-)directionand ^ay m;n(^am;n) is the creation (annihilation) operator of a particle at lattice site (m;n). The setup is detailed in [28] and summarized as follows: a linear tilt of energy
per lattice site is applied using a magnetic field gradient in thex-direction, thus suppressing normal tunneling in this direction. Resonant tunneling is restored with two far-detuned Raman beams of two-photon Rabi frequency , frequency detuning !=!1!2, and momentum transfer k=k1k2. Considering only the case of reso- nant tunneling, !=
=~, rapidly oscillating terms time average out [32], yielding an effective Hamiltonian which is time-independent [28]. H=X m;n Keim;n^ay m+1;n^am;n+J^ay m;n +1^am;n+h:c:
(2) This effective Hamiltonian describes charged particles on a lattice in a magnetic field under the tight-binding approximation [33, 34]. The gauge field arises from the spatially-varying phase m;n=k
Rm;n=mkxa+nkya whereais the lattice constant and has the form A= ~(kxx+kyy)=a^x. One can tune the flux per unit cell, , αk1,ω1k2,ω2 |á〉 Δ k1,ω1k2,ω2 |â〉 Δ-αKeiφm,n Ke-iφm,nFIG. 1. Spin-dependent tunneling in an optical lattice tilted by a magnetic field gradient. When the two spin states have opposite magnetic moments, then the role of absorbtion and emission of the two photons is exchanged. The result is that the two states have tunneling matrix elements with opposite phases, leading to opposite synthetic magnetic fields and re- alizing spin-orbit coupling and the quantum spin Hall effect. for a given spin state over the full range between zero and one by adjusting the angle between the Raman beams, and consequently ky. We now extend this scheme to the spin degree of free- dom, and assume a mixture of atoms in two hyperfine states, labeled spin up and down. If the potential en- ergy gradient is the same for the two states, then the two states experience the same magnetic field. This is the situation when the tilt is provided by gravity, a scalar AC Stark shift gradient, or a magnetic field gradient if both states have the same magnetic moment – the phase m;nis independent of z. If the two states have the same value of the magnetic moment, but opposite sign, then the potential gradient is opposite for the two states. This can be realized by using states of the same hyperfine level F, but with opposite magnetic quantum number MF(e.g. in23Na or87Rb, thejF;MFi=j2;2iandj2;2istates), or by picking another suitable pair of hyperfine states. In this case, for laser-assisted tunneling between two sites mandm+ 1, the roles of the two laser beams – absorption of a photon versus stimulated emission of a photon – for the Raman process are reversed as depicted in Fig. 1. Therefore, the two states receive opposite momentum transfer, and this sign change leads to a sign change for the enclosed phase: m;n= (mkxa+nkya)z (3) and also for the vector potential and the magnetic field. The vector potential realized by this scheme: A=~ a(kxx+kyy)^x^z (4)3 creates the spin-orbit coupling discussed in the introduc- tion, although in a different gauge. The x-dependence in thex-component of Ais necessary for a non-negligible tunneling matrix element for the laser-assisted process [28]. This system has now time reversal symmetry, in con- trast to the system with the same synthetic magnetic field for both states (since a magnetic field breaks time reversal symmetry). It therefore realizes the quantized spin Hall effect consisting of two opposite quantum Hall phases. It is protected by a Ztopological index due to fact thatzis conserved [24, 25]. When the values of the two magnetic moments are dif- ferent, and the potential energy gradient is provided by a magnetic field gradient, then the two states have differ- ent Bloch oscillation frequencies,
=h. Each state now needs two separate Raman beams for laser-assisted tun- neling (or they can share one beam). This implies that the synthetic magnetic field can now be chosen to be the same, to be opposite or to be different for the two spin states. One option is to have zero synthetic magnetic field for one of the states. Atoms in this state can still tunnel along the tilt direction by using a Raman process withouty-momentum transfer, or equivalently, by induc- ingtunnelingthroughlatticemodulation[35]. Inthecase of two different magnetic moments, one could also per- form dynamic experiments, where laser parameters are modified in such a way that one switches either suddenly or adiabatically from the quantum Hall effect to the spin quantum Hall effect. An intriguing possibility is to couple the two states. Sincezis no longer conserved, the system should be- come a topological insulator with the Z2classification [3, 36], provided that the coupling is done in a time- reversal invariant way. This can be done with a term which is not diagonal in z– ie. axpyterm – by adding spin-flip Raman lasers to induce spin-orbit coupling, or by driving the spin-flip transition with RF or microwave fields. A coherent RF drive field would not be time- reversal invariant, but it would be interesting to study the effect of symmetry-breaking in such a state [37]. A drive field where the phase is randomized should lead to a time-reversal invariant Hamiltonian. Our scheme implements the idealized scheme for a quantum spin Hall system consisting of two opposite quantum Hall phases. This is a starting point for break- ing symmetries and exploring additional terms in the Hamiltonian. Ref. [37, 38] discusses a weak quantum spin Hall phase, induced by breaking the time-reversal symmetry by a magnetic field - this can be achieved by population imbalance between the two spin states. A spin-imbalanced quantum Hall phase can turn into a spin-filtered quantum Hall phase [37, 38] where only one component has chiral edge states. This can be achieved by realizing a finite synthetic magnetic field for one com- ponent, and zero for the other. Changing the spin-orbit |á〉 |â〉 790 nm @28.7˚Δ Δ Δ2ΔΔ1064 nm, kLat kSup=kLat/3+– 2ΔA A B B C C =FIG. 2. Superlattice scheme for realizing the quantnum Hall and quantum spin Hall effect. A superlattice with three times the spatial period as the fundamental lattice leads to three distinguishablesitesA,B,C.ForthequantumspinHalleffect, the superlattice operates at a magic wavelength where the AC Stark effect is opposite for the two spin states. For rubidium, this is achieved at a wavelength of 790 nm. coupling can induce topological quantum phase transi- tions between a helical quantum spin Hall phase and a chiral spin-imbalanced quantum Hall state. This can probably be achieved in a population imbalanced system by adding additional Raman spin-flip beams [37, 38]. So far, we have discussed single-particle physics. Adding interactions, by increasing the density with deeper lattices or through Feshbach resonances, will in- duce interesting correlations and may lead to fractional topological insulators [39]. Another option are spin-drag experiments [40, 41], transport experiments where one spin component transfers momentum to the other com- ponent. For the situation mentioned above, where the synthetic magnetic field is zero for one component (e.g. spin up), a transport experiment revealing the Hall ef- fect [42] for spin down would show a non-vanishing Hall conductivity for spin up to due to spin drag. In addition, one would expect that spin-exchange interactions destroy the two opposite quantum Hall phases, and should lead to the quantum spin Hall phase with Z2topological in- dex. We now present another way of realizing the physics discussed above, using optical superlattices instead of a potential energy gradient. This has the advantage of purely optical control, and avoids possible heating due to Landau-Zener tunneling [43] between Wannier Stark states. So far, optical superlattices have allowed the ob- servation of the ground state with staggered magnetic flux [44], in contrast to experiments with magnetic tilts [28, 31]. Figure 2 summarizes the new scheme. The super- lattice has three times the period of the basic lattice,4 thus distinguishing sites A, B, C in energy. Resonant tunneling is re-established using three pairs of Raman beams with frequencies: !1+
AB=~,!2+
BC=~, and !3(
AB+
BC)=2~collinear in one arm and !1, !2, and!3collinear in another arm at an angle to the first. Consequently, there is always the same momen- tum transfer for tunneling in the y-direction, leading to the same flux as the scheme with the magnetic tilt, and Eqs. (3) and (4) apply. This is in contrast to schemes with two distinguished sites A and B (by using internal states [21, 32] or a superlattice [44]) which lead to a stag- gered magnetic field. Rectification of the magnetic flux in a staggered configuration by adding a tilt [32, 44] or a superlattice [21] has also been proposed. In the lat- ter scheme, this would result in four distinguishable sites (two internal states A, B, doubled up by the superlat- tice). Another rectification scheme uses three internal states [45]. Our scheme avoids spin-flip transitions be- tween internal states, and has the minimum number of ingredients of three different sites to provide direction- ality. Furthermore, by adjusting the spatial phase shift between the fundamental and the superlattice, one can choose the energy offsets
AB=
BC=
CA=2(see Figure 2). The scheme can then be implemented by shin- ing Raman beams from two directions, each beam having two frequencies. This scheme would realize Hofstadter’s butterfly and the quantum Hall effect. For the quantum spin Hall ef- fect, one has to choose the superlattice laser to be at the magic wavelength where the scalar AC Stark shift vanishes, and only a vector AC Stark shift remains cor- responding to a so-called fictitious magnetic field [46, 47]. By detuning the laser between the D1andD2lines, one can achieve a pure vector AC Stark shift, which is equal in magnitude, but opposite in sign when the atoms in the two hypefine states have opposite magnetic moments. In thiscase, thesuperlatticewillprovideoppositepotentials forthetwostates, resultinginoppositemomentumtrans- fers due to the Raman beams and opposite vector poten- tials. The superlattice period is =(2 sin (=2)), where is the angle between the two superlattice beams, which is adjusted to make the superlattice period three times the period of the basic lattice. This scheme realizes the quantum spin Hall effect and a topological insulator with two opposite quantum Hall phases with a purely optical scheme and no Raman spin-flip transitions. To replace the magnetic field gradient by a superlattice that generates a fictitious magnetic field, the laser detun- ing has to be on the order of the fine structure splitting, resulting in heating due to spontaneous emission. For atoms like rubidium, the lifetime is many seconds [46]. To be specific, we consider a low-density gas Rb atoms in theF= 2,MF=2states in a lattice with a depth of ten photon recoils at the wavelength of 1064 nm. A superlattice with a lattice depth of 10 kHz is created by interfering two laser beams at 790.0 nm of 1.0 mW oflaser power and a beam waist of 125 m. The result- ing offset
ABand
ACwill be approximately 4 and 8 kHz respectively, well placed in the bandgap of the basic lattice. The spontaneous scattering rate induced by the superlattice beams is less than 0.1 =s. Alternatively, the superlatticeproducingthefictitiousmagneticfieldcanbe replaced by a sinusoidal (real) magnetic field generated by an atom chip [48]. There have been several suggestions how to detect properties of the quantum Hall and quantum spin Hall phases. Time-of-flight pictures will reveal the enlarged magnetic unit cell due to the synthetic magnetic field [44, 49–51]. Hall plateaus can be discerned in the den- sity distribuion [52]. The Chern number of a filled band can be measured interferometically [53] or using ballistic expansion [54]. Topological edge states can be directly imaged [55, 56] or detected by Bragg spectroscopy [57– 59]. Our work maps out a route towards spin-orbit cou- pling, the spin Hall effect and topological insulators which does not require coupling of different internal states with spin-flipping Raman lasers. The Hamiltonian describing the system is diagonal in the zspin com- ponent. This follows closely the two original papers on the spin Hall effect [24, 25]. In addition, we have pre- sented two configurations for realizing a quantum spin Hall Hamiltonian. The scheme with the magnetic tilt completely avoids near resonant light, and the superlat- tice scheme provides a purely optical approach. This work was supported by the NSF through the Cen- ter of Ultracold Atoms, by NSF award PHY-0969731, under ARO Grant No. W911NF-13-1-0031 with funds from the DARPA OLE program, and by ONR. This work was completed at the Aspen Center for Physics (sup- ported in part by the National Science Foundation under Grant No. PHYS-1066293), and insightful discussions with Hui Zhai, Jason Ho, and Nigel Cooper are acknowl- edged. We thank Wujie Huang for a critical reading of the manuscript. After most of this work was completed [60] we became aware of similar work carried out in the group of I. Bloch in Munich [31, 61]. [1] Nature Materials Insight on spintronics, Vol. 11, May 2012. [2] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004). [3] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [4] Y.J.Lin, K.Jiménez-García, andI.B.Spielman,Nature 471, 83 (2011). [5] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B.Yan, G.-S.Pan, B.Zhao, Y.-J.Deng, H.Zhai, S.Chen, and J.-W. Pan, Phys. Rev. Lett. 109, 115301 (2012).5 [6] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [7] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [8] V. Galitski and I. B. Spielman, Nature 494, 49 (2013). [9] J. Higbie and D. M. Stamper-Kurn, Phys. Rev. Lett. 88, 090401 (2002). [10] G.Juzeli¯ unasandP.Öhberg,Phys.Rev.Lett. 93,033602 (2004). [11] G. Juzeli¯ unas, J. Ruseckas, P. Öhberg, and M. Fleis- chhauer, Phys. Rev. A 73, 025602 (2006). [12] J. Ruseckas, G. Juzeli¯ unas, P. Öhberg, and M. Fleis- chhauer, Phys. Rev. Lett. 95, 010404 (2005). [13] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein, Phys. Rev. Lett. 95, 010403 (2005). [14] B. M. Anderson, I. B. Spielman, and G. Juzeli¯ unas, Preprint, arXiv:1306.2606. [15] G. Juzeli¯ unas, J. Ruseckas, and J. Dalibard, Phys. Rev. A81, 053403 (2010). [16] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. Öhberg, Rev. Mod. Phys. 83, 1523 (2011). [17] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011). [18] B. M. Anderson, G. Juzeli¯ unas, V. M. Galitski, and I. B. Spielman, Phys. Rev. Lett. 108, 235301 (2012). [19] Z. F. Xu and L. You, Phys. Rev. A 85, 043605 (2012). [20] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi, Phys. Rev. Lett. 109, 085302 (2012). [21] F. Gerbier and J. Dalibard, New J. Phys. 12, 033007 (2010). [22] B. Béri and N. R. Cooper, Phys. Rev. Lett. 107, 145301 (2011). [23] M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, Nature 498, 201 (2013). [24] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). [25] B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006). [26] X.-J. Liu, X. Liu, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 98, 026602 (2007). [27] S.-L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan, Phys. Rev. Lett. 97, 240401 (2006). [28] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur- ton, and W. Ketterle, Preprint, arXiv:1308.1431. [29] H. Miyake, Probing and Preparing Novel States of Quan- tum Degenerate Rubidium Atoms in Optical Lattices , Ph.D. thesis, Massachusetts Institute of Technology (2013). [30] M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch, Applied Physics B (2013), 10.1007/s00340-013-5418-1. [31] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Preprint, arXiv 1308.0321. [32] D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003). [33] P. G. Harper, Proc. Phys. Soc. A 68, 874 (1955).[34] D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976). [35] V. V. Ivanov, A. Alberti, M. Schioppo, G. Ferrari, M. Ar- toni, M. L. Chiofalo, and G. M. Tino, Phys. Rev. Lett. 100, 043602 (2008). [36] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). [37] N. Goldman, W. Beugeling, and C. M. Smith, Europhys. Lett. 97, 23003 (2012). [38] W. Beugeling, N. Goldman, and C. M. Smith, Phys. Rev. B 86, 075118 (2012). [39] M. Levin and A. Stern, Phys. Rev. Lett. 103, 196803 (2009). [40] R. A. Duine and H. T. C. Stoof, Phys. Rev. Lett. 103, 170401 (2009). [41] S. B. Koller, A. Groot, P. C. Bons, R. A. Duine, H. T. C. Stoof, and P. v. d. Straten, Preprint arXiv:1204.6143. [42] L. J. LeBlanc, K. Jiménez-García, R. A. Williams, M. C. Beeler, A. R. Perry, W. D. Phillips, and I. B. Spielman, PNAS 109(27), 10811 (2012). [43] Q. Niu, X. G. Zhao, G. A. Georgakis, and M. G. Raizen, Phys. Rev. Lett. 76, 4504 (1996). [44] M. Aidelsburger, M. Atala, S. Nascimbéne, S. Trotzky, Y. A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011). [45] E. J. Mueller, Phys. Rev. A 70, 041603 (2004). [46] D. McKay and B. DeMarco, New J. Phys. 12, 055013 (2010). [47] C. Cohen-Tannoudji and J. Dupont-Roc, Phys. Rev. A 5, 968 (1972). [48] N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, Phys. Rev. Lett. 105, 255302 (2010). [49] G. Möller and N. R. Cooper, Phys. Rev. A 82, 063625 (2010). [50] S. Powell, R. Barnett, R. Sensarma, and S. DasSarma, Phys. Rev. A 83, 013612 (2011). [51] T. P. Polak and T. A. Zaleski, Phys. Rev. A 87, 033614 (2013). [52] R. O. Umucalılar, H. Zhai, and M.Ö. Oktel, Phys. Rev. Lett. 100, 070402 (2008). [53] D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 165304 (2013). [54] E. Zhao, N. Bray-Ali, C. J. Williams, I. B. Spielman, and I. I. Satija, Phys. Rev. A 84, 063629 (2011). [55] T. D. Stanescu, V. Galitski, J. Y. Vaishnav, C. W. Clark, and S. DasSarma, Phys. Rev. A 79, 053639 (2009). [56] N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, PNAS 110(17), 6736 (2013). [57] X.-J. Liu, X. Liu, C. Wu, and J. Sinova, Phys. Rev. A 81, 033622 (2010). [58] N. Goldman, J. Beugnon, and F. Gerbier, Phys. Rev. Lett. 108, 255303 (2012). [59] M. Buchhold, D. Cocks, and W. Hofstetter, Phys. Rev. A85, 063614 (2012). [60] W. Ketterle, Talk at the International Conference on Quantum Technologies Moscow, 7/20 - 7/24/2013. [61] I. Bloch, Talk at the International Conference on Quan- tum Technologies Moscow, 7/20 - 7/24/2013.

1101.4656v3.Identifying_spin_triplet_pairing_in_spin_orbit_coupled_multi_band_superconductors.pdf

arXiv:1101.4656v3 [cond-mat.supr-con] 3 May 2012Identifying spin-triplet pairing in spin-orbit coupled mu lti-band superconductors Christoph M. Puetter1andHae-Young Kee1,2 (a) 1Department of Physics, University of Toronto, Toronto, Ont ario M5S 1A7, Canada 2Canadian Institute for Advanced Research, Quantum Materia ls Program, Toronto, Ontario M5G 1Z8, Canada PACS74.20.-z – Theories and models of superconducting state PACS74.20.Rp – Pairing symmetries (other than s-wave) PACS74.70.Pq – Ruthenates Abstract – We investigate the combined effect of Hund’s and spin-orbit (SO) coupling on su- perconductivity in multi-orbital systems. Hund’s interac tion leads to orbital-singlet spin-triplet superconductivity, where the Cooper pair wave function is a ntisymmetric under the exchange of two orbitals. We identify three d-vectors describing even-parity orbital-singlet spin-tr iplet pairings among t 2g-orbitals, and find that the three d-vectors are mutually orthogonal to each other. SO coupling further assists pair formation, pins the orientat ion of the d-vector triad, and induces spin- singlet pairings with a relative phase difference of π/2. In the band basis the pseudospin d-vectors are aligned along the z-axis and correspond to momentum-dependent inter- and intr a-band pair- ings. We discuss quasiparticle dispersion, magnetic respo nse, collective modes, and experimental consequences in light of the superconductor Sr 2RuO4. Introduction. – Since its inception, standard Bardeen-Cooper-Schrieffer (BCS) theory has been con- sidered a classic example for a collective phase emerging from quantum many body effects. However, the discovery of unconventional superconducting phases near antiferro- magnetic order in heavy fermion compounds [1,2], organic materials [3], and, most recently, Fe-pnictides [4] have ex- posed the limits of a single-band BCS formulation. The origin and nature of superconductivity in complex mate- rials where multiple bands cross the Fermi level therefore remains a field of active research, harbouring intriguing challenges and mysteries. In particular, when the electronic structure near the Fermi energy is composed of different orbitals and spins mixed via spin-orbit (SO) coupling, a pairing symmetry analysis could be non-trivial. For example, a local micro- scopic interaction such as Hund’s coupling may naturally favour inter-orbital spin-triplet pairing between electrons. However,whenorbitalandspinfluctuationsaresignificant due to inter-orbital hopping and SO interaction, pairing in definite orbital and spin channels ( e.g., spin-singlet or -triplet pairing between electron in orbitals aandb) is not well defined. Equivalently, from a Bloch band per- (a)E-mail:hykee@physics.utoronto.caspective, where the kinetic Hamiltonian including SO ef- fectsisdiagonal,thedecouplingofthemicroscopicinterac- tion effectively leads to intra- and inter-band pairing with pseudospin-singlet and/or -triplet character. Below we present a systematic study of how SO and Hund’s couplings jointly give rise to superconductivity in t2g(i.e., dyz, dxz, and d xy) orbital systems. Our find- ings may apply to a number of multi-orbital d-subshell superconductors. To be specific we base our quantitative considerations on the proposed chiral spin-triplet super- conductor Sr 2RuO4. Here, despite intense investigation for more than a decade, a clear picture for the pairing symmetry, the pairing mechanism and the relevant bands involved that is consistent with all experimental observa- tions has not yet emerged [5,6]. The paper is organized as follows. In the second sec- tion we discuss Cooper pairing in multi-orbital systems. We find that superconductivity from local Hund’s ex- change can naturally be characterized by three mutually orthogonal d-vectors each describing inter-orbital even- parity spin-triplet pairing. Wethen showhowSOcoupling pins the orientation of the d-vector triad and induces and enhances pairing via coupling to spin-singlet pairing order parameters with a fixed relative phase difference of π/2. In the third section, we map these local pairing order pa- p-1Christoph M. Puetter and Hae-Young Kee rameters, defined in an orbital and spin basis, to inter- and intra-band pairing in the Bloch band basis. Pairing in the Bloch bands has a strong momentum dependence and the magnitude and direction of the d-vectors depend on the orbital composition at each k-point. In the fourth section, we present the complete self-consistent mean-field (MF) results involving 9 complex order parameters using band structure parameters that reproduce the Fermi sur- face (FS) reported on Sr 2RuO4. In addition, the resulting anisotropicquasiparticle(QP)dispersion,themagneticre- sponse and the critical pairing strengths in the presence of SO coupling are considered. We summarize our findings and discuss the relevance for SO-coupled d-orbital super- conductors such as Sr 2RuO4in the last section. Pairing in SO coupled t 2gsystems via Hund’s in- teraction. – For multi-orbital 3d-subshell systems such astheFe-pnictides, itwasrecognizedthatHund’scoupling (interaction strength denoted by J) is as important as on- site Coulomb repulsion ( U) [7,8], while SO coupling (2 λ) is relatively weak [9]. In contrast, recent x-ray measure- ments on 5d transition metal compounds such as Ir-based oxide materials found that the SO interaction of 0.6 eV is roughly comparable to the on-site Coulomb energy [10], suggesting that SO interaction is larger than Hund’s ex- change (since J < U). Given that the effective pairing interaction in the spin-triplet channel arising from Hund’s couplingandinter-orbitalHubbardrepulsion( V=U−2J) scales as V−J=U−3J(see below), we therefore ex- pect that for 4d-subshell materials such as Sr 2RuO4both SO and spin-triplet pairing interactions are intermediate in strength and of similar magnitude [11–17]. Since nei- ther interaction is negligible nor dominant, we treat both on an equal footing in the present study. While on-site Hund’s and further neighbor exchange in- teractions have been recognized to be important for spin- triplet pairing [7,18–21], the combined effect of SO and Hund’s couplings on inter-orbital spin-triplet pairing has not been investigated in t 2g-orbital systems. To under- stand superconductivity in SO coupled t 2g-orbital sys- tems,weconsideragenericHamiltonian H=Hkin+HSO+ Hintconsisting of kinetic, SO, and local Kanamori inter- action terms. In this section we leave the kinetic Hamil- tonianHkinunspecified and focus on the pairing proper- ties arising from the interplay of the atomic SO coupling HSO= 2λ/summationtext iLi·Siand the local interaction, which, pro- jected on the t 2gorbitals, are given by HSO=iλ/summationdisplay i/summationdisplay ablǫablca† iσcb iσ′ˆσl σσ′, (1) Hint=U 2/summationdisplay i,aca† iσca† iσ′ca iσ′ca iσ+V 2/summationdisplay i,a/negationslash=bca† iσcb† iσ′cb iσ′ca iσ +J 2/summationdisplay i,a/negationslash=bca† iσcb† iσ′ca iσ′cb iσ+J′ 2/summationdisplay i,a/negationslash=bca† iσca† iσ′cb iσ′cb iσ. (2) Here and in the following, summation over repeated spin Fig. 1: (Color online) The orbital-singlet spin-triplet d-vectors form a triad whose orientation is pinned along ˆx,ˆy, andˆz(or −ˆx,−ˆy, and−ˆz) in the presence of SO coupling. See main text for details. indicesσ,σ′=↑,↓is implied while the indices a,b∈ {yz,xz,xy }belong to an ordered set of t2g-orbitals. Fur- thermore, ˆ σlstands for Pauli matrices, ca† iσcreates an elec- tron on site iin orbital awith spin σ, andǫabldenotes the totally antisymmetric rank-3tensor. For transparency we have also introduced separate interaction strengths for Hund’s coupling ( J) and pair hopping ( J′), although J=J′at the atomic level. Let us apply a MF approach to study the particle- particle instabilities of the microscopic interaction Hint using the following zero momentum pairing channels ˆ∆s a/b=1 4N/summationdisplay k[iˆσy]σσ′(ca kσcb −kσ′+cb kσca −kσ′),(3) ˆdl a/b=1 4N/summationdisplay k[iˆσyˆσl]σσ′(ca kσcb −kσ′−cb kσca −kσ′),(4) whereNis the number of kpoints. Here, ∆s a/b= /an}bracketle{tˆ∆s a/b/an}bracketri}ht(= ∆s b/a) stands for intra- ( a=b) and inter- orbital ( a/ne}ationslash=b) spin-singlet pairing, which is even un- der the exchange of orbital quantum numbers ( i.e.they form “orbital triplets”). The vector order parameter da/b= (/an}bracketle{tˆdx a/b/an}bracketri}ht,/an}bracketle{tˆdy a/b/an}bracketri}ht,/an}bracketle{tˆdz a/b/an}bracketri}ht)(=−db/a)ontheotherhand parametrizes inter-orbital ( a/ne}ationslash=b) spin-triplet pairing con- sistent with the usual d-vector notation where i(d·ˆσ)ˆσy describes the spin-triplet pairing gap [2,22]. Note that da/bis odd under orbital exchange, which is characteristic ofan“orbitalsinglet”(while da/a= 0). Note alsothat the above order parameters are all even under a parity trans- formationas they are locallydefined; this feature differs in particular from conventional odd-parity spin-triplet pair- ing where orbital degrees of freedom are absent. Using the above pairing channels the interaction Hamil- tonian takes the form Hint→UN/summationdisplay aˆ∆s† a/aˆ∆s a/a+(V−J)N/summationdisplay a,b,lˆdl† a/bˆdl a/b +J′N/summationdisplay a/negationslash=bˆ∆s† a/aˆ∆s b/b+(V+J)N/summationdisplay a/negationslash=bˆ∆s† a/bˆ∆s a/b,(5) where it is clear that only Hund’s coupling can give rise to an instability in a spin-triplet channel [7,19]. We thus p-2Identifying spin-triplet pairing in spin-orbit coupled multi-band super conductors concentrate on the effective pairing interaction H′ int= (U−3J)N/summationdisplay a,b,lˆdl† a/bˆdl a/b (6) in the attractive regime U/3< J(< U). In gen- eral, orbital-singlet spin-triplet pairing can also induce spin-singlet pairing so that the remaining terms in Eq. (5) would hamper spin-singlet pairing. However, we as- sume that their effect is negligible to keep the follow- ing self-consistent calculations feasible, and since the in- duced spin-singlet pairing amplitudes are for the most part smaller than the spin-triplet pairing amplitudes (see below). For notational clarity we label in the follow- ing inter-orbital pairing only by the three combinations a/b=xz/xy,yz/xy,yz/xz . To understand the effect of SO interaction, let us re- mark on pairing in the absence of SO coupling first. In the case of the layered compound considered below (and for a rather large parameter range) the three spin- triplet d-vectors dxz/xy,dyz/xy, anddyz/xzform a triad of mutually orthogonal vectors with an arbitrary orien- tation and chirality in spin space, and no relative com- plex phase difference (hence preserving time reversal sym- metry (TRS)). This can be understood by analyzing the Ginzburg-Landau (GL) free energy, which without SO coupling is given by F ∼/summationdisplay ν/bracketleftbig Aν|dν|2+B(1) ν(dν·d∗ ν)2+B(2) ν|dν·dν|2/bracketrightbig +/summationdisplay ν/negationslash=κ/bracketleftbig C(1) νκ(dν·dν)(dκ·dκ)∗+C(2) νκ|dν|2|dκ|2(7) +C(3) νκ|dν·dκ|2+C(4) νκ|dν·d∗ κ|2+C(5) νκ(dν·d∗ κ)2/bracketrightbig up to fourth order, by analogy to He-3 [23]. Here ν,κ stand for orbital pairs a/b, while the (real) quartic mix- ing parameters obey C(i) νκ=C(i) κνand the asymmetry between in-plane and out-of-plane orbitals due to e.g. inter-orbital hopping is reflected in distinct coefficients (Ayz/xz/ne}ationslash=Ayz/xy=Axz/xy, etc.). This form is dictated by gauge symmetry, SU(2) spin rotationalsymmetry, time reversal symmetry and the underlying lattice symmetries, and shows that the C(3) νκandC(4) νκterms are sensitive to the relative orientation of the d-vectors, whereas the C(1) νκ andC(5) νκcontributions additionally depend on their rela- tive complex phases. However, once SO coupling is included, dxz/xy, dyz/xy, anddyz/xzare pinned along x,y, andz directions, respectively, as shown in fig. 1. In- version/time reversal symmetry on the other hand is still preserved and reflected in the degeneracy of the orientations/chiralities {dxz/xy,dyz/xy,dyz/xz}and {−dxz/xy,−dyz/xy,−dyz/xz}. The pinning of the d- vectors occurs due to additional terms in the free en- ergy such as ∼a(1)|dz yz/xz|2+a(2)/bracketleftbig |dz yz/xy|2+|dz xz/xy|2/bracketrightbig + b(1)/bracketleftbig |dx yz/xz|2+|dy yz/xz|2/bracketrightbig +b(2)/bracketleftbig |dx xz/xy|2+|dy yz/xy|2/bracketrightbig +c(1)/bracketleftbig dx yz/xy(dy xz/xy)∗+dy yz/xy(dx xz/xy)∗+c.c./bracketrightbig +···, where the expansion parameters depend on the SO coupling strength, naively suggesting that a(1),a(2)< b(1),b(2),c(1), etc.1SO interaction furthermore leads to a linear cou- pling between a particular component of (inter-orbital) spin-triplet pairing and (intra-orbital) spin-singlet pair- ing. For example, writing SO coupling between yzandxz orbitals in the form of −iλ[ˆσz]σσ′(cyz† kσcxz kσ′−cxz† kσcyz kσ′) the following linear coupling is allowed in the GL free energy: −i λ[ˆσz]σσ′/an}bracketle{tcyz† kσcxz kσ′−cxz† kσcyz kσ′/an}bracketri}ht (8) ×[iˆσyˆσz]σσ′/an}bracketle{tcyz kσcxz −kσ′−cxz kσcyz −kσ′/an}bracketri}ht ×/parenleftBig [iˆσy]σσ′/an}bracketle{tcyz† kσcyz† −kσ′/an}bracketri}ht+[iˆσy]σσ′/an}bracketle{tcxz† kσcxz† −kσ′/an}bracketri}ht/parenrightBig →iλdz yz/xz/parenleftBig ∆s yz/yz+∆s xz/xz/parenrightBig∗ +c.c. (9) Note that dyz/xzprefers the z-direction by coupling to spin-singlet pairing with a relative phase difference of ±π/2 depending on the sign of λ. This is consistent with our findings below that the spin-triplet order param- eters are purely real while the spin-singlet amplitudes are purely imaginary. A similar analysis can be carried out fordx xz/xyanddy yz/xy. The overall order parameter for yzandxzorbitals then is dz xz/yz+i(∆s xz/xz+ ∆s yz/yz). Since the relative phase between the orbital-triplet spin- singlet and the orbital-singlet spin-triplet order param- eters is fixed, there should be a collective mode rep- resenting a resonance of supercurrent flow between the coupled order parameters with an energy scale of order ∼/radicalBig |dz a/b|2+|∆s a/a|2+|∆s b/b|2. Note that the above result is fundamentally different from similar two orbital models, which lead to a single orbital-singletspin-triplet d-vector[19,21,24]. Thepresent model is also distinguished from other models where the momentum dependence in the band pairing usually orig- inates from nonlocal momentum dependent interactions [18], whereas here it arises from spin and orbital mixing in the Bloch bands as described next. Momentum-dependent pairing in the Bloch bands. – Despite having uniform pairing amplitudes dyz/xz,dyz/xy,dxz/xy,∆s yz/yz,...thecorrespondinginter- and intra-band pairings in the Bloch band basis (now car- rying band and pseudospin quantum numbers – η,ρ= α,β,γands=±) acquire a strong momentum depen- dence due to the mixing of orbitals through hopping and SO coupling. To understand how the above local pairing in the orbital and spin basis corresponds to pairing in the Blochbandbasis,letusintroducethekineticHamiltonian. The most generic kinetic Hamiltonian for t 2gorbitals in a 1Analyzing the energetics of a corresponding two orbital mod el one can indeed show that SO interaction tends to stabilize e.g.the dz yz/xz-component over dx yz/xzordy yz/xz. p-3Christoph M. Puetter and Hae-Young Kee (b)+ −k −|<ξ ξ >|α γ k+ −k −|<ξ ξ >|α β k+ −k −|<ξ ξ >|γ β k+ −k −|<ξ ξ >|α α k+ −k −|<ξ ξ >|γ γ k+ −k −|<ξ ξ >|β β (a) k Fig. 2: (Color online) Momentum-resolved pairing amplitud es in the Bloch band basis for 3 J−U= 0.9 andλ= 0.15. Panel (a) and (b) represent inter- and intra-band pairing, respec - tively. The grey lines indicate the β,γ, andαFS sheets (from inside to outside). Note that pairing from Hund’s coupling preferentially involves electronic states near the FS shee ts and that the intra-band pairing amplitudes are about one order o f magnitude larger than inter-band pairing amplitudes. single layer perovskite structure has the form Hkin+HSO=/summationdisplay k,σC† kσ εyz kε1d k+iλ−λ ε1d k−iλ εxz kiλ −λ−iλ εxy k Ckσ,(10) whereC† kσ= (cyz† kσ,cxz† kσ,cxy† k−σ) and the dispersions areεyz/xz k=−2t1cosky/x−2t2coskx/y−µ1,εxy k= −2t3/parenleftbig coskx+ cosky/parenrightbig −4t4coskxcosky−µ2, andε1d k= −4t5sinkxsinky. For the MF calculation below we have chosen the parameters t1= 0.5,t2= 0.05,t3= 0.5, t4= 0.2,t5= 0.05,µ1= 0.55, and µ2= 0.65 (all en- ergies here and in the following are expressed in units of 2t1= 1.0). The underlying FS obtained from diagonaliz- ingHkinwith SO coupling strength λ= 0.15 is shown in fig. 2 along with momentum-dependent band pairing am- plitudes. The FS agrees well with first principles calcula- tions[14]andthe experimentallymeasuredFS ofSr 2RuO4 [17,25,26], consisting of three bands labelled α,β, andγ. In the presence of SO coupling the bands are mix- tures of all three orbitals and different spins, e.g.ξη k+= ˜fη kcxz k↑+ ˜gη kcyz k↑+˜hη kcxy k↓(η=α,β,γ). Hence consider- ing inter- and intra-band pairing amplitudes in the band basis, it is clear that the x- andy-components of the inter-band pseudospin-triplets such as /an}bracketle{tξη k±ξρ −k±/an}bracketri}htvanish, since/an}bracketle{tdxz k↑dyz −k↑/an}bracketri}ht,/an}bracketle{tdxz k↑dxy −k↓/an}bracketri}ht, and/an}bracketle{tdyz k↑dxy −k↓/an}bracketri}htamplitudes are zero (similarly for ↑↔↓). Thus only finite z-components of the three inter-band pseudospin-triplet d-vectors and inter-band pseudospin-singlet order parameters (such as /an}bracketle{tξη k+ξρ −k−±ξρ k+ξη −k−/an}bracketri}ht) can appear. Figure 2 reveals that intra-band pairing is strongest and sharply peaked around the FS due to the mixing of all orbitals via SO interac- tion andinter-orbitalhopping, and the ideal conditionsfor zero-momentum pairing. Inter-band pairing in contrast is about an order of magnitude weaker and, in particular for0.5 1 1.5 3J-U00.05 |∆s xy/xy|00.02 |∆s yz/yz|=|∆s xz/xz| 0.5 1 1.5 3J-U00.1|dyz/xz|00.10.2|dxz/xy|=|dyz/xy| λ=0 0.075 0.15 0.225 0.3 Fig. 3: (Color online) MF solutions for different SO coupling strengths for the Sr 2RuO4based band structure. Orbital- singlet spin-triplet pairing dxz/xy,dyz/xy, anddyz/xz(purely real) induces finite intra-orbital spin-singlet pairing ∆s yz/yz, ∆s xz/xz, and ∆s xy/xy(purely imaginary). We also checked for induced inter-orbital spin-singlet pairing amplitudes, w hich, however, vanish. /an}bracketle{tξγ k+ξβ −k−/an}bracketri}ht, more spreadout in momentum space, marking Bloch band states that are energetically still close enough to the FS to participate significantly in pairing. This analysis demonstrates that inter-orbital pairing arising from Hund’s interaction leads to k-dependent inter- and intra-band pairing in pseudospin-singlet and and pseudospin-triplet (z component only) channels. Fur- thermore, the pairing instability occurs simultaneously within and between all bands ratherthan in a singleactive band with superconductivity leaking into passive bands through, e.g., pair hopping. The role of intra-band spin- triplet pairing between αandβbands in multi-orbital su- perconductors like Sr 2RuO4has also been the focus of re- cent studies, where the inter-band order parameter, how- ever, breaks TRS [27] and an intrinsic anomalous Hall effect can contribute significantly to a large TRS breaking signal in Kerr rotation experiments [28,29]. Pairing transition, QP dispersion, and magnetic response. – For concreteness we study the effect of SO coupling on spin-triplet pairing originating from Hund’s interaction, including the QP dispersion and the magnetic response. As discussed in the previous sections the quali- tative results are generic for SO coupled t2g-bands (or p- orbital systems) and can be applied to specific materials suchasthesinglelayerruthenate[5,6]andtheFe-pnictides [7,30] using the appropriate band structure. Using the kinetic Hamiltonian of eq. (10) with a pa- rameter choice mimicking the single layer ruthenate band structure, the MF solutions for various λare displayed in fig. 3. As one can see, in the absence of SO interaction an orbital-singlet spin-triplet pairing instability develops at a large coupling strength 3 J−U/greaterorsimilar1.0 fordxz/xyand dyz/xy. Although numerically difficult to resolve, we ex- pect that dyz/xzand the intra-orbital spin-singlet order parameters simultaneously become finite through quartic or higher order couplings in the Landau free energy ex- pansion. While the magnitudes of the order parameters depend on the details of the band structure, a robust fea- p-4Identifying spin-triplet pairing in spin-orbit coupled multi-band super conductors Γ X M Γ-2-1012DOS (arb. units) -0.200.2 E-µ ΓM X(a) (b) (c) Fig. 4: (Color online) QP bands for 3 J−U= 0.9 andλ= 0.15. Panel (a) is a magnification of panel (c) about the Fermi level , revealing the gaps opening up on the FS sheets. Panel (b) shows the DOS and the QP gap near the Fermi level. 0 0.002 0.004 0.00600.010.02 Mz <Si, z><Li, z>B || z λ=0.075no pairing with pairing 0 0.002 0.004 0.00600.010.02B || x λ=0.075 Mx <Li, x> <Si, x> 0 0.002 0.004 0.006 B00.010.02B || z λ=0.15 Mz <Si, z><Li, z> 0 0.002 0.004 0.006 B00.010.02 Mx <Li, x> <Si, x>B || x λ=0.15 Fig. 5: (Color online) Magnetization parallel to the applie d magnetic field Bforλ= 0.075 (top)and0 .15 (bottom) andtwo field orientations at 3 J−U= 0.9. The solid lines represent to- tal magnetization, dashed lines stand for orbital contribu tion, and dash-dotted lines for spin magnetization. For sake of co m- parison the magnetic response both in the presence (orange) and in the absence (grey) of superconductivity is displayed . (B is expressed in units of 2 t1= 1.) ture is that finite SO coupling drastically reduces the crit- ical pairing strength. This reduction is mostly facilitated by the additional hybridization provided by HSO, which helps to overcome the momentum mismatch between or- bitals/bands near the Fermi level. On the other hand the same mechanism can have a slightly detrimental effect at larger 3J−U, where the ideal inter-orbital pairing condi- tions along the diagonals are weakened by the additional hybridization. One may also wonder if the BogoliubovQP dispersionshaveanisotropicgaps. TheresultingQPbands are shown in fig. 4 and are fully gapped with a fourfold symmetric gap modulation in kspace, even though the gap minima are tiny. Note that the present superconducting state does not break TRS. The magnetic response is a combination of paramagnetic (spin-triplet) and spin-singlet behaviours, with a slightly larger out-of-plane than in-plane total magnetic susceptibility as shown in fig. 5, where M= /an}bracketle{tLi/an}bracketri}ht+ 2/an}bracketle{tSi/an}bracketri}htis the total magnetization including orbital and spin contributions and HB=B·/summationtext i(Li+ 2Si) cou-ples the orbital and spin degrees of freedom to the exter- nal field B. Both orbital and spin expectation values are finite with roughly equal contribution to the total mag- netization. For comparison, the normal state magnetiza- tions are also shown in fig. 5 and are larger than in the superconducting state, as expected for a combination of spin-singlet and -triplet pairing in the presence of SO in- teraction. In particular, note that the spin magnetization changes drastically in the superconducting state with in- creasing λ. In general,the magnitudeofthe d-vectors,and thus the magnetic response, can be modified by changing the size of the FS sheets. For instance a larger overlap between yz and xy dominated portions of the FS would enhance dyz/xycompared to dyz/xzanddxz/xy. The spin susceptibility then would be mostly dominated by dyz/xy, a situation which may be facilitated by applying uniaxial pressure. Discussion and summary. – Given that we based our MF study on the Sr 2RuO4compound to illustrate the effect of SO interaction on pairing, let us comment on the compatibility and the limitations of our results with what is known about the superconducting state in Sr 2RuO4[5, 6]. Based on the QP gap variation along the FS sheets, one expects that this modulation may also be reflected in orientation sensitive specific heat measurements. Such magnetic field dependent specific heat measurements on Sr2RuO4have indeed been carried out [31,32], but the interpretation of the experimental results is controversial, making alink to ourQPdispersiondifficult. However,due to the nature of inter-band pairing, the superconducting state presented here is sensitive to any kind of impurities associated with inter-band scattering, which is consistent with the phenomena observed in Sr 2RuO4. Our result on the magnetization indicates that the spin- susceptibility is finite and different for in-plane and out- of-plane magnetic field orientations in both the normal and the superconducting state, as reported on Sr 2RuO4. Yet below Tcthe in-plane and out-of plane susceptibilities decrease, which is in contrast to NMR Knight shift mea- surements [33,34], which revealed that a change in the spin-response across Tcis absent for any field orientation. This behaviourdiffersalsofromthe responseexpected ofa chiralp+ipsuperconductor, where the spin-susceptibility decreasesforfielddirectionsperpendiculartothe a-bplane but remains constant for parallel orientations. While the amountofchangein the presentmodel depends sensitively on the SO interaction strength, as shown in fig. 5, the question also arises as to how orbital and spin contribu- tions were separated to obtain the Knight shift data when SO interaction is significant. Besides this, we note that the magnetic field effect on vortices will be highly non- trivial as well, as it involves competition between various types of vortices including half-quantum vortices [35,36] in the presence of moderate SO coupling. Finally, the lack of TRS breakingis compatible with the absence of chiral supercurrents as observed in scanning p-5Christoph M. Puetter and Hae-Young Kee Hall probe and scanning SQUID measurements [37,38]. However, this contrasts with another proposal that the chiral states due to p+ippairing on αandβbands cancel each other leading to a topologically trivial superconduc- tor [27]. It also contradicts Kerr rotation and µSR mea- surements which have been interpreted in favour of TRS breaking [39,40]. The issue as to whether TRS is broken or not is not yet resolved in the experimental commu- nity. While the current study supports a non-TRS break- ing state, it can be modified by goingbeyondlocal interac- tions. Anaturalextensionwouldbe toinclude theeffect of further neighbour ferromagneticinteractionssuch as those discussed by Ng and Sigrist [18], which could lead to a small admixture of odd parity pairing with broken TRS in addition to the pairingfound hereand which may be re- sponsible for the broken TRS signaturesfound in µSR and Kerr experiments [39,40]. Another possibility is a finite- momentum pairing state such as a FFLO (Fulde-Ferrell- Larkin-Ovchinnikov) state [41,42]. It is plausible that a FFLO state between different bands can be stabilized over the inter-band pseudospin-triplet pairing. These studies, andmoredefinitepredictionsforSr 2RuO4orotherspecific materials, however, go beyond the scope of the current 9 complex order parameter minimization and require more detailed work. In summary, we studied the combined effect of Hund’s and SO coupling on t 2gorbital systems. Three orbital- singlet spin-triplet pairings were found to form an or- thogonal d-vector triad. A linear coupling between even- parity inter-orbital spin-triplet and even-parity intra- orbital spin-singlet pairings was allowed due to SO inter- action, determining the orientation of the three d-vectors and giving rise to a relative phase difference of π/2 be- tween spin-singlet and spin-triplet order parameters. We also showed that inter-orbital spin-triplet pairing in the orbital basis corresponds to ever-parity inter- and intra- band pairing in the Bloch band basis, and discussed how the pairing strength varies within the Bloch bands. We further found that SO coupling assists Hund’s coupling driven pairing, which generally leads to an anisotropic QP gap and an orbital dependent magnetic response. ∗∗∗ WethankS.R.Julian, A.Paramekanti,Y.-J.Kim,K.S. Burch, and C. Kallin for useful discussions. HYK thanks the hospitality of the MPI-PKS, Dresden, Germany where a part of this work was carried out. This work was sup- ported by the NSERC of Canada and Canada Research Chair. REFERENCES [1]Grewe N. andSteglich F. ,Handbook on the Physics and Chemistry of Rare Earths , Vol.14(North-Holland, Amsterdam) 1991 Chapt. “Heavy Fermions”. [2]Sigrist M. andUeda K. ,Rev. Mod. Phys. ,63(1991) 239. [3]Powell B. J. andMcKenzie R. H. ,Rep. Prog. Phys. ,74 (2010) 10301.[4]Kamihara Y. ,J. Am. Chem. Soc. ,130(2008) 3296. [5]Mackenzie A. P. andMaeno Y. ,Rev. Mod. Phys. ,75 (2003) 657 & references therein. [6]Kallin C. andBerlinsky J. ,J. Phys.: Condens. Matter , 21(2009) 164210 & references therein. [7]Lee P. A. andWen X. G. ,Phys. Rev. B ,78(2008) 144517. [8]Yang W. L. et al.,Phys. Rev. B ,80(2009) 014508. [9]Fazekas P. ,Lecture Notes on Electron Correlation and Magnetism (World Scientific, Singapore) 1999. [10]Kuriyama H. et al.,Appl. Phys. Lett. ,96(2010) 182103. [11]Liebsch A. andLichtenstein A. ,Phys. Rev. Lett. ,84 (2000) 1591. [12]Pchelkina Z. V. et al.,Phys. Rev. B ,75(2007) 035122. [13]Mravlje J. et al.,Phys. Rev. Lett. ,106(2011) 096401. [14]Pavarini E. andMazin I. I. ,Phys. Rev. B ,74(2006) 035115. [15]Malvestuto M. et al.,Phys. Rev. B ,83(2011) 165121. [16]Rozbicki E. J., Annett J. F., Souquet J. R. and Mackenzie A. P. ,J. Phys.: Condens Matter ,23(2011) 094201. [17]Haverkort M. W. et al.,Phys. Rev. Lett. ,101(2008) 026406. [18]Ng K.K. andSigrist M. ,Europhys. Lett. ,49(2000) 473. [19]Spalek J. ,Phys. Rev. B ,63(2001) 104513. [20]Han J. E. ,Phys. Rev. B ,70(2004) 054513. [21]Dai X., Fang Z., Zhou Y. andZhang F. C. ,Phys. Rev. Lett.,101(2008) 057008. [22]Leggett A. J. ,Rev. Mod. Phys. ,47(1975) 331. [23]Vollhardt D. andW¨olfle P. ,The Superfluid Phases of Helium 3 (Taylor & Francis, London) 1990, Chap. 5. [24]Werner R. ,Phys. Rev. B ,67(2003) 014505. [25]Bergemann C. et al.,Phys. Rev. Lett. ,84(2000) 2662. [26]Damascelli A. et al.,Phys. Rev. Lett. ,85(2000) 5194. [27]Raghu S., Kapitulnik A. andKivelson S. A. ,Phys. Rev. Lett. ,105(2010) 136401. [28]Taylor E. andKallin C. ,Phys. Rev. Lett. ,108(2012) 157001. [29]Wysokinski K. I., Annett J. F. andGy¨orffy B. L. , Phys. Rev. Lett. ,108(2012) 077004. [30]Daghofer M., Nicholson A., Moreo A. and Dagotto E. ,Phys. Rev. B ,81(2010) 014511. [31]Deguchi K., Mao Z. Q. andMaeno Y. ,J. Phys. Soc. Jpn.,73(2004) 1313. [32]Deguchi K., Mao Z. Q., Yaguchi H., andMaeno Y. , Phys. Rev. Lett. ,92(2004) 047002. [33]Ishida K. et al.,Nature,396(1998) 658. [34]Murakawa H. et al.,Phys. Rev. Lett. ,93(2004) 167004. [35]Salomaa M. M. andVolovik G. E. ,Phys. Rev. Lett. , 55(1985) 1184. [36]Kee H.-Y., Kim Y. B. andMaki K. ,Phys. Rev. B ,62 (2000) R9275. [37]Bj¨ornsson P. G., Maeno Y., Huber M. E. andMoler K. A.,Phys. Rev. B ,72(2005) 012504. [38]Kirtley J. R. et al.,Phys. Rev. B ,76(2007) 014526. [39]Luke G. M. et al.,Nature,396(1996) 658. [40]Xia J.et al.,Phys. Rev. Lett. ,97(2006) 167002. [41]Fulde P. andFerrell R. A. ,Phys. Rev. ,135(1964) A550. [42]Larkin A. I. andOvchinnikov Y. N. ,JETP,20(1965) 762. p-6

1312.5809v1.Effects_of_the_spin_orbital_coupling_on_the_vacancy_induced_magnetism_on_the_honeycomb_lattice.pdf

arXiv:1312.5809v1 [cond-mat.str-el] 20 Dec 2013Effects of the spin-orbital coupling on the vacancy-induced magnetism on the honeycomb lattice Weng-Hang Leong, Shun-Li Yu, and Jian-Xin Li National Laboratory of Solid State Microstructure and Depa rtment of Physics, Nanjing University, Nanjing 210093, Chi na (Dated: October 7, 2018) The local magnetism induced by vacancies in the presence of t he spin-orbital interaction is in- vestigated based on the half-filled Kane-Mele-Hubbard mode l on the honeycomb lattice. Using the self-consistent mean-field theory, we find that the spin-orb ital coupling will enhance the localization of the spin moments near a single vacancy. We further study th e magnetic structures along the zigzag edges formed by a chain of vacancies. We find that the sp in-orbital coupling tends to sup- press the counter-polarized ferrimagnetic order on the upp er and lower edges, because of the open of the spin-orbital gap. As a result, in the case of the balanc e number of sublattices, it will suppress completely this kind of ferrimagnetic order. But, for the im balance case, a ferrimagnetic order along both edges exists because additional zero modes will not be a ffected by the spin-orbital coupling. I. INTRODUCTION Graphene and related nanostructured materials have attractedmuchinterestin solidstatephysicsrecentlydue to their bidimensional character and a host of peculiar properties1. Among them, the investigation of the mag- netic properties in graphene is one of the fascinating top- ics, as no dandfelements are necessaryin the induction ofmagnetism in comparisonwith the usual magnetic ma- terials. Theoretical predictions and experimental investi- gations have revealed that a nonmagnetic defect such as animpurity oravacancycan induce the non-triviallocal- ized magnetism2–6. Similarly, a random arrangement of a large number of vacancies which are generated by the high-dose exposure of graphene to strong electron irradi- ation7can also induce magnetism theoretically8. These studies not only have the fundamental importance, but also open a door for the possibility of application in new technologies for designing nanoscale magnetic and spin electronic devices. On the other hand, the topological insulating elec- tronic phases driven by the spin-orbital (SO) interaction have also attracted much interest recently. The Kane- Mele model for the topological band insulator is defined on the honeycomb lattice9,10which is the same lattice structure as graphene. Possible realization of an appre- ciable SO coupling in the honeycomb lattice includes the cold fermionic atoms trapped in an extraordinary optical lattice11, the transition-metal oxide Na 2IrO312and the ternaty compounds such as LiAuSe and KHgSb13. Topo- logical band insulator has a nontrivial topological order andexhibitsabulkenergygapwithgapless,helicalstates at the edge14–16. These edge states are protected by the time reversal symmetry and are robust with respect to the time-reversal symmetric perturbations, such as non- magnetic impurities. It is shown that a vacancy, acting as a minimal circular inner edge, will induce novel time- reversal invariant bound states in the band gap of the topological insulator17–19. Theoretically, it is also shown that the SO coupling suppresses the edge magnetism in- duced in the zigzag ribbon of the honeycomb lattice inthe presence of electron-electron interactions20. Thus, it is expected that the SO coupling would also affect the local magnetism in the bulk induced by vacancies. In this paper, we study theoretically the effects of the SO coupling on the local magnetism induced by a sin- gle and a multi-site vacancy on the honeycomb lattice, based on the Kane-Mele-Hubbard model where both the SO coupling and the Hubbard interaction between elec- trons are taken into consideration. This model has been extensively studied to explore the effect of the strong correlation on the topological insulators21–27. Making use of the self-consistent mean field approximation, we calculate the local spin moments and their distribution around the vacancies. For a single vacancy, we find that the main effect of the SO coupling is to localize the spin moments to be near the vacancy, so that it will enhance the local spin moments. For a large stripe vacancy by taking out a chain of sites from the lattice, we find that the SO coupling tends to suppress the counter-polarized ferrimagnetic order induced along the zigzag edges, be- cause of the open of the SO gap. As a result, in the case of the balance number of sublattices (with even number of vacancies), the SO coupling will suppress completely the counter-polarized ferrimagnetic order along the up- per and lower edges. While, in the case of the imbalance number of sublattices (with odd number of vacancies), a ferrimagneticorder along both edges exists because addi- tionalzeromodeswill notbe affectedbythe SOcoupling. Wewillintroducethemodelandthemethodoftheself- consistent mean-field approximation in Sec.II. In Sec.III and IV, we present the results for a single vacancy and a multi-site vacancy, respectively. Finally, a briefsummary will be given in Sec.V. II. MODELS AND COMPUTATIONAL METHODS We start from the Kane-Mele model9, in which the intrinsic SO coupling with a coupling constant λis in-2 cluded. H0=−t/summationdisplay /angbracketleftij/angbracketright,σc† iσcjσ+iλ/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketrightσσ′vijσz σσ′c† iσcjσ′,(1) wherec† iσ(cjσ) is the creation(annihilation) operator of the electron with spin σon the lattice site i,/angbracketleftij/angbracketrightrep- resents the pairs of the nearest neighbor sites (the hop- ping ist) and/angbracketleft/angbracketleftij/angbracketright/angbracketrightthose of the next-nearest neighbors. vij= +1(−1) if the electron makes a left(right) turn to get to the second bond. The size of our system is consid- ered to be finite with periodic boundary condition. So, the position of each lattice site can be described specif- ically by i= Γ(m,n), representing that the lattice site iis in the mth column and the nth row, and Γ = A,B the sublattice labels. The number of the unit cells is denoted by Nc=L2, therefore the total number of the lattice sites is Nl= 2L2. To consider the correlation be- tween electrons, we will include the Hubbard term in the Hamiltonian, which is given by HI, HI=U/summationdisplay iˆni↑ˆni↓, (2) where ˆniσ=c† iσciσ. When vacancies are introduced, the hoppings between the vacancy and the nearest neigh- bors and the on-site interaction on that vacancy are sub- tracted from the overall Hamiltonian. Hence the corre- sponding number of the lattice sites is Nl= 2L2−Nv, whereNvis the number of vacancies. The total number of electrons Neis fixed to be at the half-filling ( Ne=Nl). The Hubbard interaction term is treated with the self- consistent mean field approximation, so that we will ob- tain an effective single-particle Hamiltonian where the electrons interact with a spin-dependent potential, HI≃U/summationdisplay i,σ/angbracketleftˆni−σ/angbracketrightˆniσ−U/summationdisplay i/angbracketleftˆni↑/angbracketright/angbracketleftˆni↓/angbracketright.(3) And the overall mean field Hamiltonian Hmfis then given by, Hmf=U/summationdisplay iσ/angbracketleftˆni−σ/angbracketrightˆniσ+H0. (4) After diagonalizing the Hamiltonian Hmf, we can de- termine the occupation number /angbracketleftˆni−σ/angbracketrightat each site with different spins using the eigenvectors of Hmf, and this process is carried out iteratively until a required accu- racy is reached. Then the magnetic moment of each site mi=/angbracketleftˆni↑−ˆni↓/angbracketrightcan be calculated. We note that a collinear magnetic texture is assumed in our system, as used before for the investigations of Kane-Mele-Hubbard model20,21. We have checked the results with the non- collinear magnetic texture and found that the collinear magnetic texture is favored. III. MAGNETISM WITH ONE VACANCY The calculation is carried out on the lattice with Nc= 14×14 unit cells in which a single vacancy is introduced/s49 /s50 /s51 /s52 /s53 /s54/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54 /s32 /s61/s48/s46/s48/s48 /s32 /s61/s48/s46/s48/s53 /s32 /s61/s48/s46/s49/s48/s32/s109 /s105 /s32 /s32/s114/s47/s97/s40/s99/s41/s40/s98/s41 /s32/s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s55 /s48/s46/s49/s49 /s48/s46/s49/s52 /s48/s46/s49/s56/s32/s40/s97/s41 FIG. 1: (color online). (a) and (b): Distribution of the spin moments mion lattice sites around asingle vacancy at A(7,7) withU= 1.0t, in which (a) corresponds to the SO coupling constant λ= 0.0 and (b) λ= 0.1t. The area and color of the hollow circles represent the magnitude of the spin moments. (c)mion theBsublattice as a function of the distance r away from the vacancy. The unit ais the distance between the nearest sites. on the site A(7,7). Figure 1 displays the distribution of the magnetic moment when the Hubbard interaction is taken to be U= 1.0t, in which the size and the color of the circle on each lattice site denote the magnitude of the local spin moment. From Fig.1(a) where the SO couplingisturnedoff, onecanseethatlocalizedmagnetic moments are induced around the vacancy in the presence of a finite Hubbard interaction U. This is in agreement with the prediction of the Lieb theorem28regarding the total spin Sof the exact ground state of the Hubbard model on bipartite lattices. It states that the total spin Sis given by the sublattice imbalance 2 S=|NA−NB|, withNAandNBthe number of atoms belonging to each sublattice. With the introducing of a single vacancy on theAsublattice, an imbalance NB−NA= 1 appears and a magnetic structure near the vacancy with the total spinS= 1/2 will form. Similar results have also been obtained in recent studies in graphene2–4. In the presence of the SO coupling, the magnitude of the magnetic moments around the vacancy increases, as shown in Fig. 1(b) for λ= 0.1t. At the meantime, if we check the distribution of the magnetic moments, as shown in Fig.1(c) where the magnitude of the magnetic moments on sublattice Bas a function of the distance r awayfromthe vacancyis presented, one will find that the magnetic moments are more localized with the increase of the SO coupling. These features demonstrate that the SOcoupling will enhancethe magneticmoments nearthe vacancy notably. In order to show the emergence of the magnetism in- duced by the vacancy in more detail, we calculate the3 spin resolved local density of state(LDOS) as defined by, Dσ(ǫ) = Σn,i|un i,σ|2δ(ǫ−ǫn), (5) whereiruns over the lattice sites surrounding the va- cancy up to the third-nearest neighbors, as those linked by the green line in Fig.1(a) and (b). un i,σis the single- particle amplitude on the ith site with spin σand the corresponding eigenvalue is ǫn. The Delta function in Eq.(2)isreplacedbytheLorentzianfunction forplotting. The results for the LDOS are presented in Fig. 2(a)-(h) for different Hubbard interaction Uand SO interaction λ. The red and blue lines represent the LDOS for the spin up and spin down components respectively, and the dash lines show the LDOS away from the vacancy for a comparison. In the case of U=λ= 0.0 as shown in Fig. 2(a), the LDOS shows a V-shape linear behavior near the Fermi level for those lattice sites far away from the vacancy (denoted by the dashed line) which is the consequence of the linear dispersion relation of the elec- trons, the so-called Dirac fermions. For those around the vacancy, a peak at the Fermi level emerges as shown by /s48/s46/s48/s48/s46/s53/s49/s46/s48 /s40/s101/s41/s76/s68/s79/s83 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50 /s40/s103/s41/s76/s68/s79/s83 /s69/s47/s124/s116/s124/s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s40/s104/s41 /s69/s47/s124/s116/s124/s40/s102/s41/s40/s100/s41 /s48/s46/s48/s48/s46/s53/s49/s46/s48 /s40/s99/s41/s76/s68/s79/s83/s40/s98/s41 /s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s76/s68/s79/s83/s40/s97/s41 FIG. 2: (color online). LDOS for λ= 0.0 [left column, includ- ing (a),(c),(e),(g)] and for λ= 0.1t[right column, including (b),(d),(f),(h)], where the Hubbard interaction U= 0.0 for (a) and (b), U= 1.6tfor (c) and (d), U= 2.6tfor (e) and (f), and U= 3.6tfor (g) and (h), respectively. LDOS for different spins is resolved, those with the spin up are denote d by the blue lines and the spin down the red lines. The grey dash lines represent the LDOS on the lattice site away from the vacancy./s49 /s50 /s51 /s52/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s40/s98/s41 /s77 /s32 /s108/s111/s99 /s85/s32/s47/s32/s116/s32 /s61/s48/s46/s48/s48 /s32 /s61/s48/s46/s48/s52 /s32 /s61/s48/s46/s49/s48 /s32 /s61/s48/s46/s49/s54 /s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56 /s48/s46/s49/s50 /s48/s46/s49/s54/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48 /s32/s32 /s32/s85/s61/s49/s46/s48 /s32/s85/s61/s50/s46/s50 /s32/s85/s61/s50/s46/s52 /s32/s85/s61/s50/s46/s56/s77 /s32 /s108/s111/s99 /s32/s47/s32/s116/s40/s97/s41 FIG. 3: (color online). Local moments Mloc(see text) are shownasafunctionoftheSOcoupling λfordifferentHubbard interaction U(a) and of Ufor different λ(b). the solid line, which corresponds to the localized states induced by the vacancy29. After turning on the SO cou- pling, such as that for λ= 0.1t[see Fig.2(b)], we can see that an energy gap opens for those lattice sites far away from the vacancy9,10, so that now a U-shape LDOS near the Fermi level occurs. In this way, the mid-gap peak is enhanced noticeably because the decay rate of the local- ized states into the continuum is reduced largely due to the open of the energy gap. This will lead to the increase in the spectral weight of the localized states around the vacancy. However, for both cases, one will find that the LDOS for the spin up and spin down components degen- erates, so that the system will not show magnetism as a whole without the Hubbard interaction. The effect of a finite Hubbard interaction Uis to split the spin degenerate LDOS, so that two peaks occur cor- responding to different spins, as shown in Fig. 2(c)-(f). Consequently, the localized spin up and down moments will not cancel out in this case, and a net magnetism around the vacancy is induced. The magnetism may be quantified by the local mo- mentMloc=/summationtext imi, where the sum runs over the lat- tice sites surroundingthe vacancy up to the third-nearest neighbors as used above in the calculation for the LDOS. The results are presented in Fig. 3(a) and (b) for dif- ferentUandλ, respectively. The local moment Mloc shows a monotonic increase with the SO coupling λ, so it reinforces our observation that the local magnetism is enhanced by the SO coupling as shown in Fig.1. On the other hand, Mlocshows a nonmonotonic dependence on the Hubbard interaction U, namely it increases with Ufirstly and then decreases with a further increase of Uafter a critical value Uc. As discussed above, the lo- cal magnetism is determined by the spin-split localized states induced by the vacancy, and it is the Hubbard in- teraction Uto split the spin-degenerate states. Because the open of the gap due to the SO coupling will decrease the decay rate of the localized states into the continuum, so it will enhance the spectral weight of the localized states[see also Fig. 2], consequently the localized mag- netism. The splitting between the two localized states with different spins is proportional to U, so the two split4 localized states will situate in the SO gap for a small U[Fig. 2(c)-(f)]. However, when U > U cthe splitting will be larger than the SO gap, and it pushes the local- izedstatestomergeintothecontinuum[Fig.2(g)and(h)], so the local magnetism will decrease. IV. THE CASE OF MULTI-SITE VACANCY The multi-site vacancy can be formed by removing the sites continuously. Here, we consider a large stripe va- cancy by taking out a chain of sites from the lattice as illuminated in Fig. 4. In this way, the stripe vacancy consists of one upper and one lower zigzag edges. As clarified by the Lieb theorem28, the sublattice imbalance between the number of atoms belonging to different sub- lattices will have significant effect on the magnetism. For the stripe vacancy considered here, the imbalance is ex- pressed by the parity of the number of vacancies, where the number is even ( NA=NB) in Fig. 4(a) and (b), and odd (NA/negationslash=NB) in Fig. 4(c) and (d), thus the total spin of the system is S= 0 and 1 /2 respectively. Inthecaseofevennumberofvacancies,aferrimagnetic spin order emerges on both the upper and lower zigzag edges around the stripe vacancies when there is no SO coupling, as shown in Fig. 4(a). The ferrimagnetic ar- rangement and the magnitude of the spin moments on these two edges are symmetric, but they are counter- polarized, so they cancel out exactly and the whole sys- tem will not show magnetism. This is consistent with the Lieb theorem28. The ferrimagnetic order on a suffi- ciently long zigzag edge around the stripe vacancies here issimilartothespin orderformedattheouteredgeofthe zigzagribbon30–33and the graphenenanoisland34. In the case of odd number of vacancies, a similar ferrimagnetic spin order is also induced with a slightly large magnitude [Fig. 4(c)]. Interestingly, this ferrimagnetic order occurs only on the upper zigzag edge, not on the lower edge. This phenomenon is ascribed to the presence of an extra spin when a sublattice imbalance NA/negationslash=NBexists, as /s40/s97/s41 /s32 /s32 /s32/s32 /s32/s32 /s40/s99/s41/s40/s98/s41 /s32 /s32/s45/s48/s46/s49/s56/s45/s48/s46/s49/s49/s45/s48/s46/s48/s52/s48/s46/s48/s52/s48/s46/s49/s49/s48/s46/s49/s56 /s40/s100/s41 /s32 /s32/s45/s48/s46/s50/s54/s45/s48/s46/s49/s54/s45/s48/s46/s48/s53/s48/s46/s48/s53/s48/s46/s49/s54/s48/s46/s50/s54 /s66/s65/s66/s65 FIG. 4: (color online). Distribution of the spin moments mi on the lattice sites surrounding the vacancies for U= 1.0tand L= 14. A cluster of vacancies is formed with the number of missing sites for (a), (b) Nv= 8 and (c), (d) Nv= 7. SO coupling is set to be λ= 0.0 for (a), (c) and λ= 0.1tfor (b), (d). The area and color of hollow circles represent the magnitude of the moments./s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s40/s98/s41 /s32/s78 /s118/s61/s51 /s32/s78 /s118/s61/s53 /s32/s78 /s118/s61/s55 /s32/s78 /s118/s61/s57 /s32/s78 /s118/s61/s50 /s32/s78 /s118/s61/s52 /s32/s78 /s118/s61/s54 /s32/s78 /s118/s61/s56 /s32/s78 /s118/s61/s49/s48/s77 /s101 /s47/s32/s116/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48 /s32/s78 /s118/s61/s51 /s32/s78 /s118/s61/s53 /s32/s78 /s118/s61/s55 /s32/s78 /s118/s61/s57 /s32/s32/s77 /s108/s111/s99 /s47/s32/s116/s40/s97/s41 FIG. 5: (color online). (a)Local moments Mlocare plotted as a function of λwhileU= 1.0tandL= 14. The function in different size of vacancy is distinguished by different color and shape of points. The cases of even Nvare not plotted as local moments are always zero obeying Lieb theorem28. (b) The function of edge moments Meversusλare given in different Nv. described by the Lieb theorem28. After turning on the SO coupling, such as for λ= 0.1t, the ferrimagnetic spin order on both the upper and lower zigzag edges around the stripe vacancies disappears com- pletely in the case of even number of vacancies[Fig. 4(b)]. However, the effect of the SO coupling on local mag- netism is quite different for the case of an odd number of vacancies. Here, a ferrimagnetic spin order similar to that on the upper edge emerges on the lower edge, though the magnitude of the individual spin moment is reduced[Fig. 4(d)]. To show variation of the total mag- netism, we plot the quantity Mlocas a function of the SO coupling λin Fig. 5(a), here Mlocis the sum of the spin moments on the sites which are on the zigzag edges around the vacancies. Since Mlocis always zero in the case of even Nv, it is not plotted here. With an odd Nv, the local moment Mlocincreases with the increase ofλ, which shows a similar behavior as that in the case of a single vacancy. This indicates that the total local magnetism shown in Fig. 4(d) is in fact enhanced with the introduction of the SO coupling and approaches the saturation value 1 finally. From Fig. 5(a), one can also find that Mlocincreases with the increase of the number of vacancies Nv. This suggests that the SO coupling will localize the induced spin moments to those lattice sites which are neighboring the vacancies. To quantify the variation of the spin moments with λon the upper zigzag edge, we also present Meas a function of λin Fig. 5(b), here Meis the sum of the spin moments only on the sites on the upper zigzag edge. Let us consider firstly the case of even number of Nv, for a small number of even vacancies, Meis always zero. Up toNv≥8, a finite Meoccurs and it increases with Nvby the formation of the zigzag edges. However, Me drops rapidly to zero after turning on the SO coupling. These results quantify the physical picture derived from Fig. 4(a) and (b). Now let us turn to the case of odd number of Nv. Without the SO coupling, Mealso shows5 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56 /s40/s98/s41 /s32/s101/s110/s101/s114/s103/s121/s47/s116 /s32/s109/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56 /s40/s97/s41/s32/s32/s101/s110/s101/s114/s103/s121/s47/s116 /s32/s109 FIG. 6: (color online). The single-particle energy levels l a- beled with m(see text) near the Fermi level of the non- interacting systems for (a) Nv= 8 and (b) Nv= 7. SO cou- pling is set to be λ= 0.0 for the red circles and λ= 0.1tfor the blue squares. an increase with Nv. With the introduction of the SO coupling, Meshows a decrease with λand saturates to near one half of Mloc. In fact, we can make an analogy between the stripe vacancy and the graphene ribbon with zigzag edges. A remarkable feature of the graphene ribbon with zigzag edges is that it has a flat band localized on the zigzag edge1. An important effect of this flat band is that a counter-polarized ferromagnetic order along the upper and lower edges will be induced when the Hubbard in- teraction between electrons is included30–33. In view of this, we plot the single-particle spectra for the systems with the stripe vacancy without the Hubbard interaction in Fig.6(a) and (b) for Nν= 8 and Nν= 7, respectively. Each energy level is labeled with m=n−Ne−1/2 in order to indicate that the energy level with m <0 is occupied by electron. For the systems without SO cou- pling, we find that there are four near-degeneracy local- izedstates[redcirclesindicatedbyarrowsinFig.6(a)and (b)] which is near the Fermi level for both Nν= 8 and Nν= 7. These states will have the same effect as the flat band in the zigzag ribbon when a suitable Hubbard U is turned on. So, a counter-polarized ferrimagnetic order as shown in Fig.4(a) will emerge. However, we note that there are two additional zero modes for Nν= 7 relative toNν= 8, due to the imbalance between the sublattices(NA> NB). Thesezeromodeswillinduceextraspinmo- ments on both edges, which counteract the antiparallel moments on the lower edge. Thus, in the case of Nν= 7, only the ferrimagnetic order on the upper edge appears. After turning on the SO coupling, those localized states [blue squares indicated by arrows in Fig.6(a) and (b)] are pushed away from the Fermi level due to the open of the SO gap. Thus, as shown in Fig.4(b), a small Hubbard U is not enough to induce the counter-polarized ferrimag- netic order on the upper and lower edges. However, the zero modes originating from the imbalance of sublattices are not affected by the SO coupling [Fig.6(b)]. So, the additional ferrimagnetic order on both edges induced by these zero modes will remain for Nν= 7. V. CONCLUSION In a summary, we have studied the local magnetism induced by vacancies on the honeycomb lattice based on the Kane-Mele-Hubbard model. It is shown that the SO coupling tends to localize and consequently enhances the local magnetic moments near a single vacancy. Further- more, along the zigzag edges formed by a chain of va- cancies, the SO coupling will suppress completely the counter-polarized ferrimagnetic order along the edges. Therefore, the system will not show any local magnetism in the case ofeven number ofvacancies. For an odd num- ber of vacancies, a ferrimagnetic order along both edges exists and the total magnetic moments along both edges will increase. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 91021001, 11190023 and 11204125) and the Ministry of Science and Technology of China (973 Project grant numbers 2011CB922101 and 2011CB605902). 1A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 2O. V. Yazyev and L. Helm, Phys. Rev. B 75, 125408 (2007). 3H. Kumazaki and D. S. Hirashima, J. Phys. Soc. Jpn. 76, 064713 (2007). 4O. V. Yazyev, Phys. Rev. Lett. 101, 037203 (2008). 5M. M. Ugeda, I. Brihuega, F. Guinea, and J. M. G´ omez- Rodr´ ıguez, Phys. Rev. Lett. 104, 096804 (2010). 6R.R.Nair, M. Sepioni, I-L.Tsai, O.Lehtinen, J.Keinonen, A. V. Krasheninnikov, T. Thomson, A. K. Geim, and I. V. Grigorieva, Nature Phys. 8, 199-202 (2012). 7J. Kotakoski, A. V. Krasheninnikov, U. Kaiser, and J. C. Meyer, Phys. Rev. Lett. 106,105505 (2011). 8J. J. Palacios, J. Fern´ andez-Rossier, and L. Brey, Phys. Rev. B 77, 195428 (2008).9C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). 10C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 11K. L. Lee, B. Gremaud, R. Han, B. G. Englert, and C. Miniatura, Phys. Rev. A 80, 043411 (2009). 12A. Shitade, H. Katsura, J. Kunes, X. -L. Qi, S. -C. Zhang, and N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009). 13H. -J. Zhang, S. Chadov, L. M¨ uchler, B. Yan, X. -L. Qi, J. K¨ ubler, S. -C. Zhang, and C. Felser, Phys. Rev. Lett. 106, 156402 (2011). 14B. A. Bernevig, T. A. Hughes, and S. C. Zhang, Science 314, 1757 (2006). 15M. K¨ onig, Wiedmann, C. Br¨ une, A. Roth, H. Buhmann, L. W. Molenkamp, X. L. Qi, and S. C. Zhang, Science 318, 766 (2007).6 16M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 17W. -Y. Shan, J. Lu, H. -Z. Lu, and S. -Q. Shen, Phys. Rev. B 84, 035307 (2011). 18J. W. Gonz´ alez and J. Fern´ andez-Rossier, Phys. Rev. B 86, 115327 (2012). 19J. He, Y. -X. Zhu, Y. -J. Wu, L. -F. Liu, Y. Liang, and S. -P. Kou, Phys. Rev. B 87, 075126 (2013). 20D. Soriano and J. Fern´ andez-Rossier, Phys. Rev. B 82, 161302(R) (2010). 21S. Rachel and K. Le Hur, Phys. Rev. B 82, 075106 (2010). 22M. Hohenadler, T. C. Lang, and F. F. Assaad, Phys. Rev. Lett. 106, 100403 (2011). 23S. L. Yu, X. C. Xie, and J. X. Li, Phys. Rev. Lett. 107, 010401 (2011). 24D. Zheng, G. M. Zhang, and C. Wu, Phys. Rev. B 84, 205121 (2011). 25W. Wu, S.Rachel, W.M. Liu, andK.LeHur, Phys.Rev.B85, 205102 (2012). 26C. Griset and C. Xu, Phys. Rev. B 85, 045123 (2012). 27J. Wen, M. Kargarian, A. Vaezi, and G. A. Fiete, Phys. Rev. B 84, 235149 (2011). 28E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989). 29V. M. Pereira, F. Guinea, J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. Lett. 96, 036801 (2006). 30M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Jpn. 65, 1920 (1996). 31K. Wakabayashi, M. Sigrist, and M. Fujita, J. Phys. Soc. Jpn. 67, 2089 (1998). 32Y. W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97, 216803 (2006). 33J. Fern´ andez-Rossier, Phys. Rev. B 77, 075430 (2008). 34J. Fern´ andez-Rossier and J. J. Palacios, Phys. Rev. Lett. 99, 177204 (2007).

1709.09811v1.Spin_orbit_interaction_of_light_induced_by_transverse_spin_angular_momentum_engineering.pdf

1 Spin-orbit interaction of light induced by transverse spin angular momentum engineering Zengkai Shao1*, Jiangbo Zhu2*, Yujie Chen1, Yanfeng Zhang1†, and Siyuan Yu1,2† 0F 1School of Electronics and Information Engineering, State Key Laboratory of Optoelectr onic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China . 2Photonics Group, Merchant Venturers School of Engineering, University of Bristol, Bristol BS8 1UB, UK . *These authors contributed equally to this work. †email: zhangyf33@mail.sysu.edu.cn ; s.yu@bristol.ac.uk We report the first demonstration of a direct interaction between the extraordinary transverse spin angular momentum in evanes cent waves and the intrinsic orbital angular momentum in optical vortex beams. By tapping the evanescent wave of in a whispering -gallery -mode -based optical vortex emitter and engineering the transverse -spin state carried therein , a conversion between the transverse -spin angular momentum and the intrinsic orbital angular momentum carried by the emitted vortex beam takes place . This unconventional interplay between the spin and orbital angular momenta allows the regulation the spin-orbital angular momentum states of the emitted vortex . In the reverse process, it further gives rise to an enhanced spin -direction coupling effect in which waveguide or surface modes are unidirectional ly excited by an incident optical vortex , with the directionality jointly control led by the spin and orbit al angular momenta states of the vortex . The identification of this previously unknown pathway between the polarization and spatial degrees of freedom of light enrich es the spi n-orbit interaction phenomena , and can enable a variety of functionalities employing spin and orbital angular momenta of light in applications such as communications and quantum information processing . Light waves possess intrinsic spin and orbital angular momentum (SAM and OAM), as determined by the polarization and spatial degrees of freedom of light1-3. These two components are separately ob servable in paraxial beams4-7, where as it is well known that fundamentally such a distinction faces difficulties in light fields with high nonparaxiali ty and/or inhomogeneit y8-11. In fact, spin -orbit interactions (SOI) can be widely observed in light through scattering or focusing12,13, propagation in anisotro pic/inhomogeneous media14,15, reflection/refractio n at optical interfaces16,17, etc. Notably, the spatial and polarization properties of light are coupled and SOI phenomena must be considered in modern optics dealing with sub-wavelength scale systems, including nano -photonics and plasmonics18-22. A variety of novel functionalities utilizing structured light and materials are underpinned by SOI of light, e.g., optical micro -manipulations23, high -resolution microscopy24, and beam shaping wi th planar structures (metasurfaces)25. On the other hand, the stud y in SOI over the past few years is accompanied by a rising interest in the transverse spin angular momentum of light, which has been revealed by recent advances in optics as a new member in the optica l angular momentum (AM) family26-29. In sharp contrast to the longitud inal SAM predicted by Poynting1, the transverse SAM exhibits spin axis orthogon al to the propagation of light28,30. Transverse SAM can be typically found in highly 2 inhomogeneo us light fields, including surface plasmon polaritons26, evanescent waves of gui ded and un -guided modes22,28 and strongly focused beams31, where longitudinal field components emerge due to the transversali ty of electromagnetic waves32. Light fields possess ing transverse SAM can enable various applications in bio -sensing, nano -photonics, etc. More interestingly, transverse spin in evanescent waves also originates from the SOI in laterally confined propagating modes11, or can also be interpreted as the quantu m spin Hall effect (QSHE) of light33-35, and thus giving rise to robust spin -controlled unidirectional coupling at opt ical interfaces18,21,22, 36. This extraordinary characteristic of transverse SAM results in the breaking of the directional symmetry in mod e excitation at any interface supporting evanescent waves, and can find applications in optical diodes37, chiral spin networks38,39, etc. The ability to simultaneously tailor light fields in the polarization and spatial degrees of freedom via SOI phenomen a has allowed for new functionalities in str uctured light manipulations40. Furthermore, combing SOI and transverse SAM control will provide a more versatile platform for processing of light fields in the full AM domain. In this paper, we presen t an enrich ment of the SOI effects revealed by the engineering of transverse spin in evanescent waves. Our method evolves from a n optical vortex emitter based on a planar integrated whispering -gallery mode (WGM) resonator , which emits beams with precisely controllabl e total angular momentum (TAM )41,42. Here we demonstrate that the engineering of transverse spin in the evanescent waves of WGMs in the resonator leads to the spin -to-orbit al AM conversion in the emitted beams. This is the first demonstration of a n SOI effe ct that features the interaction between the transverse SAM and intrinsic OAM of light , providing a promising pathway towards more sophisticated light manipulation via SOI p henomena. By reversing the emission process , we further demonstrate directional cou pling of optical vortices into this integrated photonic circuitry, with the direction of the waveguide modes jointly controlled by the spin and orbital AM states, realizing the selective reception of vector vortices without separate polarization and spatia l phase manipulation. These results can be used to bring novel functionalities to nano -photonic devices, e.g., encoding and retrieving photonic states in the SAM -OAM space, and provide the guidelines for the design of nano -photonic chiral interface between travelling and bounded vector vortices. Results Transverse spin in optic al vortex emitter . The schematic of the platform for the investigation of transverse spin engineering based SOI is shown in Figure 1a, where a single -transverse -mode ring resonator is coupled with a two -port access waveguide and embedded with periodic angular scatterers in the inner -sidewall evanescent region of the waveguide. With the sub -wavelength scatterers arranged in a second -order grating fashion, the diffracted first -order ligh t from the evanescent fields of WGMs collectively produce a vortex beam carrying optical OAM and travelling perpendicular to the resonator plane41. In addition, the emitted vortex beams exhibit cylindrically symmetric polarization and intensity distri butio ns, and thus referred to as cylindri cal vector vortices (CVVs)42,43. Generally, for the quasi -transverse -electric (TE) WGMs propagating in the high -index waveguide, a local longitudinal electric component ( Eφ) exists in the sidewall evanescent waves and is in quadrature phase with respect to the radial component ( Er) (see Figure 1b) , as a direct result of the strong lateral confinement and transversa lity condition32. Consequently, the local SAM in the evanesc ent field exhibits a ‘transverse’ sp inning axis in the z direction45, being orthogonal to the local propagation direction (+ φ or φ) of the WGM. Note that for quasi -TE WGMs , the transverse SAM at the inner - and outer - sidewalls always has opposite spin dir ections, and the transverse spin can also be flip ped by injecting light from the alternative ports 1 or 2 and exciting counter -clockewise (CCW) or clockwise (CW) WGMs, as shown in Figure 1 b. 3 Figure 1 . Illustration of the concept s. (a) Schematic of th e platform for the investigation of transverse spin induced SOI effect. A single -transverse -mode ring resonator is coup led with an access waveguide and embedded with sub -wavelength scatterers arranged as 2nd order grating in the evanescent wave region. (b) Each WGM possesses transverse spin of opposite signs in the inner - and outer -resonator evanescent waves, and clock -wise (CW) and counter clock -wise (CCW) WGMs present opposite transverse spins on each side of the resonator. (c) Illustration of the transve rse-spin-dependent geometric phase acquired by the vector evanescent wave as the WGM travels around the resonator. For CCW and CW WGMs, a rotation angle of ∓φ·z is experienced by the local coordinates to be aligned with the global reference frame (i.e., from (r’’, φ’’) to (x, y)) for phase comparison of different locations , and the geometric phase acquired by evanescent wave is ΦG = ± 2σ/(1+σ2)·φ. Interaction of transverse -spin and OAM . The emission of CVVs from such structures can be generally described in the form of transfer matrices as Eout = M2·M1·Ein. By assuming the WGM evanescent wave maintains a uniform distribution around the resonator, the generic i nput light for the matrices is the inner sidewall evanescent wave and can be written in the locally transverse and longitudinal polarization basis . Here the CCW propagating WGM is considered as an example and thus Ein ∝ eipφ[Er Eφ]T (see Supplementary Note 1 for details ), where the integer p > 0 is the azimuthal mode number and Ez is negligible at the sidewalls46. Firstly, the perturbation to WGM evanescent wave s induced by the scatterers is expressed by the matrix 1 1 20 0iδφ WeWM (1) where δ(φ) = qφ (see supplementary material of ref. 41) is the azimuthal phase acquired by the second -order grating scattering , q is the number of scatterers, and Wi (i = 1, 2) is a coefficient quantifying the sca tterers’ modulation on the field strength of the electric components. Here we define the transverse -spin state in the perturbed evanescent wave |M1|· Ein ∝ eipφ[W1Er W2Eφ]T based on the ratio of the two cylindrical components as28 1 12 2 2 12 1, ,r r φ φ φ r φ rWEW E W EiW E σiW E W E W EWE (2) 4 σ (|σ| <= 1) is a real number as Eφ and Er always oscillate in quadrature with each other at the sidewalls46, and it directly characterizes the (spatial ) transverse -spin density in the evanescent wave as S⊥ ∝ σ (refs 6,11) . For the transvers e SAM of left (right) handed spin here, σ > 0 (< 0). In addition, the vector fields of WGMs travelling along the resonator experience a rotation of local coordinate frame, which is described by the matrix 2cos sin sin cosM (3) By apply ing the transfer matrices M1 and M2, the Jones vector of the output CVV becomes TC TC11 out2211 11 2 1 2 1i l i lσσeeiiσσ E (4) where lTC = p q is defined as the topological charge (T C)41. Here the Jones vector is formulated in the global reference frame with the x- and y-polarization basis (i.e., [ Ex Ey]T). The constituent left - ([1 i]T) and right -hand ([1 i]T) circular polarized (CP) vortices are out -of- and in -phase, respectively, when following the two definitions in Equation ( 2). It is straightforward to find that the CVV possesses the SAM and OAM component s per photon as (see Supplementary Note 2) z 22 1σS= σ , z TC 22 1σL = l σ (5) where ħ is the reduced Planck constant. Note that Sz here, which should be distinguished from the spatia l transverse spin density S⊥, is the SAM in CVVs averaged over the transverse x -y plane. More profoundly, the variation in magnitude from the local density ( S⊥ ∝ σ) to the average SAM ( Sz ∝ 2σ/(1+σ2)) is associated with a transverse -spin dependent geometri c phase that stems from the rotation of local vector field . To be more specific, t he Pancharatnam phase44,47 is used to described the spatial phase variation in the CVVs of space -variant polarization state ( see Supplementary Note 3) TC 22+ 1PσΦl σ (6) where lTCφ = pφ – δ(φ) is the scattering phase solely resulted from the first -order diffraction of grating41. Meanwhile, the second term, ΦG = 2σφ/(1+σ2), has a pure geometric nature and arises from the rotation of local transverse -spin state while WGMs travel arou nd the resonator (see Figure 1c). It should be emphasized that ΦG differs from all the previous ly discussed geometric phase s of light that can be identified either in artificial anisotropic structures44 or light beams of curvilinear trajectories15, and originates essentially from the coupling between the transverse SAM of guided light and the rotation of light’s path. Nevertheless, this transverse -spin dependent geometric phase is still in accordance with the unified form of geometric phase of light ΦG = ħ-1∫S·Ωφ dφ (ref. 11), and here S = 2σ/(1+σ2)ħ·z = Sz·z is the SAM and Ωφ = z is the angular velocity of coordinate rotation with respect to coordinate φ for CCW WGMs (see Figure 1c). For CW WGMs, Ωφ = z and the geometric phase becomes ΦG = 2σφ/(1+σ2) (see Supplementary Figure 1 ). On the other hand, it’s interesting to find that the z-compnent of TAM in CVVs (Jz = Lz + S z = lTCħ) is conserved with the given WGM azimuthal mode order p and grating number q, regardless of the transverse -spin state. This is attributed to the rotationally symmetric ‘anisotropy’ orientation of the scatterer group48, and consequently the net transfer of AM betw een the WGMs (carrying TAM of pħ per photon ) and device is constantly qħ. More importantly, the transverse -spin dependen t SOI can be identified in Equation ( 5), and by engineering the transverse -spin state σ and consequently the transverse -spin dependent geometric phase ΦG, the OAM state of a CVV can be modulated and partially converted with SAM . This is a new type of SOI, and the first manifestation of spin -to-orbit al AM conversion in optical vortices directly stemming from the transverse spin of light. In addition , the left - and right -hand CP vortices in Equation ( 4) possess the topological charges of lTC1 and lTC+1, resp ectively. The composition of this ‘superposition’ is subject to the 5 transverse -spin state of WGM evanescent wave. Particularly, when the polarization at the grating scatterer locations reaches one of the CP states (i.e., σ = ± 1), this superposition reduces to a single CP scalar vortex state with a single OAM eigen -state (l = lTC ∓ 1). It should be mentioned that , by exciting WGMs from the alternative waveguide ports or scattering the evanescent waves on the other side of resonator waveguide , the sign of th e transverse spin will be flipped, and using the alternative waveguide port to excite CW propagating WGMs will also reverse the sign of lTC (see Supplementary Note 1 ). Nevertheless, the general SOI phenomena and mode decomposition described in Equations ( 4) and ( 5) still hold. Transverse spin engineering . The transverse -spin state σ in the WGM evanescent wave is dependent on the ratio of cylindrical components as shown in Equation ( 3). In contrast to the evanescent waves of WGMs in bottle micro -resonators49 and unbounded evanesce nt waves at optical interfaces26, where this ratio is largely determined by the refractive index contrast and the incident angle of light, the transverse spin of evanescent waves in highly confined waveguide modes is also significantly altered by the lateral confinement conditions, especially the waveguide co re dimensions. By modifying the mode profile of the transverse component in the core and its spatial derivative at the waveguide boundaries, the magnitude of the longitudinal component can be engineered50. In other words, by tailoring the waveguide geometr y and consequently the vector components of modes, σ can be adjusted and thus enabling the engineering of transverse spin in evanescent waves. Figure 2 . Numerically calculated field component distributions of the quasi -TE mode and the dependence of the component ratio on waveguide dimensions. (a) The cross -sectional field distribution of the transverse component Etrans in a SiN x waveguide, and the dashed rectangular indicates the waveguide of 0.6 μm width and 0.8 μm height. The results in (b) and (c) are obtained with the same waveguide. (b) The field distribution of the longitudinal component (multiplied with the imaginary unit) iElong. (c) The distribution of the component ratio iElong/Etrans over the waveguide cross -section and evanescent region. (d) T he contour map of the ratio iElong/Etrans over variable waveguide dimensions. Among all the waveguide designs calculated, 8 waveguide dimensions marked in the map are employed for device fabrication and characteri zation, consisting of two different heights (0.4 and 0.6 μm) and four widths (0.8, 1.0, 1.2, and 1.4 μm) as indicated in the subscripts. As an example, the cross -sectional maps of the fundamental quasi -TE mode components in a straight silicon nitride (SiN x) waveguide (surrounded by air and placed on a SiO 2 substrate) is depicted in Figure 2 , where the dashed rectangles indicate the waveguide cores of 0.6 μm in height and 0.8 μm in width. Apart from the transverse component Etrans (Figure 6 2a), a strong longitudinal component Elong at the core -claddi ng interface can also be observed in ±π/2 phase difference to Etrans, as shown in Figure 2b. The map of the ratio iElong/Etrans is also plotted in Figure 2c , and outside the waveguide sidewalls it remains almost constant in the decaying evanescent wave, as both components decay at the same rate. More importantly, a contour map of th is ratio is plotted in Figure 2d, in which a n effectively variable ratio of the two components can be observed over various waveguide dimensions . Variable transverse -spin state in waveguide evanescent wave can thus be achieved with routine waveguide design51. The 8 waveguide designs we choose for experimental investigation are marked in the map, and their parameters are listed in Table 1. SiN x waveguide is employed for its moderat e refractive index (~ 2.01) so that a larger range of transverse -spin state can be accessed than other materials (e.g., silicon) . Table 1. Design parameters of the fabricated devices Sample WG 4-8 WG 4-10 WG 4-12 WG 4-14 WG 6-8 WG 6-10 WG 6-12 WG 6-14 Waveguide Height (μm) 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 Waveguide Width (μm) 0.8 1.0 1.2 1.4 0.8 1.0 1.2 1.4 *These parameters apply both to the ring waveguide and access waveguide. *The ring radius of all sample devices is 80 μm, gap between ring and access waveguide i s 200 nm, and each square -shape scatterer is 100 nm by 100nm (with the same height as waveguide). For the 8 sample devices, the ring radius of 80 µ m is used. For each device, q = 517 scatterers are embedded on the inner -sidewall of ring. The ratio of evan escent cylindrical components may be perturbed by the presence of scatterers in the evanescent region , as represented by matrix M1. In this proof -of-principle study , we consider square -shape scatterers protruding from the waveguide sidewall . Each scatterer has the constant area of 100 nm by 100 nm, but is in the same height as the ring waveguide . The gap between the access waveguide and ri ng resonator is fixed at 200 nm . The calculated square of transverse -spin state (σ2) of all sample devices over the scat terer region is shown in Figure 3a. A wide σ2 range of 0.41 - 0.97 is predicted. σ > 0 holds for all cases with WGMs excited by injecting light into Port 1 and the evanescent wave at the inner sidewall is left-hand e lliptical -polarized . Especially, near -circular transverse spin is expected from the devices WG 6-8 and WG 4-10 with σ2 ≈ 0.95 and 0.97, respectively. Some scanning electron microscope (SEM) image s of device WG 6-8 are shown in Figure 3b and 3c . Figure 3. Calculated transverse -spin states of all designed devices and SEM images of fabricated device WG 6-8. (a) Calculated squared transverse -spin states in the evanescent wave of all 8 designed devices . These results are obtained considering that the WGM is excited by injection from Port 1. (b) SEM ima ge of the device WG 6-8. The inset shows a close -up of the coupling section between the access waveguide and the resonator. (c) Top: junction point of the tapered coupler consisting of a tapered SiN x waveguide and a SU8 waveguide. Bottom: c ross-section view s at various positions of the tapered coupler. The minimum width of the SiN x taper (shown in the right -hand side image) is 130 nm. 7 Polarization and t ransverse -spin state characterization . Firstly, t he average ‘cylindrical’ polarization ellipticity of the CVVs is measured to show the overall effect of near -field transverse spin on the polarization of far -field CVVs . The polarization of CVVs varies in space but ex hibits a cylindrical symmetry with respect to the propagation axis43, and therefore here the com ponents Er and Eφ are measured to characterize the average ellipticity in the cylindrical basis (i.e., ε = |Er|/|Eφ| or | Eφ|/|Er|), and compared with the calculated near-field transverse -spin state which is also defined in the same basis . A Radial Polariza tion Convertor (RPC) is used to convert Er and Eφ in far -field CVVs into x- and y -polarized fields respectively52, and the power of these two components (Pr and Pφ) is then recorded for ε2 calculation (ε2 = Pr/Pφ or Pφ/Pr) (see Supplementary Note 4 ). Figure 4. Characterization of average polarization state in CVVs. (a, b) Measured squared polarization ellipticity ε2 (solid markers) of the CVVs from the devices of height 0.4 μm and 0.6 μm, respectively. The prediction from numerical calculations (plotted in Figure 3a) is plotted with dashed lines, and the measured and calculated results for the same device are marked in the same colour. The measured ε2 in CVVs of various lTC from all devices is shown in Figure 4 as solid markers , while the corresponding pr edicted σ2 of each device from Figure 3a is plotted as the dashed line in the same color . Overall, the measured ε2 exhibits high uniformity over all lTC. CVVs of a wide range of spin state s (ε2 from ~0.4 to ~1.0) is obtained with various waveguide designs, and the agreement between the measured ε2 and calculated σ2 shows a definitive correspondence from the transverse -spin state in guided evanescent waves to the polarization in e mitted vortices (ε2 = σ2). Particularly, near -CP (|σ| ≈ 1) CVVs are observed with devices WG 4-10 and WG 6-8, indicating that the reduced sup erposition of single spin-orbital eigen -state vortices predicted by Equation ( 4) can be reached. Secondly , Stokes polarimetry is performed to characterize the local transverse -spin state distribut ion in near-field CVVs . With the Jones vector shown in Equation ( 4), the normalized Stokes parameters as a function of the azimuthal coordinate can be obtained as53 2 1 21cos 2 1 σS σ , 2 2 21sin 2 1σS σ , 3 22 1σS σ (7) For a device of a larger | σ|, the trajectory of the Stokes vector [ S1, S2, S3] on the Poincare sphere circles the pole twice at a higher latitude parallel to the equator. For | σ| = 1, the circle contracts to a single point at the poles, producing a CP CVV. The measured Stokes parameters of near -field CVVs are depicted in Figure 5, in which the results of lTC = +4 CVVs from the devices WG 6-8, WG 6-10, WG 6-12, and WG 6-14 are shown , respectively (see Supplementary Note 4 for details ). In each case, panel (i) is the measured near -field intensity. For better comparison , the calculated Stokes parameters from Equation (7) are shown as solid curves in panel (v) in each case , along with the measured values (dots) sampled from the corresponding parameter panels of (ii)-(iv). The σ values used for calculations are imported from Figure 3a, and the 8 measured S1, S2, and S3 are sampled from the pixels on the periphery of the near -field circle of 80 μm radius (i.e., the radius of ring resonator) . The agreement between the theore tical curves and measured dots shown in the S3 plots of (v) validates the overall effect of waveguide geometry on the transverse -spin state in evanescent waves. For devices of larger | σ|, e.g., WG 6-8, S1 and S2 oscillate less, indicating local polarization states of larger ellipticity. Generally, the jitters in the measured results are attributed to the non-uniformity of fabricated gratings, as well as the decaying intensity of WGMs along the resonator. The deviation of measurements from the theory is more evident with devices of smaller |S3|. This is possibly caused by the light that is scattered from the other (outer -) side of waveguide , carrying the opposite σ, due to sidewall roughness . In some devices, standing -wave -like patterns (e.g., map (iv) in Figure 5c) are introduced by the interference of scattered TE and TM modes, because in these waveguide designs these two modes are more degenerate and single -polarization -mode excitation is more crit ical to polarization control in mode launching. Figure 5. Stokes polarimetry of near-field polarization of CVVs . (a-d) Measured two -dimensional maps of near -field Stokes parameters and the comparison with theoretical prediction for devices WG 6-8, WG 6-10, WG 6-12, and WG 6-14, respectively. Each map (i) is the near -field intensity profile from the device with lTC = +4 . Maps (ii), (iii), and (iv) are the corresponding near -field profiles of the normalized Stokes parameter s S1, S2, and S3, respectively. The plots in (v) show the comparison between the measured results (dots) sampled from (ii) -(iv) and the corresponding predicti on (solid curves) from Equation (7). For each set of measured data in (v), 288 pixels intersecting with the circle of 80 μm radius along the azimuthal direction ( φ) from 0 to 2π are sampled from the corresponding map. For each solid curve of prediction, the data is calculated by substituting the transverse -spin state σ from Figure 3a into Equation (7). Transverse spin induced SOI . The OAM component carried by CVVs is measured to verify the transverse -spin induced spin-to-orbital conversion predicted by Equation ( 5) (see Supplementary Note 4 and 5 for characterization method of OAM state and emission sp ectrum from devices ). The measured OAM spectra for the CVVs from the devices WG 6-8, WG 6-10, WG 6-12, and WG 6-14 are plotted in Figure 6. In close agreement with the theory, each OAM spectrum (row) of CVV with lTC contains two dominant peaks at lTC1 and lTC+1, carried by the constituent left - and right -hand CP vortices, respectively. The intensities of all spurious modes are < 0.03. Note that each CP vortex can thus be confirmed as possessing a TAM of lTCħ (see Supplementary Note 6), and this exper imentally validates the overall TAM in each CVV is preserved as lTCħ regardless of waveguide geometries. More importantly, the average SAM in each CVV is subject to the near -field 9 transverse spin (Sz = 2σ/(1+σ2)ħ) as shown in Figure 5. Therefore, the remar kable transverse -spin dependent SOI effect is revealed, as the OAM component carried by CCVs can be partially derived out of the transverse SAM in the evanescent waves. This is the first demonstration of an SOI effect resulted from the interaction between the intrinsic OAM and transverse SAM of light. A direct and useful manifestation of this effect is that the relative intensities of the two dominant peaks, i.e., the two constituent CP vortices , can be changed by modifying σ. For example, the normalized intensities of the left - and right -hand CP vortices from WG 6-8 are around 0.93 and 0.07, respectively, while for WG 6-14 they account for about 0.62 and 0.36 of the total intensity, respectively. This variab le superposition of AM states in CVVs provides a viable pathway for information encoding in the spin -orbit space. Another implication of this SOI effect is that a vortex should appear even when lTC = 0 but σ ≠ 0 (exemplified by the square in the yellow box in Figure 6a ); that is, without introducing any spatial phase gradient that has been inherent to many optical vortex generation techniques5. This purely transverse -spin-derived vortex essentially originates from the spatially varying ‘anisotropy’ of the gratings and the rot ational symmetry of vector WGMs. In other words, this is an interesting demonstration of optical vortex generation controlled by the QSHE of light33, and the spin state in the edge modes stemming from the intrinsic SOI at optical interfaces can thus be man ipulated for spatial light modulation via the ‘extrinsic’ SOI in anisotropic structures11. Figure 6. Characterization of OAM component s in CVVs. (a -d) The measured OAM spectra for the devices WG 6-8, WG 6-10, WG 6-12, and WG 6-14, respectively. For each dev ice, the wavelengths of lTC = -5 to +5 are considered, and each column represents a spectrum of measured OAM comp onents with the corresponding lTC. Spin -orbit controlled unidirectional coupling . Given the principle of reciprocity, this device can also be used for detection of AM componen ts in an incident CVV beam54. The ring resonator supports the degenerate CW and CCW WGMs at each resonance wavelength λ0, and these two modes give rise to the emission of two CVVs of opposite T Cs, i.e., lTC = ±(p q). Mean while, these two WGMs exhibit opposite σ in the inner -side evanescent waves, and therefore the two emitted CVVs carry exactly opposi te spin and orbit al AM states , i.e., < 2σ/(1+σ2), lTC 2σ/(1+σ2)> and <2σ/(1+σ2), lTC + 2σ/(1+σ2)>. When receiving at λ0, this device can couple these two CVVs into the two opposite resonating directions and guide their power to the two access -waveguide ports , respectively . All CVVs with λ ≠ λ0, or at λ0 but with other SAM and OAM states are mismatched with this selection rule and will be denied by the device. Consequen tly, we obtain the effect of unidirectional coupling into guided modes jointly controlled by spin and orbital AM states . Although this phenomenon is essentially associated with the spin -direction locking induced by local SOI in evanescen t waves of guided modes21,22, this new spin -orbit direction locking effect incorporates the orbital degree of freedom , using the close -loop waveguide for 10 filtering in the OAM space . This spin -orbit controlled coupling provides a po tential solution for spin and orbit al AM state detection , avoiding the separate manipulations on these two degrees of freedom. Generally, the input light for ideal r eception with the device should carry the identical spin and orbital AM states as the outpu t CVVs , while exhibiting cylindrical symmetry in intensity and polarization profiles . But for this proof -of-principle study here, the special case of σ = ±1 (where the spin -orbit states of CVVs reduce to <±σ, lTC∓σ>) is demonstrated for simpler experimental configuration , using the device WG 4-10 of near-CP transverse -spin state (σ ≈ ±1) as shown in Figure 4b. Table 2. SAM and OAM states in CVVs at vari ous resonance wavelengths Wavelength (nm) 1578.61 1583.11 1587.59 1592.11 Access port 1 2 1 2 1 2 1 2 lTC 4 +4 2 +2 0 0 +2 2 SAM OAM 1 3 +1 +3 1 1 +1 +1 1 +1 +1 1 1 +3 +1 3 Figure 7. Proof -of-principle illustration of spin -orbit controlle d uni -directional coupling of waveguide modes. (a -d) Measured results for the device WG 4-10 with incident light in the wavelength of 1578.61 nm, 1583.11 nm, 1587.59 nm, and 1592.11 nm, respectively. For each wavelength, incident beams of 33 different spin -orbit states (σin = 1, 0, and +1, lin = -5, -4, …, and +5) are illuminated on the device. For each 11 incident polarization state ( σin = 1/0/+1), the received power with different incident OAM orders from the both ports are listed in a single histogram. The data in each figure (a/b/c/d) is normalized to the highest value in the group. The spin and orbital AM states of the CVVs at four resonant wavelengths from device WG 4-10 associated to Port 1 or 2 are listed in Table 2. For each wavelength, optical vortices of 3 SAM st ates (σin = 0, ± 1) and 11 OAM states ( lin from 5 to +5) are prepared and illuminated on the device (see Supplementary Note 6 for details ). The measured (and calibrated with respect to the lensed fiber coupling loss) output power P1 at port 1 and P2 at port 2 are normalized and plotted in Figure 7. The first distinct ive observation is that P 1 in blue bars (P2 in red bars) is universally negligible with incidence of σin = +1 (1), in accordance with the predefined transverse -spin state σ ≈ 1 (+1) when inputting via Port 1 (2) and the underlying prediction from the spin -direction locking effect21,22. With the incidence of an arbitrary polarization state, however, light is coupled to the both ports and the resulting ratio of P 1 and P 2 is determined by the relative i ntensity of left - and right-hand CPs in the incident CVV . For example, with the incident linear polarization (σin = 0, middle rows in Figure 7a to 7d) as an equal superposition of two CPs, P 1 and P 2 exhibit comparable values. Moreover, the coupling strength is further subject to the incident OAM state lin. For example, when measuring at Port 2 with incidence at 15 78.61 nm, a single dominant power peak at port 2 appears only at the incident state of < σin = +1, lin = +3> (upper row in Figure 7a), while it can only be observed at Port 1 with incident < σin = 1, lin = 3> (lower row in Figure 7a). This high ly directio nal and selective coupling , determined by the spin and orbital AM state <σin, lin>, is a higher -order phenomenon with respect to the basic spin-controlled coupling via evanescent waves, as both the spatial and polarization properties of light must be taken into account. This effect allows for a robust manipulation of light on the micron -scale using both the spin and orbital degrees of freedom, e.g., encoding and retrieving information, without the necessity of separate controls on polarization and spatial phase profile. Discussion To sum up, we have identified and demonstrated a new type of spin -orbit interaction of light, namely the interplay between the intrinsic OAM and the transverse spin of light. This new SOI effect originates from the manipulation of local transverse -spin-dependent geometric phase by artificially introducing a close -loop waveguide and sub -wavelength scatterers of rotational symmetry. Engineering the local transverse spin by tailoring waveguide dimensions then controls the global spin-to-orbital conversion in the generated optical vortices. Our results have both fundamental and applied importance . The interaction between the intrinsic OAM and transverse spin of light is an integral but thus far missing part of the rich SOI phenomena . The newly discovered interaction builds one more path way between the polarization and spati al degrees of freedom of light, which could provide nano -photonic technologies with additional tools of light manipulation at the subwavelength scales and of informatio n transfer over more degrees of freedom. The resulting effects , e.g., the variable superposition of spi n-orbit states in optical vortices , may find applications in optical quantum information processing. T he spin -orbit jointly controlled directional coupli ng can be used to operate on the eigen -states involving both AM components, so that the device considered here can be regarded as a prototype of a planar spin -orbit -controlled gate that interfaces propagating and bounded photons of two -dimensional entangle ment. Better performance (e.g., power efficiency and AM state purity) can be brought about by further device design and optimization. The demonstrated interaction should also exist in other systems that support evanescent modes, including surface plasmon -polaritons which can significantly min iaturize the elements. Methods Numerical simulation s. Numerical simulations are performed with the finite difference eigenmode solver (FDE, Lumerical Solutions, Inc.). For the calculation of squared transverse -spin sta te (σ2 shown in Figure. 3a) at the scatterer location of each designed device, first the distribution of the two cylindrical components (Er and Eφ) over the scatterer region is calculated, and then the average σ2 (<=1) in the scattered evanescent wave 12 is obtai ned as the ratio of integrated intensities of Er and Eφ over one scatterer region . The effect of sactterer ’s modulation on field amplitudes (W1/W2 in Equation (3)) is thus included in the calculated σ2. Fabrication . The SiN x waveguide layers are first depo sited on a 5 -μm oxidized <100> silicon wafer using inductively coupled plasma chemical vapor deposition (ICP -CVD) system (Plasmalab System 100 ICP180, Oxford ). The device structures are defined in a 450 -nm-thick negative resist using electron -beam lithogra phy (EBL , EBPG5000 ES , Vistec). Reactive -ion-etch (RIE , Plasmalab System 100 RIE180, Oxford) with a mixture of CHF 3 and O 2 gases is applied to etch through the waveguide layer to form the device . An inverse taper combined with a SU8 waveguide is used as th e coupler between external optical fiber and the access waveguide. Experimental setup s for device characterizations, SOI measurement and spin -orbit controlled unidirectional coupling are shown and explained in Supplementary Note 4 and 6. References 1. Poyntin g, J. H. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light. Proc . R. Soc. Lond . Ser. A 82, 560–567 (1909 ). 2. Beth , R. A. Mechanical detection and measurement of the angular momentum of l ight. Phys . Rev. 50, 115-125 (1936 ). 3. Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woe rdman, J. P. Orbital angular momentum of light and the transformation of Laguerre -Gaussian laser modes. Phys . Rev. A 45, 8185 -8189 (1992 ). 4. Allen, L ., Padgett, M. J . & Babiker, M. The orbital angular momentum of light. Prog . Opt. 39, 291-372 (1999 ). 5. Yao, A. M. & Padgett, M. J. Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photon 3, 161-204 (2011 ). 6. Bliokh, K. Y . & Nori, F. Transverse and long itudinal angular momenta of light. Phys . Rep. 592, 1-38 (2015 ). 7. Andrew s, D. L., & Babiker, M. The angular momentum of light (Cambridge Univ . Press, Cambridge, 2013 ). 8. Liberman, V. S. & Zel'dovich, B. Y . Spin-orbit interaction of a photon in an inhomogeneous medium. Phys . Rev. A 46, 5199 -5207 (1992 ). 9. Hasman, E., Biener, G., Niv, A. & Kleiner, V. Space -variant polarization manipulation. Prog . Opt. 47, 215-289 (2005 ). 10. Marrucci, L. et al. Spin-to-orbital conversion of the angular momentum of light and its classi cal and quantum applications. J. Opt. 13, 064001 (2011 ). 11. Bliokh, K. Y ., Rodriguez -Fortuno, F. J., Nori, F. & Zayats, A. V. Spin-orbit interactions of light. Nat. Photon 9, 796-808 (2015 ). 12. Schwartz, C. & Dogariu, A. Conservation of angular momentum of light in single scattering. Opt. Express 14, 8425 -8433 (2006 ). 13. Zhao, Y ., Edgar, J. S., Jeffries, G. D. M., McGloin, D. & Chiu, D. T. Spin-to-orbital angular momentum conversion in a strongly focused optical beam. Phys . Rev. Lett. 99, 073901 (2007 ). 14. Ciatto ni, A. , Cincotti, G. & Palma, C . Angular momentum dynamics of a paraxial beam in a uniaxial crystal. Phys . Rev. E 67, 036618 (2003 ). 15. Bliokh, K. Y ., Niv, A., Kleiner, V. & Hasman, E. Geometrodynamics of spinning light. Nat. Photon 2, 748-753 (2008 ). 16. Bliokh , K. Y . Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect. Phys . Rev. Lett. 97, 043901 (2006 ). 17. Hosten, O. & Kwiat, P. Observation of the spin Hall effect of light via weak measurements. Science 319, 787-790 (2008 ). 18. Rodriguez -Fortuno, F. J. et al. Near -Field Interference for the Unidirectional Excitation of Electromagnetic Guided Modes. Science 340, 328-330 (2013 ). 19. Lin, J. et al. Polarization -Controlled Tunable Directional Coupling of Surface Plasmon Polaritons. Scien ce 340, 331-334 (2013). 20. Shitrit, N. et al. Spin-Optical Metamaterial Route to Spin - Controlled Photonics. Science 340, 724-726 (2013 ). 21. O'Connor, D., Ginzburg, P., Rodriguez -Fortuno, F. J., Wu rtz, G. A. & Zayats, A. V. Spin-orbit coupling in surface plasmon scattering by nanostructures. Nat. Commun . 5, 5327 (2014 ). 22. Petersen, J. , Volz, J. & Rauschenbeutel, A. Chiral nanophotonic waveguide interface based on spin -orbit interaction of light. Science 346, 67-71 (2014 ). 23. Roy, B., Ghosh, N., Banerjee, A., Gupta, S. D. & Roy, S. Manifestations of geometric phase and enhanced spin Hall shifts in an optical trap. New J . Phys . 16, 083037 (2014 ). 13 24. Rodriguez -Herrera, O. G., Lara, D., Bliokh, K. Y ., Ostrovskaya, E. A. & Dainty, C. Optical Nanoprobing via Spin -Orbit Interact ion of Light. Phys . Rev. Lett. 104, 253601 (2010 ). 25. Yu, N. & Capasso, F. Flat optics with designer metasurfaces. Nat. Mater . 13, 139-150 (2014 ). 26. Bliokh, K. Y . & Nori, F. Transverse spin of a surface polariton. Phys . Rev. A 85, 061801 (2012 ). 27. Kim, K. -Y ., Lee , I.-M., Kim, J., Jung, J. & Lee, B. Time reversal and the spin angular momentum of transverse -electric and transverse -magnetic surface modes. Phys . Rev. A 86, 063805 (2012 ). 28. Bliokh, K. Y . , Bekshaev, A. Y . & Nori, F. Extraordinary momentum and spin in evan escent waves. Nat. Commun . 5, 3300 (2014 ). 29. Kim, K. -Y . & Wang, A. X. Spin angular momentum of surface modes from the perspective of optical power flow. Opt. Lett. 40, 2929 -2932 (2015 ). 30. Aiello, A., Banzer, P., Neugebaueru, M. & Leuchs, G. From transverse ang ular momentum to photonic wheels. Nat. Photon . 9, 789-795 (2015 ). 31. Banzer, P. et al. The photonic wheel - demonstration of a state of light with purely transverse angular momentum. J. Europ . Opt. Soc. Rap. Public 8, 13032 (2013 ). 32. Born, M. & Wolf, E. Princip les of optics Ch. 6 ( Pergamon Press , Oxford, 1980) . 33. Bliokh, K. Y ., Smirnova, D. & Nori, F. Quantum spin Hall effect of light. Science 348, 1448 -1451 (2015 ). 34. Ling, X. et al. Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence. Light Sci. Appl . 4, e290 (2015 ). 35. Ling, X. et al. Recent advances in the spin Hall effect of light. Rep. Prog . Phys . 80, 066401 (2017 ). 36. Young, A. B. et al. Polarization Engineering in Photonic Crystal Waveguides for Spin -Photon Entanglers. Phys . Rev. Lett. 115, 153901 (2015 ). 37. Sayrin, C. et al. Optical diode based on the chirality of guided photons. arXiv , 1502 .01549 (2015 ). 38. Pichler, H., Ramos, T., Daley, A. J. & Zoller, P. Quantum optics of chiral spin networks. Phys . Rev. A 91, 042116 (2015 ). 39. Lodahl, P. et al. Chiral quantum optics. Nature 541, 473-480 (2017 ). 40. Rubinsztein -Dunlop, H. et al. Roadmap on structured light. J. Opt. 19, 013001 (2017 ). 41. Cai, X. et al. Integrated Compact Optical V ortex Beam Emitters. Science 338, 363-366 (2012 ). 42. Zhu, J., Chen, Y ., Zhang, Y ., Cai, X. & Yu, S. Spin and orbital angular momentum and their conversion in cylindrical vector vortices. Opt. Lett. 39: 4435 -4438 (2014 ). 43. Zhu, J., Cai, X., Che n, Y . & Yu, S. Theoretical model for angular grating -based integrated optical vortex beam emitters. Opt. Lett. 38, 1343 -1345 (2013 ). 44. Bomzon, Z., Biener, G., Kleiner, V. & Hasman, E. Space -variant Pancharatnam -Berry phase optical elements with computer -generated subwavelength gratings. Opt. Lett. 27, 1141 -1143 (2002 ). 45. Neugebauer, M., Bauer, T., Aiello, A. & Banzer, P. Measuring the Transverse Spin Density of Light. Phys . Rev. Lett. 114, 06390 1 (2015 ). 46. Marcuvitz , N. Waveguide Handbook 1st edn (McGraw -Hill, New York, 1951 ). 47. Pancharatnam, S. Proc. Indian Acad. Sci. Sect. A 44, 247 (1956). 48. Marr ucci, L., Manzo, C. & Paparo, D . Optical spin -to-orbital angular momentum conversion in inhomogeneous anisotropic media. Phys . Rev. Lett. 96, 163905 (2006 ). 49. Junge, C., O'Shea, D., Volz, J. & Rauschenbeutel, A. Strong Coupling be tween Single Atoms and Nontransversal Photons. Phys . Rev. Lett. 110, 213604 (2013 ). 50. Driscoll, J. B. et al. Large longitudinal electric fields (Ez) in silicon nanowire waveguides. Opt. Express 17, 2797 -2804 (2009 ). 51. Snyder, A. W. & Love, J. Optical waveguide theory . (Springer Science & Business Media, Berlin, 2012 ). 52. Stalder, M. & Schadt, M. Linearly polarized light with axial symmetry generated by liquid -crystal polarization converters. Opt. Lett. 21, 1948 -1950 (1996 ). 53. Berry, H. G., Gabrielse, G. & Livingston , A. E . Measurement of the Stokes parameters of light. Appl . Opt. 16, 3200 -3205 (1977 ). 54. Cicek, K. et al. Integrated optical vortex beam receivers. Opt. Express 24, 28529 -28539 (2016 ). 14 Supplementary Note 1. Formulation of Cylindrical Vector Vortices Emission The evanescent wave of whispering -gallery modes (WGMs), written in the cylindrical polarization basis as Ein ∝ e±ipφ[Er Eφ]T, is perturbed by the second -order grating when circulating around the resonator. Here we denote the positive integer p as the azimuthal mode number of WGM s, and the two degenerate counter -propagating WGMs resonating in the same wavelength h ave the mode numbers of p (counter - p (clockwise, CW), respectively. The perturbation of gratings to the evanescent wave is generalized in a matrix as 1 1 20 0iδφ WeWM (1) where W1 and W2 are real numb ers that reflect the modulation on the amplitudes of local transverse ( Er) and longitudinal ( Eφ) fields, respectively, due to grating perturbation. The off -diagonal elements of M1 are vanishing as we assume the scattering does not introduce coupling betwee n orthogonal field components. δ(φ) = ∓qφ is the phase imparted on the first -order diffracted wave derived using the coupled -mode theory (cf. supplementary material of ref. [1]) and q the number of grating elements. In addition, as WGMs travel around the r esonator, the vector evanescent wave experiences a rotation of local coordinates ([ Er Eφ]T) with respect to the global laboratory frame ([ Ex Ey]T), as shown in Supplementary Figure 1. The effect of this rotation on the emitted CVVs (represented in the basi s of [ Ex Ey]T) can be written with a single matrix M2 as 2cos sin sin cosx rr yE EE E EE =M (2) The final output CVV ( Eout = M2·M1·Ein) can be obtained as TC TC11 out2211 11 2 1 2 1i l i lσσeeiiσσ E (3) where σ is the transverse -spin state defi ned in Equation (2) in the main text, and lTC = ± (p – q) the topological charge. It should be emphasized here that the constituent left - ([1 i]T) and right - i]T) circular polarized vortices are out -of- and in -phase, respectively, only when followi ng the two definitions of σ. Supplementary Figure 1. Rotation of local coordinates as WGMs circulating around the resonator . For the counter -clockwise (CCW) propagating WGM shown here, the rotation angle of local coordinates ( r, φ) from point a to b with respect to the global coordinates ( x, y) of output CVVs is φ0z, and z is the unit vector. In other words, from the perspective of light field at point b φ0z should be applied in order to align with the global reference fra me (x, y) when its spatial (Pancharatnam) phase is compared with point a in the 15 CVV field. And therefore, the angular velocity of reference frame rotation with respect to coordinate φ for CCW WGM is Ωφ z. Similarly, Ωφ = z for clockwise propagating (CW) WGMs. Supplementary Note 2. Angular Momentum in Cylindrical Vector Vortices The cylindrical vector vortices (CVVs) emitted from the angular -grating based devices considered in this paper exhibit good paraxiality, as the radius of ring resonator ( R = 80 μ m) is much larger than the wavelength ( λ = 1.5 um) [2, 3]. The angular momentum (AM) carried in paraxial optical vortex beams can be essentially considered as the sum of the spin and orbital AM components, which are associated wit h the polarization and spa tial properties of light, respectively [4, 5]. The cycle averaged z-component of the spin AM (SAM) and orbital AM (OAM) per unit length per photon of a vortex beam can be written as [5] x y y x zrdrdφ E E E E S i rdrdφ EE (4) ,,jj j x y z zrdrdφ E Eφ L i rdrdφ EE (5) By substituting the CVV shown in Equation (3) into the equations above, the SAM and OAM components carried by the CVV are 22 1zσS σ (6) TC 22 1zσLl σ (7) where σ is the transverse -spin state in the near -field evanescent wave. The total angular momentum (TAM) in a CVV ( Jz = Sz + Lz) is thus simply written as TC zJl (8) Supplementary Note 3. Geometric Phase Induced by Coordinate Ro tation As the polarization state of CVVs is space -variant [2], here the Pancharatnam phase is used to define the phase difference of light fields in different positions in CVVs [6], that ΦP = arg⟨E(r1, φ1), E(r2, φ2)⟩, where arg ⟨E1, E2⟩ is the argument of the inner product of the two Jones vectors E1 and E2. Following this definition, the Pancharatnam phase of fields at two different positions ( r1, φ1) and ( r2, φ2) in a CVV is given by TC 22arg cos sin 1PΦ l i (9) where Δφ = φ2 – φ1, and the CV the azimuthal direction is 2 TC TC 2002arctan tan2 1lim lim 1PΦll (10) Clearly, the Pancharatnam phase in CVVs scales linearly with coordinate φ, and thus we can rewrite it as TC 22 1PΦl (11) Considering the SAM component carried by CVVs shown in Supplementary Equation (6), the Pancharatnam phase can be generalized as TC1 PΦ l d S (12) where S = Sz·z is the SAM per p hoton, and Ωφ is the angular velocity of reference frame rotation with respect to the coordinate φ for Pancharatnam phase comparison (see Supplementary Figure 1). Here, Ωφ = ∓z for CCW and CW WGMs, respectively. 16 Supplementary Note 4. Techniques for Polarization and O AM States Characterization Supplementary Figure 2 . Experimental setup for device characterization and the observation of the transverse -spin induced SOI effect. The experimental characterizations of the devices are performed with the setup shown in Supplementary Figure 2. For the excitation of WGMs and hence emission of CVVs, the continuous -wave light from the tunable laser source ( 8461B, Agilent) is controlled with a fiber polarization controller ( FPC561, Thorlabs), and the quasi -TE mode in the waveguid e is excited by launching the horizontally polarized light into one of the ports (e.g., Port 1 as shown in Figure 3a) using a lensed fiber ( SMF -28E+LL, Corning ). A small fraction, 1%, of the input light is tapped using a coupler (PMC1550 -90B-FC, Thorlabs) and directed to another collimator (F240FC -1550, Thorlabs) to serve as the reference light for the interference with the emitted CVVs. For the measurement of the emission spectrum of the device, the vertically emitted beam from the device plane is collect ed and collimated with a 20X objective lens (UPlanFLN, Olympus) positioned in the working distance (1.7mm) away from the device. A power meter (PM12 2D, Thorlabs) is placed behind the collimating objective lens to record the dependence of emission power on the working wavelength, while the output wavelength of the tunable laser is swept from 1500 nm to 1640 nm with the step of 10 pm. For measuring the average cylindrical -basis polarization ellipticity of CVVs, a liquid crystal based element called Radial Po larization Converter (RPC, ARCoptix S. A., Switzerland) is used to selectively measure the power of Eφ and Er components. The RPC can be typically used for its spatially varying anisotropy to convert linearly polarized light into vector beams of azimuthal or radial polarizations [7]. Here the reversed effect of this element is employed: by injecting the light into the exit side, Eφ and Er in the CVV will be converted into x - and y -polarized light leaving the entrance side, respectively. A linear polarizer ( LPNIR100 -MP2, Thorlabs) is then used to filter out one of the components, and by detecting the power of the two orthogonal components as Pφ and Pr, the squared polarization ellipticity ( ε2) in the CVV determined by the near -field transverse spin state can be obtained as ε2 = Pr/Pφ or Pφ/Pr. For Stokes parameter s measurements, the near -field pattern of the CVV is imaged onto an InGaAs camera (C14041 -10U, Hamamastu) with an achromatic lens ( f = 250 mm, AC254 -250-C-ML, Thorlabs), and the linear - and circular -polarizations are obtained by adjusting the quarter -wave plate (QWP, AQWP 10M -1600, Thorlabs) and the linear polarizer (LP) mounted on continuous rotation mounts (CRM1, Thorlabs). For the characterization of OAM states in CVVs, a phase -only reflective spati al light modulator (PLUTO SLM, HOLOEYE Photonics AG) loaded with grey -scale fork -grating patterns is used [8]. A linear polarizer is first used to acquire one of the linear -polarized components in the CVV, which generally is a mixture of two topologically charged vortices as shown in Equation (4) in the main text. The central axis of the polarized CVVs is then aligned with the center of fork -grating patterns on the SLM. For each incident CVV, the SLM is loaded with a series of fork-grating images with conse cutive integer topological charges, e.g., lSLM = -5, -4, …, +5. The light reflected off each image is focused by an achromatic lens ( f = 150 mm, AC254 -150-C-ML, Thorlabs) followed by the InGaAs camera, and the power of the corresponding OAM component lSLM is obtained by integrating the intensity of the central Gaussian -like spot [9]. The process is repeated for the other linear -polarized component, and the measured OAM spectrum of the incident CVV is then obtained by averaging the two corresponding OA M comp onents over the two linear polarization components. Supplementary Note 5. Preliminary Characterization of Devices 17 Supplementary Figure 3. Measured emission spectral response of sample device W 6-8 as input wavelength is swept from 1500 -1640 nm. The inset shows a typical near -field intensity profile of emitted CVVs. The measured emission spectral response of sample WG 6-8, as an instance, is plotted in Supplementary Figure 3 after normalization to the output power of tunable laser. The central wavelength at which the emitted CVV has lTC = p q = 0, is λc = 1596.6 nm, and the free spectral range is around 2.2 nm. At the wavelengths longer (shorter) than λc, CVVs carry positive (negative) integer lTC at the resonance peaks. The inset shows a typical near -field intensity profile of the device a t the resonance wavelengths. The long-range variation of peak emitted power across the spectr al range is primarily caused by the fixed gap between access waveguide and ring resonator that couples varying power into the resonato r across the spectrum. Supplementary Figure 4. Far-field profiles and interferograms of left-hand circular -polarized components of CVV s from device WG 6-8. Some typical far -field intensity profiles and interferograms of CVVs are illustrated in Supplementary Figure 4, in which the d evice WG 6-8 is configured for the emission of CVVs with lTC from 2 to +4. The left -hand circular polarized (LHCP) component is obtained by filtering the far-field CVVs with a QWP and LP combination, and then interferes with the LHCP Gaussian beam. For eac h CVV of lTC, the LHCP component possesses the OAM state of lLHCP = lTC-1 (see Equation (4) in the main text), and therefore each interferogram shown in the figure clearly exhibits the spiral fringes with the number of lLHCP [1]. Supplementary Note 6. Ex perimental Setup for Spin -Orbit Unidirectional Coupling The experimental setup for the measurement of spin -orbit controlled unidirectional coupling is shown in Supplementary Figure 3. The polarized light from the tunable laser is collimated with a collimat or and then reflected by the SLM for the conversion to the vortex carrying OAM state lin. The linear -polarized vortex is imparted a certain polarization state ( σin) by the rotatable QWP. A 20X objective lens is used for focusing and illuminating the prepared vortex of spin and orbital AM states < σin, lin> onto the device. Two lensed fibers are used for collecting the received power from the waveguide Ports 1 and 2, respectively. 18 Supplementary Figure 5. Experimental setup for the measurement of spin -orbit controlled directional coupling of waveguide modes. Supplementary References [1] X. Cai, J. Wang, M. J Strain, B. Johnson -Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363-366 (2012 ). [2] J. Zhu, X. Cai, Y. Chen, and S. Yu, Opt. Lett. 38, 1343 -1345 (2013). [3] J. Zhu, Y. Chen, Y. Zhang, X. Cai, and S. Yu, Opt. Lett. 39, 4435 -4438 (2014). [4] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, an d J. P. Woerdman, Phys. Rev. A 45, 8185 -8189 (1992). [5] S. M. Barnett and L. Allen, Opt. Commun. 110, 670 -678 (1994). [6] S. Pancharatnam, Proc. Ind. Acad. Sci. 44, 247 -262 (1956). [7] M. Stalder, and M. Schadt, Opt . Lett. 21, 1948 -1950 (1996 ). [8] G. Gibson, J. Courtial , M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke -Arnold, Opt. Express 12, 5448 -5456 (2004). [9] M. J. Strain, X. Cai, J. Wang, J. Zhu, D. B. Phillips, L. Chen, M. Lopez -Garcia, J. L. O’Brien, M. G. Thompson, M. Sorel, and S. Yu, Nat. Com mun. 5, 4856 (2014) .

1411.3043v1.Fermi_Gases_with_Synthetic_Spin_Orbit_Coupling.pdf

Fermi Gases with Synthetic Spin-Orbit Coupling Jing Zhang State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P. R. China Hui Hu and Xia-Ji Liu Centre for Atom Optics and Ultrafast Spectroscopy, Swinburne University of Technology, Melbourne 3122, Australia Han Pu Department of Physics and Astronomy, and Rice Quantum Institute, Rice University, Houston, TX 77251, USA (Dated: November 13, 2014) We briefly review recent progress on ultracold atomic Fermi gases with different types of synthetic spin-orbit coupling, including the one-dimensional (1D) equal weight Rashba-Dresselhaus and two- dimensional (2D) Rasbha spin-orbit couplings. Theoretically, we show how the single-body, two- body and many-body properties of Fermi gases are dramatically changed by spin-orbit coupling. In particular, the interplay between spin-orbit coupling and interatomic interaction may lead to several long-sought exotic superfluid phases at low temperatures, such as anisotropic superfluid, topological superfluid and inhomogeneous superfluid. Experimentally, only the first type - equal weight combination of Rasbha and Dresselhaus spin-orbit couplings - has been realized very recently usingatwo-photonRamanprocess. Weshowhowtocharacterizeanormalspin-orbitcoupledatomic Fermi gas in both non-interacting and strongly-interacting limits, using particularly momentum- resolved radio-frequency spectroscopy. The experimental demonstration of a strongly-interacting spin-orbit coupled Fermi gas opens a promising way to observe various exotic superfluid phases in the near future. PACS numbers: 05.30.Fk, 03.75.Hh, 03.75.Ss, 67.85.-d Contents I. Introduction 2 II. Theory of spin-orbit coupled Fermi gas 3 A. Theoretical framework 3 1. Functional path-integral approach 3 2. Two-particle physics from the particle-particle vertex function 5 3. Many-body T-matrix theory 6 4. Bogoliubov-de Gennes equation for trapped Fermi systems 7 5. Momentum- or spatially-resolved radio-frequency spectrum 8 B. 1D equal-weight Rashba-Dresselhaus spin-orbit coupling 9 1. Single-particle spectrum 10 2. Two-body physics 12 3. Momentum-resolved radio-frequency spectrum of the superfluid phase 13 4. Fulde-Ferrell superfluidity 15 5. 1D topological superfluidity 17 C. 2D Rashba spin-orbit coupling 19 1. Single-particle spectrum 20 2. Two-body physics 20 3. Crossover to rashbon BEC and anisotropic superfluidity 22 4. 2D Topological superfluidity 24 III. Experiments 25 A. The noninteracting spin-orbit coupled Fermi gas 26 1. Rabi oscillation 26 2. Momentum distribution 28 3. Lifshitz transition 28arXiv:1411.3043v1 [cond-mat.quant-gas] 12 Nov 20142 4. Momentum-resolved rf spectrum 28 B. The strongly interacting spin-orbit coupled Fermi gas 30 1. Integrated radio-frequency spectrum 30 2. Coherent formation of Feshbach molecules by spin-orbit coupling 32 IV. Conclusion 32 Acknowledgments 33 References 33 I. INTRODUCTION Modern physical theories describe reality in terms of fields, many of which obey gauge symmetry. Gauge symmetry is the property of a field theory in which different configurations of the underlying fields — which are not themselves directly observable — result in identical observable quantities. Electromagnetism is an ideal example to illustrate this point. A system of stationary electric charges produces an electric field E(but no magnetic field). It is convenient to define a scalar potential V, a voltage, that is also determined by the charge distribution. The electric field at any position is given by the gradient of the scalar potential: E(r) =rV(r). In this system, a global symmtry is readily perceived: if the scalar potential everywhere is changed by the same amount, i.e., V(r)!V(r) +V0, the resulting electric field is unchanged. A more non-trivial example is given by a system of moving charges which produces both electric and magnetic field. In addition to the scalar potential, we now also introduce a vector potential A, the curl of which gives the magnetic field: B(r) =rA(r). This system obeys the local gauge sysmmetry: any local change in the scalar potential [ V(r)!V(r)@=@twith (r;t)being an arbitrary function of position and time] can be combined with a compensating change in the vector potential [ A(r)!A(r) +r] in such a way that the electric and magnetic fields are invariant. Maxwell’s classical theory of electromagnetism is the first gauge theory with local symmetry. A related symmetry can be demonstrated in the quantum theory of electromagnetic interactions, which describes the interaction between charged particles. From first sight, Maxwell’s theory should not directly describe the center-of-mass motion of neutral atoms. However, a beautiful series of experiments carried out at NIST [1–3] demonstrated that artificial gauge fields can be generated in cold atomic vapours using laser fields, such that neutral atoms can be used to simulate charged particles moving in electromagnetic fields [4]. How to engineer artificial gauge fields is reviewed by Spielman in an article published in the previous volume of this book series [5]. Itwouldnotbeveryinterestingifalllight-inducedgaugefieldscoulddoistomakeneutralatomsmimicthebehavior of charged particles. Indeed, artificial gauge field can be made non-Abelian, i.e., the Cartesian components of the field do not commute with each other. By contrast, the familiar electromagnetic fields are Abelian since their Cartesian components are represented by c-numbers, thus commuting with each other. A special feature of non-Abelian gauge field is that it can induce spin-orbit coupling. The concept of spin-orbit coupling (SOC) is encountered, for example, in the study of atomic structure, where the coupling between the electron’s orbital motion and its intrinsic spin gives rise to the fine structure of atomic spectrum. In the current context, SOC refers to the coupling between the internal pseudo-spin degrees of freedom and the external motional degrees of freedom of the atom. That such SOC can be induced by laser fields can be easily understood as follows: The laser light induces transitions between atomic internal states, and in the meantime imparts photon’s linear momentum to the atom. Thus the internal and the external degrees of freedom are coupled via their interaction with the photon. SOC in cold atoms was first realized in a system of87Rb condensate by the NIST group in 2011 [6]. Since then, several groups have achieved SOC in both bosonic [7–10] and fermionic quantum gases [11–15]. SOC not only dramatically changes the single-particle dispersion relation, but is also the key ingredient underlying many interesting many-body phenomena and new materials such as topological insulators [16] and quantum spin Hall effects [17]. Due to the exquisite controllability of atomic systems, one can naturally expect that SOC in cold atoms will give rise to novel quantum states of matter and may lead to a deeper understanding of related phenomena in other systems. For this reason, spin-orbit coupled quantum gases have received tremendous attention over the past few years, and they no doubt represent one of the most active frontiers of cold atom research. In this chapter, we will review the physics of spin-orbit coupled Fermi gas, both theoretically and experimentally. Although we will mainly focus on the research from our own groups, results from others will also be mentioned.3 II. THEORY OF SPIN-ORBIT COUPLED FERMI GAS We consider a spin-1/2 Fermi gas with SOC subject to attractively interaction between unlike spins. One great advantage of the atomic system is its unprecedented controllability. The interatomic interaction can be precisely tuned using the Feshbach resonance technique [18], which has already led to the discovery of the BEC-BCS crossover from a Bose-Einstein condensate (BEC) to a Bardeen-Cooper-Schrieffer (BCS) superfluid [19]. Different forms of SOC, many of which do not exist in natural materials, can also be engineered. The interplay between interatomic interactions and different forms of SOC may give rise to a number of intriguing physical phenomena. Here let us make some general remarks concerning the distinct features that can be brought out by SOC in a Fermi gas: •SOC alters the single-particle dispersion which may lead to degenerate single-particle ground state, and may render the topology of the Fermi surface non-trivial [20]. •In the presence of attractive s-wave interaction, two fermions may form pairs. In general such pairs contain both singlet and triplet components [21–26] and have anisotropic (i.e., direction-dependent) effective mass [22–24]. In the many-body setting, a spin-orbit coupled superfluid Fermi gas contains both singlet and triplet pairing correlation [20, 22, 24, 27] and therefore may be regarded as an anisotropic superfluid [22]. •SOC may greatly enhance the pairing instability and hence dramatically increases the superfluid transition temperature [22, 23, 28]. •SOC, together with effective Zeeman fields, may generate exotic pairing [29–37] and/or topologically non-trivial superfluid state [38–55]. At the boundaries of topologically trivial and non-trivial regimes, exotic quasi-particle states (e.g., Majorana mode) may be created. In the remaining part of this section, we will discuss two particular types of SOC. The first is the equal-weight Rashba-Dresselhaus SOC [56] which is the only one that has been experimentally realized so far. The second is the Rashba SOC which is of particular interest as it occurs naturally in certain semiconductor materials. However, before we do that, in the next subsection we first summarize the theoretical framework and explain the basics of momentum- or spatially-resolved radio-frequency (rf) spectroscopy, which turns out to be a very useful experimental tool for characterizing spin-orbit coupled interacting Fermi gases. For those readers who are interested in the physical consequences of a detailed type of SOC, this technical part may be skipped in their first reading. A. Theoretical framework In current experimental setups of ultracold atomic Fermi gases, the interactions between atoms are often tuned to be as strong as possible, in order to have an experimentally accessible superfluid transition temperature. With such strong interactions, there is a significant portion of Cooper pairs formed by two fermionic atoms with unlike spin. Theoretically, therefore, it is very crucial to treat atoms and Cooper pairs on an equal footing. Without SOC, a minimum theoretical framework for this purpose is the many-body T-matrix theory or pair-fluctuation theory [57–62]. In this subsection, we introduce briefly the essential idea of the pair-fluctuation theory using the functional path-integral approach and generalize the theory to include SOC [24]. Under this theoretical framework, both two- and many-body physics can be discussed in a unified fashion [24]. We also discuss the mean-field Bogoliubov-de Gennes equation, which represents a powerful tool for the study of trapped, inhomogeneous Fermi superfluids at low temperatures [42, 45, 47, 48, 50, 51]. 1. Functional path-integral approach Consider, for example, a three-dimensional (3D) spin-1/2 Fermi gas with mass m. The second-quantized Hamilto- nian reads, H= drh y ^k+VSO +U0 y "(r) y #(r) #(r) "(r)i ; (1) where ^k^k2=(2m)=r2=(2m)with the chemical potential , (r) = [ "(r); #(r)]Tdescribes collectively thefermionicannihilationoperator (r)forspin-atom, andVSO(^k)representsthespin-orbitcouplingwhoseexplicit form we do not specify here. The momentum ^ki@(=x;y;z) should be regarded as the operators in real4 space. For notational simplicity, we take ~= 1throughout this paper. The last term in Eq. (1) represents the two-body contact s-wave interaction between unlike spins. The use of the contact interatomic interaction leads to an ultraviolet divergence at large momentum or high energy. To overcome such a divergence, we express the interaction strengthU0in terms of the s-wave scattering length as, 1 U0=m 4as1 VX km k2; (2) whereVis the volume of the system. The partition function of the system can be written as [61] Z= D[ (r;); (r;)] exp S (r;); (r;) ; (3) where the action S ;  = 0d" drX  (r;)@ (r;) +H ;
# : (4) is written as an integral over imaginary time . Here= 1=(kBT)is the inverse temperature and H ;
is obtained by replacing the field operators yand with the Grassmann variables  and , respectively. We can use the Hubbard-Stratonovich transformation to transform the quartic interaction term into a quadratic form as: eU0
drd " # # "= D
;
 exp( 0d dr" j
(r;)j2 U0+
# "+
 " #
#) ; (5) from which the pairing field
(r;)is defined. Let us now introduce the 4-dimensional Nambu spinor  (r;)[ "; #; "; #]Tand rewrite the action as, Z= D[;;
;
] exp( dr0 dr 0d0 0d" 1 2(r;)G1(r0;0) +j
j2 U0(rr0)(0)# X k^k) ;(6) where the 44single-particle Green function is given by, G1= @^kVSO(^k)i
^y i
^y@+^k+VT SO(^k) (rr0)(0); (7) with the Pauli matrices ^i(i= 0;x;y;z) describing the spin degrees of freedom. The Nambu spinor representation treats equally the particle and the hole excitations. As a result, a zero-point energy appears in the last term of the action. Integrating out the original fermionic fields, we may rewrite the partition function as Z= D[
;
] exp Se
;
 ; (8) where the effective action is given by Se
;
 = 0d dr" j
(r;)j2 U0# 1 2Trln G1 +X k^k: (9) where the trace is taken over all the spin, spatial, and temporal degrees of freedom. To proceed, we restrict ourselves to the Gaussian fluctuation and expand
(r;) =
0(r)+
(r;). The effective action is then decomposed accordingly as Se=S0+S, where the saddle-point action is S0= 0d drj
0(r)j2 U01 2Trln G1 0 +X k^k (10) and the pair-fluctuating action takes the form S= 0d dr" j
(r;)j2 U0+1 21 2 Tr(G0)2# (11)5 with  = 0i
^y i
^y 0 : (12) HereG1 0is the inverse mean-field Green function and has the same form as G1in Eq. (7) with
(r;)replaced by
0(r). We note that the static pairing field
0(r)can be either homogeneous or inhomogeneous. In the latter case, a typical form is
0(r) =
0eiq
r, referred to as the Fulde-Ferrell superfluid [63], in which the Cooper pairs condense into a state with nonzero center-of-mass momentum q. Let us now focus on a homogeneous system, where the momentum is a good quantum number so that we take k=^kandVSO(k) =VSO(^k). The fluctuating part of the effective action may be formally written in terms of the many-body particle-particle vertex function (q;in)[61], S=kBTX Q=(q;in) 1(Q) 
(Q)
(Q); (13) whereQ(q;in)andnis the bosonic Matsubara frequency. By integrating out the quadratic term in S, we obtain the contribution from the Gaussian pair fluctuations to the thermodynamic potential as [61]  =kBTX q;inln 1(q;in) : (14) Within the Gaussian pair fluctuation approximation, naïvely, the vertex function may be interpreted as the Green function of “Cooper pairs”. This idea is supported by Eq. (14), as the thermodynamic potential Bof a free bosonic Green functionGBis formally given by B=kBTP q;inln[G1 B(q;in)]. At this point, the advantage of using pair-fluctuation theory becomes evident. For the fermionic degree of freedom, we simply work out the single-particle Green function G0and the related mean-field thermodynamic potential 0=kBTS0. An example will be provided lateroninthestudyoftheFulde-Ferrellsuperfluidity. WhileforCooperpairs, wecalculatethevertexfunction andthe fluctuating thermodynamic potential  . In this way, we may obtain a satisfactory description of strongly-interacting Fermi systems [59, 60, 62]. In the normal state where the pairing field vanishes, i.e.,
0= 0, we may obtain the explicit expression of the vertex function. In this case, the inverse Green function G1 0has a diagonal form and can be easily inverted to give [24]: G0(K) = [i!mkVSO(k)]10 0 i!m+k+VT SO(k)1 G0(K) 0 0 ~G0(K) ; (15) whereK(k;i!m)and!mis the fermionic Matsubara frequency. Here we have introduced the 22particle Green functionG0(K)and hole Green function ~G0(K), which are related to each other by ~G0(K) =[G0(K)]T. It is straightforward to show that, 1(Q) =1 U0+kBT 2X K=(k;i!m)h G0(K) (i^y)~G0(KQ) (i^y)i : (16) The detailed expression of the vertex function depends on the type of SOC. In the study of Rashba SOC, we will give an example that shows how to calculate the vertex function. 2. Two-particle physics from the particle-particle vertex function The vertex function can describe the pairing instability of Cooper pairs both on the Fermi surface and in the vacuum. In the latter case, it describes exactly the two-particle state. The corresponding two-body inverse vertex function 1 2b(Q)can be obtained from the many-body inverse vertex function by discarding the Fermi distribution function and by setting chemical potential = 0[64]. One important question concerning the two-body state is whether there exist bound states. For a given momentum q, the bound state energy E(q)can be determined from the two-particle vertex function using the following relation ( in!!+i0+) [22, 24]: Re 1 2b[q;!=E(q)] = 0: (17)6 A true bound state must satisfy E(q)<2EminwhereEminis the single-particle ground state energy. It is straightforward but lengthy to calculate the two-particle vertex function for any type of SOC. Here, we quote only the energy equation obtained using Eq. (17) for the most general form of SOC [34], VSO ^k =X i=x;y;z i^ki+hi ^i; (18) whereiis the strength of SOC in the direction i= (x;y;z )andhidenotes the effective Zeeman field. The eigenenergy E(q)of a two-body eigenstate with momentum qsatisfies the equation: m 4as=1 VX k2 6640 B@Ek;q4E2 k;q(
k)24hP i=x;y;ziki(iqi+ 2hi)i2 Ek;qh E2 k;qP i=x;y;z(iqi+ 2hi)2i1 CA1 +1 2k3 775; (19) whereEk;qE(q)q 2+kq 2kandk=k2=(2m). We note that, in general, the lowest-energy two-particle state may occur at a finite momentum q. That is, the two-particle bound state could have a nonzero center-of-mass momentum. Later, we shall see that this unusual property has nontrivial consequences in the many-body setting. Another peculiar feature of the two-particle bound state is that the pairs may have an effective mass larger than 2m. For example, for the bound state with zero center-of-mass momentum q= 0, it would have a quadratic dispersion for small p, E(p) =E(0) +p2 x 2Mx+p2 y 2My+p2 z 2Mz: (20) The effective mass of the bound state Mi(i=x;y;z) can then be determined directly from this dispersion relation. Another approach to study the two-particle state with SOC , more familiar to most readers, is to use the following ansatz for the two-particle wave function [21, 23, 65, 66], j2Bi=1p CX kh "#(k)cy q 2+k"cy q 2k#+ #"(k)cy q 2+k#cy q 2k"+ ""(k)cy q 2+k"cy q 2k"+ ##(k)cy q 2+k#cy q 2k#i jvaci; (21) wherecy k"andcy k#are creation field operators of spin-up and spin-down atoms with momentum kandCis the normalization factor. We note that, in the presence of SOC, the wave function of the two-particle state has both spin singlet and triplet components. Then, using the Schrödinger equation Hj2B(q)i=E(q)j2B(q)i, we can straightforwardly derive the equations for coefficients 0appearing in the above two-body wave function and then the energy equation for E(q). For the general form of SOC, Eq. (18), it leads to exactly the same energy equation (19) [34]. Each of the two approaches mentioned above has its own advantages. The vertex function approach is useful to understand the relationship between the two-body physics and the many-body physics. For example, it can be used to obtain the two-particle bound state in the presence of a Fermi surface. The latter approach of using the two-particle Schrödinger equation naturally yields the two-particle wave function. Both approaches have been used extensively in the literature. 3. Many-body T-matrix theory The functional path-integral approach gives the simplest version of the many-body T-matrix theory, where the bare Green function has been used in the vertex function. Here, for completeness, we mention briefly another partially self-consistent T-matrix scheme for a normal spin-orbit coupled Fermi gas, by taking one bare and one fully dressed Green function in the vertex function [13, 28]. In this scheme, we have the Dyson equation, G(K) = G1 0(K)(K)1; (22) where the self-energy is given by (K) =kBTX Q=(q;in)t(Q)(i^y)~G0(KQ)(i^y) (23)7 and ~G0(K)[G0(K)]T. Heret(Q)U0=[1 +U0(Q)]is the (scalar) T-matrix with a two-particle propagator (Q) =kBT 2X K=(k;i!m)Trh G(K) (i^y)~G0(KQ) (i^y)i ; (24) where the trace is taken over the spin degree of freedom only. Note that a fully self-consistent T-matrix theory may also be obtained by replacing in Eqs. (23) and (24) the bare Green function ~G0(KQ)with the fully dressed Green function ~G(KQ). We note also that Eqs. (22)-(24) provide a natural generalization of the well-known many-body T-matrix theory [62], by including the effect of SOC, where the particle or hole Green function, G(K)or~G(K), now becomes a 22matrix. In general, the partially self-consistent T-matrix equations are difficult to solve [62]. At a qualitative level, we may adopt a pseudogap decomposition advanced by the Chicago group [67] and approximate the T-matrixt(Q) = tsc(Q)+tpg(Q)to be the sum of two parts. Here tsc(Q) =(
2 sc=T)(Q)is the contribution from the superfluid with
scbeing the superfluid order parameter, and tpg(Q)represents the contribution from un-condensed pairs which give rise to a pseudogap
2 pgkBTX Q6=0tpg(Q): (25) The full pairing order parameter is given by
2 0=
2 sc+
2 pg. Accordingly, we have the self-energy (K) = sc(K) + pg(K), where sc=
2 sc(iy)~G0(K)(iy) (26) and pg=
2 pg(iy)~G0(K)(iy): (27) We note that, at zero temperature the pseudogap approximation is simply the standard mean-field BCS theory, in which (K) =
2 0(iy)~G0(K)(iy). Above the superfluid transition, however, it captures the essential physics of fermionic pairing and therefore should be regarded as an improved theory beyond mean-field. To calculate the pseudogap
pg, we approximate t1 pg(Q'0) =Z[in q+pair]; where the residue Zand the effective dispersion of pairs q=q2=2Mare to be determined by expanding (Q) aboutQ= 0in the case that the Cooper pairs condense into a zero-momentum state. The form of tpg(Q)leads to
2 pg(T) =Z1X qfB( qpair); wherefB(x)1=(ex=kBT1)is the bosonic distribution function. We finally obtain two coupled equations, the gap equation 1=U0+(Q= 0) =Zpairand the number equation n=kBTP KTrG(K), from which the superfluid order parameter
scand the chemical potential can be determined. This pseudogap method has been used to study the thermodynamics and momentum-resolved rf spectroscopy of interacting Fermi gases with different types of SOC [13, 28]. 4. Bogoliubov-de Gennes equation for trapped Fermi systems All cold atom experiments are performed with some trapping potentials, VT(r). For such inhomogeneous systems, it is difficult to directly consider pair fluctuations. In most cases, we focus on the mean-field theory by using the saddle- point thermodynamic potential Eq. (10) and minimizing it to determine the order parameter
0(r). This amounts to diagonalizing the 44single-particle Green function G1 0(r;;r0;0)with the standard Bogoliubov transformation, = drX  u(r) (r) +(r) y (r) ; (28)8 whereis the field operator for Bogoliubov quasiparticle with energy Eand Nambu spinor wave function (r) [u"(r);u#(r);v"(r);v#(r)]T, which satisfies the following Bogoliubov-de Gennes (BdG) equation,  r2=(2m)+VT(r) +VSO(^k)i
0(r) ^y i
 0(r) ^yr2=(2m) +VT(r)VT SO(^k) (r) =E(r): (29) The BdG Hamiltonian in the above equation includes the pairing gap function
0(r)that should be determined self-consistently. For this purpose, we may take the inverse Bogoliubov transformation and obtain (r) =X  u(r)+ (r)y  : (30) The gap function
0(r) =U0h #(r) "(r)iis then given by,
0(r) =U0 2X  u"(r)v #(r)f(E) +u#(r)v "(r)f(E) ; (31) wheref(E)1=[eE=(kBT)+ 1]is the Fermi distribution function at temperature T. Accordingly, the total density takes the form, n(r) =1 2X h ju(r)j2f(E) +jv(r)j2f(E)i : (32) The chemical potential can be determined using the number equation, N=
drn(r). This BdG approach has been used to investigate topological superfluids in harmonically trapped spin-orbit coupled Fermi gases in 1D and 2D [42, 45, 47, 48, 50, 51]. It will be discussed in greater detail in later sections. It is important to note that, the use of Nambu spinor representation enlarges the Hilbert space of the system. As a result, there is an intrinsic particle-hole symmetry in the Bogoliubov solutions: For any “particle” solution with wave function (p) (r) = [u"(r);u#(r);v"(r);v#(r)]Tand energy E(p) 0, we can always find a partner “hole” solution with wave function (h) (r) = [v "(r);v #(r);u "(r);u #(r)]Tand energy E(h) =E(p) 0. These two solutions correspond exactly to the same physical state. To remove this redundancy, we have added an extra factor of 1/2 in the expressions for pairing gap function Eq. (31) and total density Eq. (32). As we shall see, this particle-hole symmetry is essential to the understanding of the appearance of exotic Majorana fermions - particles that are their own antiparticles - in topological superfluids. 5. Momentum- or spatially-resolved radio-frequency spectrum Radio-frequency (rf) spectroscopy, including both momentum-resolved and spatially-resolved rf-spectroscopy, is a powerful tool to characterize interacting many-body systems. It has been widely used to study fermionic pairing in a two-component atomic Fermi gas near Feshbach resonances in the BEC-BCS crossover [68–72]. Most recently, it has also been used to detect new quasiparticles known as repulsive polarons [73, 74], which occur when “impurity” fermionic particles interact repulsively with a fermionic environment. The underlying mechanism of rf-spectroscopy is rather simple. The rf field drives transitions between one of the hyperfine states (say, j#i) and an empty hyperfine state j3iwhich lies above it by an energy !3#. The Hamiltonian describing this rf-coupling may be written as, Vrf=V0 drh y 3(r) #(r) + y #(r) 3(r)i ; (33) whereV0is the strength of the rf drive. For a weak rf field, the number of transferred atoms may be calculated using linear response theory. At this point, it is important to note that a final state effect might be present, which is caused by the interaction between atoms in the final third state and those in the initial spin-up or spin-down state. This final state effect is significant for6Li atoms; while for40K atoms, it is not important [19]. For momentum-resolved rf spectroscopy [71], the momentum distribution of the transferred atoms can be obtained by absorption imaging after a time-of-flight. This gives rise to the information about the single-particle spectral function of spin-down atoms of the original Fermi system, A##(k;!). In the absence of the final-state effect, the rf transfer strength (k;!)at a given momentum is given by, (k;!) =A##(k;k!+!3#)f(k!+!3#): (34)9 Here, we have assumed that the atoms in the third state have the dispersion relation k=k2=(2m)in free space and have taken the coupling strength V0= 1. Experimentally, we can either measure the momentum-resolved rf spectroscopy along a particular direction, say, the x-direction, by integrating along the two perpendicular directions (kx;!)X ky;kz(k;!); (35) or after integrating along the remaining direction, obtain the fully integrated rf spectrum (!)P k(k;!). We note that, in the extremely weakly interacting BCS and BEC regimes, where the physics is dominated by single- particle or two-particle physics, respectively, we may use the Fermi golden rule to calculate the momentum-resolved rf spectroscopy. This will be discussed in greater detail in the relevant subsections. We note also that momentum- resolved rf spectroscopy is precisely an ultracold atomic analogue of the well-known angle-resolved photoemission spectroscopy (ARPES) widely used in solid-state experiments. Alternatively, we may use rf spectroscopy to probe the local information about the original Fermi system. This was first demonstrated in measuring the pairing gap by using phase-contrast imaging within the local density approx- imation for a trapped Fermi gas [69]. A more general idea is to use a specifically designed third state, which has a very flat dispersion relation [75]. This leads to a spatially-resolved rf spectroscopy, which measures precisely the local density of states of the Fermi system, (r;!) =1 2X h ju(r)j2(!E) +jv(r)j2(!+E)i : (36) It could be regarded as a cold-atom scanning tunneling microscopy (STM). As we shall see, the spatially-resolved rf spectroscopy will provide a useful although indirect measurement of the long-sought Majorana fermion in atomic topological superfluids. B. 1D equal-weight Rashba-Dresselhaus spin-orbit coupling δ /c173/c175 FIG. 1: Left panel: schematic of the Raman transition that produced the equal-weight Rashba-Dresselhaus SOC. The two atomic states are labeled as j"iandj#i.is the two-photon Raman detuning. Right panel: schematic of the experimental setup where a pair of Raman beams counter-propagate along the x-axis. Right figure taken from Ref. [11]. Let us now discuss the two specific types of SOC. One simple scheme to create SOC in cold atoms is through a Raman transition that couples two hyperfine ground states of the atom, as schematically shown in Fig. 1. The Raman process is described by the following single-particle Hamiltonian in the first-quantization representation H0=^ p2 2m+1 2  ei2krx ei2krx ; (37) where ^ pis the momentum operator of the atom, 2kr^xis the photon recoil momentum taken to be along the x-axis, and arethetwo-photondetuningandthecouplingstrengthoftheRamanbeams, respectively. TheHamiltonianacts on the Hilbert space expanded by the spin-up and spin-down basis, j"iandj#i. By applying a unitary transformation with U= eikrx0 0eikrx ; (38)10 the HamiltonianH0can be recast into the following form: HSO=UyH0U= ^kx+kr^z2 2m+ ^k2 y+^k2 z 2m+ 2^x+ 2^z: (39) Here, ^k= (^kx;^ky;^kz)denotes the quasi-momentum operator of the atom: When ^kis applied to the transformed wave function, it gives the atomic quasi-momentum kthat is related to the real momentum pas^p= (^kkr^x)with for spin-up and down, respectively. From this expression, it is sometimes convenient to regard both andas the strengths of effective Zeeman fields. We note that after a pseudo-spin rotation ( z!x,x!z), Hamiltonian (39) can be cast into the general form of SOC in Eq. (18) with = (k2 r=m;0;0)andh= (=2;0; =2). It is clear that the SOC is along a specific direction. Actually, it is an equal-weight combination of the well-known Rashba and Dresselhaus SOCs in solid-state physics [56]. For this reason, hereafter we would refer to it as 1D equal-weight Rashba-Dresselhaus SOC. We may also refer to the detuning as the in-plane Zeeman field since it is aligned along the same direction as the SOC. Accordingly, we call the coupling strength as the out-of-plane Zeeman field. As we shall see, depending on and , the spin-orbit coupled Fermi system can display distinct quantum superfluid phases at low temperatures. 1. Single-particle spectrum /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s50/s48/s50/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s50/s48/s50/s52 /s40/s98/s41/s32 /s32/s61/s32 /s69 /s114 /s107 /s120/s47/s107 /s114/s107 /s120/s47/s107 /s114/s40/s107 /s121 /s61/s48/s44 /s107 /s122/s61/s48/s41/s47 /s69 /s114 /s40/s97/s41/s32 /s32/s61/s32/s48 FIG. 2: Single particle spectrum of a Fermi gas with 1D equal-weight Rashba-Dresselhaus SOC, with (a) or without detuning (b). In each panel, we increase the coupling strength of the Raman beams from Erto5Er, with a step of Er, as indicated by the arrows. The single-particle spectrum can be easily obtained by diagonalizing the Hamiltonian (39), which is given by Ek=Er+k2 2ms 22 + kx+ 22 ; (40) where we have defined a recoil energy Erk2 r=(2m)and an SOC strength kr=m. The spectrum contains two branches as shown in Fig. 2. For small , the lower branch exhibits a double-well structure. The double wells are symmetric (asymmetric) for = 0(6= 0). For large , the two wells in the lower branch merge into a single one. It is important to emphasize that in each branch atoms stay at a mixed spin state with both spin-up and down components. The single-particle spectrum can be easily measured by using momentum-resolved rf spectroscopy, as already shown at Shanxi University and MIT [11, 12]. In this case, the number of transferred atoms can be calculated by using the Fermi’s golden rule [76]: (kx;!) =X i;fjhfjVrfjiij2f(Ei)[!!3#(EfEi)]; (41) where the summation is over all possible initial single-particle states i(with energy Eiand a given wavevector kx) and final states f(with energy Ef), and the Dirac -function ensures energy conservation during the rf transition. In practice, the -function is replaced by a function with finite width (e.g., (x)!( =)(x2+ 2)1where accounts for11 /s45/s50 /s48 /s50/s45/s50/s48/s50/s47/s69 /s114 /s32/s32/s32/s61/s32/s53 /s69 /s114 /s107 /s120/s47/s107 /s114 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s52/s45/s50/s48/s50/s126 /s47/s69 /s114 /s32/s32/s32/s61/s32/s50 /s69 /s114 /s107 /s110/s120/s47/s107 /s114/s48 /s48/s46/s48/s53 /s48/s46/s49 FIG.3: Theoreticalsimulationonmomentum-resolvedrfspectroscopyofaFermigaswith1Dequal-weightRashba-Dresselhaus SOC.Leftpanel: simulatedexperimentalspectroscopy (kx;!). Rightpanel: thespectroscopy (knxkx+kr;~!=!+k2 x=2m). Here, theintensityofthecontourplotshowsthenumberoftransferredatoms, increasinglylinearlyfrom0(blue)toitsmaximum value (red). We have set !3#= 0and used a Lorentzian distribution to replace the Delta function. Figure taken from Ref. [76] with modification. the energy resolution of the measurement). The single-particle wave function iis known from the diagonalization of the Hamiltonian (39) and the transfer element hfjVrfjiiis then easy to determine. The left panel of Fig. 3 shows the predicted momentum-resolved spectroscopy (kx;!)at= 0and = 2Er. The chemical potential is tuned (= 5Er) in such a way that there are significant populations in both energy branches. The simulated spectrum is not straightforward to understand, because of the final free-particle dispersion relation in the energy conservation in Eq. (41) and also the recoil momentum shift ( kr) arising from the unitary transformation Eq. (38). Therefore, it is useful to define ~ (knx;~!) kx+kr;!+k2 x 2M ; (42) for which, the energy conservation takes the form [~!+Ei(kx)]. As shown on the right panel of Fig. 3, the single- particle spectrum is now clearly visible. /s45/s50 /s48 /s50/s45/s50/s48/s50/s47/s69 /s114 /s32/s32 /s107 /s120/s47/s107 /s114 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s52/s45/s50/s48/s50/s47/s69 /s114 /s32/s32/s114/s102 /s32/s61/s32 /s69 /s114 /s107 /s110/s120/s47/s107 /s114/s126/s48 /s48/s46/s48/s53 /s48/s46/s49 FIG. 4: Theoretical simulation on momentum-resolved rf spectroscopy of a Fermi gas with 1D equal-weight Rashba-Dresselhaus SOC and an additional spin-orbit lattice. The left and right panels show (kx;!)and(knxkx+kr;~!=!+k2 x=2m), respectively. The white lines on the right panel are the calculated energy band structure. The spin-orbit lattice depth is rf=Erand the other parameters are the same as in Fig. 3. Figure taken from Ref. [76] with modification. Experimentally, the single-particle properties of the Fermi gas can also be easily tuned, for example, by using an additional rf field to couple spin-up and down states [12]. After the gauge transformation, it introduces a term ( =2)[cos(2krx)^x+ sin(2krx)^y]in the spin-orbit Hamiltonian Eq. (39), which behaves like a spin-orbit lattice and leads to the formation of energy bands. In Fig. 4, we show the simulation of momentum-resolved rf spectroscopy12 under such an rf spin-orbit lattice. The energy band structure is apparent. We refer to Ref. [76] for more details on the theoretical simulations, in particular the simulations in a harmonic trap. The relevant measurements will be discussed in greater detail later in the section on experiments. 2. Two-body physics We now turn to consider the interatomic interaction. The interplay between interatomic interaction and SOC can lead to a number of intriguing phenomena, even at the two-particle level. Let us first solve numerically the energy E(q)of the two-particle states by using the general eigenenergy equation Eq. (19). A true bound state must satisfy E(q)<2Emin, whereEminis the single-particle ground state energy. /s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48 /s49/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s49/s46/s50/s53/s49/s46/s53/s48/s32 /s32/s32 /s32 /s49/s47/s40 /s107 /s114/s97 /s115/s41/s45/s69 /s40/s113 /s61/s48/s41/s47/s40/s50 /s69 /s114/s41/s40/s97/s41 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s46/s48/s48/s49/s46/s48/s53/s49/s46/s49/s48/s49/s46/s49/s53/s49/s46/s50/s48 /s32/s32 /s32/s49/s47/s40 /s107 /s114/s97 /s115/s41/s32/s32 /s47/s69 /s114/s61/s48/s46/s56 /s32 /s47/s69 /s114/s61/s50/s46/s48 /s32 /s47/s69 /s114/s61/s51/s46/s50/s40/s98/s41 FIG. 5: Energy E(q= 0)(a) and effective mass ratio =Mx=(2m)(b) of the two-particle ground bound state in the presence of 1D equal-weight Rashba-Dresselhaus SOC, at zero detuning = 0and at three coupling strengths of Raman beams: = 0:8Er(solid line), 2Er(dashed line), and 3:2Er(dot-dashed line). The horizontal dotted lines in (a) correspond to the threshold energies 2Eminwhere the bound states disappear. Figure taken from Ref. [66] with modification. At zero detuning = 0, the two-particle ground state has zero center-of-mass momentum q= 0[66]. In Fig. 5(a), we show its energy as a function of the dimensionless interaction parameter 1=(kras). In the presence of 1D equal- weight Rashba-Dresselhaus SOC, a two-particle bound state occurs on the BEC side with a positive s-wave scattering lengthas>0. The effective out-of-plane Zeeman field acts as a pair-breaker and pushes the threshold scattering length to the BEC limit. In other words, the position of the Feshbach resonance, originally located at as=1, now shifts to the BEC side with at lower magnetic field strengths [14]. By calculating the dispersion relation E(q)around q= 0, we are able to determine the effective mass, as shown in Fig. 5(b). It is interesting that the effective mass along the direction of SOC is greatly altered. It becomes much larger than 2mtowards the threshold scattering length. In the deep BEC limit, 1=(kras)!1, where two atoms form a tightly bound molecule, the mass is less affected by the SOC or the effective Zeeman field, as we may anticipate. 0 0.1 0.2 0.3 0.4 0.500.20.40.60.81 1234567x□10 0 0.1 0.2 0.3 0.4 0.500.20.40.60.81 00.20.40.60.81 (a) (b) / 2rE/c1002rE/c87 / 2rE/c100 FIG. 6: Binding energy Eb= 2EminEq0and the magnitude of the lowest-energy bound state momentum q0as functions of and . The coloring in (a) represents Eb=Er, and that in (b) represents q0=kr. In the upper right corner of both (a) and (b), there exist no bound states. The scattering length is given by 1=(kras) = 1. Figure taken from Ref. [26] with modifications.13 At nonzero detuning 6= 0, the result shows that the two-particle bound state will have its lowest energy at a finite center-of-mass momentum q0= (q0;0;0)[26, 30]. Fig. 6 shows the binding energy and the magnitude of q0of the lowest-energy bound state. That the two-particle ground states possessing a finite momentum implies that the Cooper pairs, which is a many-body counterpart of two-particle bound state, may acquire finite center-of-mass momentum and therefore condense into an inhomogeneous superfluid state. This possibility will be addressed in greater detail later. We note that with the typical parameters, i.e., ErandEr,q0is small and less than 1% of the recoil momentum kr, as shown in Fig. 6(b). However, its magnitude can be significantly enhanced by many-body effect. For Cooper pairs in the ground state, q0can be tuned to be comparable with kror the Fermi wavevector kF[33]. /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s48/s49/s50/s51/s52/s53/s54/s32 /s32 /s107 /s120/s47/s107 /s114/s47/s69 /s66 /s40/s97/s41/s32 /s69 /s114/s47/s69 /s66/s32/s61/s32/s48/s46/s53/s48 /s48/s46/s48/s53 /s48/s46/s49 /s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51 /s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51 /s32/s32 /s32/s32 /s69 /s114/s47/s69 /s66/s32/s61/s32/s48/s46/s53 /s48/s46/s50/s53/s69 /s114/s47/s69 /s66/s61/s49/s70 /s40 /s41 /s47/s69 /s66/s40/s98/s41/s32/s48/s46/s53 FIG. 7: (a) Momentum-resolved rf spectroscopy (a) and integrated rf spectroscopy (b) of the two-particle bound state at = 0 and = 2Er. The energy of rf photon !is measured in units of a binding energy EB1=(ma2 s)and we have set !3#= 0. In the right panel, the dashed line in the main figure plots the rf line-shape in the absence of SOC: F(!) = (2=)p!EB=!2. The inset highlights the different contribution from the two final states, as described in the text. Figure taken from Ref. [65] with modification. Ideally, momentum-resolved rf-spectroscopy can be used to probe the two-particle bound state discussed above. We can perform a numerical simulation of the spectroscopy by using again the Fermi’s golden rule. Let us assume that a bound molecule is initially at rest in the state j2Biwith energy Ei. An rf photon with energy !will break the molecule and transfer the spin-down atom to the third state j3i. In the case that there is no final-state effect, the final statejficonsists of a free atom in j3iand a remaining atom in the spin-orbit system. According to the Fermi’s golden rule, the rf strength (!)of breaking molecules and transferring atoms is proportional to the Franck-Condon factor [77], F(!) =jhfjVrfj2Bij2[!!3#(EfEi)]: (43) The integrated Franck-Condon factor satisfies the sum rule,
+1 1F(!)d!= 1. A closed expression of F(!)is derived in Refs. [65] and [66], by carefully analyzing the initial two-particle bound state j2Biand the final state jfi. Furthermore, by resolving the momentum of transferred atoms, we are able to obtain the momentum-resolved Franck-Condon factor F(kx;!). Figs. 7(a) and 7(b) illustrate respectively the momentum-resolved and the integrated rf spectrum of the two-particle ground state at zero detuning = 0. One can easily resolve two different responses in the spectrum due to two different final states, as the remaining spin-up atom in the original spin-orbit system can occupy either the upper or the lower energy branch. Indeed, in the integrated rf spectrum, we can separate clearly the different contributions from the two final states, as highlighted in the inset. This gives rise to two peaks in the integrated spectrum. We note that the lower peak exhibits a red shift as the SOC strength increases, due to the decrease of the binding energy. It is also straightforward to calculate the rf spectrum of the two-particle bound state at nonzero detuning 6= 0(not shown in the figure). However, the spectrum remains essentially unchanged, due to the fact that the center-of-mass momentum q0is quite small with typical experimental parameters. 3. Momentum-resolved radio-frequency spectrum of the superfluid phase Consider now the many-body state. As we mentioned earlier, since the two-particle wave function contains both spin singlet and triplet components, we anticipate that the superfluid phase at low temperatures would involve both s-wave pairing and high-partial-wave pairing. Therefore, in general it is an anisotropic superfluid. This is to be discussed later in detail for 2D Rashba SOC. Here, we are interested in the phase diagram and the experimental probe14 of a 3D Fermi gas with 1D equal-weight Rashba-Dresselhaus SOC. First, let us concentrate on the case with zero detuning= 0, by using the many-body T-matrix theory within the pseudogap approximation [13]. /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54 /s32/s32/s115/s47 /s70 /s84 /s47/s84 /s70 /s84/s42/s32/s61/s32/s50 /s69 /s114 /s32/s32/s84 /s47/s84 /s70 /s49/s47/s40 /s107 /s70/s97 /s115/s41/s115/s117/s112/s101/s114/s102/s108/s117/s105/s100/s84 /s99 FIG. 8: (a) Phase diagram of a spin-orbit coupled Fermi gas at the BEC-BCS crossover at = 2ErandkF=kr. The main figure and inset show the superfluid transition temperature and the superfluid order parameter at resonance, respectively, predicted by using our T-matrix theory (solid line) and the BCS mean-field theory (dashed line). Figure taken from Ref. [13] with modification. Focusing on the vicinity of the Feshbach resonance where as!1, in Fig. 8 we show the superfluid transition temperature Tcand the pair breaking (pseudogap) temperature Tof the spin-orbit coupled Fermi gas at = 2Er andkF=kr. The pseudogap temperature is calculated using the standard BCS mean-field theory without taking into account the preformed pairs (i.e.,
pg= 0) [57, 67]. We find that the region of superfluid phase is strongly suppressed by SOC. In particular, at resonance the superfluid transition temperature is about Tc'0:08TF, which is significantly smaller than the experimentally determined Tc'0:167(13)TFfor a unitary Fermi gas [78]. Thus, it seems to be a challenge to observe a novel spin-orbit coupled fermionic superfluid in the present experimental scheme. /s45/s50 /s45/s49 /s48 /s49 /s50/s40/s98/s41/s32/s49/s47 /s107 /s70 /s97 /s115/s32/s61/s32 /s32/s32/s32 /s107 /s120/s47/s107 /s114/s45/s50 /s45/s49 /s48 /s49 /s50/s40/s99/s41/s32/s49/s47 /s107 /s70 /s97 /s115/s32/s61/s32 /s32/s32/s32 /s107 /s120/s47/s107 /s114/s48/s46/s48 /s48/s46/s49/s54 /s48/s46/s51/s50 /s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s48/s50/s52/s54/s56 /s32/s32 /s107 /s120/s47/s107 /s114/s47/s69 /s114/s40/s97/s41/s32/s49/s47 /s107 /s70 /s97 /s115/s32/s61/s32 FIG.9: Zero-temperature momentum-resolvedrf-spectroscopyof aspin-orbitcoupled Fermigas acrossthe Feshbach resonance, at the parameters = 2ErandkF=kr. Figure taken from Ref. [13] with modification. In Figs. 9(a)-9(c), we show the zero-temperature momentum-resolved rf spectrum across the resonance. On the BCS side ( 1=kFas=0:5), the spectrum is dominated by the response from atoms and shows a characteristic high- frequency tail at kx<0[11, 12, 76], see for example, the left panel of Fig. 3. We note that the density of the Fermi cloud, chosen here following the real experimental parameters [11], is low and therefore only the lower energy branch is occupied at low temperatures. Towards the BEC limit ( 1=kFas= +0:5), the spectrum may be understood from the picture of well-defined bound pairs and shows a clear two-fold anisotropic distribution, as we already mentioned in Fig. 7(a) [65]. The spectrum at the resonance is complicated and might be attributed to many-body fermionic pairs. It is interesting that the response from many-body pairs has a similar tail at high frequency as that from atoms. The change of the rf spectrum across the resonance is continuous, in accordance with a smooth BEC-BCS crossover.15 4. Fulde-Ferrell superfluidity The nature of superfluidity can be greatly changed by a nonzero detuning 6= 0. As we discussed earlier in the two-body part, in this case, the Cooper pairs may carry a nonzero center-of-mass momentum and therefore condense into an inhomogeneous superfluid state, characterized by the order parameter
0(r) =
0eiq
r. This exotic superfluid has been proposed by Fulde and Ferrell [63], soon after the discovery of the seminal BCS theory. Its existence has attracted tremendous theoretical and experimental efforts over the past five decades [79]. Remarkably, to date there is still no conclusive experimental evidence for FF superfluidity. Here, we show that the superfluid phase of a 3D Fermi gas with 1D equal-weight Rashba-Dresselhaus SOC and finite in-plane effective Zeeman field is precisely the long- sought FF superfluid [33]. The same issue has also been addressed very recently by Vijay Shenoy [30]. We note that the FF superfluid can appear in other settings with different types of SOC and dimensionality [29, 31, 32, 34–36, 80]. Theoretically, to determine the FF superfluid state, we solve the BdG equation (29) with VT(r) = 0by using the following ansatz for quasiparticle wave functions k(x) =eik
x p Vh uk"e+iqx=2;uk#e+iqx=2;vk"eiqx=2;vk#eiqx=2iT : (44) The center-of-mass momentum qis assumed to be along the x-direction, inspired from the two-body solution [26]. The mean-field thermodynamic potential 0at temperature Tin Eq. (10) is then given by 0 V=1 2V2 4X k k+q=2+kq=2
X kEk3 5kBT VX kln 1 +eEk=kBT
2 0 U0; (45) whereEk(= 1;2;3;4) is the quasiparticle energy. Here, the summation over the quasiparticle energy must be restricted to Ek0because of an inherent particle-hole symmetry in the Nambu spinor representation. For a given set of parameters (i.e, the temperature T, interaction strength 1=kFas, etc.), different mean-field phases can be determined using the self-consistent stationary conditions: @ =@
= 0,@ =@q= 0, as well as the conservation of total atom number, N=@ =@. At finite temperatures, the ground state has the lowest free energy F= +N. In the following, we consider the resonance case with a divergent scattering length 1=kFas= 0and setT= 0:05TF, whereTFis the Fermi temperature. According to the typical number of atoms in experiments [11, 12], we take the Fermi wavevector kF=kr. /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s56/s52/s45/s48/s46/s56/s50/s45/s48/s46/s56/s48/s45/s48/s46/s55/s56 /s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54 /s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s45/s48/s46/s48/s48/s48/s50/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s50 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54 /s32/s32 /s32/s32 /s47/s69 /s70 /s113 /s47/s107 /s70/s32/s70 /s70 /s32/s66/s67/s83 /s32/s78/s111/s114/s109/s97/s108/s70 /s40 /s41/s47/s40 /s78/s69 /s70/s41 /s47/s69 /s70/s40/s98/s41 /s32/s43 /s43/s78/s66/s67/s83 /s47 /s69 /s70/s113/s47/s107/s70/s40/s97/s41 /s109/s102 /s70/s70 /s43 FIG. 10: (a) Landscape of the thermodynamic potential, mf= [ 0(
;q) 0(0;0)]=(NEF), at = 2EFand= 0:68EF. The chemical potential is fixed to =0:471EF. The competing ground states include (i) a normal Fermi gas with
0= 0; (ii) a fully paired BCS superfluid with
06= 0andq= 0; and (iii) a finite momentum paired FF superfluid with
06= 0and q6= 0. (b) The free energy of different competing states as a function of the detuning at = 2EF. The inset shows the detuning dependence of the order parameter and momentum of the FF superfluid state. Figure taken from Ref. [33] with modification. Ingeneral, foranysetofparameterstherearethreecompetinggroundstatesthatarestableagainstphaseseparation (i.e.,@2 0=@
2 00), as shown in Fig. 10(a): normal gas (
0= 0), BCS superfluid (
06= 0andq= 0), and FF superfluid (
06= 0andq6= 0). Remarkably, in the presence of spin-orbit coupling the FF superfluid is always more favorable in energy than the standard BCS pairing state at finite detuning (Fig. 10(b)). It is easy to check that the superfluid density of the BCS pairing state in the SOC direction becomes negative (i.e., @ 0=@q < 0), signaling the instability towards an FF superfluid. Therefore, experimentally the Fermi gas would always condense into an16 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50 /s32/s49/s69/s45/s48/s52 /s48/s46/s48/s49 /s49/s49/s48/s52 /s49/s48/s50 /s49/s70/s117/s108/s100/s101/s45/s70/s101/s114/s114/s101/s108/s108/s32/s115/s117/s112/s101/s114/s102/s108/s117/s105/s100 /s40/s103/s97/s112/s108/s101/s115/s115/s41/s47/s69 /s70 /s47/s69 /s70/s78/s111/s114/s109/s97/s108/s32/s70/s101/s114/s109/s105/s32/s103/s97/s115 /s40/s103/s97/s112/s112/s101/s100/s41 FIG. 11: Phase diagram as a function of and , atT= 0:05TF. By increasing , the Fermi cloud changes from a FF superfluid to a normal gas, via first-order (dashed line) and second-order (solid line) transitions at low and high , respectively. The FF superfluid can be either gapped or gapless, as separated by the dot-dashed line. The coloring represents the magnitude of the centre-of-mass momentum of Cooper pairs, q=kF. The BCS superfluid occurs at = 0or= 0only. Figure taken from Ref. [33] with modification. FF superfluid at finite two-photon detuning. In Fig. 11, we report a low-temperature phase diagram that could be directly observed in current experiments. The FF superfluid occupies the major part of the phase diagram. /s45/s49 /s48 /s49/s43/s43 /s48/s46/s49 /s40/s99/s41/s32 /s32/s61/s32/s48/s46/s56 /s69 /s70/s32 /s32 /s107 /s120/s47/s107 /s70/s49 /s46 /s48 /s48 /s48 /s69 /s45/s48 /s52 /s48 /s46 /s49 /s48 /s48 /s48 /s49/s48/s45 /s52 /s45/s49 /s48 /s49/s48/s49/s50/s51/s52 /s43/s32 /s32 /s107 /s120/s47/s107 /s70/s47/s69 /s70 /s40/s97/s41/s32 /s32/s61/s32/s48/s43 /s45/s49 /s48 /s49/s43/s43 /s40/s98/s41/s32 /s32/s61/s32/s48/s46/s52 /s69 /s70 /s32 /s107 /s120/s47/s107 /s70 FIG. 12: Logarithmic contour plot of momentum-resolved rf spectroscopy: number of transferred atoms (kx;!)at = 2EF and at three detunings: (a) = 0andq= 0, (b)= 0:4EFandq'0:1kF, and (c)= 0:8Eandq'0:6kF. Figure taken from Ref. [33] with modification. The experimental probe of an FF superfluid is a long-standing challenge. Here, unique to cold atoms, momentum- resolved rf spectroscopy may provide a smoking-gun signal of the FF superfluidity. The basic idea is that, since Cooper pairscarryafinitecenter-of-massmomentum q, thetransferredatomsintherftransitionacquireanoverallmomentum q=2. As a result, there would be a q=2shift in the measured spectrum. In Fig. 12, we show the momentum-resolved rf spectrum (kx;!)on a logarithmic scale. As we discussed earlier in the two-body part, there are two contributions to the spectrum, corresponding to two different final states [65]. These two contributions are well separated in the frequency domain, with peak positions indicated by the symbols “ +” and “”, respectively. Interestingly, at finite detuning with a sizable FF momentum q, the peak positions of the two contributions are shifted roughly in opposite directions by an amount q=2. This provides clear evidence for observing the FF superfluid.17 5. 1D topological superfluidity Arguably, the most remarkable aspect of SOC is that it provides a feasible routine to realize topological superfluids [38], which have attracted tremendous interest over the past few years [81]. In addition to providing a new quantum phaseofmatter, topologicalsuperfluidscanhostexoticquasiparticlesattheirboundaries, knownasMajoranafermions - particles that are their own antiparticles [82, 83]. Due to their non-Abelian exchange statistics, Majorana fermions are believed to be the essential quantum bits for topological quantum computation [84]. Therefore, the pursuit for topological superfluids and Majorana fermions represents one of the most important challenges in fundamental science. Anumberofsettingshavebeenproposedfortherealizationoftopologicalsuperfluids, includingthefractionalquantum Hall states at filling = 5=2[85], vortex states of px+ipysuperconductors [86, 87], and surfaces of three-dimensional (3D) topological insulators in proximity to an s-wave superconductor [88], and one-dimensional (1D) nanowires with strong spin-orbit coupling coated also on an s-wave superconductor [89]. In the latter setting, indirect evidences of topological superfluid and Majorana fermions have been reported [90]. Here, we review briefly the possible realizations of topological superfluids, in the context of a 1D spin-orbit coupled atomic Fermi gas [45, 47, 51, 55], which can be prepared straightforwardly by loading a 3D spin-orbit Fermi gas into deep 2D optical lattices. Later, we will discuss 2D topological superfluids with Rashba SOC. /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s113 /s47/s107 /s70/s32 /s32/s32 /s47/s69 /s70/s47/s69 /s70/s32/s61/s32/s48/s46/s54 /s69 /s70 /s32/s32/s69 /s109 /s105/s110/s47/s69 /s70 /s47/s69 /s70/s66/s47/s40/s50 /s41 FIG. 13: Theoretical examination of the topological phase transition at the detuning = 0:6EFandT= 0. The transition occurs at '2:46EF, where the energy gap of the system (solid line) close and then open. The Berry phase Bisand 0 at the topologically trivial and non-trivial regimes (circles). The insets shows the order parameter and momentum of the FF superfluid, as a function of the Rabi frequency. Figure taken from Ref. [55] with modification. Consider first a homogeneous 1D Fermi gas with a nonzero detuning 6= 0[55]. In this case, we actually anticipate a topological inhomogeneous superfluid, where the order parameter also varies in real space. Using the same theoretical technique as in the previous subsection, we solve the BdG equation (29) in 1D and then minimize the mean-field thermodynamic potential Eq. (45) to determine the pairing gap
0and the FF momentum q. In Fig. 13, we show the energy gap as a function of at= 0:6EFandT= 0. For this result, we use a Fermi wavevector kF= 0:8krand take a dimensionless interaction parameter mg1D=(n) = 3, whereg1Dis the strength of the 1D contact interaction and n= 2kF=is the 1D linear density. Topological phase transition is associated with a change of the topology of the underlying Fermi surface and therefore is accompanied with closing of the excitation gap at the transition point. In the main figure this feature is clearly evident. To better characterize the change of topology, we may calculate the Berry phase defined by [47] B=i+1 1dk W +(k)@kW+(k) +W (k)@kW(k) : (46) HereW(k)[uk"eiqz=2;uk#eiqz=2;vk"eiqz=2;vk#eiqz=2]Tdenotes the wave function of the upper (= +)and lower (=)branch, respectively. In Fig. 13, the Berry phase is shown by circles. It jumps from to0, right across the topological phase transition. It is somewhat counter-intuitive that the B= 0sector corresponds to the topologically non-trivial superfluid state. It is important to emphasize the inhomogeneous nature of the superfluid.18 Indeed, as shown in the inset, the FF momentum qincreases rapidly across the topological superfluid transition and reaches about 0:3kFat = 4EF: /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s84/s111/s112/s111/s45/s66/s67/s83/s66/s67/s83/s84/s111/s112/s111/s45/s70/s70 /s32/s32/s47/s69 /s70 /s47/s69 /s70/s70/s70 FIG.14: Zero-temperaturephasediagram. Atopologicallynon-trivialFFsuperfluidappearswhentheRamancouplingstrength is above a threshold at finite detunings . Depending on the detuning, the transition could be either continuous (solid line) or of first order (dashed line). The FF superfluid reduces to a BCS superfluid when = 0or= 0. Figure taken from Ref. [55] with modification. InFig. 14, wepresentthezero-temperaturephasediagramforthetopologicalphasetransition. Thecriticalcoupling strength cdecreases with the increase of the detuning . At zero detuning, ccan be determined analytically, since the expression for the BdG eigenenergy for single-particle excitations (after dropping a constant energy shift Er) is known [24, 47], Ek= 2 k+2k2+ 2=4 +
2 0q 42 k2k2+ 2(2 k+
2 0)1=2 ; (47) wherek=k2=(2m)and=kr=m. It is easy to see that the excitation gap closes at k= 0for the lower branch (i.e.,=), leading to the well-known result [89] c 2=p 2+
2: (48) This criterion for topological superfluids is equivalent to the condition that there are only two Fermi points on the Fermi surface [39], under which the Fermi system behaves essentially like a 1D weak-coupling p-wave superfluid. Let us now turn to the experimentally realistic situation with a 1D harmonic trap VT(x) =m!2x2=2and focus on the case with = 0[45, 47, 51]. The BdG equation (29) can be solved self-consistently by expanding the Nambu spinor wave function (x)onto the eigenfucntion basis of the harmonic oscillator. In this trapped environment, Majorana fermions with zero energy are anticipated to emerge at the boundary, if the Fermi gas stays in a topological superfluid state. The appearance of Majorana fermions can be easily understood from the particle-hole symmetry obeyed by the BdG equation, which states that every physical state can be described either by a particle state with a positive energy Eor a hole state with a negative energy E. The Bogoliubov quasiparticle operators associated with these two states therefore satisfy E= y E. At the boundary, Eq. (48) could be fulfilled at some points and give locally the states with E= 0. These states are Majorana fermions, as the associated operators satisfy 0= y 0- precisely the defining feature of a Majorana fermion [82, 83]. In Fig. 15(a), we present the zero-temperature phase diagram of a trapped 1D Fermi gas at kF= 2krand =[51]. The transition from BCS superfluid to topological superfluid is now characterized by the appearance of Majorana fermions, whose energy is precisely zero and therefore the minimum of the quaisparticle spectrum touches zero, minfjEjg= 0. In the topological superfluid phase, as shown in Fig. 15(b) with = 2:4EF, the Majorana fermions may be clearly identified by using spatially-resolved rf spectroscopy. We note that for a trapped Fermi gas with weak interatomic interaction and/or high density, the upper branch of single-particle spectrum may be populated at the trap center, leading to four Fermi points on the Fermi surface. This violates Eq. (48). As a result, we may find a phase-separation phase in which the topological superfluid occurs only at the two wings of the Fermi cloud. This situation has been discussed in Ref. [45].19 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48 /s49/s46/s49 /s49/s46/s50 /s49/s46/s51 /s49/s46/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s32 /s40/s60 /s120/s50 /s62/s41/s49/s47/s50 /s47/s120 /s70/s69 /s47/s69 /s70/s84/s111/s112/s111/s108/s111/s103/s105/s99/s97/s108/s32/s115/s117/s112/s101/s114/s102/s108/s117/s105/s100/s32/s109/s105/s110/s123/s124 /s69 /s124/s125/s47 /s69 /s70 /s47/s40/s50 /s69 /s70/s41/s66/s67/s83/s32/s115/s117/s112/s101/s114/s102/s108/s117/s105/s100 /s40/s97/s41 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s40/s98 /s41 /s32/s32/s32 /s32/s47/s69 /s70 /s120 /s47/s120 /s70/s48 /s49 /s50 FIG. 15: (a) Zero-temperature phase diagram of a trapped 1D spin-orbit coupled Fermi gas, determined from the behavior of the lowest energy in quasiparticle spectrum. The inset shows the energy spectrum at = 2:4EFas a function of the position of quasiparticles. A zero-energy quasiparticle (i.e., Majorana fermion) at the trap edge has been highlighted by a big dark circle. Here, the position of a quasiparticle is approximately characterized by: x2 =
dxx2P 2 (x) +2 (x)].xFis the Thomas-Fermi radius of the cloud. (b) Linear contour plot of the local density of state at = 2:4EF. At each trap edge, a series of edge states, including the zero-energy Majorana fermion mode, are clearly visible. Figure taken from Ref. [51] with modification. C. 2D Rashba spin-orbit coupling Let us now discuss Rashba SOC, which takes the standard form VSO=(^ky^x^kx^y)[91]. The coupling between spin and orbital motions occurs along two spatial directions and therefore we shall refer to it as 2D Rashba SOC. This type of SOC is not realized experimentally yet, although there are several theoretical proposals for its realization [92, 93]. The superfluid phase with 2D Rashba SOC at low temperatures shares a lot of common features as its 1D counterpart as we reviewed in the previous subsection. Here we focus on some specific features, for example, the two-particle bound state at sufficiently strong SOC strength - the rashbon [21, 25] - and the related crossover to a BEC of rashbons. We will also discuss in greater detail the 2D topological superfluid with Rasbha SOC in the presence of an out-of-plane Zeeman field, since it provides an interesting platform to perform topological quantum computation. We note that experimentally it is also possible to create a 3D isotropic SOC, VSO=(^kx^x+^ky^y+^kz^z), where the spin and orbital degree of freedoms are coupled in all three dimensions [94]. We note also that early theoretical works on a Rashba spin-orbit coupled Fermi gas was reviewed very briefly by Hui Zhai in Ref. [95]. /s45/s52 /s45/s50 /s48 /s50 /s52/s48/s49/s50/s51/s52 /s101/s102/s102/s61/s48/s101/s102/s102/s61/s49/s40 /s41 /s47/s69 /s70/s101/s102/s102/s61/s50 FIG. 16: Left panel: schematic of the single-particle spectrum in the kxkyplane. A energy gap opens at k= 0, due to a nonzero out-of-plane Zeeman field h. Right panel: density of states of a 3D homogeneous Rashba spin-orbit coupled system at several SOC strengths, in units of mkF. Right figure taken from Ref. [96] with modification.20 1. Single-particle spectrum In the presence of an out-of-plane Zeeman field h^z, the single-particle spectrum is given by, Ek=k2 2mq 2 k2x+k2y
+h2: (49) The spectrum with a nonzero his illustrated on the left panel of Fig. 16. Compared with the single-particle spectrum with 1D equal-weight Rashba-Dresselhaus SOC in Fig. 2, it is interesting that the two minima in the lower energy branch now extend to form a ring structure. At low energy, therefore, we may anticipate that in the momentum space the particles will be confined along the ring. The effective dimensionality of the system is therefore reduced. Indeed, it is not difficult to obtain the density of states ( h= 0) [96]: (!) = (mkF)8 >< >:0; !<2 e
; e=2; (2 e!<0);p !=EF+eh =2arctanp !=(2 eEF)i ;(!0):; (50) where we have defined a dimensionless SOC coupling strength em=kF. As can be seen from the right panel of Fig. 16, (!)with Rashba SOC becomes a constant at low energy, which is characteristic of a 2D system. This reduction in the effective dimensionality will have interesting consequences when the interatomic interaction comes into play, as we now disucss in greater detail. 2. Two-body physics We solve the two-body problem by calculating the two-particle vertex function, following the general procedure outlined in the theoretical framework (Sec. IIA). Focusing on the case without Zeeman fields, we have G0(K) =(i!mk) +(kyxkxy) (i!mk)22 k2x+k2y
: (51) By substituting it into Eq. (16), it is straightforward to obtain, 1=1 U0+kBT VX k;i!m1=2 (i!mEk;+) (ini!mEqk;+)+1=2 (i!mEk;) (ini!mEqk;)Ares ;(52) with the single-particle energy Ek;=kq k2x+k2y; (53) and Ares2k?q (qxkx)2+ (qyky)2+2kx(qxkx) +2ky(qyky) (i!mEk;+) (i!mEk;) (ini!mEqk;+) (ini!mEqk;): (54) By performing explicitly the summation over i!m, replacing kbyq=2 +kand re-arranging the terms, we find that 1=m 4as+1 2VX k" f Eq=2+k;+
+f Eq=2k;+
1 inEq=2+k;+Eq=2k;++f Eq=2+k;
+f Eq=2k;
1 inEq=2+k;Eq=2k;1 k# 1 4VX k2 41 +q2 ?=4k2 ?q (qx=2 +kx)2+ (qy=2 +ky)2q (qx=2kx)2+ (qy=2ky)23 5Cres; (55) where Cres= + f Eq=2+k;+
+f Eq=2k;+
1 inEq=2+k;+Eq=2k;++ f Eq=2+k;
+f Eq=2k;
1 inEq=2+k;Eq=2k;  f Eq=2+k;+
+f Eq=2k;
1 inEq=2+k;+Eq=2k; f Eq=2+k;
+f Eq=2k;+
1 inEq=2+k;Eq=2k;+: (56)21 The above equation provides a starting point to investigate the fluctuation effect due to interatomic interactions. Here, for the two-body problem of interest, we discard the Fermi distribution function and set q= 0, as the ground bound state has zero center-of-mass momentum in the absence of Zeeman field. The two-body vertex function is then given by, 1 2b q=0;in!!+i0+
=m 4as1 2VX k1 !+i0+2Ek;++1 !+i0+2Ek;+1 k ; (57) The energy of the two-particle bound state Ecan be obtained by solving Ref1 2b[q=0;!=E]g= 0with= 0, as we already discussed in the theoretical framework. More physically, we may calculate the phase shift 2b(q=0;!) =Imln 1 2b q=0;in!!+i0+
 : (58) Recall that the vertex function represents the Green function of Cooper pairs. Thus, the phase shift defined above is simply
d!A(q;!), whereA(q;!)is the spectral function of pairs. As a result, a true bound state, corresponding to a delta peak in the spectral function, will cause a jump in the phase shift at the critical frequency !c=E, from which we determine the energy of the bound state. /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48 /s50 /s49 /s48 /s45/s49 /s45/s50/s45/s54/s45/s52/s45/s50/s48 /s32/s32/s50 /s69 /s47/s40 /s109/s50 /s41 /s50 /s47/s40 /s109 /s97 /s115/s41 /s49 /s45/s49/s50 /s98/s40/s113 /s61/s48/s44 /s41/s47 /s50 /s40 /s41/s47/s40 /s109/s50 /s41/s50 /s47/s40 /s109 /s97 /s115/s41/s32/s61/s32 /s48/s40/s97 /s41 /s50 /s49 /s48 /s45/s49 /s45/s50/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48 /s40/s98 /s41/s50 /s49 /s48 /s45/s49 /s45/s50/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s32/s32/s84 /s66/s69/s67/s47/s84 /s66/s69/s67/s44/s48 /s50 /s47/s40 /s109 /s97 /s115/s41 /s50 /s47/s40 /s109 /s97 /s115/s41 FIG. 17: (a) Two-body bound states as evidenced by the two-body phase shift at three different scattering lengths, in the presence of Rashba SOC. The arrows indicate the position of the bound state energy. The inset shows the bound state energy as a function of the scattering length. (b) Effective mass of the two-body bound state. The inset shows the decreases of critical temperature due to the heavy mass of bound states. Figure taken from Ref. [22] with modification. In the main figure and inset of Fig. 17(a), we show the two-body phase shift and the energy of the bound state of a Rashba spin-orbit coupled Fermi gas, respectively. Interestingly, the bound state exists even in the BCS limit, where thes-wave scattering length is small and negative [20]. This is because at the low energy the effective dimensionality of the Rashba system reduces to two, as we mentioned earlier from the nature of the low-energy density of states. In 2D, we know that any weak attraction can lead to a bound state. We can calculate the effective mass of the bound state [22, 23], which is strongly renormalized by the SOC, by determining the dispersion relation of the two-body bound state E(q)at small momentum q0. The result is shown in Fig. 17(b) for Mx=(2m) =My=(2m). It is important to note that all the properties of the two-body bound state, including its energy and effective mass, depend on a single parameter 1=(mas), which is the ratio of the only two length scales 1=(m)andasin the problem. Thus, in the limit of sufficiently large SOC, the bound state becomes universal and is identical to the one obtained at 1=(mas) = 0. This new kind of universal bound state has been referred to as rashbon [21, 25]. The mass of rashbons (i.e., '1:2from Fig. 17(b)) is notably heavier than the conventional molecules 2min the BEC limit. This causes a decrease in the condensation temperature of rashbons in such a way that TBEC= 2=3T(0) BEC'0:193TF; (59) whereT(0) BEC'0:218TFis the BEC temperature of conventional molecules. In the presence of out-of-plane Zeeman field h, the two-body problem has been discussed in detail in Ref. [24].22 EEEE E FIG. 18: (a) Mean-field order parameter as a function of the Rashba SOC for a homogeneous unitary Fermi gas at zero temperature. The inset shows the chemical potential and the half of bound state energy, both in units of Fermi energy EF. (b) Momentum distribution and (c) single-particle spectral function for ==2atkF=EF= 2. Hereis the angle between kand thez-axis. The width of the curves in (c) represents the weight factor (1 k;)=4for each of the four Bogoliubov excitations. Figure taken from Ref. [22]. 3. Crossover to rashbon BEC and anisotropic superfluidity Let us now discuss the crossover to a rashbon BEC. We focus on the unitary limit with as!1and increase the 2D Rashba SOC. At the mean-field saddle-point level, the single-particle Green function Eq. (7) takes the form ( h= 0) [22], G1 0= i!mk(^ky^x^kx^y) i
0^y i
0^y i!m+k(^ky^x+^kx^y) : (60) The inversion of the above matrix can be worked out explicitly, leading to two single-particle Bogoliubov dispersions whose degeneracy is lifted by the SOC, Ek;= [(kk?)2+
2 0]1=2, and the normal and anomalous Green functions from which we can immediately obtain the momentum distribution n(k) = 1P [1=2f(Ek;)] k;and the single-particle spectral function A""(k;!) =A##(k;!) =1 4X [(1 + k;)(!Ek;) + (1 k;)(!+Ek;)]; (61) where k;= (kk?=Ek;). The chemical potential and the order parameter are to be determined by the number and the gap equations, n=P kn(k)and
0=U0
0P [1=2f(Ek;)]=(2Ek;), respectively. Fig. 18(a) displays the chemical potential and the order parameter as functions of the SOC strength. The increase of the SOC strength leads to a deeper bound state. As a consequence, in analogy with the BEC-BCS crossover, the order parameter and the critical transition temperature are greatly enhanced at kFF. In the large SOC limit, we have = (B+E)=2, whereEis the energy of the two-body bound state, and Bis positive due to the repulsion between rashbons and decreases with increasing coupling as shown in the inset of Fig. 18(a). By assuming an s-wave repulsion with scattering lengthaBbetween rashbons, where B'(n=2)4aB=M, we estimate within mean-field that in the unitarity limit, aB'3=(m), comparable to the size of rashbons. Figs. 18(b) and (c) illustrate the momentum distribution and the single-particle spectral function, respectively. These quantities exhibit anisotropic distribution in momentum space due to the SOC and can be readily measured in experiment. Another interesting feature of the crossover to rashbon BEC is that the pairing field contains both a singlet and a triplet component [97]. For the system under study, it is straightforward to show that the triplet and singlet pairing fields are given by h k" k"i=i
0ei'kP [1=2f(Ek;)]=(2Ek;)andh k" k#i=
0P [1=2 f(Ek;)]=(2Ek;), respectively, where ei'k(kxiky)=k?. The magnitude of the pairing fields are shown in Fig. 19(a) and (b). The weight of the triplet component increases and approaches that of the singlet component as the SOC strength increases. In Fig. 19(c) and (d), we plot the zero-momentum dynamic and static spin structure factor, respectively. IntheabsenceoftheSOC,boththesequantitiesvanishidentically. Henceanonzerospinstructure factor is a direct consequence of triplet pairing [97]. Note that spin structure factor can be measured using the Bragg spectroscopy method as demonstrated in recent experiments [98].23 E E EE FIG. 19: Linear contour plot for the triple pairing correlation jh k" k"ijbetween like spins (a) and the singlet pairing correlationjh k" k#ijbetween un-like spins (b) for a homogeneous unitary Fermi gas at zero temperature with kF=EF= 2. The zero-momentum dynamic and static spin structure factor are shown in (c) and (d), respectively. Figure taken from Ref. [22]. The condensate fraction and superfluid density of the rashbon system have also been studied [99, 100], and have been found to exhibit unusual behaviors: The condensate fraction is generally enhanced by the SOC due to the increase of the pair binding; while the superfluid density is suppressed because of the nontrivial effective mass of rashbons. FIG. 20: Critical temperature Tcand the dissociation temperature Tscaled by the Fermi energy eFas a function of the SOC strength=vFfor fixed gas parameters 1=(kFas) =−2and1=(kFas) = 0. Here, we measure the SOC strength in units of Fermi velocity vF=kF=m. Figure taken from Ref. [28] with modification. To understand the finite-temperature properties of rashbons, the mean-field approach becomes less reliable. So far, a careful analysis based on the pair-fluctuation theory as outlined in the theoretical framework is yet to be performed. In Fig. 20, we show the superfluid transition temperature as a function of the Rashba SOC strength, predicted by the approximate many-body T-matrix theory - pseudogap theory [28]. At sufficiently large SOC strength, Tctends to the critical temperature of a rashbon BEC given by Eq. (59) - Tc'0:193TF- regardless of the dimensionless interaction parameter 1=(kFas), as we may anticipate. For a more detailed discussion of the crossover from BCS to rashbon BEC, we refer to Ref. [25].24 4. 2D Topological superfluidity Here we consider 2D topological superfluidity with 2D Rashba SOC, in the presence of an out-of-plane Zeeman fieldh. It is of particular interest, considering the possibility of performing topological quantum computation. This is because each vortex core in a 2D topological superfluid can host a Majorana fermion. Thus, by properly interchanging two vortices and thus braiding Majorana fermions, fault-tolerant quantum information stored non-locally in Majorana fermions may be processed [84, 101]. In the context of ultracold atoms, the use of 2D Rashba SOC to create a 2D topological superfluid was first proposed by Zhang and co-workers [38], and later considered by a number of researchers [41–44, 46, 48–50]. In free space, the criterion to enter topological superfluid phase is given by h>p 2+
2, above whichthesystembehaveslikea2Dweak-coupling p-wavesuperfluid, aswealreadydiscussedintheprevioussubsection (see, for example, Eq. (48)). Here, we are interested in the nature of 2D topological superfluids for the experimentally relevant situation with the presence of harmonic traps [42]. Theoretically, we solve numerically the BdG equation (29). In the presence of a single vortex at trap center, we take
0(r) =
0(r)ei'and decouple the BdG equation into different angular momentum channels indexed by an integerm. The quasiparticle wave functions take the form, [u"(r)ei';u#(r);v"(r)ei';v#(r)]ei(m+1)'=p 2. We have solved self-consistently the BdG equations using the basis expansion method. For the results presented below, we have taken N= 400andT= 0. We have used Ea= 0:2EFandkF=EF= 1, where the binding energy Eais a useful parameter to characterize the interatomic interaction in 2D. These are typical parameters that can be readily realized in a 2D40K Fermi gas. /s45/s50/s48 /s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53 /s50/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52 /s45/s50/s48 /s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53 /s50/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s49/s48/s45 /s53/s49/s48/s45 /s52/s49/s48/s45 /s51/s49/s48/s45 /s50/s49/s48/s45 /s49/s49/s48/s48/s32 /s32/s69 /s110/s109/s47/s69 /s70/s40/s97 /s41/s32 /s104 /s32/s61/s32/s48/s46/s52 /s69 /s70 /s109/s67/s100/s71/s77 /s79/s117/s116/s101/s114/s32/s101/s100/s103/s101 /s90/s69/s83/s40/s98 /s41/s32 /s104 /s32/s61/s32/s48/s46/s54 /s69 /s70/s32 /s32 /s109 /s40/s99 /s41/s32/s80/s104/s97/s115/s101/s32/s100/s105/s97/s103/s114/s97/s109/s84/s83/s78/s83/s43/s84/s83 /s32/s32/s109/s105/s110/s91 /s69 /s110/s109/s47/s69 /s70/s93 /s104 /s47/s69 /s70/s78/s83 FIG. 21: (a) and (b) Energy spectrum at h=EF= 0:4and0:6(whereEF=k2 F=(2M) =p N!?is the Fermi energy) in the presence of a single vortex. The color of symbols indicates the mean radiusp hr2i=rF(whererF= (4N)1=4p 1=(M!?)is the Fermi radius) of the eigenstate, which is defined by r2 =
r2[ju"j2+ju#j2+jv"j2+jv#j2]dr. The color of symbols changes from blue when the excited state is localized at the trap center to red when its mean radius approaches the Thomas-Fermi radius. In (a) and (b), the CdGM states are indicated by blue squares and a solid line, respectively. (c) Phase diagram, along with the lowest eigenenergy of Bogoliubov spectrum. Figure taken from Ref. [42] with modification. Figs. 21 reports the phase diagram [Fig. 21(c)] along with the quasiparticle energy spectrum of different phases [Figs. 21(a) and 21(b)] in the presence of a single vortex. By increasing the Zeeman field, the system evolves from a non-topological state (NS) to a topological state (TS), through an intermediate mixed phase in which NS and TS coexist. The topological phase transition into TS is well characterized by the low-lying quasiparticle spectrum, which has the particle-hole symmetry Em+1=E(m+1). As shown in Fig. 21(a), the spectrum of the NS is gapped. While in the TS, two branches of mid-gap states with small energy spacing appear: One is labeled by “Outer edge” and another “CdGM” which refers to localized states at the vortex core, i.e., the so–called Caroli-de Gennes-Matricon25 (CdGM) states [102]. The eigenstates with nearly zero energy at m=1could be identified as the zero-energy Majorana fermions in the thermodynamic limit. /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s48/s46/s52/s48/s46/s54 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s48/s46/s49/s48/s46/s50/s48/s46/s51 /s78/s83/s43/s84/s83 /s78/s83/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53/s48/s46/s53/s48/s46/s54/s48/s46/s55 /s32/s32/s32 /s114 /s47/s114 /s70/s32/s110 /s40/s114 /s41/s47 /s110 /s70 /s110 /s40/s114 /s32/s61/s32/s48/s41/s47 /s110 /s70 /s32/s110 /s40/s114 /s32/s61/s32/s48/s41/s47 /s110 /s70/s40/s97 /s41 /s84/s83 /s104 /s47/s69 /s70/s78/s83/s84/s83 /s78/s83/s43/s84/s83/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53/s48/s46/s49/s48/s46/s50 /s32/s32/s32 /s114 /s47/s114 /s70/s32/s110 /s40/s114 /s41/s47 /s110 /s70/s40/s98 /s41 /s104 /s47/s69 /s70 FIG. 22: Zeeman field dependence of spin-up (a) and spin-down (b) densities at the vortex core. The density is normalized by the Thomas-Fermi density nF= (p N=)pM!?. The insets show the core density distributions at h= 0:6EF. The red dot-dashed lines show the result by excluding artificially the Majorana vortex core state, whose contribution is shown by the shaded area. Figure taken from Ref. [42] with modification. In the TS, the occupation of the Majorana vortex-core state affects significantly the atomic density and the local density of states (LDOS) of the Fermi gas near the trap center, which in turn gives a strong experimental signature for observing Majorana fermions. Fig. 22 presents the spin-up and -down densities at the trap center, n"(0)andn#(0), as a function of the Zeeman field. In general, n"(0)andn#(0)increases and decreases respectively with increasing field. However, we find a sharp increase of n#(0)when the system evolves from the mixed phase to the full TS. Accordingly, a change of slope or kink appears in n"(0). The increase of n#(0)is associated with the gradualformation of the Majorana vortex-core mode, whose occupation contributes notably to atomic density due to the largeamplitude of its localized wave function. We plot in the inset of Fig. 22(b) n#(0)ath= 0:6EF, with or without the contribution of the Majorana mode, which is highlighted by the shaded area. This contribution is apparently absent in the NS. Thus, a sharp increase of n#(0), detectable in in situabsorption imaging, signals the topological phase transition and the appearance of the Majorana vortex-core mode. This feature persists at typical experimental temperature, i.e., T= 0:1TF. In the presence of impurity scattering, topological superfluid can also host a universal impurity-induced bound state [50, 51]. That is, regardless of the type of impurities, magnetic or non-magnetic, the impurity will always cause the same bound state within the pairing gap, provided the scattering strength is strong enough. The observation of such a universal impurity-induced bound state will give a clear evidence for the existence of topological superfluids. III. EXPERIMENTS We now review the experimental work, focusing on the ones carried out at Shanxi University. The apparatus and cooling scheme in the experiment have been described in previous papers [103–107] and briefly introduced here (see Fig. 1). An atomic mixture sample of87Rb and40K atoms in hyperfine state jF= 2;mF= 2iandjF= 9=2;mF= 9=2i,respectively,arefirstprecooledto1.5 Kbyradio-frequencyevaporativecoolinginaquadrupole-Ioffe configuration (QUIC) trap. The QUIC trap consists of a pair of anti-Helmholtz coils and a third coil in perpendicular orientation. To gain larger optical access, the atoms are first transported from the QUIC trap to the center of the quadrupole coils (glass cell) by lowering the current passing through quadrupole coils and increasing the current in the Ioffe coil, and then are transferred into an crossed optical trap in the horizontal plane, created by two off-resonance laser beams, at a wavelength of 1064 nm. A degenerate Fermi gas of about N'2106 40K atoms in thej9=2;9=2i internal state at T=TF'0:3is created inside the crossed optical trap. Here Tis the temperature and TFis the Fermi temperature defined by TF=EF=kB= (6N)1=3!=kBwith a geometric mean trapping frequency !'2130Hz. A780nm laser pulse of 0:03ms is used to remove all the87Rb atoms in the mixture without heating40K atoms. To create SOC, a pair of Raman laser beams are extracted from a continuous-wave Ti-sapphire single frequency laser. The two Raman beams are frequency-shifted by two single-pass acousto-optic modulators (AOM) respectively. In this way the relative frequency difference
!between the two laser beams is precisely controlled. At the output of the optical fibers, the two Raman beams each has a maximum intensity I= 130mW, counter-propagating along the x-axis with a 1=e2radius of 200 mand are linearly polarized along the z- andy-axis, respectively, which correspond to() and() of the quantization axis ^z(^y). The momentum transferred to atoms during the Raman process is2k0= 2krsin(=2), wherekr= 2=is the single-photon recoil momentum, is the wavelength of the Raman26 beam, and is the intersecting angle of two Raman beams. Here, krandEr=k2 r=2mare the units of momentum and energy. The optical transition wavelengths of the D1 and D2-line are 770.1 nm and 766.7 nm, respectively. The wavelengths of the Raman lasers are about 772 773 nm. The two internal states involved in SOC are chosen as follows. In the case of noninteracting system, the two states are magnetic sublevels j"i=j9=2;9=2iandj#i=j9=2;7=2i. These two spin states are stable and are weakly interacting with a background s-wave scattering length as= 169a0. We use a pair of Helmholtz coils along the y-axis (as shown in Fig. 1) to provide a homogeneous bias magnetic field, which gives a Zeeman shift between the two magnetic sublevels. A Zeeman shift of !Z= 210:27MHz between these two magnetic sublevels is produced by a homogeneous bias magnetic field of 31G. When the Raman coupling is at resonance (at
!= 210:27MHz and two-photon Raman detuning =
!!Z0), the detuning between j9=2;7=2iand other magnetic sublevels like j9=2;5=2iis about 2170kHz, which is one order of magnitude larger than the Fermi energy. Hence all the other states can be safely neglected. In the case of the strongly interacting spin-orbit coupled Fermi gas, two magnetic sublevelsj#i=j9=2;9=2iandj"i=j9=2;7=2iare chosen. To create strong interaction, the bias field is ramped from 204 G to a value near the B0= 202:1G Feshbach resonance at a rate of about 0.08 G/ms. We remark that due to a decoupling of the nuclear and electronic spins, the Raman coupling strength decreases with increasing of the bias field [108]. When working at a large bias magnetic field, we have to use a smaller detuning of the Raman beams with respect to the atomic D1 transition in order to increase the Raman coupling strength. In order to control the magnetic field precisely and reduce the magnetic field noise, the power supply (Delta SM70- 45D) has been operated in remote voltage programming mode, whose voltage is set by an analog output of the experiment control system. The current through the coils is controlled by the external regulator relying on a precision current transducer (Danfysik ultastable 867-60I). The current is detected with the precision current transducer, then the regulator compares the measured current value to a set voltage value from the computer. The output error signal from the regulator actively stabilize the current with the PID (proportional-integral-derivative) controller acting on the MOSFET (metal-oxide-semiconductor field-effect transistor). In order to reduce the current noise and decouple the control circuit from the main current, a conventional battery is used to power the circuit. We use the standard time-of-flight technique to perform our measurement. To this end, the Raman beams, optical dipole trap and the homogeneous bias magnetic field are turned off abruptly at the same time, and a magnetic field gradient along the y-axis provided by the Ioffe coil is turned on. The two spin states are separated along the y- direction, and imaging of atoms along the z-direction after 12ms expansion gives the momentum distribution for each spin component. A. The noninteracting spin-orbit coupled Fermi gas In this section, we review the experiment on non-interacting system. 1. Rabi oscillation We first study the Rabi oscillation between the two spin states induced by the Raman coupling. All atoms are initially prepared in the j"istate. The homogeneous bias magnetic field is ramped to a certain value so that =4Er, that is, the k= 0component of state j"iis at resonance with k= 2kr^xstate ofj#icomponent, as shown in Fig. 23(a). Then we apply a Raman pulse to the system, and measure the spin population for different duration time of the Raman pulse. Similar experiment in bosonic system yields an undamped and completely periodic oscillation, which can be well described by a sinusoidal function with frequency [1]. This is because for bosons, macroscopic number of atoms occupy the resonant k= 0mode, and therefore there is a single Rabi frequency determined by the Raman coupling only. While for fermions, atoms occupy different momentum states. Due to the effect of SOC, the coupling between the two spin states and the resulting energy splitting are momentum dependent, and atoms in different momentum states oscillate with different frequencies. Hence, dephasing naturally occurs and the oscillation will be inevitably damped after several oscillation periods. In our case, the spin-dependent momentum distribution shown in Fig. 23(b) clearly shows the out-of-phase oscillation for different momentum states. For a non-interacting system, the population of j#icomponent is given by n#(k+ 2kr^x;r;t) =n"(k;r;0)sin2p (kxkr=m)2+ 2=4t 1 +2kxkr m
2; (62) wheretis the duration time of Raman pulse, n"(k;r;0)is the equilibrium distribution of the initial state in local density approximation. From Eq. (62) one can see that the momentum distribution along the x-axis of thej#i27 -6 -4 -2 0 2 4 6 -2 0 20510 - 4 r E δδ δδ== == Quasimomentum p x /kr Momentum kx / kr(a) (b) Fraction of |9/2,7/2> Raman pulse time (ms)| 9/2,9/2> | 9/2,7/2> 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.1 0.2 0.3 0.4 0.5 Energy/ Er (c) 1 6 µ s4 µ s 3 2 µ s 6 8 µ s 1 2 8 µ s 1 6 0 µ s FIG. 23: Raman-induced Rabi oscillation: (a) The energy dispersion with =4Er. The system is initially prepared with all atoms inj9=2;9=2istate. (b) Time-of-flight image (left) and integrated time-of-flight image (integrated along ^y) at different duration time for j"i(blue) andj#i(red). The parameters are kF= 1:35krandT=T F= 0:35. (c) The population in j9=2;7=2i as a function of duration time of Raman pulse. kF= 1:9krandT=T F= 0:30for red circles, kF= 1:35krandT=T F= 0:35for blue squares, kF= 1:1krandT=T F= 0:29for green triangles. The solid lines are theory curves with = 1:52Er. Figure taken from Ref. [11]. /s45/s50 /s48 /s50/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50 /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 /s45/s50 /s48 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s45/s50 /s48 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s40/s100/s41 /s32/s32/s110 /s40/s107 /s120/s41/s32/s45/s32 /s110 /s40/s45 /s107 /s120/s41 /s107 /s120/s32/s47/s32 /s107 /s114/s124/s57/s47/s50 /s124/s55/s47/s50 /s40/s101/s41 /s32/s32/s73 /s107 /s120/s32/s47/s32 /s107 /s114/s40/s102/s41 /s32/s32/s78 /s107 /s120/s32/s47/s32 /s107 /s114/s32 /s107 /s120/s32/s47/s32 /s107 /s114 /s32/s32/s32 /s32 /s32/s32 /s40/s97/s41 /s32 /s32/s107 /s120/s32/s47/s32 /s107 /s114/s32/s32 /s32/s40/s98/s41 /s32/s32 /s32/s107 /s120/s32/s47/s32 /s107 /s114 /s32 /s32/s32/s32 /s32/s40/s99/s41 /s32/s32/s32 /s32 /s32/s32 /s32 /s32 /s32/s32 /s32 /s32 /s32/s32 FIG. 24: Momentum distribution asymmetry as a hallmark of SOC: (a-c) time-of-flight measurement of momentum distribution for bothj"i(blue) andj#i(red). Solid lines are theory curves. (a) kF= 0:9krandT=T F= 0:8(b)kF= 1:6krandT=T F= 0:63; (c)kF= 1:8krandT=T F= 0:57. (d-f): plot of integrated momentum distribution n(k)n(k)for the case of (a-c). Figure taken from Ref. [11]. component is always symmetric respect to 2krat any time, which is clearly confirmed by the experimental data as shown in Fig. 23(b). The total population in the j#icomponent is given by N#(t) =
dkdrn#(k;r;t), and in Fig. 23(c), one can see that there is an excellent agreement between the experiment data and theory, from which we determine = 1:52(5)Er.28 2. Momentum distribution We focus on the case with = 0, and study the momentum distribution in the equilibrium state. We first transfer half of40K atoms fromj#itoj"iusing radio frequency sweep within 100ms. Then the Raman coupling strength is ramped up adiabatically in 100ms from zero to its final value and the system is held for another 50ms before time-of-flight measurement. Since SOC breaks spatial reflection symmetry ( x!xandkx!kx), the momentum distribution for each spin component will be asymmetric, i.e. n(k)6=n(k), with=";#. On the other hand, when= 0the system still satisfies n"(k) =n#(k). The asymmetry can be clearly seen in the spin-resolved time- of-flight images and integrated distributions displayed in Fig. 24(a) and (b), where the fermion density is relatively low. While it becomes less significant when the fermion density becomes higher, as shown in Fig. 24(c), because the strength of the SOC is relatively weaker compared to the Fermi energy. Although the presence of the Raman lasers cause additional heating to the cloud, we find that the temperature is within the range of 0:50:8TF, which is still below degenerate temperature. In Fig. 24(d-f), we also show n(kx)n(kx)to reveal the momentum distribution asymmetry more clearly. 3. Lifshitz transition With SOC, the single particle spectra of Eq. (39) are dramatically changed from two parabolic dispersions into two helicity branches as shown in Fig. 25(b). Here, two different branches are eigenstates of “helicity” ^sand the “helicity” operator describes whether spin pis parallel or anti-parallel to the “effective Zeeman field” h= ( ;0;krpx=m+) at each momentum, i.e. ^s=p
h=jp
hj.s= 1for the upper branch and s=1for the lower branch. The topology of Fermi surface exhibits two transitions as the atomic density varies. At sufficient low density, it contains two disjointed Fermi surfaces with s=1, and they gradually merge into a single Fermi surface as the density increases to nc1. Finally a new small Fermi surface appears at the center of large Fermi surface when density further increases and fermions begin to occupy s= 1helicity branch at nc2. A theoretical ground state phase diagram for the uniform system is shown in Fig. 25(a), and an illustration of the Fermi surfaces at different density are shown in Fig. 25(b). Across the phase boundaries, the system experiences Lifshitz transitions as density increases [109], which is a unique property in a Fermi gas due to Pauli principle. We fix the Raman coupling and vary the atomic density at the center of the trap, as indicated by the red arrow in Fig. 25(a). In Fig. 25(c1-c5), we plot the quasi-momentum distribution in the helicity bases for different atomic density. At the lowest density, the s= 1helicity branch is nearly unoccupied, which is consistent with that the Fermi surface is below s= 1helicity branch. The quasi-momentum distribution of the s=1helicity branch exhibits clearly a double-peak structure, which reveals that the system is close to the boundary of having two disjointed Fermi surfaces at s=1helicity branch. As density increases, the double-peak feature gradually disappears, indicating the Fermi surface of s=1helicity branch finally becomes a single elongated one, as the top one in Fig. 25(b). Here we define a quality of visibility v= (nAnB)=(nA+nB), wherenAis the density of s=1branch at the peak andnBis the density at the dip between two peaks. Theoretically one expects vapproaches unity at low density regime and approaches zero at high density regime. In Fig. 25(d) we show that our data decreases as density increases and agrees very well with a theoretical curve with a fixed temperature of T=T F= 0:65. Moreover, across the phase boundary between SFS and DFS-1, one expects a significant increase of population on s= 1helicity branch. In Fig. 25(e), the fraction of atom number population at s= 1helicity branch is plotted as a function of Fermi momentum kF, which grows near the critical point predicted in zero-temperature phase diagram. The blue solid line is a theoretical calculation for N+=NwithT=T F= 0:65, and the small deviation between the data and this line is due to the temperature variation between different measurements. Because the temperature is too high, the transition is smeared out. For both vandN+=Nwe observe only a smooth decreasing or growth across the regime where it is supposed to have a sharp transition, however, the agreement with theory suggests that with better cooling a sharper transition should be observable. 4. Momentum-resolved rf spectrum The effect of SOC is further studied with momentum resolved rf spectroscopy [71], which maps out the single- particle dispersion relation. A Gaussian shaped pulse of rf field is applied for 200 sto transfer atoms from j9=2;7=2i (j#i) state to the final state j9=2;5=2i, as shown in Fig. 26(a), and then the spin population at j9=2;5=2iis measured with time-of-flight at different rf frequencies. In Fig. 26(b) we plot an example of the final state population as a function of momentum pxand the frequency of rf field RF, from which one can clearly see the back-bending feature and the gap opening at the Dirac point. Both are clear evidences of SOC.29 /s32/s32/s40/s98/s41 /s68/s83/s70/s45/s49/s32/s32/s32 /s83/s70/s83/s32/s32/s68/s70/s83/s45/s50 /s32 /s32/s32 /s32/s32 FIG. 25: Topological change of Fermi surface and Lifshitz transition: (a) Theoretical phase diagram at T= 0.k0 F= (32n)1=3. “SFS” means single Fermi surface. “DFS” means double Fermi surface. (b) Illustration of different topology of Fermi surfaces. The single particle energy dispersion is drawn for small . Dashed blue line is the chemical potential. (c) Quasi-momentum distribution in the helicity bases. Red and green points are distributions for s=1ands= 1helicity branches, respectively. kF= 0:9kr,T=T F= 0:80for (c1);kF= 1:2kr,T=T F= 0:69for (c2);kF= 1:4kr,T=T F= 0:61for (c3);kF= 1:6kr, T=T F= 0:63for (c4);kF= 1:8kr,T=T F= 0:57for (c5). All these points are marked on phase diagram in (a). (d) Visibility v= (nAnB)=(nA+nB)decreases as kF=krincreases (A and B points are marked in (c1)). (e) Atom number population in s= 1helicity branch N+=Nincreases as kF=krincreases increases. In both (d) and (e), the blue solid line is a theoretical curve withT=T F= 0:65, and the background color indicates three different phases in the phase diagram. Figure taken from Ref. [11]. For an occupied state, the initial state dispersion i(k)can be mapped out by i(k) =RFEZ+f(k): (63) wheref(k) =k2=2mis the dispersion of the final j9=2;5=2istate, andEZis the energy difference between j9=2;7=2i andj9=2;5=2istate. Here, the momentum of the rf photon is neglected, thus the rf pulse does not impart momentum to the atom in the final state. In Fig. 26(c) we show three measurements corresponding to (c1), (c3) and (c5) in Fig. 25. For (c1), clearly only s=1branch is populated. For (c3), the population is slightly above the s= 1helicity branch. And for (c5), there are already significant population at s= 1helicity branch. In (c5) one can also identify the chiral nature of two helicity branches: For s=1branch, most left-moving states are dominated by j#istate; while fors= 1branch, right-moving states are mostly dominated by j#istates. The theoretical simulation of momentum-resolved rf spectroscopy has been performed and discussed in Sec. IIB1 (see, in particular, Fig. 3). We note that, the definition of momentum and rf frequency is different. These are related by,kx=pxkrand!=RF. The single-particle spectrum is also measured using the technique of spin injection spectroscopy in a spin-orbit coupled Fermi gas of6Li by the MIT group [12]. In that work, the following four lowest hyperfine states are chosen j3=2;1=2i,j3=2;3=2i,j1=2;1=2i,j1=2;1=2i, which are labelled as j"ii,j"if,j#if,j#ii. The Raman process couples j"iftoj#ifto induce SOC between these two states. For momentum-resolved rf spectroscopy, the state j#iiis coupled via rf field to the state j#if, as this connects the first and second lowest hyperfine states. Similarly, an atom in state30 Quasimomentum px [units of kr](b) Quasimomentum p x[units of kr](a) Quasimomentum px [units of kr](c1) (c3) (c5) Energy [units of Er ] Initial εR F ν |9/2,5/2> Radio frequency [units of MHz] ππππ νννν2/ RF FIG. 26: Momentum-resolved rf spectroscopy of a spin-orbit coupled Fermi gas: (a) Schematic of momentum-resolved rf spectroscopy of SO coupled Fermi gases. Green and pink solid lines are two helicity branches in which the eigenstates are all superposition ofj9=2;9=2iandj9=2;7=2i. Thus both can undergo rf transition from j9=2;7=2itoj9=2;5=2i, as indicated by dashed lines. (b) Intensity map of the atoms in j9=2;5=2istate as a function of ( RF;kx) plane. (c) Single particle dispersion and atom population measured for (c1), (c3) and (c5) in Fig. 25. Figure taken from Ref. [11]. j"iiis coupled toj"if. Since the dispersion for initial states j"iiandj#ii(i(k) =k2=2m) are known, the spectra of the final states, which is subject to the SOC, are obtained. The dispersion investigated above is the simplest case for a spin-orbit coupled system. An even richer band structure involving multiple spinful bands separated by fully insulating gaps can arise in the presence of a periodic lattice potential. This has been realized for Bose-Einstein condensates by adding rf coupling between the Raman- coupled statesj"ifandj#if[110]. Using a similar method, a spinful lattice for ultracold fermions is created, and one can use spin-injection spectroscopy to probe the resulting spinful band structure [12], see, for example, Fig. 4. B. The strongly interacting spin-orbit coupled Fermi gas We now consider the Femi gas where interaction cannot be neglected. In particular, we focus on the effect of SOC on fermionic pairing. 1. Integrated radio-frequency spectrum To create a strongly interacting Fermi gas with spin-orbit coupling, first, the bias magnetic field is tuned from high magnetic field above Feshbach resonance to a final value B(which is varied) below Feshbach resonance. Thus, Feshbach molecules are created in this process. Then, we ramp up adiabatically the Raman coupling strength in 15 ms from zero to its final value = 1:5Erwith Raman detuning = 0. The temperature of the Fermi cloud after switching on the Raman beams is at about 0:6TF[11]. The Fermi energy is EF'2:5Erand the corresponding Fermi wavevector is kF'1:6kr. To characterize the strongly-interacting spin-orbit coupled Fermi system, we apply a Gaussian shaped pulse of rf field with a duration time about 400 s and frequency !to transfer the spin-up fermions to an un-occupied third hyperfine state j3i=jF= 9=2;mF=5=2i. In Fig. 27(b), we show that the integrated rf-spectrum of an interacting Fermi gas below the Feshbach resonance, with or without spin-orbit coupling. Here, we carefully choose the one photon detuning of the Raman lasers to avoid shifting Feshbach resonance by the Raman laser on the bound-to-bound transition between the ground Feshbach molecular state and the electronically excited molecular state. We also make sure that the single-photon process does31 (a) |9/2,-9/2> |9/2,-7/2> |9/2,-5/2> R1 ωR 2 ω(b) Transferred fraction B=201.6 G 4 7 . 1 2 4 7 . 1 4 4 7 . 1 6 4 7 . 1 8 4 7 . 2 0 0 . 0 0 . 2 0 . 4 S O C N o S O C R F υ RF v (MHz) ππ ππ νν νν 2 /RF FIG. 27: (a) Energy level of a strongly-interacting Fermi gas of40K atoms with SOC. (b) The integrated rf-spectroscopy below the Feshbach resonance (at B= 201:6G andas'2215:6a0, wherea0is the Bohr radius), in the presence (solid circles) and absence (empty circles) of the spin-orbit coupling. The Raman detuning is = 0. The dimensionless interaction parameter 1=(kFas)'0:66. The fraction is defined as N5=2=(N5=2+N7=2), whereN5=2andN7=2are obtained from the TOF absorption image. Figure taken from Ref. [13]. FIG. 28: The integrated rf-spectrum for a spin-orbit coupled Fermi gas. The red solid circles (red lines) and dark empty circles show respectively the experimental data in the presence and absence of spin-orbit coupling with Raman detuning = 0. The upper panel represents experimental data and the lower panel represents the theoretical calculation. The dimensionless interaction parameter 1=(kFas)in (a), (b), and (c) are 0:89,0:66, and 0:32, respectively. Figure taken from Ref. [13]. not affect the rf spectrum. The narrow and broad peaks in the spectrum should be interpreted respectively as the rf-response from free atoms and fermionic pairs. With spin-orbit coupling, we find a systematic blue shift in the atomic response and a red shift in the pair response. The latter is an unambiguous indication that the properties of fermionic pairs are strongly affected by spin-orbit coupling [13]. The red shift of the response from the pairs may be understood from the binding energy of pairs in the two-body limit. As mentioned below Eq. (39), the Raman coupling may be regarded as an effective Zeeman field. The stronger the effective Zeeman field, the smaller the binding energy of the two-particle bound states [24, 26]. In Fig. 28, we compare the experimentally measured rf-spectrum with the many-body T-matrix prediction, which is obtained within the pseudogap approximation [13] (see the discussion in Sec. IIA3 and IIA5). In the calculation, at a qualitative level, we do not consider the trap effect and take the relevant experimental parameters at the trap center. Otherwise, there are no adjustable free parameters used in the theoretical calculations. As shown in Fig. 28, we find a qualitative agreement between theory and experiment, both of which show the red shift of the response from fermionic pairs. Note that, near Feshbach resonances our many-body pseudogap theory is only qualitatively reliable. It cannot explain well the separation of atomic and pair peaks in the observed integrated rf-spectrum. More seriously, it fails to take into account properly the strong interactions between atoms and pairs.32 2. Coherent formation of Feshbach molecules by spin-orbit coupling In a recent experiment, we studied the formation of Feshbach molecules from an initially spin-polarized Fermi gas [15]. For simplicity, let us consider two atoms both prepared in the j#istate. We label this state as j#i1j#i2, which is obviously a spin-symmetric state. Under the s-wave interaction, the Feshbach molecule is spin-antisymmetric singlet state. Hence to form Feshbach molecule from this initial state, a spin-antisymmetric coupling is required. To this end, we apply two Raman laser beams that effectively couples the hyperfine states j"iandj#i. The effective Hamiltonian arising from the Raman beams can be written as HR=H(1) R+H(2) Rwith H(j) R= 2(j) z+ 2e2ik0xj(j) ++ 2e2ik0xj(j) ; (64) forj= 1;2. Here we have (j) z= (j"ijh"jj"i jh"j)=2,(j) +=j"ijh#j, and(j) =(j)y +. In Eq. (64), is the Raman coupling intensity, xjis the position of the j-th atom in the x-direction, and k0=krsin(=2), withkr the single-photon recoil momentum and the angle between the two Raman beams. It is apparent that HRcan be written asHR=H(+) R+H() Rwith H() R= 4 ei2k0x1ei2k0x2
 (1) +(2) + +h:c:: (65) Obviously, H() RandH(+) Rareanti-symmetric andsymmetric under the exchange of the hyperfine state of the two atoms, respectively. Therefore, only H() Rcan create spin-antisymmetric state out of the initially polarized state j#i1j#i2, and as a consequence make the formation of Feshbach molecule possible. When the two Raman beams propagate along the same direction, i.e., = 0, we havek0= 0and thusH() R= 0. Then the Feshbach molecule cannot be produced from the polarized atoms. In contrast, when the angle between the two Raman beams is non-zero, we have H() R6= 0and Feshbach molecule can thus be created. This picture is exactly confirmed by our data. Our experiment is performed with the spin polarized40K gas in jF;m Fi=j9=2;9=2istate, at 201:4G, below the Feshbach resonance located at 202:1G, which corresponds to a binding energy of Eb= 230kHz (corresponding to 3.59 Er) for the Feshbach molecules and 1=(kFas)0:92for our typical density. After applying the Raman lasers for certain duration time, we turn off the Raman lasers and measure the population of Feshbach molecule and atoms in j9=2;7=2istate with an rf pulse. This rf field drives a transition fromj9=2;7=2itoj9=2;5=2i. For a mixture of j9=2;7=2iand Feshbach molecules, as a function of rf frequencyRF, we find two peaks in the population of j9=2;5=2i, as shown in Fig. 29(b). The first peak (blue curve) is attributed to free atom-atom transition and the second peak (red curve) is attributed to molecule-atom transition. Thus, in the following, we set RF=2to47:14MHz to measure Feshbach molecules. When the two-photon Raman detuning is set to=Eb=3:59Er, as shown in Fig. 29(a), we measure the population of Feshbach molecule as a function of duration time for three different angles, = 180,= 90, and = 0, as shown in Fig. 29(c), (d) and (e). We find for = 180, Feshbach molecules are created by Raman process and the coherent Rabi oscillation between atom-molecule can be seen clearly. For = 90, production of Feshbach molecules is reduced a little bit and the atom-molecule Rabi oscillation becomes invisible. For = 0, no Feshbach molecule is created even up to 40ms, which means the transition between Feshbach molecules and a fully polarized state is prohibited if Raman process imparts no momentum transfer, i.e., no SOC. In a related work, the NIST group recently carried out an experiment in which they swept a magnetic field on the BEC side of the Feshbach resonance [14]. It is shown that the number of remaining atoms exhibits a dip as a function of the magnetic field strength. This dip represents the loss of atom due to the formation of the Feshbach molecules. The position of the dip moves towards the lower field (to the BEC limit) as the Raman detuning is increased. The phenomenon can also be explained by the fact that the effective Zeeman field (in this case, the detuning ) disfavors the formation of bound molecules. Hence at larger , a largera1 s(i.e., stronger attraction between unlike spins) is required to form molecules [26]. This is in full agreement with the theoretical discussion concerning the two-body physics for the equal-weight Rashba-Dresselhaus SOC presented in Sec. IIB2. IV. CONCLUSION In this chapter, we described the properties of a spin-orbit coupled Fermi gas. Recent progress, both theoretical and experimental, were reviewed. As we have shown, spin-orbit coupled Fermi gases possess a variety of intriguing properties. The diverse configuration of the synthetic Gauge field and the extraordinary controllability of atomic systems provide new opportunities to explore quantum many-body systems and quantum topological matter. We33 9 2 , - 9 2 9 2 , - 7 2 1 ω2 ω∆ 47. 10 47. 13 47. 16 47 .19 0. 00 0. 25 0. 50 0. 75 Transferredfraction0 . 0 0 . 1 0 .0 0 .1 0 1 0 2 0 3 0 4 0 0 .0 0 .1 Fraction of Molecules Raman P luse Tim e (ms) (a) (b) (c ) (d ) (e ) Eb RF v (MH z) ππ ππ νν νν 2 /R F FIG. 29: Energy level diagram and spin-orbit coupling induced Feshbach molecules. (a)Schematic diagram of the energy levels. A pair of Raman lasers couples spin polarized state j9=2;9=2ito Feshbach molecules in Fermi gases40K.(b)Radio-frequency spectrumj9=2;7=2itoj9=2;5=2itransition applied to a mixture of Feshbach molecules and scattering atoms in j9=2;7=2i. (c-e)The population of Feshbach molecules detected by the rf pulse as a function of duration time of the Raman pulse. The angle of two Raman beams is = 180(c),= 90(d) and= 0(e). The Raman coupling strength is = 1:3Erand the two-photon Raman detuning is =Eb=3:59Er. Figure taken from Ref. [15]. note that this article by no means is a comprehensive review. For example, we only focused on a continuum system and neglected many interesting theoretical works on lattice systems. So far only one particular scheme (equal-weight Rashba-Dresselhaus) of SOC has been realized in the experiment, which is based on the Raman transition between two hyperfine ground states of the atom. One drawback of the laser-based SOC generating scheme is that the application of the laser fields inevitably induce additional heating. For certain atoms, this heating may be severe enough to prevent the system from becoming quantum degenerate. Furthermore, many interesting physics requires a strong interaction strength which is induced by applying a fairly strong magnetic field via the Feshbach resonance. Due to a decoupling of the nuclear and electronic spins in large magnetic fields, Raman coupling efficiency quickly reduces with increasing of the magnetic field [108]. This poses another severe experimental challenge. Due to these reasons, no superfluid spin-orbit coupled Fermi gas has been realized yet. As a result, many interesting theoretical proposals (e.g., topological superfluids, Majorna fermion, etc.) are still waiting to be experimentally realized. Nevertheless, we want to remark that despite the relatively high temperature of the experimental system, the effects of SOC have been clearly revealed in single-particle properties as well as the two- and many-body properties on the BEC side of the resonance, as such properties are not easily washed out by finite temperature effects. Very recently, a scheme to synthesize a general SOC is proposed, which is based on purely magnetic field pulses and involves no laser fields [111, 112]. Whether this scheme will overcome the problems mentioned above remains to be seen. Acknowledgments We are deeply appreciative for discussions with Congjun Wu, Wei Yi, Hui Zhai, Chuanwei Zhang, and many others; as well as the students and postdocs in our groups: Lin Dong, Lei Jiang, Shi-Guo Peng, Pengjun Wang, and Zhengkun Fu. JZ is supported by NFRP-China (Grant No. 2011CB921601), NSFC Project for Excellent Research Team (Grant No. 61121064), NSFC (Grant No. 11234008), Doctoral Program Foundation of Ministry of Education China (Grant No. 20111401130001). XJL and HH are supported by the ARC DP0984637 and DP0984522. HP is supported by the NSF, the DARPA OLE program and the Welch Foundation (Grant No. C-1669). [1] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102, 130401 (2009). [2] Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, and I. B. Spielman, Nature (London) 462, 628 (2009).34 [3] Y-J. Lin, R. L. Compton, K. Jiménez-García, W. D. Phillips, J. V. Porto, and I. B. Spielman, Nature Phys. 7, 531 (2011). [4] The quantum version of Maxwell’s theory is described by quantum electrodynamics, which is a dynamic gauge theory in the sense that the gauge symmetries generate dynamics by giving rise to interaction couplings between particles mediated by photons. The artificial gauge fields for cold atoms, by contrast, are static, i.e., not dynamically coupled to the atomic fields. [5] I. B. Spielman, Chapter 5 in Annual Review of Cold Atoms and Molecules , Vol. 1 (World Scientific, 2012), edited by K. Madison, Y. Wang, A. M. Rey, and K. Bongs. [6] Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Nature (London) 471, 83 (2011). [7] Z. Fu, P. Wang, S. Chai, L. Huang, and J. Zhang, Phys. Rev. A 84, 043609 (2011). [8] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 109, 115301 (2012). [9] C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels, arXiv:1301.0658. [10] J.-Y. Zhang, S.-C. Ji, L. Zhang, Z.-D. Du, W. Zheng, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, arXiv:1305.7054. [11] P. Wang, Z. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [12] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [13] Z. Fu, L. Huang, Z. Meng, P. Wang, X.-J. Liu, H. Pu, H. Hu, and J. Zhang, Phys. Rev. A 87, 053619 (2013). [14] R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jimenez-Garcia, and I. B. Spielman, Phys. Rev. Lett. 111, 095301 (2013). [15] Z. Fu, L. Huang, Z. Meng, P. Wang, L. Zhang, S. Zhang, H. Zhai, P. Zhang, and J. Zhang, arXiv:1306.4568. [16] M. Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [17] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). [18] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010). [19] S. Giorgini, L. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 80, 1215 (2008). [20] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys. Rev. B 84, 014512 (2011). [21] J. P. Vyasanakere, and V. B. Shenoy, Phys. Rev. B 83, 094515 (2011). [22] H. Hu, L. Jiang, X. -J. Liu and H. Pu, Phys. Rev. Lett. 107, 195304 (2011). [23] Z.-Q. Yu, and H. Zhai, Phys. Rev. Lett. 107, 195305 (2011). [24] L. Jiang, X.-J. Liu, H. Hu, and H. Pu, Phys. Rev. A 84, 063618 (2011). [25] J. P. Vyasanakere, and V. B. Shenoy, New J. Phys. 14, 043041 (2012). [26] L. Dong, L. Jiang, H. Hu, and H. Pu, Phys. Rev. A 87, 043616 (2013). [27] L. Han and C. A. R. Sá de Melo, Phys. Rev. A 85, 011606 (2012). [28] L. He, X.-G. Huang, H. Hu, and X.-J. Liu, Phys. Rev. A 87, 053616 (2013). [29] Z. Zheng, M. Gong, X. Zou, C. Zhang, and G.-C. Guo, Phys. Rev. A 87, 031602(R) (2013). [30] V. B. Shenoy, Phys. Rev. A 88, 033609 (2013). [31] Z. Zheng, M. Gong, Y. Zhang, X. Zou, C. Zhang, and G.-C. Guo, arXiv:1212.6826. [32] F. Wu, G.-C. Guo, W. Zhang, and W. Yi, Phys. Rev. Lett. 110, 110401 (2013). [33] X.-J. Liu and H. Hu, Phys. Rev. A 87, 051608(R) (2013). [34] L. Dong, L. Jiang, and H. Pu, New J. Phys. 15, 075014 (2013). [35] H. Hu and X.-J. Liu, arXiv:1304.0387. [36] X.-F. Zhou, G.-C. Guo, W. Zhang, and W. Yi, Phys. Rev. A 87, 063606 (2013). [37] M. Iskin, Phys. Rev. A 88, 013631 (2013). [38] C. Zhang, S. Tewari, R. Lutchyn, and S. Das Sarma, Phys. Rev. Lett. 101, 160401 (2008). [39] M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. 107, 195303 (2011). [40] L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Phys. Rev. Lett. 106, 220402 (2011). [41] S.-L. Zhu, L. B. Shao, Z. D. Wang, and L. M. Duan, Phys. Rev. Lett. 106, 100404 (2011). [42] X.-J. Liu, L. Jiang, H. Pu, and H. Hu, Phys. Rev. A 85, 021603(R) (2012). [43] X. Yang and S. Wan, Phys. Rev. A 85, 023633 (2012). [44] K. Seo, L. Han, and C. A. R. Sá de Melo, Phys. Rev. A 85, 033601 (2012). [45] X.-J. Liu and H. Hu, Phys. Rev. A 85, 033622 (2012). [46] L. He and X.-G. Huang, Phys. Rev. A 86, 043618 (2012). [47] R. Wei and E. J. Mueller, Phys. Rev. A 86, 063604 (2012). [48] M. Iskin, Phys. Rev. A 86, 065601 (2012). [49] M. Gong, G. Chen, S. Jia, and C. Zhang, Phys. Rev. Lett. 109, 105302 (2012). [50] H. Hu, L. Jiang, H. Pu, Y. Chen, and X.-J. Liu, Phys. Rev. Lett. 110, 020401 (2013). [51] X.-J. Liu, Phys. Rev. A 87, 013622 (2013). [52] C. Chen, arXiv:1306.5934. [53] C. Qu, Z. Zheng, M. Gong, Y. Xu, L. Mao, X. Zou, G.-C. Guo, and C. Zhang, arXiv:1307.1207. [54] W. Zhang and W. Yi, arXiv:1307.2439. [55] X.-J. Liu and H. Hu, Phys. Rev. A 88, 023622 (2013). [56] Rashba and Dresselhaus refer to two different types of SOC found in certain solids. The former occurs in crystals lacking inversion symmetry, while the latter arises in crystals lacking reflection symmetry. [57] C. A. R. Sá de Melo, M. Randeria, and J. R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993). [58] M. Randeria, in Bose-Einstein Condensation , edited by A. Griffin, D. W. Snoke, and S. Stringari, (Cambridge University35 Press, Cambridge, England, 1995), p. 355-392. [59] H. Hu, X.-J. Liu, and P. Drummond, Europhys. Lett. 74, 574 (2006). [60] R. B. Diener, R. Sensarma, and M. Randeria, Phys. Rev. A 77, 023626 (2008). [61] H. T. C. Stoof, K. B. Gubbels, and D. B.M. Dickerscheid, Ultracold Quantum Fields (Springer, 2009). [62] H. Hu, X.-J. Liu, and P. Drummond, New J. Phys. 12, 063038 (2010). [63] P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964). [64] Strictlyspeaking,wesetthechemicalpotentialtobetheminimumenergyinthetwo-particlecontinuum 2Emin,whereEmin is the single-particle ground state energy, such that the Fermi surface shrinks and vanishes. Thus, the Fermi distribution functions in the vertex function disappear. Note that, in the vertex function the energy !is measured with respect to the chemical potential. This procedure is exactly identical to what we have taken in the text by setting = 0and measuring the energy !from zero. [65] H. Hu, H. Pu, J. Zhang, S.-G. Peng, and X.-J. Liu, Phys. Rev. A 86, 053627 (2012). [66] S.-G. Peng, X.-J. Liu, H. Hu, and K. Jiang, Phys. Rev. A 86, 063610 (2012). [67] Q. J. Chen, J. Stajic, S. Tan, and K. Levin, Phys. Rep. 412, 1 (2005). [68] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker Denschlag, and R. Grimm, Science 305, 1128 (2004). [69] A. Schirotzek, Y. Shin, C. H. Schunck, and W. Ketterle, Phys. Rev. Lett. 101, 140403 (2008). [70] C. H. Schunck, Y. Shin, A. Schirotzek, and W. Ketterle, Nature (London) 454, 739 (2008). [71] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature (London) 454, 744 (2008). [72] Y. Zhang, W. Ong, I. Arakelyan, J. E. Thomas, Phys. Rev. Lett. 108, 235302 (2012). [73] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, P. Massignan, G. M. Bruun, F. Schreck and R. Grimm, Nature (London) 485, 615 (2012). [74] M. Koschorreck, D. Pertot, E. Vogt, B. Fröhlich, M. Feld and M. Köhl, Nature (London) 485, 619 (2012). [75] L. Jiang, L. O. Baksmaty, H. Hu, and H. Pu, Phys. Rev. A 83, 061604 (R) (2011). [76] X.-J. Liu, Phys. Rev. A 86, 033613 (2012). [77] C. Chin and P. S. Julienne, Phys. Rev. A 71, 012713 (2005). [78] M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Science 335, 563 (2012). [79] L. Radzihovsky and D. E. Sheehy, Rep. Prog. Phys. 73, 076501 (2010). [80] V. Barzykin and L. P. Gorkov, Phys. Rev. Lett. 89, 227002 (2002). [81] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). [82] E. Majorana, Nuovo Cimennto 14, 171 (1937). [83] F. Wilczek, Nature Phys. 5, 614 (2009). [84] C. Nayak, S. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). [85] G. Moore and N. Read, Nucl. Phys. B360, 362 (1991). [86] N. Read and D. Green, Phys. Rev. B 61, 10267 (2000). [87] T. Mizushima, M. Ichioka, and K. Machida, Phys. Rev. Lett. 101, 150409 (2008). [88] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010). [89] Y. Oreg. G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010). [90] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Science 336, 1003 (2012). [91] Another equivalent form of Rashba SOC, used extensively in the literature, is VSO=(^kx^x+^ky^y). These are related by a spin-rotation. [92] J. D. Sau, R. Sensarma, S. Powell, I. B. Speilman, and S. Das Sarma, Phys. Rev. B 83, 140510(R) (2011). [93] Z. F. Xu, and L. You, Phys. Rev. A 85, 043605 (2012). [94] B. M. Anderson, G. Juzeli¯ unas, V. M. Galitski, and I. B. Spielman, Phys. Rev. Lett. 108, 235301 (2012). [95] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012). [96] H. Hu and X.-J. Liu, Phys. Rev. A 85, 013619 (2012). [97] L. P. Gor’kov, and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001). [98] S. Hoinka, M. Lingham, M. Delehaye, and C. J. Vale, Phys. Rev. Lett. 109, 050403 (2012). [99] K. Zhou and Z. Zhang, Phys. Rev. Lett. 108, 025301 (2012). [100] L. He and X.-G. Huang, Phys. Rev. Lett. 108, 145302 (2012). [101] D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001). [102] C. Caroli, P. G. de Gennes, and J. Matricon, Phys. Lett. 9, 307 (1964). [103] D. Wei, D. Xiong, H. Chen, and J. Zhang, Chin. Phys. Lett. 24,679 (2007). [104] D. Xiong, H. Chen, P. Wang, X. Yu, F. Gao and J. Zhang, Chin. Phys. Lett. 25,843 (2008). [105] P. Wang, H. Chen, D. Xiong, X. Yu, F. Gao and J. Zhang, Acta. Phys. Sin. 57,4840 (2008) [106] D. Xiong, P. Wang, Z. Fu, S. Chai, and J. Zhang, Chin. Opt. Lett. 8,627 (2010). [107] D. Xiong, P. Wang, Z. Fu, and J. Zhang, Opt. Exppress. 18,1649 (2010). [108] R. Wei and E. J. Mueller, Phys. Rev. A 87, 042514 (2013). [109] I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960). [110] K. Jiménez-García, L. J. LeBlanc, R. A. Williams, M. C. Beeler, A. R. Perry, and I. B. Spielman, Phys. Rev. Lett. 108, 225303 (2012). [111] Z. F. Xu, L. You, and M. Ueda, Phys. Rev. A 87, 063634 (2013). [112] B. M. Anderson, I. B. Spielman, and G. Juzeli¯ unas, arXiv:1306.2606.

2112.01370v1.Thermal_Spin_Orbit_Torque_in_Spintronics.pdf

Thermal Spin -Orbit Torque in Spintronics Zheng -Chuan Wang The University of Chinese Academy of Sciences, P. O. Box 4588, Beijing 100049, China. Abstract Within the spinor Boltzmann equation (SBE) formalism , we derived a temperature dependent thermal spin -orbit torque based on local equilibrium assumption in a system with Rashba spin -orbit interaction. If we expand the distribution function of spinor Boltzmann equation around local equilibrium distribution, we can obtain the spin diffusion equation from SBE, then the spin transfer torque , spin orbit torque as well as thermal spin -orbit torque we seek to can be read out from this equation. It exhibits that this thermal spin-orbit torque originates from the temperature gradient of local equilibrium distribution function, which is explicit and straightforward than previous works. Finally, we illustrate them by an example of spin -polarized transport through a ferromagnet with Rashba spin -orbit coupling, in which those torques driven whatever by temperature gradient or bias are manifested quantitatively. PACS : 72.00, 65.90.+i, 85.70. -w, 85.80. -b I. Introduction In last decades, spintronics provide an alternative way to manipulate the local magnetic moment of ferromagnet in the mesoscopic device s by the magnetization switching resulting from spin transfer torque (STT)[1], in particular in the magnetoresistance random -access memory (MRAM) . However, the spin transfer torque can only be used to reverse the magnetization of noncollinear system , i.e., the Ferromagnet/ Insulator/Ferromagnet tunneling junction , where the two ferromagnetic layers have different directions of magnetization and need an additional polarizing layer. Recently, another spin torque --spin-orbit torque (SOT) had been proposed[2-6], which can be employed to switch the magnetization in a co llinear system with a broken inversion symmetry, i.e. , in the system of racetrack memory (RM). Since only a low critical current density is needed to drive the fast domain wall motion in RM, SOT had attracted more and more attentions[7-10]. Till now, a new branch of spintronics —spin-orbitronics had been established. Similar to the thermal spin transfer torque given by Hatami et al.[11], the SOT can also be driven by the temperature gradient instead of voltage bias, which is called thermal spin -orbit torque (TSOT) . The TSOT was originally presented by Freimuth et al. by Kubo’s linear response formalism , they expressed the even torque in terms of Berry phase[12-13]. Besides this TSOT initiated from the spin-polarized electronic current, the magnon -mediated TSOT was also predicted theoretically by Manchon et al. in 2014[14-15]. Nowadays, the interplay between the SOT and thermoelectric transport had been intensively explored whatever in theories or experiments[16-20]. However, Freimuth ’s Berry phase expression of TSOT need a knowledge of electronic structure of magnetic system from the first principle calculation , it is time-consuming in some complicated system, so in this manuscript, we try to give another explicit expression for TSOT by means of distribution function. The main focus of our work is to investigate the TSOT by the spinor Boltzmann equation (SBE) which was accomplished by Levy et al. in 2002[21], Sheng et al. ever proposed a similar equation at steady state in 1997[22]. At present, SBE had become a powerful tool to explore the spin -polarized transport in spintronics . In addition to calculate the magneto -resistance in the electronic transport, SBE can also be used to study the STT with respect to magnetization switching, Zhang et al. ever generaliz ed the STT to the non -adiabatic case by means of SBE[23]. SBE was also extended to the case beyond gradient approximation[24]. Subsequently, Wang et al. included the Rashba and dresslhaus spin -orbit coupling into the SBE[25], which is helpful for us to find the SOT from spin diffusion equation derived from SBE. In this manuscript, w e will show that a general ized SOT can be singled out from the spin diffusion equation, which contains not only the usual SOT , but also a term coming from the temperature gradient , it is just the TSOT we want . II. Theoretical Formalism In spintronics, t he SBE with Rashba spin -orbit coupling in a magneto -electric system under an external electric field 𝐸⃗ had been given by us in 201 9[25], (𝜕 𝜕𝑡+𝑝 𝑚⋅∇⃗⃗ −𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝)𝑓̂(𝑝,𝑥,𝑡)+𝑖𝐽 ℏ[𝑀⃗⃗ ∙𝜎 ,𝑓̂(𝑝,𝑥,𝑡)]+𝛼 2ℏ{(∇⃗⃗ ×𝑧 )⋅𝜎 ,𝑓̂(𝑝,𝑥,𝑡)} +𝑖𝛼 2ℏ2[(p⃗ ×𝑧 )⋅𝜎 ,𝑓̂(𝑝,𝑥,𝑡)]=−(𝜕𝑓̂ 𝜕𝑡) 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛, (1) where 𝑓̂(𝑝,𝑥,𝑡) is the spinor distribution function which is a 2× 2 matrix, 𝐽 is the exchange coupling constant, 𝑀⃗⃗ (x) is the unit vector of magnetization in the ferromagnet , 𝜎 denotes the Pauli matrix vector. 𝛼(∇⃗ ×𝑧 )⋅𝜎⃗ describes the Rashba spin-orbit interaction which is induced by the structure inversion asymmetry, 𝛼 is the coupling constant , 𝑧 is the direction of symmetry axis in crystal. (𝜕𝑓̂ 𝜕𝑡) 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 represents the collision term. Under the local equilibrium assumption, the spinor distribution function can be decomposed as[21] 𝑓̂(𝑝,𝑥,𝑡)=𝑓0(𝑝,𝑥)+(−𝜕𝑓0 𝜕𝜀)[𝑓(𝑝,𝑥,𝑡)+𝑔 (𝑝,𝑥,𝑡)∙𝜎 ], (2) where 𝑓0(𝑝,𝑥)=1 exp[𝜀−𝜇 𝑘𝑇(𝑥)]+1 is the local equilibrium distribution, 𝜇 is the chemical potential. The temperature 𝑇(𝑥) is position -dependent and its gradient will give rise to the thermal spin current in the magnetic system, 𝑘 is the Boltzmann constant. The scalar distribution function 𝑓(𝑝,𝑥,𝑡) and vector distribution fu nction 𝑔 (𝑝,𝑥,𝑡) in Eq.( 2) describe the deviation of spinor Boltzmann function from the local equilibrium. Substituting Eq.(2) into the SBE (1), under the relaxation approximation we can divide the SBE into the equation for the scalar distribution funct ion as (𝜕 𝜕𝑡+𝑝 𝑚⋅∇⃗⃗ −𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝)𝑓0(𝑝,𝑥)−(𝜕 𝜕𝑡+𝑝 𝑚⋅∇⃗⃗ −𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝)(𝜕𝑓0 𝜕𝜀𝑓)−𝛼 ℏ(∇⃗⃗ ×𝑧 )(𝜕𝑓0 𝜕𝜀⋅ 𝑔 (𝑝,𝑥,𝑡))=−𝜕𝑓0 𝜕𝜀𝑓 𝜏, (3) and the equation for the vector distribution function as −(𝜕 𝜕𝑡+𝑝 𝑚⋅∇⃗⃗ −𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝)(𝜕𝑓0 𝜕𝜀𝑔 (𝑝,𝑥,𝑡))+𝐽 ℏ𝜕𝑓0 𝜕𝜀𝑀⃗⃗ ×𝑔 (𝑝,𝑥,𝑡)+𝛼 ℏ(∇⃗⃗ ×𝑧 )𝑓0(𝑝,𝑥)− 𝛼 ℏ(∇⃗⃗ ×𝑧 )(𝜕𝑓0 𝜕𝜀𝑓)+𝛼 ℏ2(p⃗ ×𝑧 )×(𝜕𝑓0 𝜕𝜀𝑔 (𝑝,𝑥,𝑡))=−𝜕𝑓0 𝜕𝜀𝑔⃗ (𝑝,𝑥,𝑡)+<𝜕𝑓0 𝜕𝜀𝑔⃗ > 𝜏𝑠𝑓, (4) they are coupled together. The scalar and vector distribution function s obtained may enable us to calculate those physical observables in the spin -polarized transport . Usually, we define the charge density and charge current density by the scalar distribution function as 𝜌(𝑥,𝑡)=𝑒∫𝜕𝑓0 𝜕𝜀𝑓(𝑝,𝑥,𝑡)𝑑𝑝 and 𝑗 (𝑥,𝑡)=𝑒∫𝑣 𝜕𝑓0 𝜕𝜀𝑓(𝑝,𝑥,𝑡)𝑑𝑝, respectively, and define the spin accumulation and spin current density by the vector distribution as 𝑚⃗⃗ (𝑥,𝑡)=𝑒∫𝜕𝑓0 𝜕𝜀𝑔 (𝑝,𝑥,𝑡)𝑑𝑝 and 𝑗 𝑚(𝑥,𝑡)=𝑒∫𝑣 𝜕𝑓0 𝜕𝜀𝑔 (𝑝,𝑥,𝑡)𝑑𝑝, respectively , similarly the thermal current density can also be defined as 𝑗 𝐸(𝑥,𝑡)=∫𝜀𝑣 𝜕𝑓0 𝜕𝜀𝑓(𝑝,𝑥,𝑡)𝑑𝑝. So we can calculate these physical observables after we have solve d the scalar and vector distribution function s from Eq s. (3) and (4) . If we integrate the momentum 𝑝 over the Fermi surface on the both side s of Eq. (3), we have, 𝜕 𝜕𝑡𝜌(𝑥,𝑡)+∇⃗⃗ ∙𝑗 (𝑥,𝑡)=−∫(𝜕 𝜕𝑡+𝑝 𝑚⋅∇⃗⃗ −𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝)𝑓0(𝑝,𝑥)𝑑𝑝+𝛼 ℏ∫(∇⃗⃗ ×𝑧 )∙ (𝜕𝑓0 𝜕𝜀𝑔 (𝑝,𝑥,𝑡))𝑑𝑝, (5) which is the continuity equation for charge density and charge current defined above . Analogously, if we integrate the momentum 𝑝 on the both sides of Eq. (4), we have 𝜕 𝜕𝑡𝑚⃗⃗ (𝑥,𝑡)+∇⃗⃗ ∙𝑗 𝑚(𝑥,𝑡)=−𝐽(𝑝) ℏ(𝑀⃗⃗ (𝑥)×𝑚⃗⃗ (𝑥))−𝛼 ℏ∫(∇⃗⃗ ×𝑧 )𝑓0(𝑝,𝑥)𝑑𝑝+ 𝛼 ℏ(∇⃗⃗ ×𝑧 )∫(𝜕𝑓0 𝜕𝜀𝑓)𝑑𝑝−𝛼 ℏ2∫(p⃗ ×𝑧 )×(𝜕𝑓0 𝜕𝜀𝑔 (𝑝,𝑥,𝑡))𝑑𝑝, (6) where the spin current is a tensor. This is the continuity equation for the spin accumulation and spin current. Eq.( 6) is also called t he spin diffusion equation, which is critical for us to read out the STT from this equation[24-25]. After the spin -flip relaxation time 𝑡≫𝜏𝑠𝑓, the system will arrive at a steady state, where 𝜕 𝜕𝑡𝑚⃗⃗ (𝑥,𝑡)=0, then the spin diffusion equation (6) will reduce to 𝐽(𝑝) ℏ(𝑀⃗⃗ (𝑥)×𝑚⃗⃗ (𝑥))=−∇⃗⃗ ∙𝑗 𝑚(𝑥,𝑡)−𝛼 ℏ(∇⃗⃗ ×𝑧 )∫𝑓0(𝑝,𝑥)𝑑𝑝+𝛼 ℏ(∇⃗⃗ × 𝑧 )∫(𝜕𝑓0 𝜕𝜀𝑓)𝑑𝑝−𝛼 ℏ2∫(p⃗ ×𝑧 )×(𝜕𝑓0 𝜕𝜀𝑔 (𝑝,𝑥))𝑑𝑝, (7) where we have moved the spin torque 𝐽(𝑝) ℏ(𝑀⃗⃗ (𝑥)×𝑚⃗⃗ (𝑥))[24-25] to the left hand side and the divergence of spin current ∇⃗⃗ ∙𝑗 𝑚(𝑥,𝑡) to the right hand side of Eq.( 7). As a consequence of this rearrangement, we can see that the spin torque are contributed by several parts: the divergence of spin current which is concerned with the usual STT[23]; the temperature dependent term −𝛼 ℏ(∇⃗⃗ ×𝑧 )∫𝑓0(𝑝,𝑥)𝑑𝑝, we refer to this as the thermal SOT, because it is produced by the gradient of local equilibri um distribution function, namely the temperature gradient, so this term is just the thermal SOT we looking for, it is the central result in our manuscript. T he fourth and the fifth term s on the right hand of Eq.( 7) correspond to the usual SOT , we ever deri ved them in Ref.[25] . Our expression of TSOT is explicit, which is convenient for us to calculate it in some complex systems . In the next, we will evaluate these torques numerically. III. Numerical Results As an example, we consider a spin -polarized transport through a ferromagnet under an external electric field with a broken inversion symmetry , there exists Rashba spin -orbit coupling in this system . If we keep a temperature distribution in the system as 𝑇(𝑥)=𝑇0+𝑘𝑥, where 𝑇0 is a constant, 𝑘 is the temperature gradient, 𝑥 denote s the position, then this temperature gradient will induce a thermal spin current, which will produce the TSOT. In order to quantify th ose torques and currents defined in the above section , we should solve Eq.( 3) combining with ( 4) simultaneously , because they are the coupled differential equations about the scalar distribution function and vector distribution function. Since only the electrons near the Fermi surface mainly contribute to the transport, we can firstly approximate the 𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝(𝜕𝑓0 𝜕𝜀𝑓) in the lefe hand of Eq.( 3) as 𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝𝑓0[24,25 ] , and 𝑒𝐸⃗ ⋅∇⃗ 𝑝(𝜕𝑓0 𝜕𝜀𝑔⃗ (𝑝,𝑥)) in the left hand of Eq.(4) as 𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝𝑔 0(𝑝,𝑥), where 𝑔 0(𝑝,𝑥) is a approximated solution at steady state, 𝑔 0(𝑝,𝑥)=(exp[𝑖𝐽(𝑀𝑦−𝑀𝑧) ℏ𝑣𝑥],exp[𝑖𝐽(𝑀𝑧−𝑀𝑥) ℏ𝑣𝑥],exp[𝑖𝐽(𝑀𝑥−𝑀𝑦) ℏ𝑣𝑥]), which is analogous to the form given by Levy et al.[21], these approximations can be improved by the iteration procedure s in the latter step by step . These coupled equations may be solved by the method of Fourier transformation. After Fourier transform Eq s.(3) and ( 4) on both sides of them , we have 𝑖𝜔𝐹(𝑝,𝑘,𝜔)+𝑖𝑘⃗ ∙𝑝 𝑚𝐹(𝑝,𝑘,𝜔)−𝐹(𝑝,𝑘,𝜔) 𝜏+𝑖𝛼 ℏ(𝑘⃗ ×𝑧 )∙𝐺 (𝑝,𝑘,𝜔)=∫𝑒𝑖𝜔𝑡𝑒𝑖𝑘𝑥[− 𝑒𝐸𝜕 𝜕𝑝𝑓0(𝑝,𝑥)−(𝜕 𝜕𝑡+𝑝 𝑚⋅∇⃗⃗ −𝑒𝐸⃗ ⋅∇⃗⃗ 𝑝)𝑓0(𝑝,𝑥)+𝑓0(𝑝) 𝜏]𝑑𝑥𝑑𝑡， (8) and 𝑖𝜔𝐺 (𝑝,𝑘,𝜔)+𝑖𝑘⃗ ∙𝑝 𝑚𝐺 (𝑝,𝑘,𝜔)−𝐽 ℏ(𝑀⃗⃗ ×𝐺 (𝑝,𝑘,𝜔))−𝐺 (𝑝,𝑘,𝜔) 𝜏+𝑖𝛼 ℏ(𝑘⃗ × 𝑧 )𝐹(𝑝,𝑘,𝜔)−𝛼 ℏ2(𝑝 ×𝑧 )×𝐺 (𝑝,𝑘,𝜔)=∫𝑒𝑖𝜔𝑡𝑒𝑖𝑘𝑥[−𝑒𝐸𝜕 𝜕𝑝𝑔 0(𝑝,𝑥)+𝛼 ℏ(∇⃗⃗ × 𝑧 )𝑓0(𝑝,𝑥)+𝑔⃗ 0(𝑝,𝑥) 𝜏𝑠𝑓]𝑑𝑥𝑑𝑡， (9) where 𝐹(𝑝,𝑘,𝜔) and 𝐺 (𝑝,𝑘,𝜔) are the Fourier transformations of scalar distribution function 𝜕𝑓0 𝜕𝜀𝑓 and vector distribution function 𝜕𝑓0 𝜕𝜀𝑔⃗ (𝑝,𝑥), respectively. The advantage o f Fourier transformation lies in the fact that Eq s.(8) and ( 9) are the linear equations group about 𝐹(𝑝,𝑘,𝜔) and 𝐺 (𝑝,𝑘,𝜔), which can be easily performed by some algebra operations , then making the inverse Fourier transformation, we can obt ain the scalar distribution function and vector distribution function . After this , we further substitute these distribution functions obtained into the right hand of Eqs.(8) and ( 9) to improve the approximatio ns ∇⃗⃗ 𝑝𝑓0 and ∇⃗⃗ 𝑝𝑔 0(𝑝,𝑥) at first step , then solving Eqs.(8) and ( 9) again to get a new solutions, repeat this iteration procedure again and again until the iterated solutions converge. Finally we can evaluate the physical observables defined in the above by these converged distribution functions . In our calculation, we apply an external electric field 𝐸⃗ =36𝜇𝑉/𝑛𝑚 to the system[13], the length of ferromagnet is chosen as 20𝑛𝑚 and the coupling constant 𝐽(𝑝) for the magnetization of background ferromagnet as 1.2𝑒𝑣[24]. Meanwhile, we adopt the momentum relaxation time as 𝜏=0.01𝑝𝑠 and the spin -flip relaxation time as 𝜏𝑠𝑓=0.1𝑝𝑠[24]. The temperature distribution in the system is kept as 𝑇(𝑥)=𝑇0+𝑘𝑥, where 𝑇0=300𝐾, 𝑘=2𝐾/𝑛𝑚[13]，we adopt the spin-orbit coupling constant as 𝛼=1.0×10−11𝑒𝑉, which is a typical value in some system, i.e., the two -dimensional electron gas[25]. The average ⟨𝑓⟩ in Eq.(3) can be regarded as the local equilibrium distribution function 𝑓0(𝑝,𝑥), in this regard, ⟨𝑔 ⟩ in Eq.(4) can be approximated as 𝑔 0(𝑝,𝑥). In Fig.1, we plot the charge current density as a function of position x and y in the case of steady state. We can see that the charge current density has a slowing variation in the system , the difference of it’s maximum and minimum is small, this variatio n can be regarded as arising from the Rashba spin -orbit interaction in the system. The current is always positive because we apply a uniform electric field and temperature gradient to the system . In fact, t he variation of current with respect to position and time is governed by the continuity equation (5), here we only give out the current at stead y state. Except the charge current, there also exists thermal current. The thermal current density as a function of position is shown in Fig.1 The charge current density Fig.2 The thermal current density vs position, where vs position, where 𝑇0=300𝐾, 𝑇0=300𝐾,𝐸=36𝜇𝑉/𝑛𝑚. 𝐸=36𝜇𝑉/𝑛𝑚. Fig.2, it is similar to the charge current, which can be interpreted by its definition . The thermal current is produced not only by the temperature gradient but also by the electric field , which correspond to the first and second terms in the left hand side of Eq. (3), respectively . Here we only consider the thermal current carried by the conduction electrons , not by the phonons . In Fig.3, we draw the Fig.3 The x-component of Fig.4 The y -component of STT vs spin current density vs position, where position, where 𝐸=36𝜇𝑉/𝑛𝑚. 𝑇0=300𝐾,𝐸=36𝜇𝑉/𝑛𝑚. x-component of spin current density as a function of position x and y, it oscillate s rapidly . It’s y- and z - components have similar shapes with x -component, so we haven ’t show n them here. Since the spin Fig.5 The y-component of Fig.6 The y-component of TSOT vs SOT vs position, where position at different temperature 𝑇0=300𝐾,𝐸=36𝜇𝑉/𝑛𝑚. 𝑇0=100𝐾,200𝐾,300𝐾, where 𝐸=36𝜇𝑉/𝑛𝑚. current is a tensor, we only plot the component of spin current along x-axis. The position variation of spin current will give rise to the spin accumulation , both of them obey the spin diffusion equation (6). The spin accumulation will result in STT on the background ferromagnet. The numerical results for STT are illustrated in Fig.4, which is calculated by the expression 𝐽(𝑝) ℏ(𝑀⃗⃗ (𝑥)×𝑚⃗⃗ (𝑥))[24-25], so it is analogous to the spin accumulation. In a uniform magnetization, it is proportional to the spin accumulation. In order to address the contributions on the spin torque from the several terms in Eq.(7), we plot the usual SOT and thermal SOT in Fig.5 and Fig.6, respectively. It is obvious that the TSOT is smaller than the usual SOT, the SOT provide s a major contribution to the total spin torque . As we know the SOT is driven by the voltage bias, while the TSOT is driven by the temperature gradient , they have different origins. If we regard the SOT and the TSOT as a unified spin -orbit torque, then the unified SOT is mainly dominated by the usual SOT. For further studying the characteristic features of TSOT, we plot the TSOT at different temperatures 𝑇0=300𝐾,200𝐾,100𝐾 in Fig.6 , respectively. I t is shown that the TSOT has a relative big magnitude at low temperature 𝑇0=100𝐾, so the TSOT will play an important role at the low temperature, we can not ignore it in this case. It should be noted that our SOT and TSOT only correspond to the odd torque in Freimuth et al’s work[12-13], because it originates from the Rashba -like effect , but our expression is very explicit, it need not perform the first principle calculation, so it is helpful for us to more practical applications . IV. Summary and Discussions Under the local equilibrium assumption, we derive d a unified SOT in the framework of SBE, which consists of not only the usual SOT, but also the temperature dependent thermal SOT. The TSOT and the usual SOT have different origins, the former is induced by the temperature gradient, while the latter is induced by the voltage bias. The essence of our work lies in the fact that we express the TSOT by the local equilibrium distribution fu nction instead of the Berry phase given by Freimuth[12-13], which is helpful for us to calculate it conveniently . The numerical results indicate that the TSOT is smaller than SOT, but it will become larger at low temperature, we can not neglect it in this case. In addition to this TSOT, there exist other torques, i.e., the STT, we show it in Fig.4. The other physical observables such as charge current and thermal current are also demonstrated in the fig.1 and Fig2, respectively. It should be pointed out that we only chose a simple uniform magnetization in our calculation, in fact the magnetization usually varies with position in some systems, especially in the domain wall. However, we can not solve the SBE (3) and (4) by the meth od of Fourier transformation when the magnetization is position dependent, it will be troubled by the convolution problem, we should deal with it by other complicated methods , that is left for future exploration. Acknowledgments This study are supported by the National Key R&D Program of China (Grant No. 2018FYA0305804), and the Key Research Program of the Chinese Academy of Sciences (Grant No. XDPB08 -3). References: 1. G. E. W. Bauer, E. Saitoh and B. J. Van Wees, Nat . Mat., 11, 391 (2012). 2. A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). 3. A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009). 4. A. Matos -Abiague and R. L. Rodrıguez -Suarez, Phys. Rev. B 80, 094424 (2009). 5. K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423 (2013). 6. A. Brataas, A. D. Kent, and H. Ohno, Nat. Mat. 11, 372 (2012). 7. A. R. Mellnik et al., Nature 511, 449 (2014). 8. M. Cubukcu et al., App. Phys. Lett. 104, 042406 (2014). 9. L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D.C. Ralph and R. A. Buhman, Science 336, 5 55 (2012). 10. I. M. Miron et al., Nature 476, 189 (2011). 11. M. Hatami , G. E. W. Bauer, Q. Zhang and P. J. Kelly, Phys. Rev. Lett. 99, 066603 (2007). 12. F. Freimuth, S. Bl ügel and Y. Mokrousov, J. Phys: Condens. Matt. 26, 104202 (2014). 13. G. G éranton, F. Freimuth, S. Bl ügel and Y. Mokrousov, Phys. Rev. B 91,014417 (2015). 14. A. Manchon, P.B. Ndiaye, J. H. Moon, H. W. Lee and K. J. Lee, Phys. Rev. B 90,224403 (2014). 15. A. A. Kovalev , V. Zyuzin，Phys. Rev. B 93, 161106(2016). 16. K. S. Lee, S. W. Lee, B. C. Min and K. J. Lee, Appl. Phys. Lett.104, 072413 (2014) . 17. R. L. Conte et al., Appl . Phys . Lett. 105, 122404 (2014) . 18. C. O. Avci et al., Phys . Rev. B90, 224427 (2014 ). 19. T. H. Pham et al., Phys. Rev. Appl. 9, 064032 (2018). 20. F. Freimuth, S. Bl ügel and Y. Mokrousov, J. Phys: Condens. Matter, 28, 316001 (2016). 21. J. Zhang, P. M. Levy, S. Zhang and V. Antropov, Phys. Rev. Lett. 93, 256602 (2004). 22. L. Sheng, D. Y. Xing, Z. D. Wang, and J. Dong, Phys. Rev. B55, 5908(1997). 23. S. Zhang and Z. Li, Phys. Rev. Lett. 93, 27204 (2004). 24. Z. C. Wang, The Euro. Phys. Jour. B86, 206 (2013). 25. C. Yang, Z. C. Wang, Q. R. Zheng and G. Su, The Euro. Phys. Jour. B 92, 136 (2019).

1910.08669v2.Current_induced_spin_orbit_field_in_permalloy_interfaced_with_ultrathin_Ti_and_Cu.pdf

1 Current -induced s pin-orbit field in permalloy interfaced with ultrathin Ti and Cu Ryan W. Greening1, David A. Smith1, Youngmin Lim1, Zijian Jiang1, Jesse Barber1, Steven Dail1,2, Jean J. Heremans1, Satoru Emori1,* 1. Department of Physics, Virginia Tech, Blacksburg, VA 24061 2. Academy of Integrate d Science, Virginia Tech, Blacksburg, VA 24061 * email: semori@vt.edu How spin -orbit torques emerge from materials with weak spin -orbit coupling (e.g., light metals ) is an open question in spintronics. Here, w e report on a field -like spin -orbit torque (i.e., in-plane spin -orbit field transverse to the current axis) in SiO 2-sandwiched permalloy (Py), with the top Py-SiO 2 interface incorporating ultrathin Ti or Cu. In both SiO 2/Py/Ti/SiO 2 and SiO 2/Py/Cu/SiO 2, this spin -orbit field opposes the classical Oersted field. While the magnitude of the spin -orbit field is at least a fact or of 3 greater than the Oersted field, we do not observe evidence for a significant damping -like torque in SiO 2/Py/Ti/SiO 2 or SiO 2/Py/Cu/SiO 2. Our findings point to contributions from a Rashba -Edelstein effect or spin -orbit precession at the (Ti, Cu) -inserted interface. 2 An electric current in a material with spin -orbit coupling generally gives rise to a non -equilibrium spin accumulation [1–6], which can then exert torques – i.e., spin -orbit torques (SOTs) – on magnetization in an adjacent magnetic medium [7–9]. SOTs are often classified into two symmetries: damping -like SOT that either counte rs or enhances magnetic relaxation, and field-like SOT (or “spin -orbit field”) that acts similarly to a magnetic field . Next ge nerations of nanomagnetic computing devices may benefit from an improved understanding of mechanisms for SOTs and the discovery of new thin -film systems enabling large SOTs . While most efforts have focused on conductors known for strong spin -orbit coupling (e.g., 5d transition metals, topological insulators, etc.) [7,8] , recent reports have shown SOTs in ferromagnets interfaced with materials that are not expected to exhibit significant spin -orbit coupling [10–14]. For example, a large damping -like SOT has been reported in ferromagnetic Ni80Fe20 (permalloy, Py) interfaced with partially oxidized Cu [10,11] ; quantum -interference transport measurements have revealed that Cu with an oxidation gradient can, in fact , exhibit enhanced spin -orbit coupling comparable to that in heavier metals (e.g., Au) [15]. As another example of SOTs that emerge by incorporating seemingly wea k spin -orbit materials , Py interfaced with a Ti seed layer and Al 2O3 capping layer exhibits a sizable field -like SOT [12]. The key observed features of this spin -orbit field in Ti/Py/Al 2O3 [12] are: (1) it points in -plane and transverse to the current axis, irrespective of the magnetization orientation in Py ; (2) its magnitude scales inversely with the Py thickness , i.e., it is interfacial in origin; (3) it is modified significantly by the addition of an insertio n layer (e.g., Cu ) at the Py -Al2O3 interface. Ref. [12] claims that this spin-orbit field is governed by a Rashba - Edelstein effect (REE) [1,5,16,17] at the Py/Al 2O3 and Cu/Al 2O3 interfaces . However, the complicated stack structure s of SiO 2(substrate)/ Ti/Py/ (Cu/)Al2O3 with multiple dissimilar interfaces in Ref. [12] obscure the mechanisms of the spin -orbit field , particularly the roles played by the Ti and Cu layers . Here , by using simpler stack structures , we gain insight into the impact of ultrathin Ti and Cu interfacial insertion layers on the current -induced spin-orbit field in Py at room temperature . Specifically, we have characterized the total curre nt-induced transverse field HI,tot in SiO 2/Py/Ti/SiO 2 (Py/Ti) and SiO 2/Py/Cu/SiO 2 (Py/Cu) with the second -order planar Hall effect (PHE) [18,19] and spin-torque 3 ferromagnetic resonance (ST -FMR) [20]. From the observed HI,tot and estimated classical Oersted field HOe in each stack structure, we extract the spin -orbit field Hso via Hso = HI,tot – HOe. (1) We find that Py/Ti and Py/Cu exhibit Hso that opposes HOe with a similar magnitude, i.e., at least 3 times greater than HOe. While this field -like SOT is well above our detection limit , we observe no evidence for a significant damping -like SOT in Py/Ti or Py/Cu. We deduce that the Rashba field at the (Ti, Cu)-inserted interface plays a key role in the observed Hso. We patterned Py/Ti and Py/Cu , along with a control symmetric stack of SiO 2/Py/SiO 2 (sym -Py), by photolithography and liftoff into Hall crosses (for second -order PHE measurements ) and rectangular microstrips (for ST -FMR measurements ). The substrate was Si (001) covered with 50 -nm-thick thermally grown oxide. We used rf -sputtered SiO 2 as both the buffer and capping layers to preserve the structural symmetry of the sym-Py control stack. The metallic Py, Ti, and Cu layers were deposited by dc sputtering . The nominal deposited layer thicknesses were 3 nm for SiO 2, 3 nm for Py, and 0.5 nm for Ti and Cu. Static magnetic properties of the sym -Py, Py/Ti, and Py/Cu films are summarized in the Supplementary Material. The patterned Hall cr osses were 100 and 200 μ m wid e, with essentially identical results obtained for both device widths, where as the ST-FMR microstrips had widths of 50 μm. Both device types were contacted by thermally evaporated Cr (3 nm)/Au (100 nm) electrodes, patterned with an additional layer of photolithography and liftoff. By four-point measurements on double Hall crosses , we obtained the sheet resistance for each film stack structure: 320 Ohm/sq for sym -Py, 250 Ohm/sq for Py/Ti, and 200 Ohm/sq for Py/Cu. The smaller resistance values for Py/Ti and Py/Cu, compared to sym -Py, suggest that ultrathin Ti and Cu produce an additional conductive path. The conductance of the Py layer in Py/Ti and Py/Cu may also be higher than in sym-Py, due to the Ti and Cu insertion layer s protecting the top Py surfac e from oxidation. Both scenarios result in the top portion s of the Py/Ti and Py/Cu stacks contributing more to conductance than the bottom portion s with the direct SiO 2-Py interface s. We can therefore determine the direction of the Oersted field HOe acting on the magnetization in Py ; referring to Fig. 1(a) with the Py/Cu stack as an example , with a 4 conventional (positive) charge current along the + x direction, a higher current density in the top portion of the stack generates a net HOe along the + y direction within the Py layer . To quantify the distribution of in -plane current density, for simplicity, we treat the Ti (or Cu) and Py layers as parallel resistors and fix the resistance of Py to that found from sy m-Py. We estimate the fraction of the c urrent in Ti (Cu) to be fTi ≈ 20% ( fCu ≈ 40%). This approximation likely overestimates the current in Ti and Cu , since the Py layer in Py/Ti and Py/Cu may be more conductive than that in sym -Py. Nevertheles s, this approximation yields a useful upper bound of HOe in the stack structures via | HOe| = |Idc|f(Ti,Cu)/(2w), where Idc represents the total in -plane current through the device and w the device width. In addition to the sym -Py, Py/Ti, and Py/Cu stack s, we also used Hall cros ses and microstrips of Ta(3 )/Py(2.5)/Pt(4) from a prior study [21] as an additional control sample to validate our measurement s. In this sample , which we denote as P y/Pt, a majority of in-plane current flows through the top Pt layer (fPt ≈ 70%) ; the bottom Ta layer with high resistivity accommodates only ≈ 10% of the total current [21]. It has also been shown that the total current -induced field in Py/Pt lies along the direction of HOe [18,21] . To quantify the in -plane current -induced transverse field, we employed the second -order PHE technique (Fig. 1 ), originally developed by Fan et al. [18,19] . For Py thin films, the PHE signal from in - plane magnetization tilting dominates over any anomalous Hall effect (AHE) signal from out -of-plane tilting [18]. As such, the second -order PHE voltage VPH = VPH(+Idc)+VPH(−Idc), with VPH = V+−V− in Fig. 1(a), is related to the in-plane magnetization component transve rse to the current axis. T he second -order PHE is thus sensitive to small magnetization tilting induced by the total current -induced transverse field HI,tot, i.e., the sum of the Oersted field HOe and spin -orbit field Hso, as illustrated in Fig. 1(a) . 5 Figure 1. (a) Schematic of the second -order PHE measurement. Here, the total current -induced field HI,tot (dominated by a sizable spin -orbit field Hso) opposes the Oersted field HOe. Note that HI,tot = Hso + HOe. (b) Example second -order PHE curves for a 100 -μm-wide Py/Cu sample, obtained at | Idc| = 1 mA. We obtained HI,tot directly from the in -plane transverse calibration field Hy that nulls the second -order PHE voltage. Figure 1(b) shows exemplary second -order PHE results at a drive current of | Idc| = 1 mA in 100-μm-wide Py/Cu, measured with a probe station inside a two -axis Helmholtz coil setup. When a finite transverse calibration field Hy is applied, the second -order PHE voltage is expressed as VPH = VPH(+Idc, +Hy)+VPH(−Idc, −Hy) [18,19] . In Fig. 1(b), μ0|Hy| ≈ 6 μT along + y nulls the PHE voltage, which signifies that 1 mA in the + x-direction generates μ0|HI,tot| ≈ 6 μT in the –y direction. Our measurements near this nulled limit (e.g., μ0Hy = +6 μT in Fig. 1(b)) show that the seco nd-order Hall voltage converges to zero at large positive and negative swept fields Hx. This observation confirms the absence of any significant AHE [18] or thermoelectric contributions (e.g., spin Seebeck and anomalous Nernst effects) [22] that would produce a sizable difference in the saturate d Hall voltages at large positive and negative Hx. For results shown in the remainder of this Letter , we used transverse calibration fields μ0Hy = +100 μT and –100 μT and extrapolated HI,tot, as previously used in Refs. [12,14,19] and summarized in the Supplementary Material. We note that in Py/Cu , the observed HI,tot lies opposite to HOe (Fig. 1 ), suggesting the presence of a sizable spin-orbit field Hso (Eq. 1) as further discussed later in this Letter . The total current -induced transverse field HI,tot obtained with the second -order PHE technique is summarized in Fig. 2. In sym -Py, HI,tot is negligible as expected from the nominally symmetric current -10 -5 0 5 10-30-20-100102030 m0Hy=-6 mT m0Hy= 0 m0Hy=+6 mTVPH (mV) m0Hx (mT)|Idc| = 1 mASiO 2 Hy HxCu Py SiO 2 x y(a) (b)I- I+ V-V+ M HOeHsoHI,tot6 distribution. By contrast, HI,tot increases linearly with driving current | Idc| for Py/Ti, Py/Cu, and Py/Pt. One contribution to the observed HI,tot is the Oersted field HOe, which arises due to the higher current distribution in the top portion of the stack structure . However, as noted above and shown in Fig. 2(b,c), the direction of HOe is opposite to that of the observed HI,tot in Py/Ti and Py/Cu. We emphasize that t he calculated HOe (dashed line in Fig. 2) for each stack structure is the realistic upper bound : if the in -plane current is more uniformly distributed between the ultrathin metal and Py, then the magnitude of HOe is smaller. Figure 2. The total current -induced field HI,tot measured with the second -order PHE technique for (a) sym - Py, (b) Py/Ti, (c) Py/Cu, and (d) Py/Pt, plotted vs the dc current Idc normalized by the device width w = 100 μm. The dashed lines in (b -d) indicate the estimated Oersted field. Uncertainty of the m easured HI,tot is within the size of the dots. Evidently, the broken symmetry with an ultrathin layer of weak spin -orbit metal (i.e., Ti or Cu) gives rise to a spin-orbit field Hso (Eq. 1) , which opposes and is at least 3 times larger than HOe. While a similar Hso has been reported before [12], our present study directly shows that ultr athin insertion layers of Ti and Cu yield the same direction of Hso. This observation , in contrast to the opposite signs of the bulk spin -Hall effect in Ti and Cu [23], indicates that Hso here is unrelated to the filling of d-orbit als in Ti and Cu . The Py/Pt control sample validates our second -order PHE results. T he observed HI,tot in Py/Pt lies in the same direction as HOe (Fig. 2(d)) , consistent with prior reports [18,21] . Moreover, we confirm that the magnitude of Hso is approximately double that of HOe in Py/Pt, consistent with the dc -biased ST -FMR study on the same stack structure [21]. We remark that a prior experimental study [12] shows suppression of Hso 0 10 20 Idc/w (A/m) 0 10 20-20-1001020m0HI,tot (mT) Idc/w (A/m) 0 10 20 Idc/w(A/m) 0 10 20 Idc/w (A/m)(a) (b) (c) (d)SiO 2 SiO 2PyTiSiO 2 SiO 2PyCuSiO 2 SiO 2Py Pt Py Ta7 when Py is interfaced with 0.5 -nm-thick Pt; the origin of the significant Hso in Py with thicker Pt (or the absence of Hso with ultrathin Pt ) is unclear and will be the subject of a future investigation. To gain additional insight into the effects produced by in -plane current, we discuss ST-FMR results (Fig. 3) on Py/Ti, Py/Cu, and Py/Pt. While the dc -biased ST-FMR technique [12,20,21,24] enables straightforward quantitative analysis of the current -induced field (and damping -like SOT ), our ST -FMR setup did not yield a sufficient signal -to-noise ratio for reliable measurement of resonance field vs dc current. Nevertheless, the ST -FMR spectral shape can qualitatively reveal the types of SOTs present (or absent) in the stack structures [18,20] as discussed in the following . Figure 3. (a) Schematic of the ST -FMR measurement , driven by rf current Irf and detected via rectified dc voltage Vmix. (b-d) ST -FMR spectra at 5.5 GHz, +13 dBm microwave current excitation for (b) Py/Ti, (c) Py/Cu, and (d) Py/Pt. For each spectrum, the black solid curve indicates the antisymm etric component of the Lorentzian spectral fit, whereas the dashed curve indicates the symmetric component of the fit. Figure 3(b-d) shows example ST-FMR spectra for Py/Ti, Py/Cu, and Py /Pt, each fit with a combination of antisymmetric Lorentzian (solid black curve) and symmetric Lorentzian (dashed black curve). The antisymmetric component is related to the direction of the total current -induced field [20]. The observation that Py/Ti and Py/Cu both show a large antisymmetric component opposing that of Py/Pt confirms our second -order PHE results, i.e., there is a substantial Hso opposing HOe in Py/Ti and Py/Cu. We also observe that, while Py/Pt shows a large symmetric component , Py/Ti and Py/Cu exhibit a symmetric component about an order of magnitude smaller than the antisymmetric component. This suggests that the damping - like SOT, often related to a pronounced symmetric ST -FMR spectral component [10,11,20] , is negligibly 20 40 60 80 m0H (mT) 20 40 60 80-15-10-5051015Vmix (mV) m0H (mT)Vmix~ ref signallock-in Irf (a) (b) (c) (d) H 45oM τFLτDL SiO 2 SiO 2PyTiSiO 2 SiO 2PyCuPt Py Ta 20 40 60 80 m0H (mT)8 small in Py/Ti and Py/Cu compared to Py/Pt. Although i dentifying the origin of the small symmetric component in the ST -FMR spectra of Py/Ti and Py/Cu is beyond the scope of this Letter, it is not due to a damping -like SOT from partial oxidation of Cu, which would yield the same polarity of symmetric Lorentzian as Py/Pt [10]. We now discuss possible mechanisms responsible for the sizable spin -orbit field in Py/Ti and Py/Cu , as illustrated in Fig. 4. One candidate mechanism is the REE at metal -oxide interfaces (Fig. 4(a)) [12,17] . We first consider the top Py -(Ti, Cu)-SiO 2 interface ; we lump Py/(Cu,Ti) and (Cu,Ti)/SiO 2 into one interface , given that the (Ti, Cu) insertion layer is only 0.5 nm thick . For both Py/Ti and Py/Cu, t he spin - orbit field normalized by the estimated current density in Ti or Cu , J(Ti,Cu) = f(Ti,Cu)Idc/(wt) with t = 0.5 nm, is μ0Hso/J(Ti,Cu) ≈ 0.1 mT per 1011 A/m2 This implies essentially the same magnitude of the REE for ultrathin Ti and Cu sandwiched by Py and SiO 2. We can estimate the Rashba coefficient R from Hso/J(Ti,Cu) through R ≈ (μBMs/P)μ0Hso/J(Ti,Cu) [16,25] , where μB is the Bohr magneton, Ms ≈ 700 kA/m is the saturation magnetization of Py, and P ≈ 0.15 is t he current spin polarization (related to the strength of s-d exchange coupling [16]) in 3-nm-thick Py [26]. Our es timate of R ≈ 0.003 eV Å is an order of magnitude smaller than R from angle -resolved photoemission studies of crystalline Cu surfaces [27–29]. We remark that the interfaces of sputtered layers in our study are likely diffuse. The small ness of the estimated Rashba coefficient in our study may be due to the ill-defined interfaces of our stack structures , such that the Rashba - Edelstein field-like SOT may be enhanced with the use of highly crystalline ultrathin Ti or Cu. The bottom SiO 2-Py interface might also exhibit a REE, similar to the previous claim of a REE at Al2O3-Py [12]. However, considering that Ref. [12] shows a significant spin -orbit field even in Py sandwiched between Ti and Cu, i.e., without a direct oxide -Py interface, it appears u nlikely that the SiO 2- Py interface is the sole or dominant source. We therefore deduce that the REE at the Py -(Ti, Cu) -SiO 2 interface (Fig. 4(a)) dominates over that at the SiO 2-Py interface . 9 Figure 4. Possible mechanisms of the current -induced spin -orbit field Hso (which acts on the Py magnetization M) due to the interfacial Rashba field u. The red symbol (in (a)) and arrows (in (b)) represent the spin polarization s of the electron current je. (a) Rashba -Edelstein effect , where the electron current flowing parallel to the Py-Cu-SiO 2 interface becomes spin -polarized along u and exchange -couples to M. (b) Spin-orbit precession effect, where spin -polarized conduction electrons in Py precess about u during reflection from the Py -Cu-SiO 2 interface and then exert a torque (corresponding effective field Hso) on M. In the REE mechanism discussed above and illustrated in Fig. 4(a) , the electron current je in a quasi - two-dimensional conductor is spin-polarized by the interfacial Rashba field u ~ z × je, where z is normal to the interface; the spin -polarized electrons then generate an effective spin -orbit field Hso on the magnetization via s-d exchange coupling [16,17,25] . However, in our study with a 3 -nm-thick conductive ferromagnet , electronic transport is actually three -dimensional . In this regard, we consider an alternative mechanism [30,31] , which is illustrated in Fig. 4(b) and proceeds as follows: (1) Some c onduction electrons in Py are first spin-polarized along the magnetization M. (2) When these polarized electrons are reflected from the Py-(Ti,Cu )-SiO 2 interface with the Rashba field u, the spin polarization precesses (rotates) about u and develop s a finite component along u × M [30,31] . (3) The rotated spin polarization then dephase s in Py (i.e., ultimately aligning with M [32]) to exert a spin torque τ ~ M × Hso ~ M × [M × (u × M)], where M × (u × M) = u. Thus, t he measured spin-orbit field Hso in the Py layer points along u, irrespective of the magnetization direction. In other words , three -dimensional spin transport in Py – in concert with the interfacial Rashba field – may give rise to a magnetization -independent spin-orbit field in the ferromagnet (Fig. 4(b)) that is consistent with our experimental observations . (a) (b) SiO 2 PyCu Mjeu s Hso||u Hso||uMjeu s10 In summary, we have investigated the current -induced spin -orbit field (field -like SOT ) in SiO 2- sandwiched Py, with the top Py-SiO 2 interface incorporating an ultrathin layer of weak spin -orbit metal, Ti or Cu. In both SiO 2/Py/Ti/SiO 2 and SiO 2/Py/Cu/SiO 2, we observe a sizable spin-orbit field opposing the Oersted field , whereas no significant damping -like SOT is found. We deduce that this spin-orbit field arises from an interfacial Rashba -Edelstein effect or spin -orbit precession primarily at the Py -(Ti, Cu)-SiO 2 interface. Our findings provide further insight for engineering SOTs in ferromagnet s interfaced with weak spin-orbit materials. Acknowledgements This research was funded in part by 4 -VA, a collaborative partnership for advancing the Commonwealth of Virginia, as well as by the ICTAS Junior Faculty Award. D.A.S. acknowledges support by the Virginia Tech Graduate School Doctoral Assistantship . S.E. thanks Xin Fan and Vivek Amin for helpful discussions. References [1] V. M. Edelstein, Solid State Commun. 73, 233 (1990). [2] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). [3] R. Raimondi, C. Gorini, P. Schwab, and M. Dzierzawa, Phys. Rev. B 74, 35340 (2006). [4] A. Hoffman n, IEEE Trans. Magn. 49, 5172 (2013). [5] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, Nat. Mater. 14, 871 (2015). [6] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). [7] A. Manc hon, J. Železný, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Rev. Mod. Phys. 91, 35004 (2019). [8] R. Ramaswamy, J. M. Lee, K. Cai, and H. Yang, Appl. Phys. Rev. 5, 31107 (2018). [9] T. Wang, J. Q. Xiao, and X. Fan, SPIN 7, 1740013 (2017). [10] H. An, Y. Kageyama, Y. Kanno, N. Enishi, and K. Ando, Nat. Commun. 7, 13069 (2016). 11 [11] T. Gao, A. Qaiumzadeh, H. An, A. Musha, Y. Kageyama, J. Shi, and K. Ando, Phys. Rev. Lett. 121, 17202 (2018). [12] S. Emori, T. Nan, A. M. Belkessam, X. Wang, A. D. Matyushov, C. J. Babroski, Y. Gao, H. Lin, and N. X. Sun, Phys. Rev. B 93, 180402 (2016). [13] W. Wang, T. Wang, V. P. Amin, Y. Wang, A. Radhakrishnan, A. Davidson, S. R. Allen, T. J. Silva, H. Ohldag, D. Balzar, B. L. Zink, P. M . Haney, J. Q. Xiao, D. G. Cahill, V. O. Lorenz, and X. Fan, Nat. Nanotechnol. 1 (2019). [14] Z. Luo, Q. Zhang, Y. Xu, Y. Yang, X. Zhang, and Y. Wu, Phys. Rev. Appl. 11, 64021 (2019). [15] R. Enoki, H. Gamou, M. Kohda, and J. Nitta, Appl. Phys. Express 11, 33001 (2018). [16] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). [17] P. Gambardella and I. M. Miron, Philos. Trans. A. Math. Phys. Eng. Sci. 369, 3175 (2011). [18] X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, and J. Q. Xiao, Nat. Commun. 4, 1799 (2013). [19] X. Fan, H. Celik, J. Wu, C. Ni, K. -J. Lee, V. O. Lorenz, and J. Q. Xiao, Nat. Commun. 5, 3042 (2014). [20] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 36601 (2011). [21] T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm, J. G. Jones, B. M. Howe, G. J. Brown, and N. X. Sun, Phys. Rev. B 91, 214416 (2015). [22] C. O. Avci, K. Garello, M. Gabureac, A. Ghosh, A. Fuhrer, S. F. Alvarado, and P. Gambardella, Phys. Rev. B 90, 224427 (2014). [23] C. Du, H. Wang, F. Yang, and P. C. Hammel, Phys. Rev. B 90, 140407 (2014). [24] C. Kim, D. Kim, B. S. Chun, K. -W. Moon, and C. Hwang, Phys. Rev. Appl. 9, 54035 (2018). [25] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pi zzini, J. Vogel, and P. Gambardella, Nat. Mater. 9, 230 (2010). [26] M. Haidar and M. Bailleul, Phys. Rev. B 88, 54417 (2013). [27] A. Tamai, W. Meevasana, P. D. C. King, C. W. Nicholson, A. de la Torre, E. Rozbicki, and F. Baumberger, Phys. Rev. B 87, 751 13 (2013). [28] J. H. Dil, F. Meier, and J. Osterwalder, J. Electron Spectros. Relat. Phenomena (2014). [29] J. Jiang, S. S. Tsirkin, K. Shimada, H. Iwasawa, M. Arita, H. Anzai, H. Namatame, M. Taniguchi, I. Y. Sklyadneva, R. Heid, K. -P. Bohnen, P. M. Eche nique, and E. V. Chulkov, Phys. Rev. B 89, 85404 (2014). 12 [30] V. P. Amin, J. Zemen, and M. D. Stiles, Phys. Rev. Lett. 121, 136805 (2018). [31] A. M. Humphries, T. Wang, E. R. J. Edwards, S. R. Allen, J. M. Shaw, H. T. Nembach, J. Q. Xiao, T. J. Silva, and X. Fan, Nat. Commun. 8, 911 (2017). [32] V. P. Amin and M. D. Stiles, Phys. Rev. B 94, 104420 (2016). 13 Supplementa ry Material : Current -induced spin -orbit field in permalloy interfaced with ultrathin Ti and Cu Ryan W. Greening1, David A. Smith1, Youngmin Lim1, Zijian Jiang1, Jesse Barber1, Steven Dail1,2, Jean J. Heremans1, Satoru Emori1 1. Department of Physics, Virginia Tech, Blacksburg, VA 24061 2. Academy of Integrated Science, Virginia Tech, Blacksburg, VA 24061 I. Static Magnetic Properties We characterized the static magnetic properties of the stack structures by performing vibrating sample magneto metry (Microsense EZ9) on mm -scale squares from the same wafers as the patterned Hall crosses and ST -FMR microstrips. The saturation magnetization of ≈600 kA/m for sym -Py is slightly lower than ≈700 kA/m for Py/Ti and Py/Cu, possibly because of partial oxi dation of the top surface of Py in direct contact with SiO 2. We also find that the coercivity of sym -Py (≈0.8 mT) exceeds that of Py/Ti (≈0.5 mT) and Py/Cu (≈0.3 mT). The enhanced coercivity for sym -Py is consistent with the presence of an ultrathin antiferromagnetic oxide layer at the Py -SiO 2 interface (e.g., NiO) that is exchange -coupled to Py [S1]. Figure S1. Vibrating sample magnetometry with H applied in the film plane. -600-3000300600M (kA/m) -600-3000300600M (kA/m) -40 -20 0 20 40-600-3000300600M (kA/m) m0H (mT)(a) (b)SiO 2 SiO 2Py SiO 2 SiO 2PyTi -2 0 2 -2 0 2 -2 0 2 (c) SiO 2 SiO 2PyCu14 II. Extraction of the Total Current -Induced Trans verse Field HI,tot: Extrapolation Method Here, we summarize the extrapolation method (similar to that used in Refs. [S2–S4]) used to extract the total current -induced transverse field HI,tot. For a uniform magnetization with a small deviation from the current axis ( x-axis in Figure 1 of the main text), the se cond -order planar Hall effect (PHE) voltage VPH is proportional to the y-component of the magnetization my, and hence the sum of HI,tot and transverse calibration field Hy [S4,S5], ∆𝑉𝑃𝐻∝∆𝑚𝑦∝𝐻𝐼,𝑡𝑜𝑡+𝐻𝑦. It follows that VPH at Hy = 0 is expressed as ∆𝑉𝑃𝐻(𝐻𝑦=0)∝𝐻𝐼,𝑡𝑜𝑡. The difference of VPH at a fixed μ0|Hy| = 100 μT, which we call Vfit, is ∆𝑉𝑓𝑖𝑡(𝜇0|𝐻𝑦|=100 μT)=∆𝑉𝑃𝐻(𝜇0𝐻𝑦=+100 μT)−∆𝑉𝑃𝐻(𝜇0𝐻𝑦=−100 μT)∝2𝐻𝑦. Thus, by plotting VPH versus Vfit, we can quantify HI,tot from ∆𝑉𝑃𝐻 ∆𝑉𝑓𝑖𝑡=𝜇0𝐻𝐼,𝑡𝑜𝑡 200 μT. Figure S2. Example linear fitting in the PHE extrapolation method. Here, μ0HI,tot = –13 μT at Idc/w = 22.5 A/m. Figure S 2 shows an example of the extrapolation method . For the above -described linear fitting in our study, we only use data obtained at sufficiently large external magnetic fields (i.e., μ 0|Hx| > 2.5 mT) to ensure that the magnetization is uniform with a small -angle deviation from the x-axis. -800 -400 0 400 800-60-40-200204060VPH (mV) Vfit (mV)SiO 2 SiO 2PyCu w = 200 μm Idc= 4.5 mA15 In this extrapolation method , any anomalous Hall effect (AHE) or thermoelectric contributions in the second -order Hall voltage would appear as a significant vertical offset (discontinuity) between the separate linear fits for Vfit > 0 and Vfit < 0. Such an offset was observed in a previous study by Fan et al. (e.g., Figure 2(c) of Ref. [S4]). By contrast, no significant offsets are observed for samples in our study (e.g., Figure S2) . We therefore conc lude that AHE and thermoelectric contributions to the second -order Hall voltage are negligible in our study. S1. B. L. Zink, M. Manno, L. O’Brien, J. Lotze, M. Weiler, D. Bassett, S. J. Mason, S. T. B. Goenn enwein, M. Johnson, and C. Leighton, "Efficient spin transport through native oxides of nickel and permalloy with platinum and gold overlayers," Phys. Rev. B 93, 184401 (2016). S2. S. Emori, T. Nan, A. M. Belkessam, X. Wang, A. D. Matyushov, C. J. Babrosk i, Y. Gao, H. Lin, and N. X. Sun, "Interfacial spin -orbit torque without bulk spin -orbit coupling," Phys. Rev. B 93, 180402 (2016). S3. Z. Luo, Q. Zhang, Y. Xu, Y. Yang, X. Zhang, and Y. Wu, "Spin -Orbit Torque in a Single Ferromagnetic Layer Induced by Su rface Spin Rotation," Phys. Rev. Appl. 11, 64021 (2019). S4. X. Fan, H. Celik, J. Wu, C. Ni, K. -J. Lee, V. O. Lorenz, and J. Q. Xiao, "Quantifying interface and bulk contributions to spin -orbit torque in magnetic bilayers.," Nat. Commun. 5, 3042 (2014). S5. X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, and J. Q. Xiao, "Observation of the nonlocal spin - orbital effective field.," Nat. Commun. 4, 1799 (2013).

1101.2055v1.Multipole_correlations_of__t___rm_2g___orbital_Hubbard_model_with_spin_orbit_coupling.pdf

arXiv:1101.2055v1 [cond-mat.str-el] 11 Jan 2011Proc. Int. Conf. Heavy Electrons (ICHE2010) J.Phys. Soc. Jpn. 80(2011) SAonishi-iche2010 c/circleco√yrt2011 ThePhysical Society of Japan Multipolecorrelations of t2g-orbital Hubbard model withspin-orbit coupling HiroakiOnishi∗ Advanced Science Research Center,Japan AtomicEnergy Agen cy, Tokai, Ibaraki 319-1195, JAPAN We investigate the ground-state properties of a one-dimens ionalt2g-orbital Hubbard model including an atomic spin-orbit coupling by using numerical methods, s uch as Lanczos diagonalization and density- matrixrenormalization group. Asthe spin-orbitcoupling i ncreases, wefindaground-state transitionfrom a paramegnetic state to a ferromagnetic state. In the ferrom agnetic state, since the spin-orbit coupling mixes spin and orbital states with complex number coefficien ts, an antiferro-orbital state with complex orbitals appears. According tothe appearance ofthe comple x orbitalstate, weobserve anenhancement of Γ4uoctupole correlations. KEYWORDS: t2gorbitals,spin-orbitcoupling,multipole,density-matri xrenormalizationgroup The competition and cooperationbetween spin and orbital degrees of freedom in strongly correlated electron systems manifest itself in the emergence of various types of spin- orbital ordered and quantum liquid phases.1–3)In general, among competinginteractionsinvolvingspin and orbital, t he spin-orbit coupling is supposed to be weak in 3dtransition- metal oxides such as cupurates and manganites, while as we moveto4dand5delectrons,thespin-orbitcouplingbecomes strong and responsible for magnetic, transport, and optica l properties. When the spin-orbit coupling is dominant, spin and orbital are not independent, but instead the total angul ar momentum gives a good description of the many-bodystate. Infact,ithasbeensuggestedthatSr 2IrO4,inwhichIr4+ions have five electrons in triply degenerate t2gorbitals, exhibits a novel Mott-insulating state with an effective total angul ar momentum Jeff=1/2due to a strong spin-orbit coupling.4–7) In the limit of strong spin-orbit coupling, the ground-stat e Kramersdoubletata localioncanbedescribedbyanisospin withJeff=1/2.8,9)The exchange interaction among isospins can lead to a variety of ordering and fluctuation phenomena ofspin-orbitalentangledstates. When we move to heavy-element f-electron systems such asrare-earthandactinidecompounds,thespin-orbitcoupl ing is large comparing with other energy scales. In such a case, we usually classify the complicated spin-orbital state fro m the viewpoint of multipole, which is described by the total angular momentum. Indeed, the multipole physics has been actively discussed in the field of heavy electrons.10)A recent trendis to unveilexotic high-ordermultipoleordering.As an attempt to clarify multipole properties of f-electron systems from a microscopic viewpoint, we have numerically studied multipole correlations of an f-orbital Hubbard model on the basis of the j-jcouplingscheme.11)We believethat it is also importanttoclarifymultipolepropertiesin d-electronsystems undertheeffectofthe spin-orbitcoupling. In this paper, we investigate multipole properties in the groundstateofaone-dimensional t2g-orbitalHubbardmodel includingthespin-orbitcouplingbynumericalmethods.Wi th increasing the spin-orbit coupling, the ground state chang es fromaparamagneticstatetoaferromagneticstateintermso f ∗E-mail address: onishi.hiroaki@jaea.go.jpthe magnitude of the total spin. In the ferromagnetic phase, antiferro-dipolecorrelationsdevelopevenwhenthespins tate is ferromagneticdue to the orbital contribution. On the oth er hand,the spin-orbitcouplinginducesa complexorbitalsta te, inwhichreal xy,yz,andzxorbitalsaremixedwith complex number coefficients. According to the complex orbital state , Γ4uoctupolecorrelationsareenhanced. Let us consider triply degenerate t2gorbitals on a one- dimensional chain along the xdirection with five electrons persite.Theone-dimensional t2g-orbitalHubbardmodelwith thespin-orbitcouplingisdescribedby H=/summationdisplay i,τ,τ′,σtττ′(d† iτσdi+1τ′σ+h.c.)+λ/summationdisplay iLi·Si +U/summationdisplay i,τρiτ↑ρiτ↓+(U′/2)/summationdisplay i,σ,σ′,τ/negationslash=τ′ρiτσρiτ′σ′ +(J/2)/summationdisplay i,σ,σ′,τ/negationslash=τ′d† iτσd† iτ′σ′diτσ′diτ′σ +(J′/2)/summationdisplay i,σ/negationslash=σ′,τ/negationslash=τ′d† iτσd† iτσ′diτ′σ′diτ′σ,(1) wherediτσ(d† iτσ) is an annihilation (creation) operator for an electron with spin σ(=↑,↓) in orbital τ(=xy,yz,zx ) at sitei,andρiτσ=d† iτσdiτσ.Thehoppingamplitudeisgivenby txy,xy=tzx,zx=tand zero for other combinations of orbitals. Hereafter, tis taken as the energy unit. LiandSirepresent orbital and spin angular momentum operators, respectively , andλisthespin-orbitcoupling. U,U′,J,andJ′denoteintra- orbital Coulomb, inter-orbital Coulomb, exchange, and pai r- hopping interactions, respectively. We assume U=U′+J+J′ due to the rotation symmetry in the local orbital space and J′=Jdue to the reality of the orbital function.12)Throughout thispaper,we set /planckover2pi1=kB=1. We investigatetheground-statepropertiesofthemodel(1) by exploiting a finite-system density-matrix renormalizat ion group (DMRG) method with open boundary conditions.13) The number of states kept for each block is up to m=120, and the truncation error is estimated to be 10−4∼10−5. We remarkthatduetothethreeorbitalsinonesite,thenumbero f bases for the single site is 64, and the size of the superblock SAonishi-iche2010-1J.Phys. Soc. Jpn. 80(2011) SAonishi-iche2010 SAonishi-iche2010-2 00.20.40.60.81 0 2 4 6 8 10 λ JU=U'+J+J', J=J' U'=10 (a) ferromagnetic paramagnetic 01234 0 0.2 0.4 0.6 0.8 1 λU'=10, J=2 S tot 2(b) 012 0 0.2 0.4 0.6 0.8 1L loc S loc (c) λU'=10, J=2 23456 0 0.2 0.4 0.6 0.8 1J loc 2(d) λU'=10, J=2 11.52 0 0.2 0.4 0.6 0.8 1xy yz zx n τ(e) λU'=10, J=2 Fig. 1. Lanczosresultsforthefour-siteperiodicchain.(a )Theground-state phasediagraminthe( J,λ)planefor U′=10.(b)Themagnitudeofthetotal spin in the whole system. (c) The correlation between spin an d orbital in the single site. (d) The magnitude of the total angular momen tum in the single site. (e) Thecharge density in each orbital. Hilbert space grows as m2×642. To reduce the size of the Hilbert space, we usuallydecomposethe Hilbert space into a block-diagonalformbyusingsymmetriesoftheHamiltonian . In the present case, however, the spin-orbit coupling break s the spin SU(2) symmetry, so that we cannot utilize Sz totas a goodquantumnumber,where Sz totis thezcomponentof the total spin. Since we ignore egorbitals among dorbitals, the totalangularmomentumisnotaconservedquantity.Thetota l number of electrons can be used as a good quantum number. Thus, since DMRG calculations consume much CPU times, we supplementallyuse a Lanczosdiagonalizationmethodfor theanalysisofafour-siteperiodicchaintoaccumulateres ults withrelativelyshortCPU times. LetusfirstlookatLanczosresultsforthefour-siteperiodi c chain. In Fig. 1(a), we show the phase diagram in the ( J,λ) plane for U′=10. The phase boundary is determined by the magnitude of the total spin S2 tot. As shown in Fig. 1(b), S2 tot is almost zero for small λ, indicating a spin-singlet ground state. As λincreases, we find a transition to a ferromagnetic state with finite S2 tot. Note that even in the limit of large λ, S2 totdoes not approach the maximum value 2(2+1)=6, since the spin-orbit coupling mixes spin up and down states and thecompleteferromagneticstateisdisturbed.InFig.1(c) ,we plot the correlation between spin and orbital in the local si te Lloc·Sloc. Atλ=0, there is no correlation between spin and orbital. As λincreases, the spin-orbital correlation develops and approaches one in the limit of large λ, indicating totally parallel spin and orbital angular momenta. In Fig. 1(d), theΓγmultipole multipole operator Γ4udipole Jx,Jy,Jz Γ3gquadrupole Ou=(1/2)(2J2 z−J2 x−J2 y) Ov=(√ 3/2)(J2 x−J2 y) Γ5gquadrupole Oyz=(√ 3/2)JyJz Ozx=(√ 3/2)JzJx Oxy=(√ 3/2)JxJy Γ2uoctupole Txyz=(√ 15/6)JxJyJz Γ4uoctupole Tα x=(1/2)(2J3 x−JxJ2y−J2zJx) Tα y=(1/2)(2J3 y−JyJ2z−J2xJy) Tα z=(1/2)(2J3 z−JzJ2x−J2yJz) Γ5uoctupole Tβ x=(√ 15/6)(JxJ2y−J2zJx) Tβ y=(√ 15/6)(JyJ2z−J2xJy) Tβ z=(√ 15/6)(JzJ2x−J2yJz) Table I. Definition ofmultipole operators uptorank 3.Theov erline onthe product denotes the operation of taking all possible permut ations in terms of cartesian components, e.g., JxJy=JxJy+JyJx. magnitude of the total angular momentum in the single site J2 locis shown. At the transition point, J2 locexhibits a sudden increase, since the spin-orbit coupling stabilizes a large total angular momentum state at every local sites. Note again that J2 locdoes not reach the maximum value5 2(5 2+ 1)=35 4in the limit of large λ, since the total angular momentum is not a conserved quantity. Regarding the orbital state, we show th e charge density in each orbital in Fig. 1(e). Due to the spatia l anisotropy of orbitals, one hole is preferably accommodate d in itinerant xyorzxorbitals in each site, while localized yz orbitals are doubly occupied. Measuring charge correlatio ns, we findthat holesoccupyreal xyorzxorbitalalternatelyfor smallλ(notshown).Namely,thegroundstateisa realorbital state. Forlarge λ, however, xy,yz,andzxorbitalsaremixed with complexnumbercoefficientsbythe spin-orbitcoupling , leadingto a complexorbitalstate. Now we move on to the analysis of multipole properties to clarify the ground-state properties from the viewpoint o f multipole.We measuremultipolecorrelationfunctions χΓγ(q) =/summationdisplay j,k/angbracketleftXjΓγXkΓγ/angbracketrighteiq(j−k)/N, (2) whereXiΓγis a multipole operator with the symbol Xof multipole for the irreducible representation Γγin the cubic symmetry at site i. Here, we consider 15 types of multipoles includingthree dipoles( X=J),five quadrupoles( X=O),and sevenoctupoles( X=T),aslistedinTableI.14)Weevaluatethe multipole correlation functions by DMRG calculations with chainsof16sites. Figure 2 showsDMRG results of the multipolecorrelation functionsat U′=10,J=2,andλ=0fortheparamagneticphase. Regarding dipoles, as shown in Fig. 2(a), the Jxcorrelation has a peak at q=π, which signals an antiferromagnetic state. Here,wenoticethateachofthedipolecorrelationsexhibit sa kink atq=π/2, while the kink corresponds to a peak for the JyandJzcorrelations. This kink structure originates in the spin-orbitalSU(4)symmetrywhichrealizesataspecialpoi nt J=λ=0.15–19)At the SU(4) symmetric point, correlations of spinSiand orbital pseudospin Ti=1 2/summationtext τ,τ′,σd† iτσσττ′diτ′σ, whereσarePaulimatrices,coincidewitheachotherandhave a peak at q=π/2. With increasing J, the spin correlation of q=π/2grows and the peak of the spin correlation remains atJ.Phys. Soc. Jpn. 80(2011) SAonishi-iche2010 SAonishi-iche2010-3 00.10.20.30.4 0 0.2 0.4 0.6 0.8 1J xJ yJ zχ q / πU'=10, J=2, λ=0 (a) 012 0 0.2 0.4 0.6 0.8 1O uO vO yz O zx O xy χ q / πU'=10, J=2, λ=0 (b) 010 20 30 0 0.2 0.4 0.6 0.8 1T x α T y α T z αχ q / πU'=10, J=2, λ=0 (c) 01234 0 0.2 0.4 0.6 0.8 1T x β T y β T z β T xyzχ q / πU'=10, J=2, λ=0 (d) Fig. 2. DMRG results of multipole correlations at U′=10,J=2, andλ=0: (a)Γ4udipoles; (b) Γ3gandΓ5gquadrupoles; (c) Γ4uoctupoles; and (d) Γ5uandΓ2uoctupoles. q=π/2. On the other hand,the pseudospincorrelationof q=π is enhanced,and the peak positionof the pseudospincorrela - tionchangesto q=π.Notethatfortheorbitalangularmomen- tumLi, the correlationof the q=πcomponentis enhancedas well.Thus,theorbitalcontributiontodipoleleadstothep eak oftheJxcorrelationat q=πratherthan q=π/2. In Fig. 2(b), we find that for the OuandOvcorrelations, a sharp peak appears at q=0, since/angbracketleftOu/angbracketrightand/angbracketleftOv/angbracketrightare turned out to be finite. We also find a peak at q=πfor theOu,Ov, andOyzcorrelations, implying an antiferro-orbitalstate. For all quadrupoles, there occurs a kink at q=π/2in similar to the case of dipoles, which is a trace of the SU(4) symmetry atJ=λ=0. As shown in Fig. 2(c) and 2(d), we also observe a kinkatq=π/2foroctupoles. In Fig. 3, we present the multipole correlation functions atU′=10,J=2, andλ=0.5for the ferromagnetic phase. At a glance, we find that the kink structure at q=π/2disappears for all multipoles. Concerningdipoles,as shown in Fig. 3(a ), a peak appears at q=π. Namely, antiferro-dipole correlations become dominant even when the spin state is ferromagnetic dueto the orbitalcontribution.Infact, the spin Sicorrelation has a peak at q=0, while the orbital Licorrelation exhibits a peak atq=π(not shown). On the other hand, the quadrupole correlations are found to be almost flat, and we cannot see anyfinestructuressignalingquadrupoleordering,asshown in Fig.3(b).Asforoctupoles,weobserveasignificantenhance - ment of the Γ4uoctupole correlations of the q=πcomponent [seeFigs.2(c)and3(c)].Thegrowthoftheantiferro-octup ole correlations reflects the stabilization of the antiferro-o rbital state withcomplexorbitals. Insummary,wehavestudiedtheground-stateproperitesof thet2g-orbital Hubbard model with the spin-orbit coupling from the viewpoint of multipole, by numerical techniques. The strong spin-orbit coupling induces a transition from th e antiferromagnetic state to the ferromagnetic state. We hav e found that antiferro-dipole correlations develop even whe n00.10.20.30.4 0 0.2 0.4 0.6 0.8 1J xJ yJ zχ q / πU'=10, J=2, λ=0.5(a) 012 0 0.2 0.4 0.6 0.8 1O uO vO yz O zx O xy χ q / πU'=10, J=2, λ=0.5 (b) 010 20 30 0 0.2 0.4 0.6 0.8 1T x α T y α T z αχ q / πU'=10, J=2, λ=0.5(c) 01234 0 0.2 0.4 0.6 0.8 1T x β T y β T z β T xyzχ q / πU'=10, J=2, λ=0.5 (d) Fig. 3. DMRGresultsofmultipolecorrelations at U′=10,J=2,andλ=0.5: (a)Γ4udipoles; (b) Γ3gandΓ5gquadrupoles; (c) Γ4uoctupoles; and (d) Γ5uandΓ2uoctupoles. thespinstateisferromagnetic.Moreover,thecomplexorbi tal state appears, since the spin-orbit coupling yields the lin ear combinationsof spin andorbitalstates with complexnumber coefficients. Accordingly,we observe an enhancement of the Γ4uoctupolecorrelations.It is an interestingissue to explor e possiblemultipoleorderingin 5d-electronIrcompoundswith strongspin-orbitcoupling. TheauthorthanksG.Khaliullin,S.Maekawa,andM.Mori forusefuldiscussions. ThisworkwassupportedbyGrant-in - AidforScientificResearchofMinistryofEducation,Cultur e, Sports,Science,andTechnolotyofJapan. 1)Proc. Int. Conf. Strongly Correlated Electrons with Orbita l Degrees of Freedom (ORBITAL2001) ,J.Phys.Soc. Jpn. 71(2002) Suppl. 2) Y. Tokuraand N.Nagaosa: Science 288(2000) 462. 3) T.Hotta: Rep. Prog. Phys. 69(2006) 2061. 4) B.J.Kim,H.Jin, S.J.Moon, J.-Y.Kim,B.-G.Park, C.S.Lee m,J.Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, an d E. Rotenberg: Phys.Rev. Lett. 101(2008) 076402. 5) B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Taka gi, and T.Arima: Science 323(2009) 1329. 6) S.Chikara, O.Korneta,W.P.Crummett,L.E.DeLong,P.Sch lottmann, and G.Cao: Phys.Rev. B 80(2009) 140407. 7) S.J.Moon, H.Jin, W.S.Choi, J.S.Lee,S.S.A.Seo,J.Yu,G. Cao, T. W.Noh, and Y. S.Lee: Phys.Rev. B 80(2009) 195110. 8) H.Jin, H.Jeong, T.Ozaki, and J.Yu:Phys.Rev. B 80(2009) 075112. 9) G.Jackeli and G. Khaliullin: Phys.Rev. Lett. 102(2009) 017205. 10) Y. Kuramoto, H. Kusunose, and A. Kiss: J. Phys. Soc. Jpn. 78(2009) 072001. 11) H.Onishi and T.Hotta: J.Phys.Soc. Jpn. 75Suppl. (2006) 266. 12) E.Dagotto, T.Hotta, and A. Moreo: Phys.Rep. 344(2001) 1. 13) S.R. White: Phys.Rev. Lett. 93(1992) 2863. 14) R.Shiina,H.Shiba,andP.Thalmeier:J.Phys.Soc.Jpn. 66(1997)1741. 15) Y. Yamashita, N.Shibata, and K.Ueda: Phys.Rev. B 58(1998) 9114. 16) H.C. Lee, P.Azaria, and E.Boulat: Phys.Rev. B 69(2004) 155109. 17) J.C.Xavier,H.Onishi,T.Hotta,andE.Dagotto:Phys.Re v.B73(2006) 014405. 18) H.Onishi and T.Hotta: J.Magn. Magn. Mater. 310(2007) 790. 19) H.Onishi: Phys.Rev. B 76(2007) 014441.

2006.03384v1.Intrinsic_and_extrinsic_spin_orbit_coupling_and_spin_relaxation_in_monolayer_PtSe__2_.pdf

Intrinsic and extrinsic spin-orbit coupling and spin relaxation in monolayer PtSe 2 Marcin Kurpas Institute of Physics,University of Silesia in Katowice, 41-500 Chorz ow, Poland Jaroslav Fabian Institute for Theoretical Physics, University of Regensburg, Regensburg 93040, Germany (Dated: June 8, 2020) Monolayer PtSe 2is a semiconducting transition metal dichalcogenide characterized by an indirect band gap, space inversion symmetry, and high carrier mobility. Strong intrinsic spin-orbit coupling and the possibility to induce extrinsic spin-orbit elds by gating make PtSe 2attractive for fundamen- tal spin transport studies as well as for potential spintronics applications. We perform a systematic theoretical study of the spin-orbit coupling and spin relaxation in this material. Specically, we employ rst principles methods to obtain the basic orbital and spin-orbital properties of PtSe 2, also in the presence of an external transverse electric eld. We calculate the spin mixing parameters b2 and the spin-orbit elds for the Bloch states of electrons and holes. This information allows us to predict the spin lifetimes due to the Elliott-Yafet and D'yakonov-Perel mechanisms. We nd that b2is rather large, on the order of 102and 101, while varies strongly with doping, being about 103104ns1for carrier density in the interval 10131014cm2at the electric eld of 1 V/nm. We estimate the spin lifetimes to be on the picosecond level. I. INTRODUCTION Transition metal dichalcogenides (TMDCs) have been investigated|mainly in the bulk form but also as lay- ered slabs|for many decades1{5. The recent revival of interest in TMDCs has been fueled by a broad range of fascinating electronic, optical, and spin properties of two- dimensional (2D) samples of TMDCs, which are stable in air. The possibility of controlling physical properties of TMDCs by, e.g., stacking6{11, doping12, straining13,14, or gating15, demonstrates their potential for electronic16, optoelectronic17and valleytronic18applications. More- over, due to strong spin-orbit coupling and the presence of a semiconducting gap, TMDCs are also well suited for applications in spintronics19,20, as they can induce spin- orbit coupling (SOC) into graphene via strong proximity eect21,22. Recently demonstrated atomically thin PtSe 223,24is a distinct member of the 2D TMDC family. What sharply distinguishes this material from other TMDCs is its high room-temperature carrier mobility25, which is close to that of phosphorene26. But in contrast to phosphorene, PtSe 2exhibits good stability when exposed to air25. Like other TMDCs, a monolayer of PtSe 2consists of an atom- ically thin layer of transition metal (Pt) within two layers of chalcogen (Se) atoms [Fig. 1 (a),(b)]. It crystallizes in the centrosymmetric structure of P3 m1 space group being isomorphic with the D3dpoint group. While bulk PtSe2 is metallic, in the monolayer limit it is a semicon- ductor with a sizeable indirect gap reported to be in the range of 1:22 eV10,25,27{29. Monolayer PtSe 2also holds promise to exhibit rich spin phenomena. One of the most exciting is the hidden spin polarization30of degenerate bands near the Fermi level, recently observed in ARPES experiments31. Its origin is attributed to local site dipole elds (local Rashba eect) generating opposite helical spin textures for spin degener- (a) M K Se Pt (b)FIG. 1. Sketch of the crystalline structure of monolayer PtSe 2. (a) Top view on the unit cell and the corresponding rst Brillouin zone with indicated high symmetry points. (b) Side view of the atomic structure of monolayer PtSe 2. The center of inversion is at the Pt atom, marked red. ate states, spatially resolved with respect to dierent Se layers. The opposite dipole elds compensate each other leaving the total crystal potential inversion symmetric, and thus preserving the spin degeneracy of bands30. An- other interesting phenomenon is the defect induced mag- netism, which is reported for mono32,33and multilayer11 PtSe 2slabs. In the latter case, a magnentic phase can be switched between ferro- to antiferromagnetic by chang- ing the parity of the number of layers11. The interplay of such magnetic eects with spin-orbit coupling could lead to interesting magnetotransport phenomena. Strong spin-orbit coupling, intrinsic band gap, and high carrier mobility make PtSe 2a good candidate for building spintronic devices, such as, a spin valve or spin transistor34. Essential for these devices is a coherent (en- semble) dynamics of the electron spin. Such a dynamics is disrupted by spin dephasing and spin relaxation pro- cesses. Thus, the question concerning the electron spin lifetime in monolayer PtSe 2is of great importance for po- tential applications of this material in spintronics. ThisarXiv:2006.03384v1 [cond-mat.mes-hall] 5 Jun 20202 question has not yet been systematically addressed the- oretically. Here, we investigate the problem of the spin relax- ation in monolayer PtSe 2by employing rst principles calculations and extracting useful information about the spin-orbit coupling and spin relaxation. Two mecha- nisms dominating spin relaxation in non-magnetic mate- rials, such as PtSe 2, are considered. Namely, the Elliott- Yafet35,36and D'yakonov-Perel37,38. In the Elliott-Yafet mechanism, the intrinsic SOC mixes opposite spin com- ponents of degenerate Bloch states. In eect, an electron can ip its spin upon momentum scattering, with the probability given by the so called spin mixing parame- terb2 k. It is related to spin relaxation time s;EYvia the formula35,39,40 1 s;EY4b21 p; (1) where1 pis the momentum relaxation rate, and b2is the Fermi surface average of b2 k. In the D'yakonov-Perel mechanism spins randomize their phase via the interaction with the uctuating Rashba elds k. These elds appear due to broken space inver- sion symmetry of the structure, e.g., due to a substrate or an external electric eld. The initial phase of spins is completely randomized after the time s;DP 1 s;DP= 2 ?p; (2) where 2 ?denotes the Fermi surface average of the squared spin-orbit eld component perpendicular to the spin orientation 2 k?. In realistic systems these two mechanisms usually coexist and compete with each other. Here we show that for both mechanisms the spin re- laxation in PtSe 2is very fast, up to a few picoseconds for experimentally accessible momentum scattering time. Thus, PtSe 2does not appear to be the best material for building spintronic devices requiring long spin lifetimes. However, it should be useful for investigating spin-orbit induced transport phenomena. The paper is organized as follows. In Section II we brie y describe methods and details of calculations. Sec- tion III contains results of our rst principles calcula- tions with a discussion, including eects of the intrinsic and extrinsic SOC on the band structure, spin mixing parameter, and spin-orbit elds. Estimations of spin life- time due to Elliott-Yafet and D'yakonov-Perel relaxation mechanisms are also included here. Section IV contains nal conclusions. II. METHODS First principle calculations were performed us- ingQuantum Espresso package41,42. The norm{ conserving pseudopotential with the Perdew-Burke- Ernzerhof (PBE)43version of the generalized gradient ap- proximation (GGA) exchange{correlation potentials wasused. The kinetic energy cuto of the plane wave ba- sis sets was 50 Ry for the wave function and 200 Ry for charge density. These values were found to give con- verged results also for spin related quantities. Self consis- tency was achieved with 21 211 Monkhorst-Pack grid while for structure optimization a smaller grid 10 101 was chosen. The initial lattice constant of PtSe 2was taken from experiment23and was later optimized for the chosen pseudopotential using the variable cell and quasi-Newton schemes as implemented in the Quantum ESPRESSO package. During optimization process all atoms were free to move in all directions to minimize the internal forces below the threshold 104Ry/bohr. The calculated lattice constant is a= 3:748A, very close to the experimental value in bulk 3.73 A23, and is in a good agreement with other calculations29,32. The Fermi contour averages of spin mixing parameter b2and spin-orbit eld 2entering the formulae (1) and (2) are calculated using the formula A=1 (EF)SBZZ FCAk hjvF(k)jdk; (3) whereAkstands forb2 kor 2 k?,SBZis the area of the Fermi surface, (EF) is the density of states per spin at the Fermi level, vF(k) is the Fermi velocity and the integration takes over an iso-energy contour. III. RESULTS AND DISCUSSION We rst examine the orbital eects. The calcu- lated non-relativistic and relativistic band structures are shown in Fig. 2 (a). Monolayer PtSe 2is an indirect gap semiconductor with a sizeable band gap. Without SOC the calculated band gap is 1 :38 eV. The valence band (VB) maximum is located slightly away (0.15 A1) from the BZ center, while the conduction band (CB) minimum lies in the middle of the M path. The band edge at the point is a saddle point lying 38 meV below the global VB maximum [see the inset in Fig. 2 (a)]. The valence and conduction bands close to the band gap are formed mainly byd-electrons of platinum and p-electrons of sele- nium [Fig. 2 (b)]. In the valence band up to 1 eV below the Fermi level the dominant contribution comes from Sep-electrons, with signicant admixture of d-electrons from Pt. In the conduction band the contributions from Pt and Se atoms are almost equal. A. Intrinsic spin-orbit coupling. Spin-orbit splitting . Relativistic eects in PtSe 2are signicant. Spin-orbit coupling splits the originally two- fold (four-fold with spin) degenerate valence band at the point into two (doubly spin degenerate) bands which are separated by the spin-orbit split-o gap of
so= 350 meV. As a result the maximum of the VB3 −2−1012 K Γ M KE−EF[eV] 00.511.522.53 PDOS [states/eV]Se:p Pt:dSe:s(a) (b) (c)(d) (e) K M 1 2 3 −100−50050100150200250 K(c) Γsom,n[meV]so1,2 so2,3 so4,5 −50050100150200250300350 Γ(d) K Msom,n[meV] −200−150−100−50050100150200 K(e) Msom,n[meV]4 5 FIG. 2. Calculated non-relativistic (dashed line) and rela- tivistic (solid line) band structures along high symmetry lines in the FBZ (a). The non-relativistic band structure is mis- aligned with the Fermi energy for better transparency. The inset shows a zoom of bands close to the point. Purple rect- angles depict the range of kpoints in (c)-(e). (b) Density of states projected onto atomic orbitals. (c)-(e) Extracted spin- orbital splittings
m;n sobetween the valence bands mandn, labeled 1,2,3, and 4,5 in (a), calculated as a dierence between interband energy distances with SOC and without SOC. moves to the BZ center and the indirect band gap reduces to 1.2 eV, in agreement with earlier calculations25,29. The orbital degeneracy is also removed at the K-point. The energy splitting of the two highest valence bands [bands 1 and 2 in Fig. 2 (a)] is 170 meV. In the conduction band, the corresponding spin-orbital gaps
soare much smaller, 59 meV at the point and 5 meV at the K-point. Away from high symmetry points, we calculate the energy shifts
n;m so(k) =
n;m rel(k)
n;m nrel(k), where
n;m rel(nrel)(k) is the energy dierence between the bands nandmobtained from the relativistic (non-relativistic) calculation. It provides information about the shift of the bands upon turning on SOC, with respect to their initial energy. This can be partially translated into the strength of the direct spin-orbit interaction between bands mand n, relative to the total SOC in the band mcoming from all possible couplings. Considering that SOC leads to band repulsion, positive
n;m someans that the direct spin- orbit interaction between the bands nandmis likely dominant (with respect to couplings to other bands). Analogously, if
n;m sois negative, the spin-orbit inter- action between the bands nandmis weak enough to be overcome by couplings to others. Note that
n;m so= 0 does not meanh njHsoj mi= 0. Rather, it says that theTABLE I. Spin-orbital energy shifts
n,m soat high symmetry points extracted from rst principles calculations. k-point
1,2 so[meV]
4,5 so[meV]
2,3 so[meV] 350 59 -4 K 170 5 -24 direct SOC between bands mandnis of the same order as their couplings to the other bands, and no change in energy is observed. The results for three valence bands and two conduction bands labeled in Fig. 2 (a) respectively 1, 2, 3 and 4, 5, are shown in Fig. 2 (c)-(e). We have checked, by tracing the irreducible representations of the bands and applying the group theory methods, that for all
n;m sos shown in Fig. 2 (c)-(e) the direct SOC between bands nandm is allowed by the symmetry. For the valence bands 1-3,
n;m so(k) is strongly momentum dependent and takes sig- nicantly larger values than
4;5 soin the conduction band. In the presented k-points range it varies from -180 meV for
2;3 so(k) [Fig. 2 (e)] up to 350 meV for
1;2 so(k) [Fig. 2 (d)]. In comparison, max(j
4;5 soj) = 59 meV. The weaker k-dependence of
4;5 soresults from strong isolation of the bands 4 and 5, by 1:3eVfrom lower and upper man- ifolds (not shown), eectively limiting the possible cou- plings mainly to those two partners. Spin mixing . Apart from the spectroscopic features discussed above, the strength of SOC of inversion- symmetric crystals is measured by the spin mixing pa- rameterb2 k. Becauseb2 koriginates from the intrinsic SOC it constitutes a good measure of this interaction in the band structure44. Exceptions are spin hot spots formed around high-symmetry and accidental degeneracy points at which the value of b2 kis strongly enhanced40,44and the mixing reaches the value of one half (equal probabil- ity for spin up and down in a given state), irrespective of the strength of SOC. For an arbitrary Bloch state * n;k(r) = [an;k(r)j"i+bn;k(r)j#i]eik
r; (4) wherenis the band index, an;kandbn;kare lattice pe- riodic functions, ji,=f";#gis an eigenstate of spin one-half operator and kis the crystal momentum, the spin mixing parameter is dened as b2 k=Z jbn;k(r)j2dr; (5) where the integral is taken over the entire unit cell. Here, the amplitudes an;k(r) andbn;k(r) are chosen in a way, that bn;k(r) is the amplitude of the spin com- ponent admixed by the SOC. Such a choice is possible for any spin quantization axis (SQA). Because usually jbn;k(r)jjan;k(r)jthe state (4) can still be called a spin up state (although it is not an eigenstate of a Pauli matrix)19. For centrosymmetric systems with time rever- sal symmetry the energy degenerate spin down partner4 of * n;k(r) is + n;k(r) = a n;k(r)j#ib n;k(r)j"i eik
r;(6) and the same denition of b2 kcan be used. It is immedi- ately seen that for normalized states b2 k2[0; 0:5], where b2 k= 0 means no spin mixing and b2 k= 0:5 for fully spin mixed states. Alternatively, b2 kcan be dened as a deviation of the spin expectation value from one half45. To quantify anisotropies in the spin relaxation and spin transport in the crystal, it is instructive to study the spin admixture parameter for dierent spin quantization axes, which correspond to either the direction of an applied magnetic eld or to the orientation of the injected spin in a spin injection experiment. In Fig. 3 we show b2 k calculated in the rst Brillouin zone (FBZ) for the highest valence and rst conduction bands, and for three dierent spin quantization axes SQA= fX,Y,Zgaligned with the real space axes shown in Fig. 1. A strong anisotropy of b2 is evident. In the valence band and for SQA=X/Y [Fig. 3 (a), (c)] the region around the BZ center is a spin hot region where b2 kis close to one half. This region is very wide and extends towards the M-points, in a dierent way for SQA=X and SQA=Y. At the point spins are fully mixed, b2 k0:5; this is a witness to the lifting of the orbital degeneracy by SOC. For SQA=Z [Fig. 3 (e)], the entire FBZ is a spin hot region with the value of b2 k102101. An exception is a small circular wedge in the center of BZ corresponding to the vicinity of the valence band maximum [see Fig. 2 (a)]. In this wedge b2 k varies from 105in the center to 102at the edge. In the conduction band [Fig. 3 (b), (d), (f)] we observe much smaller variation of spin mixing parameter than for the valence band. The value of b2 kis of the order of 102 within the whole BZ, except for several spin hot spot regions localized around high symmetry and accidental degeneracy points. According to Elliott35, the spin mixing parameter can be translated into the spin relaxation rate, provided we know the momentum relaxation time. The latter strongly depends on temperature, concentration of defects and dopants and for a given sample can be determined from transport experiments. Here, we calculate the intrinsic, sample independent property of PtSe 2required to esti- mate spin lifetime { the Fermi surface averaged spin mix- ing parameter b2. It is shown in Fig. 4 (a),(b) as a func- tion of carrier density n, plotted versus Fermi energy in Fig. 4 (c). As can be seen, b2in the valence band dis- plays a qualitatively dierent behaviour for in-plane and out-of-plane SQA [Fig. 4 (a)]. For SQA=Z it is growing exponentially from b2105tob2= 101when the hole density is increasing. For in-plane spin polarization b2slowly decreases from the value of about 0.5 with in- creasingn, but never gets below 101. This unusually large value of b2is due to the very broad spin hot region. In this case the perturbative Elliott's approach to sis not valid and the spin relaxation rate is essentially the (e)SQA=Z −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] 10−410−310−210−1 (f)SQA=Z −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] 10−210−1(c)SQA=Y −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] 10−1(d)SQA=Y −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] 10−210−1(a)Valence band SQA=X −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] 10−1(b)Conduction band SQA=X −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] 10−210−1FIG. 3. Distribution of the spin-mixing parameter b2 kin the rst Brillouin zone of PtSe 2for three spin quantization axes. Left column: (a) valence band and SQA=X, (c) valence band and SQA=Y, (e) valence band and SQA=Z. Right column: same as left but for the conduction band. Blue s and ellipses encircle wedges of the BZ corresponding to the carried density up to n= 12
1013cm2. same as momentum relaxation. Therefore, if the momen- tum relaxation anisotropy (in the plane) is not very large, one should expect a giant, doping dependent anisotropy of the spin relaxation in PtSe 2for holes. In contrast, the spin relaxation anisotropy is predicted to be rather weak for conduction electrons. Indeed, as seen in Fig. 4 (b), b2in the conduction band varies very little withn. Its value is of the order of 102, meaning that out of all momentum scattering events about 1% constitute a spin ip. Moreover, there is a very weak dependence of the spin mixing probability on SQAs. The spin relaxation rate for out-of-plane spins is expected to be somewhat slower than for in-plane spins. Also, spin lifetimes of electrons should be 1-2 orders longer than for holes, for in-plane spins. B. Extrinsic spin-orbit coupling. In realistic situations monolayers are often encapsu- lated in protective layers, sit on a substrate or are studied in a gating electric eld. In any of these congurations5 1.11.151.21.251.31.351.41.45 024681012(b) b2[10−2] n [1013cm−2]10−610−510−410−310−210−1100 024681012(a) b2 n [1013cm−2]X Y Z −20−15−10−5051015 0 20 40 60 80 100 120 n [1013cm−2] EF[meV](c) valence bandconduction band FIG. 4. Calculated Fermi surface averaged spin{mixing pa- rameter b2versus carrier density nfor the valence (a) and for the conduction band (b). (c) Carrier density versus po- sition of the Fermi level given with respect to valence band maximum and conduction band minimum. the space inversion symmetry is broken, leading to the symmetry reduction D3d!C3vin the case of monolayer PtSe 2; spin degeneracy "k;"="k;#is lifted, except at the time reversal points and M. The emerging spin-orbit elds kenable the D'yakonov-Perel mechanism of spin relaxation, which coexists with the Elliott-Yafet spin- ip scattering mechanism. We model the eects of space in- version symmetry breaking by applying a uniform exter- nal electric eld Ein the direction perpendicular to the PtSe 2sheet. In this approach the spin-orbit eld depends on both the momentum and electric eld, k(E), and is related to the spin splitting as Hsoc=h 2 k(E)
; (7) where his the Planck constant, and is the vector of Pauli matrices. In Fig. 5 (a), (b) we show spin textures of the upper spin split valence band and for the lower spin split con- duction band respectively and for the external electric eld E=1 V/nm. The in-plane spin components (arrows) display a Rashba-like helical pattern, while the out-of- plane components (color) show the spin{valley locking eect. Similar spin textures have been reported to exist as a result of hidden spin polarization of spin degener-ate bands in PtSe 231. In contrast to layer-resolved spin textures picturing local Rashba eect31, Fig. 5 (a), (b) shows a global spin texture of the entire crystal structure. To answer the question about the origin of the presented spin textures we performed calculations at zero electric eld in order to preserve spin degeneracy of bands but with broken space inversion symmetry. The obtained spin textures (not shown here) resemble the same fea- tures as those shown in Fig. 5 (a), (b), with small dif- ferences due to external electric eld in the latter case. This indicates that such a texture is intrinsic to PtSe 2 crystalline structure and appears immediately once the space inversion symmetry is broken by arbitrary small crystal potential imbalance. Let us now discuss the strength of the extrinsic SOC. It can be quantied by the amplitude of the spin-orbital elds k. In Figs. 5 (c),(d) and 6 we show the k-point resolved ( k) and Fermi surface averaged ( ) spin-orbit elds, respectively, for the electric eld of 1 V/nm. In the valence band the overall value of kin the FBZ is greater that in the conduction band. This can be seen by comparing the amount of the red shaded area in Fig. 5 (a) and (b). A very characteristic hexagonal structure is formed close to the BZ center, with corners pointing towards the M points. In this region, and also along the paths M and around the K points, the spin splitting for E= 1 V/nm is less than 4 meV (the maximal value in the valence band and E= 1 V/nm is 14 meV). This gives k104ns1, which is roughly the value of for n between 8
1013cm2and 12
1013cm2[see the red line in Fig. 6 (a)]. Within the full doping range (without the rst 5 meV), in the VB varies by an order of magnitude. In the conduction band the extrinsic SOC is weaker than in the VB, resulting in smaller values of k[Fig. 5 (b)]. For the same level of doping, e.g., n= 1014cm2, the ratio of in the VB to in the CB, V B= CB3. Similarly to b2, in the conduction band is very weakly doping dependent [Fig. 6 (b)]. For E=1 V/nm it is of the order of 103ns1, and grows approximately with a step 3
103per 1 V/nm. C. Spin lifetime. To estimate spin lifetimes s;EYands;DPwe need to know the momentum relaxation time p. Taking the experimental value of the mobility for electrons = 400 cm2V1s1(100K) and eective mass m= 0:37me25, we estimate, using the Drude formula p= e=m,p80 fs. The formula (1) for the Elliott-Yafet spin relaxation rate is valid under the assumption that b2can be treated as a small parameter, b2135. For valence electrons being polarized in-plane with the PtSe 2 sheet (SQA=X/Y), b2is of the order of 1 [Fig. 4], and thus Eq. (1) cannot be applied. In such a case, due to very strong spin mixing, spin lifetime should be lim- ited by the momentum relaxation time, i.e., s;EY6 (a)Valence band −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] −0.5 0 0.5 (b)Conduction band −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] −0.5 0 0.5 (c)Ωk [1/s] −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] 108109101010111012 (d)Ωk [1/s] −0.4 0 0.4 kx [2π/a]−0.4 0 0.4ky [2π/a] 1091010101110121013 FIG. 5. Extrinsic SOC in PtSe 2: (a) In-plane spin texture (arrows) and out-of-plane spin component (color) of one spin subband of the top-most valence band plotted in the whole FBZ. (b) Same as (a) but for the bottom most conduction band. (c) The distribution of the spin-orbit eld kin the FBZ for the valence band for the electric eld E= 1 V/nm. (d) Same as (c) but for the conduction band. Black circle and ellipses encircle wedges of the BZ corresponding to the carried density up to n= 12
1013cm2. 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12(b)Ω [103 1/ns] n [1013 cm−2]10−210−1100101102(a)Ω [103 1/ns] E=1V/nm E=2V/nm E=3V/nm E=4V/nm E=5V/nm 0 0 2 4 6 8 10 12 n [1013 cm−2] FIG. 6. Calculated Fermi surface averaged spin{orbital eld versus carrier density nand several values of electric eld perpendicular to the 2D lattice. (a) valence band, (b) con- duction band. p80 fs. For spins of valence electrons being po- larized out-of-plane (SQA=Z) and for conduction elec- trons,b2102, within the perturbative limit. The corresponding spin lifetime estimated from Eq. (1) iss;EY1 ps. Two qualitatively dierent regimes of spin relaxation apply also for the D'yakonov-Perel mechanism. For small eldsE1 V/nm, we are in the motional narrow- ing regime, i.e., p 119. Taking = 3
103ns1 (E=1 V/nm) we get for out-of-plane spins s;DP s;EY1 ps. The in-plane spins in pristine paramagnetic materials usually are expected to relax slower than the out-of-plane ones due to 2-dimensional (in-plane) char- acter of extrinsic Rashba spin-orbit elds. In the Rashba limit, the in-plane spin relaxation rate is 2, which gives, roughly, the same order of magnitude as for out-of-plane spins. With increasing electric eld the condition for mo- tional narrowing breaks down ( 104ns1), and irre- versible spin randomization occurs in the time scale given by the momentum relaxation, es;DPp19, irrespective of spin polarization. IV. CONCLUSIONS We have investigated the intrinsic and extrinsic spin- orbit couplings, and their in uence on the electronic properties spin relaxation in monolayer PtSe 2using rst principles calculations. We found that the intrinsic SOC is very strong and leads to a signicant mixing of the spin states. The extrinsic SOC, characterized by spin-orbit elds , is also expected to be large, on the order of the intrinsic one. Their interplay is manifested in compa- rable contributions of the Elliott-Yafet and D'yakonov- Perel mechanisms to spin relaxation. Spin lifetime in PtSe 2is predicted to be short, on the picosecond time scale, and to a large extent governed by the momentum relaxation time especially at spin hot spots where the spins are fully mixed. This brings serious limitations for using PtSe 2in spintronic devices requiring long spin life- times. On the other hand, the strong spin-orbit coupling should be manifested prominently in spin transport and spin control phenomena. ACKNOWLEDGMENTS This work was supported by the National Science Centre under the contract DEC-2018/29/B/ST3/01892, in part by PAAD Infrastructure co-nanced by Oper- ational Programme Innovative Economy, Objective 2.3, and by the Deutsche Forschungsgemeinschaft (DFG, Ger- man Research Foundation) Project-ID 314695032 SFB 1277. V. REFERENCES marcin.kurpas@us.edu.pl1R. F. Frindt, J. Appl. Phys. 37, 1928 (1966).7 2J. Wilson and A. Yoe, Adv. Phys. 18, 193 (1969). 3L. F. Mattheiss, Phys. Rev. B 8, 3719 (1973). 4K. K. Kam and B. A. Parkinson, J. Phys. Chem. 86, 463 (1982). 5P. Joensen, R. F. Frindt, and S. Morrison, Mater. Res. Bull. 21, 457 (1986). 6S. Leb egue and O. Eriksson, Phys. Rev. B 79, 115409 (2009). 7K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 105, 136805 (2010). 8A. Kuc, N. Zibouche, and T. Heine, Phys. Rev. B 83, 245213 (2011). 9A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, Nano Lett. 10, 1271 (2010). 10A. Ciarrocchi, A. Avsar, D. Ovchinnikov, and A. Kis, Nat. Commun. 9, 1 (2018). 11A. Avsar, A. Ciarrocchi, M. Pizzochero, D. Unuchek, O. V. Yazyev, and A. Kis, Nat. Nanotechnol. 14, 674 (2019). 12S. Mouri, Y. Miyauchi, and K. Matsuda, Nano Lett. 13, 5944 (2013). 13Y. Ma, Y. Dai, M. Guo, C. Niu, Y. Zhu, and B. Huang, ACS Nano 6, 1695 (2012). 14K. Zollner, P. E. F. Junior, and J. Fabian, Phys. Rev. B 100, 195126 (2019). 15J. S. Ross, S. Wu, H. Yu, N. J. Ghimire, A. M. Jones, G. Aivazian, J. Yan, D. G. Mandrus, D. Xiao, W. Yao, and X. Xu, Nat. Commun. 4, 1474 (2013). 16B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, Nat. Nanotechnol. 6, 147 (2011). 17L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y.-J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, et al. , Science 340, 1311 (2013). 18F. Langer, C. P. Schmid, S. Schlauderer, M. Gmitra, J. Fabian, P. Nagler, C. Sch uller, T. Korn, P. G. Hawkins, Steiner, et al. , Nature 557, 76 (2018). 19I.Zuti c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 20J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I.Zuti c, Acta Phys. Slovaca 57, 565 (2010). 21A. Avsar, J. Y. Tan, T. Taychatanapat, J. Balakrishnan, G. K. W. Koon, Y. Yeo, J. Lahiri, A. Carvalho, A. S. Rodin, O'Farrell, et al. , Nat. Commun. 5, 4875 (2014). 22M. Gmitra and J. Fabian, Phys. Rev. B 92, 155403 (2015). 23Y. Wang, L. Li, W. Yao, S. Song, J. T. Sun, J. Pan, X. Ren, C. Li, E. Okunishi, Y.-Q. Wang, et al. , Nano Lett. 15, 4013 (2015). 24X. Yu, P. Yu, D. Wu, B. Singh, Q. Zeng, H. Lin, W. Zhou, J. Lin, K. Suenaga, Z. Liu, and Q. J. Wang, Nat. Commun.9, 1 (2018). 25Y. Zhao, J. Qiao, Z. Yu, P. Yu, K. Xu, S. P. Lau, W. Zhou, Z. Liu, X. Wang, W. Ji, and Y. Chai, Adv. Mater. 29 (2017). 26A. Avsar, J. Tan, M. Kurpas, M. Gmitra, K. Watanabe, T. Taniguchi, J. Fabian, and B. Ozyilmaz, Nat. Phys. 13, 888 (2017). 27H. Huang, S. Zhou, and W. Duan, Phys. Rev. B 94, 121117 (2016). 28K. Zhang, M. Yan, H. Zhang, H. Huang, M. Arita, Z. Sun, W. Duan, Y. Wu, and S. Zhou, Phys. Rev. B 96, 125102 (2017). 29A. Kandemir, B. Akbali, Z. Kahraman, S. V. Badalov, M. Ozcan, F. Iyikanat, and H. Sahin, Semicond. Sci. Tech- nol. (2018). 30X. Zhang, Q. Liu, J. W. Luo, A. J. Freeman, and A. Zunger, Nature Physics 10, 387 (2014). 31W. Yao, E. Wang, H. Huang, K. Deng, M. Yan, K. Zhang, K. Miyamoto, T. Okuda, L. Li, Y. Wang, et al. , Nat. Com- mun. 8, 14216 (2017). 32W. Zhang, H. T. Guo, J. Jiang, Q. C. Tao, X. J. Song, H. Li, and J. Huang, J. Appl. Phys. 120, 013904 (2016). 33M. Zulqar, Y. Zhao, G. Li, S. Nazir, and J. Ni, J. Phys. Chem. C (2016). 34S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). 35R. J. Elliott, Phys. Rev. 96, 266 (1954). 36Y. Yafet, in Solid State Physics , Vol. 14, edited by F. Seitz and D. Turnbull (Academic, New York, 1963). 37M. I. Dyakonov and V. I. Perel, Sov. Phys. Solid State 13, 3023 (1971). 38M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971). 39P. Monod and F. Beuneu, Phys. Rev. B 19, 911 (1979). 40J. Fabian and S. Das Sarma, Phys. Rev. Lett. 81, 5624 (1998). 41P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, et al. , J. Phys.: Condens. Matter 21, 395502 (2009). 42P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavaz- zoni, D. Ceresoli, M. Cococcioni, et al. , J. Phys.: Cond. Mat. 29, 465901 (2017). 43J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996); 78, 1396(E) (1997). 44M. Kurpas, P. E. F. Junior, M. Gmitra, and J. Fabian, Phys. Rev. B 100, 125422 (2019). 45B. Zimmermann, P. Mavropoulos, S. Heers, N. H. Long, S. Bl ugel, and Y. Mokrousov, Phys. Rev. Lett. 109, 236603 (2012).

1107.2904v2.Extending_the_effective_one_body_Hamiltonian_of_black_hole_binaries_to_include_next_to_next_to_leading_spin_orbit_couplings.pdf

arXiv:1107.2904v2 [gr-qc] 4 Nov 2011Extending the effective-one-body Hamiltonian of black-hol e binaries to include next-to-next-to-leading spin-orbit couplings Enrico Barausse1and Alessandra Buonanno1 1Maryland Center for Fundamental Physics & Joint Space-Scie nce Institute Department of Physics, University of Maryland, College Par k, MD 20742 (Dated: September 13, 2021) In the effective-one-body (EOB) approach the dynamics of two compact objects of masses m1and m2and spins S1andS2is mapped into the dynamics of one test particle of mass µ=m1m2/(m1+ m2) and spin S∗moving in a deformed Kerr metric with mass M=m1+m2and spin SKerr. In a previous paper we computed an EOB Hamiltonian for spinning black-hole binaries that (i) when expanded in post-Newtonian orders, reproduces the leading order spin-spin coupling and the leading and next-to-leading order spin-orbit couplings for any mas s ratio, and (iii) reproduces allspin-orbit couplings in the test-particle limit. Here we extend this EO B Hamiltonian to include next-to-next- to-leading spin-orbit couplings for any mass ratio. We disc uss two classes of EOB Hamiltonians that differ by the way the spin variables are mapped between th e effective and real descriptions. We also investigate the main features of the dynamics when th e motion is equatorial, such as the existence of the innermost stable circular orbit and of a pea k in the orbital frequency during the plunge subsequent to the inspiral. PACS numbers: 04.25.D-, 04.25.dg, 04.25.Nx, 04.30.-w I. INTRODUCTION Coalescing compact binaries composed of neutron stars and/or black holes are among the most promis- ing gravitational-wave sources for ground-based detec- tors, suchas the LaserInterferometerGravitational-wave Observatory (LIGO) [ 1], Virgo [ 2], GEO [ 3], the Large Cryogenic Gravitational Telescope (LCGT) [ 4], and fu- ture space-based detectors. So far, the search for gravitational waves with LIGO, GEO and Virgo detectors has focused on non-spinning compact binaries [ 5–9], although in Ref. [ 10] single-spin templates were used to search for inspiraling spinning compact objects. Within the next 4-5 years LIGO and Virgo detectors will be upgraded to a sensitivity such that event rates for coalescing binary systems will in- crease by a factor of one thousand. Thus, it is timely and necessary to develop more accurate templates that include spin effects. For maximally spinning objects, we expect that reasonably accurate templates would need to be computed at least through 3.5PN order. In the non-spinning case, studies at the interface between nu- merical and analytical relativity have demonstrated that templates computed at 3.5PN order are indeed reason- ably accurate. In the last few years, motivated by the search for grav- itational waves, the knowledge of spin effects in the two- body dynamics and gravitational-wave emission within the post-Newtonian (PN)1approximation has improved considerably. In particular, spin-orbit (SO) effects in the two-body equations of motion are currently known 1We refer to nPN as the order equivalent to terms O(c−2n) in the equations of motion beyond the Newtonian acceleration.through 3.5PN order (i.e., 2PN order beyond the lead- ing SO term) [ 11–19], and in the energy flux through 3PN order [ 12,13,20–23] (i.e., 1.5PN order beyond the leading SO term). Moreover, spin-spin (SS) effects have been computed through 3PN order (i.e., 1PN order be- yond the leading SS term) in the conservative dynam- ics[11,20,24–34]and alsoin the multipole moments[ 35]. In order to build reliable templates and search for gravitational-waves from high-mass compact binaries that merge in the detector bandwidth, it is crucial to improve the PN approximation by resumming the dy- namics and gravitational emission in a suitable way and by using numerical relativity and perturbation theory as aguidance. The effective-one-bodyapproach(EOB) [ 36– 40] offers the possibility of fulfilling this goal. The EOB approach uses the results of PN theory, not in their orig- inal Taylor-expanded form (i.e., as polynomials in v/c), but instead in a suitably resummed form. In particular, it maps the dynamics of two compact objects of masses m1andm2, and spins S1andS2, into the dynamics of one test particle of mass µ=m1m2/(m1+m2) and spinS∗moving in a deformed Kerr metric with mass M=m1+m2and spin SKerr. The deformation parame- ter is the symmetric mass ratio η=m1m2/(m1+m2)2, which ranges between 0 (test particle limit) and 1 /4 (equal-masslimit). Theanalysesandtheoreticalprogress made in Refs. [ 41–57] have demonstrated that faithful EOB templates describing the full signal (i.e., the inspi- ral, merger and ringdown) can be built and used in real searches [ 9]. Here we build on previous work[ 39,46,58,59], employ the recent results of Ref. [ 19] and extend the EOB con- servative dynamics, i.e. the EOB Hamiltonian, through 3.5PN order in the SO couplings. Since the mapping between the PN-expanded Hamiltonian (or real Hamil- tonian) and the EOB Hamiltonian is not unique, we ex-2 plore two specific classes of EOB Hamiltonians, which differ by the way the spin variables of the real and effec- tive descriptions are mapped. This paper is organized as follows. In Sec. II, af- ter reviewing the logic underpinning the construction of the EOB Hamiltonian, we proceed in steps and ex- tend the EOB Hamiltonian proposed in Ref. [ 46] through 3.5PN order in the SO couplings. In particular, in Sec.IIAwe derive the PN-expanded Arnowitt-Deser- Misner (ADM) Hamiltonian in the EOB canonical co- ordinates; then, after computing in Sec. IIBthe effec- tive Hamiltonian corresponding to the canonically trans- formed PN-expanded ADM Hamiltonian, we compare it (in Sec. IIC) to the deformed-Kerr Hamiltonian for a spinning test-particle [ 46], and work out (in Secs. IID andIIE) two classes of EOB Hamiltonians. In Sec. III we study the dynamics of these Hamiltonians for equa- torial orbits, and in Sec. IVwe summarize our main con- clusions. We use geometric units G=c= 1 throughout the paper, except when performing PN expansions, where powers of the speed of light care restored and play the role of book-keeping parameters. II. THE EFFECTIVE-ONE-BODY HAMILTONIAN FOR TWO SPINNING BLACK HOLES The main ingredient of the EOB approach is the realPN-expanded ADM Hamiltonian (or realHamilto- nian) describing two black holes with masses m1,m2 and spins S1,S2. The real Hamiltonian is then canoni- cally transformed and subsequently mappedto aneffec- tiveHamiltonian Heffdescribing a test-particle of mass µ=m1m2/(m1+m2) and suitable spin S∗, moving in adeformed Kerr metric of mass M=m1+m2and suit- able spin SKerr. The deformation is regulated by the binary’s symmetric mass-ratio parameter, η=µ/M, and therefore disappears in the test-particle limit η→0. The so-called improved real (or EOB) Hamiltonian reads Himproved real=M/radicalBigg 1+2η/parenleftbiggHeff µ−1/parenrightbigg .(1) The computation of the EOB Hamiltonian consists of several steps. We briefly review these steps and the un- derlying logic that we will follow in the next sections: (i) We apply a canonical transformation to the PN- expandedADM Hamiltonianusing the Lie method, obtaining the PN-expanded Hamiltonian in EOB canonical coordinates (see Sec. IIA); (ii) We compute the effective Hamiltonian correspond- ing to the canonically transformed PN-expanded ADM Hamiltonian (see Sec. IIB);(iii) We PN-expand the deformed-Kerr Hamiltonian for a spinning test-particle derived in Ref. [ 46] (see Sec.IIC); (iv) We compare (ii) and (iii), and work out the map- ping between the spin variables in the real and effective descriptions, and compute the improved EOB Hamiltonian (see Secs. IIDandIIE). A. The ADM Hamiltonian canonically transformed to EOB coordinates FollowingRef. [ 46], wedenote theADM canonicalvari- ables in the binary’s center-of-mass frame with r′andp′, and we introduce the following spin variables: σ=S1+S2, (2) σ∗=S1m2 m1+S2m1 m2. (3) Henceforth, to keep track of the PN orders, we rescale the spins variables as σ∗→σ∗candσ→σc. We use the spin-independent part of the ADM Hamil- tonian through 3PN order [ 38], and we include SO ef- fects through 2PN order beyond the leading-order effects (1.5PN), thus through 3.5PN order. In particular, the ADM SO Hamiltonian at 3.5PN order was computed re- cently in Ref. [ 19] (the ADM SO Hamiltonian at 1.5PN was computed in Ref. [ 60], and at 2.5PN in Ref. [ 58]). In the binary’s center-of-mass, the ADM SO Hamiltonian reads HADM SO(r′,p′,σ∗,σ) =1 c3L′ r′3·(gADM σσ+gADM σ∗σ∗),(4) where we indicate L′=r′×p′and gADM σ= 2+1 c2/bracketleftbigg19 8ηˆp′2+3 2η(n′·ˆp′)2 −(6+2η)M r′/bracketrightbigg +1 c4/bracketleftBigg 15 16η2(n′·ˆp′)4+21 2(1+η)/parenleftbiggM r′/parenrightbigg2 +1 8η(−9+22η)ˆp′4−1 16η(314+39 η)M r′ˆp′2 −1 16η(256+45 η)M r′(n′·ˆp′)2 +3 16η(−4+9η)ˆp′2(n′·ˆp′)2/bracketrightbigg , (5a) gADM σ∗=3 2+1 c2/bracketleftbigg/parenleftbigg −5 8+2η/parenrightbigg ˆp′2+3 4η(n′·ˆp′)2 −(5+2η)M r′/bracketrightbigg +1 c4/bracketleftBigg 1 8(75+82η)/parenleftbiggM r′/parenrightbigg23 +1 16(7−37η+39η2)ˆp′4 −3 16(−18+86η+13η2)M r′ˆp′2 −3 16η(32+15η)M r′(n′·ˆp′)2 +9 16η(−1+2η)ˆp′2(n′·ˆp′)2/bracketrightbigg , (5b) withn′=r′/r′, and where we have introduced the rescaled conjugate momentum ˆp′=p′/µ. In order to canonically transform the ADM Hamilto- nian to EOB coordinates, various approaches are possi- ble. A popular method, used in the previous work on the EOB model [ 37,38,58], is to use a generating function that produces a near-identity transformation, i.e. one of the form ˜G(q′,π) =q′iπi+ǫG(q′,π), where ( q,π) are the phase variables (including the angles defining the spins and their conjugate momenta, see Ref. [ 59]) andǫis a small parameter. Expressing the initial “primed” coor- dinates (the ADM coordinates) in terms of the new “un- primed” coordinates (the EOB coordinates), one gets q′i=qi−ǫ∂G(q′,π) ∂πi=qi−ǫ∂G(q,π) ∂πi +ǫ2∂2G(q,π) ∂πi∂qj∂G(q,π) ∂πj+O(ǫ3),(6a) π′ i=πi+ǫ∂G(q′,π) ∂q′i=πi+ǫ∂G(q,π) ∂qi −ǫ2∂2G(q,π) ∂qi∂qj∂G(q,π) ∂πj+O(ǫ3).(6b) Because under a time-independent canonical transforma- tion the Hamiltonian transforms as H(q,p) =H′(q′,p′), Eqs. (6) imply H(q,π) =H′(q′,π′) =H′(q,π) +ǫ/bracketleftbigg∂H′(q,π) ∂πi∂G(q,π) ∂qi−∂H′(q,π) ∂qi∂G(q,π) ∂πi/bracketrightbigg +ǫ2/bracketleftbigg∂H′(q,π) ∂qi∂2G(q,π) ∂πi∂qj∂G(q,π) ∂πj −∂H′(q,π) ∂πi∂2G(q,π) ∂qi∂qj∂G(q,π) ∂πj +1 2∂2H′(q,π) ∂qi∂qj∂G(q,π) ∂πi∂G(q,π) ∂πj +1 2∂2H′(q,π) ∂πi∂πj∂G(q,π) ∂qi∂G(q,π) ∂qj −∂2H′(q,π) ∂qi∂πj∂G(q,π) ∂πi∂G(q,π) ∂qj/bracketrightbigg +O(ǫ3).(7) The terms oforder O(ǫ) in this equation can be rewritten asǫ{G,H′}, which is very convenient because it trans- forms a sum over all the phase variables (including the angles defining the spins and their conjugate momenta) into a Poisson bracket that can be computed using onlythe commutation relations {xi,pj}=δi j,{xi,Sj (a)}= 0, {pi,Sj (a)}= 0, and {Si (a),Sj (b)}=δ(a)(b)ǫijkSk (a)(a,b= 1,2 being indices that distinguish between the two black holes). Unfortunately, the terms O(ǫ2) cannot be eas- ily expressed in terms of Poisson brackets, which makes them hard to compute (because the spin variables must be carefully taken into account in the sums). Also, the generalization of Eq. ( 7) to higher orders in ǫbecomes more and more complicated. A possible alternative to the generating function method mentioned above is given by the so-called Lie method [ 61]. This approach exploits the fact that the flux of the Hamilton equations is canonical. Therefore, onecandefineafictitiousHamiltonian H(q,π) whoseflux sends some initial data ( q,π) to (q′(q,π,ǫ),π′(q,π,ǫ)), whereǫis the “time” variable of this fictitious Hamilto- nian. The canonical transformation is then simply given by(q′(q,π,ǫ),π′(q,π,ǫ)). Theadvantageofthisapproach is that any function f(q,π) satisfies ˙f={f,H}(where we denote with ˙ = d/dǫ) under the Hamiltonian flux of H. Defining for convenience G=−H, this equation be- comes˙f={G,f}, and denoting the differential operator {G,...}byLG, a Taylor expansion yields f(q′(q,π,ǫ),π′(q,π,ǫ)) =∞/summationdisplay n=0ǫn n!Ln Gf(q,π) = exp(ǫLG)f(q,π) =f(q,π)+ǫ{G,f}(q,π) +1 2ǫ2{G,{G,f}}(q,π)+O(ǫ3). (8) Specializing to the (non-fictitious) Hamiltonian H′, we obtain the equivalent of Eq. ( 7), that is H(q,π) =H′(q′,π′) =H′(q,π)+ǫ{G,H′}(q,π) +1 2ǫ2{G,{G,H′}}(q,π)+O(ǫ3).(9) As already mentioned, the above expression allows us to account for the spin variables very easily, if necessary2, by means of the commutation relations {xi,Sj (a)}= 0, {pi,Sj (a)}= 0, and {Si (a),Sj (b)}=δ(a)(b)ǫijkSk (a). In this paper we will use the Lie method to generate the canonical transformation from ADM to EOB coordi- nates. In particular, we assume G(r,p) =r·p+GNS(r,p)+GS(r,p,σ∗,σ),(10) whereGNSis the purely orbital part of the fictitious Hamiltonian, while GSis the spin-dependent part, which 2The Poisson brackets of the spin variables with themselves d o not enter in the computations that we perform in this paper, b ut they do enter at higher PN orders.4 we assume to be linear in the spins since in this paper we focus on the SO terms only. Because the transformations (7) and (9) agree at leading order in the perturbative pa- rameter ǫ,GandGmust agree at leading PN order. In particular, since the purely orbital generating function for the transformation from ADM to EOB coordinates starts at 1PN, GNSmust start at 1PN order too, that is GNS(r,p) =GNS1PN(r,p)+GNS2PN(r,p)+O/parenleftbigg1 c6/parenrightbigg , (11) whereGNS1PNmust coincide with GNS1PN, and therefore be given by [ 37] GNS1PN(r,p) =1 c2(r·p)/bracketleftbigg −1 2ηˆp2+M r/parenleftbigg 1+1 2η/parenrightbigg/bracketrightbigg . (12) At 2PN, instead, GNSdoes not coincide with GNS, but a computation similar to the one in Ref. [ 37] easily shows that GNS2PN(r,p) =1 c4(r·p)/bracketleftbigg αˆp4+βM rˆp2 +γM r(n·ˆp)2+δ/parenleftbiggM r/parenrightbigg2/bracketrightBigg ,(13) with α=η 8, β =η 4(4−η),(14a) γ=η4+η 8, δ=1−7η+η2 4.(14b) [Note that the functional form ( 13) is the same as for GNS2PN, but the values of the parameters α,β,γandδ are different from those of Ref. [ 37].] Similarly, the spin-dependent part of the fictitious Hamiltonian, GS, must start like GSat 2.5PN order: GS(r,p,σ∗,σ) =GS2.5PN(r,p,σ∗,σ) +GS3.5PN(r,p,σ∗,σ)+O/parenleftbigg1 c9/parenrightbigg .(15) and if we restrict to functions that are linear in the spin variables, it must be [ 15] GS2.5PN(r,p,σ∗,σ) = 1 c5r3(r·ˆp) [a0(η)(L·σ)+b0(η)(L·σ∗)],(16) wherea0(η) andb0(η) are arbitrary gauge functions. [Note that restricting to functions that are linear in the spin variables is justified because here we are looking at SO effects only, but in general cubic terms in the spin may be present, see Ref. [ 46].]Themostgeneralformfor GSat3.5PNorderisinstead, if we restrict again to functions linear in the spins, GS3.5PN(r,p) =1 c7r3(r·ˆp)/braceleftbig (L·σ)/bracketleftbig a1(η)ˆp2 +a2(η)M r+a3(η) (n·ˆp)2/bracketrightbigg +(L·σ∗)/bracketleftbigg b1(η)ˆp2+b2(η)M r +b3(η)(n·ˆp)2/bracketrightBig/bracerightBig , (17) whereai(η) andbi(η) withi= 1,2,3 are other arbitrary gauge functions. To ease the notation, henceforth we drop the ηdependence in the gauge parameters, both at 2.5PN and 3.5PN order, and will denote them simply withaiandbi(withi= 0,3). Applying Eq. ( 9), we obtain that the 3.5 SO Hamilto- nian in EOB coordinates is given by HSO3.5PN=HADM SO3.5PN+{GNS2PN,HADM SO1.5PN} +{GNS1PN,HADM SO2.5PN} +{GS2.5PN,HADM 1PN} +{GS3.5PN,HADM Newt} +1 2{GNS1PN,{GNS1PN,HADM SO1.5PN}} +1 2{GNS1PN,{GS2.5PN,HADM Newt}} +1 2{GS2.5PN,{GNS1PN,HADM Newt}}. (18) A tedious but straightforward calculation gives the sev- eral terms entering the above equation: {GNS2PN,HADM SO1.5PN}=3 c7r3L·/parenleftbigg 2σ+3 2σ∗/parenrightbigg /bracketleftBigg δ/parenleftbiggM r/parenrightbigg2 +αˆp4+βM rˆp2+4αˆp2(n·ˆp)2 +(2β+3γ)M r(n·ˆp)2/bracketrightbigg , (19) {GNS1PN,HADM SO2.5PN}= 1 c7r3L·σ/bracketleftBigg −4(6+5η+η2)/parenleftbiggM r/parenrightbigg2 −95 16η2ˆp4+1 16η(382+159 η)M rˆp2 −63 8η2ˆp2(n·ˆp)2−15 2η2(n·ˆp)4 +1 8η(190+63 η)M r(n·ˆp)2/bracketrightbigg +1 c7r3L·σ∗/bracketleftBigg −2(10+9η+2η2)/parenleftbiggM r/parenrightbigg25 +5 16η(5−16η)ˆp4+1 16(−50+295η+144η2)× M rˆp2+3 8η(5−17η)ˆp2(n·ˆp)2−15 4η2(n·ˆp)4 +1 8(10+151 η+57η2)M r(n·ˆp)2/bracketrightbigg ,(20) {GS2.5PN,HADM 1PN}=1 c7r3(a0L·σ+b0L·σ∗)×/bracketleftBigg/parenleftbiggM r/parenrightbigg2 +1 2(−1+3η)ˆp4−3 2(3+η)M rˆp2 +9 2(2+η)M r(n·ˆp)2−3 2(−1+3η)ˆp2(n·ˆp)2/bracketrightbigg , (21) {GS3.5PN,HADM Newt}=1 c7r3L·σ/bracketleftBigg −a2/parenleftbiggM r/parenrightbigg2 +a1ˆp4+(a2−a1)M rˆp2−(4a2+2a1+3a3)M r(n·ˆp)2 −3(a1−a3)ˆp2(n·ˆp)2−5a3(n·ˆp)4/bracketrightbig +1 c7r3L·σ∗/bracketleftBigg −b2/parenleftbiggM r/parenrightbigg2 +b1ˆp4+(b2−b1)M rˆp2 −(4b2+2b1+3b3)M r(n·ˆp)2−3(b1−b3)ˆp2(n·ˆp)2−5b3(n·ˆp)4/bracketrightbigg , (22) {GNS1PN,{GNS1PN,HADM SO1.5PN}}=3 4c7r3L·/parenleftbigg 2σ+3 2σ∗/parenrightbigg/bracketleftBigg 4(2+η)2/parenleftbiggM r/parenrightbigg2 +5η2ˆp4−9η(2+η)M rˆp2 +8η2ˆp2(n·ˆp)2−12η(2+η)M r(n·ˆp)2+20η2(n·ˆp)4/bracketrightbigg , (23) {GNS1PN,{GS2.5PN,HADM Newt}}=1 c7r3L·(a0σ+b0σ∗)/bracketleftBigg −2(2+η)/parenleftbiggM r/parenrightbigg2 −5 2ηˆp4+1 2(10+9η)M rˆp2 −3 2ηˆp2(n·ˆp)2−1 2(22+3η)M r(n·ˆp)2+15η(n·ˆp)4/bracketrightbigg , (24) {GS2.5PN,{GNS1PN,HADM Newt}}=1 c7r3L·(a0σ+b0σ∗)/bracketleftBigg −(2+η)/parenleftbiggM r/parenrightbigg2 −2ηˆp4+3(1+η)M rˆp2 +6ηˆp2(n·ˆp)2−3 2(2+5η)M r(n·ˆp)2/bracketrightbigg . (25) Also, we have [ 46] HNewt=HADM Newt, (26a) H1PN=HADM 1PN+{GNS1PN,HADM Newt},(26b) HSO1.5PN=HADM SO1.5PN, (26c) HSO2.5PN=HADM SO2.5PN+{GS2.5PN,HADM Newt} +{G1PN,HADM SO1.5PN}, (26d) whereHADM Newt,HADM 1PNcan be found in Ref. [ 37], the explicit expressions of {GS2.5PN,HADM Newt}and {G1PN,HADM SO1.5PN}are given in Eqs. (5.20), (5.24) ofRef. [46], while {GNS1PN,HADM Newt}=µ c2/bracketleftBigg −1 2(2+η)/parenleftbiggM r/parenrightbigg2 −η 2ˆp4 +(1+η)M rˆp2+1 2(−2+η)M r(n·ˆp)2/bracketrightbigg . (27) Also, we note that Eqs. ( 26b) and (26d) immediately im- plyG2.5PN=G2.5PN, i.e. the 2.5 PN gauge parameters a0andb0appearing in Eq. ( 16) have the same mean- ing in the Lie method and in the generating function approaches.6 B. The spin-orbit terms in the effective Hamiltonian through 3.5PN order Following Refs. [ 37,39,62], we map the effective and real two-body Hamiltonians as Heff µc2=H2 real−m2 1c4−m2 2c4 2m1m2c4, (28) whereHrealis the real two-body Hamiltonian contain- ing also the rest-mass contribution M c2. Expanding Eq. (28) in powers of 1 /c, we have Heff SO3.5PN=HSO3.5PN+1 M(HSO1.5PNH1PN+HSO2.5PNHNewt). (29) Using Eqs. ( 18) and (26), we find that through 3.5PN order the SO couplings of the effective Hamiltonian are Heff SO=1 c3L r3·(geff σσ+geff σ∗σ∗), (30) where geff σ= 2+1 c2/bracketleftbigg1 8(3η+8a0)ˆp2−1 2(9η+6a0)(n·ˆp)2−(η+a0)M r/bracketrightbigg +1 c4/bracketleftBigg 1 2(−4a0−2a2−18η−a0η−3η2)/parenleftbiggM r/parenrightbigg2 +1 8(−4a0+8a1−5η−2a0η)ˆp4+1 8(−4a0 −8a1+8a2−34η+6a0η+11η2)M rˆp2+3 16(8a0−16a1+16a3+12η−20a0η−13η2)ˆp2(n·ˆp)2 +1 16(32a0−32a1−64a2−48a3+140η+48a0η−3η2)M r(n·ˆp)2+5 16(−16a3+24a0η +27η2)(n·ˆp)4/bracketrightbig , (31a) geff σ∗=3 2+1 c2/bracketleftbigg1 8(−5+4η+8b0)ˆp2−1 4(15η+12b0)(n·ˆp)2−1 4(2+5η+4b0)M r/bracketrightbigg +1 c4/bracketleftBigg 1 8(−4−16b0−8b2−55η−4b0η−13η2)/parenleftbiggM r/parenrightbigg2 +1 16(7−8b0+16b1−11η−4b0η−η2)ˆp4 +1 16(4−8b0−16b1+16b2−59η+12b0η+24η2)M rˆp2+3 16(8b0−16b1+16b3+19η−20b0η −14η2)ˆp2(n·ˆp)2+1 8(10+16b0−16b1−32b2−24b3+109η+24b0η+6η2)M r(n·ˆp)2+5 2(−2b3 +3b0η+3η2)(n·ˆp)4/bracketrightbig . (31b) C. The PN-expanded Hamiltonian of a spinning test-particle in a deformed Kerr spacetime The deformed-Kerr metric was obtained in Ref. [ 46], and it reads gtt=−Λt ∆tΣ, (32a) grr=∆r Σ, (32b) gθθ=1 Σ, (32c) gφφ=1 Λt/parenleftbigg −˜ω2 fd ∆tΣ+Σ sin2θ/parenrightbigg ,(32d)gtφ=−˜ωfd ∆tΣ. (32e) The potentials in these equations are given by ∆t=r2/bracketleftbigg A(r)+a2 r2/bracketrightbigg , (33) ∆r= ∆tD−1(r), (34) Λt= (r2+a2)2−a2∆tsin2θ, (35) Σ =r2+a2cosθ2, (36) and ˜ωfd= 2aM r+aηω0 fdM2+aηω1 fdM3 r,(37)7 whereω0 fdandω1 fdare two “frame-dragging” parameters (that we will fix later), and where A(r) = 1−2M r+2ηM3 r3+/parenleftbigg94 3−41 32π2/parenrightbiggηM4 r4, (38a) D−1(r) = 1+6ηM2 r2+2(26−3η)ηM3 r3.(38b) The Hamiltonian of a spinning test-particle in the deformed-Kerr spacetime is H=HNS+HS, (39) with HNS=βipi+α/radicalBig µ2+γijpipj+Q4(p),(40) where the term Q4(p) is quartic in the space momenta pi and was introduced in Ref. [ 62]. Moreover, we have α=1/radicalbig −gtt, (41)βi=gti gtt, (42) γij=gij−gtigtj gtt. (43) and HS=HSO+HSS, (44) whereHSOcontains the odd terms in the spins (and therefore, in particular, the SO terms) and HSScontains the even terms in the spins (and therefore, in particular, the spin-spin terms of the kind SKerrS∗). Since here we are interested in the SO couplings, we consider only HSO(HSScan be read from Eq. (4.19) in Ref. [46]): HSO=e2ν−˜µ/parenleftBig e˜µ+ν−˜B/parenrightBig (ˆp·ξr)(S·ˆSKerr) ˜B2√Qξ2+eν−2˜µ ˜B2/parenleftbig√Q+1/parenrightbig√Qξ2/braceleftBigg (S·ξ)˜J/bracketleftBig µr(ˆp·vr)/parenleftBig/radicalbig Q+1/parenrightBig −µcosθ(ˆp·n)ξ2−/radicalbig Q(νr(ˆp·vr)+(µcosθ−νcosθ)(ˆp·n)ξ2)/bracketrightBig ˜B2+e˜µ+ν(ˆp·ξr)/parenleftBig 2/radicalbig Q+1/parenrightBig × /bracketleftBig ˜J νr(S·v)−νcosθ(S·n)ξ2/bracketrightBig ˜B−˜J˜Bre˜µ+ν(ˆp·ξr)/parenleftBig/radicalbig Q+1/parenrightBig (S·v)/bracerightBigg , (45) whereˆSKerr=SKerr/SKerr,ξ=ˆSKerr×n,v=n×ξ, and where Q= 1+∆r(ˆp·n)2 Σ+(ˆp·ξr)2Σ Λtsin2θ+(ˆp·vr)2 Σ sin2θ,(46) and νr=r Σ+(r2+a2)/bracketleftbig (r2+a2)∆′ t−4r∆t/bracketrightbig 2Λt∆t,(47a) νcosθ=a2(r2+a2) cosθ(r2+a2−∆t) ΛtΣ,(47b) µr=r Σ−1√∆r, µcosθ=a2cosθ Σ, (47c) ˜B=/radicalbig ∆t,˜Br=√∆r∆′ t−2∆t 2√∆r∆t, (47d) e2˜µ= Σ, e2ν=∆tΣ Λt,˜J=/radicalbig ∆r, (47e) in which we use a prime to denote the derivative with respect to r. To obtain the SO couplings through 3.5PNorder, we expand Eq. ( 39). In particular, it is suffi- cient to consider the first term in the right-hand-side of Eq. (40), and set a= 0 (deformed-Schwarzschildlimit) in Eqs. (45), (46) and (47). Doing so, for the PN-expanded deformed-Kerr Hamiltonian we obtain HNS SO1.5PN=2 r3c3L·SKerr, (48a) HNS SO2.5PN=1 r3c5ηω0 fdM rL·SKerr, (48b) HNS SO3.5PN=1 r3c7ηω1 fd/parenleftbiggM r/parenrightbigg2 L·SKerr,(48c) and HS SO1.5PN=3 2r3c3L·S∗, (49a) HS SO2.5PN=1 r3c5L·S∗/bracketleftbigg −1 2(1+6η)M r−5 8ˆp2/bracketrightbigg , (49b) HS SO3.5PN=1 r3c7L·S∗/bracketleftBigg 1 2(−1−42η+6η2)/parenleftbiggM r/parenrightbigg28 +7 16ˆp4+1+6η 4/parenleftbiggM r/parenrightbigg ˆp2+5 4/parenleftbiggM r/parenrightbigg (n·ˆp)2/bracketrightbigg . (49c) D. The EOB Hamiltonian: spin-mapping dependent on dynamical variables We now determine the mapping between the spins σ andσ∗of the effective ADM Hamiltonian and the spins SKerrandS∗of the EOB Hamiltonian by imposing that the deformed-Kerr Hamiltonian given by Eqs. ( 48) and (49) coincides with the effective Hamiltonian given by Eqs. (30) and (31). As found in Ref. [ 46], we have to assume that the mapping depends on the orbital dynam- ical variables p2,n·pandr. The general mapping of the spins has the form S∗=σ∗+1 c2∆(1) σ∗+1 c4∆(2) σ∗,(50a)SKerr=σ+1 c2∆(1) σ+1 c4∆(2) σ.(50b) At 2.5PN order, if we assume ω0 fd= 0 [see Eq. ( 37)] and ∆(1) σ= 0, we have [ 46] ∆(1) σ∗=σ∗/bracketleftbigg1 6(−4b0+7η)M r+1 3(2b0+η)(Q−1) −1 2(4b0+5η)∆r Σ(n·ˆp)2/bracketrightbigg +σ/bracketleftbigg −2 3(a0+η)M r+1 12(8a0+3η)(Q−1) −(2a0+3η)∆r Σ(n·ˆp)2/bracketrightbigg , (51) and at 3.5PN order, assuming ω1 fd= 0 [see Eq. ( 37)] and ∆(2) σ= 0, we obtain ∆(2) σ∗=σ∗/bracketleftBigg 1 36(−56b0−24b2+353η−60b0η−27η2)/parenleftbiggM r/parenrightbigg2 +5 3(−2b3+3b0η+3η2)∆2 r Σ2(n·ˆp)4+1 72(−4b0+48b1 −23η−12b0η−3η2)(Q−1)2+1 36(−14b0−24b1+24b2−103η+66b0η+60η2)M r(Q−1)+1 12(2b0−24b1 +24b3+16η−30b0η−21η2)∆r Σ(n·ˆp)2(Q−1)+1 12(−24b0−16b1−32b2−24b3+47η−24b0η−54η2)× M r∆r Σ(n·ˆp)2/bracketrightbigg +σ/bracketleftBigg 1 9(−14a0−6a2−56η−15a0η−21η2)/parenleftbiggM r/parenrightbigg2 +5 24(−16a3+24a0η+27η2)∆2 r Σ2(n·ˆp)4+1 144(−8a0 +96a1−45η−24a0η)(Q−1)2+1 36(−14a0−24a1+24a2−109η+66a0η+51η2)M r(Q−1)+1 24(4a0 −48a1+48a3+6η−60a0η−39η2)∆r Σ(n·ˆp)2(Q−1)+1 24(−48a0−32a1−64a2−48a3−16η−48a0η −147η2)M r∆r Σ(n·ˆp)2/bracketrightbigg . (52) Note that as in Ref. [ 46], we have replaced, in the expres- sionsfor ∆(1) σ∗and∆(2) σ∗, the term ˆp2withγijˆpiˆpj=Q−1 and the term ( n·ˆp)2with ∆ r(n·ˆp)2/Σ =grrˆp2 r. Having determined the spin mappings, we can write down the real improved (or EOB) Hamiltonian for spin- ning black holes, which turns out to be Himproved real=M/radicalBigg 1+2η/parenleftbiggHeff µ−1/parenrightbigg ,(53) where Heff=HS+βipi+α/radicalBig µ2+γijpipj+Q4(p)−µ 2M r3(δij−3ninj)S∗ iS∗ j. (54) E. The EOB Hamiltonian: spin-mapping independent of dynamical variables Intheprevioussection,wehadtoassumeadependence on the orbital dynamical variables p2,n·pandrin the mapping between the spins σandσ∗of the effective ADM Hamiltonian and the spins SKerrandS∗of the deformed-Kerr Hamiltonian. To avoid this dependence9 on the dynamical variables and obtain the much simpler mapping S∗=σ∗, (55a) SKerr=σ, (55b) we need to modify the Hamilton-Jacobi equation by adding terms depending on the momenta and spins. SinceinthispaperwearedealingonlywithSOeffects, we will neglect modifications that involve spin-spin terms. We start by observing that in the presence of spins the linear momentum Pµ, which is related to the canoni- cal momentum by Pµ=pµ+Eρσ µS∗ ρσ[see Eq. (3.28) of Ref. [59]], satisfies the Hamilton-Jacobi equation [ 59] µ2+PµPµ=µ2+pµpµ+2Eρσ µpµS∗ ρσ+O(S∗)2= 0.(56) Here,S∗ µνis the spin tensor of the test-particle [see Ref. [59] and also Eqs. (2.4)–(2.7) in Ref. [ 14]]. Equa- tion (56) leads to the correct Hamiltonian for a spinning particle in curved spacetime, at linear order in the parti- cle’s spin [ 59]. To modify the Hamilton-Jacobi equation, a suitable ansatz is µ2+gµν eff(r,SKerr)pµpν+2Eρσ µpµS∗ ρσ +[Bµνλ ρσ(r)pµpνpλ +Bµνλτα ρσ(r)pµpνpλpτpα]Sρσ ∗ +Aµνλτ(r,SKerr)pµpνpλpτ +Aµνλτρσ(r,SKerr)pµpνpλpτpρpσ +···= 0. (57) If we make use at lowest order of the condition µ2+ gµν effpµpν≃0, we can replace ptwith the spatial compo- nents of the momentum, and obtain the following gener- alized form of the effective Hamiltonian Heff= βipi+α/radicalBig µ2+γijpipj+Q4(p)+QS(r,p,S∗,SKerr) −µ 2M r3(δij−3ninj)S∗ iS∗ j+HS, (58)whereQ4(p) is a quartic term in the momenta [ 62], which is due to the presence of the quartic term Aµνλτpµpνpλpτin Eq. (57), andQS(r,p,S∗,SKerr) is a term linear in S∗andSKerr QS(r,p,S∗,SKerr) =QSKerr i(r,p)Si Kerr+QS∗ i(r,p)Si ∗. (59) In particular, the term QS∗ i(r,p)Si ∗comes from the termsBµνλ ρσ(r)pµpνpλSρσ ∗andBµνλτα ρσpµpνpλpτpαSρσ ∗ in Eq. (57), while the term QSKerr i(r,p)Si Kerrcomes from AtνλτptpνpλpτandAtνλτρσptpνpλpτpρpσ(through the dependence of the tensors AµνλτandAµνλτρσon SKerr). Finally, the term HSin Eq. (58) comes from the presence of 2 Eρσ µpµS∗ ρσin Eq. (57). As already stressed, this happens because Eq. ( 56) leads to the correctHamil- tonian for a spinning particle in curved spacetime, and in particular to HS, which is the spin-dependent part of that Hamiltonian [ 59]. Through 3.5PN order the quantities QSKerr i(r,p)Si Kerr andQS∗ i(r,p)Si ∗must have the form Qs i(r,p)si=µ r2c3ǫijknjpksi× /braceleftbigg1 c2/parenleftbigg c1M r+c2ˆp2+c3(n·ˆ p)2/parenrightbigg +1 c4/bracketleftBigg c4ˆp4+c5/parenleftbiggM r/parenrightbigg2 +c6(n·ˆ p)4+ c7ˆp2M r+c8(n·ˆ p)2M r+c9(n·ˆ p)2ˆp2/bracketrightbigg/bracerightbigg , (60) wheresstands for either SKerrorS∗, while the coeffi- cientscn,n= 1,...9 are determined by the mapping of the effective to the real description. A straightforward computation leads to QS=QS2.5PN+QS3.5PN, (61) where QS2.5PN(r,p,S∗,SKerr) =µ r3c5/braceleftbigg (SKerr·L)/bracketleftbigg −2(a0+η)M r+1 4(8a0+3η)(Q−1)−3(2a0+3η)∆r Σ(n·ˆp)2/bracketrightbigg +(S∗·L)/bracketleftbigg1 2(−4b0+7η)M r+(2b0+η)(Q−1)−3 2(4b0+5η)∆r Σ(n·ˆp)2/bracketrightbigg/bracerightbigg ,(62) QS3.5PN(r,p,S∗,SKerr) =µ r3c7/braceleftBigg (SKerr·L)/bracketleftBigg (−6a0−2a2−20η−a0η−3η2)/parenleftbiggM r/parenrightbigg2 +5 8(−16a3+24a0η+27η2)× ∆2 r Σ2(n·ˆp)4+1 8(16a1−7η−4a0η)(Q−1)2+1 4(−8a1+8a2−35η+6a0η+11η2)M r×10 (Q−1)+3 8(−16a1+16a3−20a0η−13η2)∆r Σ(n·ˆp)2(Q−1)+1 8(−80a0−32a1 −64a2−48a3−64η+48a0η−3η2)M r∆r Σ(n·ˆp)2/bracketrightbigg + (S∗·L)/bracketleftBigg 1 4(−24b0−8b2+127η−4b0η−37η2)/parenleftbiggM r/parenrightbigg2 +5(−2b3+3b0η+3η2)× ∆2 r Σ2(n·ˆp)4+1 8(16b1−7η−4b0η−η2)(Q−1)2+1 8(−16b1+16b2−61η+12b0η +24η2)M r(Q−1)+3 8(−16b1+16b3+9η−20b0η−14η2)∆r Σ(n·ˆp)2(Q−1) +1 4(−40b0−16b1−32b2−24b3+27η+24b0η+6η2)M r∆r Σ(n·ˆp)2/bracketrightbigg/bracerightbigg . (63) Finally, the EOB Hamiltonian is obtained by inserting Eq. (58) into Eq. ( 53). III. THE EFFECTIVE-ONE-BODY DYNAMICS FOR EQUATORIAL ORBITS We stress that the EOB models introduced in the pre- vious sections have the correct test-particle limit, for both non-spinning and spinning black holes (for generic orbits and arbitrary spin orientations), and that the test- particle limit is recovered non-perturbatively, (i.e., at all PN orders). This is because in order to build our mod- els, in Sec. IICwe started from the Hamiltonian derived in Ref. [59], which correctly reproduces the Mathisson- Papapetrou-Pirani equation describing the motion of a classical spinning particle in a generic curved space- time [63–67]. The EOB models that we present in this papersharethisfeaturewithourearliermodel[ 46],which was valid through 3PN order in the non-spinning sec- tor and through 2.5PN order in the spinning sector, but not with other EOB models for spinning black-hole bi- naries, which recover the test-particle limit only approx- imately [ 15]. Other attractive features of our models are evident when considering configurations with spins parallel to the orbital angular momentum, which correspond, in the effective EOB dynamics, to a particle moving on equa- torial orbits. For aligned spins and equatorial orbits, in fact, both the models with dynamicaland non-dynamical spin mapping predict the existence of an innermost sta- ble circular orbits (ISCO), for all values of the system’s parameters. This feature is again shared by our earlier model [46], but not by other EOB models for spinning black-hole binaries [ 15], which do not present ISCOs for large values of the spins. While the non-existence of an ISCO is not necessarily a sign that a model is flawed, its presence helps reproduce the results of numerical- relativity simulations for binaries with aligned spins [ 68]. To calculate the radius and the orbital angular mo- mentum at the ISCO for our EOB models, we solve nu--1 -0.5 0 0.5 1χ-0.4-0.200.20.40.60.81χISCO 2.5PN, q = 1 2.5PN, q = 0.5 2.5PN, q = 0.1 3.5PN, q = 1 3.5PN, q = 0.5 3.5PN, q = 0.1 FIG. 1: The spin parameter of the binary at the ISCO given by Eq. ( 67) for the 2.5PN and 3.5PN EOB models with dy- namical mapping of the spins, for binaries having spins par- allel toL, mass ratio q=m2/m1and spin-parameter projec- tions onto the direction of Lgiven by χ1=χ2=χ. merically the following system of equations [ 36] ∂Himproved real(r,pr= 0,Lz) ∂r= 0, (64) ∂2Himproved real(r,pr= 0,Lz) ∂r2= 0,(65) with respect to randLz=pφ. The solutions can then be used to evaluate the ISCO frequency via ΩISCO=∂Himproved real(rISCO,pr= 0,LISCO z) ∂Lz,(66) which follows immediately from the Hamilton equations. The values of rISCOandLISCO zcan also be used to calculate the binding energy at the ISCO via Ebind= Himproved real−M. This quantity is interesting because it11 -1 -0.5 0 0.5 1χ02468100 x |EISCObind / M|2.5PN, q = 1 2.5PN, q = 0.5 2.5PN, q = 0.1 3.5PN, q = 1 3.5PN, q = 0.5 3.5PN, q = 0.1 FIG. 2: The same as in Fig. 1but for the binding energy of the binary at the ISCO. corresponds to the mass lost in gravitational waves dur- ing the binary’s inspiral, and is therefore a lower limit to the total mass loss, to which it reduces for η→0 (when the fluxes during the merger and the ringdown become negligible [ 46]). Similarly, one can estimate the spin of the binary at the ISCO via χISCO=Sz 1+Sz 2+Lz ISCO (M+Ebind ISCO)2. (67) This expression clearly neglects the mass and angular momentum lost during the merger and ringdown phases, but it is useful as qualitative diagnostics of our model, and it reduces to the spin of the final black-hole remnant whenη→0(again, becauseinthislimitthefluxesduring the merger and the ringdown become negligible [ 46]). We re-write the metric potentials ∆ tand ∆ rgiven in Eqs. (33), (34), using the “log-model” of Ref. [ 46] [see Eqs. (5.71) and (5.73)–(5.83) of that paper], and assume K(η) = 1.447−0.1574η−9.082η2.(68) The value of K(η) forη= 0 ensures [ 46,69] that the ISCO frequency for extreme mass-ratio non-spinning bi- naries predicted by our EOB models agrees with the ex- act result of Ref. [ 70], which calculated the shift of the ISCO frequency due to the conservative part of the self- force. The linear and quadratic terms in ηin Eq. (68) are such that our EOB models accurately reproduce numer- ical relativity simulations for non-spinning binaries with mass ratios ranging from q= 1/6 toq= 1 [68]. We fix the gauge parameters to the following values: a0=−3 2η, b0=−5 4η, (69) a1=1 2η2, b1=1 16η(9+5η), a2=1 8η(7−8η), b2=1 8η(17−5η),-1 -0.5 0 0.5 1χ00.10.20.3M ΩISCO2.5PN, q = 1 2.5PN, q = 0.5 2.5PN, q = 0.1 3.5PN, q = 1 3.5PN, q = 0.5 3.5PN, q = 0.1 FIG. 3: The same as in Fig. 1but for the ISCO frequency. a3=−9 16η2, b3=−3 8η2, (70) which we determine by requiring that all the terms in- volving∆ rˆp·n/Σcancelout in ∆(1) σ∗and∆(2) σ∗[Eqs.(51), (52)], or equivalently in QS2.5PNandQS3.5PN[Eqs.(62), (63)]. Different choices of the gauge parameters produce qualitatively similar results for the ISCO quantities that we described above. Focusing on systems with spins aligned with the or- bital angular momentum L, and denoting with S1,2= χ1,2m2 1,2the projections of the spins along the direc- tion ofL, we consider binaries with χ1=χ2=χand mass ratios q=m2/m1= 0.1, 0.5 and 1. In particular, in Figs.1–3we show how the ISCO quantities described abovechangeasaconsequenceofincludingthe3.5PNSO terms in our EOB model with dynamical spin mapping. More specifically, we calculate ΩISCOM,EISCO bind/Mand χISCOusing the Hamiltonian ( 54), with and without the 3.5PNterms givenby ∆(2) σ∗. As can be seen, the inclusion of the 3.5 PN terms does not change the ISCO quantities significantly for χ≤0, while small differences appear for χ >0. (In the caseofΩISCOM, however,these differences grow quite large when χ→1.) Overall, Figs. 1–3suggest that the model has reasonable convergence properties for radiir≥rISCO. The results for the model with non-dynamical spin mappingaresimilar[i.e., acomparisonoftheISCOquan- tities calculated using the Hamiltonian ( 58), with and without the 3.5PN term Q3.5PN, gives similar results]. In general, however, the model with non-dynamical spin mapping presents lower values for ΩISCOMat high spins and for comparable mass ratios (see Fig. 4, where we compare the 3.5PN models with dynamical and non- dynamical spin mapping). Another attractive feature of our models is the exis- tence ofa peak ofthe orbitalfrequency during the plunge starting at the ISCO. More precisely, we assume that the12 -1 -0.5 0 0.5 1χ00.10.20.3M ΩISCO3.5PN, q = 1, non-dyn 3.5PN, q = 0.5, non-dyn 3.5PN, q = 0.1, non-dyn 3.5PN, q = 1, dyn 3.5PN, q = 0.5, dyn 3.5PN, q = 0.1, dyn FIG. 4: The ISCO frequency for the 3.5PN EOB models with dynamical (dyn) and non-dynamical (non-dyn) mapping of the spins, for binaries having spins parallel to L, mass ratio q=m2/m1and spin-parameter projections ontothe direction ofLgiven by χ1=χ2=χ. -1 -0.5 0 0.5 1χ0.10.150.20.250.30.350.4M Ωmax2.5PN, q = 1 2.5PN, q = 0.5 2.5PN, q = 0.1 3.5PN, q = 1 3.5PN, q = 0.5 3.5PN, q = 0.1 FIG. 5: The same as in Fig. 1, but for the maximum of the orbital frequency during the plunge. effective particle starts off with no radial velocity at the ISCO (thus having angular momentum LISCOand energy EISCO), and we evolve the geodesic equations by calculat- ing the radial momentum prduring the plunge from en- ergy and angular momentum conservation. We then cal- culate the orbital frequency Ω = ∂Himproved real/∂Lzalong the trajectory and find that it presents a peak Ω max. This is not surprising because the same behavior was observed to be generic in our earlier model [ 46]. The values of MΩmaxfor binaries with spins parallel to L, as function of χ=χ1=χ2, are shown in Fig. 5for mass ratios q= 1,0.5 and 0.1, for the EOB model with dynamical spin mapping at 2.5PN and 3.5PN. As can-1 -0.5 0 0.5 1χ0.10.150.20.250.30.350.4M Ωmax3.5PN, q = 1, non-dyn 3.5PN, q = 0.5, non-dyn 3.5PN, q = 0.1, non-dyn 3.5PN, q = 1, dyn 3.5PN, q = 0.5, dyn 3.5PN, q = 0.1, dyn FIG.6: Thesame as inFig. 4, butfor themaximumfrequency during the plunge. be seen the differences introduced by the 3.5 PN terms, although reasonable, are larger than for the ISCO quan- tities. This may be because the plunge happens at radii that are smaller than rISCOand approach the horizon’s radius, thus making the higher order PN terms more and more important. The results for the model with non- dynamical spin mapping are generally similar, although they differ slightly at high spins. In particular, in Fig. 6 we compare the 3.5PN models with dynamical and non- dynamical spin mapping. As can be seen, for q= 0.5 and q= 1 the predictions of the two models are very close, while for q= 0.1the model with dynamicalspin mapping presents somewhat lower maximum frequencies. Also, we stress that the values of MΩmaxfor spin an- tialigned with the angular momentum (i.e., χ1=χ2= χ <0) are quite sensitive to the values of the gauge pa- rameters a0–a3andb0–b3. For instance, setting all the gauge parameters to 0 makes the behavior of MΩmax withχnon-monotonic if the 3.5PN models (both with dynamical and non-dynamical spin mapping) are consid- ered. This effect does not appear in the 2.5PN models, and can in principle be important for the calibration of our model with numerical-relativity simulations. More details on this will be given in a follow-up paper [ 68]. Even worse, when the gauge parameters are set to zero the difference in MΩmaxbetween the 2.5PN and 3.5PN models is larger than in Fig. 5, a sign that the model probably converges more slowly in this gauge. In light of this, it seems preferable to use the gauge parameters (69)–(70), which by canceling out the radial momentum ∆rˆp·n/Σ from∆(1) σ∗and∆(2) σ∗(and from QS2.5PNand QS3.5PN) provide a rather regular and monotonic behav- iorforMΩmaxandreasonabledifferencesbetween the 2.5 and 3.5PN models. Finally, in Fig. 7we show the predictions of our EOB model with dynamical spin mapping for the ISCO fre- quency of a system with q=m2/m1= 10−3,χ1=χ13 -0.5 0 0.5 1χ-20-100 cΩ3.5PN2.5PN self-force FIG. 7: The shift of the ISCO frequency cΩ, defined in Eq. (71), for the 2.5PN and 3.5PN EOB models with dynam- ical mapping of the spins, for a binary having spins parallel toL, mass ratio q=m2/m1= 10−3and spin-parameter pro- jections onto the direction of Lgiven by χ1=χandχ2= 0. andχ2= 0 (the results for the EOB model with non- dynamical spin mapping are similar). More precisely, we show the fractional deviation from the Kerr ISCO fre- quency normalized by the mass ratio, cΩ=1 q/parenleftbiggΩISCOM|q ΩISCOM|Kerr−1/parenrightbigg , (71) as a function of χ, as proposed in Ref. [ 69]. This ISCO shift is caused by the conservative part of the self-force and has been calculated exactly by Ref. [ 70] in the case of a Schwarzschild spacetime ( χ= 0). The results of Ref. [70] iscΩ= 1.2513+O(q) [see also Ref. [ 71]], and is denoted by a filled circle in Fig. 7. As can be seen, both the 2.5 and 3.5PN models predict cΩ>0, except when χ/greaterorsimilar0.83. This change of behavior of the EOB predic- tion is common also to our earlier model of Ref. [ 46], and might have important implication for configurations that might violate the Cosmic Censorship Conjecture [ 72,73]. However, the behavior of cΩ, which seems to diverge as χ approaches 1, suggests that this might simply be a spuri- ouseffectduetotheincompleteknowledgeofthefunction K[Eq. (68)] and to the fact that the EOB model only re- produces the SS coupling at leading PN order (2PN). As mentionedin Ref. [ 46],Kmayin generaldepend not only onηbut also on χ2, and these spin-dependent terms can be very important for near-extremal spins, and so will the 3PN SS couplings. It isthereforepossiblethat afterreconstructingthe full functional form of K(by comparing to future self-force calculationsinKerrortonumerical-relativitysimulations for spinning binaries) and extending the EOB model to include the 3PN SS couplings, cΩmight remain positive even at high spins.IV. CONCLUSIONS Recently, Ref. [ 19] has computed the 3.5PN SO effects in the ADM Hamiltonian. We have taken advantage of this result and extended the EOB Hamiltonian of spin- ning black holes to include these higher-order SO cou- plings. Building on previous work [ 39,58], and in particular on the EOB Hamiltonian of Refs. [ 46,59], which repro- duces the SO test-particle couplings exactly at all PN orders, we have worked out two classes of EOB Hamil- tonians, which differ by the way the spin variables are mapped between the effective and real descriptions. One class of EOB Hamiltonians is the straightforward exten- sion to the next PN order of the EOB Hamiltonian of Ref. [46]. It uses a mapping between the real and effec- tive spin variables that depends on the dynamical orbital variables p2,n·pandr. By contrast, the other class of EOB Hamiltonians uses a mapping between the real and effective spin variables that does not depend on these dynamical orbital variables. We achieved this result at the cost of modifying the Hamilton-Jacobi equation of a spinning test-particle. Quite interestingly, when restricting to spins aligned or antialigned with the orbital angular-momentum and to equatorial circular orbits, we find that the predictions of these two classes of EOB Hamiltonians for the ISCO frequency, energy and angular momentum, and for the maximum of the orbital frequency during the plunge are generallysimilar. However,for high spins the model with dynamical mapping of the spins may present somewhat lowermaximum frequencies and largerISCO frequencies. As pointed out originally in Ref. [ 58], several gauge parameters can enter the canonical transformation that maps the real and effective Hamiltonians. If the Hamil- tonian were known exactly, i.e., at all PN orders, then physical effects should not depend on these parameters. However, since we know the Hamiltonian only at a cer- tain PN order, we expect these gauge parameters to lead to non-negligible differences. In fact, we obtained that when setting all the gauge parameters to zero, the max- imum frequency during the plunge has a non-monotonic dependence on the spins, and varies quite significantly as a consequence of the inclusion of the 3.5 PN SO cou- plings. We found instead that when choosing the gauge parameterssothatthetermsdependingontheradialmo- mentum disappear from our spin mapping (in the model with dynamical spin mapping) or from the modifications to the Hamilton-Jacobi equation (in the model with non- dynamical spin mapping), the maximum frequency dur- ing the plunge has a much more regular behavior and varies by small amounts when adding the 3.5PN SO cou- plings. This suggests that such a choice of the gauge pa- rameters may accelerate the convergence of the model’s results in the strong-field region where the plunge takes place. The EOB Hamiltonians derived in this paper can be calibrated to numerical-relativity simulations with the14 goal of building analytical templates for LIGO and Virgo searches. AfirstexamplewasobtainedinRef.[ 47], where theEOBHamiltonianat2.5PNorderintheSOcouplings of Ref. [58] was calibrated to two highly-accurate numer- ical simulations. Results that use the EOB Hamiltonian at 3.5PN order developed in this paper will be reported in the near future [ 68]. Lastly, while finalizing this work, Ref. [ 74] appeared in the archives as a preprint. Both this paper and Ref. [74] derive the effective gyromagnetic coefficients [see Eq. ( 31)], but with two different methods. Our computation uses the Lie method to generate both the purely-orbitalandthespin-dependentcanonicaltransfor- mations, while Ref. [ 74] first applies explicitly the purely- orbital transformation from ADM to EOB coordinates, and then uses Eq. ( 7) to account for the effect of a spin- dependent canonical transformation. As a result of these different procedures, and as discussed in Sec. IIA, the 2.5PN gauge parameters in our spin-dependent canoni- cal transformation coincide with those of Ref. [ 74], butthe 3.5PN gauge parameters have different meanings in the two approaches and therefore do not coincide. How- ever, by suitably expressing our 3.5PN gauge parameters in terms of those of Ref. [ 74], we find that our effec- tive gyromagnetic coefficients fully agree with those of Ref. [74]. This amounts to saying that our gyromagnetic coefficients agreewith those of Ref. [ 74] up to a canonical transformation, and are therefore physically equivalent. More importantly, in this paper we have focused on and worked out two classes of EOB Hamiltonians that are different from the one considered in Ref. [ 74]. Acknowledgments E.B. and A.B. acknowledge support from NSF Grant PHY-0903631. A.B. also acknowledges support from NASA grant NNX09AI81G. [1] B. Abbott et al. (LIGO Scientific Collaboration), Rep. Prog. Phys. 72, 076901 (2009). [2] F. Acernese et al. (Virgo Collaboration), Class. Quantu m Grav.25, 184001 (2008). [3] H. Grote (GEO600 Collaboration), Class. Quant. Grav. 25, 114043 (2008). [4] K. Kuroda and the LCGT Collaboration, Class. Quan- tum Grav. 27, 084004 (2010). [5] B. Abbott et al. (LIGO Scientific Collaboration), Phys. Rev.D73, 062001 (2006), gr-qc/0509129. [6] B. Abbott et al. (LIGO Scientific Collaboration), Phys. Rev. D77, 062002 (2008). [7] B. P. Abbott et al. (LIGO Scientific Collaboration), Phys. Rev. D80, 047101 (2009), 0905.3710. [8] J. Abadie et al. (LIGO Scientific Collaboration), Phys. Rev.D82, 102001 (2010), 1005.4655. [9] J. Abadie et al. (LIGO Scientific Collaboration and Virgo) (2011), 1102.3781. [10] B. Abbott et al. (LIGO Scientific Collaboration), Phys. Rev. D78, 042002 (2008), 0712.2050. [11] L. E. Kidder, C. M. Will, and A. G. Wiseman, Phys. Rev. D47, 3281 (1993). [12] B. J. Owen, H. Tagoshi, and A. Ohashi, Phys. Rev. D 57, 6168 (1998). [13] H. Tagoshi, A. Ohashi, and B. J. Owen, Phys. Rev. D 63, 044006 (2001). [14] G. Faye, L. Blanchet, and A. Buonanno, Phys. Rev. D 74, 104033 (2006). [15] T. Damour, P. Jaranowski, and G. Sch¨ afer, Phys. Rev. D77, 064032 (2008). [16] R. A. Porto, Class. Quant. Grav. 27, 205001 (2010). [17] D. L. Perrodin, ArXiv (2010), gr-qc 1005.0634. [18] M. Levi, Phys. Rev. D82, 104004 (2010), 1006.4139. [19] J. Hartung and J. Steinhoff, Arxiv (2011), gr-qc 1104.3079. [20] L. E. Kidder, Phys. Rev. D 52, 821 (1995). [21] C. M. Will, Phys. Rev. D 71, 084027 (2005), gr- qc/0502039.[22] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74, 104034 (2006). [23] L. Blanchet, A. Buonanno, and G. Faye (2011), 1104.5659. [24] B. Mik´ oczi, M. Vas´ uth, and L. A. Gergely, Phys. Rev. D 71, 124043 (2005), astro-ph/0504538. [25] R. A. Porto, Phys. Rev. D 73, 104031 (2006), gr- qc/0511061. [26] R. A. Porto and I. Z. Rothstein, Phys.Rev.Lett. 97, 021101 (2006), gr-qc/0604099. [27] S. Hergt and G. Schaefer, Phys. Rev. D 77, 104001 (2008). [28] J. Steinhoff, S. Hergt, and G. Schaefer, Phys. Rev. D 77, 081501(R) (2008). [29] S.HergtandG.Sch¨ afer, Phys.Rev.D 78, 124004(2008). [30] J. Steinhoff, S. Hergt, and G. Sch¨ afer, Phys. Rev. D 78, 101503 (2008). [31] J. Steinhoff, G. Sch¨ afer, and S. Hergt, Phys. Rev. D 77, 104018 (2008). [32] R. A. Porto and I. Z. Rothstein, Phys.Rev. D78, 044012 (2008), 0802.0720. [33] R. A.PortoandI.Z.Rothstein, Phys.Rev.D 78, 044013 (2008), 0804.0260. [34] M. Levi, Phys.Rev. D82, 064029 (2010), 0802.1508. [35] R. A. Porto, A. Ross, and I. Z. Rothstein (2010), 1007.1312. [36] A. Buonanno and T. Damour, Phys. Rev. D 62, 064015 (2000). [37] A. Buonanno and T. Damour, Phys. Rev. D 59, 084006 (1999). [38] T. Damour, P. Jaranowski, and G. Sch¨ afer, Phys. Rev. D62, 084011 (2000). [39] T. Damour, Phys. Rev. D 64, 124013 (2001). [40] A. Buonanno, Y. Chen, and T. Damour, Phys. Rev. D 74, 104005 (2006). [41] A. Buonanno, G. B. Cook, and F. Pretorius, Phys. Rev. D75, 124018 (2007). [42] A. Buonanno, Y. Pan, J. G. Baker, J. Centrella, B. J.15 Kelly, S. T. McWilliams, and J. R. van Meter, Phys. Rev. D76, 104049 (2007). [43] Y. Pan, A. Buonanno, J. G. Baker, J. Centrella, B. J. Kelly, S. T. McWilliams, F. Pretorius, and J. R. van Meter, Phys. Rev. D 77, 024014 (2008). [44] M. Boyle, A. Buonanno, L. E. Kidder, A. H. Mrou´ e, Y. Pan, H. P. Pfeiffer, and M. A. Scheel, Phys. Rev. D 78, 104020 (2008). [45] A. Buonanno, Y. Pan, H. P. Pfeiffer, M. A. Scheel, L. T. Buchman, and L. E. Kidder, Phys. Rev. D 79, 124028 (2009). [46] E. Barausse and A. Buonanno, Phys. Rev. D81, 084024 (2010). [47] Y. Pan, A. Buonanno, L. Buchman, T. Chu, L. Kid- der, H. Pfeiffer, and M. Scheel, Phys. Rev. D81, 084041 (2010). [48] Y. Pan, A. Buonanno, R. Fujita, E. Racine, and H. Tagoshi (2010), 1006.0431. [49] T. Damour and A. Nagar, Phys. Rev. D 77, 024043 (2008). [50] T. Damour, A. Nagar, E. N. Dorband, D. Pollney, and L. Rezzolla, Phys. Rev. D 77, 084017 (2008). [51] T. Damour, A. Nagar, M. Hannam, S. Husa, and B. Br¨ ugmann, Phys. Rev. D 78, 044039 (2008). [52] T. Damour, B. R. Iyer, and A. Nagar, Phys. Rev. D 79, 064004 (2009). [53] T. Damour and A. Nagar, Phys. Rev. D 79, 081503 (2009). [54] N. Yunes, A. Buonanno, S. A. Hughes, M. Cole- man Miller, and Y. Pan, Phys. Rev. Lett. 104, 091102 (2010), 0909.4263. [55] N. Yunes et al., Phys. Rev. D83, 044044 (2011), 1009.6013.[56] S. Bernuzzi, A. Nagar, and A. Zenginoglu, Phys.Rev. D83, 064010 (2011), 1012.2456. [57] Y. Pan, A. Buonanno, M. Boyle, L. T. Buchman, L. E. Kidder, et al. (2011), 1106.1021. [58] T. Damour, P. Jaranowski, and G. Schaefer, Phys. Rev. D77, 064032 (2008), 0711.1048. [59] E. Barausse, E. Racine, and A. Buonanno, Phys.Rev. D80, 104025 (2009), 0907.4745. [60] T. Damour andG. Sch¨ afer, NuovoCimento Soc. Ital. Fis. 101 B, 127 (1988). [61] G. Benettin, in Lectures at the Porquerolles School 2001: Hamiltonian systems and Fourier analysis , edited by D. Benest, C. Froeschle, and E. Lega (Cambridge Sci- entific, 2004). [62] T. Damour, P. Jaranowski, and G. Sch¨ afer, Phys. Rev. D62, 084011 (2000). [63] M. Mathisson, Acta Phys. Pol. 6, 163 (1937). [64] A. Papapetrou, Proc. Phys. Soc. A 64, 57 (1951). [65] A. Papapetrou, Proc. R. Soc. Lond. A 209, 248 (1951). [66] E. Corinaldesi and A. Papapetrou, Proc. R. Soc. Lond. A209, 259 (1951). [67] F. Pirani, Acta Phys. Pol. 15, 389 (1956). [68] A. Taracchini et al. (2011), in preparation. [69] M. Favata, Phys.Rev. D83, 024028 (2011). [70] L. Barack and N. Sago, Phys. Rev. Lett. 102, 191101 (2009). [71] T. Damour, Phys. Rev. D81, 024017 (2010), 0910.5533. [72] E. Barausse, V. Cardoso, and G. Khanna, Phys. Rev. Lett.105, 261102 (2010). [73] E. Barausse, V. Cardoso, and G. Khanna, ArXiv (2011), gr-qc 1106.1692. [74] A. Nagar, ArXiv (2011), gr-qc 1106.4349.

1611.01840v1.Effective_magnetic_interactions_in_spin_orbit_coupled__d_4__Mott_insulators.pdf

Eective magnetic interactions in spin-orbit coupled d4Mott insulators Christopher Svoboda,1Mohit Randeria,1and Nandini Trivedi1 1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA (Dated: November 8, 2016) Transition metal compounds with the ( t2g)4electronic conguration are expected to be nonmag- netic atomic singlets both in the weakly interacting regime due to spin-orbit coupling, as well as in the Coulomb dominated regime with oppositely aligned L= 1 andS= 1 angular momenta. However, starting with the full multi-orbital electronic Hamiltonian, we show the low energy eec- tive magnetic Hamiltonian contains isotropic superexchange spin interactions but anisotropic orbital interactions. By tuning the ratio of superexchange to spin-orbit coupling JSE=, we obtain a phase transition from nonmagnetic atomic singlets to novel magnetic phases depending on the strength of Hund's coupling, the crystal structure and the number of active orbitals. Spin-orbit coupling plays a non-trivial role in generating a triplon condensate of weakly interacting excitations at antiferro- magnetic ordering vector ~k=~ , regardless of whether the local spin interactions are ferromagnetic or antiferromagnetic. In the large JSE=regime, the localized spin and orbital moments produce anisotropic orbital interactions that are frustrated or constrained even in the absence of geometric frustration. Orbital frustration leads to frustration in the spin channel opening up the possibility of spin-orbital liquids with both spin and orbital entanglement. PACS numbers: 75.30.Et,71.70.Ej,75.10.Jm I. INTRODUCTION Between weakly correlated topological insulators and strongly correlated 3d transition metal oxides lie 5 dcom- pounds that combine both strong spin-orbit coupling and correlations on an equal footing. In contrast to the well studied 5d5materials, the eect of strong spin-orbit cou- pling in 4d4and 5d4systems have been sparsely studied due to expectations that these will naturally lead to non- magnetic insulating behavior. However there are sev- eral experimental counter examples to this notion. The rst example is Ca 2RuO 4which displays a moment of 1.3B.1,2Recently both double perovskite irridates3{5 and honeycomb ruthenates6in thed4conguration have been found to show magnetism. It has been argued that partial quenching of the orbital angular momentum from the presence of lattice distortions is the root cause. Very recent developments7{10on Ba 2YIrO 6have piqued inter- est on the origin of magnetism in this 5 d4system because the compound is negligibly distorted and still shows a Curie response. In transition metal oxides with oxygen octahedra, the large crystal eld splitting puts d4ions into the t4 2gelec- tronic conguration. For materials with strong spin-orbit coupling, the j= 3=2 band is lled and the j= 1=2 band is empty leading to the conclusion that weakly correlated d4materials are non-magnetic. However when Coulomb interactions are strong, a total spin S= 1 and orbital angular momentum L= 1 lead to a total angular mo- mentumJ= 0 on every d4ion with no magnetism. Thus both jjcoupling and LScoupling schemes lead to the same conclusion that a single atom is in a J= 0 singlet state and therefore non-magnetic as shown in Fig. 1(a). We build on previous work by Khaliullin11that pro- posed an \exciton condensation" mechanism, more ac- curately a condensation of J= 1 triplon excitations,to drive the onset of antiferromagnetism in nominally non-magnetic d4systems and our previous study12show- ing that ferromagnetic superexchange interactions caused by strong Hund's coupling can precipitate ferromag- netic coupling. In this work we start with the atomic multi-orbital Hamiltonian with intra- and inter-orbital Coulomb interactions and spin-orbital coupling speci- cally fort4 2gsystems. We next allow hopping between atoms and investigate all cases of orbital geometries{ the idealized fully symmetric case when all orbitals partici- pate in hopping, as well as more realistic cases suitable for simple cubic and face-centered cubic lattices. For each case, we derive the eective spin-orbital superexchange Hamiltonian which competes with spin-orbit coupling to produce strong deviations from the non-magnetic atomic behavior. These results are obtained both using exact diagonalization on a two-site problem and perturbation theory for the eective magnetic interactions. Tuning the superexchange interactions JSErelative to spin-orbit coupling , we rst see the formation of lo- cal moments followed by a Bose condensation of weakly interacting J= 1 triplet excitations, or triplon condensa- tion. Rather remarkably, regardless of the local spin in- teractions favoring antiferromagnetic spin superexchange (spin-AF) at small Hund's coupling or ferromagnetic spin superexchange (spin-F) at large Hund's coupling, the J= 1 triplons condense at the ~k=~ point. This re- sult that the rotationally invariant spin-orbit coupling can eectively ip the sign of superexchange is unusual and unique to spin-orbital coupled systems. In the oppo- site regime where JSEdominates, the orbital interactions are frustrated even in the absence of geometric frustra- tion and can potentially lead to orbital liquid phases. Even when = 0 and the local spin interactions are simple Heisenberg FM or AFM, the frustrated orbital interactions generate frustration in the spin channel asarXiv:1611.01840v1 [cond-mat.str-el] 6 Nov 20162 well, leading to the possibility of ground states with both orbital and spin entanglement on lattices without geo- metric frustration. This is summarized schematically in Fig. 1(b). The paper is organized in the following way. In Sec- tion II we introduce the lattice Hamiltonian used as the basis for the rest of the paper which includes electron hopping, atomic spin-orbit coupling, and an eective multi-orbital Coulomb interaction that captures Hund's rules. The orbital geometries for hopping used through- out the paper include both a highly symmetric toy model to be used as a simplied diagnostic tool as well as two other more realistic cases found in perovskites. In Section III we use exact diagonalization to study a two-site specialization of the problem introduced in Sec- tion II. Ref. 13 has used a similar procedure to study transition metal systems with other electron counts. Lo- cal magnetic moments are absent when spin-orbit cou- pling is large, as expected in the atomic picture, but elec- tron hopping introduces sizeable moments when t when two or three orbitals strongly overlap between sites. Although a single orbital overlap can also promote su- perexchange which competes with spin-orbit coupling, the number of superexchange pathways is limited and local moments do not form for any reasonable ratio of t=. In Section IV we derive an eective magnetic Hamil- tonian in terms of orbital angular momentum and spin operators using second order perturbation theory. We check that the spin-orbital superexchange Hamiltonian captures both spin-AF and spin-F interactions between spins depending on the value of Hund's coupling, and the sum of spin-orbit coupling and the spin-orbital superex- change Hamiltonian reproduce the phases found in exact diagonalization of a two-site system. In Section V, we give a qualitative description of how bond-dependent spin-orbital superexchange results in or- bital frustration. However, nding solutions to orbitally frustrated models can be challenging and is outside the scope of the present paper. In Section VI we rst review the \excitonic" conden- sation mechanism where the Bose condensation of van Vleck excitations gives magnetism to d4systems with spin-orbit coupling. Although the condensation mecha- nism involves approximations to full spin-orbital models derived in the previous section, it gives valuable insight into the behavior at large spin-orbit coupling. Regardless of the nature of local interactions, only AF condensates are supported for the models studied, and we give the the critical superexchange required for AF condensation for the three orbital geometries studied. Section VII discusses potential materials realizations and experiments beyond those mentioned in the intro- duction. (a) jjcoupling LScoupling j= 3=2j= 1=2 S=L= 1 J= 0 J= 0 (b) zJSE zJSE!JH=U! van Vleck PMAF Triplon BECSpin AFSpin F FIG. 1. (a) The single site total angular momentum is zero in both thejjandLScoupling schemes. (b) Schematic phase diagram of the spin-orbital model appearing in (5) pitting spin-orbit coupling against superexchange JSEwhereis the spin-orbit coupling energy scale and JSEis the superex- change energy scale with zbeing the coordination number. Starting with a van Vleck phase with no atomic moments at largewe nd a triplon condensate at k=~ for all values of the Hund's coupling JH=U. The intermediate regime where zJSEhas not been explored. At large JSEwe obtain eective magnetic Hamiltonians that have isotropic Heisen- berg spin interactions (antiferromagnetic for small JH=Uand ferromagnetic for large JH=U) but the orbital interactions are more complex and anisotropic. We expect novel magnetic phases arising from orbital frustration in the intermediate and largeJSE=regimes. II. MODEL Our model Hamiltonian for t2gsystems is composed of three parts: (i) kinetic part, (ii) Coulomb interaction, and (iii) spin-orbit coupling. H=X hijiH(ij) t+X iH(i) int+X iH(i) so (1) The general form of the kinetic part H(ij) t=X mm0X t(ij) m0mcy im0cjm+ h:c: (2) is given in terms of matrix elements t(ij) m0mbetweent2g orbitalsm0andm(with values yz,zx, andxy) on sitesiandj. The index is for spin. We take the on-site Coulomb interaction to be the t2ginteraction Hamiltonian14 H(i) int= (U3JH)Ni(Ni1) 2+JH5 2Ni2S2 i1 2L2 i
(3)3 xyz dxydyz dxydyzpz px dxydxy(a) (b) FIG. 2. (a) The Norb= 2 model is an approximation of oxygen mediated electron hopping between t2gorbitals in a simple cubic lattice. Both dxyanddyzorbitals participate in hopping along the ydirection. (b) The Norb= 1 model is an approximation of direct hopping between t2gorbitals on the face of a face-centered cubic lattice. The dxyorbitals are most relevant for hopping in the xyplane. whereNis electron number, Sis total spin, and Lis to- tal orbital angular momentum. The on-site intra-orbital Hubbard interaction is characterized UandJHcharac- terizes the strength of Hund's coupling. We have chosen to useJHinstead ofJto avoid confusion with total an- gular momentum in the next two sections. See Appendix A for details. The atomic spin-orbit coupling has the form H(i) so=X mm0X 0cy im00(lm0m
s0)cim (4) wherelis the projection of angular momentum operators to thet2gsubspace, (lk)m0m=ikm0mso thatll=il, andsis the spin operator with s==2 where we have set~= 1. We focus on three special cases of t(ij) m0mwhich dier by the number of orbitals, Norbparticipating in hopping. Norb= 3: First we consider the orbitally symmetric case where t(ij) m0m=tm0mand all orbitals participate in hopping. While this full rotational symmetry is not usually present in material systems, the Norb= 3 case serves as a diagnostic tool where total angular momen- tum in the system is conserved and correlation func- tions have rotational symmetry. Norb= 2: The next case, t(ij) m0m=tm0m(1km), uses two orbitals participating in hopping, Norb= 2, whileone orbital is blocked. The blocked orbital kis de- termined by the direction of the line connecting sites iandj. This situation is commonly found in simple cubic lattices where tcomes from oxygen-mediated su- perexchange. See Fig. 2(a). Norb= 1: The nal case, t(ij) m0m=tm0mkm, only has one orbital contributing, Norb= 1, while two orbitals are blocked and approximates the hopping between nearest-neighbors on a face-centered cubic lattice. The active orbital kis determined by which plane the sites iandjshare. See Fig. 2(b). III. EXACT DIAGONALIZATION Before analyzing the full lattice problem which will re- quire approximations to be made, it is useful to examine exact results for a pair of interacting sites. We numeri- cally diagonalize (1) for a two-site site system, with site labelsiandj, to extract the magnetic interactions in the Mott limit. We choose the blocked orbital kto be the xyorbital for the Norb= 2 andNorb= 1 models. Fig. 3 gives ground state values of the square of the local total angular momentum, hJ2 ii, for the two-site specialization of (1). For all three types of hopping matrices, small t compared to give negligible local moments since spin- orbit coupling keeps each site in a nonmagnetic Ji= 0 spin-orbital singlet. For larger values of t, local moments may form from the tendency of superexchange to cause spin and orbital ordering which is incompatible with lo- cal spin-orbital singlet behavior on each site. For both Norb= 3 andNorb= 2, this eect is pronounced and requirest=2 at the two-site level. In a lattice, this critical ratio will be reduced due to presence of many neighboring sites contributing to superexchange, hence a smaller hopping tis able to destabilize the atomic sin- glet. ForNorb= 1, the eect is much less pronounced since the number of superexchange paths is limited. When a single orbital is active, the results do not sen- sitively depend on JH=U, however, the presence of strong Hund's coupling results in qualitatively dierent behav- ior for the Norb= 3 andNorb= 2 models. We ex- pect that antiferromagnetic superexchange between spins (spin-AF) is responsible for moment formation and can qualitatively be understood in the following way. Each site has a local total spin Si= 1 and local orbital angular momentum Li= 1 (3Pconguration) from each t2gor- bital being at least singly occupied with one of the three orbitals doubly occupied. To maximize the number of superexchange paths, the orbitals participating in anti- ferromagnetic superexchange should be singly occupied. This means the doubly occupied orbitals try to match up between neighboring sites, see Fig. 4(a). When each pair of singly occupied orbitals between sites is in a spin sin- glet, the two-site system is a spin singlet, but the orbitals are in a ferro-orbital state. The combined spin-AF and orbital-F interactions are responsible for moment forma-4 0 1 2 3 4 0 1 2 3 4 50.000.050.100.150.200.25 0 1 2 3 4 50.000.050.100.150.200.25 0 1 2 3 4 50.000.050.100.150.200.25 0 1 2 3 4 50.000.050.100.150.200.25 0 1 2 3 4 50.000.050.100.150.200.25 0 1 2 3 4 50.000.050.100.150.200.25 (a) (b) (c) (d) (e) (f)JH=U= 0:1 JH=U= 0:2hJ2 iit=U t=Norb= 3 Norb= 3 Norb= 2 Norb= 2 Norb= 1 Norb= 1 FIG. 3. The Hamiltonian in equation (1) is solved for a two- site system. The local total angular momentum squared on one site,hJ2 ii, is plotted for small and large values of Hund's coupling,JH=U= 0:1 andJH=U= 0:2, for the three types of hopping matrices used in the text. (a-b) Hopping using Norb= 3 produces sizable local moments. For small Hund's coupling, the local moment gradually forms as tis turned on. For large Hund's coupling, there is an abrupt formation of large local moments due to an energy level crossing. (c-d) Hopping using Norb= 2 produces qualitatively similar be- havior to the Norb= 3 case. (e-f) Hopping using Norb= 1 produces negligible moments. (a) (b) FIG. 4. (a) The virtual process leaves the rst site in a low spin,S=1 2, conguration and results in antiferromagnetic superexchange. (b) The virtual process leaves the rst site in a high spin, S=3 2, conguration and results in ferromagnetic superexchange.tion in Figs. 3(a) and 3(c). Large values of Hund's coupling can produce a dier- ent ground state via a level crossing at the sharp bound- aries in Figs. 3(b) and 3(d) which are not present in Figs. 3(a) and 3(c). This behavior can be understood by examining the eect of Hund's coupling on the d3d5 states during the virtual d4d4!d5d3!d4d4process. The intermediate d3may have either a maximized spin stateSi=3 2or a minimized spin state Si=1 2, and large values of Hund's coupling make the intermediate Si=3 2states energetically more favorable. Moving an electron o a doubly-occupied orbital leaves the ion in an energetically favorable Si=3 2conguration. For ex- ample, see Fig. 4(b). Since this requires electrons to move onto single-occupancy orbitals on the other site, the Goodenough-Kanamori-Anderson rules15{17indicate the the spin interactions are ferromagnetic (spin-F) but the orbitals are in a singlet state (orbital-AF). Here too, as in the small Hund's coupling case, on a lattice when an electron hops along dierent directions, the doubly occupied orbitals become bond-dependent and lead to anisotropic interactions. IV. EFFECTIVE MAGNETIC HAMILTONIAN We now begin our analysis of the full lattice problem in Eq. (1) by calculating the eective spin-orbital lattice model for each of the three Norbcases. Only the main results are presented here; the details of the calculation are presented in Appendix B. To understand the superexchange mechanisms in the three dierent Norbmodels and how they compete with spin-orbit coupling, we derive an eective magnetic spin- orbital Hamiltonian18within the local3Pspace (spec- troscopic notation2S+1LJ) on each site. This eective Hamiltonian is written as the sum of spin-orbit and su- perexchange terms. He=X iH(i) SOC+X hijiH(ij) SE (5) The rst order spin-orbit correction H(i) SOC= 2Li
Siis qualitatively correct, but we give the second order eec- tive spin-orbit interaction within the local3Pspace to numerically match the energies from exact diagonaliza- tion. H(i) SOC= 2 11 4 JH Li
Si7 402 JH(Li
Si)2(6) The spin-orbital superexchange Hamiltonian, HSE, is constructed using three dierent virtual exchange pro- cesses each dened by the energy values of intermediate multiplets.19{21In the present case, the d3electron con- guration in the virtual process d4d4!d3d5!d4d4 is used to label these superexchange pathways.22,23Each pathway yields a superexchange term which is the prod- uct of a spin interaction and a t(ij) m0m-dependent orbital5 interactionOij. Since4S,2D, and2Plabel the inter- mediated3congurations, we arrive at the three corre- sponding superexchange terms. H(ij) SE=t2 U3JH(2 +Si
Sj)OS ij t2 U(1Si
Sj)OD ij t2 U+2JH(1Si
Sj)OP ij(7) The rst pathway, OS ijcorresponding to the4Sstate, has the lowest energy of all the intermediate states since max- imizing the spin of the d3conguration minimizes the to- tal energy. We see that maximizing the total spin favors spin-F behavior as in Fig. 4(b). The other two pathways, OD ijandOP ijcorresponding to2Dand2P, minimize the total spin with Si= 1=2 in thed3conguration and will favor spin-AF as in Fig. 4(a). The spin-F and spin-AF behaviors may be veried by observing that both 2 + Si
Sjand 1Si
Sjare non- negative. Then the energy due to each pathway may be minimized by simultaneously maximizing the spin part and the orbital part separately. Since each hopping ma- trixtm0muniquely determines the resulting orbital inter- actionsOij, we will compute these orbital interactions ex- plicitly for previous three choices of tm0m(Norb= 3;2;1). We will nd that2Dand2Ppathways will together dom- inate over the4Spathway when JH=Uis small, however, this can change at larger values of JH=U. The spin-orbital models we calculate here are similar to those of d2systems.22,23This follows from the fact that a (t2g)4electronic system is the particle hole con- jugate to a ( t2g)2hole system. Formally every ( t2g)2 spin-orbital model may be transformed into a ( t2g)4spin- orbital model so long as (a) the crystal eld splitting is large enough to prevent high spin congurations from becoming energetically relevant and (b) the fundamental parameters andtm0mare negated. Before proceeding with the explicit calculations for Oij, we note the intimate connection between the type of spin state favored (ie. spin-AF or spin-F) and the orbital state pictured in Fig. 4 is now mathematically depicted in (7). While each pathway contributes a spin-orbital term which is the product of spin and an orbital term, the sum of the three pathways cannot generally be fac- tored in this way. The consequence is that even without the spin-orbit interaction, the spins and orbitals are not independent on a site in the lattice24even though they areindependent at the atomic level. This can have im- portant consequences on the types of ordering allowed in lattices when the orbital part becomes frustrated due to orbital geometry even on geometrically unfrustrated lattices.25 A.Norb= 3 For the orbitally symmetric model, tm0m=tm0m, the total orbital angular momentum, L, is conserved. We ob- 0 0.1 0.2-10.-8.-6.-4.-2.0. 0 0.1 0.20 0.1 0.2(a)Norb= 3 (b)Norb= 2 (c)Norb= 1Energy (t2=U) JH=UFAF FAFAF FIG. 5. Energy eigenvalues of the two-site superexchange Hamiltonian (7) are plotted for (a) Norb= 3 using (8), (b) Norb= 2 using (10), and (c) Norb= 1 using (12). In addi- tion to a spin-AF ground state, a spin-F ground state can be favored when Hund's coupling is large in both the Norb= 3 andNorb= 2 models. tain the eective superexchange terms for this interaction term below. OS ij=4 32 3Li
Lj2 3(Li
Lj)2(8a) OD ij=4 3+1 3Li
Lj1 6(Li
Lj)2(8b) OP ij=1 2(Li
Lj)2(8c) To understand these results, we exactly diagonalize the eective superexchange Hamiltonian in the context of a two-site system. It is useful to rewrite each pathway in terms of projection operators, P, to a particular subspace of total angular momentum L= 0;1;2. OS ij=4 3P(L= 1) (9a) OD ij=5 6P(L= 1) +3 2P(L= 2) (9b) OP ij= 2P(L= 0) +1 2P(L= 1) +1 2P(L= 2) (9c) Fig. 5(a) shows the energy levels of this superexchange Hamiltonian for dierent JH=Ufor a two-site problem. Owing to the fact that (8a) can be written as the pro- jection to a total angular momentum L= 1 shared along a bond (up to a factor of 4 =3), the ground states of the 4Spathway in (7) have total L= 1 and total S= 2 shared between the two sites. Spin-orbit coupling will split these states and make local interactions favor a to- talJ= 1 shared along a bond. For large values of JH=U, the4Spathway will dominate and the non-zero angular momentum shared between sites gives an eective Curie moment to the two-site system. Small values of JH=U will be dominated by the2Dand2Ppathways which favor total L= 2 andS= 0 in opposition to4S. The critical value of Hund's coupling where the S= 2 quintet overtakes the S= 0 singlet, as seen in Fig. 5(a), can be computed analytically as JH=U=1 54(p 50517)0:1.6 0. 0.51.-1010. 0.51.-101 0. 0.5 1.-101 0. 0.51.-1010. 0.51.-101 0. 0.5 1.-101 0. 0.51.-1010. 0.51.-101 0. 0.5 1.-101 =(=2)=(=2)=(=2) Sz iSz j Lz iLz j Jz iJz jSx iSx j Lx iLx j Jx iJx j Sz iSz jLz iLz j Jz iJz jSx iSx j Lx iLx j Jx iJx jSx iSx j Lx iLx jJx iJx j(a)Norb= 3 (b)Norb= 2 (c)Norb= 1 FIG. 6. Expectation values of dierent angular momentum correlators are plotted for the two-site eective Hamiltonian in (5) using the three dierent Norbmodels with the parame- terization= cos,t2=U= sin,JH=U= 0:1 and=JH= 1. TheNorb= 3 model features full rotational symmetry while theNorb= 2 andNorb= 1 models only have one axis of rota- tional symmetry to make the zcorrelators dierent than the xandycorrelators. The eect of increasing JH=Uis to push the crossing point from spin-AF to spin-F behavior further left in these plots. B.Norb= 2 The eective superexchange interaction for the two or- bital model, tm0m=tm0m(1xy;m), can be expressed with operators acting on the two active orbitals. Let (i;0 i) be the 3+1 Pauli matrices for the Lz i=1 sub- space corresponding to the two active orbitals. For con- venience, we dene the permutation operator on the two active orbitals as Pij=1 2(i
j+0 i0 j). Then the orbital part of the superexchange Hamiltonian can be expressed in the following way. OS ij=2 3Pij+1 3(0 i+0 j) (10a) OD ij= 11 6Pij1 6(0 i+0 j) +1 2z iz j (10b) OP ij= 1 +1 2Pij1 2(0 i+0 j)1 2z iz j (10c) WhenJH=U!0, we recover the d4spin-orbital superex- change Hamiltonian used in Ref. 11. This limit ignores the F spin interactions induced by Hund's coupling.11The above equations for the orbital part combined with both spin-AF and spin-F spin components from (7) give the complete spin-orbital interactions for the 2-orbital model. It is also worth noting that when the2Dand2Pin- termediate states are taken to have the same coecients, rotational invariance within the active orbital subspace can be restored. Since the2Dand2Ppathways have the same 1Si
Sjspin part, these two pathways may be easily combined OD ij+OP ij= 2 +1 3Pij2 3(0 i+0 j) (11) so that the z iz jIsing anisotropy has been eliminated. This allows us to draw a parallel between the Norb= 3 andNorb= 2 models. In the Norb= 3 model, the S= 0 state (spin-AF) was a maximized L= 2 (orbital-F). In theNorb= 2 model, the S= 0 state is Lz i=Lz j= 0 as seen in (11) since Oijis to be maximized so that (7) is minimized. Graphically this is shown in Fig. 4(a). This tendency for spin AF to be accompanied by aligned or- bitals is common in spin-orbital models. Similarly, spin F tends to be accompanied by o-alignment of the orbitals as in Fig. 4(b). Returning to the full Norb= 2 case whereOD ijandOP ij are not combined, we diagonalize eective superexchange Hamiltonian for a two-site system. For a critical value of Hund's coupling, JH=U=1 9(p 345)0:09, theS= 2 quintet overtakes the S= 0 singlet as shown in Fig. 5(a). C.Norb= 1 To complete the discussion, we calculate the eective superexchange Hamiltonian for a single orbital hopping modeltm0m=tm0mxy;m. OS ij=1 3(L2 i;z+L2 j;z)2 3L2 i;zL2 j;z (12a) OD ij=1 3(L2 i;z+L2 j;z)1 6L2 i;zL2 j;z (12b) OP ij=1 2L2 i;zL2 j;z (12c) The4Spathway is only active when one site is in a Li;z= 1 state and the other site is in a Li;z= 0 state re ecting that one hole needs to be shared between the sites to allow F. By combining the2Dand2Ppathways as before, OD ij+OP ij=1 3(L2 i;z+L2 j;z) +1 3L2 i;zL2 j;z (13) we see AF behavior is maximized when both sites are in theLi;z=1 state re ecting that the xyorbitals partic- ipating in superexchange should be singly occupied, and the other two may be doubly occupied. After diagonal- izing the full two-site Hamiltonian in Fig. 5(c), we nd this case is qualitatively dierent from the previous two cases in that F interactions are not supported for any reasonable value of JH=U. Fig. 4(b) shows the physi- cal mechanism for F requires active orbitals on opposing sites to share a single hole. While spin-F states are sup- ported by fewer superexchange paths compared to their7 (a)AF 12 23(b)F 1223 yz zx xy AF F unfavorable FIG. 7. Orbital frustration is graphically illustrated for the Norb= 2 model. The orbitals shown on the vertices of the plaquette are the doubly occupied orbital on each site in a square lattice. Once the rst bond, labeled as 1, is chosen to be of a particular type, either (a) AF or (b) F, the next bonds, labeled as 2, are immediately xed by this choice. The result is that the last bond on the plaquette, labeled as 3, then takes a conguration which is neither the most energetically favorable AF bond nor the most energetically favorable F bond. spin-AF counterparts, there were many paths for spin-F states to reduce their energy in both the Norb= 3 and Norb= 2 models so that Hund's coupling could still tip the balance in favor of spin-F. However, for Norb= 1 model, the energy of a spin-F state can only be reduced by a single factor of t2=Uper site, and favorable spin-F interactions require more than one orbital to be energet- ically favorable. V. ORBITAL FRUSTRATION While the orbitally symmetric Norb= 3 model features rotational symmetry, the Norb= 2 andNorb= 1 mod- els do not due to their orbital geometries. The Norb= 1 model requires the single active orbital along a bond to be singly occupied so that AF spin superexchange interac- tions can minimize the energy. In d4congurations, only two of the three orbitals can be singly occupied while one orbital must be doubly occupied. Since the doubly occupied orbital cannot participate in AF spin superex- change, one third of the bonds must be unsatised. TheNorb= 2 model extends this concept with the possibility for two dierent low energy states depending on the value of Hund's coupling. Fig. 4(a) shows that when two orbitals are active, an AF spin interaction fa- vors double occupancy on the inactive orbital. An AF bond in the Norb= 2 model then favors the orbitals perpendicular to the bond direction ( xydoubly occu- pied along a z-direction bond) to be doubly occupied. Bond 1 in Fig. 7(a) is an example of such an AF bond (green). Choosing those two doubly occupied orbitals shown in the gure immediately restricts on other bonds emanating from these two sites. Since the doubly oc- cupied orbitals are not perpendicular to the bonds la- beled 2, a dierent interaction must be favored along the bonds labeled 2. The next most energetically favorable interaction is the F bond (red) shown in Fig. 4(b). This places the double occupancies on the other two orbitals and requires the doubly occupied orbitals on each site tobe opposite (ie. xz-yzalongz-direction). However this leaves the nal bond labeled 3 (blue) matching neither the criteria for the lowest energy AF or lowest energy F bond and instead takes an energetically unfavorable AF conguration. Similarly, starting with an F bond in Fig. 7(b) as the most energetically favorable results in the same conclusion. The orbital degrees of freedom then require one of the four bonds on a plaquette to take a high energy conguration in both scenarios. Regard- less of the value of JH=U, theNorb= 2 model again naturally yields frustration due to the orbital degrees of freedom even on nominally unfrustrated lattices.25When zJSEwherezis the coordination number, these or- bital eects are very strong and, in the absence of large octahedral distortions, may lead to orbital liquid states and perhaps highly entangled spin-orbital phases of mat- ter due to quantum uctuations. VI. TRIPLON CONDENSATION Here we study the case where spin-orbit coupling  is signicantly larger than superexchange JSE, particu- larly relevant to 5 d4materials. For JSE= 0 the ground state is the product of Ji= 0 singlets and therefore non- magnetic. With increasing JSE=, a local moment starts to form continuously though long range magnetic order sets in at a nite value of JSE=. This ordered region can be described as triplon condensation of weakly interact- ingJ= 1 excitations that evolve to a strongly interacting regime. In this section, we give a detailed introduction to the mechanism of triplon condensation and then apply the formalism to the three Norbmodels considered. A. Overview of the Mechanism With zero superexchange, the energy cost to make aJi= 1 triplon excitation is =2. It was shown previously,11using the bond operator formalism,26that8 =2 Ji= 0Ji= 1Ji= 2 Ti+1 Ty i+2 k  Energy FIG. 8. The triplon condensation mechanism is graphically illustrated. When there exists a triplon excitation on a site, superexchange can move the excitation to neighboring sites. This eective hopping causes the triplon's energy to disperse ink-space. When superexchange becomes large enough, con- densation of triplon excitations occurs as the bottom of the triplon band becomes lower in energy than the original Ji= 0 level. for spin-AF superexchange interactions that are substan- tially weaker than spin-orbit coupling, the superexchange interactions allow these triplet excitations to propagate from site to site and disperse in k-space to reduce the energy cost for the excitation around the -point until condensation of these triplon excitations occur and order antiferromagnetically; (see Fig. 8). One recent work has tested this mechanism with dynamical mean eld the- ory in the limit of innite dimensions.27Here we ask the question: Can spin-F interactions from the4Spathway cause condensation for large JH=U? If so, then at which k-point does the condensation occur? We show that for spin-F interactions there is a condensate but surprisingly the condensate does not always occur at the expected k= 0 point. Whenbecomes much larger than the superexchange energy scale t2=U, the high energy Ji= 2 states become energetically unfavorable and can be ignored. We project outJi= 2 states from our spin-orbital superexchange Hamiltonians leaving just the Ji= 0 andJi= 1 parts. We utilize a set of operators Ty ito describe the triplon excitations from Ji= 0 states to Ji= 1 states. These op- erators are dened by jJi= 1;Ji;z=mi=Ty i;mjJi= 0i. We then project the superexchange Hamiltonian onto the space of triplon operators, keeping only terms which are quadratic in the triplon operators (ie. Ty iTj,Ty iTy j) and throw away terms with three and four triplon op- erators which constitute eective interactions between triplon excitations. See Appendix C for calculation de- tails. Since the projection of the magnetization operator isMi=ip 6(TiTy i)i 2Ty iTi, the quadratic part de- scribes interactions between van Vleck excitations. This approximation cannot be justied deep in the condensed phase where interactions between excitations cannot be neglected, however, it can provide a good estimate of when thet2=Uenergy scale is large enough to support condensation and qualitatively what kind of magnetic or- dering to expect. There is, however, a more subtle consequence of thequadratic approximation. Since each site may only ac- commodate at most one triplon, there is a hardcore Bo- son constraint on every site. Although neglecting this constraint and the triplon-triplon interactions is neces- sary to put the solutions in closed form, the orbital frus- tration from theNorb= 2 andNorb= 1 models is lost under these approximations. This is separate from the orbital anisotropy that will still remain present. This is notable because anisotropic interactions usually cause frustration, yet here they will not due to the approxima- tion that the triplons are non-interacting. Again, deep in the condensed phase (the unexplored region of Fig. 1(b)), the exact solutions may qualitatively dier from the pic- ture depicted here. After projecting the superexchange Hamiltonian to this quadratic subspace and making a transformation to cubic coordinates (ie. Ty 1;Ty 0;Ty 1!Ty x;Ty y;Ty z), the most general Ty iTjterm can be decomposed into the three parts given below H(ij) iso=JTy i
Tj+ h:c: (14) H(ij) skew=D
 Ty iTj + h:c: (15) H(ij) symm =Ty i

Tj+ h:c: (16) and similarly for the Ty iTy jterms. First consider the isotropic term in (14). When Jbecomes large enough, all three avors of triplons ( Ty x,Ty y,Ty z) condense si- multaneously. Negative Jcauses condensation at the kpoint corresponding to a F condensate of van Vleck excitations while positive values cause condensation at thepoint corresponding to AF van Vleck excita- tions. Like the Heisenberg term, it comes from the orbitally symmetric component of the interactions like those considered in (8). The addition of further neigh- bor interactions can cause condensation at arbitrary q- vector. Next, the skew symmetric term in (15) results in a magnetic spiral condensate at q==2 points, and the addition of further neighbor interactions along with isotropic terms can make arbitrary q-spirals pos- sible. Like the Dzyaloshinskii-Moriya interaction, this term requires broken inversion symmetry. Finally, the symmetric anisotropy in (16) picks one of the three a- vors of triplons as the favored condensate due to orbital anisotropy. This qualitative picture can be extended to nite temperature since the condensation mechanism falls into the Bose-Einstein condensation universality class. Finite temperature condensation has been extensively studied,28so we will instead focus the unique aspects of spin-orbital condensation from superexchange. B. Results We rst determine which orbital geometries allow for F condensation when the hopping matrix is diagonal. Then9 we decompose the hopping matrix into multipoles t(ij)=X ktk 0Ak 0 (17) where multipoles are dened using Wigner-3j symbols hjm0jAk qjjmi= (1)jm0 j k j m0q m (18) andtk 0are the coecients of the decomposition. Then (2) is rewritten in the new form below. H(ij) t=X ktk 0X m Ak 0
mmcy imcjm (19) This form is particularly convenient to calculate the re- sulting spin-orbital superexchange form for each of the tk 0 hopping matrices. Here we will only give the results, and details of the calculation are relegated to Appendices C and D. ForNorb= 3, the hopping matrix tm0mis simply pro- portional to t0 0in (17) and (19). We nd that isotropic hopping,t0 0, only supports AF regardless of the value of Hund's coupling. This result contradicts the claim of Ref. 12 that F condensation results for orbitally sym- metric hopping. If the hopping matrix is either skew- symmetric, t1 0, or symmetrically anisotropic, t2 0, the over- all sign of the4Spathways is the opposite to that of the2Dand2Ppathways, and large Hund's coupling can favor a F condensate. The main dierence between the isotropic term and the anisotropic terms is that both anisotropic terms feature matrix elements of dif- ferent signs while the isotropic term does not. Both the Norb= 2 andNorb= 1 models have hopping matrices described as linear combinations of t0 0andt2 0. However, only AF condensates result in these cases since their de- compositions are closer to the isotropic hopping matrix instead of the anisotropic hopping matrix. The ferro- magetic condensation of triplon excitations is possible, but, due to the isotropic term only favoring AF, special orbital geometries are required for F condensation. The lack of F condensation for most common orbital geometries has an immediate consequence on the phase diagram for d4materials. In the limit of large spin-orbit coupling for anyvalue ofJH=U, there is a PM to AF tran- sition with increasing superexchange. However the limit of smallallows for both F and AF interactions. Then there must be an additional phase transition between the AF condensate phase and a spin-orbital F phase at in- termediate values of ( t2=U)=whenJH=Uis large in the unexplored region of Fig. 1(b). In the case of AF condensation, we give the critical condensation value ( t2=U)=for each of the three models. Norb= 3: Since this model possesses rotational in- variance, each triplon avor condenses simultaneously. The eective singlet-triplet gap from on-site interac- tions is
= =2z 3(t2=U) wherezis the coordination number and the inter-site interactions give a= 5=3andb=4=3 in (C11). Condensation occurs at the (;; ) point att2=U==40. Norb= 2: This model was studied in Ref. 11 where the orbital anisotropy was averaged away so that con- densation occurred at ( ;; ) att2=U==20 for sim- ple cubic lattices; without averaging it would occur at (;; 0) and equivalent directions. We can conclude that triplon condensation is then likely to be active in both 4dand 5dtransition metal oxides. Norb= 1: In this case, condensation occurs along a degenerate set of points: at ( kx= 2=a;ky= 0;kz) wherekzcan be arbitrary for the z-boson and the other 3 degenerate lines related by C4symmetry. Here the 4 lines are parallel to the kzaxis for the z boson, and the x and y bosons condense along lines being parallel to thekxandkyaxes respectively. We nd that a= 1=6 for directions perpendicular to a bond and a= 2=3 in the direction of a bond. The values of bare just the negatives of the avalues. On a face-centered cubic lattice, we nd a critical value of t2=U= (3=32). With this value, condensation is likely to occur in 4 d compounds, but large values of spin-orbit coupling in 5dcompounds will likely prevent condensation from occurring. C. Local Interactions versus Condensation It is surprising that although local F interactions were found for large values of JH=U, F condensation did not appear even in the isotropic Norb= 3 model which was free of orbital frustration. Even more surprising was that isotropic hopping is the cause of this unexpected result despite our calculations in Figs. 3(b). This discrepancy can be resolved by examining the two site problem more carefully, and our goal is to tie the two site and lattice condensation results together. To do this, we focus on the key problem presented: the lack of F condensation in a spin-orbital superexchange Hamiltonian which is ex- plicitly F by construction. First we will rewrite the ferromagnetic part of the su- perexchange Hamiltonian for the Norb= 3 model appear- ing in (7) and (8a) HSE=JSE 2(2 +Si
Sj) 2Li
Lj(Li
Lj)2 (20) which is pitted against the lowest order spin-orbit cor- rection. HSOC= 2(Li
Si+Lj
Sj) (21) Fig. 9 shows the energy spectra for a two site system parameterized with = cosandJSE= sin. On the right hand side at ==2, the lowest energy levels are the totalS= 2 andL= 1 states. A small amount of spin- orbit coupling splits the states into total J= 1;2;3 states10 0.0 0.2 0.4 0.6 0.8 1.0-3.0-2.5-2.0-1.5-1.0 =(=2)Energy S= 2 L= 1S= 1 L= 1 J= 3 J= 2 J= 1J= 2 J= 1 J= 0 Ji= 0 Jj= 0Ji=j= 1 symm anti- symm FIG. 9. Energy levels of the eective Hamiltonian HSE+ HSOCappearing in (20) and (21) with the parameterization = cosandJSE= sin. The levels are labeled by their good quantum numbers. In the = 0 limit, the eigenstates of spin-orbit coupling are used, and, in the ==2 limit, the eigenstates of the spin-orbital superexchange Hamiltonian are used. The interpretations of the states are discussed in the main text. as already stated in Section IV A. On the left hand side at = 0, the two lowest energy levels are as follows: a non- degenerate state with both sites in the non-magnetic J= 0 singlet state and a six-fold degenerate rst excited level where one of the two sites contains an excitation. These levels correspond to the vacuum and the triplon band of excitations in the condensation picture. Introducing a small amount of superexchange splits the rst excited states into symmetric and anti-symmetric states. The lower energy states are the anti-symmetric ones which correspond to condensation at the -point from a triplon condensation Hamiltonian Ty i
Tj+ h:c:with a positive hopping coecient. Here lies the source of the discrepancy between two site results and the lattice condensation result. There are two sides to the lowest energy J= 1 (S= 2,L= 1) level: the regime where andJSEare comparable and the regime where JSE(0). Section IV A showed that the regime where the two interactions were compa- rable produced F spin interactions. However these J= 1 states in the JSEregime correspond to triplon AF. Even though the two regimes are smoothly connected, the nonmagnetic Ji= 0 states are lower in energy than the triplon excitations. Then two site exact diagonaliza- tion covers up this aspect of AF while the lattice limit allows the anti-symmetric level (triplon band) decrease enough in energy to reveal the AF nature of the 0 part of the lowest energy J= 1 states. To summarize, while from the exact diagonalization, it seemed like the J= 1 line in Fig. 9 should have beenFM for all . However, in the region where it was not FM and was actually the anti-symmetric AFM triplon condensate (at the two-site level), a dierent AFM energy level (Ji=Jj= 0) was the one exact diagonalization was measuring, not the J= 1 AFM condensate line which was the next-lowest energy level. In the full lattice (not 2 site), the anti-symmetric level is the condensate (AFM) and drops below the ( Ji=Jj= 0) energy level. This arises because on the right side for greater than the crossing,J= 1 comes from S= 2 andL= 2 (FM) while on the left side J= 1 comes from an antisymmetric splitting of Ji= 1 andJj= 0 withJi= 0 andJj= 1 (AFM). Lastly Fig. 9 highlights a dierence between local spin AF and triplon AF. The exact diagonalization results in Figs. 3 show that the two site model features a smooth transition between Ji= 0 states and AF behavior. The same behavior is found in Fig. 9 by following the Ji= 0 line to the J= 0 line which is the transition from local non-magnetic singlets to a total J= 0 singlet between sites. (For reference, the AF-F level crossing discussed in the rst two sections would be shown here by this AF J= 0 (S= 1,L= 1) level being overtaken by the F J= 1 (S= 2,L= 1) level when the other two pathways in (7) are included.) The AF from triplon condensation corresponds to the anti-symmetric level, and these two types of AF are therefore dierent. In fact, from the quantum numbers shown in Fig. 9, the two types of AF belong to dierent irreducible representations at the two site level. It is then possible that there is an additional phase boundary in Fig. 1(b) which separates AF triplon BEC and spin AF despite the fact that they are both AF phases. VII. MATERIALS AND EXPERIMENTS Thed5irridates (Ir4+) with half lled j= 1=2 bands have attracted a signicant amount of attention due to the interplay between strong correlations and spin-orbit coupling. Although experimental studies have recently been focused on these materials, many strongly corre- lated oxides with moderate to strong spin-orbit coupling haved4congurations. A well studied example1,2is Ca2RuO 4which necessarily violates Hund's rules which require non-magnetic Ji= 0 Ru sites, however there are many less well studied d4materials. There are many double perovskites of the form A 2BB'O 6where both A and B have completely lled valence shells and the B' site is in the d4conguration including La 2ZnRuO 6,29 La2MgRuO 6,30Sr2YIrO 6,3Ba2YIrO 6,7{9and a large array of compounds with the form Sr 2BIrO6.4How- ever, from our Norb= 1 results in the previous sec- tions, 5ddouble perovskites are unlikely candidates for triplon condensation due to the small superexchange en- ergy scales when compared to spin-orbit coupling. Ad- ditionally, it has been suggested that the observed mag- netism in 5ddouble perovskites is due to disorder and/or11 impurities.9However, both 4 dcompounds and com- pounds with more than one active orbital ( Norb= 2 or 3) should be good candidates. Honeycomb d4ox- ides Li 2RuO 3and Na 2RuO 3have been found to or- der antiferromagnetically.6Otherd4oxides include post- perovskite31NaIrO 3and pyrochlore32Y2Os2O7. Many probes can be used to deduce the existence of novel magnetism in d4materials. In particular, both magnetic susceptibility and x-ray absorption spec- troscopy (XAS) have the advantage that magnetic or- dering is not required to infer the existence of moments. The rst test for novel magnetism in d4systems comes at the level of magnetic susceptibility measurements. Curie Weiss ts provide a measure of the eective magnetic moment, and measuring a non-zero eective moment (of order 1B) is a direct indication that the ground state is not the product of non-magnetic Ji= 0 singlets. De- termining whether the ground state is not the product of non-magnetic singlets can also be probed by XAS.33 In the non-magnetic Ji= 0 singlet state, L2absorption edge intensity is zero while the L3edge is non-zero which leads to a diverging branching ratio B:R:=IL3 IL2=2 +r 1r(22) wherer=h(Li)
Sii=hnhi. However, the Ji= 1 and Ji= 2 states lead to nite L2edges with magnitudes comparable of that of the L3edge. Thus measuring a branching ratio of order 1 is direct evidence against the non-magnetic singlet ground state. VIII. CONCLUSIONS We have studied how superexchange opposes the eect of spin-orbit coupling in d4systems and induces local mo-ments and interactions between them. If Hund's coupling is large, the local interactions favor ferromagnetism in- stead of the expected antiferromagnetism. We also found that at least two orbitals need to be involved for this lo- cal ferromagnetic behavior to be energetically favorable. The condensation mechanism allows AF to generally be favorable in both 4 d4and 5d4compounds. However be- cause isotropic orbital interactions favor antiferromag- netic condensation regardless of how large Hund's cou- pling is, ferromagnetic condensation is unlikely in ma- terials systems. We would like to highlight again that the ability of rotationally invariant atomic spin-orbit cou- pling to ip the sign of the eective exchange constant on a superexchange interaction is a unique feature of spin- orbital systems and has no analog in pure spin systems. The eective magnetic Hamiltonians derived here for transition metal oxides with d4occupancy can be directly used for the particle-hole symmetric d2occupancy as well after changing the sign of the hopping and spin-orbit couplings. These Hamiltonians lay the foundation for spin-orbit coupled Hamiltonians in the t2gsector. Going forward, dierent analytical and numerical methods can now be applied to obtain detailed phase diagrams. IX. ACKNOWLEDGMENTS We thank Jiaqiang Yan for useful discussions. We ac- knowledge the support of the CEM, an NSF MRSEC, under grant DMR-1420451. 1S. Nakatsuji, S. ichi Ikeda, and Y. Maeno, Journal of the Physical Society of Japan 66, 1868 (1997). 2M. Braden, G. Andr e, S. Nakatsuji, and Y. Maeno, Phys. Rev. B 58, 847 (1998). 3G. Cao, T. F. Qi, L. Li, J. Terzic, S. J. Yuan, L. E. DeLong, G. Murthy, and R. K. Kaul, Phys. Rev. Lett. 112, 056402 (2014). 4M. A. Laguna-Marco, P. Kayser, J. A. Alonso, M. J. Mart nez-Lope, M. van Veenendaal, Y. Choi, and D. Haskel, Phys. Rev. B 91, 214433 (2015). 5S. Bhowal, S. Baidya, I. Dasgupta, and T. Saha-Dasgupta, Phys. Rev. B 92, 121113 (2015). 6J. C. Wang, J. Terzic, T. F. Qi, F. Ye, S. J. Yuan, S. Aswartham, S. V. Streltsov, D. I. Khomskii, R. K. Kaul, and G. Cao, Phys. Rev. B 90, 161110 (2014). 7B. Ranjbar, E. Reynolds, P. Kayser, B. J. Kennedy, J. R. Hester, and J. A. Kimpton, Inorganic Chemistry 54, 10468 (2015).8B. F. Phelan, E. M. Seibel, D. B. Jr., W. Xie, and R. Cava, Solid State Communications 236, 37 (2016). 9T. Dey, A. Maljuk, D. V. Efremov, O. Kataeva, S. Gass, C. G. F. Blum, F. Steckel, D. Gruner, T. Ritschel, A. U. B. Wolter, J. Geck, C. Hess, K. Koepernik, J. van den Brink, S. Wurmehl, and B. B uchner, Phys. Rev. B 93, 014434 (2016). 10K. Pajskr, P. Nov ak, V. Pokorn y, J. Koloren c, R. Arita, and J. Kune s, Phys. Rev. B 93, 035129 (2016). 11G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013). 12O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi, Phys. Rev. B 91, 054412 (2015). 13H. Isobe and N. Nagaosa, Phys. Rev. B 90, 115122 (2014). 14A. Georges, L. de' Medici, and J. Mravlje, Annual Review of Condensed Matter Physics 4, 137 (2013). 15J. Goodenough, Magnetism and the Chemical Bond (In- terscience Publishers, 1963). 16J. Kanamori, J. Phys. Chem. Solids 10, 87 (1959). 17P. W. Anderson, Phys. Rev. 115, 2 (1959).12 18K. I. Kugel' and D. I. Khomski, Soviet Physics Uspekhi 25, 231 (1982). 19S. Ishihara, J. Inoue, and S. Maekawa, Physica C: Super- conductivity 263, 130 (1996). 20S. Ishihara, J. Inoue, and S. Maekawa, Phys. Rev. B 55, 8280 (1997). 21L. F. Feiner and A. M. Ole s, Phys. Rev. B 59, 3295 (1999). 22G. Khaliullin, P. Horsch, and A. M. Ole s, Phys. Rev. Lett. 86, 3879 (2001). 23S. Di Matteo, N. B. Perkins, and C. R. Natoli, Phys. Rev. B65, 054413 (2002). 24Y. Chen, Z. D. Wang, Y. Q. Li, and F. C. Zhang, Phys. Rev. B 75, 195113 (2007). 25A. M. Ole s, Journal of Physics: Condensed Matter 24, 313201 (2012). 26S. Sachdev and R. N. Bhatt, Phys. Rev. B 41, 9323 (1990). 27T. Sato, T. Shirakawa, and S. Yunoki, ArXiv e-prints (2016), arXiv:1603.01800 [cond-mat.str-el]. 28V. Zapf, M. Jaime, and C. D. Batista, Rev. Mod. Phys. 86, 563 (2014). 29P. D. Battle, J. R. Frost, and S.-H. Kim, J. Mater. Chem. 5, 1003 (1995). 30R. I. Dass, J.-Q. Yan, and J. B. Goodenough, Phys. Rev. B69, 094416 (2004). 31M. Bremholm, S. Dutton, P. Stephens, and R. Cava, Jour- nal of Solid State Chemistry 184, 601 (2011). 32Z. Y. Zhao, S. Calder, A. A. Aczel, M. A. McGuire, B. C. Sales, D. G. Mandrus, G. Chen, N. Trivedi, H. D. Zhou, and J.-Q. Yan, Phys. Rev. B 93, 134426 (2016). 33M. A. Laguna-Marco, D. Haskel, N. Souza-Neto, J. C. Lang, V. V. Krishnamurthy, S. Chikara, G. Cao, and M. van Veenendaal, Phys. Rev. Lett. 105, 216407 (2010). 34J. Kanamori, Prog. Theor. Phys. 30, 275 (1963). 35A. P. Yutsis, I. B. Levinson, and V. V. Vanagas, Mathematical Apparatus of the Theory of Angular Momentum (Israel Program for Scientic Translations, 1962). 36D. Brink and G. Satchler, Angular Momentum, Oxford li- brary of the physical sciences (Clarendon Press, 1962). 37G. Racah, Phys. Rev. 63, 367 (1943). Appendix A: Coulomb Interaction The Coulomb interaction for t2gorbitals is de- rived here.14,34The three t2gorbitals are labeled byfa;b;cgwhere (a;b;c ) is a permutation of (yz;zx;xy ). While these orbitals are not required to be atomic orbitals, we still require them to be invariant un- der theOhgroup. We now give the full on-site Coulomb interaction in terms of orbital indices f;;k;gand spin indices f1;2;3;4g. Hint=X 1X 2X k3X 4y 1y 2Vk3;4 1;24k3 (A1) Vk3;4 1;2=hjVjki3 14 2hjVjki4 13 2(A2)hjVjki=Z dr1dr2(r1)(r2)V(r1r2) k(r1)(r2) (A3) We require the eective on-site Coulomb interaction to be invariant under all three mirror plane operations. The four non-zero quantities characterizing Vin thet2gsub- space are then given below. We use JHinstead ofJto avoid confusion with total angular momentum. haajVjaaiUhabjVjabiU0 habjVjbaiJHhaajVjbbiJ0 H(A4) We will expand Hintusing these four parameters. It will be convenient to group group terms based on the ve types ofVk3;4 1;2terms encountered. Va";a# a";a#=Va#;a" a";a#=U Va";b" a";b"=U0JH Vb";b# a";a#=Vb#;b" a";a#=J0 HVa";b# a";b#=U0 Va#;b" a";b#=JH(A5) The nal result is given below Hint=UN"#+U0N0 "#+ (U0JH)N0 "" JHHex+J0 HHpair (A6) with: N"#=X ana"na# (A7) N0 "#=X a6=bna"nb# (A8) N0 ""=1 2X a6=bX nanb (A9) Hex=X a6=by a"y b#b"a#(A10) Hpair=X a6=by a"y a#b#b"(A11) We will eliminate N"#,N0 "#, andN0 ""in favor of the num- ber, spin, and angular momentum operators below. N=X ana (A12) S=1 2X aX 0y a0a0 (A13) La=iX bcX abcy bc (A14)13 Using the preceding denitions, we obtain the following relations 1 2N(N1) =N"#+N0 ""+N0 "# (A15) 1 4N(N1) +3 4N+HexS2= 2N"#+N0 "# (A16) NHexHpair1 2L2=N0 "" (A17) which can be used to rewrite the Hamiltonian in the fol- lowing form. Hint=1 4(3U0U)N(N1) + (7 4U7 4U0JH)N + (U0U)S2+1 2(U0U+JH)L2 + (U0U+JH+J0 H)P(A18) RequiringJH=J0 H, the relation U=U0+ 2JHgives the rotationally invariant form of Hintused in (3). Appendix B: Eective Hamiltonian We calculate the eective spin-orbit coupling in the3P subspace. The linear correction for spin-orbit coupling is given by HSOC;(1)=h3PjjHsojj3Pi h3PjjL
Sjj3PiL
S (B1) whereh3PjjHsojj3Piandh3PjjL
Sjj3Piare the re- duced matrix elements of the two operators respectively. Evaluating this ratio, we obtain h3PjjHsojj3Pi=h3PjjL
Sjj3Pi==2. Now we calculate second order energy corrections for the3Plevels due to spin-orbit coupling. Since Hsocon- serves total angular momentum, only energy levels of the same total angular momentum Jare coupled together. Then3P2couples to1D2and3P0couples to1S0. The 3P1level remains unshifted. The second order correction for3P2requires us to calculate the matrix elements h1D2jHsoj3P2ifor the3P2energy shift, and it suces to calculate the h1D2;Jz= +2jHsoj3P2;Jz= +2imatrix element given below hvacjc1"c1# cy m(lmm0
s0)cm00 cy 0"cy 1"jvaci(B2) which is only nonzero for l10
s#"=1 2l+ 10s #"= 1=p 2. Then with an energy denominator of E(1D)E(3P) = 2JH, we have a second order energy shift of:
E(2)3P2
= =p 2
2 2JH(B3) Similarly we can calculate the matrix element h1S0jHsoj3P0i=p 2, and, with an energy denominator ofE(1S)E(3P) = 5JH, we obtain:
E(2)3P0
= p 2
2 5JH(B4)This is represented in operator form below. HSOC;(2)=2 JH1 201 8L
S7 40(L
S)2 (B5) The total eective spin-orbit interaction used in (6) is justHSOC;(1)+HSOC;(2). The eective superexchange Hamiltonian is broken into three parts based on the energy value of the inter- mediated3conguration in the process d4d4!d3d5! d4d4. Using (3), energies of the relevant states for com- puting this Hamiltonian are given below. E[d4(3P);d4(3P)] = 12U26JH (B6) E[d3(4S);d5] = 13U29JH E[d3(2D);d5] = 13U26JH E[d3(2P);d5] = 13U24JH(B7) The three energy dierences, E(d3;d5)E(d4;d4), are given below. E[d3(4S);d5]E[d4(3P);d4(3P)] =U3JH E[d3(2D);d5]E[d4(3P);d4(3P)] =U E[d3(2P);d5]E[d4(3P);d4(3P)] =U+ 2JH(B8) Using these energy dierences, HSEis computed from (3) and (2) using the following scheme. HSE=PHt1 HintE[d4(3P);d4(3P)]HtP (B9) Since the denominator must take on one of the three energy dierences in (B8), we can separate (B9) into the three pathways. HSE= t2 U3JH HSE;(4S) t2 U HSE;(2D)  t2 U+2JH HSE;(2P)(B10) Spin symmetric interactions constrain each pathway to the form H(ij) SE;()= (+Si
Sj)O ij (B11) whereandare real coecients and Oijis an orbital interaction determined by t(ij) m0m. Appendix C: Condensation Formalism Here we derive the general triplon condensation Hamil- tonian from a spin-conserving superexchange Hamilto- nian. Since each superexchange pathway can be written as the product of orbital interactions and spin interac- tions as in (7), the eective pathways can be decomposed14 into the product of orbital and spin multipole operators on each site. H(ij)=X ll0mm0xll0 mm0(Li)l m(Lj)l0 m0X ss00yss0 0(Si)s (Sj)s0 0 (C1) HereLiandSiare the multipole operators for the orbital and spin parts of site i, andxll0 mm0andyss0 0are coecients of the decomposition with l2f0;1;2gandlml, etc. We rewrite the superexchange Hamiltonian using total orbital operators Oand total spin operators S. H(ij)=X ll0LMll0L MOll0L MX ss0Sss0S Sss0S  (C2) Oll0L M=X mm0hlm;l0m0jLMi(Li)l m(Lj)l0 m0 (C3) Sss0S =X 0hs;s00jSi(Si)s (Sj)s0 0 (C4) The symbolhlm;l0m0jLMiis a Clebsch-Gordon coe- cient, andll0L Mandss0S are the new coecients of the decomposition. Since S=  = 0 for superexchange in- teractions preserving spin symmetry as in (7), we have the following superexchange interaction. H(ij)=X ll0LMll0L MOll0L MX sss0 0Sss0 0 (C5) We now project out the high energy Ji= 2 states from the Hamiltonian and only leave the Ji= 0 and 1 com- ponents. Since we are only interested in the quadratic part of the result which couples sites together (ie. Ty iTj, Ty iTy j) and captures the condensation of triplon excita- tions, we project directly to this subspace and ignore terms likeTy iTisince they only amount to energy shifts. This projection is accomplished by rst projecting the spin-orbital operators on each site to the space of triplon creation and annihilation operators. The Wigner-Eckart theorem requires this projection of the product of mul- tipole operators ( Li)l m(Si)s is proportional to Ty i;and Ti;where=m+, (Li)l m(Si)s ! 1l s  m  1l s 1 1 1 h (1)l+1Ty i;+ (1)sTi;i (C6) and the factor in braces is a Wigner-6j symbol. The ow of angular momentum due to the Ty operator from this projection is represented graphically in Fig. 10(a). Now we combine the terms from sites iandjwith the constraint from (C5). It will be more convenient to project each type of term appearing in (C5) individually, so we dene the operator H0=Oll0L MSss0 0. Note that H0 (a) 1 0 = slm 11 11 1 slm 1 11 1l s (b) 1m01m= 0LM sl0 sl 1m01mLM l0ls L1 1 FIG. 10. The ow of angular momentum is graphically shown where ingoing arrows are incoming angular momentum and outgoing arrows are outgoing angular momenta. Wigner- 3j symbols and Clebsch-Gordan coecients are vertices with three legs while the scalar contraction of four Wigner-3j sym- bols (right) is a Wigner-6j symbol.35,36(a) Equation (C6) is shown in graphical form for the Ty part of the equation. A J= 0 state is decomposed into its L= 1 andS= 1 compo- nents which are acted on by the ( Li)l mand (Si)s operators. The resulting L= 1 andS= 1 are combined together to give aJ= 1 state with quantum number =m+. (b) The projection of equation (C5) to (C7) conserves angular momentum. Equation (C8) will appear similarly except that 1m0and 1madd to yield LM instead. is not generally Hermitian, but, for every LMterm, there is a complementing L(M) term so that H(ij)in (C5) is Hermitian. Then the projection of H0takes the two forms below. H0 Ty iTj=g1X m0mTy i;m0p 2L+ 1h1m0jAL Mj1miTj;m(C7) H0 Ty iTy j=g2X m0mTy i;m0h1m0;1mjLMiTy j;m (C8) Theh1m0jAL Mj1micoecients are the multipole ma- trix elements previously dened. The coecients g1= (1)l0gandg2= (1)sgare given below. g=(1)l+L+1 p2s+ 1 1L1 l s l0 1l s 1 1 1 1l0s 1 1 1 (C9) It is important to note that the angular momentum contained in H0(and alsoH(ij)) is conserved under pro- jection. Furthermore, after making the transformation to cubic coordinates, we immediately have our result:15 theL= 0 component gives the coecient Jin (14), the L= 1 components give Din (15), and the L= 2 compo- nents give the irreducible components of kin (16). This is the advantage of writing the orbital and spin parts using their total angular momenta. While we have restricted the calculation to spin- symmetric superexchange interactions, the concept ap- plies more generally. If the total spin operators are non- trivial, further combine the total orbital and total spin operators into a total spin-orbital operator. The total angular momentum contained in the total spin-orbital operator will be that which appears in the condensation Hamiltonian. Now we determine the critical value of ( t2=U)=where condensation occurs. To simplify, we consider the con- densation of a single avor of triplon. Then the conden- sation Hamiltonian has the following form. H=
X iTy iTi+t2 UX i;a 2Ty i+Ti+b 2Ti+Ti+ h:c: (C10) Here we have assumed that the singlet-triplet gap is given by
==2+(t2=U) where a correction due to superex- change has been included. Then the criteria for triplon condensation is given by t2=U==2 min(aqbq) +(C11) a q=X acos(q
) (C12) where \min" refers to the most negative value of the ar- gument and corresponds to the lowest part of the triplon energy band in Fig. 8(a). The extra term allows for the center of the triplon band to shift in energy with superexchange. Since t2=Uis positive by denition, min(a qb q) +must be negative for condensation to be possible. Appendix D: Condensation from Spin-Orbital Superexchange We apply our formalism from Section C to diagonal hopping so that tm1m2= 0 ifm16=m2to quantitatively obtain the key result of Section VI B. This includes the three special cases ( Norb= 3;2;1) from before, but gen- erally applies to systems with corner sharing and face sharing octahedra. The previous section showed express- ing the Hamiltonian in the form of (C5) had an immedi- ate connection to the condensation Hamiltonian. In this section, we explicitly calculate the coecients ll0L Mand ss0 0in (C5) from hopping matrices to determine when the4Spathway allows a F condensate. For the scope of this section, we will make the additional simplication tothrow away single site orbital anisotropy in the superex- change Hamiltonian (ie. terms like L2 i;z+L2 j;z). Using this condition and the restriction that the Hamiltonian must preserve time reversal symmetry, we are left to cal- culatellL 0andss0 0in (C5) for arbitrary l,L, andsfor diagonal hopping matrices. These conditions allow us to correctly guess the coecients simply using conservation of angular momentum without resorting to more involved formalisms. In second order perturbation theory, there are two applications of the Htoperator. The rst application contributes an angular momentum kwith amplitude tk 0 while the second application contributes k0with ampli- tude (ty)k0 0. Together this angular momentum is shared between (Li)l mand (Lj)l m0and can be recast in terms of a total angular momentum using (C3). The resulting or- bital coecient llL 0will be proportional to hk0;k00jL0i. We denote the intermediate multiplet for the site which gives up an electron during the rst virtual hop (4S,2D, 2P) as having orbital angular momentum 1and spin2 so that the multiplet is expressed as22+1(1). Then the product of the coecients llL 0ss0 0is given below. llL 0ss0 0= 12 ~llL 0~ss0 0hp2(1;1);p(1 2;1)jp3(2;1)i2 (D1) Herehp2(S0;L0);p(S00;L00)jp3(S000;L000)iare the coe- cients of fractional parentage37for apshell and account for the total angular momenta being composed of iden- tical particles while ~ llL 0and ~ss0 0are quantities which only depend on the recoupling of orbital and spin angu- lar momentum previously described. ~llL 0= 3(1)k0+1+1(21+ 1)(2l+ 1)2hk0;k00jL0i (ty)k 0(t)k0 0 1l1 1118 < :1k1 l L l 1k019 = ; 1l1 1 1 1 (D2) The new symbol in braces is a Wigner-9j symbol. The 9j symbol and the Clebsh-Gordan coecient together con- tain the information of of how the angular momenta in the hopping matrix tadds to give the orbital angular mo- menta in the resulting orbital part OllL 0of the superex- change Hamiltonian. The two 6j symbols along with the coecients of fractional parentage contain the informa- tion of how angular moment is transferred between sites through the transfer of identical particles. Lastly, our spin part is given similarly except that the restriction S=  = 0 allows some simplication. ~ss0 0= 3(1)2+3=2(22+ 1)(2s+ 1)2 1 2s1 2 1211p2s+ 11 2s1 2 11 21 (D3) With the coecients determined, we can now use (C7) to determine when the ferromagnetic mechanism mediated by the4Schannel (1= 0,2=3 2) is active.

0811.3798v1.Dynamics_of_Black_Hole_Pairs_II__Spherical_Orbits_and_the_Homoclinic_Limit_of_Zoom_Whirliness.pdf

arXiv:0811.3798v1 [gr-qc] 24 Nov 2008Dynamics of Black Hole Pairs II: Spherical Orbits and the Homoclinic Limit of Zoom-Whirline ss Rebecca Grossman∗∗and Janna Levin∗,! ∗∗Physics Department, Columbia University, New York, NY 1002 7 ∗Department of Physics and Astronomy, Barnard College of Col umbia University, 3009 Broadway, New York, NY 10027 !Institute for Strings, Cosmology and Astroparticle Physic s, Columbia University, New York, NY 10027 becky@phys.columbia.edu and janna@astro.columbia.edu Spinning black hole pairs exhibit a range of complicated dyn amical behaviors. An interest in eccentric and zoom-whirl orbits has ironically inspired th e focus of this paper: the constant radius orbits. When black hole spins are misaligned, the constant r adius orbits are not circles but rather lie on the surface of a sphere and have acquired the name “sphe rical orbits”. The spherical orbits are significant as they energetically frame the distributio n of all orbits. In addition, each unstable spherical orbit is asymptotically approached by an orbit th at whirls an infinite number of times, known as a homoclinic orbit. A homoclinic trajectory is an in finite whirl limit of the zoom-whirl spectrum and has a further significance as the separatrix bet ween inspiral and plunge for eccentric orbits. We work in the context of two spinning black holes of c omparable mass as described in the 3PN Hamiltonian with spin-orbit coupling included. As such , the results could provide a testing ground of the accuracy of the PN expansion. Further, the sphe rical orbits could provide useful initial data for numerical relativity. Finally, we comment that the spinning black hole pairs should give way to chaos around the homoclinic orbit when spin-spin coupling is incorporated. A complete knowledge of the dynamics of black hole pairs is essential for future gravitational wave experi- ments. Yet the importance of dynamics has not always been appreciated. Although stellar mass black hole pairs are significant candidates for a first direct detection with LIGO, their detectable gravitational radiation would be emitted from nearlycircularorbits –at least that wasthe refrain. This preferential focus on quasi-circular inspiral was motivated by considerations of long-lived binaries that begin with a modest eccentricity that is gradually shed as angularmomentum is lost to gravitationalwaves. A fair assessment of known astrophysics, the claim dis- couragedanalysesoforbitaldynamics in favorofthe sim- pler analysis of circular orbits. However, black hole binaries formed by tidal capture in dense star clusters, such as globular clusters, do not conform to this story [1]. As one black hole scatters with another black hole in a dense region, a burst of radiation is emitted on close encounter. Some subset of these en- counters will leave the pair bound in a highly eccentric orbit that merges too quickly to circularize. Estimates conclude that as many as 30% of multi-black hole sys- tems will retain eccentricities >0.1 as their waves sweep into the LIGO bandwidth [2]. Most recently, a new source of eccentric mergers was predicted to have a substantial detection rate [3]. Black hole/black hole scattering in galactic nuclei would sim- ilarly lead to tidal captures and highly eccentric, short- lived black hole binaries with 90% entering the LIGO bandwidth with eccentricities >0.9 [3]. These compet- itive sources for a first detection by Advanced LIGO [3] further motivate our study of the complete dynamics of binary black holes [4, 5, 6]. In paper I in this series [7], we presented the spec- trum of orbits in the strong-field regime when only onebody spins.1There we found zoom-whirl behavior – dur- ing which an orbit zooms out in large leaves followed by nearly circular inner whirls. Significantly, this extreme form of perihelion precession is prevalent in comparable mass systems [7], just as it is in extreme-mass-ratio in- spirals [8]. Zoom-whirl behavior is not restricted to ex- treme eccentricities, but can be executed by orbits of all eccentricities in the strong-field. We should therefore be prepared to detect evidence of such black hole dynamics in gravitational waves. In this companion to paper I, we work again in the conservative Hamiltonian 3PN approximation plus spin- orbit coupling, but move beyond paper I to consider two spinning black holes in a binary. We focus on special sets of orbits, namely the spherical orbits. That might seem ironic since we have just argued that gravitational wave science will probe the full range of dynamical possibili- ties, there are several good reasons to devote some time to constant radius orbits. 1. If even one black hole spins, the constant radius or- bits are no longer circles (unless spins are exactly aligned oranti-alignedwith the orbitalangularmo- mentum) [9, 10]. As a result of spin precession, they fill a band on the surface of a sphere and have thereby been coined spherical orbits (see Fig. 1). Long-lived binaries that have shed enough angu- lar momentum to lose their eccentricity but not their spin will exhibit quasi-spherical inspiral and not quasi-circular inspiral. To detect waves from realistic binaries, we will need to understand the 1In this paper we will use the word “orbit” to mean bound, non- plunging trajectories.2 /Minus10 /Minus5 0 5 10 /Minus10/Minus50510/Minus202 /Minus10/Minus5 0 5 10/Minus10/Minus50510 FIG. 1: A spherical orbit for mass ratio m2/m1= 1/4 and spin amplitudes of 3 /4. Both spins are initially displaced from the orbital angular momentum by π/4. Notice the orbit is not closed. Upper Panel: The three-dimensional orbit fills out a strip on a sphere. If we waited long enough, the band would be solidly painted, a reflection of the aperiodicity of the or bit. Lower Panel: The path as caught by the orbital plane reveals the constant radius. orbital parameters and precessions of these spheri- cal orbits. 2. Black hole pairs that enter the LIGO bandwidth with their eccentricity intact will evolve through a sequence of zoom-whirl orbits rather than nearly spherical ones. Still, the spherical sets are special since they mark the minimum and maximum en- ergy in the strong-field spectrum of bound orbis for a given angular momentum.2The orbital de- mographic is therefore entirely determined by the spherical orbits [7]. 3. The energetically-bound unstable spherical orbits mark the divide between inspiral and plunge [11]. More specifically, black hole spacetimes harbor ho- moclinic orbits – orbits that approach the unstable spherical orbits in the infinite future or in the infi- nite past [4, 12, 13, 14]. A Homoclinic orbit, often 2As detailed in the paper, we definethe strong-field by the ap- pearance of bound, unstable spherical orbits.refered to as a separatrix, is a classic feature of a non-linear dynamical system and deserve atten- tion since they mark the transition from inspiral to plunge for allpairs. As we mention in the close of this paper, when spin-spin coupling is incorpo- rated in the PN Hamiltonian, the homoclinic set can become the locus of a transition from regular to chaotic behavior. Among the infinite list of spherical orbits, two are valuable enough to deserve names: the innermost stable spherical orbit (isso) and the innermost bound spherical orbit (ibso). The acronyms are drawn in analogy with the equatorial isco (innermost stable circular orbit) and ibco (innermost bound circular orbits). The isso is the lowest energy spherical orbital and the ibso is the highest energy, bound spherical orbit. The isco, well-known as the site of the transition from inspiral to plunge for quasi-circular inspiral, is actually the zero eccentricity homoclinic orbit [12]. All other orbits, besides the quasi-circular one, will transition to plunge through another member of the homoclinic fam- ily, hence the importance of the homoclinic set to grav- itational wave science [13, 14]. To our knowledge the homoclinic orbits have not yet been identified in the PN Hamiltonian expansion before this paper, although an earlier paper found homoclinic orbits and zoom-whirl be- havior in a hybrid PN expansion [4]. Excitingly enough, homoclinic orbits have been observed in fully relativistic, numerical treatments as well [15]. Taken together this special set – composed of spheri- calandhomoclinicorbits–demarcatesdynamicalregions and we spend time on their attributes in this paper. Due tothelackofconfidencein thePNexpansionatclosesep- arations, we do not advocate that these results be taken as quantitatively accurate descriptions of binary black hole dynamics, but rather as qualitatively descriptive.3 Ineed, we point out peculiar artifacts of the 3PN system as we go along and the results of this paper could provide a new terrain on which to test the PN expansion against, for instance, numerical relativity. Our approach has some overlap with, but is not re- dundant with, the Refs. [9, 10] and allows us to find homoclinic orbits and stability exponents for use in the periodic orbit taxonomy of paper I [7]. We also simplify the initial conditions for spherical orbits in the absence of radiation reaction. For quasi-spherical orbits with ra- diation reaction included see [10]. The outline of the paper is as follows: In §I we write out the equations of motion for two spinning bodies in an orbital basis, relying on the results of appendix A. In §II we determine the orbital parameters of spherical orbits. In§III we find the homoclinic orbits and emphasize their 3The weakness of the PN approximation famously plagues other attempts to pinpoint the transition from inspiral to plunge through the isso [10].3 connection to dynamical instability. In the conclusion, §IV, we discuss the destruction of the spherical orbits and the transition to chaos when spin-spin coupling is included. I. 3PN HAMILTONIAN + SO COUPLING We will work with a condensed and revealing set of equations of motion in a non-orthogonal orbital coor- dinate system as derived in [7]. For reference in this companion to that paper, we write out the usual 3PN Hamilton plus spin-orbit coupling for two spinning black holes. In a Hamiltonian formulation, the equations of motion are derived from ˙r=∂H ∂p,˙p=−∂H ∂r. (1)As is standard convention, we work in dimensionless co- ordinates: the dimensionless coordinatevector, r, is mea- sured in units of total mass, M=m1+m2, for a pair with black hole masses m1andm2. The canonical mo- mentum, p, is measured in units of the reduced mass, µ=m1m2/M. The dimensionless combination η=µ/M will prove useful. We write vector quantities in bold. The coordinate ris to be understood as the magnitude r=√r·r. Unit vectors such as ˆ n=r/rwill addition- ally carry a hat. Finally, we have used the dimensionless reduced Hamiltonian H=H/µin Eqs. (1), where His the physical Hamiltonian, to 3PN order plus spin-orbit terms [16, 17, 18, 19, 20, 21]. Hcan be expanded as H=HN+H1PN+H2PN+H3PN+HSO,(2) where HN=p2 2−1 r(3) H1PN=1 8(3η−1)/parenleftbig p2/parenrightbig2−1 2/bracketleftbig (3+η)p2+η(ˆ n·p)2/bracketrightbig1 r+1 2r2 H2PN=1 16/parenleftbig 1−5η+5η2/parenrightbig/parenleftbig p2/parenrightbig3+1 8/bracketleftBig/parenleftbig 5−20η−3η2/parenrightbig/parenleftbig p2/parenrightbig2 −2η2(ˆ n·p)2p2−3η2(ˆ n·p)4/bracketrightbig1 r +1 2/bracketleftbig (5+8η)p2+3η(ˆ n·p)2/bracketrightbig1 r2−1 4(1+3η)1 r3 H3PN=1 128/parenleftbig −5+35η−70η2+35η3/parenrightbig/parenleftbig p2/parenrightbig4+1 16/bracketleftBig/parenleftbig −7+42η−53η2−5η3/parenrightbig/parenleftbig p2/parenrightbig3 +(2−3η)η2(ˆ n·p)2(p2)2+3(1−η)η2(ˆ n·p)4p2−5η3(ˆ n·p)6/bracketrightbig1 r +/bracketleftbigg1 16(−27+136η+109η2)(p2)2+1 16(17+30η)η(ˆ n·p)2p2+1 12(5+43η)η(ˆ n·p)4/bracketrightbigg1 r2 +/braceleftbigg1 192/bracketleftbig −600+/parenleftbig 3π2−1340/parenrightbig η−552η2/bracketrightbig p2−1 64/parenleftbig 340+3π2+112η/parenrightbig η(ˆ n·p)2/bracerightbigg1 r3 +1 96/bracketleftbig 12+/parenleftbig 872−63π2/parenrightbig η/bracketrightbig1 r4, HSO=L·Seff r3. (4) For two spinning black holes Seffis4 Seff=δ1S1+δ2S2 (5) 4The definitions for Seffcan vary in the literature up to an overall constant although the reduced HSOmust be the same for all prescriptions.where the dimensionless reduced spins are defined as S1=a1(m2 1/µM),S2=a2(m2 2/µM).(6) and δ1≡/parenleftbigg 2+3m2 2m1/parenrightbigg η , δ 2≡/parenleftbigg 2+3m1 2m2/parenrightbigg η .(7) The dimensionless spin amplitudes are confined to the range 0≤a1,2≤1. The reduced orbital angular momen-4 tumL=r×pand the spins precess according to ˙L=Seff×L r3 ˙S1=δ1L×S1 r3 ˙S2=δ2L×S2 r3. (8) The spin precessions can be grouped together, ˙S1+˙S2=L×Seff r3. (9) Notice that the precessionofthe sum of the spins is equal and opposite to the precession of the orbital angular mo- mentum. So that J=L+S1+S2is conserved. The magnitudes L,S1andS2, the inner product Seff·L, and theenergy(theHamiltonian)arealsoconstantforagiven orbit. In general, neither ˆJ·ˆLnor the magnitude |Seff|is con- stant. However, there are notable exceptions [22]. Both ˆJ·ˆLand the magnitude |Seff|are constant (1) if one of the black holes is spinless as was the case in [7], (2) if the binaries have exactly equal mass [22], or (3) if both spins are aligned or anti-aligned with the angular momentum. Case (1) is worked out thoroughly in paper I [7]. To see that the claim is true in the equal mass case (2), notice thatSeff=δ1(S1+S2) and therefore J=L+Seff/δ1. Consequently, J·L=L2+Seff·L/δ1is conserved since both terms on the right hand side are conserved. Fur- thermore, ˙Seff=δ1(˙S1+˙S2) and it follows from Eq. (9) that the change in Seffis always perpendicular to Seff so its magnitude remains constant. In case (3), when the spins are aligned or anti-aligned with the orbital angular momentum, motion is confined to a plane and there is no precession. Therefore SeffandLare constants and the rest follows. The results of this paper will apply to a general Seff unless explicitly stated otherwise. A. Equations of Motion in the Orbital Plane In a non-orthogonalorbital basis, the equations of mo- tion assume a simple form that allows us to analyze the dynamics of the black hole pairs. The plane perpen- dicular to the precessing orbital angular momentum is spanned by the vectors ( ˆ n,ˆΦ) whereˆ n=r/rand ˆΦ=ˆL׈ n. (10) Noticeˆ n·ˆΦ= 0 so these basis vectors are orthogonal. The entire orbital plane then precesses around the con- stant total angular momentum J=L+S1+S2in the direction ˆΨdefined through ˆΨ=ˆJ×(ˆJ׈L)/vextendsingle/vextendsingle/vextendsingleˆJ׈L/vextendsingle/vextendsingle/vextendsingle. (11) FIG. 2: Upper: The orbital plane precesses around the ˆJ=ˆk axis through the angle Ψ. Lower: The orbital plane can be spanned by the vectors ( ˆ n,ˆΦ). The construction, familiar from classical celestial me- chanics [23, 24, 25, 26], is illustrated in Fig. 2. Inci- dentally, this basis is explicitly constructed for J×L/negationslash= 0. When the spin and orbital angular momentum are aligned, anti-aligned, or spin is zero then J×L= 0, mo- tion is confined to a plane, and we should use the usual equatorial planar basis. Notice that ˆΨis not orthogonal to the orbital plane and therefore our orbital basis ( ˆ n,ˆΦ,ˆΨ) is not orthogo- nal. From Eq. (1), we can find equations of motion in coordinates ( r,Φ,Ψ) and their canonical momenta (Pr,PΦ,PΨ). As in appendix A, where we follow the approach of paper I, it is convenient to first isolate the equations of motion in the orbital plane for the variables (r,Φ) and their canonical momenta ( Pr,PΦ):5 ˙r=APr+B, ˙Pr=AP2 Φ r3+CPr+D+3PΦSeff·ˆL r4 ˙Φ =APΦ r2+Seff·ˆL r3−˙Ψ(ˆJ·ˆL),˙PΦ= 0(12) whereA,B,C,D are functions of ( r,Pr) to be defined momentarily. The momentum PΦ=Lconjugate to Φ is conserved, by the justification in paper I which survives despite the addition of a second spin. In any other basis, although Lis constant, it is not a momentum conjugate to any coordinate. Instead, Lshould be interpreted in terms of the linearly independent coordinates and mo- menta appropriate for that basis. The added beauty of this non-orthogonalapproachis that Lisacanonicalmo- mentum, namely PΦ, not soin the usualspherical coordi- nate basis where Lis neither a coordinate nor a momen- tum and the Hamiltonian angular equations of motion are less transparent. The four Eqs. (12) describe motion within the orbital plane. The orbital plane itself precesses around the con- stantˆJwith variable rate ˙Ψ, derived in appendix A2 to be ˙Ψ = ˆJ×/parenleftBig Seff׈L/parenrightBig /vextendsingle/vextendsingle/vextendsingleˆJ׈L/vextendsingle/vextendsingle/vextendsingler3 ·ˆΨ,˙PΨ=/parenleftBigg Seff׈L r3/parenrightBigg ·ˆJ, (13) andPΨ=L·ˆJ=Lzis not a constant. Again,usingthemanipulationsofappendixA,particu- larlyEq.(A27)andtheidentities ˆJ·ˆL= cosθL=PΨ/PΦ,|ˆJ׈L|= sinθL, we can write the final term in the ˙Φ equation of (12) as ˙Ψ(ˆJ·ˆL) = /parenleftBig Seff·ˆJ/parenrightBig −/parenleftBig Seff·ˆL/parenrightBig PΨ/PΦ (1−(PΨ/PΦ)2)r3 PΨ PΦ. (14) Writing it in this form exploits the dependences on the coordinates, conjugate momenta, and the constant Seff·ˆL. The one term that clearly remains dependent on angles is the term Seff·J. Therefore, when both black holes spin, the angular equations will depend on the an- gular precession of the orbital plane. We saw in paper I a dramatic simplification in the case of one effective spin. As follows from the earlier discus- sion of the constants ofmotion, Seff·ˆJwould be constant if either of the spins vanished and this would remove the angular depedence in the above equations. A pair of spinning black holes of equal mass is also reducible to a system with effectively one spin (see §A3). We continue to consider the general case of two misaligned spins for arbitrary mass ratios. For completeness, and as a complement to paper I, we explicitly write out the functions A,B,C,D that wereset up in [7] as derivatives on the Hamiltonian: A= 1+1 2(3η−1)p2−(3+η)1 r+ 3 8/parenleftbig 1−5η+5η2/parenrightbig/parenleftbig p2/parenrightbig2+1 4/bracketleftBig 2/parenleftbig 5−20η−3η2/parenrightbig p2−2η2(ˆ n·p)2/bracketrightBig1 r+(5+8η)1 r2 1 16/parenleftbig −5+35η−70η2+35η3/parenrightbig/parenleftbig p2/parenrightbig3+ 1 8/bracketleftBig 3/parenleftbig −7+42η−53η2−5η3/parenrightbig (p2)2+2(2−3η)η2(ˆ n·p)2p2+3(1−η)η2(ˆ n·p)4/bracketrightBig1 r+ /bracketleftbigg1 4/parenleftbig −27+136η+109η2/parenrightbig p2+1 8(17+30η)η(ˆ n·p)2/bracketrightbigg1 r2+ 2/bracketleftbigg −25 8+/parenleftbigg1 64π2−335 48/parenrightbigg η−23 8η2/bracketrightbigg1 r3(15)6 B=−η(ˆ n·p)1 r+ 1 8/bracketleftBig −4η2(ˆ n·p)p2−12η2(ˆ n·p)3/bracketrightBig1 r+ 3η(ˆ n·p)1 r2+ 1 16/bracketleftBig 2(2−3η)η2(ˆ n·p)/parenleftbig p2/parenrightbig2+12(1−η)η2(ˆ n·p)3p2−30η3(ˆ n·p)5/bracketrightBig1 r+ /bracketleftbigg1 8(17+30η)η(ˆ n·p)p2+1 3(5+43η)η(ˆ n·p)3/bracketrightbigg1 r2+ 2/parenleftbigg −85 16−3 64π2−7 4η/parenrightbigg η(ˆ n·p)1 r3(16) C=−B r(17) D=−(ˆ n·p)C−1 r2−1 2/parenleftBig (3+η)p2+η(ˆ n·p)2/parenrightBig1 r2+1 r3+ 1 8/bracketleftBig/parenleftbig 5−20η−3η2/parenrightbig/parenleftbig p2/parenrightbig2−2η2(ˆ n·p)2p2−3η2(ˆ n·p)4/bracketrightBig1 r2+ /bracketleftBig (5+8η)/parenleftbig p2/parenrightbig +3η(ˆ n·p)2/bracketrightBig1 r3−3 4(1+3η)1 r4+ 1 16/bracketleftBig/parenleftbig −7+42η−53η2−5η3/parenrightbig (p)3+(2−3η)η2(ˆ n·p)2/parenleftbig p2/parenrightbig2+3(1−η)η2(ˆ n·p)4p2−5η3(ˆ n·p)6/bracketrightBig1 r2+ 2/bracketleftbigg1 16/parenleftbig −27+136η+109η2/parenrightbig/parenleftbig p2/parenrightbig2+1 16(17+30η)η(ˆ n·p)2p2+ 1 12(5+43η)η(ˆ n·p)4/bracketrightbigg1 r3+3/braceleftbigg/bracketleftbigg −25 8+/parenleftbigg1 64π2−335 48/parenrightbigg η−23 8η2/bracketrightbigg p2+ /parenleftbigg −85 16−3 64π2−7 4η/parenrightbigg η(ˆ n·p)2/bracerightbigg1 r4+4/bracketleftbigg1 8+/parenleftbigg109 12−21 32π2/parenrightbigg η/bracketrightbigg1 r5(18) whereˆ n·p=Prandp2=P2 r+L2/r2. Notice that A,B,C,D , which comefrom the non-spinningpartofthe Hamiltonian [7], depend only on ( r,Pr) and constants. Useful results can be drawn from a simple observation. The radialequationsin (12) haveno angulardependence. The energy, angular momentum, and radius of spherical orbits can be derived from the radial equations alone. Therefore we can find spherical orbits simply despite the precession of the orbital plane. The fact that the two equations in ( r,Pr) form a self- contained system is a restatement of the fact that the Hamiltonian itself can be viewed in a one-dimensional effective approach as a function of ( r,Pr) and constants. It is important to be cautious however when investigat- ing the angular motion. The Hamiltonian depends only on (r,Pr) and constants in time, yet those constants in time have to be carefully varied as functions of the angu- lar coordinates and their conjugate momenta in a given basis to correctly derive the remaining equations of mo- tion. This accounts for the labor in appendix A needed to derive the (Φ ,PΦ) and (Ψ ,PΨ) equations of motion. Still, thesimpledependencesoftheHamiltonianallows us to analyze the spherical orbits as one-dimensional ra- dial motion in a simple effective potential. The location of the spherical orbits was implicit in paper I to framethe distribution of all other orbits. For completeness we determine the range of spherical orbits with an eye on that companion work. II. SPHERICAL ORBITS A. Effective Potential for Spinning Black Holes Ideally, in an effective potential formulation, the radial equation could be cast in the form: 1 2˙r2+effective potential = constant (19) where the effective potential depends only on rand con- stants of the motion. Now, the Hamiltonian of Eqs. (2)- (4) does not admit a simple effective potential formula- tion since it is a complicated function of p2. We have already argued that H(r,p,Seff) can be written as an ef- fective function of ( r,Pr) and constants, yet it remains a polynomial function of Pr. However, if we only consider Veff=H(Pr= 0), (20) then we have a good representation of a pseudo effective- potential at the turning points . We cannot misuse the7 51015202530/Minus0.07/Minus0.06/Minus0.05/Minus0.04/Minus0.03/Minus0.02/Minus0.010.00 rVeff 5101520253035/Minus0.07/Minus0.06/Minus0.05/Minus0.04/Minus0.03/Minus0.02/Minus0.010.00 rVeff 51015202530/Minus0.30/Minus0.25/Minus0.20/Minus0.15/Minus0.10/Minus0.050.00 rVeff 1 2 3 4 5/Minus0.5/Minus0.4/Minus0.3/Minus0.2/Minus0.10.0 rVeff FIG. 3: An effective potential for two spinning black holes a1=a2= 3/4 of mass ratio m2/m1= 1/4 for different values oftheangular momentum. Notice thechange in scale between panels. Upper: The appearance of the ibso is marked by the effective potential touching the line H= 0. Next: As the angular momentum decreases, the potential will have both stable and unstable spherical orbits. Next: As the angular momentum is further decreased there occurs a critical value at which the unstable and stable spherical orbits merge at a saddle point, the isso. Lower: The last panel shows a differ- ence from the Schwarzschild or Kerr stories. At angular mo- menta and radii below the occurence of the isso, there occur new sets of stable and unstable spherical orbits. These occu r at radii far below which the approximation can be trusted, yet we point out their presence for completeness.Veffby trying to interpret motion away from the turning points, but it gives a perfectly valid description of the behavior at aphelia and periastra as well as on spheri- cal orbits. Hereafter we’ll shorthand the term “pseudo effective-potential” by “effective potential”. From the Hamiltonian of Eqs. (4), the effective poten- tial, Veff(r,L,Seff·ˆL,η), (21) is a function of orbital parameters ( r,L,Seff·ˆL) and the mass ratio. Again, since LandSeff·ˆLare constants of the motion, for a given ( L,Seff·ˆL) and a given mass ratio, the potential is a function of ronly. Fig. 3 shows several snapshots taken of the effective potential for a pair of spinning black holes as the mag- nitude of Ldecreases for a given Seff·ˆLvalue. (For a detailedexposition oninterpretingeffectivepotentials for black hole orbits see Refs. [27] and [13, 14].) The spher- ical orbits are simply the extrema of the potential.5An example of such an orbit was shown in Fig. 1. Although this orbit is not generally periodic, it does close in the orbital plane as shown in the lower panel of Fig. 1. ThetoppanelofFig.3marksthevalueof Lforwhicha margnially bound, unstable spherical orbit appears. An orbit is marginaly bound if its energy H= 0 and it is spherical and unstable if it is a maximum of the effective potential. The conditions are summarized as Veff(Pr= 0) = 0 ∂Veff ∂r= 0 ∂2Veff ∂r2<0 (ibso) , (22) although the first two are sufficient. We call the marg- nially bound unstable spherical orbit “ibso” in analogy with the innermost unstable circular orbit (ibco) of equa- torial orbits. For angular momenta below Libsothere will be both a stable and unstable, energetically bound spherical orbit, as in the second snapshot of Fig. 3, until the angular momentum gets so low that we reach the third snapshot from the top. Here, the unstable and stable spherical orbitshavemergedinasaddlepoint, coinedaninnermost stable spherical orbit (isso): ∂Veff ∂r= 0 ∂2Veff ∂r2= 0 (isso) . (23) 5Orbits with the same angular momentum as a stable spherical o r- bit but different energy will oscillate between two turning p oints, both of which can be read off the effective potential diagram. Again, due to spin precession, for misaligned spins these ec cen- tric orbits lift out of a plane. Their spectra was shown in pap er I for a spin/spinless black hole pair.8 The story plotted out by panels 1-3 of Fig. 3 for Lisso< L < L ibsoqualitatively follows the fully relativis- tic Schwarzaschild and Kerr stories as expected [13, 14]. However, something peculiar then happens in the PN ap- proximationatverylowvaluesoftheangularmomentum. New stable and unstable spherical orbits can appear as shown in the bottom panel of Fig. 3. Or, for some other ranges of parameters, the ibso disappears or the isso dis- appears or both disappear. Sometimes these problems occur, as in the figure, for radii far below the confidence of the PN approximation. We point out these trouble- some features in the spirit of full disclosure. More than this, the details provide a quantitative testing ground for the approximation. Despite these oddities at low rwhere the PN- approximation would make no claims of quantitative va- lidity anyway, the qualitative features of spherical orbits, homoclinic orbits, and zoom-whirl behavior should sur- vive improved approximations and full numerical treat- ments [28]. We will locate the EandLof sphericalorbits in the next subsection. B. Orbital Parameters for Spherical Orbits For a given black hole pair, that is, a given mass ra- tio andSeff·ˆL, all orbits are uniquely specified by their (E,L). Using the effective potential, we can easily gener- ate theEandLfor spherical orbits and thereby generate initial data for them. Initial conditions for spherical or- bits were also found in [9, 10]. Damour [9] noticed that when only spin-orbit terms are included that the Hamil- tonian couldbe expressedas aradialfunction. One could arrive at this conclusion, as we have in the previous sec- tion.6The constant radius orbits occur at the extrema ofH(r,Pr), in the same spirit as an effective potential method. From the vantage point of the effective potential, spherical orbits satisfy the condition ∂Veff ∂r= 0 (24) treating LandSeff·ˆLas constants. We could also take the vantagepoint ofthe equationsofmotion. Since, Pr= 0 forcesB= 0, the condition ˙ r=APr+B= 0 can be thought of as synonymous with the condition that Pr= 0. The constant radius condition is thus the requirement that ˙Pr/vextendsingle/vextendsingle/vextendsingle Pr=0= 0 (25) 6However, in another basis such as the usual basis for spheric al coordinates used in [10], projection of the vector equation s of motion will give equations of motion in component from that continue to depend on angles even when only one black hole spins, unlike the orbital basis of §A3.5 10 15 20rs45678Ls 10 15 20rs /Minus0.05/Minus0.04/Minus0.03/Minus0.02/Minus0.010.01Es FIG. 4: ( m2/m1= 10−6,Seff= 0). Upper: Angular momen- tum vsrs. Lower: Energy vs rs. in Eqs. (12) and Eq. (25) is equivalent to Eq. (24). The roots ofthe 8th-orderequation in L, Eq. (25), give the angularmomenta of the spherical orbits as a function of spherical radius, Ls(rs), where we use a subscript sto denote a quantity evaluated at a spherical orbit. (When there is no spin, the condition reduces to a quartic in L2 with only two of the four roots real.) Piecing together the real roots we find Ls’s such as the one in Fig. 4. Although the upper branch grows very quickly in Ls, these values rapidly become physically unreachable since it would require angular velocities greater than the speed of light to be that high up on the upper branch. One can think of ( Ls/rs)<1 as a crude marker of physically allowed values. To find the energy of spherical orbits, Es(rs), wesimply plug Ls(rs) intothe Hamiltonian when Pr= 0. The energy plot is also shown in Fig. 4. There are several things to notice about the Lsand Esplots, for which we chose values to illustrate the PN approximation to Schwarzschild ( m2/m1= 10−6,Seff= 0).7The large values of rscorrespond to stable spherical orbits. (Since spin is zero, these constant radius orbits are actually circular equatorial but we’ll keep the lan- 7Since the radial equation depends only on the combination Seff· ˆL, Fig. 4 should be equally valid for a non-zero effective spin t hat is orthogonal L.9 guage more general.) When Lshits a minimum, we have foundtheisso–forno L < L issoaretheresphericalorbits. To the left of that minimum are the unstable spherical orbits. A true peculiarity of the figure is the fact that the radiiofthe unstable sphericalorbitsbegin to moveout to largerr. This is simply a flaw in the PN approximation and does not occur in the Schwarzschild system. In the fully relativistic system the unstable spherical orbits al- ways move to smaller radii than the isco, hence the ibco is really innermost, earning its name. Here, the ibso is not actually innermost – due to the poor quality of the approximation – although it remains the highest energy bound spherical orbit when it exists. The ibso cannot be read off of Fig. 4 although it can be found simply as the coincident of the roots of Eqs. (22). Figs. 4 are for a non-spinning extreme-mass-ratio bi- nary and are therefore valid as an approximation to Schwarzschild. The details of these figures will be use- ful for a future test of the PN expansion. Here we note thatLibso≈4.69, which is about 17% higher than the Schwarzschild value of 4 while Lisso≈3.75, which is about 8% higher than the Schwarzschild value of√ 12. The energy of the ibso is designed to be zero so is not informative but the energy of the isso is Eisso≈ −0.0452, which is about 21% less negative, that is less energeti- cally bound, than the Schwarzschild case (2√ 2/3)−1. Due to the approximate nature of the expansion it is not necessary to take these comparisons to heart, but they indicate howthe sphericalorbitsand the periodic spectra could facillitate a test of the PN expansion. For a com- parison of the isso in different PN approaches including the resummed Kerr-like effective-one-body approach see [10, 21]. C. Dependence of Binding Energies on mass ratios and spin For completeness, we can see how the ibso and the isso vary as the mass ratio and spins of the black holes are varied; that is, as their mass ratio and spins are varied. For one, since the ibso and isso frame the distribution of orbits, they define the ranges of EandLvalues for all other orbits in the strong-field. For another, the energy at the isso gives an estimate of the energy emitted on quasi-circular inspiral up to the transition to plunge. A larger binding energy at the isso could also mean a larger signal at final coalescence so these variations attest to variouslevelsofdetectability. Wewill discussthebinding energy of the isso in this section. In the next section we will consider the transition to plunge for eccentric orbits. A black hole pair is specified by its mass ratio, m2/m1, and its spins through the particular combination Seff·ˆL. The Hamiltonian, and the radial equations, depend only on these two combinations. We will therefore consider the variations in the isso and ibso for black hole pairs distinguished only by their ( m2/m1,Seff·ˆL) values. It is important to realize that there is a great deal of degener-2.533.544.55 2 3 4 5 6 7L rm2/m1 10−6 1/10 1/4 1/3 2/5 -0.1-0.08-0.06-0.04-0.020 2 3 4 5 6 7E rm2/m1 10−6 1/10 1/4 1/3 2/5 FIG. 5: All black hole pairs represented have Seff·ˆL= 0.35355. Upper: Angular momentum vs rfor the ibso and isso for different mass ratios. The upper point is always the ibso for a given symbol while the lower point with the same symbol is always theisso. The keylists thedifferent( m2/m1). Lower: Energy vs r. acy among pairs. The ibso and isso values (their energy, angularmomenta, and radialvalues) areidentical for two physically distinct black hole pairs. For instance, a black hole with mass ratio m2/m1= 1/3andSeff·ˆL= 0.35355 could be a black hole with initial values a1= 1/4,a2= 0, and the spin of the heavier black hole aligned with the initial orbital angular momentum. However, this is not the only combination of spin amplitudes and angles that will give the combination Seff·ˆL= 0.35355. While dif- ferent black hole pairs can give degenerate isso and ibso values, they will be physically distinguishable through their angular motion. In Fig. 5 the Lof the ibso and isso is plotted in the up- per panel and the Eofthe issoand ofthe ibso areplotted in the lower panel. Qualitative conclusions can be drawn from these figures. We notice that as the mass ratio is increased towards 1, the radius of both the isso and ibso decrease, although the isso moves in faster. Therefore the isso is pushed to larger binding energies as the mass ratio is increased towards 1. Because the Hamiltonian is a high-order polynomial in r, there can be more than one marginally bound orbit and more than one saddle point for a given ( m2/m1,Seff·ˆL) pair, as demonstrated in the lowest panel of Fig. 3. The second occurence of a10 11.522.533.544.55 2 3 4 5 6 7 8 9L rSeff·ˆL −1.406 0 0.3204 0.9944 1.125 -0.2-0.15-0.1-0.050 2 3 4 5 6 7 8 9E rSeff·ˆL −1.406 0 0.3204 0.9944 1.125 FIG. 6: Upper: Angular momentum vs rfor the ibso and isso for fixed mass ratio m1/m1= 1/3 but varying Seff·ˆL. The upper point is always the ibso for a given symbol while the lower point with the same symbol is always the isso. The key listsSeff·ˆL. Lower: Energy vs r. marginally bound orbit and/or saddle point appears in the vicinity of r∼1 where the approximation is unin- terpretable. (There may even be third occurences.) It is unclear if there is any physical content to these other stable and unstable spherical orbits. Fig. 5 plots only the ibso/isso pair for rvalues>2. For the value of Seff·ˆL∼0.35355 used in the figure, either the ibso or the isso disappears (or both disappear) asm2approaches m1. There may still be very small radii (r∼1) ibso’s and/or isso’s, but the sensible ones disappear. This peculiarity is probably an artefact of the approximation, a point we return to momentarily. Fig. 6 fixes the mass ratio at m2/m1= 1/3 and varies Seff·ˆL. Increasing Seff·ˆLhas the same effect of pushing the isso to smaller separations and therefore to larger binding energies, although again the isso moves in faster – discounting any marginally bound spherical orbit or saddle points that occur in the vicinity of r∼1. (In fact, as Fig. 6 shows, at some point the isso actually occurs at a smaller radius than the ibso.) So, all other factors being equal, spins anti-aligned with the orbital angular momentum push the isso out to larger radii and smaller binding energies while aligned spins pull the isso into smaller radii and larger binding energies. For themass ratio of this figure, the ibso or the isso actually vanishes (or both vanish) as Seff·ˆLis increased much beyond the values shown. These trends are consistent with those for spherical or- bits discussed in Ref. [10]. For the equal mass case with spins aligned or anti-alignedwith the orbital angularmo- mentum, the authors of that reference also remarked on the absense of an isso (called a last stable spherical or- bit (lsso) in their lexicon). Indeed, they used this fail- ing to argue that the PN expansion could not be used to study the transition from inspirl to plunge and advo- cated instead the seemingly more reliable effective-one- body (EOB) approach [9, 29, 30]. It would be interesting to extend to the EOB method the investigation of the zoom-whirl orbits of paper I [7] and the homoclinic limit of the zoom-whirls that we turn to in the next section. We leave that to a future work and continue to use the 3PN Hamiltonian as an example of our general method. The disappearance of the ibso accompanies the disap- pearance of all unstable spherical orbits. Once this hap- pens, there can be no issosince the issois reallythe point of merger of the unstable and stable spherical orbits.8 As is already known, in the absence of spin there are no bound unstable circularorbits at 2PN[23, 29, 31, 32, 33]. At 3PN there are no bound unstable circular orbits for mass ratios bigger than about m2/m1∼1/2. (See also [21, 34].) The absence of a bound unstable circular or- bit isclearlyashortcomingofthe approximationsince we know that the Schwarzschildspacetime possess an unsta- ble circular orbit as a reflection of its high non-linearity. Furthermore, the unstable circular orbits are present in fully relativistic treatments as the equal mass numerical investigation of Ref. [28] shows. Therefore, the ibso and isso should emerge for m2→m1at higher orders. In- cidentally, their disappearance at 3PN-order implies the expansion is very likely approximating the dynamics as more stable than it really is and therefore less vulnera- ble to chaos than it really is for these comparable mass binaries. We have already warned caution to take the trends as qualitative indicators and not to invest too much in the numbers due to pressures on the PN expansion at such large values of ( m1+m2)/r. Afterall, the PN expansion is an expansion in small ( m1+m2)/rand will naturally begin to faulter for small r. We have focused on the binding energy of the isso pri- marily to fit into the wider conversation that has focused onquasi-circularinspiral. However,theeccentricbinaries formed by tidal capture in dense regions will not transi- tion from inspiral to plunge through the isso. Rather they will transition through the eccentric separatrix be- tween bound and plunging orbits. We investigate that separatrix briefly in the final section. 8This is not the only reason the isso disappears. Sometimes th e isso disappears because the potential simply never flattens out.11 III. HOMOCLINIC ORBITS – THE SEPARATRIX BETWEEN BOUND AND PLUNGING ORBITS It is worthwhile to mention another important kind of orbit that occurs in our dynamical system, the ho- moclinic orbit [4, 12]. Homoclinic orbits are intriguing for several reasons, not least of which is that they mark the orbits through which the transition from inspiral to plunge should occur. In fact, the isso itself, the transi- tion point for quasi-circular orbits, is a zero eccentricity homoclinicorbit[12, 13, 14]. Wemaketheconnectionbe- tween the energetically bound, unstable spherical orbits and the homoclinic orbit explicit in this final section. Formally, homoclinic orbits are defined as trajectories that asymptote to the same hyperbolic invariant set in the infinite future as in the inifinite past. In these black hole settings, the role of the hyperbolic invariant set is played by the energetically bound, unstable spherical or- bits. Although in the lexicon of black hole physics these orbits have been coined “unstable”, they are strictly speaking hyperoblic, which is to say they possess both a stable eigendirection and an unstable eigendirection under linear perturbations. And, the eigendirections lie along the homoclinic orbit in the local neighborhood of the unstable circle they approach. Although we won’t demonstrate that line up here, the point was emphasized in detail in Refs. [13, 14] for Kerr equatorial dynamics. The stability exponents can be found by linearizing in small perturbations around Eqs. (12). This was done for equatorial Kerr orbits in Ref. [14]. Although we will not write out the explicit procedure here, we mention that the spherical orbits have radial eigenvalues that come in plus/minus pairs, as they must in a Hamiltonian system. The radial eigenvalues are real for the unstable spherical orbits and imaginary for the stable spherical orbits. The isso occurs at the merger of the eigenvalues at zero. A direct computation of the stability exponents around cir- cular orbits confirms that the stable spherical orbits and the unstable spherical orbits are distributed around the isso as Fig. 4 shows. Through the phase space analysis we have shown that the energetically bound, unstable circular orbits are ac- tually hyperbolic – they have a positive stability expo- nent as well as a negative stability exponent. We could compute the eigenvectors and show they lie along the ho- moclinic orbit in the local neighborhood of the unstable circle as we did for Kerr in Ref. [14]. However, for our purposes it is sufficient and illuminating to consider a physical space picture. We can identify the separatrix – i.e. the homoclinic orbit – in an effective-potential picture. In particular, consider a binary with mass ratio m2/m1= 1/4 and the heavier black hole spins with amplitude a1= 1/2 offset fromˆLbyπ/4while the lighterblackholeis nonspinning. Although, again, any equivalent combination of Seff·ˆL is described by this same figure. The unstable spherical orbit,ru, at the maximum of Veffin Fig. 7 is drawn in246810/Minus0.065/Minus0.060/Minus0.055/Minus0.050 rVeff FIG. 7: An effective-potential for m2/m1= 1/4, with the spin of the heavier black hole displaced from ˆLbyπ/4 and amplitude a1= 1/2 while the lighter black hole has no spin. The straight line is the energy of the unstable spherical orb it. It is also the energy at another, larger turning point ra∼10, which identifies the apaastron of the homoclinc orbit. physical space in Fig. 8. Although the orbit is a closed circleintheorbitalplane, itfillsoutabandonaspherein three dimensions. Because of numerical instability near this orbit, we only show a few windings. The energy of this orbit, Es(ru), is indicated by a straight line across the potential of Fig. 7. Note that this energy touches the potential at the unstable radius ruand at some larger radius, roughly r∼10. This larger radius is the apastron of an orbit. If the two black holes arereleasedfromrestataninitialseparationin center-of- mass coordinates equal to this apastron, their orbit will rolldownthepotential(althoughtheshapechangeswhen Pr/negationslash= 0) and then climb back up the other side asymptot- ically approaching the spherical orbit at the top of the hill. By definition, this is a homoclinic orbit (Fig. 9). To our knowledge it is the first of its kind to be found out of the equatorial plane [8, 13, 14]. The orbit winds around the center of mass an infinite number of times as it asymptotically approaches the un- stable spherical orbit. Although not strictly periodic – the homoclinic orbitneverreturnsto apastron–it will be significant for the periodic tables [7, 8] as a maximum en- ergyorbit fora given Lin the strong-fieldregime[13, 14]. IV. CONCLUSIONS This paper, as the second in a series, provides the en- ergetic frame in which the periodic tables of paper I were set [7]. Although in support of paper I’s goals, the anal- ysis of spherical orbits could be relevant to additional tests of the PN expansion for spinning black hole pairs and could have a place in the disucssion of initial values for numerical relativity. Additionally, we find the non- equatorial homoclinic orbits that whirl an infinite num- ber of times as they asymptote to the unstable spherical orbit. The homoclinic spearatrixis important as defining the transition to plunge for all orbits, including eccentric12 /Minus4/Minus2024/Minus4 /Minus2 0 2 4/Minus1.0/Minus0.50.00.51.0 /Minus4/Minus20 2 4/Minus4/Minus2024 FIG. 8: The unstable spherical orbit that is the maximum of Vefffor Fig. 7. Unlike the effective potential, the details of the full orbit do depend on the specific combination Seff·ˆL. Left: As viewed in three dimensions. Right: As viewed in the orbital plane. /Minus50510 05/Minus1012 /Minus10/Minus50 510/Minus10/Minus50510 FIG. 9: The homoclinic orbit for Fig. 7 approaching the unsta ble spherical orbit. Left: As viewed in 3d. Right: As viewed i n the orbital plane. Because of numerical instability near th e highly unstable constant radius orbit, we only show a few wi ndings. and precessing orbits. It would be interesting to extend this study to the EOB method [9, 29, 30] in a future work. In closing, we comment on an intriguing implication of the set of spherical orbits for spinning black hole pairs. In this work, we restricted ourselves to spin-orbit cou- pling and although we allowed both black holes to spin, we found that the spherical orbits constrain the range of allowed bound orbits in the following sense. For a given angular momentum, spin initial conditions, and mass ra- tio, thestablesphericalorbitisthelowestenergygeodesic and the unstable spherical orbit is the highest energy orbit in the strong-field – barring the failures of the ap- proximationat these close separations.9As we showed in paper I [7], between these two sphericalorbits lies an infi- nite set of orbits that are closed in the orbital plane. The periodic set correspondsto a subset of the rationals, with 9We actually consider the emergence of an ibco to define the strong-field. For the equal mass cases that resist the develo p- ment of an ibco, it is as if the approximation is not effective enough to enter the strong field.the rationalidentifying a givenorbit increasingmonoton- ically between the stable spherical orbit and the unstable spherical orbit. The homoclinic orbit is the infinite whirl limit of the periodic set and would be the final entry in a periodic table of orbits correspondingto the infinite limit of the rationals. This pattern of a periodic set framed by constant ra- dius orbits and limiting to the homoclinic is consistent with a picture that has emerged for Kerr black hole or- bits [8, 13, 14]. The consistency of the picture for two spining comparable mass black holes with the Kerr case is precisely what is surprising, or at least intriguing. The geodesics in a Kerr spacetime are known to be integrable [35]. Thereareenoughconstantsofthe motionto restrict trajectories to regular tori and prohibit chaotic mixing. As Poincar´ e intuited, the structure of the periodic or- bits encodes the entire dynamics and the regularity of the system is in fact reflected in the regularity of the pe- riodic spectrum. The simplicity of the spherical orbits and the periodic set they frame suggests that even when both black holes spin and are of comparable mass, there is no chaos – at least not in physically plausible regimes13 –if only spin-orbit coupling is included.10 Put another way, homoclinic orbits are also a sign of non-linearity. They mark the intersection of the sta- ble and unstable manifolds of a hyperbolic invariant set. They are the precursor to chaos in the sense that un- der perturbation, the homoclinic orbit breaks up into a homoclinic tangle and will be the locus of a fractal set of orbits [14, 36]. The fractal set is sometimes refered to as a strange repellor and is the analog for conserva- tive systems of strange attractors in dissipative systems [5, 37, 38, 39]. Systems with a regular set of periodic orbits that cul- minate in a homoclinic limit are not chaotic. However, thespinningpairsarevulnerabletochaosasevidencedby their very possesion of a homoclinic orbit. Indeed chaos hasbynowbeenwellconfirmedintheformofafractalset when spin-spin coupling is included [5, 6, 22, 40, 41, 42]. As suspected in Ref. [43], our work suggests that the emergence of chaos must be directly tracable to the spin- spin coupling. We conjecture that the transition to chaos could be witnessed through the destruction of the cor- resondence of the periodic set with the rationals when spin-spin coupling is turned on. The additional preces- sional effects of spin-spin coupling, we suggest, must de- stroy the homoclinic orbit, replace it with a homoclinic tangle – a fractal set of orbits – and induce chaotic scat- tering among geodesics in the vicinity. 10It is possible that for Seff·ˆLmuch larger than black hole physical values chaos could develop. Afterall, one of the constants o f motionPΨhas been lost with the inclusion of spin-orbit coupling opening the door for chaos.Acknowledgments We are especially grateful to Gabe Perez-Giz for his valuable and generous contributions to this work and to Jamie Rollins for his careful reading of the manuscript. We also thank Szabi Marka for important discussions. This material is based in part upon work supported un- der a National Science Foundation Graduate Research Fellowship.14 APPENDIX A: PROJECTION THE EQUATIONS OF MOTION ONTO THE NON-ORTHOGONAL ORBITAL BASIS 1. The Orbital Plane Equations The four equations of motion in the orbital plane are obtainedby projectingHamilton’sequationsontothe ba- sis vectors, as is done in celestial mechanics. For now, consider only the projections onto the orbital basis vec- tors to generate the four equations, ˙r·ˆ n=∂H ∂p·ˆ n ˙r·ˆΦ=∂H ∂p·ˆΦ ˙p·ˆ n=−∂H ∂r·ˆ n ˙p·ˆΦ=−∂H ∂r·ˆΦ. (A1) To break down the LHS involves ˙r= ˙rˆ n+r˙ˆ n ˙p=˙Prˆ n+Pr˙ˆ n−L r2˙rˆΦ+L r˙ˆΦ.(A2) Wewillneedprojectionsof ˙ˆ nand˙ˆΦalongˆ nandˆΦ. Now, sinceˆ n·ˆ n= 1, it follows that ˙ˆ n·ˆ n= 0 and by the same reasoning˙ˆΦ·ˆΦ= 0. Also, by orthogonality, ˙/parenleftBig ˆ n·ˆΦ/parenrightBig = 0 =⇒ ˙ˆ n·ˆΦ=−˙ˆΦ·ˆ n. (A3) To obtain the final dot product above we expand the basis vectors ( ˆ n,ˆΦ,ˆΨ) in terms of an intermediate basis (ˆX,ˆY) that spans the orbital plane, and then expanding (ˆX,ˆY) in the Cartesian basis. We proceed as we did in paper I and define the intersection of the orbital plane with the equatorial plane: ˆX=ˆJ׈L/vextendsingle/vextendsingle/vextendsingleˆJ׈L/vextendsingle/vextendsingle/vextendsingle=ˆJ׈L sinθL, (A4) where cos θL=ˆL·ˆJ. The vector orthogonal to ˆXthat lies in the orbital plane is ˆY=ˆL׈X. (A5) This intermediate orbital basis will be useful in the ma- nipulations that follow. In terms of Cartesian compo- nents defined with ˆk=ˆJandˆi,ˆjspanning the equatorial plane, we can expand ˆX= cosΨˆi+sinΨˆj ˆY= sinθY(−sinΨˆi+cosΨˆj)+cosθYˆk,(A6)where cos θY=ˆY·ˆJ. SinceˆYis always orthogonal to ˆL, again by construction, this is not really a new angle but can be recast as θY=π/2−θL. Our non-orthogonal basis can then be expanded as ˆ n= cosΦˆX+sinΦˆY ˆΦ=−sinΦˆX+cosΦˆY ˆΨ=−sinΨˆi+cosΨˆj. (A7) Using ˙ˆX=˙ΨˆΨ (A8) ˙ˆΨ=−˙ΨˆX (A9) ˙ˆY=−sinθY˙ΨˆX+cosθY˙θYˆΨ−sinθY˙θYˆk.(A10) From all of the above relations we obtain for use in the projections ˙ˆ n·ˆ n= 0 (A11) ˙ˆΦ·ˆΦ= 0 (A12) ˙ˆ n·ˆΦ=˙Φ+˙ΨsinθY=˙Φ+˙ΨcosθL(A13) ˙ˆΦ·ˆ n=−˙ˆ n·ˆΦ. (A14) Conveniently, these are the same projections we found in paper I for the case in which only one black hole spins and˙θY=˙θL= 0. Now we can derive the equations of motion in the (r,Φ,Ψ) coordinates. We use the equations we con- structed in paper I [7] ˙r=Ap+Bˆ n+Seff×r r3 ˙p=Cp+Dˆ n+Seff×p r3+3L·Seff r4ˆ n,(A15) whereA,B,C,D are given by Eqs. (18). With the pro- jections (Eqs. (A1)), (A2), and the abovevectorrelations we have the radial equation from ˙r·ˆ nin (A15): ˙r=APr+B . (A16) The Φ equation follows from ˙r·ˆΦ=∂H ∂p·ˆΦ (A17) r/parenleftBig ˙Φ+˙ΨcosθL/parenrightBig =AL r+(Seff×r)·ˆΦ r3.(A18) Look at (Seff×r)·ˆΦ=r/parenleftBig Seff·ˆL/parenrightBig .(A19) The Φ equation is then ˙Φ =AL r2−˙ΨcosθL+Seff·ˆL r3(A20) whereˆS·ˆLis constant.15 The two conjugate momenta equations are next. We start with Pr: ˙p·ˆ n=˙Pr−L r(˙Φ+˙ΨcosθL) (A21) =CPr+D+2Seff·L r4 where we have used that (p×Seff)·ˆ n=Seff·L r(A22) Notice if we use Eq. (A20), we have ˙Pr=AL2 r3+CPr+D+3Seff·L r4 and last ˙p·ˆΦ=Pr(˙Φ+˙ΨcosθL)−L r2˙r(A23) =CPΦ r+PrSeff·ˆL r3 where we have used that (p×Seff)·ˆΦ=Seff·(ˆΦ×p) =−PrSeff·ˆL(A24) Notice if we use Eq. (A20), we have a cancellation and (APr−˙r)L r2=−BL r=CL r whichconfirmsatruestatementbutdoesnotprovideany new equation of motion. The final equation of motion is simply˙PΦ= 0. All four equations in the orbital basis are compiled in the boxed Eqs. (12). 2. The Precession of the Plane The plane precesses in the direction ˆΨat a rate ˙Ψ, which can be computed from the first of Eqs. (A10): ˙ˆX=˙ΨˆΨ. We can isolate ˙Ψ by projecting along ˆΨ, ˙ˆX·ˆΨ=˙Ψ. (A25) We take the time derivative of Eq. (A4) and use the con- stancy of ˆJand the precession equation for˙ˆLfrom Eq. (8) to find ˙Ψ = ˆJ×/parenleftBig Seff׈L/parenrightBig /vextendsingle/vextendsingle/vextendsingleˆJ׈L/vextendsingle/vextendsingle/vextendsingler3 ·ˆΨ.(A26)Notice that the term that would have been proportional to˙θLis killed since it is also proportional to ˆX·ˆΨ= 0. With some vector manipulations, including the general ruleA×(B×C) =B(A·C)−C(A·B), applied to both the term in parantheses and to ˆΨ=ˆJ׈XwithˆXgiven by Eq. (A4), this can be reduced to ˙Ψ =Seff·/parenleftBig ˆJ−ˆL(ˆJ·ˆL)/parenrightBig sinθ2 Lr3. (A27) Going in the other direction with the general rule, B(A· C)−C(A·B) =A×(B×C), wecanwritetheright-hand- sideas atriple crossproduct andidentify the particularly compact form ˙Ψ =Seff·ˆY sinθLr3. (A28) As in paper I, PΨ=Lz=L·ˆJ. (A29) Unlike paper I, PΨ, which can also be expressed as PΨ= LcosθL, is not conservedwhen both black holes spin and precess. 3. One Effective Spin The equations of motion simplify considerably if there is only one effective spin, such as the case of only black hole spinning [7]: ˙r=APr+B, ˙Pr=AL2 r3−B rPr+D+3δ1S1·L r4 ˙Φ =AL r2−δ1L r3,˙PΦ= 0 The orbital plane precesses with frequency ˙Ψ = Ω L=δ1J r3˙PΨ= 0. (A30) Consequently, the equations of motion above are inde- pendent ofangles. In paper I, we used these purely radial equations to study several features of the dynamical sys- tem, such as a periodic table that defined the spectrum of black hole orbits. The same simplification can be effected when the black holes are of equal mass m1=m2. Then what we really mean by S1isS1→S1+S2.16 [1] S. F. Portegeis-Zwart and a.-p. S L McMillan. [2] L. Wen, Ap. J. 598, 419 (2003). [3] R. M. O’Leary, B. Kocsis, and A. Loeb, (2008). [4] J. Levin, R. O’Reilly, and E. Copeland, Phys. Rev. D 62, 024023 (2000). [5] J. Levin, Phys. Rev. Lett. 84, 3515 (2000). [6] J. Levin, Phys. Rev. D 67, 044013 (2003). [7] J. Levin and R. Grossman, gr-qc/08093838 (2008). [8] J. Levin and G. Perez-Giz, Phys. Rev. D 77, 103005 (2008). [9] T. Damour, Phys. Rev. D 64, 124013 (2001). [10] A. Buonanno, Y. Chen, and T. Damour, Phys. Rev. D 74, 104005 (2006). [11] U. Sperhake et al., Phys. Rev. D78, 064069 (2008). [12] Bombelli and Calzetta, Class and Quant. Grav. 9, 2573 (1992). [13] J. Levin and G. Perez-Giz, Homoclinic Orbits around Spinning Black Holes I: Exact Solution for the Kerr Sep- aratrix, 2008. [14] G. Perez-Giz and J. Levin, Homoclinic Orbits around Spinning Black Holes II: The Phase Space Portrait , 2008. [15] F. Pretorius and D. Khurana, Class. Quant. Grav. 24, S83 (2007). [16] G. Schaefer, Annals Phys. 161, 81 (1985). [17] T. Damour and N. Deruelle, C. R. Acad. Sci. Paris 293, 537 (1981). [18] T. Damour and G. Schaefer, Nuovo Cim. B101, 127 (1988). [19] P. Jaranowski and G. Schaefer, erratum-ibib.d 63, 029902 (2001). [20] T. Damour, P. Jaranowski, and G. Schafer, Phys. Lett. B513, 147 (2001). [21] T. Damour, P. Jaranowski, and G. Schaefer, Phys. Rev. D62, 084011 (2000). [22] A. Gopakumar and C. K¨ onigsd¨ orffer, Phys. Rev. D 72,121501 (2005). [23] Sch¨ afer and Wex, Phys. Lett. A 174(1993). [24] N. Wex and S. Kopeikin, astro-ph/9811052 (1998). [25] B. Gong, astro-ph/0401152 (2004). [26] C. K¨ onigsdoerffer and A. Gopakumar, Phys. Rev. D 71, 024039 (2005). [27] R. Wald, General Relativity , 1984. [28] F. Pretorius, Class. Quant. Grav. 23(2006). [29] A. Buonanno and T. Damour, Phys. Rev. D 59, 084006 (1999). [30] A. Buonanno et al., Toward faithful templates for non- spinning binary black holes using the effective-one-body approach, 2007. [31] L. E. Kidder, C. M. Will, and A. G. Wiseman, Phys. Rev. D47, 4183 (1993). [32] N. Wex and G. Sch¨ afer, Class. and Quantum Grav. . [33] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev.D57, 885 (1998). [34] L. Blanchet, Phys. Rev. D65, 124009 (2002). [35] B. Carter, Phys. Rev. 174, 1559 (1968). [36] E. Ott, Chaos in Dynamical Systems , Cambridge Uni- versity Press, 2002. [37] N. J. Cornish, C. P. Dettmann, and N. E. Frankel, Phys. Rev. D50, 618 (1994). [38] C. Dettmann, N. Frankel, and N. Cornish, Phys. Rev. D.50, 618 (1994). [39] N. J. Cornish and J. J. Levin, Phys. Rev. Lett. 78, 998 (1997). [40] J. Levin, Phys. Rev. D 74, 124027 (2006). [41] X. Wu and Y. Xie, Phys. Rev. D 76, 124004 (2007). [42] X. Wu and Y. Xie, Phys. Rev. D 77, 103012 (2008). [43] M. D. Hartl and A. Buonanno, Phys. Rev. D 71, 024027 (2005).

2310.04763v1.Orbital_diffusion__polarization_and_swapping_in_centrosymmetric_metals.pdf

Orbital diffusion, polarization and swapping in centrosymmetric metals Aur´ elien Manchon1,∗A. Pezo1, Kyoung-Whan Kim2, and Kyung-Jin Lee3 1Aix-Marseille Univ, CNRS, CINaM, Marseille, France 2Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea 3Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea We propose a general theory of charge, spin and orbital diffusion based on Keldysh formalism. Our findings indicate that the diffusivity of orbital angular momentum in metals is much lower than that of spin or charge due to the strong orbital intermixing in crystals. Furthermore, our theory introduces the concept of “spin-orbit polarization” by which a pure orbital (spin) current induces a longitudinal spin (orbital) current, a process as efficient as spin polarization in ferromagnets. Finally, we find that orbital currents undergo momentum swapping, even in the absence of spin-orbit coupling. This theory establishes several key parameters for orbital transport of direct importance to experiments. Introduction - The interconversion between charge and spin currents1is one of the central mechanisms of spin- tronics and possibly its most instrumental. This mecha- nism is at the source of spin-orbit torque2and charge cur- rents induced by spin pumping3. At the core of these phe- nomena lies the spin-orbit interaction that couples the spin to the orbital angular momentum in relatively high- Z materials (5 dmetals, topological materials etc.). In re- cent years, it has been proposed that the interconversion between charge and orbital currents, via orbital Hall4–6 and orbital Rashba effects7–9for instance, might in fact be much more efficient than its spin counterpart because it arises from the orbital texture imposed by the crys- tal field rather than from spin-orbit coupling. Therefore, corresponding phenomena such as orbital torque10–13and orbital magnetoresistance14have been proposed and ex- perimentally reported. In these experiments, the scenario is based on a two-step process: orbital Hall or Rashba effect takes place in a light metal and the resulting or- bital current is converted into a spin signal once in the adjacent ferromagnet. Consequently, it is expected that the supposedly large charge-to-orbital conversion taking place in the low-Z metal compensates for the relatively low spin-orbit coupling of the ferromagnet, which seems to be confirmed by the experiments11,12,14–17. An important rational behind the promotion of orbi- tronics is twofold. First, as mentioned above, since or- bital transport is governed by the crystal field, orbital Hall and Rashba effects do not necessitate spin-orbit coupling and occur in relatively low-Z metals (e.g., 3 d metals). Second, orbital currents are expected to prop- agate over much longer distances than spin current be- cause they are immune to spin scattering. On the other hand, several questions remain open. To start with, as observed by Ref. 4, the atomic orbital moment is never a good quantum number and therefore it remains un- clear how it diffuses from one metal to another. Recent phenomenological models of orbital diffusion have been recently proposed15,18but lack quantitative predictabil- ∗aurelien.manchon@univ-amu.frity by overlooking microscopic details. In addition, sev- eral recent works have pointed out that the orbital mo- ment arises not only from intra-atomic spherical har- monics ( p, d) but also possesses substantial inter-atomic contribution19–21. Understanding the way orbital cur- rents and densities propagate in metals and accumulate at interfaces requires determining transport coefficients such as orbital conductivity or diffusivity, as well as the ability to interconvert spin currents into orbital currents via spin-orbit coupling. Indeed, when injecting an or- bital density l=⟨L⟩in a metal, it diffuses and produces an orbital current Jl=−Dl∂rll,Dlbeing the orbital diffusion coefficient (typically a tensor). In the presence of spin-orbit coupling ξsoˆσ·ˆL, this orbital current can convert into a spin current Js. FIG. 1. (Color online) Spin and orbit interconversion mechan- ims: (a) spin-to-orbit polarization and (b) orbit-to-spin po- larization mediated by spin-orbit coupling. (c) Spin and (d) orbital swapping. The former requires spin-orbit coupling whereas the latter occurs even without it. In this Letter, we derive a theory of spin and orbital diffusion in metals, and uncover several mechanisms gov- erning orbital torque and magnetoresistance phenomena, illustrated in Fig. 1. First, we find that whereas charge and spin diffusion are of about the same order of mag- nitude, the orbital diffusion is much lower. This due to the fact that the orbital moment is never a good quantum number in crystals (rotational invariance is broken). Sec- ond, we find that in the presence of spin-orbit coupling,arXiv:2310.04763v1 [cond-mat.mes-hall] 7 Oct 20232 an orbital current is systematically accompanied by a spin current that is collinear to it (and vice versa) [Fig. 1(a,b)]. This ”spin-orbit polarization” can be sizable, comparable to spin polarization in 3 dferromagnets. Fi- nally, the third class of effects uncovered by our theory is the ”angular momentum swapping”, i.e., the interchange between the propagation direction and angular momen- tum direction upon scattering [Fig. 1(c,d)]. Whereas the spin swapping was predicted by Lifshits and Dyakonov22 in the presence of spin-orbit coupling, orbital swapping arises naturally even without it. When turning on spin- orbit coupling, not only spin swapping emerges, but also spin-to-orbit and orbit-to-spin swapping. Theory - The objective of the present theory is to determine the diffusive current induced by a gradient of particle density, J=−D∂rρ. In this expression, Jcan be the charge current Jc, or the spin (orbital) current Js(l), whereas ρcan be the charge density ρc, or the spin (orbital) density s(l). In the language of nonequilibrium Green’s function, the particle current density is obtained by computing the quantum statistical expectation value of the trace of the particle current operator ˆjtaken over the lesser Green’s function G<, J=Zd3k (2π)3Zdε 2iπTrh ˆjG<i . (1) The philosophy of the present theory is to express the lesser Green’s function to the first order in the density gradient ∂rρ. We start from Keldysh-Dyson equation23 G<=GR⊗Σ<⊗GA, (2) where G<=G<(r,r′;t, t′) (Σ<) is the lesser Green’s function (self-energy), GR(A)=GR(A)(r,r′;t, t′) is the retarded (advanced) Green’s function and ⊗is the con- volution product on both time and space. In Eq. (2), we omitted the explicit time and space dependence for simplicity. In the linear response regime, we first ex- press G<to the first order in spatial gradients using Wigner transform (see, e.g., Ref. 24), i.e., we rewrite Eq. (2) in the frame of the center-of-mass, ( r,r′;t, t′)→ (r−r′,rc;t−t′, tc), with ( rc, tc) = (( r+r′)/2,(t+t′)/2), Fourier transform the small space and time coordinates (r−r′, t−t′)→(k, ω), and expand Keldysh-Dyson equation to the first order in space and time gradients (∂rc, ∂tc). In the following, the subscript cis dropped for the sake of readability. Under Wigner transform, the convolution product becomes A⊗B=AB+i 2(∂rA∂kB−∂kA∂rB), (3) and finally, the part of the lesser Green’s function that is linear in spatial gradient reads δG<=i 2 GR 0∂rΣ<∂kGA 0−∂kGR 0∂rΣ<GA 0
. (4) Here GR(A) 0 = (ℏω−H 0±iΓ) is the unperturbed retarded (advanced) Green’s function and H0is the crystal Hamil- tonian.Since we are interested in the diffusion coefficients that connect angular momentum densities (odd under time- reversal T) with angular momentum currents (even un- derT), the diffusion coefficients in nonmagnetic mate- rials are themselves odd under T. The same is true for the charge diffusivity that connects the charge density (even under T) with the charge current (odd under T). As a result, the charge, spin and orbital diffusion coeffi- cients must be dissipative, proportional to the scattering time. In the language of quantum transport, these phe- nomena are driven by Fermi surface electrons akin to the charge conductivity. This is in stark contrast with the spin and orbital Hall diffusivities, which connect charge densities (even under T) with spin and orbital currents (even under T): they are even under T, independent on scattering in the limit of weak disorder, and associated with the Berry curvature25,26. Since we focus on angu- lar momentum diffusion and spin-orbit interconversion, Eq. (4) is limited to transport at the Fermi level and disregards Fermi sea contributions. The present analysis applies to nonmagnetic materials and must be revised in the case of magnetic systems27as new terms are allowed. Considering point-like impurities, Himp=P iV0δ(r− Ri), the lesser self-energy reads Σ<=1 VX i,jZd3k (2π)3V0G< kV0eik·(Ri−Rj)=niV2 0 V⟨G< k⟩. (5) Here, niis the impurity concentration and ⟨...⟩= VRd3k (2π)3stands for momentum integration over the Bril- louin zone. Noting that ∂kGR 0=ℏGR 0vGR 0, we obtain δG< k=iℏniV2 0 VRe GR 0∂r⟨G< k⟩GA 0vGA 0 . (6) Inserting Eq. (6) into Eq. (1), we obtain the general expression of nonequilibrium properties induced by spa- tial gradients. Now, as argued above, diffusive effects are associated with Fermi surface electrons, which sug- gests (1 /V)∂r⟨G< k⟩= 2iπ∂rˆρδ(ε−εF), ˆρbeing the density matrix at Fermi level. As a result, the particle current density reads J=−ℏniV2 0ReTr kh ˆjIm[GR 0∂rˆρGA 0vGA 0]i εF.(7) For the sake of compactness, we defined Tr k=Rd3k (2π)3Tr. Equation (7) is the central result of this work and can be used to compute the diffusive charge, spin and orbital currents induced by density gradients. For instance, sub- stituting ˆ ρby the charge density ρc=−eTr[ˆρ], and the charge current operator ˆj=−eˆv, one obtains the charge diffusivity Dij=ℏniV2 0ReTr k ˆvjIm[GR 0GA 0ˆviGA 0] . (8) The validity of Eq. (8) is readily assessed by compar- ingDijwith the conductivity σijobtained from Kubo’s3 formula26. In the relaxation time approximation, niV2 0= Γ/(πNF), where NFis the density of states at Fermi level and Γ is the disorder broadening. Using this rela- tion, we confirm the Einstein relation Dij=σij/(e2NF) (not shown). In the rest of this work, we express the spin and orbital diffusivity in the units of a conductiv- ity (e2Γ/(πniV2 0))Dij, i.e., in Ω−1·m−1rather than in m2·s−1. To obtain the spin and orbital diffusivities, the re- spective densities are defined s= (ℏ/2)Tr[ ˆσˆρ] and l=ℏTr[ˆLˆρ],ˆσand ˆLbeing the dimensionless spin and orbital operators. Therefore, substituting the cur- rent operator ˆjby either the spin current operator ˆjα s,j= (ℏ/4){ˆvj,ˆσβ}or the orbital current operator ˆjβ l,j= (ℏ/2){ˆvj,ˆLβ}in Eq. (7), and ∂iˆρby ˆσα∂isαor ˆLα∂ilα, we obtain the general relation Jβ s,j Jβ l,j! =− Dsβj sαiDsβj lαi Dlβj sαiDlβj lαi! ∂isα ∂ilα (9) with the diffusion coefficients Dsβj sαi= 2niV2 0ReTr kh ˆjβ s,jIm[GR 0ˆσαGA 0ˆviGA 0]i ,(10) Dlβj lαi=niV2 0ReTr kh ˆjβ l,jIm[GR 0ˆLαGA 0ˆviGA 0]i ,(11) Dlβj sαi= 2niV2 0ReTr kh ˆjβ l,jIm[GR 0ˆσαGA 0ˆviGA 0]i ,(12) Dsβj lαi=niV2 0ReTr kh ˆjβ s,jIm[GR 0ˆLαGA 0ˆviGA 0]i .(13) Dsβj sαirepresents a spin current Jβ s,jinduced by the gra- dient of a spin density ∂isα, whereas Dlβj lαirepresents an orbital current Jβ l,jinduced by the gradient of an orbital density ∂ilα. In addition, the diffusivities Dlβj sαiandDsβj lαi represent the spin-to-orbital interconversion phenomena, i.e., spin-orbit polarization ( α=β) and spin-orbit swap- ping ( α̸=β). Spin and orbital diffusion - To quantitatively estimate the magnitude of these effects, we consider a bcc crystal with ( px, py, pz) orbitals. The tight-binding Hamiltonian is obtained using Slater-Koster parameterization, with Vσ= 0.2 eV and Vπ= 0.05 eV. Since the structure has cubic symmetry, we assume that the (charge, spin or or- bital) gradient is along x. We first compute the charge, spin and orbital diffusiv- ities in Fig. 2. Since the current diffuses in the same di- rection as the density gradient, i=j, its angular momen- tum is necessarily aligned on that of the density, α=β. In the absence of spin-orbit coupling, the charge diffusiv- ityDxxand the spin diffusivities Dsxx sxx,Dsyx syx(=Dszx szx) are all equal [black line in Fig. 2(a)]. Turning on the spin-orbit coupling ( ξso= 0.05 eV) slightly reduces the charge diffusivity and breaks the symmetry between the spin diffusion coefficients, Dsxx sxx̸=Dsyx syx(=Dszx szx). Inter- estingly, as reported on Fig. 2(b), the orbital diffusivities FIG. 2. (Color online) (a) Charge (black and gray) and spin (red) diffusivities as a function of the energy. For ξso= 0, the spin and charge diffusivities fall into one single curve (black), whereas for ξso= 0.05, the charge diffusivity is reduced (gray) and the spin diffusivity becomes anisotropic (light and dark red lines). (b) Orbital diffusivities as a function of the energy forξso= 0 (dashed) and ξso= 0.05 eV (solid). The orbital diffusivities are intrinsically anisotropic. (c) Dependence of the spin (red) and orbital (blue) diffusivities as a function of the spin-orbit coupling ξsoat transport energy ε= 0.5 eV. Dlxx lxxandDlyx lyx(=Dlzx lzx) are in fact very small (dashed blue lines), which we attribute to the strong orbital mix- ing that naturally governs the band structure of our bcc crystal. Turning on the spin-orbit coupling (solid lines) again breaks the symmetry between the diffusion coeffi- cients, Dlxx lxx̸=Dlyx lyx(=Dlzx lzx) and, remarkable, enhances the overall orbital diffusivity. This can be understood qualitatively by the fact that spin-orbit interaction cou- ples the highly conductive spin channel with the weakly conductive orbital channel, thereby reducing the spin dif- fusivity while enhancing the orbital one, as shown in Fig. 2(c). The low orbital diffusivities reported here do not nec- essarily contradict the idea that orbital momentum could be transported over much longer distance than spin momentum12,16,17. Indeed, a comprehensive theory of orbital transport requires a microscopic modeling of or- bital relaxation mechanisms, which remains out of the scope of the present work. Notice that in ferromagnets, spin dephasing severely limits the spin propagation, such that orbital diffusion naturally dominates12,13. This ef- fect is absent in nonmagnetic metals. Spin-orbit polarization - The next question we wish to address is how much orbital current can one obtain upon injecting a spin current in a heavy metal. This mechanism underlies the phenomena of orbital torque and orbital magnetoresistance10–12,14–17where a primary orbital current generated in a light metal is injected in a spin-orbit coupled material and converted into a spin current. To answer this question, we compute the so-4 called ”spin-orbit polarization”. Let us assume that a gradient of, say, spin density ∂isαdiffuses in the system. This gradient induces bothspin and orbital currents, Jα s,i andJα l,i, producing a current of total angular momentum Jα t,i=Jα l,i+Jα s,i. To quantify the relative proportion of spin and orbital currents, we define the spin-to-orbit po- larization Pα l,i=Dlαi sαi/(Dsαi sαi+Dlαi sαi), and similarly, the orbit-to-spin polarization, Pα s,i=Dsαi lαi/(Dlαi lαi+Dsαi lαi). The spin-to-orbit ( sα→lα) and orbit-to-spin ( lα→ sα) longitudinal diffusivities as well as the corresponding polarization are given in Fig. 3(a,b). Again, the diffu- sivities are anisotropic due to the presence of spin-orbit coupling. The orbital diffusivity being much smaller than the spin diffusivity, the orbit-to-spin polarization is gen- erally smaller than the spin-to-orbit polarization. The polarization increases steadily with spin-orbit coupling, as expected, and saturates at large spin-orbit coupling strength. It is worth noting that the spin-orbit polariza- tion is comparable to the spin polarization found in con- ventional 3 dmetal compounds (typically 50-70%). This observation is consistent with Ref. 5 that suggests an orbit-to-spin polarization of about 50% in Pt and Pd for Hall currents. The sizable spin-orbit polarization given in Fig. 3(c) is a crucial ingredient for the orbital torque and magnetoresistance. FIG. 3. (Color online) (a) Spin-to-orbit (red) and orbit- to-spin (blue) diffusivities as a function of the energy for ξso= 0.05 eV. (b) Corresponding spin-orbit polarizations as a function of the spin-orbit coupling for ε= 0.5 eV. Spin, orbital and spin-orbit swapping - We finally con- sider the last class of effects, the spin and orbital swap- ping. For these effects, the directions of injection and collection are perpendicular to each other, as well as the direction of the incoming and outgoing (spin/orbit) po- larization [see Fig. 1(c,d)]. The orbital diffusivity tensor has the following form Jx l,x Jy l,x Jx l,y Jy l,y =− Dlxx lxx0 0 Dlxx lyy 0Dlyx lyxDlyx lxy0 0Dlxy lyxDlxy lxy0 Dlyy lxx0 0 Dlyy lyy ∂xlx ∂xly ∂ylx ∂yly ,(14)and Onsager reciprocity imposes that Dlyx lxy=Dlxy lyxand Dlxx lyy=Dlyy lxx. Importantly, the orbital swapping does not necessitate spin-orbit coupling as it is solely governed by the orbital overlap (and hence the crystal field symme- try) of the crystal. These coefficients are reported in Fig. 4(a) in the absence of spin-orbit coupling. Turning on the spin-orbit coupling triggers spin swapping22, whose diffusivity tensor has the same form as in Eq. (14). Fig- ure 4(b) displays the spin (red) and orbital (blue) swap- ping efficiencies defined as Dsyy sxx/Dsxx sxxandDlyy lxx/Dlxx lxx, as a function of spin-orbit coupling, showing that orbital swapping is generally larger than spin swapping, which seems reasonable given the minor role of spin-orbit cou- pling in the former. In addition, spin-orbit coupling also enables the trans- fer between spin and orbital angular momenta that re- sults in spin-to-orbit (red) and orbit-to-spin (blue) swap- ping, displayed in Fig. 4(c). The diffusivity tensor has the form Jx s,x Jy s,x Jx s,y Jy s,y =− Dsxx lxx0 0 Dsxx lyy 0Dsyx lyxDsyx lxy0 0Dsxy lyxDsxy lxy0 Dsyy lxx0 0 Dsyy lyy ∂xlx ∂xly ∂ylx ∂yly ,(15) and Onsager reciprocity imposes Dsyx lxy=Dsxy lyxand Dsxx lyy=Dsyy lxx. From Fig. 4(c), it appears that spin- to-orbit swapping is larger than orbit-to-spin swapping, a feature already observed in Fig. 3 for the spin-orbit polarization. In the context of spin-orbit torque2, spin swapping, being of bulk22,28or interfacial origin29, is re- sponsible for additional torque components in magnetic multilayers. The large orbital swapping efficiencies re- ported here suggest that in systems displaying orbital torque, large deviations from the conventional field-like and damping-like torques can be expected28. FIG. 4. (Color online) (a) Orbital swapping as a function of energy for ξso= 0. (b) Spin (red) and orbital (blue) swapping as a function of the spin-orbit coupling strength. (c) Spin-to- orbit (red) and orbit-to-spin (blue) swapping as a function of the spin-orbit coupling. In (b) and (c), we set ε= 0.5 eV. Conclusion - Advancing research in orbitronics re-5 quires a proper description of spin and orbital diffusion in metals. As stated previously, whereas the vast ma- jority of theoretical studies to date focus on orbital and spin currents generated by electric currents, our theory allows us to compute the orbital and spin currents in- duced by diffusive gradients of angular momenta. It re- veals that although orbital currents do not experience ”orbital-flip” per se , their diffusivity in metals is much weaker than that of spin currents. This result seems at odds with recent experiments suggesting a long or- bital diffusion length in transition metals13,15–17,30. In diffusive transport though, the (spin or orbital) diffusion length is related to the product between the (spin or or- bital) diffusivity Dand the (spin or orbital) relaxation time τr,λ∝√Dτr. Therefore, to model the orbital diffu- sion length in transition metals, the present theory must be completed by a theory of orbital relaxation which re- mains an open question. Our theory also quantifies the spin-to-orbit and orbit-to-spin polarization, i.e., the abil- ity for a spin (orbital) current to generate a longitudinal orbital (spin) current, and finds that this effect is very efficient, potentially as efficient as conventional spin po- larization in 3 dmagnets. Finally, we show that orbital currents are subject to angular moment swapping evenin the absence of spin-orbit coupling and can be as large as spin swapping. We point out that the Green function theory proposed in this Letter is well adapted to multiband systems and in particular to realistic heterostructures computed from first principles. Indeed, systematic investigation of or- bital Hall conductivity and orbital Edelstein effects in transition metals have been recently performed31–33and extending the present work to realistic materials of inter- est to experiments could open appealing perspectives for the design of orbital devices. ACKNOWLEDGMENTS A.P. was supported by the ANR ORION project, grant ANR-20-CE30-0022-01 of the French Agence Na- tionale de la Recherche, A. M. was supported by from the Excellence Initiative of Aix-Marseille Universit´ e - A*Midex, a French ”Investissements d’Avenir” program, K.-W. K. was supported by the KIST Institutional Pro- gram and K.-J. L. was supported by the NRF (NRF- 2022M3I7A2079267). 1T. Nan, D. C. Ralph, E. Y. Tsymbal, and A. Manchon, APL Materials 9, 120401 (2021). 2A. Manchon, J. Zelezn´ y, M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Review of Modern Physics 91, 035004 (2019). 3E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Applied Physics Letters 88, 182509 (2006). 4B. A. Bernevig, T. L. Hughes, and S. cheng Zhang, Phys- ical Review Letters 95, 066601 (2005). 5H. Kontani, T. Tanaka, D. Hirashima, K. Yamada, and J. Inoue, Physical Review Letters 102, 016601 (2009). 6D. Go, D. Jo, C. Kim, and H. W. Lee, Physical Review Letters 121, 086602 (2018). 7D. Go, J. philipp Hanke, P. M. Buhl, F. Freimuth, G. Bihlmayer, H. woo Lee, Y. Mokrousov, and S. Bl¨ ugel, Scientific Reports 7, 46742 (2017). 8T. Yoda, T. Yokoyama, and S. Murakami, Nano Letters 18, 916 (2018). 9A. E. Hamdi, J. Y. Chauleau, M. Boselli, C. Thibault, C. Gorini, A. Smogunov, C. Barreteau, S. Gariglio, J. M. Triscone, and M. Viret, Nature Physics (2023), 10.1038/s41567-023-02121-4. 10D. Go and H. woo Lee, Physical Review Research 2, 13177 (2020). 11S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth, S. Becker, F. Kammerbauer, J. Yang, G. Jakob, Y. Mokrousov, and M. Kl¨ aui, Physical Review Letters 125, 177201 (2020). 12D. Lee, D. Go, H. J. Park, W. Jeong, H. W. Ko, D. Yun, D. Jo, S. Lee, G. Go, J. H. Oh, K. J. Kim, B. G. Park, B. C. Min, H. C. Koo, H. W. Lee, O. J. Lee, and K. J. Lee, Nature Communications 12, 6710 (2021). 13D. Go, D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park,S. Bl¨ ugel, H.-W. Lee, and Y. Mokrousov, Physical Review Letters 130(2023), 10.1103/physrevlett.130.246701. 14S. Ding, Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu, S. Becker, W. Yang, H. Du, C. Wang, Y. Yang, G. Jakob, M. Kl¨ aui, Y. Mokrousov, and J. Yang, Physical Review Letters 128, 067201 (2022). 15G. Sala and P. Gambardella, Physical Review Research 4 (2022), 10.1103/PhysRevResearch.4.033037. 16H. Hayashi, D. Jo, D. Go, T. Gao, S. Haku, Y. Mokrousov, H. W. Lee, and K. Ando, Communications Physics 6 (2023), 10.1038/s42005-023-01139-7. 17R. Fukunaga, S. Haku, H. Hayashi, and K. Ando, Physical Review Research 5(2023), 10.1103/PhysRevRe- search.5.023054. 18S. Han, H.-W. Lee, and K.-W. Kim, Physical Review Let- ters128, 176601 (2022). 19S. Bhowal and G. Vignale, Physical Review B 103, 195309 (2021). 20T. P. Cysne, S. Bhowal, G. Vignale, and T. G. Rappoport, Physical Review B 105, 195421 (2022). 21A. Pezo, D. G. Ovalle, and A. Manchon, Physical Review B106, 104414 (2022). 22M. B. Lifshits and M. I. Dyakonov, Physical Review Letters 103, 186601 (2009). 23S. Rammer and H. Smith, Reviews of Modern Physics 58, 323 (1986). 24X. Wang, C. O. Pauyac, and A. Manchon, Physical Review B89, 054405 (2014). 25J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Physical Review Letters 92, 126603 (2004). 26V. Bonbien and A. Manchon, Physical Review B 102, 085113 (2020).6 27H.-J. Park, H.-W. Ko, G. Go, J. H. Oh, K.-W. Kim, and K.-J. Lee, Physical Review Letters 129, 37202 (2022). 28H. Saidaoui and A. Manchon, Physical Review Letters 117, 036601 (2016). 29S. H. C. Baek, V. P. Amin, Y. W. Oh, G. Go, S. J. Lee, G. H. Lee, K. J. Kim, M. D. Stiles, B. G. Park, and K. J. Lee, Nature Materials 17, 509 (2018). 30S. Lee, M. G. Kang, D. Go, D. Kim, J. H. Kang, T. Lee, G. H. Lee, J. Kang, N. J. Lee, Y. Mokrousov, S. Kim, K. J. Kim, K. J. Lee, and B. G. Park, Communications Physics4, 234 (2021). 31D. Jo, D. Go, and H. woo Lee, Physical Review B 98, 214405 (2018). 32L. Salemi, M. Berritta, and P. M. Oppeneer, Physi- cal Review Materials 5(2021), 10.1103/PhysRevMateri- als.5.074407. 33L. Salemi and P. M. Oppeneer, Physical Review Materials 6, 095001 (2022).

1401.0101v2.Anisotropic_exchange_coupling_in_a_nanowire_double_quantum_dot_with_strong_spin_orbit_coupling.pdf

arXiv:1401.0101v2 [cond-mat.mes-hall] 11 Jul 2014Anisotropic exchange coupling in a nanowire double quantum dot with strong spin-orbit coupling Rui Li1,∗and J.Q. You1,2,† 1Beijing Computational Science Research Center, Beijing 10 0084, China 2Synergetic Innovation Center of Quantum Information and Qu antum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: April 21, 2022) A spin-orbit qubit is a hybrid qubit that contains both orbit al and spin degrees of freedom of an electron in a quantum dot. Here we study the exchange cou pling between two spin-orbit qubits in a nanowire double quantum dot (DQD) with strong spi n-orbit coupling (SOC). We find that while the total tunneling in the DQD is irrelevant to the SOC, both the spin-conserved and spin-flipped tunnelings are SOC dependent and can compete wi th each other in the strong SOC regime. Moreover, the Coulomb repulsion between electrons can combine with the SOC-dependent tunnelings to yield an anisotropic exchange coupling betwe en the two spin-orbit qubits. Also, we give an explicit physical mechanism for this anisotropic ex change coupling. PACS numbers: 73.21.La, 73.63.Kv, 71.70.Ej, 76.30.-v I. INTRODUCTION Realizing a controllable interqubit coupling is of essen- tial importance in quantum information processing (see, e.g., Refs. [ 1,2]). For the electron spin qubit defined in a semiconductor quantum dot [ 3], the two-qubit coupling can be achieved using the isotropic Heisenberg exchange interaction in a tunneling-coupled double quantum dot (DQD) [ 4,5]. Recently, a hybrid qubit, the spin-orbit qubit [6,7], was achieved in a nanowire quantum dot with strong spin-orbit coupling (SOC). A distinct advan- tage of this spin-orbit qubit is its manipulability via an electric field (an effect called electric-dipole spin reso- nance [6–15]) because a local electric field can be gener- ated in experiments much more easily than a local mag- netic field [ 16]. The key element for achieving a spin-orbit qubit is the availability of strong SOC in a quantum-dot material. The semiconductor nanowire materials, e.g., InAs [ 6,17– 19] and InSb nanowire [ 7,20], provide an ideal platform for realizing such a qubit. Indeed, a large Rabi frequency of∼100MHz was reported recently for single-qubit oper- ations [21]. Interestingly, in the presence of strong SOC, the couplingbetweenthe spin-orbitqubit andthe electric field depends nonlinearly on the SOC strength [ 22] and there is an optimal SOC where the Rabi frequency in- duced by an ac electric field becomes maximal [ 8]. Now, it becomes desirable to realize a controllablecoupling be- tween two spin-orbit qubits, in order to implement non- trivial (i.e., conditional) two-qubit operations. In this paper, we investigate the exchange coupling between two spin-orbit qubits in a gated semiconduc- tor nanowire DQD with strong SOC. Our main goal is to clarify the effect of the strong SOC on the exchange ∗rl.rueili@gmail.com †jqyou@csrc.ac.cncoupling. First, we derive a second quantized Hamilto- nian for the DQD, where the electron field operator is expanded in terms of the spin-orbit basis [ 8], other than the conventionalbasiswith separablespin andorbitalde- grees of freedom [ 23] which is valid only in the zero or weak SOC regime. We find that there exist both spin- conserved tunneling tand spin-flipped tunneling t′in the DQD, where the mentioned spin is actually a pseudo- spin (spin-orbit qubit) [ 8]. It is interesting to note that |t|2+|t′|2is irrelevant to the SOC, but t′can compete withtwhen increasing the SOC. Then, we study the exchange coupling by considering two electrons confined in this nanowire DQD. In the strong SOC regime, our results reveal that in contrast to the usual isotropic ex- change coupling, the Coulomb repulsion between elec- trons can combine with the SOC-dependent tunnelings t andt′to yield an anisotropic exchange coupling between the twospin-orbit qubits. We explicitly explain the phys- ical mechanism of this anisotropicexchange coupling and show that the obtained energy spectrum of the two cou- pled spin-orbit qubits is qualitatively in good agreement with the recent experimental results. The paper is organized as follows. In Sec. II, we give analytical expressions for the SOC-dependent tunnelings in a nanowire DQD which are valid in the strong SOC regime. In Sec. III, we study the exchange coupling be- tween two spin-orbit qubits in this nanowire DQD. Also, the impacts of the strong SOC are explicitly clarified. Finally, we conclude in Sec. IV. II. SOC-DEPENDENT TUNNELING IN A NANOWIRE DQD It is interesting to first clarify the effects of the strong SOC on the electron tunneling in a DQD, because previ- ousstudiesonlyfocusedontheweakSOCregime[ 24,25]. Figure1schematically shows the considered semiconduc- tor nanowire DQD with strong SOC, where an electron2 V x( ) tt’ 0 x -d dquE FIG. 1. A nanowire DQD modeled by a double-well potential. Both spin-conserved tunneling tand spin-flipped tunnelings t′exist in the DQD due to the presence of the SOC. Equ= El(r)⇑−El(r)⇓is the level spacing of the spin-orbit qubit and the double arrows represent the basis states of the spin-orb it qubit (i.e., the pseudospin). is confined in a double well and subjected to an external static magnetic field [ 26–28]. The Hamiltonian reads H=p2/(2me)+V(x)−ασyp+(geµBB/2)σx,(1) wherep=−i/planckover2pi1∂/∂x,V(x) is the double-well poten- tial characterizing the DQD, αis the Rashba SOC strength [ 29], and an external static magnetic field B is applied in the xdirection. For simplicity, we consider a symmetric double-well potential (see Fig. 1). In order to explicitly show the role that the strong SOC plays in a DQD, we need to derive a second quan- tized Hamiltonian for the DQD. Similarly to the deriva- tion of the tight-binding Hamiltonian, we first calculate the localizedwavefunction centered at eachdot and then expand the electron field operator in terms of these lo- calized wave functions. Near the minimum of each well, the potential can be expanded harmonically as V(x) =1 2meω2(x±d)2+···, with 2dbeing the interdot distance. Thus, we have the following Hamiltonian which describes an electron local- ized in either dot: Hl/r=p2 2me+1 2meω2(x±d)2−ασyp+geµBB 2σx.(2) In order to capture all the information of the SOC, we only treat the Zeeman term, instead of the SOC, as per- turbation [ 8,30], which is valid when geµBB/(/planckover2pi1ω)≪1. As estimated in Ref. [ 8] for an InSb nanowire quantum dot, the external static magnetic field can be as strong asB∼0.1 T, which is larger than the magnetic field usually used in a quantum device. The lowest two states of Eq. (2), up to zeroth order, are φlσ(x) =φσ(x+d), φrσ(x) =φσ(x−d),(3) whereσ=⇑and⇓describe the two pseudospin states, and the wave functions are given by [ 31] φ⇑(x) =ψ0(x)[cos(x/xso)|↑x/angbracketright−sin(x/xso)|↓x/angbracketright], φ⇓(x) =−iψ0(x)[cos(x/xso)|↓x/angbracketright+sin(x/xso)|↑x/angbracketright].(4)Herexso=/planckover2pi1/(meα) is the spin-orbit length, | ↑x /angbracketrightand|↓x/angbracketrightare the eigenstates of σx, andψ0(x) = [meω/(/planckover2pi1π)]1/4exp[−x2/(2x2 0)] is the ground state of the harmonic oscillator. The corresponding eigenvalues of φ⇑/⇓are El⇑/⇓=Er⇑/⇓= (1/2)/planckover2pi1ω/bracketleftbig 1−(x0/xso)2/bracketrightbig ±geµBBexp[−(x0/xso)2],(5) wherex0=/radicalbig /planckover2pi1/(meω) defines a characteristic length. Note that these four states are not orthogonal, because there are overlap integrations among them: sa=/integraldisplay dxφ† rσ(x)φlσ(x) = exp(−d2/x2 0)cos(2d/xso), sb=/integraldisplay dxφ† rσ(x)φl¯σ(x) =−iexp(−d2/x2 0)sin(2d/xso).(6) It can be seen that due to the SOC, the overlap integra- tionsbbecomes nonzero. This is different from the case of zero SOC [ 4], wherexso→ ∞, sosa= exp(−d2/x2 0) andsb= 0. Based on these four localized wave func- tions, we can derive an orthonormal basis φor kσ(x) via the Schmidt orthogonalization (for details see Appendix A). The electron field operator can be expanded in terms ofthe orthonormalbasis Ψ( x) =/summationtext k=l,r;σ=⇑,⇓ckσφor kσ(x), whereφor kσ(x) form the spin-orbit basis, in which both the spin and the orbital states are entangled due to the SOC. This is in sharp contrast to the usual basis where both the spin and the orbital states constitute a product state [23,24,32]. In the presence of the strong SOC, the electron spin is no longer conserved in the quantum dot. Therefore, when an electron is injected into the quantum dot, the electron should occupy spin-orbit basis states (i.e., the eigenstates of each dot) instead of the product basis states of the spin and the orbit. It should be noted that other excited orbits arenot considered here, because theyarewellseparatedfromthelowesttwoorbitsroughly by/planckover2pi1ω, i.e.,El(r)⇑−El(r)⇓≪/planckover2pi1ω. The DQD Hamiltonian can be calculated as HDQD=/integraldisplay dxΨ†(x)HΨ(x) =/summationdisplay σ=⇑,⇓/bracketleftbig εlσc† lσclσ +εrσc† rσcrσ+(tc† lσcrσ+t′c† lσcr¯σ+h.c.)/bracketrightbig ,(7) wheretis the spin-conserved tunneling amplitude and t′is the spin-flipped tunneling amplitude. In previous weak-SOC theories [ 24,25], the spin-flipped terms also exist but |t′/t| ≪1. However, in our strong-SOC theory, bothtandt′depend nonlinearly on the SOC strength α, and the ratio |t′/t|can be even larger than 1 when increasing the SOC strength α(see below). When the interdot distance is larger than the charac- teristic length (i.e., d>x0), it follows from Eq. ( 6) that |sa,b| →0. Now the parameters of the DQD have the following explicit analytical expressions (accurate to the first order of |sa,b|): εlσ=εrσ=Elσ=Erσ, t=t0cos(2d/xso), t′=−it0sin(2d/xso),(8)3 0.000 0.125 0.2500.00.51.0 |t/t0| |t'/t0| Tunneling coupling so1/x(in units of π/d) FIG. 2. The spin-conserved and spin-flipped tunnelings in a DQD as a function of the SOC, where t0corresponds to the tunneling without the SOC. wheret0is the interdot tunneling amplitude in the ab- sence of the SOC; e.g., t0=−3V0exp(−d2/x2 0) for a double-well potential V(x) =V0[(x/d)2−1]2. As we have emphasized above, the SOC is not treated as a pertur- bation in our calculations, so these expressions are valid in the strong and even ultrastrong SOC regimes. In the weak SOC limit with α→0 (i.e.,xso→ ∞), we recover thepreviousresults t≈t0andt′≈ −(2id/xso)t0[24,25]; i.e., the spin-flipped tunneling is proportionalto the SOC strengthα. There is something unexpected in the strong SOC regime. As we show in Fig. 2, when increasing the SOC, the spin-flipped tunneling |t′|can compete with the spin-conserved tunneling |t|, while the total tunnel- ing|t|2+|t′|2=t2 0is irrelevant to the SOC. The exper- imentally measured SOC length in an InSb nanowire is xso≈230±50 nm [7]. For an InSb DQD with an in- terdot distance 2 d∼50 nm,|t′/t|= tan(2d/xso)∼0.22, indicating that the spin-flipped tunneling also becomes appreciable in this device. The interesting competition between |t|and|t′|is owing to the peculiar spin-orbit basis in the strong SOC regime [see Eq. ( 4)]. The SOC can lift the Pauli spin blockade of electron tunneling in a DQD [ 33–37]. Our result explicitly shows that this reduction is due to the presence of the spin- flipped tunneling. This indicates that the existence of the spin-flipped tunneling can yield important effects on the measurements of a spin-orbit qubit when the DQD is tuned to the Pauli spin blockade regime. III. THE ANISOTROPIC EXCHANGE COUPLING Below we explore how the strong SOC affects the ex- change coupling [ 23,38,39] in the nanowire DQD. It is known that in the absence of the SOC, the spin-orbitqubit is reduced to a spin qubit, and the exchange cou- pling between two electron spins in a DQD is just the isotropic Heisenberg interaction [ 4,5]. WeconsidertwoelectronsconfinedinananowireDQD. The Coulomb interaction between these two electrons is given by HU=1 2/integraldisplay dxdx′Ψ†(x)Ψ†(x′)e2 |x−x′|Ψ(x′)Ψ(x).(9) Including both intra- and interdot Coulomb interactions, we have the Hubbard-like Hamiltonian H=/summationdisplay σ=⇑,⇓/bracketleftbig εlσc† lσclσ+εrσc† rσcrσ+(tc† lσcrσ+t′c† lσcr¯σ +h.c.)/bracketrightbig +Unl⇑nl⇓+Unr⇑nr⇓+U′/summationdisplay σσ′nlσnrσ′,(10) whereUandU′respectively represent the strengths of the intra- and interdot Coulomb repulsions. Note that σ describes the pseudospin states, i.e., the two eigenstates of the spin-orbit qubit. We consider the strong repulsion regime with ( U−U′)≫ |t|,|t′|, such that each dot con- tains only one electron. Thus, we can define a projection operator [ 41] P= [nl⇑(1−nl⇓)+nl⇓(1−nl⇑)] ×[nr⇑(1−nr⇓)+nr⇓(1−nr⇑)],(11) which retains the pseudo-spin degrees of freedom of the two electrons but reduces the Hilbert space to the sub- space with each dot occupied by one electron. The effec- tive Hamiltonian can be written as[ 42] Heff=PHP−PHQ(QHQ−E)−1QHP, (12) whereQ= 1−P. After some algebra, we obtain (for details see Appendix B) Heff=Equ(Sz l+Sz r)+(J−J[2] so)Sl·Sr +J[1] so(Sl×Sr)x+2J[2] soSx lSx r,(13) where Equ=geµBBexp[−(x0/xso)2], J=4|t|2 U−U′, J[1] so=4i(tt′∗−t∗t′) U−U′, J[2] so=4|t′|2 U−U′, (14) andSk=l,r= (1/2)/summationtext σ,σ′c† kσρσσ′ckσ′is the pseudospin operator, with ρ≡(ρx,ρy,ρz) being the Pauli matri- ces of the spin-orbit qubit. Therefore, we obtain an anisotropic Heisenberg exchange interaction between the two spin-orbit qubits. The exchange interaction con- sists of three terms: the antiferromagnetic Jterm, the anisotropic J[1] soterm, and the ferromagnetic J[2] soterm. It is known that the SOC introduces an anisotropic ex- changeJ[1] soterm in the weak SOC regime [ 23,24,38], but the isotropic antiferromagnetic Jterm dominates.4 T/c43 0T T/c45 S0T 0Tcos( / 2) sin( / 2) T T/c43 /c45 /c81 /c43 /c81 sin( / 2) cos( / 2) T T/c43 /c45 /c81 /c45 /c810,S TJ[1] so so2J/c68 /c61 Jso/c68 [1] soJ[2] soJEnergy Energy Magnetic fieldB Magnetic fieldB(a) (d)(b) (c) FIG. 3. The energy spectrum of two coupled spin-orbit qubits in a DQD with Coulomb repulsion. (a) The spectrum in the weakSOCregimewith |t′/t| ≪1, where J≫J[1] so≫J[2] so. (b) The spectrum calculated using |t′/t|= 0.3, which is chosen to fit the experimentally measured value ∆ so/J≈0.4 in Ref. [ 7]. (c) The spectrum in the strong SOC regime with |t′/t|= 1, where J=J[2] so= (1/2)J[1] so. (d) The spectrum in the ultrastrong SOC regime with |t′/t| ≫1, where J[2] so≫J[1] so≫ J, and Θ = arctan/bracketleftbig J[2] so/(2Equ)/bracketrightbig . In both (a) and (b), the energy is in units of Jand the magnetic field Bis in units ofJ/µwithµ=geµBexp[−(x0/xso)2]. In (c) and (d), the energy is in units of J[i] soandBis in units of J[i] so/µ, where i= 1 for (c) and 2 for (d). However, in the strong SOC regime, the anisotropic ex- changeJ[1] soterm becomes dominant and a ferromagnetic J[2] soterm further occurs. This ferromagnetic J[2] soterm can even play a dominant role in the ultrastrong cou- pling regime. The exchange interaction is induced by the second- order virtual tunneling in a DQD. Each exchange- coupling term in Eq. ( 13) has an explicit physical picture (for details see Appendix B): (i) The virtual tunneling involvingt2gives an antiferromagnetic exchange inter- actionJSl·Sr, (ii) the virtual tunneling involving the combination of tandt′gives an anisotropic exchange in- teractionJ[1] so(Sl×Sr)x, and (iii) the virtual tunneling involvingt′2gives a ferromagnetic exchange interaction −J[2] soSl·Sr+2J[2] soSx lSx r. In the weak SOC regime with |t′/t| ≪1,J[2] so≪ J[1] so≪Jin Eq. (13). After neglecting the second-order terms, the effective Hamiltonian ( 13) is reduced to Heff=Equ(Sz l+Sz r)+JSl·Sr+J[1] so(Sl×Sr)x.(15) The energy spectrum of this Hamiltonian is shown in Fig.3(a), where an anticrossing gap (i.e., the spin-orbit gap) ∆ so=J[1] so/√ 2 between singlet state Sand triplet stateT−occurs due to the anisotropic exchange term J[1] so(Sl×Sr)x. This energy spectrum is qualitatively ingood agreement with the experimental results in an InAs nanowire DQD [ 35,39]. It is interesting to relate these quantities to the SOC strength xso= 2d×arctan−1/bracketleftbig J[1] so/(2J)/bracketrightbig . (16) Because both the gap ∆ so=J[1] so/√ 2 at the anticrossing point and the singlet-triplet splitting Jare experimen- tally measurable quantities [ 35,39], one can use Eq. ( 16) to obtain the SOC strength αviaxso=/planckover2pi1/(meα). Figure3(b) shows the result calculated using Eq. ( 13) to fit the experimental data in an InSb nanowire DQD with ∆ so≈0.4J[7]. In this fitting, the parameter is chosen as |t′/t|= 0.3; i.e.,J[1] so= 0.6J, andJ[2] so= 0.09J. For a DQD with an interdot distance 2 d= 50 nm, our theory gives xso≈180 nm. This spin-orbit length is in good agreement with the experimental result xso= 230±50 nm in Ref. [ 7]. In the strong SOC regime with |t′/t|= 1,J=J[2] so= (1/2)J[1] soin Eq. (13). The effective Hamiltonian reads Heff=Equ(Sz l+Sz r)+J[1] so(Sl×Sr)x+J[1] soSx lSx r.(17) The energy spectrum in this case is shown in Fig. 3(c). As in Figs. 3(a) and 3(b), the triplet state T0remains uncoupledto the singletstate Sand the tripletstates T±, but the other three eigenstates become superpositions ofSandT±. Also, the level splitting at zero magnetic field changes from Jin Fig.3(a) toJ[1] soin Fig.3(c). In this regime, the spectrum is similar to that in the weak SOC regime, but the anisotropic term J[1] so(Sl×Sr)xnow dominates in the exchange coupling. In the ultrastrong SOC regime with |t′/t| ≫1,J≪ J[1] so≪J[2] soinEq.(13). Afterneglectingthesecond-order terms, the effective Hamiltonian ( 13) is reduced to Heff=Equ(Sz l+Sz r)−J[2] soSl·Sr+2J[2] soSx lSx r +J[1] so(Sl×Sr)x. (18) The energy spectrum is shown in Fig. 3(d). An apparent difference from the weak and strong SOC regimes is that both the singlet and triplet states, SandT0, become de- generate. Also, the zero-field level splitting is changed to J[2] so. Currently, this ultrastrong SOC regime is unavail- able in a semiconductor nanowire DQD, but it might be achievable in the future via quantum simulation in, e.g., ultracold-atom systems [ 40]. IV. CONCLUSION We have studied the electron tunneling in a semicon- ductor nanowire DQD with strong SOC. In addition to the usual spin-conserved tunneling, there is also appre- ciablespin-flippedtunneling. Whilethetotaltunnelingis irrelevant to the SOC, both the spin-conserved and spin- flipped tunnelings are SOC dependent and can compete5 with each other in the strong SOC regime. When two electrons are confined in this DQD, the lowest two states of each dot can be used to achieve a spin-orbit qubit. Within this DQD, the Coulomb repulsion between elec- trons can combine with the SOC-dependent tunnelings to yield an anisotropic Heisenberg exchange coupling be- tween the two spin-orbit qubits. We obtain an analytical expression for this anisotropic exchange coupling, which is valid in the strong and even ultrastrong SOC regimes. Eachexchange-couplingtermhasanexplicitphysicalpic- ture involving the second-order virtual tunneling, and its role varies in different SOC regimes. Our theory unveils some distinct properties of the nanowire DQD beyond the weak SOC regime. ACKNOWLEDGEMENTS R.L. and J.Q.Y. are supported by National Natu- ral Science Foundation of China Grant No. 91121015, National Basic Research Program of China Grant No. 2014CB921401, and NSAF Grant No. U1330201. Appendix A: The orthonormal spin-orbit basis Inthisappendix, weorthogonalizethefourstatesgiven in Eq. (3) via the Schmidt orthogonalization method. Note that these four states are not orthogonal due to the overlap integrations saandsbgiven in Eq. ( 6). For the states φl⇓(x) andφr⇓(x), using the conven- tional orthogonalization method [ 4], we obtain the fol- lowing orthogonal states: φor l⇓(x) =1√ζ/bracketleftbig φl⇓(x)−gaφr⇓(x)/bracketrightbig , φor r⇓(x) =1√ζ/bracketleftbig φr⇓(x)−g∗ aφl⇓(x)/bracketrightbig ,(A1) where ζ= 1−2Re(saga)+|ga|2, (A2) withga= (1−/radicalbig 1−s2a)/sa. Because of the overlap integration sb, bothφl⇑(x) andφr⇑(x) arenot orthogonal to the states φor l⇓(x) andφor r⇓(x). Our first step is to construct, via Schmidt orthogonal- ization, two intermediate states ˜φl⇑(x) and˜φr⇑(x) which are orthogonal to the states φor l⇓(x) andφor r⇓(x); i.e., ˜φl⇑(x) =1√1−χ/parenleftbigg φl⇑(x)−sb√ζ/bracketleftbig φor r⇓(x)−g∗ aφor l⇓(x)/bracketrightbig/parenrightbigg , ˜φr⇑(x) =1√1−χ/parenleftbigg φr⇑(x)−sb√ζ/bracketleftbig φor l⇓(x)−gaφor r⇓(x)/bracketrightbig/parenrightbigg , (A3) where χ=(|ga|2+1)|sb|2 1−2Re(saga)+|ga|2. (A4)Next, we orthogonalize these two states. It is easy to obtain the following orthogonal states: φor l⇑(x) =1√ζ′/bracketleftBig ˜φl⇑(x)−g′ a˜φr⇑(x)/bracketrightBig , φor r⇑(x) =1√ζ′/bracketleftBig ˜φr⇑(x)−g′∗ a˜φl⇑(x)/bracketrightBig , (A5) where ζ′= 1−2Re(s′ ag′ a)+|g′ a|2, (A6) withg′ a= (1−/radicalbig 1−s′2a)/s′ a, and s′ a=/integraldisplay dx˜φ† r⇑(x)˜φl⇑(x) =1 1−χ/parenleftbigg sa+2gas∗ bsb 1−2Re(saga)+|ga|2/parenrightbigg .(A7) Obviously, s′ ais the overlap integration between ˜φl⇑(x) and˜φr⇑(x). Therefore, we have derived an orthonormal spin-orbit basis φor kσ(x), wherek=l,randσ=⇑,⇓. Appendix B: The effective Hamiltonian Below we give the details for deriving the effective Hamiltonian HefffromthesecondquantizedHamiltonian of a nanowire DQD. The DQD Hamiltonian can be writ- ten asH=Hs+Ht+HU, with Hs=/summationdisplay σ(εlσc† lσclσ+εrσc† rσcrσ), Ht=/summationdisplay σ(tσc† lσcrσ+t′ σc† lσcr¯σ+h.c.), (B1) HU=Unl⇑nl⇓+Unr⇑nr⇓+U′/summationdisplay σσ′nlσnrσ′, whereσ=⇑and⇓. When the Coulomb repulsion in the DQD is so strong that (U−U′)≫ |t|,|t′|, the two electrons in the DQD have a fixed charge configuration; i.e., each dot confines one and only one electron. Thus, we can define a projec- tion operator [ 41] P= [nl⇑(1−nl⇓)+nl⇓(1−nl⇑)] ×[nr⇑(1−nr⇓)+nr⇓(1−nr⇑)],(B2) which retains the pseudospin degrees of freedom of the two electrons but reduces the Hilbert space to the sub- space with each dot occupied by one electron. The effective Hamiltonian can be written as [ 42] Heff=PHP−PHQ(QHQ−E)−1QHP, (B3) whereQ= 1−PandEis the ground-state energy. The operatorQHQ−Edescribes the energy difference be- tween the double- and single-electron occupations of the6 t’ ttU(b)(a) initial state tintermediate state t tUpossible final states t’t’t’ U(c) FIG. 4. Schematical representation of the second-order vir - tual tunnelings involved in the exchange coupling between t he two spin-orbit qubits. The virtual processes in (a) introdu ce an antiferromagnetic exchange coupling JSl·Sr, the virtual processes in (b) introduce an anisotropic exchange couplin g J[1] so(Sy lSz r−Sz lSy r), and the virtual processes in (c) introduce a ferromagnetic exchange coupling −J[2] soSl·Sr+2J[2] soSx lSx r.DQD, soQHQ−E≈U−U′. Also, it is easy to know thatPHQ=PHtQandQHP=QHtP. Therefore, we have Heff=PHP−PH2 tP U−U′. (B4) After some algebra, we obtain PHP=PHsP= (εl⇑−εl⇓)Sz l+(εr⇑−εr⇓)Sz r +εl⇑+εl⇓ 2+εr⇑+εr⇓ 2, (B5) and PH2 tP=/summationdisplay σ,σ′/bracketleftbig |t|2Pc† lσclσ′PPcrσc† rσ′P+|t|2Pc† rσcrσ′PPclσc† lσ′P+tt′∗Pc† lσclσ′PPcrσc† r¯σ′P +t∗t′Pc† rσcr¯σ′PPclσc† lσ′P+t′t∗Pc† lσclσ′PPcr¯σc† rσ′P+t′∗tPc† r¯σcrσ′PPclσc† lσ′P +|t′|2Pc† lσclσ′PPcr¯σc† r¯σ′P+|t′|2Pc† r¯σcr¯σ′PPclσc† lσ′P/bracketrightbig =/summationdisplay σ,σ′/bracketleftbig |t|2(δσσ′/2+Sl·σσ′σ)(δσσ′/2−Sr·σσσ′)+|t|2(δσσ′/2+Sr·σσ′σ)(δσσ′/2−Sl·σσσ′) +tt′∗(δσσ′/2+Sl·σσ′σ)(δσ¯σ′/2−Sr·σσ¯σ′)+t∗t′(δσ¯σ′/2+Sr·σ¯σ′σ)(δσσ′/2−Sl·σσσ′) +t′t∗(δσσ′/2+Sl·σσ′σ)(δ¯σσ′/2−Sr·σ¯σσ′)+t′∗t(δ¯σσ′/2+Sr·σσ′¯σ)(δσσ′/2−Sl·σσσ′) +|t′|2(δσσ′/2+Sl·σσ′σ)(δ¯σ¯σ′/2−Sr·σ¯σ¯σ′)+|t′|2(δ¯σ¯σ′/2+Sr·σ¯σ′¯σ)(δσσ′/2−Sl·σσσ′)/bracketrightbig = 2|t|2(1 2−2Sl·Sr)+2|t′|2(1 2+2Sl·Sr−4Sx lSx r)+(4itt′∗−4it∗t′)(Sz lSy r−Sy lSz r), (B6) whereSk=l,r= (1/2)/summationtext σ,σ′c† kσρσσ′ckσ′is the pseudospin operator, with ρ= (ρx,ρy,ρz) being the Pauli matrices of the spin-orbit qubit. Thus, we have the following ef- fective Hamiltonian describing the pseudospin degrees of freedom of the two electrons confined in the DQD: Heff=Equ(Sz l+Sz r)+(J−J[2] so)Sl·Sr +J[1] so(Sl×Sr)x+2J[2] soSx lSx r.(B7) This is the effective Hamiltonian Heffin Eq. (13), with Equ,J,J[1] so, andJ[2] sogiven in Eq. ( 14). Each exchange-coupling term in Eq. ( B7) is induced by the second-order virtual tunnelings in a DQD. Thevirtual tunneling involving t2gives an antiferromagnetic exchange interaction JSl·Sr[see Figs. 4(a)], the vir- tual tunneling involving the combination of tandt′ gives an anisotropic exchange interaction J[1] so(Sl×Sr)x [see Fig. 4(b)], and the virtual tunneling involving t′2 gives a ferromagnetic exchange interaction −J[2] soSl·Sr+ 2J[2] soSx lSx r[see Fig. 4(c)]. Note that in the absence of SOC,t′= 0, soJ[1] so=J[2] so= 0 [see Eq. ( 14)]. Therefore, without the SOC, only the isotropic antiferromagnetic termJSl·Sroccurs in the exchange coupling. Here we take Fig. 4(b) as an example. Given an initial two-qubit7 state| ⇑l⇑r/angbracketright, under the virtual tunnelings, the possible final states are | ⇓l⇑r/angbracketrightand| ⇑l⇓r/angbracketright. This virtual tun- neling can be described by the action of the operator J[1] so(Sl×Sr)xon the state | ⇑l⇑r/angbracketright: J[1] so(Sl×Sr)x| ⇑l⇑r/angbracketright=iJ[1] so 4(| ⇓l⇑r/angbracketright−| ⇑ l⇓r/angbracketright).(B8)Similarly, Figs. 4(a) and4(c) can also be explained this way. [1] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe and J. L. O’Brien, Nature (London) 464, 45 (2010). [2] J. Q. You and F. Nori, Phys. Today 58(11), No. 11, 42 (2005); Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys. 85, 623 (2013). [3] R. Hanson, J. R. Petta, S. Tarucha, and L. M. K. Van- dersypen, Rev. Mod. Phys. 79, 1217 (2007). [4] G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B59, 2070 (1999). [5] X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000). [6] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Nature (London) 468, 1084 (2010). [7] S. Nadj-Perge, V. S. Pribiag, J. W. G. van den Berg, K. Zuo, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Phys. Rev. Lett. 108, 166801 (2012). [8] R. Li, J. Q. You, C. P. Sun, and F. Nori, Phys. Rev. Lett. 111, 086805 (2013). [9] K. C. Nowack, F. H. L. Koppens, Yu.V. Nazarov, and L. M. K. Vandersypen, Science 318, 1430 (2007). [10] M. Pioro-Ladriere, T. Obata, Y. Tokura, Y.-S. Shin, T. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Nat. Phys.4, 776 (2008). [11] E. A. Laird, C. Barthel, E. I. Rashba, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 99, 246601 (2007). [12] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91, 126405 (2003). [13] Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha, Phys. Rev. Lett. 96, 047202 (2006). [14] V. N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B 74, 165319 (2006). [15] C. Echeverria-Arrondo and E. Ya. Sherman, Phys. Rev. B88, 155328 (2013). [16] F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, Nature (London) 442, 766 (2006). [17] S. Takahashi, R. S. Deacon, K. Yoshida, A. Oiwa, K. Shibata, K. Hirakawa, Y. Tokura, and S. Tarucha, Phys. Rev. Lett. 104, 246801 (2010). [18] M.D.Schroer, K.D.Petersson, M.Jung, andJ. R.Petta, Phys. Rev. Lett. 107, 176811 (2011). [19] K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung, J. M. Taylor, A. A. Houck, and J. R. Petta, Nature (Lon- don)490, 380 (2012). [20] V. S. Pribiag, S. Nadj-Perge, S. M. Frolov, J. W. G. van den Berg, I. van Weperen, S. R. Plissard, E. P. A. M.Bakkers and L. P. Kouwenhoven, Nat. Nanotechnol. 8, 170 (2013) [21] J. W. G. van den Berg, S. Nadj-Perge, V. S. Pribiag, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Phys. Rev. Lett. 110, 066806 (2013). [22] M. Trif, V. N. Golovach, and D. Loss, Phys. Rev. B 77, 045434 (2008). [23] F. Baruffa, P. Stano, and J. Fabian, Phys. Rev. Lett. 104, 126401 (2010). [24] T. Moriya, Phys. Rev. 120, 91 (1960). [25] F. Mireles and G. Kirczenow, Phys. Rev. B 64, 024426 (2001). [26] C. Flindt, A. S. Sørensen, and K. Flensberg, Phys. Rev. Lett.97, 240501 (2006). [27] D. V. Khomitsky, L. V. Gulyaev, and E. Ya. Sherman, Phys. Rev. B 85, 125312 (2012); D. V. Khomitsky and E. Ya. Sherman, Europhys. Lett. 90, 27010 (2010). [28] M. P. Nowak and B. Szafran, Phys. Rev. B 87, 205436 (2013). [29] Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984). [30] L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324 (2003). [31] E. I. Rashba, Phys. Rev. B 86, 125319 (2012). [32] K. V. Kavokin, Phys. Rev. B 64, 075305 (2001). [33] S. Nadj-Perge, S. M. Frolov, J. W. W. van Tilburg, J. Danon, Yu. V . Nazarov, R. Algra, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Phys. Rev. B 81, 201305(R) (2010). [34] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys. Rev. Lett. 99, 036801 (2007). [35] C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and D. Loss, Phys. Rev. Lett. 98, 266801 (2007). [36] L. R. Schreiber, F. R. Braakman, T. Meunier, V. Calado, J. Danon, J. M. Taylor, W. Wegscheider, and L. M. K. Vandersypen, Nat. Commun. 2, 556, (2011). [37] J. Danon, Phys. Rev. B 88, 075306 (2013). [38] S. Gangadharaiah, J. Sun, and O. A.Starykh, Phys. Rev. Lett.100, 156402 (2008). [39] A. Greilich, S. C. Badescu, D. Kim, A. S. Bracker, and D. Gammon, Phys. Rev. Lett. 110, 117402 (2013). [40] V. Galitski and I. B. Spielman, Nature (London) 494, 49 (2013). [41] R. Li, X. Hu, and J. Q. You, Phys. Rev. B 86, 205306 (2012). [42] N. Nagaosa, Quantum Field Theory in Strongly Corre- lated Electronic Systems (Springer-Verlag, Berlin, 1999), p. 79.

1512.06394v1.Magnetic_and_nematic_phases_in_a_Weyl_type_spin_orbit_coupled_spin_1_Bose_gas.pdf

arXiv:1512.06394v1 [cond-mat.quant-gas] 20 Dec 2015Magnetic and nematic phases in a Weyl type spin-orbit-coupl ed spin-1 Bose gas Guanjun Chen,1,2Li Chen,1and Yunbo Zhang1,∗ 1Institute of Theoretical Physics, Shanxi University, Taiy uan 030006, China 2Department of Physics, Taiyuan Normal University, Taiyuan 030001, China (Dated: October 27, 2018) We present a variational study of the spin-1 Bose gases in a ha rmonic trap with three-dimensional spin-orbit coupling of Weyl type. For weak spin-orbit coupl ing, we treat the single-particle ground states as the form of perturbational harmonic oscillator st ates in the lowest total angular momentum manifold with j= 1,mj= 1,0,−1. When the two-body interaction is considered, we set the tr ail orderparameter as thesuperposition ofthree degenerate si ngle-particle ground-states andtheweight coefficients are determined by minimizing the energy functio nal. Two ground state phases, namely the magnetic and the nematic phases, are identified dependin g on the spin-independent and the spin-dependent interactions. Unlike the non-spin-orbit- coupled spin-1 Bose-Einstein condensate for which the phase boundary between the magnetic and the nemati c phase lies exactly at zero spin- dependent interaction, the boundary is modified by the spin- orbit-coupling. We find the magnetic phase is featured with phase-separated density distributi ons, 3D skyrmion-like spin textures and competing magnetic and biaxial nematic orders, while the ne matic phase is featured with miscible density distributions, zero magnetization and spatially m odulated uniaxial nematic order. The emergence of higher spin order creates new opportunities fo r exploring spin-tensor-related physics in spin-orbit coupled superfluid. PACS numbers: 67.85.Fg, 67.85.Jk, 03.75.Mn, 67.85.Bc I. INTRODUCTION With the two-photon Raman coupling technique, a synthetic spin-orbit (SO) coupling has been realized in the pseudo-spin-1/2 Bose-Einstein condensates (BECs) [1, 2]. Since then, many theoretical [3–11] and experi- mental [12–16] works have been focused on this field in- cludingtheunveilingofthewell-knownplanewave,stripe and zero-momentum phases [4]. In addition, the Raman- induced SO coupled spin-1 BEC has also been realized recently [17] which is unattainable in condensed matter materials, and spatially modulated nematic order is ex- pected to appear in the stripe phase [18–21] . The Raman-induced SO coupling is a one-dimensional (1D) configuration as an equally weighted Rashba and Dresselhaus couplings. This year also witnessed the ex- perimental progress in engineering the two-dimensional (2D) Rashba-Dresselhaus-type SO coupling in cold atom gases [22, 23]. Many interesting properties have been predicted for the Rashba SO coupled Bose gases [24– 43], among which the weakly coupled BEC is found to condense into various half-quantum vortex phases [25, 26, 28, 29, 43] due to modification of the single parti- cle spectrum by the trapping potential. All these efforts make the cold atom gases with the synthetic gauge field a rapidly developing field. Researchers also go further to deal with the three- dimensional (3D) analogy of Rashba configuration, i.e. the Weyl SO coupling [44–53] and the experimental schemes for engineering the Weyl configuration in the ultracold gases have been proposed [54–56]. While the ∗ybzhang@sxu.edu.cnWeyl coupled spin-1/2 bosons in a homogeneous system reproduces the plane wave and the stripe phases [53], the 3D skyrmion mode with magnetic order [44, 46, 48, 52] spontaneously appears in the ground state of a trapped system. For a trapped Weyl coupled spin-1 BEC, one expects the emergence of topological objects with even higher spin order, i.e. the nematic order. An immediate ques- tion is that, what will the phase diagram look like, and in what a way will the 3D skyrmion and (or) the nematic order manifest themselves in the individual phases. Here we consider a spin-1 bosonic system subject to a weak WeylSO couplingof s·ptypein aharmonictrap. We im- plement a standard variational approach [3, 4, 24, 25, 52] to give a phase diagram of the system in different inter- action regimes. A magnetic phase and a nematic phase are predicted in the ground state, and the latter is en- tirely new and has no analogue in the 3D SO coupled pseudo-spin-1/2 bosonic gases [44, 46, 48, 52]. Thepaperisorganizedasfollows. In Sec. II theenergy functional for the 3D SO coupled spin-1 condensate is in- troduced in harmonic oscillator units. In the weak cou- pling limit we construct our variational order parameter in Sec. III and calculate the energy functional by means ofthe irreducibletensormethod. The groundstate phase diagramis determined by numericallyminimizing the en- ergy functional with respect to the variational parame- ters, and Sec. IV is devoted to an explicit illustration and detailed discussion of the densities and spin orders in the two ground state phases. We summarize our main results in Sec. V.2 II. MODEL We start from the mean-field Gross-Pitaevskii (GP) energy functional of spin-1 bosons with a Weyl type 3D SO coupling in the presence of a harmonic trap E=E0+Eint, (1) where the single particle part is E0=/integraldisplay d3rΨ†(r)/parenleftbiggp2 2m+1 2mω2r2+λs·p/parenrightbigg Ψ(r),(2) withmthe atomic mass and ωthe trap frequency. Ψ = (ψ1,ψ0,ψ−1)Tdenotes the spinor order parame- ters for bosons with hyperfine components 1 ,0,−1 re- spectively, and s= (sx,sy,sz) are spin-1 matrices. The Weyl SO coupling s·pwith strength λis a 3D analogy of the Rashba configuration [55]. Without the trapping potential, the momentum pand the helicity s·p/|p| are constants of motion, and the single-particle ground states are highly degenerate, i.e. the lower-energy heli- cal branch achieves a minimum along a sphere of radius pSO=mλ, called SO sphere[44]. The presenceofa trap- ping potential will partially lift this degeneracy, but at least a two-fold degeneracy related to the time-reversal symmetry will still remain [2]. The single particle Hami- tonian is also invariant under simultaneous rotation of spin and coordinate space SO(3)R+Sthat leaves the to- tal (instead of spin) angular momentum a good quan- tum number [46]. This breaking of rotational symmetry in spin space leads to spin-textured ground states with magnetic and nematic orders, as previously studied in Ref. [19, 21, 24, 25, 52]. The interaction energy functional is formulated in the standard form [57–59] Eint=1 2/integraldisplay d3r/parenleftbig c0n2+c2S2/parenrightbig , (3) wheren(r) =n1(r) +n0(r) +n−1(r) is the total par- ticle density and n1,0,−1(r) =|ψ1,0,−1(r)|2are densities for the three components, respectively, and S= Ψ†sΨ is spin density. c0,2are spin-independent and spin- dependent interaction strengthes respectively, which are related to the two-body scattering lengths in the total spin-0 and spin-2 channels as c0= 4π(a0+2a2)/3mand c2= 4π(a2−a0)/3m. The interaction is time-reversal (TR) symmetric under the operation T=e−iπsyKwith Kthe complex conjugate. Besides, this two-body inter- action is also SU(2) spin-rotation symmetric, which is different from the spin-1 /2 bosons [44, 46, 48, 52]. In the latter case the two body interaction is SU(2) sym- metric only under the condition g↑↑=g↓↓=g↑↓, i.e. c=g↑↓/g↑↑= 1. Later we will see that this highly sym- metric interaction will lead to highly degenerate ground states. In harmonic oscillator units, the system has length scale lT=/radicalbig /planckover2pi1/mω, energy scale /planckover2pi1ω, and SO cou- pling strength is in unit of/radicalbig /planckover2pi1ω/m. If we normalize theorder parameter to unity, i.e., Ψ →/radicalbig N/l3 TΨ withN the total particle number in the condensate, the energy functional per particle is obtained as ε=/integraldisplay d3rΨ†(r)/braceleftbigg −∇2 2+r2 2+λs·p/bracerightbigg Ψ(r) +/integraldisplay d3r/parenleftigc0 2n2+c2 2S2/parenrightig . (4) III. VARIATIONAL APPROACH Herewefirstintroducethesingle-particlegroundstates in the form of perturbational 3D harmonic oscillator in the weak SO coupling limit, the superposition of which constitutes the variationalorderparameter. We then cal- culate the energy functional of Eq. (4) using the irre- ducible tensor method. A. Variational Order Parameter The variational order parameter is just a spin-1 ver- sion of that presented in our recent work on the 3D SO coupled pseudo-spin-1/2 system [52]. We include some necessary contents here for completeness. It is expected that in the case of weak SO coupling the single particle energy is dominated by the 3D harmonic oscillator part, which has been verified in Ref. [44, 50, 52]. The solution of the 3D harmonic oscillator is well-known with the en- ergy eigenvalues ǫnrl= 2nr+l+3 2and eigenfunctions φnrlml(r,θ,ϕ) =Rnrl(r)Ylml(θ,ϕ). HereRnrlis the radial wave function and Ylmlis the spherical harmonics withnrthe radial quantum number, lthe orbital angu- lar momentum quantum number, and mlits magnetic quantum number. In order to take into account the SO coupling term s·plater, it is convenient to choose the coupled representation of angular momentum for spin-1 particles, i.e. the complete set of commutative operators includes l2,s2,j2,jzwherej=l+sandjzdenote the total angular momentum and its z-component, respec- tively. The eigenfunction φnrlml(r,θ,ϕ) should be com- bined with the spin wave function χms(ms= 1,0,−1) in the coupled representation as φnrljmj(r,θ,ϕ) =Rnrl(r)Yl jmj(Ω),(5) whereYl jmj(Ω) =/summationtext ml,msCjmj lml1msYlmlχmsis the vec- tor spherical harmonics [60] with j=l+ 1,l,|l−1|(if l= 0,j= 1 only) and Cjmj lml1msthe Clebsch-Gordan coef- ficients. In the coupled representation, the ground state wave function has nr=l= 0. This gives a total angu- lar momentum j= 1 withmj= 1,0,−1 and the three ground states are φ001±1(r) =R00(r)Y0 1±1(Ω) (6) and φ0010(r) =R00(r)Y0 10(Ω) (7)3 FIG. 1. (Color online). The lowest energy levels of 3D har- monic oscillator (dashed horizontal lines) and the SO coupl ed single-particle ground states (solid horizontal line). Th e selec- tion rules for transition between states with different pari ties are indicated by the two-way blue arrows. The excited states such as 1 s(gray dashed line) is neglected in the calculation. respectively. The lowest few levels of the 3D harmonic oscillator is shown in Fig. 1. The single particle spec- trum may be understood as a weak perturbation (slight level mixing) of the harmonic-oscillator energy levels in the case of weak SO coupling, which conserves the total angular momentum jandjzbutlis no longer a good quantum number. This means SO coupling term would couples,pand evendwaves into the ground state with the same jandjzas illustrated in Fig. 1. In the case of strong SO coupling, the energy spectrum is weakly dispersive or nearly flat [32, 44, 50] which will not be considered in this work. The ground state wave func- tions are thus the superposition of the lowest s,pandd wave states with j= 1. The state with mj= 1 is Φj=1,mj=1=A0φ0011+iA1φ0111−A2φ0211.(8) HereAl(l= 0,1,2) are the weight coefficients with the normalization constraint/summationtext2 l=0|Al|2= 1, andiin front ofA1comes from the pure imaginary matrix elements of SO coupling between φ0011(φ0211) andφ0111[44, 50, 52]. Explicitly this state is a spinor Φj=1,mj=1 = A0R00Y00−iA1/radicalig 1 2R01Y10−A2/radicalig 1 10R02Y20 iA1/radicalig 1 2R01Y11+A2/radicalig 3 10R02Y21 −A2/radicalig 3 5R02Y22 . (9) Moreover, in the single particle level, the energies are irrelevant to the magnetic quantum number of jz. Thus the other two states with mj= 0 and −1 are Φj=1,mj=0 = −iA1/radicalig 1 2R01Y1−1−A2/radicalig 3 10R02Y2−1 A0R00Y00+A2/radicalig 2 5R02Y20 iA1/radicalig 1 2R01Y11−A2/radicalig 3 10R02Y21 (10)and Φj=1,mj=−1 = −A2/radicalig 3 5R02Y2−2 −iA1/radicalig 1 2R01Y1−1+A2/radicalig 3 10R02Y2−1 A0R00Y00+iA1/radicalig 1 2R01Y10−A2/radicalig 1 10R02Y20 . (11) respectively. Note that the state Φ j=1,mj=−1is the time reversal of Φ j=1,mj=1, andTΦj=1,mj=0=−Φj=1,mj=0. We have neglected contribution from the excited 1 swave stateφ1011shown in Fig. 1 (gray dashed), which can be absorbedinto the lowest0 sstateφ0011owingto the same angular-spin wave function. It has been verified that in- clusion of this excited 1 sstate in the calculation will not alter our main conclusion for weak two-body interaction. In the language of Raman-induced SO coupling [1, 8, 14], thesingleparticlegroundstatehasthreeminimacor- responding to three states Φ j=1,mj=0,±1. At the many- body level, the interaction will determine which mini- mum or minima the Bose gases will condensate to by minimizing the GP energy functional [18]. We therefore set the variational wavefunction as Ψ =c1+Φj=1,mj=1+c10Φj=1,mj=0+c1−Φj=1,mj=−1 (12) withthe normalizationconstraint c2 1++c2 10+c2 1−= 1that ensuresthe conservationofthe particlenumbers. The co- efficientsc1+,c10,c1−, to be determined by the interac- tion, aregenerallycomplex. Forthe sakeofsimplicity, we restrict them to be real here. Later, we will discuss the consequences of such restriction. This variational wave function ansatz is extensively encountered in SO coupled cold atom gases. [3, 4, 18, 19, 24, 25, 27, 30, 53] B. Nematic Order Prior to the calculation of the energy functional, we first elucidate the meaning of our variational order pa- rameter ansatz. Since our single-particle hamiltonian re- spects the SO(3) R+Ssymmetry, it is convenient to in- troduce the polarization operator [60] to describe the spin order. The polarization operators for spin- ssystem T(l) ml(s) withl= 0,1,...,2sandml=−l,−l+ 1,...l, are a set of (2 s+1)2operators which act on the spin wave functions and transform under the coordinate sys- tem rotation according to the irreducible representation of SO(3) group. In this sense, it is also an irreducible tensor of rank l. For spin-1 objects, the nine polarization operators T(l) ml(s) withl= 0,1,2constitute a complete set of square 3×3 matrices and are generators of the unitary group U(3)inrotationallycovariantRacahform[61]. Therank-4 0 operator is the unit 3 ×3 matrixI T(0) 0(s) =1√2s+1I. (13) The rank-1 operators T(1) ml(s) are proportional to the ir- reducible rank-1 spin tensor with components T(1) ml(s) =√ 3/radicalbig s(s+1)(2s+1)s(1) ml,(ml= 0,±1) (14) where the spherical components of the irreducible rank-1 spin tensor s(1) mlare related to the cartesian ones as s(1) ±1=∓1√ 2(sx±isy),s(1) 0=sz.(15) The rank-2 polarization operators T(2) ml(s) are T(2) ml(s) =/summationdisplay µ+ν=mlC2ml 1µ1νs(1) µs(1) ν =/braceleftig s(1)s(1)/bracerightig(2) ml,(µ,ν= 0,±1),(16) where/braceleftbig A(m)B(n)/bracerightbig(k)defines the rank- ktensor product of rank-mirreducible tensor A(m)and rank-nirreducible tensorB(n).T(2) ml(s) are equivalent to a symmetric trace- lesscartesian tensor of rank-2, i.e. the spin nematic ten- sor or quadrupole tensor Nthrough the relation [60] T(2) ±2=1 2(Nxx−Nyy±2iNxy), T(2) ±1=∓(Nxz±iNyz), T(2) 0=/radicalbigg 3 2Nzz, where Nij=1 2/parenleftbigg sisj+sjsi−4 3δijI/parenrightbigg ,(i,j=x,y,z).(17) The unit matrix Iis a spin-rotation invariant scalar which contains important information regarding the charge(density) order, three matrices sx,sy,szform a vector which represents the local spin ( magnetic ) order, and five matrices Nijform a symmetric traceless tensor which represents the local spin fluctuations or nematic order [62, 63]. The spin-1 systems therefore support spin nematic order in addition to the charge and magnetic ones. Themagneticandthenematicorderscompetewith eachotherasincreasingoneofthemrequiresreducingthe other [62]. With the polarization operators, our variational wave- function Eq. (12) can be written as Ψ =U(r)ζ. (18) whereζ= (c1+,c10,c1−)Tis a normalized spinor and the position-dependent transformation U(r) =/radicalbigg 2s+1 4π2s/summationdisplay l=0(−i)lAlR0lC(l)·T(l)(19)on the spinor ζleads to the spin-textured ground states in the SO coupled cold atom gases. Here the dot de- fines the scalar product of the modified spherical har- monicsC(l)andT(l). This local modulation operator is expanded in series of C(l)·T(l)with the highest order l= 2s. The hamiltonian is invariant under the simulta- neous rotation in spin and coordinate space SO(3)R+S, thereforetrapped spinorcondensatewith SOcouplingin- evitably carriesangularmomentum by twisting its spinor order parameter [52, 62]. It is also intuitive to understand the modulation of spin-1/2 objects discussed in detail in Ref. [52], in which case the four polarization operators T(l) ml(s) withl= 0,1, or explicitly I,s(1) +1,s(1) 0,s(1) −1, constitute a complete set of square 2 ×2 matrices. The transformation matrix U(r) can be rewritten as U(r) =/radicalbig ˜n(r)e−iω(r)ˆr·s(20) where the density and spin modulations are apparently separated owing to the absence of nematic order. The spin-1 system thus enables to explore spin-tensor-related physics in the SO coupling superfluid, which has fun- damentally different rotation properties as in spin-1/2 system. C. Energy Functional We now have six variational parameters A0,A1,A2, c1+,c10,c1−in the trail variationalorder parameter. The energy functional of Eq. (4) can be calculated analyti- cally using the proposed order parameter Eq. (12). The single particle part of the energy functional con- sists of the kinetic energy, the trapping potential of the 3D harmonic oscillator, and the SO coupling term. As in the spin-1/2 case [52], the matrix elements for the ki- netic energyand trapping potential are non-vanishing for states with the same parity, while the SO coupling term s·pwill mix states with opposite parities. The result is /integraldisplay d3rΨ†(r)/braceleftbigg −∇2 2+r2 2+λs·p/bracerightbigg Ψ(r) =3 2A2 0+5 2A2 1+2λA0A1+∆0. (21) The contribution from the d-wave states is collected in ∆0(see Supplemental Material), and we have used /an}b∇acketle{tφ0011|s·p|φ0111/an}b∇acket∇i}ht=−i, /an}b∇acketle{tφ0111|s·p|φ0211/an}b∇acket∇i}ht=i/radicalbigg 5 6. We refer to Ref. [52] for details of the integral calcula- tion in which the method of irreducible tensor algebra is employed [60].5 The calculation of the interaction is tedious but straightforward which yields /integraldisplay d3rc0 2n2=c0 8π/radicalbigg 2 π/bracketleftbigg A4 0+A2 0A2 1+1 16(7+x)A4 1+∆n/bracketrightbigg , (22) and /integraldisplay d3rc2 2S2=c2 8π/radicalbigg 2 π/bracketleftbigg A4 0+A2 0A2 1+5 16A4 1+∆s/bracketrightbigg (1−x). (23) withx=/bracketleftig 1−(c1++c1−)2/bracketrightig2 . Here ∆ nand ∆ sagain denote the contributionsfrom the d-wavestates(seeSup- plemental Material). The energy functional per particle is simply the summation of Eqs. (21), (22) and (23). IV. GROUND STATE PHASE DIAGRAM The ground state phase diagram for a given coupling strengthλis obtained by minimizing the variational en- ergy with respect to the parameters Alandxunder two constraints/summationtext2 l=0|Al|2= 1 andc2 1++c2 10+c2 1−= 1. The latter further restricts xto the region [0 ,1]. Should the spin-independent interaction vanish c0= 0, we see from Eq. (23) that the optimized parameter xeither takes value of 0 for c2<0, or 1 forc2>0 for negligibly small contribution ∆ sfrom thed-wave states. The analysis below will show that this assumption holds generally ex- cept for extremely strong interaction c0. The variational ansatz characterizes two quantum phases: (I) the mag- netic phase with a ferromagnetic manifold ζforc2<0, asx= 0 means that ( c1++c1−)2= 1; (II) the polar or nematic phase with a polar manifold for c2>0, asx= 1 means (c1++c1−)2= 0 or 2. This is consistent with the conventional spin-1 BEC [57–59] where for c2<0 (c2>0) a ferromagnetic (polar) spinor is needed to min- imize the mean-field energies. If c0>0, the boundary between magnetic and nematic phases will drift a little to the positive c2side because the spin-independent in- teraction (22) which prefers x= 0, prevails in the c2>0 regime over the spin-dependent one (23) which prefers x= 1. For the values of optimized parameters Al, gen- erally, the weights of panddwaves becomes more and more important with increasing interaction c0, which ef- fectively diminishes the partition of swave. In the range ofc0= 0∼20 we considered here, |A0|2decreases from 0.884 to 0.723,|A1|2increases from 0 .110 to 0.260, while |A2|2from0.003to0.014whichisalwaysnegligiblysmall. A typical phase diagram is shown in Fig. 2 where the phase boundary is calculated in three successive approxi- mations: “sp” - only the lowest 0 sand 0pstates are con- sidered in the approximation; “spd” - the 0 denergy level is added; “spds” - the 1 senergy level is added further. Wecanfindthattheboundarydoesnotaltersignificantly when we include the excited swave states φ101mjin the variational order parameter. FIG. 2. Phase diagram of weakly SO coupled spin-1 bosons with coupling strength λ= 0.4. The boundary between the magnetic phase and the nematic phase is determined in suc- cessive approximations in which we include 0 s, 0p, 0dand 1s orbits step-by-step. Density distributions and spin textu res of the two phases are shown in Fig. 3 to Fig. 5 respectively. It has been pointed out that the SO coupling manifests itself in a way that the modulation of the ferromagnetic and polar spin textures in the pseduspin space could be transferred to patterned structures in orbit space even in the ground states [18]. The reason of this modulation lies in that, in the presence of SO coupling [64] or dipolar interaction [65], the spin-dependent interaction would in- evitably influence the spatial motion, which leads to rich density pattern. We discuss this in the following for the magnetic and polar phases explicitly. A. Magnetic Phase Thisphaselies in thelowerpartofthe parameterspace with the states featured as c1++c1−=±1. The cor- responding spinor ζin Eq. (18) denotes a ferromag- netic state with magnetization along any axis in the xz plane. This asymmetry between x,zandydirections is a consequence of our simplified treatment which re- stricts the coefficients c1+,c10,c1−to be real, such that the spinor ζis unable to describe the state with mag- netization along ydirection. If we relax this restric- tion to allow complex coefficients c1+,c10,c1−, all states with spinors ζmagnetized along any spatial direction belong to this magnetic phase, which leads to an in- finitelydegenerategroundstates. Twotypicalspinorsare ζz= (1,0,0)Tandζx=/parenleftig 1 2,1√ 2,1 2/parenrightigT which are longitu- dinally and transverselymagnetized ferromagnetic states respectively. This phase spontaneously breaks the time- reversal symmetry which leads to spontaneous magne- tization along the xzplane. The situation is just like spin half case we have studied before [52]. The differ- ence is that the xandzferromagnetic states are not degenerate for the spin-half case as the interaction is not SU(2) symmetric, leaving only two-fold Kramers degen- eracy originating from the time-reversal symmetry there. For spin-1 gases here, the ground states are infinitely de- generate resulted from the SU(2) symmetric interaction. We first consider the longitudinally magnetized state ζz= (1,0,0)Twith the time reversal state being ζ−z=6 FIG. 3. (Color online). Density distributions and spin tex- tures of the longitudinally magnetized state ζz. The varia- tional parameters used are ( A0,A1,A2) = (−0.85,0.51,0.12). (a) Density distributions. Three columns are densities in xy, yzandxzplanes respectively. Three rows are density dis- tributions of +1 ,0,−1 components respectively as explicitly labeled; (b) Spin texture S(r)/n(r). The figures plot the spin inxy,yz, andxzplanes respectively. The arrows indicate the in-plane components of the local spin, and the color scal e shows the magnitude of the out-plane component. (0,0,1)T. We plot the ground state density distribu- tion for the three components in Fig. 3 (a), which show clearly cylindrical symmetry. While the spin component 1 is dominantly occupied in the center, which allows the condensate to develop a longitudinal magnetization, the components 0 and −1 form two toruses surrounding the central part. It can be seen that the outermost shell of spin−1 density is negligibly small which is entirely at- tributed to the involvement of the dwave fraction in the order parameter. To visualize the dwave nature we need to zoom out 100times in the density plot. Generallyonly less populated spin component can develop d-wave char- acters in the ground states because these complex struc- ture in the high density spin components would cost too much kinetic energy [66]. For its time-reversal degener- ate state ζ−zthe spin components 1 and −1 are inversely populated. Thespintextureofthelongitudinallymagnetizedstate is plotted in Fig. 3(b) where we find asynchronousmod- ulation between the particle density and the spin density owing to the breaking of spin rotation symmetry. In the trap center, the spins are aligned along the zaxis due to the dominant occupation of spin-1 component. The suc- cessive population of the 0 and −1 components forms a local spin texture S(r)/n(r) where the spin density vec- tors deflect continuously to the xyplane away from the FIG. 4. (Color online). Density distributions and spin tex- turesofthe transverselymagnetized state ζx. The parameters A’s are the same as those of Fig. 3. center. This is a magnetic skyrmion-like texture similar to the half quantum vortex configuration in the Weyl SO coupled psedo-spin-1/2BEC[25, 46, 52]. The spin vector field lines develop into a bundle of fountain-like stream- lines close to the zaxis around which a torus is formed near thexyplane [52]. The density distribution for the transversely magne- tized state ζx=/parenleftig 1 2,1√ 2,1 2/parenrightigT with the time reversal state ζ−x=/parenleftig 1 2,−1√ 2,1 2/parenrightigT looks quite differently as shown in Fig. 4 (a). With half of the atoms filled in the spin-0 component, the components ±1 becomes equally popu- lated and spatially separated, which leads to a significant transverse magnetization [66]. The density distributions of three components are separated in an alternative way, i.e., the−1 (+1) component lies mainly in the −y(+y) half-space and its peak density center is along the direc- tion joining the III and VIII octants (I and VI octants) [52], while the 0 component is embedded between them with a density profile along the xaxis. This leads to a magnetic skyrmion-liketexture with the spins in the trap center transversely aligned along the xaxis as shown in Fig. 4 (b), and the increasing population of ±1 compo- nents away from the trap center makes the spin density vector forms a torus near the yzplane. Owing to the non-commutative nature of position-dependent transfor- mation (19) and the spin rotation, the spin texture of the longitudinally magnetized state is different from the π/2 spin rotation around yaxis of the transversely magne- tized state, though the spinor wave functions themselves ζzandζxare related by such a rotation. All other states degenerate with these two magnetic states have similar7 FIG. 5. (Color online). Density distributions and the nemat ic directors of the nematic state ζp1. (a) Density distributions; (b) Nematic director and the tensor magnetization density Nzz. The figures plot the projection of the single nematic director on the xy,yz, andxzplanes respectively. The pa- rameters A’s are the same as those of Fig. 3. properties, except that the magnetization axes may be in any direction determined by the values of parameters c’s. B. Nematic Phase Thisphaseliesintheupperpartoftheparameterspace with the states featured as c1++c1−= 0 or√ 2. The corresponding spinor ζin Eq. (18) denotes a polar state with zero magnetization S(r) =0everywhere. Two typ- ical spinors are ζp1=1√ 2(1,0,1)Tandζp2= (0,1,0)T. The time-reversal symmetry is preserved in this phase, so the ground state has no spontaneous magnetization as the non-magnetic phase in the Raman induced SO cou- pled two-component Bose gases [1, 4, 8, 14]. This is a new phase and no analogy in spin half case we consid- ered before [52]. All states belong to this phase are again infinitely degenerate. The density distribution of the state ζp1is plotted in Fig. 5 (a). A signature of the d-wave is seen in the 100 times zoomed density of 0 component in the xzplane, which comes from the d-wave contribution in the vari- ational order parameter. The phase separation among three components, which is possible only in the case of time-reversal symmetry breaking [67], i.e. the magnetic phase, is not seen for this nematic phase which preserves the time-reversal symmetry and the ±1 components are miscible. We define the nematicity density tensor N= Ψ†NΨ FIG.6. (Color online). Competingordersinthemagneticand the nematic phases along xaxis. They are1 2/vextendsingle/vextendsingleS n/vextendsingle/vextendsingle2(Red) and Tr/parenleftbigN n/parenrightbig2(Blue) respectively. Solid curves are for the magnetic phase and dashed lines for the nematic phase. to characterize the nematic order due to the absence of the magnetization. In the nematic phase ζp1orζp2, the normalized nematicity density tensor N/nhas eigenval- ues{1 3,1 3,−2 3}everywhere, therefore describing a uniax- ial nematic state. The eigenvector associated with eigen- value−2 3defines the nematic director which is plotted in Fig. 5 (b) together with the tensor magnetization densityNzz= Ψ†NzzΨ. The nematic directors form a lantern-like structure with the principle axis along the xdirection. The spatially modulation of the nematic directors, shown as headless vectors in the xy,yzand xzplanes, reflects indirectly the modulation of nematic- ity density tensor themselves. In the trap center, where the spinor wavefunction ζp1describes a transverse po- lar state, the nematic director points along the xaxis. Away from the trap center, the 0 component is gradu- ally populated and the nematic director are continuously modulated into concentric circles in the yzplane. For the state ζp2, the 0 component in the trap center will al- low a longitudinal polar state and we find an alternative modulation of the nematic directors. The tensor magne- tization/an}b∇acketle{tNzz/an}b∇acket∇i}ht[17, 20] has been adopted to resolve the order of the phase transition in the Raman-induced SO coupled spin-1 condensate, and the spatial modulation of Nzzalong direction of the SO coupling has been noticed [19, 21]. The magnetic phase is also featured with a nematic orderand the competing ordersin both phases are shown in Fig. 6. These two spin ordersare competing with each other [62] to meet the requirement 1 2/vextendsingle/vextendsingle/vextendsingle/vextendsingleS n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +Tr/parenleftbiggN n/parenrightbigg2 =2 3. (24) This prevents us from writing the transformation Eq. (19) into a local spin rotation as Eq. (20). Diagonal- izing the nematicity density tensor yields three distinct eigenvalues which are spatially modulated as well as the nematicity densitytensorthemselves [21]. Thus the mag- netic phase has both magnetic and biaxial nematic or- ders, while the nematic phase exhibits only a uniaxial nematic order. Finally, we remind that our variational order parame- ter based on the perturbation expansion may not be ap-8 plicable in the limit of strong SO coupling [28, 43, 44, 48] where a skyrmion-lattice-like ground state may appear. Moreover, when two-body interaction is strong enough, the variational order parameter starts to involve higher angular momentum jstates which will break the SO(3) rotational symmetry, and the Bose gases will condense into the plane wave or stripe phases instead [26, 44, 53]. V. SUMMARY We establish the ground state phase diagram of the weakly 3D spin-orbit coupled spin-1 bosons theoretically. The ground state may be in a magnetic or a nematic phase determined by the competing between the spin- independent and the spin-dependent two-body interac- tion. The nematic phase is a new phase that is absentin a 3D spin-orbit coupled pseudo-spin-1/2 bosonic sys- tem. We discuss the density distribution and spin orders of the two phases in detail. The magnetic phase permits both a magnetic order and a biaxial nematic order, while the nematic phase is featured by a uniaxial nematic or- der. Thesenovelphasesareincurrentexperimentalreach benefiting from the rapid progress of cold gases with ar- tificial gauge field. ACKNOWLEDGMENTS This work is supported by NSF of China under Grant Nos. 11234008 and 11474189, the National Basic Re- searchProgramofChina(973Program)underGrantNo. 2011CB921601, Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT)(No. IRT13076). [1] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Natu re (London) 471, 83 (2011). [2] V. Galitski and I. B. Spielman, Nature (London) 494, 49 (2013). [3] T.-L. Ho and S. Z. Zhang, Phys. Rev. Lett. 107, 150403 (2011). [4] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett . 108, 225301 (2012). [5] G. I. Martone, Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. A 86, 063621 (2012). [6] Y. Li, G. I. Martone, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 110, 225302 (2013). [7] Q.-Q. L¨ u and D. E. Sheehy, Phys. Rev. A 88, 043645 (2013). [8] W. Zheng, Z.-Q. Yu, X. L. Cui, and H. Zhai, J. Phys. B: At. Mol. Opt. Phys. 46, 134007 (2013). [9] T. Ozawa, L. P. Pitaevskii, and S. Stringari, Phys. Rev. A87, 063610 (2013). [10] G. I. Martone, Y. Li, and S. Stringari, Phys. Rev. A 90, 041604(R) (2014). [11] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015). [12] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 109, 115301 (2012). [13] C. L. Qu, C. Hamner, M. Gong, C. W. Zhang, and P. Engels, Phys. Rev. A 88, 021604(R) (2013). [14] S.-C. Ji, J.-Y. Zhang, L. Zhang, Z.-D. Du, W. Zheng, Y.- J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Nature Phys. 10, 314 (2014). [15] M. A. Khamehchi, Y. P. Zhang, C. Hamner, T. Busch, and P. Engels, Phys. Rev. A 90, 063624 (2014). [16] C. Hamner, C. L.Qu, Y.P.Zhang, J. J. Chang, M. Gong, C. W. Zhang, and P. Engels, Nature Commun. 5, 4023 (2014). [17] D. L. Cambell, R. M. Price, A. Putra, A. Vald´ es-Curiel, D. Trypogeorgos, and I. B. Spielman, arXiv:1501.05984 (2015). [18] Z. H. Lan and P. ¨Ohberg, Phys. Rev. A 89, 023630 (2014).[19] S. S. Natu, X. P. Li, and W. S. Cole, Phys. Rev. A 91, 023608 (2015). [20] K. Sun, C. L. Qu, Y. Xu, Y. P. Zhang, and C. W. Zhang, arXiv:1510.03787 (2015). [21] G. I. Martone, F. V. Pepe, P. Facchi, S. Pascazio, and S. Stringari, arXiv:1511.09225 (2015). [22] L. H. Huang, Z. M. Meng, P. J. Wang, P. Peng, S.-L. Zhang, L. C. Chen, D. H. Li, Q. Zhou and J. Zhang, arXiv:1506.02861 (2015). [23] Z. Wu, L. Zhang, W. Sun, X. -T. Xu, B. -Z. Wang, S. -C. Ji, Y. J. Deng, S. Chen, X. -J. Liu, and J.-W. Pan, arXiv:1511.08170 (2015). [24] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett.105, 160403 (2010). [25] C. J. Wu, M. S. Ian and X. F. Zhou, Chin. Phys. Lett. 28, 097102 (2011). [26] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401 (2011). [27] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012). [28] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys. Rev. Lett. 108, 010402 (2012). [29] B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D. Drummond, and H. Hu, Phys. Rev. A 85, 023606 (2012). [30] Z.-Q. Yu, Phys. Rev. A 87, 051606(R) (2013). [31] Q. Zhou and X. L. Cui, Phys. Rev. Lett. 110, 140407 (2013). [32] B. Ramachandhran, H. Hu, and H. Pu, Phys. Rev. A 87, 033627 (2013). [33] T. Ozawa and G. Baym, Phys. Rev. A 85, 013612 (2012). [34] T. Ozawa and G. Baym, Phys. Rev. A 85, 063623 (2012). [35] Y. P. Zhang, L. Mao, and C. W. Zhang, Phys. Rev. Lett. 108, 035302 (2012). [36] X. L. Cui and Q. Zhou, Phys. Rev. A 87, 031604(R) (2013). [37] Z.-F. Xu, S. Kobayashi, and M. Ueda, Phys. Rev. A 88, 013621 (2013). [38] P. Nikoli´ c, Phys. Rev. A 90, 023623 (2014). [39] H. Hu and X.-J. Liu, Phys. Rev. A 85, 013619 (2012).9 [40] G. Juzeli¯ unas, J. Ruseckas, and J. Dalibard, Phys. Rev . A81, 053403 (2010). [41] L. Wen, Q. Sun, H. Q. Wang, A. C. Ji, and W. M. Liu, Phys. Rev. A 86, 043602 (2012). [42] S.-W. Song, Y.-C. Zhang, H. Zhao, X. Wang, and W.-M. Liu, Phys. Rev. A 89, 063613 (2014). [43] X. Chen, M. Rabinovic, B. M. Anderson, and L. Santos, Phys. Rev. A 90, 043632 (2014). [44] Y. Li, X. Zhou, and C. Wu, arXiv:1205.2162 (2012). [45] J. P. Vyasanakere and V. B. Shenoy, New J. Phys. 14, 043041 (2012). [46] T. Kawakami, T. Mizushima, M. Nitta, and K. Machida, Phys. Rev. Lett. 109, 015301 (2012). [47] X. L. Cui, Phys. Rev. A 85, 022705 (2012). [48] X. F. Zhou, Y. Li, Z. Cai, and C. J. Wu, J. Phys. B: At. Mol. Opt. Phys. 46, 134001 (2013). [49] D.-W. Zhang, J.-P. Chen, C.-J. Shan, Z. D. Wang, and S.-L. Zhu, Phys. Rev. A 88, 013612 (2013) [50] B. M. Anderson and C. W. Clark, J. Phys. B: At. Mol. Opt. Phys. 46, 134003 (2013). [51] R. Gupta, G. S. Singh, and J. Bosse, Phys. Rev. A 88, 053607 (2013). [52] G. J. Chen, T. T. Li, and Y. B. Zhang, Phys. Rev. A 91, 053624 (2015). [53] R. Y. Liao, O. Fialko, J. Brand, and U. Z¨ ulicke, Phys. Rev. A92, 043633 (2015). [54] Y.Li, X.F. Zhou, andC. J. Wu, Phys. Rev.B 85, 125122 (2012).[55] B. M. Anderson, G. Juzeli¯ unas, V. M. Galitski, and I. B. Spielman, Phys. Rev. Lett. 108, 235301 (2012). [56] B. M. Anderson, I. B. Spielman, andG. Juzeli¯ unas, Phys . Rev. Lett. 111, 125301 (2013). [57] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998). [58] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998). [59] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. 81, 5257 (1998). [60] D.Varshalovich, A.Moskalev, andV.Khersonskii, Quan- tum Theory of Angular Momentum (World Scientific, Singapore, 1988). [61] F. Iachello, Lie Algebras and Applications (Springer, Berlin, 2006) [62] E. J. Mueller, Phys. Rev. A 69, 033606 (2004). [63] Y. Kawaguchia and M. Ueda, Phys. Rep. 520, 253 (2012). [64] Z. F. Xu, R. L¨ u, and L. You, Phys. Rev. A 83, 053602 (2011). [65] R. M. Wilson, B. M. Anderson, and C. W. Clark, Phys. Rev. Lett. 111, 185303 (2013). [66] Y. Deng, J. Cheng, H. Jing, C.-P. Sun, and S. Yi, Phys. Rev. Lett. 108, 125301 (2012). [67] S. Gautam and S. K. Adhikari, Phys. Rev. A 91, 013624 (2015).arXiv:1512.06394v1 [cond-mat.quant-gas] 20 Dec 2015Supplemental Material: Magnetic and nematic phases in a Wey l type spin-orbit-coupled spin-1 Bose gas Guanjun Chen,1,2Li Chen,1and Yunbo Zhang1,∗ 1Institute of Theoretical Physics, Shanxi University, Taiy uan 030006, China 2Department of Physics, Taiyuan Normal University, Taiyuan 030001, China (Dated: October 27, 2018) ∆0=7 2A2 2−2λA1A2/radicalbigg 5 6. (S1) ∆n=−/radicalbigg 1 12000(10A0A2 1A2−7A0A3 2)(1+3x)+3 10A2 0A2 2(2+x)+7 240A2 1A2 2(19−3x)+63 1600A4 2(7+x).(S2) ∆s=/radicalbigg 1 480(10A0A2 1A2−7A0A3 2)+35 48A2 1A2 2+63 320A4 2. (S3) ∗ybzhang@sxu.edu.cn

1307.1607v1.Spin_orbit_coupled_fermions_in_ladder_like_optical_lattices_at_half_filling.pdf

arXiv:1307.1607v1 [cond-mat.quant-gas] 5 Jul 2013Spin-orbit coupled fermions in ladder-like optical lattic es at half-filling G. Sun,1J. Jaramillo,1L. Santos,1and T. Vekua1 1Institut f¨ ur Theoretische Physik, Leibniz Universit¨ at H annover, 30167 Hannover, Germany We study the ground-state phase diagram of two-component fe rmions loaded in a ladder-like lattice at half filling in the presence of spin-orbit couplin g. For repulsive fermions with unidirectional spin-orbit coupling along the legs we identify a N´ eel state which is separated from rung-singlet and ferromagnetic statesbyIsingphasetransition lines. Thes elines cross for maximal spin-orbitcoupling anda direct Gaussian phase transition between rung-single t and ferro phases is realized. For the case of Rashba-like spin-orbit coupling, besides the rung singl et phases two distinct striped ferromagnetic phases are formed. Incase of attractive fermions with spin- orbitcoupling at half-filling for decoupled chains we identify a dimerized state that separates a single t superconductor and a ferromagnetic states. I. INTRODUCTION The possibility of inducing synthetic electromagnetism in ultra-cold gases has attracted recently a large deal of attention. In spite of the electric charge neutrality of an atom, synthetic magnetic field may be induced by a proper laser arrangement [1]. Interestingly, uni- directional spin-orbit coupling (USOC) resulting from an equal superposition of Rashba [2] and linear Dres- selhaus [3] terms, has been realized for both spinor Bose [4] and Fermi gases [5, 6] with the help of counter- propagating Raman lasers. Recently this technique has allowed for the observation of superfluid Hall effect [7], Zitterbewegung[8], and the spin-Halleffect in aquantum gas [9]. Several theory works have discussed the creation of pure Rashba or Dresselhaus SOC with optical [10] and magnetic means [11], and even proposed methods to gen- erate a three-dimensional SOC [12]. The presence of a synthetic SOC is expected to lead to a rich physics for atoms loaded in optical lattices. For two-dimensional Hubbard models at half filling the ef- fects of a Rashba-like SOC were studied both for two- component bosons and fermions, for which exotic spin texturesin the groundstatesuchascoplanarspiralwaves and stripes as well as non-coplanar vortex/antivortex configurations have been predicted [13–16]. Note, how- ever, that the SOC introduces frustration, invalidating quantum Monte Carlo (MC) approaches, and hence most studies have relied on classical MC calculations. In this paper we analyze the effects of SOC in a two- component Fermi gas loaded in an optical lattice in the Mott-insulator regime. Since we are interested in the quantum spin-1 /2 phases in the presence of SOC, we can not rely on classical MC, and must hence employ exact diagonalization or density-matrix renormalization group (DMRG) techniques. We employ the latter in our paper, restricting our analysis to the minimal system where the non-Abelian character of the vector potential may be manifested allowing non-trivial effects of SOC without the need of breaking the time-reversal invari- ance, namely a two-leg ladder-like optical lattice, which may be created by incoherently combining a 1D lattice and a two-well potential. By a combination of numerical1,1S S 2,1S S S S1,2 1,3 2,3 2,2ty Ayt ,xAx FIG. 1: Two-leg ladder lattice of s= 1/2 spinsSα,j, where α= 1,2 enumerates the ladder legs, and j= 1,2,3,···,L labels the ladder rungs. DMRG results, bosonization techniques and strongrung- coupling expansions, we obtain the spin quantum phases for both a USOC with different orientations with respect to the ladder, and the isotropic SOC. The paper is organized as follows. In Sec. II we in- troduce the effective spin model for a Mott state of two- component fermions with USOC in a ladder-like lattice. In Sec. III we review the phases for the case of decou- pled one-dimensional lattices. Section IV deals with the quantum phases of an USOC discussing the different ori- entations between the USOC and the ladder legs. In Sec. V we analyze the case of an isotropic SOC. We fi- nally summarize in Sec. VI. II. EFFECTIVE SPIN MODEL FOR TWO-COMPONENT FERMIONS WITH USOC Recent experiments have realized an USOC character- ized by a Hamiltonian of the form [4]: HUSOC=1 2m(pσ0−A)2+δ 2σz−h 2σx,(1) whereσz,xare Pauli matrices, σ0is the identity matrix, andthe effectivevectorpotentialfor counter-propagating Raman lasers on the xyplane is given by A=−/planckover2pi1k0σz, withk0= (kx 0,ky 0,0). Here the eigenvectors of σzcor- respond to atomic hyperfine components, the termδ 2σz is due to detuning from resonance, and his the Rabi2 coupling. Crucially, Acannot be completely gauged out, since it does not commute with the scalar poten- tial Φ =δ 2σz−h 2σx. We consider a two-component Fermi gas loaded in a ladder-like optical lattice of inter-site spacing a, with the ladder legs oriented along xand the rungs along y. Pro- jecting on the lowest lattice band [17] one obtains, in ab- senceofSOC,the two-componentFermi-Hubbardmodel: HFH=−/summationdisplay (i,i′),σ,σ′ti,i′σ0 σ,σ′a† i,σai′,σ′+U 2/summationdisplay ini(ni−1),(2) whereai,σis the annihilation operator of fermions with spinσ=↑,↓on sitei,ni=/summationtext σa† i,σai,σ,Ucharacter- izes the on-site interaction, and ti,i′are the hopping amplitudes along the bonds connecting nearest-neighbor sites (i,i′), with the hopping along the legs (the rungs) given by ti,i′=tx(ty). The presence of SOC results in the Peierls substitution ti,i′σ0→ti,i′eiA(ri′−ri) /planckover2pi1. In the strong coupling limit, U→ ∞, and considering half- filling (i.e. we consider a Mott phase with one fermion per site [18, 19]), the Fermi-Hubbard model may be re- written as an effective spin-1 /2 model of the form: H=J/bardbl/summationdisplay α,j/braceleftBig cos(2kx 0a)Sα,jSα,j+1 +2sin2(kx 0a)Sz α,jSz α,j+1+sin(2kx 0a)[Sα,j×Sα,j+1]z/bracerightBig +J⊥/summationdisplay j/braceleftBig cos(2ky 0a)S1,jS2,j +2sin2(ky 0a)Sz 1,jSz 2,j+sin2ky 0a[S1,j×S2,j]z/bracerightBig +δ/summationdisplay α,jSz α,j−h/summationdisplay α,jSx α,j. (3) whereJ/bardbl= 4t2 x/U,J⊥= 4t2 y/U, and Sα,j= (a† α,j,↑,a† α,j,↓)σ 2/parenleftbigg aα,j,↑ aα,j,↓/parenrightbigg . (4) are the spin operators associated to the leg α= 1,2 and the rung j(see Fig. 1), with the site index iin Eq. (2) split into leg and rung indices: i→(α,j). The value ofkx 0andky 0is provided by the orientation between the Raman lasers creating the USOC and the ladder axis. Note that the scalar potential Φ produces the last two termsinEq.(3), whereasthevectorpotential Aproduces Dzyaloshinskii-Moriya (DM) terms [21, 22], ∼[Sα,j× Sα′,j′]z, as well as easy-axis anisotropy (EAA) along ez. h 0ππ/2 3π/4 π /4012 LL LLF k0xa FIG. 2: (Color online) Ground states of a 1D spin-1/2 chain with USOC and transverse magnetic field obtained us- ing DMRG for 96 sites. The magnetic field is in units of J/bardbl. LL denotes a luttinger liquid phase and F stands for ferro- magnetic state. III. DECOUPLED CHAINS A. Repulsive interactions We first discuss the case of decoupled chains, J⊥= 0 (i.e.ty= 0), which results in the 1D Hamiltonian H1D=J/bardbl/summationdisplay j/parenleftBig Sz jSz j+1+cos2kx 0a(Sx jSx j+1+Sy jSy j+1) +sin2kx 0a(Sx jSy j+1−Sy jSx j+1)/parenrightBig −h/summationdisplay jSx j.(5) Fork0 x= 0 Eq. (5) describes an SU(2)-symmetric spin- 1/2 antiferromagnetic chain in external magnetic field, which is exactly solvable by means of Bethe ansatz [23]. The ground state is a gapless Luttinger liquid (LL) for h <2J/bardbl, and a fully polarized state for h >2J/bardbl. These two phases are separated by a commensurate- incommensurate (C-IC) phase transition. In order to discuss the effects of the USOC it is con- venient to introduce a gauge transformation that ren- ders exchange interactions explicitly SU(2) invariant, H1D→UH1DU†=¯H1D, where U=/producttext je−2ikx 0ajSz j. The spin operators transform as ¯Sx j= cos(2 kx 0aj)Sx j−sin(2kx 0aj)Sy j, ¯Sy j= cos(2 kx 0aj)Sy j+sin(2kx 0aj)Sx j.(6) and¯Sz j=Sz j, and the Hamiltonian becomes ¯H1D=J/bardbl/summationdisplay j¯Sj¯Sj+1−/summationdisplay jhj(k0)¯Sj,(7)3 where the effect of the USOC is entirely ab- sorbed into an external magnetic field, hj(k0) = h(cos(2kx 0aj),sin(2kx 0aj),0), that spirals on the xyplane. Forkx 0a=π/2,hj= (−1)jhex, i.e. a staggered effec- tive magnetic field. A staggered field constitutes a rele- vantperturbation(inthe renormalizationgroupsense)as itcouplestotheN´ eelorder,whichisoneoftheleadingin- stabilities in a 1D antiferromagnetic chain. As a result of that, agapin the excitationspectrum, ∆ E∼h2/3, opens for any arbitrary coupling h. The low-energy behavior is described by a massive sine-Gordon model where one of the breather modes is degenerate with soliton and anti- soliton excitations [24]. In the gauge transformed vari- ables the ground state developes N´ eel order, which after un-doing the gauge transformation results for the orig- inal spin operators into an uniformly magnetized state, i.e. a ferromagnetic (F) state, although magnetization is never fully saturated for kx 0/ne}ationslash= 0. For 0< kx 0a < π/2,hj(k0) is incommensurate and hence the gapless LL phase survives up to a finite hvalue at which the F phase is reached. We have employed the matrix product formulation [26] of DMRG method [27, 28] to obtain numerically the phase diagram for ar- bitrary values of the USOC (see Fig. 2). This phase dia- gramconfirmsthe existenceofa gaplessLL anda gapped F phase separated by a C-IC transition. Note that cor- relation functions, which decay algebraically in the LL phase and exponentially in the F phase, are generically incommensurate due to the DM anisotropy and the vec- tor product of two neighbouring spins has finite expec- tation value /an}bracketle{t[Sj×Sj+1]z/an}bracketri}ht ∼ −sin(2kx 0a) as depicted in Fig. 3(a). Its magnetic field dependence is presented in Fig. 3(b). B. Attractive interactions For the decoupled chains we have also studied the case of two-component fermions with attractive inter- actions. The most interesting ground-state physics oc- curs at half-filling in the vicinity of the maximal USOC, kx 0a≃π/2. In this case, after particle-hole transforma- tion the 1D Fermi-Hubbard model becomes dual to the repulsive ionic-Hubbard model [25], being characterized by the existence of a dimerized (D) phase between a su- perconducting (SC) phase and the F state. With increas- ing magnetic field the SC phase undergoes a Kosterlitz- Thouless (KT) transition into the D state, where trans- lational symmetry is spontaneously broken. Further in- creasing the magnetic field results in a D-F Ising tran- sition. We characterized the D phase in our numerical simulations by means of the dimerization order parame- ter, which in a chain with Lsites is defined as: D=/summationdisplay j(−1)j L/an}bracketle{ta† j,↑aj+1,↓−a† j,↓aj+1,↑+h.c./an}bracketri}ht.(8) The phase diagram of the 1D attractive Fermi-Hubbard modelwith kx 0a=π/2athalf-fillingispresentedinFig.4.0−0.4−0.200.20.4 k0x aπ/4π/23π/4πj j+1S S X<[ ] > (a)z 00.5 11.5−0.3−0.2−0.10 h/J||(b)<[ ] >S SjXj+1z LL F FIG. 3: Expectation value of the vector product of two neighboring spins as a function of: (a) the USOC parameter, kx 0a, forh= 0; and (b) magnetic field for kx 0a=π/4. IV. TWO-LEG LADDER WITH USOC We consider now the case of coupled chains with nonzero hoppings tx,y. As mentioned above, the value ofk0 xandk0 ydepends on the orientation of the USOC lasers and the ladder axis. In the following we consider separatelythe case in which the USOC is along the rungs and that in which the USOC is along the legs. A. USOC along the ladder rungs We analyze first the case of an USOC along the ladder rungs, i.e. kx 0= 0 in Eq. (3). For ky 0= 0 the magnetic field introduces two C-IC phase transitions: from a rung- singlet(RS) intoaLL andthen from the LLinto the fully polarized F state. As in our discussion of Sec. III it is4 0246012 |U|/thF D SC x FIG. 4: Phase diagram, for an attractive two-component Fermi Hubbard model on a chain at half filling with maximal USOC, where D denotes a dimerized phase, and SC stands for a1Dsuperconductor. The magnetic fieldis in unitsof tx. The phase boundaries are obtained after finite-size extrapolat ion from data obtained for 128, 256, 512 and 1024 sites. FIG. 5: Raman lasers counter-propagating along ladder rungs result in USOC as discussed in subsection IVA. convenient to introduce the gauge transformation ¯Sx α,j= cos(2 ky 0aα)Sx α,j−sin(2ky 0aα)Sy α,j, ¯Sy α,j= cos(2 ky 0aα)Sy α,j+sin(2ky 0aα)Sx α,j(9) and¯Sz α,j=Sz α,j. For the case of the maximal USOC, ky 0a=π/2, the gaugetransformed Hamiltonian becomes: ¯H=J/bardbl/summationdisplay α=(1,2),j¯Sα,j¯Sα,j+1+J⊥/summationdisplay j¯S1,j¯S2,j −h/summationdisplay α=(1,2),j(−1)α¯Sx α,j. (10) In the strong rung-coupling limit, J⊥≫J/bardbl, the ground state becomes a rung-product state of the form: |¯RS/an}bracketri}ht=/productdisplay j/parenleftBig |¯↑1,j/an}bracketri}ht⊗|¯↓2,j/an}bracketri}ht−β|¯↓1,j/an}bracketri}ht⊗|¯↑2,j/an}bracketri}ht/parenrightBig //radicalbig 1+β2, (11)where{¯↑,¯↓}refertotheeigenstatesof ¯Sx. Forh= 0,β= 1and the ground-stateis aproduct stateofsingletsalong the rungs. With increasing magnetic field βdecreases gradually tending to zero. For β= 0 the ground-state after undoing the gauge transformation translates into the F state. Hence, for ky 0a=π/2 the magnetic field just results in an adiabatic evolution of |¯RS/an}bracketri}htinto the F state. To address the general case 0 < ky 0< π/2 we consider the case of weak USOC, ky 0a≪1, closely following the strong rung-coupling derivation of Ref. [29]. For h= 0 the ground state is well approximated by a direct prod- uct of singlets along the rungs, and the energy gap to the lowest rung triplet excitation is ∼J⊥. The external magnetic field splits linearly the rung triplet excitations, and the energy of the state where both spins of the rung point in the direction of the field approaches that of the RS state for h∼J⊥. Identifying the RS state on a rung with an effective spin-1 /2 pointing down, and the Sx= 1 component of the rung triplet state with the spin-1 /2 pointing up, the effective pseudo-spin-1 /2 model in the strong rung-coupling limit for h∼J⊥takes the form of an XXZ model in a tilted uniform magnetic field: Hτ=J/bardbl/summationdisplay j(1 2τx jτx j+1+τy jτy j+1+τz jτz j+1) −hx/summationdisplay jτx j−hy/summationdisplay jτy j (12) whereτx,y,zare the pseudo-spin-1 /2 operators, hx= h−J⊥cos2ky 0a+J⊥(1−cos2ky 0a)/4−J/bardbl/2, andhy= J⊥sin2ky 0a/√ 2. With varying hxthe model (12) under- goes changes in three ground-state phases [30]: two F phases separated by Ising transitions from an intermedi- ate N´ eel phase in τzstate. One of the F phases of the effective model (12) translates to the RS phase of the ladder, whereas the N´ eel phase and the second F phase of (12) translate into identical ladder phases. Note that it is the DM interaction that in the leading order breaks in Eq. (12) the U(1) rotation symmetry in the yzplane allowing for the N´ eel ordering. We consider at this point weaklycoupled chains, J⊥≪ J/bardbl, again for weak USOC, ky 0a≪1. For this case we can use bosonization mapping [31] with the convention: Sx α,j→∂xφα√ 2π+(−1)jsin√ 2πφα, Sy j,α→(−1)jsin√ 2πθα+··· (13) Sz α,j→(−1)jcos√ 2πθα+··· wherex=ja, the dots denote sub-leading fluctuations of uniform components, and we have introduced two pairs of dual bosonic fields, [ θα(x),∂yφα′] =iδα,α′δ(x−y). It is convenient to introduce the symmetric and anti- symmetric combinations of the original bosonic fields, θ±= (θ1±θ2)/√ 2,φ±= (φ1±φ2)/√ 2. We treat J⊥as a perturbation of the two decoupled chains retaining only the relevant contributions that it generates. We obtain5 ez xh||e FIG. 6: N´ eel state configuration for USOC along rungs. The inter-leg correlation functions are also antiferromag netic, /angbracketleftSz 1,iSz 2,j/angbracketright ∼(−1)i−j+1. the following Hamiltonian density: HB=/summationdisplay ν=±vν 2/bracketleftbig (∂xφν)2+(∂xθν)2/bracketrightbig −h∂xφ+/√π +˜J⊥(2cos√ 4πθ−−cos√ 4πφ++cos√ 4πφ−) +˜d⊥/parenleftbig cos√πφ+sin√πθ+sin√πφ−cos√πθ− −sin√πφ+cos√πθ+cos√πφ−sin√πθ−/parenrightbig +d⊥(cos√ 4πθ−+cos√ 4πθ+), (14) where v±≃J/bardblaπ 2/parenleftbigg 1±J⊥cos(2ky 0a) J/bardblπ2/parenrightbigg and˜J⊥∼J⊥cos(2ky 0a). The DM anisotropy generates the terms with the coupling constant ˜d⊥∼J⊥sin(2ky 0a), whereas the EEA induces the terms with the pre factor d⊥∼J⊥(1−cos(2ky 0a)). All three factors ˜J⊥,˜d⊥andd⊥ depend as well on a short-distance cut-off. Note that the magnetic field, h, just couples to the symmetric sector. Forky 0a≪1,d⊥,˜d⊥≪˜J⊥, and the antisymmetric sector remains gapped with /an}bracketle{tθ−/an}bracketri}ht=√π/2. Note that when the magnetic field suppresses the dominant cou- pling in the symmetric sector, ˜J⊥cos√ 4πφ+, the EEA term induces the leading instability. The symmetric sector can be solved by means of the Jordan-Wigner mapping and a subsequent Bogoliubov transformation. It supports two Ising phase transitions with increasing magnetic field that separate three differ- ent ground-state phases. In the original spin variables, these phases translate into the RS phase, a N´ eel state with order parameter n= (−1)j+α/an}bracketle{tSz α,j/an}bracketri}ht /ne}ationslash= 0, and the F phase. In N´ eel state spins are canted uniformly along the applied field as depicted in Fig. 6. We recall that for ky 0a=π/2 a growing magnetic field does not introduce any phase transition but rather adia- batically connects RS and F phases. Since in the vicinity ofk0 ya= 0 an intermediate N´ eel phase occurs, we hence expectasafunctionof ky 0aandhthepresenceofaN´ eelis- land inside an overall RS-F state. Our numerical results confirm this expectation, as depicted in Fig. 7. Since N´ eel order is spontaneous, in our numerical calculations we monitor n2≡lim |i−j|≫1|(−1)i−j/an}bracketle{tSz α,iSz α,j/an}bracketri}ht|. (15)001234 k0y ah −π/2−π/4 π/4π/2RSF NeelNeel FIG. 7: Phase diagram for the USOC along ladder rungs. The phase transition curve into the N´ eel state is determine d fromtheclosingofthegap betweenthetwolowest eigenstate s, see Fig. 8(b). For ky 0= 0 a LL line is realized between the RS and F phases. The magnetic field is in units of J/bardbl=J⊥. The magnetic field dependence of n2is illustrated in Fig. 8(a). We have studied as well the behavior of the excita- tion gap. The N´ eel state is characterized by a doubly- degenerate ground-state in the thermodynamic limit, whereasthe RS and F stateshaveunique gapped ground- states. Hence a simple wayto obtain the boundary of the N´ eel state is to follow the closing of the gap between the ground-state and the first excited state (that becomes degenerate with the ground state in the thermodynamic limit in the N´ eel phase). We plot the behavior of the gap in Fig 8(b). The gaps close linearly with the mag- netic field when approaching the quantum phase transi- tion points, as expected from the Ising character. We note finally, that the vector product of two neigh- boring spins has a finite expectation value along rungs, /an}bracketle{t[S1,j×S2,j]z/an}bracketri}ht ∼ −sin(2ky 0a), in all phases, becoming zero only deep in the F phase for large values of h. B. 2-leg ladder with USOC along legs We consider now that the USOC is oriented along the ladder legs, as depicted in Fig. 9, and hence ky 0= 0 in Eq. (3). For kx 0= 0, the external magnetic field induces two consecutive C-IC phase transitions: a first one from RS into the gapless LL phase, and a second one from LL into the fully polarized F state. For k0 x/ne}ationslash= 0 we may perform a gauge transformation similar to the ones dis- cussed above, which for the maximal USOC, kx 0a=π/2, results in a model similar to Eq. (10) but in this case with a field that couples uniformly to spins belonging to the same rung and it alternates from rung to rung, −h/summationtext α,j(−1)j¯Sx α,j. Using bosonization in the weak rung- coupling limit, J⊥≪J/bardbl, and in opposite limit J⊥≫J/bardbl employing strong rung-coupling expansion it has been6 0 1 2 300.050.10.15 hn2 RS Neel F(a) 0 1 2 300.20.40.6 h∆E RS Neel F(b) FIG. 8: (a) Square N´ eel order as a function of the magnetic field along a cut through RS-N´ eel-F phases for kx 0= 0 and ky 0=±3π/16; (b)Behavior of the energy gap between the two lowest eigenstates as a function of the magnetic field across the RS-N´ eel-F phases. The N´ eel state is characterized by doubly degenerate ground states. The magnetic field and the gap are both measured in units of J/bardbl=J⊥. The depicted results correspond to DMRG simulations with L= 48 rungs. Finite size-effects near both phase transitions are very sim ilar to the behavior depicted on Figs. 10 (a) and (b) close to the N´ eel to F transition. FIG. 9: Raman lasers counter- propagating along the ladder legs result in an USOC as that discussed in subsection IVB. The value of k0 xcan be controlled by the angle between the laser propagation direction and the ladder legs. determined that such magnetic field introduces Gaussian criticality between two gapped phases of the antiferro- magnetic spin ladder [33], that for our original spin vari- ables corresponds to RS and F states. The difference at maximal USOC between the case with USOC along the ladder legs and that with USOC along the rungs can be easily understood in the limit J⊥≫J/bardbl. For the case of USOC along the ladder legs the magnetic field couples uniformly to the spins on the samerung,andhenceitfavorsatripletstateoneachrung with both spins pointing in the same direction, which al- ternates from one rung to the next. This state is orthog- onal to the RS configuration. In contrast, in Eq. (10) the magnetic field couples in a staggered way to the spins in the same rung, and the ground-state favored by a strong00.511.522.500.10.20.3 hhn2 L1/4L=24 L=36 L=48 L=60 −50500.2 (h−h cc )Ln2 L1/4 Neel F RS(a) 00.511.522.500.20.40.6 hEL=24 L=36 L=48 L=60 RS Neel F(b)∆ FIG. 10: (Color online) (a) Square N´ eel order, n2, as a function of the magnetic field along a cut through RS-N´ eel- F phases for kx 0=π/4 andk0 y= 0 and for different system sizes. The inset shows the collapse of our numerical results for different system sizes on a single curve according to the Ising scaling. (b) Behavior of the energy gap between the two lowest eigenstates as a function of the magnetic field across the RS-N´ eel-F phases. The N´ eel state is characterized by doubly degenerate ground states. The magnetic field and the gap are measured in units of J/bardbl=J⊥. The results displayed correspond to DMRG simulations for L= 24,36,48 and 60 rungs. magnetic field is not orthogonal to the RS state. As a result, for the USOC along rungs the RS state can be adiabatically connected to the F state, whereas for the USOC along legs this is not possible. Based on the previous discussion we hence expect that the two C-IC phase transition points for kx 0a= 0 have to evolve into a single Gaussian point for kx 0a=π/2. As for the case of USOC along rungs, we can employ bosonization to understand this evolution of the critical points in the limit J⊥≪J/bardbl. The leading instability once the magnetic field suppresses the RS phase is again the EAA, however now in exchange interactions along the chains and DM anisotropy induces incommensura- bility [34]. In contrast to the relevant couplings pro- duced by the USOC along the rungs, in Eq. (14) the USOC along ladder legs produces a marginal pertur- bation,∼cos√ 4πθ−cos√ 4πθ+. After mean-field de- coupling between the symmetric and antisymmetric sec- tors the weak rung-coupling bosonic Hamiltonian for the USOC along legs is equivalent to Eq. (14), and hence the ground-states and phase transitions will be similar to the previous case of USOC along rungs. Thus C-IC phase transition points evolve into Ising lines for kx 0a >0 and atk0 xa=π/2 these two Ising lines merge in a Gaussian7 kx 0 ah 0π/4π/23π/4π01234 NeelF Neel R S FIG. 11: (Color online) Numerical phase diagram for the USOC along the ladder legs, ky 0= 0. For kx 0= 0(π) there are two C-IC transitions with increasing magnetic field: a first one from RS to LL, and a second one from LL into the fully polarized state (both LL and fully polarized states ar e indicated by bold lines). For 0 < kx 0< π/2 instead of the LL state a N´ eel state is realized, being separated from the R S and F states by Ising phase transition lines. These lines cro ss atkx 0=π/2 resulting in a direct Gaussian transition from RS to F. The magnetic field is in units of J/bardbl=J⊥. Phase boundaries are obtained for the system with L= 48 rungs. criticality due to the enhanced symmetry. Our numericalresults for n2and the excitation gapare depicted in Figs. 10. Note that finite size effects are more pronounced at the RS to N´ eel transition, whereas for the N´ eel to F transition finite-size effects are negligible. For the RS to N´ eel transition we have carefully performed finite-size scaling of the order parameter and determined the critical field hccorresponding to the phase transition from the intersection of the order parameter curves for different system sizes. The collapse of order parameter fordifferentsystemsizesinthe vicinityof hconthe single curve according to the Ising law is depicted in the inset of Fig. 10(a). Note finally that the vector product of two neighbor- ing spins has finite expectation value along the chains /an}bracketle{t[Sα,j×Sα,j+1]z/an}bracketri}ht ∼ −sin(2kx 0a). Its magnetic field de- pendence is similar to the curve of Fig. 3(b), and it van- ishes quickly in the F phase. The ground-state phase diagram for the USOC along the ladder legs is depicted in Fig. 11. As mentioned above, the C-IC phase transition points (corresponding toU(1) symmetry at kx 0= 0) transform into Ising transi- tions (for 0 < kx 0< π/2 the system does not have contin- uous symmetry), and then they combine into a Gaussian point at kx 0a=π/2 (where U(1) symmetry is revived). So in both cases, USOC either along ladder rungs or along ladder legs, the system presents three possible phases, RS, N´ eel, and F. For a general orientation of the USOC and the ladder legs, kx 0/ne}ationslash= 0 and ky 0/ne}ationslash= 0, we henceexpect these three phases as well. V. TWO-LEG LADDER WITH NON-ABELIAN VECTOR POTENTIAL We consider at this point a non-Abelian vector poten- tial of the form, A= (−/planckover2pi1kx 0σx,−/planckover2pi1ky 0σy). Contrary to the case of USOC the magnetic field, h, is not necessary to ensure the non-trivial character of SOC. We hence consider the time-reversal symmetric case, h= 0, and a balanced mixture of up and down spin fermions. The effective spin model in this case acquires the form: H=J/bardbl/summationdisplay α,j/braceleftBig cos(2kx 0a)Sα,jSα,j+1 +2sin2(kx 0a)Sx α,jSx α,j+1+sin(2kx 0a)[Sα,j×Sα,j+1]x/bracerightBig +J⊥/summationdisplay j/braceleftBig cos(2ky 0a)S1,jS2,j +2sin2(ky 0a)Sy 1,jSy 2,j+sin(2ky 0a)[S1,j×S2,j]y/bracerightBig .(16) kx 0 aky0 a 0 π π/4 3π/4 π/20π/4π/23π/4π RSRS St StSt Sty yx x~ FIG.12: Groundstatesfor spin-ladderwithRashbaSOC,see text. Thenumerical resultscorrespond toDMRGcalculation s forL= 48 rungs. Our numerical results for the ground-state phase di- agram are presented in Fig. 12. In the vicinity of zero SOC,kx 0a=ky 0a= 0 the system is in the RS state. For the case of a maximal SOC, kx 0a=ky 0a=π/2, we may employ the canonical transformation Sα,i→˜Sα,i= USα,iU†, with U=/productdisplay je−iπSy 1,j/productdisplay α,k=2je−iπSx α,k, (17) which transforms the Hamiltonian (16) into an SU(2) symmetric antiferromagnetic spin ladder Hamiltonian of the form of Eq. (10) with h= 0.8 ex FIG. 13: St ystate configuration of a two-leg ladder with Rashba SOC. Spins are oriented along ±exdirection on odd rungs and along ∓exon even rungs. In the St ystate the vector products of nearest-neighbour spins behave as /angbracketleft[Sα,j×Sα,j+1]x/angbracketright ∼ −sin(2kx 0a) and/angbracketleft[S1,j×S2,j]y/angbracketright ∼0. In the cartoon of St ystate a nonzero value of /angbracketleft[Sα,j×Sα,j+1]x/angbracketright quantity is not reflected. eyex FIG. 14: St xphase configuration of two-leg ladder with Rashba SOC. Spins are oriented along ±eydirection on one leg and along ∓eyon another. In St xphase the vector prod- ucts of nearest-neighbour spins behave as /angbracketleft[Sα,j×Sα,j+1]x/angbracketright ∼ 0 and along rungs /angbracketleft[S1,j×S2,j]y/angbracketright ∼ −sin(2ky 0a). Thus for kx 0a=ky 0a=π/2 the system is in the RS phase but in the gauge transformed spins (and hence we denote it as ˜RS), with no long-range order and exponen- tially decaying correlation functions. In the strong rung coupling limit the ground state in gauge transformed variables is the rung-singlet product state Eq. (11) with β= 1, that for original variables transforms via Uto the direct product of Sz= 0 components of the rung- triplets, alsooftheformofEq.(11)howeverwith β=−1. Since the ˜RS state is gapped it will occupy a finite region aroundkx 0a=ky 0a=π/2 point. In the RS phase the vector product of two neigh- bouring spins has finite expectation value: along the chains/an}bracketle{t[Sα,j×Sα,j+1]x/an}bracketri}ht ∼ −sin(2kx 0a) and along rungs /an}bracketle{t[S1,j×S2,j]y/an}bracketri}ht ∼ −sin(2ky 0a). In˜RS phase the vector product of two neighboring spins is negligeably small. Our numerical results reveal as well the appearance of striped phases with long range order where spins are ferromagnetically ordered in one direction and antifer- romagnetically in the other. The case of ferromagnetic order along the rung (St yphase) is best understood in the vicinity of ( kx 0a,ky 0a) = (π/4,π/2) point, where cos(2kx 0a) = 0, and cos(2 ky 0a) =−1. For these param- eters the coupling along the rung Sx 1,jSx 2,jis ferromag- netic, whereas intra-leg coupling Sx α,jSx α,j+1is antiferro- magnetic, which results in the St yconfigurationobserved in our numerical calculations (Fig. 13).We may understand in a similar way the appearance of the St xphase, see Fig. 14, analyzing the behavior in the vicinity of ( kx 0a,ky 0a) = (π/2,π/4), which is char- acterized by a ferromagnetic Sy α,jSy α,j+1coupling along legs, and an antiferromagnetic Sy 1,jSy 2,jexchange along rungs. St xand St ystates are dual to each other with respect to the interchange of leg and rung directions and SxandSycomponents. Our numerical simulations sug- gest that similarly to the USOC case all phase transi- tions for the case of the non-Abelian vector potential are of second-order Ising nature. This is natural, since the system does not enjoy in general any continuous symme- try, and striped phases break spontaneously discrete Z2 symmetries: St ybreaks translation symmetry along the chains direction whereas St xbreaks the parity symmetry associatedwith the exchange of ladder legs. Both striped phases break as well time reversal symmetry. The RS and ˜RS phases present different parity symme- try for an odd number of rungs, whereas the RS phase is antisymmetric the ˜RS is symmetric; As a result both phases cannot connect adiabatically. We could not de- termine numerically whether RS and ˜RS states can be connected adiabatically for an even number of rungs in the parameter space ( kx 0,ky 0) in Fig. 12. In particular, the string order, defined for the pair of spins across the ladder diagonal, is finite for both RS and ˜RS states and vanishes in the striped phases. However in the thermo- dynamic limit we expect the behavior of odd and even number of rungs to converge, and hence it is most likely that in the model given in Eq. (16) RS and ˜RS states are always separated by a phase transition (indicated by dashed lines in Fig. 12). VI. CONCLUSIONS Inourworkwehavediscussedthequantumspinphases and the associated quantum phase transitions for a two- component Fermi lattice gas, focusing on the case of a two-legladder-likelatticeathalf-filling,aminimalsystem to study the non-Abelian character of the vector poten- tial. We have shown that for an USOC along the ladder rungsan N´ eel state phaseis locatedwithin a RS-F phase, in which a rung-singlet may be adiabatically connected to a ferromagnetic phase in the parameter space of hand the SOC. In contrast, for the USOC along the ladder legs the RS and F states cannot be adiabatically con- nected, and are separated by an intermediate N´ eel state, which disappears at a maximal SOC to lead to a direct Gaussian RS-F quantum phase transition. The case of a Rashba-like SOC is characterized by the appearance of rung-singlet and striped phases. Compared to the clas- sical spin phases predicted for fermions on a square lat- tice with SOC [15] only the striped configurations of the 2D lattice have identical quantum counterparts on the ladder. On the contrary, the N´ eel and spiral waves are substituted by gapped rung-singlet states, whereas non- coplanar configurations such as vortex/antivortex tex-9 tures are not stabilized. Acknowledgments We thank A. K. Kolezhukand S. Manmamafor discus- sions. This work has been supported by QUEST (CenterforQuantumEngineeringand Space-TimeResearch)and DFGResearchTrainingGroup(Graduiertenkolleg)1729. [1] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. Porto, and I. B. Spielman, Nature 462, 628 (2009). [2] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984). [3] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [4] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011). [5] P. Wang et al., Phys. Rev. Lett. 109, 095301 (2012). [6] L. W. Cheuk et al., Phys. Rev. Lett. 109, 095302 (2012). [7] L. J. LeBlanc et al., Proc. Natl. Acad. Sci. 109, 10811 (2012). [8] L. J. LeBlanc et al., arxiv: 1303.0914. [9] M. C. Beeler et al., Nature498, 201 (2013). [10] D. L. Campbell, G. Juzeliunas, and I. B. Spielman, Phys. Rev. A84, 025602 (2011). [11] B. M. Anderson, I. B. Spielman, and G. Juzeliunas, arXiv:1306.2606. [12] B. M. Anderson, G. Juzeliunas, I. B. Spielman, and V. M. Galitski, Phys. Rev. Lett. 108, 235301 (2012). [13] Z. Cai, X. Zhou, and C. Wu, Phys. Rev. A 85, 061605(R) (2012). [14] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi, Phys. Rev. Lett. 109, 085302 (2012). [15] J. Radic, A. Di Ciolo, K. Sun, and V. Galitski,Phys. Rev. Lett.109, 085303 (2012). [16] M. Gong, Y. Qian, V. W. Scarola, Chuanwei Zhang, arXiv:1205.6211. [17] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998). [18] R. J¨ ordens, N. Strohmaier, K. G¨ unter, H. Moritz, and T . Esslinger, Nature (London) 455, 204 (2008). [19] U. Schneider et al., Science 322, 1520 (2008). [20] R. J¨ ordens et al., Phys. Rev. Lett. 104, 180401 (2010). [21] I. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958).[22] T. Moriya, Phys. Rev. 120, 91 (1960). [23] M. Takahashi, Thermodynamics of One-Dimensional Solvable Models , Cambridge University Press, Cam- bridge, UK (1999). [24] M. Oshikawa and I. Affleck, Phys. Rev. Lett. 79, 2883 (1997). [25] M. Fabrizio, A. O. Gogolin, and A. A. Nersesyan, Phys. Rev. Lett. 83, 2014 (1999); A. P. Kampf, M. Sekania, G. I. Japaridze, and P. Brune, J. Phys.: Condens. Mat- ter15, 5895 (2003); S. R. Manmana, V. Meden, R. M. Noack, and K. Sch¨ onhammer, Phys. Rev. B 70, 155115 (2004). [26] F. Verstraete, J.J. Garcia-Ripoll, and J.I. Cirac, Phy s. Rev. Lett. 93, 207204 (2004). [27] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B48, 10345 (1993). [28] U. Schollw¨ ock, Rev. Mod. Phys. 77, 259 (2005). [29] K. Penc, J.-B. Fouet, S. Miyahara, O. Tchernyshov, and F. Mila, Phys. Rev. Lett. 99, 117201 (2007). [30] A. A. Ovchinnikov, D. V. Dmitriev, V. Y. Krivnov, and V. O. Cheranovskii, Phys. Rev. B 68, 214406 (2003); D. V. Dmitriev and V. Y. Krivnov, Phys. Rev. B 70, 144414 (2004); J.-S. Caux, F. H. L. Essler, and U. L¨ ow, Phys. Rev. B68, 134431 (2003). [31] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization and Strongly Correlated Systems , Cam- bridge University Press, Cambridge, UK (1998). [32] Y.-J. Wang, arXiv:cond-mat/0306365v1 (unpublished) ; F. H. L. Essler and I. Affleck, J. Stat. Mech. P12006 (2004). [33] Y.-J. Wang, F. H. L. Essler, M. Fabrizio, and A. A. Ners- esyan Phys. Rev. B 66, 024412 (2002). [34] R. CitroandE. Orignac, Phys.Rev.B 65, 134413 (2002).

1809.10859v2.Engineering_long_spin_coherence_times_of_spin_orbit_systems.pdf

arXiv:1809.10859v2 [cond-mat.mes-hall] 1 Oct 2018Engineering long spin coherence times of spin-orbit system s T. Kobayashi,1,2,∗J. Salfi,1J. van der Heijden,1C. Chua,1M. G. House,1D. Culcer,3,4W. D. Hutchison,5B. C. Johnson,6J. C. McCallum,6H. Riemann,7 N. V. Abrosimov,7P. Becker,8H.-J. Pohl,9M. Y. Simmons,1and S. Rogge1,† 1Centre for Quantum Computation and Communication Technolo gy, School of Physics, University of New South Wales Sydney, NSW 2052, Australia 2Department of Physics, Tohoku University, Sendai 980-8578 , Japan 3School of Physics, University of New South Wales Sydney, NSW 2052, Australia 4Australian Research Council Centre of Excellence in Low-En ergy Electronics Technologies, The University of New South Wales, Sydney 2052, Australia 5School of Physical, Environmental and Mathematical Scienc es, The University of New South Wales Canberra, Canberra, ACT 26 00, Australia 6Centre for Quantum Computation and Communication Technolo gy, School of Physics, University of Melbourne, VIC 3010, Austr alia 7Leibniz-Institut f¨ ur Kristallz¨ uchtung, 12489 Berlin, G ermany 8PTB Braunschweig, 38116 Braunschweig, Germany 9VITCON Projectconsult GmbH, 07745 Jena, Germany (Dated: October 2, 2018) 1Abstract Spin-orbitcouplingfundamentallyalters spinqubits, ope ningpathways toimprovethescalability of quantum computers via long distance coupling mediated by electric fields, photons, or phonons. It also allows for new engineered hybrid and topological qua ntum systems. However, spin qubits with intrinsic spin-orbit coupling are not yet viable for qu antum technologies due to their short (∼1µs) coherence times T2, while qubits with long T2have weak spin-orbit coupling making qubit coupling short-ranged and challenging for scale-up. Here we show that an intrinsic spin-orbit coupled “generalised spin” with total angular momentum J=3 2, which is defined by holes bound to boron dopant atoms in strained28Si, hasT2rivalling the electron spins of donors and quantum dots in28Si. Using pulsed electron paramagnetic resonance, we obtai n 0.9 ms Hahn-echo and 9 ms dynamical decoupling T2times, where strain plays a key role to reduce spin-lattice r elaxation and the longitudinal electric coupling responsible for decohe rence induced by electric field noise. Our analysis shows that transverse electric dipole can be explo ited for electric manipulation and qubit coupling while maintaining a weak longitudinal coupling, a feature of J=3 2atomic systems with a strain engineered quadrupole degree of freedom. These res ults establish single-atom hole spins in silicon with quantised total angular momentum, not spin, as a highly coherent platform with tuneable intrinsic spin-orbit coupling advantageous to bu ild artificial quantum systems and couple qubits over long distances. ∗kobayashi20131124@gmail.com †s.rogge@unsw.edu.au 2Spin-orbit coupling fundamentally alters spin qubits, opening pathwa ys to improve the scalability of quantum computers via long distance coupling mediated b y electric fields, photons, or phonons [1, 2]. It also allows for new engineered hybrid a ndtopological quantum systems [3–5]. However, spin qubits with intrinsic spin-orbit coupling a re not yet viable for quantum technologies due to their short ( ∼1µs) coherence times T2[6–9], while qubits with long T2have weak spin-orbit coupling [10–12] making qubit coupling short-ra nged and challenging for scale-up. Here we show that an intrinsic spin-orb it coupled “generalised spin” with total angular momentum J=3 2, which is defined by holes bound to boron dopant atoms in strained28Si, hasT2rivalling the electron spins of donors and quantum dots in28Si [10–12]. Using pulsed electron paramagnetic resonance (EPR), we o btain 0.9 ms Hahn-echo and 9 ms dynamical decoupling T2times, where strain plays a key role to reduce spin-lattice relaxation and the longitudinal electric coupling responsible for deco herence induced by electric field noise [13]. Our analysis shows that transverse electric d ipole can be exploited for electric manipulation and qubit coupling [13] while maintaining a weak lo ngitudinal coupling, a feature of J=3 2atomic systems with a strain engineered quadrupole degree of freedom. These results establish single-atom hole spins in silicon wit h quantised total angular momentum, not spin, as a highly coherent platform with tune able intrinsic spin- orbit coupling advantageous to build artificial quantum systems and couple qubits over long distances. Hole spins bound to group-III acceptors in Si have compelling prope rties for building qubits with spin-orbit coupling. These properties derive from the Γ 8symmetry of valence- band holes where the L= 1 angular momentum of the atomic orbitals |px,y,z/an}bracketri}htcouples to spinS=1 2. As a result, the total angular momentum J=3 2is a good quantum number and the Bloch states (Fig. 1d), described by projected angular mo mentum mJ=±1 2(light holes) and mJ=±3 2(heavy holes), can be written as [14] /vextendsingle/vextendsingle3 2,±1 2/angbracketrightbig =1√ 6(|px/an}bracketri}ht±i|py/an}bracketri}ht)⊗/vextendsingle/vextendsingleSz=∓1 2/angbracketrightbig ∓2√ 6|pz/an}bracketri}ht⊗/vextendsingle/vextendsingleSz=±1 2/angbracketrightbig ,and /vextendsingle/vextendsingle3 2,±3 2/angbracketrightbig =1√ 2(|px/an}bracketri}ht±i|py/an}bracketri}ht)⊗/vextendsingle/vextendsingleSz=±1 2/angbracketrightbig , where|Sz=±1/2/an}bracketri}htare spin up/down. For group-III acceptors, different types of q ubits can be defined by using the low-energy J=3 2manifold. In a perfect silicon crystal, the two lowest-energy states form a charge-like subsystem {/vextendsingle/vextendsingle3 2,+3 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig }(Fig. 1a). Lowering the crystal symmetry by mechanical strain results in a gap ∆ (Figs. 1b and c), defining 3another qubit subsystem {/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,−1 2/angbracketrightbig }. This qubit is referred to as generalised spin because it is time-reversal symmetric while spin Sis not a good quantum number. Coupling to electric and elastic fields for hole spins (blue arrows, Fig. 1c) take s place via quadrupolar tensor operators ˆQij(see Supplemental material) that are represented by quadratic f orms of spin-3 2matrices ˆJx,y,zand have no analogue in the conduction band [15]. Combined with the J=3 2Zeeman interaction (orange arrows, Fig. 1c), the quadrupolar co uplings endow the generalised spin with intrinsic spin-orbit coupling (red arro w, Fig. 1c). This contrasts recent work on electron quantum dot systems in silicon w here spin-orbit coupling is induced with extrinsic sources such as charge degrees of freedo m [16] and micron-scale magnets [17–19]. Recently it has been predicted that the quadrupo les allow longitudinal electric couplings responsible for qubit decoherence to be minimized w hile maintaining spin- orbit qubit functionality via large transverse electric coupling [20]. Fu rther advantages of acceptors in Si compared to conventional spin qubits include remov al of the nuclear spin bath by28Si purification [10], single-atom addressability [21], confinement of sp in without gate electrodes [11, 21], and reduced influence from charge traps at interfaces. Nevertheless, the coherence of acceptor-bound hole spins has received little att ention [7]. Hereweexperimentallyexploretheeffectofastrain-inducedgapon theelectricquadrupole and coherence of holes bound to boron acceptors in silicon 28 (28Si:B). To study effects of strain in coherence, we use two bulk28Si crystal samples with and without mechanical strain. The mechanically relaxed sample provides the energy level co nfiguration in Fig. 1a (∆ = 0), where we investigate the {/vextendsingle/vextendsingle3 2,+3 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig }subsystem. Another sample is sub- jected to biaxial tensile strain (Fig. 2a) to obtain ∆ exceeding the qu bit energy splitting /planckover2pi1ω0(Figs. 1b and c). In this gapped configuration, we investigate the {/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,−1 2/angbracketrightbig } generalised spin-subsystem (Fig. 1c), which has been predicted to have enhanced immunity to decoherence from electrical noise [20] and has a reduced longitu dinal relaxation rate [22, 23]. For both the strained and relaxed samples we use low-tempe rature (base tem- perature Tb≈25 mK) pulsed EPR (see Supplemental material) to measure the qubit T2 with and without dynamical decoupling, and the longitudinal relaxatio n timeT1for an Si:B ensemble with concentration nB≈1015cm−3. A static magnetic field− →B0is aligned to the [110] axis of Si crystal for all measurements. Figures 2b and 2c show spin-echo spectra (see Supplemental mate rial) measured by a Hahn-echo sequence ( π/2)X–τ–(π)Y(Fig. 2b top right) with ( π/2)Xand (π)Ypulses sep- 4arated by a time interval τ, whereX,Ysubscripts indicate the rotation axes in a qubit subsystem. In the relaxed28Si:B sample a narrow spin-echo signal appears at |− →B0| ≈383 mT with a microwave frequency ωMW/2πof 6.255 GHz (Fig. 2b). We obtain an effective g-factor|g∗|of 1.17 by equating ωMWwith|g∗|µB|− →B0|//planckover2pi1whereµBis the Bohr magneton and /planckover2pi1is thereduced Planck constant. This isconsistent with |g∗|= 1.13reported inEPR studies of28Si:B for− →B0/bardbl[110] [24, 25], for the {/vextendsingle/vextendsingle3 2,+3 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig }subsystem (black arrow, Fig. 1a). In the strained28Si:B sample we observe a spin-echo signal over a broad range of |g∗|from 2.4 to 2.6 atωMW/2π= 6.331 GHz (Fig. 2c). No signal is found at |g∗|= 1.17 in contrast with the relaxed sample (not shown), ensuring that the sample is pro perly strained. We attribute the broad spin-echo signal in the strained sample to the {/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,−1 2/angbracketrightbig }gener- alised spin (black arrow, Fig. 1b), since |g∗|in this range is expected for the configuration of strain and static magnetic field (Fig. 2a) [22, 26], as supported by th eory (see Supplemental material). The broadening of the spin-echo spectrum can be due to the distribution of the g∗value induced by strain inhomogeneity (see Supplemental material). An additional spin- echo peak attributed to dangling bond surface defects (P bcentres) appears at |g∗| ≈2.0 in both samples. The relative sharpness of the P bsignal compared to the strained Si:B signal (Fig. 2c) provides evidence that magnetic field inhomogeneity is neglig ible. Figure 3 displays qubit coherence (Fig. 3a) and longitudinal relaxatio n (Fig. 3b) mea- surements in the relaxed and strained samples (black and red symbo ls, respectively). We obtainT2measured by Hahn-echo decay, T2H, by fitting to a compressed exponential Aexp{−(2τ/T2H)β}, whereβreveals temporal noise characteristics [10, 27]. For the relaxed sample, fitting results in T2Hof 23±1µs andβ= 1.05 (blue curve). For the strained sample, we find enhanced T2Hof 0.92±0.01 ms with β= 2.45 at|− →B0|= 175.7 mT (green curve). We observeanimprovement in T2Hoverthefullrange g∗= 2.4–2.6;T2H= 0.93±0.02,0.97±0.02, and 0.78±0.03 ms and β= 3.5,3.0,and 2.9, for|− →B0|= 190.5,182.0 and 171 .7 mT, respec- tively (see Supplemental material). This improvement compared to t he relaxed sample is attributed to the strain-engineered quadrupole coupling as discus sed later. We also find a noticeable component of fast decay (2 τ/lessorsimilar300µs, black dashed line) for the strained sample. Because of the random distribution of boron atoms in the sample, th e local environments and thus T2of each generalised spin may differ. We posit that the fast decay is du e to a generalised-spin subset strongly coupled to decoherence sourc es such as another closely placed boron atom, impurities, or surface defects, which could be r educed by lower boron 5concentration, higher crystal purity, or a standard annealing pr ocedure. The generalised- spin subset well isolated from such decoherence sources is thus re sponsible for the slow decay with T2Hof 0.92±0.01 ms.T1is measured with an inversion-recovery pulse sequence (π)Y–t′–(π/2)X–τ–(π)Y(Fig. 3b, top left) consisting of a first inversion pulse, a subsequen t Hahn echo pulse sequence and a time interval t′between them. Inversion-recovery signals as a function of t′are well fitted by an exponential function A−Bexp(−t′/T1) in both samples (solid curves in Fig 3b); we obtain T1of 85±9µs for the relaxed sample and 5 ±1 ms for the strained sample. The T1improvement in strained Si:B is consistent with the previous experimental report [22], attributed to quadrupolar elastic couplin g suppressed by strain [23]. We note that T2HandT1of the relaxed sample are also longer than those observed in natSi:B (∼2µs and 4µs, respectively) [7], which could be explained by a lower temperature than the previous report. Remarkably, T2Hof the generalised spin {/vextendsingle/vextendsingle3 2,−1 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig }in the strained sample is comparable to T2H∼3 ms obtained for 1015-cm−3concentration P donors in28Si [10]. The spin-echo decay curves of the strained sample have β≈2.5–3.5 such that decoherence is induced by slow fluctuations (spectral diffusion) [10, 27]. In contra st, an28Si:P ensemble hasβ≈1 derived from instantaneous diffusion where spin re-focusing is disr upted by EPR- driven flips of neighbouring spins [10]. For strained28Si:B, this process has a negligible effect because of g-factor inhomogeneity much larger than the28Si:P ensemble. Without instantaneous diffusion, spectral diffusion in the28Si:P ensemble provides T2Hof∼100 ms. Importantly, T2His also comparable to measured values for state-of-the-art elect ron-spin qubits defined by28Si quantum dots without spin-orbit coupling [12], and exceeds measu red values for electron-spin qubits with extrinsic spin-orbit coupling indu ced by integrated mi- cromagnets [17–19]. This demonstrates that the28Si:B generalised spin can be as coherent as electron spins in28Si. Decoherence due to spectral diffusion can be ameliorated by dynam ical decoupling. Fig- ure 4 shows therefocused echo intensity inthe strained sample asa function of time after the first (π/2)Xpulse, measured by the Carr–Purcell–Meiboom–Gill (CPMG) pulse seq uence (topright). Byfitting anexponential function(solidcurve) to the spin-echo decay, weobtain T2CPMGof 9.2±0.1 ms, that is 10-times longer than T2Hof the strained sample, 400-times longer than T2Hin the relaxed sample and over four orders of magnitude longer than other solid-state systems with intrinsic spin-orbit coupling in previous repo rts [6, 8, 9]. Notably, 6we find that T2CPMGis very close to the upper bound of T2set by longitudinal relaxation, 2T1, for the strained sample. Magnetic field direction and further stra in engineering could be used to further improve T1[20, 28] and therefore improve T2CPMG(see Supplemental material). We now discuss the improvement in hole generalised spin coherence in s trained28Si:B by analysing electric coupling and the observed Hahn-echo decay cu rves. Generally, qubit dynamics are described by a Hamiltonian ˆHqbt=1 2/planckover2pi1(ω0+ω)ˆσZ+/summationtext α=X,Y/planckover2pi1Ωαˆσα, where Ω α is the Rabi frequency and ˆ σα(α=X,Y,Z) are the Pauli matrices of the qubit subsystem. ωis the change in resonance frequency induced by electric field− →E, expressed as ω=− →χ· − →E//planckover2pi1with the longitudinal electric dipole moment− →χ[13]. In any electrically active qubit, decoherence arises from fluctuations in ω,δω, induced by electric field fluctuations δ− →E, thus mitigated by suppressing− →χ. Indeed,− →χfor strained Si:B is predicted to be reduced in magnitude by /planckover2pi1ω0/2∆ compared to relaxed Si:B for our strain configuration and our magnetic field in the xyplane (see Supplemental material). Furthermore, coupling to first order of δ− →Eonly occurs for the z-oriented electric fields associated with the bold blue quadrupolar couplings in Fig. 1c. This− →χsuppression is independently estimated to be /planckover2pi1ω0/2∆≈1 10by comparing the measured T1in the relaxed and strained samples with perturbation theory (see Supplemental material). Then, the sup pression of− →χby strain reducesδωattributed to electric fluctuations originating from (i) background electric dipoles by/planckover2pi1ω0/2∆∼10−1and (ii) flips of neighbouring Si:B hole qubit by ( /planckover2pi1ω0/2∆)2∼10−2 originating from the longitudinal part of the electric dipole-dipole inte raction∼(1/R3)|− →χ|2 for Si:B atoms a distance Rapart. Another clue in the decoherence processes is a change in the echo-decay exponent from β= 1.05 toβ= 2.5–3.5, which implies that the characteristic time scale of dominant fluctuations τcchanges from τc> T2H, toτc/lessorsimilarT2H[27]. The T2H improvement arisesthenfromsuppressioninlongitudinalcouplingto fastelectricfluctuators, suchthatslowerfluctuatorsemergeasthedominantdecoherenc emechanism. Oneprominent possibility is that the dominant fluctuators in the strained sample bec ome magnetic in origin under the reduced longitudinal electric coupling. An increase in τcfor the dominant electric fluctuators in the strained sample cannot be ruled out; give n that the dominant electric fluctuator is the Si:B hole qubit subsystem not excited by EPR pulses, their spin-flip processes induced by longitudinal relaxation and spin-pair flip-flop a re both slowed down by reducedT1−1and transverse dipole-dipole coupling, respectively. In either case , the reduced 7longitudinal electric dipole is a key ingredient to improve coherence. Here we discuss two key properties, electrical controllability and th e intrinsic EPR linewidth, of the generalised spin qubit based on a theoretical model developed in Sup- plemental material. Electrical controllability of a qubit is characteris ed by the transverse electric dipole moment− →vα[13], which provides Rabi frequency Ω α=− →vα·− →E//planckover2pi1. In the lowest order of electric field, |− →vα|is reported to be 0.26 Debye for the relaxed Si:B system [29], suppressed by√ 3/planckover2pi1ω0/2∆ in our strained Si:B system. Consequently, Ω α/2πof 10 MHz will be available in the strained system with realistic oscillating electric fields of 40 kV/m. The intrinsic linewidth is determined by electric field noise and the longitudina l dipole coupling − →χ. The magnitude of− →χis suppressed by a factor of /planckover2pi1ω0/2∆ for strained Si:B compared to relaxed Si:B (0.26 Debye [29]) in a magnetic field along the [110] direction. An amplitude of electric field noise has been reported ∼10 V/m in interface-defined silicon quantum dots [18], thus implying the intrinsic linewidth of ∼2 kHz in strained Si:B comparable to EPR linewidths of state-of-the-art28Si electron spin qubits without external sources of spin-orbit coupling [11, 12]. We also note that smaller |− →χ|and thus longer T2can be available by aligning the magnetic field to the [100] direction without influencing− →vα. These observations imply that acceptor-bound holes embedded into silicon field-effect tr ansistor devices [21] could offer long T2and spin-orbit functionality usable for quantum manipulations. Together with long T2, several mechanisms could be employed to realise hybrid spin- photon systems or long-range spin-spin interactions via photons in superconducting mi- crowave resonators. The key strategy will be to engineer the spin -photon interaction to realise the strong coupling regime without increasing decoherence. This could be achieved by periodically modulating the transverse or longitudinal couplings to enhance the spin- photon interaction [30]. Indeed, an oscillating control electric field− →Ec(t) can be used to periodically modulate− →vα(− →Ec(t)) and− →χ(− →Ec(t)) via the second-order effect of electric field attributed to quadrupolar coupling of hole generalised spins (see Su pplemental material). Alternatively, the transverse coupling to x- andy-oriented electric fields could be statically enhanced using a z-oriented electric field and interface via a Rashba-like quadrupolar in ter- action at a sweet spot with long T2[20]. We note that− →Ealong the zaxis enhances− →vα(− →E) without changing the− →χ(− →E) up to higher order terms for generalised spins (see Supplemen- tal material). This means that the generalised-spin qubit allows to co ntrol the transverse coupling without increasing thelongitudinal coupling by− →Ealongz. Phononcoupling stands 8out as an alternative qubit coupling mechanism [23], and the coherenc e properties we have verified make Si:B hole generalised spins interesting candidates for ph onon coupled hybrid systems using silicon mechanical resonators. Systems where holes are allowed to tunnel be- tween neighbouring quantum dot or Si:B sites [21] will experience spin- orbit coupling, which could be useful to realise exotic spin-orbit coupled states [4, 5]. It s hould be possible to com- bine these spin-orbit functionalities with long T2because the transverse coupling could be engineered without increasing decoherence induced by the longitud inal coupling to electric fields. We have experimentally established long T2in generalised hole spins bound to acceptor atoms in mechanically strained Si where total angular momentum J, and not real spin S, is a good quantum number. Our measured T2times are similar to state-of-the-art results for systems such as electrons bound to Si:P donors and Si quantum dots, and three to four orders of magnitude longer than other solid-state systems with sp in-orbit coupling. These observations open up a new and promising pathway to use spin-orbit coupling available in holes to engineer new kinds of highly coherent hybrid quantum syste ms and to achieve long distance spin qubit coupling for single atom qubits in silicon. We acknowledge that this work was supported by the ARC Centre of Excellence for QuantumComputation andCommunication Technology (CE17010001 2), inpartby theU.S. Army Research Office (W911NF-08-1-0527). T.K. acknowledges su pport from the Tohoku University Graduate Program in Spintronics. J.S. acknowledges sup port from an ARC DE- CRA fellowship (DE160101490). M.Y.S. acknowledges a Laureate Fello wship. The authors thank Mike Thewalt for the28Si sample. 9BJ=3/2 -10010Energy (GHz) 0 0.5 Magnetic field (T)Relaxed Si:Ba d c 42 2Gap ∆ Electric / elastic coupling Magnetic couplingpy +- pz+ -px +- y xz iipx-ipy iipz px-ipy Generalised spin ii px+ipy iipz px+ipyAtomic orbitals 0.5h-ω0h-ω0 h-ω0Strained Si:B 0246 243 10 -100 Magnetic field (T)Energy (GHz)b Gap ∆ Fig.1.a,b,Si:B hole spin levels in relaxed ( a) and strained silicon ( b) in an applied magnetic field, assuming a biaxial tensile strain of 0.02 %. Black soli d arrows indicate transitions addressed in this work. c,Energy level diagram and couplings for a strained (gapped) s ystem. Spin-orbit coupling induced quadrupolar electric and elastic couplin gs are shown (blue arrows) as well as magnetically induced couplings (orange arrows). In a const ant magnetic field, the quadrupolar coupling introduces an electric transition dipole (red arr ow) to the {/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,−1 2/angbracketrightbig }generalised spin subsystem. d,Schematic images of Bloch wavefunctions for J=3 2hole states. 10Magnetic field (mT) Magnetic field (mT)Pb centre g* = 2.01Pb centre Boron g* = 1.17Boron g* = 2.63 -2.38 200 300 400 160 180 200 2201 1.01.4 1.2 0.8 02Integra ted amplitud e (mV)c bRelaxed Si Strained Si τ τ(π/2)X (π)YEcho Timeτ = 5 µs τ = 50 µsa EpoxyFused silica (1 mm) Biaxial tensile strain28Si:B (50 µm) B0//[110]z x y Nb CPWG cavity on Si substrateMicrowave pulsePress downOscillating magnetic field Fig.2.a,Schematic figure of the sample and the cavity. A 50 µm thin28Si (001) chip is bonded to a 1 mm-thick fused-silica chip by two-component epoxy res in. A biaxial tensile strain is induced in the28Si chip at cryogenic temperatures due to thermal expansion m ismatch of Si and fused silica. The magnetic field is applied along the [110] crystal direction of the28Si chip. To perform EPR experiments, the sample stack is pressed down to a superc onducting Nb coplanar waveguide resonator. b,c,Spin-echo spectra for the mechanically relaxed ( b) and strained ( c) samples, measured with τ= 5µs and 50 µs respectively. 11Expo nential fit Expo nential fit T1 = 5 ± 1 msT1 = 85 ± 9 µsτ τ t’(π/2)X(π)Y (π)YEcho t’ (ms)Quadrature si gnal ( µV) 0.01 0.1 1 10100 0Fit with exp{ -(2τ/T2H)β} T2H = 0.92 ± 0.01 ms , β = 2.45τ τ(π/2)X (π)YEcho Fit with exp{ -(2τ/T2H)β} T2H = 23 ± 1 µs, β = 1.05 2τ (ms)Integrated amplitud e (µV) 0.1 0.01 110100 TimeTimeFast decay component Strained Si Strained SiRelaxed Si Relaxed Si τ = 5 µs τ = 100 µsba Fig.3. Spin-echo signals for the strained (red) and relaxed (bla ck) samples measured by the standard Hahn echo sequence as a function of τ(a) and measured by the recovery pulse sequence as a function of t′(b). The solid curves show fitting functions. The dashed line sh ows the fast decay component in the strained sample. 12τ τ(π/2)X (π)YEcho(-π)Y (π)Y (π)Y(-π)Y TimeIntegrated amplitude ( µV)τ = 10 µs Exponential fit T2CPMG = 9.2 ± 0.1 ms Time after first pulse (ms)004080120 0 2 0 1Strained Si Fig.4. Decay of spin echo refocused by the CPMG pulse sequence (to p right diagram) in the strained sample as a function of elapsed time after the first ( π/2)Xpulse, and exponential fit to the data (solid line). 13[1] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe , and J. L. O’Brien, Nature 464, 45 (2010). [2] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Ph ys.85, 623 (2013). [3] I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys. 86, 153 (2014). [4] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science306, 1910 (2004). [5] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010). [6] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L. P. K ouwenhoven, Nature 468, 1084 (2010). [7] Y. P. Song and B. Golding, Europhys. Lett. 95, 47004 (2011). [8] A. P. Higginbotham, T. W. Larsen, J. Yao, H. Yan, C. M. Lieb er, C. M. Marcus, and F. Kuemmeth, Nano Lett. 14, 3582 (2014). [9] R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bohus lavskyi, R. LaviChan ville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer, and S. De Francesc hi, Nature Commun. 7, 13575 (2016), article. [10] A. M. Tyryshkin, S. Tojo, J. J. L. Morton, H. Riemann, N. V . Abrosimov, P. Becker, H.-J. Pohl, T. Schenkel, M. L. W. Thewalt, K. M. Itoh, and S. A. Lyon, Nature Mater. 11, 143 (2012). [11] J. T. Muhonen, J. P. Dehollain, A. Laucht, F. E. Hudson, R . Kalra, T. Sekiguchi, K. M. Itoh, D. N. Jamieson, J. C. McCallum, A. S. Dzurak, and A. Morello, N ature Nanotechnol. 9, 986 (2014). [12] M. Veldhorst, J. C. C. Hwang, C. H. Yang, A. W. Leenstra, B . de Ronde, J. P. Dehollain, J. T. Muhonen, F. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dz urak, Nature Nanotech 9, 981 (2014). [13] F. Beaudoin, D. Lachance-Quirion, W. A. Coish, and M. Pi oro-Ladri` ere, Nanotech. 27, 464003 (2016). [14] J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955). [15] R. Winkler, Phys. Rev. B 70, 125301 (2004). [16] D. Kim, Z. Shi, C. B. Simmons, D. R. Ward, J. R. Prance, T. S . Koh, J. K. Gamble, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, and M. A . Eriksson, Nature 511, 70 14(2014). [17] E. Kawakami, P. Scarlino, D. R. Ward, F. R. Braakman, D. E . Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson, and L. M. K. Van dersypen, Nature Nan- otechnol. 9, 666 (2014). [18] J. Yoneda, K. Takeda, T. Otsuka, T. Nakajima, M. R. Delbe cq, G. Allison, T. Honda, T. Kodera, S. Oda, Y. Hoshi, N. Usami, K. M. Itoh, and S. Taruch a, Nature Nanotech- nol.13, 102 (2018). [19] D. M. Zajac, A. J. Sigillito, M. Russ, F. Borjans, J. M. Ta ylor, G. Burkard, and J. R. Petta, Science359, 439 (2018). [20] J. Salfi, J. A. Mol, D. Culcer, and S. Rogge, Phys. Rev. Let t.116, 246801 (2016). [21] J. van der Heijden, T. Kobayashi, M. G. House, J. Slafi, S. Barraud, R. Lavieville, M. Y. Simmons, and S. Rogge, arXiv: , 1703.03538 (2017). [22] P. Dirksen, A. Henstra, and W. T. Wenckebach, J. Phys.: C ondens. Matter 1, 8535 (1989). [23] R. Ruskov and C. Tahan, Phys. Rev. B 88, 064308 (2013). [24] H. Neubrand, Phys. Status Solidi B 86, 269 (1978). [25] A. R. Stegner, H. Tezuka, T. Andlauer, M. Stutzmann, M. L . W. Thewalt, M. S. Brandt, and K. M. Itoh, Phys. Rev. B 82, 115213 (2010). [26] G. Feher, J. C. Hensel, and E. A. Gere, Phys. Rev. Lett. 5, 309 (1960). [27] W. B. Mims, Phys. Rev. 168, 370 (1968). [28] J. C. Abadillo-Uriel, J. Salfi, X. Hu, S. Rogge, M. J. Cald er´ on, and D. Culcer, Appl. Phys. Lett.113, 012102 (2018). [29] A. K¨ opf and K. Lassmann, Phys. Rev. Lett. 69, 1580 (1992). [30] N. Lambert, M. Cirio, M. Delbecq, G. Allison, M. Marx, S. Tarucha, and F. Nori, Phys. Rev. B97, 125429 (2018). [31] G. Bir, E. Butekov, and G. Pikus, Journal of Physics and C hemistry of Solids 24, 1467 (1963). [32] G. Bir, E. Butikov, and G. Pikus, Journal of Physics and C hemistry of Solids 24, 1475 (1963). [33] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Pergamon, London, 1981). [34] G. K. White, J. Phys. D: Appl. Phys. 6, 2070 (1973). [35] G. H. Olsen and M. Ettenberg, J. Appl. Phys. 48, 2543 (1977) 15SUPPLEMENTAL MATERIAL Acceptor Hamiltonian. Si:B holes couple to uniform magnetic, electric and elastic fields as described in Refs. 31 and 32. The coupling Hamiltonian to the fields is expressed as follows: ˆH′=ˆH′ B+ˆH′ E,ion+ˆH′ E+ˆH′ ε, (1) ˆH′ B=µB/summationdisplay i=x,y,z/parenleftBig g′ 1ˆJi+g′ 2ˆJ3 i/parenrightBig B0,i, ˆH′ E,ion=2p√ 3/parenleftBig ExˆQyz+EyˆQzx+EzˆQxy/parenrightBig , ˆH′ ε=b′/summationdisplay i=x,y,zεiiˆQii+2d′ √ 3/parenleftBig εxyˆQxy+εyzˆQyz+εzxˆQzx/parenrightBig , ˆH′ E=b/summationdisplay i=x,y,zE2 iˆQii+2d√ 3/parenleftBig ExEyˆQxy+EyEzˆQyz+EzExˆQzx/parenrightBig , wherex,yandzaxes correspond to the [100], [010] and [001] axes of silicon crystal, re- spectively, and ˆIis the identity operator. Quadrupole operators ˆQij(i,j=x,y,z) for J=3 2are expressed by angular momentum operators ˆJiasˆQij=1 2{ˆJi,ˆJj}−5 4δijˆI, where {ˆJi,ˆJj}=ˆJiˆJj+ˆJjˆJiandδijis Kronecker’s delta. Eiare electric field and εijare normal (shear) strain for i=j(i/ne}ationslash=j). Notice that all interactions with electrical andelastic degrees of freedom are proportional to ˆQij[15, 33] and as such they couple the light and heavy holes. WhileH′ εgenerally includes a hydrostatic term like aˆI, it does not cause any relative energy change of J=3 2hole states and thus is dropped off here. For the coefficients, we use g-factors g′ 1=−1.07 andg′ 2=−0.03 [24], linear electric-field coupling coefficient p= 0.26 Debye [29] and deformation potentials b′=−1.42 eV and d′=−3.7 eV [24]. banddare cubic electric-field coupling coefficients. While there is no ex- perimental information on the values of bandd, we estimate them at ∼ −3 Debye/(MV/m) and∼ −5 Debye/(MV/m), respectively, based on the effective mass appro ach in Ref. 32. By numerically diagonalising H′with certain sets of− →Eandεijas a function of− →B0, we obtain eigenenergy spectra as shown in Figs. 1a and b: εxx=εyy=εzz= 0 and− →B0/bardbl[110] for Fig. 1b, and εxx=εyy= 0.02 %,εzz=−0.0156 % and− →B0/bardbl[110] for Fig. 1c. ( |− →E|and εij(i/ne}ationslash=j) are kept zero.) Experimental setup. Both of the mechanically relaxed and strained samples are prepared from pieces of the same boron-doped28Si wafer with dimensions of 4.0 mm ×3.5 mm in 16area and 500 µm in thickness,28Si-purity of 99.99+ % and boron concentration nBof 1.0– 1.5×1015cm−3. A diced piece of this crystal is used for the relaxed sample. To prep are the strained sample, another piece is thinned down to 50 µm and glued to a fused silica chip (5.0 mm×5.0 mm in area and 1 mm in thickness) by two-component epoxy adhesiv e (Fig. 2a). While both the28Si and fused silica chips in this stack are mechanically relaxed at room temperature, biaxial tensile strain is applied to the28Si chip at low temperature owing to the difference in thermal expansion coefficient of these two materia ls [34]. The magnitude of the strain applied to the strained sample is discussed in the section Strain analyses . ToperformEPRspectroscopyatmillikelvintemperature, weuseasu perconductingcopla- nar waveguide cavity with a small mode volume. The cavity is fabricate d from 100-nm-thin niobium film and consists of a 20- µm-wide centre conductor and 12- µm-wide separations between the centre conductor and ground plates, which yield the c haracteristic impedance of 50 Ω on 400- µm-thick highly resistive silicon substrate. Each sample is mounted to t he cavity in an independent experimental run to avoid overlapping signa ls from different sam- ples. To couple to the cavity modes, the samples are closely fitted to the cavity so that the polished silicon surface faces the cavity structure and fixed by GE v arnish (Extended Data Fig. 5a). A cavity chip together with a sample is mounted to the lowest tempera ture stage of a dilution refrigerator with EPR experimental set up as shown in Exten ded Data Fig. 5b. Extended Data Fig. 5c shows a typical transmission spectrum thro ugh the refrigerator with thecavitycooleddownto ∼25mKatzeromagneticfield. Cavityresonancemodesappearat each∼2.1 GHz. For EPR experiments, we use the cavity mode at ∼6.3 GHz (green arrow), which has a linewidth of ∼1 MHz and a quality factor of ∼6,000. The magnetic field B0is applied parallel to the resonator surface to maintain the resonato r’s quality factor, which is more than 3,000 at the magnetic fields used for experiments. The [11 0] axis of the samples is nearly aligned to the magnetic field. For the strained sample, misalignm ent of the magnetic field with respect to the [110] axis is approximately 10 deg .(Extended Data Fig. 5a). In the spin-echo measurements, output from the refrigerator is de modulated into quadrature signalsI(t) andQ(t), and recorded by a digitiser. Spins are manipulated by microwave pulses modulated by the modulation block containing microwave switch es and an IQ mixer controlled by arbitrary waveform generators. ( π/2)Xand (π)Ypulses used in this work are microwave modulated by 300-ns-long square pulse without phas e shift and 600-ns-long 17square pulse with phase shift of π/2, respectively. Extended Data Fig. 5d displays a typical output signal in the form of time-domain amplitude V(t) =/radicalbig I(t)2+Q(t)2after a standard Hahn-echo pulse sequence consisting of ( π/2)Xand (π)Ypulses and τ= 6µs. A spin-echo signal emphasised by red lines appears just after large pulsed signa ls corresponding to the (π/2)Xand (π)Ypulses, assuring that we can detect spin signal by using this experim ental setup. Strain analyses. A previous report shows that bulk silicon (silica) contracts by 0 .021 % (∼ −0.002 %) for each direction when it is cooleddown from293 K to 4 K [34]. Th is thermal expansion mismatch induces biaxial tensile strain of up to ∼0.023 % in the silicon layer of the stack structure used for the strained sample. Strain in the ac tual sample is less than this estimation and also distributed owing to the finite thickness of th e silicon layer. Based on a more sophisticated calculation of stress in a stack struc ture [35], we expect biaxial tensile stress of ∼30 MPa is induced to the silicon layer (Extended Data Fig. 6a). Assuming Young’s modulus 130 GPa and Poisson’s ratio 0 .28 along the [100] axis in silicon, we find that tensile biaxial strain of ∼0.017 % is induced to the silicon chip. Numerical simulation of the strain distribution in the stack structure also show s biaxial tensile strain of 0.032 % at the centre of the sample (Extended Data Fig. 6b). This simu lation also implies that, while strain is uniform around the centre of the silicon layer, de creases and eventually turns to compressive strain with approaching the sample edge. This strain variation induces difference between strain along xandyaxis and thus could result in the distribution in g∗value of boron spins in the strained sample (see the section Land´ e g-factor distribution induced by strain anisotropy ). We also perform X-ray diffraction analysis at low temperature by usin g an identical stack structure by usingnatSi wafer instead of28Si. While the silicon (400) diffraction angles taken from the stack structure and a reference silicon sample well coincid e at room temperature, they gradually move apart as temperature decreases (Extended Data Figs. 6c and d). This indicates that the silicon layer in the stack structure is contracted perpendicularly to the surface more than normal thermal contraction observed in the r eference sample. This extra contraction in the stack structure is associated with stretch alon g the silicon surface as expected. From these measurements, we estimate biaxial strain in the silicon layer of the stackstructureat0 .028%tensilewhichisconsistent withtheexpectationfromthediffere nce in thermal contraction. 18Detail of spin-echo measurements. In Hahn echo measurements, recorded signals I(t) andQ(t) are converted to an time-domain amplitude V(t) and subsequently integrated over a time span around the echo signal as shown by the red arrow in Exte nded Data Fig. 5d. Data plotted in Fig. 3 are taken by repeat of this process with chang ingτort′at certain |− →B0|andωMW/2π. For Fig. 2, we first take integrated amplitude signals with changing n ot only|− →B0|but also ωMW/2π, since the resonant frequency of the superconducting resonat or is slightly shifted as |− →B0|is changed. Extended Data Figs. 7a and b display typical inte- grated amplitude spectra taken with sweeping |− →B0|andωMW/2π(top panels), showing that microwave frequency providing spin-echo signal changes gradually with changing magnetic field. By integrating these data over ωMW/2π, we finally obtain spin-echo spectra as a func- tion of|− →B0|(bottom panels) as shown in Fig. 2. Note that, experiments for the relaxed and strained samples are carried out in different experimental runs , the cavity resonance and thus spin echo signal appear at slightly different frequencies be tween the relaxed and strained samples. In CPMG measurements, the time-domain echo sig nal appears after each of (π)Yand (−π)Ypulses as schematised in the inset of Fig. 4. By integrating each of th em over time independently, we obtain the CPMG-echo decay signal as a function of elapsed time after the first ( π/2)Xpulse. Since, in general, spin-echo signals that appear after ( π)Y and (−π)Ypulses are not equivalent, we plot echo-signals accompanying ( π)Ypulses only in Fig. 4 and use them for fitting. Extended Data Fig. 7c shows the fine structure of the boron spin- echo spectrum in the relaxed sample. The spectrum consists of a sharp line and backgrou nding broad line rather than simple single peak, indicating that two different spin transitions o ccur at almost the same magnetic field. This observation is reasonable for the relaxed s ample because the |mJ|=3 2↔1 2andmJ= +1 2↔ −1 2transitions (orange and blue arrows in the inset) have almost same transition frequency. As reported in Refs. 24–26, |mJ|=3 2↔1 2transition linewidth is inhomogeneously broadened by random strain in the cryst al more strongly than themJ= +1 2↔ −1 2transition. Hence, we attribute the broad spin-echo peak to the mJ= +3 2↔+1 2andmJ=−1 2↔ −3 2transitions, while the sharp peak to the mJ= +1 2↔ −1 2 transition. We fit a sum of two Gaussian functions f1(|− →B0|) +f2(|− →B0|) (red curve), where fn(|− →B0|) =Aexp{−(|− →B0| −Bc,n)2/δn2}(n= 1,2), to the data, obtaining a linewidth δnof 3.7 mT (1.0 mT) for the broader (sharper) spin-echo peak as shown by the o range (blue) curve. These linewidths well coincide with previously measurements in relaxed28Si:B [25]. 19We note that, while28Si:B resonance lines are reported to be well fitted by Lorentzian functions [25], Lorentzian fit of our peak shape implies unrealistic the rmal population of hole spin states and thus we use Gaussian functions for fitting. Land´ e g-factor distribution induced by strain anisotropy .As implied by Extended Data Fig. 6b, strain in the stack structure for the strained sample is not perfectly biaxial (εxx=εyy) but strain along the [100] and [010] axes can be different ( εxx/ne}ationslash=εyy). Extended Data Fig. 8a shows eigenenergy spectra of an acceptor-bound ho le for three different sets ofεxxandεyywith same εxx+εyy: (εxx,εyy) = (0.02 %,0.02 %), (0 .025 %,0.015 %) and (0.015 %,0.025 %). A marked difference appears in the magnetic field dependenc e, while energy splitting at zero magnetic field shows only a little change. The m agnetic field de- pendence in the low magnetic field regime (Extended Data Fig. 8b) is re levant to this work, characterised by the effective g-factor g∗. We calculate g∗value from such energy spectra and plot them as a function of εxxandεyyas shown in Extended Data Fig. 8c. Here we assume− →B0is not perfectly aligned to the [110] axis but misaligned by ∼10 degrees. The g∗value is changed by ∼0.1 when strain anisotropy ( εxx−εyy)/2(εxx+εyy) is∼20 %. Since Extended Data Fig. 6b implies that strong strain anisotropy ap pears near the edges of the silicon layer, the observed distribution in g∗value is presumably attributed to strain anisotropy in the real sample. Magnetic field used to measure data in Figs. 3 and 4. While the |mJ|=3 2↔1 2 transitions have electric dipole moment as mentioned in the main text, themJ= +1 2↔ −1 2 transitiondoesnotinrelaxedsilicon. Todiscusseffectofelectricdipo lemomentincoherence, we need to address the mJ= +3 2↔+1 2transition without exciting the mJ= +1 2↔ −1 2 transition. By measuring at |− →B0|= 384.4 mT (black arrow in Extended Data Fig. 7c) and ωMW/2π= 6.255GHz, we eliminate signal attributed to the mJ= +1 2↔ −1 2transition from spin-echo decay measurements. In addition the spin temperature is estimated at ∼300 mK from ratio of the Hahn-echo intensity between the |mJ|=3 2↔1 2transitions and the mJ= +1 2↔ −1 2transition. This assures that the thermal population of the/vextendsingle/vextendsingle3 2,−1 2/angbracketrightbig state is quite small and thus the mJ=−1 2↔ −3 2transition does not contribute to the observed Hahn-echo signal. In contrast, spin-echo decay in the strained sa mple does not show clear dependence in magnetic field (Extended Data Figs. 9a-c). We also co nfirmed that the CPMG decay is also not influenced to a peak-like substructure aroun d|− →B0|= 175.7 mT in thespinecho spectrumasshowninExtended DataFigs.9d-i. While T2CPMG’sobtainedfrom 20these measurements are slightly shorter than presented in the ma in text, this is presumably attributed to the signal to noise ratio in Extended Data Figs. 9d-i lo wer than Fig. 4. Electric dipole moment and coupling to phonons. The effective Hamiltonian that describes the response of the qubit to electric fields− →Eis ˆHqbt=1 2/planckover2pi1(ω0+ω)ˆσZ+/summationdisplay α=X,Y/planckover2pi1Ωαˆσα, /planckover2pi1ω≡− →χ(− →E)·− →E=/summationdisplay i=x,y,z/parenleftBigg χi+/summationdisplay j=x,y,zχijEj/parenrightBigg Ei, (2) /planckover2pi1Ωα≡− →vα(− →E)·− →E=/summationdisplay i=x,y,z/parenleftBigg vαi+/summationdisplay j=x,y,zvαijEj/parenrightBigg Ei. (3) Here,ω0is the qubit Larmor precession frequency without electric fields. ωis the change in Larmor frequency with electric fields. Fluctuations of ω,δω=− →χ(− →E)·δ− →E, induced by electric field fluctuations δ− →Ecause decoherence. Ω αis the frequency of nutation around the α=X,Yaxes, enabling quantum manipulations of the qubit. The effective elec tric dipole moment of the qubit is split to longitudinal component− →χ(− →E) and transverse component − →vα(− →E) in accordance with the effect in ωand Ω α, respectively. We note that the XYZ coordinate system, in general, coincides neither the real-space xyzcoordinate nor the simple rotating frame coordinate commonly used to discuss electron spin r esonance. Finite strain deviates the spin precession trajectory of holes from the plane pe rpendicular to the magnetic field, requiring complicated rotations to obtain a frame where the ho le spin is static. We have calculated χi,χij,vαiandvαijin Eqs. (2) and (3) for the charge-like subsystem {/vextendsingle/vextendsingle3 2,+3 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig }where ∆ = 0 and for the generalised spin {/vextendsingle/vextendsingle3 2,+1 2/angbracketrightbig ,/vextendsingle/vextendsingle3 2,−1 2/angbracketrightbig }where ∆ > /planckover2pi1ω0using the Schrieffer–Wolff transformation described in ref. 20. For relaxed Si:B (∆ = 0) subjected to a magnetic field in the (001) plane , the following non-zero electric dipole matrix elements are obtained by directly eva luating elements of 21Eq. 1: χzEz=√ 3sin(2θ0)/parenleftBig pEz/parenrightBig , χ xyExEy=√ 3sin(2θ0)/parenleftBig dExEy/parenrightBig , χxxE2 x=/parenleftBig −3cos(2θ0)−1/parenrightBig1 2bE2 x, χyyE2 y=/parenleftBig +3cos(2 θ0)−1/parenrightBig1 2bE2 y, χzzE2 z=−bE2 z, vXxEx= sin(θ0)/parenleftBig pEx/parenrightBig , v XyzEyEz= sin(θ0)/parenleftBig dEyEz/parenrightBig , vXyEy= cos(θ0)/parenleftBig pEy/parenrightBig , v XxzExEz= cos(θ0)/parenleftBig dExEz/parenrightBig , vYzEz= cos(2θ0)/parenleftBig pEz/parenrightBig , v YxyExEy= cos(2θ0)/parenleftBig dExEy/parenrightBig , vYxxE2 x=√ 3 2sin(2θ0)/parenleftBig bE2 x/parenrightBig , vYyyE2 y=−√ 3 2sin(2θ0)/parenleftBig bE2 y/parenrightBig .(4) Hereθ0is the angle of magnetic field to the [100] direction. pis the T dsymmetry linear coupling to electric fields and dis the cubic symmetry second-order couplings to electric fields, which both introduce spin-orbit coupling and are equivalent to those in the original acceptor Hamiltonian ˆH′defined by Eq. (1). The longitudinal relaxation rate T1−1is obtained by using Fermi’s golden rule together with the Bir–Pikus deformation potential b′andd′[24, 31, 32] T1−1=(/planckover2pi1ω0)3 20πρ/planckover2pi14/bracketleftBig b′2sin2(2θ0)/parenleftBig2 v5 l+3 v5 t/parenrightBig +d′2(1+cos2(2θ0))/parenleftBig2 3v5 l+1 v5 t/parenrightBig/bracketrightBig , while for θ0=π/4 as in the experiment we have T1−1=(/planckover2pi1ω0)3 20πρ/planckover2pi14/bracketleftBig b′2/parenleftBig2 v5 l+3 v5 t/parenrightBig +d′2/parenleftBig2 3v5 l+1 v5 t/parenrightBig/bracketrightBig . (5) Here,ρ= 2990 kg /m3is the mass density of silicon and vl= 8.99×103m/s (vt=vl/1.7) is the longitudinal (transverse) speed of sound in silicon crystal. Whe n the gap ∆ dominates /planckover2pi1ω0we obtain the following non-zero electric dipole matrix elements to lowe st order in 22/planckover2pi1ω0/2∆ using a Schrieffer–Wolff transformation: χzEz=√ 3/planckover2pi1ω0 2∆sin(2θ0)/parenleftBig pEz/parenrightBig , χ xyExEy=√ 3/planckover2pi1ω0 2∆sin(2θ0)/parenleftBig dExEy/parenrightBig , χxxE2 x=√ 3/planckover2pi1ω0 2∆cos(2θ0)/parenleftBig −√ 3bE2 x/parenrightBig , χyyE2 y=√ 3/planckover2pi1ω0 2∆cos(2θ0)/parenleftBig√ 3bE2 y/parenrightBig , vXxEx=√ 3/planckover2pi1ω0 2∆sin(θ0)/parenleftBig pEx/parenrightBig , v XyzEyEz=√ 3/planckover2pi1ω0 2∆sin(θ0)/parenleftBig dEyEz/parenrightBig , vXyEy=√ 3/planckover2pi1ω0 2∆cos(θ0)/parenleftBig pEy/parenrightBig , v XxzExEz=√ 3/planckover2pi1ω0 2∆cos(θ0)/parenleftBig dExEz/parenrightBig , vYzEz=√ 3/planckover2pi1ω0 2∆cos(2θ0)/parenleftBig pEz/parenrightBig , v YxyExEy=√ 3/planckover2pi1ω0 2∆cos(2θ0)/parenleftBig dExEy/parenrightBig , vYxxE2 x=√ 3/planckover2pi1ω0 2∆sin(2θ0)/parenleftBig√ 3 2bE2 x/parenrightBig , vYyyE2 y=√ 3/planckover2pi1ω0 2∆sin(2θ0)/parenleftBig −√ 3 2bE2 y/parenrightBig .(6) All other terms are zero to linear order in /planckover2pi1ω0/∆. Please note that the bJ2 zE2 zterm inH′ E does not contribute to the longitudinal dipole that causes decoher ence, that is χzzE2 z= 0, in contrast to the relaxed case. This fact holds to any order n >1 in (/planckover2pi1ω0/∆)nfor the generalised spin two-level system because J2 zis diagonal in the J=3 2basis and it is the identity matrix in the generalised spin subspace. Also note that this m eans that logic gates implementing a periodic transverse coupling− →vα(t) using a periodic modulation of Ez(t) would not cause a shift in Larmor frequency during the gate, only an oscillating component that could easily be made to average to zero with no effect on qubit co herence. This is convenient since a sinusoidal control field Ez(t) could be implemented with a top gate to control the transverse dipole. The longitudinal relaxation rate T1−1for ∆>/planckover2pi1ω0is calculated by projecting the elastic interactions into the qubit subspace by a Schrieffer–Wolff transfor mation. We obtain the result: T1−1=(/planckover2pi1ω0)3 20πρ/planckover2pi14/parenleftBig/planckover2pi1ω0 ∆/parenrightBig2/bracketleftBigb′2(72sin2(2θ0)+9) 128/parenleftBig4 3v5 l+2 v5 t/parenrightBig +d′2(24cos2(2θ0)+33) 64/parenleftBig2 3v5 l+1 v5 t/parenrightBig/bracketrightBig , while for θ=π/4 like in the experiment we have T1−1=(/planckover2pi1ω0)3 20πρ/planckover2pi14/parenleftBig/planckover2pi1ω0 ∆/parenrightBig2/bracketleftBig81b′2 128/parenleftBig4 3v5 l+2 v5 t/parenrightBig +33d′2 64/parenleftBig2 3v5 l+1 v5 t/parenrightBig/bracketrightBig . (7) 23c d46.33 6.34 6-60-40-20-10 -30 8Frequency (GHz)S21 (dB)B0 = 0 T Q~6,000 bDilution refrigerator CPWG Resonator SampleDigitiser20 3 6Modulation blockDemodulation blockCW in LO inOutI ch. Q ch. Pulse outSignal in 4 K 2 K 50 mKStill 25 mKNbTi coax.NbTi coax.MW synthesiser Modulation block AWG 10106 AWG CW in Pulse out 3 Demodu lation blockLO inOutI ch Q ch. Signal inxLow pass filter High pass filter x-dB attenuator DC blockSplitter Isolator IQ mixer Microwave switchAmplifier B0Integrate 0 100.1110100 20 Time (µs)Ampl itude (mV)(π/2)X (π)YEcho Timea a Fused silica28Si:B [110] B0 ~104.0 mm x 3.5 mm Fig.5.a,Photo image of the strained28Si:B sample on the niobium coplanar waveguide resonator. b,Schematic figure of the experimental setup. c,Microwave transmission spectrum of the experi- mental setup measured at zero magnetic field by a network anal yser. The sold arrow indicates the resonance mode used for spin-echo experiments. Inset: Deta iled transmission spectrum around the resonance used for spin-echo experiments. d,Typical output signal for input of a Hahn-echo pulse sequence in time domain. Taking ratio between Eqs. (5) and (7) and substituting parameter s other than /planckover2pi1ω0and ∆, we obtain the relation /planckover2pi1ω0/2∆ = 0.765/radicalbig T1,Relaxed/T1,Strained. Substituting T1,Relaxed= 85µs andT1,Strained= 5 ms to this equation, we obtain /planckover2pi1ω0/2∆≈1 10as shown in the main text. We note that, while this estimation is made on the assumption /planckover2pi1ω0/∆≪1, well coincides with a result obtained by exact diagonalisation approach with an accu racy of/lessorsimilar1 %. 24bc 0 100 200 300Diffraction angle 2 θ (deg.) Temperature (K)Si/Epoxy/SiO 2 Si reference 69.15 69.2 69.25 Copper K α1 0.019 deg.0.017 deg. 69.184 degdDiffraction angle 2 θ (deg.) Temperature (K)0 100 200 300Si/Epoxy/SiO 2 Si reference 69.35 69.4 69.45 Copper K α2 0.019 deg.0.016 deg. 69.381 dega Depth from silicon surface ( µm)20 10 -10 0 100 200030Stress a long int erfac e (MPa)28Si:B Fused silica Tensile stress in Si Biaxial strain 0.03 0.01 0 -0.04 -1 10 00.1 -0.1 -0.2 -2 20.04 -0.01 -0.03 Normal stratin along x (%)z (mm) x (mm)Silicon Fused silica Fig.6.a,Theoretical analysis of the stress distribution. b,Numerical analysis of the strain ( εxx) distribution. c,d,Temperature dependence of X-ray diffraction angle of Si(004) measured by using the copper K α1 emission line ( c) and the K α2 emission line ( d). 25Magnetic field (mT)200 300 400 160 180 200 2201.2 0.8 02 02040606.256.266.27 6.3266.3286.3306.332Amplit ude ( µV)Integra ted amplit ude ( mV)Integra ted amplitud e (µV) Microwave frequency (GHz) 250 0τ = 5 µsa c cbRelaxed Si Strained Si Amplit ude ( µV) 100 20τ = 50 µs 384.4 mT175.7 mT 380 384 388τ = 10 µs Fig.7.a,b,Spin-echo spectra as a function of magnetic field and microwa ve frequency (top panels) for the relaxed ( a) and strained samples ( b). Each data point is obtained by integrating time domain signals as shown in Fig. 5d. For comparison, spin -echo spectra same as Figs. 2b and c, obtained by integrating data in the top panels along the mi crowave frequency axis, are shown in the bottom panels. c,Detailed spin-echo spectrain thebox in the bottom panel of a measured with τ= 10µs. The black arrows in the c and the bottom panel of b indicate t he magnetic field used to measure T2andT1in the relaxed (384.4 mT) and strained samples (175.7 mT), re spectively. The red curve shows the fitting function, which is composed of two Gaussian functions shown by the blue and orange curves. Inset: Level spectrum of light-h ole and heavy-hole states in relaxed Si:B. Blue and orange arrows show transitions that provide s pin-echo spectra fitted by the same coloured curves in the main panel. 26a b 0.0 0.25 0.5 Magnetic field (T) Magnetic field (T)0.0 5.0 100.0 -8.08.0Energy (GHz)0.0 -200200400 |g*|µB|B0|εxx = εyy = 0.02 %εxx = 0.025, εyy = 0.015 % εxx = 0.015, εyy = 0.025 % 2.6 2.0g* valuec Strain along x (%)0.002 0.01 0.02 2.72.62.52.42.32.2 2.1 g* = 2.0Strain along y (%) 0.0020.010.02B0 10 -misaligned from [110] in the (001) plane Fig.8.a,b,Eigenenergy spectra as a function of magnetic field for three different sets of εxx andεyywith same εxx+εyy: (εxx,εyy) = (0.02 %,0.02 %) (black), (0 .025 %,0.015 %) (red) and (0.015 %,0.025 %) (green). We estimate g∗value from the magnetic field dependence of level splitting as shown in b. c,εxxandεyydependence of g∗for magnetic field of 175.7 mT misaligned by 10 deg. from the [110] axis in the (001) plane. 27a c b d f e g i h160 180 200 220 m T 177.4 mT174.0 mT182.0 mT T2CPMG = 8.2 ± 0.5 ms T2CPMG = 7.8 ± 0.4 ms Time (ms)2τ (ms) 0 10 200.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0 10 20 0 10 20 0 10 20 0 10 20176.3 mT T2CPMG = 8.1 ± 0.4 ms175.7 mT T2CPMG = 8.2 ± 0.4 ms0 10 20175.1 mT T2CPMG = 7.9 ± 0.3 ms174.6 mT T2CPMG = 8.8 ± 0.4 msT2H = 0.97 ± 0.02 ms , β = 3.0171.7 mT T2H = 0.78 ± 0.03 ms , β = 2.9190.5 mT T2H = 0.93 ± 0.02 ms , β = 3.5Amplitude ( µV) CPMG measurements1060 Fig.9.a-c,Hahn-echo decay measurements at three different magnetic fiel d.d-i,CPMG-echo decay measurements at six different magnetic field. Magnetic fi eld used for these measurements are indicated by symbol corresponding to each of them in a spi n-echo spectrum in the top panel. 28

1107.0578v1.Anomalous_Hall_conductivity_from_the_dipole_mode_of_spin_orbit_coupled_cold_atom_systems.pdf

Anomalous Hall conductivity from the dipole mode of spin-orbit-coupled cold-atom systems E. van der Bijl and R.A. Duine Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands (Dated: October 25, 2018) Motivated by recent experiments [Lin et al. , Nature 417, 83 (2011)] that engineered spin-orbit coupling in ultra-cold mixtures of bosonic atoms, we study the dipole oscillation of trapped spin- orbit-coupled non-condensed Bose and Fermi gases. We nd that dierent directions of oscillation are coupled by the spin-orbit interactions. The phase dierence between oscillatory motion in orthogonal directions and the trapping frequencies of the modes are shown to be related to the anomalous Hall conductivity. Our results can be used to experimentally determine the anomalous Hall conductivity for cold-atom systems. PACS numbers: 05.30.Fk, 03.75.-b, 67.85.-d,71.70.Ej Introduction. | Transport phenomena play a cru- cial role in understanding and characterizing condensed- matter systems. Two of these phenomena, the Hall eect and the anomalous Hall eect (AHE) were both discov- ered in the late 19thcentury. That the Hall eect is due to the Lorentz force has been understood since those days. The AHE, a transverse voltage or current present in fer- romagnets in the absence of a magnetic eld, is related to spin-orbit (SO) coupling and has proven much more chal- lenging to understand (for a review see Ref. [1]). Since SO coupling is responsible for the AHE, anomalous-Hall- like eects should also be present for particles that do not carry charge, and, indeed, such eects are observed for magnons [2], phonons [3, 4], and photons [5]. Al- though these eects were observed using heat currents, and they should thus be called anomalous Righi-Leduc eects, their physical mechanism is similar to that of the AHE. In this Letter we consider the AHE in homogeneous and harmonically-trapped cold-atom systems. (The AHE was considered in cold-atom systems in the presence of an optical lattice in two dimensions by Dudarev et al. [6].) As the atoms are neutral, the AHE here refers to a mass current perpendicular to an applied force in the absence of a Coriolis force. (For cold-atom systems rotation and the resulting Coriolis force play the role of a magnetic eld and the Lorentz force.) Our investigation is mo- tivated by the recent experiment by Lin et al. [7] who engineered spin-orbit coupling in a Bose-Einstein con- densate [8, 9] with lasers. This experiment is one of the latest achievements in studying phenomena known from solid-state physics in a cold-atom setting. Other exam- ples are the Mott-insulator-to-super uid phase transition [10], Bardeen-Cooper-Schrieer super uidity [11], the Berezinskii-Kosterlitz-Thouless phase transition [12, 13], and Anderson localization [14]. Important features of cold-atom systems are that new regimes of physics (as compared to solid-state systems) can be explored, and their great amount of tunability. Furthermore, cold-atom systems are in principle disorder free and have a well-known microscopic description making it worthwhile to undertake a detailed comparison between theory and experiment, whereas in solid-state materials typically a multitude of eects play a role which makes modeling harder. In the case of the AHE, for example, the diculty in understanding the eect lies in part in the inter- play between so-called intrinsic andextrinsic contribu- tions. Intrinsic contributions come from spin-orbit cou- pling eects in the bandstructure, whereas extrinsic con- tributions arise from disorder. A recent theoretical ad- vancement in the understanding of the AHE is the semi- classical description in terms of equations of motion for Bloch wavepackets [15, 16]. In this description the in- trinsic contribution to the AHE stems from anomalous- velocity contributions to these semi-classical equations of motion [17]. In modern language, this anomalous velocity results from the Berry-phase curvature of the Bloch bands that in turn is determined by the topology of the band structure. The relation between bandstructure topology and the Hall conductivity was rst emphasized by Thouless et al. [18], and has regained interest with the very recent discovery of topological insulators [19]. In a typical cold-atom experiment steady-state cur- rents are not readily created and transport coecients can be measured only indirectly. In this Letter we show that the anomalous Hall conductivity can be obtained from the properties of the dipole oscillation of a cloud of spin-orbit coupled cold atoms that is trapped in an ex- ternal harmonic trapping potential. The dipole mode is a collective oscillation of the center-of-mass of the cloud. According to Kohn's theorem [20], the frequencies of the dipole oscillation are equal to the trap frequencies. The SO coupling, however, breaks the harmonic nature of the system and as a result Kohn's theorem for the dipole modes does not hold. We nd that spin-orbit coupling modies the oscillation frequencies and that dierent di- rections of oscillation are coupled by the spin-orbit in-arXiv:1107.0578v1 [cond-mat.quant-gas] 4 Jul 20112 teractions. The phase dierence between oscillatory mo- tion in dierent directions and the mode frequencies turn out to be related to the anomalous Hall conductivity. This result can be used to experimentally determine the anomalous Hall conductivity for cold-atom systems. Be- low we detail the semi-classical Boltzmann approach on which our ndings are based, determine the anomalous Hall conductivity for homogeneous non-condensed Bose and Fermi gases, and show how this conductivity can be obtained from the dipole oscillation of trapped atomic gases. Semi-classical equations of motion | We consider spin-1=2 atoms with mass mtrapped in an external po- tentialVex(x) in the presence of a generic spin-orbit cou- pling. The Hamiltonian is ^H=^p2 2m+Vex(^x)M(^p)
; (1) with ^pand ^xthe momentum and position operators of the particles and the vector of Pauli matrices. The last term describes the SO coupling, that for spin one- half particles is without loss of generality given in terms of a momentum-dependent eective magnetic eld M. At the semi-classical level we consider in rst instance the dynamics of the expectation values of the position x=h^xi, momentum, p=h^pi, and spins=~hi=2 degrees of freedom. We obtain the Heisenberg equations of motion _x=p m2 ~@M @p
s; (2) _p=@Vex @x; (3) _s=sM ~: (4) We proceed by assuming that the spin degree of freedom is much faster than the motion of the particles. Thus we let the spin follow the eective magnetic eld Madi- abatically, and only allow for a small misalignment be- tween the spin and the eective magnetic eld that is rst order in time-derivatives of the orbital dynamics. This approach is essentially exact in the linear-response regime. Hence, we solve the equation for the spin degree of freedom Eq. (4) up to rst order in time-derivatives bys/P im+~ jMj(m@m @pi)
_pi, withm(p(t)) the unit vector in the direction of M. For spins opposite to the eld the result is s. Insertion of the result for sinto Eq. (2) gives [15, 16] _xk=@p;k @p+k_pB(p); (5) where the band index kdistinguishes between atoms with spin parallel (+) or antiparallel ( ) to the eldM. Furthermore, the dispersion is given by p;k=p2=2mkjM(p)j, and the vector eldBc(p) = ~P a;b2fx;y;zgabc(@m=@pa@m=@pb)
m=2 determines the anomalous velocity contribution. We re- fer toBas the Berry magnetic eld. Boltzmann equation and anomalous Hall conductivity. |We proceed by calculating the anomalous Hall conduc- tivity for a homogeneous gas from the Boltzmann equa- tion for the distribution function fk(x;p;t) for atoms in bandk, that is given by @fk @t+_xk
@fk @x+_p
@fk @p= 0; (6) where we ignored collisions as the intrinsic anomalous Hall conductivity does not depend on relaxation [1]. In the above, _xkand _pare given by Eq. (5) and Eq. (3) respectively. We consider a steady-state situation with a constant applied force F=@Vex=@xacting equally on atoms in both bands, and dene the conductivity tensor byj=
F, wherejis the particle current density which is given by jP kRd3p (2~)3fk(p)_xk. The anoma- lous Hall conductivity AHis the o-diagonal component of this conductivity tensor. The solution of the Boltz- mann equation leads to the anomalous Hall conductivity AH=X k2f+;gZd3p (2~)3kN(p;k)Bz(p); (7) whereN() = [e()=kBT1]1withkBTthe thermal energy and the chemical potential, is the Fermi-Dirac (+) or Bose-Einstein ( ) distribution function that ap- plies for fermions or bosons, respectively. The above ex- pression for the anomalous Hall conductivity is the intrin- sic contribution due to SO coupling eects in the band structure. In cold-atom atom systems there is, unless en- gineered [14], no disorder and thus extrinsic contributions are absent. So far we have considered a generic SO coupling. In order to make a connection with experiments we will now consider a Rashba-Dresselhaus [21] form of the SO cou- pling so that the eective magnetic eld reads M(p) = ~py ~px; ~px+ ~py;
2T (8) whereandare the coupling constants for Rashha and Dresselhaus SO-coupling respectively, and
is a spin- splitting energy. We then nd for the Berry magnetic eld B(p) =4(22)
~2 (
2~2+ 4(2+2)p216pxpy)3=2^z :(9) Note that in the experiments by Lin et.al. [7] an equal amount of Rashba and Dresselhaus coupling was realized, i.e.,=. It follows that in this specic case B(p) = 0 and thatAH= 0 [22]. It is however experimentally straightforward to consider a more general SO coupling3 024680.000.050.100.150.200.250.30 nL3LÑsAH FIG. 1. (Color online) The anomalous Hall conductivity for bosons (solid) and fermions (dashed) as a function of n3. The lines correspond from top to bottom to == (1;2;1:5) where2+2=m=~2. The spin splitting energy
= 0:2kBT. [7]. The anomalous Hall conductivity vanishes in the absence of a spin splitting
, in agreement with the fact that the AHE occurs in ferromagnets. In Fig. 1 we show results for the anomalous Hall con- ductivity of bosons and fermions. We only show the re- sults for >  sinceAH(;) =AH(;) and AH(;) =AH(;). The results shown for bosons are above the critical temperature for Bose-Einstein con- densation (this temperature depends on ; and
). The anomalous Hall conductivity is independent of tem- perature in the degenerate ( n31) limit for fermions, withnthe density and  (2~2=mk BT)1=2the de Broglie wavelength, as expected. Collective Modes. | We now study the dipole oscilla- tion of an atomic cloud of Naatoms in an anisotropic harmonic trapping potential of the form Vex(x) = m 2 !2 r(x2+y2) +!2 zz2
, with!rand!zthe trapping fre- quencies. The dipole ( l= 1) oscillations are pure transla- tions of the cloud with no changes in its internal structure that are described by the equations of motion for the cen- ter of mass position x01 NaP kR dxR dp=(2~)3fkxk and velocityv01 NaP kR dxR dp=(2~)3fk_xk. Hence, we make the following ansatz for the distibution func- tion,fk(x;p;t) =nk(xx0(t);pmv0(t)), where nk(x;p) =N(p;k(x)) is the Bose-Einstein or Fermi- Dirac distribution function in the local-density approx- imation, with (x) =Vex(x). From the Boltz- mann equation we obtain the equations of motion for the center-of-mass coordinates. For small oscillations, we linearize these equations of motion resulting in _x0=H
v0rVex(x0)B; (10) m_v0=rVex(x0) (11) where His proportional to the Hessian matrix of the 0.00.51.01.52.02.53.00.700.750.800.850.900.951.00 amÑ2Lw+wr 0.51.01.52.02.53.0amÑ2L 0.000.010.020.030.04sinj+FIG. 2. (Color online) Dipole mode frequencies as given by Eq. (12). The solid line is calculated for = 0, the dashed lines correspond to = 0:5m=~2. The spin splitting is
= 0:2kBT. The number of particles Na= 1:8105and temperature T= 200 nK. The inset shows the sin 'as a function of  dispersion and Bis the Berry magnetic eld averaged over the trap, which are given by B=1 NaX k2f+;gZ d3xZd3p (2~)3knk(x;p)B(p); Hij=1 NaX k2f+;gZ d3xZd3p (2~)3nk(x;p)Hij;k(p); withHij;km@2p;k=@pi@pj. For the mode in the z- direction we obtain the result !=!zas predicted by Kohn's theorem and expected since the SO coupling only aects the dynamics in the xyplane. The modes in this plane have frequencies given by !=!rr Aq (H2xyHxxHyy) +A2; (12) withA= (Hxx+Hyy+B2 zm2!2 r)=2. When there is no SO coupling,;= 0 we nd !=!ras predicted by Kohn's theorem. For nonzero SO coupling the double degeneracy of this mode is lifted. The eigenmodes of oscillation are given byx 0(t) = (x1sin(!t+');x2sin(!t))Twith sin'=mBz!=(H2 xy+m2B2 z!2 )1=2. In Fig. 2 we show the mode frequency !+and angle'+as a function of, in the special case =we have Bz= 0 and nd '= 0 as shown in the inset. Another special case occurs when the SO coupling is of the pure Rashba or Dressel- haus form. Then Hxyvanishes which results in '==2. The phase dierence between the two dierent directions of oscillations is determined by the Berry magnetic eld. We can relate the average of the Berry magnetic eld over the trap to the anomalous Hall conductivity for a homo- geneous gas with a density equal to the central density n0 of the trapped cloud Bz'AH(n0)=n0, whereAH(n0)4 is given by Eq. (7). This conductivity can therefore be experimentally determined by measuring the frequencies !or the phase dierences 'of the modes. Discussion and conclusions | We have studied the dipole oscillation of a trapped gas of spin-orbit coupled cold-atoms, and found that these oscillations can be used as an experimental probe for the anomalous Hall ef- fect. In the experiments by Lin et.al. [7] the SO coupling strength is ;'m=~2and the Zeeman spin-splitting is
'0:2kBT, using a temperature of T'200 nK. Taking values for andof this order, we nd that the angle'+'0:03 and that ( !!r)=!r'10% which appear to be observable. Up to this point we have not considered collisions be- tween the atoms leading to damping of collective oscil- lations. The harmonic nature of our system is explicitly broken by the SO coupling leading to relaxation of the center-of-mass motion of the cloud (such relaxation is ab- sent when the SO coupling is zero and Kohn's theorem prevails). This can be described phenomenologically by adding a termv0=on the right-hand-side of Eq. (11) which would lead to damping of the dipole modes but does not aect the anomalous Hall conductivity. The frequencies of the damped system are =!+i  with the damping rate , up to rst order in 1 =, given by =1 !2 +Hxx!2 r 2!2 + 2Hxx!2r+B2zm2!4r1 : We note that the relaxation time can in principle be calculated from the Boltzmann equation but considering this, given the above remarks regarding its importance, is beyond the scope of the present paper. In the adiabatic approximation that leads to the semi- classical equations of motion, spin directions transverse to the magnetic eld M(p) are taken into account ap- proximately as they give rise to the anomalous veloc- ity terms. One could go beyond this adiabatic approx- imation and consider the (2 2)-distribution function f0(p) that allows for all possible spin directions. We have checked, by solving the Boltzmann equation for this distribution function in the collisionless limit, that our re- sults for the anomalous Hall conductivity and the phases 'are not altered. Possible extensions of this work are to consider the partially Bose-Einstein condensed phase for bosons, and the situation without Zeeman spin splitting
. In the latter case the AHE is absent, but there will be a spin Hall eect [23, 24] that can be probed via the spin-dipole mode. (The spin Hall eect for cold atoms was proposed by Zhu et al. [25] for a cloud falling due to gravita- tion.) We also intend to investigate the eects of spin- orbit coupling on other collective modes, in particular the quadrupole oscillation. We would like to thank Henk Stoof for carefullyreading the manuscript. This work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), the Netherlands Organization for Scientic Re- search (NWO), and by the European Research Council (ERC). [1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010). [2] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Science 329, 297 (2010). [3] C. Strohm, G. L. J. A. Rikken, and P. Wyder, Phys. Rev. Lett. 95, 155901 (2005). [4] L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 96, 155901 (2006). [5] M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004). [6] A. M. Dudarev, R. B. Diener, I. Carusotto, and Q. Niu, Phys. Rev. Lett. 92, 153005 (2004). [7] Y. J. Lin, K. Jim enez-Garc a, and I. B. Spielman, Nature 471, 83 (2011). [8] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A 78, 023616 (2008). [9] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett. 105, 160403 (2010). [10] M. Greiner, O. Mandel, T. Esslinger, T. H ansch, and I. Bloch, Nature 415, 39 (2002). [11] M. Greiner, C. Regal, and D. Jin, Nature 426, 537 (2003). [12] Z. Hadzibabic, P. Kr uger, M. Cheneau, B. Battelier, and J. Dalibard, Nature 441, 1118 (2006). [13] V. Schweikhard, S. Tung, and E. A. Cornell, Phys. Rev. Lett. 99, 030401 (2007). [14] J. Billy, V. Josse, B. A. Zuo, Z., B. Hambrecht, L. P., D. Cl ement, L. Sanches-Palencia, P. Bouyer, and A. As- pect, Nature 453, 891 (2008). [15] M.-C. Chang and Q. Niu, Phys. Rev. Lett. 75, 1348 (1995); T. Jungwirth, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 88, 207208 (2002). [16] N. A. Sinitsyn, Journal of Physics: Condensed Matter 20, 023201 (2008). [17] R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954). [18] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). [19] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005); A. Roth, C. Br une, H. Buhmann, L. W. Molenkamp, J. Maciejko, X.-L. Qi, and S.-C. Zhang, Science 325, 294 (2009); M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [20] W. Kohn, Phys. Rev. 123, 1242 (1961). [21] E. I. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960); G. Dresselhaus, Phys. Rev. 100, 580 (1955). [22] J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003). [23] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003). [24] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung- wirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004). [25] S.-L. Zhu, H. Fu, C.-J. Wu, S.-C. Zhang, and L.-M. Duan, Phys. Rev. Lett. 97, 240401 (2006).

1701.04591v1.Vortex_pairs_in_a_spin_orbit_coupled_Bose_Einstein_condensate.pdf

arXiv:1701.04591v1 [cond-mat.quant-gas] 17 Jan 2017Vortex pairs in a spin-orbit coupled Bose-Einstein condens ate Masaya Kato,1Xiao-Fei Zhang,2,3and Hiroki Saito1 1Department of Engineering Science, University of Electro- Communications, Tokyo 182-8585, Japan 2Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China 3University of Chinese Academy of Sciences, Beijing 100049, China (Dated: September 18, 2018) Static and dynamic properties of vortices in a two-componen t Bose-Einstein condensate with Rashba spin-orbit coupling are investigated. The mass curr ent around a vortex core in the plane- wave phase is found to be deformed by the spin-orbit coupling , and this makes the dynamics of the vortex pairs quite different from those in a scalar Bose-E instein condensate. The velocity of a vortex-antivortex pair is much smaller than that without s pin-orbit coupling, and there exist stationary states. Two vortices with the same circulation m ove away from each other or unite to form a stationary state. I. INTRODUCTION Topological excitations in superfluids originate from the intertwining between internal and external degrees of freedom in the order parameters. The simplest exam- ple is a quantized vortex in a scalar superfluid, in which the complex order parameter with the U(1) manifold winds around the vortex core, producing azimuthal su- perflow[1, 2]. Forthe orderparameterswith spin degrees of freedom, a rich variety of topological excitations are possible; these include skyrmions [3, 4], monopoles [5], half-quantum vortices [6], and knots [7]. Because of the close relationship between the spin and motional degrees of freedom in the topological excitations, we expect that their static and dynamic properties are significantly al- tered if there exists coupling between them, that is, if there exists spin-orbit coupling (SOC). Recently, Bose-Einstein condensates (BECs) of ultra- cold atomic gases with SOC have been realized exper- imentally [8–12]; in these experiments, the atomic spin or quasispin was coupled with the atomic momentum us- ing Raman laser beams. Numerous theoretical studies have been performed to evaluate the static properties of topological excitations in spin-orbit (SO) coupled BECs, e.g., vortex arrays [13], vortices in rotating systems [14– 17], half-quantum vortices [18, 19], skyrmions [20–24], topologicalspintextures [25–29], dipole-induced topolog- ical structures [30–32], and solitons with vortices[33, 34]. However, there have been only a few studies on their dy- namics in SO-coupled BECs. The dynamics of a single quantized vortex in a harmonic trap was considered in Refs. [35, 36]. In this paper, we investigate the dynamics of a quan- tized vortex pair in a quasispin-1 /2 BEC with Rashba SOC. When a singly quantized vortex is created in a uni- form plane-wave state, the phase distribution around the vortex core is significantly altered by the SOC; this in- dicates that the mass current around the vortex is quite different from that without SOC and affects the dynam- ics of a vortex pair. As a result, a vortex-antivortex pair will be stationary or will travel much more slowly thanone without SOC. The dynamics of a vortex-vortex pair with the same circulation are also quite different from those without SOC; the vortices move away from each other, or they approach each other and unite to form a stationary state. This paper is organized as follows. The problem is formulated in Sec. II. The static properties of a single vortex are studied in Sec. III. The dynamics of a vortex- antivortexpair and those ofa vortex-vortexpair with the same circulation are investigated in Secs. IVA and IVB, respectively. Conclusions are presented in Sec. V. II. FORMULATION OF THE PROBLEM We consider a two-dimensional (2D) quasispin-1/2 BEC in a uniform system with Rashba SOC. Within the framework of mean-field theory, the system can be de- scribed by the order parameter Ψ(r) = [ψ1(r),ψ2(r)]T, whereTdenotes the transpose. The kinetic and SOC energies are given by E0[Ψ] =/integraldisplay drΨ†/parenleftbiggp2 2m−/planckover2pi1k0 mp·σ⊥/parenrightbigg Ψ,(1) wheremis the atomic mass, k0is the strength of the SOC, and σ⊥= (σx,σy) are the 2 ×2 Pauli matrices. Thes-wave contact interaction energy is written as Eint[Ψ] =/integraldisplay dr g0 22/summationdisplay j=1|ψj|4+g12|ψ1|2|ψ2|2 ,(2) where g0and g12are the intra- and inter-component in- teraction coefficients, respectively. The total energy is given by E[Ψ] =E0[Ψ]+Eint[Ψ]. (3) In this paper, we consider an infinite system in which the atomic density Ψ†Ψfar from vortices is a constant, n0. In the following, we normalize the length, veloc- ity, time, and energy by the healing length /planckover2pi1/√mg0n0,2 the sound velocity/radicalbig g0n0/m, the characteristic time scale/planckover2pi1/(g0n0), and the chemical potential g0n0. The dimensionless coupled Gross-Pitaevskii (GP) equations, i∂Ψ/∂t=δE[Ψ]/δΨ, have the form i∂ψ1 ∂t=−1 2∇2ψ1+iκ∂−ψ2+/parenleftbig |ψ1|2+γ|ψ2|2/parenrightbig ψ1,(4a) i∂ψ2 ∂t=−1 2∇2ψ2+iκ∂+ψ1+/parenleftbig γ|ψ1|2+|ψ2|2/parenrightbig ψ2,(4b) where∂±=∂/∂x±i∂/∂y,κ=/planckover2pi1k0/√mg0n0, and the ratiobetweentheinter-andintra-componentinteractions isγ= g12/g0. The ground state is the plane-wave state forγ <1 and the stripe state for γ >1 [13], which breaks the rotational symmetry of the system. In the following discussion, we will focus on the miscible case, γ <1, and the ground state is given by the plane-wave state, Ψ(r) =1√ 2/parenleftbigg eiκx eiκx/parenrightbigg , (5) where the wave vector is chosen to be in the xdirection. The velocity field is useful for understanding the dy- namics of vortices. From the equation of continuity ∂ρ/∂t+∇·(ρv) = 0withatomicdensity ρ=|ψ1|2+|ψ2|2, we obtain the velocity field as vξ(r) =1 2iρ(r)/bracketleftbig Ψ†(r)∇ξΨ(r)−Ψ(r)T∇ξΨ∗(r)/bracketrightbig −κSξ(r),(ξ=x,y) (6) where Sξ(r) =1 ρ(r)Ψ(r)†σξΨ(r)(ξ=x,y,z) (7) is the pseudospin density. The first term in Eq. (6) cor- responds to the canonical part related to the superfluid velocity, and the second term corresponds to the gauge part induced by the SOC. The velocity field vanishes for the vortex-freeground state in Eq. (5), since the first and second terms in Eq. (6) cancel each other. We numerically solve Eq. (4) by the pseudospectral method with the fourth-order Runge-Kutta scheme. In the imaginary-time propagation, on the left-hand side of Eq. (4),iis replaced with −1. The numerical space is taken to be 400 ×400, which is sufficiently large, and the effect of the periodic boundary condition can be ne- glected. III. SINGLE VORTEX Webeginwith asinglevortexstate, inwhicheachcom- ponent contains a singly quantized vortex. The initial state of the imaginary-time propagation is Ψ(r) =1√ 2/parenleftbigg ei[Φ(r)+κx] ei[Φ(r)+κx]/parenrightbigg , (8)02 -2 4 -4 1 -1 0(c) spin










 (d) dependence of 05 -5 0(b) phase 05 -5 (a) density 01 FIG. 1. (a)-(c) Stable stationary state of a single vortex wi th counterclockwise circulation for κ= 1 andγ= 0.8. Pan- els (a) and (b) show the density and phase profiles of each component, where the unit of density is n0. In (b),δis the distance between the phase defects in the two components. Panel (c) shows the spin distribution S(r) defined in Eq. (7). The arrows indicate the transverse direction of the spin vec - tor, and the background color indicates the value of Sz. The dashed square region in (a) is shown magnified in (c). (d) κ dependence of the vortex shift δ. The solid curve shows 1 /κ for comparison. where Φ( r) = tan−1(y/x). After sufficiently long imaginary-time propagation, we obtain the stable sta- tionary state, as shown in Fig. 1. Figures 1(a) and 1(b) show the density and phase distributions of the station- ary state. We note that the phase defect in component 1 (2) is shifted in the + y(−y) direction. We define the distance between the phase defects as δ. The vortex core in each component is occupied by the other com- ponent. This structure can therefore be regarded as a pair of half-quantum vortices; nevertheless, we will refer to it as a “single vortex” in this paper. In the absence of SOC, such a pair of half-quantum vortices repel each other and cannot form a stationary state [37]. A similar structure is also found in a one-dimensional SOC sys- tem [36]. Figure 1(c) shows the spin distribution, and we can see a spin vortex near the origin. The dependence of the vortex shift δon the SOC strength κis shown in Fig. 1(d), which implies δ≃1/κ. Figure 2 shows the velocity field v(r) of the single- vortexstate. Thevelocityfieldisgreatlydeformedbythe SOC, compared with the rotationally symmetric velocity field without SOC. We note that the deformation of the velocity field extends over a wide range, and the upper region (y>∼10) exhibits a uniformly leftward velocity field, while the lower region ( y<∼10) is rightward. In these regions, |v|<∼0.01, which is much smaller than that without SOC, |v|= 1/r. This effect of SOC is also seen in Fig. 1(b), where the phase in the upper and lower regions is almost ∝eiκx, i.e., the 2πphase rotation aroundthe vortexcoreisstronglycompressedaroundthe3 FIG. 2. Velocity field v(r) of the single-vortex state for γ= 0.8 and (a)κ= 0.5 and (b)κ= 1. The arrows indi- cate the directions of the velocity, and the background colo r indicates the value of |v|. The regions in the dashed squares are magnified in the right-hand panels. The red arrows indi- cate the direction of the vortex. x-axis. The velocity field near the vortex core exhibits complicated structures containing multiple circulations, as shown in the right-hand panels in Fig. 2. Due to the symmetry of the GP equation in Eq. (4), the single-vortex state with clockwise circulation can be obtained from that with counterclockwise circulation by the following transformation: ψ1(x,y)→ψ2(x,−y), ψ2(x,y)→ψ1(x,−y).(9) Bythistransformation,thewindingnumberofthevortex is inverted without changing the direction of the plane waveeiκx. Applying the transformation to the state shown in Fig. 1, we find that the vortex core in com- ponent 1 (2) shifts in the + y(−y) direction also for the clockwise vortex. The velocity field and the pseu- dospin density are transformed as vx(x,y)→vx(x,−y), vy(x,y)→ −vy(x,−y),Sx(x,y)→Sx(x,−y), and Sy(x,y)→ −Sy(x,−y). For a better understanding of the numerical results, we perform variational analysis. The variational wave function is Ψ(r) =1√ 2/parenleftbigg ei[Φ1(r)+κx] ei[Φ2(r)+κx]/parenrightbigg . (10)Substitution of this wave function into Eq. (1) yields E0=/integraldisplay dr/bracketleftbigg1 2(∇χ)2+1 8(∇φ)2+κ∂χ ∂x −κ/parenleftbigg∂χ ∂x+κ/parenrightbigg cosφ+κ∂χ ∂ysinφ/bracketrightbigg ,(11) whereχ= (Φ1+Φ2)/2,φ= Φ1−Φ2, and the constant term is neglected. The first and second lines in Eq. (11) correspond to the kinetic and SOC energies, respectively. From the numerical results that the cores are shifted by δand that the 2 πphase rotation around the vortex core is compressed in the ydirection, the phases in Eq. (10) are assumed to be Φ1(r) = tan−1/parenleftbigg λy−δy/2 x−δx/2/parenrightbigg , (12a) Φ2(r) = tan−1/parenleftbigg λy+δy/2 x+δx/2/parenrightbigg ,(12b) whereλandδare variational parameters. We substitute these phases into Eq. (11) and integrate with respect to θ. Because of the complicated structure near the vortex cores, we consider the region in which r≫1. The energy is E0=/integraldisplay rdr/braceleftBigg −2πκ2+π 2λr2 +πκ2 2r2/bracketleftBigg δ2 x+λ2/parenleftbigg δy−1 κ/parenrightbigg2/bracketrightBigg +O(r−3)/bracerightBigg ,(13) which is minimized by δx= 0 andδy= 1/κ. Thus, the energy is lowered by the displacement of the vortex cores intheydirection,andthedisplacement δyisestimatedto be 1/κ; this is in good agreement with the numerical re- sultsshowninFig.1(d). TheenergyinEq.(13)decreases asλincreases, and this accounts for the compressed 2 π phase rotation. A better variational wave function will allow us to determine the value of λ. We note that the term∝δyin Eq. (13) originates from the last term in the integrand of Eq. (11), which thus plays an important role in the vortex deformation due to the SOC. IV. VORTEX PAIR First, for clarity, we define the positions of the vortices and the distances between them, as shown in Fig. 3. The position of the phase defect of the jth vortex in compo- nentiis denoted by ( xij,yij). As shown in Fig. 1(b), in each vortex, the cores in the two components are shifted byδin theydirection, and then x1j=x2jand y1j−y2j=δ. The position of the single vortex is de- fined by (xj,yj) = ((x1j+x2j)/2,(y1j+y2j)/2). For a vortex pair, the index jis taken in such a way that y1> y2. The distance between the vortices is defined by (dx,dy) = (x1−x2,y1−y2) andd= (d2 x+d2 y)1/2. The center of the vortex pair is defined by ( xc,yc) =4 plane wave single vortex vortex pair vortex shift vortex of vortex of vortex
 vortex
 center of vortex pair FIG. 3. Schematic illustration of the vortex pair for /angbracketleft1,−1/angbracketright. The black points are the vortex positions ( x1,y1) and (x2,y2), which are defined as the midpoints between the phase defects in the two components. The open circle is the center of the vortex pair ( xc,yc) = ((x1+x2)/2,(y1+y2)/2). ((x1+x2)/2,(y1+y2)/2). The winding numbers of the first and second vortices are denoted by ∝angb∇acketleftn1,n2∝angb∇acket∇ight. In the following subsections, we will consider the vortex pairs ∝angb∇acketleft±1,∓1∝angb∇acket∇ightand∝angb∇acketleft±1,±1∝angb∇acket∇ight, which we call vortex-antivortex pairs and vortex-vortex pairs, respectively. A. Vortex-antivortex pair In the absence of SOC, a vortex-antivortexpair travels at a constant velocity oris annihilated [38, 39]. A vortex- antivortex pair is stationary only in a trap potential [40], and there is no stationary state in a uniform system. In the presence of the SOC, our numerical results show that stable stationary vortex-antivortex pairs can be formed with a proper choice of the distance between vorticesd; an example is shown in Fig. 4. We prepare the initial state in Eq. (8) with Φ(r) =2/summationdisplay j=1njtan−1y−yj x−xj, (14) where, for this example, n1=±1,n2=∓1,x1=x2= 0, andy1=−y2=di/2 withdibeing the initial distance between vortices. From this initial state, the imaginary- time propagation is performed sufficiently. The sta- tionary state is always reached if the initial distance is di>∼20. ItcanbeseeninFig.4thatthedistancebetween vortices in the stationary state is dy≃4.1 for∝angb∇acketleft1,−1∝angb∇acket∇ightand dy≃6.5 for∝angb∇acketleft−1,1∝angb∇acket∇ight. To understand the stabilization mechanism of the sta- tionaryvortex-antivortexpairs, we calculatethe total en- ergy using a model function given by Ψ(r) =/parenleftbigg/radicalbig ρ1(r)ei[Φ1(r)+κx]/radicalbig ρ2(r)ei[Φ2(r)+κx]/parenrightbigg ,(15) FIG. 4. Stable stationary states of vortex-antivortex pair s for κ= 1 andγ= 0.8. The winding combinations are (a) /angbracketleft1,−1/angbracketright and (b) /angbracketleft−1,1/angbracketright. Panels (a1) and (b1) show the density pro- files, and (a2) and (b2) show the spin distributions. The regions indicated by dashed squares in (a1) and (b1) corre- spond to (a2) and (b2), respectively. The arrows indicate th e directions of the transverse spin vector, and the backgroun d color indicates the value of Sz.














 
 








    






 

 








     local minumum barrier Energy (a) (b) (c) (d) Energy FIG. 5. Total energy of avortex-antivortexpair as afunctio n of the distance between vortices dyfor (a)κ= 0, (b)κ= 0.5, (c)κ= 1, and (d) κ= 1.5. In (b)-(d), local energy minima appear; these correspond to the stationary states shown in Fig. 4. with phases Φ1(r) = tan−1/parenleftbiggy−δ/2 x/parenrightbigg −tan−1/parenleftbiggy−δ/2−dy x/parenrightbigg , (16a) Φ2(r) = tan−1/parenleftbiggy+δ/2 x/parenrightbigg −tan−1/parenleftbiggy+δ/2−dy x/parenrightbigg , (16b)5 05 -5 05 -5 center of vortex pair (a) (b) (c) (d) FIG. 6. Trajectories of vortex-antivortex pairs for κ= 1 and γ= 0.8. Red and blue circles indicate the positions of the vortex cores in ψ1andψ2, respectively. The directions of the circulations of the vortices are indicated by black arrows. The initial distance between vortices is d= 10. Black circles indi- cate the center of the vortex pairs ( xc,yc) att= 0, 400, and 800, and green arrows indicate the direction of motion. See the Supplemental Material for movies of the dynamics [41]. and densities ρ1(r) =1 2N(r)ν(x,y−δ/2)ν(x,y−δ/2−dy), (17a) ρ2(r) =1 2N(r)ν(x,y+δ/2)ν(x,y+δ/2−dy), (17b) whereN(r) is the normalization factor to ensure ρ1(r)+ ρ2(r) = 1 andν(x,y) = (x2+y2)/(x2+y2+w2). We setδ= 1/κ, and from the numerical results, the radius of the vortex wis estimated to be 2 w≃1/κ. Figure5 showsthe total energyasa function of dy; this is obtained by substituting Eq. (15) into Eq. (3). It can be seen in Fig. 5 that local energy minima appear on either side of the global minimum and form the energy barriers, that stabilize the vortex-antivortex pair. We note that without SOC, there are no such barriers, as can be seen in Fig. 5(a) for κ= 0. We also note that the barriers do not appear for uniform densities ρ1=ρ2= 1/2, and the inhomogeneousdensities of Eq. (17) are necessary for the barriers to form. Hence, we conclude that this is the combined effect of SOC and the nonlinear interaction. We now turn our attention to the dynamics of the vortex-antivortex pair. Figure 6 shows the trajectories of the vortex cores, where the initial state is prepared as


           (b)(a)


           unstable FIG. 7. Velocity vxversus the distance dyof a vertically aligned vortex-antivortex pair for (a) κ= 1 and (b) κ= 0.5 withγ= 0.8. The red and blue plots are for /angbracketleft1,−1/angbracketrightand /angbracketleft−1,1/angbracketright, respectively. The configurations of the vortex pairs are illustrated in the insets. In (b), there is an unstable re gion (see text). follows. We first prepare the state in Eq. (8) with the phase in Eq. (14), and then we allow the imaginary-time evolution for a short period (typically, t≃80). From this state, the real-time evolution begins. Figures 6(a) and 6(b) show the dynamics of the vertically aligned vortex pair; the distance d≃10 is larger than that of the stationary states shown in Fig. 4. The vortex- antivortex pair moves in the −xand +xdirections at constant velocity with a fixed distance between vortices. These directions for the propagation agree with those for a scalar BEC. However, the velocities vx≃ −0.006 in Fig. 6(a) and vx≃0.011 in Fig. 6(b) are much slower thanvx= 1/d≃0.1, which is that seen in a scalar BEC for the same d. Figures 6(c) and 6(d) show the cases of oblique and horizontal alignments. The propagation di- rections of these vortex pairs are different from those in a scalarBEC.Thiscanbe understoodbyinspecting theve- locity field shown in Fig. 2. For example, on the negative x-axisintheleft-handpanelofFig.2(b), thevelocityfield is towards the lower right, which indicates that a vortex located on the left-hand side of the counterclockwise vor- tex will feel a mass current in this direction. Similarly, a vortex located on the right-hand side of the clockwise vortex will feel a mass current towards the lower right; this results in the dynamics shown in Fig. 6(d). Figure 7 shows the velocity vxof the vertically aligned6 FIG. 8. Time evolution of the (a) density and (b) phase of the unstable vortex-antivortex pair for κ= 0.5 andγ= 0.8, where the vertical gauges indicate the distance dybetween the vortex cores. See the Supplemental Material for a movie of the dynamics [41]. vortex pair (i.e., dx= 0) as a function of the vortex dis- tancedy, which is obtained by a method similar to that used to obtain Fig. 6. Such vortex pairs always travel in the±xdirection. The dydependence of the velocity is quite different from that in a scalar BEC. For κ= 1 (Fig. 7(a)), the velocity vxof the∝angb∇acketleft1,−1∝angb∇acket∇ightpair (red cir- cles) changes from negative to positive as dyincreases, andvx= 0 atdy≃5, which corresponds to the sta- tionary state seen in Fig. 4(a). The velocity vxof the ∝angb∇acketleft−1,1∝angb∇acket∇ightpair (blue circles) also crosses the vx= 0 axis at dy≃7, which corresponds to the stationary state seen in Fig. 4(b). For 6 <∼dy<∼8, the velocity changes from negative to positive and from positive to negative as dy increases. For a relatively large distance between vor- tices (d>∼10), the propagation directions are the same as those of a scalar BEC, but the dydependence of vx is weak; this can be understood from the fact that the velocity field is almost uniform far from the vortex core, as shown in Fig. 2. The velocity |vx|is always smaller than that in a scalar BEC for both ∝angb∇acketleft1,−1∝angb∇acket∇ightand∝angb∇acketleft−1,1∝angb∇acket∇ight pairs. There is no stable vortex-antivortex pair for small dy; the vortices are unstable against pair annihilation. The∝angb∇acketleft−1,1∝angb∇acket∇ightpair exhibits interesting dynamics when κ is small. As shown in Fig. 7(b), there is no stable ∝angb∇acketleft−1,1∝angb∇acket∇ight pair in the region 5 .5<∼dy<∼9.6. Figure 8 shows the dynamicsofthe ∝angb∇acketleft−1,1∝angb∇acket∇ightpairwiththeinitialdistance dy= 9.6, where the initial state is prepared by the imaginary- time propagation for a short duration from the initial05 -5 (a) (b) (c) (d) center of vortex pair 05 -5 FIG. 9. Trajectories of vortex-vortex pairs for κ= 1 andγ= 0.8. The initial vortex distance is (a) d= 8.0, (b)d= 11.6, (c)d= 18.0, and (d)d= 12.9. Red and blue circles show the positions of the vortex cores in ψ1andψ2, respectively. Black arrows indicate the direction of circulation. Black circle s are the center of the vortex pairs ( xc,yc) att= 0, 400, and 800, and green arrows show the directions of motion. See the Supplemental Material for a movie of the dynamics [41]. phase in Eq. (14) with dy>10. There is no stable state fordy= 9.6 according to Fig. 7(b). As the vortex pair travels in the −xdirection, the distance dydecreases, and eventually the pair settles into a stable state with dy≃5.7; the excess energy is released from the vortex pair as density and spin waves. B. Vortex-vortex pair In a scalar BEC, two quantized vortices with the same circulation move around each other. In contrast, the dy- namics of vortex-vortex pairs with SOC are significantly different from those in a scalar BEC. Figure 9 shows the trajectories of vortices for the ∝angb∇acketleft1,1∝angb∇acket∇ightpair, where the ini- tial state is prepared by the same method as in Fig. 6. When the initial positions are those shown in Fig. 9(a), they moveawayfrom each other. In the case ofFig. 9(b), the two vortices pass each other. The dynamics shown in Figs. 9(c) and 9(d) are more interesting. The two vor- tices approach each other and unite to form a stationary state, and the excess energy is released as waves. The re- sultant stationary state is stable and remains at rest, and the two vortices lie in a line perpendicular to the plane wave. In all cases, the center of the pair initially moves in the direction of + x. The transformation in Eq. (9) gives the dynamics of ∝angb∇acketleft−1,−1∝angb∇acket∇ight.7 (a) (b) plane wave Welocity of center relative velocity Denter of vortex pair FIG. 10. Schematic illustration of the dynamics of (a) vortex-antivortex pairs and (b) vortex-vortex pairs when t he distance is d= 10. The red arrows indicate the direction of the velocity of ( xc,yc), when one vortex is located at the origin. In (a), the relative position of the vortices remain s nearly constant. In (b), the relative velocity (the velocit y of the vortex in the moving frame in which the other vortex is fixed to the origin) is indicated by blue arrows. Figures 10(a) and 10(b) summarize the directions of the vortex motion when d≃10 for the ∝angb∇acketleft1,−1∝angb∇acket∇ightand∝angb∇acketleft1,1∝angb∇acket∇ight pairs, respectively. InFig. 10(a), the motionofthe center of the vortex-antivortex pair ( xc,yc) is indicated by the red arrows, and the relative position of the two vortices is nearly constant. In Fig. 10(b), the relative motion of the vortex-vortexpairis indicated by the blue arrow, and the center of the pair always shifts in the direction of the plane wave. V. CONCLUSIONS We have investigated the behaviors of quantized vor- tices in quasispin-1/2 BECs with Rashba SO coupling in a uniform 2D system, where the atomic interactions sat- isfy the miscible condition and the ground state is theplane-wave state. We found that the static and dynamic propertiesofvorticesaresignificantlydifferentfromthose of a scalar BEC. For a single vortex state, we found that the vortex cores in two components are shifted in the ±ydirections by≃1/κ(Fig. 1). We also found that the phase distri- bution and velocity field around the vortex are greatly deformed compared with those of a scalar BEC (Figs. 1 and 2), which affects the dynamics of the vortex pairs. The vortex-antivortex pairs have stable stationary states at rest (Fig. 4), and this is in marked contrast to the vortex-antivortex pairs in a scalar BEC, which always travel. The stationary states can be explained by varia- tional analysis (Fig. 5). Other than when in a station- ary state, the vortex-antivortex pair travels at a velocity much slower than that for a scalar BEC with the same vortex distance. The dependence of the velocity and moving direction on the vortex location is also quite dif- ferent from that in the case of a scalar BEC (Figs. 6 and 7). The vortex-vortex pair exhibits interesting dynam- ics: the vortices pass and move away from each other, or approach each other and combine into a stationary state (Fig. 9). In experiments, the vortex states shown in this pa- per may be produced by the phase imprinting tech- nique [42, 43] and the ensuing relaxation. The dynam- ics of vortices can be observed by the destructive imag- ing [44] or the real-time imaging [45]. We hope that our numerical results presented in this paper can provide in- sight into a range of topics in the nonlinear dynamics of SO-coupled BECs. ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Numbers JP16K05505,JP26400414,and JP25103007,by the NMFSEID under Grant No. 61127901, and by the Youth Innovation Promotion Association of CAS under Grant No. 2015334. [1] L. Onsager, Nuovo Cimento Suppl. 6, 279 (1949). [2] R. P. Feynman, Prog. Low Temp. Phys. 1, 17 (1955). [3] L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, Phys. Rev. Lett. 103, 250401 (2009). [4] J.-Y. Choi, W. J. Kwon, and Y.-I. Shin, Phys. Rev. Lett. 108, 035301 (2012). [5] M. W. Ray, E. Ruokokoski, S. Kandel, M. M¨ ott¨ onen, and D. S. Hall, Nature (London) 505, 657 (2014); M. W. Ray, E. Ruokokoski, K. Tiurev, M. M¨ ott¨ onen, and D. S. Hall, Science348, 544 (2015). [6] S. W. Seo, S. Kang, W. J. Kwon, and Y.-I. Shin, Phys. Rev. Lett. 115, 015301 (2015); S. W. Seo, W. J. Kwon, S. Kang, and Y.-I Shin, Phys. Rev. Lett. 116, 185301 (2016). [7] D. S. Hall, M. W. Ray, K. Tiurev, E. Ruokokoski, A.H. Gheorghe, and M. M¨ ott¨ onen, Nature Phys. 12, 478 (2016); [8] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Natu re (London) 471, 83 (2011). [9] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 109, 115301 (2012). [10] D. L. Campbell, R. M. Price, A. Putra, A. Vald´ es-Curiel , D. Trypogeorgos, and I. B. Spielman, Nature Comm. 7, 10897 (2016). [11] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Science 354, 83 (2016). [12] J. Li, W. Huang, B. Shteynas, S. Burchesky, F. C ¸. Top, E. Su, J. Lee, A. O. Jamison, and W. Ketterle, Phys.8 Rev. Lett. 117, 185301 (2016). [13] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett.105, 160403 (2010). [14] X.-Q. Xu and J. H. Han, Phys. Rev. Lett. 107, 200401 (2011). [15] J. Radi´ c, T. A. Sedrakyan, I. B. Spielman, and V. Galit- ski, Phys. Rev. A 84, 063604 (2011). [16] X.-F.Zhou, J.Zhou, andC.Wu,Phys.Rev.A 84, 063624 (2011). [17] C.-F. Liu, H. Fan, Y.-C. Zhang, D.-S. Wang, and W.-M. Liu, Phys. Rev. A 86, 053616 (2012). [18] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401 (2011). [19] V. E. Lobanov, Y. V. Kartashov, and V. V. Konotop, Phys. Rev. Lett. 112, 180403 (2014). [20] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys. Rev. Lett. 108, 010402 (2012). [21] T. Kawakami, T. Mizushima, M. Nitta, and K. Machida, Phys. Rev. Lett. 109, 015301 (2012). [22] C.-F. Liu and W. M. Liu, Phys. Rev. A 86, 033602 (2012). [23] G. Chen, T. Li, and Y. Zhang, Phys. Rev. A 91, 053624 (2015). [24] Y. Li, X. Zhou, and C. Wu, Phys. Rev. A 93, 033628 (2016). [25] T. Kawakami, T. Mizushima, and K. Machida, Phys. Rev. A84, 011607(R) (2011). [26] Z. F. Xu, Y. Kawaguchi, L. You, and M. Ueda, Phys. Rev. A86, 033628 (2012). [27] E. Ruokokoski, J. A. M. Huhtam¨ aki, and M. M¨ ott¨ onen, Phys. Rev. A 86, 051607(R) (2012). [28] W. Han, G. Juzeli¯ unas, W. Zhang, and W.-M. Liu, Phys. Rev. A91, 013607 (2015). [29] W. Han, X.-F. Zhang, S.-W. Song, H. Saito, W. Zhang, W.-M. Liu, and S.-G. Zhang, Phys. Rev. A 94, 033629 (2016).[30] Y. Deng, J. Cheng, H. Jing, C.-P. Sun, and S. Yi, Phys. Rev. Lett. 108, 125301 (2012). [31] R. M. Wilson, B. M. Anderson, and C. W. Clark, Phys. Rev. Lett. 111, 185303 (2013). [32] M. Kato, X.-F. Zhang, D. Sasaki, and H. Saito, Phys. Rev. A94, 043633 (2016). [33] H. Sakaguchi, B. Li, and B. A. Malomed, Phys. Rev. E 89, 032920 (2014). [34] Y.-C. Zhang, Z.-W. Zhou, B. A. Malomed, and H. Pu, Phys. Rev. Lett. 115, 253902 (2015). [35] A. L. Fetter, Phys. Rev. A 89, 023629 (2014). [36] K. Kasamatsu, Phys. Rev. A 92, 063608 (2015). [37] M. Eto, K. Kasamatsu, M. Nitta, H. Takeuchi, and M. Tsubota, Phys. Rev. A 83, 063603 (2011). [38] T. Aioi, T. Kadokura, T. Kishimoto, and H. Saito, Phys. Rev. X1, 021003 (2011). [39] C. Rorai, K. R. Sreenivasan, and M. E. Fisher, Phys. Rev. B88, 134522 (2013). [40] L.-C. Crasovan, V. Vekslerchik, V. M. P´ erez-Garc´ ıa, J. P. Torres, D. Mihalache, and L. Torner, Phys. Rev. A 68, 063609 (2003). [41] See Supplemental Material at http://... for movies of t he dynamics of vortex pairs. [42] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sen- gstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewen- stein, Phys. Rev. Lett. 83, 5198 (1999). [43] J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schnei- der, and W. D. Phillips, Science 287, 97 (2000). [44] T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis, andB. P.Anderson, Phys.Rev.Lett. 104, 160401 (2010). [45] D. V. Freilich, D. M. Bianchi, A. M. Kaufman, T. K. Langin, and D. S. Hall, Science 329, 1182 (2010).

1205.6211v2.Dzyaloshinskii_Moriya_Interaction_and_Spiral_Order_in_Spin_orbit_Coupled_Optical_Lattices.pdf

Dzyaloshinskii-Moriya Interaction and Spiral Order in Spin-orbit Coupled Optical Lattices Ming Gong1;2, Yinyin Qian1, Mi Yan3, V. W. Scarola3,and Chuanwei Zhang1y 1Department of Physics, the University of Texas at Dallas, Richardson, Texas, 75080 USA 2Department of Physics and Center for Quantum Coherence, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China 3Department of Physics, Virginia Tech, Blacksburg, Virginia 24061 USA We show that the recent experimental realization of spin-orbit coupling in ultracold atomic gases can be used to study dierent types of spin spiral order and resulting multiferroic eects. Spin-orbit coupling in optical lattices can give rise to the Dzyaloshinskii-Moriya (DM) spin interaction which is essential for spin spiral order. By taking into account spin-orbit coupling and an external Zeeman eld, we derive an eective spin model in the Mott insulator regime at half lling and demonstrate that the DM interaction in optical lattices can be made extremely strong with realistic experimental parameters. The rich nite temperature phase diagrams of the eective spin models for fermions and bosons are obtained via classical Monte Carlo simulations. PACS numbers: 67.85.Hj, 03.75.Lm, 67.85.Fg Introduction The interplay between ferroelectric and ferromagnetic order in complex multiferroic materials presents a set of compelling fundamental condensed matter physics problems with potential multifunctional device applications [1{4]. Ferroelectric and ferromagnetic order compete and normally cannot exist simultaneously in conventional materials. While in some strongly correlated materials, such as the perovskite transition metal oxides [5{10], these two phe- nomena can occur simultaneously due to strong correlation. Nowadays construction and design of high- Tcmagnetic ferroelectrics is still an open and active area of research [11]. These materials incorporate dierent types of interac- tions, including electron-electron interactions, electron-phonon interactions, spin-orbit (SO) couplings, lattice defects, and disorder, making the determination of multiferroic mechanisms a remarkable challenge for most materials [12, 13]. In this context an unbiased and direct method to explore multiferroic behavior in an ideal setting is highly appealing. On the other hand, the realization of a super uid to Mott insulator transition of ultracold atoms in optical lattices [14] opens fascinating prospects [15] for the emulation of a large variety of novel magnetic states [16{18] and other strongly correlated phases found in solids because of the high controllability and the lack of disorder in optical lattices. For instance, it has been shown [16, 17] that the eective Hamiltonian of spin-1/2 atoms in optical lattices is the XXZ Heisenberg model in the deep Mott insulator regime. On the experimental side, superexchange interactions between two neighboring sites have already been demonstrated [19] and quantum simulation of frustrated classical magnetism in triangular optical lattices has also been realized [20]. These experimental achievements mark the rst steps towards the quantum simulation of possible magnetic phase transitions in optical lattices. In this paper, we show that the power of optical lattice systems to emulate magnetism can be combined with recent experimental developments [21{24] realizing SO coupling to emulate multiferroic behavior. Recently, SO coupled optical lattices have been realized in experiments for both bosons [25] and fermions [26], where interesting phenomena such as at bands [26{28] can be observed. The main ndings of this work are the following: (I) We incorporate spin-orbit and Zeeman coupling into an eective Hamiltonian for spin-1/2 fermions and bosons in optical lattices in the large interaction limit. We show that SO coupling leads to an eective in-plane Dzyaloshinskii-Moriya (DM) term, an essential ingredient in models of spiral order and multiferroic eects in general. The DM term is of the same order as the Heisenberg coupling constant. (II) We study the nite temperature phase diagram of the eective spin model using classical Monte Carlo (MC). We nd that competing types of spiral order depend strongly on both SO and eective Zeeman coupling strength. (III) We nd that the critical temperature for the spiral order can be of the same order as the Heisenberg coupling constant. Thus, if magnetic quantum phase transitions can be emulated in optical lattices, then spiral order and multiferroic-based models can also be realized in the same setup with the inclusion of SO coupling. Corresponding author, email: scarola@vt.edu yCorresponding author, email: chuanwei.zhang@utdallas.eduarXiv:1205.6211v2 [cond-mat.quant-gas] 29 May 20152 Results Eective Hamiltonian. We consider spin-1/2 ultracold atoms loaded into a two-dimensional (2D) square optical lattice. We restrict ourselves to the deep Mott insulator regime where the charge/mass degree of freedom is frozen while the spin degree of freedom remains active. Here the atomic hyperne levels map onto eective spin states. The scattering length between atoms in optical lattices can be controlled by a Feshbach resonance. Certain atoms, e.g., 40K, exhibit considerable tunability[29]. To derive the inter-spin interaction in this regime we rst consider a two-site tight-binding model, H=X tcy 1c2+Vso+Vz+1 2X i;0U0:nini0:; (1) wherecy icreates a particle (either a boson or a fermion) in a Wannier state, wi;, localized at a site iand in a spin state2f";#g.ni=cy iciis the number operator. The tunneling and interaction matrix elements are t=R dxw i;2=2m+V(x)]wi+1;andU0=g0R dxjwi;j2jwi;0j2, respectively, where g0is the interaction strength between species and0,mis the mass of the atom, and V(x) is a lattice potential. Here :: denotes normal ordering. For a general theory the tunneling is assumed to be spin dependent, which is a feature unique to ultracold atom systems [17, 18]. The second term is the Rashba SO coupling [30], written in the continuum as (pxypyx). But on a lattice it can be written as Vso=icy iez
(d)cj+h:c:; (2) wherecy i= (cy i";cy i#),denotes Pauli matrices, and =i R dxw ipxwi+exis the SO coupling strength. d(dx;dy ) is the vector from a site at position rjto a site atri, wheredx= (rirj)
exanddy= (rirj)
ey. Eq. 2 describes the tunneling between neighboring sites paired with a spin ip. The magnitude and sign of can be tuned in experiments using coherent destructive tunneling methods [31]. The third term is the external Zeeman eld Vz=P i;niwith ==2. In the deep Mott insulator regime, the degeneracy in spin congurations is lifted by second order virtual processes. The eective Hamiltonian Hecan be obtained using perturbation theory. We take the Mott insulator as the unper- turbed state and derive the corrections of the eective Hamiltonian by the standard Schrieer-Wolf transformation [17, 32]. The Schrieer-Wolf transformation applies a canonical transformation He=eiSHeiSto obtain the sec- ond order Hamiltonian He=H0+1 2[iS;V ] by eliminating the rst order term using V=[iS;H 0]. In the spin representation we dene Si=P sscy isss0cis0, and extend the two-site model to the whole lattice, yielding He=X hi;jiX =x;y;zJS iS j+X iB
Si+ X ijDij
(SiSj) +Si
ij
Sj: (3) The rst two terms are Heisenberg exchange and Zeeman terms, respectively, while the last two terms arise from SO coupling. In solid state systems the third term is called the DM interaction [33, 34], which is believed to drive multiferroic behavior. The denition of the Dvector and the tensor will be presented below. The structure of these terms can be derived from basic symmetry analyses but the coecients must be computed microscopically. In the following we derive the coecients in Eq. 3 by considering the coupling between four internal degenerate ground states ji2fj" ;"i;j";#i;j#;"i;j#;#igthrough the spin independent and dependent tunnelings tand. The couplings are dierent for fermions and bosons, as illustrated in Fig. 1. Fermionic atoms. For fermionic atoms, there are only two possible excited states jexi=j"#; 0iandj0;"#i, as shown schematically in Fig. 1 (a). We nd ( Jx+Jy)=2 = 4t"t#=U, (JxJy)=2 = 8(dx2+dy2)U2=(U22), and Jz= 2(t"t#)2=U4d2U2=(U22), withd2=dx2+dy2. The DM interaction coecient is D= 2(t"+t#)(2 2U2)=(U(2U2))(dy;dx; 0), and the eective Zeeman eld contains B= 4(2d22=(2U2))(0;0;1). Note that without SO coupling the model reduces to the well-known XXZ Heisenberg model with rotational symmetry [16, 17]. However, this symmetry is broken by the SO coupling, yielding an XYZ-type Heisenberg model. Similar results are also observed for bosons. Bosonic atoms. For bosonic atoms, there are six excited states jexi=j""; 0i,j"#; 0i,j##; 0i,j0;""i,j0;"#i,j0;##i, as shown in Fig. 1 (b). Without SO coupling, the only allowed inter-state second-order transition is between j2i andj3i, similar to the fermionic case. The presence of SO coupling permits other inter-state transitions, therefore the bosonic case is much more complex than the fermionic case. For simplicity we only show the results for U""=3 FIG. 1: Transition processes due to dierent tunneling mechanisms. Spin-conserving tunneling (solid lines, tterms) and SO coupling mediated tunneling (dashed lines, terms) are plotted for spin-1/2 fermions (a) and spin-1/2 bosons (b). is the chemical potential. The lowest 4 levels are ground states, and the higher energy levels are the excited states. FIG. 2: Tunable parameters in an optical lattice. (a) Tunneling amplitudes as a function of lattice depth. tis the hopping due to the kinetic energy, tappr. is the analytic expression derived in the deep lattice regime, and is the SO mediated hopping strength. (b) Plot of jDj=jJjas a function of =tforU0=U,t=t. U##=U"#=U, which yields ( Jx+Jy)=2 =4t"t#=U, (JxJy)=2 = 4(d2 xd2 y)U2=(U22),Jz=4t"t#=U+ 2 (2U2)(t"t#)2+ 2U2d22 =U(U22),D=2(t"+t#)(22U2)=U(2U2)(dy;dx; 0), and B= (0;0;4). The last term in Eq. 3 reads as Si
ij
Sj=8dxdyU2 (U22)(Sx iSy j+Sy jSx i), where= +1(1) for fermions (bosons). This term arises from the coupling between states j1iandj4i,j1ih4j=Sx iSx jSy iSy j+i(Sx iSy j+Sy iSx j). Here the real part contributes asymmetric terms to the Heisenberg model, while the imaginary part contributes to ij. In a square lattice with dxdy= 0, this term vanishes. However, for tilted lattices, such as triangular and honeycomb, this term should be signicant. Lattice parameters. We estimate the possible parameters that can be achieved in a square optical lattice V(x;y) = V(x)V(y), whereV(x) =VLsin2(kLx). We dene the lattice depth s=VL=ERin units of he recoil energy ER= ~2k2 L=2m, wherekLis the wavevector of the laser. The SO coupling coecient is given by ~kR=m,kRis the wavevector of the external Raman lasers, and kRkLin most cases. The Raman lasers are pure plane waves, and serve as a perturbation to the hopping between adjacent sites. We use the Wannier functions of the lowest band without SO coupling to calculate the tight binding parameters tand. In a square lattice, coordinates decouple and the Bloch functions are Mathieu functions. The Wannier4 functions can be obtained from the Fourier transform of the Bloch functions. Our numerical results are presented in Fig. 2 (a). The large slimit,ttappr. = 4ER=ps3=4exp(2ps), is also plotted for comparison. Note that U=ER(8=)1=2kLass3=4is in general much larger than tand can be controlled through a Feshbach resonance independently. In Fig. 2 (b) we plot jDj=jJjas a function of ==tforU0=U,t=t.jDj=jJjreaches the maximum value of 1.0 at=t. This is in sharp contrast to models of weak multiferroic eects in solids with D=J =jDj=jJj0:0010:1, which is generally induced by small atomic displacements [35]. Optical lattices, by contrast, can be tuned to exhibit either weak or strong DM terms. This enhanced tunability enables optical lattice systems to single out the eects of strong DM interactions and study the impact of the DM term. There are notable dierences between our model and corresponding models in solids ( i) In solids the SO coupling arises from intrinsic (atomic) SO coupling and Dis generally along the zdirection (out of plane). However, in our model Dis in the plane and the out of plane component is zero. ( ii) In our eective spin model, J ijdepends on the direction of the bond ( dx;dy) and the SO coupling strength, while in solids J ijis independent of SO coupling due to its negligible role. Spiral order and multiferroics in 2D optical lattices. We now explore the rich phase diagrams of the eective spin Hamiltonian using classical MC simulations. Classical MC has been widely used to explore the phase diagrams of the Heisenberg model with DM interactions in the context of solids [11, 36{38] (thus weak DM interactions). This method may not be used to determine the precise boundaries between dierent phases but can be an ecient tool to determine dierent possible phases. Due to the unique features of our eective model (e.g., strong DM interactions) the phase diagrams we present here are much more rich and comprehensive than those explored in the context of solids. We focus on the regime where t=t,U0=U(spin independent), and  U, and dene J0= 4t2=Uas the energy scale. The rescaled eective Hamiltonian becomes H=X ijX a=x;y;zjaSa iSa j+D
SiSj+hX iSz i; (4) wherejx=1 + (d2 xd2 y)2,jy=1(d2 xd2 y)2,jz=1 +2,D=2(dy;dx;0), and==t. Eq. 4 hosts a variety of magnetic and spin spiral phases, which are generally characterized by the magnetic and spiral order parameters [39, 40] M=N1 sX iSz iand P=N1 sX hi;jidijSiSj; (5) whereNsis the number of sites. However, these two order parameters do not fully characterize the phase diagrams because in some cases there are still local magnetic or spiral orders although both MandP=jPjare vanishingly small. In these cases, we also take into account the spin structure factor: S(k) =N2 sX i;jhSi
Sjiexp(ik
(RiRj)): (6) FIG. 3: Phase diagrams of 2D optical lattices. Classical Monte Carlo simulations are performed for an 8 8 lattice with fermions (a) and bosons (b) at temperature T= 0:05J0. The phases diagrams are determined by the magnetization order, the spiral order, and the spin structure factor. Dierent regions correspond to: M= 0,P= 0 for green, M6= 0,P= 0 for grey, M= 0;P6= 0 for cyan, and M6= 0,P6= 0 for red. The abbreviations are: (a) AF: antiferromagnetic phase with zero total magnetization; MAF: antiferromagnetic phase with non-zero total magnetization; NMS: zero magnetization spiral order; MS: magnetic spiral order; NMFS: nonmagnetic ux spiral phase; MFS: magnetic ux spiral phase. In (b), SM: simply magnetic order; SMS: simply magnetic spiral order: Other abbreviations are the same as in (a). The dashed lines are guides to the eye. The spin structure factors of the points marked by plus signs are shown in Fig. 4.5 FIG. 4: Spin structure factors for dierent quantum phases marked by plus signs in Fig. 3 . The upper panels show the results for fermions at h=J0= 1:1, while the lower panels show the results for bosons at h=J0= 0:218. FIG. 5: Spin congurations and phase transitions. (a) The spin conguration of fermions in an 8 8 lattice at T= 0:05J0, = 1:0 andh=J0= 1:5. The corresponding magnetization and spiral order as a function of temperature is shown in (b). The inset plots p(P)2=Tvs. temperature, which indicates a phase transition at Tc0:5J0. Similar features can also been found for bosons with the same parameters. S(k) shows peaks at dierent positions in momentum space for dierent phases. For instance, the peak of the spin structure factor is at k= (0;0) for ferromagnetic phases, k= (;) for antiferromagnetic phases, and ( ;0) (or (0;)) for the ux spiral phase ( P= 0 but with nontrivial local spin structure). General spiral orders correspond to other k. We obtain the phase diagrams by analyzing both the order parameters and spin structure factors. We have not checked for long range order in the spin structure factor. We expect quasi-long range order to accompany magnetized phases at low h, e.g., a ferromagnetic phase for 1. The phase diagrams of an 8 8 lattice in Fig. 3 show a rich interplay between magnetic orders and spin spiral orders. For instance, for fermions with small SO coupling (  <0:25), the ground states are anti-ferromagnetic states with zero (non-zero) magnetization for a Zeeman eld h=J0<0:8 (h=J0>0:8). While for large SO coupling (  >1:45), the ground states are either nonmagnetic or magnetic ux spiral phases (similar to the ux phase with a small spiral orderP). For1 the DM term is not important because D=J1=, therefore the pure ux phase with zero spiral order can be observed. Similarly, the increasing SO coupling for bosonic atoms gives rise to a series of transitions from simply magnetic (ferromagnetic at small h) order to simply magnetic spiral order (with zero total spiral order but local spiral structure), then to magnetic spiral orders (or non-magnetic spiral orders) and nally to ux spiral orders. The emergence of the spiral order and ux order with increasing SO coupling can be clearly seen from the change of the spin structure factors in Fig. 4, which shift from k= (0;0) or (;) to (;0) and (0;). The spin spiral order phase transition temperature is comparable to the magnetic phase transition temperature, J0. In Fig. 5 (a), we plot the spin conguration of fermions at T= 0:05J0,= 1:0 andh= 1:5 (MS phase), which shows clear spiral ordering. The corresponding order parameters PandMare plotted in Fig. 5 (b) as a function of temperature. The inset shows the susceptibility p(P)2=T. We see a phase transition at Tc0:5J0, which is comparable to the magnetic critical temperature [17] (In 2D, the Heisenberg model has a critical temperature Tc=J0 in mean-eld theory). Note that spiral order can also exist in the frustrated model without SO coupling, however, the critical temperature is generally much smaller than the magnetic phase transition temperature [11, 41]. Our results therefore show that SO coupling in the absence of frustration provides an excellent platform to search for spiral order and multiferroics-based states in optical lattices.6 Discussion Finally we note that dierent spiral orders may be observed using optical Bragg scattering methods [42], which probe dierent spin structure factors for dierent spiral orders. Similar methods have been widely used in solid state systems. Furthermore, in optical lattices, the local spin magnetization at each lattice site (thus the magnetic order M) as well as the local spin-spin correlations (thus the spiral order P) can be measured directly [43, 44], which provides a powerful new tool for understanding the physics of spiral orders and multiferroic eects in optical lattices. Note added. During the preparation of this manuscript (the initial version is available at arXiv:1205.6211) we became aware of work [45{47] on similar topics. Methods The phase diagrams of an 8 8 lattice are computed by classical MC methods for both fermions and bosons. The results are obtained after 106thermalization steps followed by 106sampling steps in each MC run at low temperature (T= 0:05J0). We have checked that for lower temperatures the phase diagrams do not change quantitatively. We also verify that similar phase diagrams can be obtained for larger system sizes, however, the spiral orders in a larger optical lattice become more complicated, and the boundary between dierent quantum phases is shifted. [1] Fiebig, M. Revival of the magnetoelectric eect. J. Phys. D: Appl. Phys. 38, R123 (2005). [2] Dawber, M., Rabe, K. M. & Scott, J. F. Physics of thin-lm ferroelectric oxides. Rev. Mod. Phys. 77, 1083 (2005). [3] Basov, D. N. et al. Electrodynamics of correlated electron materials. Rev. Mod. Phys. 83, 471 (2011). [4] Catalan, G., Seidel, J., Ramesh, R. & Scott, J. F. Domain wall nanoelectronics. Rev. Mod. Phys. 84, 119 (2012). [5] Tokura, Y. & Seki, S. Multiferroics with spiral spin orders. Adv. Mater. 22, 1554 (2010). [6] Kimura, T. Spiral magnets as magnetoelectrics. Annu. Rev. Mater. Res. 37, 387-413 (2007). [7] Cheong, S.-W. & Mostovoy, M. Multiferroics: a magnetic twist for ferroelectricity. Nature Materials 6, 13 (2007). [8] Ramesh, R. & Spaldin, N. A. Multiferroics: progress and prospects in thin lms. Nature Materials 6, 21 (2007). [9] Eerenstein, W., Mathur, N. D. & Scott, J. F. Multiferroic and magnetoelectric materials. Nature 442, 759 (2006). [10] Tokura, Y. Multiferroics as quantum electromagnets. Science 312, 1481 (2006). [11] Jin, G., Cao, K., Guo, G.-C. & He, L. Origin of ferroelectricity in high- Tcmagnetic ferroelectric CuO. Phys. Rev. Lett. 108, 187205 (2012). [12] Sergienko, I. A. & Dagotto, E. Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites. Phys. Rev. B 73, 094434 (2006). [13] Katsura, H., Nagoasa, N. & Balatsky, A. V. Phys. Rev. Lett. 95, 057205 (2005). [14] Greiner M., Mandel, O., Esslinger, T., H ansch, T. W. & Bloch, I. Quantum phase transition from a super uid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39 (2002). [15] Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008). [16] Kuklov, A. B. & Svistunov, B. V. Counter ow super uidity of two-species ultracold atoms in a commensurate optical lattice. Phys. Rev. Lett. 90, 100401 (2003). [17] Duan, L. -M., Demler, E. & Lukin, M. D. Controlling spin exchange interactions of ultracold atoms in optical lattices. Phys. Rev. Lett. 91, 090402 (2003). [18] Altman, E., Hofstetter, W., Demler, E. & Lukin, M. D. Phase diagram of two-component bosons on an optical lattice. New Journal of Physics 5, 113, (2003). [19] Trotzky, S. et al. Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices. Science 319, 295 (2008). [20] Struck, J. et al. Quantum simulation of frustrated classical magnetism in triangular optical lattices. Science 333, 996 (2011). [21] Lin, Y.-J., Jimenez-Garcia, K. & Spielman, I. B. Spinorbit-coupled BoseEinstein condensates. Nature 471, 83 (2011). [22] Chen, S. et al. Collective dipole oscillations of a spin-orbit coupled Bose-Einstein condensate. Phys. Rev. Lett. 109, 115301 (2012). [23] Fu, Z., Wang, P., Chai, S., Huang, L. & Zhang, J. Bose-Einstein condensate in a light-induced vector gauge potential using 1064-nm optical-dipole-trap lasers. Phys. Rev. A 84, 043609 (2011). [24] Wang, P. et al. Spin-orbit coupled degenerate Fermi gases. Phys. Rev. Lett. 109, 095301 (2012). [25] Jim enez-Garc a, K. et al. Peierls Substitution in an Engineered Lattice Potential. Phys. Rev. Lett. 108, 225303 (2012). [26] Cheuk, L. W. et al. Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas. Phys. Rev. Lett. 109, 095302 (2012) [27] Zhang, Y. & Zhang, C. Bose-Einstein condensates in spin-orbit-coupled optical lattices: Flat bands and super uidity. Phys. Rev. A 87, 023611 (2013)7 [28] Lin, F., Zhang, C., & Scarola, V.W. Emergent Kinetics and Fractionalized Charge in 1D Spin-Orbit Coupled Flatband Optical Lattices Phys. Rev. Lett. 112, 110404 (2014). [29] K ohl Khl, M., Moritz, H., Stferle, T., G unter, K. & Esslinger, T. Fermionic atoms in a 3D optical lattice, Phys. Rev. Lett. 94, 080403 (2005). [30] Zhang, Y. & Zhang, C. Mean-eld dynamics of spin-orbit coupled Bose-Einstein condensates. Phys. Rev. Lett. 108, 035302 (2012). [31] Zhang, Y., Chen, G. & Zhang, C. Tunable spin-orbit coupling and quantum phase transition in a trapped Bose-Einstein condensate. Sci. Rep. 3, 1937 (2013) [32] Hewson, A. C. The Kondo Problems to Heavy Fermions. (Cambridge University Press, Cambridge, England, 1997). [33] Dzyaloshinskii, I. E. Theory of helicoidal structures in antiferromagnets. I. nonmetals. Sov. Phys. JETP 19, 960 (1964). [34] Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91 (1960). [35] Sergienko, I. A. & Dagotto, E. Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites. Phys. Rev. B 73, 094434 (2006). [36] Sen, C., Liang, S. & Dagotto, E. Complex state found in the colossal magnetoresistance regime of models for manganites. Phys. Rev. B 85, 174418 (2012). [37] Liang, S., Daghofer, M., Dong, S., Sen, C. & Dagotto, E. Emergent dimensional reduction of the spin sector in a model for narrow-band manganites. Phys. Rev. B 84, 024408 (2011). [38] Dong, S. et al. Exchange bias driven by the Dzyaloshinskii-Moriya interaction and ferroelectric polarization at G-type antiferromagnetic perovskite interfaces. Phys. Rev. Lett. 103, 127201 (2009). [39] Mostovoy, M. Ferroelectricity in Spiral Magnets. Phys. Rev. Lett. 96, 067601 (2006). [40] Katsura, H., Nagaosa, N. & Balatsky, A. Spin Current and Magnetoelectric Eect in Noncollinear Magnets. Phys. Rev. Lett.95, 057205 (2005). [41] Blake, G. R. et al. Spin structure and magnetic frustration in multiferroic RMn 2O5(R = Tb, Ho, Dy). Phys. Rev. B 71, 214402 (2005). [42] Corcovilos, T. A., Baur, S. K., Hitchcock, J. M., Mueller, E. J., & Hulet, R. G. Detecting antiferromagnetism of atoms in an optical lattice via optical Bragg scattering Phys. Rev. A 81, 013415 (2010). [43] Bakr, W. S., Gillen, J. I., Peng, A., F olling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 74 (2009). [44] Weitenberg, C. et al. Single-spin addressing in an atomic Mott insulator. Nature 471, 319 (2011). [45] Radic, J., Di Ciolo, A., Sun, K. & Galitski, V. Exotic quantum spin models in spin-orbit-coupled Mott insulators Phys. Rev. Lett. 109, 085303 (2012) (also available at arXiv:1205.2110). [46] Cole, W. S., Zhang, S. Z., Paramekanti, A. & Trivedi, N. Bose-Hubbard models with synthetic spin-orbit coupling: Mott insulators, spin textures, and super uidity. Phys. Rev. Lett. 109, 085302 (2012) (also available at arXiv:1205.2319). [47] Cai, Z., Zhou, X., Wu, C. Magnetic phases of bosons with synthetic spin-orbit coupling in optical lattices. Phys. Rev. A 85, 061605(R) (2012) (also available at arXiv:1205.3116). Acknowledgements M.G. thanks S. Liang for numerical assistance with classical MC simulations. This work is supported by AFOSR (FA9550-11-1-0313), DARPA-YFA (N66001-10-1-4025, N66001-11-1-4122), ARO (W911NF-09-1-0248), and the Jeress Memorial Trust (J-992). M.G. is also supported by Hong Kong RGC/GRF Projects (No. 401011, No. 401213 and No. 2130352), University Research Grant (No. 4053072) and The Chinese University of Hong Kong (CUHK) Focused Investments Scheme. Author contributions M.G. and C.Z. conceived the idea, M.G., Y.Q., M.Y. performed the calculation, with input from V.W.S. and C.Z.. V.W.S. and C.Z. supervised the whole research project. All authors analyzed and discussed the results and contributed in writing the manuscript. All authors have given approval to the nal version of the manuscript. Additional information Competing nancial Interests: The authors declare no competing nancial interests.

0911.3206v2.Large_polaron_formation_induced_by_Rashba_spin_orbit_coupling.pdf

arXiv:0911.3206v2 [cond-mat.str-el] 9 Feb 2010Large polaron formation induced by Rashba spin-orbit coupl ing C. Grimaldi LPM, Ecole Polytechnique F´ ed´ erale de Lausanne, Station 1 7, CH-1015 Lausanne, Switzerland Here the electron-phonon Holstein model with Rashba spin-o rbit interaction is studied for a two dimensional square lattice in the adiabatic limit. It is dem onstrated that a delocalized electron at zero spin-orbit coupling localizes into a large polaron sta te as soon as the Rashba term is nonzero. This spin-orbit induced polaron state has localization len gth inversely proportional to the Rashba coupling γ, and it dominates a wide region of the γ-λphase diagram, where λis the electron-phonon interaction. PACS numbers: 71.38.-k, 71.70.Ej I. INTRODUCTION Spin manipulation and control is at the core of spin- tronics, a technology that uses the spin of the elec- trons, rather than their charge, to transfer and/or pro- cess information.1,2The Rashba spin-orbit (SO) cou- pling arising in materials lacking structural inversion symmetry3plays a leading role in this field because its strength can be tuned by an applied electric field and by specific material engineering methods. The SO induced lifted spin degeneracy may then be used in spin filtering devices and spin transistors. Whether the main effect of SO coupling is limited to the spin splitting or it is accompanied by substantial modifications in other electronic properties, which could be detrimental for the spin propagation, is of course cru- cial for the functioning of spin-based devices. In this respect, an important issue calls into play the role of the SO interaction on the coupling of electrons to the lattice vibrations(phonons). In particular, a sensible problem is whether the polaron, that is the quasiparticle composed by the electron and its phonon cloud, is strengthened or weakened by the Rashba SO interaction. In previous works, an enhancement of the polaronic character has been obtained for a two-dimensional (2D) electron gas with linear Rashba coupling for both short- range (Holstein model Ref.[4]) and long-range (Fr¨ ohlich model Ref.[5]) electron-phonon (el-ph) interactions.6–8 On the contrary, a recent calculation on the 2D tight- binding Holstein-Rashba model on the square lattice has shown that a large el-ph interaction gets effectively sup- pressed by the Rashba SO coupling.9At present there- fore the role of the Rashba SO coupling on the polaron properties is not clear, and different models and approx- imations appear to give quite contradicting results. In thisarticlethe tight-bindingHolstein-Rashbamodel for one electron coupled to adiabatic phonons is consid- ered and the corresponding non-linear Schr¨ odinger equa- tion for the polaron wave function is solved numerically. It is shown that, for el-ph couplings such that the elec- tron is delocalized in the zero SO limit, the Rashba term creates a large polaron state, with polaron localization length inversely proportional to the SO strength. Fur- thermore, the small polaron regime appearing at largeel-ph couplings and zero SO gets weakened (or even sup- pressed) for sufficiently strong SO couplings. Hence, the Holstein-Rashba polaron is strengthened or weakened by the SO interaction depending on whether the el-ph cou- pling is respectively weak or strong, thereby reconciling the different trends reported in Refs.[6,9] into one single picture. II. MODEL By presenting the spinor operator Ψ† R= (c† R↑,c† R↓), wherec† Rαcreates an electron with spin α=↑,↓on site R,thetight-bindingHolstein-RashbaHamiltonianonthe square lattice can be written as H=H0+Hph+Hel−ph, where10 H0=−t/summationdisplay R/parenleftBig Ψ† RΨR+ˆx+Ψ† RΨR+ˆy/parenrightBig −iγ 2/summationdisplay R/parenleftBig Ψ† RσyΨR+ˆx−Ψ† RσxΨR+ˆy/parenrightBig +H.c., (1) is the lattice Hamiltonian for a free electron with trans- fer integral tand SO coupling γ.σxandσyare Pauli matrices. The lattice constant is taken to be unity, andˆxandˆyare unit vectors along the xandydirec- tions, respectively. The Hamiltonian (1) is easily di- agonalized in momentum space, and the resulting elec- tron dispersion is composed of two branches: E± k= −2t[cos(kx)+cos(ky)]±γ/radicalbig sin(kx)2+sin(ky)2. The low- est branch, E− k, has a four-fold degenerate minimum E0=−4t/radicalbig 1+γ2/(8t2) for momenta k= (±k0,±k0) withk0= arctan[γ/(√ 8t)].9The Hamiltonian for Ein- stein phonons with mass Mand frequency ω0is given by: Hph=/summationdisplay R/parenleftbiggP2 R 2M+1 2Mω2 0X2 R/parenrightbigg , (2) wherePRandXRare impulse and displacement phonon operators. Finally, the el-ph Hamiltonian contribution is Hel−ph=/radicalbig 2Mω0g/summationdisplay RΨ† RΨRXR,(3)2 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48 /s32/s32/s69/s32/s47/s32/s116/s40/s97/s41 /s42/s61/s48/s46/s56/s51/s53/s32/s47/s32/s116/s32/s61/s32/s48 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s32/s47/s32/s116/s32/s61/s32/s49 /s42/s42/s61/s48/s46/s51/s56/s56 /s42/s61/s48/s46/s56/s54/s53/s40/s98/s41 /s32/s32 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s32/s47/s32/s116/s32/s61/s32/s50 /s42/s42/s61/s48/s46/s54/s52/s56 /s42/s61/s48/s46/s57/s51/s53/s40/s99/s41 /s32/s32 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s32/s47/s32/s116/s32/s61/s32/s51 /s42/s42/s61/s48/s46/s57/s49 /s42/s61/s49/s46/s48/s50/s40/s100/s41 /s32/s32 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s32/s32/s48/s50 /s42 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s42/s42 /s42 /s32/s32/s48/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s42/s42/s42 /s32/s32/s48/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s42/s42/s42 /s32/s32/s48/s50 FIG. 1: (Color online) Total energy difference ∆ E=E−E0for the adiabatic Holstein-Rashba model as a function of the el-ph coupling λand for different values of the SO Rashba interaction γ.E0is the ground state energy for λ= 0. Different symbols refer to different solutions of the nonlinear Schr¨ odinger e quation (5), and the ground state is given by the solution wit h lower ∆Evalues. Insets: corresponding electron density probabili ty atR=0. wheregis the el-ph interaction matrix element. The (quasi-) 2D materials and heterostructures which display non-zero Rashba couplings (semiconductor quan- tum wells, surface states of metals and semimetals) are wide electron bandwidth systems with tof the order of 1 eV, while the typical phonon energy scale is of the order of few to tens meV.11These systems are expected there- fore to be well within the adiabatic regime ω0/t≪1. In the following, however, only the strict adiabatic limit ω0/t= 0 is considered, which simplifies considerably the problemand,asshownbelow,permitstoidentifythecrit- ical parameters governing the electron localization tran- sitions. The adiabatic limit ω0/t= 0 is obtained formally from Eqs. (2) and (3) by setting M→ ∞and keep- ingK=Mω2 0finite. Since for M→ ∞the phonon kinetic energy is zero, the ground state in the adiabatic limit is obtained by finding the displacement configura- tionX0 Rwhich minimizes the total energy E=/angbracketleftH/angbracketright, where the brackets mean the expectation value with re- spect to the electron wave function and the lattice dis- placement. Hence, since by Hellmann-Feynman theorem X0 R=√2Mω0g/angbracketleftψ|Ψ† RΨR|ψ/angbracketright/K, the ground state en- ergy becomes EGS=/angbracketleftψ|H0|ψ/angbracketright−EP/summationdisplay R/angbracketleftψ|Ψ† RΨR|ψ/angbracketright2,(4) whereEP=g2/ω0is independent of M, and|ψ/angbracketright=/summationtext R,αφRαc† Rα|0/angbracketright. The ground state electron wave func- tionφRαcan be found from Eq.(4) by applying the variational principle, leading to the following non-linear Schr¨ odinger equation: εΦR=−t/summationdisplay n=±(ΦR+nˆx+ΦR+nˆy)−2EP|ΦR|2ΦR −iγ 2/summationdisplay n=±n(σyΦR+nˆx−σxΦR+nˆy), (5)where Φ R= (φ∗ R↑,φ∗ R↓)+andε=EGS+EP/summationtext R|ΦR|4. Finally, the ground state energy EGSis obtained by solv- ing Eq.(5) iteratively, with/summationtext R,α|φRα|2= 1, and by in- serting the resulting wave function into Eq.(4). III. RESULTS Solutions of (5) for lattices of N= 101×101 sites are plotted in Fig. 1 as a function of the el-ph cou- pling constant λ=EP/(4t) =g2/(4tω0) and for four different values of γ. Forγ= 0, Fig. 1(a), we re- cover the well-known behavior of the adiabatic Holstein model in two-dimensions:12a delocalized solution with EGS=E0=−4t(filled circles) extending to the whole range ofλvalues considered, and a localized one (filled squares)havingenergylowerthan E0forλ≥λ∗= 0.835. The delocalized/localized nature of the solutions is illus- trated in the inset of Fig. 1(a) where the electron density probability |ΦR|2=/summationtext α|φRα|2is plotted for R=0. The solution having lower energy for λ≥λ∗is a small polaron state, with more than 90% of its wave function localized at the origin. Let us now consider the γ >0 case. As shown in Figs. 1(b)-(c), a nonzero Rashba term gives rise to a new feature absent for γ= 0. Namely, besides the two solu- tionsalreadydiscussedforthe γ= 0case,athirdsolution appears (filled triangles), which has lower energy than the delocalized and small polaron states in a region of intermediate values of λ. It is thus possible to identify a second critical coupling, λ∗∗, such that for λ∗∗≤λ≤λ∗ the ground state is given by this third solution. Further- more, the transition to the small polaron state (identi- fied byλ∗) gets shifted to larger el-ph couplings as γ/t increases, thereby confirming the results of Ref.[9] ob- tained by a different method and for ω0/t/negationslash= 0. A map of the behavior of λ∗andλ∗∗asγis varied is reported in3 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53 /s52/s69/s45/s48/s52/s54/s69/s45/s48/s52/s57/s69/s45/s48/s52/s49/s69/s45/s48/s51/s50/s69/s45/s48/s51/s51/s69/s45/s48/s51/s53/s69/s45/s48/s51/s56/s69/s45/s48/s51/s49/s69/s45/s48/s50/s50/s69/s45/s48/s50/s51/s69/s45/s48/s50/s52/s69/s45/s48/s50/s55/s69/s45/s48/s50/s49/s69/s45/s48/s49/s50/s69/s45/s48/s49/s50/s69/s45/s48/s49/s52/s69/s45/s48/s49/s54/s69/s45/s48/s49/s57/s69/s45/s48/s49/s49/s69/s43/s48/s48 /s118/s97/s114 /s32/s32/s47/s32/s116/s100/s101/s108/s111/s99/s97/s108/s105/s122/s101/s100/s32/s101/s108/s101/s99/s116/s114/s111/s110 /s42/s42/s108/s97/s114/s103/s101 /s112/s111/s108/s97/s114/s111/s110/s42/s42 /s42 /s115/s109/s97/s108/s108 /s112/s111/s108/s97/s114/s111/s110/s49/s46/s48/s32/s32 /s49/s48/s45/s49 /s49/s48/s45/s50 /s49/s48/s45/s51 /s49/s48/s45/s52 FIG. 2: (Color online) Phase diagram of the 2D adiabatic Holstein-Rashba model. The λ∗∗andλ∗transition lines are the phase boundaries separating the different states of the polaron. The dashed curve has been obtained from the max- imum of d2E/λ2and identifies a smooth crossover from large to small polaron for large γ/tvalues. The solid line is the variational result of Eq.(12). The graded gray (violet) sca le refers to the polaron density probability at R=0. theγ/t-λphase diagram of Fig. 2, where the filled circles are the calculated values of λ∗∗, while the filled squares mark the onset of the small polaron regime ( λ∗).13The resulting diagram is therefore composed of three sepa- rate regions: a delocalized electron with EGS=E0for γ/t>λ∗∗(white region), a small polaron state for large el-ph couplings ( λ>λ∗), and a new ground state in the region comprised between the λ∗∗andλ∗lines. As it can be inferred from the insets of Fig. 1 and from the gray (violet) scale of Fig. 2, in this region the den- sity probability at R=0,|Φ0|2, is lower than the small polaron solution, but substantially larger than zero as long asγ/negationslash= 0, and increasing with γ/t. The region be- tween theλ∗∗andλ∗lines identifies therefore a large po- laronstatecreatedbytheSOinteraction,withalocalized wave function which may extent over severallattice sites. The large polaron nature of this solution is substantiated in Fig. 3, where the polaron localization radius RP, ex- tracted from a fit of |ΦR|2to exp(−|R|/RP) (see inset), is plotted as a function of γ/tforλ= 0.4, 0.6, and 0.8. Although a numerical evaluation of RPforγ/t→0 is hampered by the finite size ofthe lattice, RPturns out to be approximately proportional to t/γ, suggesting there- fore that the large polaron evolves continuously towards a delocalized electron as γ/t→0. Further insight on the large polaron state, and in par- ticular on its behavior as γ/t→0, can be gained by a simple variational calculation in the continuum. In fact, as long as RPis much larger than the lattice constant (RP≫1) then an upper bound for EGScan be obtained/s48/s46/s49 /s49 /s49/s48/s49/s49/s48 /s32/s32/s82 /s80 /s32/s47/s32/s116/s32 /s32/s61/s32/s48/s46/s52 /s32 /s32/s61/s32/s48/s46/s54 /s32 /s32/s61/s32/s48/s46/s56/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s49/s48/s45/s49/s50/s49/s48/s45/s49/s48/s49/s48/s45/s56/s49/s48/s45/s54/s49/s48/s45/s52/s49/s48/s45/s50/s49/s48/s48 /s49/s46/s48/s48/s46/s53/s48/s46/s50/s53 /s32/s32/s82/s50 /s47/s32 /s48/s50 /s82 FIG. 3: (Color online) Polaron radius RPof the large polaron state as a function of γ/tand for different el-ph couplings λ. Inset: Density probability (symbols) of the large polaron f or λ= 0.4 andγ/t= 0.25, 0.5, and 1.0 as a function of distance R=|R|along the (1 ,0) direction. The solid lines are fits to exp(−|R|/RP). from a minimization of the energy functional E[Φ(r)] =/integraldisplay drΦ†(r)/bracketleftbig tˆp2+γ(σyˆpx−σxˆpy)/bracketrightbig Φ(r) −EP/integraldisplay dr|Φ(r)|4, (6) where ˆpq=−i∂/∂qis the electron momentum operator (q=x,yand/planckover2pi1= 1) and ˆp2= ˆp2 x+ ˆp2 y. In the above expression, Φ( r) is a suitable ansatz for the ground state spinor, which is assumed to vary slowly over distances comparabletothe latticespacing. InwritingEq.(6), only the lowest order terms in the lattice constant have been retained, which amounts to consider a parabolic band with a Rashba coupling linear in the momentum oper- ators. One can then use for Φ( r) an ansatz which has been already introduced in studying the effects of a lin- ear Rashba term on the 2D Fr¨ ohlich polaron and the 2D hydrogen atom:8,14 Φ(r) =Aexp(−ar)/bracketleftbigg J0(br) J1(br)eiϕ/bracketrightbigg . (7) Here,r=|r|andϕis the azimuthal angle, Ais a nor- malization constant, J0andJ1are Bessel functions, and aandbare variational parameters. By using (7) and the properties of the Bessel functions, equation (6) reduces to E=t(a2+b2)−γb−EP 2π/integraltext∞ 0drre−4arF(br)2 /bracketleftbig/integraltext∞ 0drre−2arF(br)/bracketrightbig2 ≃t(a2+b2)−γb−2EPa2 πln/parenleftbiggb√ea/parenrightbigg , (8) whereF(br) =J0(br)2+J1(br)2. The second equality stems from assuming a≪b, which is the relevant limit of4 the large polaron regime. Minimization of Ewith respect toaandbleads to two possible solutions: b=γ/(2t) and a= 0, which corresponds to a delocalized electron with Emin=E0=−4t−γ2/(4t), and a=bexp/parenleftBig −1−π 8λ/parenrightBig , b=γ/(2t) 1−4λexp[−2−π/(4λ)]/π, (9) which represents the large polaron solution with Emin−E0=−λ πγ2 texp/parenleftBig −2−π 4λ/parenrightBig ,(10) forλsmall. Since Eq.(10) is an upper bound for ∆ EGS= EGS−E0, then that the large polaron state has energy always lower than the delocalized electron. Furthermore, by realizing that the variational parameter arepresents the polaron radius through a= 1/(2RP), it turns out from Eq.(9) that RPscalesast/γ, in agreementtherefore with the results of Fig. 3. The finding that a large polaron is formed for γ/t/negationslash= 0 is in accord with the observation of Ref.[6] that per- turbation theory breaks down in the adiabatic limit for any finite λ. This breakdown basically stems from the one-dimensional-like divergence of the density of states (DOS) of a parabolic band with linear Rashba coupling.6,7 Although the variational result presented above cor- rectly predicts the appearance of the large polaron state as soon asγ/t/negationslash= 0, it fails nevertheless in describing the λ∗∗transition line of Fig. 2 separating the large polaron state from the delocalized solution. This is because the lowest order expansion in the lattice constant of Eq. (6) neglects higher order powers of the momentum operator arising from the lattice Rashba term, which shift the van Hove divergence of the DOS from E0to higher energies,9 therebymakingtheperturbationtheorynon-singular. To investigate this point within the variational method, it suffices to expand the discrete Hamiltonian up to the third order in the lattice constant. This corresponds to add to the energy functional (6) the following contribu- tion E′[Φ(R)] =γ 6/integraldisplay drΦ†(r)/parenleftbig σxˆp3 y−σyˆp3 x/parenrightbig Φ(r),(11) which, by using again the ansatz (7) and for a≪b, leads to the third order correction term E′= (γ/8)(b3+3a2b− πa3) to Eq.(8). It is then easy to shown that [ E+E′]min− E0is negative [with E0=−4t−γ2/(4t)+γ4/(128t3)] as long asγ/t<λ∗∗ var, where for λsmall λ∗∗ var= 8/radicalbigg 2λ πexp/parenleftBig −1−π 8λ/parenrightBig . (12) Although Eq. (12) provides only a lower bound for λ∗∗ (solid line in Fig. 2), it shows nevertheless that, as γ/tis enhanced for fixed λ, the transition from the large po- laron to the delocalized electron state originates from higher orderofthe SO interactionthan the linear Rashba coupling.IV. DISCUSSION AND CONCLUSIONS Let us discuss now the significance of the results re- ported above for materials of interest and possible con- sequences for spintronics applications. First of all, it is important to identify the region in the phase diagram of Fig. 2 where realistic values of γ/tandλare expected to fall. This is easily done by realizing that the largest Rashba SO coupling to date is that found in the surface stats of Bi/Ag(111) surface alloys15for whichγ/t≈1.4 can be estimated. Other 2D systems and heterostruc- tures have lower or much lower γ/tvalues. Concerning the coupling to the phonons, a survey16on the el-ph in- teraction at metal surfaces evidences that λis usually lower than 0 .6-0.7 (see also Ref.[17]), at least for the sur- face states with large SO splittings ( i.e.Ag, Cu, Bi). It is therefore a rather conserving assumption to confine to γ/t/lessorsimilar1 andλ/lessorsimilar1 the region of interest for the micro- scopic parameters which, as shown in Fig. 2, is substan- tially dominated by the SO induced large polaron state. Hence, upontuningoftheRashbaSOcoupling,adelocal- izedelectronat γ/t= 0caninprinciplebechangedintoa self-trapped large polaron state for γ/t>0, with obvious consequences on the spin propagation in the system. In passing,itisworthnoticingthatthesmallpolaronregime instead is affected rather weakly by the SO interaction forγ/t/lessorsimilar1, while its weakening gets pronounced only for unrealistically large values of γ/t(see also Fig. 1). Before concluding, it is important to discuss a last im- portant point. Although the adiabatic limit employed here allows for a clear identification of the λ∗andλ∗∗ transition lines, the energy gain associated to the large polaron formation becomes very small in the weak cou- pling and small SO limits [see Eq. (10)]. In this regime, the inclusion of quantum fluctuations which arise as soon asω0/t/negationslash= 0 may wash out completely any signature (like e.g. an anomalous enhancement of the electron effective massm∗) of the large polaron state, even for ω0/tsmall, while they should remain visible for larger λandγ/tval- ues. For a more complete description of the SO effects on the Holstein-Rashba polaron, it is therefore necessary to extend the study to the non-adiabatic regime ω0/t/negationslash= 0, by keeping however in mind that, as discussed above, relevant materials have ω0/t≪1. In summary, the complete phase diagram of the 2D adiabatic Holstein el-ph Hamiltonian in the presence of Rashba SO coupling has been calculated. It has been shown that a self-trapped large polaron state is created by the SO interaction in a wide region of the phase dia- gram, and that its localization radius can be modulated by the SO coupling. This result implies that, for realistic values of the microscopic parameters, the appearance of a self-trapped large polaron state is a potentially detri- mental factor for spin transport.5 Acknowledgments The author thanks E. Cappelluti, S. Ciuchi, and F. Marsiglio for valuable comments. 1D. Awschalom and N. Samarth, Physics 2, 50 (2009). 2I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 3E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). 4T. Holstein, Ann. Phys. 8, 325 (1959); 8, 343 (1959). 5H. Fr¨ ohlich, Adv. Phys. 3, 325 (1954). 6E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev. B76, 085334 (2007). 7E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev. Lett.98, 167002 (2007).. 8C. Grimaldi, Phys. Rev. B 77, 024306 (2008). 9L. Covaci and M. Berciu, Phys. Rev. Lett. 102, 186403 (2009). 10L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 94, 016602 (2005). 11It should be notes also that large SO splittings are ex- pected in systems whose constituting elements have large atomic number Z, and so large mass number. As a ruleof thumb therefore, larger values of γare accompanied by lower phonon frequencies ω0. 12A. Lagendijk and H. De Raedt, Phys. Lett. A 108, 91 (1985); V. V. Kabanov and O. Yu. Mashtakov, Phys. Rev. B47, 6060 (1993). 13Despite that lattices up to 1001 ×1001 sites have been considered in compiling Fig. 2, it has not been possible to identify with sufficient accuracy the delocalized electron / large polaron transition line λ∗∗forγ/t <0.2, because of the tiny energy differences involved. 14C. Grimaldi, Phys. Rev. B 77, 113308 (2008). 15C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D. Pacil´ e, P. Bruno, K. Kern, and M. Grioni, Phys. Rev. Lett.98, 186807 (2007). 16J. Kr¨ oger, Rep. Prog. Phys. 69, 899 (2006). 17Ph. Hofmann, Prog. Surf. Sci. 81, 191 (2006).

2106.04874v2.Chiral_control_of_quantum_states_in_non_Hermitian_spin_orbit_coupled_fermions.pdf

Chiral control of quantum states in non-Hermitian spin-orbit-coupled fermions Zejian Ren,1Dong Liu,1Entong Zhao,1Chengdong He,1Ka Kwan Pak,1Jensen Li,1, 2,and Gyu-Boong Jo1, 2,y 1Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 2IAS Center for Quantum Technologies, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China Spin-orbit coupling is an essential mechanism underly- ing quantum phenomena such as the spin Hall effect and topological insulators [1]. It has been widely studied in well-isolated Hermitian systems, but much less is known about the role dissipation plays in spin-orbit-coupled sys- tems [2]. Here, we implement dissipative spin-orbit- coupled bands filled with ultracold fermions, and observe parity-time symmetry breaking as a result of the competi- tion between the spin-orbit coupling and dissipation. Tun- able dissipation, introduced by state-selective atom loss, enables us to tune the energy gap and close it at the criti- cal dissipation value, the so-called exceptional point [3]. In the vicinity of the critical point, the state evolution exhibits a chiral response, which enables us to tune the spin-orbit coupling and dissipation dynamically, revealing topologi- cally robust chiral spin transfer when the quantum state encircles the exceptional point. This demonstrates that we can explore non-Hermitian topological states with spin- orbit coupling. An open quantum system that does not conserve energy is effectively described by a non-Hermitian Hamiltonian [2] and exhibits various counterintuitive phenomena that cannot exist when the system is Hermitian. One such example is the fun- damental understanding of non-Hermitian topological matter that may require subtle classification in contrast to the Her- mitian topological system, such as iconic topological insula- tors [1]. Although extensive theoretical research [2, 4–6] and experimental works [7–9] have been carried out on the non- Hermitian topological band, how the non-Hermitian topolog- ical state can be classified remains elusive. In particular, spin- orbit coupling (SOC), a key mechanism driving the non-trivial band topology, has not been investigated in non-Hermitian quantum systems. Recently, considerable efforts have been made in ultracold atoms to explore synthetic SOCs [10–13] and associated topological bands [14–17], and therefore ultra- cold atoms offer the unprecedented possibility of studying the non-Hermitian SOC mechanism, a critical step toward realiz- ing non-Hermitian topological phases [2]. In this work, we make an important step in this direction by realizing non-Hermitian spin-orbit-coupled quantum gases and oberving a parity-time ( PT) symmetry-breaking transi- tion as a result of the competition between SOC and dissipa- tion. We implement synthetic SOC for ultracold fermions [10] together with non-Hermiticity tunable in time. This control- Electronic address: jensenli@ust.hk yElectronic address: gbjo@ust.hklability allows us to investigate how the energy spectrum of a spin-orbit-coupled system changes with dissipation and ex- plore thePTsymmetry-breaking transition across the excep- tional point (EP) and its topological nature [3]. Exploring the parameter regime from SOC-dominated to loss-dominated be- havior, we identify an EP where the PTsymmetry-breaking transition occurs in a fully quantum regime. Finally, we exper- imentally probe the chiral property of the EP by encircling it in a parameter space and showing chiral quantum state transfer due to the breakdown of adiabaticity. Our work sets the stage for the experimental study of many-body states in the com- plex energy bands across the PT symmetry-breaking transi- tion. Recently, the feasibility of realizing unprecedented phe- nomena in non-Hermitian atomic systems was noted, includ- ing enhanced fermionic superfluids [18], the generalized Flo- quet time crystal with dissipation [19] and higher-order topo- logical phases [20]. This work complements the non-Hermitian phenomena observed in classical systems including topological energy transfer [21–23], enhanced sensing [24], PT symmetric las- ing [25], and novel topological entities such as exceptional rings [26] and non-Hermitian topological edge states [27]. Nevertheless, the role that non-Hermiticity plays in the gen- uine quantum regime ranging from few-body to many-body systems remains largely unexplored. Although recent works have explored non-Hermitian systems within a quantum framework [28] and have demonstrated the PT symmetry- breaking transition in non-Hermitian quantum systems, such as photons [29, 30], superconducting qubits [31], single elec- tronic spins [32], exciton-polaritons [33], or atoms [34, 35], the dynamic evolution of the quantum state near the EP has been limited to the single particle quantum system [36]. Here, we demonstrate the chiral control of the quantum state with ul- tracold fermions showcasing the many-particle quantum sys- tems. Our experiment begins by loading ultracold fermions of 173Yb atoms into the engineered energy band [15], in which two hyperfine states (labeled as the j"iandj#i) are coupled by a pair of Raman transition beams resulting in synthetic SOC with equal strengths of the Rashba and Dresselhaus SOC fields [10]. In a typical energy dispersion, the energy gap be- tween two dressed bands is opened by Raman coupling R. Adding the spin-dependent atom loss ";#, we realize a tun- able non-Hermitian Hamiltonian, H= ~2 2m(qxkr)2+=2 R=2 R=2~2 2m(qx+kr)2=2! +Hloss whereHloss=i 2( "j"ih"j + #j#ih#j ),qxis the quasi-arXiv:2106.04874v2 [cond-mat.quant-gas] 16 Dec 20212 mF=3/2 5/2Raman5/2 3/2mF’=1/2 1.2MHz6.9MHz δ/2 5 0 -5 q y(k r) 024 -2-4 qx(kr) q y(k r) qX(kr)-5 0 55 0 -5 a b c d e 8 6 4 2 0 -2 1 22015 10 5 qx(kr)γ¼(Er)Energy band(Er) 8 6 4 2 0 1 22015 10 5 qx(kr) γ¼(Er)Damping rate band(Er)0 5 10 15 (Er)01234Energy band(Er) 0 5 10 15 (Er)02468Damping rate band(Er) -0.500.5 -0.500.5 Ω γ» γ ¼γ ¼ Raman RamanYb xzyB |↑/angbracketright |↓/angbracketright |↑/angbracketright |↓/angbracketrightExceptional poin t Exceptional poin t FIG. 1: Non-Hermitian spin-orbit-coupled system of ultracold atoms a , Our quantum system comprises ultracold fermions with two spin states,j"i(red) andj#i(blue), coupled by Raman beams (green arrow). Atom loss is controlled by additional light (not shown), resulting in spin-dependent loss ";#.b, Schematic energy level diagram with relevant transitions. The atom loss beam is de- tuned by 1.2 MHz (6.9 MHz) from the jmF= 3=2i!jmF0= 1=2itransition (jmF= 5=2i!jmF0= 3=2itransition). c, Den- sity distribution of j"i(red) andj#i(blue) atoms with SOC and its cross-sectional profile along the xdirection after 6 ms time-of-flight expansion. d, Complex energy bands shown by the real and imag- inary parts of in the parameter space of quasimomentum ( qx) and dissipation ( #) for= 4Erand R= 4:5Er.e, Energy band (Re()) and damping rate (-Im ()) showing the EP at #=9.75Er alongqx= 1:0kr(gray plane in (D)) when 2 R= ( # "). momentum of atoms along the ^x-direction,mis the mass of the ytterbium atom and is the two-photon detuning. Here, we define natural units of momentum and energy as ~kr= sin( 2)2~ 556andEr=~2k2 r 2m=h1.41 kHz, where 556=556 nm. Then, the real momentum kxis related to the quasimomentum as kx=qxkrfor spin-up and spin-down, respectively. In our experiment, atom loss is induced by thesingle near-resonant beam (Fig. 1b), resulting in the fixed ratio of #= "= 13 . The quantum dynamics of two dressed energy bands fj+i;jig , corresponding to the eigenvalues ofH, are governed by the eigenvalue difference
= (+). In theqx- #parameter space , the PTsymmetry-breaking tran- sition occurs at the EP where both eigenvalues and eigenvec- tors coalesce (Fig. 1d and e). When the system is Hermitian with ";#=0, the gap is opened at qx= ( 4Er)kr[10]. With finite atom loss ( ";#6=0), however, the energy gap is re- duced and eventually closed at the critical value (i.e., the EP), above which the eigenvalue difference becomes complex as Im(
)6=0. We now investigate how the spin-orbit-coupled band is affected by dissipation via momentum-sensitive Rabi spec- troscopy. We begin with a spin-polarized degenerate Fermi gas of 2104atoms inj"iwithout the Raman transition and loss (i.e., R= ";#= 0). Subsequently,j"iatoms are trans- ferred to thej#istate when the Raman coupling is switched on (see Fig. 1c). Here, brief pulses of the Raman coupling and loss beams are applied for a variable time duration fol- lowed by 6 ms ballistic expansion. The time-of-flight expan- sion maps momentum to real space, allowing direct momen- tum resolution of the energy band. One of the features associated with dissipation is the closing of the energy gap at the EP [26]. Fig. 2a shows typical spin- orbit-coupled energy bands at different dissipation strengths for=4Erand their Rabi oscillations pulsing in the Raman field with dissipation for a variable duration (Fig. 2b), which reveals the nonuniform energy gap of the dressed bands. In the absence of atom loss ( ";#=0), the state undergoes Rabi oscillation subject to the energy of the gap at each quasimo- mentum. The Rabi coupling is resonant at qx= 1:0kr, reveal- ing the smallest energy gap. The Rabi oscillation becomes slower with increasing dissipation. By fitting the spin oscil- lation with a damped sinusoidal function, we determine the energy gap and damping rate at each quasimomentum as de- scribed in Fig. 2b. A Fermi sea collectively undergoes Rabi oscillations in contrast to the non-Hermitian classical [3] or single-particle quantum [36] system, revealing a nonuniform band gap in time-dependent spin textures (see Fig. 3a). From this quasimomentum-dependent Rabi oscillation, we extract the energy gap, Re (
), between two dressed bands for different dissipation strengths, as shown in Fig. 3b, in good agreement with the theoretical expectation. The dissipation gives rise to gap closing at the resonant coupling at qx=1.0kr, while the other quasimomenta still reveal a finite energy gap. Fig. 3d and e show a quantitative measurement of the band gap (Re (
)) and damping rate (Im (
)) of the non-Hermitian system at qx=1.0kr, manifesting the PT symmetry-breaking transition. In the PT symmetric phase, the initial quantum state oscillates between two eigenstates of the non-Hermitian Hamiltonian (see Fig. 3c). Above the EP, however, the strong dissipation completely closes the en- ergy gap with finite Im (
)6=0, giving rise to monotonic be- havior, as shown in Fig. 3c. Both the band gap and damping rate show thePT symmetry-breaking transition near the EP3 -2 0 2 4 6 qx(kr)-202468Energy band(Er) -2 0 2 4 6 qx(kr)-2 0 2 4 6 qx(kr) -10.5 01 -0.5 050100 150 200 250300 Time(us)050100 150 200 250300 Time(us)050100 150 200 250300 Time(us) Spin polarization qx= 0.2 qx= 0.5 qx= 1 qx= 0.2 qx= 0.5 qx= 1 -0.500.5|↑/angbracketright |↓/angbracketright a bγ¼= 0Er 8Er 10Er qx= 0.2 qx= 0.5 qx= 1 FIG. 2: Momentum-resolved Rabi spectroscopy for observing band merging a , Single-particle energy dispersion of dressed states with increasing dissipation. The energy gap at the resonant quasimomentum qx= 1:0kris closed above the EP at #=9.75Er.b, Rabi oscillation for=+4Erand R=4.5Er. Atoms, prepared in j"i, are suddenly projected into a superposition of eigenstates revealing a nonuniform energy gap. The spin polarization is averaged taking into account the finite optical resolution over 0.15kr. at #=9.75Er(see Fig. 3d,e). We can trace the energy gap at a nonresonant momentum, qx=0.5kr, as shown in the inset of Fig. 3d. In this case, the energy gap saturates to a finite energy gap. With large dissipation, the spin-orbit-coupled band struc- ture becomes similar to free particle dispersion, which may be related to the quantum Zeno effect (see Methods) [37]. We now explore topological features near the EP by dy- namically encircling it with fermions. Near the EP, the state evolution is direction dependent, resulting in intriguing chi- ral behavior when encircling the EP due to the breakdown of adiabaticity in non-Hermitian systems [3]. While this topo- logical chirality has been observed in classical systems [21– 23, 33, 38–41], such as photonics and acoustics, it remains largely unexplored in quantum systems [36], especially in many-body quantum systems. In contrast to classical systems, the quantum system with time-varying dissipation may allow for full Hamiltonian engineering capability. Here, we demonstrate full time-varying control of a non- Hermitian Hamiltonian such that the EP is effectively encir- cled by ultracold fermions in different directions. This is enabled by the EP occurring at the quasimomentum qx= ( 4Er)krwhere SOC is resonant. This leads to control of the EP position along the qxaxis, effectively with respect to the Fermi sea in the dressed band, by tuning . In our experi- ment, we simultaneously tune the loss ( #) and two-photon detuning () and dynamically explore arbitrary paths in the complex band. We begin with adiabatic loading of fermions into the low- est energy band at = +3Er. Initially, a Fermi sea is adi- abatically formed at approximately qx' 0.8krwith the dominant spin component of j#i(blue color of energy band in Fig. 4), whereas the EP occurs at qx= ( 4Er)kr=0.75krin theqx #parameter space. We now tune from +3Er to -6Er, which shifts the EP to qx=-1.5kr. Subsequently, the loss is increased from zero to different max #, followed by consecutive control of the two-photon detuning and loss. This results in counterclockwise encircling along a closed loop withinT=10.1 ms in the qx #parameter space, as depicted in Fig. 4a (left), where Tis the total encircling time (see the Supplementary Materials). By reversing the aforementioned process, we can encircle the EP in a clockwise manner. Then, the spin polarization of the quantum state after the encircling is determined via an optical Stern-Gerlach pulse followed by ballistic expansion. When the atom loss #is increased to max #=10 or 8Er, we observe the initial quan- tum statej#iis selectively switched to spin up depending on the encircling direction (Fig. 4b). This observation reveals the evolution of the quantum state switches the state to a different eigenstate near the EP [23] depending on the encircling direc- tion. The chiral spin transfer disappears for max #=2Erif the EP is located far from the loop (Fig. 4c). To better understand the quantum dynamics during the nonadiabatic evolution, we perform numerical calculations in a larger domain of accessible parameters in max # and encir- cling duration. The state conversion efficiencies Cfor encir- cling in the two directions are numerically calculated with considering each momentum sector integrated over the Fermi sea, and are plotted as two color maps in Fig. 4a. The exper- imental results in Fig. 4b and c (labeled 1 to 4) are indicated by circles at T=10.1 ms, in good agreement with C 0 for the case 1 andC=0 for the other cases. In fact, a smaller encir- cling speed (i.e., a larger T) increases the chance of nonadia- batic transitions as long as the trajectory runs near the EP and further lowers the transition threshold of max # [23]. The chi-4 0 0.5 1 1.5 qx(kr)01234567Energy gap(Er) 0.5 -0.50 d 0123456Band gap(Er) 0Er 5Er 8Er 13ErLoss γ¼ |↑/angbracketright |↓/angbracketright 0 100 200 300 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 qx(kr) 0 0.5 1 1.50Er 5Er 8Er 13Er0 0.5 1 1.5Loss γ¼Time(us) 10Er a bc 0 100 200 Time(us)-0.50.5 Spin polarization 0 100 200 300 Time(us)-101 0 2 4 6 8 10 12 14 16 18 γ¼(Er) 01234567Damping rate(Er)d e300qx=1 =0Er qx=1 =10Er Exceptional poin tγ¼(Er)PT symmetric PT broken 0246Band gap(Er) 0 510 15 FIG. 3:PT symmetry-breaking transition and band closing at the EP a , Time-dependent spin texture obtained for different dissipation strengths. The quasimomentum-resolved spin polarization is monitored after switching on SOC fields with dissipation for a variable time. Each spin texture is averaged over 10 measurements. b, Energy band gap (circles) measured via Rabi spectroscopy, which is in good agreement with theory (dashed lines). We estimate the uncertainty of the theoretical band gap (shaded region) based on calibrated atom loss and its measurement uncertainty. c, the unbroken phase with prominent Rabi oscillations (left) and the broken phase showing a monotonic spin polarization (right). d,ePhase diagram of the PT symmetry-breaking transition. Energy gap and damping rate measured from the Rabi oscillation shown with the real ( d), and imaginary ( e), eigenvalues ofH(solid lines), respectively. The shaded region indicates the uncertainty of band gap and damping rate associated with the uncertainty of atom loss. The error bars in all panels represent standard fitting errors (vertical) and the experimental uncertainty related to the calibration (horizontal). ral behavior is also observed for T=5 ms. On the other hand, we find the inability of the system to adapt to the rapidly vary- ing parameters when T0.04 ms is small as Fig. 4d (labeled 5 and 6). The chiral behavior originates from the asymme- try between the (imaginary) dynamical phase factors acquired during the CCW and CW encirclings, which results preferen- tial amplification leading to the imbalanced spin populations. A more detailed classification of the behavior of Cin different phases is given in the Supplementary Information (Fig. S4). Our system indeed provides an intriguing platform for the study of interplay between many-body statistics and non- Hermiticity. However, the current observations can be ef- fectively described within the non-Hermitian picture by ig- noring a quantum jump operator in the Lindblad equation.This is an appropriate approximation for a large number of nearly non-interacting atoms, smearing out the effect of the quantum jumps [28, 42]. Furthermore, hole excitations of the Fermi sea, which requires a full master equation approach, are rapidly relaxed in the time scale of E1 F(whereEFis the Fermi energy) due to the decoherence heating process induced by the loss beam. This rapid relaxation allow us to treat the system as the 22 non-Hermitian effective Hamiltonian. In this work, we have experimentally demonstrated how non-Hermiticity affects the dispersion relation of spin-orbit- coupled fermions inducing the PT symmetry-breaking tran- sition. The topological nature near the EP indicates that non- Hermiticity can fundamentally modify the physical proper- ties of the spin-orbit-coupled quantum system. In contrast5 ccw cw-1-0.50 0 10 20 00.51 024681012 0246810120 5 10 15 T(ms) 0 5 10 15 T(ms)γ¼NBY(Er) γ¼NBY(Er) -1-0.50 ccw cw -1-0.50 Exceptional poin ta b c d1 35 2 46 -0.500.5|↑/angbracketright |↓/angbracketright 3 4 5 6 Exceptional poin t1 2Spin polarizationChirality OccurrenceCounterclockwise(ccw): Clockwise(cw): Spin polarization Spin polarization Band energy(Er) γ¼(Er) qx(kr) Band energy(Er) γ¼(Er) qx(kr)-1 0 148 0.5 -0.5Encirling paths 48 -1 0 10.5 -0.5 48 -1 0 10.5 -0.5γ¼(Er) γ¼(Er) γ¼(Er) κ FIG. 4: Topological chiral spin transfer by dynamically encircling an EP a , A Fermi gas is first adiabatically prepared in the initial j#i-dominated state (spin polarization indicated by blue) in the parameter space with #= 0Erand= 3Er(left panel). We effectively encircle the EP with fermions by tuning the loss and two-photon detuning (green arrow). The spin polarization before dynamic encircling is approximately -0.7 in the initial state. The right panel of (a) shows expected state conversion efficiency Cfromj#itoj"iagainst loop duration time T and the maximum atom loss max # for the counterclockwise (top) and clockwise (bottom) encircling in the parameter space. The dominant spin polarization of the quantum state measured after the encircling is indicated by the color of the circles and stars. The final spin polarization after encircling is experimentally measured for different Tand max # as indicated by the colour of the circle or star, after following the trajectory along clockwise (CW) and counterclockwise (CCW) closed loops. The horizontal dashed line indicates the EP. b-c, show the chiral behavior of final spin polarization for the experimental configurations indicated as circles with the same indices in (a). For other values ofT, we keep the same fraction of time for each edge of the trajectory as in the experiment. d, No chirality is observed when dynamic encircling is instantaneous within 0.04 ms corresponding to solid star in the right panel of (a). For all cases of (b-d), corresponding histograms for the 100 consecutive measurement series with a binning width of 0.02 for spin polarization are shown together with the encircling paths of the Fermi sea (green region) in the parameter space where the exceptional point is indicated by the yellow circle and =qx=4(see the Supplementary Information for more deatils). The error bar represents the standard deviation of the measurements. to classical systems where only single (bosonic) particle dy- namics are considered, our system sets the stage for investi- gating many-body interacting fermions with dissipation [43]. Furthermore, the possibility of exploring non-equilibrium dy- namics, quantum thermodynamics [44] and information criti- cality [45] across the PTsymmetry-breaking transition by en- gineering the non-Hermitian Hamiltonian in a dynamic man- ner is conceivable. Acknowledgments G.-B.J. acknowledges support from the RGC, the Croucher Foundation and Hari Harilela foundation through 16305317, 16304918, 16306119, 16302420, C 6005- 17G and N-HKUST601/17. J.L. acknowledges support from the RGC through 16304520 and C6013-18G. Competing interests The authors declare no competing inter- ests Data availability The data that support the finding of this work are available from the corresponding authors upon reasonable request.Author contributions Z.R., E.Z., C.H. and K.K.P. carried out the experiment and data analysis and helped with numerical calculations. D.L. performed theoretical calculations. G.-B.J. and J.L. and supervised the research.6 [1] M. Hasan and C. Kane, Colloquium: Topological insulators. Reviews of Modern Physics 82, 3045 (2010). [2] Y . Ashida, Z. Gong and M. Ueda, Non-Hermitian physics. Adv. Phy. 69249-435 (2020). [3] M.-A. Miri and A. Alu, Exceptional points in optics and pho- tonics . Science 363, eaar7709 (2019). [4] Y . Xu, S.-T. Wang, and L.-M. Duan, Weyl, Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas. Phys. Rev. Lett. 118, 045701 (2017). [5] Z. Gong, Y . Ashida, K. Kawabata, K. Takasan, S. Higashikawa and M. Ueda, Topological Phases of Non-Hermitian Systems. Physical Review X 8, (2018). [6] S. Yao, F. Song and Z. Wang, Non-Hermitian Chern Bands. Physical review letters 121, 136802 (2018). [7] T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter and R. Thomale, Generalized bulk-boundary correspondence in non-Hermitian topoelectrical circuits. Nature Physics 16, 747 (2020). [8] L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi and P. Xue, Non-Hermitian bulk-boundary correspondence in quantum dy- namics. Nature Physics 16, 761 (2020). [9] S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale and A. Szameit, Topolog- ical funneling of light . Science 368, 311 (2020). [10] Y .-J. Lin, K. Jimenez-Garcia and I. B. Spielman, Spin-orbit- coupled Bose-Einstein condensates. Nature 471, 83 (2011). [11] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai and J. Zhang, Spin-Orbit Coupled Degenerate Fermi Gases. Physical Review Letters 109, 095301 (2012). [12] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr and M. W. Zwierlein, Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas. Physical Review Letters 109, 095302 (2012). [13] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental realization of two- dimensional synthetic spin-orbit coupling in ultracold Fermi gases. Nat.Phys 12, 540 (2016). [14] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y . Deng, S. Chen, X.-J. Liu and J.-W. Pan, Realization of two- dimensional spin-orbit coupling for Bose-Einstein condensates. Science 354, 83 (2016). [15] B. Song, L. Zhang, C. He, T. F. J. Poon, E. Hajiyev, S. Zhang, X.-J. Liu and G.-B. Jo, Observation of symmetry-protected topological band with ultracold fermions. Science advances 4, eaao4748 (2018). [16] B. Song, C. He, S. Niu, L. Zhang, Z. Ren, X.-J. Liu and G.-B. Jo, Observation of nodal-line semimetal with ultracold fermions in an optical lattice. Nature Physics 15, 911 (2019). [17] Z.-Y . Wang, X.-C. Cheng, B.-Z. Wang, J.-Y . Zhang, Y .-H. Lu, C.-R. Yi, S. Niu, Y . Deng, X.-J. Liu, S. Chen et al., Realization of an ideal Weyl semimetal band in a quantum gas with 3D spin-orbit coupling. Science 372, 271 (2021). [18] K. Yamamoto, M. Nakagawa, K. Adachi, K. Takasan, M. Ueda and N. Kawakami, Theory of Non-Hermitian Fermionic Super- fluidity with a Complex-Valued Interaction. Physical review let- ters123, (2019). [19] A. Lazarides, S. Roy, F. Piazza and R. Moessner, Time crys- tallinity in dissipative Floquet systems. Physical Review Re- search 2, 022002 (2020). [20] X.-W. Luo and C. Zhang, Higher-Order Topological CornerStates Induced by Gain and Loss. Physical review letters 123, (2019). [21] H. Xu, D. Mason, L. Jiang and J. G. E. Harris, Topological energy transfer in an optomechanical system with exceptional points. Nature 537, 80 (2016). [22] J. Doppler, A. A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev and S. Rotter, Dy- namically encircling an exceptional point for asymmetric mode switching. Nature 537, 76 (2016). [23] A. U. Hassan, G. L. Galmiche, G. Harari, P. LiKamWa, M. Kha- javikhan, M. Segev and D. N. Christodoulides, Chiral state con- version without encircling an exceptional point. Physical Re- view A 96, 052129 (2017). [24] J. Wiersig, Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection. Physical Review Letters 112, 203901 (2014). [25] M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci and S. Rotter, Pump-Induced Exceptional Points in Lasers. Physical Review Letters 108, 173901 (2012). [26] B. Zhen, C. W. Hsu, Y . Igarashi, L. Lu, I. Kaminer, A. Pick, S. L. Chua, J. D. Joannopoulos and M. Soljacic, Spawning rings of exceptional points out of Dirac cones. Nature 525, 354 (2015). [27] H. Zhao, X. Qiao, T. Wu, B. Midya, S. Longhi and L. Feng, Non-Hermitian topological light steering. Science 365, 1163 (2019). [28] F. Minganti, A. Miranowicz, R. W. Chhajlany and F. Nori, Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps. Physical Re- view A 100, 062131 (2019). [29] L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B.C. Sanders and P. Xue, Observation of topological edge states in parity-time-symmetric quantum walks. Nature Physics 13, 1117 (2017). [30] F. E. Ozturk, T. Lappe, G. Hellmann, J. Schmitt, J. Klaers, F. Vewinger, J. Kroha and M. Weitz, Observation of a non- Hermitian phase transition in an optical quantum gas. Science 372, 88 (2021). [31] M. Naghiloo, M. Abbasi, Y . N. Joglekar and K. W. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit. Nature Physics 15, 1232 (2019). [32] Y . Wu, W. Liu, J. Geng, X. Song, X. Ye, C.-K. Duan, X. Rong and J. Du, Observation of parity-time symmetry breaking in a single-spin system. Science 364, 878 (2019). [33] T. Gao, E. Estrecho, K. Y . Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Hofling, Y . Yamamoto et al., Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard. Nature 526, 554 (2015). [34] J. Li, A. K. Harter, J. Liu, L. d. Melo, Y . N. Joglekar and L. Luo, Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms. Nature Com- munications 10, 855 (2019). [35] Y . Takasu, T. Yagami, Y . Ashida, R. Hamazaki, Y . Kuno and Y . Takahashi, PT-symmetric non-Hermitian quantum many-body system using ultracold atoms in an optical lattice with con- trolled dissipation. Progress of Theoretical and Experimental Physics 2020 , ptaa094 (2020). [36] W. Liu, Y . Wu, C.-K. Duan, X. Rong and J. Du, Dynamically Encircling an Exceptional Point in a Real Quantum System.7 Physical Review Letteres 126, 170506 (2021). [37] A. G. Kofman and G. Kurizki, Acceleration of quantum decay processes by frequent observations. Nature 405, 546 (2000). [38] A. A. Mailybaev, O. N. Kirillov and A. P. Seyranian, Geometric phase around exceptional points. Physical Review A 72, 014104 (2005). [39] R. Uzdin, A. Mailybaev and N. Moiseyev, On the observability and asymmetry of adiabatic state flips generated by exceptional points. Journal of Physics A: Mathematical and Theoretical 44, 435302 (2011). [40] J. H. Wu, M. Artoni and G. C. L. Rocca, Non-hermitian degen- eracies and unidirectional reflectionless atomic lattices. Physi- cal Review Letters 113, 123004 (2014). [41] J. W. Yoon, Y . Choi, C. Hahn, G. Kim, S. H. Song, K.-Y . Yang, J. Y . Lee, Y . Kim, C. S. Lee, J. K. Shin, H.S. Lee and P. Berini, Time-asymmetric loop around an exceptional point over the full optical communications band.. Nature 562, 86 (2018). [42] A. J. Daley, Daley. Advances in Physics 63, 77 (2014). [43] L. Zhou, W. Yi and X. Cui, Dissipation-facilitated molecules in a Fermi gas with non-Hermitian spin-orbit coupling. Physical Review A 102, 043310 (2020). [44] S. Deffner and A. Saxena, Jarzynski Equality in PT-Symmetric Quantum Mechanics. Physical Review Letters 114, 150601 (2015). [45] K. Kawabata, Y . Ashida and M. Ueda, Information Retrieval and Criticality in Parity-Time-Symmetric Systems. Physical Review Letters 119, 190401 (2017).8 Methods Preparation of the sample We slow down173Yb atoms through the two-stage pro- cess and pre-cool them in the intercombination magneto- optical trap. We then begin with the experiment for a two- component degenerate173Yb Fermi gas of 2104atoms and prepared at T=TF.0.3(1), where TF'kB400 nK, following forced evaporative cooling in an optical dipole trap formed by 1064 nm and 532 nm laser beams with a trap frequency of != (!x!y!z)1=3=1122Hz. Here j";#i=jmF= 5=2;mF= 3=2irepresent hyperfine states of the ground manifold with a negligible interaction strength ofkFas'0.1 wherekFis the Fermi wave vector and asde- notes the s-wave scattering length. A quantized axis is fixed by the bias magnetic field of 13.6 G along the z direction. To create a non-Hermitian SOC Hamiltonian, two hyperfine states of the ground state manifold of173Yb (labeled as the j"iandj#istates) are coupled by a pair of Raman transition beams intersecting at =76and blue-detuned by 1 GHz. Typically, before the spin-orbit coupling is switched on by a pair of two-photon Raman beams, two hyperfine levels ( j"i,j# i) are isolated from other states (i.e. mF= 1=2;1=2;3=2 and5=2) using the spin-dependent light shift induced by the polarized beam (referred to a lift beam), blue-detuned by 1 GHz, which lifts the degeneracy of the ground manifold. Within the experimental resolution, no atoms are observed in the hyperfine states other than j"iandj#i. The background heating arising from the Raman transition is negligible within the time scale of the experiment. Control of atom loss To control the dissipation of the system, we use a ded- icated plane-wave laser beam (referred to loss beam) at a small detuning around the 556 nm narrow intercom- bination transition1S0(F=5 2)$3P1(F0=7 2)with the natural linewidth of 182 kHz. The loss beam is  polarized, and is detuned by 1.2 MHz and 6.9 MHz with respect tojmF= 3=2i ! jmF0= 1=2iand jmF= 5=2i!jmF0= 3=2itransitions, respectively, which results in spin-sensitive photon scattering. The detuning of the loss beam is chosen such that the light is essentially resonant for atoms in the jmF= 3=2i=j#istate with the fixed ratio of #= "=13. Although the absorption and reemission of a photon can leave an atom back in its original state, for example, with the probability of 14%for atoms inj#i(see the relative optical transition strength betweenhyperfine states in Supplementary Information), we define the effective photon scattering rate as the genuine one-body dissipation ignoring the heating effect which is not significant in our current experimental regime. Nevertheless, the sample heats up within the time scale of E1 Fin the strong dissipation regime, for example, at the loss rate of #=10Er, resulting inT'140 nK. We expect the heating associated with the photon scattering can be further suppressed by pumping excited atoms in the3P1manifold to3S1with 680 nm light. We calibrate the photon scattering rate by fitting a functione tto the atom number of the state j=";#iafter the loss beam is suddenly switched on (see Supplementary Information for more details). We achieve tunable loss rate by controlling the power of the loss beam. The change in the two-photon detuning ( ) due to the energy shift induced by the loss beam is less than h0.1 kHz at #= 1Er. PTsymmetric Hamiltonian The HamiltonianHcan be decomposed as H=HPT i #+ " 4Iwhere we define the PT-symmetric Hamiltonian: HPT=~2 2m(qxkr)2j"ih"j +~2 2m(qx+kr)2j#ih#j +( 2+i # " 4)z+ R 2x HereP=xrepresents spin exchange operation and T= IKdenotes pseudo time reversal operation where Kis the complex conjugate operation which results in [HPT;PT] = 0whenqx= ( 4Er)kr. The constant decay term i #+ " 4Idoes not affect thePT-symmetry breaking transition. Alterna- tively, the gain and loss can be understood as being balanced at the EP by gauging outi 4( "+ #)IfromH. Near the gap where spin-orbit coupling is resonant (i.e. qx= ( 4Er)kr), the dissipative two-level system is effectively described by HPT. In the strong-dissipation limit (i.e. parity-time symmetry broken phase), the slowly-decaying eigenstate (i.e. complex energy indicated by red color in the manuscript) has a near- unity overlap with the spin-up state without spin-orbit cou- pling while the rapidly-decaying eigenstate becomes similar to the bare spin-down state. When the dissipation is extremely strong, the quantum dynamics of slowly-decaying state is ef- fectively projected onto the low-loss manifold, which is a bare energy band of spin-up atoms. Therefore, the spin-orbit cou- pling effectively disappears in this limit, closing the band gap.

1610.02375v1.Spin_orbit_coupling_controlled__J_3_2__electronic_ground_state_in_5_d__3___oxides.pdf

Title: Spin-orbit coupling controlled J= 3/2electronic ground state in 5d3oxides Authors: A. E. Taylor,1S. Calder,1R. Morrow,2H. L. Feng,3M. H. Upton,4 M. D. Lumsden,1K. Yamaura,3P. M. Woodward,2and A. D. Christianson*1, 5 1Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 2Department of Chemistry, The Ohio State University, Columbus, Ohio 43210-1185, USA 3Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan 4Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA 5Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USA *To whom correspondence should be addressed: Email: christiansad@ornl.gov Abstract: Entanglement of spin and orbital degrees of freedom drives the for- mation of novel quantum and topological physical states. Discovering new spin- orbit entangled ground states and emergent phases of matter requires both ex- perimentally probing the relevant energy scales and applying suitable theoret- ical models. Here we report resonant inelastic x-ray scattering measurements of the transition metal oxides Ca 3LiOsO 6and Ba 2YOsO 6. We invoke an in- termediate coupling approach that incorporates both spin-orbit coupling and electron-electron interactions on an even footing and reveal the ground state of 5d3based compounds, which has remained elusive in previously applied models, is a novel spin-orbit entangled J=3/2 electronic ground state. This work reveals the hidden diversity of spin-orbit controlled ground states in 5d systems and introduces a new arena in the search for spin-orbit controlled phases of matter. Main Text: The electronic ground state adopted by an ion is a fundamental determinant of manifested physical properties. Recently, the importance of spin-orbit coupling (SOC) in creating the electronic ground state in 5 d-based compounds has come to the fore andarXiv:1610.02375v1 [cond-mat.str-el] 7 Oct 20162 revealed novel routes to a host of unconventional physical states including quantum spin liquids, Weyl semimetals, and axion insulators (1, 2). As a result, major experimental and theoretical efforts have been undertaken seeking novel spin-orbit physics in various 5d systems, and the influence of SOC has now been observed in the macroscopic properties of numerous systems. However, beyond the Jeff= 1/2case such as that found in Sr 2IrO 4(3) — which is a single-hole state that applies only to idealised 5 d5ions in cubic materials — questions abound concerning the electronic ground states which govern 5dion interactions. In this context 5d3materials present a particularly intriguing puzzle, because octahedral d3configurations are expected to be orbitally-quenched S= 3/2states — in which case SOC enters only as a 3rd order perturbation (4) — yet there is clear experimental evidence that SOC influences the magnetic properties in 5 d3transition metal oxides (TMOs). This includes the observations of large, SOC-induced spin-gaps in their magnetic excitation spec- tra (5–7) and x-ray absorption branching ratios which deviate from BR =IL3/IL2= 2(8, 9). Despite this, no description beyond the S= 3/2state had been established. The emergent phenomena in 4 d3and 5d3systems, such as incredibly high magnetic transition tempera- tures (10–13), Slater insulators (14), and possible Mott-insulators (15) are therefore poorly understood. We selected 5d3TMOS Ca 3LiOsO 6and Ba 2YOsO 6as model systems in which to inves- tigate the influence of spin-orbit coupling on the electronic ground states via RIXS mea- surements on polycrystalline samples. Both materials have relatively-high magnetic ordering temperatures, TN= 117and 67K, respectively, despite large separation of nearest-neighbour Os ions of 5.4–5.9 Å(5, 10, 16). The relative isolation of Os-O octahedra allows us to un- ambiguously access the ground state, because only extended superexchange interactions are present. Stronger, close-range interactions can mask the effective single-ion levels we wish to observe (17). In Ca 3LiOsO 6the oxygen octahedra surrounding Os5+ions are very close to ideal, despite the overall non-cubic symmetry — hexagonal R¯3c(10). As previously re- ported (5), we find that Ba 2YOsO 6has the ideal cubic double perovskite structure Fm¯3m to within experimental measurement limits, see supplementary material for high-resolution3 10.874 10.878012345 Incident energy (keV) Energy loss (eV) 012345 -5 ×10Intensity (a. u.) Figure 1: Incident energy dependence of electronic excitations in Ca 3LiOsO 6. Measurements were performed at 300K. synchrotron x-ray and neutron diffraction. Figure 1 presents the x-ray energy loss, E, versus incident energy, Ei, RIXS spectra of Ca 3LiOsO 6at 300K. Four lines are present at E < 2eV, which are enhanced at Ei= 10.874keV, whereas the feature at E≈4.5eV is enhanced at Ei= 10.878keV. This indicates that theE < 2eV features are intra- t2gexcitations, whereas the higher energy feature is fromt2gtoegexcited states, as has been observed in many 5doxides (7, 18–20). Subsequent measurements were optimised to probe the t2gexcitations by fixing Ei= 10.874keV. Figure 2 presents the detailed RIXS spectra of Ca 3LiOsO 6and Ba 2YOsO 6at tempera- tures of 300 K and 6 K. In each spectrum there are 5 peaks in addition to the elastic line: four peaks with E < 2eV, labeled a, b, c and d (Fig. 2c and d) which we associated with intra-t2gexcitations, and a broad peak, e, centered at E∼4.5eV (Fig. 2a and b) associ- ated witht2gtoegexcited states. The qualitative similarity of the spectra indicates that these features are not controlled by non-cubic structural distortions, as splittings would be heightened in Ca 3LiOsO 6. At 300K the peaks a–d are resolution limited, as determined from least-squares fitting of Gaussian peaks on a flat background to the data, Fig. 2c and d. The peak energies for Ca 3LiOsO 6areaCa= 0.760(7)eV,bCa= 0.992(5)eV,cCa= 1.470(5)eV anddCa=4 0.5 1 1.5 2024x 10−6Intensity (a. u.) Energy loss (eV)Γ, Γ 86Γ7Γ8Γ80 1 2 3 4 501234x 10−5 Energy loss (eV)Intensity (a. u.) 6 K 300 K 0 1 2 3 4 50123456x 10−6 Energy loss (eV)Intensity (a. u.) 6 K 300 K 0.5 1 1.5 201234x 10−5Intensity (a. u.) Energy loss (eV)BaYOsO2 6CaLiOsO3 6 CaLiOsO3 6 Γ, Γ 86Γ7Γ8Γ8e a bc dea b c d BaYOsO2 6ab c d Figure 2: RIXS excitation spectra measured in Ca 3LiOsO 6and Ba 2YOsO 6. Panels a and b show the data for Ca 3LiOsO 6and Ba 2YOsO 6, respectively. Black lines indicate the full-width half maximum of the elastic line. Panels c and d show the energy range 0.4–2eV, i.e. the t2gmanifold, with the results of Gaussian peak fitting to the data shown as solid lines. 1.72(1)eV, and for Ba 2YOsO 6areaBa= 0.745(7)eV,bBa= 0.971(7)eV,cBa= 1.447(9)eV anddBa= 1.68(1)eV. At 6K, well below TNin both materials, the peaks appear broadened, although maintain the SOC-induced four peak character, see Fig. 2. The low temperature broadening is most pronounced in peak e — this peak is due to 20 different excited levels in the Coulomb plus SOC regime, so no discrete levels can be resolved with current RIXS capabilities. The broadening could be due to splittings from non-cubic structural distortions occurring below the magnetic ordering temperatures - some distortion should be expected from magnetoelastic coupling. However, if a purely structural distortion were the origin we5 would expect to see broadening in Ca 3LiOsO 6compared to Ba 2YOsO 6at all temperatures. It is possible that at low temperatures there is dispersion of the levels (17), or that increased hybridisation influences the accessible distribution of excited states — this would be more pronounced for eglevels which show greater oxygen overlap. Here, we focus on what the splitting of the t2gcharacter peaks reveals about the electronic ground state. The fourt2g-character peaks we observe, Fig. 2, are incompatible with the multiplets expected in standard crystal field theory developed for 3 dions which leads to the S=3/2 ground state (4). Here, we identify that a method first proposed by Kamimura et al.(21) for transition metal halides is relevant in this case, in which the assumption that inter-electron interaction energies (including intra- and inter-orbital Coulomb and exchange interactions) are much larger than SOC is dropped (a similar method was also recently derived in Ref. (22)). We identify that such an approach can be utilised to model our TMO RIXS data, with the primary assumption that the hybridisation with the surrounding oxygen ligands leaves the transformation properties of the free ion wavefunctions unaltered. This therefore promotesthebreakingofthe S= 3/2singletandstrongentryofSOC.Thewavefunctionsare therefore described in terms of irreducible representations in the Odouble group determined by the octahedral symmetry (4, 23, 24), Fig. 3. This formulation does not necessitate approximations that the cubic crystal field is infinite or that Coulomb or SOC must be neglected, and is not restricted to use for 5d3configurations (21). To determine the wavefunctions for the 5 d3case, as in crystal field theory we use initial basis states that describe the ways in which three electrons can occupy the pure t2gand eglevels (i.e. terms such as4A2g- which is one electron in each of dxy,dyz,dxzwith total spin 3/2), and then apply inter-electron interaction and SOC between these states. The inter-electron interactions are expressed in terms of the Racah parameters BandC. Due to hybridisation, the radial form of the dlevels are unknown, i.e. BandCdeviate from pure ionic values, but the Racah parameters are formulated such that they can be easily extracted from experiment (4, 24, 25). The same interactions can be expressed in terms of intra- and inter-orbital Coulomb interactions, UandU/prime, and the effective Hund’s coupling6 4 A2g2 Eg2 T2g 2 T1g 5+ Os t2g Γ8Γ8Γ, Γ 86Γ7Γ8 Inter-electron interactions and spin-orbit coupling~0.75 eV~0.25 eV~0.45 eV~0.20 eV abcd S=3/2 L=0J=3/2ζ = 0.3 eVSO Inter-electron interactions3 Figure 3: Structure of t3 2glevels for Os5+ions in octahedral environment with strong spin-orbit interaction. The irreducible representations for the states without SOC, i.e. ζSO= 0, are labeled by the appropriate Mulliken Symbols, full spatial forms are tabulated in many textbooks e.g. Ref. (4). Here2Egand2T2gappear degenerate as we determine B= 0.00(4)eV within resolution, see main text. The irreducible representations describing the SOC-induced states are four-fold degenerate, Γ8, and two-fold degenerate, Γ6andΓ7. Dashed lines link the final states with the ζSO= 0states which provide the greatest contribution to them, however each final state is an intermixing of all ζSO= 0states, which allows SOC to enter the ground state Γ8. The labels a, b, c and d indicate the excited-state energies that are observed in the RIXS spectra of Ba 2YOsO 6and Ca 3LiOsO 6. Jh, which have the relationships to the Racah parameters Jh= 3B+C,U=A+ 4B+ 3C andU/prime=A−2B+C(25). The Racah parameter A, however, only appears in the diagonal matrix elements of the interactions matrices and causes a constant shift in all eigenvalues, including the ground state, (23, 26) so cannot be determined by the energies of the excited states probed by RIXS. The full interaction matrices are given in Ref. (23). By numerically diagonalising the matrices we determine the eigenstates illustrated in Fig. 3, and are able to fit the resulting eigenvalues to the determined energies aBa–dBaandaCa–dCa, Fig. 2, see supplementary material for further details. We fix the value of the crystal field to the peak value of peak e, 10Dq= 4.5eV for Ca 3LiOsO 6and10Dq= 4.3eV for Ba 2YOsO 6, because the positions of levels a–dare insensitive to small changes in this term, and the resulting levels include negligible mixing of egstates, as expected for a strong cubic crystal field splitting. The resulting parameters provide direct insight into the dominant interactions in the7 materials. For Ca 3LiOsO 6we findζSOC= 0.35(7)eV,B= 0.00(5),C= 0.3(2)andJh= 3B+C= 0.3(2)eV, and for Ba 2YOsO 6we findζSO= 0.32(6)eV,B= 0.00(5)eV,C= 0.3(2)eV andJh= 0.3(2)eV. The fact that ζSOis of similar size to C(andJh) clearly demonstrates that perturbative approaches are not appropriate for the treatment of SOC in5d3systems. The ratio of C/Bis commonly used to indicate the scale of deviation from pure ionic wavefunctions (the nephelauxetic effect) by comparison to the same ratio in 3 d materials where SOC is weak, with C/B = 4for Cr3+(3d3) ions (4, 24). We find Bto be zero within the error, indicative of a large nephelauxetic effect; improved resolution of RIXS measurements would be advantageous for this comparison. We can, however, look at the eigenvector we determined for the Γ8ground state - which is a linear combination of the 21 initial basis states which describe 3 electrons occupying the t2glevels, or two occupying the t2glevels and one in the eglevels (the latter ultimately form a negligible contribution, see supplementary material). The largest component is, as expected, from the S= 3/2|4A2/angbracketrightstate, but the next major contribution is from the S= 1/2|2T2g/angbracketrightstate. For Ca3LiOsO 6(Ba 2YOsO 6), these|4A2/angbracketrightand|2T2g/angbracketrightterms appear in the normalised eigenvector with weights of 0.95 (0.95) and 0.27 (0.25), respectively — the complete eigenfunctions are given in the supplementary material. This latter component (plus smaller terms) directly explains the entry of SOC physics and the observations of small orbital moments for 5d3 ions (5, 6, 8, 9, 27). We finally explore how the framework presented provides insight about the physical man- ifestation of SOC. An x-ray absorption near-edge spectroscopy plus x-ray circular dichroism study of 5d3Ir6+double perovskites found strong coupling between orbital and spin mo- ments despite small orbital moments, and suggested this should be due to some deviation from the pure t3 2glevels (8), with similar results reported in Os5+materials (9). The wave- function we determine explains these results, with a J= 3/2state which has only a small orbital moment. The observation of a large spin gap in the magnetic excitation spectra of Ba 2YOsO 6and related double perovskites is also explained by the intermediate coupling framework, asthespingapresultsfromstrongSOC-inducedanisotropywhichisunexplained8 in aS= 3/2picture. In these double perovskites and the pyrochlore Cd 2Os2O7anisotropy is held responsible for stabilisation of the observed magnetic ground states (6, 7, 28). Our results therefore show that intermediate coupling electronic ground states are central in dic- tating the macroscopic physical properties of such materials. We speculate that a similar approach should be utilised for 5 d2and 5d4materials, as they are expected to show larger SOC effects (29), alongside increased hybridisation in the d2case. Thed4materials have attracted interest for hosting magnetic order (30) despite the d4configuration leading to non-magnetic singlets in either LS or jj limits (29). Acknowledgements We thank A. Huq and M. J. Kirkham for assistance with high-resolution neutron and x-ray diffraction experiments. We thank G. Pokharel for useful discussions. Use of the Advanced Photon Source at Argonne National Laboratory was supported by the U. S. De- partment of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. The research at ORNL’s Spallation Neutron Source and High Flux Isotope Reactor was supported by the Scientific User Facilities Division, Office of Basic En- ergy Sciences, U.S. Department of Energy (DOE). This research was supported in part by the Center for Emergent Materials an National Science Foundation (NSF) Materials Re- search Science and Engineering Center (DMR-1420451). This research was supported in part by the Japan Society for the Promotion of Science (JSPS) through a Grant-in-Aid for Scientific Research (15K14133, 16H04501). The authors declare no competing financial interests. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access9 Plan (http://energy.gov/downloads/doe-public-access-plan). 1. G. Jackeli, G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). 2. W. Witczak-Krempa, G. Chen, Y. B. Kim, L. Balents, Annual Review of Condensed Matter Physics 5, 57 (2014). 3. B. J. Kim, et al.,Science 323, 1329 (2009). 4. S. Sugano, Y. Tanabe, H. Kamimura, Multiplets of Transition-Metal Ions in Crystals (Aca- demic Press, New York and London, 1970). 5. E. Kermarrec, et al.,Phys. Rev. B 91, 075133 (2015). 6. A. E. Taylor, et al.,Phys. Rev. B 93, 220408 (2016). 7. S. Calder, et al.,Nat Commun 7, 11651 (2016). 8. M. A. Laguna-Marco, et al.,Phys. Rev. B 91, 214433 (2015). 9. L. S. I. Veiga, et al.,Phys. Rev. B 91, 235135 (2015). 10. Y. Shi, et al.,J. Am. Chem. Soc. 132, 8474 (2010). 11. G. J. Thorogood, et al.,Dalton Trans. 40, 7228 (2011). 12. E. E. Rodriguez, et al.,Phys. Rev. Lett. 106, 067201 (2011). 13. Y. Krockenberger, et al.,Phys. Rev. B 75, 020404 (2007). 14. S. Calder, et al.,Phys. Rev. Lett. 108, 257209 (2012). 15. O. N. Meetei, O. Erten, M. Randeria, N. Trivedi, P. Woodward, Phys. Rev. Lett. 110, 087203 (2013). 16. S. Calder, et al.,Phys. Rev. B 86, 054403 (2012). 17. L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, J. van den Brink, Rev. Mod. Phys. 83, 705 (2011). 18. X. Liu, et al.,Phys. Rev. Lett. 109, 157401 (2012). 19. S. Boseggia, et al.,Phys. Rev. B 85, 184432 (2012). 20. M. Moretti Sala, et al.,Phys. Rev. Lett. 112, 176402 (2014). 21. H. Kamimura, S. Koide, H. Sekiyama, S. Sugano, J. Phys. Soc. Jpn. 15, 1264 (1960). 22. H. Matsuura, K. Miyake, J. Phys. Soc. Jpn. 82, 073703 (2013). 23. J. C. Eisenstein, The Journal of Chemical Physics 34, 1628 (1961). 24. Brian N. Figgis, M. A. Hitchman, Ligand field theory and its applications , Special topics in inorganic chemistry (Wiley, New York, 2000). 25. A. Georges, L. d. Medici, J. Mravlje, Annual Review of Condensed Matter Physics 4, 137 (2013). 26. P. B. Dorain, R. G. Wheeler, The Journal of Chemical Physics 45, 1172 (1966). 27. A. E. Taylor, et al.,Phys. Rev. B 91, 100406 (2015). 28. E. V. Kuz’min, S. G. Ovchinnikov, D. J. Singh, Phys. Rev. B 68, 024409 (2003). 29. G. Chen, L. Balents, Phys. Rev. B 84, 094420 (2011). 30. G. Cao, et al.,Phys. Rev. Lett. 112, 056402 (2014). 31. T. Gog, et al.,Synchrotron Radiation News 22, 12 (2009). 32. M. Wakeshima, Y. Hinatsu, Solid State Communications 136, 499 (2005). 33. B. H. Toby, J. Appl. Cryts. 34, 210 (2001). 34. J. Rodríguez-Carvajal, Physica B: Condensed Matter 192, 55 (1993). 35. C. J. Howard, B. J. Kennedy, P. M. Woodward, Acta Crystallographica Section B Structural Science 59, 463 (2003).10 36. J. Kanamori, Prog. Theor. Phys. 30, 275 (1963). 37. M. S. Dresselhaus, G. Dresselhaus, A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). 38. G. F. Koster, J. O. Dimmock, R. G. Wheeler, H. Statz, The Properties of the Thirty-Two Point Groups (The MIT Press, Cambridge (Mass.), 1963).11 Supplementary Material for: Spin-orbit coupling controlled J= 3/2electronic ground state in 5 d3oxides Resonant inelastic x-ray scattering RIXS measurements were performed at the Advanced Photon Source (APS) at Sector 27 using the MERIX instrumentation (31). A closed-cycle refrigerator was used to control the sampletemperature. TheOsL 3-edgeincidentenergywasaccessedwithaprimarydiamond(1 1 1) monochromator and a secondary Si(4 0 0) monochromator. A Si(4 6 6) diced analyzer was used to determine the energy of the beam scattered from the sample, and a MYTHEN strip detector was utilised. All measurements were performed with 2θ= 90◦in horizontal geometry. The RIXS energy resolution was 150meV FWHM. The raw data counts are normalised to the incident beam intensity via an ion chamber monitor. To compare the temperatures, the 6K data is normalised so that the featureless energy gain side overlaps with the 300K data – this accounts for deviations of the beam position on the sample. Synthesis The synthesis route and properties of Ca 3LiOsO 6are reported in Ref. (10). Characteri- sation of the powder sample utilised here, including diffraction measurements and magnetic structure determination, are reported in Ref. (16). The powder sample of Ba 2YOsO 6of mass 1.5 g was synthesised by grinding together stoichiometric quantities of barium peroxide, yt- trium oxide, and osmium metal. The reactants were loaded into an alumina tube which was placed into a silica tube along with a secondary alumina vessel containing PbO 2. The silica tube was sealed under dynamic vacuum, and heated to 1000◦C for a period of 48hours in a box furnace located in a fume hood. The PbO 2decomposed into PbO at elevated tempera- ture, acting as a source of O 2gas for the reaction. A calculated excess was used resulting in 1 4mole excess O 2per mole of product in order to ensure full oxidation of the reactants. Care12 must be taken when heating Os or Os containing compounds due to the potential formation of highly toxic OsO 4gas. X-ray and neutron diffraction and susceptibility measurements were consistent with those previously reported (5), and are detailed in the supplementary material. Ca3LiOsO 6Characterisation PolycrystallineCa 3LiOsO 6wasusedinthiswork, forwhichsynthesisrouteandproperties are reported in Ref. (10). Neutron powder diffraction measurements and magnetic structure determination are reported in Ref. (16). Ba2YOsO 6Characterisation The temperature dependence of the magnetization of the Ba 2YOsO 6powder was mea- sured with a Quantum Design MPMS SQUID magnetometer. The powder was contained in a gel capsule and mounted in a straw for insertion into the device. Data were collected between 2.5 and 400K under an applied field of 1kOe under field cooled (FC) and zero-field cooled(ZFC)conditions. Nocorrectionswereappliedtothedata. Asharptransitionat70K was observed in both data sets, see Fig.4, associated with the antiferromagnetic ordering of Os5+as previously observed(5). The rise of the magnetization at very low temperatures is attributed to the presence of ferromagnetic impurity Ba 11Os4O24, which has TC= 6.8K(32). A Curie-Weiss fit was conducted in the temperature range 300–400K resulting in a Weiss constant, Θ, of -785K and an effective moment of 4.27 B. These parameters result in a large frustration index, |Θ/TN|, of 11.2. However, the effective moment is larger than the predicted spin-only moment of 3.87 Bsuggesting that the true paramagnetic regime is not achieved below 400K, consistent with Ref. (5). Ba2YOsO 6Structural Study High resolution x-ray and neutron diffraction experiments were performed on Ba 2YOsO 6 in an attempt to identify any non-cubic distortion. The synchrotron x-ray experiments13 Figure 4: Temperature dependence of the zero field cooled (blue open circles) and field cooled (red closed circles) magnetization of Ba 2YOsO 6under 1kOe measurement field. The low temperature riseofthemagnetizationisattributedtothepresenceofferromagneticimpurityBa 11Os4O24, which hasTC= 6.8K(32). are optimised for identifying non-cubic lattice parameter splittings, and for identifying Y- Os site mixing, whereas the neutron scattering experiments are more likely to be able to identify distortions and tilting of the oxygen octahedra. Space group Fm¯3mwas used for the refinements, in which the ions are on Wyckoff sites: Ba 8 c(1 4,1 4,1 4), Y 4 a(0,0,0), Os 4 b (1 2,1 2,1 2)and O1 24 e,(0,0,z). The Y and Os ions contribute to the same Bragg reflections, therefore for both x-ray and neutron refinements the displacement parameters for these ions were constrained to be equal. Synchrotron x-ray diffraction measurements were conducted on 11-BM at the Advanced Photon Source with wavelength λ= 0.4593 Åat temperature of 295K. The 0.012g of sample wasmixedwithgroundquartzina1:1massratiotominimizeabsorption, andthiswasplaced in a 0.4mm radius capillary for the measurement. The data were analyzed via Rietveld refinement as implemented in GSAS(33). Y/Os cation ordering refinements were performed indicating full cation ordering, similar to previous work(5). No non-cubic peak splittings, nor additional peaks, could be identified. The results of refinements based on space group Fm¯3mis illustrated in Fig.5 and summarized in Table I. Neutron powder diffraction measurements were conducted on POWGEN at the Spalla- tion Neutron Source at Oak Ridge National Laboratory (ORNL). 1.4g of Ba 2YOsO 6was14 1 2 3 4 5 6 7 8 9 10 11−10123456x 104 Q (Å−1)Intensity (Arb. units) Figure 5: X-ray powder diffraction pattern of Ba 2YOsO 6measured at 295K. The black circles are observed data, the red line is the calculated pattern, and the blue line is the difference of the two. Allowed reflections of the phase are given as green hashes. Table I: Results from Rietveld refinements against XRPD data from BM-11 using GSAS. Errors quoted in parenthesis are the standard deviations determined by GSAS. Temperature a(Å)χ2RwpO1zBaBisoY/OsBisoO1Biso 295K 8.35518(1) 2.5348.75%0.2379(2) 0.709(4) 0.248(3) 1.00(3) contained in a vanadium can, and measured at 10 and 100K. High-resolution settings were chosen,λ= 1.066 Åandf= 60Hz, tooptimizechancesofidentifyinganon-cubicdistortion. The data were analyzed via Rietveld refinement as implemented in Fullprof(34). No peaks associated with octahedral rotations in non-cubic symmetries could be identified(35). The results of refinements based on space group Fm¯3mis illustrated in Fig.6 and summarized in Table II. In this refinement all ion occupancies were kept at 100% as indicated by the x-ray data. Using anisotropic displacement parameters for the oxygen positions did not produce an improvement to the fit qualities, so we quote the results with isotropic displacement parameter results for all ions. Temperature a(Å)χ2RBragg RFO1zBaBisoY/OsBisoO1Biso 10K 8.34383(8) 9.916.763.770.23446(5) 0.104(7) 0.107(5) 0.293(6) 100K 8.34596(8) 10.16.834.100.23438(5) 0.183(7) 0.132(6) 0.342(7) Table II: Results from Rietveld refinements against NPD data from POWGEN using Fullprof. Errors quoted in parenthesis are the standard deviations determined by Fullprof.15 2 4 6 8 10 12 14 16 18 20−1000100200300400500600 Q (Å−1)Intensity (Arb. units) Figure 6: Neutron powder diffraction pattern of Ba 2YOsO 6measured at 10K. The black circles are observed data, the red line is the calculated pattern, and the blue line is the difference of the two. Allowed reflections of the phase are given as green hashes. Spin-Orbit Calculation As referenced in the main text, the complete Hamiltonian describing cubic-crystal field, Coulomb interactions (including Hund’s coupling) and spin-orbit effects is given by Eisenstein(23) in the form of a 21 ×21 matrix for Γ8and a 9×9matrix for each of the Γ6 andΓ7representations, describing the interactions between each of the basis states utilized in the standard Coulomb-only model(4, 25, 36). These basis states are the |4A2/angbracketright,|2T2g/angbracketrightetc. states which describe all the ways in which three electrons can occupy the t2gandeglevels, and are documented in full in textbooks, for example Ref. (4). The interaction matrices are formulated in terms of the Racah parameters, An,BnandCn(n= 0to4is symmetry allowed) but we follow the formulation in Ref. (26) and assume only one parameter of each type is required (i.e. An=A,Bn=B,Cn=C∀n) as is commonly adopted(25), and found to be reasonable in Ref. (23) for Re4+in Cl octahedra. The term Ais only found on the diagonal elements, adding 3 Ato each eigenvalue, and therefore the differences in energy between states is independent of Aand we can set it to any arbitrary value in describing our data. We diagonalise the matrices to find the eigenvalues and eigenvectors, and shift all eigenvalues by the energy of the lowest term, so that the ground state is at E= 0. We then fit the first four excited state eigenvalues to the observed RIXS excitations as described in16 the main text. The basis vectors of the Γ8representation in the O double group are angular momentum wavevectors φ(J,m J)withJ= 3/2andmJ=3 2,1 2,−1 2and−3 2(37, 38). We determine the eigenvector for the Γ8ground state in terms of a linear combination of 21 basis states describing the three electron occupations of the t2gandeglevels, including t2g-egexcited states, although as 10Dq∼4eV, the contributions from these states are extremely small. The complete normalized eigenfunction of the ground state for Ba 2YOsO 6from a numer- ical diagonalisation is, to 3 decimal places: |Γ8g.s./angbracketright=−0.953/vextendsingle/vextendsingle/vextendsingle4A2(t2g3)/angbracketrightBig + 0.069/vextendsingle/vextendsingle/vextendsingle2Eg(t2g3)/angbracketrightBig + 0.017/vextendsingle/vextendsingle/vextendsingle2Eg(t2g2eg1)/angbracketrightBig + 0.015/vextendsingle/vextendsingle/vextendsingle2Eg(t2g2eg1)/angbracketrightBig −0.001/vextendsingle/vextendsingle/vextendsingle2Eg(e3)/angbracketrightBig −0.081/vextendsingle/vextendsingle/vextendsingle2T1g(t2g3)/angbracketrightBig −0.007/vextendsingle/vextendsingle/vextendsingle2T1g(t2g2eg1)/angbracketrightBig + 0.013/vextendsingle/vextendsingle/vextendsingle2T1g(t2g2eg1)/angbracketrightBig −0.004/vextendsingle/vextendsingle/vextendsingle2T1g(t2g1eg2)/angbracketrightBig + 0.002/vextendsingle/vextendsingle/vextendsingle2T1g(t2g1eg2)/angbracketrightBig −0.024/vextendsingle/vextendsingle/vextendsingle4T1g(t2g2eg1)/angbracketrightBig + 0.006/vextendsingle/vextendsingle/vextendsingle4T1g(t2g1eg2)/angbracketrightBig −0.007/vextendsingle/vextendsingle/vextendsingle4T1g(t2g2eg1)/angbracketrightBig −0.002/vextendsingle/vextendsingle/vextendsingle4T1g(t2g1eg2)/angbracketrightBig −0.252/vextendsingle/vextendsingle/vextendsingle2T2g(t2g3)/angbracketrightBig + 0.069/vextendsingle/vextendsingle/vextendsingle2T2g(t2g2eg1)/angbracketrightBig −0.013/vextendsingle/vextendsingle/vextendsingle2T2g(t2g2eg1)/angbracketrightBig −0.011/vextendsingle/vextendsingle/vextendsingle2T2g(t2g1eg2)/angbracketrightBig + 0.006/vextendsingle/vextendsingle/vextendsingle2T2g(t2g1eg2)/angbracketrightBig −0.038/vextendsingle/vextendsingle/vextendsingle4T2g(t2g2eg1)/angbracketrightBig −0.098/vextendsingle/vextendsingle/vextendsingle4T2g(t2g2eg1)/angbracketrightBig . (.1) The complete normalized eigenfunction of the ground state for Ca 3LiOsO 6from a numerical diagonalisation is, to 3 decimal places: |Γ8g.s./angbracketright=−0.947/vextendsingle/vextendsingle/vextendsingle4A2(t2g3)/angbracketrightBig + 0.076/vextendsingle/vextendsingle/vextendsingle2Eg(t2g3)/angbracketrightBig + 0.019/vextendsingle/vextendsingle/vextendsingle2Eg(t2g2eg1)/angbracketrightBig + 0.016/vextendsingle/vextendsingle/vextendsingle2Eg(t2g2eg1)/angbracketrightBig −0.001/vextendsingle/vextendsingle/vextendsingle2Eg(e3)/angbracketrightBig −0.090/vextendsingle/vextendsingle/vextendsingle2T1g(t2g3)/angbracketrightBig −0.007/vextendsingle/vextendsingle/vextendsingle2T1g(t2g2eg1)/angbracketrightBig + 0.014/vextendsingle/vextendsingle/vextendsingle2T1g(t2g2eg1)/angbracketrightBig −0.004/vextendsingle/vextendsingle/vextendsingle2T1g(t2g1eg2)/angbracketrightBig + 0.002/vextendsingle/vextendsingle/vextendsingle2T1g(t2g1eg2)/angbracketrightBig −0.025/vextendsingle/vextendsingle/vextendsingle4T1g(t2g2eg1)/angbracketrightBig + 0.006/vextendsingle/vextendsingle/vextendsingle4T1g(t2g1eg2)/angbracketrightBig −0.008/vextendsingle/vextendsingle/vextendsingle4T1g(t2g2eg1)/angbracketrightBig −0.002/vextendsingle/vextendsingle/vextendsingle4T1g(t2g1eg2)/angbracketrightBig −0.266/vextendsingle/vextendsingle/vextendsingle2T2g(t2g3)/angbracketrightBig + 0.070/vextendsingle/vextendsingle/vextendsingle2T2g(t2g2eg1)/angbracketrightBig −0.014/vextendsingle/vextendsingle/vextendsingle2T2g(t2g2eg1)/angbracketrightBig −0.011/vextendsingle/vextendsingle/vextendsingle2T2g(t2g1eg2)/angbracketrightBig + 0.006/vextendsingle/vextendsingle/vextendsingle2T2g(t2g1eg2)/angbracketrightBig −0.038/vextendsingle/vextendsingle/vextendsingle4T2g(t2g2eg1)/angbracketrightBig −0.100/vextendsingle/vextendsingle/vextendsingle4T2g(t2g2eg1)/angbracketrightBig . (.2)

1310.3916v1.Vanishing_of_interband_light_absorption_in_a_persistent_spin_helix_state.pdf

arXiv:1310.3916v1 [cond-mat.mes-hall] 15 Oct 2013Vanishing of interband light absorption in a persistent spi n helix state Zhou Li1, F. Marsiglio2, and J. P. Carbotte1,3 1Department of Physics, McMaster University, Hamilton, Ont ario,Canada,L8S 4M1 2Department of Physics, University of Alberta, Edmonton, Al berta,T6G 2E1 3Canadian Institute for Advanced Research, Toronto, Ontari o, Canada M5G 1Z8 (Dated: September 18, 2018) Spin-orbit coupling plays an important role in various prop erties of very different materials. More- over efforts are underway to control the degree and quality of spin-orbit coupling in materials with a concomitant control of transport properties. We calculat e the frequency dependent optical con- ductivity in systems with both Rashba and Dresselhaus spin- orbit coupling. We find that when the linear Dresselhaus spin-orbit coupling is tuned to be eq ual to the Rashba spin-orbit coupling, the interband optical conductivity disappears. This is tak en to be the signature of the recovery of SU(2) symmetry. The presence of the cubic Dresselhaus spin- orbit coupling modifies the dispersion relation of the charge carriers and the velocity operator. T hus the conductivity is modified, but the interband contribution remains suppressed at most but not a ll photon energies for a cubic coupling of reasonable magnitude. Hence, such a measurement can serv e as a diagnostic probe of engineered spin-orbit coupling. PACS numbers: 73.25.+i,71.70.Ej,78.67.-n2 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51 /s49/s61/s48/s46/s50/s44 /s49/s61/s48/s46/s51/s44 /s51/s61/s48/s46/s51 /s49/s61/s48/s46/s53/s44 /s49/s61/s48/s46/s53/s44 /s51/s61/s48/s49/s61/s48/s44 /s49/s61/s49/s44 /s51/s61/s48/s32/s32/s107/s121/s47/s107 /s48 /s107/s120/s47/s107 /s48/s49/s61/s49/s44 /s49/s61/s48/s44/s32 /s51/s61/s48 /s32/s32/s32/s32 /s32/s32 Fig.1. Spin texture in the conduction band as a function of mo mentum kx/k0,ky/k0for various values of Rashba ( α1), Dresselhaus ( β1), and cubic Dresselhaus ( β3) spin-orbit coupling. In the case of purely Rashba coupling (upper left frame), the spin is locked in the direction perpendicular to momentu m, while for linear Dresselhaus coupling (upper right frame ) the y-component of spin is of opposite sign to that of its momentu m. For the persistent spin helix state (lower left frame) all spins are locked in the 3 π/4 direction and oppositely directed on either side of this cr itical direction. The lower right frame shows the spin texture for a case with all three kinds of coupl ing. Spin-orbit coupling in semiconductors [1] and at the surface of thre e dimensional topological insulators [2–8] where protected metallic surface states exist, plays a crucial role in their fundamental physical properties. Similarly pseu- dospin leads to novel properties in graphene [9–11] and other two d imensional membranes, such as single layer MoS2 [12–17] andsilicene [18–22]. In particular MoS2has been discussed within the contextofvalleytronicswherethe va lley degree of freedom can be manipulated with the aim of encoding inform ation in analogy to spintronics. Spin-orbit coupling has also been realized in zincblende semiconductor quantum w ells [23–25] and neutral atomic Bose-Einstein condensates [26] at very low temperature [27]. In some systems both Rashba [28] and Dresselhaus [29] spin-orbit c oupling are manipulated, the former arising from an inversion asymmetry of the grown layer while the latter come s from the bulk crystal. In general spin-orbit coupling will lead to rotation of the spin of charge carriers as they ch ange their momentum, because SU(2) symmetry3 -2-1 0 1 2-2-1 0 1 2 0 1 2 -2-1 0 1 2-2-1 0 1 2 0 1 2 -2-1 0 1 2-2-1 0 1 2 0 1 2 -2-1 0 1 2-2-1 0 1 2 0 1 2 Fig.2. Band structure of the conduction and valence band ( Eq . (10)) as a function of momentum kx/k0,ky/k0for various values of Rashba ( α1), Dresselhaus ( β1), and cubic Dresselhaus ( β3) spin-orbit coupling. The left two panels are for pure Rashbaα1= 1.0,β1= 0.0,β3= 0.0 (top panel) and Rashba equals to Dresselhaus α1= 0.5,β1= 0.5,β3= 0.0 (bottom panel). The right two panels are for α1= 0.4,β1= 0.4,β3= 0.3 (top panel) and α1= 0.2,β1= 0.8,β3= 0.3 (bottom panel). The dispersion curves are profoundly changed from the familiar Dirac cone of the pure Rashba case when β1andβ3are switched on. In the contour plots, red refers to energy 0 .2E0and dark green refers to energy −0.2E0. is broken. In momentum space this has been observed by angle-res olved photoemission spectroscopy (ARPES) as the phenomenon of spin momentum locking. In a special situation whe n the strength of Rashba and Dresselhaus spin-orbit coupling are tuned to be equal, SU(2) symmetry is recove red and a persistent spin helix state is found [23–25]. This state is robust against any spin-independent scatter ing. However it will be potentially destroyed by the cubic Dresselhaus term which is usually tuned to be negligible. To describe these effects we consider a model Hamiltonian describing a free electron gas with kinetic energy given simplyby /planckover2pi12k2/(2m), whichdescribeschargecarrierswith effectivemass m. Wealsoincludespin-orbitcouplingterms, with linear Rashba ( α1) and Dresselhaus ( β1) couplings, along with a cubic Dresselhaus ( β3) term. The Hamiltonian is ˆH0=/planckover2pi12k2 2mˆI+α1(kyˆσx−kxˆσy)+β1(kxˆσx−kyˆσy)−β3(kxk2 yˆσx−kyk2 xˆσy). (1) Here ˆσx,ˆσyand ˆσzare the Pauli matrices for spin (or pseudospin in a neutral atomic Bo se-Einstein condensate) and ˆIis the unit matrix. For units we use a typical wave vector k0≡mα0//planckover2pi12with corresponding energy E0=mα2 0//planckover2pi12, whereα0is a representative spin-orbit coupling which has quite different value s for semiconductors ( α0//planckover2pi1≈105m/s, estimated from Ref. [25]) and cold atoms ( α0//planckover2pi1≈0.1m/s, estimated from Ref. [26]). The mass of a cold atom is at least 1000 times heavier than that of an electron and the wavelengt h of the laser used to trap the atoms is at least 1000 times (estimated from Ref. [26]) larger than the lattice spacing in semiconductors. In this report we study the dynamic longitudinal optical conductivit y of such a spin-orbit coupled 2D electron gas. We find that the interband optical absorption will disappear when th e Rashba coupling is tuned to be equal to the Dresselhaus coupling strength. We discuss the effect of nonlinear ( cubic) Dresselhaus coupling on the shape of the interband conductivity and the effect of the asymmetry between t he conduction and valence band which results from a mass term in the dispersion curves.4 /s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53 /s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48 /s50/s46/s52 /s50/s46/s56/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s49 /s49/s51/s61/s48/s46/s48 /s32 /s49/s61/s48/s44 /s49/s61/s49/s46/s48 /s32 /s49/s61/s48/s46/s50/s44 /s49/s61/s48/s46/s56 /s32 /s49/s61/s48/s46/s52/s44 /s49/s61/s48/s46/s54 /s32 /s49/s61 /s49 /s32/s47/s69 /s48/s61/s48/s46/s50/s32/s40 /s41/s32/s105/s110/s32/s117/s110/s105/s116/s32/s111/s102/s32/s40/s50 /s101/s50 /s47/s104/s41 /s47/s69 /s48/s49 /s49 /s32/s47/s69 /s48/s61/s32/s45/s48/s46/s50 Fig.3. The interband contribution to the longitudinal opti cal conductivity of Eq. (2) for various values of α1andβ1as labeled, with β3set to zero. In the top frame the chemical potential was set at µ/E0= 0.2 and in the bottom µ/E0=−0.2. RESULTS We compute the optical conductivity (see Methods section) as a fu nction of frequency, for various electron fillings and spin-orbit coupling strengths. In all our figures we will use a dime nsionless definition of spin-orbit coupling; for example, the choice of values designated in the lower right frame of F ig. 1,α1= 0.2,β1= 0.3, andβ3= 0.3, really meansα1/α0= 0.2,β1/α0= 0.3, andβ3k2 0/α0= 0.3. In Fig. 1 we plot the spin direction in the conduction band as a function of momentum for several cases. The top left frame is for pure Rashba coupling, in which case spin is locked t o be perpendicular to momentum [2] as has been verified in spin angle-resolved photoemission spectroscopy st udies [30–33]. The top right frame gives results for pure linear Dresselhaus coupling (no cubic term β3= 0). The spin pattern is now quite different; the direction of the spin follows the mirror image of the momentum about the x-axis. T he lower left frame for equal linear Rashba and Dresselhaus coupling is the most interesting to us here. All spins are locked in one direction, namely θ= 3π/4 with those in the bottom (upper) triangle pointing parallel (anti-par allel) to the 3 π/4 direction, respectively. This spin arrangement corresponds to the persistent spin helix state o f Ref. [23–25]. The condition α1=β1andβ3= 05 /s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48 /s50/s46/s52 /s50/s46/s56/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57 /s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52 /s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48 /s50/s46/s52 /s50/s46/s56/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48 /s47/s69 /s48/s47/s69 /s48/s32 /s49/s61/s48/s44 /s49/s61/s49/s46/s48/s44 /s51/s61/s48/s46/s51 /s32 /s49/s61/s48/s46/s50/s44 /s49/s61/s48/s46/s56/s44 /s51/s61/s48/s46/s51 /s32 /s49/s61/s48/s46/s52/s44 /s49/s61/s48/s46/s52/s44 /s51/s61/s48/s46/s51 /s32 /s49/s61/s48/s46/s48/s44 /s49/s61/s49/s46/s48/s44 /s51/s61/s48/s46/s50 /s32/s47/s69 /s48/s61/s48/s46/s50 /s32/s47/s69 /s48/s61/s32/s45/s48/s46/s50/s32/s40 /s41/s32/s105/s110/s32/s117/s110/s105/s116/s32/s111/s102/s32/s40/s50 /s101/s50 /s47/s104/s41 /s32 /s47/s69 /s48/s61/s48/s46/s50/s47/s69 /s48/s61/s32/s45/s48/s46/s50/s68/s40 /s41 /s32 Fig.4. Joint density of states D(ω) (top two panels) defined in Eq. (3) which involves the same tr ansitions as does the interband conductivity (bottom two panels) of Eq. (2) but wi thout the critical weighting(VxS2+VyS1)2 (S2 1+S2 2)ω. Left column is for positive chemical potential µ/E0= 0.2 and the right for -0.2. is a state of zero Berry phase [34] and was also characterized by Li et al.[35] as a state in which the spin transverse “force” due to spin-orbit coupling cancels exactly. Finally the right lo wer frame includes a contribution from the cubic Dresselhaus term of Eq. (1) and shows a more complex spin arrange ment. Spin textures have been the subject of many recent studies [30–33, 36]. In Fig. 2 we present results for th e dispersion curves in the conduction and valence bandE+/−(k) of Eq. (10) as a function of momentum k. The two left panels are pure Rashba (top) and Rashba equals to Dresselhaus (bottom, see also Fig.1 of Ref.[37] where only t he contour plots of the valence band is shown). The two right panels include the Dresselhaus warping cubic term which profoundly affects the band structure. The optical conductivity is obtained through transitions from one e lectronic state to another. In general these can be divided into two categories — transitions involving states within the same band, and interband transitions. Here6 Fig.5. Color contour plot of the energy difference 2/radicalbig S2 1+S2 2≡E+−E−, as a function of momentum ( kx,ky) in units of k0 forα1= 0.4,β1= 0.4,β3= 0.3 (top panel) and α1= 0.2,β1= 0.8,β3= 0.3 (bottom panel). we focus on interband transitions; the interband optical conduct ivity is given by σxx(ω) =e2 iω1 4π2/integraldisplaykcut 0kdkdθ(VxS2+VyS1)2 S2 1+S2 2/bracketleftbiggf(E+)−f(E−) /planckover2pi1ω−E++E−+iδ−f(E+)−f(E−) /planckover2pi1ω−E−+E++iδ/bracketrightbigg , (2) wheref(x) = 1/(e(x−µ)/kBT+1)istheFermi-Diracdistributionfunction with µthechemicalpotential. For β3= 0and β1=α1, we haveacancellation in the opticalmatrix element, VxS2+VyS1= 0; remarkablythe interband contribution vanishes. This result is central to our workand shows that in the pe rsistent spin helix state the interband contribution7 to the dynamic longitudinal optical conductivity vanishes. This is the optical signature of the existence of the spin helix state which exhibits remarkable properties. With β3= 0 the optical matrix element is ( β2 1−α2 1)ky//planckover2pi1. Thus, pure Rashba or pure (linear) Dresselhaus coupling will both lead to ex actly the same conductivity although the states (and spin texture) involved differ by a phase factor of π. When they are both present in equal amounts this phase leads to a cancelation which reduces the interband transitions to ze ro as the two contributions need to be added before the square is taken. Of course the joint density of states, widely u sed to discuss optical absorption processes, remains finite. It is given by D(ω) =1 4π2/integraldisplaykcut 0kdkdθ[f(E+)−f(E−)]Im/bracketleftbigg1 /planckover2pi1ω−E++E−+iδ−1 /planckover2pi1ω−E−+E++iδ/bracketrightbigg (3) and will be contrasted with the interband optical conductivity below . We first focus on the case β3= 0. The interband conductivity is shown in Fig. 3 as a function of freq uency for positive (top frame) and negative (bottom frame) chemical po tential (µ/E0=±0.2). It is clear that there is a considerable difference between the two cases, and there is also co nsiderable variation with the degree of Rashba vs. Dresselhaus coupling. This will be discussed further below. Most impo rtant is that for equal amounts of Rashba and Dresselhaus coupling, the interband conductivity is identically zero f or all frequencies. What is the impact of a finite value of β3? In Fig. 4 we show both the joint density of states (top two panels) and the interband conductivity (bottom two panels) for non-zero β3forµ/E0= 0.2 (left panels) and µ/E0=−0.2 (right panels). Various combinations of α1,β1andβ3are shown as labeled on the figure. There is a striking asymmetry between positive and negative values of the chemical potential. This asymmetry has its origin in the quadratic term /planckover2pi12k2/(2m) of the Hamiltonian (1) which adds positively to the energy in both vale nce and conduction band while the Dirac like contribution is negative ( s=−1) and positive ( s= +1) respectively [see Eq. (10)]. While the quadratic piece drops out of the energy denominator in Eq. (2) it remains in the Fermi factors f(E+) andf(E−). Several features of these curves are noteworthy. They all hav e van Hove singularities which can be traced to extrema in the energy difference E+−E−= 2/radicalbig S2 1+S2 2. Taking β3= 0 for simplicity, this energy becomes 2k/radicalbig α2 1+β2 1+2α1β1sin(2θ) which depends on the direction ( θ) of momentum k, but has no minimum or maxi- mum as a function of |k|=k. To get an extremum one needs to have a non-zero cubic Dresselha us term. This gives dispersion curves which flatten out with increasing values of k. The dependence of the energy E+−E−on momentum is illustrated in Fig. 5 where we provide a color plot for this energy as a f unction of kx/k0andky/k0for two sets of spin-orbit parameters α1= 0.4,β1= 0.4,β3= 0.3 (top panel) and α1= 0.2,β1= 0.8,β3= 0.3 (bottom panel). Note the saddle points correspond to the most prominent van Hove singularities in the joint density of states (and conductivity) in Fig. 4. The van Hove singularities are at about 1 .4E0(kx=kyin the momentum space) in the top frame of Fig. 5 and at about 2 E0(kx=ky) and 0.9E0(kx=−ky) in the bottom. DISCUSSION The optical conductivity is often characterized by the joint densit y of states, D(ω), which has a finite onset at small energies. This is well known in the graphene literature where interba nd transitions start exactly at a photon energy equal to twice the chemical potential. Here this still holds approxima tely in all the cases considered in Fig. 4 except for the solid red curve in the two left side frames. In this case α1=β1= 0.4 andβ3is non zero. If β3is small the energy/radicalbig S2 1+S2 2would be approximately equal to√ 2kα1√ 1+sin2θ, which is zero for θ= 3π/4, the critical angle in the spin texture of the lower left frame of Fig. 1 for which all spins a re locked in this direction. This means that only the quadratic term /planckover2pi12k2/(2m) and cubic Dresselhaus term contribute to the dispersion curve in t his direction and there is no linear (in k) graphene-like contribution. Thus, the onset of the interband op tical transition no longer corresponds to ω= 2µ. Considering the case of positive µ, for the direction θ= 3π/4, (k/k0)2/2+β3(k/k0)3is the dominant contribution to the energy which is equal to µ/E0and the minimum photon energy is now 2 β3(k/k0)3, which could be very small as is clear from the figure. For negative values of µthe onset is closer to 2 |µ|/E0because in this case the momentum at which the chemical potential crosses the band dispersion is given by (k/k0)2/2−α1(k/k0) =−µ/E0(the cubic term is ignored because it is subdominant for small k/k0compared to the linear term). Now the photon energy onset will fall above 2 |µ|/E0, at a value dependent on α1. While the optical conductivity Eq. (2) requires a non-zero joint den sity of states Eq. (3), the additional weighting of (VxS2+VyS1)2inσxx(ω) can introduce considerable changes to its ωdependence [38] as we see in Fig. 3 and Fig. 4. In the top frame of Fig. 3, β3= 0 and there are no van Hove singularities because the Dirac contrib ution8 to the dispersion curves simply increases with increasing k. The solid black and dashed red curves both reduce to the pure graphene case with onset exactly at 2 µand flat background beyond. The dotted red curve for mixed linear Dresselhaus and Rashba is only slightly different. The onset is near bu t below 2 µand the background has increased in amplitude. It is also no longer completely flat to high frequency; inste ad it has a kink near /planckover2pi1ω/E0≈1.7 after which it drops. The dash-dotted black curve for α1= 0.4 andβ1= 0.6 has changed completely with background reduced to near zero but with a large peak corresponding to an onset which h as shifted to a value much less than 2 µ. Finally forα1=β1the entire interband transition region is completely depleted as we kn ow from Eq. (2). In Fig. 4 there is (non-zero) cubic Dresselhaus coupling present. T he solid red curves, for which α1=β1but with β3= 0.3 illustrate that the conductivity on the left (positive µ) is non-zero, and β3= 0 is necessary for a vanishing interband conductivity at all photon energies. We see, however, t hat these transitions have been greatly reduced below what they would be in graphene for all photon energies except for a narrow absorption peak at ωmuch less than 2µ. For negative values of µ, on the other hand, even with β3/negationslash= 0 the conductivity is zero. The experimental observation of such a narrow low energy peak to gether with high energy van Hove singularities could be taken as a measure of nonzero β3. It is interesting to compare these curves for the conductivity wit h the joint density of states (lower frames). The color and line types are the same for both panels. The onset energy as well as energies of the van Hove singularities are unchanged in going from the joint density of states to the conductivity. Also, as is particularly evident in the dotted black and short dashed r ed curves the 1 /ωfactor in σxx(ω) leads to a nearly flat background for the conductivity as compared with a reg ion of nearly linear rise in the density of states. This is true for both positive and negative values of µ. In conclusion we have calculated the interband longitudinal conduct ivity as a function of photon energy for the case of combined Rashba and Dresselhaus spin-orbit coupling. We ha ve also considered the possibility of a cubic Dresselhaus contribution. We find that in the persistent spin helix st ate when the spins are locked at an angle of 3 π/4 independent of momentum, which arises when the linear Rashba coup ling is equal to the linear Dresselhaus coupling, the interband optical transitions vanish and there is no finite energ y absorption from these processes. Only the Drude intraband transitions will remain. When the cubic Dresselhaus term is nonzero the cancelation is no longer exact but we expect interband absorption to remain strongly depressed for photon energies above 2 µas compared, for example, to the universal background value found in single layer graphene. W e propose interband optics as a sensitive probe of the relative size of Rashba and Dresselhaus spin orbit coupling as w ell as cubic corrections. METHODS The optical conductivity is given by σxx(ω) =e2 iω1 4π2/integraldisplaykcut 0kdkdθT/summationdisplay lTr/angbracketleftˆvx/hatwideG(k,ωl)ˆvx/hatwideG(k,ωn+ωl)/angbracketrightiωn→ω+iδ. (4) HereTis the temperature and Tris a trace over the 2 ×2 matrix, and ωn= (2n+ 1)πTandωl= 2lπTare the Fermion and Boson Matsubara frequencies respectively with nandlintegers. To get the conductivity which is a real frequency quantity, we needed to make an analytic continuation fr om imaginary iωntoω+iδ, whereωis real and δ is an infinitesimal. The velocity operators ˆ vxand ˆvyare given by ˆvx=∂H0 /planckover2pi1∂kx=VIˆI+Vxˆσx+Vyˆσy ˆvy=∂H0 /planckover2pi1∂ky=V′ IˆI+V′ xˆσx+V′ yˆσy. (5) HereVI=/planckover2pi1kx/m,Vx= (β1−β3k2 y)//planckover2pi1,Vy= (−α1+ 2β3kykx)//planckover2pi1,V′ I=/planckover2pi1ky/m,V′ x= (α1−2β3kykx)//planckover2pi1and V′ y= (−β1+β3k2 x)//planckover2pi1. The Green’s function can be written as [39] /hatwideG(k,ωn) =1 2/summationdisplay s=±(ˆI+sFk·ˆσ)G0(k,s,ωn) (6) whereFk= (S1,−S2,0)//radicalbig S2 1+S2 2, G0(k,s,ωn) =1 i/planckover2pi1ωn+µ−/planckover2pi12k2 2m−s/radicalbig S2 1+S2 2(7)9 and S1= (α1ky+β1kx−β3kxk2 y) S2= (α1kx+β1ky−β3kyk2 x) (8) The wave function is given by Ψk,±|0>=1√ 2/bracketleftbigg c† k,↑|0>±S1−iS2/radicalbig S2 1+S2 2c† k,↓|0>/bracketrightbigg , (9) with corresponding eigenvalues E±=/planckover2pi12k2 2m±/radicalBig S2 1+S2 2. (10) Herec† k,↑(c† k,↓) creates a particle with momentum kand spin up (down). The spin expectation values work out to be Sx=/planckover2pi1 2/angbracketleftΨk,±|σx|Ψk,±/angbracketright=±/planckover2pi1 2S1/radicalbig S2 1+S2 2 Sy=/planckover2pi1 2/angbracketleftΨk,±|σy|Ψk,±/angbracketright=±/planckover2pi1 2−S2/radicalbig S2 1+S2 2 Sz=/planckover2pi1 2/angbracketleftΨk,±|σz|Ψk,±/angbracketright= 0. (11) These formulas allow us to calculate the spin texture, as well as the o ptical conductivity as given in Eq. (2). ACKNOWLEDGMENTS This work was supported by the Natural Sciences and Engineering R esearch Council of Canada (NSERC), the Canadian Institute for Advanced Research (CIFAR), and Alberta Innovates. AUTHOR CONTRIBUTIONS ZL carried out the calculations, and all authors, ZL, FM, and JPC co ntributed equally to the development of the work. COMPETING FINANCIAL INTERESTS The authors declare no competing financial interests. [1] Wolf, S. A. et al., Spintronics: A Spin-Based Electronics Vision for the Futu re.Science294, 1488-1495, (2001). [2] Hasan, M. Z. and Kane, C. L., Colloquium: Topological ins ulators.Rev. Mod. Phys. 82, 3045C3067 (2010). [3] Moore, J. E., The birth of topological insulators. Nature464, 194-198 (2010). [4] Qi, X.-L. and Zhang, S.-C., The quantum spin Hall effect an d topological insulators. Physics Today 63, 33-38 (2010). [5] Bernevig, B. A., Hughes, T. L. and Zhang, S.-C., Quantum S pin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science314, 1757-1761 (2006). [6] Fu, L., Kane, C. L. and Mele, E. J., Topological Insulator s in Three Dimensions. Phys. Rev. Lett. 98, 106803 (2007). [7] Chen, Y.-L. et al., Experimental Realization of a Three-Dimensional Topolog ical Insulator, Bi2Te3.Science325, 178-181 (2009). [8] Li, Zhou and Carbotte, J. P., Hexagonal warping on optica l conductivity of surface states in Topological Insulator Bi2Te3. Phys. Rev. B 87, 155416 (2013). [9] Pesin, D. and MacDonald, A. H., Spintronics and pseudosp intronics in graphene and topological insulators. Nature Mate- rials11, 409-416 (2012).10 [10] Novoselov, K. S. et al., Electric field effect in atomically thin carbon films. Science306, 666-669 (2004). [11] Zhang, X., Tan, Y.-W., Stormer, H. L. and Kim, P., Experi mental observation of the quantum Hall effect and Berry’s phase in graphene. Nature438, 201-204 (2005). [12] Mak, K. F., Lee, C., Hone, J., Shan, J. and Heinz, T. F., At omically Thin MoS2: A New Direct-Gap Semiconductor. Phys. Rev. Lett. 105, 136805 (2010). [13] Splendiani, A., et al., Emerging Photoluminescence in Monolayer MoS2.Nano Lett. 10, 1271-1275 (2010). [14] Leb´ egue, S. and Eriksson, O., Electronic structure of two-dimensional crystals from ab initio theory. Phys. Rev. B 79, 115409 (2009). [15] Lee, C. et al., Anomalous Lattice Vibrations of Single- and Few-Layer MoS2.ACS Nano 4, 2695-2700 (2010). [16] Li, Zhou and Carbotte, J. P., Longitudinal and spin/val ley Hall optical conductivity in single layer MoS2.Phys. Rev. B 86, 205425 (2012). [17] Li, Zhou and Carbotte, J. P., Impact of electronCphonon interaction on dynamic conductivity of gapped Dirac fermio ns: Application to single layer MoS2.Physica B 421, 97-104 (2013), http://dx.doi.org/10.1016/j.physb.201 3.04.030 [18] Drummond, N. D., Z´ olyomi, V., and Fal’ko, V. I., Electr ically tunable band gap in silicene. Phys. Rev. B .85, 075423 (2012). [19] Aufray, B. et al., Graphene-like silicon nanoribbons on Ag(110): A possible formation of silicene. Appl. Phys. Lett. 96, 183102(2010). [20] Stille, L., Tabert, C. J. and Nicol, E. J., Optical signa tures of the tunable band gap and valley-spin coupling in sil icene. Phys. Rev. B .86, 195405 (2012). [21] Ezawa, M., A topological insulator and helical zero mod e in silicene under an inhomogeneous electric field. New J. Phys. 14, 033003 (2012). [22] Ezawa, M., Spin-Valley Optical Selection Rule and Stro ng Circular Dichroism in Silicene. Phys. Rev. B .86, 161407(R) (2012). [23] Koralek, J. D. et al., Emergence of the persistent spin helix in semiconductor qu antum wells. Nature458, 610-613(2009). [24] Walser, M. P., Reichl, C., Wegscheider W. and Salis, G., Direct mapping of the formation of a persistent spin helix. Nature Phys.8, 757–762 (2012). [25] Bernevig, B. A., Orenstein, J. and Zhang, S.-C., Exact S U(2) Symmetry and Persistent Spin Helix in a Spin-Orbit Coup led System. Phys. Rev. Lett. 97, 236601 (2006). [26] Lin, Y. J., Jimenez-Garcia K. and Spielman, I. B., Spin- orbit-coupled Bose-Einstein condensates. Nature, 471, 83-86 (2011). See also Ozawa, T. and Baym, G., Ground-state phases of ultracold bosons with Rashba-Dresselhaus spin-orbit coupling. Phys. Rev. A 85, 013612 (2012). [27] Bloch, I., Dalibard, J. and Zwerger, W., Many-body phys ics with ultracold gases. Rev. Mod. Phys. 80, 885-964 (2008). [28] Rashba, E.I., Properties ofsemiconductors with an ext remumloop. 1. Cyclotron and combinational resonance in ama gnetic field perpendicular to the plane of the loop. Sov. Phys. Solid State 2, 1109 (1960). [29] Dresselhaus, G., Spin-Orbit Coupling Effects in Zinc Bl ende Structures. Phys. Rev. 100, 580–586 (1955). [30] Jozwiak, C. et al., Widespread spin polarization effects in photoemission fro m topological insulators. Phys. Rev. B 84, 165113 (2011) [31] Jozwiak, C. et al., Photoelectron spin-flipping and texture manipulation in a topological insulator. Nature Phys. 9, 293-298 (2013) [32] Xu, S.-Y. et al., Topological Phase Transition and Texture Inversion in a Tu nable Topological Insulator. Science332, 560 (2011). [33] Wang, Y. H. et al., Observation of a Warped Helical Spin Texture in Bi2Se3from Circular Dichroism Angle-Resolved Photoemission Spectroscopy. Phys. Rev. Lett. 107, 207602 (2011) [34] Shen, S.-Q., Spin Hall effect and Berry phase in two-dime nsional electron gas. Phys. Rev. B 70, 081311 (R) (2004) [35] Li, J., Hu, L. and Shen, S.-Q., Spin resolved Hall effect d riven by spin-orbit coupling. Phys. Rev. B 71, 241305 (R) (2005) [36] Khomitsky, D. V., Electric-field induced spin textures in a superlattice with Rashba and Dresselhaus spin-orbit co upling. Phys. Rev. B .79, 205401 (2009). [37] Li, Zhou, Covaci, L. and Marsiglio, F., Impact of Dresse lhaus versus Rashba spin-orbit coupling on the Holstein pol aron. Phys. Rev. B 85, 205112 (2012). [38] Maytorena, J. A., Lopez-Bastidas, C. and Mirele, F., Sp in and charge optical conductivities in spin-orbit coupled systems. Phys. Rev. B .74, 235313 (2006). [39] Grimaldi, C., Cappelluti, E. and Marsiglio, F., Off-Fer mi surface cancellation effects in spin-Hall conductivity o f a two- dimensional Rashba electron gas. Phys. Rev. B 73, 081303R (2006); Spin-Hall Conductivity in Electron-Phon on Coupled Systems. Phys. Rev. Lett. 97, 066601 (2006).

1711.07041v1.SU__N___spin_wave_theory__Application_to_spin_orbital_Mott_insulators.pdf

SU(N) spin-wave theory: Application to spin-orbital Mott insulators Zhao-Yang Dong, Wei Wang, and Jian-Xin Li National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China (Dated: November 21, 2017) We present the application of the SU( N) (N > 2) spin-wave theory to spin-orbital Mott insulators whose ground states exhibit magnetic orders. When taking both the spin and orbital degrees of freedom into account rather than projecting onto the Kramers doublet, the lowest spin-orbital locking energy levels, due to the inevitable spin-orbital multipole exchange interactions, the SU( N) spin-wave theory should take the place of the SU(2) one. To implement the application, we introduce an ecient general local mean eld approach which involves all the local uctuations into the SU( N) linear spin-wave theory. Our approach is tested rstly by calculating the multipolar spin-wave spectra of the SU(4) antiferromagnetic model. Then we apply it to spin-orbital Mott insulators. It is revealed that the Hund's coupling would in uence the eectiveness of the isospin-1 =2 representation when the spin orbital coupling is not large enough. Besides, we also calculate the spin-wave spectra based on the rst principle calculations for two concrete materials, -RuCl 3and Sr 2IrO4. The SU(N) spin-wave theory appropriately depicts the low-energy magnons and the spin-orbital excitations qualitatively. I. INTRODUCTION The physics of transition-metal oxides (TMOs) with 4dor 5dorbitals occupied has drawn considerable atten- tion recently. One reason is that the spin-orbital cou- pling (SOC), which was considered as a small pertur- bation until recently, entangles the spin and orbital de- grees of freedom. This eect in cooperation with elec- tronic correlations could give rise to a novel type of in- sulators (spin-orbital Mott insulators) in which the local moments are spin-orbital entangled Je= 1=2 Kramers doublets1{3. Another is their crystal structures with a special bond geometry formed by edge-shared octahe- dra, which will result in the anisotropy and the frustra- tion of the eective Hamiltonian4, because the exchange coupling between the local moments depends highly on the spatial direction of the exchange path. The Hamilto- nian with such a novel symmetry could lead to unconven- tional magnetism, including spin liquids, multipolar or- ders and uncommon magnetic orders1. In real materials, the zigzag (Na 2IrO35and 4dTMOs-RuCl 36{8), spiral (Li2IrO39{11) type magnetic orderings, and a canted an- tiferromagnetic (AF) structure (Sr 2IrO4)12,13have been proved. Generally, 4 dand 5dstates are spatially so extended that the Hubbard interaction is reduced compared to that of 3dstates. However, owing to the large crystal eld and SOC, a separate band with a reduced bandwidth allows for the opening of a Mott gap. The underlying picture for this process is as following. For a d5electronic con- guration, when the two egorbitals split o due to the crystal eld of octahedrons, the ve electrons loaded on thet2gorbitals results in a s= 1=2 hole residing in an eectivel= 1 orbitals. A strong SOC leads to a sys- tem with a fully lled Je= 3=2 band and a half-lled Je= 1=2 band. Thus, the so-called spin-orbital Mott insulators emerge even with a relatively small electronic correlation. In this case, the Je= 1=2 states presentthe essential physics and eectively behavior as spin-1 =2 pseudo spins. The resulting spin-exchange model can be obtained by projecting the electronic Hamiltonian onto theJe= 1=2 Kramers doublet which consists of only dipole-dipole interaction terms. To study the low-energy excitations of this spin-1 =2 system with a magnetically ordered ground state, one can resort to the famous SU(2) linear spin-wave theory14. However, in many real materi- als the mixing between the egandt2gorbitals are always presented and the deviation from the spherical symme- try drags some composition of Je= 3=2 states into the Kramers doublet15. In addition, the Hund's coupling in the multi-orbital system will induce electrons to orbit in the same direction. All of these would weaken the va- lidity of the picture of a half-lling Je= 1=2 Kramers doublet, and complicate the spin exchange Hamiltonian by introducing the interactions between spin-orbital mul- tipolar momentum1. Thus, the spin-orbital multipolar orders and excitations are needed to be considered. Generally, to study a spin-1 =2 system with a magnet- ically ordered ground state and small quantum uctu- ations, the famous SU(2) linear spin-wave theory14are used, in which the spins are regarded as a classical three- components vector and its uctuations are described by rotations of the vector. However, when the degrees of freedom of both spins and orbitals are involved, it is in- sucient to treat the local states as the rotations of a classical three-components angular momentum. There- fore, a generalization of the SU(2) linear spin-wave theory is needed16. Recently, the SU( N) spin-wave theory based on the multi-boson approach has been introduced17{20. Since the generators of the SU( N) group can be repre- sented as bilinear forms in N- avored bosons, instead of two bosons in the SU(2) spin-wave theory, the low-energy modes of the SU( N) spin-wave theory are described with N1 dierent bosons, which would provide a more ac- curate description of the low-energy excitations for un- conventional magnetic orders.arXiv:1711.07041v1 [cond-mat.str-el] 19 Nov 20172 In this paper, we will use the SU( N) spin-wave theory to study the magnetic excitations in spin-orbital Mott insulators. In the SU( N) spin-wave theory, the local or- der parameter is dened in the space of SU( N) unitary transformations of the local spin states, instead of the SU(2) space of local spin rotations, and it consists of N21 components of the SU( N) order parameter in the most general form. Therefore, a universal local mean eld theory facilitating the SU( N) spin-wave theory is re- quired. Here, we introduce a general ecient local mean eld theory based on the supercoherent state21, which fully includes the on-site quantum uctuations essential for multipolar states. As an illustration, we rst apply the SU(N) spin-wave theory to a toy three-band Hubbard model on a hexagon lattice, and focus on the examination of the eect of Hund's coupling by calculating the weights ofJe= 1=2 stats in the ground state and spin-wave spectra. If the SOC is not large enough to lift the spin- orbital excitations across the Je= 1=2 andJe= 3=2 states away from those within the Je= 1=2 doublets, the Hund's coupling will compel the angular momentum Lto parallel the spin momentum. Therefore, the low energy physics is not governed only by the Je= 1=2 ef- fective Hamiltonian. We then study the spin excitations in two systems of TMOs, -RuCl 3and Sr 2IrO4where the eective Hamiltonian include both spin and orbital degrees of freedom, by using the SU ( N) linear spin-wave theory. Our results for the magnetic ground states and their low-energy spin dynamics in two systems are consis- tent with recent experiments3,7,8,13. In addition, we can obtain the high-energy spin-orbital excitations across the gap in the presence of the spin-orbital coupling. The paper is organized in the following manner. In section II, we brie y review the Schwinger bosons rep- resentation and SU( N) spin-wave theory, then intro- duce the general local mean eld theory. In section III, based on the SU(4) antiferromagnetic Hamiltonian22{24, we calculate its magnon excitations and spin-3 =2'sl= 2 multipole-multipole correlation function. In section IV, we apply the SU( N) spin-wave theory to spin-orbital Mott insulators. First, we derive an eective Hamilto- nian from a three-band Hubbard model with the SOC in the hexagon lattice and study the magnetic dynamics by the SU(N) spin-wave theory. Then we calculate the spin correlation function of -RuCl 3with the ve-band Hub- bard model and correlation function of resonant inelastic X-Ray scattering (RIXS) operators25of Sr 2IrO4with a three-band Hubbard model. II. SU(N) LINEAR SPIN-WAVE THEORY Muniz et al present a mathematical framework of the multi-boson approach to generalize the traditional spin- wave theory from SU(2) to SU( N)20. As we know, the eective exchange models from the electron models in thestrong interaction limit would always be written as H0=Jrr0 00S rS00 r0+hr S r; (1) where the repeated index r;r0;;;00is summed up, andS rare the generators of SU( N) group, which obey the commutation relations [S r;S00 r0] =r;r0(S0 r0S0 r0): (2) Then, they can be represented by Schwinger bosons. In the spin-wave theory, one of the bosons will be condensed depending on a given magnetic order and the rest N1 dierent bosons will be used to describe the low-energy modes of systems. In this section, we will rst review the multi-boson approach based on the Schwinger bosons representation. Then, a general local mean eld theory will be introduced and applied to the SU( N) linear spin- wave theory. A. Schwinger bosons representation It is often useful to map a spin model into a bosonic one, which may be easier to study since bosons have sim- ple commutation relations. Also, the common magnons are bosonic excitations which are proper to be repre- sented in bosonic language. In the Schwinger bosons representation, the SU( N) generators are written as26, S r=by rb r; (3) n1X =0by rb r=nb; (4) whereby randb r(= 0;1;:::;n1) are bosonic creation and annihilation operators on the local site r, respec- tively. Eq. (4) is a constraint on the bosonic operators in the physical space. nbis the number of bosons on the local site, denoting the order of the irreducible rep- resentations of SU( N) group. For the well known SU(2) linear spin-wave theory, we set nb= 2S. Here we use nb=1 for simplicity. Thus, nbindicates the dimensions of the local state and there is an one-to-one match between each boson and each local dimension. Furthermore, the space of local operators is a n2-dimensional linear space, which could be expanded on the basis of the identity and then21 generators of SU( N) group. Correspondingly, the identity is the constraint Eq. (4) and n21 genera- tors are bilinear forms byb. So, any local operator can be expressed as a linear combination of bosonic bilinear forms. To sum up, all local uctuations are described by bosonic particle-hole forms byb. For instance, if there is a local spin S= 3=2, then local uctuations can be expanded by the multipole expansion, which has 16 = (2S+ 1)2dierent scattering channels classied by the total spin of a pair of particle and hole. Ml;m=X m1(1)s2+mm1Cs1;s2;l m1;mm1;mbs1;m1ybs2;m1m; (5)3 whereCs1;s2;l m1;mm1;mare Clebsch-Gordan coecients, and (s1;m1);(s2;mm1) are the spin quantum numbers of the particle and hole, respectively. Ml;mis multipole spin operators. Ml;mis the identity when l= 0, the dipolar operatorsS+,SandSzwhenl= 1, and the quadrupo- lar and octupolar operators when l= 2;3. There are totally 16 =P3 l=02l+ 1 multipole spin operators, which are equal to the dimensions of the space of local operators and can also be expanded by SU( N) generators. There- fore, SU(N) spin-wave theory based on this multi-boson approach includes all of bosonic multipolar excitations. B. Local mean eld theory It is necessary to construct a general local mean eld theory to utilize all advantages of the SU( N) spin-wave theory. As we known, the parameter manifold of a n- dimensional ( n-D) state is ( n1)-D complex projective space CP(n1) when the overall phase is neglected. There aren1 complex parameters, which are 2( n1) real parameters. The local mean-eld state should travel all over the space, so according to the supercoherent states constructed by Fatyga et al21, we assume the test local wave function to be generated from a unitary trans- formation acting on an given state, jTir=U(xr)b0y rj0i: (6) U(xr) is the unitary transformation and j0iis the vac- uum without any bosons: U(xr) = exp[iX 6=0(x21 r(b0y rb r+by rb0 r); +x2 r(iby rb0 rib0y rb r))] (7) j0i= (0;0;0;:::;0|{z} n)T; (8) where x2R2(n1), the 2(n1)-D real space. Obviously, U(xr) is particle conserved, so the test state complies with the constraint Eq. (4). It is arduous to nd the minimum in such a plain space. Thus, we will utilize the structure of CP( n1) to convert the x2R2(n1) parameter space to the rotation space in the n-D complex space, x1=1cos(2)cos(1); x2=1cos(2)sin(1); x3=1sin(2)cos(3)cos(2); x4=1sin(2)cos(3)sin(2); :::; x2n3=1sin(2):::sin(n1)cos(n1); x2(n1)=1sin(2):::sin(n1)sin(n1); j2f0;g;j2f0;2g: Whenn= 2, it is the well known state of spin-1 =2,jTi= (cos(1);ei1sin(1))T, where (1;1) are Euler angles. Itcorresponds to a rotation in 2-D complex space or 3-D real space. The mean eld ground state of the system is the di- rect product state of local wave function, jGi=NjTir, which would minimize the energy of hGjHjGi. Due to the translational symmetry of the ground state, gener- ally only the magnetic cell is considered in the spin-wave theory. C. SU(N) Linear spin-wave approximation It is known that the spin-wave approximation is based on the Holstein-Primako (HP) bosons which dene the spin-deviation operators. Its generalization can be ob- tained by extending the HP representation from SU(2) to SU(N)20. To obtain the SU( N) HP bosons, we should rst determine the condensed boson which creates the local state minimizing the mean-eld energy. According to the variational form of the mean eld ground state in- troduced in the last subsection, the condensed boson is the one minimizing hGjHjGi, withjGi=Q r~b0y rNj0ir. It is related to the Schwinger boson brvia the unitary transformation Eq. (7), ~b0y r=X U0(xr)by r: (9) Namely, ~b0y ris the= 0 component of ~br, and the corre- sponding creation and annihilation operator are replaced by a number according to the constraint of Eq. (4), ~b0y r'~b0 r'vuut1n1X =1~by r~b r: (10) Then, the N1 bosons ~b6=0 rbecome the HP bosons, which describe the spin waves originating from uctua- tions around the ordered spin state created by the con- densed boson ~b0y r. Substituting Eq. (10) into the Hamil- tonian Eq. (1) and retaining only the quadratic terms, we get, H'X hr;r0iJrr0 0000+ (Jrr0 000by rb0 r0+Jr;r0 000b rb0 r0+H:c) +X rhr 00+hr 00b0y rb0 r+X hr;r0i[(Jrr0 00Jrr0 0000)b0y rb0 r +(Jrr0 0000Jrr0 000000)b0y r0b0 r0]; (11) where the index ;;0;06= 0 and will be summed up when appear twice in a single term, and the tilde ~ on Jrr0 00andb r, which denotes the expressions after the unitary transformation that minimizes the mean eld variational energy, is omitted for simplicity. Now Eq. (11) is a free bosonic Hamiltonian and can be solved by performing the Fourier transformation, b k=1p LX rb reik
r; (12)4 withLthe lattice number of the system. It leads to, H=X k y kh(k) k; k= (b1 k;:::;bM(n1) k;b1y k;:::;bM(n1)y k)T;(13) whereMis the size of magnetic cell. There are two di- agonalization methods for a bosonic Hamiltonian as pro- posed by White27and Colpa28. After diagonalization, we get the spin-wave dispersion (k) as expressed by, H=M(n1)X =1(k) y k  k;  k=T 0b0 k; (14) withT 0the element of the matrix used to diagonalize the Hamiltonian. As noted, the SU( N) spin-wave theory includes not only the dipole-dipole correlations, but also the multipole-multipole correlations. In general, the cor- relation function of two SU( N) generators can be written by, S00(k;!) =1 2M(n1)Z dtei!t r;r0eik
(rr0)hS rS00 r0(t)i:(15) As same as the SU(2) linear spin-wave theory, only the quadratic forms of the dynamical part of correlation func- tions are calculated. Therefore, the correlation function is expanded inhbybi, which describes the probability to excite one of bosonic excitations. It is clear that there areM(n1) spin-wave modes. III. SU( 4) ANTIFERROMAGNETISM As an example, we rst calculate the spin-wave spec- trum for the SU(4) antiferromagnetic model in a square lattice. The model can be generated from the generic one-band Hubbard model loaded with spin-3 =2 fermions. Due to Paulis exclusion principle, the wave functions of two on-site fermions have to be antisymmetric. The total spin of two on-site spin-3 =2 fermions can only be either singlet (S= 0) or quintet ( S= 2). So the eective model at quarter-lling will have only two exchange channels, and the spin singlet channel results in the SU(4) antifer- romagnetic Hamiltonian: H=JX hi;ji2 4X 1a<b5ab iab j5X a=1a ia j3 5; (16) where aare Dirac matrices which form Cliord alge- bra,fa;bg= 2aband ab= a;b =(2i). Specif- ically, the ve Dirac matrices can be expressed as ten- sor products of tow Pauli spin-1 =2 matrices ( ;), orrepresented by symmetric bilinear combinations of the components of a spin-3 =2 operator, Sx;Sy;Sz: 1=zy=1p 3fSy;Szg; 2=zx=1p 3fSx;Szg; 3=y0=1p 3fSx;Syg; 4=x0=1p 3 (Sx)2(Sy)2 ; 5=zz= (Sz)25 4: First of all, the spin exchange Hamiltonian stems from a SU(2) symmetrical one-band Hubbard model with spin- 3=2 fermions, so it has the genetic SU(2) symmetry. Also, all 15 Gamma operators together span the SU(4) alge- bra. Among them, the 10 aboperators are SO(5) anti- symmetric tensors, while the ve aare SO(5) vectors. Thus the Hamiltonian Eq. (16) obviously possesses SO(5) symmetry. Moreover it also has a hidden SU(4) symme- try in the bipartite lattice23. We can dene a particle- hole transformation b!Jbywith an antisymmetric matrixJ=ixy. With this operation, the fundamen- tal representation transforms to a conjugate representa- tion where ab= aband a=a. If transforming allBsublattices into the conjugate representation, then we have, H=JX hi;ji2 4X 1a<b5ab iab j+5X a=1a ia j3 5: (17) One should note that Eq. (16) is invariant under SU(4) rotations and conjugate rotations on sublattices Aand B, respectively, rather than under uniform SU(4) trans- formations. In a square lattice, the SU(4) linear spin wave theory shows a long-range Neel order which is consistent with the quantum Monte Carlo simulations29. There are three local order parameters of SU(4) Neel order in the square lattice: 12;34;5
= ((1)x+ym;(1)x+ym;m ). In the case of spin-3 =2, they can be expanded in multipole orders as dened in Eq. (5): 12=2p 5(2M1;0M3;0); 34=2p 5(M1;0+ 2M3;0); 5= 2M2;0: Therefore, we choose to calculate a quadrupolar- quadrupolar correlation function along high symmetry directions, M2(k;!)/X r;r0eik
(rr0)Z dtei!t*X mM2;m(r)My 2;m(r;t)+ ;5 0 01 12 E FIG. 1. (Color online) Spin waves of the SU(4) antiferro- magnetic model in a square lattice along high symmetry di- rections. The dashed lines denote the dispersions, and the size and color of the marks indicate the intensity of the quadrupolar-quadrupolar correlation function. The numerical results are shown in Fig. 1. The Gold- stone manifold is CP(3) = U(4) =[U(1)NU(3)] with 6 branches of spin waves, which are degenerated and look like the dispersion of the SU(2) antiferromagnetic spin waves in a square lattice. However, the quadrupolar- quadrupolar correlation exhibits a noticeable intensity at the = (0;0) point as shown in Fig. 1. It is in sharp con- trast to the behavior of the antiferromagnetic spin-spin correlation, which vanishes at that point. IV. SU(N) SPIN WAVE STUDY OF TMOS As we know, most of TMOs have a magnetic ordered ground state. Considering that these magnetic ordered states can be described by isospins which are the entan- gled states of spin and orbital degrees of freedom, we use the SU(N) spin wave theory to investigate excitations from the ordered state. We will rst present a general method to derive the eective exchange model from an electron model in the strong interaction limit. We con- sider the multi-band Hubbard model which is suitable to describe properties of TMOs, H=X hiji;0tij 0cy icj0+X iHi: Here the rst term is hopping terms with tij 0the element of hopping integrals, and indicates all the local degrees of freedom, such as orbitals and spins. Hiare the local interactions which include the multi-band Hubbard term Vi, SOCOi, and local potential eld Wi, Vi=1 2X mm0nn0X fUm=m0=n=n0(1) +U0mn0m0n(1mm0) +Jhmnm0n0(1mm0) +J0mm0mn0(1mn)(1)g
cy imcy im0cincin0; (18) Oi=Si
Li; (19) Wi=X wicy ici: (20)whereU(U0) is the intra-orbital (inter-orbital) Coulomb interaction, JhandJ0are the Hund's coupling and the pairing hopping, respectively. In this paper, we employ U=U0+ 2JhandJ0=Jhas used usually. By means of the perturbation theory, we treat the hop- ping terms as the perturbation in the strong interaction limit and obtain the eective exchange model which can be generally written as, He=X iP0 iHiP0 i+X hi;ji[Hi!j+Hj!i]; (21) Hi!j=X (lre) 01
lrethiji 0h s00 ii (lre)thjii 0h ~s ji (lre):(22) The rst term in Eq. (21) is the zero and rst order perturbation term, and the second is the second order perturbation term accounting for the virtual hoppings of electrons contributing to spin exchanges. P0 iis the op- erator projecting the Hamiltonian Hiinto its low-energy subspace.s i=cy iciand ~s i=cicy iare SU(N) gen- erators and their conjugate representation, respectively. (lre) denotes various scattering channels related to the virtual processes from a low energy state j ri=Q ijrii to a high onej ei=Q ijeii, and back to the low one j li=Q ijlii, whereQ ijriiis the eigenstate of Hamil- tonianP iHi. 1=
lre= 1=2(Eli+EljEeiEej) + 1=2(Eri+ErjEeiEej), in which Emi(m=l;e;r ) is the eigenenergy of the local state jmiion the site i. [ ](lre)indicates a special representation of s iand ~s j in the states (jlii;jrii;jeii) h s ii (lre)=jliihlijcy ijeiiheijc ijriihrij; =hlijcy ijeiiheijc ijriiSliri i;h ~s ii (lre)=jliihlijc ijeiiheijcy ijriihrij; =hlijc ijeiiheijcy ijriiSliri i; whereSliri i=jliihrijis the SU(N) generator in the fun- damental representation dened on the low-energy space ofHi. We note the symmetry of Hamiltonian Eq. (22) is related to the symmetry of ( jlii;jrii;jeii) andtij 0, which are determined by the symmetry of the crystal structure. Now with Eq. (21), we will carry out the SU( N) spin wave calculation. A. Three band Hubbard model with an SOC on the hexagon lattice As an illustration of the application of the SU( N) spin wave theory, let us rst consider a simple three band Hub- bard model with one spin-1 =2 particle per site and SOC, ~ s
~l(The minus sign is due to that lis a mirror angular momentum) on the hexagon lattice. The Hubbard term6 λ J 1.2 0.8 0.4 000.40.81.2 00.20.40.60.81.0 A BCh FIG. 2. (Color online) Weights of the Je= 1=2 states in ground states vary with andJh, calculated based on the three band Hubbard model with an SOC on the hexagon lat- tice. The intra-orbital Coulomb interaction is U= 5:0 presents SU(2) and SO(3) symmetry with U=U0+ 2Jh. Focusing on the eect of Hund's coupling and SOC, we suppose a simply isotripic hopping term, tij 0=t0 only among the nearest neighbours. If SOC is absent, its eective exchange model is com- paratively explicit. Because the wave functions of two on-site fermions have to be antisymmetric, there are only three exchange channels. The initial and nal low energy states are singly occupied states with zero en- ergy, and three intermediate states which are vacuum states on one site and doubly occupied states on the other site with 1) total spins are S= 1, total orbital momentums L= 1 and
lre=U+ 3Jh, 2) total spins areS= 0, total orbital momentums L= 2 and
lre=U+Jhand 3) total spins are S= 0, total or- bital momentums L= 0 and
lre=U2Jh. However, when SOC is comparable to the Hubbard term, U, there will be 20 = 2 52 channels due to the in- terplay of the SOC and Hund's coupling: two kinds of initial and nal states with energy =2 andrespec- tively, and ve kinds of intermediate states with energy U3Jh=2;(2UJhp 25J2 h+ 10Jh+ 92)=2 and (4U8Jh+p 16J2 h+ 8Jh+ 92)=4. Substitut- ing
lrewith the corresponding ( jlii;jrii;jeii) andtij 0 into Eq. (22), we can easily obtain the exchange model numerically. IfJh= 0;= 0,Hihas SU(6) symmetry, so does (jlii;jrii;jeii) andtij 0, but the symmetry of eigenstates will be broken into SU(2) by either SOC or Hund's cou- pling. Furthermore, when tij 0is SU(2) symmetrical, the eective Hamiltonian must be SU(2) symmetrical too. If Jh, only the lowest energy channel is active. In this case, the Hamiltonian can be further approximated to be an eective isospin-1 =2 model. However, the Hund's cou- pling will lower the energy of the spin parallelling states of two electrons, while the SOC will lower the energy of single electron Je=sl= 1=2 states. This would in uence the validity of the isospin Je= 1=2 model. Therefore, we intend to take both andJhinto account to examine the SU(6) spin-wave spectrum of the system. 01 0201a) b) c)E E EFIG. 3. (Color online) Spin waves of three band Hubbard model on the hexagon lattice with SOC and Hund's coupling Jh, which are: a) = 0;Jh= 0, b) > 0;Jh= 0 and c) = 0;Jh>0. Firstly, the local mean eld theory suggests a mag- netic cell with two sites, so we suppose the local mean eld wave function in two sublattices of the hexagon lat- tice arejTAiandjTBi. In order to verify the validity of the isospin Je= 1=2 model, we calculate the weight (hJe= 1=2jTAi+hJe= 1=2jTBi)=2 ofJe= 1=2 states in ground states as shown in Fig. 2. We use the hopping termt= 1 as unit, set U= 5:0 and change andJh from 0 to 1:2. There are roughly three regions: A. Right- side region in which the ground states are dominated by Je= 1=2 states; B. A bump in the area of small and Jhwhere ground states are also dominated by Je= 1=2 states; C.Jhis so large that the ground states are mixed by theJe= 3=2 states. The blue discontinuous region on the right top is due to the divergence of the second order perturbation, which means the SOC gap is compa- rable to the Hubbard gap. Thus the low energy physics can certainly be described by the Je= 1=2 doublet in the region beyond this discontinuous region (where the SOC is dominated). Let us rst consider some extreme situations. The calculated dispersions for spin excitations in three band Hubbard model based on the spin wave theory for several cases are shown in Fig. 3. When Jh= 0 and= 0, there are highly degenerated zero energy spin waves suggesting that the magnetic order are unstable, as shown in Fig. 3 a). This is because the ground state is the SU(6) plaque- tte state30,31in this situation, where SU(6) spins form local singlets on a hexagon plaquette. There is no long- range ordering on which the SU( N) spin wave theory is based, so the spin wave theory fails in this case. As  increases, the zero energy spin waves are lifted [see Fig. 3 b)], and the system approaches ordered phases because the uctuations become weak gradually as the system departs the SU(6) symmetry due to SOC. On the other7 00.511.5 01234a) c)b) d) 1 1 3 3 1 3E EE 0123Region A Region B Region C e) FIG. 4. (Color online) Spin waves with paremeters: a) = 0:9;Jh= 0:6, b)= 0:4;Jh= 0:4 and c)= 0:2;Jh= 1:1. The dashed lines denote dispersions. The size and saturation of makers indicate the intensity of correlation function, and three dierent channels are indicated by three dierent col- ors. e)Reciprocal lattices and high symmetry directions of a hexagon lattice. d)The legend indicating the compositions of the correlation function. hand, there is a ferromagnetic-like spin wave emerging when turning on the Hund's coupling Jhinstead of SOC , as shown in Fig. 3 c). However, there is still some zero energy degeneracies. Thus, the ground state may be still an SU(6) plaquette state or some RVB states. Then, we study the correlation functions in three re- gions A,BandC, respectively. In the dipole-dipole ap- proximation, the correlation function consists of three parts of contributions: spin ippings within either Je= 1=2 or 3=2 states and spin ippings across the Je= 1=2 andJe= 3=2 states, which are denoted by 1 1, 33 and 13, respectively. In Figs 4 a)-c), we present the dispersions of spin waves denoted by the dashed lines and intensities of the correlation functions indicated by the saturation of three dierent colors and size of mark- ers. The colors will mix as shown by the legend in Fig. 4 e), when spin wave excitations includes more than two types of contributions. In region A, the result suggests an antiferromagnetic-like spin wave at low energies, which is linear around point and the intensity diverges at 0 but vanishes at point, and a ferromagnetic-like spin wave at high energies above 2, which is parabolic around point and the intensity is higher at than 0point. At the meantime, the result calculated by using the local mean eld theory shows the system has a Je= 1=2 anti- ferromagnetic ordered ground state, conrming that theexcitations at low energies are indeed antiferromagnetic spin waves. As shown by the cyan-blue color in Fig. 4 a), these low-energies excitations comes basically from spin ippings within the Je= 1=2 states, so the low-energy physics in region Ais dominated by isospin-1 =2 states. Furthermore, the excitations arising from the spin ip- pings across the Je= 1=2 andJe= 3=2 states as de- noted by the magenta color are far beyond the low-energy excitations due to the suciently large SOC. Thus, we arrive at the conclusion that an eective isospin Heisen- berg model can depict the low-energy physics in region A, which is also consistent with the calculation of weights ofJe= 1=2 states in ground states as shown in Fig. 2. When the SOC is decreased, we will enter gradually into region B. In this progress, the gap between the low- energy antiferromagnetic spin wave and the high-energy ferromagnetic spin wave decreases gradually. However, as long asJhis not large enough, although the dispersion of ferromagnetic spin waves overlaps with the low energy one, the two spin waves do not entangle each other, as indicated by Fig. 4 b) where the colors representing two dierent kinds of spin waves do not mix. Thus, apart from the eective isospin Heisenberg terms in the Hamil- tonian, which describes the antiferromagnetic spin waves, there have to be another term to describe the ferromag- netic spin waves at least. Starting from region B, one can increaseJhto enter into region C. In this region, the an- tiferromagnetic and ferromagnetic spin waves are entan- gled, so that there is no well-dened antiferromagnetic- like spin waves or ferromagnetic-like spin waves, and the local test wave functions of ground state in two dier- ent sublattices are not completely orthogonal, namely hTAjTBi0:016. Because the ground state consists of bothJe= 1=2 andJe= 3=2 states now, the multipolar orders are inevitable to be taken into account. Its dipo- lar order parameters hJ eiare almost antiferromagnetic, but quadrupolar order parameters hJ eJ e+J eJ eiare ferromagnetic. In this case, all degrees of freedom have to be taken into account and there is no so-called isospin eective Hamiltonian, so the SU( N) spin wave theory rather than the traditional SU(2) one is applicable. B.-RuCl 3and Sr 2IrO 4 In this subsection, we will use the SU( N) spin-wave theory to study spin dynamics in -RuCl 3and Sr 2IrO4. Both-RuCl 3and Sr 2IrO4have ad5conguration and have an octahedral crystal eld. Their dierences are that the active electrons residing in 4 dorbitals of Ru has a smaller SOC than that in 5 dof Ir, and-RuCl 3is a honeycomb lattice while Sr 2IrO4is a square lattice. -RuCl 3has a layered crystal structure with Ru3+ forming the honeycomb lattice layers and the energy bands near the Fermi level are dominated by the dor- bitals of Ru. We consider a ve band tight-binding model with ve electrons per site and the on-site crystal elds to describe the 4 d5conguration of Ru3+. The tight-8 binding parameters include the nearest-, next-nearest- and third-nearest-neighbour hopping integrals, which are obtained by tting to the energy-band dispersions cal- culated by the rst principle calculations and given in our previous paper Ref. [32]. We take U= 2:7 eV;Jh= 0:13U;U0=U2Jh;and= 0:14 eV7,15,32{35in the fol- lowing calculations. Then, an eective exchange model is obtained numerically according to Eq. (21). Due to the large crystal eld potential on the egorbitals, there are isolated six lowest energy states, onto which we will project the initial and nal states. Using the local mean eld theory and the SU( N) Linear spin-wave approxi- mation introduced in Sec. II, we investigate numerically the magnetic ground state and spin dynamics. Numer- ical results show that the magnetic ground state has a zigzag type order of which the magnetic unit cell con- tains four sites (two cells), in agreement with experiments in-RuCl 36{8The spin-spin correlation functions calcu- lated by Eq. (15) is shown in Fig. 5 a). Below 30 meV, four zigzag spin waves are evident, and the other sixteen excitations around 200 meV come from the spin-orbital excitations across the Je= 1=2 andJe= 3=2 states. Though there is long-range zigzag spin order, the results in Fig. 5 a) show that the low-energy spin waves have a gap of about 2 meV at Mpoint and the spin-spin correla- tion function has a maximum magnitude also at Mpoint. These results are consistent with the recent experiments of inelastic neutron scatterings on -RuCl 37,8. On the other hand, the gap between the zigzag spin waves and the spin-orbital excitations is of about 210 meV, thus suggests that the low energy physics of -RuCl 3could be captured by an eective isospin-1 =2 model. We have found in our previous paper Ref. [32] that the minimum eective isospin-1 =2 model is the K- model containing a ferromagnetic nearest-neighbor Kitaev interaction (K) and a nearest-neighbor o-diagonal exchange interaction (). Now let us turn to Sr 2IrO4. We start our investigations from a three band Hubbard model with a single hole per site to t the band dispersion around the Fermi level36,37, and choose U= 3:6 eV;Jh= 0:18U;and= 0:37 eV in the calculation. Because iridium is a strong absorber of neutrons, it is more useful to calculate the resonant inelastic X-ray scattering (RIXS) spectrum for the pur- pose of a comparison with experiments. RIXS involves a second order process that includes an absorption and an emission of a photon. In the fast collision approximation, the direct RIXS spectrum is proportional to the correla- tion function of spin-orbital moment operators25. Due to the two scattering progresses (absorption and emis- sion), the total angular momentum of spin-orbital mo- ment operators is equal to the coupling of two l= 1 angular momenta (angular momentum exchange of the two scatterings is one in the dipole limit). Thus, there exists multipole-multipole correlations in RIXS besides the usual dipole-dipole correlations. It is known that the RIXS spectrum of Sr 2IrO4is dependent on the inci- dent angle3. So, we calculate the correlation function for 200250300 0510 0.20.40.60.81 0 200400600700 0.250.500.751.00a) b)FIG. 5. (Color online) Spin-spin correlation functions for - RuCl 3a), and correlation functions of RIXS operators25for Sr2IrO4b) along the high-symmetry lines, calculated by the SU(N) spin-wave theory. two dierent incident angles = 8;85using the SU(6) spin theory, and the results are presented in Fig. 5 b) where the left hand one is for = 8and right hand for = 85. Below 200 meV, both results exhibit the gapless antiferromagnetic spin waves dispersing up linearly from the point, which are consistent with experiments in Sr2IrO43,12,13. Above 200 meV, a gap of 180 meV exists arising from the SOC, and the spin-orbital excitations across the gap are ferromagnetic-like spin waves that are parabolic around point. Moreover, there is a small gap in the spin-orbital exciton resulting from the splitting in thet2gorbital. We also notice that these spin-orbital exciton modes correspond to a type of SU( N) bosons in the framework of the SU( N) spin-wave theory. As for the incident-angle dependence of the spectrum, one can see that the scattering intensity of the low-energy Je= 1=2 antiferromagnetic magnon is suppressed heavily, and at the same time the spin-orbital excitations are strongly enhanced for a small incident angle such as = 8, as shown in the left-hand side in Fig. 5 b). While, an oppo- site behavior of the spectrum is observed for a large inci- dent angle such as = 85(the right-hand side in Fig. 5 b)). Around the 0point, the intensity vanishes and only the dispersion of the spectrum is reserved, because the resolution is in uenced due to the antiferromagnetic di- vergence at 0. The results presented above demonstrate a good per- formance of the SU( N) spin wave theory in the study of magnetic orders and dynamics in TMOs. Compared with the SU(2) spin wave theory, the SU( N) theory con-9 tains more than one type of uncondensed bosons, so that the spin-orbital or multipolar orders and excitations can be captured. Of course, the linear approximation used here involves only single magnon excitations and does not take their interactions into account. So, the broad- ening and renormalization of the magnonic spectrum are not captured. To study other spin dynamics, such as magnon decay eects38,39, one should goes beyond the linear order approximation. We note that some modi- cations of the spin-wave theory40,41have been developed in the SU(2) case, their generalizations to the SU( N) case deserve further study. V. CONCLUSION In summary, we implement the application of the SU(N) spin wave theory by introducing an ecient local mean eld method based on the supercoherent state. The approach is tested rstly by applying to the investigation of magnetic properties in the SU(4) antiferromagnetic model in a square lattice. We nd a long-range Neel order which is consistent with the quantum Monte Carlo sim- ulations, and this order can be interpreted by multipolar orders of 3=2 spins. We have also calculated the multipo- lar spin waves of the SU(4) antiferromagnetic model, to demonstrate the application of SU( N) spin wave theoryin the description of multipolar orderings. Due to the entanglement of spin and orbital degrees of freedom, the multipole-multipole exchange terms are also present in the eective exchange models of spin-orbital Mott insu- lators. Only if the spin-orbital coupling is large enough that the low-energy physics is conned in Kramers dou- blet, the eective Hamiltonian will be described by an isospin-1=2 model. In this aspect, we examine a toy three-band Hubbard model on a hexagon lattice and nd that the Hund's coupling also aects the validity of the isospin-1=2 picture when the spin-orbital coupling is be- low a critical value. Finally, we apply the SU( N) spin wave theory to two systems of spin-orbital Mott insula- tors,-RuCl 3and Sr 2IrO4. Our results for the magnetic ground states and their low-energy spin dynamics in both systems are consistent with recent experiments. We also obtain the high-energy spin-orbital excitations across the gap in the presence of the spin-orbital coupling. ACKNOWLEDGMENTS This work was supported by the National Natural Sci- ence Foundation of China (11374138 and 11774152) and National Key Projects for Research and Development of China (Grant No. 2016YFA0300401). jxli@nju.edu.cn 1W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annu. Rev. Condens. Matter Phys. 5, 57 (2014). 2B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, Phys. Rev. Lett. 101, 076402 (2008). 3J. Kim, M. Daghofer, A. H. Said, T. Gog, J. van den Brink, G. Khaliullin, and B. J. Kim, Nat. Commun. 5, 4453 (2014). 4G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). 5X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim, H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill, Phys. Rev. B 83, 220403 (2011). 6J. S. Sears, M. Songvilay, K. W. Plumb, J. P. Clancy, Y. Qiu, Y. Zhao, D. Parshall, and Y.-J. Kim, Phys. Rev. B91, 144420 (2015). 7A. Banerjee, J. Yan, J. Knolle, C. A. Bridges, M. B. Stone, M. D. Lumsden, D. G. Mandrus, D. A. Tennant, R. Moess- ner, and S. E. Nagler, arXiv:1609.00103. 8K. Ran, J. Wang, W. Wang, Z.-Y. Dong, X. Ren, S. Bao, S. Li, Z. Ma, Y. Gan, Y. Zhang, J. T. Park, G. Deng, S. Danilkin, S.-L. Yu, J.-X. Li, and J. Wen, Phys. Rev. Lett. 118, 107203 (2017). 9A. Bin, R. D. Johnson, I. Kimchi, R. Morris, A. Bom- bardi, J. G. Analytis, A. Vishwanath, and R. Coldea, Phys. Rev. Lett. 113, 197201 (2014). 10S. C. Williams, R. D. Johnson, F. Freund, S. Choi, A. Jesche, I. Kimchi, S. Manni, A. Bombardi, P. Manuel,P. Gegenwart, and R. Coldea, Phys. Rev. B 93, 195158 (2016). 11A. Bin, R. D. Johnson, S. Choi, F. Freund, S. Manni, A. Bombardi, P. Manuel, P. Gegenwart, and R. Coldea, Phys. Rev. B 90, 205116 (2014). 12B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, Science 323, 1329 (2009). 13J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F. Mitchell, M. Van Veenendaal, M. Daghofer, J. Van Den Brink, G. Khaliullin, and B. J. Kim, Phys. Rev. Lett. 108, 177003 (2012). 14J. T. Haraldsen and R. S. Fishman, J. Phys.: Condens. Matter 21, 216001 (2009). 15H.-S. Kim, V. Shankar V., A. Catuneanu, and H.-Y. Kee, Phys. Rev. B 91, 241110 (2015). 16A. Joshi, M. Ma, F. Mila, D. N. Shi, and F. C. Zhang, Phys. Rev. B 60, 6584 (1999). 17B. Bauer, P. Corboz, A. M. L auchli, L. Messio, K. Penc, M. Troyer, and F. Mila, Phys. Rev. B 85, 125116 (2012). 18K. Penc, J. Romh anyi, T. R~ o om, U. Nagel, A. An- tal, T. Feh er, A. J anossy, H. Engelkamp, H. Murakawa, Y. Tokura, D. Szaller, S. Bord acs, and I. K ezsm arki, Phys. Rev. Lett. 108, 257203 (2012). 19J. Romh anyi and K. Penc, Phys. Rev. B 86, 174428 (2012). 20R. A. Muniz, Y. Kato, and C. D. Batista, Prog. Theor. Exp. Phys. 2014 , 83I01 (2014). 21B. W. Fatyga, V. A. Kosteleck y, M. M. Nieto, and D. R. Truax, Phys. Rev. D 43, 1403 (1991). 22Y. Qi and C. Xu, Phys. Rev. B 78, 014410 (2008).10 23C. Wu, J.-P. Hu, and S.-C. Zhang, Phys. Rev. Lett. 91, 186402 (2003). 24H.-H. Hung, Y. Wang, and C. Wu, Phys. Rev. B 84, 054406 (2011). 25J. Luo, G. Trammell, and J. Hannon, Phys. Rev. Lett. 71, 287 (1993). 26D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988). 27R. M. White, M. Sparks, and I. Ortenburger, Phys. Rev. 139, A450 (1965). 28J. Colpa, Physica A 93, 327 (1978). 29K. Harada, N. Kawashima, and M. Troyer, Phys. Rev. Lett. 90, 117203 (2003). 30P. Nataf, M. Lajk o, P. Corboz, A. M. L auchli, K. Penc, and F. Mila, Phys. Rev. B 93, 201113 (2016). 31H. H. Zhao, C. Xu, Q. N. Chen, Z. C. Wei, M. P. Qin, G. M. Zhang, and T. Xiang, Phys. Rev. B 85, 134416 (2012). 32W. Wang, Z.-Y. Dong, S.-L. Yu, and J.-X. Li, Phys. Rev. B96, 115103 (2017).33A. Koitzsch, C. Habenicht, E. M uller, M. Knupfer, B. B uchner, H. C. Kandpal, J. van den Brink, D. Nowak, A. Isaeva, and T. Doert, Phys. Rev. Lett. 117, 126403 (2016). 34L. J. Sandilands, Y. Tian, A. A. Reijnders, H.-S. Kim, K. W. Plumb, Y.-J. Kim, H.-Y. Kee, and K. S. Burch, Phys. Rev. B 93, 075144 (2016). 35S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent , Phys. Rev. B 93, 214431 (2016). 36H. Watanabe, T. Shirakawa, and S. Yunoki, Phys. Rev. Lett. 105, 216410 (2010). 37H. Wang, S.-L. Yu, and J.-X. Li, Phys. Rev. B 91, 165138 (2015). 38A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. B 79, 144416 (2009). 39S. M. Winter, K. Riedl, A. Honecker, and R. Valenti, arXiv:1702.08466. 40M. Takahashi, Phys. Rev. B 40, 2494 (1989). 41Z. Z. Du, H. M. Liu, Y. L. Xie, Q. H. Wang, and J.-M. Liu, Phys. Rev. B 92, 214409 (2015).

1211.6221v2.Short_range_asymptotic_behavior_of_the_wave_functions_of_interacting_spin_half_fermionic_atoms_with_spin_orbit_coupling__a_model_study.pdf

Short range asymptotic behavior of the wave-functions of interacting spin-half fermionic atoms with spin-orbit coupling: a model study Yuxiao Wu and Zhenhua Yu Institute for Advanced Study, Tsinghua University, Beijing 100084, China (Dated: June 22, 2021) We consider spin-half fermionic atoms with isotropic Rashba spin-orbit coupling in three direc- tions. The interatomic potential is modeled by a square well potential. We derive the analytic form of the asymptotic wave-functions at short range of two fermions in the subspace of zero net mo- mentum and zero total angular momentum. We show that the spin-orbit coupling has perturbative eects on the short range asymptotic behavior of the wave-functions away from resonances. We argue that our conclusion should hold generally. I. INTRODUCTION The study of dilute unitary atomic Fermi gases has sig- nicantly and substantially extended our knowledge on strongly interacting many-body systems [1]. Theoretic advances in this direction have been propelled by ob- serving the short-range asymptotic behavior of the wave- function of the systems [2{4]. In such ultra-cold dilute gases, when the separation rbetween two fermions of dierent species is much smaller than the mean inter- particle spacing d, but bigger than r0, the range of the interatomic potential, the pair wave-function of the two fermions is mainly s-wave and has the asymptotic form s(r) = 1as=r; when a similar situation happens to two fermions of same species, the pair wave-function is mainlyp-wave and its radial part has the asymptotic form p(r) =k(rup=r2). Herekis the relative wave vector between the two fermions and of order 1 =d,asis thes-wave scattering length and upis thep-wave scat- tering volume. Thus, when considering the short-range correlations, the p-wave part can be safely neglected com- pared to the s-wave part unless upbecomes divergent. Based on this observation, a collection of remarkable re- lations regarding the short-range correlations and various physical observables have been derived [2{4]. On the other hand, recent successful engineering of synthetic gauge eld adds another important dimension to atomic gases [5{10]. Raman processes couple dierent hyper-ne states of atoms and realizes a model with an eective spin-orbit coupling for spin-half particles when the hyper-ne states other than the lowest two are adi- abatically eliminated. This success raises the prospect of using ultra-cold atomic systems to study and simulate spin-orbit coupling physics [11], which has excited great interest in condensed matter physics [12, 13]. In the con- text of Fermi gases, subsequent theoretical studies inves- tigated the eects of the spin-orbit coupling on the scat- tering and bound states of two fermionic atoms [14{20], the BEC-BCS crossover in two and three dimensions [21{ 29], collective motions in the fermionic super uid phase Electronic address: huazhenyu2000@gmail.com[30{33], the equation of state in the high temperature regime [34], and its joint eects with polarization [35{ 40] and mass imbalance [36], and possible emergence of majorana fermions [41, 42]. The introduction of the spin-orbit coupling to ultra- cold atomic gases also raises another interesting question: How would the short-range asymptotic behavior of the wave-function of atomic gases be modied? Attempts to answer this question include: Cui studied the two- body problem of fermions with symmetric Rashba spin- orbit coupling in three directions and discovered that the usuals-wave pseudo-potential is still applicable in certain regimes. This indicates that the eects of the spin-orbit coupling on the short-range asymptotic behavior of the wave-function can be perturbative [15]. Later Ref. [34] presented a general argument based on the magnitudes of relevant length scales. Since the spin-orbit coupling strength, as usually realized in the experiments by Ra- man processes, corresponds to a length scale 500 nm, which is much larger than, e.g., the interatomic poten- tial range50 nm, the wave-function inside the range of the interatomic potential should remain intact to zero order. The asymptotic behavior of the wave-function out- side the interatomic potential range which is determined by the one inside the range should stay unchanged as well. Reference [18] reached a similar conclusion by an argument in which unitary transformations are imple- mented to eliminate the spin-orbit coupling from the ki- netic part of the Hamiltonian. However, besides the case with spin-orbit coupling in one direction [18], the gen- eral arguments mentioned above await justication from explicitly worked out examples. In this paper, we study the problem of two spin-half fermions with isotropic spin-orbit coupling in three di- rections. The single-particle Hamiltonian of the fermions is H1=p2 2m+ mp
+2 2m(1) withmthe mass of the atoms. Without loss of gener- ality, we assume the spin-orbit coupling strength to be positive,>0. Possible experimental realization of the spin-orbit coupling of the form p
has been proposed in Ref. [43]. We model the interatomic potential by an attractive square well potential, V(r) =V0(r0r),arXiv:1211.6221v2 [cond-mat.quant-gas] 17 Mar 20132 wherer0is the potential range and V0>0. As of ex- perimental interest, we assume r01. We calcu- late the scattering and bound state wave-functions of the two interacting fermions in the subspace of zero net mo- mentum and zero total angular momentum (sum of the spin and orbital angular momentum). We derive analyt- ically the asymptotic behavior of the wave-functions out- side the range of the square well potential. Away from resonances, the modication of the asymptotic behavior due to the spin-orbit coupling is shown to be perturba- tive. We explain this perturbative eect by inspecting the wave-functions inside the potential range and argue that our conclusion is valid for generic situations with spin-orbit coupling. II. THE SCHR ODINGER EQUATION The Hamiltonian of two interacting fermions can be cast into the form H2=HK+Hk (2) HK=K2 4m+K 2m
(2+1) (3) Hk=k2 m+k m
(21) +2 m+V(r): (4) HereKis the net momentum of the two fermions and kis the relative one, and rare the relative coordinates. The subscript of labels for the ith-fermion. From Eqs. (3) and (4), the motion of the center of mass and the relative one are coupled together via the spin operators. In the following, we focus on the subspace of K= 0 and J L+S= 0, with Lthe orbital angular momentum and S the total spin, where analytic results can be derived. The relative wave-function ( r) in the subspace K= 0 satises the Schr odinger equation "^k2+2 m+M+V(r)# (r) =E (r); (5) with M= 2 m2 666640^kx+i^kyp 2^kxi^kyp 2^kz ^kxi^kyp 20 0 0 ^kx+i^kyp 20 0 0 ^kz 0 0 03 77775(6) when represented in the spin basis [( "##" )=p 2;"";## ;("#+#")=p 2)] for the two spin-half fermions. Since each elements of Mare proportional to the spherical harmonic functions Y1;m( k) or their complex conju- gates, with further contraint J= 0 we have (r) = 0(r)2 64Y0;0( r) 0 0 03 75ip 3 1(r)2 640 Y1;1( r) Y1;1( r) Y1;0( r)3 75:(7)The new spinor wave-function ( r) = [ 0(r); 1(r)]Tsat- ises  1 r2d dr r2d dr +2+m[V(r)E] + 02(d=dr + 2=r) 2(d=dr) 2=r2 (r) = 0:(8) III. SCATTERING STATES It is straightforward to show that for the scattering states, the wavefunctions are <(r) =A j0(q1r) j1(q1r) +B j0(q2r) j1(q2r) (9) forr<r 0, and >(r) =C" h(2) 0(k1r) h(2) 1(k1r)# +D" h(1) 0(k1r) h(1) 1(k1r)# +E" h(1) 0(k2r) h(1) 1(k2r)# +F" h(2) 0(k2r) h(2) 1(k2r)# (10) forr>r 0. Herejiare the spherical Bessel functions and h(1;2) iare the spherical Hankel functions, and k1k; k 2k+; (11) q1q; q2+q; (12) kp mE; qp m(E+V0): (13) Since there are six coecients ( AtoF) and four con- nection conditions at r=r0for the wavefunctions and an overall normalization factor, the wavefunctions can be determined up to a coecient; there are two lin- early independent solutions for each energy E. Let us choose the ith linearly independent solution  ias < i= [j0(qir);j1(qir)]Tforr < r 0. By requiring  i and its rst derivative continuous at r=r0, we nd the corresponding coecients for > i Ci=k2qi k1+k2h(1) 0(~k1)j1(~qi) +h(1) 1(~k1)j0(~qi) h(1) 1(~k1)h(2) 0(~k1)h(1) 0(~k1)h(2) 1(~k1)(14) Ei=k1+qi k1+k2h(2) 0(~k2)j1(~qi)h(2) 1(~k2)j0(~qi) h(2) 0(~k2)h(1) 1(~k2)h(2) 1(~k2)h(1) 0(~k2)(15) Di=C i (16) Fi=E i; (17) with ~kikir0and ~qiqir0. The physical meaning of the above coecients can be understood in the following way. Without interaction, V(r) = 0, either [ j0(k1r);j1(k1r)]Tor [j0(k2r);j1(k2r)]T is the free particle solution to Eq. (8). The interaction V(r) realizes mutual scattering between the two waves.3 We can construct two new linearly independent solutions as > k1=E 2> 1E 1> 2 E 2C1E 1C2=" h(2) 0(k1r) h(2) 1(k1r)# +S11" h(1) 0(k1r) h(1) 1(k1r)# +S21k2 k1" h(1) 0(k2r) h(1) 1(k2r)# (18) > k2=C2> 1C1> 2 C2E 1C1E 2=" h(2) 0(k2r) h(2) 1(k2r)# +S22" h(1) 0(k2r) h(1) 1(k2r)# +S12k1 k2" h(1) 0(k1r) h(1) 1(k1r)# ;(19) whereSijare the elements of the matrix S=1 E 2C1E 1C2 " E 2C 1E 1C 2k2 k1(C 2C1C 1C2) k1 k2(E 2E1E 1E2)C1E2C2E1# :(20) Sinceh(1) i(x)eix=xandh(2) i(x)eix=xwhenx! 1, the wave-function > k1(> k2) describes the process that the incoming wave of wave vector k1(k2) is scattered by V(r) into the outgoing waves of wave vectors k1andk2. The matrix Sformed by the coecients Sijis the scat- teringS-matrix. It is straightforward to show that S satises the unitary condition SyS= 1. The appearance of the ratios between k1andk2in Eq. (20) is because the two wave vectors are of dierent magnitude. By diagonalizing the unitary S-matrix, we obtain the standing-wave solutions >  =v 1k1> k1+v 2k2> k2 =v 1k1" h(2) 0(k1r) h(2) 1(k1r)# +v 2k2" h(2) 0(k2r) h(2) 1(k2r)# +e2i( v 1k1" h(1) 0(k1r) h(1) 1(k1r)# +v 2k2" h(1) 0(k2r) h(1) 1(k2r)#) ; (21)where [v 1;v 2]Tare the eigenvectors of Sgiven by Eq. (20) and e2iare the corresponding eigenvalues. From Eqs. (14) and (15), one can prove k2 1(E 2E1 E 1E2) =k2 2(C 2C1C 1C2). Since the S-matrix can be expanded as S=~S01+~Sxx+~Szzand the ra- tio between the coecients ~Sxand ~Szis real, we can choose the eigenvectors [ v 1;v 2]Tto be real. Note that [> ]is proportional to > apart from a phase. This complies with the expectation that solutions to the one- dimensional dierential equation (8) can be chosen to be real. Equation (21) corresponds to the ansatz used in Ref. [15] [cf. Eqs. (31) and (32) therein]. The two new phase shifts , characterizing the scattering eects, are the counterparts of the s-wave andp-wave phase shifts sandpin the absence of the spin-orbit coupling. We plot(kr0) in Fig. (1) and +(kr0) in Fig. (2) for~= 0:01 and= 1 with ~r0and p mV0r2 0. Generically we nd exp[2 i(0)] =1 and exp[2i+(0)] = 1. The unitarity of (0) is a reminiscent of one-dimensional scattering [15]. When both and +are dened in the domain [ =2;=2], to zero order of~,(0) jumps by whereasswitches between 1to +1, and the slope of (k) in the small klimit changes sign where up ips between1 to +1. To analyze the properties of , we expand the S-matrix to rst order ofk S=x+M~k+O(~k2); (22) where ~kkr0. The elements Mijof the matrix Mare listed in the Appendix. Correspondingly in such limit, we have=sgn(0 )=2 +0 ~k,+=0 +~kand v=1p 2 1 1 +v~k 1 1 ; (23) v+=1p 2 1 1 v~k 1 1 ; (24) with 0 =i 4(M11+M22M12M21) =f2~422~2+[2+~222~2+ 2~4+ (2+~2) cos(2 ~)] sin(2) + 22~[~cos(2 ~)sin(2 ~)] + 2~cos(2)[22~+~32sin(2 ~)]g=f22~2[cos(2)cos(2 ~)] + 2~2(2~2) sin(2)g; (25)4 0 +=i 4(M11+M22+M12+M21) =f[2+~222~2+ 2~4(2+~2) cos(2 ~)] sin(2) + 2 ~cos(2)[~322~+2sin(2 ~)] 2~[~32~2~cos(2 ~) +2sin(2 ~)]g=f22~2[cos(2)cos(2 ~)] + 2~2(2~2) sin(2)g;(26) v=(M11M22)=4 =f2~[cos(2) cos(2 ~)1] + (2+~2) sin(2) sin(2 ~)g=f2~2[cos(2)cos(2 ~)] + 2 ~2(2~2) sin(2)g:(27) 0 5/Multiply10/Minus410/Minus3kr0Π/Slash14Π/Slash12∆/Minus FIG. 1: Phase shift (kr0) vskr0for~= 0:01 and= 1. The three quantities, 0 ,0 +andv, diverge when  ~[cos(2)cos(2 ~)] + (+~) sin(2) = 0;(28) where a bound state emerges [cf. Eq. (38)]. This diver- gence re ects that bound state energies are the poles of theS-matrix. Away from the singularities, these quanti- ties have the asymptotic expansions in ~as 0 =r0 ~2ash 1 +O(~2)i ; (29) 0 +=2up 3r0h 1 +O(~2)i ; (30) v=1 ~2up 3asr0 +O(~3): (31) For our square well potential model, the s-wave scat- tering length asis given by r0=as==(tan), and thep-wave scattering volume upbyup=r3 0= 13(1 cot)=2. Equations (29) and (30) explain the jumps of(0) and the sign change of the slope of +(k) for smallk. Exceptions occur at resonances: at the rst set of res- onances given by Eq. (41), we nd (0) =+(0) ~=2,v= [0;1]T, and v+= [1;0]T; at the second set, (0) =+(0)2~3=3,v= [0;1]T, and v+= [1;0]T. The asymptotic form of Eq. (21) in the region r&r0 0 0.01 0.02kr04/Multiply10/Minus88/Multiply10/Minus8∆/PlusFIG. 2: Phase shift +(kr0) vskr0for~= 0:01 and= 1. is > =v 1+v 2 r+ (v 1k1+v 2k2) cot 1 0 +v 2 k2v 1 k11 r2+1 2(v 2k2v 1k1) +1 3cot(k2 2v 2k2 1v 1)r 0 1 : (32) Note that an overall factor 2 esinhas been omit- ted. In the low energy limit k!0, when away from resonances, using Eqs. (29) to (31), we nd > 1 r1 as 1 0 + 2up 3asr2+ 1 +r 3as 0 1 ; (33) > +1 r2 as 1 0 +1 r2+ 2r up 0 1 :(34) At the rst set of resonances given by Eq. (41), > = > + 1=r2=r0 1=r2+=22r=3r0 ; (35) at the second set > = > + 1=r+ 3=22r3 0 1=r2+=2 +r=2r3 0 ; (36) Note that Eqs. (33) to (36) satisfy Eq. (8) with E= 0 to the leading order of .5 IV. BOUND STATES The bound state solutions to Eq. (8) with energy E are Eq. (9) for r<r 0and >(r) =C0" h(1) 0(k0 1r) h(1) 1(k0 1r)# +D0" h(2) 0(k0 2r) h(2) 1(k0 2r)# (37) forr > r 0. Herek0 1+i,k0 2i, andp mE. The requirement that the wave-function and itsrst derivative are continuous at r=r0gives rise to the equation determining the energy E: 0 =(~2+ ~2+ ~)(~q2~2)[j0(~q1)j1(~q2)j0(~q2)j1(~q1)] + 2~~~q[j0(~q2)j1(~q1) +j0(~q1)j1(~q2)] 2~~q[(~+ 1)2+~2]j0(~q1)j0(~q2) 2~~q(~2+ ~2)j1(~q1)j1(~q2); (38) with ~r0and ~qqr0. Assuming D0= 1, we have C0e2i=2~qh(2) 1(~k2)j0(~q1)j0(~q2)h(2) 0(~k2)[j0(~q1)j1(~q2)(~qi~) +j0(~q2)j1(~q1)(~q+i~)] 2~qh(1) 1(~k1)j0(~q1)j0(~q2)h(1) 0(~k1)[j0(~q1)j1(~q2)(~q+i~) +j0(~q2)j1(~q1)(~qi~)]: (39) In the weak attraction limit !0, we nd ~=22 41 sin~ ~!23 5; (40) there is always a bound state no matter how weak the attraction is, in contrast to the case in the absence of the spin-orbit coupling. Furthermore, in the limit ~1, Eq. (40) gives the bound state energy E=a2 s4=m, agreeing with a previous pseudopotential calculation [14]. Equation (40) ts the numerical results given in Ref. [15]. The energies of bound states other than in the weak at- traction limit have also been studied in Ref. [15]. According to Eq. (38), a bound state emerges at thresholdE= 0 when Eq. (28) is satised. In the limit ~!0, Eq. (28) reduces to j0() = 0 orj1() = 0. In the absence of the spin-orbit coupling, the former condition is where a bound state emerges in the p-wave channel, while the latter is where the s-wave scattering length as is zero. The introduction of the spin-orbit coupling has the nonperturbative eect of shifting the onset of a series of bound states from where 1 =as= 0 to where as= 0. For small ~, the critical values of the depth of square well potential Vcwhere a bound state forms at the zero energy threshold are given by cq mVcr2 0=( n~2=n+O(~4); x`+~2=x`+O(~4);(41) withn= 1;2;3;:::and nonzero x`(`= 1;2;3;:::) being the solutions of j1(x`) = 0. The change of Vcdue to small agrees with the numerical results obtained in Ref. [15]. To obtain the asymptotic behavior of the wave- function of the most shallow bound state for r&r0when V0is close to a critical value Vc, let us rst look at the limit!c= 0 where there is only one bound state. From Eq. (40), in the limit ~!0, for the bound state,we have ~=2~2=3. Substituting this into Eq. (39) and expanding in the order 0 <~~1, we obtain = 2 arctan~ ~ 1 +O(~2) '2~ ~; (42) and thus for r&r0the wave-function of the bound state asymptotically is >(r) 1=r1=as 2r2 0=15r2+ ; (43) which coincides with the !0 limit of Eq. (33). WhenV0approaches one of the rst set of critical val- ues of Eq. (41), '1 2~; (44) and >(r) 1=r2=r0 1=r2+=2 ; (45) which agrees with Eq. (35) to the corresponding order ofr. ForV0close to the second set of critical values of Eq. (41), '2 3~3(46) with the asymptotic wave-function >(r) 1=r+ 3=22r3 0 1=r2+=2 ; (47) which is the same as Eq. (36) to the corresponding order ofr.6 V. DISCUSSION Equations (33) and (34) indicate that away from res- onances the modication of the asymptotic forms of the wave-functions at short range due to the spin-orbit cou- pling is perturbative. To see this point clearly, we multi- ply Eq. (34) by and have > 1 r1 as 1 0 + 2up 3asr2+ 1 +r 3as 0 1 ; (48) > +1 r2r up 0 1 +1 r2 as 1 0 ; (49) apart from certain factors, to lowest order of , > have the same asymptotic form as the s-wave-function sand > +as thep-wave-function p. The perturbative modication can be understood by inspecting the wave- functions inside the range of the potential. For r < r 0, in the limit k!0 and expanded to the rst order of ~, < (r) = < 1(r) + 1 + 41 cot ~ < 2(r);(50) and < +(r) = < 1(r) 1 +4 332 +2 tan ~ < 2(r): (51) Thus to zero order of ~, < +(r) and < (r) are the same as thes- andp-waves inside the square well potential in the absence of the spin-orbit coupling respectively. Since the coecients of the terms linear in are well-behaved away from the resonances, nonzero ~adds corrections pertur- batively. Given that >(r) and <(r) need to connect atr=r0, the change of the asymptotic form of >(r) is bound to be perturbative in ~consequently. The above analysis is essentailly the same as the pre- vious argument based on the comparison of energy scales given in Ref. [34]: Within the potential range, the spin- orbit coupling strength is much smaller than 1 =r0andpmV0and therefore generally has perturbative eectson the wave-functions in this regime, and so does on the wave-functions in the regime r&r0. According to this general consideration, we expect the same conclusion ap- plies to the cases with nonzero pair net momentum Kand J6= 0, and to the ones with generic forms of spin-orbit coupling, given additionally K1=r0;pmV0. If one takes the zero range limit r0!0 while with asxed and the scattering volume up!0, to lowest order, Eq. (49) becomes the noninteracting form and Eq. (48) reduces to (1=r1=as)[1;0]T+[0;1]T, the same as found for zero range interactions in Ref. [18]. Nevertheless, Eqs. (50) and (51) signal that the per- turbation expansions in ~break down when as!0 (tan= 0) and 1=up!0 (tan= 0 for6= 0), which are the conditions for the emergence of a new bound state in the limit ~!0. At the rst set of resonances, the asymptotic forms (35) suggests that ~can still be treated perturbatively. At the second set of resonances, extra powers ofin the denomenators of (36) become singular as!0 there. This coincides with the perturbative and nonperturbative shifts of the onset of bound states compared to the cases without the spin-orbit coupling mentioned in Sec. IV. Our calculations suggest that for generic spin-orbit coupling, no matter for bosons or fermions, the correc- tions to the asymptotic form of the wave-functions at short range should be perturbative except for at ne tuned points where resonances occur. Consequently the relations derived based on the asymptotic form [2{4] should stand valid correspondingly. Acknowledgments We thank Xiaoling Cui, Peng Zhang and Qingrui Wang for helpful discussions. This work is supported in part by Tsinghua University Initiative Scientic Research Pro- gram, and NSFC under Grant Numbers 11104157 and 11204152. Appendix A: Elements of the Mmatrix We list the elements of the Mmatrix here: M11=e2i(~)(+~)2e2i(+~)(~)2+ 4i~[2(i+~)~3] 2~2f[cos(2)cos(2 ~)] + (2~2) sin(2)g; (A1) M12=i2~2(22~2) cos(2)22~2cos(2 ~) +[(221)~22~42] sin(2) + 22~sin(2 ~) ~2f[cos(2)cos(2 ~)] + (2~2) sin(2)g; (A2) M21=M12; (A3) M22=M 11: (A4)7 [1] S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 80, 1215 (2008). [2] S. Tan, Ann. Phys. 323, 2952 (2008); 323, 971 (2008). [3] E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008). [4] S. Zhang and A.J. Leggett, Phys. Rev. A 79, 023601 (2009). [5] Y.-J. Lin, R.L. Compton, A.R. Perry, W.D. Phillips, J.V. Porto, and I.B. Spielman, Phys. Rev. Lett. 102, 130401 (2009). [6] Y.-J. Lin, R.L. Compton, K. Jim enez-Garc a, J.V. Porto, and I.B. Spielman, Nature 462, 628 (2009). [7] Y.-J. Lin, R.L. Compton, K. Jim enez-Garc a, W.D. Phillips, J.V. Porto, and I.B. Spielman, Nat. Phys. 7, 531 (2011). [8] Y.-J. Lin, K. Jim enez-Garc a, and I.B. Spielman, Nature 471, 83 (2011). [9] P. Wang, Z. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [10] L.W. Cheuk, A.T. Sommer, Z. Hadzibabic, T. Yef- sah, W.S. Bakr, M.W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [11] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Ohberg, Rev. Mod. Phys. 83, 1523 (2011). [12] M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [13] X. Qi and S. Zhang, Rev. Mod. Phys. 83, 1057 (2011). [14] J.P. Vyasanakere, and V.B. Shenoy, Phys. Rev. B 83, 094515 (2011). [15] X. Cui, Phys. Rev. A 85, 022705 (2012). [16] S. Takei, C.-H. Lin, B.M. Anderson, and V. Galitski, Phys. Rev. A 85, 023626 (2012). [17] J.P. Vyasanakere, and V.B. Shenoy, New J. of Phys. 14, 043041 (2012). [18] P. Zhang, L. Zhang, and Y. Deng, Phys. Rev. A 86, 053608 (2012). [19] P. Zhang, L. Zhang, and W. Zhang, Phys. Rev. A 86, 042707 (2012). [20] L. Zhang, Y. Deng, and P. Zhang, arXiv: 1211.6919. [21] J.P. Vyasanakere, S. Zhang, and V.B. Shenoy,Phys. Rev. B 84, 014512 (2011). [22] M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. 107, 195303 (2011). [23] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett. 107, 195304 (2011). [24] Z. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305 (2011). [25] L. Jiang, X.-J. Liu, H. Hu, and H. Pu, Phys. Rev. A 84, 063618 (2011). [26] L. Han and C.A.R. S a de Melo, Phys. Rev. A 85, 011606 (2012). [27] G. Chen, M. Gong, and C. Zhang, Phys. Rev. A 85, 013601 (2012). [28] L. He and X. Huang, Phys. Rev. Lett. 108, 145302 (2012). [29] L. He and X. Huang, Phys. Rev. B 86, 014511 (2012). [30] J.P. Vyasanakere, and V.B. Shenoy, Phys. Rev. A 86, 053617 (2012). [31] E. Doko, A.L. Subasi, and M. Iskin, Phys. Rev. A 85, 053634 (2012). [32] M. Iskin, Phys. Rev. A 85, 013622 (2012). [33] L. He and X. Huang, arXiv: 1207.2810. [34] Z. Yu, Phys. Rev. A 85, 042711 (2012). [35] W. Yi and G.-C. Guo, Phys. Rev. A 84, 031608 (2011). [36] M. Iskin and A.L. Subasi, Phys. Rev. A 84, 041610 (2011). [37] J. Zhou, W. Zhang, and W. Yi, Phys. Rev. A 84, 063603 (2011). [38] R. Liao, Y. Xiang, and W. Liu, Phys. Rev. Lett. 108, 080406 (2012). [39] X. Yang and S. Wan, Phys. Rev. A 85, 023633 (2012). [40] K. Seo, L. Han, and C.A.R. S a de Melo, Phys. Rev. A 85, 033601 (2012). [41] M. Gong, G. Chen, S. Jia, and C. Zhang, Phys. Rev. Lett. 109, 105302 (2012). [42] K. Seo, L. Han, and C.A.R. S a de Melo, Phys. Rev. Lett. 109, 105303 (2012). [43] B.M. Anderson,G. Juzeliunas, V.M. Galitski, and I.B. Spielman, Phys. Rev. Lett. 108, 235301 (2012).

1012.0973v2.Electron_spin_diffusion_and_transport_in_graphene.pdf

arXiv:1012.0973v2 [cond-mat.mes-hall] 12 May 2011Electron spin diffusion and transport in graphene P. Zhang and M. W. Wu∗ Hefei National Laboratory for Physical Sciences at Microsc ale and Department of Physics, University of Science and Technology of China, Hefei, Anhui , 230026, China (Dated: October 29, 2018) We investigate the spin diffusion and transport in a graphene monolayer on SiO 2substrate by means of the microscopic kinetic spin Bloch equation approa ch. The substrate causes a strong Rashba spin-orbit coupling field ∼0.15 meV, which might be accounted for by the impurities initially present in the substrate or even the substrate-in duced structure distortion. By surface chemical doping with Au atoms, this Rashba spin-orbit coupl ing is further strengthened as the adatoms can distort the graphenelattice from sp2tosp3bondingstructure. By fittingtheAudoping dependence of spin relaxation from Pi et al.[Phys. Rev. Lett. 104, 187201 (2010)], the Rashba spin-orbit coupling coefficient is found to increase approxi mately linearly from 0.15 to 0.23 meV with the increase of Au density. With this strong spin-orbit coupling, the spin diffusion or transport length is comparable with the experimental values. In the st rong scattering limit (dominated by the electron-impurity scattering in our study), the spin di ffusion is uniquely determined by the Rashba spin-orbit coupling strength and insensitive to the temperature, electron density as well as scattering. With the presence of an electric field along the s pin injection direction, the spin transport length can be modulated by either the electric field or the ele ctron density. It is shown that the spin diffusion and transport show an anisotropy with respect to th e polarization direction of injected spins. The spin diffusion or transport lengths with the injec ted spins polarized in the plane defined by the spin-injection direction and the direction perpendi cular to the graphene are identical, but longer than that with the injected spins polarized vertical to this plane. This anisotropy differs from the one given by the two-component drift-diffusion model, wh ich indicates equal spin diffusion or transport lengths when the injected spins are polarized in t he graphene plane and relatively shorter lengths when the injected spins are polarized perpendicula r to the graphene plane. PACS numbers: 72.25.Rb, 75.40.Gb, 72.80.Vp, 71.70.Ej I. INTRODUCTION Graphene is considered to be a promising candidate for the spintronic applications recently,1–22partly due to the perfect two dimensionality, gate-voltage-tunable charge carrier type and density,3,4high mobility5–7,23,24 as well as the potentially long spin relaxation time limited by the small intrinsic spin-orbit and hyperfine couplings.8–12,25,26From the high mobility and long spin relaxation time, a long spin relaxation length, favor- able to the spin information transport and manipula- tion, is anticipated. However, both the spin relaxation time and transport length were experimentally found to be much smaller than expected.2,25,27–32This suggests that the spin relaxation in the experiments is most likely to be contributed by extrinsic factors such as the possi- ble impurity-enhanced spin relaxation28,30via the Elliot- Yafet33mechanism or the enhanced Rashba spin-orbit coupling field9,11from the impurities.26,34,35The former case may exist in a highly dirty graphene sample and causes the spin relaxation time τsto be proportional to the momentum relaxation time τp.28,30However, for the latter case, the Dyakonov-Perel (DP) spin relaxation mechanism36dominates and the relation τs∝τpis ab- sent. In fact, recently Pi et al.reported that τsincreases with decreasing τpin the surface chemical doping exper- iment with Au atoms on graphene,31indicating that the DP spin relaxation mechanism is important there. How-ever, the relation τs∝1/τp, valid when the DP spin relaxation mechanism is dominant and the scattering is strong enough, is not obeyed in their experiment.31Nev- ertheless, we will show that this deviation can be un- derstood by taking account of the strengthening of the Rashba spin-orbit coupling with the increasing coverage of Au adatoms. The Rashba spin-orbit coupling, referred toasanextrinsicone,isduetothebrokenoftheinversion symmetry which can be caused by either a perpendicu- lar electric field, the interaction with substrate, or the atoms adsorbed on the surface.9,11,26,34,35The contribu- tion of the electric field to the Rashba spin-orbit coupling is small ( ∼µeV under a perpendicular electric field as large as 1 V/nm),26,32while the adatoms can effectively enhance the Rashba spin-orbit coupling to be of order of 10 meV by distorting the graphene lattice from sp2to sp3bonding structure.26,34,35 In this work, we investigate the spin diffusion and transport limited by the DP mechanism in a graphene monolayer on SiO 2substrate as presented by Pi et al..31 To account for the short spin relaxation time ( ∼70 ps) before Au doping in the experiment,31we assume that the impurities inevitably present in the substrate, as well as the other effects such as the substrate-induced struc- ture distortion, cause a strong Rashba spin-orbit cou- pling. When the surface chemical doping by Au atoms31 is performed, the Rashba spin-orbit coupling coefficient αRis further strengthened. By fitting the chemical dop-2 ing dependence of spin relaxation time from the exper- imental data,31we obtain the chemical doping depen- dence of αR. It is found that αRincreases approximately linearly with the density of adatoms when the latter is not too high. With this essential information obtained, we then study the spin diffusion and transport in the graphene layer. The method utilized in our study is the kinetic spin Bloch equation (KSBE) approach which has been successfully applied to the study of spin dynamics in semiconductors.37In the framework of this approach, the spatial spin precession frequency during the steady- state scattering-free spin diffusion (assumed to be along thex-axis) is37–41 ωk= (2Ωk+gµBB)/∂kxεk. (1) HereΩkis the DP term, Bis the external magnetic field andεkis the electron energy spectrum. The momentum dependenceof ωkleadstotheinhomogeneousbroadening in spin precession, with which any scattering (including the Coulomb scattering) can cause an irreversible spin relaxation along with spin diffusion and transport.37–41 It is noted that different DP terms as well as different energy spectra lead to distinct momentum dependences ofωk. For graphene, εk=/planckover2pi1vFkwithvF= 106m/sbeing the Fermi velocity and Ωk=αR(−sinθk,cosθk,0) (2) withθkbeing the polar angle of momentum k. Therefore in the absence of any external magnetic field ωk= 2αR(−tanθk,1,0)/(/planckover2pi1vF), (3) which depends on the angle θkrather than the magni- tude ofk. This indicates that the inhomogeneous broad- ening isinsensitive to temperature and electron density as long as αRis fixed. Therefore the spin diffusion is only possible to be modulated effectively by the scattering.42 However, in this work it is revealed that when the scat- tering is strong enough (just as in the graphene layer under study), the spin diffusion becomes insensitive to the scattering. As a result, the spin diffusion is uniquely determined by αR. Moreover, the mean spin precession frequency ∝an}bracketle{tωk∝an}bracketri}ht=2αR /planckover2pi1vF(0,1,0) shows a strong anisotropy which can also lead to the anisotropy of spin diffusion with respect to the spin polarization direction. This anisotropy is found to be quite different from the widely believed one predicted from the two-component drift- diffusion model.43–47The discrepancy reveals the inade- quacy of the two-component drift-diffusion model, espe- ciallyforthecaseswith spinprecessioninspatialdomain. This paper is organized as follows. In Sec. II, we present the model and introduce the KSBEs. In Sec. III, we first investigate the spin relaxation by fitting the ex- perimental data from Pi et al.31to obtain essential pa- rameters and then study the spin diffusion and transport in graphene. Both the analytical and numerical investi- gations are performed. By comparing the results from the analytical and numerical studies, we find that theanalytical model depicts the zero-electric-field spin dif- fusion perfectly and the nonzero-electric-field spin trans- port with a small discrepancy which increases with the strength of the electric field. At last we summarize in Sec. IV. II. MODEL AND KSBES Then-doped graphene monolayer under investigation is on a SiO 2substrate perpendicular to the z-axis. The depth of the SiO 2substrate is assumed to be a= 300 nm and the electric field along the z-axis isEz=Vg/awith Vgbeing the gatevoltage. Thespins areinjected at x= 0 and diffuse or transport along the x-axis. The external electric field, if applied, is along the x-axis, i.e., E= Eˆ x. Under the basis laid out in Refs. 25 and 32, the Hamiltonian of electrons can be written as25 H=/summationdisplay µkss′/bracketleftbig (εk−λI+eEx)δss′+Ωk·σss′/bracketrightbig cµks†cµks′ +Hint. (4) Hereµlabels the valley located at KorK′point,σ denote the Pauli matrices and cµks(cµks†) is the an- nihilation (creation) operator of electron in µvalley with momentum k(relative to the valley center) and spins(s=±1 2).λIis the intrinsic spin-orbit cou- pling constant and −eis the electron charge ( e >0). The coefficient in the Rashba term Ωk[Eq. (2)] reads αR=ζEz+η, with the first term contributed by the electric field along the z-axis and the second term by the substrate (including the impurities initially present in- side) as well as the adatoms from surface chemical dop- ing. The coefficient ζis 5×10−3meV·nm/V (Refs. 26 and 32). The Hamiltonian Hintconsists of the electron- impurity, electron-phonon as well as electron-electron Coulomb interactions.25We adopt the minimal model proposed by Adam and Das Sarma48to depict the electron-impurity scattering. Within this model, only the intravalley electron-impurity scattering is important while the intervalley electron-impurity scattering is neg- ligible due to the large momentum transfer from one valley to the other and the finite distance between the impurity layer and the graphene plane. The intraval- ley electron-impurity scattering matrix element reads |Uk−k′|2=Ni|Vk−k′|2e−2d|k−k′|,48whereNiis the effec- tiveimpuritydensity, disthe effectivedistanceofimpuri- ties from the graphene layer48andVk−k′is the Coulomb potential under the random phase approximation.49 The electron-phonon scattering includes the intraval- ley electron-acoustic phonon scattering,50the intervalley and intravalley electron-optical phonon scattering,51as well as the intravalley electron-optical surface phonon scattering.52The electron-electron Coulomb scattering includes both the intervalley and intravalley scattering, with the screening under random phase approximation given in Ref. 49.3 Byusingthe nonequilibriumGreenfunction method,53 the KSBEs are constructed as37 ∂ρµk(x,t) ∂t=∂ρµk(x,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle dri+∂ρµk(x,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle dif +∂ρµk(x,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle coh+∂ρµk(x,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat.(5) Hereρµk(x,t) represent the density matrices of electrons with relative momentum kin valley µat position xand timet.∂ρµk(x,t) ∂t/vextendsingle/vextendsingle/vextendsingle dri=eE /planckover2pi1∂ρµk(x,t) ∂kxare the driving terms fromtheexternalelectricfield (the fluctuationofelectron density is neglected and thus the total electric field is taken to be the external one). The diffusion terms due to the spatial gradient are ∂ρµk(x,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle dif=−∂εk /planckover2pi1∂kx∂ρµk(x,t) ∂x =−vFcosθk∂ρµk(x,t) ∂x.(6) ∂ρµk(x,t) ∂t/vextendsingle/vextendsingle/vextendsingle cohand∂ρµk(x,t) ∂t/vextendsingle/vextendsingle/vextendsingle scatare the coherent and scat- tering terms, respectively. Their expressions can be found in Ref. 25. In the steady-state scattering-free spin diffusion, the spatial spin precession frequency, given by Eq. (3), is immediately obtained according to the KSBEs.39–41 III. SPIN RELAXATION AND SPIN DIFFUSION AND TRANSPORT In the following, we first study the spin relaxation in graphene by fitting the experimental data from Pi et al.31to obtain information on impurities (including the effective density as well as the distance from the graphene layer) and the chemical doping dependence of the Rashba spin-orbit coupling coefficient. We then use the information to study the spin diffusion and transport in graphene, first analytically for the case with strong electron-impurity scattering only, and then numerically withallthe scattering explicitly included. A. Spin relaxation time We fit the chemical doping dependence of spin re- laxation time and diffusion coefficient from Pi et al. [Fig. 3(c) in Ref. 31] to establish: (i) the density and typical distance from the graphene layer of charged im- purities initially present in the substrate and those of the chemical doping adatoms; and (ii) the dependence ofαRon chemical doping. The electron density Neis 2.9×1012cm−2and the temperature Tis 18 K.31The electrons are initially polarized in the x-yplane31with the polarization P0assumed to be 0.05. To perform thefitting, the KSBEs are solved in the time domain un- der spatial uniform case, as carried out recently by Zhou and Wu in the ultraclean graphene monolayer.25(An an- alytical study of spin relaxation time in graphene is also given in Appendix A.) The diffusion coefficient Dgiven by Piet al.is actually for spin instead of charge, al- though it is treated as the charge diffusion coefficient in the experiment.31In fact, these two coefficients are usually close to each other and J´ ozsa et al.found this most likely to be the case in graphene when the elec- tron density is high ( ∼3×1012cm−2)30due to the weak electron-electron Coulomb scattering.25Therefore we fit the experimental data with the charge diffusion coeffi- cientD=√πNe 2e/planckover2pi1vFµe, whereµeis the electron mobility obtained under a small in-plane electric field.25 1 2 3 4 5 6 0 1 2 3D (10-2 m2/s) NAu (1012 cm-2)(a) 70 80 90 100 0 2 4 6 8 0.15 0.2τs (ps) αR (meV) Au deposition (s)(b) 0.15 0.2 0.25 0 1 2 3αR (meV) NAu (1012 cm-2) FIG. 1: (Color online) (a) Chain curve with triangles: depos i- tion time dependence of calculated diffusion coefficient, wit h the Au density growing linearly with the deposition time wit h a fixed rate of 5 ×1011atom/(cm2·s).31Dashed curve with closed squares: deposition time dependence of calculated d if- fusion coefficient, with the deposition time dependence of Au density given by the solid curve (the scale is on the right- hand side of the frame). Crosses: experimental data from Piet al..31(b) Dashed curve with closed squares: deposition time dependence of calculated spin relaxation time, with th e deposition time dependence of αRshown by the solid curve with open squares (the scale is on the right-hand side of the frame). Crosses: experimental data from Pi et al..31Inset of (b): dependence of αRon Au density.4 We first make use of the value of D≈0.059 m2/s for the case without surface chemical doping given in Ref. 31 to explore the information of impurities initially present in the substrate. This single value of Dis not sufficient for us to fix both the effective density and distance from the graphene layer of these impurities. However, these details are not essential and we just choose two proper parameters, e.g., Ns= 2.1×1012cm−2andds= 0.7 nm, to ensure D≈0.059 m2/s. The surface chemical doping depositsAu atomson the graphenesurfacewith agrowth rate of 5×1011atom/(cm2·s).31By fitting the deposition time (adatom density) dependence of D,31the distance of adatoms from the graphene layer dAuis obtained to be about 0.2 nm. Nevertheless, the fitting does not confirm withtheexperimentaldatawellwhenthedepositiontime exceeds 4 s [compare the fitting data (chain curve with triangles)tothe experimental data(crosses)in Fig. 1(a)]. This indicates that the effective density of adatoms does not increase linearly with time any more when the dop- ing has been performed for several seconds. Therefore, when the doping time is longer than 4 s, we choose the proper density of adatoms to reproduce the experimen- tal diffusion coefficient. In Fig. 1(a), the deposition time dependence of Au density is plotted by the solid curve with open squares (the scale is on the right-hand side of the frame) and that of the calculated diffusion coefficient is shown by the dashed curve with closed squares. With the parameters for two kinds of impurities ob- tained, we then fit the spin relaxation time τsto obtain αRunder different deposition times. In Fig. 1(b), the deposition time dependence of fitted αRis shown by the solid curve with open squares (the scale is on the right- hand side of the frame) and that of the calculated spin relaxation time is shown by the dashed curve with closed squares. The crosses represent the experimental spin re- laxation times under different deposition times. In the inset of Fig. 1(b), we also plot the dependence of αRon Au density NAu. It is shown that αRincreases approxi- mately linearly with Au density when the latter is not so high. The fitted value of αRis comparable to the value estimated by Ertler et al.when taking account of the adatoms, i.e., 0.3 meV.32It is noted that αRτp//planckover2pi1has the largest value 0.027 ≪1 withτp=√Neπ/planckover2pi1 evFµe= 0.12 ps (Ref. 54) when NAu= 0. Therefore, the electron system is in the strong scattering limit (the electron-impurity scattering is dominant), let alone when the tempera- ture is increased or the chemical doping is performed. It is necessary to point out that in the experiment the gate voltage Vgis adjusted to keep Neconstant during chemical doping as adatoms also donate electrons to the graphenelayer.31However, Vgdoes not exceed 200V and the term ζEz=ζVg/ais at leasttwo ordersof magnitude smaller than αR. Therefore αR≈ηand is solely deter- mined by the impurities. When NAu= 0,η= 0.153 meV and is contributed by the impurities in the substrate.B. Spin diffusion and transport: analytical study 1. Spin diffusion In this section we study the spin diffusion in graphene analytically for the case with only the electron-impurity scattering. No external electric field is present. We first perform the Fourier transformation of the steady-state KSBEs with respect to the polar angle θkand then re- tain the equations involving the lowest three orders.41 The neglect of the higher orders will not lose much infor- mation in the strong scattering limit where the electron distributionapproachesisotropyinthemomentumspace. As a result the following second-order differential equa- tion about ρ0 µk(x) [ρl µk(x) =1 2π/integraltext2π 0dθkρµk(x)e−ilθkand ρµk(x)≡ρµk(x,+∞)] is obtained: ∂2 xρ0 µk(x)+i2αR /planckover2pi1vF[σy,∂xρ0 µk(x)]−α2 R /planckover2pi12v2 F[σx,[σx,ρ0 µk(x)]] −α2 R /planckover2pi12v2 F[σy,[σy,ρ0 µk(x)]] = 0. (7) Itisnotedthatwithonlythelowestthreeordersof ρl µk(x) considered from the beginning, the electron-impurity scattering time is actually absent from the above equa- tion (refer to Appendix B for detail). This indicates that in the strong scattering limit the spin diffusion be- comes insensitive to scattering in graphene. We define the “spin vector”as S0 µk(x) = Tr[ρ0 k(x)σ] andS0 µk(x) can be solved from Eq. (7) with boundary conditions (refer to Appendix B for detail). Then one can calculate the total spin signal contributed by all the different electron states in two valleys as S(x) =1 4π2/summationdisplay µ/integraldisplay+∞ 0dk/integraldisplay2π 0dθkkTr[ρk(x)σ] =1 π/integraldisplay+∞ 0dkkS0 µk(x). (8) In the following we present the solutions of S(x) under three typical boundary conditions. For boundary condition (I) S0 µk(0) = (S0 µk(0),0,0) and S0 µk(+∞) = 0, which corresponds to the case with the injected spins polarized along the x-axis, S(x) =S(0)e−x/lx √ 1+∆2sin(ωx+φ) 0 c1sin(ωx) .(9) For boundary condition (II) S0 µk(0) = (0,S0 µk(0),0) and S0 µk(+∞) = 0, S(x) =S(0)e−x/ly 0 1 0 . (10)5 For boundary condition (III) S0 µk(0) = (0,0,S0 µk(0)) and S0 µk(+∞) = 0, S(x) =S(0)e−x/lz c2sin(ωx) 0 −√ 1+∆2sin(ωx−φ) .(11) In the above equations S(0) =1 π/integraldisplay+∞ 0dkkS0 µk(0) (12) and lx=lz=√ 7 (2√ 2−1)/radicalbig 1+2√ 2/planckover2pi1vF αR,(13) ly=/planckover2pi1vF 2αR, (14) ω=/radicalBig 1+2√ 2αR /planckover2pi1vF. (15) c1=−4 (1+√ 2)√ 1+2√ 2,c2=(20√ 2−24)√ 1+2√ 2 7, ∆ = 8√ 2−11√ 7andφ= arctan1 ∆. It is noted that the spin precession frequency given by the simplified model is ω≈1.96αR /planckover2pi1vF, a little smaller than |∝an}bracketle{tωk∝an}bracketri}ht|=2αR /planckover2pi1vFdue to the approximations made here. From Eqs. (9)-(11) one notices that in the strong scat- tering limit, the spin diffusion is not only insensitive to the scattering, but also unrelated to temperature Tand electron density Ne. Nevertheless, the coefficient αRmay depend on Tand/orNe, with the relation unclear so far. For simplicity we assume αRto be independent of TandNein this work. As a result, the spin diffusion in the strong scattering limit is uniquely determined by αR, which is only modulated by chemical doping. Eqs. (9)- (11) indicate a strong anisotropy of spin diffusion with respect to the spin-polarization direction. For the cases with the injected spins polarized along the x- andz-axis, both the spin signals show an exponential decay in the magnitude accompanying with the precession in the x- zplane. The spin precessions have the same frequency ωexcept for a phase difference. However, when the in- jected spins are polarized along the y-axis, the spin sig- nal decays exponentially without any precession, i.e., it is bound along the y-axis. The above phenomena are understood by noticing that the mean effective magnetic field felt by the diffusing electrons is along the y-axis as ∝an}bracketle{tωk∝an}bracketri}ht=2αR /planckover2pi1vF(0,1,0). In the non-local spin valve experi- ments, the spin diffusion length is usually measured from the exponential decay of spin signal with the increasing spacing between the central spin-injector and -detector ferromagnetic electrodes.27,28In these experiments, the ferromagnetic electrodes happen to be magnetized along they-axis and therefore the injected and detected spin polarizations are both along the y-axis. With such con- figuration, the exponential decay of spin signal with in- creasing spacing between the electrodes can be well ob- served. However, if the injected spins are polarized inthex-zplane, the spatial spin precession is expected to be detected. Besides the anisotropy of spin precession, the spin dif- fusion length also shows an anisotropy as lx=lz≈1.48ly (16) withly=/planckover2pi1vF/(2αR). In fact, when the injected spins are polarized along any other direction in the x-zplane, the spin diffusion length is all the same as lx(lz) [for this case the solution of S(x) is the combination of Eqs. (9) and (11)]. However, based on the widely utilized two- component drift-diffusion model43–47which gives ls=√Dτs[Eq. (D11) in Appendix D], one may expect that the spin diffusion lengths satisfy lx=ly=√ 2lz=/planckover2pi1vF/(2αR) (17) as the spin relaxation times in time domain follow (refer to Appendix A) τx=τy= 2τz=/planckover2pi12/(2α2 Rτp) (18) andD=v2 Fτp/2. It is noted that only when the in- jected spins are polarized along the y-axis, for which no spin precession exists, the two-component drift-diffusion model gives the result in consistence with that from the KSBEs, i.e., ly=/planckover2pi1vF/(2αR). (19) The discrepancy in the anisotropies given by the KSBEs and the two-component drift-diffusion model strongly indicates the inadequacy of the two-component drift- diffusion model. Due to the loss of the off-diagonal spin components, i.e., the spin coherence, the two-component drift-diffusion model not only fails to predict the spin precession in spatial domain in the absence of an ex- ternal magnetic field, but also incorrectly inherits the anisotropy from the spin relaxation in time domain. We emphasize that the reason for the different anisotropic properties of spin diffusion in spatial domain and spin relaxation in time domain is that the inhomogeneous broadening is quite different in these two cases. In spa- tial domainthe inhomogeneousbroadeninggoverningthe spin diffusion arises from the kdependence of ωk, while in time domain from that of Ωk. Popinciuc et al.re- ported the relationship between the in-plane and out-of- plane spin relaxation times directly from the anisotropy of spin diffusion via the two-component drift-diffusion model.28However, based on the above discussion, one may realize that studying the anisotropy of spin relax- ation time in such a way can be incorrect. Finally, from another point of view, if the two-component drift- diffusion model is still used, then in order to reflect the correct anisotropy of spin diffusion, the spin diffusion co- efficient has to differ from the charge diffusion coefficient and shows an anisotropy as Dx= 0.5Dz≈2.2Dywith Dy=v2 Fτp/2.6 It should be pointed out that all the above analy- sis and conclusion also apply to the electron system where the energy spectrum is parabolic in momentum and the linear Rashba spin-orbit coupling term Ωk∝ k(−sinθk,cosθk,0) is dominant, such as that in the asymmetric InAs quantum wells.42That is because the steady-state scattering-free spatial spin precession fre- quencyωkin this system has the similar momentum de- pendence as shown in Eq. (3).42However, for electron system in the absence of the DP term but under a mag- netic field perpendicular to both the spin polarization andspintransportdirectionssuchasinbulksilicon55and symmetric silicon quantum wells,41or with the Dressel- haus term56containing the cubic dependence on momen- tum such as in GaAs quantum wells,39,40the situation is quite different as ωkdepends on the magnitude of mo- mentum. In fact, it has been revealed in the symmetric silicon quantum wells under an in-plane magnetic field that the scattering can suppress spin diffusion effectively in the strong scattering limit.41 2. Spin transport We further take account of the electric field along the x-axis to study the spin transport. Still only the strong electron-impurity scattering is included. The second- order differential equation about ρ0 µk(x), corresponding to Eq. (7) but including the driving term, reads (refer to Appendix C) ∂2 xρ0 µk(x)+i2αR /planckover2pi1vF[σy,∂xρ0 µk(x)]−α2 R /planckover2pi12v2 F[σx,[σx,ρ0 µk(x)]] −α2 R /planckover2pi12v2 F[σy,[σy,ρ0 µk(x)]]−eE∂x∂εkρ0 µk(x) −iαReE /planckover2pi1vF[σy,∂εkρ0 µk(x)] = 0. (20) It should be pointed out that when the electric field is so large that the electron density matrices ρµk(x,+∞) become strongly anisotropic due to the driving of the electric field, retaining only the lowest three orders of ρl µk(x) toobtain the aboveequationof ρ0 µk(x) maynot be sufficient. The second-order differential equation about S0 µk(x) is obtained from the above equation and that aboutS(x) can be obtained by further summing over k andµ(refer to Appendix C). With the same three differ- enttypicalboundaryconditionspresentedintheprevious section,S(x) is solved to have the same form as Eqs. (9)- (11) except that the parameters are now electric-field de-pendent. Explicitly, l′ x=l′ z=1 E/2+F(E)/planckover2pi1vF αR, (21) l′ y=1 E/2+/radicalbig 4+E2/4/planckover2pi1vF αR, (22) ω′=G(E)αR /planckover2pi1vF, (23) c′ 1=−1 2√ E4+48E2+512√ E2+7F(E)+5G(E),c′ 2=1 2√ E4+48E2+512√ E2+7F(E)+3G(E), ∆′=5F(E)−√ E2+7G(E)√ E2+7F(E)+5G(E)andφ′= arctan1 ∆′. In the above equations, E=eE S(0)παRβ/planckover2pi1vFln1+eβµ↑ 1+eβµ↓, (24) F(E) =√ E4+48E2+512+ E2−8 16√ 2√ E2+7 ×/radicalBig/radicalbig E4+48E2+512−E2+8,(25) G(E) =/radicalBig 1−E2/8+/radicalbig E4+48E2+512/8.(26) Hereβ= 1/(kBT) andµ↑(µ↓) is the chemical potential of electrons with spin parallel (antiparallel) to the spin- polarization direction. It is noted that when the electric field is absent, i.e., E= 0, all the above solutions recover those presented in the previous section. In most conditions (such as in the present work) elec- trons in graphene are highly degenerate. In the de- generate limit with small spin polarization, E≈eE αRkF, wherekF=√πNeis the magnitude of the Fermi mo- mentum of unpolarized electrons with density being Ne (Appendix C). Differing from the spin diffusion without electricfield, the spintransportbecomessensitivetoelec- tron density as Edepends on the electron density. In the nondegenerate limit, E≈eEβ/planckover2pi1vF αR(Appendix C) and the spin transport becomes sensitive to temperature rather than electron density. Moreover, with this value of E, Eq. (22) becomes l′ y=/bracketleftBig eEβ/2+/radicalBig e2E2β2/4+1/l2y/bracketrightBig−1 ,(27) wherelyis the spin diffusion length without electric field [Eq. (14)]. This result recovers that from the two- component drift-diffusion model, which apparently fails to correctly reflect the anisotropy of spin transport.13,44 Therefore, our investigation again indicates that only when the spatial spin precession is absent, the two- component drift-diffusion model gives the appropriate depiction of spin transport. In Fig. 2 we plot the dependence of l′ x,y,z,ω′andφ′ onE. From Fig. 2(a), one notices that the spin trans- port length decreases with increasing E(E≈eE αRkF). On one hand, this means that when the electron density is fixed (e.g., Ne= 1012cm−2, for which the variation of E from−8 to 8 corresponds to a variation of Efrom about7 −2.2to 2.2kV/cm), the spin transportis suppressed(en- hanced) by increasing the electric field parallel (antipar- allel) to the spin injection direction. On the other hand, this also means that when the non-zero electric field par- allel (antiparallel) to the spin injection direction is fixed, the spin transportis enhanced(suppressed) byincreasing electron density. Fig. 2(b) and (c) indicate that the spin precession frequency ω′and the phase angle φ′vary with Emarginally (with a variation ∼2 %). In fact, when |E| becomes even larger, both ω′andφ′quickly saturate ( ω′ approaches2αR /planckover2pi1vFandφ′approaches π/2). Therefore, the spin precession pattern in spatial domain is insensitive to the electric field or the electron density. (c) Eφ′ 86420-2-4-6-81.46 1.44(b)ω′(αR//planckover2pi1vF) 1.98 1.96l′ yl′ x(l′ z) (a)l′ s(/planckover2pi1vF/αR)5 4 3 2 1 0 FIG. 2: (Color online) The dependence of (a) spin transport lengthl′ x,y,z, (b) spin precession frequency ω′and (c) phase angleφ′onE. C. Spin diffusion and transport: numerical results The KSBEs need to be solved numerically in order to take full account of all the different kinds of scattering as well as the large electric field. To numerically solve the KSBEs, the initial conditions are set as ρµk(0,0) =F0 k↑+F0 k↓ 2+F0 k↑−F0 k↓ 2ˆ n·σ,(28) ρµk(x >0,0) =FL k↑+FL k↓ 2, (29) /summationdisplay µkTr[ρµk(0,0)ˆ n·σ]//summationdisplay µkTr[ρµk(0,0)] =P0,(30)and the two-side injection boundary conditions39,40are ρµk(0,t)|kx>0=F0 k↑+F0 k↓ 2+F0 k↑−F0 k↓ 2ˆ n·σ,(31) ρµk(L,t)|kx<0=FL k↑+FL k↓ 2. (32) Here the injected spins at left boundary x= 0 are as- sumed to be polarized along ˆ nwith polarization P0= 0.05.x=Lstands for the right boundary with Lmuch longer than the spin diffusion or transport length. F0,L k↑,↓ arethe Fermi distributions ofelectronsat the twobound- aries when the external electric field is absent. When the electric field is present, F0,L k↑,↓then stand for the drifted Fermi distributions of hot electrons.25In the previous analytical study the boundary conditions are in fact ap- proximated as the single-side injection case. This ap- proximation works well when the scattering is strong.39 By numerically solving the KSBEs, the steady-state dis- tribution of spin polarization along ˆ nis obtained as P(x) =/summationtext µkTr[ρµk(x,+∞)ˆ n·σ]//summationtext µkTr[ρµk(x,+∞)] and then the spin diffusion or transport length is deter- mined from the exponential decay of P(x) (or its enve- lope) along the x-axis. 10-710-610-510-410-310-210-1 0 5 10 15 20 25|P| x (µm)T=18 K, Ne=2.9×1012 cm-2 Ns=2.1×1012 cm-2, NAu=0Numerical: x y z Analytical: x y z FIG. 3: (Color online) The absolute value of steady-state spin polarization |P|versus position xwith the injected spins polarized along the x-,y- andz-axis, respectively. The squares, circles and triangles are obtained by numerically solving the KSBEs with T= 18 K, Ne= 2.9×1012cm−2, Ns= 2.1×1012cm−2andNAu= 0. The solid, dashed and chain curves are calculated from Eqs. (9)-(11) with P(x) = Sx(x)/Ne,Sy(x)/NeandSz(x)/Ne, respectively. 1. Anisotropic spin diffusion As revealed by the analytical model, the spin diffusion shows anisotropic properties with respect to the polar- ization direction of injected spins. In Fig. 3, we show the8 spatial distribution of the absolute value of the steady- state spin polarization |P|for the cases with the injected spins polarized along the x-,y- andz-axis, respectively. NAu= 0, with which αR= 0.153 meV. The squares, circles and triangles are obtained by numerically solving the KSBEs while the solid, dashed and chain curves are calculated by Eqs. (9)-(11). When αR= 0.153 meV the analytical model gives lx=lz≈3.18µm,ly≈2.16µm andω≈0.45µm−1. The anisotropy of spin diffusion is clearlyshownin thisfigure. Itisnotedthatthe simplified analytical model almost perfectly recovers the numerical results [except that the spin precession frequencies for both cases with the injected spins polarized along the x- andz-axis are numerically shown to be closer to 2αR /planckover2pi1vF rather than 1 .96αR /planckover2pi1vFgiven by the analytical study (the difference is expected from the approximations made in the analytical analysis)]. In fact, further numerical cal- culations show that varying Tfrom 18 to 300 K and/or Nefrom 0.5 to 2 .9×1012cm−2changes the numerical results marginally. This is consistent with the conclu- sion from the analytical model, i.e., the spin diffusion of electrons in graphene is insensitive toTandNein the strong scattering limit. As a result, in the strong scat- tering limit, one can depict the spin diffusion quite well with the singleparameter αRvia Eqs. (9)-(11). 2. Chemical doping dependence of spin diffusion In Fig. 4, we plot the deposition time dependence of spin diffusion length with ˆ n=ˆ x,ˆ yandˆ zrespectively by the solid curves. The spin diffusion lengths are di- rectly obtained from Eqs. (13)-(14). It is shown that with the increase of chemical doping time, αRincreases and the spin diffusion length decreases. For comparison, we also plot the deposition time dependence of spin dif- fusion length given by the two-component drift-diffusion model (chain curves), i.e., lx=ly=√ 2lz=√Dτx, with Dandτxgiven in Fig. 1. The comparison between these two sets of results shows that, only when the injected spins are polarized along the y-axis, the two-component drift-diffusion model yields the same result as that from the KSBEs, just as revealed in the analytical study [refer to Eq. (19) and the discussion there]. 3. Effect of scattering on spin diffusion The electron system under investigation is always in the strong scattering limit and therefore the spin dif- fusion becomes insensitive to scattering. However, the properties of spin diffusion in the weak scattering limit can be different. In order to investigate the spin diffusion with scattering strength ranging from the weak to strong scattering limit, we artificially vary the impurity density in the substrate from 0 to 1012cm−2. At the same time, the chemicaldopingisabsent(no adatom)and αRis kept as a constant, e.g., 0.153 meV. We choose T= 50 K, 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8ls (µm) Au deposition (s)T=18 K Ne=2.9×1012 cm-2, Ns=2.1×1012 cm-2KSBEs: x (z) y Drift-Diffusion: x (y) z FIG. 4: (Color online) Deposition time dependence of spin diffusion length with the injected spins polarized along the x-,y- andz-axis, respectively. The results from the KSBEs (solid curves) and the two-component drift-diffusion model (chain curves) are both plotted for comparison. Ne= 5×1011cm−2andˆ n=ˆ y. In Fig. 5 we plot the de- pendence of spin diffusion length lyon the impurity den- sity by the dashed curve. For comparison, we also plot the corresponding dependence of spin relaxation time τy on the impurity density by the solid curve (the scale is on the right-hand side of the frame). It is seen that with the increase in Ns, whilelydecreases obviously in the weak scattering limit ( Ns/lessorsimilar0.05×1012cm−2) and then saturates in the strong scattering limit, τyfirst decreases in the weak scattering limit (refer to the inset for detail) and then increases almost linearly in the strong scatter- ing limit.37The two-component drift-diffusion model is able to capture the dependence of spin diffusion length onNsby means of the relation ly=/radicalbig Dτy: while D∝τp∝1/Ns,τydecreaseswith Nsintheweakscatter- ing limit and ∝Nsin the strong scattering limit; there- forelyfirst decreases with Nsand then becomes insen- sitive to Ns(the insensitivity of lytoNsin the strong scattering limit is revealed previously by the analytical study). It should be emphasized that in Fig. 5 the results are shown with αRbeing a constant. In reality, when one further takes account of the increase of αRwith increas- ingNs,lyshould always decrease with increasing Ns, from the weak scattering limit to the strong scattering limit. 4. Spin transport under the electric field At last we investigatethe spin transportunder an elec- tric field along the x-axis.T= 300 K, Ne= 1012cm−2, Ns= 2.1×1012cm−2andNAu= 0. The injected9 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 0 50 100ly (µm) τy (ps) Ns (1012 cm-2)T=50 K, αR=0.153 meV Ne=5×1011 cm-2, NAu=0 0 0.05 2 7 τy (ps) Ns (1012 cm-2) FIG. 5: (Color online) The impurity(inthesubstrate) densi ty dependence of spin diffusion length (dashed curve) and spin relaxation time (solid curve with the scale on the right-han d side of the frame). The inset shows the detail of the solid curve in the small density regime. The injected spins are polarized along the y-axis.T= 50 K, Ne= 5×1011cm−2 andαR= 0.153 meV. spins are polarized along the z-axis. In Fig. 6 the po- sition dependence of |P|under different electric fields as well as the Edependence of lz(squares with the scale on the right-hand side and top of the frame) are plot- ted. It is shown that while the spin-precession pattern almost keeps the same with varying E, the spin trans- port length is increased (decreased) by increasing the electric field along the −x(x)-direction.39,44These re- sults are in consistence with the analytical study pre- sentedinSec.IIIB2. Forcomparison,wefurtherplotthe Edependence of lzfrom Eq. (21) by the double-dotted chain curve with the scale also on the right-hand side and the top of the frame. It is shown that the analytical model depicts the spin transport in the low electric-field regime well except when the electric field antiparallel to the spin-injection direction is large (e.g, a discrepancy reaches 20% when Ereaches−2 kV/cm). The electron density dependence of spin transport is also investigated. In Fig. 7, we plot the density de- pendence of spin transport length under the electric field parallel ( E= 0.3 kV/cm) and antiparallel ( E= −0.3 kV/cm) to the spin transport direction in (a) and (b), respectively. The squares are from the numerical calculation and the curves are from Eq. (21). It is clearly shown that for the cases with opposite directions of the electric field, the density dependences of spin transport length have opposite tendencies.10-1210-1010-810-610-410-2100102 0 5 10 15 20 25 0 5 10 15 20-2 -1 0 1 2|P| lz (µm) x (µm)E (kV/cm) T=300 K, Ne=1012 cm-2 Ns=2.1×1012 cm-2, NAu=0E= − 0.9 kV/cm − 0.3 kV/cm 0 kV/cm 0.3 kV/cm 0.9 kV/cm FIG. 6: (Color online) The absolute value of the steady-stat e spin polarization |P|versus position xunder different electric fields. The electric field dependence of spin transport lengt h lzis also plotted with the scale on the right-hand side and top of the frame, where the squares and double-dotted chain curve are obtained from the numerical calculation and from Eq. (21), respectively. T= 300 K, Ne= 1012cm−2,Ns= 2.1×1012cm−2andNAu= 0. IV. CONCLUSION In conclusion, we have investigated the spin diffusion and transport in graphene monolayer on SiO 2substrate as presented by Pi et al.,31by means of the KSBE ap- proach. The substrate (including the impurities initially present) contributes a Rashba spin-orbit coupling field much stronger than the one modulated by the electric field perpendicular to the graphene layer. By surface chemical doping with Au adatoms, the Rashba spin-orbit coupling coefficient αRis increased. By fitting the chem- ical doping dependence of diffusion coefficient and spin relaxation time,31we obtain the information on impu- rities as well as the chemical doping dependence of αR. Our fitting finds that αRincreases linearly from 0.15 to 0.23 meV with increasing Au density when the latter is not so high. With the necessary parameters obtained from fitting, we investigate the spin diffusion and trans- port in graphene both analytically and numerically. The analytical study with only the electron-impurity scattering included reveals that in the strong scatter- ing limit (just as the situation under investigation in the present work), the spin diffusion is uniquely deter- mined by αR. When the injected spins are polarized along the x-,y- andz-axis, the spin diffusion lengths are given by the analytical study with an anisotropy as lx=lz≈0.74/planckover2pi1vF/αRandly= 0.5/planckover2pi1vF/αR. Meanwhile, the spatial spin precession is present when the injected spins are polarized in the x-zplane but absent when the injected spins are polarized along the y-axis. Further10 4.2 4.7 5.2 5.7 0.5 1 1.5 2lz (µm) Ne (1012 cm-2)(b) E= − 0.3 kV/cm T=300 K Ns=2.1×1012 cm-2, NAu=0 1.7 1.9 2.1 2.3 (a) E=0.3 kV/cm Analytical Numerical FIG. 7: (Color online) Electron density dependence of spin transport length lzunder electric fields with opposite direc- tions: (a) E= 0.3 kV/cm and (b) E=−0.3 kV/cm. The squares are from the numerical calculation while the curves are from Eq. (21). T= 300 K, Ns= 2.1×1012cm−2and NAu= 0. numerical calculations with all the scattering explicitly included show that the analytical model depicts the spin diffusion pretty well. It is noted that the anisotropy of spin diffusion length from the KSBEs differs from the one from the two- component drift-diffusion model where lx=ly=√ 2lz= 0.5/planckover2pi1vF/αR. The qualitative discrepancy indicates the in- adequacyofthe two-componentdrift-diffusion model due to the neglect of the off-diagonal spin components, i.e., the spin coherence. In fact, only when the injected spins arepolarized alongthe y-axisand the spatial spin preces- sion is absent, the two-component drift-diffusion model gives the same spin diffusion length as the KSBE ap- proach does. The analytical and numerical study of spin transport under an electric field parallel or antiparallel to the spin injection direction is also investigated. In the presence of the electric field, the analytical model depicts the spin transport with a small discrepancy which increases with the strength of the electric field. It is shown that when the electric field is applied, the spin precession in spatial domain for the cases with the injected spins polarized along the x- andz-axis remains almost unchanged. How- ever, the spin transport length is increased (decreased) by increasing the magnitude of the electric field when it is antiparallel (parallel) to the spin transport direction. Moreover, in the presence of the electric field, the spin transport becomes sensitive to the electron density, dif- fering from the case of spin diffusion. The spin transportis enhanced (suppressed) by increasing electron density when the electric field is parallel(antiparallel) to the spin injection direction. Acknowledgments This work was supported by the National Natural Sci- ence Foundation of China under Grant No. 10725417. One of the authors (MWW) acknowledges valuable dis- cussions with J. Fabian and B. J. van Wees. Appendix A: Spin relaxation in graphene We considerspin relaxationin grapheneunder the spa- tial uniform case in the absence of the electric field. We only include the electron-impurity scattering. The KS- BEs, Eq. (5), are then simplified to be ∂tρµk(t) =−i /planckover2pi1[Ωk·σ,ρµk(t)]−2π /planckover2pi1/summationdisplay k′Mk−k′Ikk′ ×δ(εk−εk′)[ρµk(t)−ρµk′(t)].(A1) HereMk−k′=|Us k−k′|2+|UAu k−k′|2is the total electron- impurity scattering matrix element contributed by im- purities in the substrate and Au adatoms. Ikk′=1 2[1+ cos(θk−θk′)] is the form factor.25By expanding ρµk(t) asρµk(t) =/summationtext lρl µk(t)eilθk, one comes to ∂tρl µk(t) =−αR 2/planckover2pi1[σ+,ρl+1 µk(t)]+αR 2/planckover2pi1[σ−,ρl−1 µk(t)] −ρl µk(t) τl k, (A2) whereσ±=σx±iσy, and 1 τl k=k(1−δl0) 4π/planckover2pi12vF/integraldisplay2π 0dθMq(1+cosθ)(1−coslθ) (A3) withMqdepending only on |q|= 2ksinθ 2. It is noted that1 τl k=1 τ−l k. Retaining the lowest three orders of ρl µk(t), i.e.,l= 0, ±1, and using the initial conditions ρl µk(0) =δl0ρ0 µk(0), one obtains the second-order differential equation about ρ0 µk(t) as ∂2 tρ0 µk(t)+1 τ1 k∂tρ0 µk(t)+α2 R 2/planckover2pi12[σx,[σx,ρ0 µk(t)]] +α2 R 2/planckover2pi12[σy,[σy,ρ0 µk(t)]] = 0 (A4) with an affiliated initial condition ∂tρ0 µk(0) = 0. Defining the spin vector as S0 µk(t) = Tr[ρ0 µk(t)σ], one can obtain an equation satisfied by S0 µk(t) directly from the above one, which reads /bracketleftBig ∂2 t+1 τ1 k∂t+2α2 R /planckover2pi12(1+δαz)/bracketrightBig S0 µkα(t) = 0 (A5)11 withα=x,y,z. With the initial condition ∂tS0 µkα(0) = 0,S0 µkα(t) is solved to be S0 µkα(t) =S0 µkα(0) 2/bracketleftBig/parenleftBig 1+1/radicalbig 1−c2α/parenrightBig e−t 2τ1 k(1−√ 1−c2α) +/parenleftBig 1−1/radicalbig 1−c2α/parenrightBig e−t 2τ1 k(1+√ 1−c2α)/bracketrightBig ,(A6) wherecα= 2/radicalbig 2(1+δαz)αRτ1 k//planckover2pi1. When the scattering is strong enough and hence cα≪1, S0 µkα(t)≈S0 µkα(0)e−t 4τ1 k/c2α ≡S0 µkα(0)e−t τα. (A7) As a result, for spins polarized along the x- and y-axis the spin relaxation times are τx=τy= /planckover2pi12/(2α2 Rτ1 k), while for spins polarized along the z-axis τz=/planckover2pi12/(4α2 Rτ1 k). From Eq. (A3) one notices that τ1 k is in fact the momentum relaxation time τp(k). For the highly degenerate electron system in graphene, τp(k)≈ τp(kF)≈τp. Therefore we have τx=τy= 2τz= /planckover2pi12/(2α2 Rτp). Appendix B: Spin diffusion in graphene The spin diffusion in the absence of an electric field is also investigated for the case with only the electron- impurity scattering included. Performing angle expan- sion on the steady-state KSBEs in a way similar to that shown in Appendix A, one arrives at ∂x/summationdisplay l0=±1ρl+l0 µk(x)+γ[σ+,ρl+1 µk(x)]−γ[σ−,ρl−1 µk(x)] +2 vFρl µk(x) τl k= 0 (B1) withγ=αR/(/planckover2pi1vF). Retaining the lowest three orders of ρl µk(x) one obtains three equations involving ρ0,±1 µk(x) as ∂x/summationdisplay l0=±1ρl0 µk(x)+γ[σ+,ρ1 µk(x)]−γ[σ−,ρ−1 µk(x)] = 0,(B2) ∂xρ0 µk(x)−γ[σ−,ρ0 µk(x)]+2 vFρ1 µk(x) τ1 k= 0, (B3) ∂xρ0 µk(x)+γ[σ+,ρ0 µk(x)]+2 vFρ−1 µk(x) τ1 k= 0. (B4)From these equations one immediately arrives at Eq. (7) withτ1 kbeing irrelevant. By multiplying σand perform- ing trace on both sides of Eq. (7), one gets the equation satisfied by S0 µk(x) which can be written as ∂2 x−4γ20−4γ∂x 0∂2 x−4γ20 4γ∂x0∂2 x−8γ2 S0 µkx(x) S0 µky(x) S0 µkz(x) = 0.(B5) With specified boundary conditions, S0 µk(x) is solvedand the total spin signal S(x) is obtained by Eq. (8), as pre- sented in Sec. IIIB1. Explicitly, taking the boundary condition (I) given in Sec. IIIB1 as an example, one ob- tainsS0 µk(x) as S0 µk(x) =S0 µk(0)e−x/lx √ 1+∆2sin(ωx+φ) 0 c1sin(ωx) ,(B6) with the parameters lx,ω, ∆,φandc1given in Sec. IIIB1. Byfurther summing over kandµonearrives at Eq. (9). Appendix C: Spin transport in graphene The analytical study of spin transport is carried out analogly. The driving term from the electric field in the steady state is approximated as eE /planckover2pi1∂ρµk(x) ∂kx=eE /planckover2pi1∂ρµk(x) ∂εk∂εk ∂kx ≈eEvFcosθk∂ρ0 µk(x) ∂εk.(C1) Then the Fourier transformation of the steady-state KS- BEs reads ∂x/summationdisplay l0=±1ρl+l0 µk(x)+γ[σ+,ρl+1 µk(x)]−γ[σ−,ρl−1 µk(x)] −eE∂ρ0 µk(x) ∂εk(δl−1+δl1)+2 vFρl µk(x) τl k= 0.(C2) From this equation one comes to Eq. (20) by retaining the lowest three orders of ρl µk(x). The equation satisfied byS0 µk(x) is then ∂2 x−eE∂x∂εk−4γ20 −4γ∂x+2eEγ∂εk 0 ∂2 x−eE∂x∂εk−4γ20 4γ∂x−2eEγ∂εk 0 ∂2 x−eE∂x∂εk−8γ2 S0 µkx(x) S0 µky(x) S0 µkz(x) = 0. (C3)12 Having the experience of solving Eq. (B5), we assume thatS0 µk(x) has the solution as S0 µk(x) =S0 µk(0)T(x) and therefore S(x) =S(0)T(x). Performing summation overµandkon both sides of the above equation and using the trick /integraldisplay+∞ 0dkk[∂εkS0 µk(0)]T(x) =−/integraltext+∞ 0dεkS0 µk(0) /planckover2pi12v2 FT(x) =−1 βln1+eβµ↑ 1+eβµ↓ S(0)/planckover2pi12v2 FS(x),(C4) one obtains the equation satisfied by S(x) as ∂2 x+γE∂x−4γ20 −4γ∂x−2γ2E 0 ∂2 x+γE∂x−4γ20 4γ∂x+2γ2E 0 ∂2 x+γE∂x−8γ2 × Sx(x) Sy(x) Sz(x) = 0, (C5) in which Eis given by Eq. (24). With the three differ- ent typical boundary conditions presented in Sec. IIIB1, S(x) is solved to have the same form as Eqs. (9)-(11) except that the parameters are now given in Sec. IIIB2. We now calculate Ein both the degenerate and nondegenerate limits. In the degenerate limit, ln1+eβµ↑ 1+eβµ↓≈lneβ(εF↑−εF↓)=β/planckover2pi1vF(kF↑−kF↓) = β/planckover2pi1vF√πNe(√1+P0−√1−P0)≈β/planckover2pi1vF√πNeP0. Mak- ing use of the relation S(0) =NeP0, one has E≈eE√πNeαR=eE αRkF, (C6) wherekF=√πNe, the magnitude of Fermi momentum of unpolarized electrons with density being Ne. In the nondegenerate limit, ln1+eβµ↑ 1+eβµ↓≈eβµ↑−eβµ↓and S(0)≈1 π/integraldisplay+∞ 0dkk[e−β(εk−µ↑)−e−β(εk−µ↓)] =1 π(β/planckover2pi1vF)2(eβµ↑−eβµ↓), (C7) therefore E≈eEβ/planckover2pi1vF αR. (C8) Appendix D: Derivation of two-component drift-diffusion equation from KSBEs The two-component drift-diffusion equation can be de- rived from the KSBEs in the collinear spin space57with thez-axis along the initial spin-polarization direction ˆ n, by neglecting the spin coherence (i.e., the off-diagonalcomponents of the density matrices). The density ma- trices then have the diagonal form as1 2[fµk↑(x,t) + fµk↓(x,t) +(fµk↑(x,t)−fµk↓(x,t))σz]. In the following we present a brief derivation of the two-component drift- diffusion equation from the KSBEs with only the strong electron-impurity scattering considered. Other kinds of scattering can also be incorporated similarly under elas- tic scattering approximation. The spin relaxation time isτˆ nand the momentum relaxation time is τp(both are given in Appendix A) and we neglect their momentum dependencehereafter. Thetwo-componentdrift-diffusion equation is obtained from the equation of continuity and the equation of current, both to be derived from the KS- BEs. The equation of continuity is derived as follows. By multiplying the KSBEs [Eq. (5)] with1 2(1±σz) and then performing the trace, one obtains the simplified KSBEs for each spin band ( σ=↑,↓) as ∂fµkσ(x,t) ∂t−eE /planckover2pi1∂fµkσ(x,t) ∂kx+vFcosθk∂fµkσ(x,t) ∂x =−fµkσ(x,t)−fµk−σ(x,t) 2τˆ n. (D1) The right-hand side of the above equation comes from the term Tr {1 2(1±σz)[∂tρµk(x,t)|coh+∂tρµk(x,t)|scat]}, which can be calculated with the aid of the KSBEs in the time domain [Eq. (A1)] and the corresponding solution [Eq. (A7)]. Performing summation over µandkon both sides of Eq. (D1) in the steady state, one comes to −eE /planckover2pi1/summationdisplay µk∂fµkσ(x) ∂kx+∂ ∂x/summationdisplay µkvFcosθkfµkσ(x) =−Nσ(x)−N−σ(x) 2τˆ n, (D2) whereNσ(x) is the electron density with spin σat posi- tionx. Up to the first order of the electric field E, the first summation over kin the above equation leads to zero when fµkσis approximated by f0 µkσ, the distribu- tion in equilibrium. Defining the charge current along thex-axis with spin σas Jσ(x) =−/summationdisplay µkevxfµkσ(x) (D3) withvx=vFcosθk, one has the equation of continuity −1 e∂Jσ(x) ∂x=−Nσ(x)−N−σ(x) 2τˆ n.(D4) We then calculate the current Jσfrom the diagonal part of the KSBEs ∂fµkσ(x,t) ∂t−eE /planckover2pi1∂fµkσ(x,t) ∂kx+vFcosθk∂fµkσ(x,t) ∂x =−fµkσ(x,t)−f0 µkσ(x,t) τp, (D5)13 where the right-hand side of the equation comes from the electron-impurityscattering(Appendix A). Inthesteady state, multiplying −evxonboth sides ofthe equation and then summing over µandk, one comes to e2EvF /planckover2pi1/summationdisplay µkcosθk∂fµkσ(x) ∂kx−ev2 F/summationdisplay µkcos2θk∂fµkσ(x) ∂x =−Jσ(x) τp. (D6) Again, up to the first order of the electric field, one has Jσ(x) =eµσENσ(x)+eD∂xNσ(x),(D7) wherethemobility µσ=evFτp /planckover2pi1√2πNσandthe chargediffusion coefficient D=1 2v2 Fτp. For the case with small spin po- larization, µ↑≈µ↓=µe≡evFτp /planckover2pi1√πNeinwhich Ne=N↑+N↓ is the total electron density. Finally, the two-component drift-diffusion equation is obtained by combining Eqs. (D4) and (D7) −µeE∂Nσ(x) ∂x−D∂2Nσ(x) ∂x2=−Nσ(x)−N−σ(x) 2τˆ n. (D8)This equation is consistent with that in the literature.44–47The equation of ∆ N=N↑−N↓ then reads −µeE∂∆N(x) ∂x−D∂2∆N(x) ∂x2=−∆N(x) τˆ n.(D9) With boundary condition ∆ N(+∞) = 0, ∆ N(x) is solved as ∆ N(x) = ∆N(0)e−x/lˆ nwhere the spin trans- port length44 lˆ n=/bracketleftBigg µeE 2D+/radicalbigg/parenleftbigµeE 2D/parenrightbig2+1 Dτˆ n/bracketrightBigg−1 .(D10) When the electric field is zero, lˆ n=/radicalbig Dτˆ n. (D11) As indicated by Eqs. (D10) and (D11), in the frame of the two-component drift-diffusion model, the anisotropy of the spin transport or diffusion is solely determined by that of the spin relaxation and no spatial spin precession can be obtained. ∗Author to whom correspondence should be addressed; Electronic address: mwwu@ustc.edu.cn. 1A.K.GeimandK.S.Novelia, NatureMater. 6, 183(2007). 2N. Tombros, C. J´ ozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature (London) 448, 571 (2007). 3A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 4E. H. Hang, S. Adam, and S. Das Sarma, Phys. Rev. Lett. 98, 186806 (2007). 5K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature 438, 197 (2005). 6K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science306, 666 (2004). 7K. I. Bolotin, K.J. Sikes, Z. Jiang, M. Klima, G. Fuden- berg, J. Hone, P. Kim and H. L. Stormer, Sol. State Com- mun.146, 351 (2008). 8J. Fischer, B. Trauzettel, and D. Loss, Phys. Rev. B 80, 155401 (2009). 9C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 10D. H. Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74, 155426 (2006). 11H. Min, J. E. Hill, N. A.Sinitsyn, B. R.Sahu, L. Kleinman, and A. H. MacDonald, Phys. Rev. B 74, 165310 (2006). 12M. Gmitra, S. Konschuh, C. Ertler, C. A. Draxl, and J. Fabian, Phys. Rev. B 80, 235431 (2009). 13C. J´ ozsa, M. Popinciuc, N. Tombros, H. T. Jonkman, and B. J. van Wees, Phys. Rev. Lett. 100, 236603 (2008). 14H. Goto, A. Kanda, T. Sato, S. Tanaka, Y. Ootuka, S. Odaka, H. Miyazaki, K. Tsukagoshi, and Y. Aoyagi, Appl.Phys. Lett. 92, 212110 (2008). 15E. W. Hill, A. K. Geim, K. Novoselov, F. Schedin, and P. Blake, IEEE Trans. Mag. 42, 2694 (2006). 16S. Cho, Y. F. Chen, and M. S. Fuhrer, Appl. Phys. Lett. 91, 123105 (2007). 17H. Haugen, D. H. Hernando, and A. Brataas, Phys. Rev. B77, 115406 (2008). 18W. H. Wang, K. Pi, Y. Li, Y. F. Chiang, P. Wei, J. Shi, and R. K. Kawakami, Phys. Rev. B 77, 020402(R) (2008). 19M. Ohishi, M. Shiraishi, R. Nouchi, T. Nozaki, T. Shinjo, and Y. Suzuki, Jpn. J. Appl. Phys. 46, L605 (2007). 20L. Brey and H. A. Fertig, Phys. Rev. B 73, 195408 (2006). 21V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005). 22C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 23S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. C. Elias, J. A. Jaszczak, and A. K. Geim, Phys. Rev. Lett. 100, 016602 (2008). 24X. Du, I. Skachko, A. Barker, and E. Y. Andrei, Nat. Nan- otechnol. 3, 491 (2008). 25Y. Zhou and M. W. Wu, Phys. Rev. B 82, 085304 (2010). 26S. Abdelouahed, A. Ernst, and J. Henk, Phys. Rev. B 82, 125424 (2010). 27W. Han, K. Pi, W. Bao, K. M. McCreary, Y. Li, W. H. Wang, C. N. Lau, and R. K. Kawakami, Appl. Phys. Lett. 94, 222109 (2009). 28M. Popinciuc, C. J´ ozsa, P. J. Zomer, N. Tombros, A. Veligura, H. T. Jonkman, and B. J. van Wees, Phys. Rev. B80, 214427 (2009). 29N. Tombros, S. Tanabe, A. Veligura, C. J´ ozsa, M. Popin- ciuc, H. T. Jonkman, and B. J. van Wees, Phys. Rev. Lett.14 101, 046601 (2008). 30C. J´ ozsa, T. Maassen, M. Popinciuc, P. J. Zomer, A. Veligura, H. T. Jonkman, and B. J. van Wees, Phys. Rev. B80, 241403(R) (2009). 31K. Pi, W. Han, K. M. McCreary, A. G. Swartz, Y. Li, and R. K. Kawakami, Phys. Rev. Lett. 104, 187201 (2010). 32C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B80, 041405(R) (2009). 33R. J. Elliott, Phys. Rev. 96, 266 (1954). 34A. H.C. NetoandF. Guinea, Phys. Rev.Lett. 103, 026804 (2009). 35A. Varykhalov, J. S. Barriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O. Rader, Phys. Rev. Lett. 101, 157601 (2008). 36M. I. D’yakonov and V. I. Perel’, Zh. ´Eksp. Teor. Fiz. 60, 1954 (1971) [Sov. Phys. JETP 33, 1053 (1971)]. 37M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Rep. 493, 61 (2010). 38M. Q. Weng and M. W. Wu, Phys. Rev. B 66, 235109 (2002). 39J. L. Cheng and M. W. Wu, J. Appl. Phys. 101, 073702 (2007). 40J. L. Cheng, M. W. Wu, and I. C. da Cunha Lima, Phys. Rev. B75, 205328 (2007). 41P. Zhang and M. W. Wu, Phys. Rev. B 79, 075303 (2009). 42B. Y. Sun, P. Zhang, and M. W. Wu, Semicond. Sci. Tech- nol.26, 075005 (2011). 43Spin Electronics , edited by M. Ziese and M. J. Thornton (Springer, Berlin, 2001).44Z. G. Yu and M. E. Flatt´ e, Phys. Rev. B 66, 201202(R) (2002). 45I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Phys. Rev. Lett. 88, 066603 (2002). 46J. Fabian, I. ˇZuti´ c, S. Das Sarma, Phys. Rev. B 66, 165301 (2002). 47J. Fabian, A. M. Abiague, C. Ertler, P. Stano, I. ˇZuti´ c, Acta Phys. Slov. 57, 565 (2007). 48S. Adam and S. Das Sarma, Solid State Commun. 146, 356 (2008). 49M. R. Ramezanali, M. M. Vazifeh, R. Asgari, M. Polini, and A. H. MacDonald, J. Phys. A: Math. Theor. 42, 214015 (2009). 50E. H. Hwang and S. Das Sarma, Phys. Rev. B 77, 115449 (2008) 51M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. Lett. 95, 236802 (2005). 52S. Fratini and F. Guinea, Phys. Rev. B 77, 195415 (2008). 53H. Haug and A. P. Jauho, Quantum kinetics in Transport and Optics of Semiconductors (Springer, Berlin, 1998). 54Y. W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, Phys. Rev. Lett. 99, 246803 (2007). 55I. Appelbaum, B. Huang, and D. J. Monsma, Nature (Lon- don)447, 295 (2007). 56G. Dresselhaus, Phys. Rev. 100, 580 (1955). 57J. L. Cheng and M. W. Wu, J. Appl. Phys. 99, 083704 (2006).

2006.02114v2.Magnetic_reorientation_transition_in_a_three_orbital_model_for___rm_Ca_2_Ru_O_4_____Interplay_of_spin_orbit_coupling__tetragonal_distortion__and_Coulomb_interactions.pdf

arXiv:2006.02114v2 [cond-mat.str-el] 31 Aug 2020Magnetic reorientation transition in a three orbital model for Ca2RuO4— Interplay of spin-orbit coupling, tetragonal distortion, and Coulomb interactions Shubhajyoti Mohapatra and Avinash Singh∗ Department of Physics, Indian Institute of Technology, Kanpu r - 208016, India (Dated: September 1, 2020) Includingtheorbital off-diagonal spinandcharge condensat es in theself consistent determination of magnetic order within a realistic three-o rbital model for the 4 d4 compoundCa 2RuO4, reveals ahostofnovel featuresincludingstrongandaniso tropic spin-orbit coupling (SOC) renormalization, coupling of st rong orbital magnetic mo- ments to orbital fields, and a magnetic reorientation transi tion. Highlighting the rich interplay between orbital geometry and overlap, spin- orbit coupling, Coulomb interactions, tetragonal distortion, and staggered octah edral tilting and rotation, our investigation yields a planar antiferromagnetic (AFM) ord er for moderate tetragonal distortion, witheasy a−bplaneandeasy baxis anisotropies, along withsmall canting of the dominantly yz,xzorbital moments. With decreasing tetragonal distortion, w e find a magnetic reorientation transition from the dominantl y planar AFM order to a dominantly caxis ferromagnetic (FM) order with significant xyorbital moment.2 I. INTRODUCTION The interplay of spin-orbit coupling (SOC) with electronic correlation s and crystal field splittings has been found to drive various topologically nontrivial pha ses in condensed mat- ter systems such as topological Mott insulators, quantum spin liquid s, and superconducting states.1,2The 4dand 5dtransition metal oxides containing Ru4+, Os4+, Ir4+, Ir5+ions have emerged as promising candidates exhibiting SOC-induced exotic grou nd states, magnetic anisotropy effects, and intriguing collective excitations. SOC effect s in thed5systems are moretransparent and well understoodin terms ofthe spin-orbita l entangled electronic states with nominally filled J= 3/2 quartet and half-filled magnetically active J= 1/2 doublets.3 The isospin dynamics involving Jstates provides insight into the experimentally observed magnetic behavior in perovskite iridates as well as iridate heterostr uctures which are gain- ing interest as their magnetic properties are much more sensitive to structural distortion compared to pure spin systems due to spin-orbital entanglement.4–7 However, the situation is very different in d4systems with four electrons per metal ion. For strong SOC, all four electrons fill the J= 3/2 sector, leaving the J= 1/2 sector empty and naturally leading to non-magnetic insulating behavior.8Similarly, for strong Hund’s coupling, total spin moment S= 1 antiparallel to the orbital moment L= 1 leads to total angular momentum J= 0 on every metal ion with no magnetism. Thus, both scenarios lead tothenon-magnetic J= 0singletgroundstatefor d4systems. However, magnetismhasbeen revealed in some double perovskite iridates and ruthenates with d4electronic configuration, and the origin of magnetism is under investigation.9–13 Amongd4systems, the quasi-two-dimensional antiferromagnet Ca 2RuO4has attracted strong interest. With decreasing temperature, it undergoes a pe culiar non-magnetic metal- insulator transition (MIT) at 356 K, and a magnetic transition at TN≈113 K with observed magnetic moment of 1 .3µB.14–17Under high pressure and at low temperature, Ca 2RuO4 undergoes a transition to a ferromagnetic (FM) metallic phase, with maximumTC≈30 K at 5 GPa pressure,18and the existence of a FM quantum critical point at pressures abov e 10 GPa is indicated. The MIT is associated with a structural transitio n from L-phase (long octahedral c-axis) to S-phase (short c-axis) due to continuous flattening of octahedra till the onset of antiferromagnetic (AFM) order at TN.19Compared to the isoelectronic member Sr 2RuO4,17,20this system has severe structural distortions due to the small Ca2+3 size, resulting in compression, rotation, and tilting of the RuO 6octahedra. Thus, the low- temperature phase is characterized by highly distorted RuO 6octahedra and canted AFM order with moments lying along the crystal baxis.21,22Such transitions have been identified in temperature,23hydrostatic pressure,24epitaxial strain,25chemical substitution,18,20,26and electrical current27,28studies of Ca 2RuO4. In the isoelectronic series Ca 2−xSrxRuO4, the ground state has been successively driven from the AFM insulator ( x <0.2) to an AFM correlated metal (0 .2< x <0.5), a nearly FM metal ( x∼0.5), and finally to a non-magnetic two-dimensional Fermi liquid ( x∼2). Since the substitution is isovalent, the dominant effects are struct ural modifications due to larger Sr ionic size.23With increasing x, the distortion occurs in steps, resulting in removal of first the flattening of the octahedra, then the tilting, and finally the rotation around thecaxis.17,21,29Although the substitution is isovalent, the magnetism of Ca 2−xSrxRuO4is affected in the sequence given above by the changes in orbital hybr idization resulting from substitution induced structural distortions. In the literature, mainly two different scenarios have been discusse d for classifying the magnetisminCa 2RuO4. Inthefirst, octahedralcompression inducedlargetetragonal crystal field (≈0.3 eV) lifts the degeneracy of the t2gorbitals by lowering the xyorbital energy. BasedonDFTcalculations,19,29–31whichagreewithX-rayscatteringaswell asangleresolved photoemission spectroscopy (ARPES) studies,32,33thexyorbital is nominally filled, and the half-filledyz,xzorbitals form a spin S= 1 state. Further, low octahedral symmetry around the Ru ion is believed to quench the orbital moment completely. Thus, the ordering of S= 1 spins supports a more conventional explanation for the magnetism with a negligible role of SOC. However, the presence of the strong in-plane anisotropy in t he magnon dispersion indicates the importance of SOC in tuning the magnetic anisotropy in t he system.22 Inthesecond scenario, Ca 2RuO4, withonlymoderateSOCstrength, hasbeen arguedasa possible candidate for excitonic antiferromagnetism. If the super exchange involving excited magnetic states (triplet J= 1) is strong enough to compete with the singlet-triplet splitting caused by SOC, the on-site wave function becomes a superposition ofJ= 0,1 states and acquires a magnetic moment.34–38This picture is supported by the observed unconventional magnetic excitation spectra from the recent inelastic neutron sca ttering (INS) and resonant inelasic X-ray scattering (RIXS) experiments.32,39–41Spin-wave dispersion in the INS study has revealed a global maximum at the Brillouin zone center, which is in sh arp contrast to4 (a) (b) (c) FIG. 1: Summaryof the physical properties of (a) layered Cu 2RuO4showingtransitions from AFM insulator (AFM-I) to non-magnetic metal or ferromagnetic m etal (FM-M) induced by different agents, and (b) isoelectronic series Ca 2−xSrxRuO4showing successive transition from AFM-I to non-magnetic Fermi liquid (NM-FL), through AFM-M and FM-M s tates. With increasing x, the distortions occur in steps. (c) Tetragonal field ǫxyand Hund’s coupling stabilized L=0,S=1 state inabsence of SOC(scenario-1). StrongSOCpicture(scenari o-2) showingnonmagnetic J=0ground state andJ=1 triplet excited state, which further splits into singlet (Tz) and degenerate doublet (Tx,y) in presence of ǫxy. The energy difference [ E(Tx,y)−E(J=0)] comparable to the exchange energy (Jex) induces magnetic ordering. theS= 1 quantum Heisenberg antiferromagnet (QHAF), and has been int erpreted as a sign of such excitonic magnetism in Ca 2RuO4.39The various properties of this system and theoretical scenarios as discussed above are summarized in Fig. 1. While numerous computational and experimental techniques have b een applied,33,42–47 very little is known about the electronic band structure of Ca 2RuO4in the low-temperature AFM state. Earlier numerical calculations within three orbital models have adopted simpli- fiedHamiltonianstodiscussthemechanism ofmetal-insulatortransit ionandmagnetism.48,49 However, realistichoppings, structural distortions, SOC, andele ctroniccorrelationswere not considered on an equal footing in these simplistic models. Earlier work s have also lacked in fully accounting for the Coulomb interaction effects, especially tho se associated with or- bital off-diagonal spin and charge correlations. Indeed, the effec tive SOC strength ∼200 meV extracted from ARPES and RIXS studies32,50indicates a strong correlation-induced enhancement compared to the predicted theoretical value ∼100 meV.34,35 The richness and complexity displayed in structural, magnetic, and t ransport properties of this system, along with intimate couplings between lattice, spin, an d charge degrees of5 freedom, have led to difficulty in realistic modeling of these phenomena . Classification of the nature of magnetic ground state and the role of SOC and distor tion effects in tuning the magnetic behavior of Ca 2RuO4therefore remains far from being well understood. A delicate interplay of different Coulomb interaction terms with SOC may lead to complex and nontrivial behavior of orbital and spin degrees of freedom. In vestigation of magnetic ordering, anisotropy, and electronic band structure in Ca 2RuO4by incorporating the SOC, structural distortions, and multi-orbital Coulomb interaction ter ms on an equal footing is therefore of strong interest. For a multi-orbital interacting electron system, a general treatm ent of the various Coulomb interaction terms in the Hartree-Fock (HF) approximation yields, besides the con- tributionsfromthenormal(orbitaldiagonal)spinandchargedens itycondensates, additional contributions involving orbital off-diagonal condensates. Since th e SOC and orbital angular momentum terms involve orbital off-diagonal one-body operators , due to interplay between strong SOC-induced spin-orbital correlations and Coulomb interac tions, Ca 2RuO4presents a case where the off-diagonal condensates should play an importan t role in determining the magnetic order and anisotropy. However, these aspects hav e not been systematically investigated within the itinerant electron picture. In this work, all orbital off-diagonal spin /angbracketleftψ† µσψν/angbracketrightand charge /angbracketleftψ† µ1ψν/angbracketrightcondensates will therefore be included, and a self consistent determination of magn etic order and anisotropy will be carried out within a realistic three-orbital interacting electro n model for Ca 2RuO4 in thet2gmanifold of the µ,ν=yz,xz,xy orbitals. The orbital off-diagonal spin and charge condensates will be seen to result in strong and anisotropic SOC ren ormalization and strong orbital magnetic moments /angbracketleftLx,y,z/angbracketrightin the magnetic ground state. We will first focus on the planar AFM order with dominantly yz,xzmoments, which is realized for moderate tetragonal distortion. However, with decreasing tetragonal dis tortion, we find a magnetic reorientation transition to a dominantly caxis ferromagnetic (FM) order, as seen in high- pressure investigations of Ca 2RuO4.18 The structure of this paper is as follows. After introducing the thr ee-orbital model and Coulomb interaction terms in Sec. II, the SOC-induced easy-plane a nisotropy and the octahedral tilting induced easy-axis anisotropy are discussed in Se cs. III and IV. Results of the self-consistent determination of magnetic order including all orbital off-diagonal spin and charge condensates in the HF approximation are presented in S ec. V, together with6 the orbital resolved electronic band structure. The orbital magn etic moments and Coulomb interaction induced anisotropic SOC renormalization are discussed in Sec. VI, and the magnetic reorientation transition in Sec. VII. After some observa tions on the strongly coupled spin-orbital fluctuations in Sec. VIII, conclusions are fina lly presented in Sec. IX. II. THREE ORBITAL MODEL AND COULOMB INTERACTIONS In the three-orbital ( µ=yz,xz,xy ), two-spin ( σ=↑,↓) basis defined with respect to a common spin-orbital coordinate axes (Fig. 2), we consider the Ham iltonianH=HSOC+ Hcf+Hband+Hintwithin the t2gmanifold. The spin-orbit coupling term HSOC, which explicitly breaks SU(2) spin rotationsymmetry and thereforegene rates anisotropic magnetic interactions from its interplay with other Hamiltonian terms, will be int roduced in the next section. For the band and crystal field terms together, we consider: Hband+cf=/summationdisplay kσsψ† kσs ǫyz k′0 0 0ǫxz k′0 0 0ǫxy k′+ǫxy δss′+ ǫyz kǫyz|xz kǫyz|xy k −ǫyz|xz kǫxz kǫxz|xy k −ǫyz|xy k−ǫxz|xy kǫxy k δ¯ss′ ψkσs′ (1) in the composite three-orbital, two-sublattice ( s,s′= A,B) basis. Here the energy offset ǫxy(relative to the degenerate yz/xzorbitals) represents the tetragonal distortion induced crystal field effect, and the band dispersion terms in the two group s, corresponding to hopping terms connecting the same and opposite sublattice(s), ar e given by: ǫxy k=−2t1(coskx+cosky) ǫxy k′=−4t2coskxcosky−2t3(cos2kx+cos2ky) ǫyz k=−2t5coskx−2t4cosky ǫxz k=−2t4coskx−2t5cosky ǫyz|xz k=−2tm1(coskx+cosky) ǫxz|xy k=−2tm2(2coskx+cosky) ǫyz|xy k=−2tm3(coskx+2cosky). (2) Heret1,t2,t3are respectively the first, second, and third neighbor hopping ter ms for7 (a)(b) FIG. 2: (a) The common spin-orbital coordinate axes ( x−y) along the Ru-O-Ru directions, shown along with the crystal axes a,b. (b) Octahedral tilting about the crystal aaxis is resolved along thex,yaxes, resulting in orbital mixing hopping terms between the xyandyz,xzorbitals. thexyorbital. For the yz(xz) orbital,t4andt5are the NN hopping terms in y(x) and x(y) directions, respectively, corresponding to πandδorbital overlaps. Octahedral ro- tation and tilting induced orbital mixings are represented by the NN h opping terms tm1 (betweenyzandxz) andtm2,tm3(betweenxyandxz,yz). We have taken hopping pa- rameter values: ( t1,t2,t3,t4,t5)=(−1.0,0.5,0,−1.0,0.2), and for the orbital mixing terms: tm1=0.2 andtm2=tm3=0.15 (≈0.2/√ 2), all in units of the realistic hopping energy scale |t1|=200meV.34,35,38The choice tm2=tm3corresponds to the octahedral tilting axis oriented along the ±(−ˆx+ ˆy) direction, which is equivalent to the crystal ∓adirection (Fig. 2). Thetm1andtm2,m3values taken above approximately correspond to octahedral rot ation and tilting angles of about 12◦(≈0.2 rad) as reported in experimental studies.24 For the on-site Coulomb interaction terms in the t2gbasis (µ,ν=yz,xz,xy ), we consider: Hint=U/summationdisplay i,µniµ↑niµ↓+U′/summationdisplay i,µ<ν,σniµσniνσ+(U′−JH)/summationdisplay i,µ<ν,σniµσniνσ +JH/summationdisplay i,µ/negationslash=νa† iµ↑a† iν↓aiµ↓aiν↑+JP/summationdisplay i,µ/negationslash=νa† iµ↑a† iµ↓aiν↓aiν↑ =U/summationdisplay i,µniµ↑niµ↓+U′′/summationdisplay i,µ<νniµniν−2JH/summationdisplay i,µ<νSiµ.Siν+JP/summationdisplay i,µ/negationslash=νa† iµ↑a† iµ↓aiν↓aiν↑(3) including the intra-orbital ( U) and inter-orbital ( U′) density interaction terms, the Hund’s coupling term ( JH), and the pair hopping interaction term ( JP), withU′′≡U′−JH/2 = U−5JH/2 from the spherical symmetry condition U′=U−2JH. Herea† iµσandaiµσare the electron creation and annihilation operators for site i, orbitalµ, spinσ=↑,↓, and the8 density operator niµσ=a† iµσaiµσ, total density operator niµ=niµ↑+niµ↓=ψ† iµψiµ, and spin density operator Siµ=ψ† iµσψiµ, whereψ† iµ= (a† iµ↑a† iµ↓). All interaction terms above are SU(2) invariant and thus possess spin rotationsymmetry inreal-sp in space. Inthe following, we will take U= 8 in the energy scale unit (200 meV) and JH=U/5, so thatU= 1.6eV, U′′=U/2 = 0.8eV, andJH= 0.32eV. These are comparable to reported values extracted from RIXS ( JH= 0.34eV) and ARPES ( JH= 0.4eV) studies.33,41 For moderate tetragonal distortion ( ǫxy≈ −1), thexyorbital in the 4 d4compound Ca2RuO4is nominally doubly occupied and magnetically inactive, while the nominally h alf- filledandmagnetically active yz,xzorbitalsyieldaneffectively two-orbitalmagneticsystem. Hund’s coupling between the two S= 1/2 spins results in low-lying (in-phase) and apprecia- bly gapped (out-of-phase) spin fluctuation modes. The in-phase m odes of the yz,xzorbital S= 1/2 spins correspond to an effective S= 1 spin system. However, the rich interplay be- tween SOC, Coulomb interaction, octahedral rotations, and tetr agonal distortion results in complex magnetic behaviour which crucially involves the xyorbital and is therefore beyond the above simplistic picture. Before proceeding with the self-consis tent determination of magnetic order (Sec. V), some of the important physical elements are individually discussed below. III. SOC INDUCED EASY PLANE ANISOTROPY The bare spin-orbit coupling term (for site i) can be written in spin space as: HSOC(i) =−λL.S=−λ(LzSz+LxSx+LySy) = /parenleftig ψ† yz↑ψ† yz↓/parenrightig/parenleftig iσzλ/2/parenrightig ψxz↑ ψxz↓ +/parenleftig ψ† xz↑ψ† xz↓/parenrightig/parenleftig iσxλ/2/parenrightig ψxy↑ ψxy↓ +/parenleftig ψ† xy↑ψ† xy↓/parenrightig/parenleftig iσyλ/2/parenrightig ψyz↑ ψyz↓ +H.c. (4) which explicitly shows the SU(2) spin rotation symmetry breaking. He re we have used the matrix representations: Lz= 0−i0 i0 0 0 0 0 , Lx= 0 0 0 0 0−i 0i0 , Ly= 0 0i 0 0 0 −i0 0 , (5)9 for the orbital angular momentum operators in the three-orbital (yz,xz,xy ) basis. As the orbital “hopping” terms in Eq. (4) have the same form as spin -dependent hopping termsiσ.t′ ij, carrying out the strong-coupling expansion51for the−λLzSzterm to second order inλyields the anisotropic diagonal (AD) intra-site interactions: [H(2) eff](z) AD(i) =4(λ/2)2 U/bracketleftbig Sz yzSz xz−(Sx yzSx xz+Sy yzSy xz)/bracketrightbig (6) betweenyz,xzmoments inthese nominallyhalf-filled orbitals. Corresponding toaneff ective single-ion anisotropy (SIA), this term explicitly yields preferential x−yplane ordering for parallelyz,xzmoments, as enforced by the relatively stronger Hund’s coupling. For later reference, we note here that condensates of the orbit al off-diagonal one-body operators as in Eq. (4) directly yield physical quantities such as orb ital magnetic moments and spin-orbital correlations: /angbracketleftLα/angbracketright=−i/bracketleftbig /angbracketleftψ† µψν/angbracketright−/angbracketleftψ† µψν/angbracketright∗/bracketrightbig = 2 Im/angbracketleftψ† µψν/angbracketright /angbracketleftLαSα/angbracketright=−i/bracketleftbig /angbracketleftψ† µσαψν/angbracketright−/angbracketleftψ† µσαψν/angbracketright∗/bracketrightbig /2 = Im/angbracketleftψ† µσαψν/angbracketright λint α≈U′′/angbracketleftLαSα/angbracketright (7) where the orbital pair ( µ,ν) corresponds to the component α=x,y,z, and the last yields the interaction induced SOC renormalization, as discussed in Sec. VI . IV. OCTAHEDRAL TILTING AND EASY-AXIS ANISOTROPY While SOC directly induces an easy x−yplane anisotropy, interplay between the stag- gered octahedral tilting in Ca 2RuO4and SOC yields an easy-axis anisotropy along the ˆ x+ˆy direction, which is same as the crystal bdirection. Octahedral tilting generates orbital mix- ing hopping terms between xyandyz,xzorbitals (Eq. 2). These normal NN hopping terms, together with the local spin-flip SOC mixing terms between xyandyz,xzorbitals, lead to effective spin-dependent NN hopping terms: H′ eff=/summationdisplay /angbracketlefti,j/angbracketright,µψ† iµ[−iσ.t′]ψjµ+H.c. (8) for the magnetically active ( µ=yz,xz) orbitals. The hopping terms are bond dependent, with only finite t′ x(t′ y) betweenxz(yz) orbital in the x(y) direction. Within the usual strong-coupling expansion, the combination of the normal ( t) and spin-dependent ( t′ x,t′ y)10 FIG. 3: Spin cantings about the (a) crystal aaxis and (b) crystal caxis, due to the effective DM interactions induced by the staggered octahedral tilting a nd rotation, respectively. Octahedral tilting about crystal aaxis yields the perpendicular (crystal b) direction as the magnetic easy axis. hopping terms generates Dzyaloshinski-Moriya (DM) interaction te rms in the effective spin model: [H(2) eff](x,y) DM=8tt′ x U/summationdisplay /angbracketlefti,j/angbracketrightxˆx.(Si,xz×Sj,xz)+8tt′ y U/summationdisplay /angbracketlefti,j/angbracketrightyˆy.(Si,yz×Sj,yz) ≈8t|t′ x| U/summationdisplay /angbracketlefti,j/angbracketright(−ˆx+ ˆy).(Si×Sj) (9) fort′ x=−t′ y=−ive andSi,xz≈Si,yzdue to the relatively much stronger Hund’s coupling. The effective DM axis ( −ˆx+ ˆy) is along the octahedral tilting axis, which is same as the crystal−aaxis (Fig. 2). The easy-axis anisotropy as well as spin canting in the zdirection follow directly from the above DM interaction, which induces spin canting about the DM ax is and favors spins lying in the perpendicular plane. Intersection of the perpendicular p lane (φ=π/4,z) and the SOC-induced easy x−yplane yields φ=π/4 as the easy-axis direction, and canting about the DM axis yields spin canting in the zdirection, as shown in Fig. 3(a). In close analogy with the above effects of octahedral tilting, the st aggered octahedral rotation about the crystal caxis leads to orbital mixing hopping terms between yz,xz orbitals on NN sites, and hence to effective spin-dependent NN hopp ing termst′ zin Eq. (8). The resulting effective DM term −(8tt′ z/U)ˆz.(Si×Sj) causes spin canting about the crystal caxis, as shown in Fig. 3(b). The easy-axis anisotropy as well as the t wo spin cantings of the dominant yz,xzmoments are confirmed in the full self-consistent calculation discus sed below. Also, the effective spin dependent hopping terms discussed a bove are explicitly11 confirmed from the electronic band structure features in the self consistent state. Before continuing with the other important physical elements, it is c onvenient to first systematically introduce the different Coulomb interaction contribu tions in the HF theory. Contributions involving the orbital off-diagonal spin and charge con densates naturally lead to interaction induced SOC renormalization and coupling of orbital ma gnetic moments to orbital fields. V. SELF-CONSISTENT DETERMINATION OF MAGNETIC ORDER We consider the various Coulomb interaction terms in Eq. (3) in the HF approximation, focussing first on the terms with normal (orbital diagonal) spin and charge condensates. The resulting local spin and charge terms can be written as: [HHF int]normal=/summationdisplay iµψ† iµ[−σ.∆iµ+Eiµ1]ψiµ (10) where the spin and charge fields are self-consistently determined f rom: 2∆α iµ=U/angbracketleftσα iµ/angbracketright+JH/summationdisplay ν<µ/angbracketleftσα iν/angbracketright(α=x,y,z) Eiµ=U/angbracketleftniµ/angbracketright 2+U′′/summationdisplay ν<µ/angbracketleftniν/angbracketright (11) in terms of the local charge density /angbracketleftniµ/angbracketrightand the spin density components /angbracketleftσα iµ/angbracketright. For /angbracketleftnyz/angbracketright=/angbracketleftnxz/angbracketright, the Coulomb renormalized tetragonal splitting is obtained as: ˜δtet= ˜ǫxz,yz−˜ǫxy= (ǫxz,yz−ǫxy)+[Eyz,xz−Exy] =δtet+/bracketleftbiggU/angbracketleftnyz,xz/angbracketright 2+U′′/angbracketleftnyz,xz+nxy/angbracketright/bracketrightbigg −/bracketleftbiggU/angbracketleftnxy/angbracketright 2+2U′′/angbracketleftnyz,xz/angbracketright/bracketrightbigg =δtet+(U′′−U/2)/angbracketleftnxy−nyz,xz/angbracketright (12) which shows that the Coulomb renormalization identically vanishes for the realistic relation- shipU′′=U/2 for 4dorbitals, as discussed in Sec. II. There are additional contributions in the HF approximation resulting from orbital off- diagonal spin and charge condensates which are finite due to the SO C induced spin-orbital correlations. The contributions corresponding to different Coulom b interaction terms are summarized in the Appendix, and can be grouped in analogy with Eq. (1 0) as: [HHF int]OOD=/summationdisplay i,µ<νψ† iµ[−σ.∆iµν+Eiµν1]ψiν (13)12 where the orbital off-diagonal spin and charge fields are self-cons istently determined from: ∆iµν=/parenleftbiggU′′ 2+JH 4/parenrightbigg /angbracketleftσiνµ/angbracketright+/parenleftbiggJP 2/parenrightbigg /angbracketleftσiµν/angbracketright Eiµν=/parenleftbigg −U′′ 2+3JH 4/parenrightbigg /angbracketleftniνµ/angbracketright+/parenleftbiggJP 2/parenrightbigg /angbracketleftniµν/angbracketright (14) in terms of the corresponding condensates /angbracketleftσiνµ/angbracketright ≡ /angbracketleftψ† iνσψiµ/angbracketrightand/angbracketleftniνµ/angbracketright ≡ /angbracketleftψ† iν1ψiµ/angbracketright. The spin and charge condensates in Eqs. 11 and 14 are evaluated using t he eigenfunctions ( φk) and eigenvalues ( Ek) of the full Hamiltonian in the given basis including the interaction contributions [ HHF int] (Eqs. 10 and 13) using: /angbracketleftσα iµν/angbracketright ≡ /angbracketleftψ† iµσαψiν/angbracketright=Ek<EF/summationdisplay k(φ∗ kµs↑φ∗ kµs↓)[σα] φkνs↑ φkνs↓ (15) for siteion thes=A/Bsublattice, and similarly for the charge condensates /angbracketleftniµν/angbracketright ≡ /angbracketleftψ† iµ1ψiν/angbracketright, with the Pauli matrices [ σα] replaced by the unit matrix [ 1]. The normal spin and charge condensates correspond to ν=µ. Results of the full self consistent calculation including all spin and cha rge condensates (orbital diagonal and off-diagonal) are presented below. For each orbital pair ( µ,ν) = (yz,xz), (xz,xy), (xy,yz), there are three components ( α=x,y,z) for the spin condensates /angbracketleftψ† µσαψν/angbracketrightand one charge condensate /angbracketleftψ† µ1ψν/angbracketright. This is analogous to the three-plus-one normal spin and charge condensates for each of the three orbita lsµ=yz,xz,xy . The magnetization and density values for the three orbitals are presen ted in Table I, all off- diagonal spin and charge condensates in Table II, and the renorma lized SOC values and orbital magnetic moments in Table III. Here U= 8,ǫxy=−0.8, the bare SOC strength λbare= 1, and the staggered octahedral rotation ( tm1= 0.2) and tilting ( tm2=tm3= 0.15) have been included. As seen from Table I, the dominant yz,xzmoments show the expected cantings in and about thezdirection due to the octahedral tilting and rotation (Sec. IV). How ever, there is anadditionalsmallrelativecantingbetween the yz,xzmoments. Tounderstandtheoriginof this effect, we consider the real part of the off-diagonal charge c ondensate /angbracketleftψ† xzψyz/angbracketrightas given in Table II. The corresponding charge term in Eq. (13) yields a norma l “hopping” term −(λ0/2)ψ† yzψxz, and the combination of this normal and spin-dependent ψ† yz(iσzλz/2)ψxz “hopping” terms yields an effective intra-site DM interaction: [H(2) eff](z) DM(i) =−8(λ0/2)(λz/2) Uˆz.(Syz×Sxz) (16)13 TABLE I: Self consistently determined magnetization and de nsity values for the three orbitals ( µ) on the two sublattices ( s). µ(s)mx µmy µmz µnµ yz(A) 0.472 0.578 0.153 1.177 xz(A) 0.459 0.647 0.163 1.133 xy(A) 0.113 0.179 0.101 1.690 yz(B)−0.647−0.459 0.163 1.133 xz(B)−0.578−0.472 0.153 1.177 xy(B)−0.179−0.113 0.101 1.690 which leads to relative canting between the yzandxzmoments about the zaxis. The overall−ive sign of the DM term favors canting of Syztowardsxaxis and Sxztowards yaxis. Repeating the calculation with the same parameters as above b ut without the octahedral rotation, so that the overall canting about the zdirection is suppressed, yields magnetization values mx yz=my xz=±0.56 andmy yz=mx xz=±0.52 on A and B sublattices, which clearly show this relative canting effect. Fig. 4 shows the orbital resolved electronic band structure in the s elf consistent AFM state calculated for the two cases: (a) including only normal conde nsates, and (b) including all off-diagonal spin and charge condensates along with octahedra l rotation and tilting. The band structure shows the narrow AFM sub bands for the magn etically active yz,xz orbitals above and below the Fermi energy due to the dominant exch ange field splitting. The relatively smaller splitting between the xysub bands (both below EF) is due to the weaker effect of yz,xzmoments through the Hund’s coupling. The octahedral tilting and rotation are seen to introduce fine splittings due to the orbital mixin g hopping terms. VI. ORBITAL MAGNETIC MOMENT AND SOC RENORMALIZATION The off-diagonal charge condensates /angbracketleftψ† µψν/angbracketrightdirectly yield the orbital magnetic moments: /angbracketleftLx/angbracketright=/angbracketleftψ† xz(−i)ψxy/angbracketright+/angbracketleftψ† xy(i)ψxz/angbracketright =−i/angbracketleftψ† xzψxy/angbracketright+i/angbracketleftψ† xzψxy/angbracketright∗= 2Im/angbracketleftψ† xzψxy/angbracketright (17)14 -12-10-8-6-4-2 0 2 4 6 (0,0) ( π,0) (π,π) (0,0) (0, π) (π,0)(a)yz xz xy Ek - EF -12-10-8-6-4-2 0 2 4 6 (0,0) ( π,0) (π,π) (0,0) (0, π) (π,0)(b)yz xz xy Ek - EF FIG. 4: Calculated electronic band structure in the self-co nsistent AFM state for moderate tetrag- onal distortion: (a) without and (b) with all off-diagonal spi n and charge condensates included, along with octahedral tilting and rotation. Colors indicat e dominant orbital weight: red ( yz), green (xz), blue (xy). HereU= 8,ǫxy=−0.8, and bare SOC = 1. and similarly for the other components. Accordingly, the charge te rm in Eq. (13), of which only the anti-symmetric part is non-vanishing (see Appendix), can be represented as TABLE II: Self consistently determined off-diagonal spin and charge condensates for the three orbital pairs on the two sublattices. Orbital pair /angbracketleftψ† µσxψν/angbracketright /angbracketleftψ† µσyψν/angbracketright /angbracketleftψ† µσzψν/angbracketright /angbracketleftψ† µ1ψν/angbracketright yz−xz(A) (0.066,0.030) (0.071,0.025) (0.018,0.169) −(0.089,0.067) xz−xy(A) (0.026,0.281) (0.057,0.126) (0.079,0.039) −(0.061,0.245) xy−yz(A) (0.042,0.108) (0.053,0.333) (0.081,0.034) −(0.073,0.289) yz−xz(B)−(0.071,0.025) −(0.066,0.030) (0.018,0.169) −(0.089,0.067) xz−xy(B) (0.053,0.333) (0.042,0.108) −(0.081,0.034) (0.073,0.289) xy−yz(B) (0.057,0.126) (0.026,0.281) −(0.079,0.039) (0.061,0.245)15 a coupling of orbital angular momentum operators to orbital fields: [HHF int]charge OOD(i)|anti−sym=−U′′ c|a 2/summationdisplay µ<ν/angbracketleftnµν/angbracketrightIm/bracketleftbig ψ† µ(−i)ψν+H.c./bracketrightbig =−U′′ c|a 4[/angbracketleftLx/angbracketrightLx+/angbracketleftLy/angbracketrightLy+/angbracketleftLz/angbracketrightLz] (18) which corresponds to a weak effective isotropic interaction −(U′′ c|a/8)L.Lbetween orbital moments, and will therefore weakly enhance the /angbracketleftLα/angbracketrightvalues in the HF calculation. Turning now to the spin part of Eq. (13), the anti-symmetric part ( see Appendix) can be represented in terms of the spin-orbital operators: [HHF int]spin OOD(i)|anti−sym=−(U′′ s|a/2)/summationdisplay µ<ν/angbracketleftσµν/angbracketrightIm./bracketleftbig ψ† µ(−iσ)ψν+H.c./bracketrightbig =−/summationdisplay α=x,y,z/bracketleftigg λint αLαSα+/summationdisplay β/negationslash=αλint αβLαSβ/bracketrightigg (19) where the interaction-induced SOC renormalization terms: λint α=U′′ s|aIm/angbracketleftψ† µσαψν/angbracketright=U′′ s|a/angbracketleftψ† µ(−iσα)ψν/angbracketrightRe=U′′ s|a/angbracketleftLαSα/angbracketright (20) for the orbital pair µ,νcorresponding to component α. Although the off-diagonal SOC terms (LαSβ) are smaller than the diagonal terms ( λint αβ<λint α), they are still significant. For example, with Im /angbracketleftψ† xzσyψxy/angbracketright= 0.126 from Table II, we obtain λint xy≈U′′×0.126≈0.5 on the A sublattice, whereas the bare SOC = 1.0. Similarly, for the symmetric part we obtain: [HHF int]spin OOD(i)|sym=−(U′′ s|s/2)/summationdisplay µ<ν/angbracketleftσµν/angbracketrightRe./bracketleftbig ψ† µσψν+H.c./bracketrightbig (21) representingthecouplingoftheorbitaloff-diagonalspinoperato rstorealspinfieldsinvolving the enhanced effective interaction U′′ s|s=U′′+3JH/2. In the limit of bare SOC →0, since Im/angbracketleftψ† µψν/angbracketrightand Im/angbracketleftψ† µσαψν/angbracketrightare identically zero, the above term is the only surviving orbital off-diagonal contribution, and that too only for finite octahedral tilting and rotation which generate orbital mixing. We summarize here the results obtained above for moderate tetra gonal distortion ( ǫxy∼ −1.0), withallorbitaloff-diagonalspinandchargecondensatesinclud edintheselfconsistent calculation. With nearly half filled yz,xzorbitals and nearly filled xyorbital, the AFM insulating state is characterized by dominantly yz,xzmoments lying in the SOC induced16 TABLE III: Self consistently determined renormalized SOC v aluesλα=λbare+λint αand the orbital magnetic moments /angbracketleftLα/angbracketrightforα=x,y,zon the two sublattices. Bare SOC strength λbare= 1.0 s λ xλyλz/angbracketleftLx/angbracketright /angbracketleftLy/angbracketright /angbracketleftLz/angbracketright A 1.898 2.065 1.540 −0.490−0.578−0.134 B 2.065 1.898 1.540 0.578 0.490 −0.134 easy (a−b) plane and aligned along the octahedral tilting induced easy ( b) axis, with small canting of moments in and about the crystal caxis. The spin cantings become negligible when octahedral tilting and rotation are set to zero. Spin canting in thecdirection has been recently observed in resonant elastic X-ray scattering experimen ts.52The SOC induced spin- orbital correlations lead to strong orbital moments /angbracketleftLx/angbracketrightand/angbracketleftLy/angbracketrightand strongly anisotropic Coulomb renormalized SOC values ( λx,λy>λz), as shown in Tables II and III. VII. MAGNETIC REORIENTATION TRANSITION With decreasing tetragonal distortion, we find a sharp magnetic re orientation transition from the dominantly a−bplane AFM order to a dominantly caxis FM order, as shown in Fig. (5). The two orbital averaged magnetic orders shown in this plo t are defined as: mx−y AFM= (1/3)/summationdisplay µ/bracketleftigg/parenleftbiggmx µ(A)−mx µ(B) 2/parenrightbigg2 +/parenleftbiggmy µ(A)−my µ(B) 2/parenrightbigg2/bracketrightigg1/2 mz FM= (1/3)/summationdisplay µmz µ (22) The planar AFM order decreases sharply across the transition, wh ile the FM ( z) order (whichissameforbothsublattices)increasessharply. Theelectro nicstateremainsinsulating throughout the range of ǫxyshown, with filling n= 4. AFM correlations are seen to persist after the transition to the FM ( z) order. The reorientation transition is even stronger for bare SOC = 0.5 whic h corresponds to the realistic value of 100 meV. Results for the FM ( z) order obtained for ǫxy=−0.5 with no octahedraltiltingorrotationareparticularlyinteresting, withiden ticalmagnetization( mz µ= 0.65) and density ( nµ= 4/3) for all three orbitals, and very small planar components mx,y µ.17 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -1.2-1-0.8-0.6-0.4-0.2 0mx-yAFM and mzFM εxymx-y AFM mz FM 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -1.2-1-0.8-0.6-0.4-0.2 0 FIG. 5: The reorientation transition with decreasing tetra gonal distortion, as reflected in the sharp drop in the orbital averaged planar AFM order mx−y AFMand the sharp rise in the FM order mz FM. Here bare SOC = 1.0. The renormalized SOC and orbital moment values obtained are: λx,y,z= (0.78,0.78,1.28) and/angbracketleftLx,y,z/angbracketright= (∓0.26,∓0.26,−0.48) on A/B sublattice. The electronic band structure in the self consistent state is shown in Fig. 6 for this case. We find that the indirect band gap between valence band top at ( π/2,π/2) and conduction band bottom at ( π,π), (0,0) is reduced to nearly zero for slightly enhanced yz,xzNN hopping term corresponding to no octahedral tilting. Fig. 6 also shows the small orbital gap near the Fermi energy highligh ting the orbital physics. Band splittings near ( π/2,0), (π,π/2), and (0,π/2) arise from the orbital moment interaction term (Eq. 18). Finite /angbracketleftLx/angbracketrightand/angbracketleftLy/angbracketrightgenerate orbital fields which couple to the orbital angular momentum operators involving mixings between xyandyz,xzorbitals. The consequent orbital field induced splitting is analogous to the usual e xchange field splitting of spin sub bands. The small orbital gap vividly illustrates the crucial role of the orbital off-diagonal charge condensates in the insulating behaviour. With in creasingǫxypushing up thexybands, the upper xysub-band is now seen to be straddling the orbital gap, reflecting an important interplay between orbital physics and decreasing tet ragonal distortion. The orbital gap is maintained even as the xyspectral weight is transferred across the Fermi energy. We also find a robust FM metallic phase for electron filling n/greaterorsimilar4. Results of the self consistent cacluation obtained for the same set of parameters as above (bare SOC = 0.5 and18 -12-10-8-6-4-2 0 2 4 6 (0,0) ( π,0) (π,π) (0,0) (0, π) (π,0)yz xz xy Ek - EF FIG. 6: Orbital resolved electronic band structure for the F M (z) order (n= 4), obtained for reduced tetragonal distortion, with no octahedral tilting or rotation. Here bare SOC = 0.5 and ǫxy=−0.5. ǫxy=−0.5) are shown in Table IV. The FM metallic phase is characterized by iden tically vanishing planar magnetization components. All orbital off-diagona l condensates except for the SOC renormalization terms /angbracketleftLαSα/angbracketrightare also identically zero. Driven by switching of the dominant rolefrom yz,xzorbitals (AFM interaction) to the xyorbital (FM interaction), the magnetic reorientation transition with decreasing tetragonal dist ortion as discussed above provides a unified understanding of the planar AFM order as well as t he low-temperature FM metallic phase found in Ca 2RuO4under high pressure18and also in Ca 2−xSrxRuO4for x∼0.5 in neutron and DFT studies.17,21,23,29 Forhigher valuesofbareSOC,theplanarAFMorderisstableeven fo rreduced tetragonal TABLEIV:Selfconsistently determinedmagnetization andd ensityvalues, alongwithrenormalized SOC and orbital magnetic moment values in the FM metallic pha se, with bare SOC = 0.5, ǫxy= −0.5, and no octahedral rotation and tilting. µ mx µmy µmz µnµ yz0 0 0.57 1.42 xz0 0 0.57 1.42 xy0 0 0.66 1.33λxλyλz/angbracketleftLx/angbracketright /angbracketleftLy/angbracketright /angbracketleftLz/angbracketright 0.73 0.73 1.55 0 0 −0.6519 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4FM (z)AFM (x-y)-εxy SOC 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 FIG. 7: The magnetic phase boundary between the two regions w ith dominantly planar AFM and FM (z) orders. distortion, which is expected from the SOC induced easy a−bplane anisotropy. However, in the weak SOC regime (bare SOC /lessorsimilar0.5), the FM ( z) order is stabilized with increasing SOC, as seen in Fig. 7, which shows the phase boundary between the two m agnetic orders. The two axes here represent increasing bare SOC and tetragonal dist ortion. For realistic value of bare SOC= 0.5 andslightly above the magnetic phase boundary ( ǫxy=−1.0), we also find a stable AFM metallic state for n/lessorsimilar4, suggesting persistence of AFM correlations even if long range AFM order is destroyed by quantum spin fluctuations as in cup rate antiferromagnets. This is in agreement with the antiferromagnetically correlated metallic state reported for Ca2−xSrxRuO4in the range 0 .2<x<0.5. Spin resolved electronic density of states (DOS) is shown in Fig. 8 for (a) planar AFM and (b) FM ( z) order, with same parameters as in Figs. 3(b) and 5. For FM ( z) order, Fig. 8(b) shows that states near the Fermi energy are purely mino rity (down) spin states, highlighting the orbital character of the small gap as discussed for Fig. 5. Also, spin down spectral weight for the xyorbital is transferred above the Fermi energy, whereas for yz,xz orbitals it is transferred below, reversing the dominant orbital weig ht in the sub band just below the Fermi energy from xy(planar AFM order) to yz,xz(FM order). The importance of the orbital off-diagonal spin and charge conden sates in determining the self consistent magnetic order is illustrated by the strongly anis otropic SOC renormal- ization and strong orbital magnetic moments, which are both magne tic order dependent. We also note here that without the off-diagonal condensates includ ed in the self consistent20 -0.4-0.2 0 0.2 0.4 -12-10-8-6-4-2 0 2 4 6(a) AFM(x-y)spin up spin downyz xz xy Density of States ω-0.4-0.2 0 0.2 0.4 -12-10-8-6-4-2 0 2 4 6(b) FM(z)spin up spin downyz xz xy Density of States ω FIG. 8: The spin-resolved electronic density of states for t he (a) planar AFM order and (b) FM (z) order, with same parameters as in Fig. 3(b) and Fig. 5. calculation, the planar AFM order is obtained even for reduced tetr agonal distortion (down toǫxy=−0.3). The off-diagonal condensates are therefore responsible for the reorientation transition from planar AFM order to FM ( z) order. The reduced tetragonal distortion induced reorientation transit ion as found here provides a microscopic understanding of the pressure-induced stabilization of FM order in Ca 2RuO4 and the chemical substitution induced stabilization of FM correlation s in the isoelectronic series Ca 2−xSrxRuO4. Another candidate for the theory presented here is the Ca 2RuO4thin film where the tetragonal distortion and octahedral tilting are tun ed by the film thickness, as found in the recently synthesized nanofilm single crystal,53which shows robust FM cor- relations and significantly higher Curie temperature ( TC= 180 K) due to the suppression of lattice distortion. Other possible candidates could be ruthenate heterostructures where lattice distortions are tuned by synthesizing layered superlattices , as in the recently studied bilayer iridate heterostructure.54 VIII. COUPLED SPIN-ORBITAL FLUCTUATIONS Spin orientation in the AFM state affects orbital densities due to str ong spin-orbital coupling in Ca 2RuO4. Fig. 9(a) shows the variation of yz,xzorbital densities (summed over both sublattices) with iterations in the self-consistency proc ess, starting with spins oriented towards the xdirection. Also shown are the sublattice magnetization components mx avandmy avaveraged for yz,xzorbitals. Initially, we find that nxz>nyz, whereas the two21 FIG. 9: Variation of the (a) yz,xzorbital densities (upper panel) and x,ycomponents of the sublattice magnetization (lower panel), (b) x,ycomponents of the interaction induced SOC renor- malizations (upper panel) and orbital magnetic moments (lo wer panel), with iteration in the self consistency process for the planar AFM order. Here bare SOC = 1.0 andǫxy=−0.8. densities converge as the spin orientation approaches the self-co nsistent easy-axis ( φ=π/4) direction. This implies that the planar Goldstone mode, correspondin g to rigid spin rotation away from the easy axis towards x(y) axis, will be associated with ferro orbital fluctuation due to density transfer between orbitals. In contrast, the out- of-phase (zone boundary) fluctuation mode, with spin twistings towards x(y) and−y(−x) directions on A and B sublattices, respectively, will be associated with antiferro orbital fluctuation with opposite signofnxz−nyzonthetwosublattices. Thephysical quantities relatedtoorbitalo ff-diagonal condensates also show [Fig. 9(b)] strong dependence on the spin o rientation. Quite generally, since the self consistent determination of magnetic order requires all spin and charge condensates to be included, investigation of the flu ctuation propagator must therefore necessarily involve the generalized spin ( ψ† µσψν) and charge ( ψ† µψν) operators including both orbital diagonal and off-diagonal parts. This require s consideration of the generalized time-ordered fluctuation propagator: [χ(q,ω)] =/integraldisplay dt/summationdisplay ieiω(t−t′)e−iq.(ri−rj)×/angbracketleftΨ0|T[Oα µν(i,t)Oα′ µ′ν′(j,t′)]|Ψ0/angbracketright (23) in the self-consistent AFM ground state |Ψ0/angbracketright, where the generalized spin-charge operators at lattice sites i,jare defined as Oα µν=ψ† µσαψν, which include both the orbital diagonal (µ=ν) and off-diagonal ( µ/negationslash=ν) cases, and the spin ( α=x,y,z) and charge ( α=c)22 operators, with σαdefined as Pauli matrices for α=x,y,zand unit matrix for α=c. Investigation of the generalized fluctuation propagator can reve al if the planar Goldstone mode acquires a finite mass due to the coupled spin-orbital fluctuat ions, as reflected in the ferro and antiferro orbital fluctuations associated with in-phase and out-of-phase spin twist- ing modes. The coupling between spin and orbital fluctuations clearly highlights the strong deviation from conventional Heisenberg behaviour in effective spin m odels, as discussed recently to account for the magnetic excitation measurements in I NS experiments.39 IX. CONCLUSIONS Including the orbital off-diagonal spin and charge condensates in t he self consistent de- termination of magnetic order illustrates the rich interplay between the different physical elements in the 4d4compound Ca 2RuO4. These include SOC induced easy-plane anisotropy, octahedral tilting induced easy-axis anisotropy, spin-orbital cou pling induced orbital mag- netic moments, Coulomb interaction induced anisotropic SOC renorm alization, decreasing tetragonal distortion induced magnetic reorientation transition f rom planar AFM order to FM (z) order, and orbital moment interaction induced orbital gap. Stab le FM and AFM metallic states were also obtained near the magnetic phase boundar y separating the two magnetic orders. Since the orbital off-diagonal condensates con tribute on the same foot- ing as the normal condensates, the coupled spin-orbital fluctuat ions must be investigated within a unified formalism involving the generalized spin and charge oper ators including orbital off-diagonal terms.23 Appendix: Orbital off-diagonal condensates in the HF approximation TheadditionalcontributionsintheHFapproximationarisingfromthe orbitaloff-diagonal spin and charge condensates are given below. For the density, Hun d’s coupling, and pair hopping interaction terms in Eq. 3, we obtain (for site i): U′′/summationdisplay µ<νnµnν→ −U′′ 2/summationdisplay µ<ν[nµν/angbracketleftnνµ/angbracketright+σµν./angbracketleftσνµ/angbracketright]+H.c. −2JH/summationdisplay µ<νSµ.Sν→JH 4/summationdisplay µ<ν[3nµν/angbracketleftnνµ/angbracketright−σµν./angbracketleftσνµ/angbracketright]+H.c. JP/summationdisplay µ/negationslash=νa† µ↑a† µ↓aν↓aν↑→JP 2/summationdisplay µ<ν[nµν/angbracketleftnµν/angbracketright−σµν./angbracketleftσµν/angbracketright]+H.c. (A.1) in terms of the orbital off-diagonal spin ( σµν=ψ† µσψν) and charge ( nµν=ψ† µ1ψν) oper- ators. The orbital off-diagonal condensates are finite due to the SOC-induced spin-orbital correlations. These additional terms in the HF theory explicitly pres erve the SU(2) spin rotation symmetry of the various Coulomb interaction terms. Collecting all the spin and charge terms together, we obtain the orb ital off-diagonal (OOD) contributions of the Coulomb interaction terms: [HHF int]OOD=/summationdisplay µ<ν/bracketleftbigg/parenleftbigg −U′′ 2+3JH 4/parenrightbigg nµν/angbracketleftnνµ/angbracketright+/parenleftbiggJP 2/parenrightbigg nµν/angbracketleftnµν/angbracketright −/parenleftbiggU′′ 2+JH 4/parenrightbigg σµν./angbracketleftσνµ/angbracketright−/parenleftbiggJP 2/parenrightbigg σµν./angbracketleftσµν/angbracketright/bracketrightbigg +H.c. (A.2) Separating the condensates /angbracketleftnµν/angbracketright=/angbracketleftnµν/angbracketrightRe+i/angbracketleftnµν/angbracketrightIminto real and imaginary parts in order to simplify using /angbracketleftnνµ/angbracketright=/angbracketleftnµν/angbracketright∗, and similarly for /angbracketleftσµν/angbracketright, allows for organizing the OOD charge and spin contributions into orbital symmetric and anti-s ymmetric parts: [HHF int]OOD=−U′′ c|s 2/summationdisplay µ<ν/angbracketleftnµν/angbracketrightRe[nµν+H.c.]−U′′ c|a 2/summationdisplay µ<ν/angbracketleftnµν/angbracketrightIm[−inµν+H.c.] −U′′ s|s 2/summationdisplay µ<ν/angbracketleftσµν/angbracketrightRe.[σµν+H.c.]−U′′ s|a 2/summationdisplay µ<ν/angbracketleftσµν/angbracketrightIm.[−iσµν+H.c.] (A.3) where the effective interaction terms above are obtained as: U′′ c|a=U′′ s|a=U′′−JH/2 =U−3JH U′′ s|s=U′′+3JH/2 =U−JH U′′ c|s=U′′−5JH/2 =U−5JH (A.4)24 usingJP=JH. While the effective interaction U′′ s|s(spin term, symmetric part) is enhanced relative toU′′, the corresponding charge term interaction U′′ c|svanishes for JH=U/5. ∗Electronic address: avinas@iitk.ac.in 1W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated Quantum Phenomena in the Strong Spin-Orbit Regime , Ann. Rev. Condens. Mat. Phys. 5, 5782 (2014). 2G. Cao and P. Schlottmann, The Challenge of Spin–Orbit-Tuned Ground States in Iridates: A Key Issues Review , Rep. Prog. Phys. 81, 042502 (2018). 3B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, NovelJeff= 1/2Mott State Induced by Relativistic Spin-Orbit Coupling in Sr2IrO4, Phys. Rev. Lett. 101, 076402 (2008). 4S. Mohapatra, J. van den Brink, and A. Singh, Magnetic Excitations in a Three-Orbital Model for the Strongly Spin-Orbit Coupled Iridates: Effect of Mixing Between the J= 1/2and3/2 Sectors, Phys. Rev. B 95, 094435 (2017). 5S. Mohapatra and A. Singh, Pseudo-Spin Rotation Symmetry Breaking by Coulomb Interacti on Terms in Spin-Orbit Coupled Systems , arXiv:2001.00190 (2020). 6S. Mohapatra and A. Singh, Correlated Motion of Particle-Hole Excitations Across the R enor- malized Spin-Orbit Gap in Sr2IrO4, J. Magn. Magn. Mater 512, 166997 (2020). 7S. Mohapatra, S. Aditya, R. Mukherjee, and A. Singh, Octahedral Tilting Induced Isospin Reorientation Transition in Iridate Heterostructures , Phys. Rev. B 100, 140409(R) (2019). 8S. Fuchs, T. Dey, G. Aslan-Cansever, A. Maljuk, S. Wurmehl, B . B¨ uchner, and V. Kataev, Unraveling the Nature of Magnetism of the 5d4Double Perovskite Ba2YIrO6, Phys. Rev. Lett. 120, 237204 (2018). 9J. C. Wang, J. Terzic, T. F. Qi, Feng Ye, S. J. Yuan, S. Aswartha m, S. V. Streltsov,D. I. Khom- skii, R. K. Kaul, and G. Cao, Lattice-Tuned Magnetism of Ru4+(4d4)Ions in Single Crystals of the Layered Honeycomb Ruthenates Li2RuO3andNa2RuO3, Phys. Rev. B 90, 161110 (2014). 10G. Cao, T. F. Qi, L. Li, J. Terzic, S. J. Yuan, L. E. DeLong, G. Mu rthy, and R. K. Kaul, Novel Magnetism of Ir5+(5d4)Ions in the Double Perovskite Sr2YIrO6, Phys. Rev. Lett. 112, 056402 (2014). 11M. A. Laguna-Marco, P. Kayser, J. A. Alonso, M. J. Martnez-Lo pe, M. van Veenen-daal, Y.25 Choi, and D. Haskel, Electronic Structure, Local Magnetism, and Spin-Orbit Effe cts of Ir(IV)-, Ir(V)-, and Ir(VI)-Based Compounds , Phys. Rev. B 91, 214433 (2015). 12S. Bhowal, S. Baidya, I. Dasgupta, and T. Saha-Dasgupta, Breakdown of J= 0Nonmagnetic State ind4Iridate Double Perovskites: A First-Principles Study , Phys. Rev. B 92, 121113 (2015). 13T. Dey, A. Maljuk, D. V. Efremov, O. Kataeva, S. Gass, C. G. F. B lum, F. Steckel, D. Gruner, T. Ritschel, A. U. B. Wolter, J. Geck, C. Hess, K. Koepernik, J . van den Brink, S. Wurmehl, and B. B¨ uchner, Ba 2YIrO6: A Cubic Double Perovskite Material with Ir5+Ions, Phys. Rev. B 93, 014434 (2016). 14S. Nakatsuji, S.-i. Ikeda, and Y. Maeno, Ca 2RuO4: New Mott Insulators of Layered Ruthenate , J. Phys. Soc. Jpn 66(7), 1868 (1997). 15M.Braden, G.Andr´ e, S.Nakatsuji, andY. Maeno, Crystal and Magnetic Structure of Ca2RuO4: Magnetoelastic Coupling and the Metal-Insulator Transition , Phys. Rev. B 58, 847 (1998). 16C. S. Alexander, G. Cao, V. Dobrosavljevic, S. McCall, J. E. C row, E. Lochner, and R. P. Guertin, Destruction of the Mott Insulating Ground State of Ca2RuO4by a Structural Transi- tion, Phys. Rev. B 60, R8422 (1999). 17O. Friedt, M. Braden, G. Andr´ e, P. Adelmann, S. Nakatsuji, a nd Y. Maeno, Structural and Magnetic Aspects of the Metal-Insulator Transition in Ca2−xSrxRuO4, Phys. Rev. B 63, 174432 (2001). 18F. Nakamura, T. Goko, M. Ito, T. Fujita, S. Nakatsuji, H. Fuka zawa, Y. Maeno, P. Alireza, D. Forsythe, and S. R. Julian, From Mott Insulator to Ferromagnetic Metal: A Pressure Study of Ca2RuO4, Phys. Rev. B 65, 220402(R) (2002). 19E. Gorelov, M. Karolak, T. O. Wehling, F. Lechermann, A. I. Li chtenstein, and E. Pavarini, Nature of the Mott Transition in Ca2RuO4, Phys. Rev. Lett. 104, 226401 (2010). 20S. Nakatsuji and Y. Maeno, Quasi-Two-Dimensional Mott Transition System Ca2−xSrxRuO4, Phys. Rev. Lett. 84, 2666 (2000). 21Z. Fang and K. Terakura. Magnetic Phase Diagram of Ca2−xSrxRuO4Governed by Structural Distortions , Phys. Rev. B 64, 020509 (2001). 22S. Kunkem¨ oller, D. Khomskii, P. Steffens, A. Piovano, A. A. Nu groho, and M. Braden, Highly Anisotropic Magnon Dispersion in Ca2RuO4: Evidence for Strong Spin-Orbit Coupling , Phys. Rev. Lett. 115, 247201 (2015).26 23S. Nakatsuji and Y. Maeno, Switching of Magnetic Coupling by a Structural Symmetry Chang e Near the Mott Transition in Ca2−xSrxRuO4, Phys. Rev. B 62, 6458 (2000). 24P. Steffens, O. Friedt, P. Alireza, W. G. Marshall, W. Schmidt, F. Nakamura, S. Nakatsuji, Y. Maeno, R. Lengsdorf, M. M. Abd-Elmeguid, and M. Braden, High-Pressure Diffraction Studies onCa2RuO4, Phys. Rev. B 72, 094104 (2005). 25C. Dietl, S. K. Sinha, G. Christiani, Y. Khaydukov, T. Keller , D. Putzky, S. Ibrahimkutty, P. Wochner, G. Logvenov, P. A. van Aken, B. J. Kim, and B. Keimer, Tailoring the Electronic Properties of Ca2RuO4via Epitaxial Strain , Appl. Phys. Lett. 112, 031902 (2018). 26P. Steffens, O. Friedt, Y. Sidis, P. Link, J. Kulda, K. Schmalzl , S. Nakatsuji, M. Braden, Mag- netic Excitations in the Metallic Single-Layer Ruthenates Ca2−xSrxRuO4Studied by Inelastic Neutron Scattering , Phys. Rev. B 83, 054429 (2011). 27F. Nakamura, M. Sakaki, Y. Yamanaka, S. Tamaru, T. Suzuki, an d Y. Maeno, Electric-Field- Induced Metal Maintained by Current of the Mott Insulator Ca2RuO4, Sci. Rep. 3, 2536 (2013). 28R. Okazaki, Y. Nishina, Y. Yasui, F. Nakamura, T. Suzuki, and I. Terasaki, Current-Induced Gap Suppression in the Mott Insulator Ca2RuO4, J. Phys. Soc. Jpn. 82, 103702 (2013). 29Z. Fang, N. Nagaosa, and K. Terakura, Orbital-Dependent Phase Control in Ca2−xSrxRuO4 (0/lessorsimilarx/lessorsimilar0.5), Phys. Rev. B 69, 045116 (2004). 30A. Liebsch and H. Ishida, Subband Filling and Mott Transition in Ca2−xSrxRuO4, Phys. Rev. Lett.98, 216403 (2007). 31G. Zhang and E. Pavarini, Mott transition, Spin-Orbit Effects, and Magnetism in Ca2RuO4, Phys. Rev. B 95, 075145 (2017). 32C. G. Fatuzzo, M. Dantz, S. Fatale, P. Olalde-Velasco, N. E. S haik, B. Dalla Piazza, S. Toth, J. Pelliciari, R. Fittipaldi, A. Vecchione, N. Kikugawa, J. S. Brooks, H. M. Rønnow, M. Grioni, Ch. R¨ uegg, T. Schmitt, and J. Chang, Spin-Orbit-Induced Orbital Excitations in Sr 2RuO4and Ca2RuO4: A Resonant Inelastic X-ray Scattering Study , Phys. Rev. B 91, 155104 (2015). 33D. Sutter, C. G. Fatuzzo, S. Moser, M. Kim, R. Fittipaldi, A. V ecchione, V. Granata, Y. Sassa, F. Cossalter, G. Gatti, M. Grioni, H. M. Rønnow, N. C. Plumb, C . E. Matt, M. Shi, M. Hoesch, T. K. Kim, T.-R. Chang, H.-T. Jeng, C. Jozwiak, A. Bostwick, E . Rotenberg, A. Georges, T. Neupert, and J. Chang, Hallmarks of Hund’s Coupling in the Mott Insulator Ca2RuO4, Nat. Comm.8, 15176 (2017). 34G. Khaliullin, Excitonic Magnetism in Van Vleck–type d4Mott Insulators , Phys. Rev. Lett.27 111, 197201 (2013). 35A. Akbari and G. Khaliullin, Magnetic Excitations in a Spin-Orbit-Coupled d4Mott Insulator on the Square Lattice , Phys. Rev. B 90, 035137 (2014). 36D. I. Khomskii, Transition Metal Compounds (Cambridge University Press, Cambridge, Eng- land, 2014). 37O. Meetei, C. Nganba , S. William, M. Randeria, and N. Trivedi ,Novel Magnetic State in d4 Mott Insulators , Phys. Rev. B 91, 054412 (2015). 38T. Feldmaier, P. Strobel, M. Schmid, P. Hansmann, and M. Dagh ofer,Excitonic Magnetism at the Intersection of Spin-Orbit Coupling and Crystal-Field Spl itting, arXiv:1910.13977 (2019). 39A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen, D. L. A bernathy, J. T. Park, A. Ivanov, J. Chaloupka, G. Khaliullin, B. Keimer, and B. J. Kim ,Higgs Mode and Its Decay in a Two-Dimensional Antiferromagnet , Nat. Phys. 13, 633 (2017). 40L. Das, F. Forte, R. Fittipaldi, C. G. Fatuzzo, V. Granata, O. Ivashko, M. Horio, F. Schindler, M. Dantz, Yi Tseng, D. E. McNally, H. M. Rønnow, W. Wan, N. B. Ch ristensen, J. Pelliciari, P. Olalde-Velasco, N. Kikugawa, T. Neupert, A. Vecchione, T. S chmitt, M. Cuoco, and J. Chang, Spin-Orbital Excitations in Ca2RuO4Revealed by Resonant Inelastic X-Ray Scattering , Phys. Rev. X8, 011048 (2018). 41H. Gretarsson, H. Suzuki, H. Kim, K. Ueda, M. Krautloher, B. J . Kim, H. Yava, G. Khaliullin, andB. Keimer, Observation of Spin-Orbit Excitations and Hund’s multiple ts inCa2RuO4, Phys. Rev. B100, 045123 (2019). 42A. V. Puchkov, M. C. Schabel, D. N. Basov, T. Startseva, G. Cao , T. Timusk, and Z.-X. Shen, Layered Ruthenium Oxides: From Band Metal to Mott Insulator , Phys. Rev. Lett. 81, 2747 (1998). 43K. Park, Electronic Structure Calculations for Layered LaSrMnO 4andCa2RuO4, J. Phys.: Condens. Matt. 13(41), 9231 (2001). 44G.-Q. Liu, Spin-Orbit Coupling Induced Mott Transition in Ca2−xSrxRuO4(0≤x≤0.2), Phys. Rev. B84, 235136 (2011). 45S. Acharya, D. Dey, T. Maitra, and A. Taraphder, Quantum Criticality Associated with Di- mensional Crossover in the Iso-electronic Series Ca2−xSrxRuO4, J. Phys. Commun. 2, 075004 (2018). 46J. Bertinshaw, N. Gurung, P. Jorba, H. Liu, M. Schmid, D. T. Ma ntadakis, M. Daghofer,28 M. Krautloher, A. Jain, G. H. Ryu, O. Fabelo, P. Hansmann, G. K haliullin, C. Pfleiderer, B. Keimer, and B. J. Kim, Unique Crystal Structure of Ca2RuO4in the Current Stabilized Semimetallic State , Phys. Rev. Lett. 123, 137204 (2019). 47H. Hao, A. Georges, A. J. Millis, B. Rubenstein, Q. Han, and H. Shi,Metal-Insulator and Mag- netic Phase Diagram of Ca2RuO4from Auxiliary Field Quantum Monte Carlo and Dynamical Mean Field Theory , arXiv:1911.02702 (2019). 48N. Kaushal, J. Herbrych, A. Nocera, G. Alvarez, A. Moreo, F. A . Reboredo, and E. Dagotto, Density Matrix Renormalization Group Study of a Three-Orbit al Hubbard Model with Spin-Orbit Coupling in One Dimension , Phys. Rev. B 96, 155111 (2017). 49T. Sato, T. Shirakawa, and S. Yunoki, Spin-Orbital Entangled Excitonic Insulator with Quadrupole Order , Phys. Rev. B 99, 075117 (2019). 50T. Mizokawa, L. H. Tjeng, G. A. Sawatzky, G. Ghiringhelli, O. Tjernberg, N. B. Brookes, H. Fukazawa, S. Nakatsuji, and Y. Maeno, Spin-Orbit Coupling in the Mott Insulator Ca2RuO4, Phys. Rev. Lett. 87, 077202 (2001). 51S.MohapatraandA.Singh, Spin Waves and Stability of Zigzag Order in the Hubbard Model with Spin-Dependent Hopping Terms: Application to the Honeycom b Lattice Compounds Na2IrO3 andα-RuCl3, J. Magn. Magn. Mater 479, 229 (2019). 52D. G. Porter, V. Granata, F. Forte, S. Di Matteo, M. Cuoco, R. F ittipaldi, A. Vecchione, and A. Bombardi, Magnetic Anisotropy and Orbital Ordering in Ca2RuO4, Phys. Rev. B 98, 125142 (2018). 53H. Nobukane, K. Yanagihara, Y. Kunisada, Y. Ogasawara, K. Is ono, K. Nomura, K. Tanahashi, T. Nomura, T. Akiyama, and S. Tanda, Co-appearance of superconductivity and ferromagnetism in aCa2RuO4nanofilm crystal , Sci. Rep. 10, 3462 (2020). 54D. Meyers, Y. Cao, G. Fabbris, N. J. Robinson, L. Hao, C. Frede rick, N. Traynor, J. Yang, J. Lin, M. H. Upton, D. Casa, J.-W. Kim, T. Gog, E. Karapetrova, Y . Choi, D. Haskel, P. J. Ryan, L. Horak, X. Liu, J. Liu, and M. P. M. Dean, Magnetism in iridate heterostructures leveraged by structural distortions , Sci. Rep. 9, 4263 (2019).

1311.1778v1.Spin_Orbit_Torques_and_Anisotropic_Magnetization_Damping_in_Skyrmion_Crystals.pdf

arXiv:1311.1778v1 [cond-mat.mes-hall] 7 Nov 2013Spin-Orbit Torques and Anisotropic Magnetization Damping in Skyrmion Crystals Kjetil M. D. Hals and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway The length scale of the magnetization gradients in chiral ma gnets is determined by the relativistic Dzyaloshinskii-Moriya interaction. Thus, even conventio nal spin-transfer torques are controlled by the relativistic spin-orbit coupling in these systems, and additional relativistic corrections to the current-induced torques and magnetization damping become important for a complete understand- ing of the current-driven magnetization dynamics. We theor etically study the effects of reactive and dissipative homogeneous spin-orbit torques and anisotrop ic damping on the current-driven skyrmion dynamics in cubic chiral magnets. Our results demonstrate t hat spin-orbit torques play a significant role in the current-inducedskyrmion velocity. The dissipa tive spin-orbit torque generates a relativis- tic Magnus force on the skyrmions, whereas the reactive spin -orbit torque yields a correction to both the drift velocity along the current direction and the trans verse velocity associated with the Magnus force. The spin-orbit torque corrections to the velocity sc ale linearly with the skyrmion size, which is inversely proportional to the spin-orbit coupling. Cons equently, the reactive spin-orbit torque correction can be the same order of magnitude as the non-rela tivistic contribution. More impor- tantly, the dissipative spin-orbit torque can be the domina nt force that causes a deflected motion of the skyrmions if the torque exhibits a linear or quadratic re lationship with the spin-orbit coupling. In addition, we demonstrate that the skyrmion velocity is de termined by anisotropic magnetization damping parameters governed by the skyrmion size. I. INTRODUCTION The manipulation of submicron-scale magnetic el- ements via electric currents has paved the way for a promising new class of magnetoelectronic devices with improved scalability, faster execution time, and lower power consumption.1,2The dominant mechanism of current-driven magnetic excitations in conventional metallic ferromagnets is the spin-transfer-torque (STT) effect,3,4in which spin angular momentum is transferred from spin-polarized currents to the magnetization, gen- erating the torque τ(r,t) on the magnetization:1,2 τ(r,t) =−(1−βm×)(vs·∇)m. (1) Here, the vector vsis proportional to the out-of- equilibrium current density Jand the spin polarization of the current. The first term in Eq. (1) describes the reactive STT, whereas the second term, which is propor- tional to β, describes the dissipative STT. In systems that lack spatial-inversion symmetry, an alternative manner of generating current-induced mag- netization torques is via relativistic intrinsic spin- orbit coupling (SOC). Orbital momentum is (via SOC) transferred to the spins, causing a so-called spin- orbit torque (SOT) on the magnetization.5–21Recently, SOTs have been observed to lead to remarkably effi- cient current-driven magnetization dynamics in ultra- thin magnetic films and strained ferromagnetic semicon- ductors.7–13,18–20,22,23Similar to the STT in Eq. (1), such SOTs also have reactive and dissipative contribu- tions.21,24Although the reactive homogeneous SOT is known to scale linearly with the SOC to the lowest order, 6,7there are only a few theoretical works regarding the dissipative SOT that typically predict it to be smaller.24 However, a recent experiment and theory demonstrate thatthedissipativehomogeneousSOTcanbeofthesameorder of magnitude as the reactive part.21SOTs can en- able the design of significantly simpler devices because the torques originate from a direct conversion of orbital angularmomentum intospinexcitationsandappeareven in homogenous ferromagnets with no external sources of spin-polarized currents. Magnetic skyrmions are vortex-like spin configura- tions that cannot be continuously deformed into a ho- mogeneous magnetic state.25Experimentally, magnetic skyrmion phases were first reported in bulk chiral mag- nets,26–32and more recently, they also have been ob- served in magnetic thin films.33–38Under the applica- tion of a weak external magnetic field, a chiral mag- netic system crystallizes into a two-dimensional lattice of skyrmions in which the magnetic moment at the core of each vortex is antiparallel to the applied field, whereas the peripheral magnetic moments are parallel. In ultra-thin magnetic films, the skyrmion phase be- comes more energetically favorable with decreasing film thickness,34–36,38,39and for a single atomic layer of Fe, a skyrmion structure at the atomic length scale has been observed even in the absence of an external magnetic field.34 In the context of spintronic applications, promising characteristics of skyrmions are the extremely low depin- ningcurrents31,32thatarerequiredtomovethem andthe fact that they avoid pinning centers.40A proposed expla- nation of the latter feature is that the Magnus force acts on the skyrmions, leading to a deflection of their mo- tion.40,41This force is closely linked to the topological Hall effect and can be viewed as the reaction of the fic- titious Lorentz force experienced by the itinerant quasi- particles as their spins adiabatically align with the local magnetization direction. The underlying physical mechanism that gives rise to the skyrmion phase is the Dzyaloshinskii-Moriya inter-2 action (DMI).42,43The DMI has the same relativistic origin as the homogeneous SOTs observed in ferromag- netic heterostructures and strained ferromagnetic semi- conductors. They both arise from the combined effect of spin-orbit coupling and broken spatial-inversion sym- metry. Neglected in earlier works concerning current- induced skyrmion motion, a reactive SOT in chiral mag- nets was predicted and studied for systems in the helical phase in Ref. 44. However, the effect of reactive and dis- sipative SOTs on the skyrmion dynamics in bulk chiral magnets remains unknown. Such studies are important becauseSOTs canbe equallyasimportantas the conven- tional STTs in Eq. (1). This equal importance is because the typical length scale of the spatial variations of the magnetization is determined by the DMI, i.e., the mag- netization gradients ∂imjscale as ∂imj∼D/J, where Dis the DMI parameter and Jis the spin stiffness. Be- causeDis linear in the SOC, even the non-relativistic STT in Eq. (1) is proportional to the SOC to the lowest order in the relativistic corrections. Thus, the SOTs in chiral magnets can be of the same order of magnitude as the conventional STT. Thus, a complete understanding of the current-drivendynamics of chiral magnets requires a correct treatment of SOC effects on both the current- induced torques and magnetization damping. The magnetization dynamics driven by currents or ex- ternal fields strongly depends on dissipation. In isotropic systems, the damping can typically be assumed to be de- coupled fromthe magnetizationdirectionand itstexture. However, this model does not necessarily extend to chi- ral magnets. First, the broken spatial-inversion symme- try allows for terms that are linear, not only quadratic, in the magnetization gradients. Second, chiral magnets have a preferred direction, and thus, the dissipation is likely not isotropic and independent of the texture struc- ture. In this paper, we present a theoretical study of the current-induceddynamicsofskyrmionsthat correctlyac- counts for the effects of SOC on both current-induced torques and magnetization damping. Our results demon- strate that SOC generates reactive and dissipative ho- mogeneous SOTs that lead to important corrections to the drift velocity along the current direction and to the Magnus force. Another essential consequence of the SOC is that the skyrmions experience effective damping and torque parameters that depend on the current direction relative to the crystallographic axes. This paper is organized as follows. Section II intro- duces the phenomenology of SOTs that was presented in Ref. 45 and performs a similar phenomenological expan- sion for the Gilbert damping tensor. In Section III, we applythe phenomenologytothe studyofcurrent-induced skyrmion dynamics in cubic chiral magnets and derive a collective coordinate description for the skyrmion veloc- ity. Our results are summarized in Section IV.II. PHENOMENOLOGICAL EXPANSION We include the SOC effects phenomenologically by de- riving an equation for the magnetization dynamics that satisfies the symmetry of the underlying crystal struc- ture. In deducing expressions for the torques and dis- sipation, we perform a phenomenological expansion of the magnetization-damping tensor and current-induced torques in terms of the magnetization and its gradients. The magnetic system is assumed to satisfy the local ap- proximation, in which the magnetization dynamics de- pends only on the local properties of the system. In this approximation, the magnetization dynamics can be phenomenologically expressed in terms of the Landau- Lifshitz-Gilbert-Slonczewski (LLGS) equation: ˙m=−γm×Heff+m×α˙m+τ. (2) Here,m(r,t) is the unit vector of the magnetization M(r,t) =Msm(r,t),γis (minus) the gyromagnetic ra- tio, and Heff(r,t) =−δF[M]/δMis the effective field determined by the functional derivative of the magnetic free-energy functional F[M] =/integraltext drF.F(r,t) is the free-energy density. The second term on the right side of Eq. (2) describes the magnetization damping, where α(r,t) is a symmetric, positive-definite second-rank ten- sor that depends on the local magnetization direction and local magnetization gradients. The torque τ(r,t) in Eq. (2) represents the current-induced torques. The free-energy density of a magnetic system is, to the second order in the magnetization gradients, F=Jij∂iM·∂jM+DijkMi∂jMk+K(1) ijMiMj+ K(2) ijklMiMjMkMl−M·B. (3) The tensor Jijis the spin stiffness, Dijkdescribes the DMI, and Bis an external magnetic field. In an inversion-symmetric system, Dijk= 0. The two terms proportional to K(1) ijandK(2) ijklrepresent the two first harmonics in the phenomenological expansion of the magnetocrystalline anisotropy energy. Jij,Dijk,K(1) ij, andK(2) ijklare polar tensors that are invariant under the point group of the system.46In the above equation and in what follows, we assume summation over repeated in- dices, and ∂iis a shorthand notation for ∂/∂ri. A similar phenomenological expansion can be per- formed for the Gilbert damping tensor α(r,t). To the second order in the spatial magnetization gradients, we obtain αij=α(0) ij+α(1) ijklmkml+α(2) ijklpmk∂lmp+ α(3) ijklpq∂kml∂pmq+α(4) ijklpqmk∂l∂pmq.(4) Again, the tensors α(0) ij,α(1) ijkl,α(2) ijklp, andα(3,4) ijklpqare in- variant under the point group of the system, and the tensorα(2) ijklponlyappearsinsystemswith brokenspatial- inversion symmetry, such as chiral magnets, the focus of3 this study. Terms that are odd under time reversal do not appear in the expansion because such terms do not represent dissipative processes. Recently, in Ref. 45, we carried out a phenomenologi- cal expansion of the current-induced torque τ(r,t). The starting point of the derivation is to write down the most general form of the torque in the linear-response regime and in the local approximation: τ(r,t) =m×ηJ. (5) The second-rank tensor ( η[m,∇m])ij, which we refer to as thefieldance tensor , depends on the local magnetiza- tion direction and local gradients of the magnetization. The fieldance tensor contains all information concerning the local torque. The general form of the fieldance ten- sor is not known in systems with strong SOC, but recent experiments indicate that it has a complicated struc- ture.18–21Therefore, we expand the fieldance tensor in powers of miand∂jmito find simplified expressions for the allowed torques: ηij= Λ(r) ij+Λ(d) ijkmk+βijkl∂kml+Pijklnmk∂lmn+...(6) The first and second terms in Eq. (6) represent the re- active and dissipative homogeneous SOTs, respectively, that recently have been observed experimentally.7,8,21 These terms are only present in systems with broken spatial-inversionsymmetry. ThelasttwotermsinEq.(6) represent a generalization of the torque in Eq. (1). In the non-relativisticlimit, in which the systemis invariantun- der separate rotations of the spin space and coordinate space, the βijklterm and Pijklnterm are equal to the dissipative and reactive STTs in Eq. (1), respectively.45 When SOC is included, the forms of the tensors in Eq. (6) are, as for the free energy and magnetization damping, determined by the crystal symmetry: Λ(r) ijand Pijklnbecome invariant axial tensors of the point group, whereas Λ(d) ijkandβijklbecome invariant polar tensors of the point group. Higher-order terms in the expansion of the fieldancetensorrepresenttorqueswith higherdegrees of anisotropy. However, the leading-order terms explic- itly written in Eq. (6) provide a sufficient description of the current-driven dynamics for many materials. III. CHIRAL MAGNETS AND SKYRMION DYNAMICS We now focus on bulk magnets with cubic B20-type crystal structures. Examples of such magnets include MnSi, FeGe, and (Fe,Co)Si. The symmetry of these sys- tems is described by the tetrahedral point group T (in the Sch¨ onflies notation). We begin by writing down the specific form of the in- variant tensors in systems described by the point group T.47,48The invariant tensors lead to phenomenological expressions for the free energy, Gilbert damping tensor,and current-induced torque given in Sec. II. A collec- tive coordinate description is then applied to model the current-driven dynamics of the magnetic system. A. Invariant tensors The point group T is the smallest of the five cubic point groups and consists of 12 proper rotation opera- tors: the identity operator; three two-fold rotation op- erators about the x, y, and z axes; and eight three-fold rotation operators about the body diagonals of the cube. Because the group only consists of proper rotations, the invariant axial and polar tensors have the same form.46 In the present study, expressions are required for all ten- sors written explicitly in Section II, with the exception of α(3)andα(4)in Eq. (4) because we expand the Gilbert damping tensor to the lowest order in the magnetization gradients. Thus, we will consider invariant tensors up to the fifth rank. Invariant second-rank tensors Tijare described by one independent tensor coefficient: Tij=Tδij. (7) Here,δijis the Kronecker delta. Invariant third-rank tensors are described by two independent tensor coeffi- cients, and their non-vanishing elements satisfy the two symmetry relations Txyz=Tyzx=Tzxy, (8) Txzy=Tyxz=Tzyx. (9) Invariant fourth-rank tensors are described by seven in- dependent coefficients that satisfy the relations Txxxx=Tyyyy=Tzzzz, (10) Txxyy=Tyyzz=Tzzxx(p= 3), (11) Txxzz=Tyyxx=Tzzyy(p= 3), (12) wherep= 3 indicates the three different symmetry relations obtained from the expression by holding the first index constant and permuting the last three in- dices. For example, Eq. (11) also yields the relations Txyyx=Tyzzy=TzxxzandTxyxy=Tyzyz=Tzxzx. Invariant fifth-rank tensors are determined by 20 inde- pendent tensorcoefficients whose non-vanishingelements satisfy the symmetry relations Txxxyx=Tyyyzx=Tzzzxy(p= 20).(13) Here,p= 20 refers to the 20 independent symmetry re- lations obtained from Eq. (13) via permutations of the five indices. There are 68 independent tensor coefficients that de- termine the required free energy, torques, and damping to describe current-induced skyrmion motion. Whereas the number of parameters might appear overwhelming, we demonstrate below that only certain combinations of these parameters appear in the final results, which are more transparent.4 B. Collective coordinate description We apply a collective coordinate description to model the current-driven magnetization dynamics.49The mag- netic state is assumed to be parameterized by a set of time-dependent collective coordinates {ai(t)|i= 1,2,...} such that m(r,t) =m(r,{ai(t)}). The equations of mo- tion for the collective coordinates are then given by (Γij−Gij) ˙aj=γFi+Li. (14) Here, the matrices Γ ijandGijareGij=/integraltext m· [(∂m/∂ai)×(∂m/∂aj)]dV and Γ ij=/integraltext (∂m/∂ai)· α(∂m/∂aj)dV, and Fiis the force attributable to the ef- fective field, Fi=/integraltext Heff·(∂m/∂ai)dV. The force on the collectivecoordinatesattributable tothe current-induced torque is represented by Li=/integraltext (∂m/∂ai)·[m×τ]dV. We compute the above matrices and vectors, which are governed by the 68 independent tensor coefficients dis- cussed in the previous section, and determine the rate of change of the collective coordinates. C. Equations of motion A current density J= [Jx,Jy,Jz] is applied to the system, and an external magnetic field is applied along thezaxis such that a lattice of skyrmions forms in the xyplane. The skyrmion lattice is assumed to be undis- torted during the current-driven dynamics. In this ap- proximation, we can disregard rotations of the magnetic texture50, and the skyrmions can be considered a lattice of non-interacting particles, where each skyrmion is de- scribed by mx=2qR(y−ry) (x−rx)2+(y−ry)2+R2,(15) my=−2qR(x−rx) (x−rx)2+(y−ry)2+R2,(16) mz=−q(x−rx)2+(y−ry)2−R2 (x−rx)2+(y−ry)2+R2.(17) The two-dimensional center-of-mass coordinates rxand ryarethecollectivecoordinatesthatdescribethedynam- ical evolution of each skyrmion. Distortions introduce an additional collective coordinate, which describes the ro- tational motion of the skyrmions.50The parameters R andq∈ {1,−1}are the size and topological charge of the skyrmion, respectively. Disregarding terms that are of second order in the Gilbert damping parameters, the collective coordinate formulas presented in Section IIIB generate the veloc-ity of the center of mass: /parenleftbigg ˙rx ˙ry/parenrightbigg =−/parenleftbigg/parenleftbig Peff y+RΛeff r/parenrightbig Jx /parenleftbig Peff x+RΛeff r/parenrightbig Jy/parenrightbigg + (18) q/parenleftbigg−/parenleftbig Peff yβeff y−Peff xαeff y/parenrightbig Jy /parenleftbig Peff xβeff x−Peff yαeff x/parenrightbig Jx/parenrightbigg + qR −/parenleftBig Λeff d,y−Λeff rαeff y/parenrightBig Jy/parenleftBig Λeff d,x−Λeff rαeff x/parenrightBig Jx . Here,Peff xandPeff y(Peff xβeff xandPeff yβeff y) areeffective re- active (dissipative) STT parameters that are linear com- binationsofthe tensorcoefficients Pijkln(βijkln) andαeff x andαeff yare effective Gilbert damping parameters that are linear combinations of the tensor coefficients α(0) ij, α(1) ijkl, andα(2) ijklp. Theparameters βeff xandβeff yaredimen- sionless and determined by the ratio between the dissipa- tive and reactive STT. Their magnitude is proportional to the spin-flip rate, which is of second order in the SOC. The effective Gilbert damping parameters depend on the skyrmion size Rin combination with the α(2) ijklptensor, i.e., the parameters can be decomposed into two terms, αeff x,y=α′ x,y+R−1α′′ x,ywhereα′′ x,ydepends on α(2) ijklp. The effective reactive and dissipative homogeneous SOT pa- rameters Λeff r, Λeff d,x, and Λeff d,yare proportional to the ten- sorcoefficientsofΛ(r) ijandΛ(d) ijk. The effective parameters in Eq. (18) determine the current-driven magnetization dynamics and can be extracted from experiments. Their explicit forms and relations to the invariant tensors are given in Appendix A. To revealthe effects ofthe SOC, let us comparethe ex- pression for the velocity of the center of mass in Eq. (18) with the conventional expression for the velocity in the non-relativistic limit, in which the homogeneous SOTs vanish, and the STT and the Gilbert damping parame- ters satisfy the symmetry relations Peff≡Peff x=Peff y, βeff≡βeff x=βeff y, andαeff≡αeff x=αeff y: ˙r=−PeffJ+qPeff(βeff−αeff)ˆz×J.(19) ComparingEqs.(18)andEq.(19)indicatesthattheSOC introduces several important effects on the skyrmion dy- namics. First, the equations of motion in Eq. (18) are no longer rotationally symmetric about the zaxis. Clearly, the ef- fective torque and damping parameters that govern the motion along the xandyaxes differ because there are no symmetry operations that relate the two axes. Thus, dif- ferent current-driven velocities can be observed for cur- rents applied along the two directions, and a measure- ment of this velocity anisotropy provides a simple test for investigating the importance of the SOC. A current along the zaxis does not influence the velocity in the linear-response regime. Second, the SOTs strongly affect the skyrmion mo- tion along the current direction. The reactive homoge- neousSOTleadstoarenormalizationofthe drift velocity5 along the current direction that scales linearly with the skyrmion size R. The reason for this linear dependency is that the homogeneous SOTs do not depend on the magnetization gradients. Thus, the homogeneous SOTs couple more strongly to largerskyrmions, whose textures are distributed over a larger spatial region. Because Λeff r is linear in the SOC, whereas Rscales as the inverse of the SOC (and thus the product RΛeff ris independent of the SOC), the reactive SOT contribution to the drift ve- locity can be of the same orderof magnitude as the terms that are induced by the reactive STT, i.e., the terms pro- portional to Peff xandPeff y. Third, SOTs also strongly influence the Magnus force. The terms proportional to qin Eqs. (18)-(19) repre- sent the transverse drift velocity induced by the Mag- nus force. Both the reactive and dissipative homogenous SOTs produce corrections to the Magnus-force-induced motion that scale linearly with R. Using the same ar- guments as above, the reactive SOT yields a transverse drift velocity ∼RΛeff rαeff ithat can be of the same or- der of magnitude as the non-relativistic terms ∼Peff iβeff i and∼Peff iαeff i(i∈ {x,y}) ifβeff i<<1. The most in- teresting observation from Eq. (18) is the contribution of the dissipative homogeneous SOT to the transverse velocity. In contrast to the terms that arise from the STTs and reactive SOT, the dissipative SOT produces a transverse velocity that depends neither on damping parameters nor the dimensionless effective βparameters. The velocity is solely determined by the values of RΛeff d,x andRΛeff d,y. There is little knowledge regarding the mag- nitude of the dissipative homogeneous SOT and how it depends on the SOC in chiral magnets. However, a re- cent experiment concerning (Ga,Mn)As indicated that the dissipative part can be comparable in magnitude to the reactive part. If the same result is applicable to chi- ral magnets, the dissipative homogeneous SOT provides the largest contribution to the transverse drift velocity and is the dominant driving force that causes deflected motion of the skyrmions. This relativistic Magnus force is not linked to the fictitious magnetic field generated by the spin texture but instead arises from the dissipative part of the out-of-equilibrium spin density generated by the SOC combined with an applied electric field. The three independent tensor coefficients that de- scribe the reactive and dissipative homogeneous SOTs can be extractedfrom spin-orbitferromagneticresonance (FMR) measurements.11,21An external magnetic field is used to align the magnetization along different directions relative to the bar direction, and an alternating current is applied to produce microwave SOTs within the sample that resonantly drive the magnetization. The reflected direct current contains information regarding the magni- tudeoftheSOTs. Webelievethatsuchameasurementof the SOTs will be one of most interesting tasks for future experimental work concerning chiral magnets. Just prior to the submission of our paper, a relatedtheoreticalworkconcerningSOTsandskyrmionsin mag- netic thin films waspresented.51However,that workcon- siders a different symmetry class that is intended for the description of ultra-thin ferromagnetic heterostructures, in which there is complete rotational symmetry and bro- ken spatial-inversion symmetry along a transverse direc- tion. Both reactive and dissipative Rashba SOTs are considered in their study, and they demonstrate that the SOTs also play a significant role in the skyrmion dy- namics of these systems. However, only isotropic and spatially independent damping is considered. IV. SUMMARY In summary, we studied the effects of SOC on the current-driven dynamics of skyrmions in cubic chiral magnets. We performed a phenomenological expan- sion of the Gilbert damping tensor and current-induced torques that accounts for the relativistic SOC effects. A collective-coordinate description was applied to model the current-induced motion of the skyrmions. Our re- sults demonstrated that the skyrmion velocity depends on the direction of the applied current relative to the crystallographic axes and that the SOTs contribute sig- nificantly to the current-induced velocity. The reactive SOT induces a correction to both the parallel and trans- verse drift velocities of the skyrmions that is of the same order of magnitude as the non-relativistic contributions. If the dissipative SOT exhibits a linear or quadratic rela- tionship with the SOC, it produces a relativistic Magnus- force motion that is larger than the transverse drift ve- locity induced by conventional STTs. The SOTs cannot be neglected in the modeling of current-driven skyrmion dynamics because they do not depend on the gradients of the magnetization and couple more strongly to larger skyrmions. V. ACKNOWLEDGMENTS K.M.D.H. wouldliketothankRembertDuine forstim- ulating discussions of SOTs in chiral magnets. K.M.D.H. and A.B. would like to thank Jairo Sinova for notifying us of Ref. 5 and its important contribution to the theory of SOTs. Appendix A: Effective Parameters Below, we provide the expressions for the effective torque and damping parameters in terms of the tensor coefficients of the invariant tensors given in Section II. The effective damping parameters are6 αeff x=1 60(60a(0)+12a(1) xxxx+19a(1) xxyy+29a(1) xxzz−10a(1) xyxy−10a(1) xyyx−2a(1) xzxz−2a(1) xzzx)− 1 30R(3a(2) xxxyz−5a(2) xxxzy−9a(2) xxyxz+10a(2) xxyzx+7a(2) xxzxy−6a(2) xxzyx+8a(2) xyxxz−6a(2) xyxzx− 2a(2) xyyyz+4a(2) xyyzy−2a(2) xyzxx+2a(2) xyzzz+2a(2) xzxxy−2a(2) xzxyx−2a(2) xzyxx+2a(2) xzyyy), (A1) αeff y=1 60(60a(0)+12a(1) xxxx+29a(1) xxyy+19a(1) xxzz−6a(1) xyxy−6a(1) xyyx−6a(1) xzxz−6a(1) xzzx)− 1 30R(5a(2) xxxyz−3a(2) xxxzy−7a(2) xxyxz+6a(2) xxyzx+9a(2) xxzxy−10a(2) xxzyx+2a(2) xyxzx−2a(2) xyyyz+ 2a(2) xyzxx−2a(2) xyzzz−6a(2) xzxxy+6a(2) xzxyx+2a(2) xzyxx−2a(2) xzyyy−4a(2) xzzyz). (A2) The effective polarizations are Peff x=−1 12(Pxxxzy+3Pxxyxz−Pxxyzx−3Pxxzxy−Pxyxxz+Pxyyyz+Pxyzxx−Pxyzyy−Pxzyyy+ Pxzyzz+Pxzzyz−3Pxzzzy), (A3) Peff y=1 12(Pxxxyz−3Pxxyxz+3Pxxzxy−Pxxzyx−3Pxyyyz+Pxyyzy+Pxyzyy−Pxyzzz−Pxzxxy+ Pxzyxx−Pxzyzz+Pxzzzy). (A4) The effective βfactors are defined via the expressions Peff xβeff x=−1 6(βxxxx−βxxyy+2βxyyx−βxzxz+3βxzzx), (A5) Peff yβeff y=1 6(−βxxxx+βxxzz+βxyxy−3βxyyx−2βxzzx), (A6) and finally, the effective SOT parameters are given by Λeff r=−Λ(r) xx/2,Λeff d,x=−Λ(d) xyz/2,Λeff d,y= Λ(d) xzy/2. (A7) These expressions are used to quantify the center-of-mass velo city in Eq. (18). 1D. C. Ralph and M. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008). 2A. Brataas, A. D. Kent, and H. Ohno, Nature Materials 11, 372 (2012). 3L. Berger, Phys. Rev. B 54, 9353 (1996). 4J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 5B. A. Bernevig and O. Vafek, Phys. Rev. B 72, 033203 (2005). 6A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). 7A. Chernyshov et al., Nat. Phys. 5, 656 (2009). 8I. M. Miron et al., Nat. Mat. 9, 230 (2010). 9I. Garate and A. H. MacDonald, Phys. Rev. B 80, 134403 (2009). 10K. M. D. Hals, A. Brataas, and Y. Tserkovnyak, Europhys. Lett.90, 47002 (2010). 11D. Fang et al., Nat. Nanotech. 6, 413 (2011). 12I. M. Miron et al., Nature (London) 476, 189 (2011). 13I. M. Miron et al., Nat. Mat. 10, 419 (2011). 14D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86,014416 (2012). 15E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406 (2012). 16X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201 (2012). 17L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). 18S. Emori, U. Bauer, S. M Ahn, E. Martinez, and G. S. D. Beach, Nat. Mat. 12, 611 (2013). 19K. S. Ryu, L. Thomas, S. H. Yang, and S. Parkin, Nat. Nanotech. 8, 527 (2013). 20K. Garello et al., Nat. Nanotech. 8, 587 (2013). 21H. Kurebayashi et al., arXiv:1306.1893. 22T. A. Moore et al., Appl. Phys. Lett. 93, 262504 (2008). 23A. Thiaville, S. Rohart, E. Jue, V. Cros, and A. Fert, Eu- rophys. Lett. 100, 57002 (2012). 24G. Tatara, N. Nakabayashi, and K. J. Lee, Phys. Rev. B 87, 054403 (2013). 25A. Fert, V. Cros, and J. Sampaio, Nat. Nanotech. 8, 152 (2013).7 26U. K. R¨ oßler, A. N. Bogdanov, and C. Pfleiderer, Nature 442, 797 (2006). 27S. M¨ uhlbauer et al., Science 323, 915 (2009). 28A. Tonomura et al., Nano Lett. 12, 1673 (2012). 29S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 (2012). 30N. Kanazawa et al., Phys. Rev. B 86, 134425 (2012). 31F. Jonietz et al., Science 330, 1648 (2010). 32X. Z. Yu et al., Nat. Commun. 3, 988 (2012). 33X. Z. Yu et al., Nature 465, 901 (2010). 34S. Heinze et al., Nat. Phys. 7, 713 (2011). 35X. Z. Yu et al., Nat. Mat. 10, 106 (2011). 36S. X. Huangand C. L. Chien, Phys. Rev.Lett. 108, 267201 (2012). 37N. Romming, Science 341, 636 (2013). 38J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, accepted in Nat. Nanotech. (2013). 39N. S. Kiselev, A. N. Bogdanov, R. Sch¨ afer, and U. K. R¨ ossler, J. Phys. D 44, 392001 (2011). 40J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Com. 4, 1463 (2013). 41J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys. Rev. Lett. 107, 136804 (2011).42I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). 43T. Moriya, Phys. Rev. 120, 91 (1960); Phys. Rev. Lett. 4, 228 (1960). 44K. M. D. Hals and A. Brataas, Phys. Rev. B 87, 174409 (2013). 45K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423 (2013). 46Apolar tensor Tij...kis atensor thattransforms as Tij...k= RiαRjβ...RkγTαβ...γunder a coordinate transformation x′=Rx. An axial tensor is a tensor that transforms as Tij...k=|R|RiαRjβ...RkγTαβ...γ, where|R|is the deter- minant of R. 47R. R. Birss, Symmetry and Magnetism (North-Holland, Amsterdam, 1966). 48R. Fieschi and F. G. Fumi, IL Nuovo Cimento 10, 865 (1953). 49O. A. Tretiakov, D. Clarke, Gia-Wei Chern, Ya. B. Baza- liy, and O. Tchernyshyov, Phys. Rev. Lett. 100, 127204 (2008). 50K. Everschor, M. Garst, R. A. Duine, and A. Rosch, Phys. Rev. B84, 064401 (2011). 51M. E. Knoester, J. Sinova, and R. A. Duine, arXiv:1310.2850.

2311.13191v1.Effects_of_magnetic_fields_and_orbital_angular_momentum_on_excitonic_condensation_in_two_orbital_Hubbard_model.pdf

Effects of magnetic fields and orbital angular momentum on excitonic condensation in two-orbital Hubbard model Ryota Koga1and Joji Nasu1 1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan (Dated: November 23, 2023) We investigate the magnetic-field e ffects on a two-orbital Hubbard model that describes multiple spin states. The spin-state degrees of freedom in perovskite-type cobalt oxides have been explored due to their characteristic nature, where low-spin, intermediate-spin, and high-spin states play a crucial role in their magnetic properties caused by the competition between Hund coupling and crystalline field e ffects. Recent findings have suggested that this interplay leads to quantum mechanical hybridization of these spin states, which is interpreted as exci- tonic condensation. When the ground state comprises primarily low-spin states because of dominant crystalline fields, fluctuations between di fferent spin states are anticipated to arise from Zeeman splittings in higher-spin states. This o ffers an alternative approach to induce excitonic condensation using an external magnetic field. To understand magnetic-field e ffects on excitonic condensation in multi-orbital systems, it is crucial to account for contributions from both spin and orbital degrees of freedom to magnetic properties. Here, we study field-induced phenomena in the two-orbital Hubbard model by focusing on the role of the orbital angular momentum. We comprehensively analyze this model on a square lattice employing the Hartree-Fock approximation. Omitting contributions from the orbital moment, we find that an applied magnetic field gives rise to two excitonic phases, besides the spin-state ordered phase, between the nonmagnetic low-spin and spin-polarized high-spin phases. One of these excitonic phases manifests a staggered-type spin-state order, interpreted as an excitonic supersolid state. Conversely, the other phase is not accompanied by it and exhibits only a spin polarization due to the applied magnetic field. When spin-orbit coupling is present, this phase displays a ferrimagnetic spin alignment attributed to spin anisotropy. Our analysis also reveals that incorporating the contribution of the orbital magnetic moment to the Zeeman term significantly alters the overall structure of the phase diagram. Notably, the orbital magnetization destabilizes the excitonic phase in contrast to scenarios without this contribution. We also discuss the relevance of our findings to real materials, such as cobalt oxides. I. INTRODUCTION Excitonic insulating states, proposed nearly half a century ago, are characterized by the spontaneous condensation of ex- citons, pairs of electrons in conduction bands and holes in va- lence bands [1–3]. Since an exciton is a pair of particles with anS=1/2 spin, excitonic condensation is roughly classi- fied into two types: spin-triplet and spin-singlet. Recent stud- ies on excitonic condensation primarily focused on the latter. Amongst others, the transition metal chalcogenide Ta 2NiSe 5 has been intensively investigated as a candidate of excitonic insulators [4, 5]. This material undergoes a structural phase transition at TS≃328K [6]. Angle-resolved photoemission spectroscopy experiments have observed a flattening of the valence band below this temperature [4, 5]. This band struc- ture deformation is considered a manifestation of spin-singlet excitonic condensation associated with lattice distortion [7, 8]. Spin-triplet excitonic condensation is also predicted to manifest in real materials due to Hund coupling interac- tion rather than electron-lattice interactions. Perovskite-type cobalt oxides have attracted considerable attention as can- didate materials exhibiting spin-triplet excitonic condensa- tion [9, 10]. These materials, typified by LaCoO 3, have been studied as strongly correlated electron systems with spin-state degrees of freedom in cobalt ions, which govern the electric and magnetic properties [11, 12]. In a trivalent cobalt ion, six electrons occupy the 3 dorbitals. This ion takes three distinct electronic configurations depending on the balance between Hund coupling interaction and crystalline electric- field e ffects: low-spin (LS) state with the ( t2g)6configura-tion, intermediate-spin (IS) state with the ( t2g)5(eg)1config- uration, and high-spin (HS) state with the ( t2g)4(eg)2config- uration. Severe competition between these states can lead to spin-state transitions or crossovers when external conditions, such as temperature or pressure, are altered. It is well estab- lished that LaCoO 3shows a spin crossover with increasing temperature [11, 12]. The competition among the multiple spin states leads to more exotic phenomena in the presence of quantum fluctua- tions. When the t2gandegorbitals are regarded as valence and conduction bands, respectively, the IS state corresponds to an exciton with a triplet spin S=1 and the LS state is con- sidered the vacuum of excitons [8–10, 13–15]. In this con- sideration, excitonic condensation corresponds to a quantum hybridization between the LS and IS states with long-range coherence. This phenomenon is expected in the vicinity of the phase boundary of the LS and IS states. As the candidate ma- terial fulfilling this condition, Pr 0.5Ca0.5CoO 3has been pro- posed. This material undergoes a phase transition around 90 K without magnetic orders [16–18]. The first-principles and model calculations in Refs. [9, 10] suggested that the phase transition is accounted for by the spin-triplet excitonic con- densation, which is also regarded as a higher-order magnetic multipole [14, 19, 20]. Magnetic fields o ffer another route to bring about competi- tions between the LS and IS states. While the IS and HS states have nonzero spin moments, the LS state does not. An ap- plied magnetic field lowers the energy of the former but does not that of the latter, promoting a competition between the IS and LS states when the ground state without magnetic fields isarXiv:2311.13191v1 [cond-mat.str-el] 22 Nov 20232 the LS state. Thus, it has been argued that an strong magnetic field applied to LaCoO 3induces excitonic condensation and spin-state ordered states with di fferent spin states arranged al- ternately [13, 21–24]. The appearance of these states is also supported by theoretical calculations [25–27]. More interest- ingly, a recent study suggested that ultra-high magnetic fields possibly cause a phase where both order parameters of the ex- citonic condensation and staggered-type spin-state alignment are nonzero simultaneously [13]. This is known as an exci- tonic supersolid (ESS). Nonetheless, the mechanism for sta- bilizing the ESS owing to applying magnetic fields remains elusive. Moreover, orbital angular momentum and spin-orbit cou- pling are also expected to play a key role in discussing magnetic-field e ffects on electrons in 3 dorbitals. In cobalt ox- ides, both the HS and IS states retain orbital angular momen- tum owing to partially filled t2gorbitals [28–30]. The spin- orbit coupling is also suggested to be crucial for interpret- ing the experimental results obtained from soft x-ray absorp- tion spectroscopy and magnetic circular dichroism in LaCoO 3 [31, 32]. Although the e ffects of spin-orbit coupling have been studied from the aspect of exciton condensation [33], compre- hensive understanding including contributions of orbital mo- ments remains elusive. In this paper, we investigate magnetic-field e ffects on ex- citonic condensation realized in correlated electron systems with spin-state degrees of freedom. We introduce a half-filled two-orbital Hubbard model with a crystalline field splitting on a square lattice and examine magnetic field e ffects on this model comprehensively by applying the Hartree-Fock approx- imation. This is a simplified model that cannot address the three spin states but can deal with competitions between LS and HS states. In the absence of magnetic fields, we find an excitonic supersolid phase in addition to a spin-triplet exci- tonic phase without spin-state orderings between the LS and HS phases. These phases with excitonic condensation are also induced by applying magnetic fields. We also examine e ffects of the orbital angular momentum intrinsic to multi-orbital sys- tems and focus on the spin-orbit coupling and contributions of orbital magnetic moments to the Zeeman terms. We reveal that the spin-orbit coupling slightly modifies spin configura- tions in the spin-triplet excitonic phase but the overall struc- ture of magnetic-field phase diagrams remains largely intact. On the other hand, the introduction of orbital magnetic mo- ments dramatically alters the phase boundary of the magnetic- field phase diagrams. An applied magnetic field does not in- duce spin-triplet excitonic condensation for the LS phase. In the spin-triplet excitonic phase, the magnetic field suppresses the excitonic order parameter and changes this phase to a dis- ordered phase connected continuously with the LS phase. We clafity that this phenomenon originates from the orbital Zee- man e ffect inducing a spin-singlet excitonic order parameter, which compete with the spin-triplet excitonic phase. This paper is organized as follows. In the next section, we introduce the two-orbital Hubbard model with a crystalline field splitting under magnetic fields. The orbital angular mo- mentum and spin-orbit coupling are also given in this sec- tion. In Sec. III, we describe the method used in the present LSHSSpin state𝑎𝑏Exciton pictureVacuumExciton(𝑆=1)EC=+=−(a)(b)FIG. 1. (a) Correspondence between the two spin states and their exciton pictures in the half-filled two-orbital Hubbard model [13]. The LS and HS states correspond to vacuum and an exciton state, respectively. (b) Schematic pictures of excitonic condensation. The information on the relative phase of the superposition between the LS and HS states is presented as the direction of a semicircular shape. study. We review the Hartree-Fock approximation applied to the two-orbital Hubbard model in Sec. III A. Order parameters in this model are defined in Sec. III B. The results are given in Sec. IV. We show the phase diagram in the absence of mag- netic fields in Sec. IV A. Magnetic-field e ffects on the system neglecting contributions from the orbital angular momentum are examined in Sec. IV B. Next, we consider the impact of the spin-orbit coupling. After briefly discussing the phase diagram without magnetic fields in Sec. IV C, we examine magnetic-field e ffects and the results are shown in Sec. IV D. In Sec. IV E, the contribution of the orbital moment to the Zee- man term are investigated. In Sec. V, we compare the present results with previous theoretical studies and discuss the rel- evance to real materials. Finally, Sec. VI is devoted to the summary. II. MODEL In the present study, we address magnetic-field e ffects on excitonic condensation in correlated electron systems with orbital degrees of freedom. In particular, we focus on the perovskite-type cobalt oxides, such as LaCoO 3and Pr0.5Ca0.5CoO 3, which are the promising candidates of spin- triplet excitonic insulators. To extract the physics based on the spin-state degrees of freedom in these cobalt oxides, the half- filled two-orbital Hubbard model with a crystalline electric- field splitting has been investigated [34–38]. This model can describe two distinct spin states: the LS state with S=0 where two electrons occupy the lower-energy orbital, and the HS state with S=1 where each electron occupies a di fferent orbital. Furthermore, the appearance of excitonic condensa- tion has been discussed within the two-orbital Hubbard model where the average number of electrons is two per atom [8– 10, 14, 15, 39, 40]. When one regards the LS state as vacuum, the HS state is interpreted as an exciton, where an electron and a hole are present in high-energy and low-energy orbitals, re- spectively, as shown in Fig. 1(a). Hybridizations between the LS and HS states [Fig. 1(b)] are regarded as excitonic con- densation. Note that it has been pointed out that the mixing3 between dx2−y2in the higher-energy egorbitals and dxyin the lower-energy t2gorbitals are crucial for the emergence of the excitonic condensation in the cobalt oxides [9, 10, 20]. Thus, we examine the properties of the two-orbital Hubbard model consisting of dx2−y2anddxyorbitals. Although candidate materials for the spin-triplet excitonic condensation are three-dimensional compounds, we consider the two-orbital Hubbard model on a square lattice for simplic- ity. This model is written as ˆHHubbard =ˆH0+ˆHU. Here, ˆH0 represents the one-body term and given by ˆH0=−X ⟨i j⟩ησtη ˆc† iησˆcjησ+H.c. + ∆X iˆnia, (1) where⟨i j⟩denotes nearest-neighbor (NN) sites, ˆ c† iησis the cre- ation operator of an electron occupying orbital η(=a,b) at site iwith spinσ(=↑,↓), and ˆ niηis the number operator of elec- trons occupying orbital η, which is given by ˆ niη=P σˆc† iησˆciησ. For simplicity, we only consider the electron hopping between the same orbital; tηis the transfer integral between orbital η. The second term of Eq. (1) stands for the crystalline field ef- fect with the energy ∆(>0). The onsite interaction term of the two-orbital Hubbard model is written as ˆHU=UX iηˆniη↑ˆniη↓+U′X iˆniaˆnib +JX iσσ′ˆc† iaσˆc† ibσ′ˆciaσ′ˆcibσ+IX iη,η′ˆc† iη↑ˆc† iη↓ˆciη′↓ˆciη′↑,(2) where ˆ niησ=ˆc† iησˆciησ. The terms in Eq. (2) with U,U′,J, and Iare the intra-orbital and inter-orbital Coulomb interactions, Hund coupling interaction, and pair-hopping interaction, re- spectively. Next, we introduce the Zeeman term to examine magnetic- field e ffects. This term is given by ˆHZeeman =−X ih·ˆMi, (3) where his an applied magnetic field and ˆMirepresents the local magnetic moment at site i, which is written as ˆMi= −ˆLi−2ˆSiwith ˆLiand ( ˆSi) being orbital (spin) angular mo- mentum. The local spin operator ˆSiis given as ˆSγ i=X ησσ′(sγ)σσ′ˆc† iησˆciησ′, (4) where sγ=1 2σγis the 2×2 matrix with σγ(γ=x,y,z) be- ing theγcomponent of the Pauli matrices. On the other hand, the orbital angular momentum depends on the choice of the orbital basis. In this study, we assume that aandborbitals correspond to dx2−y2anddxyorbitals, respectively. Since these orbitals are represented by linear combinations of the lz=±2 states, only the zcomponent of ˆLiis nonzero, which is repre- sented as, ˆLz i=X ηη′σ(lz)ηη′ˆc† iησˆciησ=−2i ˆc† ia↑ˆcib↑+ˆc† ia↓ˆcib↓ +H.c.,(5)where lzis the 2×2 matrix given by lz=2σy. Because of the presence of orbital angular momentum, we need to consider the spin-orbit coupling. This contribution is given by [33] ˆHSOC=λX iηη′σσ′(lz i)ηη′(sz i)σσ′ˆc† iησˆciη′σ′ =λX ih −i ˆc† ia↑ˆcib↑−ˆc† ia↓ˆcib↓ +H.c.i . (6) Here, the total Hamiltonian that we should address in the presence of magnetic fields and spin-orbit coupling is ˆH= ˆHHubbard +ˆHZeeman +ˆHSOC. We examine this Hamiltonian for the half-filled case, where average electron number is two per site. In addition, we focus on the case with tatb>0 cor- responding to the indirect gap, which is expected in the per- ovskite cobalt oxides [20, 41]. III. METHOD In this section, we show the detail of the Hartree-Fock ap- proximation that we apply to the two-orbital Hubbard model and introduce the order parameters to characterize symmetry- broken states. A. Hartree-Fock approximation In the Hartree-Fock approximation, we apply the following decouplings to the interaction term ˆHU: ˆc† il1ˆcil2ˆc† il3ˆcil4 ≃⟨ˆc† il1ˆcil2⟩ˆc† il3ˆcil4+ˆc† il1ˆcil2⟨ˆc† il3ˆcil4⟩−⟨ ˆc† il1ˆcil2⟩⟨ˆc† il3ˆcil4⟩ −⟨ˆc† il1ˆcil4⟩ˆc† il3ˆcil2−ˆc† il1ˆcil4⟨ˆc† il3ˆcil2⟩+⟨ˆc† il1ˆcil4⟩⟨ˆc† il3ˆcil2⟩,(7) where we introduce index l=(ησ) to represent orbital and spin indices together. Here, ⟨ˆc† ilˆcil′⟩is a local mean field. This quantity is assumed to depend on the sublattice the site ibe- longs to, where the number of sublattices is defined as Ns. The mean field on sublattice Xis written as⟨ˆc† lˆcl′⟩X. By ap- plying the above decoupling, the original Hamiltonian is ap- proximated to a bilinear form of the fermionic operators as follows: ˆHHF=X kll′XX′(Mk)(lX),(l′X′)ˆc† klXˆckl′X′+C0, (8) where ˆ c† klX=√1/N0P i∈Xˆc† ileik·riwith N0andribeing the number of unit cells and the position of site i, respectively, Mkis the 4 Ns×4NsHermitian matrix determined by the pa- rameters of the Hamiltonian and the set of mean fields ⟨ˆc† lˆcl′⟩X, andC0is a constant. The matrixMkis diagonalized by a unitary matrix Vkwith the eigenvalues{λk1,λk2,···}. By introducing the fermionic operator ˆαkn=P lX(Vk)∗ (lX),nˆck(lX), the Hartree-Fock Hamilto- nian is rewritten as ˆHHF=X knλknˆα† knˆαkn+C0. (9)4 (a) ABCD++++Γ−++−M++−−X(b) FIG. 2. (a) Four sublattices assumed in the present Hartree-Fock calculations for a square lattice. The red dashed line represents a unit cell and A,B,C, and Dstand for the labels of sublattices. (b) Schematic pictures of three ordered structures defined in Eqs.(15), (16), and (17), respectively. The sign ±at each site represents that of the order parameter at each sublattice. The self-consistent equation determining the mean fields is given by ⟨ˆc† lˆcl′⟩X=1 N0X k⟨ˆc† klXˆckl′X⟩=1 N0X knV∗ k,(lX),nVk,(l′X),nf(λkn), (10) where f(λ)=1/(e(λ−µ)/T+1) is the Fermi distribution function with the chemical potential µ. We assume that the Boltzmann constant is unity. In the calculations, we impose that there are four sites in a unit cell ( Ns=4), as shown in Fig. 2(a), and the chemi- cal potential is determined such that the system is half-filled. Moreover, we assume U=U′−2JandI=J. We focus only on the case at T=0 and find a stable solution with a minimal internal energy from several initial states. B. Order parameters To characterize the realized state obtained by Hartree-Fock calculations, we introduce order parameters constructed from local mean fields ⟨ˆc† lˆcl′⟩X. In the present case, there are 16 mean fields for each sublattice. As the diagonal parts with re- spect to orbital η, we introduce the number of electrons and three components of the spin moment for each orbital at sub- lattice X: nη,X=X σ⟨ˆc† ησˆcησ⟩X,Sγ η,X=1 2X σσ′σγ σσ′⟨ˆc† ησˆcησ⟩X.(11) We also define the quantities summed over the orbitals as nX=P ηnη,XandSγ X=P ηSγ η,Xand the di fference of the occupancy between the aandborbitals as ∆nX=na,X−nb,X. Furthermore, we introduce the o ff-diagonal parts for the or- bitals, which characterize excitonic condensation. The order parameter describing the spin-triplet excitonic condensation at sublattice Xis defined as [3, 42], ϕγ t,X=X σσ′σγ σσ′⟨ˆc† aσˆcbσ′⟩X, (12)and that for the spin-singlet one is also given by ϕs,X=X σ⟨ˆc† aσˆcbσ⟩X. (13) While the spin-triplet excitonic condensation corresponds to the spontaneous hybridization between the LS and spin-triplet (S=1) HS states, the spin-singlet excitonic condensation in- dicates that between the LS and spin-singlet states, the lat- ter of which is the S=0 state composed of two electrons singly occupying in the aandborbitals. This implies that the Hund coupling prefers the spin-triplet excitonic condensation rather than the spin-singlet one in the the two-orbital Hub- bard model. On the other hand, it has been pointed out that the spin-singlet excitonic condensation is stabilized when the electron-lattice coupling is taken into account [8, 15]. There- fore, we mainly focus on the spin-triplet excitonic condensa- tion in the present calculations. The excitonic order parameters given in Eqs. (12) and (13) take complex values. Note that, in the subspace with the num- ber of electrons being two at each site, the spin moment in Eq. (12) is related to the excitonic order parameters as Sγ X∼Reϕγ′ t,X×Imϕγ′′ t,X, (14) where (γ,γ′,γ′′)=(x,y,z) and its cyclic permutations [33, 42]. As mentioned in the previous section, we perform four- sublattice calculations on the square lattice, where the sub- lattices are denoted as A,B,C, and D, as shown in Fig. 2(a). To make it easier to distinguish between distinct ordered states that can be composed from these sublattice moments, we in- troduce three order parameters as OΓ=|OA+OB+OC+OD|, (15) OM=|OA−O B−O C+OD|, (16) OX=|OA−O B+OC−O D|, (17) whereOXis an order parameter at sublattice X, such as Sγ η,X and Reϕγ t,X. Note thatOΓ,OM, andOXcorrespond to the ferro- type, antiferro-type and ( π,0)-type orders for O, respectively, as shown in Fig. 2(b). Moreover, for simplicity, we introduce the absolute value with respect to the spin component γas follows: [S]Γ,M,X=sX γ=x,y,z[Sγ]2 Γ,M,X. (18) We also define [Re ϕt]Γ,M,Xand [Imϕt]Γ,M,Xin a similar man- ner. IV . RESULT A. Phase diagram at zero magnetic field Before discussing magnetic-field e ffects on the two-orbital Hubbard model, we examine the properties of this model5 0.00.51.01.52.0(b)naMX(c)n(d)n 0.51.01.52.02.5J/ta0.00.20.40.60.81.0(e)S 0.51.01.52.02.5J/ta(f)Ret 0.51.01.52.02.5J/ta(g)Imt 5.05.56.06.57.0/ta0.51.01.52.02.5J/taLSHS(AFM)ECESS(4sub)tb/ta=0.1(a) FIG. 3. (a) Ground-state phase diagram on the plane of the Hund coupling Jand crystalline field ∆fortb/ta=0.1 and U=2U′=4J in the absence of magnetic fields. Schematic pictures of four states in the phase diagram are given in Figs. 4(a)–4(d). (b)–(g) Jdependence of the Γ, M and X configurations of (b) the number of electrons in the aorbital, (c) di fference between the number of electrons in the aand borbitals, (d) total number of electrons, (e) spin moment, and (f) real and (g) imaginary parts of the spin-triplet excitonic order parameters along the dotted line in (a) with ∆/ta=6. without magnetic fields. Figure 3(a) shows the phase dia- gram on the plane of the crystalline field and Hund coupling attb/ta=0.1 where we assume U/J=4 and U′/J=2. In the region with the strong crystalline field ∆≫J, we find the LS phase where na=0 and nb=2 are satisfied for all sites. This is understood from the Jdependence of the num- bers of electrons in aandborbitals, as shown in Figs 3(b)– 3(d). At ∆/ta=6, [na]Γvanishes while [ ∆n]Γ=[n]Γ=2 be- lowJ/ta≃1.2, which characterizes the LS phase [Fig. 4(a)]. On the other hand, the Hund coupling is strong enough, we find the HS phase with the antiferromagnetic (AFM) order [Fig. 4(b)], where na=1 and nb=1 are satisfied for all sites [14, 34, 36, 37, 43], which are confirmed from the re- sults in Figs 3(b)– 3(d). The AFM order in the HS phase is determined from the fact that [ S]Mtakes a nonzero value in this region [Fig 3(e)]. In between the LS and HS phases, we find two phases with excitonic condensation due to the competition between the HS(AFM)EC(FM)LSEC+ESS(4sub)ECLS/HS(4sub)(b)(c)(d)(e)(a)FIG. 4. Schematic pictures of the ordered states emerging in the absence of magnetic fields for (a) the LS phase, (b) HS phase with an antiferromagnetic order, (c) spin-triplet excitonic phase, (d) four- sublattice excitonic supersolid phase, and (e) spin-triplet excitonic phase with a ferromagnetic order. crystalline-field e ffect and Hund coupling interaction. One is the spin-triplet excitonic phase characterized by the nonzero Reϕt, where the LS and HS states are spontaneously hy- bridized as illustrated in Fig. 1(b). The Jdependence of this quantity at ∆/ta=6 is presented in Fig. 3(f). In the phase termed EC in Fig. 3(a), [Re ϕt]Mtakes a nonzero value, indi- cating an antiferro-type excitonic order, as shown in Fig. 4(c). The antiferro-type order is attributed to the indirect-gap band structure with tatb>0 [8, 10, 15, 20, 39, 42, 44]. When the direct gap with tatb<0 is assumed, a ferro-type exci- tonic order is expected as discussed in the previous studies [10, 14, 42, 45]. Note that Im ϕtis zero, as shown in Fig. 3(g), in the EC phase. This originates from the pair-hopping inter- action, which fixes the phase of ϕtto zero [8, 10, 14]. In addition to the EC phase, we find another excitonic phase with nonzero ϕt[see Fig. 3(f)] near the HS phase. This excitonic phase is distinguished from the EC phase by the presence of nonzero [ ∆n]M[see Fig. 3(c)] indicating the staggered-type spin-state order corresponding to a superlat- tice structure formed by excitons. Since this excitonic phase is characterized by condensation and superlattice formation of excitons emergent simultaneously, it is regarded as a su- persolid of excitons, termed ESS in Fig. 3(a). In addition to nonzero [ ∆n]Mand [Reϕt]M, we find the ( π,0)-type spin order characterized by the appearance of [ S]X, as shown in Fig. 3(e). The ( π,0)-type spin order comes from the anti- ferromagnetic correlation between HS states, which aligned alternatively [see Fig. 4(d)]. Because of the coexistence of the antiferro-type excitonic and ( π,0)-type spin orders, this phase is characterized as a four-sublattice order and denoted as ESS(4sub) in Fig. 3(a). From Eq. (14), the ( π,0)-type spin order accompanied by the antiferro-type excitonic order with nonzero [Re ϕt]Mresults in nonzero [Im ϕt]X, as shown in Fig. 3(g). Since the magnitude of the quantum hybridization between the LS and HS states for HS dominant sites is di ffer- ent from that for LS dominant sites, the uniform component of the excitonic-order parameter, [Re ϕt]Γ, becomes nonzero in the ESS phase. Furthermore, the superlattice formation of excitons results in a charge disproportionation, which is characterized by [ n]Mas shown in Fig. 3(d). Note that the phase transition between HS and ESS phases is of first order6 0.00.51.01.52.0(b)naMX(c)n(d)n 0.51.01.52.02.5J/ta0.00.20.40.60.81.0(e)S 0.51.01.52.02.5J/ta(f)Ret 0.51.01.52.02.5J/ta(g)Imt 5.05.56.06.57.0/ta0.51.01.52.02.5J/taLSHS(AFM)ECESS(4sub)EC(FM)tb/ta=0.4(a) FIG. 5. Similar plots to Fig. 3 for the case with tb/ta=0.4. The J dependence of the order parameters in (b)–(g) is calculated along the dotted line in (a) with ∆/ta=6.5. but those betwen the LS and EC phases and the EC and ESS phases appear to be of second order. The origin of the first- order transition is ascribed to the di fferent spin-order patterns in the ESS and HS phases. The emergence of a phase similar to ESS has been sug- gested by dynamical mean-field theory (DMFT) at finite tem- peratures in the two orbital Hubbard model where the spin exchange in the Hund coupling and pair-hopping terms are ne- glected for simplicity [40]. The previous study have revealed that the ESS appears in a very narrow region at the boundary between the EC and spin-state ordered phase, which is di ffer- ent from the present result; we do not find the spin-state or- dered phase without excitonic condensation. We believe that the di fference is due to the simplification of the previous study because the pair-hopping interaction fixing the phase of ϕtsta- bilizes excitonic condensation. Indeed, we have confirmed that neglecting the pair-hopping term stabilizes the spin-state ordered order without excitonic condensation. This result is consistent with the previous study using a variational cluster approximation, which clarified that the pair-hopping term sta- bilizes the spin-triplet excitonic condensation [8, 15]. Next, we examine the e ffect of the ratio between the band- widths of the two orbitals. Figure 5(a) shows the phase dia-gram at tb/ta=0.4 and other parameters are the same as that in Fig. 3. The region of the EC phase is larger than that at tb/ta=0.1 presented in Fig. 3(a). This tendency can be un- derstood from the fact that the itinerancy of excitons is propor- tional to the product of taandtb[10, 14]. Moreover, we find a ferromagnetic phase accompanied by excitonic condensa- tion between the EC and ESS phases, which is schematically depicted in Fig. 4(e). Figures 5(b)–5(g) present the Jdepen- dence of the order parameters along the dotted line in Fig. 5(a) at∆/ta=6.5. Around J/ta=2.2, there is a region where both [S]Γand [Reϕt]Mare nonzero simultaneously, which corre- sponds to the ferromagnetic EC phase [Figs. 5(e) and 5(f)]. In this phase, [Im ϕt]Mis also nonzero because of the relation given in Eq. (14) [Fig. 5(g)]. The previous DMFT studies for the two-orbital model without the pair-hopping interaction and spin exchange part of the Hund coupling interaction sug- gested the presence of a ferromagnetic instability in the exci- tonic phase but did not find an excitonic phase with a ferro- magnetic order under the half-filled condition [40, 46]. The difference from the previous study may be due to the artifact of the Hartree-Fock approximation in the present calculations, which could overestimate e ffects of the Hund coupling inter- action leading to ferromagnetism. It is di fficult to settle the presence of a ferromagnetic excitonic phase in the half-filled Hubbard model, and this issue remains a future work. B. Magnetic-field e ffect without orbital angular momentum Here, we present the results under magnetic fields. In this section, we omit the e ffects originating from the orbital an- gular momentum given in Eq. (5), which corresponds to ne- glecting the contribution of the orbital magnetic moment to the Zeeman term in Eq. (3) and the spin-orbit coupling in Eq. (6). This assumption results in the absence of the angle dependence of an applied magnetic field and parameterize the magnetic-field e ffect using its magnitude h. Figure 6(a) shows the phase diagram on the plane of the Hund coupling Jand field magnitude h, where tb/ta=0.1 and ∆/ta=6. In the ab- sence of the magnetic field, the LS, EC, ESS, and HS phases appear by changing J, as illustrated in Fig. 3. Note that the EC state possesses no local spin moments, but ESS and HS phases exhibit magnetic orders with no total magnetization. Applying the magnetic field to the LS phase brings about the phase transition into an EC phase [see Fig. 6(b)–(g)]. In the field-induced EC phase, a nonzero total spin moment emerges [Fig. 6(e)] in addition to [Re ϕt]M,0, as shown in Fig. 6(f). The schematic picture of this state is depicted in Fig. 7(c). As the result of the spin moment induced by the magnetic field, Im ϕtis also nonzero, and three vectors h(par- allel to the local spin moment), Re ⃗ϕtand Im⃗ϕtare orthogonal to each other at each site owing to Eq. (14). Further increase of the magnetic field results in the state with nonzero [ ∆n]M, cor- responding to the staggered alignment of the LS and HS states [see Fig. 6(c)]. The continuous phase transition occurs from the EC to ESS where both [Re ϕt]Mand [ ∆n]Mare nonzero. Unlike the four-sublattice ESS phase appearing in the absence of magnetic fields, this field-induced ESS phase is described7 0.00.51.01.52.0(b)naM(c)n(d)n 0.00.51.01.52.02.5h/ta0.00.20.40.60.81.0(e)S 0.00.51.01.52.02.5h/ta(f)Ret 0.00.51.01.52.02.5h/ta(g)Imt 1.01.21.41.61.8J/ta0.00.51.01.52.02.5h/taLSHSECESSLS/HS(a) FIG. 6. (a) Ground-state phase diagram on the plane of the Hund coupling Jand magnetic field hfortb/ta=0.1,∆/ta=6, and U= 2U′=4J. Schematic pictures of four states in the phase diagram are given in Fig. 7. (b)–(g) Magnetic-field dependence of the Γand M configurations of (b) the number of electrons in the aorbital, (c) difference between the number of electrons in the aandborbitals, (d) total number of electrons, (e) spin moment, and (f) real and (g) imaginary parts of the spin-triplet excitonic order parameters along the dotted line in (a) with J/ta=1.1. as a two-sublattice order where spin moments in HS sites are in the same direction, as shown in Fig. 7(e). With increasing the magnetic field, the excitonic order parameters vanish, and the phase characterized by nonzero [∆n]Mappears, indicated by the yellow region in Fig. 6. This phase is regarded as an LS-HS superlattice corresponding to a staggered order of excitons [Fig. 7(d)], which is denoted by LS/HS in Fig. 6(a). We find that the staggered-type charge disproportionation emerges in the LS /HS phase in addition to the ESS phase [Fig. 6(d)]. The detailed analysis clarifies that the number of electrons in LS sites is more than that in HS sites. This is understood from the kinetic-energy gain of spin- polarized electrons in the aorbital while the system remains insulating. Thus, an applied magnetic field causes charge dis- proportionations in the phases with an LS-HS order. The dis- proportionation in the field-induced ESS phase is also under- stood in a similar manner. LSHSECLS/HSESS+(a)(b)(c)(d)(e)FIG. 7. Schematic pictures of the ordered states emerging in the presence of magnetic fields for (a) the LS phase, (b) spin-polarized HS phase, (c) spin-triplet excitonic phase, (d) staggered-type LS-HS ordered phase, and (e) two-sublattice excitonic supersolid phase. 0.0000.0010.002 (a)Sxλ/ta=0 Γ M (d)Sxλ/ta=0.01 0.00.10.2 (b)Reφy t (e)Reφy t 1.0 1.1 1.2 J/ta0.0000.0050.010 (c)Imφz t 1.0 1.1 1.2 J/ta (f)Imφz t FIG. 8. (a)–(c) Jdependence of (a) Sx, (b) Reϕy t, and (c) Im ϕz tfor the ferro-type ( Γ) and antiferro-type (M) configurations without the spin-orbit coupling. (d)–(f) Similar plots for the system with λ/ta= 0.01. The other parameters are set to tb/ta=0.1,∆/ta=6, and U=2U′=4J. Further increase of the magnetic field results in reentry into the ESS and EC phases successively, and finally, the system becomes the fully spin-polarized HS phase, as depicted in Fig. 7(b). The magnetization curve is presented in Fig. 6(e). It changes continuously as a function of an applied magnetic field, and we find a plateau at half of the maximum value in the LS/HS phase. When the magnetization takes its maximum, a fully spin-polarized HS phase appears, as shown in Fig. 7(b).8 C. E ffect of spin-orbit coupling at zero magnetic field In this section, we examine the properties of systems with the spin-orbit coupling and consider the indirect-gap system with tatb>0, which is di fferent from previous calculations in Ref. [33] where tatb<0 is assumed. The systems with di- rect and indirect gaps are connected via transformation with cibσ→(−1)icibσin the absence of the spin-orbit coupling [42]. Nonzero spin-orbit coupling violates this connection, implying distinct phase diagrams for these two cases. It is also worth noting that ˆHSOCis expected to induce [Im ϕz t]Γfrom Eq. (6). A change in the sign of λaffects only the induced [Imϕz t]Γand does not alter the overall phase diagram. Figure 8 shows the Jdependence of relevant order parameters in the absence and presence of the spin-orbit coupling. In this ab- sence, the EC phase characterizing nonzero [Re ϕz t]Mdoes not possess any magnetic orders [see Fig. 8(a)–8(c)]. When the spin-orbit coupling is introduced, [Im ϕz t]Γis always nonzero even in the LS phases [see Fig. 8(e) and 8(f)]. We find that the EC phase is accompanied by an antiferromagnetic order with tiny spin moments, as shown in Fig. 8(d). The magnetic order along the Sxdirection arises from the coexistence of the two nonzero parameters [Re ϕy t]Mand [Imϕz t]Γ[see Eq. (14)]. Note that there is a rotational symmetry around the Szaxis, and therefore, the EC phase with [Re ϕx t]Mresults in the anti- ferromagnetic order along the Sydirection. On the other hand, if the EC phase appears with only the zcomponent of Re ϕt, the spin-orbit coupling does not induce local spin moments, which is understood from Eq. (14). Since the emergence of spin moments gives the energy gain of antiferromagnetic in- teractions between neighboring HS sites, the excitonic order parameter of the EC phase prefers lying on the Re ϕx t-Reϕy t plane and is associated with local magnetic moments on the Sx-Syplane. D. Magnetic-field e ffect with spin-orbit coupling As discussed in the previous section, the inplane spin anisotropy on the Sx-Syplane is present in the EC phase in the absence of magnetic fields. This suggests that magnetic fields applying along the hxorhydirections cause di fferent results from those in fields along the hzdirection. Here, we focus on the e ffects of the magnetic field along the hxdirec- tion. Figures 9(a) and 9(b) show the phase diagrams in the absence and presence of the spin-orbit coupling, respectively. The phase boundary between the LS and the EC phases re- mains largely una ffected by the introduction of the spin-orbit coupling. However, their magnetic properties are critically different from each other. Figure 10 shows the hxdependence of order parameters obtained along the dotted lines in Fig. 9. In the scenario with λ=0, the lower phase is composed only of the pure LS states as indicated by [ na]Γ=0 in Fig. 10(a). Conversely, a nonzero λleads to a hybridization between the LS and HS states, and [Im ϕz t]Γand [ na]Γbecome nonzero, as seen in Figs. 10(e) and 10(h). Due to the spin-state hybridiza- tion by the spin-orbit coupling, the applied magnetic field in- duces a spin polarization [Fig. 10(f)], which is distinctly dif- 0.000.050.100.150.20hx/ta(a)LSEC(nonmag)EC(FF)/ta=0 1.081.101.121.141.16J/ta0.000.050.100.150.20hx/ta(b) LSFFEC(AFM)EC(Ferri)/ta=0.01FIG. 9. Ground-state phase diagrams on the plane of the Hund coupling Jand magnetic field hxfor (a)λ/ta=0 and (b)λ/ta=0.01. The other parameters are set to tb/ta=0.1,∆/ta=6, and U=2U′= 4J. ferent from the case with λ=0 [Fig. 10(b)]. The spin polar- ization leads to ferro-type Re ϕy tas shown in Fig. 10(g) [see Eq. (14)]. Therefore, the low-field phase in Fig. 9(b) is re- garded as the forced ferromagnetic (FF) phase, and the mag- netic susceptibility is nonzero, unlike the LS phase without spin-orbit coupling. This result is consistent with the previous study where tatb<0 is assumed [33]. Next, we focus on the higher-field EC phase, which is de- fined by a nonzero [Re ϕy t]Mindependent of the existence of the spin-orbit coupling [Figs. 10(c) and 10(g)]. In the pres- ence of the spin-orbit coupling, the coexistence of [Im ϕz t]Γand [Reϕy t]Myields the antiferromagnetic order of the Sxcompo- nent despite the magnetic field applied along the hxdirection [see Figs. 10(f)–(h)]. Thus, the applied magnetic field induces a ferrimagnetic state aligned with the field direction, rather than spin canting. Note that the field-induced ferrimagnetic EC phase also possesses a staggered-type spin-state order in the presence of the spin-orbit coupling, as shown in Fig. 10(e). We comment on the results under magnetic fields along the hzdirection. Similar to the results with the field along the hx direction shown in Fig. 9(b), the phase boundary on the plane ofJandhzremains largely intact even in the presence of the spin-orbit coupling. Nevertheless, the spin alignments in the9 0.00000.00050.00100.00150.0020 (a)naλ/ta=0 Γ M (e)naλ/ta=0.01 0.00000.00010.00020.0003 (b)Sx (f)Sx 0.000.010.020.030.04 (c)Reφy t (g)Reφy t 0.00 0.05 0.10 hx/ta0.0000.0050.0100.0150.020 (d)Imφz t 0.00 0.05 0.10 hx/ta (h)Imφz t FIG. 10. (a)–(d) Magnetic-field dependence of (a) na, (b) Sx, (c) Reϕy t, and (d) Im ϕz tfor the ferro-type ( Γ) and antiferro-type (M) con- figurations without the spin-orbit coupling. The other parameters are set to tb/ta=0.1,∆/ta=6, and U=2U′=4J. (e)–(h) Similar plots forλ/ta=0.01. The calculations are made along the dotted lines in Fig. 9 with J/ta=1.11. two phases di ffer from those in magnetic fields along the hx direction. The in-plane anisotropy, which arises from the spin- orbit coupling, prevents the emergence of spin moments due to the magnetic field in the lower-field phase. On the other hand, in the higher-field EC phase, the magnetic field along thehzdirection induces the spin moment while this EC phase is associated with antiferromagnetic order on the Sx-Syplane. Thus, a spin canting is observed in the EC phase. E. E ffect of orbital magnetic moment Finally, we examine the e ffect of the orbital angular mo- mentum in the Zeeman term given in Eq. (3) in addition to the spin-orbit coupling. Since only the zcomponent is nonzero, we here consider the case with magnetic fields applied along thehzdirection. We have confirmed that the results obtained for the other field directions are identical to those in Fig. 9. The Zeeman e ffect on the orbital moment induces the spin- singlet excitonic order parameter given in Eq. (13) because ⟨ˆLz⟩∝ Imϕs. In other words, this e ffect mixes the LS state with the spin-singlet state, although the spin-orbit coupling 0.0000.0010.0020.003(b)SJ/ta=1.11z,x,M(f)SJ/ta=1.14 0.000.050.100.150.20(c)Rety,M(g)Ret 0.0000.0050.0100.0150.020(d)Imtz,x,M(h)Imt 0.00.10.2hz/ta0.00.10.20.3(e)Ims0.00.10.2hz/ta(i)Ims 1.081.101.121.141.16J/ta0.000.050.100.150.20hz/taLSLS+Singlet EC (AFM)EC (canted)(a)FIG. 11. (a) Ground-state phase diagram on the plane of the Hund coupling Jand magnetic field hzforλ/ta=0.01, where the con- tribution of the orbital angular momentum to the magnetic moment is taken into account. The other parameters are set to tb/ta=0.1, ∆/ta=6, and U=2U′=4J. (b)–(e) Magnetic-field dependence of (b) [Sz]Γand [ Sx]M, (c) [Reϕy t]M, (d) [Imϕz t]Γand [Imϕx t]M, and (e) [Imϕs]ΓforJ/ta=1.11 along the dotted line in (a). (f)–(i) Similar plots to (b)–(e) for J/ta=1.14 along the dashed-dotted line in (a). results in the hybridization between the LS state and the HS spin-triplet states. Thus, the orbital contribution to the Zee- man e ffect competes with the appearance of the spin-triplet excitonic condensation. Figure 11(a) presents the magnetic- field phase diagram when the spin-orbit coupling and the ef- fect of the orbital angular momentum are taken into account.10 In the absence of magnetic fields, the LS and EC phases with antiferromagnetic order appear depending on the Hund cou- pling, as in Fig. 9(b). While the magnetic field changes the LS phase to the excitonic phase in the system without the or- bital angular momentum, such a field-induced transition does not occur for the LS phase, as shown in Fig. 11(a). The spin moment is not induced in the LS phase, as shown in Fig. 11(b). Instead, the orbital magnetic moment proportional to Im ϕsap- pears as shown in Fig. 11(e). This is attributed to the fact that thedx2−y2anddxyorbitals possess a large orbital moment with lz=±2, suggesting that the doubly occupied lz=−2 state with Mz=4 is preferred by strong magnetic fields along hz direction rather than the S=1 spin triplet state with Mz=2. Next, we discuss magnetic-field e ffects on the EC phase. Figures 11(f)–11(i) show the hzdependence of the order pa- rameters. In the absence of the magnetic field, the EC phase is associated with the antiferromagnetic order along the Sx direction, as shown in Fig. 11(f). An applied magnetic field induces the spin moment along the Szdirection, and a canted spin structure is realized, similar to the case without the or- bital angular momentum. We also find that the EC order pa- rameter is suppressed with increasing hzand vanishes above a critical field, as shown in Fig. 11(g). Above the critical field, the spin moment also disappears, and the higher-field phase is continuously connected with the LS phase. Although the spin moment vanishes, Im ϕscorresponding to the orbital mo- ment monotonically increases with increasing the magnetic field [Fig. 11(i)]. This is ascribed to the dominant contribu- tion of the orbital moment to the magnetization. V . DISCUSSION Here, we compare the current results with the previous studies and discuss their relevance to real materials such as perovskite cobalt oxides. In the previous study, the impact of magnetic fields on the two-orbital Hubbard model has been examined using DMFT, where the pair hopping interaction and spin-exchange terms were neglected, as mentioned in Ref. [26]. This study has revealed that an excitonic phase is induced by applying a magnetic field to the LS phase, con- sistent with our findings where orbital magnetic moments are omitted. Meanwhile, the field-induced ESS phase was not ob- served in the earlier study. Given the detailed DMFT calcula- tions conducted without magnetic fields, which indicated the presence of the ESS phase as discussed in Sec. IV A, we ex- pect that this phase would emerge in a region of higher mag- netic fields. The e ffects of magnetic fields on the two-orbital Hubbard model in the strong correlation limit have also been exam- ined so far [25]. In this approach, a localized quantum model is introduced using a second-order perturbation theory, where the electron hopping between neighboring sites are treated as a perturbation term. This study suggests that applying mag- netic fields induces excitonic and LS-HS-ordered phases from the LS phase. Additionally, another excitonic phase associ- ated with the LS-HS order is found between the LS-HS and HS phases in a lower-field region. This phase is believedto correspond to the ESS shown in Fig. 6(a). Note that the strength of external magnetic fields inducing excitonic states in the strong correlation limit is significantly lower than that obtained by the Hartree-Fock approximation. This discrep- ancy is attributed to the overestimation of on-site interactions within this approximation. When electron correlations are more precisely considered beyond the Hartree-Fock theory, the critical field required to induce excitonic states is expected to be lower. Recent experimental studies on cobalt oxides have sug- gested the presence of two types of excitonic states under ultra-high magnetic fields, reaching up to 600 T [13]. More- over, theoretical analysis of the localized model within the strong correlation limit, which is based on the five-orbital Hubbard model for the 3 dorbitals, has discussed the possi- bility that these states are an excitonic condensation of the IS states and a supersolid. This is consistent with the find- ings of this study, which employs an approach based on weak correlations in the two-orbital Hubbard model. Hence, it is inferred that, even beyond the strong correlation regime, the five-orbital Hubbard model could support not only excitonic states but also supersolid states. Here, we compare the magnitude of the critical magnetic field observed experimentally with the predictions of our study. The transfer integral tarepresents the electron hopping between egorbitals via porbitals of an oxygen ion, typically estimated to be ta∼1 eV [26, 47, 48]. Based on this consid- eration, our results suggest that the critical magnetic field for the transition from the LS phase to the EC phase is ∼1000 T. On the other hand, field-induced phase transitions in LaCoO 3 have been experimentally observed at approximately 70 T at 5 K and 170 T at 78 K, significantly lower than the predictions of the current calculations. We attribute this discrepancy to the limitations of the Hartree-Fock approximation, which tends to overestimate the e ffects of the Coulomb interactions, as pre- viously discussed. Properly incorporating many-body e ffects is expected to reduce the required field intensity inducing the EC phase from the LS phase. Finally, we address how the e ffects of spin-orbit coupling and orbital magnetic moments manifest in real materials. Let us consider the case where the dx2−y2anddxyorbitals are hy- bridized to realize the EC phase. As indicated in Ref. [33], it is expected that lattice distortions accompany this EC phase because the z-axis becomes inequivalent to the other two axes. Indeed, in Pr 0.5Ca0.5CoO 3, lattice distortions accompanying a phase transition at Ts∼90 K have been observed [16, 17]. On the other hand, no magnetic order below Tshas been re- ported in this material. Our calculation results with spin-orbit coupling [Fig. 8(e)] indicate that the EC phase accompanies a magnetic ordering, which appears to be inconsistent with the experimental results. Nevertheless, the experimental results have also reported the presence of a small spin splitting below the transition temperature [18, 49], which may not contradict the presence of magnetic order with very small ordered mo- ments obtained in this study. Furthermore, experimental stud- ies have also been conducted on epitaxially grown LaCoO 3 thin films on substrates, which enables applying pseudo pres- sure in a specific direction to control the spin and orbital11 states [50–53]. In this context, the realization of the EC phase through hybridization between dx2−y2anddxyis expected due to the two-dimensionality of the system. The superposition of these two orbitals can create significant orbital angular mo- mentum in the out-of-plane direction, which could strongly influence the magnetism due to the orbital magnetic moment in the presence of a magnetic field. Consequently, reducing the dimensionality of cobalt oxides might make it feasible to experimentally observe the directional dependence of applied magnetic fields due to spin-orbit coupling and orbital mag- netic moments, as demonstrated in the present calculations. VI. SUMMARY In summary, we have examined the two-orbital Hubbard model with a crystalline field splitting under magnetic fields, employing the Hartree-Fock approximation. Our findings re- veal that when spin-orbit coupling and orbital magnetic mo- ments are omitted, applying magnetic fields to the low-spin state induces spin-triplet excitonic and excitonic supersolid states. The latter is characterized by the coexistence of spin-triplet excitonic and a staggered-type spin-state order. Intro- ducing spin-orbit coupling leads to an antiferromagnetic order with notably small ordered moments in the excitonic phase. Moreover, we have revealed that the Zeeman e ffect originat- ing from the orbital magnetic moments destabilizes the spin- triplet excitonic state and leads to the low-spin state, which suggests that the orbital Zeeman e ffect brings about a substan- tial change to the magnetic phase diagram. This phenomenon arises due to a substantial orbital moment generated by the dx2−y2anddxyorbitals, which are considered active in exci- tonic condensation in perovskite cobalt oxides. ACKNOWLEDGMENTS The authors thank M. Naka, A. Ono, and A. Hariki for fruitful discussions. Parts of the numerical calculations were performed in the supercomputing systems in ISSP, the Uni- versity of Tokyo. This work was supported by Grant-in- Aid for Scientific Research from JSPS, KAKENHI Grant No. JP23H04859, JP23H04865, and JP23H01129. [1] N. F. Mott, Philos. Mag. 6, 287 (1961). [2] D. Jérome, T. M. Rice, and W. Kohn, Phys. Rev. 158, 462 (1967). [3] B. I. Halperin and T. M. Rice, Rev. Mod. Phys. 40, 755 (1968). [4] Y . Wakisaka, T. Sudayama, K. Takubo, T. Mizokawa, M. Arita, H. Namatame, M. Taniguchi, N. Katayama, M. Nohara, and H. Takagi, Phys. Rev. Lett. 103, 026402 (2009). [5] K. Seki, Y . Wakisaka, T. Kaneko, T. Toriyama, T. Kon- ishi, T. Sudayama, N. L. Saini, M. Arita, H. Namatame, M. Taniguchi, N. Katayama, M. Nohara, H. Takagi, T. Mi- zokawa, and Y . Ohta, Phys. Rev. B 90, 155116 (2014). [6] F. Di Salvo, C. Chen, R. Fleming, J. Waszczak, R. Dunn, S. Sunshine, and J. A. Ibers, J. Less-Common Met. 116, 51 (1986). [7] T. Kaneko, T. Toriyama, T. Konishi, and Y . Ohta, Phys. Rev. B 87, 035121 (2013). [8] T. Kaneko, B. Zenker, H. Fehske, and Y . Ohta, Phys. Rev. B 92, 115106 (2015). [9] J. Kuneš and P. Augustinský, Phys. Rev. B 90, 235112 (2014). [10] J. Kuneš and P. Augustinský, Phys. Rev. B 89, 115134 (2014). [11] Y . Tokura, Y . Okimoto, S. Yamaguchi, H. Taniguchi, T. Kimura, and H. Takagi, Phys. Rev. B 58, R1699 (1998). [12] K. Asai, A. Yoneda, O. Yokokura, J. Tranquada, G. Shirane, and K. Kohn, J. Phys. Soc. Jpn. 67, 290 (1998). [13] A. Ikeda, Y . H. Matsuda, K. Sato, Y . Ishii, H. Sawabe, D. Naka- mura, S. Takeyama, and J. Nasu, Nat. Commun. 14, 1744. [14] J. Nasu, T. Watanabe, M. Naka, and S. Ishihara, Phys. Rev. B 93, 205136 (2016). [15] T. Kaneko and Y . Ohta, Phys. Rev. B 90, 245144 (2014). [16] S. Tsubouchi, T. Kyômen, M. Itoh, P. Ganguly, M. Oguni, Y . Shimojo, Y . Morii, and Y . Ishii, Phys. Rev. B 66, 052418 (2002). [17] T. Fujita, T. Miyashita, Y . Yasui, Y . Kobayashi, M. Sato, E. Nishibori, M. Sakata, Y . Shimojo, N. Igawa, Y . Ishii, K. Kakurai, T. Adachi, Y . Ohishi, and M. Takata, J. Phys. Soc.Jpn. 73, 1987 (2004). [18] J. Hejtmánek, Z. Jirák, O. Kaman, K. Knížek, E. Šantavá, K. Nitta, T. Naito, and H. Fujishiro, Eur. Phys. J. B 86, 305 (2013). [19] T. Kaneko and Y . Ohta, Phys. Rev. B 94, 125127 (2016). [20] T. Yamaguchi, K. Sugimoto, and Y . Ohta, J. Phys. Soc. Jpn. 86, 043701 (2017). [21] M. M. Altarawneh, G.-W. Chern, N. Harrison, C. D. Batista, A. Uchida, M. Jaime, D. G. Rickel, S. A. Crooker, C. H. Mielke, J. B. Betts, J. F. Mitchell, and M. J. R. Hoch, Phys. Rev. Lett. 109, 037201 (2012). [22] M. Rotter, Z.-S. Wang, A. T. Boothroyd, D. Prabhakaran, A. Tanaka, and M. Doerr, Sci. Rep. 4, 7003 (2014). [23] A. Ikeda, T. Nomura, Y . H. Matsuda, A. Matsuo, K. Kindo, and K. Sato, Phys. Rev. B 93, 220401 (2016). [24] A. Ikeda, Y . H. Matsuda, and K. Sato, Phys. Rev. Lett. 125, 177202 (2020). [25] T. Tatsuno, E. Mizoguchi, J. Nasu, M. Naka, and S. Ishihara, J. Phys. Soc. Jpn. 85, 083706 (2016). [26] A. Sotnikov and J. Kuneš, Sci. Rep. 6, 30510 (2016). [27] K. Kitagawa and H. Matsueda, J. Phys. Soc. Jpn. 91, 104705 (2022). [28] J. Kanamori, Prog. Theor. Phys. 17, 177 (1957). [29] J. Kanamori, Prog. Theor. Phys. 17, 197 (1957). [30] K. Tomiyasu, M. K. Crawford, D. T. Adroja, P. Manuel, A. Tominaga, S. Hara, H. Sato, T. Watanabe, S. I. Ikeda, J. W. Lynn, K. Iwasa, and K. Yamada, Phys. Rev. B 84, 054405 (2011). [31] M. W. Haverkort, Z. Hu, J. C. Cezar, T. Burnus, H. Hartmann, M. Reuther, C. Zobel, T. Lorenz, A. Tanaka, N. B. Brookes, H. H. Hsieh, H.-J. Lin, C. T. Chen, and L. H. Tjeng, Phys. Rev. Lett. 97, 176405 (2006). [32] K. Tomiyasu, J. Okamoto, H. Y . Huang, Z. Y . Chen, E. P. Sinaga, W. B. Wu, Y . Y . Chu, A. Singh, R.-P. Wang, F. M. F. de Groot, A. Chainani, S. Ishihara, C. T. Chen, and D. J. Huang,12 Phys. Rev. Lett. 119, 196402 (2017). [33] J. Nasu, M. Naka, and S. Ishihara, Phys. Rev. B 102, 045143 (2020). [34] P. Werner and A. J. Millis, Phys. Rev. Lett. 99, 126405 (2007). [35] R. Suzuki, T. Watanabe, and S. Ishihara, Phys. Rev. B 80, 054410 (2009). [36] Y . Kanamori, H. Matsueda, and S. Ishihara, Phys. Rev. Lett. 107, 167403 (2011). [37] Y . Kanamori, J. Ohara, and S. Ishihara, Phys. Rev. B 86, 045137 (2012). [38] J. Kuneš and V . K ˇrápek, Phys. Rev. Lett. 106, 256401 (2011). [39] T. Kaneko, K. Seki, and Y . Ohta, Phys. Rev. B 85, 165135 (2012). [40] A. Niyazi, D. Ge ffroy, and J. Kuneš, Phys. Rev. B 102, 085159 (2020). [41] J. Nasu and M. Naka, Phys. Rev. B 103, L121104 (2021). [42] J. Kuneš, J. Phys.: Condens. Matter 27, 333201 (2015). [43] S. Hoshino and P. Werner, Phys. Rev. B 93, 155161 (2016). [44] H. Li, J. Otsuki, M. Naka, and S. Ishihara, Phys. Rev. B 101, 125117 (2020). [45] S. Yamamoto, K. Sugimoto, and Y . Ohta, Phys. Rev. B 101, 174428 (2020).[46] D. Ge ffroy, J. Kaufmann, A. Hariki, P. Gunacker, A. Hausoel, and J. Kuneš, Phys. Rev. Lett. 122, 127601 (2019). [47] T. Mizokawa and A. Fujimori, Phys. Rev. B 54, 5368 (1996). [48] T. Saitoh, T. Mizokawa, A. Fujimori, M. Abbate, Y . Takeda, and M. Takano, Phys. Rev. B 55, 4257 (1997). [49] J. Hejtmánek, E. Šantavá, K. Knížek, M. Maryško, Z. Jirák, T. Naito, H. Sasaki, and H. Fujishiro, Phys. Rev. B 82, 165107 (2010). [50] J. Fujioka, Y . Yamasaki, H. Nakao, R. Kumai, Y . Murakami, M. Nakamura, M. Kawasaki, and Y . Tokura, Phys. Rev. Lett. 111, 027206 (2013). [51] J. Fujioka, Y . Yamasaki, A. Doi, H. Nakao, R. Kumai, Y . Mu- rakami, M. Nakamura, M. Kawasaki, T. Arima, and Y . Tokura, Phys. Rev. B 92, 195115 (2015). [52] Y . Yamasaki, J. Fujioka, H. Nakao, J. Okamoto, T. Sudayama, Y . Murakami, M. Nakamura, M. Kawasaki, T. Arima, and Y . Tokura, J. Phys. Soc. Jpn. 85, 023704 (2016). [53] Y . Yokoyama, Y . Yamasaki, M. Taguchi, Y . Hirata, K. Takubo, J. Miyawaki, Y . Harada, D. Asakura, J. Fujioka, M. Nakamura, H. Daimon, M. Kawasaki, Y . Tokura, and H. Wadati, Phys. Rev. Lett. 120, 206402 (2018).

1701.07269v1.Direct_Mapping_of_Spin_and_Orbital_Entangled_Wavefunction_under_Interband_Spin_Orbit_coupling_of_Rashba_Spin_Split_Surface_States.pdf

Direct Mapping of Spin and Orbital Entangled Wavefucntion under Interband Spin-Orbit coupling of Rashba Spin-Split Surface States Ryo Noguchi,1Kenta Kuroda,1,K. Yaji,1K. Kobayashi,2M. Sakano,1 A. Harasawa,1Takeshi Kondo,1F. Komori,1and S. Shin1 1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan 2Department of Physics, Ochanomizu University, Bunkyo-ku, Tokyo 112-8610, Japan (Dated: August 19, 2021) We use spin- and angle-resolved photoemission spectroscopy (SARPES) combined with polarization-variable laser and investigate the spin-orbit coupling eect under interband hybridiza- tion of Rashba spin-split states for the surface alloys Bi/Ag(111) and Bi/Cu(111). In addition to the conventional band mapping of photoemission for Rashba spin-splitting, the dierent orbital and spin parts of the surface wavefucntion are directly imaged into energy-momentum space. It is un- ambiguously revealed that the interband spin-orbit coupling modies the spin and orbital character of the Rashba surface states leading to the enriched spin-orbital entanglement and the pronounced momentum dependence of the spin-polarization. The hybridization thus strongly deviates the spin and orbital characters from the standard Rashba model. The complex spin texture under inter- band spin-orbit hybridyzation proposed by rst-principles calculation is experimentally unraveled by SARPES with a combination of p- ands-polarized light. PACS numbers: A realization of functional capabilities to generate spin- splitting of electronic states without any external mag- netic eld is a key subject in the research of spintron- ics [1]. A promising strategy exploits the in uence of spin-orbit (SO) interaction that can give rise to the lift- ing of spin degeneracy under broken space inversion sym- metry, the so-called Rashba eect [2]. In the conven- tional Rashba model, an eigenstate of the SO-induced spin-splitting is treated with an assumption of a pure spin state fully chiral spin-polarized which protects elec- trons from backscattering [2{5]. However, in real mate- rials, the assumption can be usually broken because the SO coupling mixes dierent states with dierent orbitals and orthogonal spinors in a quasiparticle eigenstate [6{ 8]. The SO entanglement can permit the spin- ip elec- tron backscattering [9] and moreover orbital mixing in the eigenstate can play a signicant role in an emergence of the large spin-splitting [10{14]. Therefore, it is essen- tially important to experimentally explore the SO cou- pling not only in the lifting spin degeneracy but also in the spin and orbital wavefunction as eigenstates. Beyond the conventional Rashba model, a well-ordered surface alloy BiAg 2grown on Ag(111) provides an ideal case to study the SO entanglement in Rashba surface states. In the surface alloy, an occupied spz-like band and a mostly unoccupied pxy-like band show signicant Rashba spin splitting [15, 16] and cross each other at the specickjj[17{19] as shown in Fig. 1 (b). In particular, density functional theory (DFT) calculations showed the strong SO entanglement [8, 9] and predicted the com- plex spin texture that is signicantly dierent from the conventional Rashba model; the spzband switches spin- polarization at the crossing through SO-induced inter- band hybridization [19] which is in contrast to the similarsystem BiCu 2/Cu(111) as shown in Fig. 1 (b) [14, 20, 21]. While the presence of the spin-polarized electronic bands has been demonstrated by spin and angle-resolved pho- toemission spectroscopy (SARPES) [21{23] and inverse- SARPES [8], the SO entanglement and particularly the spin texture due to the interband SO coupling is still un- der discussion, because of the lack of the orbital selectiv- ity in the previous experiments. Up to now, the complex spin texture of these surface states was only indirectly de- tected by quantum interference mapping through scan- ning tunneling spectroscopy [9, 24, 25]. Thus, no con- clusive understanding of the phenomenon apart from the conventional Rashba model has been achieved yet. In this Rapid Communication, we directly investi- gate the SO entanglement in the Rashba surface states of BiAg 2/Ag(111) surface alloy by using a combina- tion of polarization variable laser with SARPES (laser- SARPES) and compare to those of BiCu 2/Cu(111) as a simple case. In contrast to the previous experi- ments [8, 21{23], our laser-SARPES deconvolves the or- bital wavefucntion and the coupled spin, and the surface wavefunctions are directly imaged into momentum-space through orbital-selection rule. It is shown that the inter- band SO coupling modies the spin and orbital character of the Rashba surface states leading to the spin-orbital entanglement and the kjjdependence. The resulting spin texture thus shows a large deviation from the conven- tional Rashba model. The full spin information is ex- perimentally unraveled only by a combination of p- and s-polarized light in accordance with a view of the SO entanglement. The laser-SARPES measurement was performed at the Institute for Solid State Physics (ISSP), the Univesity of Tokyo, with a high- ux 6.994-eV laser and ScientaOmi-arXiv:1701.07269v1 [cond-mat.mtrl-sci] 25 Jan 20172 0.3 0.2 0.1 0 0.3 0.2 0.1 0 0.3 0.2 0.1 0-0.6-0.4-0.20.0 0.3 0.2 0.1 0 -0.1 -0.2-0.6-0.4-0.20 0.2 0.1 0 -0.1 -0.2 kx(Å-1)-0.9-0.6-0.30 E-EF(eV)(b) (c)Detection plane =Mirror plane Crystal surfacep-pol. s-pol.light x yzphotoelectrons kxkyΓΚSurface Brillouin zone(a) pxy spzkE 0.3 0.2 0.1-0.6-0.4-0.20 E-EF(eV)(d) (e) 0.3 0.2 0.1 0.3 0.2 0.1spz spzpxypxypxy pxypxypxyspzspz spzspzBiAg2 BiCu2BiAg2 BiCu2BiAg2 BiCu2spzpxy BiAg2/Ag(111)spzpxy Low High Low Odd Even HighBiCu2/Cu(111) kx(Å-1) kx(Å-1) kx(Å-1)kx(Å-1) LowHighEFpxy spzEFSpin-up Spin-down50 BiAg2BiCu2 E-EF(eV) FIG. 1: (Color online) (a) The experimental conguration for p- ors-polarization with an angle incidence of 50. A mir- ror plane of the surface coincides with the plane of incidence (x-zplane). Surface Brillonin zone of the surface alloys [un- derlying fcc(111) substrate of Ag(111) and Cu(111)] is shown by solid (dashed) line. (b) Schematic of the energy disper- sion of Rashba spin-split bands in (left) BiAg 2and (right) BiCu 2surface alloys. (c) ARPES intensity maps for BiAg 2 and BiCu 2surface alloys with p-polarization along high sym- metry -K line of the surface Brillouin zone. The dashed lines indicate the edge of the projected bulk bands. (d) and (e) The magnied ARPES intensity maps with (left) p- and (middle)s-polarization, and (right) the dierential intensity maps, which are obtained by IpIswhereIpandIsare the photoelectron intensity obtained by p- ands-polarized light without normalization, respectively. cron DA30L photoelectron analyzer [26]. The experi- mental conguration is shown in Fig. 1(a). The p- and s-polarizations ( pands, respectively) were used in the experiment. The photoelectrons were detected along - K line of the surface Brillouin zone. The spectrometer resolved the spin component along y, which is perpen- dicular to the mirror plane of the surface. The sam-ple temperature was kept at 15 K. The instrumen- tal energy (angular) resolutions of the setup is 2 meV (0.3) and 20 meV (0.7) for spin-integrated ARPES and SARPES, respectively. The BiAg 2and BiCu 2surface al- loys were obtained by the procedures presented in the literatures [14, 15]. Low-energy electron diraction mea- surements conrmed the (p 3p 3)R30reconstruction of the surface alloys. First-principles calculations were performed using the VASP code [27]. The projector augmented wave method [28] is used in the plane-wave calculation. The generalized gradient approximation by Perdew, Burke, and Ernzerhof [29] is used for the exchange-correlation potential. The spin-orbit interaction is included. The atom positions of BiAg 2are optimized. Those of BiCu 2 are taken from the experimental data of Ref. [30]. Let us start with showing a brief overview of observed electronic structure of BiAg 2and BiCu 2surface alloys in Fig. 1(c). The spz-derived bands and the higher-lying pxybands disperse downwards in energy with a large Rashba spin-splitting in the both materials. Compared with the surface bands in BiAg 2, the most part of the sur- face bands in BiCu 2is above the Fermi level ( EF). These results are in good agreement with previous works [14{ 17]. Considering the orbital selection rule in the dipole ex- citation [31], pandsenable us to draw dierent orbital symmetry. Since the surface states near EFare com- posed of Bi 6 sand 6porbitals [19], pselectively detects weight ofeven-parity orbital with respect to the mirror plane, mainly from s,pzandpxcomponents, while sis sensitive to the odd-parity orbital mainly derived from py. Figure 1(d) summarizes the linear polarization depen- dence in BiAg 2. For the result obtained by p[see the left panel], we observe the strong intensity for the spz and inner pxybands. The data particularly displays the band crossing of the outer spzand innerpxybands aroundkjj=0.15 A1[Fig. 1(d)], where the spectral inten- sity of the outer spzis strongly suppressed. Surprisingly, switching the light polarization ptos, the spectral in- tensity is dramatically changed [see the middle panel in Fig. 1(d)]. The dispersion of the outer spzis clearly ob- served even at lager kjj>0.15A1together with the outer pxyband. Consequently, the overall parabolic dispersion of the outer spzband is clearly seen, which was absent in previous experiments [14, 15, 22]. The right panel of Fig. 1(d) shows the dierential intensity map [see gure caption of Fig.1]. The red-blue color contrast re ects the contribution of the even- and odd-orbital components in the surface wavefunction. It is immediately found that the orbital character of spzband changes the orbital char- acter at the band crossing. In contrast, the result for BiCu 2is found to be simple [see Fig. 1(e)]: the spzandpxybands comprise mainly3 0.2 0.1 0 0.2 0.1 0-0.6-0.4-0.20 0.3 0.3p-polarization p-polarization s-polarization (e) (g) (h)(a) E-EF(eV) -0.6-0.4-0.20 E-EF(eV)p-polarization s-polarization p-polarization s-polarizationBiAg2BiCu2 (c) E-EF(eV)(b) E-EF(eV)(d) E-EF(eV)Spin-resolved intensity (arb. units) Spin-resolved intensity (arb. units) E-EF(eV)kx(Å-1) -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24kx(Å-1) 0.07 0.10 0.13 0.16 0.19 0.22 0.25 0.28 0.31 0.34 -0.6 -0.4 -0.2 0 -0.6 -0.4 -0.2 0 kx(Å-1) kx(Å-1) kx(Å-1) kx(Å-1)0.3 0.3 0.2 0.1 0.2 0.1Sy -101Sy -101(f) s-polarization Low HighLow High-0.6 -0.4 -0.2 0 -0.6 -0.4 -0.2 0 FIG. 2: (Color online) (a)-(d) SARPES spectra with spin quantum axis along yfor BiAg 2and BiCu 2by usingpands. Spin-up and spin-down spectra are plotted with red and blue lines. The peak positions in the spin-resolved spectra for spzand pxybands are indicated by open and closed triangles, respectively. (e)-(h) The corresponding spin-polarization and intensity maps with the two-dimensional color codes [32]. The dashed lines indicate the edge of the projected bulk bands. even- and odd-parity orbitals, respectively. These two bands in BiCu 2are separated in momentum-space away from the band crossing. Nevertheless, we see the reduc- tion of the spectral intensity of the outer spzband when it overlaps with the projected bulk-band, as observed also in the even-part of the outer spzband in BiAg 2[see the left panel in Fig. 1(d)]. This common feature suggests the interaction of the the outer spzband with the bulk spzprojection from the substrate [14, 16] modies the spectral weight of the orbital wavefunction particularly for the even-orbitals. Apparently, the signicant kjjdependence of the or- bital symmetry is unique for the outer spzof BiAg 2. This result indicates the presence of the interband SO hy- bridization that allows to mix the even- and odd-orbital components in the surface wavefunction of the outer spz band. Previously, it has been believed that the hybridiza- tion associates with gap opening [17{19, 25]. However, in our laser-SARPES experiment, there is no clear gap observed around the crossing point. To get further insight into the in uence of the inter- band SO coupling, we carried out laser-SARPES mea- surements collaborated with the orbital selection rule ofpands. Figures 2(a)-(d) show SARPES spectra ob- tained by using pandsfor BiAg 2and BiCu 2. The corresponding spin-polarization and intensity maps [32] are shown in Figs. 2(e)-(h). For p, the inner and outer spz-bands around  point in the both materials show negative and positive spin-polarization, respectively, dis- playing a conventional Rashba-type spin-texture [22, 23]. The observed spin polarization is found to be large up to nearly 80 and 60 % for BiAg 2and BiCu 2, respectively. Most remarkably, we nd that the sign of the spin polarization sensitively depends on the linear polariza- tion [33]. This can be seen in the SARPES spectra par- ticularly for kx=0.12 A1in BiAg 2. The spectral weight of the spin-up is considerably larger than the spin-down forpand achieves nearly +80 % spin-polarization. For s, the intensity relation turns to the opposite and the re- sulting spin-polarization is found to be 70 %. Since the linear polarization is sensitive to dierent orbital symme- try, our laser-SARPES umambiguously reveals that the spin direction strongly depends on the orbital character. Figures 3(a) and (b) represent the calculated spin tex- ture for the both materials, which is consistent with pre- vious theoretical results [14, 19, 20]. In BiAg 2, the hy-4 Spin-textureEven Odd Even -0.9-0.6-0.30 0.3 0.2 0.1 0 (d) Spin-texture(a) (c) (b) -0.8-0.400.40.8 E-EF(eV)E-EF(eV) -0.9-0.6-0.30 -0.8-0.400.40.8 0.4 0.2 0 kx(Å )-1kx(Å )-10.3 0.2 0.1 0 0.3 0.2 0.1 0 0.4 0.2 0 0.4 0.2 0 kx(Å )-1kx(Å )-1 OddBiAg2 BiCu2BiAg2 BiCu2 FIG. 3: (Color online) (Color online) (a) and (b) Calculated total spin texture for BiAg 2and BiCu 2surface alloys, respec- tively. (c) and (d) The coupled spin textures decomposed into (left)even- and (right) odd-orbital characters. The size of the circles is proportional to the total spin polarizations. The red and blue color indicate spin-up and spin-down quan- tized along y, respectively. bridization of the spzandpxybands is found in a gap opening at the band crossing where the spin polariza- tion changes its sign [Fig. 3(a)]. Due to the hybridiza- tion, one may not assign whether the two branches at the gapped point originate from either the spzor thepxy derived states. Nevertheless let us refer the spzandpxy bands, since our experimental result shows the hybridiza- tion avoids the gap opening. The complex spin textures are decomposed into the even and odd orbital contributions in Figs. 3(c) and (d). These results clearly show the SO entanglement in which the dierent orbital components are coupled with op- posite spin. The calculated SO entangled texture re- produces our experimental results for the spin mapping of the surface wavefunction (see Fig. 2). By general group-theoretical analysis for the mirror symmetry [6], the wavefucntion under SO coupling is generally repre- sented as: j i>=jeven;"(#)>+jodd;#(")>; (1) where the spinors j">,j#>are quantized along y, which is perpendicular to the mirror plane, and the index i is representation for the mirror symmetry. This explains not only that the even- andodd-parity orbitals couple with opposite spins but also that the SO entanglement is a general consequence of the SO coupling. Indeed the similar SO-coupled states have been recently con- rmed in surface states of topological insulators [34{39] and Rashba states in BiTeI [40, 41]. (a) (b) E-EF(eV) kx(Å-1) kx(Å-1)-0.6-0.4-0.20p-pol. +s-pol. p-pol. +s-pol. 0.3 0.2 0.1-0.4-0.20 0.3 0.2 0.1 0Sy -101 -0.6Low HighBiAg2BiCu2FIG. 4: (Color online) (a) and (b) Experimentally obtained orbital-integrated spin-texture of the Rashba surface states in BiAg 2and BiCu 2, which is in good agreement with that theoretically predicted as shown in Figures 3(a) and (b). The color arrangement indicates the total photoelecton intensity and spin-polarization [32]. In BiAg 2[Figs. 3(a) and (c)], the spin-polarization cou- pled to the even-orbital component predominates the spz state around the  point, and nally the opposite spin coupled to the odd-orbital component becomes dominant at higherkjj. This indicates that the weight of the even- and odd-orbital components in the surface wavefunction play a signicant role in the total spin texture through the SO entanglement [Fig. 3(a)]. Our experiment indeed demonstrates the signicant kjj-dependence of the orbital wavefunction [Fig. 1(d)], which shows a good agreement with DFT calculation [42]. Therefore, the hybridization through the interband SO coupling modies the orbital component and induces the SO entanglement in the spz, which considerably deviates the spin texture from the conventional Rashba model. Apparently, the mapping of the spin in our experi- ment [Fig. 2(e)-(h)] does not re ect the predicted spin texture [Figs. 3(a) and (b)]. This is because photomie- sion measurement by using linearly polarized light in our experimental set-up selects the specic orbital symme- try [8]. Indeed, the orbital-dependent spin texture in theory [Figs. 3(c) and (d)] shows good agreement with our laser-SARPES results for pands. We now show that the total spin information can be traced back only by using a combination of the spin map- ping withpandslasers. Owing to selective detection of the pure orbital-symmetry in our experimental set-up of Fig. 1(b), the orbital-dependence in the spin polar- ization is eliminated by integrations of spin-polarization maps (Ptotal) in Figs. 4(a) and (b) as follows: Ptotal=(I";p+I";s)(I#;p+I#;s) (I";p+I";s) + (I#;p+I#;s); (2) whereI";p(I";s) andI#;p(I#;s) indicates the spin-up and spin-down intensities obtained by p(s). The mapping ofPtotal clearly demonstrates the complex spin texture5 of thespzband under interband hybridization, which is obviously comparable to the theoretical predictions in Figs. 3(a) and (b). Since the SO entanglement is gen- erally expected in materials as long as the SO coupling plays a signicant role, this technique therefore demon- strates a general advantage to investigate the unconven- tional spin textures in strong SO-coupled states although the results could be infuenced by the cross-section for p andsand require the photoemission calculation to con- sider photon energy dependence [43, 44]. The question remains as to why a hybridization gap is absent in our experiment while the calculation predicts the gap. We attribute the absence of the gap to an inter- action of the conned surface states with the electronic state in the substrate. It was recently shown that the size of the hybridization gap sensitively depends on the thick- ness of Ag(111) quantum well and decreases with increas- ing substrate thickness [17]. Hence, one can expect the gap absence in the case of the bulk substrate with contin- uum electronic states. The similar hybridization recon- structed by the bulk interaction is reported in image po- tential resonances [45]. This fact indicates that the pres- ence/absence of the gap does no give a direct evidence of the interband SO-coupling but the pronounced recon- struction of the surface wavefunction directly displays. In particular, the knowledge of the interband SO cou- pling is critically important for emergence of Dirac and Weyl fermions in semiconductors and semimetals [46{51], related to non-trivial band topology. In conclusion, we have deconvolved the spin and or- bital wavefuntion of the Rashba spin-split surface states for the Bi-based surface alloys, and directly mapped these wavefucntions into momentum-space by combin- ing orbital-selective laser-SARPES and rst principles calculations. The interband SO hybridization strongly in uence the spin and orbital character in the surface wavefunction leading to the kjjdependence of the SO en- tanglement. The resulting spin texture thus considerably deviates from the conventional Rashba model. Although the measured spin texture by using porsdoes not give the full spin information in this case, the full spin-texture is experimentally unraveled by SARPES with a combi- nation of the both linear polarization. Our ndings can be widely applied for clarifying the complex spin infor- mation in the SO-entangled surface states. We gratefully acknowledge funding JSPS Grant-in- Aid for Scientic Research (B) through Projects No. 26287061 and for Young Scientists (B) through Projects No. 15K17675. kuroken224@issp.u-tokyo.ac.jp [1] A. Mabchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.Duine, Nature Mat. 20, 871 (2015).[2] Y. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984). [3] S. LaShell, B. A. McDougall and E. Jensen, Phys. Rev. Lett. 77, 3419 (1996). [4] G. Nicolay, F. Reinert, S. H ufner and P. Blaha, Phys. Rev. B 65, 033407 (2001). [5] M. Hoesch, M. Muntwiler, V. N. Petrov, M. Hengsberger, L. Patthey, M. Shi, M. Falub, T. Greber and J. Oster- walder, Phys. Rev. B 69, 241401 (2004). [6] J. Henk, A. Ernst, and P. Bruno, Phys. Rev. B 68, 165416 (2003). [7] K. Miyamoto, H. Wortelen, H. Mirhosseini, T. Okuda, A. Kimura, H. Iwasawa, K. Shimada, J. Henk, and M. Donath, Phys. Rev. B 93, 161403 (2016). [8] S. N. P. Wissing, A. B. Schmidt, H. Mirhosseini, J. Henk, C. R. Ast and M. Donath, Phys. Rev. Lett. 113, 116402 (2014). [9] S. Schirone, E. E. Krasovskii, G. Bihlmayer, R. Piquerel, P. Gambardella and A. Mugarza, Phys. Rev. Lett. 114, 166801 (2015). [10] M. Nagano, A. Kodama, T. Shishidou and T. Oguchi, J. Phys.: Condens. Matter 21, 064239 (2009). [11] H. Ishida, Phys. Rev. B 90, 235422 (2014). [12] S.-R. Park, C.-Y. Kim, J. Electron Spectrosc. Rel. Phen. 201, 6 (2015). [13] I. Gierz, B. Stadtm 'uller, J. Vuorinen, M. Lindroos, F. Meier, J. H. Dil, K. Kern, and C. R. Ast, Phys. Rev. B81, 245430 (2010). [14] H. Bentmann, F. Forster, G. Bihlmayer, E. V. Chulkov, L. Moreschini, M. Grioni and F. Reinert, Europhys. Lett. 87, 37003 (2009). [15] C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D. Pacile, P. Bruno, K. Kern, and M. Grioni, Phys. Rev. Lett. 98,186807 (2007). [16] K. He, T. Hirahara, T. Okuda, S. Hasegawa, A. Kakizaki, and I. Matsuda, Phys. Rev. Lett. 101, 107604 (2008). [17] H. Bentmann, S. Abdelouahed, M. Mulazzi, J. Henk, and F. Reinert, Phys. Rev. Lett. 108, 196801 (2012). [18] G. Bian, X. Wang, T. Miller and T. C. Chiang, Phys. Rev. B 88, 085427 (2013). [19] G. Bihlmayer, S. Blugel and E. V. Chulkov, Phys. Rev. B75, 195414 (2007). [20] H. Mirhosseini, J. Henk, A. Ernst, S. Ostanin, C.-T. Chi- ang, P. Yu, A. Winkelmann, and J. Kirschner, Phys. Rev. B79, 245428 (2009). [21] H. Bentmann, T. Kuzumaki, G. Bihlmayer, S. Blugel, E. V. Chulkov, F. Reinert and K. Sakamoto, Phys. Rev. B84, 115426 (2011). [22] F. Meier, H. Dil, J. Lobo-Checa, L. Patthey and J. Os- terwalder, Phys. Rev. B 77, 165431 (2008). [23] K. He, Y. Takeichi, M. Ogawa, T. Okuda, P. Moras, D. Topwal, A. Harasawa, T. Hirahara, C. Carbone, A. Kakizaki, and I. Matsuda, Phys. Rev. Lett. 104, 156805 (2010). [24] H. Hirayama, Y. Aoki, and C. Kato, Phys. Rev. Lett. 107, 027204 (2011). [25] L. E. El-Kareh, P. Sessi, T. Bathon, and M. Bode, Phys. Rev. Lett. 110, 176803 (2013). [26] K. Yaji, A. Harasawa, K. Kuroda, S. Toyohisa, M. Nakayama, Y. Ishida, A. Fukushima, S. Watanabe, C. Chen, F. Komori, S. Shin, Rev. Sci. Instrum. 87, 053111 (2016). [27] G. Kresse and J. Furthm 'uller, Phys. Rev. B 54, 111696 (1996). [28] P. E. Bl 'ochl, Phys. Rev. B 50, 17953 (1994). [29] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). [30] D. Kaminski, P. Poodt, E. Aret, N. Radenovic, and E. Vlieg, Surf. Sci. 575, 233 (2005). [31] A. Damascelli, Z. Hussain, Z. X. Shen, Rev. Mod. Phys. 75, 473 (2003). [32] C. Tusche, A. Krasyuk, J. Kirschner, Ultramicroscopy, 159, 520 (2015). [33] The inner spzband in BiCu 2does not show the orbital- dependence in Figs. 2(g) and (h), which is con ict with the theoretical result in Fig. 3(d). We expect that this may indicate an interaction with the bulk projection bands. However, it is strongly required further theoretical supports to fully explain this behavior. [34] H. Zhang, C.-X. Liu, and S.-C. Zhang, Phys. Rev. Lett. 111, 066801 (2013). [35] Z.-H. Zhu, C. N. Veenstra, G. Levy, A. Ubaldini, P. Syers, N. P. Butch, J. Paglione, M. W. Haverkort, I. S. Elmov, and A. Damascelli, Phys. Rev. Lett. 110, 216401 (2013). [36] Y. Cao, J. A. Waugh, X.-W. Zhang, J.-W. Luo, Q. Wang, T. J. Reber, S. K. Mo, Z. Xu, A. Yang, J. Schneeloch, G. Gu, M. Brahlek, N. Bansal, S. Oh, A. Zunger, and D. S. Dessau, Nat. Phys. 9, 499 (2013). [37] Z.-H. Zhu, C. N. Veenstra, S. Zhdanovich, M. P. Schnei- der, T. Okuda, K. Miyamoto, S.-Y. Zhu, H. Namatame, M. Taniguchi, M. W. Haverkort, I. S. Elmov, and A. Damascelli, Phys. Rev. Lett. 112, 076802 (2014). [38] Zhuojin Xie, Shaolong He, Chaoyu Chen, Ya Feng, Hemian Yi, Aiji Liang, Lin Zhao, Daixiang Mou, Jun- feng He, Yingying Peng, Xu Liu, Yan Liu, Guodong Liu, Xiaoli Dong, Li Yu, Jun Zhang, Shenjin Zhang, Zhimin Wang, Fengfeng Zhang, Feng Yang, Qin- jun Peng, Xiaoyang Wang, Chuangtian Chen, Zuyan Xu, and X. J. Zhou, Nature Comm. 5, 3382 (2014). [39] K. Kuroda, K. Yaji, M. Nakayama, A. Harasawa,Y. Ishida, S. Watanabe, C.-T. Chen, T. Kondo, F. Ko- mori, and S. Shin, Phys. Rev. B 94, 165162 (2016). [40] L. Bawden, J. M. Riley, C. H. Kim, R. Sankar, E. J. Monkman, D. E. Shai, H. I. Wei, E. B. Lochocki, J. W. Wells, W. Meevasana, T. K. Kim, M. Hoesch, Y. Ohtsubo, P. L. Fevre, C. J. Fennie, K. M. Shen, F. Chou, P. D. C. King, Sci. Adv. 1, e1500495 (2015). [41] H. Maa, H. Bentmann, C. Seibel, C. Tusche, S. V. Ere- meev, T. R. F. Peixoto, O. E. Tereshchenko, K. A. Kokh, E. V. Chulkov, J. Kirschner and F. Reinert, Nat. Com- mun. 7, 1162 (2016). [42] See Supplemental Material at [http://...] for more details. [43] Ulrich Heinzmann and J. Hugo Dil, J. Physics: Cond. Mat. 24, 173001 (2012). [44] E. E. Krasovskii, J. Phys. Cond. Matt. 27, 49 (2015). [45] M. Winter, E. V. Chulkov, and U. H ofer, Phys. Rev. Lett. 107, 236801 (2011). [46] Z. H. Hasan, C. L. Kane, Rev. Mod. Phys. 82, 82, 3045 (2010). [47] Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013). [48] X. Wan, A. M. Turner, A Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011). [49] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011) [50] S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, M. Z. Hasan, Science, 349, 6428 (2015). [51] G. Bian, T. R. Chang, R. Sankar, S. Y. Xu, H. Zheng, T. Neupert, C. K. Chiu S. M. Huang, G. Chang, I. Be- lopolski, D. S. Sanchez, M. Neupane, N. Alidoust, C. Liu, B. Wang, C. C. Lee, H. T. Jeng, C. Zhang, Z. Yuan, S. Jia, A. Bansil, F. Chou, H. Lin and M. Z. Hasan, Na- ture Commun. 7, 10566 (2016).

1704.06788v1.Observation_of_Spin_Nernst_effect_in_Platinum.pdf

Observation of spin Nernst effect in Platinum Arnab Bose1, Swapnil Bhuktare1, Hanuman Singh1, Venu Gopal Achanta 2 and Ashwin A. Tulapurkar *1 1 Dept. of Electrical Engineering, I ndian Institute of Technology - Bombay, Mumbai 400076, India 2 Department of Condensed Matter Physics and Material Sciences , Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India The spin direction of a nano -magnet can be efficiently manipulated by spin current injection. Several mechanisms are implemented to create spin current like electrical injection, spin Hall effect [1-6], spin Seebeck effect [ 7-8], spin pumping [ 9] and many more [ 10-11]. By virtue of spin Seebeck effect (SSE) pure spin current is generated in ferromagnet while temperature gradient is applied. In contrast heavy metals ha ving large spin orbit coupling can convert charge current i nto pure spin current via the spin Hall effect (SHE) . Spin current generated by these mechanism s can produce torque on nanomagnet which can be useful in memory and logic application s [12-15]. But the fundamental question: what happens if non magnet with large spin orbit coupling carries heat current, has not been studied experimentally. In this paper we report a new way of generation of spin current in heavy metal like Platinum (Pt) by applying temperature grad ient which can be interpreted as spin Nernst Effect (SNE) [16-20]. We have demonstrated that non -magnetic Pt can convert heat current into pure spin current by virtue of SNE , which can be injected to nearby magnetic contact to obtain measurable voltage. We have used Pt Hall bar structure with ferromagnetic Ni detectors which allow us to compare the r elative strength of SNE and SHE in the same sample. In ordinary Hall effect [OHE] [21] while electric current is passed perpendicular to the applied magnetic field , Hall voltage is generated normal to the direction of both electric current and magnetic field since electron is deflected by the Lorentz force. Likewise if heat current is passed instead of charge current, open circuit voltage is developed normal to both heat current and applied field direction. This is known as Nernst -Ettingshausen Effect [NE] [ 22]. Interestingly when heavy metal (HM) like Pt carries electric current , up and down spins separate in opposite direction orthogonal to the direction of current flow (Figure 1.a) even without application of any external magnetic field. This is known as spin Hall effect (SHE). SHE was theoretically predicted long back [1-2] and experimentally observed in past few years [ 3-5]. This effect arises due to the coupling between electrons spin angular momentum and its orbital mo tion, which originates from the relativistic Dirac equation . Heavy metals like Pt are good candidates for observation of SHE due to the large spin -orbit coupling. There are two possible mechanisms of SHE. It can arise from the internal band structure of a mate rial where scattering plays a minor role (intrinsic SHE) [23] or it can arise from spin dependent scattering of electrons with the impurities present in the material (extrinsic SHE) [24]. Now question is : if heavy metal like Pt is set between two temperature baths, can it generate pure spin current . If thermal gradient is created in a metal , in open circuit condition Seebeck voltage is generated across it . In this condition, internally electrons can flow maintain ing net charge current equal to zero. If we look at the energy resolved electron current, the electrons below the Fermi level (cold electrons) flow along the temperature gradient, while electrons above the Fermi level (hot electrons) flow in opposite direction. As shown in fig 1.b, the hot electrons are scattered sideways due to the spin orbit interaction (SOI) , resulting in spin current J spin1. Similarly cold electrons give rise to spin current J spin2, which is opposite to J spin1 as the cold electrons flow opposite to hot electrons (Fig 1.b ). If the SOI scattering rate is the same for hot and cold electrons, the net spin current wold be zero. However, if the scattering rates are different (J spin1Jspin2), a non -zero spin current can be created in heavy metals perpendicular to the flow of heat current (Fig 1 .c). This effect can be interpreted as spin Nernst effect or thermally driven spin Hall effect. It is to be noted that metals with large spin orbit coupling is not a sufficient condition to observe SNE , the scattering should have large energy dependence at Fermi level . Since last few years there were predictions [17- 18] of SNE but it was lacking proper experimental evidence. We have employed multi -terminal Ni/Pt junctions to compare the strength and relative sign of SHE and SNE of Pt. Our result is consistent with very recent report of SNE [19-20]. With the disco very of SNE two separate fields viz. spin- orbitronics [1-6] and spin-calortitron ics [25-30] can be merged together to form spin -orbito - caloritronics (Fig 1.d ). Figure1 | Conception of spin Hall effect (SHE) and spin Nernst effect (SNE) . a, Schematic diagram of SHE (charge current to spin current conversion in heavy metals (HM) ). The electrons flow along –X direction, spins oriented along +Y ( -Y) are accumulated at top (bottom) surface of HM. b, Microscopic origin of SNE . When temperature gradient is applied along HM , hot electrons (shown by red colour) and cold electrons (shown by deep blue colour) flow in opposite directions. The spin current produced by hot (Jspin1) and cold (Jspin2) electrons is in opposite direction. A non - zero net spin current (J spin1-Jspin2) would flow transverse to thermal gradient if the spin-orbit scattering rate is different for hot and cold electrons. c. Conceptual picture of SNE (heat current to spin current conversion) which is equivalent to figure b when scattering rate for hot and cold electrons are differnt . d, New emerging field of spin -orbito -caloritronics which considers interplay of electrons spin and orbital interaction in presence of temperature gradient. Experimental procedure is described in figure 2 . Figure 2.a shows the coloured Scanning Electron M icroscopic (SEM) image of fabricated device and figure 2.b, 2.c show the schematic diagram s of SNE and SHE experiment s respectively. A Hall cross -bar structure (blue cross in fig 2.a ) is prepared by electron beam lithography (EBL) and sputter deposition of 10 nm thick Pt. Then Ni lines (thickness 10 nm) are deposited by EBL, sputtering and lift off technique (green lines as shown in figure 2.a ). Before deposition of Ni, top surface of Pt is cleaned in-situ by Argon ions to make transparent contact. Final contacts are made by Ti/Au (shown as yellow colour in fig 2.a) . All depositions are d one at base vacuum better than 8 E-8 Torr. Separation of Pt line and Ni line is approximately 2 μm (Fig 2.a). Typical length and width of Hall bar is 4 0 μm and 6 μm. Figure 2 | Experimental description. a, Coloured SEM image of the fabricated device. Hall bar (blue colo ur) is made of Pt. Two Ni lines (green colo ur) are deposited on Pt. b, schematic representation of SNE experiment. Electric c urrent (ILR) flows through Pt line between two terminals ‘L’ and ‘R’ along X axis and Joule h eat is gener ated at centre of Hall bar and flows towards Ni line along Y axis . VSNE is measured between two different Ni contacts sweeping field along X axis ). c, schematic description of SHE experiment. Charge c urrent flows in Pt Hall bar such that two Pt lines below Ni contacts carry current in opposite direction . V SHE is measured betw een contacts of two different Ni terminals as a function of field (H x). For both the experiments of SHE and SNE spins are injected from Pt to Ni along Z axis (indicated as blue arrow pointing out of plane direction) . So if up spins (+X orientated) are accumulated in top Ni line then down spins (- X orientated) will be accumulated in bottom Ni line since direction of heat current (in SNE experiment) and electric current (in SHE experiment) is opposite. d, Summary of the experiment s: SNE, SNE+ ANE,ANE and SHE. For SNE experiment large dc electric current (I heater) is passed along heater line (between terminals ‘L’ and ‘R’ of Pt ). While Pt carries current, centre of the Hall bar is heated due to Joule heating and heat flows towards the Ni lines ( Fig 2.b). Hence temperature gra dient is created along Ni/Pt bi layer along Y direction. Pt converts heat current into spin current by SNE which is injected in to Ni. Since Ni is fabricated on top of Pt, spatial direction of spin flow from Pt to Ni is out of plane (+Z axis, shown with blue arrows in figure 2.b,c ) but heat flow direction is opposite (along Y) in two different Ni wires ( Fig 2.b ). Hence spins of opposite polarity (along the direction of X) are accumulated into Ni at non equilibrium condition. If magnetization of the Ni contacts point along same direction (which happens for magnetic field greater than coercivity ), we expect to see differential voltage step between the contacts of top and bottom Ni lines (VTR-VBR or V TL-VBL) as a function of magnetic field sweep along X axis as shown in figure 2.b . The signal (kinks) near zero field is due to the domains present in the Ni lines and hence we focus on the voltage signal for large magnetic fields. In this configuration we’ll get only contribution from SNE . All figures shown in figure 3 are obtained after averaging positive and negative heater current so that we get pure signal generated by heating effect and any sort of electrical voltage will be cancelled. We measured clear voltage st ep of 120 nV (approx.) between top and bottom Ni contact ( Fig 3.a ) while applied heater current (I heater -average) is 15 mA. Further we verified that this observed voltage step is proportional to square of average heater current and voltage step sign is independent of the polarity of applied current ( see Fig S3.2 in supplementary information for more details ). In this configuration -1 we can only observe contribution from SNE as mentioned in Fig 2.d. Differential measurement of voltage enables us to detect SNE voltage step with higher sensitivity since it reduces background volta ge significantly. For maximum heater current (I LR=15mA) estimated temperature gradient in Pt/Ni bilayer is appr oximately 8.5 K/μm (see Section S1 in supplementary information ). During the experiment resistance of Pt (RLR) heater line and Ni line (RLR (top) and R LR (bottom) , (see figure 2.a ) are monitored which provides on -chip temperature calibration. Further it is compared with COMSOL simu lation . There could be unintentional temperature gradient along out of plane direct ion (Z axis) which turns out to be quite small as compared to the in -plane temperature gradient (see S1 in supplementary material ). We have performed additional experiment s to observe the impact of Z direction temperature gradient on SNE. Out of plane temperature gradient can contribute to anomalo us Nernst voltage [29-30] according to the relation ˆ ( ) ( ) ( )ANEV y M x T z where M, T, V represent magnetization, temperature and voltage respectively. We can refer configuration -2 to compare voltage step due to combination of SNE and ANE. In configuration -2 voltage is measured between top (‘T’) and bottom (‘B’) contact (Fig 2.a) with external field sweep along X. In this configuration observed voltage step is approximately 150 nV which is quite comparable to voltage step observed in configuration -1 (ΔV SNE~120 nV) (see Fig 3.a,b ). In this way we can argue that ANE signal is quite small compared to SNE signal. Even in the same device geometry we can measure the pure signal of ANE by measuring voltage along same Ni line (VTL-VTR or V BL-VBR) as a function of external field sweep along Y. This is shown configuration -4 (see Fig 2 .d). In configuration -4 magnetic field is swept along Y which is orthogonal to the direction of spin accumulation ( X). Hence no SNE signal is generated but ANE voltage can be measured along same Ni line due to the relation: ˆ ( ) ( ) ( )ANEV x M y T z , where voltage drop is in Ni (along X), magnetization is swept along Y and thermal gradient is along Z. Even in this configuration ANE voltage is quite negligible (~20nV) compared to SNE signal ( ~120 nV) (see supplementary information S3.1 for more details) . Ni-Pt junction size is typically (5x5) μm2 (Fig1.a ). Experimental configuration 1 -3 involves measurement of differential voltage which is highly sensitive compared to configuration -4. Addition al experiments are done to rule out possibility of other spurious effects. In configuration -1 it is shown that SNE voltage step is observed when magnetic field is parallel to the direction of spin accumulation (X axis). In contrast when voltage is m easured between two different Ni terminals (VTL-VBL or VTR-VBR) sweeping magnetic field along Y we notice that voltage step disappears (Fig 3.d). It is due to spin polarization direction (along X) is orthogonal to the field sweep direction (Y). Finally when Pt is substituted by Al (low spin orbit coupling) measured SNE voltage is significantly reduced ( Fig 3 .e). More experimental results are shown with details of power variation and sample variation in supplementary information [ S3]. It indicates that we have consistent ly observed finite step in SNE voltage with same sign in all experiments which cannot be ascribed by other spurious effects like asymmetry in differential magnetic thermopower , non saturating domain activity difference in different Ni contacts and even magnetic proximity effect (see section S4 in supplementary information) . All of the experimental observation supports the fact that heat current can be converted into spin current by SNE in nonmagnetic Platinum. Figure 3|Experimental result of spin Nernst effect . a, VSNE measured between two different Ni lines for 15 mA average current . b, Measured ( VSNE+V ANE) volta ge as a function of external field sweep along X when voltage is measured between top (T) and bottom (B) contacts. c, Step of SNE voltage as function of average heater current. d, Measured voltage between different Ni line s as a function of field sweep along Y . e, measured V SNE voltage when Pt is replaced by Al. Now we measure SHE in the same device (configuration -3 in figure 2.d) . Electric current is passed between terminals L -T (ILT) and L -B (ILB) as shown in figure 2.c . In this arrangeme nt current flow direction in each of Pt/ Ni is opposite ( Y). While Pt conducts electric current along Y direction it injects spins with orientation X direction in to Ni by SHE. This is also similar to earlier situation where spins of different polarity are accumulated in Ni detector (Fig 2.b,c ). Only difference is that here spin current is generated in Pt by SHE whereas in previous case spin current was created by SNE. So we measure voltage between two different Ni lines (top and bottom Ni lines as shown in fig 2.c ) as a function of external field sweep along X direction. We observe clear step in voltage (ΔV SHE) for saturating magnetic field and step-sign reverses with changing the polarity of electric current flow direction (Fig 4 .a, b). In our control experiment we replace Pt by Al (low spin orbit coupling) which also does not show any step in measured voltage (Fig 4 .d). So it clearly indicates that Pt converts charge curren t into spin current and injects into Ni which floats to measurable voltage as a function of external field ( along X direction ). It is notable that additional peaks or dips are observed in measured volt age near the zero magnetic field in figure 2 and 3 . It can occur due to combination of plana r Nernst and thermal AMR (planar Hall and AMR ) effect in the experiment of SNE (SHE) ( Fig 2.b,c ). PNE (PHE) corresponds to generation of transverse voltage (along X ) while it carries heat (electric) current (along Y ) [29,30 ]. In ideal condition PNE (PHE) generated in each of the Ni bran ch should get cancel led but there is always some asymmetry in device fabrication and they do not cancel each other resulting kinks in near the field where domains of Ni rotate. According to the relation ()ˆˆ PNE PHEV M M T , PNE (PHE) voltage shows sin2θ dependence while magnetic field is rotated with respect to the current (heat or charge) flow direction. So PHE and PNE do not exhibit any step in measured voltage but they can manifest as kinks due to rotation of magnetic domain while field is swept [29,30 ]. Similar arguments apply to thermal AMR and electrical AMR (VAMR α cos2θ), and these effects also do not result in any step in the voltage [29, 30]. Hence steps observed while voltage is measured b etween different Ni contacts have to come from SNE or SHE. We have tested different sets of samples which reproduced similar behaviour of voltage step for SHE and SNE . The error bars shown in fig 3c include the sample variation (see section S3 .2 in supplementary information) . Unlike the previous study of SHE [ 3,4] where non -local detection method was used, we implement local detection method with multi -terminal device structure which further reduces the background (ANE/SNE) significantly. Hence this local detection method enables us to observe SNE directly and compare with SHE. Figure 4| Local detection of spin Hall effect (SHE) . a, b Voltage step (ΔV) sign of SHE changes with reversing the polarity of passed current (2.5 mA) in Pt/Ni while field is swept along X axis . c, when Pt is replaced by Al step becomes negligible. Now we compare the results of spin Hall effect and spin Nernst effect. We e xtract spin Hall angle (θSH) of Pt from the experiment shown in figure 4 . We o bserved approximately 1.9 μV step (ΔV SHE) when each of Ni/Pt br anch carries 2.5 mA current. We estimated how much spin current density is needed to obtain voltage step of 1.9μV. Considering electrical conductivities of Ni and Pt 2(ohm-μm)-1 and 6 (ohm-μm)-1, spin flip length of Ni 3 nm, spin polarization of Ni ( FM) 0.2 6 we obtain spin Hall angle of Pt (θ SH) 0.07 which is very close to other reported values [1-5]. In contrast to SHE when Pt carries thermal gradient of order 8.5 K/μm (see S1 in supplementary materials for temperature calibration ) we observe step (ΔV SNE) of 12 0 nV between two Ni lines from which we can estimate that Pt converts 8.5 K/μm temperature gradient in to pure spin current density of 1.43x108 A/m2. To get dimensionless spin Nernst angle (θ SN) we have to convert thermal gradient into equivalent charge current density [19]. So the equivalent charge current density driven by thermal gradient in short circuit condition can be represented by 'Q dTJQdY where σ, Q, dT dY represent the electrical conductivity, Seebeck coefficient and the rmal gradient along Pt. Assu ming Seebeck coefficient of Pt to be -6.25 μV/K at 340K [31] spin Nernst angle ( θSN=Js/JQ’) turns out to be close to -0.45. In this calculation we have negle cted the interfacial resistance effect [see supplementary information ]. Magnitude of Spin Nernst angle can slightly vary based on the thermal interfacial resistance and values of different parameters used. Magnitude of reported value of spin Nernst efficiency is more (-0.45) than spin Hall efficiency (0.07) but opposite in sign. It is to be noted that equivalent charge current density in short circuit condition may not be directly related to generation of spin current density from thermal gradient which depends on asymmetry of spin scattering around Fermi energy level (EF)as argued earlier (Fig 1). On this argument we may have some material with finite Seebeck coefficient and spin orbit coupling but exactly symmetric scattering rate around EF will not convert heater current to spin current by SNE. However this definition of dimensionless spin Nernst angle will be useful to compare various experimental results . In conclusion we have demonstrated that heavy metal , Pt can convert heat current into spin current by spin Nernst effect. We comparatively studied spin Hall effect and spin Nernst effect in Ni/Pt multi -terminal bi layer heterostructure which reveals that spin Hall angle and spin Nernst angle in Pt are opposite in polarity . Our method to detect spin current by measuring differ ential voltage is very sensitive. Generation of spin current by temperature gradient in heavy metals involves interplay of spin, orbit -coupling and thermal gradient which opens a new field of spin orbito -caloritronics. References [1] Dyakonov, M. I. & Perel, V. I . Current -Induced Spin Orientation of Electrons in Semiconductors . Phys. Lett. A 35, 459 (1971). [2] J. E. Hirsch. Phys. Rev. Lett. 83, 1834 (1999) [3] Valenzuela, S. O. and Tinkham, M. Direct electronic measurement of the spin Hall Effect. Nature . 442, 176 –179 (2006) . [4] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa. Room -Temperature Reversible Spin Hall Effect . Phys. Rev. Lett . 98, 156601 (2007). [5] Sinova, J., Valenzuela, S. O., Wunderlich, J., Back, C. H. & Jungwirth, T. Spin Hall effects . Rev. Mod. Phys . 87, 1213 (2015). [6] Nakayama, H., Althammer, M., Chen, Y. -T., Uchida, K., Kajiwara, Y., Kikuchi, D., Ohtani, T., Gepr¨ags, S., Opel, M., Takahashi, S., Gross, R., Bauer, G. E. W., Go ennenwein, S. T. B., and Saitoh. Spin Hall Magnetoresistance Induced by a Nonequilibr ium Proximity Effect . E. Phys. Rev. Lett. 110, 206601 (2013) . [7] Uchida, K., Takahashi, S., Harii, K., Ieda, J., Koshibae, W., Ando, K., Maekawa, S., and Saitoh, E. Observation of the spin Seebeck effect . Nature 455, 778 – 781 (2008). [8] Uchida, K., Adachi, H., Ota, T., Nakayama, H., Maekawa, S., and Saitoh, E. Observation of longitudinal spin -Seebeck effect in magnetic insulators . Appl. Phys. Lett. 97, 172505 (2010) . [9] Arne Brataas, Yaroslav Tserkovnyak , Gerrit E. W. Bauer, and Bertrand I. Halperin. Spin battery operated by ferromagnetic resonance . Phys. Rev. B 66, 060404R (2002) [10] Le Breton, J. -C., Sharma, S., Saito, H., Yuasa, S. & Jansen, R. Thermal spin current from a ferromagnet to silicon by See beck spin tunnelling . Nature 475 , 82 (2011) [11] Handbook of Spin Transport and Magnetism. Edited by Evgeny Y. Tsymbal, Igor Zutic. 2011, CRC Press. [12] Arnab Bose, Amit Kumar Shukla, Katsunori Konishi, Sourabh Jain, Nagarjuna Asam, Swapnil Bhuktare , Hanuman Singh, Duc Duong Lam, Yuya Fujii, Shinji Miwa, Yoshishige Suzuki, and Ashwin A. Tulapurkar. Appl. Phys. Lett . Observation of thermally driven field -like spin torque in magnetic tunnel junctions . 109, 032406 (2016). [13] A. Pushp, T. Phung, C. Ret tner, B. P. Hughes, S. H. Yang, and S. S. P.Parkin, Giant thermal spin-torque -assisted magnetic tunnel junction switching . Proc. Natl. Acad. Sci. U.S.A . 112, 6585 – 6590 (2015). [14] Luqiao Liu, Chi -Feng Pai, Y. Li, H. W. Tseng, D. C. Ralph and R. A. Buhrman . Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum . Science 335, 556. [15] Dinesh Kumar, K. Konishi, Nikhil Kumar, S. Miwa, A. Fukushima, K. Yakushiji, S. Yuasa, H. Kubota, C. V. Tomy, A. Prabhak ar, Y. Suzuki and A. Tulapurkar. Coherent mi crowave generation by spintronic feedback oscillator . Scientific Reports 6, 30747 (2016) [16] Cheng, S. -g., Xing, Y., Sun, Q. -f. & Xie, X. C. Spin Nernst effect and Nernst effect in two - dimensional electron systems . Phys. Rev. B 78 , 045302 (2008). [17] Wimmer, S., Koedderitzsch, D., Chadova, K. & Ebert, H. First -principles linear response description of the spin Nernst effect . Phys. Rev. B 88, 201108 (2013). [18] T.Seki, ,I.Sugai, Y.Hasegawa , S.Mitani ,K.Takanashi . Spin Hal leffect and Nerns teffect in FePt/Au multi -terminal devices with different Au thicknesses . Solid State Comm . 150, 496 (2010). [19] Meyer et al . arXiv:1607.02277 (2016) [20] Peng Sheng et al. arXiv:1607.06594 (2016) [21] Hall, E.American Journal of Mathematics2, 286 (1879). [22] Von Ettingshausen, A. and Nernst, W.Annalen der Physik und Chemie 265, 343 (1886). [23] Introduction to Spintronics . Supriyo Bandyopadhyay . March 20, 2008 by CRC Press . [24] Concepts in Spin Electronics . Edited by Sadamichi Maekawa . Oxford Science Publication (2007). [25] N. Liebing, S. Serrano -Guisan, K. Rott, G. Reiss,J. Langer, B. Ocker, and H. W. Schumacher. Tunneling Magnetother mopower in Magnetic Tunnel Junction Nanopillars . Phys. Rev. Lett . 107, 177201 (2011). [26] Flipse, J. , Bakker, F. L., Slachter, A., Dejene, F. K., and van Wees, B. J. Direct observation of the spin -dependent Peltier effect . Nature Nanotechnology 7, 166 –168 (2012) [27] Bauer, G. E. W., Saitoh, E. & van Wees, B. J. Spin caloritronics. Nat. Mater . 11, 391 (2012). [28] Flipse, J., Dejene, F. K., Wagenaar, D., Bauer, G. E. W., Ben Youssef, J. and van Wees, B. J. Observation of the Spin Peltier Effect for Magnetic Insulators . Phys. Rev. Lett. 113, 027601 (2014). [29] S. L. Yin, Q. Mao, Q. Y. Meng, D. Li, and H . W. Zhao. Hybrid anomalous and planar Nernst effect in permalloy thin films . Phys. Rev. B . 88, 064410 (2013) [30] M. Schmid, S. Srichandan, D. Meier, T. Kuschel, J. -M. Schmalhorst, M. Vogel, G. Reiss, C. Strunk, and C. H. Back. Transverse Spin Seebeck Effect versus Ano malous and Planar Nernst Effects in Permalloy Thin Films . Phys . Rev. Lett. 111, 187201 (2013). [31] Moore J. P. and Graves R. S. Absolute Seebeck coefficient of platinum from 80 to 340 K and the thermal and electrical conductivities of lead from 80 to 400 K . Jrnal. App. Phys. 44, 1174 (1973). AUTHOR INFORMATION Corresponding Author * Email: ashwin@ee.iitb.ac.in Author Contributions The device fabrication and measurements were carried out by AB. SB and H S helped AB in experiments. AB analysed the data and wrote manuscript with help from AT. VGA gave inputs and experimental support for local heating experiments. AT supervised the project. All authors contributed to this work and commented on this paper. Note: The authors declare no financial interests. ACKNOWLEDGMENT We would like to acknowledge the support of Centre of Excellence in Nanoelectronics (CEN) at IIT -Bombay Nanofabrication facility (IITBNF), Indian Institute of Technology Bombay, Mumbai, India. Supplementary Information S1. Temperature Calibration S2. Estimation of spin Hall angle and spin Nernst angle S3. More experimental data S3.1 Comparison of measured voltage from various terminals S3.2 Details of power variation with positive and negative heater current S4. Effect of current spreading near Ni/Pt bilayer S5. Proximity effect and other possible source of volt age step vs SNE signal S1. Temperature calibration : Figure S1 | Temperature calibration. a, Experimental device set up. b, c estimation of average temperature of Pt and Ni line from its resistance value as a function of temperature . d, 3D temperature profile of the device. e, estimated temperature profile along Y axis from COMSOL simulation. f, Estimated temperature gradient along Y axis. Blue stripes in dicate the position of Ni line. As shown in figure S1 .a two Ni lines are fabricated on Pt Hall bar. While current flows in Pt heater line between terminal ‘L’ and ‘R’ (along X direction) centre of Hall bar is heated up and heat flows towards Ni terminals (along Y). During the experiment resistance of Pt (between terminals ‘L’ and ‘R’) is monitored which shows linearly increasing with square of applied current (I LR) (right Y axis of Fig S1.b). Resistance of Pt (between ‘L’ and ‘R’) as a function of temperature is also measured (left Y axis of Fig S1.b ). From this we obtain the temperature of hot spot created by Joule heating. For 15 mA applied current (I LR) maximum temperature reaches close to 37 0 K (left axis of Fig S1.b) which is close to the predicted value from COMSOL simulation ( Fig S1.e ). Similarly resistance of each Ni line (between terminals {‘TL’ and ‘TR’} or {‘BL’ and ‘BR’}) is monitored while passing current in Pt heater line ( ILR, between terminals ‘L’ and ‘R’) (right Y axis of Fig S1.c ). Resistance of same Ni line is also measured as a function of temperature from (left Y axis of Fig S1.c). From this measurement we get average temperat ure of Ni line to be approximately 332 K which is close to the predicted temperature value by COMSOL simulation ( Fig S 1.e). From figure S1.e we see that approximately 40 K temperature difference along Ni/Pt bilayer of width 5 μm (along Y axis). We can assume that average in-plane temperature gradient (dT/dy) in Ni/Pt bilayer is roughly 8.5 K/μm (fig S1.f). Now we try to find the out of plane temperature gradient in Ni experimentally . For this we measure Anomalous Nernst (ANE) [ ˆ ANEV x M y T z ] voltage at same Ni terminal (either between {‘TL’ and ‘ TR’} or between {‘BL’ and ‘ BR’}) as shown in the figure S1.a sweeping external field along Y axis . Figure 3.e in the main manuscript shows the measured ANE voltage step (~20 nV) corresponding to 15 mA heater current (I LR)). From the previous reports [30] assuming ANE coefficient to be 2.6 μV/K we calculate the average out of plane temper ature gradient in Ni to be roughly 1 mK/μm which further exponentially decays in Ni along Y axis as we go away from hot spot. This value is also comparable from COMSOL prediction ( Fig S1.f ). Ni-Pt junction size is typically 5μm x 5μm. S2. Estimation of spin Hall angle and spin Nernst angle We estimated voltage considering Valvet -Fert equation [S1]: Let’s co nsider Valvet –Fert equation for electrons spins in magnet: 2 22 (1) sf zl where is the difference between electrochemical potential for up (+) spin and down spin ( -), sfl is the spin flip length of ferromagnet. Now we consider conservation of charge current in the system: 22 220chJ z z z where J ch is the charge current and σ + (σ-) is the conductivity of electron in up channel and down channel. We can rewrite it in following way: 2 20 (2)z Solving equation (1) and (2) we get following expressions: 23 1 2 3 23 *2 (3) (4)2 1sf sf sf sf sfzzll zzll zzll spin sfK e K e Je z K K e K e J J J J K e K eel * 21 , (5)(1 )sf Where, J+ (J-) is the up (down) spin current density, K 1, K 2, K 3 are constants, MAJORITY minority MAJORITY minority is spin polarization in ferromagnet. At Pt/Ni interface (Z=0) there is no charge current but Pt injects spin current into ferromagnetic Ni contact. So J ch (z=0) =0 but Jspin (z=0)0. From equation (4 ) we get : 1 3 3 ( ) ( ), assuming ( ) 0sf sfzzllK K e K e z From equation (3) and (5) : 32( )sfzlKe , * 33 *1( ) (0)sfzl spin spin sf sfJ J J K e K e l Jel . From this equation we can write the expression of voltage generated in Ni while absorbed spin current: * 3 ( 0) ( ) (0)spin sf z z K e l J ' For sp in Hall effect For spin Nernst effect spin SH ch Q SN SNJJ dTJQdY Thickness of Pt and Ni is 10nm. Conductivity of Pt and Ni= 6x106 and 2x106 (in SI unit) respectively. Assuming β=0.26 and l sf=3 nm we get the magnitude of step to be approximately 1.9 5 μV while 2.5 mA current is passed in each of Ni /Pt branch (Figure 2.c in main paper) which corresponds to spin Hall ang le of Pt 0.07. It implies that 2.4 x109 A/m2 spin current density is injected from Pt to Ni by SHE. This result is quite close to the pre vious experimental values [ 3-5]. Now we want to quantify spin Nernst angle comparing the data of SHE. As shown in SNE experiment ( Figure 3.a in the main paper) we o bserved approximately 12 0 nV step while in-plane temperature gradient is created in Pt (8.5K/μm) along Y axis. From the above expressions and material parameters we can ge t that an estimate that 1.43 x108A/m2 spin current has to be injected from Pt to Ni to obtai n 120 nV of voltage step. So 1.43 x108A/m2 spin current density is converted by Pt by SNE. From this we calculate that spin Nernst coefficient is approximately -0.45. It is to be noted that sign of spin Hall angle and spin Nernst angle is different. The exact value of θ SH and θ SN depends on the various material parameter s. However in these calculations we have neglected interfacial resistance effect . If interfacial heat resistance is different than the interfacial charge resistance we may obtain different values of spin Nernst angle . Figure S2 | Spin injection from Pt to Ni by SNE or SHE. S3. More experimental data S3.1 Comparison of measured voltage from various terminals Figure S3.1 | Comparative study of voltage for various configurations. a, VSNE by measuring voltage between two different Ni lines sweeping field along X. b, VANE by measuring voltage between two ends of same Ni terminals sweeping field along Y. c, Measured voltage between top and bottom contacts as a function of external field sweep along X. d, Conventional AMR measurement of Ni line. e, MR measurement of Pt heater line between contact ‘L’ and ’R’ Figure S3.1 describes complete analysis of experiment including SNE, ANE, AMR and MR measurements. Cross bar shown at centre of all the figures of S3.1. a-e consists of Pt and additional wires show Ni lines. Black (red) arrows indicate the direction of charge (heat) current flow in the system. Fig S3.1.a and S3.1.c are also described in the main article which demonstrate s the measurement of SNE and ANE respectively . For 15 mA heater current calculated temperature gra dient along Y direction is 8.5 K /μm. While voltage is measured between two different Ni lines ( ‘TR’ and ‘BR’ or ‘TL’ and ‘B L’ etc.) sweeping magnetic field along X , we obtain contribution of SNE in form of voltage step (120 nV in fig S3.1.a). Now if we measure voltage between two ends of same Ni terminals (‘TR’ and ‘TL’ or ‘BR’ and ‘BL’ in fig S 3.1.b) sweeping magnetic field along Y, we measure contribution from ANE in form of voltage step since ( ) ANE z YV x T M where Tz is unintentional temperature gradient along Z and M Y is magnetization of Ni along Y. We can clearly see that ANE voltage step is much smaller than SNE voltage step (Fig S3.1. a-b). Now if voltage is measured between top (‘T’) and bottom (‘B’) contact in pr esence of external field sweep along X, we expect to see combination of both ANE and SNE signal s (Fig S3.1.c ). SNE signal should be there since contacts are made on Ni. ANE signal should also be observed because Ni is magnetized along X , unintentional grad ient is along Z and finite length of Ni along Y (5μm on top and 5μm at bottom ). But magnitude of observed voltage step is almost same for configuration shown in figure S3.1.a and S3.1.c (120 nV to 140 nV) . It is because ANE voltage steps are ten times less er compared to SNE (Fig S3.1.a -b). In configuration shown in figure S3.1.a contact is made on Ni from sideways and hence ANE signal is short to zero (since length along Y becomes short due to Au contact s) which is not the case in t he configuration shown in S3.1.c . Additionally we measure resistance of Ni as a function of external field along X axis (Fig S3.1.d ) which shows typical behaviour of AMR signal . Figure S3.1.e shows MR of Pt heater line when 0.1 mA is passed. It clearly shows negligible MR within our measurement sensitivity (0.01%). It is also possible that some minor fraction of heater current may spread through Ni line (see S4 section) but it has negligible effect since MR of Pt line as the current spread is very less. If significant amount of heater current flows through Ni due to current spreading we would observe additional AMR signal in figure S3.1.e which w ould be quite similar to Figure S3.1.d. But we do not observe any such signature (see S4) . Hence we can neglect the effect of current spreadin g around Ni in context of SNE voltage step. S3.2 Details of power variation with positive and negative heater current Figure S3.2 | Detailed analysis of power dependence and heater current polarity alteration. a-d show data for 15 mA heater current. e-h show data for 10 mA heater current . In this section we shall discuss details of power variation, sample variation and effect of po sitive and negative heater current (i.e. I LR and I RL). We shall focus on the configuration wh ere voltage is measured between two different Ni terminals (for example ‘TR’ & ‘BR’) while field is swept along X . This is typical configuration to measure SNE voltage. Figure S3.2.a,b show the data for +15 mA (I LR) and -15 mA (I RL) heater current respectively . In both these cases we see signal consists with kinks around centre and a step for higher applied field. Impo rtantly voltage step does not change sig n on reversal of heater current but background voltage shifts by some DC value s. It clearly indicates that the observed step is originated by heating effect which can be justified by spin Nernst Effect (SNE). Figure S3.2.c is average of positi ve and negative heater current which gives us sole information of heating effect or SNE . Figure S3.2.a-c look almost similar since heating effect mostly dominates electrical signal. However we cannot rule out the effect from smal l electrical current leakage at Ni line since background voltage is slightly different for different polarity of applied heater current. On subtraction of S3.2.a and S3.2.b we get contribution of small leakage of electric current. It only contributes to peaks at centre but not to step which is consistent to AMR and PHE (see SI-3.1). Importantly we observe that figure S3.2.c also consists of combination of peaks -dips and step. The step in measured voltage is attributed to SNE whereas kinks are due to thermal AMR and PNE (Planner Nernst effect) . Figure S3.2 .e-h also show the same for 10 mA heater current. Since we measure differential voltage between two different Ni lines the background voltage and kinks at centre are taking random values below 100 μV depending upon the asymmetry of device structure. For example in device -2 background voltage for +10 mA applied current is negative ( -45.6μV) but for -10 mA current it is +5.3 μV . But convincingly we measure the step in voltage which i s deterministic and it does not depend on current polarity. So it is clear indication of SNE voltage step. Current (mA) V(back ground in μV) V_SNE (nV) devices +15 -60 -83.6 -90.6 150 115 125 D1 D2 D3 -15 -29.5 -10.2 -40.3 130 120 110 D1 D2 D3 +10 -33.2 -45.6 -45.2 60 52 62 D1 D2 D3 -10 -10.3 +5.3 -15.1 70 55 58 D1 D2 D3 S4. Effect of current spreading near Ni/Pt bilayer Figure S2| Effect of current spreading around Ni contacts In Pt heater line maximum applied current is 15 mA and corresponding current density is 3x1011 A/m2. When it passes near the centre of Pt cross bar it can spread towards Ni contact as shown in si mulated result of Fig S4. From figure S4 we can estimate that a t least 100 times less current flows through Ni line (1x109 A/m2). Now we have to address whether this minor leakage of current adds any signal to observed SNE voltage . It can contribute in two different ways: (i) It can add additional electrical signal in measured voltage due to asymmetry of the sample . (ii) It can cause local heating around Ni to create out of plane gradient. First of all data shown in figure 3 .a, S3.1.a -c and S3.2.c,g are average of positive and negative heater current. So our measured signal is free from any electrical spurious contribution. Secondly , current density under Ni is 100 times smaller compared to actual heater current density . So heating effect will be 104 times lesser which can be negligible. If some unintentional temperature gradient is created in Ni along Z axis that will add to ANE signal which turns out to be very small ( Fig S3.1.a and Fig S3.1.b ) compared to SNE. Simulated values of current spreading around Ni contacts are consistent with ex periments shown in Fig S3.1.e . If significant amount of current leaks through Ni then we could observe magnetoresistance in Pt heater line between terminals ‘L’ and ‘R’ itself due to AMR effect of Ni (Fig S3.1.e ). But in our experim ent magnetoresistance in Pt heater line is not observed ( Fig S3.1.e ). Additionally in S3.2 section we have individually shown the results while positive and negative heater current is passed ( Fig S3.2.a -b.e-f). These results indicate that voltage step is invariant of polarity of heater current which can be only explained by SNE. However there can be additional DC background of measured voltage due to current spreading around Ni and asymmetry of device structure . We can figure out the direct effect of minor leakage of current by su btracting the data of positive heater current and negative heater current. It is shown in figure S3.2.d and S3.2.h . It clearly shows that additional background appears along with small kinks in the signal due to current leakage but we never got any evidenc e of voltage step due to this (Fig S3.2.d and S3.2.h ). Current spreading effect is less since Ni contacts are quite away from centre of the hot spot (5μm). S5. Proximity effect and other possible source of voltage step vs SNE signal There is a possibility that few monolayers of Pt can become magnetized when it comes in contact of ferromagnet. It is known as magnetic proximity effect [s2]. It is very important issue for YIG/Pt bilayer since YIG is insulator and Pt is conductive. Origin of such observe d magnetoresistance in YIG/Pt can be due to magnetized Pt [s2] or it can be simply coexistence of SHE and ISHE [s3-5]. Different groups have ruled out the possibly of proximity effect by rotating magnetic field in three different directions and putt ing Cu in between YIG and Pt [s3-7]. Our device consists of Pt(10nm)/Ni(10nm) in which both layers are conductive and magnetic property of the system is completely dominated by the magnetization of Ni. Apart from t hat 10 nm thicker Pt can never b e completely magnetized by proximity effect. So we always have heterostructure of heavy metal (HM) and ferromagnet (FM) in which HM will inject spin current and FM will detect. So our justification of SNE behind the observed voltage step in figure 3.a is quite valid despite magnetic proximity effect. Due to this reason magnetic proximity effect is neglected for FM/HM bilayer in previous studies [s8-11]. It is already established that t hermopower of magnetic material saturate for higher values of magnetic field since it obeys either sin2θ or cos2θ dependence depending on relative direction of current (heat) and magnetization []. Hence step observed in our experiment between higher values of field (+300Oe and -300 Oe) it has to come from SNE. Simultaneously we observe negligible step when Pt is replaced by Al or field swept norm al to accumulated spins (Fig 3.d in main article). References [s1] Theory of the perpendicular magnetoresistance in magnetic multilayers, T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993) [s2] Transport Magnetic Proximity Effects in Platinum, S. Y. Huang et al. Phys. Rev. Lett. 109, 107204 (2012). [s3] Spin Hall Magne toresistance Induced by a Nonequilibrium Proximity Effect. H. Nakayama et al. Phy. Rev. Lett. 110, 206601 (2013). [s4] Quantitative study of the spin Hall magnetoresistance in ferromagnetic insulator/normal metal hybrids , Phys. Rev. B 87 , 224401 (2013) . [s5] Unidirectional spin Hall magnetoresistance in ferromagnet/normal metal bilayers , Can Onur Avci et al. Nat Phys. 11, 570 (2015) . [s6] Longitudinal Spin Seebeck Effect Free from the Proximity Nernst Effect , Phys. Rev. Lett. 110, 067207 (2013) . [s7] Separation of longitudinal spin Seebeck effect from anomalous Nernst effect: Determination of origin of transverse thermoelectric voltage in metal/insulator junctions , T. Kikkawa et al. Phys. Rev . B 88 , 214403 (2013) . [s8] Room -Temperature Reversible Spin Hall Effect , T. Kimura et al. Phys. Rev . Lett. 98, 156601 (2007) . [s9] Spin-Torque Ferromagnetic Resonance Induced by the Spin Hall Effect , Luqiao Liu et al. Phys. Rev . Lett. 106, 036601 (2011) . [s10] Perpendicula r switching of a single ferromagnet ic layer induced by in -plane current injection , I. M. Miron et al. Nature 476, 189 (2011) . [s11] Symmetry and mag nitude of spin –orbit torques in ferromagnetic heterostructures , Kevin Garello et al. Nature Nanotech 8, 587 (2013). [s12] Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum , L. Liu et al. Science 336, 555 (2012)

1104.1684v1.Spin_Distribution_in_Diffraction_Pattern_of_Two_dimensional_Electron_Gas_with_Spin_orbit_Coupling.pdf

Spin Distribution in Diraction Pattern of Two-dimensional Electron Gas with Spin-orbit Coupling Cheng-Ju Lin and Chyh-Hong Chern Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan (Dated: May 27, 2022) Spin distribution in the diraction pattern of two-dimensional electron gas by a split gate and a quantum point contact is computed in the presence of the spin-orbit coupling. After diracted, the component of spin perpendicular to the two-dimensional plane can be generated up to 0.42  h. The non-trivial spin distribution is the consequence of a pure spin current in the transverse direction generated by the diraction. The direction of the spin current can be controlled by tuning the chemical potential. I. INTRODUCTION Separating spin up and spin down electrons in materi- als is a basic but very important procedure in spintron- ics. It can be achieved by the generation of a spin current so that electrons of dierent spins propagate in opposite directions. There have been many proposals of gener- ating spin current. They can be roughly classied into two categories. One is by means of magnetic methods1,2 and the other is by pure electrical sources3{6. The rst one usually acquires loops of electric current to generate magnetic eld to switch electron spins. Inevitably, heat always comes as a by-product that is notoriously unfa- vored, especially when many loops are squeezed in a small volume. Therefore, the pure electrical methods bring the ultimate hope for the sake of practical applications. All of the electrical methods to generate spin current share the same mechanism, namely the spin-orbital in- teraction (SOI). In the presence of the SOI, up spin and down spin are no longer degenerate at each point in the momentum space. The spin orientation and mo- mentum lock with each other in dierent ways for dif- ferent SOI systems. Due to the time-reversal symmetry, an electron at a particular momentum can always nd its Kramer's partner at the momentum of the opposite direction. When the SOI is very strong, some systems form topological insulators, where there is a gap in the bulk bands and there are odd numbers of the Kramer's pairs of the edge modes7{9. Although SOI plays an important role in the electrical means to control electron spin, it is the major contribu- tion to the spin relaxation at the same time10,11. The stronger SOI is, the shorter spin relaxation time will be. The reason is the following. In understanding the spin- momentum locking, one can eectively associate a cti- tious magnetic eld with each point in the momentum space so that the spin orientation prefers to be parallel to the ctitious magnetic eld. Suppose that an electron moves in a certain momentum, its spin is not necessary in the spin eigenstate of that momentum. Consequently, the electron spin precesses and the information of elec- tron spin is lost in the diusive transport. Unfortunately, the pros and cons of the SOI are always accompanyingwith one another. Therefore, one of the most important questions in spintronics for real applications is to com- promise the advantages and the disadvantages. In this regard, a robust eect of spin splitting is needed for the spin relaxation time to be long enough for a practical purpose. In this paper, we propose a new eect of spin-splitting from the coherent transport by the electron diraction. Electron diraction by the quantum point contact in the two-dimensional electron gas (2DEG) has been observed by the scanning probe microscopy technique. We will further point out that there is a non-trivial spin distri- bution in the diraction pattern if SOI is present. It can be measured if the spin-resolved experimental technique is performed, for example, Kerr rotation spectroscopy. The physical picture can be summarized as the fol- lowing. Because of the wave nature, electrons pick up a transverse momentum when scattered by the quantum point contact, which plays a role as a selector of propaga- tion direction. Because of the spin-momentum locking, electron spin passing through the quantum point con- tact is determined due to the selection of propagation direction. After diracted, the initial electron spin will certainly not be in the spin eigenstate of the diracted momenta. Therefore, the electron spin precesses. Elec- trons in dierent propagation directions precess in dier- ent ways resulting in a spin distribution in the diraction pattern. Dierent from the spin precession in the spin- relaxation mechanism, the precession in our case is co- herent due to the coherent transport of the diraction as long as there is no mixing between spin states in dierent bands at the same momentum. This paper serves to provide many calculation details in our previous Letter12. The structure of the paper is given by the following. The model of the theoretical pro- posal and the formalism will be given in the section II. In particular, we illustrate our method using the Rashba system. In section III A, we show the numerical results of the diraction by a single slit. In section III B, we improve the eciency of the spin splitting by consider- ing the diraction by a grating. In section IV, physical origin of the new spin-splitting eect is provided The rel- evancy for being realized in experiments is discussed. InarXiv:1104.1684v1 [cond-mat.mes-hall] 9 Apr 20112 the appendix, the results for the Dresselhaus system are given. Some formula of the special function used in the computation is included in the last appendix. II. THE MODEL The system we are considering is a 2DES with SOI diracted by a single slit. In experiments, a single slit may be realized by a quantum point contact. The prob- lem we want to solve is the spin distribution in the dirac- tion pattern. The eective Hamiltonian of a 2DEG with SOI is given by the following H=p2 2m+( ^xpy^ypx) +( ^xpx^ypy);(1) wheremis the eective mass of the electron, is the chemical potential, and ^ kare the Pauli spin matri- ces, andandare the SOI strength corresponding to the Rashba and the Dresselhaus couplings, respec- tively. Eq. (1) is a 2 2 Hamiltonian. The energy bands can be obtained easily by E p=p2 2m
p, where
p=p (2+2)p2+ 4pxpy. Apparently, SOI cre- ates a band crossing. E+()denotes the upper (lower) band. The eigenstates are v= (ei;1)T=p 2, where = tan1(px+py px+py). For the upper band, += tan1(px+py px+py), and for the lower band =++. contains the information of the spin orientation. The orientation of the spin eigenstate is locking with the di- rection of momentum. Therefore, in the system with SOI, there is no longer spin degeneracy. Moreover, the spin orientations of the eigenstate vlie in thexyplane. Without perturbations, the electron spin in the 2DEG has onlyxycomponent. Spin orientations are opposite for the upper band and lower band at each momentum. The situation we consider here is similar to the case of the single slit diraction in optics. A slit of width dis located aty0= 0. We compute the diraction pattern of the electron wave after propagating to the screen at distanceLaway in the x-direction. The diraction pat- tern is the superposition of the quantum waves from the slit, according to the Huygens principle. The wave am- plitude on the screen is the quantum superposition of all the spherical wave emitted from the slit. Since the wave nature of electron is identical to photon, the diraction pattern should be the same. However, it is not trivial of the spin distribution in the diraction pattern when the SOI interaction is considered. We compute the quantum amplitude using the Green's function method pioneered by Feynman. The quantum amplitude of the electrons from ( x0;y0) att= 0 to (x;y) at time t is denoted by hx;y;tjx0;y0;0i. Att= 0, elec- trons pass through the slit. Suppose that the slit locates atx= 0, and the screen is placed at x=Laway, thewave function on the screen is given by (L;y;t ) =Zd=2 d=2dy0hL;y;tj0;y0;0i(0;y0;0); (2) where(x0;y0;0) is the initial wavefunction at the slit. In the following, we compute Eq. (2) in the pure Rashba and pure Dresselhaus cases. We shall use the pure Rashba case to illustrate our calculation procedure. The one for the Dresselhaus case is given in the Appdendix A The pure Rashba case is the one for 6= 0 and= 0. Eq. (1) can be written by H= Hdiagy, where Hdiag= (p2 2mp) 0 0 (p2 2m+p)! ;  =1p 2 ieiiei 1 1 ; = tan1(py=px): (3) The propagator in the momentum space is given by U(t) =ei hHt= ei hHdiagty = cos(p ht)eisin(p ht) eisin(p ht) cos(p ht) ei h(p2 2m)t = U11U12 U21U22 : (4) The propagator in real space is the Fourier transforma- tion of Eq. (4) giving as the following hx;y;tjx0;y0;0i=Z1 1Z1 1dpxdpy (2h)2ei h~xpxei h~ypyU(t) =1 (2h)2Z2 0dpZ1 0pdp
ei hp(~xcosp+~ysinp)U(t);(5) where ~x=xx0, and ~y=yy0. Eq. (5) is a 22 matrix. The matrix element can be computed in terms of the special functions. In Appendix B, we include the useful integration formulae of the special function. Using Eq.(B3) and Eq.(B2), we obtain hx;y;tjx0;y0;0i11=hx;y;tjx0;y0;0i22 (6) =m 2ith ei ht1X n=0n! (2n)!2im2t hn 1F1(n+ 1;1;im~r2 2ht); where 1F1(a;b;z) is the hypergeometric function and ~r2= ~x2+ ~y2. Using Eq. (B4), Eq. (B5), and Eq. (B6), we can compute the other elements given by hx;y;tjx0;y0;0i12 =m 2ithm h ei ht(~xi~y) 1X n=0(n+ 1)! (2n+ 1)!2im2t hn 1F1(n+ 2; 2;im~r2 2ht) (7)3 Similarly, hx;y;tjx0;y0;0i21 =m 2ithm h ei ht(~x+i~y) 1X n=0(n+ 1)! (2n+ 1)!2im2t hn 1F1(n+ 2; 2;im~r2 2ht):(8) The propagation time tis determined by t=mL=pFx, wherepFxis thex-component of the momentum of the electron at the Fermi energy. For small angle diraction as usually considered in optics, it is a good approxima- tion. Furthermore, we introduce the following dimension- less quantities for computational convenience. We dene mL= hand mL= h, which are the dimension- less strength of the couplings, d=d=Land y=y=L, and kpFxd=2h. The computation can be further simplied as the fol- lowing. There are four terms to compute in the ma- trix product in Eq. (2). In order to make the numerical results more convergent, we express the hypergeometric functions in the integral form. Given the initial wave- function(0;y0;0) = (1;2) = (i;1)T=p 2dfrom the lower Rashba band, we obtain Zd=2 d=2hx;y;tjx0;y0;0i111(0;y0;0) =ip 2dm 2ith ei ht1X n=0n! (2n)!i2d kn 1 2iI ds(s1)(n+1)snZd=2 d=2dy0exp(sim~r2 2ht) =ip 2dmd 2ith ei htF(y); (9) where ~r2=x2+y22yy0+y02r22yy0andr2=x2+y2 in they0integration are used, and F(y) andfn(y) are given by F(y) =1X n=01 (2n)!i2d kn fn(y); (10) fn(y) =d dsn sn1exp(simr2 2ht)sin(sky) ky s=1: Similarly, Zd=2 d=2hL;y;tjx0;y0;0i122(0;y0;0) =1p 2dmd 2ith ei ht (1iy)G(y)dH(y) ;(11)where G(y) =1X n=01 (2n+ 1)!i2d kn gn(y); gn(y) =d dsn snexp(simr2 2ht)sin(sky) ky s=1; H(y) =1X n=01 (2n+ 1)!i2d kn hn(y); hn(y)=d dsn sn1esimr2 2htskycos(sky)sin(sky) 2k2y2 s= 1: The other two terms can be also simplied in the same way. Zd=2 d=2hx;y;tjx0;y0;0i222(0;y0;0) =ip 2dmd 2ith ei htF(y) Zd=2 d=2hL;y;tjx0;y0;0i211(0;y0;0) =ip 2dmd 2ith ei ht (1 +iy)G(y) +dH(y) (12) Combining Eq. (9), Eq. (10), Eq. (11), Eq. (12), the wave function = ( () 1; () 2) on the screen can be com- puted as the following () 1=A iF(y)(1iy)G(y) + dH(y) ; () 2=A F(y)i(1 +iy)G(y)idH(y) ;(13) whereA=1p 2dmd 2ith
ei ht. The diraction pattern for the electrons from the upper Rashba band can be also obtained in the same way. Given the initial wave function+(0;y0;0) = (i;1)T=p 2d, the wavefunction += ( (+) 1; (+) 2) on the screen is given by (+) 1=A iF(y) + (1iy)G(y)dH(y) ; (+) 2=A F(y)i(1 +iy)G(y)idH(y) ;(14) We will compute Eq. (13) and Eq. (14) numerically in the next section. j j2=j 1j2+j 2j2is probability dis- tribution of electron on the screen, which is nothing but the diraction pattern. As 1and 2are the spin com- ponents, we will show later that the spatial distribution ofj 1j2andj 2j2on the screen are dierent in Eq. (13) and Eq. (14). The dierence implies a nontrivial spin distribution in the diraction pattern that we hope to investigate in this paper. The pure Dresselhaus case can be computed in the sim- ilar way. The Hamiltonian now becomes H= Hdiagy, where Hdiag= (p2 2mp) 0 0 (p2 2m+p)! ;(15)  =1p 2 eiei 1 1 ; = tan1(py=px)4 1.0  0.5 1.0Probability Density 1.0 0.51.52.0 -1.0 -0.5 1.0 0.5 FIG. 1. (color online) The probability density of j j2,j 1j2 and j 2j2in the dimensionless parameters  = 0;= 0;k= 7:5, and d= 0:01. The diraction pattern for spin-up electron is the same as for spin-down electron. Since there is spin de- generacy in the momentum space, the total spin of the central peak is zero. Although the energy dispersion is the same as the Rashba case, the orientation of the spin eigenstates diers. It gives rise to dierent spin distribution. Using the same computational procedure, we obtain () 1=A F(y) +(iy)G(y) +idH(y) ; () 2=A F(y)(i+ y)G(y) +idH(y) ;(16) where ()= ( () 1; () 2) are the wavefunction on the screen for the lower Dresselhaus band. (+) 1=A F(y)(iy)G(y)idH(y) ; (+) 2=A F(y)(i+ y)G(y) +idH(y) ;(17) where (+)= ( (+) 1; (+) 2) are the wavefunction on the screen for the upper Dresselhaus band. This completes all of our analytical results in this paper. In the next section, we will solve them numerically and investigate the magnetic property. III. NUMERICAL RESULTS A. Single-slit diraction We rst plotj j2and the spin components without SOI in Fig. (1). Since we work with dimensonless pa- rameters, this is the diraction pattern equivalently at L= 1. Furthermore, we consider the wave propagating to the positive xdirection.j j2we obtained in this case is exactly the same as the diraction pattern of photon. Dark fringes locate at the same position as the formula given in all textbooks of general physics. Without sur- prise,j 1j2andj 1j2are the same. These results imply that the electron spin remain lies in the xyplane af- ter diraction. Actually, without SOI, upper and lower bands are degenerate. Given a chemical potential, bothspin up and spin down pass through the slit. Therefore, the spin distribution for each component dened by <Si(y)>=< (y)jSij (y)> < (y)j (y)>fori=x;y;z (18) are trivial. Namely <Si(y)>= 0 for all components. 1.0 0.5 0.5 1.00.51.01.52.0Probability Density(a) 2 1 1 2 0.60.40.20.20.40.6 (b) 2 1 1 2 0.60.40.20.20.40.6 (c) 2 1 1 2 0.60.40.20.20.40.6 (d) FIG. 2. (color online) The diraction pattern and spin dis- tribution for the Rashba lower band in the dimensionless pa- rameters = 0:5,k= 7:5, and d= 0:01. (a)The yellow line indicates the distribution of probability densities j j2. The blue line is the distribution of j 1j2, and the red line is the one of j 2j2. (b)(c)(d) are the spin distributions <Si(y)>. When SOI is considered, < Si(y)>become nontriv- ial. In Fig. (2) and Fig. (3), we show the results for the lower and upper bands in the Rashba system respec- tively. If the chemical potential is tuned above the band- crossing point, the electrons from the upper band has larger diraction eect, because they have longer wave- length. If the chemical potential is tuned below the band- crossing point, the diraction pattern comes from the electron of the lower band. Fig. (2a) is the diraction pattern, namelyj j2drawn in a yellow line, for the lower band. It is exactly the same as the one without SOI. If an experimental method that is not spin-resolved, for exam- ple scanning probe microscopy, is used, one can not dis- tinguish the dierence between the system with or with- out SOI. However, the spin components j 1j2andj 2j2 have asymmetric ydependence. Their dark fringes lo- cate at the same points. "The bright fringes" distribute, however, asymetrically. The asymmetry results in a new eect of spin splitting. in Fig. (2b) to Fig. (2d), we com- pute< Si(y)>for dierent components. At y= 0, <Sz(0)>=<Sx(0)>= 0. The spin orientation on the screen aty= 0 is the same as the one of the initial wave- function at the slit. At y6= 0, the other two components start to develop in a way that is the odd function of y. The spin distribution for the electrons in the upper band is shown in Fig. (3b) to Fig. (3d). Since the spin compo- nentsj 1j2andj 2j2are asymmetric in the opposite way to the lower band, the spin distribution distribute oppo- sitely. If we dene the spin current as Ii j=< Si> vi,5 our results imply the existence of the spin current in the transverse direction, namely Ix y6= 0 andIz y6= 0. 1.0 0.5 0.5 1.00.51.01.52.0Probability Density (a) 2 1 1 2 0.60.40.20.20.40.6 (b) 2 1 1 2 0.60.40.20.20.40.6 (c) 2 1 1 2 0.60.40.20.20.40.6 (d) FIG. 3. (color online) The diraction pattern and spin dis- tribution for the Rashba upper band in the dimensionless pa- rameters = 0:5,k= 7:5, and d= 0:01. (a)The yellow line indicates the distribution of probability densities j j2. The blue line is the distribution of j 1j2, and the red line is the one of j 2j2. (b)(c)(d) are the spin distributions <Si(y)>. One important feature in those results is that the < Sz(y)>component becomes nite at y6= 0. One thing we learn in the Rashba system is that the electron spin lies in the xyplane. After scattered by a slit, a com- ponent perpendicular to the xyplane develops! Its mag- nitude grows asjyjincreases. Its maximum value can be as large as 0.42  hwhich is 84% of the Sz= h=2. Another interesting feature in the spin distribution is that there existsy=y0so that electron spin at y=y0andy=y0 are antiparallel. It is the position where <Sy(y0)>= 0, since only< Sy(y)>is an even function in y. We fur- ther note that the spin distribution is independent of the chemical potential. It only depends on the SOI strength. We also compute the spin distribution for the Dressel- haus case. We provide the numerical results in Fig. (5) and Fig. (6). The strength of the signal to detect spin distribution in experiments should be proportional to < (y)jSij (y)>, which is the product of the spin < Si>and the wave amplitude. If the chemical potential is tuned below the band-crossing point, the electrons at the Fermi energy come from the lower band. In this case, the central maximum of the diraction peak is spin-polarized. If the chemical potential is tuned above the band-crossing point, the electron taken part in the diraction come from both upper and lower bands. Since the electron spin is polarized in the opposite direction between up- per and lower bands. In the central peak, the signal of spin polarization is almost zero. In some applications, the spin component perpendicular to the plane is useful. Although this new spin-splitting eect gives rise to the Szcomponent, the signal is weak. In the rst dirac- tion peak,< Sz>reaches its maximum. However, the wave amplitude is much smaller than the central peakmaximum. Therefore, new idea is needed to enhance the signal ofSz. In the next subsection, we consider the grating device to achieve this goal. B. Diraction by grating The optical grating is usually made of hundreds or thousands slits that make diraction peaks more local- ized. It also increases the separation distance between the diraction peaks of the photons of dierent frequen- cies. Therefore, it is one of the basic optical devices to separate photons of dierent frequencies. Here, we bor- row those beautiful features to design an electronic device to enhance the spin polarization. The computational procedure is the same as the one for the single-slit problem. The only dierence is the in- tegration range. Here, we consider 10 slits and 20 slits with equal distance. The quantum amplitude on the screen comes from the initial wavefunction at all point sources in all slits. In Fig. (4), we show the results of the calculation for N= 10 andN= 20 for the lower Rashba band, where Nis the number of slits. The pa- rameters () = 0:5 corresponding to the real parameter h(h) = 7:621013eV
mare used and the screen isL= 1maway. We reproduce the correct dirac- tion pattern for grating. The peak for N= 20 are much localized and sharper than the one for N= 10. The asymmetry of j 1j2andj 2j2gives rise to the same spin distribution as the single-slit problem. It is indepen- dent of not only the wavelength of electron but also the number of slits. We will provide the explanation in the next section. The position of the diraction peaks makes no dierence from the one of photons. Dierent from the case without SOI, the spin orientation of diraction peaks are dierent. One can tune the chemical potential to change the electron wavelength and the position of the diraction peaks. Since <Sz(y)>is independent of the electron wavelength, one can tune the spin orientation of the diraction peaks. In optics, it is the rst diraction peak used to distinguish photons of dierent frequency. Here, the rst diraction peak at positive yand negative yare dierent. At y=y0mentioned in the single-slit case, the spin orientation of two rst diraction peaks isantiparallel . When the chemical potential is tuned so that the rst diraction peaks locates at y=jy0j, one can observe opposite spin orientations on the screen, mimick- ing the Stern-Gerlach experiment. IV. DISCUSSION AND CONCLUSION As mentioned in the introduction, the spin-splitting eect proposed in the paper is due to the precession of electron spin subjected to the ctitious magnetic eld by the SOI. After scattered by the slit, electron acquires a transverse momentum, namely the ydirection. The elec- trons propagating in the positive yand negative yprecess6 Probability Density(a) 20 15 10 5 -1.0 -0.5 0.5 1.0 1040 30 10Probability Density (b) -1.0 -0.5 0.5 1.020 FIG. 4. (color online) Distributions of probability density of j j2,j 1j2, and j 2j2for grating devices N= 10 and 20 in the dimensionless parameters  = 0:5;k= 7:5,d= 0:01, andb= 0:03, whereNis number of slits, b=b=L, andbis the separation distance between nearest slits. They are the results of the Rashba lower band. (a) N= 10. (b)N= 20. in opposite directions, and hence the spin distribution arises. Due to the precession mechanism, the spin distri- bution depends on the distance of the screen from the slit. However, it does not depend on the electron wavelength. The reason is giving as the following. Changing the wave- length means to change the electron momentum. For example, if the electron has larger momentum, it takes shorter time to reach the screen. On the other hand, the magnitude of the ctitious magnetic eld is proportional to the electron momentum. The larger the momentum is, the larger the ctitious magnetic eld. Even though the electron with larger momentum reaches the screen faster, it also has larger rate of precession. Therefore, the - nal spin orientation becomes independent of the electron wavelength. The precession mechanism also provides the reason why the spin distribution does not depend on the num- ber of slits, namely gratings. It depends only on the strength of SOI and where to measure it, namely dis- tance of screen. Within spin coherent length, one can use this new eect to manipulate electron spin. Usually, the coherent transport can be observed more easily in the low temperature. In our system, the spin coherent trans- port can be achieved more easily if there is no interband mixing, that mixes electrons of opposite spins. Interbandmixing can happen by the thermal excitation. In usual cases, energy to cause interband mixing is around few meV, which corresponds to few tens of kelvin. A good feature of current proposal is that the dirac- tion pattern is exactly the same as the one of photon even when the SOI is present. The diraction nature depend hundred percents on the wave nature of electron. The new thing we discovered is the non-trivial spin distribu- tion in the diraction pattern due to the spinor nature of electrons and the SOI. It brings huge convenience to control the peak of the diraction peaks without wor- rying the strength of SOI. The experimental feasibility is already discussed in our precious Letter and will not repeat here. In conclusion, we propose a new eect of spin-splitting in the 2DEG in the presence of SOI. It is based on the electron spin precession due to the ctitious magnetic eld generated by the SOI. Diracted by slits, electron gains transverse momentum and processes in opposite ways in the dierent transverse momentum. Spin current Ix yandIz yis generated for the Rashba case and Iy yand Iz yis generated for the Dresselhaus case. The presence of the spin current leads to nontrivial spin distribution in the diraction pattern. Most importantly, the new eect shed lights on the potential applications in spintronics. We appreciate the stimulating discussion with Chi-Te Liang for various experimental aspects. This work was supported by the National Science Council of Taiwan un- der NSC 97-2112-M-002-027-MY3. 1.0 0.5 0.5 1.00.51.01.52.0Probability Density(a) 2 1 1 2 0.60.40.20.20.40.6 (b) 2 1 1 2 0.60.40.20.20.40.6 (c) 2 1 1 2 0.60.40.20.20.40.6 (d) FIG. 5. (color online) The diraction pattern and spin dis- tribution for the Dresselhaus lower band in the dimensionless parameters = 0:5,k= 7:5, and d= 0:01. (a)The yellow line indicates the distribution of probability densities j j2. The blue line is the distribution of j 1j2, and the red line is the one of j 2j2. (b)(c)(d) are the spin distributions <Si(y)>.7 1.0 0.5 0.5 1.00.51.01.52.0Probability Density(a) 2 1 1 2 0.60.40.20.20.40.6 (b) 2 1 1 2 0.60.40.20.20.40.6 (c) 2 1 1 2 0.60.40.20.20.40.6 (d) FIG. 6. (color online) The diraction pattern and spin distri- bution for the Dresselhaus upper band in the dimensionless parameters = 0:5,k= 7:5, and d= 0:01. (a)The yellow line indicates the distribution of probability densities j j2. The blue line is the distribution of j 1j2, and the red line is the one of j 2j2. (b)(c)(d) are the spin distributions <Si(y)>. Appendix A: Spin distribution in the Dresselhaus case The diagonalization of Eq. (1) with = 0 and6= 0 givesH= Hdiagy, where Hdiag= (p2 2mp) 0 0 (p2 2m+p)! ;(A1)  =1p 2 eiei 1 1 ; = tan1(py=px): (A2) The propagator in the momentum space is given by U(t) =ei hHt= ei hHdiagty = cos(p ht)ieisin(p ht) ieisin(p ht) cos(p ht) ei h(p2 2m)t = U11U12 U21U22 : (A3) The kernels in the Dresselhaus case can be written in terms of hypergeometric function given by hx;y;tjx0;y0;0i11=hx;y;tjx0;y0;0i22 =m 2ith ei ht1X n=0n! (2n)!2im2t hn 1F1(n+ 1; 1;im~r2 2ht); (A4) hx;y;tjx0;y0;0i12=m 2ithm h ei ht(i~x+ ~y) 1X n=0(n+ 1)! (2n+ 1)!2im2t hn 1F1(n+ 2; 2;im~r2 2ht); (A5)and hx;y;tjx0;y0;0i21=m 2ithm h ei ht(i~x+ ~y) 1X n=0(n+ 1)! (2n+ 1)!2im2t hn 1F1(n+ 2; 2;im~r2 2ht): (A6) For the electron comes from the lower band, the ini- tial wave function is (0;y0;0) = (1;1)T=p 2d. We compute the matrix product in Eq. (2) as the following Zd=2 d=2hx;y;tjx0;y0;0i111(0;y0;0) =1p 2dm 2ith ei ht1X n=0n! (2n)!i2d kn 1 2iI ds(s1)(n+1)snZd=2 d=2dy0exp(sim~r2 2ht) =1p 2dmd 2ith ei htF(y); (A7) where we used an approximation ~ r2=x2+y22yy0+ y02r22yy0,r2=x2+y2in they0integration and the functions F(y) =1X n=01 (2n)!i2d kn fn(y); (A8) fn(y) =d dsn sn1exp(simr2 2ht)sin(sky) ky s=1:(A9) Furthermore, Zd=2 d=2hL;y;tjx0;y0;0i122(0;y0;0)=1p 2dm 2ith ei ht m h1X n=0(n+ 1)! (2n+ 1)!i2d kn(n+ 1) (n+ 2) 1 2iI ds(s1)(n+1)sn+1 Zd=2 d=2dy0(iL+yy0) exp(sim~r2 2ht) =1p 2dmd 2ith ei ht (iy)G(y) +idH(y) ; (A10)8 where G(y) =1X n=01 (2n+ 1)!i2d kn gn(y); gn(y) =d dsn snexp(simr2 2ht)sin(sky) ky s=1; H(y) =1X n=01 (2n+ 1)!i2d kn hn(y); hn(y)=d dsn sn1esimr2 2htskycos(sky)sin(sky) 2k2y2 s=1:(A11) For 2, we have Zd=2 d=2hx;y;tjx0;y0;0i222(0;y0;0)=1p 2dmd 2ith ei htF(y); (A12) and Zd=2 d=2hL;y;tjx0;y0;0i211(0;y0;0)=1p 2dm 2ith ei ht m h1X n=0(n+ 1)! (2n+ 1)!i2d kn(n+ 1) (n+ 2) 1 2iI ds(s1)(n+1)sn+1 Zd=2 d=2dy0(iL+yy0) exp(sim~r2 2ht) =1p 2dmd 2ith ei ht (i+ y)G(y)idH(y) ; (A13) whereF(y),G(y), andH(y) are given by Eq. (34), Eq. (36), and Eq.(38), respectively. Therefore,, the electron wave function on the screen ()= ( () 1; () 2) in the Dresselhaus lower band can be simplied as the following () 1=A F(y) +(iy)G(y) +idH(y) ; (A14) () 2=A F(y)(i+ y)G(y) +idH(y) :(A15) Similarly, for Dresselhaus upper band (+) 1=A F(y)(iy)G(y)idH(y) ;(A16) (+) 2=A F(y)(i+ y)G(y) +idH(y) ;(A17)whereA=1p 2dmd 2ith
ei ht. Appendix B: Special functions In this section, we provide some useful integral for- mula for the computation of the kernel in Eq. (2). The hypergeometric function 1F1(a;b;z) can be expressed in an integral form given by 1F1(a;c;z)=(1+ac) (a)(c1)! 2iI ds(s1)ca1sa1esz; (B1) where the integral contour is around the pole s= 1 and c is an integer. It is involved in the integration n! 2an+11F1(n+1; 1;b2 4a)=Z1 0eay2J0(by)y2n+1dy; (B2) where J0(rp h) =1 2Z2 0dei hp(xcos+ysin)(B3) is the Bessel function of the rst kind, and  r=p x2+ y2. In the following, we list some formula used in the com- putation. Z2 0dcosei hp(xcos+ysin)=i2x rJ0 0(rp h) (B4) Z2 0dsinei hp(xcos+ysin)=i2y rJ0 0(rp h) (B5) Z1 0eay2J1(by)y2n+2dy=b(n+1)! 4an+21F1(n+2; 2;b2 4a) (B6) chchern@ntu.edu.tw 1H. Ohno, H. Munekata, T. Penney, S. von Molnar, and L. L. Chang, Phys. Rev. Lett. 68, 2664 (1992). 2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka- nova, and D. M. Treger, Science 294, 1488 (2001). 3S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003).4S. Murakami, N. Nagaosa, and S.-C. Zhang, Phys. Rev. Lett. 93, 156804 (2004). 5G. Y. Guo, Y. Yao, and Q. Niu, Phys. Rev. Lett. 94, 226601 (2004). 6G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa, Phys. Rev. Lett. 100, 096401 (2008). 7C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005).9 8L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007). 9D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature 452, 970 (2008).10M. Dyakonov and V. Perel, Sov. Phys. Solid State 13, 3023 (1972). 11C. Tahan and R. Joynt, Phys. Rev. B 71, 075315 (2005). 12C.-H. Chern, C.-J. Lin, and C.-T. Liang, Phys. Rev. Lett. 105, 217205 (2010).

1106.3613v1.Evolution_from_BCS_to_BEC_superfluidity_in_the_presence_of_spin_orbit_coupling.pdf

arXiv:1106.3613v1 [cond-mat.quant-gas] 18 Jun 2011Evolution from BCS to BEC superfluidity in the presence of spi n-orbit coupling Li Han and C. A. R. S´ a de Melo School of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332, USA (Dated: August 15, 2018) We discuss the evolution from BCS to BEC superfluids in the pre sence of spin-orbit coupling, and show that this evolution is just a crossover in the balanced c ase. The dependence of several thermo- dynamic properties, such as the chemical potential, order p arameter, pressure, entropy, isothermal compressibility and spin susceptibility tensor on the spin -orbit coupling and interaction parameter at low temperatures are analyzed. We studiedboth the case of equal Rashbaand Dresselhaus (ERD) and the Rashba-only (RO) spin-orbit coupling. Comparisons between the two cases reveal several striking differences in the corresponding thermodynamic qu antities. Finally we propose measuring the spin susceptibility as a means to detect the spin-orbit c oupling effect. PACS numbers: 03.75.Ss, 67.85.Lm, 67.85.-d Superfluidity is a ubiquitous phenomenon that is en- countered in nearly every area of physics including con- densed matter, nuclear, astro, and atomic and molec- ular physics. Superflow results from strong correla- tions between particles, which for any given interacting Fermi system could not be controlled externally until re- cently with the advent of ultra-cold atoms. In standard condensed matter there is a continuous search for new charged superfluids (superconductors) since the type and strength of interactions can not be tuned even within the same class of materials. In the case of nuclear matter the issue of tunability of interactions is even worse, be- ing hopeless for neutron stars. However, the situation is much more favorable for ultra-cold Fermi atoms, where the ability to control interactions between particles, via Feshbach resonances, has been demonstrated in exper- imental studies of the so-called crossover from BCS to BEC superfluidity. Further control of interactions is now possible through newly developed experimental techniques that allow the production of fictitious magnetic fields which couple to neutral bosonic atoms [1, 2]. These fictitious magnetic fields are generated through an all optical process, but produce real effects like the creation of vortices in the superfluid state of bosons. Furthermore, artificial spin- orbit coupling has also been produced in neutral bosonic systems[3]wherethestrengthofthecouplingcanbecon- trolled optically. In principle the same techniques can be applied to ultracold fermions [3, 4], which, when coupled with the control over the interaction using Feshbach res- onances, allows for the exploration of superfluidity not only as a function of interactions, but also as a function of fictitious magnetic fields [5], or as a function of spin- orbit coupling discussed here. An introduction to the effects of controllable fictitious magnetic and spin-orbit fields can now be found in the literature [6]. It is in anticipation of experiments involving spin-orbit couplingin fermionicatomssuch as6Li,40Kand isotopes of Ytterbium, that we discuss here the evolution from BCS to BEC superfluidity in the presence of controllablespin-orbit couplings for balanced fermions in three di- mensions. We investigate spin-orbit effects with Dressel- haus [7] and/or Rashba [8] terms, and analyze several thermodynamic quantities including the order param- eter, chemical potential, thermodynamic potential, en- tropy, pressure, isothermal compressibility, and spin sus- ceptibility tensor as a function of spin-orbit coupling and interaction parameter at low temperatures. We conclude that the BCS-to-BECevolutionfor balanced fermions in- cluding spin-orbit effects is just a crossover. Hamiltonian: To address the problem of the evolu- tion from BCS to BEC superfluidity in the presence of spin-orbit fields for balanced or imbalanced Fermi-Fermi mixtures, we start with the generic Hamiltonian density H(r) =H0(r)+HI(r). (1) The single-particle Hamiltonian density is H0(r) =/summationdisplay αβψ† α(r)/bracketleftig ˆKαδαβ−hi(r)σi,αβ/bracketrightig ψβ(r),(2) whereˆKα=−∇2/(2mα)−µαis the kinetic energyin ref- erencetothechemicalpotential µα, andhi(r)isthespin- orbit field along the i-direction (α=↑,↓,i=x,y,z). The interaction term is HI(r) =−gψ† ↑(r)ψ† ↓(r)ψ↓(r)ψ↑(r), wheregis a contact interaction. In this paper we set ¯h=kB= 1. Effective Action: The partition function at tempera- tureTisZ=/integraltext D[ψ,ψ†]exp/parenleftbig −S[ψ,ψ†]/parenrightbig with action S[ψ,ψ†] =/integraldisplay dτdr/bracketleftigg/summationdisplay αψ† α(r)∂ ∂τψα(r)+H(r,τ)/bracketrightigg .(3) Using the standard Hubbard-Stratanovich transfor- mation that introduces the pairing field ∆( r,τ) = g/an}b∇acketle{tψ↓(r,τ)ψ↑(r,τ)/an}b∇acket∇i}htwe can write the intermediate action Sint[ψ,ψ†,∆,∆†] =Sno[ψ,ψ†] +SI[ψ,ψ†,∆,∆†],where the no-interaction action is Sno[ψ,ψ†] =/integraldisplay dτdr/bracketleftigg/summationdisplay αψ† α(r)∂ ∂τψα(r)+H0(r,τ)/bracketrightigg ,2 and the action due to the auxiliary field is SI=/integraldisplay dτdr/bracketleftbigg|∆(r,τ)|2 g−∆ψ† ↑ψ† ↓−∆†ψ↓ψ↑/bracketrightbigg . Using the four-dimensional vector Ψ†= {ψ† ↑,ψ† ↓,ψ↑,ψ↓},the intermediate action becomes Sint=/integraldisplay dτdr/bracketleftbigg|∆(r,τ)|2 g+1 2Ψ†MΨ+1 2(/tildewideK↑+/tildewideK↓)/bracketrightbigg . The 4×4 matrix Mis M= ∂τ+/tildewideK↑−h⊥0−∆ −h∗ ⊥∂τ+/tildewideK↓∆ 0 0 ∆†∂τ−/tildewideK↑h∗ ⊥ −∆†0h⊥∂τ−/tildewideK↓ ,(4) whereh⊥=hx−ihycorresponds to the transverse com- ponent of the spin-orbit field, hzto the parallel com- ponent with respect to the quantization axis z,/tildewideK↑= ˆK↑−hz, and/tildewideK↓=ˆK↓+hz. Integration over the fields Ψ and Ψ†leads to the effective action Seff=/integraldisplay dτdr/bracketleftbigg|∆(r,τ)|2 g−T 2VlndetM T+/tildewideK+δ(r−r′)/bracketrightbigg , (5) where/tildewideK+= (/tildewideK↑+/tildewideK↓)/2. Saddle Point Approximation: To proceed we use the saddle point approximation ∆( r,τ) = ∆ 0+η(r,τ),and separate the matrix Minto two parts. The first one is the saddle point matrix M0, where the transformation ∆(r,τ)→∆0takesM→M0. The second one is the fluctuation matrix MF=M−M0, which depends only onη(r,τ) and its Hermitian conjugate. Using the saddle point approach we write the effective action asSeff=S0+SF, where S0=/integraldisplay dτdr/bracketleftbigg|∆0|2 g−T 2VlndetM0 T+/tildewideK+δ(r−r′)/bracketrightbigg is the saddle point action and SF=/integraldisplay dτdr/bracketleftbigg|η(r,τ)|2 g−T 2Vlndet/parenleftbig 1+M−1 0MF/parenrightbig/bracketrightbigg is the fluctuation action in all orders in the fluctuation field. Theeffectsoffluctuationsatbothzerotemperature and near the critical temperature will be discussed later. A transformation to the momentum-frequency coordi- nates (k,iωn), whereωn= (2n+1)πT, leads to S0=V T|∆0|2 g−1 2/summationdisplay k,iωn,jln/bracketleftbiggiωn−Ej(k) T/bracketrightbigg +/summationdisplay k/tildewideK+ T, whereEj(k) are the eigenvalues of the matrix H0= /tildewideK↑(k)−h⊥(k) 0 −∆0 −h∗ ⊥(k)/tildewideK↓(k) ∆ 0 0 0 ∆† 0−/tildewideK↑(−k)h∗ ⊥(−k) −∆† 00h⊥(−k)−/tildewideK↓(−k) , (6)which describes the Hamiltonian of the elementary ex- citations in the four-dimensional vector basis Ψ†= /braceleftig ψ† ↑(k),ψ† ↓(k),ψ↑(−k),ψ↓(−k)/bracerightig .The spin-orbit field is h⊥(k) =hR(k) +hD(k),where the first term is of the Rashba-type hR(k) =vR(−kyˆx+kxˆy),and the sec- ond is of the Dresselhaus-type hD(k) =vD(kyˆx+kxˆy). We assume, without loss of generality, that vR>0 and vD>0. The magnitude of the transverse field is then h⊥(k) =/radicalig (vD−vR)2k2y+(vD+vR)2k2x.In the lim- iting cases of pure Rashba (R) with vD= 0 and for equal Rashba-Dresselhaus (ERD) couplings with vR= vD=v/2, the transverse fields are h⊥(k) =vR/radicalig k2x+k2y (vR>0) andh⊥(k) =v|kx|(v>0), respectively. Order parameter and number equations: The saddle point thermodynamic potential Ω 0=TS0is obtained by integrating out the fermions leading to Ω0=V|∆0|2 g−T 2/summationdisplay k,jln{1+exp[−Ej(k)/T]}+/summationdisplay k¯K+, with¯K+=/bracketleftig /tildewideK↑(−k)+/tildewideK↓(−k)/bracketrightig /2.The order parame- ter is determined via the minimization of Ω 0with respect to|∆0|2leading to V g=−1 2/summationdisplay k,jnF[Ej(k)]∂Ej(k) ∂|∆0|2, (7) wherenF[Ej(k)] = 1/(exp[Ej(k)/T] + 1) is the Fermi function for energy Ej(k). We replace the contact in- teractiongby the scattering length asthrough the rela- tion 1/g=−m+/(4πas) + (1/V)/summationtext k[1/(2ǫk,+)],where m+= 2m↓m↑/(m↓+m↑) is twice of the reduced mass, ǫk,α=k2/(2mα) are the kinetic energies, and ǫk,+= [ǫk,↑+ǫk,↓]/2.The number of particles at the saddle point is obtained by Nα=−∂Ω0/∂µα, leading to Nα=1 2/summationdisplay k 1−/summationdisplay jnF[Ej(k)]∂Ej(k) ∂µα .(8) The self-consistent relations shown in Eqs. (7) and (8) aregeneralforarbitrarymassandpopulationimbalances. However,next, weparticularizeourdiscussiontothecase of a balanced system with equal masses. Balanced Populations: In the case of mass and pop- ulation balanced systems, the four eigenvalues of the matrixH0areE1(k) =/radicalig [ε1(k)]2+|∆0|2, E2(k) =/radicalig [ε2(k)]2+|∆0|2, E3(k) =−E1(k), andE4(k) = −E2(k).Here, the auxiliary energies are ε1(k) =ξ(k) + h⊥(k),andε2(k) =ξ(k)−h⊥(k). The corresponding order parameter equations at the saddle point level is V g=1 2/summationdisplay k/bracketleftbiggX1(k) 2E1(k)+X2(k) 2E2(k)/bracketrightbigg , (9)3 whereXm(k) = tanh[Em(k)/2T] (m= 1,2). Since the mixture of equal mass fermions is balanced, the chemical potentials are the same µ↑=µ↓=µ,and the associated number equation is N=−∂Ω/∂µthat reduces to N=/summationdisplay k/bracketleftbigg 1−X1(k) 2E1(k)ε1(k)−X2(k) 2E2(k)ε2(k)/bracketrightbigg .(10) In Fig. 1, we show the zero temperature behavior of |∆0|andµas a function of 1 /(kFas) for various values of spin-orbit coupling in the equal-Rashba-Dresselhaus (ERD) andfor Rashba-only(RO)cases. In the ERDcase the order parameter |∆0|is independent of v, and the chemical potential µ(v) is simply µ(v) =µ(0)−mv2/2, since the transverse field h⊥(k) =v|kx|can be elimi- nated by momentum shifts along the x-direction, effec- tively gauging away spin-orbit effects in the charge or momentum sector. This symmetry also implies that the criticaltemperature Tcasafunctionof1 /(kFas)forfinite vis the same as that for v= 0. However, in the RO case, shifts in momentum can not gauge away the spin-orbit coupling, and |∆0|increases with increasing vR, whileµ decreases as vRincreases, exhibiting the same tendency as in the ERD case. In the BCS regime, the increase of |∆0|withvRalso leads to an increase of Tcwith increas- ingvR. /Minus2/Minus101200.511.5 1/Slash1kFas/VertBar1/CapDΕlta0/VertBar1a/RParen1 /Minus2/Minus1012/Minus4/Minus202 1/Slash1kFasΜb/RParen1 FIG. 1: Order parameter |∆0|and chemical potential µ(in units of the Fermi energy ǫF) as a function of interaction pa- rameter 1 /(kFas) for different spin-orbit couplings vR/vF= 0 (solid),vR/vF= 0.8 (dashed), vR/vF= 1.0 (dotted), and vR/vF= 1.2 (dot-dashed) at T= 0 in the RO case. Here vF=kF/mis the Fermi velocity. Momentum distribution and excitation spectrum: The momentum distribution n(k) is obtained from Eq. (10) using the definition N=/summationtext kn(k). At fixed momentum component kz= 0 and fixed interaction strength, the momentum distribution n(k) shifts continuously with in- creasing spin-orbit coupling in the BCS [1 /(kFas)≪ −1] or unitarity regimes [1 /(kFas)→0]. For zero spin-orbit coupling,n(k) is that of a superfluid degenerate Fermi system with identical single-particle bands ξ(k) and has a nearly flat momentum distribution until the Fermi mo- mentum is reached. However, as the spin-orbit cou- pling is turned on, non-identical single-particle bands ξ⇑(k) =ξ(k)−h⊥(k) andξ⇓(k) =ξ(k) +h⊥(k) in the helicity basis |⇑/an}b∇acket∇i}ht,|⇓/an}b∇acket∇i}htemerge and produce a doublestructure with a reasonably flat momentum distribution centered around finite momenta in the ( kx,ky) plane. In the BEC regime [1 /(kFas)≫1] the momentum distribu- tions for weak and strong spin-orbit coupling broadens substantially due to the loss of degeneracy in the Fermi system when the chemical potential goes below the min- ima of the helicity bands and becomes large and nega- tive. Even though there is a substantial change in the momentum distribution as a function of the spin-orbit coupling, we notice that the excitation energies E1(k) andE2(k) is always gapped for all values of the inter- action parameter 1 /(kFas) or the spin-orbit field h⊥(k), immediately suggesting that thermodynamic properties, which depend on the excitationenergies, evolvesmoothly from the BCS to the BEC regime in the balanced case for fixed values of spin-orbit coupling. The omnipres- ence of a gap in the excitation spectrum shows that the evolution from BCS to BEC superfluidity at finite spin- orbit coupling for balanced systems is just a crossover. The situation is different for imbalanced systems, where gapless regions emerge in the excitation spectrum and topological phase transitions occur, so long as the sys- tem is stable [9, 10]. A thermodynamic signature of this crossover for balanced systems is seen in the isothermal compressibility discussed next. Isothermal compressibility: An important thermody- namic property, which can now be measured experimen- tally using the fluctuation-dissipation theorem, is the isothermal compressibility κT=−1 V/parenleftbigg∂P ∂V/parenrightbigg T=V N2/parenleftbigg∂N ∂µ/parenrightbigg T.(11) As shown in Fig. 2a, for the RO case, the isother- mal compressibility κTat fixed interaction parameter 1/(kFas) increases with increasing spin-orbit coupling vR, astheFermisystembecomeslessdegeneratereducing the Pauli pressure, and thus more compressible. How- ever, in the ERD case, the isothermal compressibility for fixedinteractionparameterdoes notchangewith increas- ing spin-orbit coupling v. In this high symmetry situa- tion the momentum shift in the energy spectrum and the accompanied shift in the chemical potential do not affect the degeneracy of the Fermi system or the Pauli pres- sure, leading to an isothermal compressibility which is independent of the spin-orbit coupling v. Equation of State and Entropy: Since the thermody- namic potential Ω = −PV, the saddle point pressure isP0(T,µα) =−Ω0/V,which can be shown to be al- ways positive for arbitrary spin-orbit coupling. The gen- eral trend of the pressure for fixed interaction param- eter (from the BCS to the unitarity regimes) is to de- crease with increasing spin-orbit coupling for both ERD and RO cases. The situation in the BEC regime re- quires the inclusion of quantum fluctuations to recover the corresponding Lee-Yang corrections in the presence of spin-orbit effects. The entropy is then calculated from4 /Minus2/Minus101204812 1/Slash1kFasΚTa/RParen1 00.20.40.60.80123 T/Slash1ΕFS0/Slash1Nb/RParen1 FIG. 2: a) Compressibility κT(in units of 1 /(nǫF)) as a function of interaction parameter 1 /(kFas) atT= 0 and b) entropy per particle S0/Nas a function of temperature T(in units of ǫF) at unitarity in the RO case, for the same group of spin-orbit coupling values vR/vFas in Fig. 1. S=−(∂Ω/∂T)V,µα.InFig.2b, weshowthesaddlepoint entropyS0for the RO case at unitarity. For fixed T,S0 decreases with increasing spin-orbit coupling due to the stabilization of superfluidity by the spin-orbit field. Spin Susceptibility Tensor: A rotation of the matrix H0into the helicity basis |⇑/an}b∇acket∇i}ht,|⇓/an}b∇acket∇i}htintroduces order pa- rameters ∆ 0,⇑⇑and ∆ 0,⇓⇓, which are controlled by the spin-orbit coupling. The emergence of the triplet com- ponent affects dramatically the spin susceptibility of the system. Using standard linear response theory [11], the uniform spin susceptibility tensor per unit volume is χij=−µ2 B V/summationdisplay k[aij(k)−bij(k)],(12) where the spin-spin correlations in the single-particle channel are aij(k) =/summationtext iωTr[σiG(k,iω)σjG(k,iω)] and in the pair (anomalous) channel are bij(k) =/summationtext iωTr/bracketleftbig σiF(k,iω)σT jF†(k,iω)/bracketrightbig .The matrices GandF are the block matrices appearing in the inverse of Mde- fined in Eq. (4), /tildewiderM−1(k,iω) =/parenleftbiggG F F†G/parenrightbigg . In Fig. 3a, we show plots of χzzfor the ERD case at T= 0asafunction of1 /(kFas) forvariousvaluesofspin- orbit coupling, and the behavior of χzzfor the RO case is qualitatively similar. In Fig. 3b, we show χzzversusv in the unitary limit 1 /(kFas) = 0. The maximum in χzz corresponds to the maximum in the triplet component of ∆0. For small and large vthe triplet component is small. In the ERD case χzz=χxx/ne}ationslash=χyy, and in the zero temperature limit χyy(T→0) = 0, while χzz=χxx remains finite for non-zero spin-orbit coupling. In the RO caseχzz/ne}ationslash=χxx=χyy, and in the T→0 limit χxx(T→0) =χyy(T→0) =χzz(T→0)/2. Lastly, forh⊥(k) = 0 (no spin-orbit coupling) the spin sus- ceptibilty tensor becomes χij=χδij, where the scalar χ=/bracketleftbig µ2 B/(2VT)/bracketrightbig/summationtext ksech2/bracketleftig/radicalbig ξ2 k+|∆0|2/(2T)/bracketrightig is the Yoshida function, which vanishes at zero temperature,i.e.,χ(T→0) = 0. The existence of non-zero spin re- sponse even at T= 0 is a direct measure of the induced tripletcomponent ofthe orderparameterduetothe pres- ence of spin-orbit coupling, since that a pure singlet su- perfluid atT= 0 must have zero spin susceptibility since all fermions are paired into a zero-spin state. /Minus2/Minus101200.51 1/Slash1kFasΧzza/RParen1 00.511.5200.10.20.3 v/Slash1vFΧzzb/RParen1 FIG. 3: a) Spin susceptibility χzz(in units of µ2 Bn/ǫF) as a function of 1 /(kFas) atT= 0 in the ERD case for v/vF= 0 (solid),v/vF= 0.8 (dashed), v/vF= 1 (dotted), and v/vF= 1.2 (dot-dashed). b) Spin susceptibility χzzas a function of v/vFatT= 0 at unitarity in the ERD case. Conclusions: We have studied the effects of spin-orbit coupling in the evolution from BCS to BEC superfluidity at low temperatures, and concluded that this evolution is just a crossover. We discussed effects of spin-orbit cou- pling on thermodynamic properties including the order parameter,chemical potential, pressure,entropy, isother- mal compressibility and spin susceptibility tensor to sup- port the crossover picture. We also proposed way to experimentally detect the spin-orbit coupling effect by measuring the spin susceptibility. We would like to thank NSF (Grant No. DMR- 0709584) and ARO (Contract No. W911NF-09-1-0220). [1] Y. J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. Porto, and I. B. Spielman, Nature (London) 462, 628 (2009). [2] I. B. Spielman, Phys. Rev. A 79, 063613 (2009). [3] Y. J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011). [4] M. Chapman and C. S´ a de Melo Nature 471, 41 (2011). [5] M. Iskin and C. A. R. S´ a de Melo, Phys. Rev. A 83, 045602 (2011). [6] J. Dalibard, F. Gerbier, G. Juzeli˜ unas, P. ¨Ohberg, arXiv:1008.5378v1 (2010). [7] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [8] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6029 (1984). [9] M. Iskin and C. A. R. S´ a de Melo, Phys. Rev. Let. 97, 100404 (2006). [10] M. Gong, S.Tewari, C.Zhang, arXiv:1105.1796v1(2011) . [11] Lev P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001).

2002.07779v2.Majorana_like_localized_spin_density_without_bound_states_in_topologically_trivial_spin_orbit_coupled_nanowires.pdf

Majorana-like localized spin density without bound states in topologically trivial spin-orbit coupled nanowires Lorenzo Rossi,1,Fabrizio Dolcini,1and Fausto Rossi1 1Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, 10129 Torino, Italy In the topological phase of spin-orbit coupled nanowires Majorana bound states are known to localize at the nanowire edges and to exhibit a spin density orthogonal to both the magnetic eld and the spin-orbit eld. By investigating a nanowire exposed to a uniform magnetic eld with an interface between regions with dierent spin-orbit couplings, we nd that the orthogonal spin density is pinned at the interface even when both interface sides are in the topologically trivial phase, and even when no bound state is present at all. A trivial bound state may additionally appear at the interface, especially if the spin-orbit coupling takes opposite signs across the interface. However, it can be destroyed by a smoothening of the spin-orbit prole or by a magnetic eld component parallel to the spin-orbit eld. In contrast, the orthogonal spin density persists in various and realistic parameter ranges. We also show that, while the measurement of bulk equilibrium spin currents has been elusive so far, such robust orthogonal spin density peak may provide a way to detect spin current variations across interfaces. I. INTRODUCTION Topological materials have been under the spotlight of experimental and theoretical research for years by now, due to their relevance in terms of fundamental physics and their broad spectrum of applications, from spin- tronics to quantum computing[1{3]. One of the most remarkable features of a topological phase is that edge states localize at the interface with a topologically triv- ial phase. Indeed several theoretical analysis have shown that such interface states emerge at the boundaries of topological insulators (TIs), like the one-dimensional Su- Schrieer-Heeger model for polyacetylene [4{6] or the two-dimensional quantum spin Hall systems [7{11]. Sim- ilarly, as rst predicted by Kitaev[12], at the edges of topological superconductors[13{15], realized in proxi- mized nanowires (NWs) with Rashba spin-orbit coupling (RSOC)[16,17], in ferromagnetic atomic chains deposited on a superconductor[18], or in two-dimensional TIs prox- imized by superconductors and magnets[19{21], Majo- rana quasi-particles (MQPs) appear. These exotic quasi- particles, which are equal to their anti-particles, are cur- rently considered a promising platform for quantum com- puting in view of their non-trivial braiding properties and their robustness to charge decoherence eects[22{26]. While in theoretical models a topological phase is characterized by a well specied range of parameters in the Hamiltonian, when it comes to nding an ex- perimental evidence of such phase in a given material, the dicult question is \how to distinguish signatures of a topological from a trivial bound state?" As a general criterion, a topological bound state is stable to perturbations that do not close the gap of the topological phase, while a trivial bound state is not. However, because in a given experimental setup the actual parameter range characterizing the topological phase is not known a priori and/or may be relatively narrow, the search for such stable signatures is ingeneral not a trivial task. For instance, although it is by now commonly accepted that MQPs exist in RSOC nanowires[27{34], the early observations of a zero-bias conductance peak stable to magnetic eld and Fermi energy variations were cautiously claimed to be compatible with the existence of MQPs. The remark that (a)(b)(c)xzhRSO <latexit sha1_base64="j/sLwretYzmE5t6woKQO5Xaj070=">AAAB73icdVDLTgIxFO3gC/GFunTTiCauJh2IijsSNu4AlUcCI+mUAg2dzth2TMiEn3DjQmPY+hX+gzv/xjJgokZPcpOTc+7Nvfd4IWdKI/RhpZaWV1bX0uuZjc2t7Z3s7l5DBZEktE4CHsiWhxXlTNC6ZprTVigp9j1Om96oPPOb91QqFogbPQ6p6+OBYH1GsDZSa9iNryuT26tuNofsU+RcnCGIbJQgIUWn4EBnoeRK8KgyrZXfqt3se6cXkMinQhOOlWo7KNRujKVmhNNJphMpGmIywgPaNlRgnyo3Tu6dwGOj9GA/kKaEhon6fSLGvlJj3zOdPtZD9dubiX957Uj3i27MRBhpKsh8UT/iUAdw9jzsMUmJ5mNDMJHM3ArJEEtMtIkoY0L4+hT+Txp52ynY+ZpJowTmSIMDcAhOgAPOQQlcgiqoAwI4eABP4Nm6sx6tF2s6b01Zi5l98APW6yfN9JLB</latexit>hLSO <latexit sha1_base64="N+0tmmcmR6cpxgaZ+Oa3L4FC6rg=">AAAB73icdVDLSgMxFM3UV62vqks3wSq4GjItat0VunEhtEX7gHYsmTRtQzOZMckIZehPuHGhSLd+hf/gzr8xnVZQ0QMXDufcy733eCFnSiP0YaWWlldW19LrmY3Nre2d7O5eQwWRJLROAh7IlocV5UzQumaa01YoKfY9TpveqDzzm/dUKhaIGz0OqevjgWB9RrA2UmvYja8rk9urbjaH7FPkXJwhiGyUICFFp+BAZ6HkSvCoMq2V36rd7HunF5DIp0ITjpVqOyjUboylZoTTSaYTKRpiMsID2jZUYJ8qN07uncBjo/RgP5CmhIaJ+n0ixr5SY98znT7WQ/Xbm4l/ee1I94tuzEQYaSrIfFE/4lAHcPY87DFJieZjQzCRzNwKyRBLTLSJKGNC+PoU/k8aedsp2PmaSaME5kiDA3AIToADzkEJXIIqqAMCOHgAT+DZurMerRdrOm9NWYuZffAD1usnxNySuw==</latexit>hxhz <latexit sha1_base64="gk2xoj1OGm663i3H4rmq7nq+5fI=">AAAB7XicdVDLSgMxFM3UV62vVlfSTbAIrkpSwbbgouDGZQv2Ae1QMmmmjWYyQ5IR6tB/cONCEbf+jQt3+jVmWgUVPXDhcM693HuPFwmuDUJvTmZpeWV1Lbue29jc2t7JF3Y7OowVZW0ailD1PKKZ4JK1DTeC9SLFSOAJ1vWuzlK/e82U5qG8MNOIuQEZS+5zSoyVOpNhcjODw3wJlRFCGGOYElw9QZbU67UKrkGcWhalRmFQfG/tvzSH+dfBKKRxwKShgmjdxygybkKU4VSwWW4QaxYRekXGrG+pJAHTbjK/dgYPrTKCfqhsSQPn6veJhARaTwPPdgbETPRvLxX/8vqx8WtuwmUUGybpYpEfC2hCmL4OR1wxasTUEkIVt7dCOiGKUGMDytkQvj6F/5NOpYyPy5WWTeMULJAFRXAAjgAGVdAA56AJ2oCCS3AL7sGDEzp3zqPztGjNOJ8ze+AHnOcPneKSGQ==</latexit>xy↵L <latexit sha1_base64="y5vy+0a4fUvf6sjD8yADbfCjPbk=">AAAB73icdVC7SgNBFJ31GeMr0UrSDAbBaplNyAssAjYWFgmYB2SXMDuZTYbMPpyZFcKSn7CxUMTWn7Gw069xNlFQ0QMXDufcy733uBFnUiH0Zqysrq1vbGa2sts7u3v7ufxBV4axILRDQh6Kvosl5SygHcUUp/1IUOy7nPbc6Xnq926okCwMrtQsoo6PxwHzGMFKS30b82iCh5fDXBGZqFpplBFEZgVZtUZDE4Sq9XIJWpqkKDbzduG9ffTSGuZe7VFIYp8GinAs5cBCkXISLBQjnM6zdixphMkUj+lA0wD7VDrJ4t45PNHKCHqh0BUouFC/TyTYl3Lmu7rTx2oif3up+Jc3iJVXdxIWRLGiAVku8mIOVQjT5+GICUoUn2mCiWD6VkgmWGCidERZHcLXp/B/0i2ZVtkstXUaZ2CJDCiAY3AKLFADTXABWqADCODgFtyDB+PauDMejadl64rxOXMIfsB4/gAZ+ZL0</latexit>s <latexit sha1_base64="YjbbkXog/h3S8sEYe8EPOcyQX4Y=">AAAB8HicbVC7SgNBFL3rM8ZXopWkGQyCVdiNhRYWARvLBMxDskuYnZ1NhszMLjOzQgj5ChsLRWz9GAs7/Ronj0ITDwwczjmXufeEKWfauO6Xs7a+sbm1ndvJ7+7tHxwWikctnWSK0CZJeKI6IdaUM0mbhhlOO6miWISctsPhzdRvP1ClWSLvzCilgcB9yWJGsLHSvc9tNMI93SuU3Yo7A1ol3oKUa0W/9N04+aj3Cp9+lJBMUGkIx1p3PTc1wRgrwwink7yfaZpiMsR92rVUYkF1MJ4tPEFnVolQnCj7pEEz9ffEGAutRyK0SYHNQC97U/E/r5uZ+CoYM5lmhkoy/yjOODIJml6PIqYoMXxkCSaK2V0RGWCFibEd5W0J3vLJq6RVrXgXlWrDtnENc+SgBKdwDh5cQg1uoQ5NICDgEZ7hxVHOk/PqvM2ja85i5hj+wHn/AZ1ckzc=</latexit>x<latexit sha1_base64="MhnwFn71Pg2jG6I8ipZUXInB7T4=">AAAB6HicbZC7SgNBFIbPxluMt6ilIINBsAq7sdBCMGBjmYC5QLKE2clJMmb2wsysGJaUVjYWitj6FLa+gp3PoA/h5FJo4g8DH/9/DnPO8SLBlbbtTyu1sLi0vJJezaytb2xuZbd3qiqMJcMKC0Uo6x5VKHiAFc21wHokkfqewJrXvxjltRuUiofBlR5E6Pq0G/AOZ1Qbq3zbyubsvD0WmQdnCrnz96+7/bfyd6mV/Wi2Qxb7GGgmqFINx460m1CpORM4zDRjhRFlfdrFhsGA+qjcZDzokBwap006oTQv0GTs/u5IqK/UwPdMpU91T81mI/O/rBHrzqmb8CCKNQZs8lEnFkSHZLQ1aXOJTIuBAcokN7MS1qOSMm1ukzFHcGZXnodqIe8c5wtlO1c8g4nSsAcHcAQOnEARLqEEFWCAcA+P8GRdWw/Ws/UyKU1Z055d+CPr9Qf2V5F5</latexit>↵R <latexit sha1_base64="DPd8pjcvZazxgG/I2v+nNtk2yEE=">AAAB73icdVDLSgMxFM34rPXV6kq6CRbB1ZBp6QtcFNy4bMU+oDOUTJppQzMPk4xQhv6EGxeKuPVnXLjTrzHTKqjogQuHc+7l3nvciDOpEHozVlbX1jc2M1vZ7Z3dvf1c/qArw1gQ2iEhD0XfxZJyFtCOYorTfiQo9l1Oe+70PPV7N1RIFgZXahZRx8fjgHmMYKWlvo15NMHDy2GuiExUrTTKCCKzgqxao6EJQtV6uQQtTVIUm3m78N4+emkNc6/2KCSxTwNFOJZyYKFIOQkWihFO51k7ljTCZIrHdKBpgH0qnWRx7xyeaGUEvVDoChRcqN8nEuxLOfNd3eljNZG/vVT8yxvEyqs7CQuiWNGALBd5MYcqhOnzcMQEJYrPNMFEMH0rJBMsMFE6oqwO4etT+D/plkyrbJbaOo0zsEQGFMAxOAUWqIEmuAAt0AEEcHAL7sGDcW3cGY/G07J1xficOQQ/YDx/ACMRkvo=</latexit>+Z <latexit sha1_base64="9m94K9yXAV8NXVz3l+CM4GGCkH8=">AAAB8HicbVC7SgNBFL0bXzG+Eq0kzWIQBCHsxkILi4AWlgmYh2aXMDuZTYbMzC4zs0JY8hU2ForY+jEWdvo1Th6FJh64cDjnXu69J4gZVdpxvqzMyura+kZ2M7e1vbO7ly/sN1WUSEwaOGKRbAdIEUYFaWiqGWnHkiAeMNIKhlcTv/VApKKRuNWjmPgc9QUNKUbaSHen3jVhGnXvu/mSU3amsJeJOyelasErftcPP2rd/KfXi3DCidCYIaU6rhNrP0VSU8zIOOclisQID1GfdAwViBPlp9ODx/axUXp2GElTQttT9fdEirhSIx6YTo70QC16E/E/r5Po8MJPqYgTTQSeLQoTZuvInnxv96gkWLORIQhLam618QBJhLXJKGdCcBdfXibNStk9K1fqJo1LmCELRTiCE3DhHKpwAzVoAAYOj/AML5a0nqxX623WmrHmMwfwB9b7DwHSktI=</latexit>Z <latexit sha1_base64="z7nvOfjxyjKwlE+VDpaJme4as6g=">AAAB8HicbVC7SgNBFL0bXzG+Eq0kzWIQbAy7sdDCIqCFZQLmodklzE5mkyEzs8vMrBCWfIWNhSK2foyFnX6Nk0ehiQcuHM65l3vvCWJGlXacLyuzsrq2vpHdzG1t7+zu5Qv7TRUlEpMGjlgk2wFShFFBGppqRtqxJIgHjLSC4dXEbz0QqWgkbvUoJj5HfUFDipE20t2pd02YRt37br7klJ0p7GXizkmpWvCK3/XDj1o3/+n1IpxwIjRmSKmO68TaT5HUFDMyznmJIjHCQ9QnHUMF4kT56fTgsX1slJ4dRtKU0PZU/T2RIq7UiAemkyM9UIveRPzP6yQ6vPBTKuJEE4Fni8KE2TqyJ9/bPSoJ1mxkCMKSmlttPEASYW0yypkQ3MWXl0mzUnbPypW6SeMSZshCEY7gBFw4hyrcQA0agIHDIzzDiyWtJ+vVepu1Zqz5zAH8gfX+AwTqktQ=</latexit> FIG. 1. (Color online) (a) Top view of a Rashba nanowire deposited on a substrate: the Rashba eective magnetic eld hSOis directed along z, whereas an actual magnetic eld, externally applied in the substrate plane, has components in thex-zsubstrate plane. The NW contains an interface between two regions with dierent RSOC values. (b) The spatial prole of the RSOC across the interface of the NW, ranging from the bulk values LtoRover a smoothening lengthscale s. (c) Examples of electronic bands related to the bulks of the two interface sides, the left-hand side in the Zeeman dominated regime, and the right-hand side in the Rashba-dominated regime.arXiv:2002.07779v2 [cond-mat.mes-hall] 15 May 20202 such scenario may also be caused by Kondo eect[35], disorder[36,37] or inhomogeneities[38] has recently spurred further investigations, which pointed out that in the topological phase also trivial bound states may be present[38{44]. Furthermore, a quite recent analysis[45], carried out on a nanowire with homogeneous RSOC and with inhomogeneous magnetic eld, showed that at the interface between two magnetic domains with opposite magnetization directions, bound states appear that are unrelated to the Jackiw-Rebbi topological states. A more clear evidence of topological bound states re- quires a spatially resolved analysis. This was done, for instance, in ferromagnetic atomic chains deposited on a superconductor[46], where the combined use of spa- tially resolved spectroscopic and spin-polarized measure- ments showed that zero-bias conductance peaks are due to states localized at the ends of the chain. Yet, the smoking gun enabling one to identify such states with MQPs is their disappearance in the normal state, when superconductivity is suppressed. As far as NWs are con- cerned, it has been pointed out that MQPs in the topo- logical phase exhibit an orthogonal spin density, i.e., a component perpendicular to both the magnetic and spin- orbit elds, localized at the NW ends [47{49]. In order to identify a topological phase in a given system, it is thus particularly important to understand whether and when the topologically trivial phase may exhibit observ- ables that are spatially localized at the interfaces and that may mistakenly be interpreted as a topological sig- nature. So far, this aspect has been analyzed far less than the topological bound states. This paper is meant to bridge this gap. Specically, we consider the case of a RSOC NW exposed to a uni- form magnetic eld, and we analyze the spatial prole of charge and spin densities at the interface between two regions with dierent values of RSOC, as sketched in Fig.1(a). Such type of interfaces emerge quite natu- rally in any realistic setup, since metallic electrodes or gates are typically deposited on top of a portion of the NW, thereby altering the underneath structure inversion asymmetry characterizing the very RSOC. Furthermore, the recent advances in various gating techniques, includ- ing gate-all-around approaches, allow a large tunability of the RSOC constant, possibly even changing the RSOC sign [50{57]. Importantly, on both sides of the interface, the NW that we consider is in the topologically trivial phase , since no superconducting coupling is included. Furthermore, as the gap depends only on the strength of the magnetic eld, it never closes at the interface, since the magnetic eld is assumed to be uniform . Thus, under these conditions the existence of bound states of topological origin is ruled out a priori. Our analysis unveils various noteworthy aspects. In the rst instance, a bound state may appear at the in- terface. Importantly such bound state, while being nottopological, is nota customary interface state merely arising from the inhomogeneity of the RSOC. Indeed it can only exist if an external magnetic eld is applied orthogonally to the RSOC eld direction, and if its in- tensity fullls specic conditions with respect to the two spin-orbit energies characterizing the two NW regions. The conditions of existence and the robustness of the bound state are analyzed in details in terms of dier- ent values of RSOC across the interface, including the smoothening length characterizing the crossover between these two values and the presence of a magnetic eld component parallel to the RSOC eld direction. Second, we nd for realistic values of chemical poten- tial and temperature that the orthogonal spin density exhibits a peak pinned at the interface. Despite the NW is in the topologically trivial phase, such a peak is rela- tively robust to other parameter variations. In fact, we show that it persists even when the bound state is ab- sent, indicating that in such a case also the continuum states locally modify their spin-texture to maintain such eect. Furthermore, by considering the case of two interfaces, we show that the peaks of the orthogonal spin density are opposite at the two ends of the inner NW region, similarly to what occurs for MQPs in the topological phase. These results imply that a localized orthogonal spin- density can neither be taken as a unique signature of a MQP, nor of a topologically trivial bound state. How- ever, we argue that it can represent a useful way to in- directly detect spin current dierences. Indeed, while the detection of a bulk equilibrium spin current, which emerges in a homogeneous NW from the correlations be- tween spin and velocity induced by the magnetic and spin-orbit elds[58], has been elusive so far, any variation of equilibrium spin current occurring at the interface is precisely related to the orthogonal spin-density peak pre- dicted here. The paper is organized as follows. In Sec. II we intro- duce the model and describe the involved energy scales. In Sec. III we present the results concerning the bound state, discussing rst the case of a sharp RSOC interface prole in the presence of a magnetic eld applied along the NW axis. Then we analyze the more realistic case of a nite smoothening length in the prole, and address the eect of a magnetic eld component parallel to the spin-orbit eld direction. In Sec. IV we investigate the spatial prole of the charge and spin densities, and ana- lyze specically the bound state contribution to them. In Sec. V we discuss the interpretation of our main results, we include the case of two interfaces and we propose some possible experimental realizations. Finally, in Sec. VI we draw our conclusions.3 II. THE MODEL FOR A SOC INTERFACE A. Nanowire Hamiltonian Letxdenote the longitudinal axis of a NW deposited on a substrate. The NW is characterized by a RSOC, which is assumed to take two dierent values LandR on the left and on the right side of an interface, respec- tively [see Figs. 1(a) and 1(b)]. This inhomogeneity in the RSOC prole (x) may result e.g. from the presence of a gate covering only one portion of the NW, or by two dierent gate voltage values applied to top/bottom gates or to the substrate. The crossover between Land Roccurs over a smoothening length s. Denoting by z the direction of the spin-orbit eld hSO, i.e., the eective \magnetic" eld generated by the RSOC [see Fig.1(a)], the NW Hamiltonian is ^H=Z ^ y(x)H(x)^ (x)dx ; (1) where H(x) =p2 x 2m0f(x);pxg 2~zh
:(2) Here ^ (x) = ( ^ "(x);^ #(x))Tis the electron spinor eld, with";#corresponding to spin projections along z,px= i~@xis the momentum operator, mthe NW eective mass,0the 22 identity matrix, and = (x;y;z) are the Pauli matrices. For deniteness, we take the location of the interface at x= 0. The anticommuta- tor in Eq.(2) is necessary since pxdoes not commute with the inhomogeneous RSOC (x) [59,60]. The last term in Eq.(2), where h=gBB=2, describes the Zee- man coupling with an external uniform magnetic eld B= (Bx;0;Bz) applied in the substrate plane, with B denoting the Bohr magneton and gthe NW Land e factor. It is useful to decompose the magnetic gap energy vector ash=hxix+hziz, wherehxandhzdenote the compo- nents parallel and perpendicular to the nanowire axis x, i.e., perpendicular and parallel to the Rashba spin-orbit eld direction z, respectively [see Fig.1(a)]. Although for most of our analysis we shall focus on the case of the magnetic eld directed along the nanowire axis x, we shall also discuss the eects of the component hzparallel tohSO. B. Energy scales In order to describe the results about the inhomo- geneous RSOC prole at the interface, it is rst worth pointing out the energy scales involved in the problem. 1. The homogeneous NW Let us start by brie y summarizing the case of a ho- mogeneous prole (x)in Eq.(2), for an innitelylong NW. In such case the Hamiltonian (2) commutes withpx, and the spectrum reads [58{60] E(k) ="0 kp h2x+ (k+hz)2; (3) where"0 k=~2k2=2mis the customary parabolic spec- trum in the absence of RSOC and magnetic eld. The spectrum (3) describes two bands separated by a minimal gap 2
Z, where the quantity
Z=jhxj (4) shall be henceforth called the magnetic gap energy. Moreover, the RSOC identies the spin-orbit wavevec- torkSO=mjj=~2, which characterizes, in the absence of external magnetic eld, the two degenerate minima E(kSO) =ESOof the spectrum, where ESO=m2 2~2=~2k2 SO 2m(5) is called the spin-orbit energy. In the case hz= 0 the magnetic eld is directed alongx, i.e., orthogonal to the RSOC eld, the spec- trum (3) is symmetric E(k) =E(+k). Two regimes can be identied: (a) in the Zeeman-dominated regime (
Z>2ESO) both bands have a minimum at k= 0, which takes values Emin =
Z, respectively. (b) in the Rashba-dominated regime (
Z<2ESO), the upper band still has a minimum Emin += +
Zatk= 0, while the lower band acquires two lower and degener- ate minima Emin =ESO
2 Z=4ESOoccurring at k=kmin, with kmin=kSOq 1
2 Z=4E2 SO: (6) When a component hz6= 0 parallel to the RSOC eld is also present, the minimal gap 2
Zbetween the two bands occurs at k=hz=and the spectrum is no longer symmetric E(k)6=E(+k). The eigenfunctions related to the spectrum (3) read k(x) =wkexp[ikx]=p ; (7) with denoting the system length. They describe plane waves with spinors wk=0 @cosk 2 sink 21 Awk+=0 @sink 2 cosk 21 A; (8) whose spin orientation n(k)(sink;0;cosk) lies on thex-zsubstrate plane and forms with the z-axis an anglek2[;]. The latter, dened through 8 >>< >>:cosk=k+hzp (k+hz)2+h2x sink=hxp (k+hz)2+h2x; (9)4 depends on the wavevector k, the magnetic eld and the RSOC. In particular, it is worth recalling that in the case of a magnetic eld along the NW axis ( hz= 0) and in the deep Rashba-dominated regime (
Z2ESO) the states with energy inside the magnetic gap mimic the he- lical edge states of the quantum spin Hall eect. Indeed their spin orientation, determined mainly by the RSOC, is opposite for right- and left-moving electrons, whose helicity is determined by the sign of the RSOC . This is precisely the most suitable regime for the topological phase to be induced by an additional s-wave supercon- ducting coupling [16,17,61,62]. 2. The NW with a RSOC interface When an interface separates two portions of a NW characterized by two dierent values LandRof RSOC [see Fig.1(b)], the momentum pxdoes not commute with the Hamiltonian characterized by an inhomogeneous (x)-prole, and the spectrum cannot be labeled by a wavevector k. Before attacking the inhomogeneous prob- lem in the next section, it is worth identifying the energy scales and the possible scenarios one can expect in the in- terface problem from a preliminary analysis of the bulks of the two regions across the RSOC interface. To begin with, the two bulk values LandRof the two NW regions lead to two spin-orbit energies (5) ESO; =m2  2~2=R=L : (10) Without loss of generality, we shall choose the RSOC with higher magnitude jjon the right-hand side, and we can set it to a positive value, R>0, whereas the RSOC on the left-hand side is allowed to take any value in the rangeRLR[63]. Correspondingly, one hasESO;LESO;R. The fact that the magnetic eld is uniform has important consequences, which are easily illustrated in the case hz= 0: First, in the bulk of each region the gap between the bands is always given by 2
Z, regardless of the regime (Rashba- or Zeeman- dominated) of each interface side. Secondly, the overall minimum of the two energy band bottoms is determined by the band bottom of the side with higher spin-orbit energy, i.e., the right-hand side, and is thus given by Emin band=8 >< >:
Z if
Z>2ESO;R ESO;R 1 +
2 Z 4E2 SO;R if
Z<2ESO;R (11) With these notations, if the right-side is in the Zeeman-dominated regime, so is the left-hand side, whereas if the right-side is in the Rashba-dominated regime the left-hand side can be either in the Rashba- or in the Zeeman-dominated regime. There can thus be only three possible regime combinations: (i)ESO;LESO;R
Z=2, where both sides areZeeman-dominated; (ii)
Z=2ESO;LESO;R, where both sides are Rashba-dominated; (iii) ESO;L
Z=2ESO;R, where the left-side is Zeeman-dominated while the right-side is Rashba- dominated. The bands of the latter case are illustrated as an example in Fig.1(c). III. BOUND STATE AND ITS STABILITY In this section we focus on the inhomogeneous interface problem. By diagonalizing the inhomogeneous Hamilto- nian, with methods to be described here below, we nd that its spectrum always exhibits a continuum branch, whose bottom Emin cont coincides with the minimal band energy obtained in Eq.(11) from the comparison of bare bulk spectra. However, for some parameter range (see below), the spectrum also displays an additional eigen- valueEbs, lying below the continuum spectrum Emin cont. The related eigenfunction exhibits an evanescent behav- ior forjxj!1 . When such bound state exists, we dene its positive `binding energy' as Eb=Emin contEbs>0: (12) Here below we now analyze the conditions for its exis- tence. A. The case of a sharp interface Let us start by analyzing the existence of the bound state in the case of a sharp interface, where the smoothen- ing length s!0 vanishes and the prole can be as- sumed as (x) =L(x) +R(x) (13) withdenoting the Heaviside function. In this case the eigenfunctions of the inhomogeneous problem can be ob- tained analytically by combining the eigenstates (7) of the homogeneous problem in each side and by matching them appropriately at the interface. In particular, since bound states are eigenstates with evanescent wavefunc- tion forjxj!1 , they are obtained requiring that the wavevector kacquires an imaginary part. Details of such calculation can be found in Appendix A. By keeping one side of the junction as a reference, e.g. the right-hand side where the bulk spin-orbit en- ergy is maximal, the problem can be formulated in terms of dimensionless parameters, namely the RSOC ratio L=R2[1;1] and the energy ratios Eb=ESO;R and h=ESO;R to the maximal spin-orbit energy ESO;R. We shall focus here below on the case where the applied mag- netic eld is directed only along the nanowire axis x, h=hxix, i.e., orthogonally to the Rashba spin-orbit eld, while the eects of a parallel magnetic eld compo- nenthzwill be discussed later.5 The results are presented in Fig.2. In particular, Fig.2(a) displays the phase diagram of the existence of the bound state. For a suciently strong magnetic eld,
Z>2ESO;R, i.e., when both NW sides are in the Zeeman-dominated regime, the bound state always ex- ists, while for
Z<2ESO;R, where the NW right side is in the Rashba-dominated regime, the bound state may or may not exist. In particular, for
Z= 0 (no external magnetic eld), the bound state never exists, regardless of the ratio of the two RSOC values across the interface. This shows that the bound state, although it has no topo- logical origin, it is not an intrinsic interface state like the ones occurring at a customary semiconductor interface. The thick black in Fig.3(a) denotes the transition curve for the existence of the bound state, and corresponds to the vanishing of the binding energy, Eb= 0. In particu- lar, the parabolic curve for
Z=2ESO;R<1 is described by the equation
? Z 2ESO;R=r 1 +L=R 2; (14) while the upper horizontal line corresponds to the ho- mogeneous NW in the Zeeman-dominated regime, where the bound state does not exist, as is obvious to expect. Then, Fig.2(b) shows, for four dierent values of the ra- tioL=R, the behavior of the binding energy Ebas a function of the ratio
Z=2ESO;R. Several features are noteworthy. First, in all cases the binding energy exhibits a non- monotonic behavior as a function of the magnetic gap en- ergy, with a maximum Emax boccurring for a magnetic gap energy slightly below the transition value
Z= 2ESO;R between the Rashba- and Zeeman-dominated regime of the right-hand side, highlighted by the vertical dashed line as a guide to the eye. Secondly, the bound state energy strongly depends on the ratioL=Rof the two RSOC values, and is typi- cally much higher when the RSOC changes sign across the interface. In particular, the optimal condition for the existence of the bound state is L=R=1, i.e., when the RSOC takes equal and opposite values of two sides: In this situation not only the bound state always exists, its binding energy is also higher than any other case. For these reasons, we shall henceforth term such case the `optimal conguration'. In particular, it can be shown that, for weak applied eld (
Z2ESO;R) the binding energy of the optimal conguration behaves as Eb'
2 Z=4ESO;R while for strong eld (
Z2ESO;R) one ndsEb'E2 SO;R=2
Z. Third, for all other cases ( 1<L=R<1) the bound state exists only if the magnetic gap energy overcomes a minimal threshold value, which precisely corresponds to the transition curve of Fig.2(a) described by Eq.(14). The threshold of the magnetic gap energy increases as the RSOC ratio L=Rincreases from the negative value 1 to the value +1, corresponding to the homogeneous case. Furthermore, the following `rule of thumb' can be in- ferred: when the band bottoms of the two interface sides (a) (b) 0,00,51,01,52,02,53,03,54,00,000,020,040,060,080,100,12 Eb/2ESO,R ΔZ / 2ESOαL /αR -1 -1/2 0 +1/2 123-1-0.50.51↵L↵R<latexit sha1_base64="rLi5dWJoEzS+rmB8D7JOD3Z9vf8=">AAACAnicbZDLSsNAFIYnXmu9RV2Jm8EiuCqJFHRZcOPCRRV7gSaEk+mkHTqZhJmJUEJx46u4caGIW5/CnW/jtI2grT8MfPznHM6cP0w5U9pxvqyl5ZXVtfXSRnlza3tn197bb6kkk4Q2ScIT2QlBUc4EbWqmOe2kkkIcctoOh5eTevueSsUScadHKfVj6AsWMQLaWIF96EUSSO4BTwcQXI9/6HYc2BWn6kyFF8EtoIIKNQL70+slJIup0ISDUl3XSbWfg9SMcDoue5miKZAh9GnXoICYKj+fnjDGJ8bp4SiR5gmNp+7viRxipUZxaDpj0AM1X5uY/9W6mY4u/JyJNNNUkNmiKONYJ3iSB+4xSYnmIwNAJDN/xWQAJhNtUiubENz5kxehdVZ1a9XaTa1Sd4o4SugIHaNT5KJzVEdXqIGaiKAH9IRe0Kv1aD1bb9b7rHXJKmYO0B9ZH9/LIpei</latexit> no bound statebound stateZ2ESO,R <latexit sha1_base64="Rc/TcV8TiHaTtn519ePe1MTR3bA=">AAACBHicbVC7SgNBFL3rM8ZXfDSSZjAIFhJ3Y6FlwAh2xkcemIRldjKbDJl9MDMrhGULG3/FxkIRWz/CztqvsHPyKDTxwIXDOfdy7z1OyJlUpvlpzMzOzS8sppbSyyura+uZjc2qDCJBaIUEPBB1B0vKmU8riilO66Gg2HM4rTm904Ffu6NCssC/Uf2Qtjzc8ZnLCFZasjPZpiswiZslyhW2b5O4gM7s+Pri4CpJ7EzOzJtDoGlijUmueFja+YLt77Kd+Wi2AxJ51FeEYykblhmqVoyFYoTTJN2MJA0x6eEObWjqY4/KVjx8IkF7WmkjNxC6fIWG6u+JGHtS9j1Hd3pYdeWkNxD/8xqRck9aMfPDSFGfjBa5EUcqQINEUJsJShTva4KJYPpWRLpYp6J0bmkdgjX58jSpFvLWUb5wqdMwYYQUZGEX9sGCYyjCOZShAgTu4RGe4cV4MJ6MV+Nt1DpjjGe24A+M9x+vH5pY</latexit> Z/2ESO,R <latexit sha1_base64="kJ5b2RL3dJKCQgrn4r6Z4b3SG2A=">AAAB/HicbVDLSsNAFJ3UV62vaN25GSyCC6lJXOiyoII766MPbEOYTKft0MkkzEyEEOqvuHGhiCvRD3HnL7h37/Sx0NYDFw7n3Mu99/gRo1JZ1qeRmZmdm1/ILuaWlldW18z1jaoMY4FJBYcsFHUfScIoJxVFFSP1SBAU+IzU/N7xwK/dEiFpyK9VEhE3QB1O2xQjpSXPzDdPCFPIu9l34KmXXp3vXfY9s2AVrSHgNLHHpFDa/H6TX6/dsmd+NFshjgPCFWZIyoZtRcpNkVAUM9LPNWNJIoR7qEMamnIUEOmmw+P7cEcrLdgOhS6u4FD9PZGiQMok8HVngFRXTnoD8T+vEav2kZtSHsWKcDxa1I4ZVCEcJAFbVBCsWKIJwoLqWyHuIoGw0nnldAj25MvTpOoU7YOic6HTcMAIWbAFtsEusMEhKIEzUAYVgEEC7sEjeDLujAfj2XgZtWaM8Uwe/IHx/gPpGZgL</latexit>FIG. 2. (Color online) The case of a sharp prole interface Eq.(13). (a) The phase diagram for the existence of the bound state is shown as a function of the magnetic gap energy (in units of twice the maximal spin-orbit energy 2 ESO;R) and of the ratio between the two RSOC values across the interface. The thick black line identies the transition curve, where the binding energy vanishes. The vertical thin dashed line indi- cates the crossover value from the Rashba-dominated to the Zeeman-dominated regime for the right-side of the interface. (b) The binding energy Ebof the bound state as a function of
Z=2ESO;R for four dierent values of the RSOC ratio across the interface. are equal, the bound state certainly exists. Indeed a close inspection of Fig.2 shows that this certainly occurs in these two situations: (i) when
Z=2ESO;R>1, i.e., when both sides are in the Zeeman-dominated regime and their band bottoms are both equal to
Z; (ii) when L=R, i.e., when the two spin-orbit energies (10) are equal, both sides are in the same regime (Rashba- or Zeeman-dominated) and thus have the same band bot- toms. In all other cases the existence of the bound state depends on the specic energy ratios. Finally, even when the bound state exists, its binding energy can be quite small. For instance, the maximal binding energy in the case where L=R= 1=2 is about 25 times smaller than the maximal value in the opti- mal caseL=R=1. Similarly, even in the regime
Z=2ESO;R>1 the binding energy decreases with in- creasing magnetic eld.6 B. Eects of smoothening length In any realistic system the crossover between two RSOC bulk values occurs over a nite smoothening lengths. To include such eect we now assume the following prole function (x) =R+L 2+RL 2Erf p 8x s! ; (15) which varies from LtoRup to 2% within the length- scales. In Eq.(15) Erf denotes the error function. Al- though in the presence of such smoothened prole the model cannot be solved analytically, it can be approached by an exact numerical diagonalization of the Hamilto- nian (2), with a method similar to the one introduced in Ref.[58], whose details specic to the prole (15) are summarized in App.B. Instead of expressing the results in terms of dimensionless parameters, we now choose to x the parameters to realistic setup values. For denite- ness, we consider the case of a InSb NW, with an eective massm= 0:015meand a maximal spin-orbit energy ESO;R = 0:25 meV. Furthermore, in order to appreciate the eects of the smoothening length, we focus on the case of the optimal conguration R=L=1. The re- sults, displayed in Fig.3(a), show the binding energy as a function of the magnetic gap energy
Zfor four dif- ferent values of the smoothening length. As one can see, while for the ideal case s!0 (sharp prole) the bound state always exists, for any nite smoothening length the bound state only appears above a threshold value for the Zeeman eld. For suciently strong applied magnetic eld (Zeeman-dominated regime) the bound state always exists. However, the binding energy exhibits an overall suppression for increasing s. These eects can be un- derstood be realizing that a crossover from RtoR in the RSOC prole occurring over a nite smoothening length can, to a rst approximation, be considered as a stair-like sequence of smaller sharp -steps. As the anal- ysis carried out above on the sharp prole indicates (see Fig.2), in the case of a non-optimal jump L>R, a threshold value for
Zdoes exist and the binding energy is reduced. In summary, a nite smoothening length s broadens the white portion of the sharp-prole phase di- agram Fig.2(a) where the bound state does not exist, and suppresses the binding energy. C. Eects of a parallel eld component So far, we have analyzed cases where the magnetic eldhxis directed along the NW. Here we want to discuss the eect of a magnetic eld component hzparallel to the spin-orbit eld. We rst point out that, for hz6= 0 and hx= 0, i.e., for a magnetic eld directed purely along the spin-orbit eld direction z, the eigenvalue problem for the Hamiltonian (2) completely decouples in the two spin-"and spin-#components, and it can be shown that (a) (b)0,00,51,01,52,00,000,010,020,030,040,050,060,07 Eb [meV] ΔZ [meV]λs [nm] 0 50 100 300-α to α (smooth)ESO,R = 0.25 meVαL / αR = -1 0,00,51,01,52,00,000,010,020,030,040,05Eb [meV]ΔZ [meV]hz [meV] 0 0.10 0.25 0.50 0.75 1.00ESO,R = 0.25 meVαL/αR = -1λs = 50 nmFIG. 3. (Color online) The binding energy as a function of the magnetic gap energy, for an interface with L=R, with ESO;R =ESO;L = 0:25 meV. (a) The eects of a smoothening length. (b) Eects of a magnetic eld component hzparal- lel to the spin-orbit eld on the binding energy, for a xed smoothening length s= 50 nm. the bound state does not exist (see App.A). The orthog- onal eld component hxis thus a necessary, though not sucient, condition for the bound state to exist. One can then analyze how the parallel eld component hzmod- ies the existence of the bound state, for a xed value ofhx6= 0. To this purpose, we focus again on a InSb NW, with an optimal conguration R=L>0, and we take a realistic smoothening length s= 50 nm. The result, displayed in Fig.3(b), shows that the presence of an additional parallel eld component hzmodies the dependence of the binding energy Ebas a function of the magnetic gap energy
Z, especially by increasing the threshold value
? Zat which the bound state starts to exist. Similarly to the case of the smoothening length, the binding energy values are quite reduced as compared to the case hz= 0. IV. CHARGE AND SPIN DENSITY SPATIAL PROFILES In the previous section we have discussed the existence and the robustness of the bound state, which is a spec-7 tral feature. Here we wish to analyze spatial behavior of physical observables, namely the charge and spin densi- ties, described by the operators ^n(x) = e ^ y(x)^ (x) (16) ^S(x) =~ 2^ y(x)^ (x); (17) respectively, where e denotes the electron charge. The presence of the interface makes the NW an inhomoge- neous system, and we aim to investigate the spatial pro- le of the equilibrium expectation values (x)1 eh^n(x)i (18) s(x)2 ~h^S(x)i (19) with a particular focus on their behavior near the inter- face. Details about the computation of such expectation values can be found in App.B. Before presenting our results, a few general comments are in order. Chemical potential and Temperature. The equilibrium distribution determining the expectation values (18) and (19) is characterized by a well dened value of chemical potentialand temperature T. As pointed out above, the whole spectrum of the inhomogeneous Hamiltonian (2), which we obtain by an exact numerical diagonaliza- tion, consists of a continuum spectrum, related to ex- tended propagating states, and possibly (if present) a bound state, energetically lying below the continuum and corresponding to a state localized at the interface. At equilibrium, and ideally at zero temperature, all states (localized or extended) with energy up to the chemical potentialare lled up, and contribute to determine the equilibrium expectation values (x) and s(x), while at nite temperature the Fermi function is smeared over a rangekBTaround the chemical potential. We shall choose forTandrealistic values of low-temperature ex- perimental setups involving NWs, namely T= 250 mK and= 0, corresponding to the energy value in the middle of the magnetic gap [see Fig.1(c)]. This is the situation, for instance, where the Fermi energy states of a NW in the Rashba-dominated regime mimic the helical states of a quantum spin Hall system. Orthogonal spin density. Concerning the spin den- sitys(x) in Eq.(19), we shall specically focus on sy component, which we shall refer to as the orthogonal spin density , since it is orthogonal to the x-zplane identied by the applied magnetic eld and the spin-orbit eld. The interest in analyzing the prole of sy(x) stems from a comparison with the topological phase. Indeed it has been predicted [47{49] that the MQPs appearing at the ends of a proximitized NW in the topological phase, are precisely characterized by a non-vanishing expectation value sy. However, we shall show here below that such orthogonal spin density already appears in the NW interface problem, where the NW is certainly in the topologically trivial phase, so that it cannot beconsidered as a signature of a MQP. Full vs. bound state contribution. Bound states and orthogonal spin density syshare two properties. First, both can only exist at an interface, i.e., in the presence of inhomogeneities. Indeed, in the bulk of a homogeneous NW,syvanishes since the spin orientation of each elec- tron lies in the x-zplane [see Eqs.(8)-(9)]. Second, just like the bound state, symay only exist if both a magnetic eld component hxand the spin-orbit eld are present. Indeed ifhx= 0 (or= 0) the electron spin is directed, alongz(orx) for all states. In view of such common fea- tures, one is naively tempted to conclude that an orthog- onal spin density is necessarily ascribed to the presence of the bound state. However, this is not the case. To this purpose, we shall illustrate below two types of spa- tial proles. First, we shall show the actual equilibrium values(x) andsy(x) [see Eqs.(18) and (19)], which can be referred to as the `full' density and orthogonal spin density proles, as they result from contributions of all states, with the customary weight given by the Fermi function. In particular, since we focus on the low tem- perature regime, the latter essentially amounts to the contribution of all states occupied up to the chemical po- tential. Then, we shall also provide the proles bs(x) andsy;bs(x) describing the contribution to (x) andsy(x) due to the localized bound state only [see App.B for de- tails]. This distinction enables us to show that an orthogonal spin density peak, besides being no evidence for a MQP, may also not originate from any bound state. A. The case of a sharp prole with an orthogonal magnetic eld Let us start our analysis from the case of a sharp prole interface and a magnetic eld applied along the NW axis. As an illustrative example, we con- sider an interface with L=R=1=2, which implies ESO;L =ESO;R=4 [see Eq.(10)], and we choose a value ofESO;R = 0:25 meV for the maximal spin-orbit energy. Figure 4(a) shows the full equilibrium density Eq.(18), for four dierent values of the magnetic gap energy
Z of the applied magnetic eld hx. Its spatial prole (x) exhibits a crossover at the interface x= 0 between two dierent bulk density values. The density increases to- wards the right-hand side, namely the region with higher spin-orbit energy, whose band bottom is lower than on the left-hand side with lower spin-orbit energy, as ob- served above in Sec.II B 2. This indicates that a higher spin-orbit energy has a similar eect on the density as a lower gate voltage bias. In Fig.4(b) we have singled out the contribution bs due to the bound state only. Dierently from (x), the prole ofbs(x) is localized only around the interface and is dramatically sensitive to the value of
Z. Indeed,8 (a)(b)(c) (d) -1,0-0,50,00,51,00123αL/αR= -1/2ESO,R = 0.25 meVλs = 0/s61508Z [meV] 0.1 0.3 0.5 1.0bound statecontribution ρbs [µm-1]x [µm]-1,0-0,50,00,51,023456789all states up to µ=0αL/αR= -1/2ESO,R = 0.25 meVλs = 0/s61508Z [meV] 0.1 0.3 0.5 1.0 ρ [µm-1]x [µm] -1.0-0.50.00.51.0-1.5-1.0-0.50.00.51.01.52.0bound statecontribution sy,bs [µm-1]x [µm]/s61508Z [meV] 0.1 0.3 0.5 1.0αL/αR= -1/2ESO,R = 0.25 meVλs = 0-1.0-0.50.00.51.0-0.50.00.51.01.52.02.5αL/αR= -1/2ESO,R = 0.25 meVλs = 0hz=0µ=0T=250 mKZ [meV] 0.1 0.3 0.5 1.0 sy [µm-1]x [µm]all states up to µ=0 -0.040.000.040.51.01.52.0sy [µm-1]x [µm] FIG. 4. (Color online) Spatial proles of charge density and orthogonal spin density for a sharp interface prole Eq.(13) with L=R=1=2 andESO;R = 0:25 meV. The four dierent curves in each panel refer to four dierent values of the magnetic gap energy
Z= (0:1;0:3;0:5;1:0) meV. (a) The actual equilibrium density prole (x) [see Eq.(18)]. (b) The bound state contribution bs(x) to the density (x). For
Z= 0:1 meV the bound state does not exist and yields a vanishing contribution (black dashed curve). Panel (c) describes the full orthogonal spin density syEq.(19) (with the inset magnifying the peaks) while panel (d) describes the related bound state contribution sy;bs. as can be deduced from Eq.(14), the minimal threshold for the appearance of the bound state is, for the cho- sen parameters,
? Z=ESO;R = 0:25 meV. For values
Z>
? Z[red, blue and green curves in Fig.4(b)], where the bound state exists, a comparison of the height of the peak ofbswith the prole of the full [Fig.4(a)] sug- gests that the increase of across the interface is mainly due to the presence of the bound state. However, for a magnetic gap energy
Z<
? Z[black dashed curve in Fig.4(b)]bsis vanishing because the bound state is ab- sent. Note the striking dierence from the behavior of the full(x) across the interface [Fig.4(a)], which is instead qualitatively very similar for all values of the magnetic gap energy
Z. In conclusion, the increase of the pro- le ofat the interface is not necessarily ascribed to a bound state. This sounds reasonable, since the electron density is a bulk property receiving contributions from all states up to the chemical potential, and the boundstate is just one of such contributions. The same reason- ing holds for the sxcomponent of the spin density [see Eq.(19)], which is also a bulk quantity, due to the applied magnetic eld hx. Let us now turn to consider the spin density sy. Dierently from and fromsx, the orthogonal spin densitysyis vanishing in the bulk of a homogeneous NW, as observed above. Thus, sycan only exist (if it does) in the presence of inhomogeneities, and one could naively expect that it is the hallmark of the presence of a bound state localized at the interface. The prole of the full sy, plotted in Fig.4(c), provides two important insights. First, a peak of the orthogonal spin density sydoes exist, even if the NW is in the topologically trivial phase, implying that it cannot be a unique signature of MQP. Second, the central peak at the interface is weakly sensitive to the values of the magnetic gap energy
Z. This is in striking contrast9 to the behavior of the bound state contribution sy;bs, shown in Fig.4(d), which is again strongly dependent on the magnetic eld. In particular, just like the density bs, for weak Zeeman eld sy;bsvanishes since the bound state is absent (dashed curve), while for higher magnetic eld its broadening depends on
Z. These results show that a localized peak of orthogonal spin density syis not necessarily ascribed to the presence of a bound state, neither topological nor trivial. Before concluding this subsection, a few further com- ments about Fig.4 are in order. We observe that, while the spatial prole of the bound state density bs[panel (b)] is smooth, the prole of sy;bs[panel (d)] exhibits a cusp at the interface. This dierence originates from the boundary conditions induced by the sharp prole (13), which cause spin-diagonal observables like and szto have continuous derivatives, while spin o-diagonal observables like sxandsyto exhibit a cusp at the in- terface (see App.A). Moreover, for
Z= 0:3 meV, i.e., slightly above the threshold
? Z= 0:25 meV, the pro- les of the bound state contributions exhibit a slowly decaying oscillations on the right-hand side, since the bound state wavefunction is characterized by a complex wavevector kon such side. In contrast, for
Z= 0:5 meV and
Z= 1:0 meV the wavevector is purely imaginary, and the bound state density prole has an exponential decay without oscillations. Finally, the peak of the or- thogonal spin density sy;bshas a narrower extension than the one of bs. This is due to the fact that, since on each interface side the bound state wavefunction is a lin- ear combination of two elementary spinorial waves [see Eq.(7)],bsandsy;bsare determined by dierent combi- nations ofw-spinor components of the wavefunctions, re- sulting also into dierent weights for the space-dependent proles. B. Eects of a smoothened prole and parallel magnetic eld on the orthogonal spin density In the previous subsection we have shown that the peak of the orthogonal spin density is far more robust than the bound state. In order to test how general such ef- fect is, we now extend the previous analysis including the presence of a nite smoothening length in the RSOC prole and a magnetic eld component hzparallel to the spin-orbit eld. For simplicity, we focus on the optimal conguration L=R=1 andESO;R = 0:25 meV, with a smoothening length s= 50 nm. These are the pa- rameters also used in Fig.3(b), whence we observe that, keeping a xed value of the magnetic gap energy
Z, and varying the additional parallel eld component hzrepre- sents a natural physical knob to control the weight and the existence of the bound state. Figure 5 shows the spatial prole of the orthogonal spin density for
Z= 0:50 meV and for various values ofhz. In particular, panel (a) displays the full sy, while (a)(b) -0,75-0,50-0,250,000,250,500,75-0,50,00,51,01,52,0 sy,bs [µm-1]x [µm]hz [meV] 0 0.10 0.25 0.50bound state contribution-0.75-0.50-0.250.000.250.500.75-0.50.00.51.01.52.0ESO,R = 0.25 meVΔZ = 0.50 meVαL / αR= -1λs = 50 nmµ=0T=250 mKall statesup to µ=0 sy [µm-1]x [µm]hz [meV] 0 0.10 0.25 0.50-0.020.000.021.41.61.82.0sy [µm-1]x [µm]FIG. 5. (Color online) Spatial prole of the orthogonal spin density for a NW interface with L=R=1 and a smoothen- ing length of s= 50 nm. The maximal spin-orbit energy is ESO;R = 0:25 meV, and the magnetic gap is
Z= 0:50 meV. Dierent curves refer to dierent values of the magnetic eld component hzparallel to the spin-orbit eld. (a) The ac- tualsydue to all states, with the inset magnifying the peaks. (b) The bound state contribution to sy. panel (b) shows the bound state contribution sy;bs. Two features are noteworthy. In the rst instance, as com- pared to the cuspid peaks obtained at the interface in the case of the sharp prole [Fig.4(c)-(d)], the peaks of Fig.5 are rounded o by the nite smoothening length s. Secondly, while the peak of the full sy[Fig.5(a)] is very weakly aected by the parallel magnetic eld component hz, the bound state peak shown in Fig.5(b) rapidly de- creases and eventually disappears when the parallel mag- netic eld component hzis ramped up, yielding a van- ishing contribution (dashed line). This is in agreement with the binding energy behavior previously shown in Fig.3(b), where one can see that, at
Z= 0:50 meV, the bound state disappears for hz= 0:50 meV. The compar- ison between panels (a) and (b) of Fig.5 clearly indicates that, when the bound state exists and has a relatively high binding energy, the peak of syis mainly due to it. However, when the binding energy decreases, the bound10 state contribution to the peak is replaced by the one of the excited states, so that the orthogonal spin density peak remains present. V. DISCUSSION We have demonstrated that the peak of the orthogonal spin density localized at the interface does not necessarily stem from a localized bound state, and appears to be a quite general feature. Two natural questions then arise, namely i) what parameters characterizing the interface determine such peak? ii) can one explain its presence on some general principle? Here we wish to address these two questions. A. General features of the orthogonal spin density To answer the rst question, we consider for denite- ness the case of magnetic gap energy
Z= 0:50 meV and a maximal spin-orbit energy ESO;R = 0:25 meV. Two parameters characterize the interface, namely the ratio L=Rof the two RSOC, and the smoothening length of the prole. In Fig.6(a) we show, for a xed smoothening lengths= 50 nm, the orthogonal spin density prole for dierent values of the RSOC ratio L=Racross the interface. As one can see, the height of the peak grows with the relative RSOC jump, in a roughly linear way. In Fig.6(b), keeping now the ratio of the two RSOC bulk values toL=R=1, we vary the smoothening length sof the prole. The peak decreases and broadens with increasings. Importantly, one can verify by a numerical integration that the area underneath each sy(x) prole is to a very good approximation independent of the value of the smoothening length s. B. Origin of the orthogonal spin density Keeping in mind the two features described in the pre- vious subsection, let us now discuss the origin of the or- thogonal spin density peak. As is well know, a mag- netic moment exposed to a magnetic eld experiences a magnetic torque [64]. So is the case for spin magnetic moments of electrons moving in a NW, where both the externally applied magnetic eld hand the eective spin- orbit eld hSOgive rise to corresponding torques, dened as ^Th^ y(h)^ ; (20) ^TSO1 2 ^ y(hSO)^ + H:c: ; (21) respectively, where hSO(x;t) =f(x);pxg 2~(0;0;1) (22) (a)(b)-0,75-0,50-0,250,000,250,500,75-0,50,00,51,01,52,0λs = 50 nmsy [µm-1]x [µm]αL /αR -1 -1/2 0 +1/2ESO,R = 0.25 meVEZ = 0.50 meVλs = 50 nm -0,75-0,50-0,250,000,250,500,75-0,50,00,51,01,52,02,53,0ESO = 0.25 meVEZ = 0.50 meVµ = 0-α to α with smoothening: all states up to µλs [nm] 0 50 100 300 sy [µm-1]x [µm]αL/αR= -1FIG. 6. (Color online) Spatial prole of the orthogonal spin density for an interface with ESO;R = 0:25 meV, and a mag- netic gap energy
Z= 0:50 meV. (a) The eects of the ratio between the two values of RSOC, for a xed smoothening lengths= 50 nm. (b) Eects of the smoothening length, for the conguration L=R=1. is the spin-orbit eld. Note that, by denition Eqs.(21)- (22), the spin-orbit torque ^TSO= (^TSO x;^TSO y;0) has no component along the Rashba eld direction z. Importantly, the torques determine the spin-dynamics through the operator identity @t^S+@x^Js=^Th+^TSO(23) where ^Sis the spin density operator in Eq.(17), and ^Js=~ 2 i~ 2m ^ y(x)@x^ (x)@x^ y(x)^ (x) (x) ~^ y(x)f;zg 2^ (x) (24) is the spin current density operator [64,65]. Dierently from the continuity equation for charge, in Eq.(23) the torques on the right-hand side play the role of sources and sinks of spin.11 At equilibrium the expectation values of ^Sis time- independent, while the one of the magnetic torque is straightforwardly related to the equilibrium spin-density Eq.(19), through Th=h^Thi=sh. Thus, taking the equilibrium expectation value Eq.(23) one has @xJs=s(x)h+TSO(x) (25) where TSO=h^TSOi. Let us focus on the most custom- ary situation where the magnetic eld is directed along the NW axis x(h=hxix), i.e., orthogonal to the spin- orbit eld. In this case, one can show that the spin-orbit torque TSO(x) vanishes, and that the spin current is ori- ented along z, so that Eq.(25) reduces to @xJs z=hxsy(x): (26) We shall now argue that this equation, derived under quite general hypotheses, is the key to interpret the ap- pearance of the orthogonal spin density at the interface, even when the bound state is absent. Indeed, as has been demonstrated in Ref.[58], when uniform spin-orbit and magnetic elds are present in a NW, an equilibrium spin current Js z ows in its bulk. Such bulk spin current arises from the interplay between spin-orbit eld and a magnetic eld orthogonal to it, which induce non-trivial quantum correlation between spin and velocity, in close similarity to what happens in the helical states of a quantum spin Hall system. The bulk equilibrium spin current is odd in and even in hx. For example, for = 0 and in the regime
ZESO, one hasJs z=sgn()p
ZESO=3. Equilibrium spin currents have been predicted for other RSOC systems as well [64{79] and, in fact, they can be regarded to as the diamagnetic color currents associated to the non-abelian spin-orbit gauge elds [80]. However, its measurement in actual experiments has not been achieved thus far. In this respect, Eq.(26) suggests that, while the equilib- rium spin current itself is perhaps elusive, its variation in the presence of inhomogeneities could be detected, as it is straightforwardly connected to the orthogonal spin density. Indeed, when two regions with dierent RSOC are connected, a kink @xJs zmust arise at the interface to match the dierent spin current values in the two bulks. In view of Eq.(26), a peak in the orthogonal spin density synecessarily appears. This is the reason why the peak ofsyshown in Fig.6(a) is the more pronounced the higher the dierence in the RSOC of the two regions. Further- more, integrating both sides of Eq.(26), one can see that the integral of the syprole equals the dierence between the two bulk spin currents, which is independent of the smoothening length. This is precisely what we found in Fig.6(b). Finally, this argument is quite general and is not based on the existence of a bound state at the in- terface. This explains why the peak shown in Fig.4(d) persists even when the bound state is absent, and shows that the naive interpretation of an orthogonal spin den- sity localized peak in terms of a bound state is in general wrong.C. The case of two interfaces Thus far, we have considered the case of one single in- terface along the NW. Here we wish to brie y discuss the case of two interfaces, modeling a NW inner region characterized by a RSOC parameter insandwiched be- tween two outer regions, where the RSOC shall be taken for simplicity equal to outin both. This corresponds to a prole (x) =out+inout 2(27) " Erf p 8 s(x+L 2)! Erf p 8 s(xL 2)!# ; sketched in Fig.7(a), where Ldenotes the length of the inner NW region, supposed to be much bigger than the smoothening length ( Ls), so that the notion of in- terfaces still makes sense. When the distance Lis much larger than the typical variation lengthscale for observ- ables in the single interface problem, the two interfaces act independently. However, when such two scales be- come comparable, noteworthy aspects emerge, which are illustrated in Fig.7. First, if the interface bound states exist, they over- lap across the distance L, causing a splitting of their degeneracy. The density prole of the resulting lowest eigenstate is mainly peaked at the interfaces, but is non vanishing also in the center of the inner region, as illus- trated by the black curve in Fig.7(b). Second, even if the interface bound states are not present, another type of bound states may appear. Indeed, when the inner re- gion is Rashba-dominated and jinj>joutj, the band bottom of the inner region is lower than in the outer regions. Thus, for short Lthe two interfaces give rise to an eective Rashba quantum dot[59,60], with discrete bound states localized within the connement length L. This is the case depicted by the red curve in Fig.7(b), where the density prole of the lowest eigenstate corre- sponds to a Rashba dot bound state. Such quantum dot bound states thus have a completely dierent origin from the interface bound states. In particular, while the inter- face bound states are present only in the presence of an applied magnetic eld, the Rashba dot bound states are intrinsic, as they may also be present without magnetic eld[81]. The third interesting feature of the double interface problem is that, in all cases, pronounced orthogonal spin density peaks appear at the interfaces, regardless of whether interface bound states exist or not. Remarkably, the signs of the peaks are opposite at the two interfaces, as shown in Fig.7(c). This is because the opposite jump in the RSOC across the two interfaces causes two oppo- site kinks in the equilibrium spin current, as observed in Sec.V B. Thus, despite the NW is in the topologically trivial phase, the emerging scenario is identical to the one occurring in a NW in the topological phase, where the spin density of the MQPs is orthogonal to both the12 zx<latexit sha1_base64="BPOi5UFDQJfpY75XGGUjzEAe88Y=">AAAB6HicbZC5TgMxEIZnwxXCFY6OxiJCoop2oYCOSBRQJhI5pGQVeZ3ZxMR7yPYiwipPQEMBQrQ8AE/BE9BR8iY4RwEJv2Tp0//PyDPjxYIrbdtfVmZhcWl5JbuaW1vf2NzKb+/UVJRIhlUWiUg2PKpQ8BCrmmuBjVgiDTyBda9/McrrtygVj8JrPYjRDWg35D5nVBurctfOF+yiPRaZB2cKhfOP++/L97203M5/tjoRSwIMNRNUqaZjx9pNqdScCRzmWonCmLI+7WLTYEgDVG46HnRIDo3TIX4kzQs1Gbu/O1IaKDUIPFMZUN1Ts9nI/C9rJto/c1MexonGkE0+8hNBdERGW5MOl8i0GBigTHIzK2E9KinT5jY5cwRnduV5qB0XnZPiccUulEowURb24QCOwIFTKMEVlKEKDBAe4AmerRvr0XqxXielGWvaswt/ZL39ACCNkNw=</latexit>(a)(b) (c)↵out <latexit sha1_base64="3aoy/TRrKB93ONarRlImmBXv4wE=">AAAB83icbVC7SgNBFJ31GeMrmtJmMAgWEnZjoWXAxjKCeUB2CbOT2WTI7MwwDyEs2/gRNhaK2PozFoJ/4DdYOXkUmnjgwuGce7n3nlgyqo3vf3orq2vrG5uFreL2zu7efungsKWFVZg0sWBCdWKkCaOcNA01jHSkIiiNGWnHo6uJ374jSlPBb81YkihFA04TipFxUhgiJoeolwlr8l6p4lf9KeAyCeakUi9/fN9/5WeNXuk97AtsU8INZkjrbuBLE2VIGYoZyYuh1UQiPEID0nWUo5ToKJvenMMTp/RhIpQrbuBU/T2RoVTrcRq7zhSZoV70JuJ/Xtea5DLKKJfWEI5nixLLoBFwEgDsU0WwYWNHEFbU3QrxECmEjYup6EIIFl9eJq1aNTiv1m5cGnUwQwEcgWNwCgJwAergGjRAE2AgwQN4As+e9R69F+911rrizWfK4A+8tx85T5Y3</latexit>↵out <latexit sha1_base64="3aoy/TRrKB93ONarRlImmBXv4wE=">AAAB83icbVC7SgNBFJ31GeMrmtJmMAgWEnZjoWXAxjKCeUB2CbOT2WTI7MwwDyEs2/gRNhaK2PozFoJ/4DdYOXkUmnjgwuGce7n3nlgyqo3vf3orq2vrG5uFreL2zu7efungsKWFVZg0sWBCdWKkCaOcNA01jHSkIiiNGWnHo6uJ374jSlPBb81YkihFA04TipFxUhgiJoeolwlr8l6p4lf9KeAyCeakUi9/fN9/5WeNXuk97AtsU8INZkjrbuBLE2VIGYoZyYuh1UQiPEID0nWUo5ToKJvenMMTp/RhIpQrbuBU/T2RoVTrcRq7zhSZoV70JuJ/Xtea5DLKKJfWEI5nixLLoBFwEgDsU0WwYWNHEFbU3QrxECmEjYup6EIIFl9eJq1aNTiv1m5cGnUwQwEcgWNwCgJwAergGjRAE2AgwQN4As+e9R69F+911rrizWfK4A+8tx85T5Y3</latexit>↵in <latexit sha1_base64="V17LSI752Bv08sod4P7+8kbi8zo=">AAAB8nicbVDLSgMxFM34rPVV7dJNsAgupMzUhS4LblxWsA+YDiWTZtrQTDIkd4QyzMZ/cONCEbd+jQvBP/AbXJk+Ftp6IHA4515yzwkTwQ247qezsrq2vrFZ2Cpu7+zu7ZcODltGpZqyJlVC6U5IDBNcsiZwEKyTaEbiULB2OLqa+O07pg1X8hbGCQtiMpA84pSAlfwuEcmQ9DIu816p4lbdKfAy8eakUi9/fN9/5WeNXum921c0jZkEKogxvucmEGREA6eC5cVualhC6IgMmG+pJDEzQTY9OccnVunjSGn7JOCp+nsjI7Ex4zi0kzGBoVn0JuJ/np9CdBnYPEkKTNLZR1EqMCg8yY/7XDMKYmwJoZrbWzEdEk0o2JaKtgRvMfIyadWq3nm1dmPbqKMZCugIHaNT5KELVEfXqIGaiCKFHtATenbAeXRenNfZ6Ioz3ymjP3DefgBMHZWs</latexit>-1.0-0.50.00.51.0 0.00.51.01.52.02.5 αout /αin -1 0ESO,in= 0.25 meVΔZ = 0.25 meVλs = 50 nmL = 1µmx [µm]lowest [µm-1]αout /αin -1 0 -1.0-0.50.00.51.0-2-1012 sy [µm-1]x [µm]Rashba dot bound stateinterface bound states FIG. 7. (Color online) (a) Sketch of a double interface prob- lem, modeled by the RSOC prole (27). The parameters are L= 1m,s= 50 nm, the value in>0 in the inner re- gion corresponds to ESO;in = 0:25 meV, while the magnetic gap energy is
Z= 0:25 meV. (b) The density prole of the lowest electron state, for two values out=in(black curve) and out= 0 (red curve), showing the dierence be- tween interface and Rashba dot bound states. (c) The total orthogonal spin density, for the same two values of in, shows two opposite peaks at the interfaces. magnetic eld and the RSOC eld direction, and takes opposite signs at the two NW ends[47{49]. This explic- itly demonstrates that such orthogonal spin polarization pinned at the NW ends can neither be taken as a hall- mark of the topological phase, nor as an evidence of bound states. Note also that the orthogonal spin po- larization peaks are typically narrower than the interface bound state and are thus more robust to nite length L eects too.D. Possible setup realizations Several experiments in topological systems are based on InSb [30, 34, 52, 82{84] or InAs [33, 50, 53, 85{87] NWs deposited on a substrate. In the case of InSb the eective mass and the g-factor arem'0:015meandg' 50, respectively, while the value of the RSOC depends on the specic implementation and experimental conditions and can be widely tunable, ranging from 0:03 eV A to1 eVA [27,30,50,52,82,83]. The spin-orbit energy ESOresulting from these values [see Eq.(5)] is a fraction of meV. The same order of magnitude is obtained for the magnetic gap energy
Zin a magnetic eld range of some hundreds of mT. These are the values adopted in our plots. Similarly, in the case of InAs nanowires m'0:022me,g'20 and the RSOC ranges from  0:05 eV A to0:3 eVA [29,85,50,86]. The temperature value of 250 mK used in our plots is state of the art with modern refrigeration techniques. Interfaces between regions with dierent RSOC emerge quite naturally in typical NW setups, where a portion of the NW is covered by e.g. a superconductor or by a normal metal to induce proximity eect, to measure the current, or to locally vary the potential. The resulting SIA is inhomogeneous along the NW, and can be con- trolled e.g. by the application of dierent gate voltage values applied to top/bottom gates or to the substrate, similarly to the case of constrictions in quantum spin Hall systems[88{90]. In particular, covering one portion with the gate-all-around technique and by applying a su- ciently strong gate voltage, it is reasonable to achieve an inversion of the sign of the RSOC as compared to the un- covered NW portion, as has already been done in similar setups[51, 57, 91{93]. Finally, the orthogonal spin polarization predicted here can be measured by spatially resolved detection of spin orientation. In particular, nanometer scale resolution can be reached with various methods such as magnetic resonance force microscopy [94,95], spin-polarized scan- ning electron microscopy [96,97], by using quantum dots as probes [98,99], or also electrically by potentiomet- ric measurements exploiting ferromagnetic detector con- tacts [100,101]. VI. CONCLUSIONS In conclusions, in this paper we have considered a NW with an interface between two regions with dier- ent RSOC values, as sketched in Fig.1, when proximity eect is turned o and the NW is in the topologically trivial phase. In Sec.III we have shown that at the interface bound states may appear, whose energy is located below the continuum spectrum minimum. Such bound states are neither topological (since proximity eect is absent), nor intrinsic interface bound states (since they only exist if an external magnetic eld is applied along the13 NW axis). Analyzing rst the case of a sharp interface RSOC prole Eq.(13), we have obtained the phase diagram determining the existence of the bound state [see Fig.2(a)], as well as the dependence of its binding energy on the magnetic gap energy [see Fig.2(b)]. While the bound state always exists if the RSOC takes equal and opposite values across the interface (optimal conguration), for all other situations it only exists if the magnetic eld overcomes a minimal threshold value. Furthermore, even in the optimal conguration, it can be suppressed by either a nite smoothening length in the RSOC prole or a magnetic eld component parallel to the spin-orbit eld (see Fig.3). In Sec.IV we have then investigated the spatial prole of the charge density and the spin density, with a spe- cial focus on the spin density component sy, orthogonal to both the applied magnetic eld and the RSOC eld direction, which is known to characterize the MQPs lo- calized at the edges of a NW in the topological phase. By analyzing both the full equilibrium values andsy due to all occupied states, and the bound state contribu- tionsbsandsy;bs, we have been able to gain two useful insights. First, the orthogonal spin density appears also in the topologically trivial phase as a quite general eect characterizing any interface between two dierent RSOC regions under a magnetic eld. This extends our previ- ous results of Ref.[58] related to NW contacted to nor- mal leads without RSOC. Second, for realistic and typical values of chemical potential and temperature, the orthog- onal spin density peak is relatively robust to parameter changes, and persists even when the bound state is absent (see Figs.4 and 5). This means that also the propagating states of the continuum spectrum modify their spin tex- ture around the interface to preserve the peak, so that a localized orthogonal spin-density cannot be considered a signature of a bound state. Furthermore, in Sec.V, after analyzing in Fig.6 the peak dependence on the single interface parameters, we have addressed the case of two interfaces [see Fig.7]. While for a large distance Lbetween the interfaces the single-interface scenario is merely doubled, for a shorter L the interface bound states may overlap and additional Rashba quantum dot states may appear. In all cases, and independently of the presence of interface bound states, the spin density sy,orthogonal to both the magnetic eldand the Rashba spin-orbit eld, exhibits relatively ro- bust peaks taking opposite signs at the two interfaces [see Fig.7(c)]. Remarkably, these are the same features predicted for the spin density of the MQPs emerging at the ends of a NW in the topological phase, despite the NW considered here is in the topologically trivial phase. Our results thus show that such orthogonal spin polar- ization pinned at the NW ends can neither be taken as a hallmark of the topological phase, nor as an evidence of bound states. However, we have also shown in Sec.V that such sta- ble peaks may in fact have an impact on the detection of spin currents. Indeed a spin current ows in the bulk of a NW as a result of quantum correlations between spin and velocity induced by the interplay between magnetic and spin-orbit eld, similarly to the case of quantum spin Hall helical states. Despite various proposals in the liter- ature, the measurement of equilibrium spin currents has not been achieved yet. Our results suggest that, while the equilibrium spin current itself may be elusive, its varia- tions can be detected through the orthogonal spin den- sitysy, which is instead experimentally observable with spin-resolved detection techniques. Indeed the orthogo- nal spin density peak is precisely related to the kink of the spin current localized at the interface. With the provided description of possible implementations in realistic NW setups, the predicted eects seem to be at experimental reach. ACKNOWLEDGMENTS Fruitful discussions with M. Sassetti, F. Cavaliere, and N. Traverso Ziani are greatly acknowledged. Com- putational resources were provided by hpc@polito at Politecnico di Torino. Appendix A: Calculation for sharp prole interface In this Appendix we provide details about the calcula- tion for a sharp prole interface (13). In such a situation the eigenvalue equation stemming from the Hamiltonian (2) at energy Ereads ~2 2m@2 x+i(x)@x+iRL 2(x)hz hx hx ~2 2m@2 xi(x)@xiRL 2(x) +hz! (E) "(x) (E) #(x)! =E (E) "(x) (E) #(x)! (A1) equipped with the boundary conditions at the interface 8 >>>< >>>: "(0) = "(0+) #(0) = #(0+) @x "(0) =@x "(0+)im ~2(RL) "(0) @x #(0) =@x #(0+) +im ~2(RL) #(0)(A2)A few remarks about the boundary conditions (A2) are in order. First, the discontinuity in the derivative of the wavefunction involves an imaginary unit too, making such boundary conditions intrinsically dierent from14 the ones of the well known problem of a particle in a scalar-potential. Second, as a consequence of such imaginary unit, it can straightforwardly be shown that, despite the derivative @x sis discontinuous ( s=";#), the derivative @xsof the quantity s(x)  s(x) s(x) iscontinuous at the interface x= 0. For this reason, both the density (x) ="+#[see Eq.(18)] and the spin density component sz="#[see Eq.(19)] do not exhibit any cusp in their spatial prole. In contrast, o-diagonal spin density components sxandsy, which cannot be expressed in terms of the s's, do exhibit a cusp due to the discontinuity of the derivative implied by the boundary conditions (A2). This dierence becomes apparent by comparing e.g. panels (b) and (d) in Fig.4. Let us now proceed with the calculation of the en- ergy spectrum. As observed above, we have assumed R>0 andjLj  jRjwithout loss of generality. As a consequence ESO;R is the higher spin-orbit energy, ESO;RESO;L [see Eq.(10)]. By denoting the ratio be- tween the two RSOC values rL R2[1;1 ] (A3) one hasESO;L =r2ESO;R. One can introduce the mo- mentum space Hamiltonian H k="0 kkzhxx hzzdescribing the homogeneous bulk of each side = R=L of the interface, and match the related eigenfunc- tions with the boundary conditions (A2). The energy spectrum characterizing the NW on the right-hand side and on the left-hand side of the interface can be suitably rewritten as ER (K) =K2 4ESO;Rq
2 Z+ (K+hz)2 (A4) EL (K) =(rK)2 4ESO;Lq
2 Z+ (rK+hz)2(A5) respectively, where K=Rkhas the dimension of an energy, while
Zis the magnetic gap energy Eq.(4). The eigenstates of the momentum Hamiltonian in each side can be written, for arbitrary complex wavevector K,in the following explicit form forx>08 >>>>< >>>>:w(K) =1p
2 Z+jz(K)j2 z(K)
Z w+(K) =1p
2 Z+jz(K)j2
Z z(K)(A6) forx<08 >>>>< >>>>:w(rK) =1p
2 Z+jz(rK)j2 z(rK)
Z w+(rK) =1p
2 Z+jz(rK)j2
Z z(rK)(A7) wherez(K) =p
2 Z+ (K+hz)2+ (K+hz). In order to determine the energy Ebsof the bound state, the crucial point is to correctly re-express Eqs.(A6)-(A7) as a function of the energy E, and then to impose the boundary conditions (A2). To this purpose, the rst step is to invert the dispersion relation in each side =R=L. This can be done analytically in two specic cases, namely for hz= 0 or forhx= 0. Here below we shall discuss these two situations, while the general case hx;hz6= 0 will be approached numerically as described in App.B. 1. The case hz= 0 In this case the dispersion relation can be inverted yielding four possible K-values K ;0(E) = (A8) r 4ESO;Rh E+ 2ESO; +0q
2 Z+ 4E2 SO; + 4ESO;Ei where;0=1. Note that K2C, and we have adopted the conventionpz=p jzjei 2for the square root of a complex number z=jzjeiwith2(;  ]. One then inserts the four possible values (A8) of K ;0into the two eigenvectors Eqs.(A6)-(A7). In do- ing that, some caution must be taken, since for a given energyEand each side of the interface a seeming re- dundancy of eigenstates appears. However, only half of the possible eigenstates actually fulll the equation Hk[K(E)]w[K(E)] =Ew[K(E)], as it should be. Their explicit expressions depend on the regime of the involved energy scales E,
ZandESO;, Focusing e.g. on the right hand side of the interface, one can identify three regimes where, for a given energy Elower than the overall minimum of the bulk bands, the corresponding 4 correct eigenspinors are given in Table Eq.(A9). Regime 2 diers from regime 3 because in the former wave vectors turn out to be strictly imaginary, while in the latter they exhibit a real part as well. The expression for the eigenspinors on the left hand side, together with their corresponding domain, can be directly obtained from the ones in Table (A9) by simply replacingESO;R!ESO;L andKR (E)!rKL (E).15 regime eigenvectors 1)
Z>4ESO;R and
2 Z+ 4E2 SO;R 4ESO;R<E <
Zw KR ;+(E) w+ KR ;+(E)=1 2)
Z>2ESO;R and
2 Z+ 4E2 SO;R 4ESO;R<E < min
2 Z 4ESO;R;
Z w KR ;0(E) ;0=1 3)
Z<4ESO;R andE <
2 Z+ 4E2 SO;R 4ESO;Rw KR ;0(E) ;0=1(A9) Once the four eigenspinors wand momenta Kare iden- tied, the wavefunction is constructed as a linear super- position of each spinor wmultiplied by the related phase factoreiKx=R. In doing that, the requirement that does not diverge at x!1 reduces the four terms to two in each side. Let thus w j(E) andK j(E) withj= 1;2 denote such two eigenspinors and momenta related to non-divergent wavefunctions in the region =R=L at energyEin a given regime. Then, the eigenfunction (E)(x) can be written as a linear superposition (E)(x) =8 >< >:P2 j=1ljwR j(E)eiKR j(E) Rxx>0 P2 j=1rjwL j(E)eiKL j(E) Rxx<0: (A10) Thus, the boundary condition Eq.(A2) leads to a homo- geneous system of 4 linear equations in 4 unknowns l1, l2r1andr2. Imposing the solvability of the system one obtains an equation for the energy Ewhose solutions, if they exist, correspond to the energy Ebof the bound state for given values of
Z,ESO;R andr. The binding energy (12) is then straightforwardly obtained. 2. The case hx= 0 In this case the eigenvalue problem (A1) decouples into two separate problems for the spin- "and spin-#compo- nents of the wave function, and the magnetic gap energy
Z=jhxjvanishes. Accordingly, the eigenvectors (A6) acquire the simple form w(K)j
Z=0= 1 0! ; w+(K)j
Z=0= 0 1! (A11) both forx >0 andx <0, while the eigenvalues have a quadratic dependence on K, 8 >>>>< >>>>:ER "(K) =K2 4ESO;R(K+hz)x>0 EL "(K) =(rK)2 4ESO;L(rK+hz)x<0 ER #(K) =K2 4ESO;R+ (K+hz)x>0 EL #(K) =(rK)2 4ESO;L+ (rK+hz)x<0:(A12) Without loss of generality, we can focus on the spin- " component of the wave function. The dispersion relationcan be easily inverted ( KR (E) = 2ESO;Rp (2ESO;R)2+ 4ESO;R(hz+E) KL (E) = 2rESO;Rp (2rESO;R)2+ 4ESO;R(hz+E) (A13) In order for K (E) to exhibit an imaginary part, one has to consider energies in the range E <hzESO; and the most general eigenfunction of energy Ecan thus be written as (E)(x) =8 >< >:aeiKR +(E) Rx+beiKR (E) Rxx>0 ceiKL +(E) Rx+deiKL (E) Rxx<0(A14) wherea; b; c; d are complex coecients to be determined. The regularity at x!1 and the continuity in x= 0 reduce the wavefunction to the form (E)(x) =8 >< >:aeiKR +(E)x Rx>0 aeiKL (E)x Rx<0(A15) while the matching condition (A2) on the rst derivative inx= 0 implies KL (E) =KR +(E)2ESO;R(1r) (A16) whose only possible solution is: ( r2= 1 E=hzESO;R(A17) However, this corresponds to the lowest energy eigen- function of the continuum, demonstrating that no bound state exists in such case. Appendix B: Diagonalization strategy in the presence of a smoothening length Here we describe how to numerically approach the problem in the presence of the RSOC prole (15) char- acterized by a nite smoothening length s, and when both perpendicular and parallel magnetic eld compo- nentshx;hz6= 0 are present. To this end, we impose periodic boundary conditions onto the NW, and express16 the electron spinor eld in terms of discretized Fourier components k= 2n= , namely ^ (x) =X keikx p ^ck" ^ck#! ; (B1) where is the (large) NW periodicity length and ^ ck;s denotes the Fourier mode operators for spin s=";#. The Hamiltonian (1) is thus rewritten in terms of the dis- cretizedk-basis introduced in Eq. (B1) as ^H=X k1;k2X s1;s2=";#^cy k1;s1Hk1;s1;k2s2^ck2;s2;(B2) where Hk1;s1;k2s2= "0 k10h

k1;k2 (B3) k1k2k1+k2 2z s1;s2; whereqis the (discretized) Fourier transform of the RSOC prole (x). Specically, taking for qthe follow- ing expression q=8 < :L+R 2for q = 0 eq22 s 32L eiq 21 R eiq 21 iq otherwise: (B4) one obtains the (periodic version) of the prototypical pro- le Eq.(15) as Fourier series (x) =P qqeiqx. Then, we have performed an exact numerical diagonal- ization of the Hamiltonian matrix Eq.(B3), thereby ob- taining diagonalizing operators ^ddened through ^ ca=P Ua;^d, wherea= (k;s) is a compact quantum num- ber notation for the original basis, and Uis the matrix of the eigenvectors of Eq.(B3). Denoting by Ethe eigen- values, the NW Hamiltonian can be rewritten as ^H=X E^dy ^d (B5) Finally, to compute the equilibrium expectation values h:::iof the operators (16), (17), one can re-express the electron eld operator s(x) with spin component s=";# in terms of the diagonalizing operators ^d's, ^ s(x) =1p X k;eikxUks;^d (B6) and to exploith^dy ^d0i=0f(E), withf(E) = f1 + exp [(E)=kBT]g1denoting the Fermi distribu- tion function. For instance, the density Eq.(18) is ob- tained as(x) =P (x), where (x) =1 X s=";#X k1;k2ei(k1k2)xU k1s;Uk2s;f(E) (B7) is the contribution arising from the -th eigenstate. In this way, the contribution of each eigenstate (in particu- lar the bound state) can be singled out. lorenzo.rossi@polito.it 1M.Z. Hasan, and C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 2X.-L. Qi, and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). 3M. Sato and Y. Ando, Rep. Prog. Phys. 80, 076501 (2017). 4W. P. Su, J. R. Schrieer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979). 5W. P. Su, J. R. Schrieer, and A. J. Heeger, Phys. Rev. B22, 2099 (1980). 6C. L. Kane and T. C. Lubensky, Nature Phys. 10, 39 (2014). 7C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). 8C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 9B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006). 10C. Liu, T. L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang, Phys. Rev. Lett. 100, 236601 (2008). 11M. K onig, H. Buhmann, L. W. Molenkamp, T. Hughes, C.-X. Liu, X.-L. Qi, and S.-C. Zhang, J. Phys. Soc. Jpn77, 031007 (2008). 12A. Y. Kitaev, Phys. Usp. 44, 131 (2001). 13J. Alicea, Rep. Progr. Phys 75, 076501 (2012). 14M. Sato, and S. Fujimoto J. Phys. Soc. Jpn 85, 072001 (2016). 15R. Aguado, Rivista del Nuovo Cimento, 40, 523 (2017). 16Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010). 17R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010). 18S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani, Phys. Rev. B 88, 020407(R) (2013). 19L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008). 20J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. Lett. 101, 120403 (2008). 21F. Cr epin, B. Trauzettel, and F. Dolcini, Phys. Rev. B 89, 205115 (2014). 22C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). 23D. Aasen, M. Hell, R.V. Mishmash, A. Higginbotham, J. Danon, M. Leijnse, T.S. Jespersen, J.A. Folk, C.M. Marcus, K. Flensberg, and J. Alicea, Phys. Rev. X 6,17 031016 (2016). 24T.D. Stanescu, and S. Das Sarma, Phys. Rev. B 97, 045410 (2018). 25T. E. O'Brien, P. Ro_ zek, and A.R. Akhmerov, Phys. Rev. Lett. 120, 220504 (2018). 26H. Zhang, D. E. Liu, M. Wimmer, and L.P. Kouwenhoven, Nat. Comm. 10, 5128 (2019). 27V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven, Science 336, 1003 (2012). 28L.P. Rokhinson, X. Liu, and J. K. Furdyna, Nat. Phys. 8, 795 (2012). 29A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Nat. Phys. 8, 887 (2012). 30M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caro, and H. Q. Xu, Nano Lett. 12, 6414 (2012). 31E.J.H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, Nat. Nanotechnol. 9, 79 (2014). 32S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nyg ard, P. Krogstrup, and C. M. Marcus, Nature 531, 206 (2016). 33M.T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nyg ard, P. Krogstrup, and C. M. Marcus, Science 354, 1557 (2016). 34O. G ul , H. Zhang, J.D.S. Bommer, M.W. A. de Moor, D. Car, S.R. Plissard, E.P.A.M. Bakkers, A. Geresdi, K. Watanabe, T. Taniguchi, and L.P. Kouwenhoven, Nat. Nanotechnol. 13, 192 (2018). 35E.J.H. Lee, X. Jiang, R. Aguado, G. Katsaros, C.M. Lieber, and S. De Franceschi, Phys. Rev. Lett. 109, 186802 (2012). 36J. Liu, A.C. Potter, K.T. Law, and P.A. Lee, Phys. Rev. Lett. 109, 267002 (2012). 37H. Pan, S. Das Sarma, Phys. Rev. Research 2, 013377 (2020). 38C.-X. Liu, J.D. Sau, T.D. Stanescu, and S. Das Sarma, Phys. Rev B 96, 075161 (2017). 39J. Klinovaja, and D. Loss, Eur. Phys. J. B 88 , 62 (2015). 40C. Fleckenstein, F. Dom nguez, N. Traverso Ziani, and B. Trauzettel Phys. Rev B 97, 155425 (2018). 41J. Chen, B.D. Woods, P. Yu, M. Hocevar, D. Car, S.R. Plissard, E.P.A.M. Bakkers, T. D. Stanescu, and S.M. Frolov, Phys. Rev Lett. 123, 107703 (2019). 42L. S. Ricco, M. de Souza, M. S. Figueira, I. A. Shelykh, and A. C. Seridonio, Phys. Rev B 99, 155159 (2019). 43T.D. Stanescu, and S. Tewari, Phys. Rev. B 100, 155429 (2019). 44E. Prada, P. San-Jos e, M.W. A. de Moor, A. Geresdi, E.J.H. Lee, J. Klinovaja, D. Loss, J. Nyg ard, R. Aguado2, L.P. Kouwenhoven, Cond-mat arXiv:1911.04512 45F. Ronetti, K. Plekhanov, D. Loss, J. Klinovaja, Cond- mat arXiv:1911.03133 46S. Nadj-Perge, I.K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A.H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 347, 602 (2014). 47D. Sticlet, C. Bena, and P. Simon, Phys. Rev. Lett. 108, 096802 (2012). 48K. Bj ornson, S. S. Pershoguba, A. V. Balatsky, and A. M. Black-Schaer, Phys. Rev. B 92, 214501 (2015). 49M. M. Ma ska, and T. Doma nski, Sci. Rep. 7, 16193 (2017). 50D. Liang and X. P.A. Gao, Nano Lett. 12, 3263 (2012).51B. Slomski, G. Landolt, S. Mu, F. Meier, J. Osterwalder, and J.H. Dil, New J. Phys. 15, 125031 (2013). 52I. van Weperen, B. Tarasinski, D. Eeltink, V. S. Pribiag, S. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven, and M. Wimmer, Phys. Rev. B 91, 201413(R) (2015). 53Z. Scher ubl, G. F ul op, M. H. Madsen, J. Nyg ard, and S. Csonka, Phys. Rev. B 94, 035444 (2016) 54K. Takase, Y. Ashikawa, G. Zhang, K. Tateno, and S. Sasaki, Sci. Rep. 7, 930 (2017). 55J. Borge and I. V. Tokatly, Phys. Rev. B 96, 115445 (2017). 56Ch. Kloeel, M. J. Ran ci c, and D. Loss, Phys. Rev. B 97, 235422 (2018). 57H.Tsai, S. Karube, K. Kondou, N. Yamaguchi, Y. Otani, Sci. Rep. 8, 5564 (2018). 58F. Dolcini and F. Rossi, Phys. Rev. B 98, 045436 (2018). 59D. S anchez, L. Serra, Phys. Rev. B 74153313 (2006). 60D. S anchez, L. Serra, and M.-S. Choi, Phys. Rev. B 77, 035315 (2008). 61M. Cheng, and R. M. Lutchyn, Phys. Rev. B 86, 134522 (2012) 62P. Szumniak, D. Chevallier, D. Loss, and J. Klinovaja, Phys. Rev. B 96, 041401(R) (2017). 63The case where R<0 can easily be mapped into the one considered here, since the case with a (x) prole can be mapped into the case (x) by space parity ( px!px), as is clear from the Hamiltonian (2). 64E. B. Sonin, Adv. Phys. 59, 181 (2010). 65E. I. Rashba, Phys. Rev. B 68241315(R) (2003). 66J. Splettstoesser, M. Governale, and U. Z ulicke, Phys. Rev. B 68, 165341 (2003). 67G. Usaj and C. A. Balseiro, Europhys. Lett. 72, 631 (2005). 68E. B. Sonin, Phys. Rev. B 76, 033306 (2007). 69E. B. Sonin, Phys. Rev. Lett. 99, 266602 (2007). 70Q.-F. Sun, X. C. Xie, J. Wang, Phys. Rev. Lett. 98, 196801 (2007). 71Q.-F. Sun, X. C. Xie, J. Wang, Phys. Rev. B 77, 035327 (2008). 72V. A. Sablikov, A. A. Sukhanov, and Y. Ya. Tkach, Phys. Rev. B 78, 153302 (2008). 73F. Liang Y. G. Shen, Y. H. Yang, Phys. Lett. A 372, 4634 (2008). 74B. Berche, C. Chatelain, E. Medina, Eur. J. Phys. 31, 1267 (2010). 75E. Nakhmedov, and O. Alekperov, Phys. Rev. B 85, 153302 (2012). 76H. Zhang, Z. Ma, and J.F. Liu, Sci. Rep. 4, 6464 (2014). 77F. Liang, B.-L. Gao, G. Hu, Y. Gu, and N. Xu, Phys. Lett. A 379, 3114 (2015). 78T.-W. Chen, C.-M. Huang, and G. Y. Guo, Phys. Rev. B 73, 235309 (2006). 79F. Meier, and D. Loss, Phys. Rev. Lett. 90, 167204 (2003). 80I.V. Tokatly, Phys. Rev. Lett. 101, 106601 (2008). 81Without magnetic eld the problem Eqs.(1)-(2) can be mapped into the problem of a particle in a quantum well with a potential ESO(x) =m2(x)=2~2by per- forming the spin-dependent gauge transformation ^ (x) = exp imzRx 0(x0)dx0=~2^ 0(x). 82H. A. Nilsson, Ph. Caro, C. Thelander, M. Larsson, J. B. Wagner, L.-E. Wernersson, L. Samuelson, and H. Q. Xu, Nano Lett. 9, 3151 (2009). 83S. Nadj-Perge, V. S. Pribiag, J.W. G. van den Berg, K.18 Zuo, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Phys. Rev. Lett. 108, 166801 (2012). 84O. G u, H. Zhang, F.K. de Vries, J. van Veen, K. Zuo, V. Mourik, S. Conesa-Boj, M.P. Nowak, D.J.van Wo- erkom, M. Quintero-P erez, M. C.Cassidy, A. Geresdi, S. Koelling, D. Car, S.R. Plissard, E.P. A. M. Bakkers, and L.P. Kouwenhoven, Nano Lett. 17, 2690 (2017). 85P. Roulleau, T. Choi, S. Riedi, T. Heinzel, I. Shorubalko, T. Ihn, and K. Ensslin, Phys. Rev. B 81, 155449 (2010) 86H. J. Joyce, C. J. Docherty, Q. Gao, H. H. Tan, C. Ja- gadish, J. Lloyd-Hughes, L. M. Herz and M. B. Johnston, Nanotechnology 24, 214006 (2013). 87S. Heedt, N. Traverso Ziani, F. Cr epin, W. Prost, St. Trellenkamp, J. Schubert, D. Gr utzmacher, B. Trauzettel and Th. Sch apers, Nat. Phys. 13, 563 (2017). 88F. Romeo, R. Citro, D. Ferraro, and M. Sassetti, Phys. Rev. B 86, 165418 (2012). 89P. Sternativo, and F. Dolcini, Phys. Rev. B 89, 035415 (2014). 90J. Strunz, J. Wiedenmann, C. Fleckenstein, L. Lunczer, W. Beugeling, V.L. M uller, P. Shekhar, N.T. Ziani, S. Shamim, J. Kleinlein, H. Buhmann, B. Trauzettel, and L.W. Molenkamp, Nat. Phys. 16, 83 (2020). 91O. Krupin, G. Bihlmayer, K. Starke, S. Gorovikov, J. E. Prieto, K. D obrich, S. Bl ugel, and G. Kaindl, Phys. Rev.B71, 201403(R) (2005). 92W. Wang, X.M. Li, J.Y. Fu, J. Magn. Magn. Mat. 411, 84 (2016). 93F. Nagasawa, A.A. Reynoso, J.P. Baltan as, D. Frustaglia, H. Saarikoski, and J. Nitta, Phys. Rev. B 98, 245301 (2018). 94D. Rugar, R. Budakian, H. J. Mamin and B. W. Chui, Nature 430, 329 (2004). 95J. Cardellino, N. Scozzaro, M. Herman, A J. Berger, C. Zhang, K. C. Fong, C. Jayaprakash, D. V. Pelekhov and P. C. Hammel, Nature Nanotech. 9, 343 (2015). 96K. Koyke, H. Matsuyama, H. Todokoro, and K. Hayakawa, Jpn. J. Appl. Phys. 24, 1078 (1985). 97T. Kohashi, J. Magn. Soc. Jpn., 39, 131 (2015). 98T. Otsuka, E. Abe, Y. Iye, and S. Katsumoto, Phys. Rev. B79, 195313 (2009). 99T. Otsuka, Y. Sugihara, J. Yoneda, S. Katsumoto, and S. Tarucha, Phys. Rev B 86, 081308(R) (2012). 100C. H. Li, O. M. J. van't Erve, J. T. Robinson, Y. Liu, L. Li, and B. T. Jonker, Nature Nanotech. 9, 218 (2014). 101J. Tang, L.-T. Chang, X. Kou, K. Murata, E. S. Choi, M. Lang, Y. Fan, Y. Jiang, M. Montazeri, W. Jiang, Y. Wang, L. He, and K. L. Wang, Nano Lett. 14, 5423 (2014).

2403.07188v1.Spin_orbit_coupling_in_symmetric_and_mixed_spin_symmetry.pdf

Spin-orbit coupling in symmetric and mixed spin-symmetry Ayaka Usui,1, 2, 3, ∗Abel Rojo-Franc` as,1, 2James Schloss,4and Bruno Juli´ a-D´ ıaz1, 2 1Departament de F´ ısica Qu` antica i Astrof´ ısica, Universitat de Barcelona, Mart´ ı i Franqu´ es, 1, E08028 Barcelona, Spain 2Institut de Ci` encies del Cosmos (ICCUB), Universitat de Barcelona, Mart´ ı i Franqu´ es, 1, E08028 Barcelona, Spain 3Grup d’ `Optica, Departament de F´ ısica, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Spain 4Massachusetts Institute of Technology, Cambridge, MA, United States Synthetically spin-orbit coupling in cold atoms couples the pseudo-spin and spatial degrees of freedom, and therefore the inherent spin symmetry of the system plays an important role. In systems of two pseudo-spin degrees, two particles configure symmetric states and anti-symmetric states, but the spin symmetry can be mixed for more particles. We study the role of mixed spin symmetry in the presence of spin-orbit coupling and consider the system of three bosons with two hyper-fine states trapped in a harmonic potential. We investigate the ground state and the energy spectrum by implementing exact diagonalization. It is found that the interplay between spin-orbit coupling and repulsive interactions between anti-aligned pseudo-spins increases the population of the unaligned spin components in the ground state. The emergence of the mixed spin symmetric states compensates for the rise of the interaction energy. With the aligned interaction on, the avoided crossing between the ground state and the first excited state is observed only for small interaction, and this causes shape changes in the spin populations. Furthermore, we find that the pair correlation of the ground state shows similarly to that of Tonks-Girardeau gas even for relatively small contact interactions and such strong interaction feature is enhanced by the spin-orbit coupling. I. INTRODUCTION Spin-orbit coupling (SOC) was originally discussed in the system of charged particles and, for instance, studied in the context of the spin Hall effect [1] or topological insulators states [2]. However, by using cold atomic sys- tems it is possible to create synthetic SOC in neutral (pseudo) spin-1/2 bosons [3, 4], spin-1 Bose gases [5] and also in Fermi gases [6, 7]. In contrast to condensed mat- ter systems, cold atoms can provide tunable and clean platforms and allow us to explore all possible states gen- erated by SOC. SOC in cold atoms is often discussed in the context of the mean-field regime, e.g. [8–10]. While the mean-field approximation has revealed interesting results, for exam- ple discovering the phase diagram [8], it imposes classical fields and ignores the quantum effects. To bridge single- particle physics and many-body physics in the SOC sys- tem, a mapping between the SOC cold atomic system and the Dicke model has been proposed [11]. However, the validity of the mapping is not obvious. The assump- tion imposed for the mapping is that all particles occupy the same real space state, which automatically leads to the pseudo-spin state spanned in only symmetric spin space. This assumption is commonly used for Bose gases with two internal degrees of freedom [12], however SOC couples pseudo-spin states and real space states and al- lows pseudo-spin states to get out of the symmetric spin space. It is not clear how even the lowest energy state is confined in symmetric spin space. The spin symmetry of the ground state in the two- particle system with SOC has been investigated by per- ∗ayaka.usui@uab.catforming exact diagonalization [13]. Competition be- tween SOC and contact interaction has been studied, and the emergence of anti-symmetric pseudo-spin states for strong interactions has been found. Despite us- ing a bosonic system, anti-symmetric pseudo-spin states are preferable for the ground state. This work has also revealed the parameter regime where anti-symmetric pseudo-spin states or only symmetric pseudo-spin states are contained. However, these results cannot be extended to many-particle systems straightforwardly. One of the reasons is because two-pseudo-spin states have unique spin symmetry. Particles with two internal degrees of freedom are classified into symmetric and anti-symmetric pseudo-spin states, but the spin symmetry can be mixed in more-particle systems and anti-symmetry under all permutation of pseudo-spins no longer exist. It is un- clear how the mixed spin symmetry affects the property of the ground state. In this work, we investigate the role of mixed spin sym- metry in the SOC system. To this end, we consider a three-particle system with two internal states as it is the smallest system that has mixed spin symmetry and focus on comparison with a two-particle system. We have built the Hamiltonian of three bosons with pseudo-spin-1/2 in the presence of SOC by considering second quantisa- tion and computed the ground state. Our numerical re- sults confirm the appearance of mixed symmetric pseudo- spin states in the ground state for strong interaction be- tween anti-aligned pseudo-spins and reveal the parameter regime where it is observed. Furthermore, the emergence of the mixed spin symmetry states reduces the interaction energy. This is also seen in the two-particle system when the anti-symmetric pseudo-spin states emerge. This im- plies that the mixed spin symmetric state mimics the anti-symmetric pseudo-spin state to suppress the energy. In addition, we study the spatial structure of the groundarXiv:2403.07188v1 [cond-mat.quant-gas] 11 Mar 20242 state by looking at the pair correlation and find that SOC assists contact interaction in inducing a strong interac- tion effect. Even for relatively small contact interaction, the pair correlation of the ground state has similarly to the strongly correlated Tonks-Girardeau gas. This is not observed in the two-particle system. The structure of this paper is as follows. Section II explains the basis of our system. First, we introduce the Hamiltonian of our system in the first quantization repre- sentation and then transform it into the second quantiza- tion representation. This is necessary to implement exact diagonalization. Next, we explain the spin symmetry in pseudo-spin-1/2 systems and clarify the existence of non- symmetric pseudo-spin states, which are often ignored. Section III shows our numerical results of the ground state and the energy spectrum. We consider two cases, (i) repulsive interaction between anti-aligned pseudo-spins and (ii) repulsive interaction between aligned pseudo- spins. The conclusion is given in Sec. IV. II. FEW INTERACTING BOSONS WITH SOC We present our formalism here. First, we introduce the Hamiltonian of our system in the first quantization basis and then rewrite it in the second quantization basis to induce the bosonic symmetry. This approach is often used to present the Hamiltonian of few particle systems, e.g. [14–16], and we refer interested readers to a detailed reference [17]. Second, we explain the spin symmetry of two spins and more spins respectively. While bosonic sys- tems with two hyper-fine states are well studied, usually only pseudo-spin symmetric states are focused on, and the existence of anti-symmetry and mixed spin symme- try does not attract attention. We review the categories of spin symmetry before showing our numerical results. A. Hamiltonian in the first quantisation basis We consider a few bosons trapped in a one-dimensional harmonic potential with two internal degrees of freedom in the presence of SOC. The Hamiltonian reads ˆH=ˆH0+ˆHint. (1) The single-particle Hamiltonian is given by ˆH0=NpX j=1 ˆp2 j 2m+mω2ˆx2 j 2+ℏk mˆpjˆσ(j) z+ℏΩ 2ˆσ(j) x! ,(2) where the third term is the SOC term and describes cou- pling between real space and pseudo-spin space and the fourth term is coherent coupling between pseudo-spins. The SOC term gives positive momentum to down spins and negative momentum to up spins. Without either of these terms, the single-particle Hamiltonian can be di- agonalised with the basis of ˆ σzor ˆσx. The interactionHamiltonian can be decomposed into pseudo-spin basis as ˆHint=ˆH↓↓+ˆH↓↑+ˆH↑↑. (3) Each of the components describes contact interaction and is given by ˆH↓↓=X i<jg↓↓δ(xi−xj)|↓⟩i|↓⟩j⟨↓|i⟨↓|j, (4) ˆH↑↑=X i<jg↑↑δ(xi−xj)|↑⟩i|↑⟩j⟨↑|i⟨↑|j, (5) ˆH↓↑=X i<jg↑↑δ(xi−xj)  |↓⟩i|↑⟩j⟨↓|i⟨↑|j+|↑⟩i|↓⟩j⟨↑|i⟨↓|j .(6) B. Hamiltonian in the second quantization basis The above formalism is called the first quantization and describes distinguishable particles. To express indis- tinguishable particles simply, we implement the second quantization representation in a truncated space. Con- sider that Nparticles occupy Nof first Meigenstates of the harmonic oscillator for down spins and up spins. Defining the creation and annihilation operators for the jth eigenstate as ˆ ajand ˆa† j, the second quantised version of the single-particle part (2) is given by ˆH0=2MX i,j=1ˆa† iˆajϵi,j, (7) where the indices i, jinclude the label of the eigen- states of the harmonic oscillator and the label of the pseudo-spin degrees. Specifically, we take numbers in [1, M] for the eigenstates for down spins and numbers in [M+ 1,2M] for the eigenstates for up spins. The single- particle energy is represented as [15] ϵi,j=ℏω nx(i) +1 2 δi,j+ikξ√ 2ms(j)δms(i),ms(j) ×p nx(j) + 1δnx(i),nx(j)+1−p nx(j)δnx(i),nx(j)−1 +ℏΩ 2δms(i),−ms(j)δnx(i),nx(j), (8) where ξ=p ℏ/mω is the trap length, nx(j) represents the label of eigenstate of the harmonic oscillator, and ms(j) represents the pseudo-spin state: down spins give ms(j) =−1, and up spins give ms(j) = 1. The interac-3 tion part (3) is given by ˆHint=1 22MX i,j,k,l =1ˆa† iˆa† jˆakˆalVi,j,k,l × g↓↓δms(i),−1δms(j),−1δms(k),−1δms(l),−1 +g↑↑δms(i),1δms(j),1δms(k),1δms(l),1 +g↓↑ δms(i),1δms(j),−1δms(k),1δms(l),−1 +δms(i),−1δms(j),1δms(k),−1δms(l),1
 ,(9) where Vi,j,k,l =Z∞ −∞dx ϕ nx(i)(x)ϕnx(j)(x)ϕnx(k)(x)ϕnx(l)(x) (10) with ϕnthenth eigenstate of harmonic oscillator. The above integral (10) can be solved analytically but is hard to compute for large indices as binomial coefficients ap- pear. See Ref. [16] for an efficient calculation of it. C. Spin symmetry of bosons with two internal states Here, we review the symmetry of pseudo-spin-1/2 sys- tems, and let us start with two-particle states. Since we consider bosons, every state is symmetric under permu- tation. The considered system has the pseudo-spin de- gree and the spatial degree of freedom, and the pseudo- spin basis are classified into three symmetric pseudo-spin states, |↓↓⟩, (|↓↑⟩+|↑↓⟩)/√ 2,|↑↑⟩and one anti-symmetric pseudo-spin state ( |↓↑⟩−|↑↓⟩ )/√ 2. To satisfy the bosonic symmetry, the spatial states of symmetric pseudo-spin states are symmetric, i.e. even functions in the rela- tive coordinate x1−x2. Similarly, the spatial states of anti-symmetric pseudo-spin states are anti-symmetric, i.e. odd functions in the relative coordinate x1−x2. Since the contact interaction occurs at a point and is described as the delta function δ(x1−x2), the anti-symmetric spa- tial states do not feel the contact interaction. By adding more particles to the system, the spin sym- metry is mixed, and the anti-symmetry under all per- mutation does not exist anymore. Specifically, states of three bosons with pseudo-spin-1/2 can be constructed with the following symmetric spin basis, |S1⟩=|↓↓↓⟩ |S2⟩=1√ 3(|↓↓↑⟩ +|↓↑↓⟩ +|↑↓↓⟩ ) |S3⟩=1√ 3(|↓↑↑⟩ +|↑↓↑⟩ +|↑↑↓⟩ ) |S4⟩=|↑↑↑⟩ , (11)and the following mixed spin symmetric basis, |M1⟩=−1√ 6(|↓↑↑⟩ +|↑↓↑⟩ − 2|↑↑↓⟩ ) |M2⟩=1√ 6(|↑↓↓⟩ +|↓↑↓⟩ − 2|↓↓↑⟩ ) |M3⟩=1√ 2(|↑↓↑⟩ − |↓↑↑⟩ ) |M4⟩=1√ 2(|↑↓↓⟩ − |↓↑↓⟩ ). (12) For example, switching the first spin and the second spin in the pseudo-spin basis |M1⟩changes nothing (symmet- ric), but by switching the second spin and the third spin, the resultant state cannot be described with |M1⟩any- more. Therefore, these basis do not belong to symmetry or anti-symmetry but are called mixed symmetry. To sat- isfy symmetry under all permutation, the spatial states of the mixed spin symmetric states are neither symmetric nor anti-symmetric but formed such that the total state is symmetric. When the spatial states and the pseudo-spin states are decoupled in systems of pseudo-spin-1/2 bosons, the pseudo-spin space is confined to the symmetric spin space. For instance, a pseudo-spin-1/2 BEC in a har- monic trap can be modelled with the collective spin op- erators, and the pseudo-spin Hamiltonian is expressed as ˆH=gˆS2 z[12]. The influence of the spatial state is in- cluded in the interaction strength g, and the strength gcan be tuned by modifying the potential for exam- ple. However, SOC couples the spatial degree and the pseudo-spin degree, and the pseudo-spins can be non- symmetric. For instance, in the two-particle system, rel- ative motion is induced by coupling between symmetric and anti-symmetric spin states (see Eq. (3) in Ref. [13]). Non-symmetric spin states play as an important role as symmetric spin states in the system with SOC. III. NUMERICAL RESULTS We construct the Hamiltonains (7)(9) and diagonalize them in a truncated space numerically. In this work, we fixg↓↓=g↑↑=gto keep the symmetry between down and up spins. Moreover, to study competition between SOC and contact interaction, we turn on the anti-aligned pseudo-spin interaction g↓↑or the aligned pseudo-spin interaction g. The study of the two particle system has shown that these two cases give clear different properties in the ground state [13]. We use the trap energy ℏωas the energy unit and normalise the parameters with the trap energy ℏωand the trap length ξ=p ℏ/mω, and display normalised parameters in all the figures. Also, we fix kξ= 4 and set the cutoff of the truncated space asM= 50. We discuss the justification of the cutoff M in Appendix A, and our numerical code can be viewed in Ref. [18].4 𝑔↓↑=0𝑔=0𝑔↓↑=1𝑔=0(a)Ω𝐸#−𝐸$(b)Ω𝑔↓↑=10𝑔=0(c)Ω FIG. 1. Energy difference Ej−E0between the first to fifth excited states and the ground state for (a) the non-interacting case, (b)g↓↑/ℏωξ= 1, and (c) g↓↑/ℏωξ= 10. The dashed line is the critical point Ω0 c/ω= 2k2ξ2in free space, and the dotted line is when E1andE0start to deviate, ( E1−E0)/ℏω >0.01. A. Zero interactions (g=g↓↑= 0) It is known that SOC systems can have three different ground state phases [8]: the stripe phase, the magne- tised phase, and the single minimum phase. In the stripe phase, the ground state is a superposition of positive and negative momenta and thus displays an interference pat- tern. In the magnetised phase, the ground state acquires positive or negative momentum, and in the single mini- mum phase the spectrum only possess a single minimum, leading to zero momentum usually. Without contact in- teractions, only the stripe phase and the single minimum phase exist, and at the critical point some degeneracies are resolved. In free space the critical point between these two phases is given analytically by Ω0 c/ω= 2k2ξ2, and in trapped systems the density modulation modifies the critical value a lower value [19]. We plot the energy difference Ej−E0between some excited states and the ground state for the noninteract- ing case ( g=g↓↑= 0) for changing the coherent cou- pling strength Ω and reconfirm that the degeneracy in the ground state is resolved off from Ω0 c(see the dashed line in Fig. 1(a)). Therefore, we define another critical point Ω cfor the trap system as a point where the energy difference ( E1−E0)/ℏωbetween the ground state and the first excited state is larger than 10−2(see the dotted line). In the absence of coherent coupling, the ground states are four-fold degenerate, and their pseudo-spin states are aligned pseudo-spins such as (i) three down- spins and (ii) three up-spins and upaligned pseudo-spins such as (iii) two up-spins and one down-spin (iv) one up- spin and two down-spins. Positive (negative) momentum is induced to down (up) spins due to SOC. The coherent coupling mixes these states, and in the strong coherent coupling limit an equally-weighted superposition of all pseudo-spin states obtains the lowest energy. By adding contact interactions between pseudo-spins, the above four degenerate ground states (i-iv) obtain dif- ferent energy. We discuss the ground state in the case of repulsive anti-aligned interactions ( g↓↑>0 and g= 0)first and in the case of repulsive aligned interactions (g >0 and g↓↑= 0) later. B. Anti-aligned interactions ( g↓↑>0andg= 0) We consider non-zero positive anti-aligned interaction g↓↑>0, i.e. when the unaligned pseudo-spin compo- nents suffer from repulsive interaction. As a result, in the absence of the coherent coupling only the aligned pseudo-spin states |↓↓↓⟩ ,|↑↑↑⟩ obtain the lowest energy. We have computed the energy spectrum for relatively weak interaction g↓↑/ℏωξ= 1 and for strong interac- tiong↓↑/ℏωξ= 10. In the former (latter) case, the en- ergy shift due to the contact interaction is less (more) than ℏω. It is found that the degeneracy between the ground state and the first excited state is resolved at a different value of Ω from the noninteracting case, and it depends on the interaction strength g↓↑(see the dotted lines in Fig. 1(b,c)). Such deviation from the noninter- acting case is also seen in the two-particle system but slightly smaller than in the three-particle system. This is because in the three-particle system the contact inter- action energy is larger than in the two-particle system, and the maximum energy shift due to contact interac- tion is ℏωfor two particles and 3 ℏωfor three particles. For Ω /ω≃40, the energy spectrum for different g↓↑look similar to each other because the coherent coupling is so strong that the contact interaction contribution to the energy is negligible. 1. Pseudo-spin population We study the pseudo-spin component in the ground state. Here, we categorise the pseudo-spin population of the ground state into the aligned pseudo-spin com- ponents and the unaligned pseudo-spin components, i.e. the population paligned of|↓↓↓⟩ and|↑↑↑⟩ and the pop- ulation punaligned of the rest of the pseudo-spin basis,5 𝑝!"#$%&'𝑝($!"#$%&'𝑝)*+),𝑝-Ω(a)𝑔↓↑=1𝑔=0(b)𝑔↓↑=10𝑔=0ΩΩ𝐸!"#𝑔↓↑=1𝑔↓↑=2𝑔↓↑=10(c) FIG. 2. (a,b) Pseudo-spin population paligned ,punaligned in the ground state for weak or strong contact interactions in the three-particle system. The dashed and dotted lines represent the population pS2+S3of|S2⟩and|S3⟩and the population pMof the mixed spin symmetry states, respectively. Note that punaligned =pS2+S3+pM. The dotted vertical lines represent Ω c. (c) Interaction energy for different g↓↑. therefore paligned +punaligned = 1. Note that the popula- tions of |↓↓↓⟩ and|↑↑↑⟩ are the same due to the sym- metry of the contact interactions we set ( g↓↓=g↑↑). We have plotted these pseudo-spin components paligned , punaligned as a function of coherent coupling strength Ω (see Fig. 2(a,b)). For weak coherent coupling Ω, the aligned pseudo-spin population paligned is dominant be- cause of repulsive anti-aligned interaction. By increasing Ω, the unaligned pseudo-spin component punaligned also grows. Eventually, the population paligned is approach- ing 3 /4 while punaligned is approaching 1 /4, because the lowest energy state in the limit Ω → ∞ is (|↑⟩ − |↓⟩ )⊗3=|↑↑↑⟩ − |↑↑↓⟩ − |↑↓↑⟩ − |↓↑↑⟩ +|↑↓↓⟩ +|↓↑↓⟩ +|↓↓↑⟩ − |↓↓↓⟩ . In the intermediate regime, these populations paligned , punaligned grow differently depending on g↓↑. While for the weak interaction case (a) in Fig. 2 both populations increase monotonically, for the strong interaction case (b) in Fig. 2 the population paligned (punaligned ) reaches an extreme value and overcomes 1 /4 (3/4), i.e. paligned is concave, and punaligned is convex. Furthermore, we have computed the population pM1, pM2, pM3, pM4of the mixed spin symmetric ba- sis (12) (see the details of the calculation to Appendix B). Since these populations are the same due to our setting g↓↓=g↑↑, we have plotted the sum pM=pM1+pM2+ pM3+pM4(see the dotted lines in Fig. 2(a,b)). We have found that a large amount of the mixed spin symmet- ric states appear in the ground state for the strong in- teraction case (b) in Fig. 2. Considering punaligned = pS2+S3 +pMwith pS2+S3 the population of |S2⟩and |S3⟩, this leads to the non-monotonic growth of paligned , punaligned . This is also seen in the two-particle system, and in that case it is originated from the emergence of the anti-symmetric pseudo-spin states [13]. The interac- tion energy increases for increasing the contact interac- tion strength in general, but the anti-symmetric pseudo- spin states do not feel contact interactions. Thus, the appearance of the anti-symmetric pseudo-spin states forstrong contact interactions suppresses the interaction en- ergy. Therefore, it is a question whether the mixed spin symmetric states also reduce the interaction energy in the case we consider, and the answer is that they do. We have calculated the interaction energy Eint=⟨ˆHint⟩ and plotted it as a function of Ω, which displays a dent for strong interaction (see Fig. 2(c)). Finally, we have inspected the non-monotonic growth of punaligned for a wide range of interaction g↓↑(see Fig. 3) and confirmed that the excess over 3 /4 is larger for stronger g↓↑. 2. Pair correlation So far we have shown that the property of the ground state in the three particle system is similar to that in the two particle system regardless of the difference of the spin symmetry. The spatial structure of the ground state discussed in this section rather shows the opposite and reveals that the interplay between SOC and contact in- teractions brings a distinct feature from the two-particle Ω𝑔↓↑ FIG. 3. Excess of punaligned over the population at the limit of Ω→ ∞ as function of Ω /ωandg↓↑/ℏωξ, i.e. punaligned −3/4. Negative values are not shown for clear presentation.6 𝑔↓↑=1𝑔↓↑=0.5 𝑔↓↑=3TG gas Ω=10Ω=20Ω=26Ω=30Fix 𝑔=0 FIG. 4. Pair correlation of the ground state when changing g↓↑and Ω while fixing g= 0. The right panel displays the pair correlation of TG gas, and notice that the scale of the right panel is different from the other panels. system. The pair correlation of the ground state ΨGS(x1, x2, x3) is defined as ρ2(x, y) =Z dx3|ΨGS(x, y, x 3)|2, (13) where two of the spatial degrees are kept and the other is integrated out. In the second quantisation representation it is given by ρ2(x, y) =1 N(N−1)2MX i,j,p,q =1⟨ΨGS|ˆa† iˆa† pˆajˆaq|ΨGS⟩ ϕ∗ i(x)ϕ∗ p(y)ϕj(x)ϕq(y) (14) with ϕi(x) the ith eigenstate of the harmonic oscillator andN= 3 particle number [14]. The pair correlation of the free particle system is given by a two-dimensional Gaussian function, ρ2(x, y) =|ϕ0(x)|2|ϕ0(y)|2. In the limit of strong interactions g↓↑, g→ ∞ , the system corresponds to Tonks-Girardeau gas (TG) gas and be- haves as (spinless) fermions [20–22], and particularly the density profile corresponds to that of the free fermions, |ΨTG(x1, x2, x3)|=|ΨF(x1, x2, x3)|. Therefore, the pair correlation in the limit can be calculated analytically by using the wavefunction of three free fermions trapped in the harmonic potential, given by ΨF(x1, x2, x3) =1√ 3!ϕ0(x1)ϕ1(x1)ϕ2(x1) ϕ0(x2)ϕ1(x2)ϕ2(x2) ϕ0(x3)ϕ1(x3)ϕ2(x3),(15)and is plotted for reference in the right panel in Fig. 4. We study the pair correlations, and first let us focus on g↓↑/ℏωξ= 1 and change the coherent coupling strength Ω (see the middle panels at g↓↑= 1 in Fig. 4). For relatively small coherent coupling Ω /ω= 10, the pair correlation is close to a Gaussian distribution, and the contact inter- action affects the pseudo-spin populations but does not change the spatial structure of the ground state from the free particle case. By increasing the coherent coupling strength (Ω /ω= 20), the shape is squished along x=y. At Ω/ω= 26, a dent emerges at the centre ( x=y= 0). For increasing Ω more, a slit at x=ybecomes deeper, and two bumps remain. This slit is originated solely from the contact interaction, and the pair correlation remains in the same shape even for larger Ω. It is consistent to the pseudo-spin populations paligned ,punaligned , which reach plateau when Ω /ω≃30. To dig into the interplay between the contact interac- tion and the SOC, we fix the coherent coupling strength to Ω/ω= 26 and change the contact interaction strength (see the panels at Ω = 26 in Fig. 4). By decreas- ing the contact interaction strength, the dent seen for g↓↑/ℏωξ= 1 becomes invisible. On the other hand, for increasing gthe pair correlation shows three bumps in x > y and in x < y . This is a qualitatively similar fea- ture to the TG gas other than the size of these bumps. It is interesting that such similarly to TG gas is seen even for relatively small interaction such as g↓↑/ℏωξ= 3. For larger g↓↑, the same structure remains. Also, it is worth noting that there is no interaction between the same pseudo-spins. In the two-particle system, the coun- terpart of the pair correlation is density distribution7 𝑔=0.48𝑔↓↑=0(a)Ω𝐸#−𝐸$𝑔=2𝑔↓↑=0(b)Ω FIG. 5. Energy difference Ej−E0between some excited states and the ground state for g/ℏωξ= 0.48 (a) and g/ℏωξ= 2 (b). The dashed line is the critical point Ω0 c/ω= 2k2ξ2in free space, and the dotted line is when E1andE0start to deviate, ( E1−E0)/ℏω >0.01. ρ2(x1, x2) =|ΨGS(x1, x2)|2, and it was investigated in our previous work [13]. However, such strong interaction feature enhanced by SOC is not observed. C. Aligned interaction ( g >0andg↓↑= 0) Now we turn off anti-aligned interaction g↓↑= 0 and set non-zero aligned interaction strength g > 0. Any pseudo-spin configuration of three particle states has the aligned pseudo-spin components and is affected by the aligned interaction. For Ω /ω= 0, the two un- aligned pseudo-spin states (the states (iii,iv) mentioned in Sec. III) obtain the lowest energy. For finite Ω, the coherent coupling mixes those states and the two pseudo-spin aligned components |↓↓↓⟩ ,|↑↑↑⟩ . In the limit Ω→ ∞ , the contact interaction contribution is not visi- ble, and the ground state approaches that of no interac- tions. Figure 5 reveals the energy difference Ej−E0in the intermediate regime of Ω for two different interaction strengths g/ℏωξ= 0.48,2. It is similar to the anti-aligned interaction case ( g↓↑>0) that the critical points Ω cvary in different interaction strengths. One different feature is that the avoided crossing between the ground state and the first excited states is seen for g/ℏωξ= 0.48 when Ω/ω≃25 but not for g/ℏωξ= 2. This is caused by the interplay between the contact interaction and the coher- ent coupling that have attempt to shift energy in the op- posite directions [23]. By increasing g, the excited states are pushed up, and accordingly the avoided crossing is shifted up and shades out eventually. The existence of this avoided crossing affects the pseudo-spin population as see below. 1. Pseudo-spin population As shown, in the anti-aligned interaction case ( g↓↑> 0), the pseudo-spin populations paligned ,punaligned be- come larger than the values in the limit Ω → ∞ for larger Ω𝑝!"#$%&'for 𝑔=0.48𝑝($!"#$%&'for 𝑔=0.48𝑝!"#$%&'for 𝑔=2𝑝($!"#$%&'for 𝑔=2(a) Ω𝑔(b)FIG. 6. (a) Pseudo-spin population paligned ,punaligned in the ground state for g/ℏωξ= 0.48 and g/ℏωξ= 2. (b) Excess of paligned over the population at the limit of Ω → ∞ as function of Ω/ωandg/ℏωξ, i.e. paligned−1/4. Negative values are not shown for clear presentation. interaction strength. In the aligned interaction case ( g > 0), the aligned pseudo-spin population paligned jumps up and the unaligned pseudo-spin population punaligned drops down for relatively weak interaction (see Fig. 6(a)). The emergence of this sharp change matches the appear- ance of the avoided crossing, and for large gthese extreme values of paligned andpunaligned disappear (see Fig. 6(b)). This is contrasted with the anti-aligned interaction case, where the pseudo-spin populations paligned andpunaligned obtain the extreme values when Ω = Ω c, i.e. when the degeneracy between the first excited state and the ground state is resolved. This non-monotonic behaviour survives only for small values of gcompared to the anti-aligned interaction case because the emergence of aligned pseudo- spin states does not reduce the interaction energy. We note that such population jump has been also observed in the two-particle system [13]. 2. Pair correlation We have computed the pair correlations for different g and Ω. First, let us focus on g/ℏωξ= 1 and change Ω (see the middle panels at g= 1 in Fig. 7). Overall, the pair correlation behaves similar to the anti-aligned inter- action case but is more sensitive to Ω. Even for relatively small coupling, the pair correlation starts deviating from Gaussian distribution. For Ω /ω= 5, one bump appears at the centre ( x=y= 0), and for increasing Ω four bumps come out. At Ω /ω= 26, where the populations paligned ,punaligned approach the extreme values as shown8 Ω=1Ω=10Ω=5Ω=20Ω=30Ω=26𝑔=1𝑔=2𝑔=0.48 TG gas Fix 𝑔↓↑=0 FIG. 7. Pair correlation of the ground state when changing for gand Ω while fixing g↓↑= 0. The right panel displays the pair correlation of TG gas, and notice that the scale of the right panel is different from the other panels. in Fig. 5, a dent emerges at the centre. For increasing Ω more, a slit at x=ycaused by the contact interaction appears, and the effect of SOC vanishes. Now we fix the coherent coupling strength to Ω /ω= 26 and change the contact interaction strength to g/ℏωξ= 0.48,2. For g/ℏωξ= 0.48, the dent at x=y= 0 van- ishes, and for g/ℏωξ= 2 three bumps emerge in x > y and in x < y respectively, which is similar to the anti- aligned interaction case. The aligned interaction and the anti-aligned interaction give similar effects to the struc- ture of the pair correlations even though they affect the energy spectrum and the pseudo-spin population differ- ently. IV. CONCLUSIONS We have studied the system of three bosons trapped in a harmonic potential with two internal states in the pres- ence of SOC and investigated the ground state and the energy spectrum by implementing exact diagonalization. Particularly, we focus on how the mixed spin symmet- ric states contribute to the ground state. We have found that the interplay between the anti-aligned contact inter- action and SOC increases the population of the unaligned pseudo-spin components in the ground state. The emer- gence of the mixed spin symmetric states compensates for the rise of the interaction energy. This is similar to the two-particle system which has different spin symme- try. With the aligned interaction on, the avoided crossing between the ground state and the first excited state is ob- served only for small interaction, and this causes shape changes in the pseudo-spin populations. However, the emergence of aligned pseudo-spin states does not con- tribute to the reduction of the interaction energy, and therefore it decays out for strong interaction. Further-more, we have found that the pair correlation of the ground state shows similarly to the TG gas even for rel- atively small contact interactions due to the interplay between contact interaction and SOC. A question arises whether the same behaviour is seen in the more-particle systems. Our results cannot guaran- tee but implies that this would be the case for the anti- aligned interaction case. In the anti-aligned interaction case, the mixed spin symmetry affects the pseudo-spin populations, and the more-particle systems also have the same spin symmetry. The mixed spin symmetric states may act in the same way. The mapping between Dicke model and the SOC sys- tem with contact interactions was proposed [11, 24, 25], assuming that the pseudo-spin space of the SOC system is only symmetric. However, as shown already, there is significant amount of mixed spin symmetric components even in the ground state for some parameter regime. Therefore, the mapping between these two models are limited. In this work, we focus on the low energy states, but it is possible to study excited states and quench dynamics in a wide parameter range with our numerical method. As future work, it would be interesting to study trans- port via SOC [26] in the presence of strong interaction and utilise SOC, which gives different directions of mo- mentum to different pseudo-spins, to create spatial en- tanglement [27–29]. V. ACKNOWLEDGEMENT Discussions with Thomas Busch, Karol Gietka, Yongping Zhang, Peter Engels, Pere Mujal, and Ian B. Spielman are appreciated. We acknowledge funding from Grant No. PID2020-114626GB-I00 by9 𝑘𝑔=𝑔↓↑=0𝑔=𝑔↓↑=1𝑔=𝑔↓↑=10𝐸!"!#$ FIG. 8. Total energy for different interaction strengths g= g↓↑, where Ω = 0 and M= 50 are set. The dotted lines show the analytical results of the energy for k= 0. MCIN/AEI/10.13039/5011 00011033 and “Unit of Ex- cellence Mar´ ıa de Maeztu 2020-2023” award to the In- stitute of Cosmos Sciences, Grant CEX2019-000918-M funded by MCIN/AEI/10.13039/501100011033. We also acknowledge financial support from the Generalitat de Catalunya (Grant 2021SGR01095). A.U acknowledges further support from the Agencia Estatal de Investi- gaci´ on and the Ministerio de Ciencia e Innovaci´ on. A.R.- F. acknowledges funding from MIU through Grant No. FPU20/06174. Appendix A: Justification of cut off M In Sec. III, we set the SOC strength kξ= 4 and the cut offM= 50. Here, we show that the cut off Mis large enough. As discussed in Sec. II B, the cut off Mis the number of the eigenstates of the harmonic oscillator we take. The SOC couples these eigenstates such that the momentum of up spin or down spin is boosted. Although larger SOC needs larger M, the deviation from the results when k= 0 is negligible if Mis large enough. Without coherent coupling Ω, the Hamiltonian (1) can be diagonalised in the basis of σz, and the lowest energy can be obtained anaalytically and is the same as whenk= 0. For instance, the kinetic term of the single particle Hamiltonian (2) can be written asPN j=1(pj−ℏkσz)2/2m. The SOC acts as a moving frame, and the lowest energy does not change. Thus, by looking at the lowest energy for different k, we see whether the cut off Mis large enough. Figure 8 shows the energy computed with M= 50 as function of k. For large k, the energy obtained numerically deviates from the energy with k= 0. For strong interactions, the deviation is worse because strong interactions also couple highly excited eigenstates. For kξ= 4, the error is small and at worst about 2 .7% for g/ℏωξ=g/ℏωξ= 10, and we have taken kξ= 4 in the main text. Appendix B: Population of mixed spin symmetry In this work, we investigate the population of mixed spin symmetry and here explain how to compute it briefly. The mixed spin symmetry basis are described in the first quantization representation in Eq. (12). On the other hand, we adopt the second quantization represen- tation for exact diagonalization (see Sec. II B). Therefore, we transform the second quantization representation of the Hamiltonians (7)(9) to the first quantization repre- sentation, i.e., label the particles and treat the system as distinguishable particles (see Fig. 9). This enables us to obtain the quantum number and the pseudo-spin state of each particles, i.e., have |n1, σ1;n2, σ2;n3, σ3⟩for n1, n2, n3= 0,1, . . . , M −1 and σ1, σ2, σ3=↓,↑. We transform the pseudo-spin basis using ↓,↑to the classified pseudo-spin basis Sj,Mjby using the relations (11)(12). For more concrete details, see our code [18]. +132231+321 FIG. 9. Example of transforming the second quantization representation to the first quantization representation. The left hand shows a state of three indistinguishable particles, and the right hand shows superposition between three states of three distinguishable particles. These are summed up due to bosons. [1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall effect, Rev. Mod. Phys. 82, 1539 (2010). [2] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010). [3] Y. J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spin– orbit-coupled bose–einstein condensates, Nature 471, 83 (2011). [4] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Collective dipole oscillations of a spin- orbit coupled bose-einstein condensate, Phys. Rev. Lett.109, 115301 (2012). [5] D. L. Campbell, R. M. Price, A. Putra, A. Vald´ es-Curiel, D. Trypogeorgos, and I. B. Spielman, Magnetic phases of spin-1 spin–orbit-coupled bose gases, Nature Communi- cations 7, 10897 (2016). [6] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Spin-orbit coupled degenerate fermi gases, Phys. Rev. Lett. 109, 095301 (2012). [7] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Spin-injection spec- troscopy of a spin-orbit coupled fermi gas, Phys. Rev. Lett. 109, 095302 (2012).10 [8] Y. Li, L. P. Pitaevskii, and S. Stringari, Quantum tri- criticality and phase transitions in spin-orbit coupled bose-einstein condensates, Phys. Rev. Lett. 108, 225301 (2012). [9] Y. Zhang, L. Mao, and C. Zhang, Mean-field dynamics of spin-orbit coupled bose-einstein condensates, Phys. Rev. Lett. 108, 035302 (2012). [10] Y. Zhang, M. E. Mossman, T. Busch, P. Engels, and C. Zhang, Properties of spin–orbit-coupled bose–einstein condensates, Frontiers of Physics 11, 118103 (2016). [11] C. Hamner, C. Qu, Y. Zhang, J. Chang, M. Gong, C. Zhang, and P. Engels, Dicke-type phase transition in a spin-orbit-coupled bose–einstein condensate, Nature Communications 5, 4023 (2014). [12] A. Sørensen, L. M. Duan, J. I. Cirac, and P. Zoller, Many-particle entanglement with bose–einstein conden- sates, Nature 409, 63 (2001). [13] A. Usui, T. Fogarty, S. Campbell, S. A. Gardiner, and T. Busch, Spin–orbit coupling in the presence of strong atomic correlations, New Journal of Physics 22, 013050 (2020). [14] P. Mujal, E. Sarl´ e, A. Polls, and B. Juli´ a-D´ ıaz, Quan- tum correlations and degeneracy of identical bosons in a two-dimensional harmonic trap, Phys. Rev. A 96, 043614 (2017). [15] P. Mujal, A. Polls, and B. Juli´ a-D´ ıaz, Spin-orbit-coupled bosons interacting in a two-dimensional harmonic trap, Phys. Rev. A 101, 043619 (2020). [16] A. Rojo-Franc` as, A. Polls, and B. Juli´ a-D´ ıaz, Static and dynamic properties of a few spin 1/2 interacting fermions trapped in a harmonic potential, Mathematics 8, 10.3390/math8071196 (2020). [17] D. Ravent´ os, T. Graß, M. Lewenstein, and B. Juli´ a-D´ ıaz, Cold bosons in optical lattices: a tutorial for exact di- agonalization, Journal of Physics B: Atomic, Molecular and Optical Physics 50, 113001 (2017). [18] A. Usui, https://github.com/ayaka-usui/SOCED (2023). [19] C. Zhu, L. Dong, and H. Pu, Harmonically trapped atoms with spin–orbit coupling, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 145301 (2016). [20] L. Tonks, The complete equation of state of one, two and three-dimensional gases of hard elastic spheres, Phys.Rev.50, 955 (1936). [21] E. H. Lieb and W. Liniger, Exact analysis of an interact- ing bose gas. i. the general solution and the ground state, Phys. Rev. 130, 1605 (1963). [22] M. A. Garc´ ıa-March, B. Juli´ a-D´ ıaz, G. E. Astrakharchik, T. Busch, J. Boronat, and A. Polls, Quantum correla- tions and spatial localization in one-dimensional ultra- cold bosonic mixtures, New Journal of Physics 16, 103004 (2014). [23] Q. Guan, X. Y. Yin, S. E. Gharashi, and D. Blume, Energy spectrum of a harmonically trapped two-atom system with spin–orbit coupling, Journal of Physics B: Atomic, Molecular and Optical Physics 47, 161001 (2014). [24] J. Lian, L. Yu, J. Q. Liang, G. Chen, and S. Jia, Orbit- induced spin squeezing in a spin-orbit coupled bose- einstein condensate, Scientific Reports 3, 3166 (2013). [25] Y. Huang and Z.-D. Hu, Spin and field squeezing in a spin-orbit coupled bose-einstein condensate, Scientific Reports 5, 8006 (2015). [26] X. Chen, R.-L. Jiang, J. Li, Y. Ban, and E. Y. Sherman, Inverse engineering for fast transport and spin control of spin-orbit-coupled bose-einstein condensates in moving harmonic traps, Phys. Rev. A 97, 013631 (2018). [27] M. Fadel, T. Zibold, B. D´ ecamps, and P. Treutlein, Spatial entanglement patterns and einstein-podolsky-rosen steering in bose- einstein condensates, Science 360, 409 (2018), https://www.science.org/doi/pdf/10.1126/science.aao1850. [28] P. Kunkel, M. Pr¨ ufer, H. Strobel, D. Linne- mann, A. Fr¨ olian, T. Gasenzer, M. G¨ arttner, and M. K. Oberthaler, Spatially distributed multipartite entanglement enables epr steer- ing of atomic clouds, Science 360, 413 (2018), https://www.science.org/doi/pdf/10.1126/science.aao2254. [29] K. Lange, J. Peise, B. L¨ ucke, I. Kruse, G. Vitagliano, I. Apellaniz, M. Kleinmann, G. T´ oth, and C. Klempt, Entanglement between two spatially separated atomic modes, Science 360, 416 (2018), https://www.science.org/doi/pdf/10.1126/science.aao2035.

2311.09303v1.Chirality_induced_emergent_spin_orbit_coupling_in_topological_atomic_lattices.pdf

Chirality-induced emergent spin-orbit coupling in topological atomic lattices Jonah S. Peter,1, 2,∗Stefan Ostermann,1and Susanne F. Yelin1 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2Biophysics Program, Harvard University, Boston, Massachusetts 02115, USA Spin-orbit coupled dynamics are of fundamental interest in both quantum optical and condensed matter systems alike. In this work, we show that photonic excitations in pseudospin-1/2 atomic lattices exhibit an emergent spin-orbit coupling when the geometry is chiral. This spin-orbit coupling arises naturally from the electric dipole interaction between the lattice sites and leads to spin polarized excitation transport. Using a general quantum optical model, we determine analytically the conditions that give rise to spin-orbit coupling and characterize the behavior under various symmetry transformations. We show that chirality-induced spin textures are associated with a topologically nontrivial Zak phase that characterizes the chiral setup. Our results demonstrate that chiral atom arrays are a robust platform for realizing spin-orbit coupled topological states of matter. I. INTRODUCTION The spin-orbit (SO) interaction refers to the coupling of a particle’s intrinsic angular momentum to its motional degrees of freedom. For an electron moving in an elec- trostatic potential, this phenomenon can be interpreted as arising from a Lorentz transformation of the electric field in the laboratory frame to a momentum dependent magnetic field that couples directly to the electron’s spin in its rest frame. The resulting spin-momentum locking and associated spin textures can result in nontrivial topo- logical properties [1, 2] and are of considerable interest in the development of new spintronics devices [3–5]. In par- ticular, the generation of long-lived nonequilibrium spin currents is crucial for designing efficient spin transistors, spin diodes, and other related technologies [6]. Recently, various photonic and quantum optical sys- tems have emerged as attractive candidates for realizing SO coupled dynamics. Those involving photonic nanos- tructures have demonstrated the SO coupling of pho- tons using the circular polarizations of light [7–10]. In cold atoms, pairs of internal hyperfine states can act as pseudospin-1/2 systems that resemble electronic spin de- grees of freedom [11]. Coupling these states to coher- ent laser fields can produce synthetic SO potentials in ultracold Fermi gases [12–14] and Bose-Einstein conden- sates [15]. In addition to ultracold gases comprised of moving atoms, the hyperfine levels of atoms or atom-like emitters arranged in ordered lattices can also be used as pseudospin-1/2 states. Such atomic arrays support the transport of optical excitations in a manner analogous to electrons in traditional crystal lattices [16–18] and can be leveraged as quantum simulation platforms to study spin dependent transport processes in a highly tunable environment. In a related Letter, we demonstrate that the collective Bloch modes of pseudospin-1/2 atomic arrays experience an emergent SO coupling when the lattice geometry is chiral [19]. In this case, the associated photonic band ∗jonahpeter@g.harvard.edustructure exhibits a finite spin texture and a topologi- cally nontrivial Zak phase. In turn, these spin bands result in spin dependent dynamics which manifest as po- larization selective photon transport and superradiant emission. In this work, we present a general description of the emergent SO coupling and topological properties associated with chiral atomic lattices. Our findings are distinct from those of previous works in that, here, the SO coupling results from the phase dependence of the electric dipole field that arises naturally from the chiral- ity of the geometry itself. We determine the analytical conditions that give rise to SO coupling in propagating optical excitations and characterize the behavior under different symmetry transformations. Our results demon- strate that chiral atom arrays are a robust platform for realizing SO coupled topological states of matter. II. THEORETICAL FORMALISM In order to study SO coupled transport, we con- sider ordered arrays of dipole-coupled quantum emitters (e.g., atoms, molecules, quantum dots) at fixed positions within the laboratory frame. The pseudospin degree of freedom for each emitter, i, is encoded in a V-type level structure consisting of a single ground state, |gi⟩, and two hyperfine states, |↑i⟩and|↓i⟩, corresponding to the two orthogonal polarizations of circularly polarized light [Fig. 1(a)]. The bare hyperfine states are assumed to have identical resonance frequencies, ω0= 2πc/λ 0, where λ0is the wavelength of each optical transition and cis speed of light in vacuum. A. Dipole-dipole interactions The transport of optical excitations between quantum emitters in free space involves long-range interactions me- diated by a radiation field. It is convenient to trace out the field degrees of freedom in the Born and Markov ap- proximations to obtain an effective descriptive in terms of the matter operators only [20, 21]. In free space, thearXiv:2311.09303v1 [quant-ph] 15 Nov 20232 effective interactions between quantum electric dipoles at points riandrjare determined by the dyadic Green’s tensor G(rij, ω0) =eik0rij 4πk2 0r3 ijh (k2 0r2 ij+ik0rij−1)11 −(k2 0r2 ij+ 3ik0rij−3)rij⊗rij r2 iji ,(1) where rij=ri−rj,rij=|rij|, and k0=ω0/c(see also Appendix A). The coherent and dissipative parts of the dipole-dipole interaction are then given by Jσσ′ ij=−3 2λ0Γ0εεε† iσ·ReG(rij, ω0)·εεεjσ′ (2) Γσσ′ ij= 3λ0Γ0εεε† iσ·ImG(rij, ω0)·εεεjσ′, (3) where εεεiσis the circular polarization vector for orbital σ∈ {↑,↓}on emitter i, Γ0≡Γσσ ii=ω3 0|℘℘℘iσ|2/(3πℏϵ0c3) is the spontaneous emission rate associated with each ex- cited state orbital, ℘℘℘iσis the transition dipole matrix el- ement vector, and ϵ0is the vacuum permittivity. The interactions therefore depend only on the scalar distance between the emitters and on the relative orientations of the polarization vectors. These polarization vectors are defined with respect to a quantization axis, ˆq, about which the optically excited orbitals are circularly polar- ized. The orientation of the quantization axis relative to the symmetry planes of the lattice is directly related to the emergence of SO coupling in chiral systems (Sec- tion III). For general ˆq, the polarization vectors for left and right circularly polarized excitations are given by εεε↑↓=1√ 2(ˆd1±iˆd2), (4) where ↑(↓) corresponds to + ( −),ˆd1andˆd2denote the orthonormal vectors defining the polarization plane, and ˆd1׈d2=ˆq. B. Open system dynamics The unitary dynamics for an arbitrary arrangement of V-type quantum emitters interacting via the Greeen’s tensor formalism are generated by the Hamiltonian H=NX i=1X σω0|σi⟩⟨σi|+NX i,j̸=i=1X σ,σ′Jσσ′ ij|σi⟩⟨σ′ j|(5) (we set ℏ≡1 here and throughout). The non-unitary contributions of collective dissipation and single emitter spontaneous emission are included via the Lindbladian L[ρ] =NX i,j=1Γσσ′ ij 2 2|g⟩⟨σ′ j|ρ|σi⟩⟨g| − |σi⟩⟨σ′ j|, ρ
. (6) FIG. 1. (a) Schematic of the V-type level structure consid- ered in this work. The excited state orbitals have resonant frequency ω0and spontaneous emission rate Γ 0. Emitters are coupled with coherent and dissipative hopping rates Jσσ′ ijand Γσσ′ ij, respectively. (b) Interactions between emitters iandj are mediated through photon exchanges that excite orbitals with either the same (left) or opposite (right) polarization. When the photon spin is changed, conservation of angular momentum requires that the interaction picks up an addi- tional position-dependent phase. Here,|g⟩denotes the collective ground state of the multi- atom system, and we assume that thermal effects are neg- ligible. The full open system dynamics for the state ρ(t) are then given by the quantum optical master equation ˙ρ=−i[H, ρ] +L[ρ]. Throughout this work, we focus on the single- excitation subspace which is sufficient to observe SO cou- pled transport. For single-excitation states, the master equation dynamics are equivalent to those evoked by the non-Hermitian effective Hamiltonian Heff=NX i=1X σω0b† iσbiσ−3 2λ0Γ0NX i,j̸=i=1X σ,σ′Gσσ′ ijb† iσbjσ′, (7) where b† iσ(biσ) denotes a bosonic creation (annihila- tion) operator satisfying [ biσ, b† iσ] = 1, and Gσσ′ ij= εεε† iσ·G(rij, ω0)·εεεjσ′. We may further trace over the ground state of each emitter and denote the excited states using the basis vector mapping |↑⟩=1 0
T,|↓⟩=0 1
T such that the circularly polarized excitations at each emitter site behave as pseudospin-1/2 degrees of freedom characterized by the 2 ×2 Pauli matrices. The operator b† ↑b↑−b† ↓b↓=|↑⟩⟨↑| − |↓⟩⟨↓| =σzthen quantifies the relative spin population in each emitter. Finally, it is useful to define a set of collective opera- tors that act on the spin indices of each emitter simulta-3 neously. We denote Sα≡11⊗σα (8) forα∈ {x, y, z}, where 11 is the identity matrix acting on the spatial indices. The pseudospin operator for a delocalized state extending across multiple emitters then follows simply as Sz. C. Photonic band structures In order to characterize the spin properties of pseudospin-1/2 atomic lattices, we will assess the pho- tonic band structures obtained by transforming the real space Hamiltonian into momentum space. For simplicity, we limit the discussion to lattices that are periodic only along one direction. In this case, the site index for a non- Bravais lattice composed of Nsublattices can be decom- posed into i= (m, µ) where m∈[1, M] indexes the unit cell along the axis of periodicity and µ∈[1,N] denotes the sublattice index. In the limit of large M, the sub- stitution biσ= (1/√ M)P kexp (ik·rmµ)bkµσfor quasi- momentum kyields Heff=P kH(k), where the Bloch Hamiltonian H(k) =X µ,νX σ,σ′hµσ,νσ′(k)b† kµσbkνσ′ (9) has matrix elements [17, 18] hµσ,νσ′(k) =ω0δµνδσσ′+χI µσ,νσ′+χII µσ,νσ′ (10) for χI µσ,νσ′=−3 2λ0Γ0X Rµ̸=0e−ik·RµGσσ′(Rµ)δµν (11) χII µσ,νσ′=−3 2λ0Γ0X Rµe−ik·RµGσσ′(Rµ+nµν)(1−δµν) (12) (see also Appendix B). Here, the (infinite) set of Rµ denotes vectors of the underlying Bravais lattice and nµν=nµ−nνis the basis vector pointing from sub- lattice νtoµwithin a given unit cell. The off-diagonal terms χIandχIIdescribe interactions between emitters on the same and different sublattices, respectively. Like the real-space Hamiltonian, it is important to note that H(k) is, in general, non-Hermitian. III. CONDITIONS FOR SPIN-ORBIT COUPLING The emergence of SO coupling and finite spin polar- izations within pseudospin-1/2 atomic lattices require a nontrivial spin texture for the Bloch modes. Put differ- ently, the spin ⟨Sz⟩must be nonzero at some point in theBrillouin zone in order to observe SO coupling and spin dependent transport. As a main result, we now deter- mine analytically the conditions for ⟨Sz⟩ ̸= 0. In partic- ular, we will show that SO coupling emerges in systems that lack inversion symmetry about axes in the polar- ization plane, which motivates a generalized definition of chirality for pseudospin-1/2 bosons. Quite generally, the spin of each Bloch mode is con- strained to be zero if there exists a symmetry of the Bloch Hamiltonian that reverses the spin of each mode for all k. We can write this symmetry as W≡ SxV, where V|R, µ, σ⟩=X νˆVµν|R, ν, σ⟩ (13) andˆVis a 2N × 2Nunitary matrix acting on the sub- lattice indices. Note that Vpreserves the sign of R. In other words, Wis an operator that reverses the spin but leaves the lattice geometry—including the Bravais lattice vectors—invariant, up to a unitary transformation of the sublattice indices. To see that this symmetry prohibits spinful Bloch bands, consider a general Bloch Hamiltonian H(k) with orthonormal eigenstates |un(k)⟩and corresponding eigenvalues εn(k). We do not require H(k) to be Hermi- tian or time reversal ( T) invariant (see Section IV B). By construction, [ H(k), W] = 0 such that WH(k)|un(k)⟩=H(k)W|un(k)⟩=εn(k)W|un(k)⟩. (14) The states |un(k)⟩and|um(k)⟩ ≡W|un(k)⟩are there- fore both eigenstates of H(k) with the same eigenvalue [22]. Noting that [ V,Sz] = 0 and S† xSzSx=−Sz, the spins of these Bloch states satisfy ⟨um(k)|Sz|um(k)⟩=⟨un(k)|W†SzW|un(k)⟩ =⟨un(k)|V†S† xSzSxV|un(k)⟩ =−⟨un(k)|Sz|un(k)⟩. (15) Now, by orthonormality, we must have ⟨um(k)|un(k)⟩= δmn. If⟨um(k)|un(k)⟩= 1, then there is no degener- acy and |um(k)⟩=eiϕ(k)|un(k)⟩forϕ(k)∈R. That is,|um(k)⟩and|un(k)⟩represent the same state up to aU(1) gauge ambiguity. In this case, it follows triv- ially from Eq. (15) that ⟨Sz⟩= 0. If, on the other hand, ⟨um(k)|un(k)⟩= 0, then the states are orthogo- nal with equal and opposite spin (see, e.g., the N= 1 case of Fig. 4 in Ref. [19]). However, because the states are degenerate, the linear combinations |u±(k)⟩ ≡ (1/√ 2)(|un(k)⟩ ± |um(k)⟩) are also eigenstates of H(k) with the same eigenvalue. These superposition states sat- isfy⟨Sz⟩= 0 by construction. It follows that if Wis a symmetry of the Bloch Hamiltonian, then one may al- ways construct a basis such that all Bloch modes have zero spin. The remaining task is to relate this result back to the geometrical properties of the system. If the lattice con- tains an inversion center at position R= 0, then the4 parity ( P) operator may be written as [23], P |R, µ, σ⟩=X νˆPµν|−R, µ, σ⟩. (16) That is, Pacts by inverting the spatial coordinates of the unit cell, up to a unitary transformation on the sublat- tice indices. We require that Pleave the spin indices invariant because angular momentum is an axial vec- tor. In this case, the Bloch bands are inversion sym- metric about k= 0, and the Bloch Hamiltonian satisfies PH(k)P−1=H(−k). Analogously, we may write W|R, µ, σ⟩=X νˆVµν|R, ν,¯σ⟩, (17) where |¯σ⟩=σx|σ⟩denotes the opposite spin state. The operator Wtherefore acts as a “spin inversion” (as op- posed to spatial inversion) while preserving the real space coordinates of the unit cell. Heuristically, Pacts to negate the sign of the lattice vectors, R(or equivalently the quasimomentum, k), whereas Wnegates the sign of the quantization axis, q. Thus, for Wto be a symme- try of the Bloch Hamiltonian, there must exist a unitary transformation that relates the original and spin-flipped configurations while preserving the mutual orientation of the basis vectors together with kandq. If such a trans- formation does not exist, then the spin inversion symme- try of the combined lattice-quantization axis system is broken. We may therefore take this as the definition of chirality for pseudospin-1/2 bosonic systems. In the fol- lowing section, we will demonstrate how this generalized definition of chirality has a straightforward interpretation in terms of orthogonal transformations of the real space lattice geometry. IV. SYMMETRY ANALYSIS OF SPIN BANDS A. Orthogonal group symmetries In order to study how the condition for nontrivial spin textures relates back to the geometry of the system, we now consider transformations under representations of the orthogonal group O(3). Each unit cell associated with Bravais lattice vector Rcontains 2 Nstates labeled by|R, n⟩, where n= (µ, σ) is a composite index denoting the subblatice and spin. If the unit cell is invariant un- der an orthogonal transformation U, then the operator corresponding to this symmetry may be written as U|R, n⟩=X mˆUnm|UR, m⟩, (18) where ˆUis a 2N ×2Nunitary matrix. Indeed, the parity operator (16) is one such operator. The particular form ofˆUnmdepends on the symmetry operation in question and on the structure of the unit cell. Nevertheless, anumber of general relations can be deduced that apply to all Hamiltonians of the form (7). We are particularly interested in transformations that satisfy Eq. (17) for spin inversion symmetry. The simplest case occurs for lattices that possess a mirror plane. In this case, the basis vectors nµexist in mirror- symmetric pairs such that nν=RMnµfor reflection op- erator RM(note also that µ=νwhen the lattice vectors Rµlie in the mirror plane). If this mirror plane also contains the quantization axis, then the corresponding operator Rq,Macts on the basis states as Rq,M|R, µ, σ⟩=|Rq,MR, ν,¯σ⟩. (19) This transformation satisfies Eq. (17), provided Rq,MR=R. That is, if the lattice possesses a mirror plane that contains the quantization axis and the Bravais lattice vectors, then the spin of each Bloch mode is guaranteed to be zero. The transformation (19) is not the only form of W that imposes trivial spin textures. Rotoreflections are also possible, so long as the combined operation satisfies Eq. (17). To flip the spin, the rotation should be by an angle πabout an axis lying in the polarization plane. Denoting this rotation as R⊥(π), the transformation R⊥(π)Rq|R, µ, σ⟩=|R⊥(π)RqR, ν,¯σ⟩ (20) also fulfills Eq. (17) but for a broader class of lat- tice geometries that satisfy R⊥(π)RqR=Rand R⊥(π)Rqnµ=nν. The physical interpretation of this result is that the Bloch modes have zero spin when the polarization plane contains a symmetry axis of improper rotation. The considerations above justify the notion of spin in- version symmetry breaking as a form of generalized chi- rality for pseudospin-1/2 systems. Whereas true chirality is usually defined as a lack of anyaxis of improper ro- tation, here we only require that such an axis not lie in the polarization plane. The latter definition naturally encompasses the former, but also includes additional con- figurations where the chirality stems from the mutual ori- entation of the lattice vectors and the quantization axis, rather than from the lattice geometry alone (Section VI). B. Time-reversal symmetry In addition to the geometry of the system, the behav- ior under time-reversal also influences the spin properties of the system. The dipole-dipole interaction present in Eq. (5) describes the creation (destruction) of an excita- tion at site i(j) with a rate determined by the Green’s tensor (1). This process neglects electronic exchange interactions, which is a good approximation when the spacing between adjacent emitters is much larger than the spatial extent of the atomic wavefunctions. In this case, the circularly polarized excitations at each emitter site can be described using bosonic statistics and with5 a bosonic time-reversal operator. Generally, a Hamilto- nian HisT-invariant if and only if there exists a uni- tary operator UTsuch that THT−1=HforT=KUT, where Kis the anti-unitary complex conjugation opera- torK:i→ −i[24]. For Hamiltonians of the form (5), the time-reversal operator is given by T=KSx (21) and satisfies T2= 1. This represents an important dis- tinction from traditional electronic systems and signifi- cantly influences the spin textures of the resulting pho- tonic bands. In traditional band theory, spin-1/2 electrons obeying fermionic statistics exist as degenerate Kramers’ pairs whenTsymmetry is preserved. If inversion symmetry is also present, this constraint requires at least a two-fold degeneracy at every point in the Brillouin zone. If in- version symmetry is broken, Tinvariance still requires this degeneracy be preserved at all Tinvariant quasimo- menta. In either case, the spin of each excitation may be interpreted as a vector on the Bloch sphere. In bosonic bands, no such degeneracy is required. For the pseudospins described by Eq. (4), the polarization of each excitation is instead given by a vector on the Poincare sphere [25, 26] with left and right circular po- larizations residing at the north and south poles, re- spectively. In contrast to the fermionic case (where an equal superposition of ↑and↓spins results in an equal magnitude spin-1/2 excitation pointing in the orthog- onal plane), vectors residing along the equator of the Poincare sphere do not carry angular momentum. That is, an equal superposition of left and right circularly po- larized excitations yields a linearly polarized excitation of pseudospin-0 [see Eq. (4)]. Whereas the Toperator acting on a fermionic system corresponds to a complete inversion of the Bloch vector through the origin, the bosonic Tresults in a reflection through the Poincare sphere equatorial plane and leaves vectors residing in this plane unchanged. In later sections, we will consider the influence of bro- kenTsymmetry as induced by the anti-Hermitian part of the effective Hamiltonian (7). To this end, it is in- structive to present a more detailed description of the considerations above. In particular, direct application of Eq. (21) demonstrates that the pseudospin operator is time-odd, obeying T SzT−1=−Sz. If the Hamiltonian is time-reversal invariant, then [ H,T] = 0 and THT−1=KSxH(KSx)−1=SxH∗S−1 x=H. (22) A general Bloch Hamiltonian H(k) =e−ik·rHeik·rthen satisfies TH(k)T−1=eik·rSxH∗S−1 xe−ik·r=H(−k),(23) where H(−k) =SxH∗(k)S−1 x. The Brillouin zone con- tains a set of time-reversal invariant momenta, k= Γi, where −Γi= Γi+GµandGµis a reciprocal lattice vector, Gµ·Rµ= 2π(to avoid ambiguity with otherquantities denoted by Γ, we will always use a superscript to denote time-reversal invariant momenta). At these points, the Bloch Hamiltonian satisfies [ H(Γi),T] = 0. To see how the bosonic Tinfluences the degeneracy of the resulting Bloch bands, consider |un(k)⟩as a general eigenstate of H(k) with eigenvalue εn(k). As in the pre- vious section, the spin associated with this state is given by⟨Sz⟩. Away from the Γipoints, the spin of this state is, in general, nonzero (Section III). Applying Eq. (23), the Bloch Hamiltonian for a time-reversal invariant system satisfies TH(k)|un(k)⟩=H(−k)T |un(k)⟩=εn(k)T |un(k)⟩ (24) such that |um(−k)⟩ ≡ T | un(k)⟩is an eigenstate of H(−k) with the same energy, εm(−k) = εn(k). The states at kand−knecessarily have opposite spin: ⟨um(−k)|Sz|um(−k)⟩=⟨un(k)|T†SzT |un(k)⟩ =⟨u∗ n(k)|SxSzS−1 x|u∗ n(k)⟩ =−⟨un(k)|Sz|un(k)⟩,(25) where the second step follows from Sx=S† x=S−1 xand the third step from the Hermiticity of Sz. However, be- cause T2= 1,|um(k)⟩and|un(k)⟩need not represent orthogonal states. In the absence of accidental degenera- cies, these states are instead equal (up to a phase) such that each band nis symmetric about Γiin energy and antisymmetric in spin. The requirement at k= Γithen follows simply as ⟨un(Γi)|Sz|un(Γi)⟩=−⟨un(Γi)|Sz|un(Γi)⟩= 0 (26) for each band individually, and no two-fold spin degen- eracy is required. In position space, the eigenstates are equal superpo- sitions of states at kand−k. The antisymmetric spin textures required by Tinvariance therefore dictate that all position space eigenstates have zero spin. However, this does notpreclude the emergence of nontrivial spin textures, which requires only that the spin be nonzero at a single point in the Brillouin zone. If this point has nonzero dispersion, then the antisymmetric nature of the spin bands—together with the symmetric nature of the energy bands—implies spin-momentum locking for all T- invariant systems with nonzero spin. If Tsymmetry is broken, this antisymmetric condition need not apply, and the position space eigenstates may also exhibit nonzero spin (Section V). Finally, it should be noted that we have defined V explicitly as a unitary operator. However under certain conditions, the symmetry [ H(k), W] = 0 may be satisfied by an anti-unitary operator, W=Sx˜V. In this case, WH(k)W−1=VSxH∗(k)S−1 xV−1=VTH(k)T−1V−1, (27) and SO coupling can only emerge by breaking T- invariance. In analogy with Barron [27], we denote sys- tems obeying [ H(k), W]̸= 0 for both unitary and anti- unitary Was “truly chiral,” and those for which this condition holds only for unitary Was “falsely chiral.”6 C. Inversion and anti-inversion symmetries When the Hamiltonian is parity invariant, the Bloch bands are fully symmetric about k= Γi. Similarly, when the Hamiltonian is time-reversal invariant, the Bloch bands are symmetric in energy and antisymmetric in spin. Suppose now that the Hamiltonian is invariant not under PorTbut under the unitary operator ¯P |R, µ, σ⟩=X νˆPµν|−R, µ,¯σ⟩. (28) The matrix representationˆ¯P=ˆP ⊗ σxthen acts as the parity operator but treats the spin degree of free- dom as a polar vector that is odd under inversion (for a 1D lattice, we may also interpret this operation as a series of rotations). Consequently, [ H,¯P] = 0 implies ¯PH(k)¯P−1=H(−k) such that ¯PH(k)|un(k)⟩=H(−k)¯P |un(k)⟩=εn(k)¯P |un(k)⟩. (29) In analogy with Eqs. (24) and (25), the states |un(k)⟩ and ¯P |un(k)⟩have equal energy and opposite spin on opposite sides of the Brillouin zone. The operator ¯P therefore acts as a sort of “anti-inversion” by enforcing a spin antisymmetry about k= Γi. Importantly, this symmetry is a property of the lattice geometry and the orientation of the quantization axis, and is independent of the behavior under T. V. CHIRAL LATTICES We are now in a position to study specific lattice ge- ometries. We first consider “chiral lattices,” which we define as those exhibiting either true or false chirality for all orientations of q. The quintessential chiral structure is a right circular helix, which serves as a paradigmatic example. This structure is periodic along its longitudinal helical axis which, without loss of generality, we choose to be parallel to the z-axis (Fig. 2). The lattice vectors of the underlying Bravais lattice are then given by Rµ=alˆz for lattice constant aandl∈Z. For a right-handed helix periodic along the z-axis with radius r0, pitch a, andN emitters per unit cell, the emitter positions are given by rj=r0cos(ϕj)ˆx+r0sin(ϕj)ˆy+aϕj 2πˆz, (30) where ϕj= 2π(j−1)/N. The relative coordinate be- tween emitters iandjcan then be written as rij= xijˆx+yijˆy+zijˆzfor xij=−2r0sinh (µ−ν)π Ni sinh (µ+ν)π Ni yij= 2r0sinh (µ−ν)π Ni cosh (µ+ν)π Ni zij=h (µ−ν)a N+ali . (31) r0r02π/N (a)(b)FIG. 2. (a) Schematic of the helical lattice geometry for pitch aandN= 3. The dashed grey rectangle denotes one unit cell, whereas the solid yellow rectangle denotes one sublat- tice. (b) Top-down view of the helical lattice. The sublattices are arranged with radial position r0and azimuthal separation 2π/N. Here, µ(ν) =i(j) mod N+ 1 for integers i, j∈[1, N] andldenotes the number of unit cells between iandj (i.e., l= 0 if iandjare in the same unit cell). We note that with this choice of coordinates, the µ= 1 sublattice lies on the positive x-axis. A. Longitudinal circular polarization As a demonstration of true chirality, we first orient the quantization axis to lie along the axis of periodicity (ˆq=ˆz). In this case, r0=ϱi=ϱjand the spin-flip interaction is found to be G↑↓ ij=3r2 0λ0Γ0 4πk2 0r5 ijsin2ϕi−ϕj 2 e−i(ϕi+ϕj) ×eik0rij 3−k2 0r2 ij−i3k0rij
,(32) where ϕi= tan−1(yi/xi). A reflection through the y- zplane then takes ϕi→ −ϕisuch that G↑↓ ij→G↓↑ ij(see also Appendix C). The spin dynamics are therefore inter- changed by reflections perpendicular to the polarization plane that transform the right-handed helix to its left- handed mirror image. The photonic band structures for left- and right- handed helices are shown in Fig. 3(a). Because the helix geometry lacks an axis of improper rotation, the sym- metry [ H(k), W]̸= 0 is broken for all unitary Wand the Bloch bands exhibit nontrivial spin textures. As a consequence of this broken symmetry, the longitudinally polarized helix supports bulk helical modes that mediate the spin-momentum locking of propagating wave pack- ets. This dynamical effect is illustrated by the spin an- tisymmetry of the band structures. Away from the Γi points, each Bloch mode acquires a finite group velocity vn(k) =∂εn(k)/∂kalong the axis of periodicity. Because7 ←Ryz→ 312 132⊙ˆk ⇔⊗ˆk⇔T312 ˆq→ˆq→ˆq→⊙ˆkPπφ0zy⊙x(a) (b) FIG. 3. (a) Band structures for the left-handed (left) and right-handed (right) helices when ˆq=ˆz. The spin bands are nontrivially antisymmetric about k= Γi, irrespective of T-invariance. The system exhibits true chirality with a quan- tized nontrivial Zak phase. Additional parameters: N= 3, r0= 0.05λ0,a= 0.175λ0. (b) Top-down view of the heli- cal unit cell for ˆq=ˆz. The µ= 2 sublattice serves as an anti-inversion for the lattice, and the corresponding transfor- mation ¯P: (ˆk,ˆq)→(−ˆk,−ˆq) enforces antisymmetric Bloch bands. If dissipation is neglected, the spin antisymmetry is also protected by Tsymmetry. of the spin antisymmetry, these modes exhibit a helicity η=⟨Sz⟩v/|⟨Sz⟩v|that is symmetric about Γi. When dissipative interactions are neglected [i.e., the Hamilto- nian is of the form (5)], this spin-momentum locking is protected by time-reversal symmetry. Invariance of the group velocity under reflection through the y-zplane then dictates that the two chiralities support equal and opposite helical modes. Specifically for ˆq=ˆz, the broken spin inversion symmetry persists for anti-unitary Was well. Con- sequently, the band structures exhibit nontrivial spin textures regardless of whether the Hamiltonian is T- invariant. When dissipative interactions are included, the Hermitian Hamiltonian (5) is replaced by the non- Hermitian effective Hamiltonian (7). In this case, Tsym- metry is broken and Eq. (25) no longer holds. However, the spin antisymmetry of the Bloch bands persists be- cause the non-Hermitian Hamiltonian remains invariant under ¯P. In other words, the z-polarized helix contains a 1D anti-inversion center at the center of the unit cell. To verify that the Hamiltonian is invariant under this transformation, we construct explicitly the associated matrix representation. The anti-inversion center lies at the azimuthal angle ϕ0=π(N −1)/N(as measured from thex-axis) along an axis that bisects the angle between sublattices µ= 1 and µ=N. With this point chosen as the origin, the unit cell is symmetric under the com-bined operation of 1D spatial inversion and spin-flip. As demonstrated in Fig. 3(b), this operation is equivalent to aπrotation about the ϕ0axis. ¯Ptherefore reverses the direction of qandk, while simultaneously exchang- ing the positions of the three sublattices (for odd N, the central sublattice remains fixed). As such, the symmetry operation can be represented asˆ¯P=X⊗σx, where Xis theN × N antidiagonal exchange matrix acting on the sublattice indices. With the above choice of origin, the basis states of each unit cell then transform as ¯P |R, µ, σ⟩=X⊗σx|−R, µ, σ⟩. (33) One may verify explicitly that [ Heff,¯P] = 0 and that the spin antisymmetry follows accordingly as in Eq. (29). The net result is that the longitudinally polarized helix exhibits antisymmetric spin textures that are protected by anti-inversion symmetry (or equivalently, rotational symmetry). This property manifests even in the absence of time-reversal invariance: the system is truly chiral. To examine the topological properties of the Bloch bands, we compute the Zak phase, φ=I CTr [A(k)]·dk. (34) Here, Amn(k) =i⟨umk|∇kunk⟩is the non-Abelian Berry connection matrix evaluated over a closed loop Caround the first Brillouin zone (see Appendix D). The Zak phase is defined modulo 2 πand is quantized to either 0 (trivial) or±π(topologically nontrivial) when the Hamiltonian commutes with Por¯P. The finite spin textures exhib- ited by the longitudinally polarized helix induce a non- trivial topology in the energy bands. The associated SO coupled dynamics are therefore topologically protected by the chirality of the geometry. Because of the anti- inversion center at ϕ0, the Zak phase is quantized to ±π. B. Transverse circular polarization The symmetry properties of the non-Hermitian Hamil- tonian are altered when the quantization axis is oriented perpendicular to the axis of periodicity. As an example, we consider the helix of Eq. (30) but with qpointing along the ϕ0axis [Fig. 4(a)]. Because the lattice geome- try is unchanged, it remains true that there is no unitary Wenforcing ⟨Sz⟩= 0. However, the two-fold rotational symmetry of the lattice no longer corresponds to ¯P. In- stead, the πrotation that swaps the µ= 1 and µ= 3 sublattices leaves the quantization axis invariant. Conse- quently, the system exhibits a true inversion center and is parity symmetric with P |R, µ, σ⟩=X⊗11|−R, µ, σ⟩. (35) The spin bands are therefore symmetric about k= Γi [Fig. 4(b)], and the Zak phase is quantized to ±π. If dissipative interactions are neglected, then the Hamiltonian also commutes with T, and the spin bands8 [H,T]=0[Heff,T]/negationslash=0312 132⊙ˆk ⇔⊗ˆk⇔T312 ˆq→ˆq→ˆq→⊙ˆkPπφ0zy⊙x(a) (b) FIG. 4. (a) Top-down view of the helical unit cell for qori- ented along the ϕ0axis. The µ= 1 and µ= 3 sublattices are separated by an angle πin the polarization plane such that the µ= 2 sublattice serves as a true inversion center. Inversion symmetry transforms P: (ˆk,ˆq)→(−ˆk,ˆq), enforc- ing spin symmetric Bloch bands. If the Hamiltonian is also T-invariant, the additional relation T: (ˆk,ˆq)→(−ˆk,−ˆq) requires ⟨Sz⟩= 0. (b) Band structures for the ϕ0-polarized helix. The system is P-invariant with a quantized Zak phase. When Tsymmetry is broken (left), the spin bands are sym- metric about k= Γiand the Zak phase is topologically non- trivial. If Tsymmetry is restored (right), the combined PT symmetry causes both the spin textures and the Zak phase to vanish. The system is therefore falsely chiral. Helix parame- ters are the same as in Fig. 3. must also be antisymmetric [Eq. (25)]. The combined PTsymmetry then forces ⟨Sz⟩= 0 for all kand the SO coupling is lost. In turn, the system becomes topolog- ically trivial and the Zak phase is zero for each band. With this choice of quantization axis, the helix is only falsely chiral. For completeness, we also present the case where q does not correspond to a symmetry axis of the unit cell (e.g., lies at an angle π/4 from the x-axis in the x-y plane). This configuration is not invariant under Por ¯P. Hence, when Tsymmetry is broken, the spin bands are neither symmetric nor antisymmetric (Fig. 5). If however dissipation is neglected, then the Hamiltonian isT-invariant and the antisymmetric spin textures are restored. In either case, the Zak phase is not quantized. VI. CHIRAL SETUPS The spin inversion symmetry breaking required for fi- nite SO coupling need not come from the lattice geometry alone. If the lattice contains a mirror plane, there may still exist a choice of quantization axis such that the Bloch [H,T]=0[Heff,T]/negationslash=0FIG. 5. Band structures for the transversely polarized helix in the absence of both Pand ¯Pinvariance. When Tsym- metry is broken (left) the spin bands are neither symmetric nor antisymmetric about k= Γi. When Tsymmetry is re- stored (right) the spin bands become antisymmetric. In both cases, the Zak phase is not quantized. Helix parameters are the same as in Figs. 3 and 4. bands exhibit nontrivial spin textures. We refer to this scenario as a “chiral setup” because not all orientations ofqsatisfy [ H(k), W]̸= 0 for unitary W. A concrete example is demonstrated by a lattice ar- ranged into an oblique triangular prism. If the triangu- lar faces are isosceles, then the lattice possesses a mirror plane along the axis of periodicity [Fig. 6(a)]. For simplic- ity, we choose the emitter positions to be circumscribed about a right circular cylinder such that xijandyijare given by Eq. (31), and zij= 0, µ, ν ∈ {2,3} a/2, µ > ν −a/2, µ < ν(36) where µandνare defined as before. For ˆq=ˆx, the quantization axis lies in the mirror plane [Fig. 6(b)], and the reflection operator Rxzacts as in Eq. (19). This mirror reflection commutes with both the Hermitian and non-Hermitian Hamiltonians such that the geometry is spin inversion ( W) symmetric irrespective of the behavior under T. It follows that ⟨Sz⟩must vanish for all k, and the system is not chiral at all. By contrast, if the quantization axis is instead ori- ented along the y-axis, then Rxzleaves the quantization axis invariant ( qis an axial vector). The SO coupling is therefore preserved. Because the unit cell does not contain a center of (anti-)inversion, the Bloch bands are not fully (anti)symmetric and the Zak phase is not quan- tized. Nevertheless, the antisymmetry of the nontrivial spin textures is consistent with Tsymmetry, and thus the system is truly chiral. VII. SPIN POLARIZED DYNAMICS Thus far, we have discussed the emergence of a finite SO coupling in pseudospin-1/2 atomic lattices based on9 ⇔⊙ˆk132⊙ˆk ˆq→ 123 ˆq→ ⇔132⊙ˆk ˆq→ Rxz⊙ˆk123 ˆq→ Rxzzy⊙x ˆq=ˆxˆq=ˆy(a)(b)(c) FIG. 6. (a) Lattice geometry for an oblique triangular prism. Each triangular face represents a single unit cell. The faces are assumed to be isosceles with the x-zplane chosen as the mirror plane. (b) Top-down view of the oblique triangular prism unit cell for ˆq=ˆx(top) and ˆq=ˆy(bottom). (c) Band structures for the oblique triangular prism. When qis oriented in the mirror plane (left), the spin textures vanish irrespective of the behavior under T. Ifqis instead oriented orthogonal to the mirror plane, then the SO coupling can be nonzero. In both cases, the Zak phase is not quantized. Additional geometrical parameters are the same as in Figs. 3, 4, and 5. symmetry properties alone. In order to relate these re- sults to potential future experiments, we now demon- strate the influence of this SO coupling on the time evo- lution of propagating photonic excitations. The results presented in this section represent a set of testable pre- dictions that could be verified with modern day platforms (Section VIII). In the single excitation regime, the time dynamics of an arbitrary state ρ(t) are given by the no-jump quantum master equation, ˙ρ=−i Heffρ−ρH† eff . (37) We consider the evolution of an initially unpolarized mixed state localized at emitter j= 1 on the helix defined by Eq. (30). The initial state is given by ρ(0) =1 2(|↑1⟩⟨↑1|+|↓1⟩⟨↓1|). (38) In order to quantify the magnitude of spin dependent transport, we define the spin polarization, ∆, as the pop- ulation difference between the ↑and↓spin manifolds in- tegrated until the wave packet reaches the opposite end of the helix at time τ: ∆ =Zτ 0dtX i ⟨b† i↑bi↑⟩ − ⟨b† i↓bi↓⟩  ⟨b† i↑bi↑⟩+⟨b† i↓bi↓⟩. (39) The transport time is monitored by discretizing the helix into chunks of N/10 emitters, and we define τas thetime at which the total emitter population is largest in the final chunk. Fig. 7(a) shows the value of ∆ for helices with ˆq=ˆz, N= 3, and varying radius and pitch. As discussed in Section V A, the nonzero spin polarization demonstrates the preferential transport of optical excitations with a particular helicity. Extremely large spin polarizations— upwards of 40%—are easily achievable across a wide pa- rameter range, and we suspect that even larger spin po- larizations could be achieved by further optimizing the geometry. Interestingly, for a given chirality, both the magnitude and sign of the spin polarization depend non- trivially on the geometric proportions of the helix. This fact may be traced back to the nontrivial positional de- pendence of the electromagnetic Green’s tensor and to the presence of long-range all-to-all couplings that in- troduce multiple frequency scales to the dynamics (see Fig. A1). A reflection through the y-zplane preserves the direc- tion of motion but reverses the spin and transforms the left-handed lattice geometry to its right-handed mirror image. It follows from the same symmetry analysis ap- plied to the band structures that the spin polarization is equal and opposite between the left- and right-handed geometries. If the helix is rotated by an angle πabout an axis in the polarization plane, then the chirality is preserved and both the spin polarization and the direc- tion of motion change sign. The simultaneous reversal of the spin polarization together with the change in prop- agation direction demonstrates chirality dependent spin-10 R⊥(π)←Ryz→←Ryz→←Ryz→(a)(b) FIG. 7. (a) Spin polarization as a function of helix radius and pitch for ˆq=ˆz. Colors show the spin polarization, ∆, calculated for left-handed (left) and right-handed (right) helices. A πrotation about an axis in the polarization plane reverses the spin polarization and the direction of motion (black arrows). Black crosses denote the radius and pitch used in previous figures. Additional parameters for both panels: N= 3 emitters per 2 πturn and M=N/N= 20 turns total. (b) Dependence of the spin polarization on helix length and the number of atoms per unit cell. Additional parameters: r0= 0.05λ0,a= 0.175λ0, ˆq·ˆk= 1. momentum locking. Notably, for ˆq=ˆz, this πrotation is also equivalent to a change of initial condition (38) from emitter j= 1 to j=Nfollowed by a trivial azimuthal rotation about q[19]. The dependence of the spin polarization on the he- lix length and the number of emitters per unit cell is shown in Fig. 7(b). The N= 1 and N= 2 configura- tions correspond to the uniform chain and the staggered chain, respectively. These geometries are not chiral and do not exhibit any spin polarization. For the true helices (N ≥ 3), there is a modest increase in the spin polar- ization with increasing helix length. This trend has also been reported with electron spin polarizations in chiral molecules [28] and may be a common feature of helicity dependent transport. VIII. TOWARDS EXPERIMENTAL REALIZATIONS The results described here should be experimentally observable with readily available techniques. A standard approach would be to realize the helical geometry using neutral atoms in a 3D optical tweezer array. Coherent oscillations between atoms with V-type level structures have been demonstrated, e.g., by isolating the 60 S1/2 and 60 P3/2Rydberg manifolds of87Rb [29, 30]. The 17.2 GHz transition frequency between these manifolds allows for subwavelength dynamics at µm-scale tweezer separations. Although the influence of dissipation may not be observable with this platform due to long Ryd- berg excitation lifetimes, the chirality dependent trans- port and topological properties of the system could stillbe achieved. Dissipative dynamics could, however, be observed in a similar setup using the3P0and3D1manifolds of 88Sr [31], which is commonly used in 3D optical lattice clocks [32]. Helical arrangements of atoms could be built by selectively loading particular sites of the optical lattice using an extension of the tweezer-based programmable loading scheme recently demonstrated for 2D optical lat- tices [33], or by employing holographic optical traps made from dielectric metasurfaces [34]. Techniques for measur- ing topological Bloch bands simulated with optical lat- tices are well-established [35–37]. An alternative setup based on a Laguerre-Gauss mode optical trapping potential may also be realized with read- ily available techniques [38–40]. Laguerre-Gauss modes are eigenmodes of the paraxial wave equation with or- bital angular momentum quantum number, l. The l= 1 mode exhibits a cylindrical geometry with a phase ad- vance that winds once per wavelength. Interfering this mode with an orthogonally polarized plane wave shapes the field intensity into a helix. Intersecting this field with a cloud of red-detuned atoms would trap some of these atoms at intensity maxima determined by the helical po- tential. An additional long-range interaction potential (which could be imposed by Rydberg dressing the atoms via an additional laser [41, 42]) would result in a periodic arrangement of atoms along the helix. Finally, the reported effects might also be observ- able at optical or UV wavelengths using ultra-cold chiral molecules. One option is to use artificial fluorophores conjugated to a helical molecular scaffold. Alternatively, the monomers that comprise the larger helical polymer could themselves act as individual quantum emitters and11 facilitate excitation transport. Such a setup could be rel- evant to the ongoing development of molecular spintron- ics devices or biomimetic light-harvesting complexes. IX. CONCLUSIONS In this paper, we have presented a complete symme- try analysis of the SO coupling and topological proper- ties associated with pseudospin-1/2 atomic lattices. Our results describe a general photonic excitation transport process that is unique to chiral systems and depends on the orientation of the quantization axis. These findings introduce an exciting new avenue for cold atom quantum simulators, which could be used to study the governing principles of chirality dependent photon transport in a well-controlled environment. Because the emergence of SO coupling is dictated by the symmetries of the Bloch bands, the symmetry anal- ysis given above is complete within the single excitation regime. Nevertheless, an interesting follow-up would be to analyze the dynamics in the multi-excitation regime.In this case, the V-type level structure imposes a hard- core boson constraint in which each atomic orbital is at most singly occupied. The ensuing non-linearity in- duced by the hard-core interaction is expected to result in modifications of the transport phenomena. In this re- gard, chiral atom arrays could serve as a promising plat- form for new photonics devices and for studying photonic analogues of many-body topological physics in condensed matter systems. X. ACKNOWLEDGEMENTS The authors are grateful to Mikhail D. Lukin and Jonathan Simon for suggestions on potential experimen- tal implementations. S.O. is supported by a postdoctoral fellowship of the Max Planck-Harvard Research Center for Quantum Optics. All authors acknowledge funding from the National Science Foundation (NSF) via the Cen- ter for Ultracold Atoms (CUA) Physics Frontiers Centers (PFC) program and via PHY-2207972, as well as from the Air Force Office of Scientific Research (AFOSR). Appendix A: The electromagnetic Green’s tensor In free space, the effective interactions between quantum emitters are determined by the dyadic Green’s tensor, G(r,r′, ω), which is the solution to the wave equation ∇2G(r,r′, ω)−ω2 c2G(r,r′, k) =δ(r−r′)11 (A1) for observational coordinates r, source coordinates r′, and frequency ω. For the case where the emitters located at positions rjare well approximated by point electric dipoles, the Green’s tensor between emitters iandjdepends only on the relative coordinate rij=ri−rj. In the Born and Markov approximations, the Green’s tensor may further be regarded as dispersionless and is given by [16] G(rij, ω0) =eik0rij 4πk2 0r3 ijh (k2 0r2 ij+ik0rij−1)11−(k2 0r2 ij+ 3ik0rij−3)rij⊗rij r2 iji , (A2) with rij=|rij|andk0=ω0/c. The σcomponent of the positive frequency electric field calculated at position ris then calculated (in the time domain) as [16, 43] E+ σ(r) =µ0ω2 0NX j=1G(r−rj, ω0)·℘℘℘jσ|g⟩⟨σj|, (A3) where µ0is the vacuum permeability. Consequently, the polarization (or photon “spin”) dependent dipole-dipole interaction between emitters iandjis given by Jσσ′ ij−(i/2)Γσσ′ ij, where Jσσ′ ij=−µ0ω2 0℘℘℘† iσ·ReG(rij, ω0)·℘℘℘jσ′ (A4) Γσσ′ ij= 2µ0ω2 0℘℘℘† iσ·ImG(rij, ω0)·℘℘℘jσ′ (A5) describe the coherent and dissipative parts of the interaction, respectively. The geometrical dependence of the coherent interaction for atoms12 (a)(b) FIG. A1. Coherent coupling strengths between atoms iandjlocated on a helix with ˆq=ˆzand radius r0= 0.05λ0. The x-axis denotes the relative distance between the emitters in the ˆzdirection (along the longitudinal helical axis). The y-axis denotes the relative azimuthal coordinate between the emitters. (a) Spin-preserving interactions. Colors denote the interaction strength Jσσ ij/Γ0with σ∈ {↑,↓}. (b) Spin-flipping interactions. Colors denote the interaction strength |Jσσ′ ij|/Γ0with σ, σ′∈ {↑,↓}for σ̸=σ′. Appendix B: Photonic band structures for quasi-1D non-Bravais lattices Beginning with the real-space Hamiltonian (7), we expand the site index i= (m, µ) to yield Heff=MX m=1NX µ=1X σω0b† mµσbmµσ−3 2λ0Γ0MX m,n=1NX µ,ν=1X σ,σ′(1−δmnδµν)Gσσ′ mµ,nν b† mµσbnνσ′, (B1) where mandnindex the unit cells along the axis of periodicity, Mis the number of unit cells, µandνare the sublattice indices, and Ndenotes the number of sublattices. Note that in this notation, Gσσ′ mµ,mµ = 0. We now make the discrete Fourier transform bmµσ= (1/√ M)P kexp (ik·rmµ)bkµσto arrive at Heff=ω0 MMX m=1NX µ=1X σX k,k′ei(k′−k)·rmµb† kµσbk′µσ−3 2λ0Γ0 MMX m,n=1NX µ,ν=1X σ,σ′X k,k′e−ik·rmµeik′·rnνGσσ′ mµ,nν b† kµσbk′νσ′(B2) fork=kˆkandk=πj/(Ma). Here, a=|rm+1,µ−rmµ|is the lattice spacing between adjacent unit cells and j= 0, ...,2Mis an integer. Noting that the Green’s tensor depends only on the relative coordinate Rl µν≡rmν−rnν forl=m−n, the second term can be written as −3 2λ0Γ0 M∞X l=−∞MX m=1NX µ,ν=1X σ,σ′X k,k′ei(k′−k)·rmµe−ik′·Rl µνGσσ′ mµ,lν b† kµσbk′νσ′. (B3) In the limit of large M, the identity MX m=1ei(k′−k)·rmµ=Mδkk′ (B4) yields the partially diagonalized Hamiltonian Heff=NX µ=1X kX σω0b† kµσbkµσ−3 2λ0Γ0X kNX µ,ν=1X σ,σ′∞X l=−∞e−ik·Rl µνGσσ′ mµ,lν b† kµσbkνσ′. (B5) Changing notation slightly, we may drop the superscript on Rl µνby replacing the sum over lwith a sum over the entire set of Rµ=Rµν−nµν. Here, Rµdenotes a Bravais lattice vector on sublattice µandnµνdenotes the basis vector pointing from sublattice νto sublattice µ. Writing Gσσ′ mµ,lν asGσσ′(Rµ+nµν), the Hamiltonian takes the form13 Heff=P kH(k), where the Bloch Hamiltonian H(k) =NX µ,ν=1X σ,σ′hµσ,νσ′(k)b† kµσbkνσ′ (B6) has matrix elements hµσ,νσ′(k) =ω0δµνδσσ′+χI µσ,νσ′+χII µσ,νσ′for χI µσ,νσ′=−3 2λ0Γ0X Rµ̸=0e−ik·RµGσσ′(Rµ)δµν (B7) χII µσ,νσ′=−3 2λ0Γ0X Rµe−ik·(Rµ+nµν)Gσσ′(Rµ+nµν)(1−δµν). (B8) Finally, to enforce periodicity of the Brillouin zone, it is necessary to apply the local gauge transformation [44] bkµσ→e−ik·nµbkµσ. A redefinition of H(k) to include this k-dependent phase transforms χII µσ,νσ′→ −3 2λ0Γ0X Rµe−ik·RµGσσ′(Rµ+nµν)(1−δµν) (B9) and ensures that H(k+Gµ) =H(k) for any reciprocal lattice vector, Gµ·Rµ= 2π. Appendix C: Spin dynamics under rotations and reflections Because the elements of O(3) preserve spatial distances and the spin-preserving interaction Gσσ ijdepends only on rij, it is sufficient to consider the effects of group multiplication on the spin-flip interaction alone. It is convenient to work in cylindrical coordinates and in the ( ˆd1,ˆd2,ˆq) basis such that ri=ϱicos(ϕi)ˆd1+ϱisin(ϕi)ˆd2+rqˆq, where ϱi is the radial coordinate of emitter iandϕiis the corresponding azimuthal coordinate measured in the polarization plane. Substituting the circular polarization vectors of Eq. (4) into Eq. (2), the spin-flip amplitude is then G↑↓ ij=3λ0Γ0 16πk2 0r5 ije−2i(ϕi+ϕj) eiϕjϱi−eiϕiϱj
2eik0rij −3 +k2 0r2 ij+i3k0rij
, (C1) with ϕi= tan−1(ri·ˆd2/ri·ˆd1). (C2) The inverse process G↓↑ ijis given by taking ϕi→ −ϕiandϕj→ −ϕj. Eq. (C1) demonstrates that for rij∥ˆq, the spin-flip interaction vanishes ( ϱi=ϱjandϕi=ϕj). In this case, Eq. (5) is diagonal in spin space and the dynamics are those of two uncoupled bosonic subspaces. If, on the other hand, the emitters are not collinear with the quantization axis ( rij∦ˆq), then the spin-flip amplitude can be nonzero and lead to spin mixing. Note, however, that this is not a sufficient condition for nontrivial spin textures, which require the breaking of spin inversion symmetry (Section III). The sign of (C2) is the only quantity in the Hamiltonian that distinguishes ↑from↓. Thus, any spin dependent dynamics must be encoded in this phase, and orthogonal transformations that change the sign of this phase must map spin↑dynamics to spin ↓dynamics and vice versa. We first consider proper rotations R, which satisfy det( R) = 1. Definition 1. LetRb(α) denote the orthogonal operator specifying azimuthal rotation by an angle αabout an axis b. In the ( ˆd1,ˆd2,ˆq) basis, a rotation about the quantization axis has matrix representation ˆRq(α) = cos(α)−sin(α) 0 sin(α) cos( α) 0 0 0 1 . (C3) Acting with this operator on an arbitrary lattice geometry yields the transformed position vectors ˆRq(α)ri= cos(α)ri·ˆd1−sin(α)ri·ˆd2 sin(α)ri·ˆd1+ cos( α)ri·ˆd2 ri·ˆq ≡r′ i. (C4)14 Invoking the identity tan−1(u)±tan−1(v) = tan−1((u±v)/(1∓uv)), the phase (C2) transforms as ϕ′ i= tan−1 sin(α)ri·ˆd1+ cos( α)ri·ˆd2 cos(α)ri·ˆd1−sin(α)ri·ˆd2! = tan−1 tan(α) +ri·ˆd2 ri·ˆd1 1−ri·ˆd2 ri·ˆd1tanα =ϕi+α. (C5) Hence, rotations about qcontribute only an overall phase to the spin-flip interaction and can be gauged away by a suitable redefinition of the ϕi= 0 reference value. We now consider improper rotations (reflections) R, satisfying det( R) =−1. Definition 2. LetRq(α) denote the orthogonal operator specifying reflection through a plane containing qthat makes an angle αwith the d1axis. The corresponding matrix representation is ˆRq(α) = cos(2 α) sin(2 α) 0 sin(2α)−cos(2 α) 0 0 0 1 . (C6) Because rotations about qleave ϕiinvariant (up to an arbitrary constant), it follows that all reflections Rq(α) have an equivalent effect on the spin dynamics. In more detail, we may write the combined rotoreflection operation as Rq(α)Rq(α′) =Rq(α′+α/2)≡ R q(α′′). (C7) Then for r′′ i≡ R q(α′′)ri, the corresponding azimuthal coordinate in the polarization plane is given by ϕ′′ i= tan−1 sin(2α′′)ri·ˆd1−cos(2 α′′)ri·ˆd2 cos(2 α′′)ri·ˆd1+ sin(2 α′′)ri·ˆd2! = tan−1 −ri·ˆd2 ri·ˆd1! =−ϕi, (C8) where the second step follows by setting α′=−α/2. It follows that any reflection Rq(α′) reverses the spin dynamics. In addition to reflections through planes containing the quantization axis, the sign of (C2) can also be reversed under rotations about orthogonal axes. For simplicity, we consider here only rotations about the ˆd1axis, though those about other axes in the polarization plane follow similarly. In this case, the transformed vector r′′′ i≡Rd1(β)ri has azimuthal coordinate ϕ′′′ i= tan−1 cos(β)ri·ˆd2−sin(β)ri·ˆq ri·ˆd1! , (C9) andϕ′′′ i=−ϕiforβ=π. These geometrical considerations may be summarized more succinctly in terms of orthogonal transformations acting on the quantization axis itself. By virtue of the cross product, qis an axial vector that changes sign under parallel reflections but is invariant under orthogonal reflections. As discussed in the main text, this property is crucial for determining the spin dynamics in both chiral lattices and chiral setups. Appendix D: Topological classification For a non-Bravais lattice with Nsublattices, the Hamiltonian (9) gives rise to 2 NBloch modes of the form |ψnk⟩=eik·r|unk⟩, where nis the band index. If the system admits a band gap, then the isolated Mbands on one side of the gap obey the U(M) gauge freedom |unk⟩ →MX m=1Umn(k)|umk⟩, (D1)15 where the M × M unitary matrix Umndescribes an equivalence class of physically identical Bloch manifolds. The non-Abelian Berry connection for each isolated manifold then follows as [45, 46] Amn(k) =i⟨umk|∇kunk⟩. (D2) For a closed loop Caround the first Brillouin zone, the Berry phase is given by φ=I CTr [A(k)]·dk=−Im{ln [det ( WC)]}, (D3) where WCis the Wilson loop for the path Ctraversed in reciprocal space. In 1D and for discretized k=k0, ..., k L, the Wilson loop may be written as [47, 48] WC=PexpI C−iAmn(k)dk =L−1Y i=0exp{−iA(ki,ki+1) mn dk}, (D4) where Pis the path-ordering operator and exp{−iA(ki,ki+1) mn dk} ≈δmn−iA(ki,ki+1) mn dk =δmn+⟨umki|(|unki+1⟩ − |unki⟩) =M(ki,ki+1) mn (D5) for overlap matrix M(ki,ki+1) mn =⟨umki|unki+1⟩. Eq. (D4) holds provided the cell-periodic functions are specified in the periodic gauge where |unk0⟩=|unkL⟩. In this case, the Wilson loop is easily computed as WC=L−1Y i=0M(ki,ki+1) mn . (D6) Importantly, while Amn(k) is gauge-dependent, the 1D Berry phase (or Zak phase) is gauge invariant modulo 2 π. In general, the Zak phase for the 1D Bloch bands can assume any value, but is quantized to either 0 (trivial) or π (nontrivial) in the presence of either inversion or anti-inversion symmetry. [1] C. L. Kane and E. J. Mele, Physical Review Letters 95, 226801 (2005). [2] C. L. Kane and E. J. Mele, Physical Review Letters 95, 146802 (2005). [3] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, Nature Materials 14, 871 (2015). [4] R. Naaman and D. H. Waldeck, Annual Review of Physical Chemistry 66, 263 (2015). [5] J. M. Abendroth, D. M. Stemer, B. P. Bloom, P. Roy, R. Naaman, D. H. Waldeck, P. S. Weiss, and P. C. Mondal, ACS Nano 13, 4928 (2019). [6] I. ˇZuti´ c, J. Fabian, and S. Das Sarma, Reviews of Modern Physics 76, 323 (2004). [7] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Nature 541, 473 (2017). [8] K. Y. Bliokh, F. J. Rodr´ ıguez-Fortu˜ no, F. Nori, and A. V. Zayats, Nature Photonics 9, 796 (2015). [9] K. Y. Bliokh and F. Nori, Physics Reports 592, 1 (2015). [10] K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, Physical Review Letters 119, 073901 (2017). [11] V. Galitski and I. B. Spielman, Nature 494, 49 (2013). [12] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Physical Review Letters 109, 095301 (2012). [13] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Physical Review Letters 109, 095302 (2012). [14] X.-J. Liu, M. F. Borunda, X. Liu, and J. Sinova, Physical Review Letters 102, 046402 (2009). [15] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Nature 471, 83 (2011). [16] A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. Kimble, and D. Chang, Physical Review X 7, 031024 (2017). [17] J. Perczel, J. Borregaard, D. E. Chang, H. Pichler, S. F. Yelin, P. Zoller, and M. D. Lukin, Physical Review A 96, 063801 (2017). [18] J. Perczel, J. Borregaard, D. E. Chang, H. Pichler, S. F. Yelin, P. Zoller, and M. D. Lukin, Phys Rev Lett 119, 023603 (2017).16 [19] J. S. Peter, S. Ostermann, and S. F. Yelin, Phys. Rev. Lett. (submitted). [20] R. H. Lehmberg, Physical Review A 2, 883 (1970). [21] R. H. Lehmberg, Physical Review A 2, 889 (1970). [22] For simplicity, we ignore the subtleties arising from multi-band gauge ambiguities that are resolved via the sewing matrix formalism. [23] L. Fu and C. L. Kane, Physical Review B 76, 045302 (2007). [24] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, New Journal of Physics 12, 065010 (2010). [25] H. Poincar´ e, M. Lamotte, D. Hurmuzescu, and C. A. Chant, Th´ eorie math´ ematique de la lumi` ere II. Nouvelles ´ etudes sur la diffraction. Th´ eorie de la dispersion de Helmholtz. Le¸ cons profess´ ees pendant le premier semestre 1891-1892 (Paris, G. Carr´ e, 1892). [26] E. Collett, Field guide to polarization , SPIE field guides No. v. FG05 (SPIE Press, Bellingham, Wash, 2005). [27] L. D. Barron, Journal of the American Chemical Society 108, 5539 (1986). [28] S. Mishra, A. K. Mondal, S. Pal, T. K. Das, E. Z. B. Smolinsky, G. Siligardi, and R. Naaman, The Journal of Physical Chemistry C 124, 10776 (2020). [29] S. de L´ es´ eleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. B¨ uchler, T. Lahaye, and A. Browaeys, Science 365, 775 (2019). [30] V. Lienhard, P. Scholl, S. Weber, D. Barredo, S. de L´ es´ eleuc, R. Bai, N. Lang, M. Fleischhauer, H. P. B¨ uchler, T. Lahaye, and A. Browaeys, Physical Review X 10, 021031 (2020). [31] B. Olmos, D. Yu, Y. Singh, F. Schreck, K. Bongs, and I. Lesanovsky, Physical Review Letters 110, 143602 (2013). [32] S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. D. Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, and J. Ye, Science 358, 90 (2017). [33] A. W. Young, W. J. Eckner, N. Schine, A. M. Childs, and A. M. Kaufman, Science 377, 885 (2022). [34] X. Huang, W. Yuan, A. Holman, M. Kwon, S. J. Masson, R. Gutierrez-Jauregui, A. Asenjo-Garcia, S. Will, and N. Yu, Progress in Quantum Electronics 89, 100470 (2023). [35] M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Nature Physics 9, 795 (2013). [36] D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Physical Review Letters 110, 165304 (2013). [37] T. Li, L. Duca, M. Reitter, F. Grusdt, E. Demler, M. Endres, M. Schleier-Smith, I. Bloch, and U. Schneider, Science 352, 1094 (2016). [38] L. W. Clark, N. Schine, C. Baum, N. Jia, and J. Simon, Nature 582, 41 (2020). [39] N. Schine, M. Chalupnik, T. Can, A. Gromov, and J. Simon, Nature 565, 173 (2019). [40] N. Schine, A. Ryou, A. Gromov, A. Sommer, and J. Simon, Nature 534, 671 (2016). [41] J. E. Johnson and S. L. Rolston, Physical Review A 82, 033412 (2010). [42] J. Honer, H. Weimer, T. Pfau, and H. P. B¨ uchler, Physical Review Letters 105, 160404 (2010). [43] H. T. Dung, L. Kn¨ oll, and D.-G. Welsch, Physical Review A 66, 063810 (2002). [44] C. Bena and G. Montambaux, New Journal of Physics 11, 095003 (2009). [45] F. Wilczek and A. Zee, Physical Review Letters 52, 2111 (1984). [46] M. V. Berry, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45 (1984). [47] D. Gresch, G. Aut` es, O. V. Yazyev, M. Troyer, D. Vanderbilt, B. A. Bernevig, and A. A. Soluyanov, Physical Review B 95, 075146 (2017). [48] D. Vanderbilt, Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators , 1st ed. (Cambridge University Press, 2018).

1111.4072v2.Intrinsic_Spin_Swapping.pdf

arXiv:1111.4072v2 [cond-mat.mes-hall] 27 Mar 2012Intrinsic Spin Swapping Severin Sadjina and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway A. G. Mal’shukov Institute of Spectroscopy, Russian Academy of Sciences, 14 2190, Troitsk, Moscow oblast, Russia Here, we study diffusive spin transport in two dimensions and demonstrate that an intrinsic analog to a previously predicted extrinsic spin swapping effect, wh ere the spin polarization and the direction of flow are interchanged due to spin-orbit coupling at extrin sic impurities, can be induced by intrinsic (Rashba) spin-orbit coupling. The resulting accumulation of intrinsically spin-swapped polarizations is shown to be much larger than for the extrinsic effect. Intri nsic spin swapping is particularly strong when the system dimensions exceed the spin-orbit precessio n length and the generated transverse spin currents are of the order of the injected primary spin cu rrents. In contrast, spin accumulations and spin currents caused by extrinsic spin swapping are prop ortional to the spin-orbit coupling. We present numerical and analytical results for the secondary spin currents and accumulations generated by intrinsic spin swapping, and we derive analytic expressi ons for the induced spin accumulation at the edges of a narrow strip, where a long-range propagation o f spin polarizations takes place. I. INTRODUCTION Understanding the spin-orbit interaction is essential to the development of spintronics and gives rise to various spin transport mechanisms. Effects of the spin-orbit in- teraction can be intrinsic or extrinsic. Intrinsic effects a re caused by the spin-orbit interaction in the band struc- ture. Extrinsic contributions arise from spin-orbit cou- pling at impurities. The spin Hall effect, where a trans- verse spin current is generated via a longitudinal charge current, is one of the effects resulting from the spin- orbit coupling and has attracted much attention, both theoretically1–8and experimentally9–15. In a sample, this transverse spin current generates opposite spin accumu- lations at the lateral boundaries. While the spin Hall effect provides coupling between charge and spin, another spin-orbit-induced transport mechanism has recently been introduced in which only spins couple and which emerges even in the absence of charge currents. Primary longitudinal spin currents give rise to secondary transverse spin currents due to spin- orbit coupling at extrinsic impurities.16The generated secondary spin currents are proportional to the extrinsic spin-orbit coupling strength. The effect has been coined ‘spin swapping’ because, in its simplest manifestation, it interchanges the spin polarization direction and the spin flow.17It has been suggested that any mechanism inducing a spin Hall effect should also give rise to spin swapping. However, it has not yet been clear how the intrinsic mechanism could produce this effect. In this paper, we demonstrate that an intrinsic (Rash- ba spin-orbit-induced) spin swapping effect exists in two- dimensional diffusive metals and that it is drastically dif- ferent from its extrinsic analog. The main distinction between these two effects is that the extrinsic effect is of the same order as the spin-orbit coupling strength and is thus small, irrespective of the system size. In contrast, the intrinsic spin swapping effect is large for system di-mensions exceeding the spin-orbit precession length, and the secondary spin currents generated by this effect are then of the same order as the primary spin currents. If, however, the system width is small compared to the spin- orbit precession length, the effect is small but leads to a long-range propagation of spin polarizations closely re- lated to the increase of the D’yakonov-Perel spin relax- ation time in narrow strips18. Furthermore, the symme- try of intrinsic spin swapping is more complex and richer than that of the extrinsic spin swapping effect resulting in a non-trivial dependence on the relative orientation of the injected spin flow and the spin polarization. We present numerical and analytical results for the transverse sec- ondary spin currents and accumulations induced by pri- mary spin currents in two-dimensional diffusive metals, and we compare the intrinsic and extrinsic spin swapping effects. This paper is organized as follows. We first provide a review of the previously discussed extrinsic spin swapping effect in Sec. II, and we compute the spin accumulations and spin currents induced by an injected primary spin current in a two-dimensional diffusive metal. In Sec. III, we discuss the intrinsic spin swapping effect, numerically evaluate the spin densities and spin currents generated through intrinsic spin swapping, and derive analytical results for the resulting spin currents far away from the lateral edges of a sample. Next, in Sec. IV, we treat the case of a narrow strip whose width is small compared to the spin-orbit precession length and find analytical ex- pressions for the spin accumulations at the lateral edges of a sample stemming from the intrinsic spin swapping effect. In Sec. V, we briefly discuss how the spin swap- ping effects could be observed in experiment. Finally, we give our conclusions in Sec. VI.2 II. EXTRINSIC SPIN SWAPPING First, we review the extrinsic spin swapping effect introduced in Ref. 16 and present its features in two- dimensional diffusive metals in order to compare it to the intrinsic spin swapping effect to be discussed later. The Hamiltonian of the system under consideration reads H(ρ) =−1 2m∂2 ρ+Vimp(ρ) +Vso(ρ), (1) whereρ= (x,y) is a two-dimensional coordinate, Vimp(ρ) =1 A/summationdisplay ρi/summationdisplay kv(k) eik·(ρ−ρi)(2a) is the elastic impurity scattering potential, and Vso(ρ) =−iγ/bracketleftBig σ×∇Vimp(ρ)/bracketrightBig ·∂ρ (2b) is the spin-orbit coupling. ρiis the position of the ith impurity,Ais the area, v(k) is the Fourier transformed scattering potential, σ= (σx,σy,σz)Tis a vector of Pauli matrices, and γis the dimensionless spin-orbit coupling strength. Considering transport in the diffusive limit, the spin diffusion equation reads ∂2 ρfb−1 l2 sffb= 0, (3) wherefbis thebcomponent of the spin density, b∈ {x,y,z }, andlsfis the spin-flip length. In order to study spin transport, one also needs to define the spin current. In the leading approximation, while neglecting spin-orbit effects, the spin current is given by the spin diffusion currentj(0) ab=−D∂afbflowing along aand polarized alongb, whereDis the diffusion constant. The spin- orbit interaction gives rise to additional terms in the spin current. To first order in the spin-orbit coupling strength γ, when there is no charge current giving rise to the spin Hall effect, the spin current is16 jab=j(0) ab+χ/parenleftBig j(0) ba−δabj(0) cc/parenrightBig . (4) The term proportional to the swapping constant χrelates the spin polarization to the direction of flow and results in the induction of secondary spin currents, i.e., a ‘spin swapping’ effect.16For example, a primary spin current directed along xwill induce transverse spin currents that arise as follows, j(0) xb⇒jbx, ifb/nequalx, and j(0) xx⇒ −jyy−jzz.The first of these transformations swaps the current’s flow direction and its polarization. In general, this causes spin accumulations at the lateral edges of a sample, as we shall see shortly. The swapping constant is linear in the spin-orbit coupling strength and can be calculated explicitly,16 χ= 2γp2 F, (5) wherepFis the momentum at the Fermi level and short- ranged scattering potentials are assumed. Extrinsic spin swapping arises from the additional terms in the spin current (4) that are proportional to χ, whereas the spin diffusion equation (3) is unaltered. Extrinsic spin swapping therefore affects the boundary conditions for an unaltered, conventional spin diffusion differential equation. We will see later that the spin dif- fusion equation for the intrinsic spin swapping effect is altered as well, giving rise to a richer class of phenom- ena. (a) Spin density and spin current polarized along x (b) Spin density and spin current polarized along y FIG. 1. (Color online) The extrinsic spin swapping effect in a semi-infinite two-dimensional diffusive metal of width L= 4lsf. Shown are the scaled spin densities and the scaled spin currents according to Eqs. (3) and (4). A primary spin current j(0) xxinjected at x= 0 in (a) induces a transverse spin current jyythrough the spin swapping effect in (b). Note that the secondary spin accumulation and spin currents in (b) are linear with respect to the small swapping constant χ. If, in- stead, a primary spin current j(0) xypolarized along yis injected, fyandjxyare illustrated by (a) and the resulting secondary spin density and spin current ( fxandjyx, respectively) only differ from (b) by a sign. In order to compare the extrinsic spin swapping ef- fect with its intrinsic analog to be discussed in the next section, we first study the spin polarizations generated via the extrinsic spin swapping effect beyond the dis- cussion given in Ref. 16. We consider a semi-infinite two-dimensional diffusive metal of width Linto which a spin current j(0) xxdirected along xand carrying spins polarized along xis injected at x= 0. We assume that the injected current is homogeneous along yat the in- jection edge. Further, we assume impenetrable lateral sample edges such that no spin current flows through,3 withjyb(y=±L/2) = 0 for any spin polarization b. The spin-orbit coupling at extrinsic impurities generates a transverse spin current jyyon length scales larger than the mean free path according to Eq. (4). In turn, this gives rise to an accumulation of spins at the lateral edges of the sample polarized along ythat is anti-symmetric in the transverse coordinate y. The spin accumulation and spin current are plotted in Fig. 1: Fig. 1(a) shows the polarization along x, and Fig. 1(b) shows the polar- ization along y. In the two-dimensional case considered here, no transformation into spins polarized along ztakes place. Note that the extrinsic spin swapping effect and, therefore, the resulting secondary spin accumulations and spin currents are of the order of the small swapping con- stantχ. Solving the spin diffusion equation with the above-mentioned boundary conditions, the accumulationof spins at the lateral edges of a sample can be obtained analytically and may be probed experimentally, fy(y=±L/2) = ±2 πj(0) xxχ DxK1(x/lsf),forL≫lsf, ±L 2j(0) xxχ De−x/l sf, forL≪lsf, whereK1is the modified Bessel function of the second kind and first order. This coincides with the numerical result illustrated in Fig. 1. If, instead, a primary spin currentj(0) xypolarized along yis injected, the resulting secondary spin densities and spin currents ( fxandjyx, respectively) differ only by a sign according to Eq. (4) and can also be illustrated as shown in Fig. 1. (a) Spin density and spin current polarized along x (b) Spin density and spin current polarized along y (c) Spin density and spin current polarized along z FIG. 2. (Color online) The intrinsic spin swapping effect in a two-dimensional diffusive metal of width L= 4lsand length Lx= 16 ls. A primary spin current j(0) xxinjected at x= 0 in (a) induces an oscillating transverse spin current jyyin (b) through coupling with the zcomponents of the spins in (c). The resulting accumulation o fycomponents of the spins at the sample edges in (b) is a signature of the intrinsic spin swapping effe ct. Shown are the spin densities and spin current densities a ccording to Eqs. (8) and (9) on a relative scale for each plot. Note that all quantities are of the same order of magnitude. III. INTRINSIC SPIN SWAPPING We now elucidate the nature of the intrinsic spin swap- ping effect. The Hamiltonian of a two-dimensional metal with intrinsic spin-orbit coupling reads as H(k) =/planckover2pi12k2 2m∗+σ·hk+v(k), (6) wherem∗is the effective electron mass, kis the electron wave vector, and v(k) is the Fourier transformed scatter- ing potential. We assume Rashba spin-orbit coupling,19 hk=/parenleftbig αky,−αkx,0/parenrightbigT, (7) whereαdefines the spin-orbit coupling strength. When αis sufficiently small, such that the spin-orbit precessionlengthls= (αm∗)−1is much larger than the elastic mean free path, the spin diffusion equation reads20 ∂2 ρfx−4 l2sfx=4 ls∂xfz, (8a) ∂2 ρfy−4 l2sfy=4 ls∂yfz, (8b) ∂2 ρfz−8 l2sfz=−4 ls/parenleftbig ∂xfx+∂yfy/parenrightbig . (8c) The spin current is given by20,21 jab=−D∂afb+2 lsD/parenleftbig δabfz−δbzfa/parenrightbig . (9) The diffusion equations (8) for the case of intrinsic spin- orbit coupling are more difficult to solve analytically than4 for the extrinsic case because the x,y, andzspin compo- nents are coupled. Therefore, we numerically study the spin currents and the accumulations of spins resulting from intrinsic spin swapping in a two-dimensional sys- tem. Before presenting the numerical results, we discuss the simple analytical expressions that can be derived for the spin accumulations and spin currents induced by in- trinsic spin swapping far away from the lateral edges of a sample. The problem can also be treated analytically for a narrow strip system whose width is small compared to the spin-orbit precession length (see Sec. IV). We first consider a case analogous to that given for ex- trinsic spin swapping. A spin current j(0) xx=jxx(x= 0) carrying spins polarized along the xdirection and di- rected along xis injected at x= 0. Again, we assume that the injected current is homogeneous along yat the injection edge and that the lateral edges of the sample are impenetrable, i.e., jyb(y=±L/2) = 0 for any spin polarization b. The situation is, to some extent, similar to the extrinsic case depicted in Fig. 1. However, while the swapping effect in this scenario is straightforward for the extrinsic case, it is much more complex and rich for intrinsic spin swapping. As mentioned before, analytical expressions can be found for the spin currents and accu-mulations far away from the lateral boundaries, at dis- tances much larger than ls. In this region, the influence of the boundaries is weak, and the expressions approach the limit of a system that is infinite in the ydirection. We thus find that a transverse spin current jyyflowing along theydirection carrying spins polarized along yis induced, jyy(x) j(0) xx= e−krx/l s/bracketleftBig/parenleftbig√ 2−1/parenrightbig cos/parenleftbig kix/ls/parenrightbig −3 +√ 2√ 7sin/parenleftbig kix/ls/parenrightbig/bracketrightBig , (10) wherekr/i=/radicalbig 2√ 2∓1. This is the intrinsic spin swap- ping effect. The induced spin current reaches its maxi- mum, |jyy(xmax)|/j(0) xx≈61%, within one spin-orbit pre- cession length from the injection edge at x= 0. The injected spin current itself decays away from the spin current source at x= 0, jxx(x) j(0) xx= e−krx/l s/bracketleftBig cos/parenleftbig kix/ls/parenrightbig +k2 r√ 7sin/parenleftbig kix/ls/parenrightbig/bracketrightBig . (11) (a) Spin density and spin current polarized along x (b) Spin density and spin current polarized along y (c) Spin density and spin current polarized along z FIG. 3. (Color online) The intrinsic spin swapping effect in a two-dimensional diffusive metal of width L= 4lsand length Lx= 16 ls. A primary spin current j(0) xyinjected at x= 0 in (b) induces a transverse spin current jyzin (c). In turn, this leads to an accumulation of zspins at the sample edges, which is a signature of the intrins ic spin swapping effect. In (a), an oscillating spin current polarized along xis only generated close to the lateral edges of the system. Sh own are the spin densities and spin current densities according to Eqs. (8) and (9) on a relative scale for each plot. Note that all quantities are of the same o rder of magnitude. While extrinsic spin swapping in general directly cou- plesx-polarized and y-polarized spins, in intrinsic spin swapping the conversion between x-polarized and y- polarized spin currents occurs via spins polarized along zas can be seen from Eqs. (8) and (9). In addition, spin currents and spin accumulations oscillate as a functionof the distance from the injection edge. The situation is depicted in Fig. 2 for a system with width L= 4ls and length Lx= 16ls. In Fig. 2(a), we see that the spin current carrying spins polarized along x, which is given by Eq. (11) in the bulk, as well as the spin accumula- tion decay away from the spin current source at x= 0.5 Thexcomponents of the spins are converted to zcom- ponents, as shown in Fig. 2(c), which in turn gives rise to a swapped transverse spin current jyy, shown in Fig. 2(b), that is polarized along y. In the bulk, this current is given by Eq. (10). We also see that this swapped spin current causes an oscillating spin accumulation at the lateral edges, which is a signature of the intrinsic spin swapping effect that may be probed experimentally (see Sec. IV for an explicit expression of this spin swapping induced spin accumulation in a narrow strip system). Next, we turn to the case in which a homogeneous spin currentj(0) xy=jxy(x= 0) carrying spins polarized along yis injected at x= 0. To analyze this situation, we first find an analytic expression for the transverse spin current induced through spin swapping far from the lateral edges of the system. We find that the primary spin current jxy is directly transformed into a transverse spin current, jyz(x) j(0) xy=−e−2x/l s=−jxy(x) j(0) xy, (12) that gives rise to an accumulation of zspins at the lat- eral edges of the sample (again, refer to Sec. IV for an explicit expression for the induced spin accumulation in a narrow strip system). In contrast to the case of extrinsic spin swapping, Eqs. (8) and (9) provide a direct coupling between the yandzspins, with the resulting spin current having polarization along z(rather than x): again, the resulting current is of the same order as the primary spin current. It is only near the lateral boundaries that spin currents polarized along xare generated as well. This spin current leads to an oscillating spin accumulation at the sample edges. This situation is depicted in Fig. 3. In both scenarios of injected spin currents discussed above, intrinsic spin swapping is a much stronger effect than extrinsic spin swapping. IV. INTRINSIC SPIN SWAPPING IN A NARROW STRIP In this section, we will consider the special case of a strip whose width Lis much less than ls. This case is interesting because, in such a system, a long-range spin swapping effect can be realized, such that the spin- swapped accumulation can extend far along the strip, over a length much greater than ls. This long-range be- havior is closely related to the increase of the D’yakonov- Perel spin relaxation time in narrow strips18. Due to the small parameter L/l s, the spin-swapping problem can be treated analytically. Following Ref. 18, we introduce new spin density variables, ψ±1=1√ 2/parenleftbig ±fx−ify/parenrightbig , ψ 0=fz. (13) In terms of these variables, Eq. (8) can be transformed into /parenleftBig i∂x+2 lsJy/parenrightBig2 ψ+/parenleftBig i∂y−2 lsJx/parenrightBig2 ψ= 0, (14)whereψis a 3-vector ( ψ1,ψ0,ψ−1)TandJi,i∈ {x,y,z }, are the corresponding 3 ×3 angular momentum opera- tors for spin 1. Using Eq. (9), the boundary conditions can be expressed as /parenleftBig i∂x+2 lsJy/parenrightBig ψ|x=0=I, (15a) /parenleftBig i∂y−2 lsJx/parenrightBig ψ|y=±L/2= 0, (15b) whereIis determined by the spin current injected at x= 0, I±1=i√ 2D/parenleftBig ∓j(0) xx+ ij(0) xy/parenrightBig , I 0=−i Dj(0) xz.(15c) The unitary transformation ψ= eiJx(π/2−2y/l s)φ (16) further simplifies Eq. (14) to /parenleftBig i∂x+2 lsJy(y)/parenrightBig2 φ−∂2 yφ= 0, (17) whereJy(y) = e−iJx(π/2−2y/l s)JyeiJx(π/2−2y/l s), and the boundary conditions at the lateral edges of the sys- tem then read ∂yφ|y=±L/2= 0. (18) The transformed differential equations (17) and the boundary conditions (18) are exact equivalent represen- tations of the original problem. For the case of a narrow strip, L≪ls, one can ex- pandJy(y) up to second order in y/lsto obtainJy(y) = Jz+2(y/ls)Jy−2(y/ls)2Jzand consider the last two terms in this expression as a perturbation. Due to Eq. (18), the solution of Eq. (17) can be represented as a Fourier ex- pansion in sin/parenleftbig (2n+ 1)πy/L/parenrightbig and cos/parenleftbig 2nπy/L/parenrightbig , where nis an integer. Further analysis reveals that only a term uniform in yis relevant because the other Fourier com- ponents decay very quickly along the xdirection. The equation for ¯φ, that is,φaveraged over −L/2≤y≤L/2, can then be derived from Eq. (17) as18 /parenleftBig i∂x+2 lsJz/parenrightBig2¯φ+Γ l2s/parenleftBig 2−J2 z/parenrightBig ¯φ= 0, (19) where Γ = 2 L2/3l2 s. The general solution of this equa- tion that converges for x→ ∞ has the form ¯φ±1= A±1e±2 ix/l se−√ Γx/l sand ¯φ0=A0e−√ 2Γx/l s. The co- efficientsAcan be found from the boundary condition (15a). If we consider a case analogous to that presented for extrinsic spin swapping in Sec. II, where a spin current j(0) xxcarrying spins polarized along xis injected, we find I±1=∓i√ 2Dj(0) xxandI0= 0. Applying the unitary op- erator (16) to this boundary condition we obtain in the leading approximation ¯φ±1(x= 0) = ±j(0) xxls√ 2ΓD,¯φ0(x= 0) = 0. (20)6 From this it follows that ¯φ±1=±j(0) xxls√ 2ΓDe±2 ix/l s−√ Γx/l s,¯φ0= 0. (21) Using Eqs. (16) and (13), we finally obtain the spin den- sities fx=j(0) xxls√ ΓDe−√ Γx/l scos/parenleftbig 2x/ls/parenrightbig , (22a) fy=−2j(0) xxls√ ΓDy lse−√ Γx/l ssin/parenleftbig 2x/ls/parenrightbig , (22b) fz=−j(0) xxls√ ΓDe−√ Γx/l ssin/parenleftbig 2x/ls/parenrightbig , (22c) to first order in y/ls. The accumulation of yspins at the lateral edges of the narrow strip caused by the intrinsic spin swapping effect reads fy(y=±L/2) = ∓/radicalbigg 3 2j(0) xxls De−√ Γx/l ssin/parenleftbig 2x/ls/parenrightbig . (23) Since√ Γ/ls≪1, the spin accumulation oscillates and slowly decreases along x. Considering the second case treated in Sec. III, where a spin current j(0) xycarrying spins polarized along yis injected, a similar calculation yields fx= 0, (24a) fy=j(0) xyls√ 2ΓDe−√ 2Γx/l s, (24b) fz=−2j(0) xyls√ 2ΓDy lse−√ 2Γx/l s. (24c) Again, the spin densities slowly decay along xbut, analo- gous to the previous discussion, no oscillation takes place . V. EXPERIMENTAL OBSERVATION OF SPIN SWAPPING In order to observe spin swapping, a primary spin cur- rent needs to be injected. This can be achieved in a two terminal setup where a spin current is electrically in- jected into a two-dimensional diffusive metal from a fer- romagnetic electrode.16As discussed here, spin swapping then gives rise to spin accumulations at the lateral sam- ple edges that could be detected experimentally, for ex- ample, by optical means10or by measuring the interface voltage at weak contacts between the lateral boundaries and ferromagnets12,14,15. However, in such a setup, an electric current is present in the system as well and addi- tional spin currents therefore emerge from the coupling of charge and spin via the spin Hall effect. In a two- dimensional system with extrinsic spin-orbit coupling, the spin accumulations resulting from spin swapping atthe lateral sample edges are polarized in-plane while those generated by the electric current via the spin Hall effect are polarized out-of-plane.2,3This makes it possi- ble to experimentally distinguish the two effects. On the other hand, in a diffusive system with intrinsic Rashba spin-orbit coupling, a uniform electric field gives rise to a uniform in-plane spin polarization via the Edelstein effect (while it does not produce spin currents).22,23In contrast, the resulting in-plane accumulation of swapped spins generated by a primary spin current with in-plane polarization is opposite at the lateral boundaries, as dis- cussed above. This difference allows to distinguish the intrinsic spin swapping effect and the Edelstein effect in experiment. Another possibility is the use of a non-local geometry14 where the spin swapping effects could be observed in a part of the system where there is no charge current. There, an electric current is injected from a ferromag- netic electrode on top of a diffusive metal towards a sec- ond electrode. A tunnel barrier between the electrodes and the metal assures that the current is injected uni- formly and it optimizes the polarization of the injected electrons. A pure spin current is thus generated in the system, propagating in opposite direction of the injected charge current and away from the electrodes. This spin current will give rise to spin accumulations at the lateral sample edges through the spin swapping effect that could be detected experimentally. VI. CONCLUSION In conclusion, we have demonstrated that there is an intrinsic analog to the extrinsic spin swapping effect in two-dimensional diffusive metals with Rashba spin-orbit coupling. We found that the intrinsic effect is drasti- cally different because it is large for system dimensions exceeding the spin-orbit precession length and gives rise to secondary spin currents and accumulations that are of the same order of magnitude as the injected primary spin currents while leading to a long-range propagation of spin polarizations in narrow strip systems. In con- trast, the extrinsic spin swapping effect is proportional to the spin-orbit coupling strength for any system size and is therefore small. Moreover, intrinsic spin swapping is more complex and richer than its extrinsic counterpart, resulting in a non-trivial dependence on the relative ori- entation of the injected spin flow and the spin polariza- tion. We derived explicit expressions for the transverse spin currents in the bulk and numerically computed the re- sulting spin accumulations at the lateral boundaries. In addition, we derived explicit expressions for the spin ac- cumulations in a narrow strip when L≪lsand found that the exponential decay of spin polarizations along thexdirection is greatly reduced in such systems. We further gave a brief discussion on how the spin swapping effect could be observed in experiment.7 ACKNOWLEDGMENTS This work was partially supported by the Research Council of Norway. 1R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954) 2M. I. Dyakonov and V. I. Perel, Sov. Phys. JETP Lett. 13, 467 (1971) 3M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971) 4J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999) 5S. Zhang, Phys. Rev. Lett. 85, 393 (2000) 6S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003) 7J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004) 8H.-A. Engel, E. I. Rashba, and B. I. Halperin, in Handbook of Magnetism and Advanced Magnetic Materials , Vol. 5, edited by H. Kronm¨ uller and S. Parkin (Wiley, Chichester, 2007) pp. 2858–2877 9Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004) 10V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gossard, and D. D. Awschalom, Nat. Phys. 1, 31 (2005) 11J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005) 12E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006)13N. P. Stern, S. Ghosh, G. Xiang, M. Zhu, N. Samarth, and D. D. Awschalom, Phys. Rev. Lett. 97, 126603 (2006) 14S. Valenzuela and M. Tinkham, Nature 442, 176 (2006) 15T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007) 16M. B. Lifshits and M. I. Dyakonov, Phys. Rev. Lett. 103, 186601 (2009) 17The terms responsible for extrinsic spin swapping were al- ready indicated in Refs. 2 and 3. However, their physical origin was not understood at the time16 18A. G. Mal’shukov and K. A. Chao, Phys. Rev. B: Condens. Matter 61, R2413 (2000) 19Y. A. Bychkov and E. I. Rashba, J. Phys. C: Solid State Phys. 17, 6039 (1984) 20C. S. Tang, A. G. Mal’shukov, and K. A. Chao, Phys. Rev. B: Condens. Matter 71, 195314 (2005) 21A. Brataas, A. G. Mal’shukov, and Y. Tserkovnyak, New J. Phys. 9, 345 (2007) 22V. Edelstein, Solid State Commun. 73, 233 (1990) 23J.-i. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B: Condens. Matter 67, 033104 (2003) 24J. Rammer, Quantum Field Theory of Non-Equilibrium States (Cambridge University Press, Cambridge, 2007) 25R. Raimondi and P. Schwab, Physica E 42, 952 (2010)

1211.0762v1.Spin_orbital_Texture_in_Topological_Insulators.pdf

arXiv:1211.0762v1 [cond-mat.mes-hall] 5 Nov 2012Spin-orbital Texture in Topological Insulators Haijun Zhang1, Chao-Xing Liu2& Shou-Cheng Zhang1 1Department of Physics, McCullough Building, Stanford Univ ersity, Stanford, CA 94305-4045 2Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802-6300 (Dated: November 6, 2012) Relativistic spin-orbit coupling plays an essential role i n the field of topological in- sulators and quantum spintronics. It gives rise to the topol ogical non-trivial band structure and enables electric manipulation of the spin deg ree of freedom. Because of the spin-orbit coupling, rich spin-orbital coupled textur es can exist both in momentum and in real space. For three dimensional topological insula tors in the Bi 2Se3family, topological surface states with p zorbitals have a left-handed spin texture for the upper Dirac cone and a right-handed spin texture for the lower Dira c cone. In this work, we predict a new form of the spin-orbital texture associated wi th the p xand p yorbitals. For the upper Dirac cone, a left-handed (right-handed) spin texture is coupled to the “radial” (“tangential”) orbital texture, whereas for the l ower Dirac cone, the coupling of spin and orbital textures is the exact opposite. The “tang ential” (“radial”) orbital texture is dominant for the upper (lower) Dirac cone, leadin g to the right-handed spin texture for the in-plane orbitals of both the upper and lower Dirac cones. A spin- resovled and photon polarized angle-resolved photoemissi on spectroscopy experiment is proposed to observe this novel spin-orbital texture. PACS numbers: 71.20.-b,73.43.-f,73.20.-r I. INTRODUCTION Three-dimensionaltopologicalinsulators(TIs) arenew states of quantum matter with helical gapless surface states consisting of odd number of Dirac cones inside the bulk band gap protected by time-reversal symmetry (TRS).1–4The underlying physical origin of the topo- logical property of TIs is the strong spin-orbit coupling (SOC), which plays a similar role as the Lorentz force in the QuantumHallstate. Due tothe SOCinteraction, the spin and momentum are locked to each other, forming a spintextureinthemomentumspaceforthesurfacestates of TIs5–7. The spin texture has been directly observed in the spin-resolved angle-resolved photon emission spec- troscopy (spin-resolved ARPES)8–12. The spin texture gives rise to a non-trivial Berry phase for the topological surface states and suppresses the backscatterings under TRS, leading to possible device applications in spintron- ics. Besides the spin texture, it has also been shown re- cently that the atomic porbitals of the Bi 2Se3family of topological insulators form a pattern in the momentum space, dubbed as the orbital texture, for the topological surface states.13,14In this work, we predict a coupled spin-orbital texture for the topological surface states. Based on both the effective k·ptheory and ab-initio cal- culations, we find, besides the usual locking between the electron spin and the crystal momentum, the spin tex- ture is also locked to the atomic orbital texture, which is dubbed as “spin-orbital texture”. We show that pzor- bitals have left-handed spin texture for the upper Dirac cone and right-handed spin texture for the lower Dirac cone, sharing the same feature as the total spin texture of the surface states. In contrast, the in-plane orbitalsa kxkyb kxky FIG. 1. (color online) a, b, The tangential orbital texture with the right-handed helical spin texture (a) and the radia l orbital texture with the left-handed helical spin texture ( b) for the upper Dirac cone. (pxandpyorbitals) reveal more intriguing features: for the upper Dirac cone of surface states, a “radial” orbital texture is coupled to a left-handed spin texture and a “tangential” orbital texture is coupled to a right-handed spin texture. For the lower Dirac cone, the coupling be- tween spin and orbital textures is exactly opposite. An electron spin-resolved and photon polarized ARPES ex- periment is proposed to observe this novel spin-orbital texture of the surface states of TIs. II. EFFECTIVE THEORY OF THE SPIN-ORBITAL TEXTURE The surface states of TIs are described by the Dirac type of effective Hamiltonian5,15 Hsurf(kx,ky) =/planckover2pi1vf(σxky−σykx),(1) with the Fermi velocity vfand Pauli matrix σ. The salient feature of this effective Hamiltonian is the “spin-2 a b cc FIG. 2. (color online) a, The Dirac cone of Bi 2Se3on the surface with the normal direction [0001] with the spin texture marked by blue arrows. b, c, The projection of pzorbital and the related in-plane spin texture for upper (b) a nd lower (c)Dirac cones. More red means more pzcharacter. The red arrows represent the in-plane spin textu re related to the pzorbitals. The insets are the schematics of the spin texture marked by green arrows. momentum locking”, which means for a fixed momentum k, the “spin”, denoted by the Pauli matrix σ, has a fixed direction for the eigenstate of the Hamiltonian. Since the “spin” is always perpendicular to the momentum, we can introduce a helicity operator, defined as ˆh=1 kˆz·(/vectork×/vector σ) which commutates with the Hamiltonian, to determine the handness of the “spin” texture. For the upper Dirac cone of surface states, the helical operator ˆh=−1, lead- ing to a left-handed “spin” texture in the momentum space while for the lower Dirac cone, ˆh= 1 yields a a right-handed “spin” texture. However, one should note that here the “spin” is not the real spin, but the total angular momentum /vectorJ=/vectorS+/vectorL, which is a combination of the real spin /vectorSand the orbital angular momentum /vectorL due to SOC. Consequently, the basis of the surface effec- tive Hamiltonian (1) are denoted as |ΨJz=±1 2∝angbracketrightwith the lower indices ±1 2representing the total angular momen- tum along zdirection. In order to understand what is the texture for the real spin /vectorS, it is necessary to write down the explicit form of the basis wavefunction |Ψ±1 2∝angbracketright. The form of the basis |Ψ±1 2∝angbracketrightcan be constructed bysymmetry considerations. Generally the basis |Ψ±1 2∝angbracketrightde- pends on the momentum kand we can expand it up to the first order in kas|Ψ±1 2∝angbracketright=|Ψ(0) ±1 2∝angbracketright+|Ψ(1) ±1 2∝angbracketright. Here we areonly interested in the porbitalsof Bi and Se atoms in the topological insulator Bi 2Se3and can decompose the zeroth-order wavefunction as |Ψ(0) ±1 2∝angbracketright=/summationdisplay α[u0,α|α,pz,↑(↓)∝angbracketright+v0,α|α,p±,↓(↑)∝angbracketright] (2) and the first-order wavefunction as |Ψ(1) ±1 2∝angbracketright=/summationdisplay α[±k±(iu1,α|α,p∓,↑(↓)∝angbracketright+iv1,α|α,pz,↓(↑)∝angbracketright) ∓iw1,αk∓|α,p±,↑(↓)∝angbracketright](3) wherek±=kx±iky,| ↑∝angbracketrightand| ↓∝angbracketrightdenote the spin, |pz∝angbracketright and|p±∝angbracketright=∓1√ 2(|px∝angbracketright±i|py∝angbracketright) denote different porbitals, andαdenotes indices other than the spin and orbital, such as atom indices. Here u0(1),α,v0(1),αandw1,αare material-dependent parameters. By comparing with the ab-initio calculations, we find that we can take them to be real. |Ψ1 2∝angbracketrightand|Ψ−1 2∝angbracketrightare related to each other by TRS. The expressions of the basis (2) and (3) can be3 a cb c d FIG. 3. (color online) a, b, c, d, The pxprojection on the states of upper (a) and lower (c) Dirac cone s, and the pyprojection of upper (b) and lower (d) Dirac cones. More red means more pxcharacter in (a) and (c), and more red means more py character in (b) and (d). The red arrows indicate the in-plan e spin texture related to the orbitals. The insets are the sch ematics of the spin texture. substituted into the eigen wavefunctions of the Hamil- tonian (1), |Φ±∝angbracketright=1√ 2/bracketleftBig ±ie−iθk|Ψ1 2∝angbracketright+|Ψ−1 2∝angbracketright/bracketrightBig , yielding the following forms of the wavefunctions |Φ+∝angbracketright=/summationdisplay α[(u0,α−v1,αk)|α,pz,↑θ∝angbracketright −i√ 2(v0,α−u1,αk−w1,αk)|α,pr,↑θ∝angbracketright +1√ 2(v0,α−u1,αk+w1,αk)|α,pt,↓θ∝angbracketright](4) |Φ−∝angbracketright=/summationdisplay α[(u0,α+v1,αk)|α,pz,↓θ∝angbracketright +i√ 2(v0,α+u1,αk+w1,αk)|α,pr,↓θ∝angbracketright −1√ 2(v0,α+u1,αk−w1,αk)|α,pt,↑θ∝angbracketright].(5) Here| ↑θ(↓θ)∝angbracketright=1√ 2(+(−)ie−iθk| ↑∝angbracketright+| ↓∝angbracketright) stands for the left-handed(right-handed)helicalspintextureand |pr∝angbracketright= cosθk|px∝angbracketright+sinθk|py∝angbracketright,|pt∝angbracketright=−sinθk|px∝angbracketright+cosθk|py∝angbracketrightare the radial and tangential orbital textures, as shown inFig. 1a,b, respectively. From the expressions (4) and (5), we can clearly see that |pz∝angbracketrightorbital is coupled to the left- handed spin texture | ↑θ∝angbracketrightfor the upper Dirac cone and the right-handed spin texture | ↓θ∝angbracketrightfor the lower Dirac cone. Furthermore, for the upper Dirac cone, the radial orbital texture |pr∝angbracketrightis always coupled to the left-handed spin texture | ↑θ∝angbracketrightand the tangential orbital texture |pt∝angbracketright is always coupled to the right-handed spin texture | ↓θ∝angbracketright. ThesituationisexactlyoppositeforthelowerDiraccone. The expressions (4) and (5) are the main analytical results of this paper, which show explicitly the spin- orbital texture. To confirm our analytical results, ab- initiomethod is adopted to calculate the projection of surface states on the spin and orbital basis, defined by the quantity D± i,η=∝angbracketleftΦ±|(|pi∝angbracketright∝angbracketleftpi|⊗sη)|Φ±∝angbracketright, (6) wherepi=px,py,pzfor the three porbitals, s0=12×2 denotes the charge part and sx,y,zdenote the three Pauli matrices for the spin. In the following, we will compare the analytical calculation of the intensity D± i,ηwithab- initiocalculations.4 ab c d 0 1/2 ππ3/2π2π θ0.020.030.040.050.06Character Densitypx upper Dirac cone py upper Dirac cone px lower Dirac cone py lower Dirac cone 0 0.04 0.08 0.12 Energy (eV)00.10.20.30.4 Polarization FIG. 4. (color online) a, b, The tangential orbital texture w ith the related in-plane spin texture for the upper Dirac con e (a) and the radial orbital texture for the lower (b) from ab-initio calculations. More red means more pxcharacter, and more blue means more pycharacter. The red arrows represent the total in-plane spin texture related to pxandpyorbitals. c, The px andpyprojections onto upper and lower Dirac cones. The solid curv es are the pxandpyprojections for the upper Dirac cone at energy level 0 .10eV, and the dashed for the lower Dirac cone at the energy −0.07eV. The basic feature is the πperiod which exactly agrees with the prediction by the effective model. d, Orbital polarization Ppx. The energy of Dirac point is shifted to be zero. The positive value of the orbital polarization repr esents the radial orbital texture, and the negative value re presents the tangential orbital texture. In order to plot more visual ly, thePpxfor the upper Dirac cone is inversed, marked by the red color.Ppxis exact zero at Dirac point, which indicates the transition point between the tangential and radial orbital textures. III. SPIN-ORBITAL TEXTURE FROM AB-INITIO CALCULATIONS The Vienna Ab-initio Simulation Package (VASP)16,17 is employed to carry out ab-initio calculations with the framework of the Perdew-Burke-Ernzerhof-type(PBE)18 generalized gradient approximation (GGA) of density functional theory19. Projector augmented wave (PAW) pseudo-potentials are used for all of the calculations in this work20. 10×10×10and 10 ×10×1are used for k-grid of bulk and free-standing calculations, respectively. The kinetic energy cutoff is fixed to 450eV. 6 quintuple layers (QLs) are fixed in the supercell for free-standing calcu- lations, and the thickness of vacuum is taken to be 50 ˚A. The lattice constant and the atomic parameters are di- rectly obtained from experiments. SOC is included with the non self-consistent calculation. In order to compare with the result of ARPES experiments, the projectionsof all the orbitals are only for the first Se and Bi atoms on the top surface of the free-standing model. The surface states of Bi 2Se3consist of a single Dirac cone at Γ point on one surface inside the large bulk band gap (∼0.3eV)5,21, which provides an ideal material to study the coupling of spin and orbital textures of sur- face states. As the starting point, we compare the bulk band structure of Bi 2Se3with the previous calculation5 and find good agreements. The surface states are ob- tained from the calculation of a free-standing structure with the normal direction [0001], as shown in Fig. 2a. The blue arrows represent the spin texture, where the spin is mainly lying in plane near the Dirac point. The spin texture is left-handed for the upper Dirac cone and right-handed for the lower one, the same as the total angular momentum texture. To understand the under- lying physics, we calculate the spin texture for different atomic orbitals. For pzorbitals, a left-handed helical5 spin texture is found for the upper Dirac cone and a right-handed texture for the lower Dirac cone, as shown in Fig. 2b,c. The schematic of the spin texture are shown in the inset. Here the background color indi- cates the projection of pzorbitals, which is isotropic, and the red arrows represent the corresponding in-plane spin texture. The spin texture of pzorbitals can be re- produced with the expressions [ D± pz,σx,D± pz,σy,D± pz,σz] = ±/summationtext α(u0,α∓v1,αk)2[sinθk,−cosθk,0] with ‘ ±’ for the upper and lower Dirac cone and /vectork= (k,θk) in the polar coordinate. The spin textures for in-plane orbitals are shown in Fig. 3a,b for the upper Dirac cone and in Fig. 3c,d for the lower Dirac cone, respectively. We find that the asso- ciated spins for pxandpyorbitals don’t rotate clockwise or anti-clockwise around the Dirac point as in the case ofpzorbitals, but instead, they take the form [D± px,x,D± px,y,D± px,z] =∓/summationdisplay αv2 0,α 2[sinθk,cosθk,0] (7) [D± py,x,D± py,y,D± py,z] =±/summationdisplay αv2 0,α 2[sinθk,cosθk,0] (8) for small karound the Γ point. The corresponding spin textures are shown schematically in the inset of Fig. 3a,c forpxorbitals in upper and lower Dirac cones and in the inset of Fig. 3b,d for pyorbitals. Unlike pzor- bitals, the amplitude of pxandpyorbitals for the sur- face states is not isotropic, but has 2 θkangular depen- dence around the Fermi surface, as shown in Fig. 4c. We may take the difference of the amplitudes between pxandpyorbitals, as shown by colors in Fig. 4a,b. Here more red means more pxcharacter, and more blue means more pycharacter. The angular dependence in- dicates a tangential orbital texture for the upper Dirac cone and a radial orbital texture for the lower Dirac cone, as schematically shown by the inset of Fig. 4a,b, respectively. This orbital texture was experimentally ob- served recently14. Furthermore, we also plot the total spin textures for in-plane orbitals on the same figure, which show a right-handed texture for both upper and lower Dirac cones. All these salient feature can be un- derstood by the wavefunctions (4) and (5). For the up- per Dirac cone, although both | ↑θ∝angbracketright|pr∝angbracketrightand| ↓θ∝angbracketright|pt∝angbracketright terms exist in the wavefunction(4), their associated coef- ficients are unequal. When/summationtext α(v0,α−u1,αk+w1,αk)2>/summationtext α(v0,α−u1,αk−w1,αk)2,| ↓θ∝angbracketright|pt∝angbracketrightterm dominates over | ↑θ∝angbracketright|pr∝angbracketrightterm, dominantlygivingatangentialorbitaltex- turecoupledtoaright-handedspintexture. Similarly,for the lower Dirac cone, when/summationtext α(v0,α+u1,αk+w1,αk)2>/summationtext α(v0,α+u1,αk−w1,αk)2,| ↓θ∝angbracketright|pr∝angbracketrightterm in the wave- function (5) is dominant, yielding a radial orbital texture coupled to a right-handed spin texture. The difference between pxandpyorbitals can be calculated directly asD± px,0−D± py,0=∓2cos2θk/summationtext α[(v0,α∓ku1,α)kw1,α], which indeed shows a 2 θkangular dependence, and the total spin textures for in-plane orbitals can be ob- tained as [ D± x,D± y] = [D± px,x+D± py,x,D± px,y+D± py,y] =4[−sinθk,cosθk]/summationtext α(v0,α∓ku1,α)kw1,α, which shows a right-handedspintexture when/summationtext α(v0,α∓ku1,α)kw1,α> 0. Especially, if kgets close zero, both the total spin tex- ture [D± x,D± y] of the in-plane orbitals and the difference between pxandpyorbitalsD± px,0−D± py,0approacheszero, also as shown in Fig. 4a,b. Therefore, there is a transition from a tangential or- bital texture in the upper Dirac cone to a radial orbital texture in the lower Dirac cone, switching exactly at the Dirac point. To quantitatively describe this transition, we introduce a polarization quantity Ppx(±) =Dpx,0(±,θ= 0)−Dpx,0(±,θ= 90) Dpx,0(±,θ= 0)+Dpx,0(±,θ= 90)(9) with ‘±’ for upper and lower Dirac cones. The plot of Ppx(±) is shown in Fig. 4d where the energy level of the Dirac point is shifted to zero. In order to show the plot more visually, we reverse the value of the Ppx(+) for the upper Dirac cone plotted with the red. The feature of Ppx(±) undoubtedly indicates that the state of the lower Dirac cone forms a radialorbital texture, and the state of the upper Dirac cone forms a tangential orbital texture. The Dirac point is shown to be the exact transition point from the tangential to radial orbital texture. This is ex- actly the behavior observed in a recent experiment and explained within the first principle calculations14. The numerical results fit well to the analytical calculation, with the expression Ppx(±) =∓2/summationtext α(v0,α∓Eu1,α//planckover2pi1vf)Ew1,α//planckover2pi1vf /summationtext α/bracketleftBig (v0,α∓Eu1,α//planckover2pi1vf)2+E2w2 1,α//planckover2pi12v2 f/bracketrightBig (10) with the energy E. For small Earound Dirac point, Ppx(±)∝ ∓/summationtext αv0,αw1,α/summationtext α(v0,α)22E /planckover2pi1vfshows the linear dependence on energy, as found in Fig. 4d. Although in-plane orbitals show different spin textures compared to pzorbitals, we stress the pzorbitals (50%) dominate the states near the Dirac point with pxand pyonly around 30%. Therefore, the spin texture for the whole states show left-handed for the upper Dirac cone and right-handed for the lower Dirac cone, the same as thatofpzorbitals,aswellasthetotalangularmomentum texture. IV. DISCUSSION In order to detect the spin texture of electrons, the spin-resolved ARPES technology has been developed by taking advantage of spin-dependent scatting pro- cesses and precisely measuring the magnitude of the asymmetry in the spin-dependent intensity with per- fect spin-polarimeters.12The non-trivial spin texture of surface states of TIs has been clearly observed by experiments.8–12In addition, the orbital character can be detected through the photon polarization selection6 rules22based on the symmetry analysis. With this tech- nology, the orbital texture of surface states of Bi 2Se3was reported recently by a polarized ARPES experiment.14 Therefore,itispossibletocombinethesetwotechnologies together in an electron spin-resolved and photon polar- ized ARPES experiment, with both the spin and orbital textures extracted in the same measurement. The pre- dicted spin-orbital texture can be directly confirmed in this type of experiment, which can explicitly reveal how SOC plays a role in the real material at the atomic level. V. ACKNOWLEDGMENTS We would like to thank Dr. Dan Dessau for sharinghis experimental data and for useful discussions, which par-tiallymotivated this work. This workis supportedby the Defense Advanced Research Projects Agency Microsys- tems TechnologyOffice, MesoDynamic Architecture Pro- gram (MESO) through the contract number N66001-11- 1-4105 and by the DARPA Program on ”Topological Insulators – Solid State Chemistry, New Materials and Properties”, under the award number N66001-12-1-4034. 1X.-L. Qi and S.-C. Zhang, Physics Today 63, 33 (2010). 2J. E. Moore, Nature 464, 194 (2010). 3M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 4X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). 5H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Nature Physics 5, 438 (2009), 10.1038/nphys1270. 6W. Zhang, R. Yu, H.-J. Zhang, X. Dai, and Z. Fang, New Journal of Physics 12, 065013 (2010). 7O. V. Yazyev, J. E. Moore, and S. G. Louie, Phys. Rev. Lett.105, 266806 (2010). 8D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R.J.Cava, andM.Z.Hasan,Science 323,919(2009), http://www.sciencemag.org/content/323/5916/919.full .pdf. 9S. Souma, K. Kosaka, T. Sato, M. Komatsu, A. Takayama, T. Takahashi, M. Kriener, K. Segawa, and Y. Ando, Phys. Rev. Lett. 106, 216803 (2011). 10S.-Y. Xu, L. A. Wray, Y. Xia, F. v. Rohr, Y. S. Hor, J. H. Dil, F. Meier, B. Slomski, J. Osterwalder, M. Neupane, H. Lin, A. Bansil, A. Fedorov, R. J. Cava, and M. Z. Hasan, arXiv: cond-mat/1101.3985 (2011). 11Z.-H. Pan, E. Vescovo, A. V. Fedorov, D. Gardner, Y. S. Lee, S. Chu, G. D. Gu, and T. Valla, Phys. Rev. Lett. 106, 257004 (2011). 12C. Jozwiak, Y. L. Chen, A. V. Fedorov, J. G. Analytis, C. R. Rotundu, A. K. Schmid, J. D. Denlinger, Y.-D.Chuang, D.-H. Lee, I. R. Fisher, R. J. Birgeneau, Z.-X. Shen, Z. Hussain, and A. Lanzara, Phys. Rev. B 84, 165113 (2011). 13S. R. Park, J. Han, C. Kim, Y. Y. Koh, C. Kim, H. Lee, H. J. Choi, J. H. Han, K. D. Lee, N. J. Hur, M. Arita, K. Shimada, H. Namatame, and M. Taniguchi, Phys. Rev. Lett.108, 046805 (2012). 14Y. Cao, J. A. Waugh, X.-W. Zhang, J.-W. Luo, Q. Wang, T. J. Reber, S. K. Mo, Z. Xu, A. Yang, J. Schneeloch, G. Gu, M. Brahlek, N. Bansal, S. Oh, A. Zunger, and D.S. Dessau, ArXiv e-prints (2012), arXiv:1209.1016 [cond- mat.mtrl-sci]. 15C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Phys. Rev. B 82, 045122 (2010). 16G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). 17G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 18J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.77, 3865 (1996). 19P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 20P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994). 21Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H.Lin, A.Ban- sil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature Physics 5, 398 (2009), 10.1038/nphys1274. 22A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys.75, 473 (2003).

1511.04917v3.Drude_weight_and_optical_conductivity_of_a_two_dimensional_heavy_hole_gas_with__k__cubic_spin_orbit_interactions.pdf

arXiv:1511.04917v3 [cond-mat.mes-hall] 29 Jan 2016Drude weight and optical conductivity of a two-dimensional heavy-hole gas with k-cubic spin-orbit interactions Alestin Mawrie and Tarun Kanti Ghosh Department of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India (Dated: October 8, 2018) We present detailed theoretical study on zero-frequency Dr ude weight and optical conductiv- ity of a two-dimensional heavy-hole gas(2DHG) with k-cubic Rashba and Dresselhaus spin-orbit interactions. The presence of k-cubic spin-orbit couplings strongly modifies the Drude wei ght in comparison to the electron gas with k-linear spin-orbit couplings. For large hole density and st rong k-cubic spin-orbit couplings, the density dependence of Dru de weight deviates from the linear be- havior. We establish a relation between optical conductivi ty and the Berry connection. Unlike two-dimensional electron gas with k-linear spin-orbit couplings, we explicitly show that the o ptical conductivity does not vanish even for equal strength of the t wo spin-orbit couplings. We attribute this fact to the non-zero Berry phase for equal strength of k-cubic spin-orbit couplings. The least photon energy needed to set in the optical transition in hole gas is one order of magnitude smaller than that of electron gas. Types of two van Hove singularitie s appear in the optical spectrum are also discussed. PACS numbers: 78.67.-n, 72.20.-i, 71.70.Ej I. INTRODUCTION Spin-orbitcoupling1–4playsavitalroleinseveralphys- ical properties of various systems because it breaks the spin degeneracy even at zero magnetic field. There are mainly two kinds of spin-orbit interaction in condensed matter systems, namely Rashba spin-orbit coupling due to inversion asymmetry of the confining potential5,6and Dresselhaus spin-orbit coupling generated by the asym- metry of the host bulk crystals7–9. The Rashba spin- orbit coupling has been realized in various systems such as zincblende semiconductor quantum wells10, carbon nanotubes11, two-dimensional materials12,13, and neu- tral atomic Bose-Einstein condensates14,15. The spin- orbit interaction (SOI) is essential in controlling the spin degree of freedom of the charge carrier in spin-based devices. In most of the studies systems with k-linear Rashba and Dresselhaus spin-orbit interactions(RSOI and DSOI) are of much interest. However higher order momentum-dependent spin-orbit interactions have also been seen to dominate in many physical systems. For example,k-cubic RSOI dominates in two-dimensional heavy-hole gas formed at the p-type GaAs/AlGaAs heterostructure16–18, in two-dimensional electron gas formed at the surface of the inversion symmetric oxide SrTiO 319and in strained Ge/SiGe quantum wells20. The optical conductivity is due to the transitions from one energy level to another energy level, whereas the zero-frequencyDrude weight is associated with the intra- level transitions. The real part of the complex frequency- dependent longitudinal optical conductivity provides the absorption as a function of photon energy. Its measure- ment through optical spectroscopy is an important tool for probing shape of the spin-split energy levels. Typi- cal order of the spin-split energy is the same as that of an electromagnetic radiation with terahertz (THz) fre- quency. Thehigh-frequencyradiationplaysanimportantrole to control the spinors of the spin-split energy levels due to the SOIs. Optical response studies will be useful for the high-speed electronic devices since the radiation with THz frequency flips the spin in a very short time. It opens the possibility of seeing the resonance effects through the optical transition between spin-split levels and leads to unique spectral features. Several theoretical studies of the optical conductivity have been carried out on spin-orbit coupled electron sys- tems formed at the semiconductor heterojunctions21,22 as well as in t2gbands of an oxide with perovskite structure23. The optical spectrum of hole gas with k- cubic RSOI has been studied24partially. The connection between optical longitudinal charge conductivity and op- tical spin Hall conductivity in electron and hole systems have been established in Refs.22,25. In Ref.26, it is shown that the optical conductivity disappears when the linear DSOI is same as the RSOI of two-dimensional electron systems. This optical conductivity has also been stud- ied in various single layer two-dimensional materials like graphene27, MoS228, silicene29andsurfacestatesoftopo- logical insulators30. In this work we present zero-frequency Drude weight and optical conductivity of a two-dimensional heavy-hole gas withk-cubic RSOI and DSOI. The Drude weight is stronglymodified due to the presence ofthe k-cubic spin- orbit couplings. We obtain an analytical expression of the Drude weight when only the k-cubic RSOI is present. It deviates from the linear density dependence for large hole density and for strong spin-orbit couplings. It de- creases with the increase of the spin-orbit couplings. We show that the optical conductivity is directly related to the Berry connection and does not vanish even for equal strength of the two spin-orbit couplings. The minimum photon energy required for the onset of the optical tran- sition in hole gas is one order of magnitude smaller than that of electron gas. We also identify the nature of the2 two van Hove singularities appear in the optical spec- trum. We organise this paper as follows. In section II, we present basic ground state properties of the heavy-hole system with the k-cubic RSOI and DSOI. In section III, we present the effect of spin-orbit interactions on the Drude weight and study various aspects of the optical conductivity for the same system at zero as well as non- zero temperature. Types of the van Hove singularities are tabulated in this section. A summary of our main results are provided in section IV. II. PHYSICAL SYSTEM The Hamiltonianofaheavy-holewith the both k-cubic RSOI and DSOI is given by31–33 H=p2 2m∗+iα 2/planckover2pi13/parenleftbig p3 −σ+−p3 +σ−/parenrightbig −β 2/planckover2pi13/parenleftbig p−p+p−σ++p+p−p+σ−/parenrightbig ,(1) wherem∗is the effective mass of the hole, p±=px±ipy andσ±=σx±iσy, withσxandσyare the Pauli’s matri- ces. Also,αis the strength of RSOI which measures the structure inversion asymmetry-induced splitting and βis the strength of DSOI which measures the bulk inversion asymmetry-induced spin splitting in the system. Typical value of Rashba strength in narrow gap semiconductor is α∼10−22eV-cm3andβis always less than α. The energy spectrum and the corresponding eigen- states are given by Eλ(k) =/planckover2pi12k2 2m∗+λk3∆(θ) (2) and Ψλ k(r) =exp(ik·r)√ 2Ω/parenleftbigg1 λei(2θ−φ)/parenrightbigg , (3) whereλ=±denotes spin-split energy levels, Ω is the surface area of the two-dimensional system, and ∆(θ) =/radicalbig α2+β2−2αβsin2θis the angular anisotropic term with θ= tan−1(ky/kx) andφ= tan−1(αcosθ− βsinθ)/(αsinθ−βcosθ). The presence of both the SOIs is responsible for the angular anisotropy of the energy spectra. The spin splitting energy Eg(k) = E+(k)−E−(k) = 2k3∆(θ) between two branches is also anisotropic. The maximum and minimum spin splitting occurs atθ= 3π/4 or 7π/4 andθ=π/4 or 5π/4; and the corresponding spin splitting energy values are Eg= 2k3(α±β), respectively. The Berry connection for this system is given by Ak=i/an}bracketle{tψλ k|∇k|ψλ k/an}bracketri}ht=/parenleftbigg3α2+β2−4αβsin2θ ∆2(θ)/parenrightbiggˆθ k,(4)whereˆθ=−sinθˆx+cosθˆyis the unit polar vector. The corresponding Berry phase is given as γ=/contintegraldisplay Ak·dk=π+2πα2−β2 |α2−β2|.(5) The Berry connection and Berry phase do not vanish for α=βcase, in complete contrast to the electron gas with equal strength of the spin-orbit couplings26,30. First we shall calculate density of states (DOS) of the spin-split energy branches, required for the calculation of the Fermi energy ( Ef) and the associated anisotropic Fermi wave vectors kλ f(θ). The density of states of the spin-split energy branches are obtained from Dλ(E) =1 (2π)2/integraldisplay d2kδ(E−Eλ(k)) =D0 2π/integraldisplay d2kδ(k−kλ E(θ)) |kλ E(θ)+λ6πD0∆(θ)(kλ E(θ))2|,(6) whereD0=m∗/(2π/planckover2pi12) is the DOS of spin-polarized hole gas without SOIs and kλ Eis the real solution of the cu- bic equation ( /planckover2pi1kλ E)2/2m∗+λ∆(θ)(kλ E)3−E= 0. The energy dependence of the density of states is shown in Fig. 1 for two different values of SOIs. The DOS of the heavy-hole spin-split energy branches change asymmet- rically with respect to D0, whereas it changes symmet- rically with respect to D0fork-linear spin-orbit coupled electron systems1. E(meV)0 0.2 0.4 0.6 0.8 1 1.2Dλ(D0) 0.40.450.50.550.60.650.70.75 D+D+D−D− FIG. 1: (color online) Plots of D±(E) (in units of D0) vs energyEforα= 0.08 eV nm3(dashed) and α= 0.1 eV nm3 (solid) with β= 0.6α. For a given set of system parameters ( nhbeing the heavy-holedensity, αandβ), the Fermi energy Efcan be evaluated numerically from the normalization condition: nh=/summationtext λ/integraltextEf 0Dλ(E)dE. Forβ= 0, theexactanalyticalexpressionsoftheFermi wave vectors34are given by k0,± f=/radicalbig 3πnh−L/(8l2α)∓3 L/(4lα), whereL= (1−/radicalbig 1−16πnhl2α) andlα= m∗α//planckover2pi12. Whenβ/ne}ationslash= 0, it is not possible, due to the anisotropic nature of the spectrum, to derive exact an- alytical expressions of the spin-split anisotropic Fermi wave vectors kλ f(θ). Therefore, we numerically calculate kλ f(θ) from the following cubic equation: ( /planckover2pi1k)2/2m∗+ λk3∆(θ)−Ef= 0, for given values of α,β,nhandEf. The anisotropic Fermi contours kλ f(θ), shown in Fig. 6, are symmetric with respect to the lines ky±kx= 0. III. DRUDE WEIGHT AND OPTICAL CONDUCTIVITY Consider a two-band system of charge carriers, elec- tron/hole, subjected to an oscillating electric field ( E∼ ˆxE0eiωt). The complex charge current conductivity is given by Σxx(ω) =σD(ω)+σxx(ω), whereσD(ω) =σd/(1−iωτ) is the dynamic Drude con- ductivity due tothe intra-bandtransitions, with σdbeing the static Drude conductivity and σxx(ω) is the complex optical conductivity due to the inter-band optical tran- sitions between two branches. Also, τis the momentum relaxation time. It is to be noted here that real part of σDandσxxcorrespond to the absorptive parts of the optical transition. It implies that absorption peaks in real parts of the conductivities will display dips in the experimentally measured transmission. The real part of the Drude conductivity is Re σD(ω) = Dwδ(ω), whereDw=πσd/τis known as Drude weight. It shows that the peak appears around ω= 0. On the otherhand, Re σxx(ω)isafunctionofphotonenergywith vanishing momentum ( q→0). Here the vanishing mo- mentum limit displays the fact that the momentum of the charge carrier is not altered by the electron-photon interaction. An optical absorption occurs through inter- band transition from λ=−branch toλ= + branch and helps to make spin-flip transition from one spin branch to another spin branch. A. Drude weight Usingthesemi-classicalBoltzmanntransporttheory35, the Drude weight at very low temperature can be written as Dw=e2 4π/summationdisplay λ/integraldisplay d2k/an}bracketle{tˆvx/an}bracketri}ht2 λδ(Eλ(k)−Ef),(7) where ˆvxis thex-component of the velocity operator and Efis the Fermi energyfora given system. Using Eq. ( 7), we have calculated (see Appendix for detail calculation) the Drude weight ( De w) of a two-dimensional electron gaswithk-linear RSOI, and it is given by De w=πe2 me/parenleftBig ne−α2 em2 e 2π/planckover2pi14/parenrightBig . (8) Heremeistheeffectivemassofanelectron, αeisthespin- orbit coupling strength and neis the density of electrons. This result exactly matches with the result obtained in Ref.36. For the present problem, the x-component of the ve- locity operator is given by ˆvx=vxI+V1σx−V2σy (9) wherevx=vcosθ,V1= 3vαsin2θ−vβ(2 + cos2θ), V2= 3vαcos2θ+vβsin2θ, withv=/planckover2pi1k/m,vα=αk2//planckover2pi1 andvβ=βk2//planckover2pi1. Also, /an}bracketle{tˆvx/an}bracketri}htλis the average value of the operator ˆ vxwith respect to the state ψλ k(r). After simplification, it reduces to Dw=e2 4π/summationdisplay λ/integraldisplay dθBλ(k) |/planckover2pi12kλ f/m∗+λ3∆(θ)(kλ f)2|,(10) whereBλ(k) = [(/planckover2pi1kλ f/m∗)cosθ+λ((kλ f)2//planckover2pi1){3αsinφ− βcosφ−2βcos(2θ−φ)}]2.Forβ= 0, we have the fol- lowing analytical expression of Dw: DR w=3e2 16m∗l2α/bracketleftBig 1−56 3πnhl2 α−(1−16πnhl2 α)3/2/bracketrightBig .(11) Forα= 0 andβ/ne}ationslash= 0, the Drude weight can be obtained from Eq. ( 11) by replacing lαbylβ=m∗β//planckover2pi12. The variations of Dwwithnhandαare shown in Fig. 2. It is known that the Drude weight varies linearly with the carrierdensityforafreefermionaswellasfor2DEGwith k-linear spin-orbit couplings35–37. Equation ( 11) and the left panel of Fig. 2 clearly show deviation from the linear density dependence of Dwfor large density and strong RSOI. On the other hand, the Drude weight decreases with the increase of the spin-orbit couplings as shown in the right panel of Fig. 2. It should be mentioned here that the effect of the Coulomb interaction is not taken into account in the above discussion. There may or may not be a signif- icant effect of the Coulomb interaction on Dwwhen the interaction parameter rsis very large (i.e. at the very low densities nh<0.5×1015m−2). For exam- ple, the strong effect of the Coulomb interaction in low- density 2DHG leads to the negative compressibility ob- served in Ref.38. On the other hand, based on electron measurements39, one would expect an enhanced spin sus- ceptibilityatverylowdensitiesfor2DHGbutitisnotob- served experimentally40. The behavior of electrons and holescanbequitedifferent, duetodifferenteffectivemass and different form of spin-orbit coupling, as revealed in several theoretical studies41–43. B. Optical conductivity The Kubo formula for the xxcomponent of the optical conductivity in terms of the Matsubara Green’s function4 0.050.10.150.2 α(eV nm3)Dw(Dn) Solid:β= 0, Dashed: β= 0.04 eV nm3 123456 nh(×1015m−2)Dw(Dα)Solid:β= 0, Dashed: β=α/2 0.4× 10−3 1.32.2 0.81 FIG. 2: (color online) Left panel: Plots of Dw(in units of Dα=e2π/m∗l2 α) vsnhforα= 0.12 eV nm3. Right panel: Plots of Dw(in units of Dn=e2nhπ/m∗) vsαfornh= 2.0×1015m−2. is given by σxx(ω) =−e2 iω1 (2π)2/integraldisplay∞ 0/integraldisplay2π 0kdkdθ ×T/summationdisplay lTr/an}bracketle{tˆvxˆG(k,ωl)ˆvxˆG(k,ωs+ωl)/an}bracketri}htiωs→ω+iδ.(12) HereTis the temperature, ωs= (2s+ 1)πTandωl= 2lπTare the fermion and boson Matsubara frequencies, respectively, with sandlare integers. The matrix Green’s function for the two-level system associated with the Hamiltonian ( 1) is given by ˆG(k,iωs) =/bracketleftBig i/planckover2pi1ωs+µ−(/planckover2pi1k)2/2m∗−S1σx−S2σy/bracketrightBig−1 =i/planckover2pi1ωs+µ−/planckover2pi12k2 2m∗+S1σx+S2σy (i/planckover2pi1ωs+µ−/planckover2pi12k2 2m∗)2−S2 1−S2 2(13) withS1=k3(−αsin3θ+βcosθ) andS2=k3(αcos3θ+ βsinθ). It is convenient to write the Green’s function as follows ˆG(k,iωs) =1 2/summationdisplay λ/bracketleftBig I+λF·σ/bracketrightBig G0(k,λ,ωs),(14) whereF= (S1,S2)//radicalbig S2 1+S2 2, andG0(k,λ,ωs) = 1/(i/planckover2pi1ωs+µ−/planckover2pi12k2/2m∗−λ/radicalbig S2 1+S2 2). Now we can write down ˆ vxˆG(k,ωl) as below ˆvxˆG(k,ωl) =1 2/summationdisplay λMλG0(k,λ,ωl), (15) whereMλ= (v+V1σx−V2σy)(I+λF·σ), which willgive us Tr/an}bracketle{tˆvxˆG(k,ωl)ˆvxˆG(k,ωs+ωl)/an}bracketri}ht= 1 2/summationdisplay λλ′/bracketleftBig (1+λλ′)v2+V2 1+V2 2+λλ′{(V1Fx−V2Fy)2 −(V1Fy+V2Fx)2}+2(λ+λ′)v(V1Fx−V2Fy)/bracketrightBig ×G0(k,λ,ωl)G0(k,λ′,ωs+l). Using the identity T/summationdisplay l/bracketleftBig1 (i/planckover2pi1ωl+µ−Eλ)·1 i/planckover2pi1(ωs+ωl)+µ−Eλ′/bracketrightBig =/braceleftBiggf(Eλ)−f(Eλ′) i/planckover2pi1ωs−Eλ′+Eλλ/ne}ationslash=λ′ 0 λ=λ′(16) with the Fermi-Dirac distribution function f(E) = [e(E−µ)/(kBT)+ 1]−1, one can see that the contribution of the intraband transition ( λ=λ′) to the optical con- ductivity is simply zero. This happens as a result of the momentum conservation. The non-zero contribution is coming only from the interband transitions ( λ/ne}ationslash=λ′). With this we can simplify further as T/summationdisplay lTr/an}bracketle{tˆvxˆG(k,ωl)ˆvxˆG(k,ωs+ωl)/an}bracketri}ht =(S2V1+S1V2)2 S2 1+S2 2/bracketleftBigf(E−)−f(E+) i/planckover2pi1ωs−E++E−+(E−↔E+)/bracketrightBig , thereby we obtain the expression for the optical conduc- tivity as follows σxx(ω) =−e2 i(2π/planckover2pi1)2ω/integraldisplay∞ 0/integraldisplay2π 0k5H(θ)dkdθ ×/bracketleftBigf(E−)−f(E+) /planckover2pi1ω+iδ−E++E−+(E−↔E+)/bracketrightBig ,(17) where the explicit expression of the optical matrix el- ementH(θ) is given by H(θ) = sin2θ/parenleftBig 3α2+β2− 4αβsin2θ/parenrightBig2 /∆2(θ). It is interesting to note that the above equation can be re-written in terms of the x- component of the Berry connection ( Akx), which is given by σxx(ω) =−e2 i(2π/planckover2pi1)2ω/integraldisplay∞ 0/integraldisplay2π 0k7∆2(θ)A2 kxdkdθ ×/bracketleftBigf(E−)−f(E+) /planckover2pi1ω+iδ−E++E−+(E−↔E+)/bracketrightBig .(18) Similar connection has been established for MoS 2 system28. We have also calculated other components of the opti- cal conductivity i.e. σyy(ω) andσxy(ω). We find that σyy(ω) =σxx(ω) andσxy(ω) = 0 =σyx(ω). It im- plies that the anisotropic Fermi contours do not lead to5 anisotropic optical conductivity. This is similar to the isotropic charge conductivity of a two-dimensional elec- tron gas with combined RSOI and DSOI37. Moreover, σyy(ω) canalsobe expressedintermsofthe y-component of the Berry connection, similar to the σxx(ω) case. Keeping in mind that ω>0 the absorptive part of the optical conductivity σxx(ω) simplifies to Re/bracketleftbig σxx(ω)/bracketrightbig =e2 4π/planckover2pi11 /planckover2pi1ω/integraldisplay∞ 0/integraldisplay2π 0k5H(θ)dkdθ ×/bracketleftbig f(E−)−f(E+)/bracketrightbig δ/parenleftbig /planckover2pi1ω−2k3∆(θ)/parenrightbig =e2 16π/planckover2pi1/integraldisplay2π 0dθsin2θ/bracketleftbig 3+η2−4ηsin2θ/bracketrightbig2 3[1+η2−2ηsin(2θ)]2 ×/bracketleftbig f(E−(kω))−f(E+(kω))/bracketrightbig , (19) withk3 ω=/planckover2pi1ω/2∆(θ) andη=β/α. Pure Rashba ( β= 0):In the absence of DSOI ( β= 0),theabsorptivepartoftheopticalconductivityatfinite temperature is given by Re[σxx(ω)] =3e2 16/planckover2pi1/bracketleftbig f(E−(kω))−f(E+(kω))/bracketrightbig ,(20) withk3 ω=/planckover2pi1ω/2α. At zero temperature we have Re[σxx(ω)] =3e2 16/planckover2pi1/bracketleftbig Θ(E−(kω)−µ)−Θ(E+(kω)−µ)/bracketrightbig , where Θ(x) is the usual unit step function. Depending on the carrier density ( nh) and spin-orbit coupling strength ( α), there must be an upper and lower limits of the photon energy ( Ep=/planckover2pi1ω) in order to have transitions from the initial state λ=−1 to the final stateλ= +1. We use the following parameters for var- ious plots: charge carrier density nh= 2.4×1015m−2 and heavy hole mass m= 0.41m0withm0is the bare electron mass. In Fig. 3, we plot the optical conductivity σxx(ω) vs photon energy Epfor fixedα= 0.1 eV nm3 at four different temperatures. At T= 0, the interband transitions take place only when photon energy satisfies the following inequality: 2 α(k0,− f)3≤/planckover2pi1ω≤2α(k0,+ f)3 and the optical conductivity becomes box function with the edges at E± edge= 2α(k0,± f)3. The width of the opti- cal absorption is then ∆ b= 2α[(k0,+ f)3−(k0,− f)3], whose variation with nhandαare shown in Fig. 4. At finite temperature, the optical conductivity deviates from the box function and smears beyond the box edges. More- over, the conductivity at the box edges ( E± edge) is always σ0/2becauseofthenatureoftheFermidistributionfunc- tion. The peaks in the optical conductivity at finite T is located near the center of the box and it is given by Epeak≃(E+ edge+E− edge)/2 =α[(k0,+ f)3+(k0,− f)3]. Figure 4 shows that ∆ bincreases with nhandα. We mention here that similar analysis can be done for the opposite case i.e.α= 0 butβ/ne}ationslash= 0. The optical conductivity at zero temperature will be σ(ω) =e2/(48/planckover2pi1) which is 9 times less than β= 0case. Otherresultswill be the same as forα/ne}ationslash= 0 butβ= 0 case.Ep(meV)0.2 0.3 0.4 0.5 0.6Reσxx(ω) 00.20.40.60.81T=0 0.3 0.5 1 FIG. 3: (color online) Plots of σ(ω) (in units of σ0= 3e2/16/planckover2pi1) vsEpat four different temperatures. 00.511.5200.050.10.150.20.25 α= 0.18 eV nm3 α= 0.16 eV nm3 α= 0.12 eV nm3 nh(×1015m−2)∆b (meV) 00.05 0.1 0.1500.050.10.150.20.25 nh= 2.0×1015m−2nh= 2.2×1015m−2nh= 2.4×1015m−2 α(eV nm3)∆b(meV) FIG. 4: (color online) (a) Bandwidth ∆ bvsαat various hole density and (b) ∆ bvsnhfor various α. Non-zero Rashba and Dresselhaus SOIs: Now we discuss how the simultaneous presence of both the spin-orbit interactions modifies the behavior of the op- tical conductivity. Similar to the previous case, the op- tical transitions between the initial state λ=−1 and the final state λ= +1 can take place only when photon energy satisfies the following inequality: ǫ−(θ)≤/planckover2pi1ω≤ ǫ+(θ) withǫ±= 2(k± f(θ))3∆(θ). The values of k± f(θ) are the numerical solutions of the two cubic equations (/planckover2pi1k± f)2/2m∗±(k± f)3∆(θ)−Ef= 0, where Efis the Fermi energy for given values of α,βandnh. In the top panel of Fig. 5, ǫ±(θ) vs photon energy are plotted. The shaded angular region contribute to the optical transitions. The interband optical conductivity σxx(ω) vsEpis displayed in the middle panel of Fig.6 0.2 0.4 0.6 0.8 1 1.2θ/π 00.511.52 0.2 0.4 0.6 0.8 1 1.2Reσxx(ω) 00.10.20.30.4 Ep(meV)0.2 0.4 0.6 0.8 1 1.24παk0D(ω) 00.050.10.15cba ǫ+(θ)ǫ-(θ) FIG. 5: (color online) Top panel: ǫ±= 2(k± f(θ))3∆(θ), Mid- dle panel: σxx(ω) (in units of σ0) vsEpand bottom panel: joint density of states D(ω) withk0=√2πnh. Hereα= 0.12 eV nm3,η= 0.6. 5. We see that the optical transition begins and ends atEp= 0.162 meV and Ep= 1.226 meV, respectively. Looking at the top panel, one can see that these values correspond to ǫ+(π/4) =ǫ+(5π/4) =ǫ1andǫ−(3π/4) = ǫ−(7π/4) =ǫ4, respectively. The minimum (maximum) photon energy ǫ1(ǫ4) needed for interband optical tran- sitions correspond to the excitation of a heavy hole with the Fermi wave vector k+ f(θ)(k− f(θ)) atθ=π/4 or 5π/4 (θ= 3π/4or7π/4). TheopticalabsorptionedgesofFig.5 are exactly ǫ1andǫ4. Moreover, two peaks of the optical conductivity occur at Ep= 0.196 meV and Ep= 0.522 meV. It is easy to see from the top panel of Fig. 5 that these values correspond to ǫ−(π/4) =ǫ−(5π/4) =ǫ2and ǫ+(3π/4) =ǫ+(7π/4) =ǫ3, respectively. In order to understand these behaviors we plot a con- stant energy-difference curve Eg(k) = 2∆(θ)k3=/planckover2pi1ωfor /planckover2pi1ω=ǫ1(C1: dotted-blue), /planckover2pi1ω=ǫ2(C2: solid-blue), FIG. 6: (color online) This figure shows the Fermi con- tourk+ f(θ) (dotted-black), k− f(θ)) (solid-black), the constant energy-difference C(/planckover2pi1ω) =/planckover2pi1ω= 2∆(θ)k3for/planckover2pi1ω=ǫ1 (dashed-blue), /planckover2pi1ω=ǫ4(solid-blue), /planckover2pi1ω2=ǫ2(dashed-red) and/planckover2pi1ω4=ǫ3(solid-red). /planckover2pi1ω=ǫ3(C3: dashed-red) and /planckover2pi1ω=ǫ4(C4: solid-red) in Fig. 6. Because of the angular anisotropy in the disper- sion relation, the optical conduction becomes k-selective as shown in the shaded portions of Fig. 6 where the Ci’s (i= 1,2,3,4) intersect with the two Fermi lines k+ f(θ) (dotted) and k− f(θ) (solid), respectively. Twopeaksin the opticalconductivitycan be explained by analyzing the joint density of states. Usually, the absorptive part of the optical conductivity is character- ized by the joint density of states between the spin-split branches, which is given by D(ω) =1 (2π)2/integraldisplay d2k[f(E+)−f(E−)]δ(Eg(k)−/planckover2pi1ω). Usingthe standardapproach,we canwrite the joint DOS as D(ω) =1 (2π)2/integraldisplaydC[f(E+(kω))−f(E−(kω))] |∂kEg(k)|Eg=/planckover2pi1ω,(21) wheredCis the line element along the contour and kω= (/planckover2pi1ω/2∆(θ))1/3. The peaks appear in Fig. 5 when- ever|∂kEg(k)|in thejointDOSattainsaminimumvalue. Therefore, the two peaks correspond to the van Hove singularities in the joint density of states. The first (sec- ond) peak is at a photon energy /planckover2pi1ω2(/planckover2pi1ω3) for which the longer(shorter) axis of the curve Cicoincides with the Fermi linek− f(θ)(k+ f(θ)). The joint density of states for finiteηis shown in the bottom panel of Fig. 5. Locations of the two peaks as well as the optical absorption edges7 in the optical conductivity are exactly described by the joint density of states. The asymmetric splitting at the Fermi level along the symmetry axis ky=±kxis thus responsible for the peaks at ǫ2andǫ3, respectively. The magnitude and the non-symmetric shape of the optical conductivity is controlled by the factor H(θ). For two-dimensional systems, van Hove singularities are classified into three types based on the nature of change of the energy gap Eg(k) as we go away from the singularpoints44. Thiscanbe obtainedby usingthe Tay- lor series expansion of Eg(k) around the singular points ksat which the energy difference attains minimum value. Here the singularpoints are at ks= (k,π/4 or 5π/4) and ks= (k,3π/4 or 7π/4). Expanding Eg(k) around ksas Eg(k) =Eg(ks)+/summationtext ibi(ki−ksi)2withi=x,yand the expansion coefficients are 2 bi=∂2Eg(k) ∂k2 i|ks. The classi- fication of the van Hove singularities are based on how many coefficients ( bi) are negative. For the present sys- tem, the coefficients correspond to the expansion about the singular point ks= (k,π/4) are given by bx=αk9(1−η)2+12η 2(1−η), by=αk9(1−η)2−4η 2(1−η) and the coefficients correspond to the expansion about the singular point ks= (k,3π/4) are bx=αk9(1+η)2+4η 2(1+η), by=αk9(1+η)2−12η 2(1+η). The type of singularities that arise are summarized in the table below: Singular pointη=β/αbx byType of singularity (k,π/4)η <1>0>0 forη < ηlM0 <0 forη > ηlM1 η >1<0>0 forη < ηhM1 <0 forη > ηhM2 (k,3π/4)–>0 >0 M0 TABLE I: Table showing the type of singularity. Here, ηl= (11−2√ 10)/9 andηh= (11+2√ 10)/9) are the solutions of the quadratic equation 9 η2−20η+9 = 0. The optical conductivity versus Epat different values ofηat zero temperature is shown in Fig. 7. Similarly, σ(ω) versusEpatdifferent temperaturesforagivenvalue ofηis shown in Fig. 8. Now we shall point out here the main differences be- tween the electron and heavy hole systems. Unlike two- dimensional electron gas with k-linear RSOI and DSOI, the optical conductivity of two-dimensionalhole gas with k-cubic RSOI and DSOI does not vanish for α=β case. This is related to non-zero Berry phase of two- dimensionalholegaswith equalstrengthof k-cubic RSOIEp(meV)0.2 0.4 0.6 0.8 1Reσxx(ω) 00.20.40.60.81 η=0.65η=0.1 η=0.5η=0.3 FIG. 7: (color online) Optical conductivity Re σxx(ω) in units ofσ0for several values of ηwithα= 0.1 eV nm3. Ep(meV)0 0.2 0.4 0.6 0.8 1 1.2Reσxx(ω) 00.050.10.150.20.250.30.350.4 T=0K T=1K T=0.5K FIG. 8: (color online) Optical conductivity Re σxx(ω) in units ofσ0for several values of temperature Twithα= 0.1 eV nm3 andη= 0.6. andDSOI.Forrealisticsystems,theminimumphotonen- ergy needed to trigger the optical transition in hole gas is one order of magnitude smaller than that of electron gas. In electron systems, heights of the two peaks are unequal, whereas they are more or less same for hole gas at very low temperature.8 IV. SUMMARY AND CONCLUSION We have presented detailed analysis of zero-frequency Drude weight and optical conductivity of a two- dimensional heavy-hole gas with k-cubic RSOI and DSOI, at both zero and non-zero temperature. We ob- tained an analytical expression of the Drude weight for Rashba interaction only. It is shown that the Drude weight deviates from the linear density dependence, in contrast to the case of electron gas with and without k-linear spin-orbit interactions. It decreases with the in- crease of the spin-orbit couplings. We have identified a connection between the optical conductivity and the Berry connection. On contray to the electron case, for equal strength of the Rashba and Dresselhaus couplings, theopticalconductivityremainsfinite. Thisisduetofact that Berry phase is not zero for equal strength of k-cubic spin-orbit couplings. The bandwidth increases with in- crease of the hole density as well as spin-orbit couplings. It is seen that the minimum photon energy required to set in the optical transition in hole gas is one order of magnitude smaller than that of electron gas. We have classified the type of the two van Hove singular points. Appendix: Calculation of Drude weight for system with linear RSOI Here we consider two-dimensional electron gas with k-linear spin-orbit interaction and calculate the Drudeweight using Eq. ( 7). The Hamiltonian for this system is given by He=p2 2me+αe /planckover2pi1/parenleftbig σxpy−σypx/parenrightbig , (22) wheremeis the electron’s effective mass and αeis the strength of the Rashba SOI. The corresponding energy eigenvalues and eigenstates are Eλ(k) =/planckover2pi12k2/2me+ λαekandψλ k(r) =eik·r/parenleftbig 1,−iλeiθ/parenrightbigT/√ 2Ω, respectively. Hereθ= tan−1(ky/kx) andTdenotes the transpose operation. The Fermi energy ( Ef) can be obtained as Ef=π/planckover2pi12ne/me−meα2 e//planckover2pi12withnebeing the electron density. Thex-component of the velocity operator is ˆ vx= px/me−(αe//planckover2pi1)σy. Its expectation value is /an}bracketle{tˆvx/an}bracketri}ht= (/planckover2pi1/me)(k+λkα)cosθwithkα=meαe//planckover2pi12. Following Eq. ( 7), we have the Drude weight as below De w=e2 4πme/summationdisplay λ/integraldisplay d2k(k+λkα)2cos2θδ(k−kλ f) |kλ f+λkα|, wherekλ f=−λkα+/radicalbig 2πne−k2αare the spin-split Fermi wave-vectors. The final expression of the Drude weight is now De w=πe2 me/parenleftBig ne−m2 eα2 e 2π/planckover2pi14/parenrightBig , which is the same as given in Eq. (44) of Ref.36. 1R. Winkler, Spin-orbit Coupling Effects in Two- Dimensional Electron and Hole systems (Springer, Berlin 2003). 2S. Bandyopadhyay and M. Cahay, Introduction to Spin- tronics (CRC press-2008). 3I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 4J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, Acta Physica Slovaca 57, 565 (2007). 5E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). 6Y. A. Bychkov and E. I. Rashba, J. Phys. C: Solid State Phys.17, 6039 (1984). 7J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955). 8G. Dresselhaus, Phys. Rev. 100, 580 (1955). 9J. M. Luttinger, Phys. Rev. 102, 1030 (1956). 10B. Das, D. C. Miller, S. Datta, R. Reifenberger, W. P. Hong, P. K. Bhattachariya, J. Sing, and M. Jaffe, Phys. Rev. B39, 1411 (1989); J. Luo, H. Munekata, F. F. Fang, and P. J. Stiles, Phys. Rev. B 38, 10142 (1988); 41, 7685 (1990); J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997). 11F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen, Nature (London) 452, 448 (2008). 12A.Varykhalov, J.Snchez-Barriga, A.M.Shikin, C. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O. Rader Phys. Rev. Lett. 101, 157601 (2008).13D. Marchenko, J. Snchez-Barriga, M. R. Scholz, O. Rader, and A. Varykhalov Phys. Rev. B 87, 115426 (2013). 14Y. J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011). 15T. Ozawa and G. Baym, Phys. Rev. A 85, 013612 (2012). 16E. I. Rashba and E. Ya. Sherman, Phys. Lett. A 129, 175 (1988). 17R. Winkler, Phys. Rev. B 62, 4245 (2000). 18R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys. Rev. B65, 155303 (2002); G. M. Minkov, A. A. Shersto- bitov, A. V. Germanenko, O. E. Rut, V. A. Larionova, and B. N.Zvonkov, Phys.Rev.B 71, 165312 (2005); F. Nichele, A. N. Pal, R. Winkler, C. Gerl, W. Wegscheider, T. Ihn, and K. Ensslin, Phys. Rev. B 89, 081306(R) (2014). 19H. Nakamura, T. Koga, and T. Kimura, Phys. Rev. Lett. 108, 206601 (2012). 20R. Moriya, K. Sawano, Y. Hoshi, S. Masubuchi, Y. Shi- raki, A. Wild, C. Neumann, G. Abstreiter, D. Bougeard, T. Koga, and T. Machida, Phys. Rev. Lett. 113, 086601 (2014). 21D. W. Yuan, W. Xu, Z. Zeng, and F. Lu, Phys. Rev. B 72, 03320 (2005). 22J. A. Maytorena, C. Lpez-Bastidas, and F. Mireles, Phys. Rev. B74, 235313 (2006). 23M. Xie, G. Khalsa, and A. H. Macdonals, Phys. Rev. B 89, 245417 (2014).9 24C. H. Yang, W. Xu, Z. Zeng, F. Lu, and C. Zhang, Phys. Rev. B74, 075321 (2006). 25A. Wong and F. Mireles, Phys. Rev. B 81, 085304 (2010). 26Z. Li, F. Marsiglio, and J. P. Carbotte, Sci. Rep. 3, 2828 (2013). 27V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. B75, 165407 (2007); T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008). 28Z. Li and J. P. Carbotte, Phys. Rev. B 86, 205425 (2012); I. Miloevic, B. Nikolic, E. Dobardzic, M. Damnjanovic, I. Popov, and G. Seifert, Phys. Rev. B 76, 233414 (2007). 29L. Stille, C. J. Tabert, and E. J. Nicol, Phys. Rev. B 86, 195405 (2012); L. Matthes, P. Gori, O. Pulci, and F. Bech- stedt, Phys. Rev. B 87, 035438 (2013); C. J. Tabert and E. J. Nicol, Phys. Rev. B 87, 235426 (2013). 30Z. Li and J. P. Carbotte, Phys. Rev. B 87, 155416 (2013); X. Xiao and W. Wen, Phys. Rev. B 88, 045442 (2013). 31T. Ma and Q. Liu, Appl. Phys. Lett. 89, 112102 (2006). 32M. Zarea and S. E. Ulloa, Phys. Rev. B 73, 165306 (2006). 33D. V. Bulaev and D. Loss, Phys. Rev. Lett. 95, 076805 (2005). 34J. Schliemann and D. Loss, Phys. Rev. B 71, 085308 (2005). 35N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Harcourt College Publishes-2001).36A. Agarwal, S. Chesi, T. Jungwirth, J. Sinova, G. Vignale, and M. Polini, Phys. Rev. B 83, 115135 (2011). 37M. Trushin and J. Schliemann, Phys. Rev. B 75, 155323 (2007); P. M. Krstaji, M. Pagano, and P. Vasilopoulos, Physica E: Low-dimensional Systems and Nanostructures 43, 893 (2011). 38S. Shapira, U. Sivan, P. M. Solomon, E. Buchstab, M. Tischler, and G. Ben Yoseph, Phys. Rev. Lett. 77, 3181 (1996)). 39J. Zhu, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 90, 056805 (2003); E. Tu- tuc, S. Melinte, E. P. De Poortere, M. Shayegan, and R. Winkler, Phys. Rev. B 67, 241309(R) (2003). 40R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M. Shayegan, D. Wasserman, and S. A. Lyon, Phys. Rev. B 72, 195321 (2005). 41J. Schliemann, Phys. Rev. B 74, 045214 (2006). 42S. Chesi and G. F. Giuliani, Phys. Rev. B 75, 155305 (2007). 43T. Kernreiter, M. Governale, R. Winkler, and U. Zlicke, Phys. Rev. B 88, 125309 (2013). 44C. Hamaguchi, Basic Semiconductor Physics, (Springer- Verlag-2010).

1108.1899v1.Nonlinear_spin_Hall_effect_in_GaAs__110__quantum_wells.pdf

arXiv:1108.1899v1 [cond-mat.mtrl-sci] 9 Aug 2011Nonlinear spin Hall effect in GaAs (110) quantum wells V. I. Ivanov1, V. K. Dugaev2,3, E. Ya. Sherman4,5, and J. Barna´ s6,∗ 1Insitute for Problems of Materials Science, Ukrainian Acad emy of Sciences, Vilde 5, 58001 Chernovtsy, Ukraine 2Department of Physics, Rzesz´ ow University of Technology, al. Powsta´ nc´ ow Warszawy 6, 35-959 Rzesz´ ow, Poland 3Department of Physics and CFIF, Instituto Superior T´ ecnic o, TU Lisbon, Av. Rovisco Pais 1049-001 Lisbon, Portugal 4Department of Physical Chemistry, Universidad del Pa´ ıs Va sco, Bilbao, Spain 5IKERBASQUE Basque Foundation for Science, 48011, Bilbao, S pain 6Institute of Molecular Physics, Polish Academy of Sciences , ul. Smoluchowskiego 17, 60-179 Pozna´ n, Poland (Dated: September 7, 2018) We consider stationary spin current in a (110)-oriented GaA s-based symmetric quantum well due to a nonlinear response to an external periodic electric field. The model assumed includes the Dresselhaus spin-orbit interaction and the random Rashba s pin-orbit coupling. The Dresselhaus term is uniform in the quantum well plane and gives rise to spi n splitting of the electron band. The external electric field of frequency ω– in the presence of random Rashba coupling – leads to virtual spin-flip transitions between spin subbands, generating st ationary pure spin current proportional to the square of the field amplitude. PACS numbers: 72.25.Rb, 72.25.Hg I. INTRODUCTION Spin-orbit(SO) interactionsinsemiconductorsreveala variety of fundamental spin related phenomena.1Forma- tion of stable spin helices with nontrivial temporal and spatial dynamics,2–4spin optics,5and spin-dependent sound radiation6represent only some of these effects. Spin currents,7solely attributed to the spin-orbit cou- pling, provide a possibility of inducing and controlling spin motion by electrical and optical fields, and there- fore became one of the key elements of the modern spin- tronics oriented at new device applications of semicon- ductor based structures. Thorough investigations of re- alistic systems enlarge the variety of both fundamental phenomena and possible applications. As an example we mention that disorder – always present in real systems – plays a crucial role in the spin-Hall effect8, as the spin Hall conductivity can be totally suppressed by any finite concentration of impurities. Recently, symmetricGaAs(110)quantumwells(QWs) became the subject of extensive experimental and the- oretical investigations. This is related to the expec- tation of the longest spin relaxation times in these structures,9–16which in turn can lead to interesting spin dynamics.17,18A stationary pure spin current accompa- nying an electric current in (110) QWs was observed as reported in Ref. [19]. SO interaction in these systems, described by the Dresselhaus term in the corresponding Hamiltonian,20–22conserves the electron spin along the axis normal to the QW plane for any electron momen- tumk. As a result, random motion of an electron does not lead to a random direction of the spin-orbit field and therefore does not lead to spin relaxation. In re- ality, however, this spin component relaxes very slowly, and its analysis provides a test for the rapidly developing low-frequency spin-noise spectroscopy23suitable for themeasurements of the long spin evolutions. In the case of perfect z→ −zsymmetry (the axis zis perpendicular to the QW plane), the Rashba SO inter- action is zero. In real structures, however, the Rashba coupling still exists in the form of a spatially fluctuat- ing SO field (though being zero on average).24–26This interaction induces spin-flip processes leading to the spin relaxation,16and can be also responsible for generation of a nonequilibrium spin density due to the absorption of an external electromagnetic field.26Recently, it was proposed that this random SO coupling can result in the spin orientation by an external current27and also can play a role in the formation of the stripe structure of spin current distribution.28 In this paper we propose a new possibility of exciting a steady pure spin current by a periodic external filed, extending thus the abilities of spin manipulation in real situations. In contrast to the conventional spin Hall ef- fect, which is linear in the external electric field, the pro- posed spin current is quadratic in the external periodic field. The effect is a result of the interplay of constant Dresselhaus and spatially random Rashba terms, and is not related to the spin currents produced by gate ma- nipulation of the Rashba coupling29or adiabatic pump- ing in graphene.30Exact mechanism of the effect does not necessarily involve real spin-flip transitions of elec- trons between the spin-split subbands in (110)-oriented GaAs QW, but relies on virtual spin-flip processes which renormalize the wave functions of electrons in a nonequi- librium state. This makes such a nonlinear current a physically new phenomenon, which appears if one ac- counts for more realistic effects than those described by the conventional Rashba and Dresselhaus models. In Section 2 we describe the model and Hamiltonian of the system. Spin current is calculated in Section 3. Summary and final conclusions are presented in Section2 4. II. MODEL Hamiltonianofatwo-dimensionalelectrongaswiththe constant Dresselhaus term HDand spatially fluctuating Rashba spin-orbit interaction HR, subjected to external electromagnetic field described by the vector potential A(r,t), takesthefollowingform(weuseunitswith¯ h= 1) H=H0+HD+HR, (1) where the first two terms are H0=−1 2m/parenleftbigg ∇−ieA c/parenrightbigg2 , (2) HD=−iασz/parenleftbigg ∇x−ieAx c/parenrightbigg . (3) The Dresselhaus constant α=γπ2/2w2, whereγis the corresponding bulk Dresselhaus coupling parameter, is inversely proportional to the square of the QW width w. The other components of the Dresselhaus interac- tion vanish due to the specific symmetry of the (110) orientation.16,21 The last term in Eq. (1) stands for the effects of the spatially nonuniform Rashba SO interaction, which can be written as HR=H0 R+V, whereH0 Ris the Rashba term for A(r,t) = 0, H0 R=−i 2σx{∇y, λ(r)}+i 2σy{∇x, λ(r)},(4) with{,}denoting the anticommutator and λ(r) being the randomRashbaSO interaction. The term V, in turn, describes coupling of the electron spin to the external fieldA(r,t)viathe Rashba field, V=−e cλ(r)(σxAy−σyAx). (5) Due to the assumed symmetry with respect to z- inversion, the spatially averaged Rashba interaction van- ishes,/an}b∇acketle{tλ(r)/an}b∇acket∇i}ht= 0.We assume that the random Rashba field can be described by the correlation function related to fluctuating density of impurities near the QW,24,26 Cλλ(r−r′)≡ /an}b∇acketle{tλ(r)λ(r′)/an}b∇acket∇i}ht=/angbracketleftbig λ2/angbracketrightbig F(r−r′),(6) where the range function F(r−r′) depends on the type of disorder. We assume the correlator of random Rashba interaction in the momentum space in the form26,31 |λq|2= 2π/angbracketleftbig λ2/angbracketrightbig R2e−qR, (7) whereRis the spatial scale of the fluctuations. In the absenceofexternalfield and randomRashbaSO interaction, the Hamiltonian H0+HDdescribesthe spec- trum of spin-polarized electrons, εkσ=/parenleftbig k2 x+k2 y/parenrightbig /2m+σαkx. The energy bands of spin up and spin down elec- trons are thus shifted in opposite directions along the kx axis. The corresponding Green function is then diagonal in the spin subspace, G(0) kε=/parenleftbigg Gkε+0 0Gkε−/parenrightbigg , Gkεσ=1 ε−εkσ+µ+iδkσsign(ε),(8) whereσ= + for spin up ( ↑) electrons and σ=−for spin down (↓) electrons, whereas δkσis the momentum and spin dependent relaxation rate. III. NONLINEAR SECOND-ORDER SPIN CURRENT In the following we consider the z-component of a pure spin current flowing along the xaxis, that is the only component allowed by symmetry of the system under consideration. The operators of the electron velocity ˆ vx and the corresponding spin current tensor component ˆ z x are ˆvx=i[H0+HD,x] =kx m+ασz−λσy,(9) ˆz x=1 2{ˆvx,σz}=kx mσz+α, (10) where the α-related terms correspond to the anomalous contribution to the velocity. The macroscopic spin cur- rent density is then given by jz x=iTr/summationdisplay k/integraldisplaydε 2πˆz xGkε, (11) whereGkεis the Green’s function of the system inter- acting with the external electromagnetic field. Upon substituting (10) into Eq. (11) one can note that the second term describes the current caused only by the electron density, n=iTr/summationdisplay k/integraldisplaydε 2πGkε, (12) conserved under any external perturbation. This conser- vation is achieved in calculations by an appropriate shift of the chemical potential µ. In equilibrium, however, there is no spin current in the system (expected, e.g., for QWs with other crystallographic orientations) since the integrated contributions from kx/mandαterms in Eq. (10) exactly cancel each other. As a result, this type of structures does not demonstrate the Rashba paradox of the non-zero equilibrium pure spin current.32This can beseendirectlybycalculatingspincurrentusingEq.(11) with the equilibrium Greenfunction in Eq.(8), orby tak- ing into account the fact that Hamiltonian in Eq. (3) can be transformed by the SU(2) rotation to the form that does not have spin dependent terms.333 FIG. 1: Feynman diagrams for the excited spin current in Eq. (11). The circles correspond to the matrix elements Vkk′ andVk′k. However,anonzeropurespincurrent,whichisthesub- ject ofinterest here, can be generatedby an external field in the presence of random Rashba coupling, as presented schematically by the Feynman graph in Fig. 1. In this graph we introduced the following notations: Vkk′=λkk′(σxAy(ω)−σyAx(ω)),(13) Vk′k=λk′k(σxAy(−ω)−σyAx(−ω)),(14)for the transition matrix elements due to the external field and spin-orbit coupling. With the Feynman graph shown in Fig. 1 we find the spin current density omitting the αterm in the velocity operator since its contribution is conserved if no addi- tional electrons are injected to the system. The resulting expressionforthecontributionofasinglespincomponent to the total spin current becomes jxσ=iA2 m/summationdisplay kk′/integraldisplaydε 2πkx|λkk′|2GkεσGk′ε+ωσ′Gkεσ, (15) whereAis the vector potential amplitude, making the injected current independent on the external field orien- tation. The total spin current is given by jz x=jx,σ=1− jx,σ=−1withjx,σ=−1=−jx,σ=1. Using Eqs. (8) and (15), after rather tedious integra- tion of the product of the three Greens functions in the complex εplane, one obtains jxσ=−A2 m/summationdisplay kk′kx|λkk′|2/bracketleftBigg −f(εkσ) (εkσ−εk′σ′+ω+iδkσ+iδk′σ′sign(εkσ+ω−µ))2 +f(εk′σ′) (−εkσ+εk′σ′−ω+iδk′σ′+iδkσsign(εk′σ′−ω−µ))2/bracketrightBigg =A2 m/summationdisplay kq|λq|2/bracketleftBigg kx[f(εkσ)−f(εkσ+ω)] (εkσ−εk−qσ′+ω+iδ+)2−(kx+qx)f(εkσ′) (εk+qσ−εkσ′+ω+iδ−)2 +kxf(εkσ+ω) (εkσ−εk−qσ′+ω+iδ−)2/bracketrightBigg . (16) Here we introduced the notation: δ+≡δkσ+δk′σ′and δ−≡δkσ−δk′σ′. Equation (16) shows that the injec- tion of spin current is a coherent effect arising due to the change in electron wave function under the resonant electromagnetic radiation rather than the injection due to the two-photon absorption typical in nonlinear semi-conductor optics. Since the single electron energy εkσcan be written as [(kx+σαm)2+k2 y]/2m−mα2/2, one can shift the chemical potential, µ→µ+mα2/2. Then one can write jxσin the form, jxσ=A2 m/summationdisplay kq|λq|2/bracketleftBigg (kx−σαm)f(εk)−f(εk+ω) (k·q/m−q2/2m+2σα(kx−qx)−2mα2+ω+iδ+)2 +(kx−σαm)f(εk+ω) (k·q/m−q2/2m+2ασ(kx−qx)−2mα2+ω−iσδ−)2 −(kx+qx+σαm)f(εk+ω) (k·q/m+q2/2m+2ασ(kx+qx)+2mα2+ω−iσδ−)2/bracketrightBigg . (17)4 0.2 0.4 0.6 0.8-40-2002040jx /j0 ω/µkFR=1kFR=2kFR=4z FIG. 2: Total spin current, calculated by using Eq. (17) for indicated values of kFR. The parameters used in numerical calculations are given in the main text. We emphasize here that the calculated spin current is a stationary coherent nonlinear effect proportional to the intensity of incident radiation,34in contrast to the spin current generated by pulse excitations, where the result is proportional to the total fluence in the pulse.25 Thetransitionprocessesproducerealholesintheinitially occupied subbands and electrons in those initially empty, changing the real occupation of the spin-up and spin- down states. The calculated current is also not related to the Drude-likelinearresponseat frequency ω, suppressed by the factor of the order of ( δ+/ω)2. The results of numerical calculation of the spin cur- rent (taking part of Eq. (17)) are presented in Fig. 2 for different values of the parameter kFR. The parameters typical for the (110) quantum wells are: 2 αm/kF= 0.1 andδ+/µ= 0.1. For the momentum-dependent δ−, we assume a typical value, δ−/µ= 0.05. Furthermore, we used for GaAs: m= 0.067m0, wherem0is the free elec- tronmass, Fermi momentum kF= 1.8×106cm−1(corre- sponding to electron concentration 5 .2×1011cm−2) and µ= 18.5 meV. The spin current in Fig. 2 is presented in the units of j0, withj0defined as j0=2m2αe2 c2π3A2/angbracketleftbig λ2/angbracketrightbig k2 F. (18) Taking into account the relation A2= (c/ω)2E2,where Eis the electric field amplitude, we obtain j0= 2m2α/an}b∇acketle{tλ2/an}b∇acket∇i}ht(eE/ω)2/π3k2 F,witheE/ωbeing the ampli- tude of the momentum oscillation of a classical electron in a periodic electric field. It is interesting to mention that the maximum value of E,which still can be consid- eredasaperturbation,isdeterminedby eE/ω∼kF,and, therefore, the maximum of j0is ofthe orderof m2α/angbracketleftbig λ2/angbracketrightbig , having the physical meaning of the equilibrium spin cur- rent induced by the random Rashba spin-orbit coupling. To understand better the physical mechanism of the nonlinear spin-current generation we consider a schematic picture presenting the electron energy bands as a function of kxwithout Rashba random SO interac- tion and without external field, see Fig. 3. As we haveFIG. 3: Schematic presentation of the light-induced resona nt formation of spin holes in the energy bands occupied with electrons: (a) spin-split energy bands in GaAs (110) quantu m well; (b) due to the coupling Vkk′(cf. Eq.(13)) of the spin-up and spin-down states, the effective spin /angbracketleftSz/angbracketrightin each subband decreases. alreadymentioned above, the DresselhausSO interaction leads to spin splitting of the electron states of a free elec- tron gas, which results in the energy bands εkσshown in Fig. 3(a) as a function of kx(forky= 0). Even though the states |kσ/an}b∇acket∇i}htof these bands are spin polarized, the spin current in equilibrium is exactly zero. This is related to the zero current associated with each of the subbands, j↑,↓, calculated as the flux of electrons in each subband. Obviously, vanishing current jσin the subband σmeans that the spin current js σis also zero. Distortions of the energy subbands either due to the random Rashba inter- action in Eq. (4) or due to the external field in Eq. (2) do not break the condition j= 0. Our calculations, however, showed that nonzero ma- trixelements offield-induced spin-flip intersubbandtran- sitions appear in the presence of random Rashba cou- pling. Accordingly, in the nonequilibrium situation the electron states in each subband are a superposition of spin up and down states, so that the resulting state |k±/an}b∇acket∇i}ht (Fig. 3(b)) has a smaller effective spin. Such mixing of |kσ/an}b∇acket∇i}htand|k′σ′/an}b∇acket∇i}htstates effectively depends on |k−k′|and on|εkσ−εk′σ′±ω|,sothattheabove-mentionedspinmix- ing is different at different parts of the dispersion curves εkσ. This is shownschematicallyin Fig. 3(b) for different spin subbands. Thus, even though in nonequilibrium the current in each subband εkσis zero, the associated spin current is not zero anymore. For example, in the εk↑band more up spins flow in + xthan−xdirection. Correspondingly, in theεk↓band more down spins flow in −xdirection. This results in the net spin-up current in + xdirection. Obviously, the direction of spin current is related to the sign of Dresselhaus coupling constant α. A remarkable change of the wave function by a strong electric field causes the injected pure spin current of the order of the equilibriumspincurrentsarisingasaresultoftheRashba5 paradox.25,32 IV. SUMMARY AND CONCLUSIONS To conclude, we have proposed a new effect of the co- herentnonlineargenerationofasteadypurespincurrents in GaAs (110) quantum wells by electromagnetic wave. The injected spin current is proportional to the intensity of the external radiation, strongly depends on the fre- quency, and can be injected in the frequency range up to the Fermi energy of the two-dimensional electron gas. Physical mechanism of the effect is related to the virtualspin reorientation of electrons filling the spin subbands split by the Dresselhaus interaction in the presence of a randomly varying Rashba coupling. The latter may be introduced, e.g. by random doping of the quantum well. Asaresult, a’spinhole’virtuallyappearsinthesubband, leading to the light-induced spin current. Acknowledgements. This work is partly supported by FCT Grant PTDC/FIS/70843/2006 in Portugal and by National Science Center in Poland as a research project in years 2011 – 2014. The work of EYS was supported bytheMCINNofSpaingrantFIS2009-12773-C02-01and ”Grupos Consolidados UPV/EHU del Gobierno Vasco” grant IT-472-10. * Permanent position: Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Pozna´ n, Poland. 1I. Zuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004); M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Reports, 493, 61 (2010). 2J. D. Koralek, C. Weber, J. Orenstein, A. Bernevig, S.- C. Zhang, S. Mack, and D. Awschalom, Nature 458, 610 (2009). 3Yu. V. Pershin and V. A. Slipko, Phys. Rev. B 82, 125325 (2010), V. A. Slipko, I. Savran, and Yu. V. Pershin, Phys. Rev. B83, 193302 (2011). 4M. C. L¨ uffe, J. Kailasvuori, and T. S. Nunner, arXiv:1103.0773. 5M.Khodas, A.Shekhter,andA.M. Finkelstein, Phys.Rev. Lett.92, 086602 (2004). 6S. Smirnov, Phys. Rev. B 83, 081308 (2011). 7J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004); J. Wunderlich, B. G. Park, A. C. Irvine, L. P. Zarbo, E. Rozkotova, P. Nemec, V. Novak, J. Sinova, and T. Jung- wirth, Science 330, 1801 (2010). 8O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005); R. Rai- mondi and P. Schwab, Phys. Rev. B 71, 033311 (2005); A. Khaetskii, Phys. Rev. Lett. 96(2006) 056602; T. S. Nun- ner, N. A. Sinitsyn, M. F. Borunda, V. K. Dugaev, A. A. Kovalev, Ar. Abanov, C. Timm, T. Jungwirth, J.-I. Inoue, A. H. MacDonald, and J. Sinova, Phys. Rev. B 76, 235312 (2007). 9Y. Ohno, R. Terauchi, T. Adachi, F. Matsukura, and H. Ohno, Phys. Rev. Lett. 83, 4196 (1999). 10S. Dohrmann, D. H¨ agele, J. Rudolph, M. Bichler, D. Schuh, and M. Oestreich, Phys. Rev. Lett. 93, 147405 (2004). 11E. M. Hankiewicz, G. Vignale, and M. E. Flatt´ e, Phys. Rev. Lett. 97, 266601 (2006). 12L. Schreiber, D. Duda, B. Beschoten, G. G¨ untherodt, H.-P. Sch¨ onherr, andJ. Herfort, Phys. Rev.B 75, 193304 (2007). 13V. V. Bel’kov, P. Olbrich, S. A. Tarasenko, D. Schuh, W. Wegscheider, T. Korn, C. Schuller, D. Weiss, W. Prettl, and S. D. Ganichev, Phys. Rev. Lett. 100, 176806 (2008). 14S. Crankshaw, F. G. Sedgwick, M. Moewe, C. Chang- Hasnain, H. Wang, and S.-L. Chuang, Phys. Rev. Lett. 102, 206604 (2009). 15S. A. Tarasenko, Phys. Rev. B 80, 165317 (2009). 16Y. Zhou and M. W. Wu, Solid State Communs. 49, 2078 (2009); Y. Zhou and M.W. Wu, EPL 89, 57001 (2010).17R. V¨ olkl, M. Griesbeck, S. A. Tarasenko, D. Schuh, W. Wegscheider, C. Schller, and T. Korn, Phys. Rev. B 83, 241306 (2011). 18J. H¨ ubner, S. Kunz, S. Oertel, D. Schuh, M. Pochwala, H. T. Duc, J. F¨ orstner, T. Meier, and M. Oestreich, Phys. Rev. B84, 041301 (2011). 19V. Sih, R. C. Myers, Y. K. Kato, W. H. Lau, A. C. Gossard and D. D. Awschalom, Nature Physics 1, 31 (2005). 20G. Dresselhaus, Phys. Rev. 100, 580 (1955). 21R. Winkler, Spin-Orbit Coupling Effects in Two- dimensional Electron and Hole Systems (Springer, New York, 2003). 22M. I. D’yakonov and V.Yu. Kachorovskii, Sov. Phys. Semi- conductors 20, 110 (1986). 23G. M. M¨ uller, M.L. R¨ omer, D. Schuh, W. Wegscheider, J. H¨ ubner, and M. Oestreich, Phys. Rev. Lett. 101, 206601 (2008); G. M. M¨ uller, M. Oestreich, M.L. R¨ omer, and J. H¨ ubner, Physica E 43, 569 (2010). 24E. Ya. Sherman, Phys. Rev. B 67, 161303(R) (2003). 25E. Ya. Sherman, A. Najmaie, and J. E. Sipe, Appl. Phys. Lett.86, 122103 (2005); H. Zhao, X. Pan, A. L. Smirl, R. D. R. Bhat, A. Najmaie, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 72, 201302 (2005). 26M. M. Glazov, E. Ya. Sherman, and V.K. Dugaev, Physica E42, 2157 (2010). 27L. E. Golub and E. L. Ivchenko, arXiv:1107.1109. 28Y. B. Lyanda-Geller, arXiv:1107.3121. 29C. S. Tang, A. G. Mal’shukov, and K. A. Chao, Phys. Rev. B71, 195314 (2005), C.-H. Lin, C.-S. Tang, and Y.-C. Chang, Phys. Rev. B 78, 245312 (2008). 30Q. Zhang, K. S. Chan, and Z. Lin, Appl. Phys. Lett. 98, 032106 (2011). 31V.K.Dugaev, E.Ya.Sherman, V.I.Ivanov,andJ. Barna´ s, Phys. Rev. B 80, 081301(R) (2009). 32E. I. Rashba, Phys. Rev. B 68, 241315 (2003). 33I. V. Tokatly, Phys. Rev. Lett. 101, 106601 (2008); I. V. Tokatly and E. Ya. Sherman, Ann. Physics 325, 1104 (2010). 34This second-order process is qualitatively different from thenonlinearphotogalvaniceffectingyrotropiccrystals( E. L. Ivchenko and G. E. Pikus, JETP Lett. 27, 604 (1978)) and from the frequency doubling in the spin-Hall effect (Yu.V.PershinandM.DiVentra,Phys.Rev.B 79, 153307 (2009)).

1408.2261v1.Strong_interactions__narrow_bands__and_dominant_spin_orbit_coupling_in_Mott_insulating_quadruple_perovskite_CaCo3V4O12.pdf

arXiv:1408.2261v1 [cond-mat.str-el] 10 Aug 2014Strong interactions, narrow bands, and dominant spin-orbi t coupling in Mott insulating quadruple perovskite CaCo 3V4O12 H. B. Rhee and W. E. Pickett Department of Physics, University of California Davis, Dav is CA 95616 (Dated: June 11, 2021) We investigate the electronic and magnetic structures and t he character and direction of spin and orbital moments of the recently synthesized quadruple p erovskite compound CaCo 3V4O12us- ing a selection of methods from density functional theory. I mplementing the generalized gradient approximation and the Hubbard Ucorrection (GGA+U), ferromagnetic spin alignment leads to half-metallicity rather than the observed narrow gap insul ating behavior. Including spin-orbit cou- pling (SOC) leaves a half-semimetallic spectrum which is es sentially Mott insulating. SOC is crucial for the Mott insulating character of the V d1ion, breaking the dm=±1degeneracy and also giving a substantial orbital moment. Evidence is obtained of the la rge orbital moments on Co that have been inferred from the measured susceptibility. Switching to the orbital polarization (OP) func- tional, GGA+OP+SOC also displays clear tendencies toward v ery large orbital moments but in its own distinctive manner. In both approaches, application of SOC, which requires specification of the direction of the spin, introduces large differences in the or bital moments of the three Co ions in the primitive cell. We study a fictitious but simpler cousin comp ound Ca 3CoV4O12(Ca replacing two of the Co atoms) to probe in more transparent fashion the inte rplay of spin and orbital degrees of freedom with the local environment of the planar CoO 4units. The observation that the underlying mechanisms seems to be local to a CoO 4plaquette, and that there is very strong coupling of the size of the orbital moment to the spin direction, strongly su ggest non-collinear spins, not only on Co but on the V sublattice as well. PACS numbers: I. INTRODUCTION Recently the Mott insulating quadruple perovskite CaCo3V4O12(CCVO) was reported by Ovsyannikov et al.1having been synthesized under pressure. Several other quadruple perovskites with the formula AA′ 3B4O12 have been studied, and this class of compounds with two magnetic sublattices has been found to exhibit a wide range of intriguing phenomena, suggesting that its unusual structure may be playing a crucial role in the complex behavior that emerges. CaCu 3Mn4O12 (CCMO) gained attention for its colossal magnetore- sistance (CMR),2and thus was compared to the man- ganatesLn1−xDxMnO3(Ln=lanthanide, D=alkali earth metal), which also exhibit CMR. Unlike the struc- turally simpler manganates, whose magnetoresistance (MR) becomes negligible at low applied magnetic fields, CCMOhasalargelow-fieldMRresponsespanningawide temperature range below a high ferrimagnetic transition temperature Tc= 355 K, a characteristic that is very attractive for magnetic sensor applications. CCMO was found to possess no Jahn-Teller Mn3+ions at the Bsite and hence no double-exchangemechanism, which is what givesrisetoCMR inthe manganateperovskites. The ori- ginofCMR(40%)inCCMO,andverylargelow-fieldMR in BiCu 3Mn4O12(Ref. 3) and LaCu 3Mn4O12(Ref. 4) as well, appears to be due to spin tunneling or scattering at grain boundaries, rather than double exchange in the bulk. Another extreme property was found in the antiferro- magnetic insulator CaCu 3Ti4O12, which displays a gi- ant dielectric constant ∼105that remains fairly con-stant from 100 to 500 K (Refs. 5,6). This dielectric re- sponse was established to be extrinsic in nature, with the structure perhaps playing a central role in the de- fects that produce very large polarization. Unusual magnetic transitions have been observed when gradu- ally doping from ferromagnetic CaCu 3Ge4O12to antifer- romagnetic CaCu 3Ti4O12, and in turn to ferromagnetic CaCu3Sn4O12, the mechanism of which is yet unknown but is apparently not due to bond angle-dependent su- perexchange interactions.7 The structure of these materials provides important insight into, and restrictions on, the magnetic behavior, where a magnetic transition (apparently antiferromag- netic) is seen at TN= 98 K.1Both the Co2+and V4+ ions in this cobaltovanadate are expected to be magnetic and Mott insulating, and seemingly carries a substantial orbital moment to account for the observed Curie-Weiss moment. The Co ion symmetry is mmmin the rectan- gular CoO 4plaquettes that promote, as we will show, strong local anisotropy. Our calculations reveal that the Mott insulating character of the open-shell Co ion arises through the dz2orbital in the local coordinate system. However, the overall cubic symmetry of the compound dictates that there are CoO 4plaquettes perpendicular to each of the cubic axes, as can be seen in Fig. 1(a). The very narrow Co 3 dbands – very nearly localized orbitals – will promotestrong magnetic anisotropy, with spin mo- ment either perpendicular to, or within the plane of, the O4plaquette. The V4+d1ion, with its ¯3 site symmetry, suggests a related conundrum. The four V ions have their individ- ual symmetry axes along one of the four (111) directions.FIG. 1: (Color online) (a) Global coordinates and crystal structure of quadruple perovskite CCVO, containing two for - mula units per unit cell. The Co atoms (dark blue) are each surrounded by a plaquette of oxygens (pink), and the tilted VO6octahedraare nearlyregular. Thebreakingofcubicsym- metry due to spin-orbit coupling makes the three Co atoms (labeled 1 to 3) in the formula unit inequivalent. The mag- netization quantization axis for our calculations, unless noted otherwise, is the ˆ zaxis. To get the structure of fictitious com- pound Ca 3CoV4O12, replace Co atoms 1 and 2 with Ca. (b) Local coordinate system of a CoO 4plaquette. Images were generated with VESTA.8 The occupied orbitals will be symmetry adapted linear combinations of the t2gorbitals. Crystal field consid- erations suggest the a1gsymmetric combination will lie highestin energy, leaving the doublet usually denoted e′ g singly occupied and therefore prime for orbital ordering. With the local (111) direction dictating the rotational symmetry, the t2gorbitals can be expressed in the form ψm=1√ 3(ζ0 mdxy+ζ1 mdyz+ζ2 mdzx), (1) whereζm=e2πmi/3is the associated phase factor for threefold rotations. There is vanishing orbital moment for thea1gmemberm= 0, but potentially large orbital momentsforthe e′ gpairm=±1. Theseorbitalmoments, hence the spin and net moments, should tend to lie along the (111) axis of symmetry of each V ion. Hence the V moments will be non-collinear among themselves, and also not collinear with the Co moments. In such a situa- tion, interpretation of the magnetization data in the or- dered state will become a challenging problem. We note that the non-collinearity that we are mentioning results from single-ion anisotropy rather than from competing, frustrated exchange coupling, which is beyond the scope of the present study.CCVO is currently the only quadruple perovskite re- ported that has Co exclusively occupying the Bsite. Most known quadruple perovskites are half-metallic and semiconducting in the majority-spin channel; half- metallicity is often seen in perovskite oxides and other transition-metal oxides displaying CMR.9–11In this pa- per, we perform density functional studies on CCVO, and our methods indicate how CCVO becomes a Mott insulator. We also study a related fictitious system, Ca3CoV4O12, in which two of CCVO’s three Co atoms are replaced by Ca, so that spin and orbital effects may be better understood on a local scale. II. STRUCTURE OF CCVO AND CALCULATION METHODS CCVO takes on a structure that can be pictured as a variant of the cubic perovskite oxide ABO3. The super- structureAA′ 3B4O12, withIm¯3spacegroup,isformedby quadrupling the parent unit cell and replacing 3 /4 of ele- mentAwithA′. Due to the introduction of A′, the sym- metryofthestructureisloweredbyalargerotationofthe BO6octahedra, which brings four oxygen ions closer to theA′(Co) site to form a seemingly nearly square-planar environment, but we will showthereis animportant non- squarecomponentofthepotential. Surprisingly,thispar- ticular quadruple perovskite houses VO 6octahedra that are virtually regular: all V-O distances are identical, and the O-V-O angles deviate from 90◦by only 0.04◦. The natural local axes of course do not align with the global crystal axes. The CoO 4plaquettes are not as regular, with the O-Co-O angles being 93 .6◦and 86.4◦. Because the non-square aspect becomes important, we will refer to the unit as the CoO 4plaquette. Both Co and V ions are anticipated to be magnetic with Mott insulating character, soconsiderationsofmag- netic coupling arise. V ions lie on a simple cubic sublat- tice separated by a/2 = 3.67˚A, while Co ions lie on a bcc sublattice with the same nearest-neighborCo-Co dis- tance. The two perovskite AandBsublattices form a CsCl configuration, making it likely that nearest neigh- bor Co-V exchange interactions (versus Co-Co or V-V) are the driving force for magnetic order. In oxides, Co and V often display strongly correlated behavior, so orbitally independent treatments such as GGAand localdensity approximation(LDA) donot pro- vide the flexibility to handle a compound like CCVO. We have therefore employed the GGA+U method,12,13 in which the intra-shell Coulomb repulsion Uand inter- orbitalHund’s Jmagneticcouplingswereappliedtoboth Co and V with the following strengths: UCo= 5 eV, JCo= 1 eV,UV= 3.4 eV,JV= 0.7 eV. All-electron calculations of CCVO and C3CVO were done with the WIEN2k14,15program, which is based on a full potential, linearized augmented planewave (FP- LAPW) method within the density functional theory for- malism. The Perdew-Burke-Ernzerhof flavor16of the 2GGA was chosen for the exchange-correlationfunctional. We used a 17 ×17×17k-point mesh for the cubic unit cell, outlined in Fig. 1(a). The sphere radii Rwere set to 2.44, 2.04, 1.90, and 1.72 a.u. for Ca, Co, V, and O, respectively. RKmax= 7.0 was the cutoff for the planewave expansion of eigenstates. We used the CCVO experimentallatticeconstant( a= 7.3428˚A)andinternal coordinates,1and a collinear ferromagnetic (FM) config- uration was adopted for this first study, to obtain in- sight into the intricate electronic and magnetic nature of CCVO. The orbital moment on Co, and thus the magnetocrys- talline anisotropy in CCVO have been suggested to be large,1reflecting the presence of strong spin-orbit cou- pling (SOC). It then is important to include SOC in the calculations. Without SOC, the symmetry of CCVO is cubic (even with spin polarization) so each of the three Co atoms in the stoichiometric formula are equivalent. Taking into account SOC, with the direction of magneti- zationMin any one of the three axial directions, lowers the symmetry, and the three Co ions in their individually oriented CoO 4plaquettes are no longer equivalent, and we find very large differences in the orbital and even the spin moments. In this paper the magnetization direction for all of our calculations will be along the ˆ zaxis unless noted otherwise, and we will refer to the three different Co atoms as Co1, Co2, and Co3, labeled in Fig. 1(a) as 1, 2, and 3, respectively. The respective CoO 4pla- quettes are perpendicular to the ˆ z, ˆx, and ˆydirections respectively. Even with the addition of U, the orbital moment in Co perovskite oxides may be underestimated. In a fully rel- ativistic treatment there is an additional orbital correc- tion, referred to as orbital polarization (OP),17,18which is an attractive energy that is proportional to the square of the orbital angular moment L. This OP correction has successfully been applied to intermetallics and espe- cially to Co compounds that exhibit a large Co orbital moment.19–24The OP correction can be implemented in a phenomenological approach analogous to Hund’s sec- ond rule that is implicit in GGA. Our results will show that orbital effects are indeed much larger within the OP scheme than within conventional GGA+U. One one hand, the symmetry lowering due to SOC raises questions about the character of magnetic order- ing that occurs in CCVO. On the other hand, orbital mo- ment formation and orientation seems to be a local phe- nomenon, almost dictating non-collinear magnetic order. To simplify some of the local questions, we have stud- ied a theoretical cousin of CCVO, in which the Co2 and Co3 ions are replaced by simple Ca2+, thereby isolating the Co1 ion for study. We present results for this model compound Ca 3CoV4O12, which wereferto as C3CVO,to demonstratethatthelocalenvironmentbyandlargegov- erns the magnetic character of the CoO 4unit in CCVO.III. ELECTRONIC STRUCTURE ANALYSIS A. CCVO The progression of the electronic and magnetic struc- ture of FM CCVO with inclusion of Uand then SOC is displayed in the various panels of Fig. 2. The right-hand panels allow identification of the most relevant orbitals— those of the partially filled 3 dshell of Co. When SOC is included, the angular momentum basis orbitals dmare used for the projection, using the local coordinatesystem pictured in the Fig. 1 inset. Electronic structure within the GGA approach. The collinear FM CCVO ground state within GGA is metal- lic, asrevealedinthe density ofstates(DOS) in Fig.2(a). In all results to follow, occupation of the majority Co states is subject to only one uncertainty: whether all majority 3dstates are fully occupied, or whether the dx2−y2orbitals, whose density lobes are directed toward the nearby O2−orbitals and thereby form the highest ly- ing crystal field orbital, are occupied (high spin [HS]) or unoccupied (low spin [LS]). The minority projected DOS (PDOS) shown in Fig. 2(d) allows identification of the crystal field splittings of Co, since all bands are narrow and peaks are evident. Relative to the GGA Fermi level, the crystal field picture of orbital energies is dx2−y22.25 eV dz20.75 eV dyz0.00 eV dxy-0.20 eV dxz-0.50 eV Itwaspreviouslysuggested1basedonthe bond valence method that CCVO would have formal charges Ca2+, Co2+d7, V4+d1, and O2−. Our results are consistent withthisassignment,whichalsoistheonlyreasonableas- signmentconsistent with integralvaluesofformalcharge. Within GGA, the spectral density of the majority dx2−y2 orbital (see Fig. 2(d)) is split above and below εF—i.e., an itinerant band. The unfilled majority states’ contri- bution to the spin moment is drastically reduced from the HS value of 3 µBto 1.77µB, but far too large for LS. The majorityV t2gbands extend from just below εFto 1eV.The Vspinof0.77 µBis consistentwith a d1config- uration; however,partialfillingoftheessentiallylocalized minority Co dxzstate suggests some charge is taken from V. The very narrowCo minority dyz,dxystates are filled. The localized dyzstate with veryhigh density ofstates at εF,N(εF), pins the Fermi level and implies, e.g., charge density wave or lattice instabilities. However, correlation effects and SOC have not been taken into account yet. Electronic structure within the GGA+U approach. Adding an onsite Coulomb repulsion on each of the Co and V ions results in a half-metallic electronic struc- ture. The minority gap is 1.25 eV, the change evident in Figs. 2(a) and (b), and the corresponding GGA+U 3-6-3036 Co dz2 Co dx2-y2 Co dxy Co dxz Co dyz -3036PDOS (states/spin/eV)Co dz2 Co dx2-y2 Co dxy Co dxz Co dyz -2 -1 0 1 2 3 4 Energy (eV)-30369 Co1 d-2 Co1 d-1 Co1 d0 Co1 d+1 Co1 d+2-40-20020 Total Co V O -2002040DOS (states/spin/eV)Total Co V O -2 -1 0 1 2 3 4 Energy (eV)-40-2002040 Total Co1 Co2 Co3 V O1 - O3 -0.25 00.2504(b) GGA+U (c) GGA+U+SOC(a) GGA (d) GGA (e) GGA+U (f) GGA+U+SOC↑ ↓ FIG. 2: (Color online) Spin-up and -down DOSs of CCVO resulti ng from FM alignment for [left panels] (a) GGA, (b) GGA+U, and (c) GGA+U+SOC calculations, and [right panels] (d)–(f) , the corresponding orbital- and spin-projected DOSs (in lo cal coordinates) of Co. The top left (right) legend box applies t o the two top left (right) graphs. The pseudogap mentioned in the text can be better seen in the blow-up of (c), inset in the bott om left panel. Ca does not contribute to the DOS within the featured range of energies. For the SOC-included calculati ons, the moment is along the ˆ zaxis and the projected DOS plot is for Co1. band structure displayed in Fig. 3. Calculated moments are tabulated in Table I. One effect of Uhas been to remove all Co states from near the Fermi level, leaving the clear Co2+formal charge with dx2−y2of both spins and minority dz2unoccupied. With four majority and three minority electrons on Co, Uhas driven Co through a spin-state transition, from roughly HS to clearly LS, with moment 1.0 µB, as is common for square-planar geometry. The Co ion is Mott insulating in character. Plotted in Fig. 2(d) and (e) are the PDOSs of Co in CCVO from GGA and GGA+U calculations, respec- tively. If the O 4plaquette around Co were perfectly reg- ular, thedxzanddyzstates would be degenerate. Ap- plyingUColowers the sharp dxzanddyzpeaks in the mi- nority channel, initially centered at εF, to 1.5 eV below εF. Thedz2orbital is affected most by UCo, the minority state being displaced higher in energy by almost 3 eV. In the majority channel, most all of the states are shifted up; the formerly divided dx2−y2states have coalesced to comprise the single hole in the majority channel. Vanadium, on the other hand, drives the half-metallic nature by declining to open a Mott gap, retaining a ma- jority band crossing εFwithN(εF) = 6.1 eV−1spin−1. FIG. 3: Majority (left) andminoritybandstructuresresult ing from a FM GGA+U calculation of CCVO. The lack of a Mott insulating gap is due to V bands, as discussed in the text. However, the symmetry used to this point retains the nearly cubic local symmetry of V and more specifically thee′ gdegeneracy; symmetry-breaking of some kind is required to break this symmetry and allow a Mott insu- 4TABLE I: Spin and orbital moments of the Co and V ions in CCVO, obtained from GGA, GGA+U and GGA+U+SOC calculations, as well as moments in C3CVO obtained from a GGA+U+SOC calculation. Moment are given in µB. µspin µorb CCVOGGACo 1.77 - V 0.79 - GGA+UCo1–Co3 1.00 - V 0.87 - GGA+U+SOCCo1 0.94 −0.75 Co2 0.93 −0.05 Co3 0.93 +0 .07 V 0.84 −0.36 C3CVO GGA+U+SOCCo 0.97 −0.08 V 0.75 −0.07 lating V configuration. However, SOC has not yet been taken into account and its effect is crucial. Electronic structure from GGA+U+SOC. Incorporating SOC lowers the symmetry, one outcome being that the Co ions become inequivalent. Unexpected behavior was encountered at the GGA+SOC level when directing the spin along the (say) ˆ zaxis: the three Co ions developed strongdifferences, violatingtheobservedcubicsymmetry (although the differences might be difficult to see in stan- dard x-ray diffraction). Due to the very narrow Co 3 d bands near or at the Fermi level, self-consistency became difficult and the behavior was suggestiveof multiple local minima with similar energies. Related computational (mis)behavior ariseseven when lowering the symmetry by adding SOC afterGGA+U, when (our subsequent results show) all Co states are in the process of being removed from the Fermi level. Large orbitals moments arise, which is surprising at first glance becaused+1,d−1ord+2,d−2occupations (relative to the spin direction) must become unbalanced while their real and imaginary parts are nondegenerate and thus are not so readily mixed. This choice of order in including Uand SOCresultsinantiferromagnetic(AFM) tendencies—the spin of one of the three Co atoms reverses direction, and orbital moments are vastly differing. To control the behavior of the Co moments, we de- vised the following procedure. Before adding Uor SOC, the symmetry was broken (for example, by displacing an atom by a physically negligible amount), which al- lowed the calculation to proceed from the GGA solution rather than to begin anew. In addition we enforced a total fixed spin moment (FSM) equal to that obtained from a calculation before the symmetry breaking; other- wise the artificial symmetry-breaking again makes self- consistency difficult. With the total moment fixed, the electronic and magnetic structures were relaxed. Then with the moment freed, Uwas added and taken to self- consistency, and finally SOC was incorporated. The or- der of these last two steps was found to be important. Carried out in this way, the unrealistically large differ-ences between the Co ions was lessened. All results in this paper ofGGA+U+SOC calculationsforCCVO were obtained from this “infinitesimal brokensymmetry + ini- tial FSM” protocol. We mention in passing that, since the experimentalevidence suggestedAFM order(at least not ferromagnetic order), we attempted some AFM cal- culations, which require further doubling of the unit cell. These calculations encountered additional difficulties, in- cluding resulting in considerably differing Co spins. We did not succeed in finding a procedure to avoid this in the AFM case, and we discuss in the final section why doing so might not be any more realistic than the FM ordered results that we present. Including SOC dramatically changes the V-derived majority DOS at and around the Fermi energy, as shown in Fig. 2(c). At the value of UVthat we have chosen (3.4 eV), the V manifold centered at εFopens, leaving a pseudogap. With our chosen values of U, this result leavesCCVOasahalf-semimetalwith a slightbandover- lap in the majority channel; a somewhat larger value of UVresults in a FM Mott insulator state. The minority spin spectrum, on the other hand, changes very little. The gap is a result of relativity – accounting for SOC – which breaks the degeneracy of the V d±1orbitals and creates a substantial orbital moment of −0.36µBthat cancels 40% of the spin moment. Despite the significant changes to the V ion when SOC is included, the formal valence remains unchanged. Though the changes to the Co1 spectrum in Fig. 2(f) seem less important, they are such to drive an orbital moment of 0.75 µBon the Co1 ion. The unoccupied minoritydz2orbital becomes the d0peak and contributes nothingtothe orbitalmoment. The unoccupied majority dx2−y2orbital becomes about 85% d+2and 15%d−2, thus providing the net orbital moment. This effect is enhanced (perhaps, one might say, enabled) by the non- square symmetry of the CoO 4unit. Beyond GGA+U+SOC. A fully relativistic treatment of the electronic structure reveals an orbital polariza- tion (OP) energy and associated potential analogous to that of spin polarization. The OP method developed by Brooks17and by Eriksson et al.18is an orbitally depen- dent correction to energy functionals such as GGA and LDA. Hund’s second rule is not taken into account in these functionals, leading to an underestimation of the orbital moment of many dmetals.25OP adds to the en- ergy functional a correlation energy functional modeled as EOP=−1 2/summationdisplay σBσL2 σ, which acts to energetically favor larger orbital moments, while the resulting potential makes mℓ-dependent shifts in bands. Here, B=F2−5F4is the Racah parameter in terms of the Slater integrals F2andF4,Lis the orbital moment in units of µB, andσdenotes spin. For Co, B= 0.079 eV; for V, 0.071 eV. The spin dependence ( <1%) 5TABLE II: Spin and orbital moments of Co1, Co2, Co3, and V in CCVO, and Co and V for each ˆ z, ˆy, and ˆxmagnetization direction in fictitious C3CVO, all from GGA+OP+SOC cal- culations. Units are in µBand magnetization directions are in parentheses next to the atom. µspin µorb CCVOCo1 (ˆz) 1.69 0.25 Co2 (ˆz) 1.83 0.87 Co3 (ˆz) 2.05 1.54 V (ˆz) 0.75 −0.02 C3CVOCo (ˆz) 1.52 0.27 Co (ˆy) 1.51 0.43 Co (ˆx) 1.77 1.11 V (ˆx,ˆy,ˆz)0.71–0.73 −0.01 of these paramters is included but is unimportant. The moments from the OP method (GGA+OP+SOC) are listed at the top of Table II. The largest Co orbital moment in the OP scheme is 1 .5µB, more than double the value obtained from the GGA+U+SOC calculation. However, this time it is Co3, whose O 4plaquette is per- pendicularto ˆ y, thatpossessesthelargemoment; thespin and orbital moments lie with the CoO 4plane rather than perpendicular to it. Co2 also has a large orbital moment, 0.9µB, again lying with the plane of the CoO 4unit. The three Co spin moments range from 1.7 to 2.0 µB; these more nearly HS values are (almost) twice the size of the LS moments from the GGA+U+SOC method. Thus OP enhances orbital moments but, unexpectedly, feeds back to enhance greatly the spin moments while complicating the atomic configurations of the Co ions. The three-dimensional isosurface of the GGA+U+SOC valence spin density of CCVO is provided in Fig. 4. The spin isosurfaces centered on each of the three cobalts take roughly a dz2-orbital shape, due to the Mott-Hubbard spitting of the Co dz2orbitals. They do not align collinearly but along each O 4plaque- tte’s normal axis. The three show certain differences; the “ring” around the Co1 atom, when compared to that of Co2 and Co3, is more square-like, some dx2−y2-like character in addition to the stronger dz2character. This difference in spin density is a consequence of SOC – without it, all three Co isosurfaces would be identical – and its magnetization direction with respect to the direction of the O 4plaquette. The spin isosurface around V, shown also in Fig. 4, is essentially identical to its total valence chargedensity isosurface, since the occupied V 3 dstates originate from the majority channel only. These V dstates are con- centrated between −0.45 eV and the Fermi energy, and feature lobes corresponding to the linear combination in Eq. 1 and maintaining ¯3 symmetry along the local (111) symmetry axis. Discussion of results for CCVO. Inclusion of SOC, and moresowithGGA+OP+SOC,indicatesthelikelihoodof very large Co orbital moments, as suggested by Ovsyan- FIG. 4: (Color online) Spin-density isosurface of the valen ce state of CCVO obtained from a GGA+U+SOC calculation. The V isosurface is the central one. Differences of the Co spin densities are discussed in the text. nikovet. alfrom analysis of the susceptibility.1Further- more, the extreme narrowness of the Co bands suggest the physics is local: the magnetic anisotropy is deter- mined by the CoO 4configuration. The perpendicular axis (the local ˆ zaxis) seems the natural direction for a large orbital moment, and thus the spin as well. While GGA+U+SOCgavethatresult, includingOPshowsthat largeorbitalmomentsmaywellliewithin the planeofthe CoO4unit. In any case, if the physics is local and each unit behaves the same, the Co moments will be non- collinear. Similarly, the natural axis for the V orbital moment is along its (111) symmetry direction, a differ- ent direction for each V ion. The strong indications are that the magnetic order is non-collinear, possibly reinforced by competing exchange interaction and suffering canting tendencies due to the Dzyaloshinskii-Moriya interaction between Co-V and V- V pairs. This combination of strong correlation, impor- tant spin-orbit coupling in which the orbital moments feeds back on the spin density, and intricate geometrical arrangement presents a daunting challenge for an ab ini- tio calculation. With substantial orbital moments, the magnetic coupling between total moments becomes ten- sorial rather than scalar. Treating and understanding the coupling would be a challenge even if the exchange tensors were known, and it has recently been shown that obtaining this tensor coupling when SOC coupling has large effects requires special technology.26 CCVO’s effective paramagnetic moment µeffderived from magnetic susceptibility measurements1is 9.3µB. Assuming the S=1 2moment on V4+, the effective net moment of Co is 4.4 µB, implying a considerable orbital moment of Co up to ∼2.5µB, depending on the Co spin moment. Only recently have unusually large orbital mo- 6ments for Co in perovskites been reported, yet not ex- ceeding 1.8 µB. The possibility that V alsohas amoment does not help to account for the observed value, since by Hund’s rules it would cancel the spin moment, as shown in Table I. We do observe a large orbital moment of 0 .7µBon Co1 in the GGA+U+SOC result, but anti-parallel to the spin. The other two cobalts have small, typically sized orbital moments. For the (001) spin direction, we have presumed, Co1 is the distinctive Co, since its O 4 plaquette is the one that is perpendicular to the mag- netization quantization axis, and its large moment lies along that axis. Orbital polarization, analogous to spin polarization (Hund’s rule), has been found to improve calculated orbital moments in several magnetic materi- als, and it produces large orbital moments for CCVO. While 0.7µBis an impressive value for a 3 dorbital mo- ment, OP more than doubles this to 1.5 µB. The unique Co in the GGA+OP+SOC scheme is Co3 however, not Co1, such that the large orbital moment lies parallel to the O4plaquette. B. C3CVO Continuing with the theme that the CoO 4behavior in- volved primarily local physics, we chose to adopt a sim- pler model that contains only one Co ion. Two of the Co2+ions are replaced with Ca2+ions, greatly simpli- fying both the conceptual issues and the self-consistency process. We proceed to investigate some of the behavior of this model compound Ca 3CoV4O12(C3CVO). GGA+U+SOC. We have set the magnetization direction for C3CVO to be perpendicular to the O 4plane on which the remaining Co lies; this is equivalent to keeping Co1, and replacing Co2 and Co3 with Ca in Fig. 1(a). Com- paring the PDOS in Fig. 5 for C3CVO to the right-hand panels of Fig. 2 for CCVO, it is apparent that the elec- tronic environment of Co in C3CVO is similar to that of Co1 in CCVO when SOC is included. This confirms that C3CVO represents reasonably the local planar envi- ronmentofCo1inCCVOwithoutthecomplicationsfrom the otherCo ions. The generalcharacteristicsthat GGA, U, and SOC produced in C3CVO are very much like those seen in CCVO. With the GGA+U+SOC method, C3CVO, like CCVO, remains metallic in the majority channel, and semimetallic in the minority. A pseudogap, near Mott insulating, feature made up of V states at the Fermi energy (not shown), similar to that in CCVO, is also present in C3CVO. Unexpectedly, the orbital moments of V and Co in C3CVO are much smaller that on the V and analogous Co1 in CCVO, as displayed at the bottom of Table I. In C3CVO, the orbital moment of V of 0.07 µBis only one fifth of its value in CCVO. We note we have not oriented the spin along a (111) direction, where the V orbital moment may be larger. The difference in theCo orbital moment is even larger, with C3CCVO’s Co moment (0.08 µB) having only 10% of the strength that it has in CCVO (0.75 µB), due perhaps to the absence of the Co-V network of exchange. -4-202468 Co dz2 Co dx2-y2 Co dxy Co dxz Co dyz -4-202468DOS (states/spin/eV)Co dz2 Co dx2-y2 Co dxy Co dxz Co dyz -2 -1 0 1 2 3 4 Energy (eV)-6-4-202468 Co d-2 Co d-1 Co d0 Co d+1 Co d+2(a) GGA (b) GGA+U (c) GGA+U+SOC FIG. 5: (Color online) Spin-upand -down, orbitally project ed PDOSsofCoinCCVOresultingfrom FM alignmentusing(a) GGA,(b)GGA+U,and(c)GGA+U+SOC(001)calculations. The top legend box applies to the top two graphs. GGA+OP+SOC. Referring back to Fig. 1(a), when CCVO’s quantization axis is the ˆ zaxis, Co1 is analo- gous to the sole Co in C3CVO when itsquantization axis is also the ˆ zaxis. Similarly, Co2 (Co3) is analogous to the only Co in C3CVO when quantization is in the ˆ y (ˆx) direction. The analogies are reflected by the order- ing of the entries in Table II. The trends in CCVO and C3CVO are common, both in spin and orbital moments, which increase from Co1 to Co2 to Co3, and also as the magnetization axis is rotated in C3CVO from ˆ zto ˆyto ˆx. In the GGA+U+SOC scheme, the difference between the orbital moment of Co1 in CCVO and that of Co in C3CVO was ten-fold. OP howeverlessens this difference: the largest orbital moment of Co in C3CVO is 1 .1µB, which is 75% of the maximum orbital moment of 1 .5µB in full structure of CCVO. This difference arises from the differences in Co 3 doccupation. 7-1001020 Co1 Co2 Co3 V -1001020DOS (states/spin/eV) -1001020 -2 -1 0 1 2 3 4 Energy (eV)-1001020(a) CCVO OP+SOC ( M // z) (b) C3CVO OP+SOC ( M // z) (c) C3CVO OP+SOC ( M // y) (d) C3CVO OP+SOC ( M // x) FIG. 6: (Color online) GGA+OP+SOC DOSs of cobalt(s) and V in (a) CCVO and (b)-(d) C3CVO. Mis parallel to the (a)-(b) ˆz, (c) ˆy, and (d) ˆ xaxes. Co in C3CVO has the largest orbital moment when M/bardblˆx, with an analogous result in CCVO—i.e., Co3 has the largestµorb. The uniqueness of Co3 in the OP+SOC method is curious, since it is the Co1 plaquette that is perpendicular to the quantization axis. The strong cou- pling of the size of µorbto the local spin direction can be understood by studying the Co DOSs presented in Fig. 6. In Fig. 6(a), the three Co DOSs are compared in one plot. Panels (b), (c), and (d) below it are the Co states of C3CVO with quantization in the various orien- tations, and it is clear that the DOSs are similar when M/bardblˆzandM/bardblˆywhile different when M/bardblˆx. The spin- up occupied Co states are lower in energy when M/bardblˆx, and the down-spin occupied peak, instead of butting up against the edge of the Fermi level, has states that pass aboveεF. Theseadditionalunoccupiedstatesprovidethe distinction between M/bardblˆxandMalong the other two axialdirections. Theorbitalpotential evidentlyproduces important shifts in positions of Co 3 dbands. In Fig. 6 providing the PDOSs of CCVO and C2CVO for the OP+SOC results, one can observe that the Co1 electronic states in CCVO in panel (a) are very similar to those of C3CVO in (b) and (c). Meanwhile, Co3 ofCCVO, like Co ofC3CVO in Fig. 6(d) ( M/bardblˆx), has more stronglyboundmajorityoccupiedstates, andunoccupied states“leak”outoftheminoritypeakcenteredjust below εF. Co2 in CCVO and Co in C3CVO when M/bardblˆydo not share as many similarities as the other two analogous pairs,andthatisreflectedinthedifferingspinandorbital values in Table II. TABLE III: EOPvalues of spin-down cobalts in CCVO and C3CVO, from OP+SOC calculations. Direction of Mis in parentheses next to the atom. Eorb(eV) CCVOCo1 (ˆz) 0.01 Co2 (ˆz) 0.06 Co3 (ˆz) 0.19 C3CVOCo (ˆz) −0.01 Co (ˆy) −0.02 Co (ˆx) 0.10 IV. DISCUSSION AND SUMMARY Our several investigations into the electronic and mag- netic structure of CCVO reveal a great deal of complex- ity, even considering the three Co and four V ions in the primitive cell, each with its own natural axis of exact (V) or approximate (Co) symmetry. This complexity is the result of (1) two types of open shell cations, (2) strong correlation in narrow bands, (3) important and in some casesdominatingspin-orbitcoupling, and(4)anintricate three dimensional network. To account for the insulat- ing nature, both Co and V must display Mott insulating character. While the crystal field splittings on the Co site – a span of 2.75 eV – are typical for a 3 dion in an oxide, the subsplittings become very important given the very narrow Co 3 dbands. Though these bands are nar- row and separate above and below the Fermi level, the Co spin is not very representative of either the HS nor LS state, a reflection of some charge transfer character of this cobaltovanadate. The O 4plaquette has an im- portant non-square components of the potential, so spin directed in the ˆ xand ˆydirections produce very different orbital moments. We have not given to the V ion the attention that it maydeserve. EachoftheVionshasitsownthreefoldaxis (symmetry related, of course), which provides a natural axis for the orbital moment, and hence the spin moment. Thed1configuration in the e′ gdoublet requires SOC, orbital ordering, or structural distortion to provide the splitting necessary for a Mott gap. In addition to the widely applied GGA+U+SOC method, we have explored the application of the little used orbital polarization potential via GGA+OP+SOC. This OP is an analog of the spin polarization poten- tial within GGA but is roughly an order of magnitude smaller, but it arises in a fully relativistic theory of elec- 8tronic structure and should be applied more widely and studied. Both methods giveatendency towardverylarge orbital moments, but the behavior is different in the two approaches. Whichever approach is used, only one of the three Co ions acquires a large µorbwhen the spin moments (hence orbital moments) are restricted to be collinear. This observation, together with the large orbital moment in- ferred from experiment and the fact that the mechanisms of magnetization appear to be local, suggest that the Co moments will be non-collinear, each directed along its own largeµorbaxis. Analogously, the V moments are likely to be non-collinear as well, and in different direc- tions [(111) axes] than the Co moments. CaCo3V4O12thus presents a very challenging elec- tronic and magnetic structure problem. After starting from the GGA starting point, both the Hubbard Uand spin-orbit coupling—together with the required great re- duction in symmetry—are required to produce the ob- served Mott insulating state. As we have mentioned, acorrect calculation also includes the orbital polarization term, one whose evident effect (from its form of the en- ergy) is to enhance the orbital moment. However, the re- sulting orbital-dependent potential shifts bands and the final outcome can be more complicated than a simple enhancement. Finally, all moments (spin plus orbital) should be treated non-collinearly. A calculation of this type – non-collinear GGA+U+OP+SOC – might be the first of its kind. V. ACKNOWLEDGMENTS We acknowledge many cogent observations from A. S. Botana throughout the course of this study, and thank V. Pardo for many discussions on the impact of spin- orbit coupling and the origin of orbital moments. This research was supported by U.S. Department of Energy grant DE-FG02-04ER46111. 1S. V. Ovsyannikov, Y. G. Zainulin, N. I. Kadyrova, A. P. Tyutyunnik,A.S.Semenova, D.Kasinathan, A.A.Tsirlin, N. Miyajima, and A. E. Karkin, Inorg. Chem. 52, 11703 (2013). 2Z. Zeng, M. Greenblatt, M. A.Subramanian, andM. Croft, Phys. Rev. Lett. 82, 3164 (1999). 3K. Takata, I. Yamada, M. Azuma, M. Takano, and Y. Shi- makawa, Phys. Rev. B 76, 024429 (2007). 4J. A. Alonso, J. Snchez-Bentez, A. De Andrs, M. J. Martnez-Lope, M. T. Casais, and J. L. Martnez, Appl. Phys. Lett. 83, 2623 (2003). 5M. Subramanian, D. Li, N. Duan, B. Reisner, and A. Sleight, J. Solid State Chem. 151, 323 (2000). 6Y. Lin, Y. B. Chen, T. Garret, S. W. Liu, C. L. Chen, L. Chen, R. P. Bontchev, A. Jacobson, J. C. Jiang, E. I. Meletis, et al., Appl. Phys. Lett. 81, 631 (2002). 7Y. Shimakawa, H. Shiraki, and T. Saito, J. Phys. Soc. Jpn. 77, 113702 (2008). 8K. Momma and F. Izumi, J. Appl. Crystallogr. 44, 1272 (2011). 9S. J. Youn and B. I. Min, Phys. Rev. B 56, 12046 (1997). 10a. K.-I. Kob, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature 395, 677 (1998). 11J.-H. Park, S.-W. Cheong, and C. T. Chen, Phys. Rev. B 55, 11072 (1997). 12V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B44, 943 (1991). 13E. R. Ylvisaker, K. Koepernik, and W. E. Pickett, Phys. Rev. B79, 035103 (2009).14P. Blaha, K. Schwarz, P. Sorantin, and S. Trickey, Comp. Phys. Comm. 59, 399 (1990). 15K. Schwarz and P. Blaha, Comput. Mater. Sci. 28, 259 (2003). 16J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.77, 3865 (1996). 17M. Brooks, Physica B+C 130, 6 (1985). 18O. Eriksson, B. Johansson, R. C. Albers, A. M. Boring, and M. S. S. Brooks, Phys. Rev. B 42, 2707 (1990). 19Y. Wu, J. St¨ ohr, B. D. Hermsmeier, M. G. Samant, and D. Weller, Phys. Rev. Lett. 69, 2307 (1992). 20J.Trygg, B.Johansson, O.Eriksson, andJ.M. Wills, Phys. Rev. Lett. 75, 2871 (1995). 21M. Sargolzaei, M. Richter, K. Koepernik, I. Opahle, H. Es- chrig, and I. Chaplygin, Phys. Rev. B 74, 224410 (2006). 22S. Wurmehl, G. H. Fecher, K. Kroth, F. Kronast, H. A. Drr, Y. Takeda, Y. Saitoh, K. Kobayashi, H.-J. Lin, G. Schnhense, et al., J. Phys. D: Appl. Phys. 39, 803 (2006). 23A. Sa´ ul, D. Vodenicarevic, and G. Radtke, Phys. Rev. B 87, 024403 (2013). 24N. Hollmann, S. Agrestini, Z. Hu, Z. He, M. Schmidt, C.- Y. Kuo, M. Rotter, A. A. Nugroho, V. Sessi, A. Tanaka, et al., Phys. Rev. B 89, 201101 (2014). 25G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, Phys. Rev. B 44, 12054 (1991). 26S.-T. Pi, R. Nanguneri, and S. Savrasov, Phys. Rev. Lett. 112, 077203 (2014). 9

1906.11939v2.Extrinsic_Spin_Orbit_Coupling_and_Spin_Relaxation_in_Phosphorene.pdf

Extrinsic Spin-Orbit Coupling and Spin Relaxation in Phosphorene S. M. Farzanehand Shaloo Rakheja Department of Electrical and Computer Engineering, New York University, Brooklyn, NY 11201 An eective Hamiltonian is derived to describe the conduction band of monolayer black phos- phorus (phosphorene) in the presence of spin-orbit coupling and external electric eld. Envelope function approximation along with symmetry arguments and Lowdin partitioning are utilized to de- rive extrinsic spin-orbit coupling. The resulting spin splitting appears in fourth order perturbation terms and is shown to be linear in both the magnitude of the external electric eld and the strength of the atomic spin-orbit coupling, similar to the Bychkov-Rashba expression but with an in-plane anisotropy. The anisotropy depends on the coupling between conduction band and other bands both close and distant in energy. The spin relaxation of conduction electrons is then calculated within the Dyakonov-Perel mechanism where momentum scattering randomizes the polarization of a spin ensemble. We show how the anisotropic Fermi contour and the anisotropic extrinsic spin splitting contribute to the anisotropy of spin-relaxation time. Scattering centers in the substrate are considered to be charged impurities with screened Coulomb potential. I. INTRODUCTION Extrinsic spin-orbit coupling induced by an external electric eld in two-dimensional electron systems lifts the spin degeneracy while it preserves time-reversal symme- try. The induced spin splitting, which is proportional to the magnitude of the eld and the crystal wavevec- tor, enables control of spin through movement of charge and vice versa. This eect, which has enabled several phenomena and ideas in spintronics and beyond [1], was originally derived by Ohkawa and Uemura [2] for an in- version layer of zinc-blende crystals. Later, Vasko [3], and Bychkov and Rashba [4], generalized the spin splitting for a two-dimensional electron system with an isotropic in-plane eective mass. Unlike Ohkawa and Uemura's derivation based on the Kane's model of zinc-blende crys- tals [5], the Vasko and Bychkov-Rashba spin splittings are phenomenological. In this paper, we utilize envelope function approximation and symmetry arguments to de- rive the spin splitting for monolayer black phosphorus, which demonstrates a highly anisotropic in-plane eec- tive mass and, therefore, makes the phenomenological description inapplicable. Black phosphorus, the most stable allotrope of phos- phorus, is a layered material similar to graphite where van der Waals interaction binds individual layers to- gether. Each monolayer, dubbed phosphorene, is a two- dimensional crystal with a puckered honeycomb struc- ture which shares the symmetry properties of its bulk form denoted by the orthorhombic space group Cmca [6, 7]. A century after black phosphorus was discov- ered [8], phosphorene and its multilayer thin lms were isolated [9{14] using mechanical exfoliation, which had been utilized earlier to isolate graphene [15]. Similar to graphene, phosphorene consists of light atoms producing a spin-orbit coupling of 1 meV, which is weaker than that of conventional zinc-blende crystals. Hence, both farzaneh@nyu.edu;graphene and phosphorene are expected to have a long spin-relaxation time 1 ns [16, 17], which could allow spin-polarized currents to ow macroscopic distances in these materials. Unlike graphene, which is gapless, phos- phorene is a semiconductor with a direct band gap of 1.73 eV [18], which enables a wide control over its car- rier density. Phosphorene also exhibits a large anisotropy in its band structure: the ratio of the in-plane eective mass of carriers along the armchair and zigzag directions is0:1 [11]. This band structure anisotropy is expected to result in anisotropic extrinsic spin-orbit coupling, also conrmed through rst-principles calculations [19, 20]. It is shown [2] that in conventional semiconductors with a zinc-blende crystal the three upper valence bands, which are made of only porbitals at the band edge, induce the spin splitting in the conduction band made of sor- bital. However, in the case of phosphorene, it is not clear which bands couple to the conduction band to produce spin splitting. While previous works [19, 20] on extrin- sic spin-orbit coupling in phosphorene are solely based on rst-principles calculations, this work focuses on analytic derivation of the spin-orbit splitting and demonstration of the bands involved from a group theoretical aspect. Utilizing the symmetry analysis of phosphorene, devel- oped in Ref. 21, we derive an eective Hamiltonian for the extrinsic spin-orbit coupling in the conduction band by Lowdin partitioning and then quantify the contribu- tions of dierent bands in the coecients of spin-orbit coupling in zigzag and armchair directions. Using the eective Hamiltonian we then study how the anisotropy impacts spin lifetime, a key measure of spin transport properties. Spin-relaxation time charac- terizes the decay of the polarization of a non-equilibrium spin ensemble due to random uctuations of a mag- netic eld. Extrinsic spin-orbit coupling acts as an ef- fective momentum-dependent magnetic eld and causes the spins of conduction electrons undergoing momen- tum scattering to relax. This mechanism, introduced by Dyakonov and Perel [22], has been used to develop closed- form solutions of the spin-relaxation time in isotropic semiconductors [23, 24]. However, when the eectivearXiv:1906.11939v2 [cond-mat.mtrl-sci] 8 Nov 20192 mass of carriers is anisotropic, momentum scattering be- comes anisotropic as well and must be accounted for nu- merically to determine spin-relaxation time accurately. The anisotropic spin relaxation time has been calculated before [20] considering a constant momentum scatter- ing. Here, we assume that the momentum scattering is anisotropic and, therefore, angle dependent. Generaliz- ing the Dyakonov-Perel mechanism in the case of phos- phorene, we account for the anisotropy of momentum scattering and calculate the spin-relaxation time for spin ensembles with dierent initial polarization. Our calcu- lations assume that the temperature is much lower than the Fermi energy but much greater than the spin split- ting i.e.EFT
ESO. Therefore, only the electrons at the Fermi energy are taken into account and the spin- orbit coupling is treated as a perturbation. II. EXTRINSIC SPIN-ORBIT COUPLING Considering the two-dimensional crystal of phospho- rene with a periodic lattice potential V0(r) lying on the xyplane with the armchair edge along the x-direction and the zigzag edge along the y-direction, the Hamilto- nian in the presence of Pauli spin-orbit coupling, HSO, and a perpendicular electric eld V(z) =eEzis H=p2 2m0+V0(r) |{z} H0+~2 4m2 0c2p
rV0(r) |{z} HSO+V(z)
(1) Here,pis the momentum operator, is the vector of Pauli matrices, and m0is the mass of free electron . We note that the contribution of V(z) inHSOis neglected. Using envelope function approximation [25] and Bloch's theorem in the xysubspace, we can describe the solutions to the Schrodinger's equation as n(r) =eikk
rkX l;fnl(z)ul0(r)ji; (2) wherekk= (kx;ky;0) andrk= (x;y;0) are the in- plane wavevector and position respectively, fnl(z) are the envelope functions and ul0(r)jiare the lattice- periodic Bloch functions at the band edge (i.e. point, kk= 0) which provide a complete and orthonormal ba- sis. Plugging Eq. 1 and Eq. 2 into the Schrodinger's equation, multiplying with h0ju l000(r), and integrat- ing over the primitive unit cell of the lattice, we arrive at the eigenvalue equation for the envelope functions, Hf(z) =Ef(z), X l; El(0) +~2(k2 kd2 dz2) 2m0+V(z)! ll00+
l00l +~ m0(kkid dz^z)
Pl00l fnl(z) =Enfnl00(z); (3)whereEl(0) =hljH0jliare the energies at the band edge andPl00l=hl00jp+~ 4m0c2rV0jlihl0jpjli0 are approximated by the matrix elements of momentum operator which couple dierent bands at the edge. This approximation is valid for light atoms such as phospho- rus where the contribution of spin-orbit coupling is or- ders of magnitude smaller than that of momentum op- erator. The matrix elements of HSOare denoted by
l00l=hl00jHSOjli. We rewrite the Pauli spin-orbit couplingHSO= (~=2)L
, whereLis the angular momentum operator, and the parameter represents the strength of atomic spin-orbit coupling and includes the average radial contribution of rV0(r). Therefore, we rewrite
l00l= (~=2)hl00jL
jli. The innite-dimensional Hilbert space in Eq. 3 can be reduced to a nite-dimensional one by considering only the bands that are in the vicinity of the Fermi energy. These bands are usually made up of orbitals of the va- lence electrons namely sandpatomic orbitals in phos- phorus. Using atomic orbitals as a basis, Li et al. [21] studied the symmetry properties of phosphorene and the orbital composition of the bands at the point. Follow- ing their study, we derive an eective Hamiltonian for the extrinsic spin-orbit coupling in the conduction band utilizing the theory of invariants and Lowdin partition- ing. There are eight irreducible representations (IRs) in the space group of phosphorene, Cmca , denoted by  i at the point of the Brillouin zone. Each band at the point can be labeled by one of the IRs of the space group. Table I in the appendix lists the characters of these IRs. The invariants corresponding to each IR is also listed in this table. According to the theory of in- variants, the energy of each band should be an invariant of + 1. Therefore, various products of kx,ky,x, and ycould appear in the diagonalized eective Hamilto- nian as long as the direct product of their corresponding IRs results in + 1. For instance, from Table I we can see that terms such as kzkxyandkzkyxwill be present in the eective Hamiltonian but terms such as kzkxxand kzkyywill not. Since the external electric eld V(z) breaks the inversion symmetry in the z-direction, it does not commute with kzand, therefore, after replacing kz withid=dz , their commutator [ kz;V(z)] =idV(z)=dz leads to the extrinsic spin-orbit coupling. To derive the eective Hamiltonian for the conduction bandHcc, we rst represent the Hamiltonian in Eq. 3 asH=H0+H0whereH0contains band-edge and free- electron energies and H0contains the electric eld and the o-diagonal parts of Hnamely k.p and spin-orbit terms. By applying Lowdin partitioning up to the fourth order perturbation, the coupling of conduction band with other bands is incorporated in the eective Hamiltonian. Thez-dependence is averaged out afterwards to obtain the nal two-dimensional eective Hamiltonian. The de- tails of Lowdin partitioning and the derivation of the eective Hamiltonian is provided in Appendix A. The3 resulting eective Hamiltonian is Hcc=~2k2 x 2mx+~2k2 y 2my+xkxy+ykyx; (4) wheremxandmyare the in-plane eective masses writ- ten as 1 mx(y)=1 m0 1 +2 m0X ljPx(y);clj2 EcEl! ; (5) where the sum is over l= + 2andl= + 3forxand ydirections, respectively. The coecients of extrinsic spin-orbit coupling term, xandy, are of the following form x(y)=(~=2)dV(z) dz~2 m2 0 X l;l0Im PzLy(x)Px(y) (EcE0 l)(EcEl) (EcEl)2(EcE0 l)2; (6) wherePandLare matrix elements of momentum and angular momentum operators that couple c,l, andl0 bands according to the invariants and their correspond- ing IRs. For instance, for l= + 1andl0= + 2(c= 4 denotes the conduction band), the matrix elements ap- pearing in xarePz;cl=hcjpzjli,Ly;ll0=hljLyjl0i, andPx;l0c=hl0jpxjci. Equation A6 contains all possible terms that produce extrinsic spin-orbit coupling which is linear in spin-orbit strength , electric eld dV(z)=dz, and in-plane crystal momentum kx(y). First principle calculations within density functional theory (DFT) using projected augmented plane wave method implemented in Quantum ESPRESSO package [26] were performed to verify the symmetry of the bands and also calculate the k.p parameters namely dierent P andLmatrix elements appearing in Eq. A6. Details of the DFT setup are provided in Table II. Parameters of the crystal structure of phosphorene are listed in Table III. Figure 1 illustrates the band structure of phospho- rene for the lowest 24 bands for a path along the high symmetry points. The bands are labeled with their cor- responding IRs which are consistent with previous cal- culations [21, 27]. Results show that mx= 0:16m0and my= 1:13m0which is similar to what has been reported before [19, 28]. Figure 2a plots xandyversus the ex- ternal electric eld. Quadratic terms with respect to the electric eld appear only at high magnitudes, i.e. E>0:4 V/A(not shown in the gure). The expectation value of spin, over the Fermi contour, is illustrated in Fig. 2b. As seen from the gure, the spin is not tangential to the Fermi contour in contrast to the isotropic case. To verify Eq. 6 produces the same anisotropy as that of Fig. 2a, the various PandLmatrix elements for dierent bands were calculated. The contributions of dierent bands in the spin-orbit coecients are listed in Table IV. The band number corresponding to each IR is mentioned as well to Γ SY ΓX−16−12−8−4048Energy (eV) Γ+ 1Γ− 4Γ+ 2Γ+ 1Γ− 3,Γ− 4Γ− 1Γ+ 4,Γ+ 1Γ+ 2Γ− 4Γ+ 2Γ− 3Γ− 2Γ+ 1,Γ− 4Γ+ 1,Γ+ 3Γ− 4,Γ− 3Γ+ 1Γ− 3,Γ− 1Γ− 4FIG. 1. Fully relativistic band structure of phosphorene. The lowest 24 bands are labeled at the point with respect to the irreducible representation of the space group of phosphorene. 0.00 0.01 0.02 0.03 0.04 E(V/˚A)−0.8−0.6−0.4−0.20.00.20.4λ(meV·˚A) λx λy −0.05 0.00 0.05 kx(2π a)−0.10−0.050.000.050.10ky(2π b) FIG. 2. (a) Coecients of extrinsic spin splitting versus ex- ternal electric eld. For an electric eld of E= 0:1 V/ A, we obtainx=1:79 meV
Aandy= 1:03 meV
A. (b) Expec- tation value of spin over the Fermi contour ( Ef= 0:1 eV) at E= 0:36 V/ A. uniquely specify the rst 24 diagonalized bands ordered from the lowest energy to the highest. For instance, the conduction band is labeled as 4and is the 11thband from the bottom in Fig. 1. As seen from the table, only few of the bands contribute signicantly to the spin-orbit coecients. Also contributions from dierent bands pos- sess dierent signs which cancel each other out. The anisotropy of spin-orbit coupling is determined by the x=yratio which requires the contributions from dif- ferent bands to be summed up. As seen from Table IV4 this ratio is x=y1:4 which is close to the full rst principle result of x=y1:7 plotted in Fig. 2. The small discrepancy can be resolved as one adds other con- tributions from weakly coupled bands to the sum in Eq. 6. III. SPIN RELAXATION Utilizing the eective Hamiltonian in Eq. 4, we study the spin relaxation of conduction electrons. Decompos- ing the Hamiltonian Hcc=H+H0, we rewrite the ex- trinsic spin-orbit coupling term as H0= k
where k=yky^x+xkx^yis an eective magnetic eld which isk-dependent. This eective magnetic eld causes the electrons with dierent momenta to precess around dif- ferent axes. Therefore, scattering between dierent mo- menta randomizes the precession of a polarized spin en- semble and consequently leads to spin relaxation. This is the aforementioned Dyakonov-Perel mechanism. To calculate the spin-relaxation time, we follow a sim- ilar procedure as in Refs. 23 and 29, but we specically analyze an anisotropic Fermi contour with an anisotropic extrinsic spin-orbit coupling. Considering a polarized spin ensemble which is spatially-homogeneous and is de- scribed by a k-dependent density matrix k, the time evolution is given as the following kinetic equation [24] @k @t=1 i~[k; k
]X k06=kWkk0(kk0);(7) where we used [ k;H] = 0. Here Wkk0is the probabil- ity density of transition between kandk0states. The rst term on the right-hand side represents spin preces- sion about k, and the second term represents momen- tum scattering between incoming wavevector kand out- going wavevector k0. We assume that the density ma- trix can be decomposed as k=+0 k, whereis the average of density matrix over the Fermi contour, i.e. =`1R d`k, where`is the perimeter of the Fermi contour and d`=dj@k/@jis the dierential arc length. We assume that 0 kis a small perturbation with zero av- erage, i.e.0 k= 0. Taking the average of Eq. 7 over the Fermi contour, we obtain @ @t=1 i~[0 k; k
]; (8) where we used the fact that kis zero. The reason is that for each point kon the Fermi contour, kis also on the Fermi contour. Since kis linear inkand therefore an odd function of k, i.e. k= k, it averages to zero over the Fermi contour. Applying the decomposition to Eq. 7 and dropping the terms containing product of k and0 k, we can nd the quasistatic value of 0 k, by setting @0 k/@tto zero, assuming that momentum relaxation is much faster than spin relaxation. Therefore, 1 i~[; k
] =X k06=kWkk0(0 k0 k0)
(9)Equations 8 and 9 are coupled and must be solved self- consistently. To do so, rst we assume that the average spin polarization is in ^sdirection. Therefore, we can write= (1+ ^s
)=2. It can be shown that [ ; k
] = i(^s k)
. Using Eq. 9, we can solve for 0 kiteratively using the following equation: 0 k=1 ~(^s k)
+P k06=kWkk00 k0P k06=kWkk0
(10) Plugging0 kinto Eq. 8, we calculate the rate of decay @/@tor correspondingly d ^s/dt=^s=swhich results in the spin-relaxation time s. The collision sum in the continuum limit becomes an integral, i.e.P k06=kWkk0!AR d2k0(2)2Wkk0, whereAis the area of the system. Using Fermi's golden rule, the probability density of transition is given as Wkk0=2 ~NjUkk0j2(E(k)E(k0)), whereNis the num- ber of scatterers and Ukk0is the matrix element of the scattering potential. For long-range scattering potential varying slowly compared to the periodic lattice poten- tial,Ukk0=U(kk0)=A=U(q)=A, whereU(q) is the Fourier transform of U(r). In two-dimensional electron systems, the eective Coulomb potential in the Fourier domain is [30] U(q) =2e2 (q+qs)eqd; (11) whereis the average relative permittivity, dis the depth of the scattering center in the substrate, and qs2pmxmye2=~2is the Thomas-Fermi screening constant. The delta function in Wkk0reduces the k-space integral to an integral over the Fermi contour. Therefore, X k06=kWkk0!n 2~Z d`0jU(q)j2 jrE(k0)j; (12) wheren=N=A is the density of scatterers. Thek-dependent momentum scattering time, k= 1=P k0Wkk0, is depicted in Fig. 3a as a function of the polar angle for a typical value of charged impurity den- sity [31, 32], i.e. n= 1012cm-2. We assume that the monolayer is deposited on an SiO 2substrate [9, 33] with relative permittivity of r= 3:9. As seen from the g- ure, the momentum scattering time shows a signicant anisotropy which consequently aects the spin relaxation. Figure 3b depicts the energy dependence of both aver- age momentum scattering time, k, and spin-relaxation time for two ensembles initially polarized in the x- direction,s;xx, and the y-directions, s;yy. We as- sume that the external electric eld is xed at E= 0:1 V/A and the spin-orbit coecients are x=1:79 meV
A andy= 1:03 meV
A (from Fig. 2). Spin life- time of the x-polarized ensemble is about 2-3 times longer than that of the y-polarized ensemble. This anisotropy increases with an increase in the Fermi energy. The spin- relaxation time decreases with Fermi energy, while the5 0π/2π3π/22π θ200220240260280τk(fs) (a) 0.0 0.2 0.4 E(eV)10−410−310−210−1100τ(ns)τs,xx τs,yy τk (b) FIG. 3. (a) k-dependent momentum scattering time for n= 1012cm2,= 2:45, andE= 0:2 eV. (b) Spin-relaxation time along with the average momentum scattering time versus Fermi energy. The external electric eld is assumed to be xed atE= 0:1 V/A and the spin-orbit coecients are x=1:79 meV
A andy= 1:03 meV
A. average momentum scattering time increases with Fermi energy. This opposite energy dependence is a signature of the Dyakonov-Perel mechanism. We note that our as- sumptions in deriving Eqs. 8 and 9 are valid as long as the average momentum scattering occurs on a faster timescale compared to spin relaxation, i.e. for E < 0:4 eV. We also note that 1 =s;xy= 1=s;yx= 0. The spin relaxation for an ensemble polarized along the ^zaxis is always faster than in-plane directions (not shown in the gure). Replacing ^swith ^zin Eq. 10, we can see that 0 kobtains both xandycomponents. Therefore, the corresponding spin relaxation rate is the sum of relax- ation rates along the in-plane directions, i.e. 1 =s;zz= 1=s;xx+ 1=s;yy. We note that as the energy increases, the ratio of in-plane spin-relaxation times, s;yy=s;xx, increases as well. The reason is that at higher energies screening becomes less eective as we can see from the screened Coulomb potential in Eq. 11 which leads to a greater anisotropy in k. Finally, for an isotropic two- dimensional system, i.e. mx=myandx=y, we obtains;xx=s;yy= 2s;zzwhich has been reported previously in the literature [23]. IV. SUMMARY Using envelope function approximation along with symmetry arguments and Lowdin partitioning an eec- tive Hamiltonian was derived to describe the extrinsic spin-orbit coupling for the conduction electrons in phos- phorene. Based on the theory of invariants, we deter- mined the bands that are involved in generating extrin- sic spin splitting. In contrast to the isotropic Bychkov- Rashba and Vasko spin splittings, phosphorene shows an anisotropic spin splitting which is characterized by two coecients. The coecients are determined by the coupling between the conduction band and other bands. First-principles calculations were performed to verify the symmetry of the bands and also quantify the contribution of dierent bands in spin splitting of conduction electronsby calculating the parameters of the eective Hamilto- nian. Given the eective Hamiltonian in the conduction band, we calculated the spin-relaxation time for a ho- mogeneous polarized spin ensemble within a generalized Dyakonov-Perel mechanism. Our results show that spin- relaxation time is highly anisotropic in the plane of phos- phorene. A spin ensemble polarized in the armchair di- rection (x-direction) relaxes about 2-3 times longer than a spin ensemble polarized in the zigzag direction (y- direction). The calculated spin lifetimes are compara- ble in magnitude to the recent experiment on phospho- rene [17] which is shown to be dominated by Elliott-Yafet mechanism. However, in order for this anisotropy to be detected experimentally, the Dyakonov-Perel mechanism needs to be dominant. Therefore, a signicant electric eldE0:1 V/ A and a highly disordered sample with charged impurity density n= 1012cm2would be re- quired. ACKNOWLEDGMENTS The authors acknowledge the funding support from the MRSEC Program of the National Science Foundation un- der Award Number DMR-1420073. Appendix A: Lowdin Partitioning Lowdin partitioning [34], also known as quasi- degenerate perturbation theory, is a method for block diagonalizing a Hamiltonian with two sets of bands that weakly interact with each other. Here we are interested in the conduction band only and therefore calculate the eect of all other bands perturbatively as a separate set interacting with the conduction band. We assume that the Hamiltonian of the conduction band is denoted by Hccand the Hamiltonian in Eq. 3 is decomposed as H=H0+H0where H0;ll0= (El+~2(k2 k(d=dz)2 2m0)ll0; H0 ll0=V(z)ll0+
ll0+~ m0(kkid dz)
Pll0s
(A1) Based on Lowdin partitioning, the leading four terms that comprise Hccwhich are given as follows[25]. H(0) cc=H0;cc; (A2a) H(1) cc=H0 cc; (A2b) H(2) cc=X lH0 clH0 lc EcEl; (A2c)6 TABLE I. Character table of the space group of phosphorene ( Cmca : 64) for the point. The plus and minus signs denote the parity under spatial inversion i. fEj0g fC2xjg fC2yj0g fC2zjg fij0g fRxjg fRyj0g fRzjg Basis Invariant + 1 1 1 1 1 1 1 1 1 k2 x+k2 y + 2 1 -1 1 -1 1 -1 1 -1 xz  y + 3 1 1 -1 -1 1 1 -1 -1 yz  x + 4 1 -1 -1 1 1 -1 -1 1 xy  z 1 1 1 1 1 -1 -1 -1 -1 2 1 -1 1 -1 -1 1 -1 1 y k y 3 1 1 -1 -1 -1 -1 1 1 x k x 4 1 -1 -1 1 -1 1 1 -1 z k z=id=dz H(3) cc=1 2X lH0 ccH0 clH0 lc (EcEl)21 2X lH0 clH0 lcH0 cc (EcEl)2(A2d) +X l;l0H0 clH0 ll0H0 l0c (EcEl)(EcEl0) H(4) cc=1 2X lH0 ccH0 ccH0 clH0 lc+H0 clH0 lcH0 ccH0 cc (EcEl)3(A2e) 1 2X l;l0(H0 clH0 ll0H0 l0cH0 cc+H0 ccH0 clH0 ll0H0 l0c) (EcEl0) + (EcEl) (EcEl)2(EcEl0)2 1 2X l;l0H0 clH0 lcH0 cl0H0 l0c(EcEl0) + (EcEl) (EcEl)2(EcEl0)2 +X l;l0;l00H0 clH0 ll0H0 l0l00H0 l00c (EcEl)(EcEl0)(EcEl00) The zeroth order term is similar to the energy disper- sion of a free electron conned in two-dimensions i.e. H(0) cc=H0 cc=Ec+~2(k2 k(d=dz)2)=2m0. The rst order term is similarly given by H0asH(1) cc=V(z) which vanishes upon taking the expectation value over thezdirection. The expectation value is hH(1) cci=R dzf c(z)V(z)fc(z) = 0 which is a direct consequence of inversion symmetry of the unperturbed envelope function fc(z). More generally, higher order terms that contain odd powers of V(z) orkz=id=dz vanish upon averag- ing overz. This makes the calculation of the higher order terms more convenient as many terms become zero. The second order term contains the eective mass terms, sub- band energy in the zdirection, and other terms quadratic in spin-orbit strength which we discard. H(2) cc=~2 m2 0" k2 xX l=+ 2jPx;clj2 EcEl+k2 yX l=+ 3jPy;clj2 EcEl d2 dz2X l=+ 1jPz;clj2 EcEl# +O(2)0
(A3)The rst sum in the third order term vanishes upon av- eraging over zfor it contains H0 cc=V(z) in all of its terms. The same is true for the second sum although it is more subtle as the terms with l= + 1are op- erators aecting H0 cc=V(z). In other words, since c l= 4 + 1= 4,Hclcontainskzinvariant, i.e. Hcl= (~=m0)(id=dz )Pz;cl. Therefore, H0 clH0 lcH0 cc=~2jPz;clj2 m2 0 d2V(z) dz2+ 2dV(z) dzd dz +V(z)d2 dz2!
(A4) All the terms on the right hand side vanish upon av- eraging out z. Similarly, the third sum contains only odd terms in V(z) andd=dz which vanish as well. It also produces some high order terms in momentum and spin-orbit strength, i.e. hH(3) cci=O(kxky)z+O(3)0, which are negligible. The extrinsic spin orbit coupling, which couples electric eld, in-plane momentum, and spin-orbit together, emerges from fourth order terms. The rst sum in the fourth order term produces terms that are quadratic in electric eld ( dV(z)=dz)2which are negligible and appear only at high electric elds. The third sum produces other negligible high order terms i.e.O(k4 x) +O(k4 y) +O(k2 xd2=dz2) +O(k2 yd2=dz2) + O(d4=dz4) +O(2k2 x) +O(2k2 y) +O(2d2=dz2) +O(4). In the second sum, only terms containing both V(z) and d=dz survive. Other terms are either quadratic in V(z) or zero after averaging out z. First order terms in V(z) can be found in the fourth sum where either l=l0orl0=l00. Other terms stemming from the fourth sum are negligible for they contain higher orders of electric eld, spin-orbit, or momentum. Therefore, we can approximate the fourth sum as X l;l00H0 clH0 llH0 ll00H0 l00c (EcEl)2(EcEl00)+X l;l0H0 clH0 ll0H0 l0l0H0 l0c (EcEl)(EcEl0)2
(A5) SinceHcc=Hll=V(z) for alll, we can combine the second and the fourth sum together. It might seem that they would cancel each other completely. However, since7 V(z) andd=dz do not commute with each other, their commutator, i.e. [ d=dz;V (z)] =dV(z)=dz, survives the summation. We can see that this is the direct conse- quence of the broken inversion symmetry by V(z) which leads to the extrinsic spin-orbit term. Finally, based on the multiplication of group elements we obtain the non- zero terms that couple electric eld, momentum in the z direction, an in-plane momentum, and the angular mo- mentum perpendicular to the rest. Therefore, we obtain the eective Hamiltonian for extrinsic spin-orbit coupling averaged over zdirection. D H(4) ccE (~=2)dV(z) dz~2 m2 0" kxyX l=+ 1 l0=+ 2ImfPz;clLy;ll0Px;l0cg(EcEl0) + (EcEl) (EcEl)2(EcEl0)2 +kxyX l=+ 1 l0= 3ImfPz;clPx;ll0Ly;l0cg(EcEl0) + (EcEl) (EcEl)2(EcEl0)2 kxyX l=+ 2 l0= 3ImfPx;clPz;ll0Ly;l0cg(EcEl0)(EcEl) (EcEl)2(EcEl0)2 +kyxX l=+ 1 l0=+ 3ImfPz;clLx;ll0Py;l0cg(EcEl0) + (EcEl) (EcEl)2(EcEl0)2 +kyxX l=+ 1 l0= 2ImfPz;clPy;ll0Lx;l0cg(EcEl0) + (EcEl) (EcEl)2(EcEl0)2 kyxX l=+ 3 l0= 2ImfPy;clPz;ll0Lx;l0cg(EcEl0)(EcEl) (EcEl)2(EcEl0)2# = (~=2)dV(z) dz~2 m2 0X l;l0x;ll0kxy+y;ll0kyx
(A6) As expected, this term is linear in spin-orbit strength , electric eld dV(z)=dz, and in-plane momentum kx,ky. Using rst principle calculations we quantify the terms in Eq. A6 by calculating the matrix elements of momen- tum and angular momentum operator. The importance of each term is determined by the strength of coupling between the bands involved and also the energy dier- ence between them. Table IV lists all signicant nonzero terms and their corresponding coecients. The sum over all contributions (last row) determines the anisotropyPy;ll0=Px;ll01:4 . tTABLE II. First principle calculations setup Pseudopotential Type Ultrasoft, Fully Relativistic Exchange-Correlation FunctionPBE Kinetic Energy Cuto 34.0 Ry Charge Density Cuto 136.0 Ry Convergence Threshold 106Ry k-Point Grid Monkhrost 12 121 # of bands 24 Interlayer Spacing 31 :22 = A TABLE III. Parameters of the crystal structure of bulk and monolayer black phosphorus based on the denition in Ref. 27. a(A)b(A)d1(A)d2(A)12 Bulk 4.376 3.314 2.224 2.244 96.34102.09 Monolayer 4.552 3.306 2.224 2.262 93.89102.98 TABLE IV. Contributions of dierent bands in the extrinsic spin-orbit coupling quantied by coecients x;ll0andy;ll0 according to Eq. A6. The values are obtained from rst principle calculations and are in atomic units. The last row lists the total contributions of the bands which determines the anisotropy between extrinsic spin-orbit coecients. l band# l0band# x;ll0y;ll0 + 14 + 210 -71.24 4 12 2.95 9 10 10.08 15 10 -6.39 17 10 -5.68 + 117 3 13-2.58 21 -5.75 + 2 10 35 32.84 13 -2.76 20 46.36 22 -8.77 + 1 + 3 <1 + 14 2 142.10 17 -2.76 21 -7.02 + 3 2 <1P l;l0 -10.95 7.678 [1] A. Manchon, H. C. Koo, J. Nitta, S. Frolov, and R. Duine, Nature materials 14, 871 (2015). [2] F. J. Ohkawa and Y. Uemura, Journal of the Physical Society of Japan 37, 1325 (1974). [3] F. Vasko, JETP Lett 30, 541 (1979). [4] Y. A. Bychkov and E. Rashba, JETP lett 39, 78 (1984). [5] E. O. Kane, Journal of Physics and Chemistry of Solids 1, 249 (1957). [6] R. Hultgren, N. Gingrich, and B. Warren, The Journal of Chemical Physics 3, 351 (1935). [7] A. Brown and S. Rundqvist, Acta Crystallographica 19, 684 (1965). [8] P. Bridgman, Journal of the American Chemical Society 36, 1344 (1914). [9] L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, Nature nanotechnology 9, 372 (2014). [10] S. P. Koenig, R. A. Doganov, H. Schmidt, A. Cas- tro Neto, and B. Oezyilmaz, Applied Physics Letters 104, 103106 (2014). [11] H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tom anek, and P. D. Ye, ACS nano 8, 4033 (2014). [12] M. Buscema, D. J. Groenendijk, S. I. Blanter, G. A. Steele, H. S. Van Der Zant, and A. Castellanos-Gomez, Nano letters 14, 3347 (2014). [13] A. Castellanos-Gomez, L. Vicarelli, E. Prada, J. O. Is- land, K. Narasimha-Acharya, S. I. Blanter, D. J. Groe- nendijk, M. Buscema, G. A. Steele, J. Alvarez, et al. , 2D Materials 1, 025001 (2014). [14] F. Xia, H. Wang, and Y. Jia, Nature communications 5, 4458 (2014). [15] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, science 306, 666 (2004). [16] N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. Van Wees, Nature 448, 571 (2007). [17] A. Avsar, J. Y. Tan, M. Kurpas, M. Gmitra, K. Watan- abe, T. Taniguchi, J. Fabian, and B. Ozyilmaz, Nature Physics 13, nphys4141 (2017). [18] L. Li, J. Kim, C. Jin, G. J. Ye, D. Y. Qiu, H. Felipe, Z. Shi, L. Chen, Z. Zhang, F. Yang, et al. , Nature nan- otechnology 12, 21 (2017). [19] Z. Popovi c, J. M. Kurdestany, and S. Satpathy, Physical Review B 92, 035135 (2015).[20] M. Kurpas, M. Gmitra, and J. Fabian, Physical Review B94, 155423 (2016). [21] P. Li and I. Appelbaum, Physical Review B 90, 115439 (2014). [22] M. Dyakonov and V. Perel, Soviet Physics Solid State, USSR 13, 3023 (1972). [23] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, Acta Physica Slovaca 57, 565 (2007). [24] N. Averkiev and L. Golub, Physical Review B 60, 15582 (1999). [25] R. Winkler, Spin-orbit coupling eects in two- dimensional electron and hole systems , Vol. 191 (Springer Science & Business Media, 2003). [26] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. D. Jr, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawa- mura, H.-Y. Ko, A. Kokalj, E. Kkbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.- V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Ponc, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitso- nen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, Journal of Physics: Con- densed Matter 29, 465901 (2017). [27] Y. Takao, H. Asahina, and A. Morita, Journal of the Physical Society of Japan 50, 3362 (1981). [28] J. Qiao, X. Kong, Z.-X. Hu, F. Yang, and W. Ji, Nature communications 5, 4475 (2014). [29] N. Averkiev, L. Golub, and M. Willander, Journal of physics: condensed matter 14, R271 (2002). [30] T. Ando, A. B. Fowler, and F. Stern, Reviews of Modern Physics 54, 437 (1982). [31] S. Yuan, A. Rudenko, and M. Katsnelson, Physical Re- view B 91, 115436 (2015). [32] Y. Liu, T. Low, and P. P. Ruden, Physical Review B 93, 165402 (2016). [33] X. Wang, A. M. Jones, K. L. Seyler, V. Tran, Y. Jia, H. Zhao, H. Wang, L. Yang, X. Xu, and F. Xia, Nature nanotechnology 10, 517 (2015). [34] P.-O. L owdin, The Journal of Chemical Physics 19, 1396 (1951).

1603.07450v1.Semiclassical_Landau_quantization_of_spin_orbit_coupled_systems.pdf

Semiclassical Landau quantization of spin-orbit coupled systems Tommy Li1, Baruch Horovitz2, Oleg P. Sushkov1 1School of Physics, University of New South Wales, Sydney 2052, Australia and 2Department of Physics, Ben Gurion University, Beer-Sheva 84105, Israel A semiclassical quantization condition is derived for Landau levels in general spin-orbit coupled systems. This generalizes the Onsager quantization condition via a matrix-valued phase which describes spin dynamics along the classical cyclotron trajectory. We discuss measurement of the matrix phase via magnetic oscillations and electron spin resonance, which may be used to probe the spin structure of the precessing wavefunction. We compare the resulting semiclassical spectrum with exact results which are obtained for a variety of spin-orbit interactions in 2D systems. PACS numbers: 73.21.Fg,73.43.Qt,76.30.Pk,75.70.Tj I. INTRODUCTION Two-dimensional (2D) semiconductor systems oer strong and tunable intrinsic spin-orbit interactions1,2 which have been exploited in recently proposed spintronic devices3{6. In these systems, the close relationship be- tween charge and spin dynamics produces strongly modi- ed electronic transport properties which are exhibited in a range of eects including the anomalous Hall eect7{9, the spin Hall eect10{13and weak antilocalization14. In the past, quantum interference measurements have been proposed as a probe of the spin-orbit interaction in low- dimensional semiconductor heterostructures due to their sensitivity to quantum phases arising from coherent spin precession accompanying ballistic transport15{19. In par- ticular, the role of adiabatic and non-adiabatic phases in magnetic oscillations20,21is of high interest due to the fact that oscillatory magnetotransport experiments have provided crucial measurements of the spin-orbit coupling in these systems1,2,22{31. Furthermore, recent experimen- tal and theoretical studies of 2D Dirac systems such as graphene and surface states of three-dimensional topolog- ical insulators have highlighted the role of the geometric phase in particular in magnetotransport32{41. In this work we derive an expression for the Lan- dau level spectrum of a 2D system with spin-orbit in- teraction via a generalization of the Onsager quantiza- tion condition42to account for non-trivial spin dynamics. Spin evolution is encoded in the SU(2) phase represent- ing the total rotation of an initial spin state around a period of cyclotron motion. This SU(2) phase is nec- essary to describe non-adiabatic spin dynamics which is present when the eective magnetic eld in momentum space describing the spin-orbit interaction is not su- ciently strong to locally polarize the spin of the particle along the orbit19,43. We evaluate the semiclassical spec- trum for the cases when the spin-orbit eective magnetic eld is simply rotating in momentum space with a single winding number and compare to the exact solutions for a variety of spin-orbit interactions in semiconductor sys- tems, including several cases which have not been previ- ously mentioned in the literature. In addition, we show that magnetic oscillations and electron spin resonance(ESR) serve as eective probes of the precessing spin structure of Landau level states. This paper is organized as follows: in Section II we de- rive the semiclassical quantization condition and a gen- eral expression for the Landau level spectrum of spin- orbit coupled system, accounting for spin dynamics via a matrix valued phase. In Section III we discuss magnetic oscillations and derive the expression for the oscillatory density of states in terms of the matrix-valued phase. In Section IV we evaluate the level spectrum and eigenstates for a rotating spin-orbit interaction with xed winding number. We also calculate exact results for a variety of interactions in p-type systems and present a comparison of the semiclassical and exact results for these as well as for previously discussed results in n-type systems.44{47 In Section V we discuss ESR as a probe of the spin-orbit interaction type and evaluate the ESR matrix elements for the cases discussed in Section IV. Our summary and concluding remarks are presented in Section VI. II. SPECTRUM OF LANDAU LEVELS WITH SPIN-ORBIT INTERACTION We consider a 2D electron or hole gas in perpendicular magnetic eld, described by the Hamiltonian H=2 x+2 y 2m+Hs(x;y); Hs=(x;y)
(1) where= (x;y) =peAare the operators of kinetic momentum48(eis the charge of the electron or hole) and mis the eective mass. For electron systems, the Pauli matricesact on spin, while for hole systems, acts on the doublet of heavy hole states49. The spin-dependent interaction Hs, accounting for the spin-orbit interaction in addition to Zeeman coupling to the external magnetic eld, will be expressed in terms of the eective magnetic eld in momentum space, = (x;y;z). Since the kinetic momenta satisfy a commutation re- lation [x;y] =ieBz, we may construct annihilation and creation operators a;aywitha=x+iy j2eBzjwhere is the sign of the charge, with the corresponding numberarXiv:1603.07450v1 [cond-mat.mes-hall] 24 Mar 20162 operator N=aya=2 x+2 y j2eBzj1 2: (2) It is possible to diagonalize the Hamiltonian (1) in the number representation, as has been done in previous ap- proaches to the problem44{47. We consider however the semiclassical picture, in which the spectrum is related to the dynamics of wavepackets moving along the cy- clotron trajectory. Due to the spin-orbit interaction, a wavepacket in some initial polarization state will precess along the orbit, and generally undergo a rotation after a complete revolution, which is described by an SU(2) matrix. Thus for the purposes of semiclassical quantiza- tion the phase is matrix-valued, and the spectrum will be determined by the eigenvalues and eigenvectors of the matrix-valued phase. In order to rigorously derive this result, we introduce the spinor wavefunction () which varies along the angle in momentum space. Explicitly, this is given by () =hj iwhere the states jiare related to number eigenstates via50 ji=1X n=0einjni; = sgn(e): (3) wherejniare eigenstates of the number operator. Here we assumeBz>0. The basis states jiare eigenstates of the operators ei;eiwhich are related to the momen- tum operators via a=eip N=x+iy j2eBzj: (4) Note that the number operator corresponds to the clas- sical action coordinate, while is simply the angle in momentum space, ( x;y) = (jjcos;jjsin). Thus in the semiclassical limit the wavefunction () represents the motion of a particle in momentum space as a function of the angle . In order to obtain the Schr odinger equation for () then obtained from the Hamiltonian (1), we note that the classical coordinates ( ;N) are canonically conjugate, which implies that the operator Ntakes the form of a derivative operator in the representation. Explicitly acting on the basis (3) with Nshows that hjNj i=id dhj i=id () d: (5) Thus the rst term in (1) may be replaced with2 x+2 y 2m! !(id d+1 2) where!=jeBz mjis the cyclotron frequency, and the eective magnetic eld may be regarded as a function of the coordinates ( ;N!id d). Thus the Schr odinger equation for () reads:  i!d d+(;id d)
E+! 2 () = 0; !=jeBzj m: (6)In the absence of spin-orbit interaction, = 0, the wave- function satises the equation id d= ;  =E !1 2(7) which yields the wavefunction () =ei; (8) corresponding to a circular orbit in momentum space, with= sgn(e) indicating the direction in which the circle is traversed (clockwise for  > 0 and anticlock- wise for <0). (This is simply the wavefunction of the harmonic oscillator in the phase representation.50) Sin- gle valuedness of the wavefunction then requires to be an integer, which of course yields the usual Landau level spectrumEn= (n+1 2)!. However, is also related to the average momentum of the orbit via h2i=j2eBzj(hNi+1 2) =j2eBzj(+1 2):(9) Thus single valuedness of the wavefunction in the - representation is equivalent to Onsager's quantiztion rule42, that the area of the momentum space orbit must be quantized: 1 j2eBzjZ2 02d= 2(n+1 2): (10) In the presence of spin-orbit coupling the spinor () generally precesses as a function of under the in uence of the eective magnetic (;N). In the semiclassical r egime,hNi 1 the wavefunction takes the form of a Born-Oppenheimer product of orbital and spin factors, () =ei(): (11) The rst factor in (11) corresponds to an orbital trajectory in momentum space with radius jj=q j2eBzj(+1 2) (and we assume y= 1). When () is slowly varying compared to the orbital factor, we may replace the action of the derivative with the semiclassi- cal variable in(;id d)!(;). This requires that the spin-orbit eective magnetic eld be much smaller than the total energy, j(;)jE, as well asd d. Nevertheless, this does not require j(;)j!(this inequality is violated, e.g. in the r egime of double mag- netic focusing51). Thus the Schr odinger equation for spin reads i!d d= [(;)
!] (12) whereis a parameter dened by E=!(+1 2+): (13) Equation (12) is identical to the equation of motion for a precessing wavepacket moving along the classi- cal cyclotron orbit, ( =const:; =!t) if the left3 hand side is replaced by the time along the trajectory, i!d d=id dt. We may divide the evolution of spin into two parts,() =eiU()(0) whereU() is an SU(2) matrix, U() =Pei!1R 0[(;)
]d; (14) wherePindicates path-ordering. Over a complete or- bit, the spin wavefunction accumulates a complex phase factor, as well as a rotation generated by the matrix- valued phase U(2); however in a stationary state, the spin polarization must return to its initial value after a complete orbit, implying that the spinors (0) and (2) =e2iU(2)(0) dier at most by a phase. It follows that (0) =is an eigenvector of U(2). Since U(2) is an SU(2) matrix, its eigenvalues e+i;eiare complex conjugate, and the phase accumulated due to unitary transformations of spin over a complete orbit for an initial spin state is equal to ei. The wavefunc- tion () must be single valued, implying the quantiza- tion condition 2(+) = 2n ; (15) Recalling the denition (13), this gives a relationship be- tween the spectrum and the eigenvalues of the matrix phaseU(2) in the semiclassical limit, En;=!(n+1 2 2): (16) The matrix phase is fully determined by the path ordered exponential (14) which depends on the radius of the or- bital trajectory,jj=q j2eBzj(+1 2). Nevertheless, the quantization condition (15) does not x andindivid- ually, but only the combination +, withbeing a free parameter corresponding to an arbitrary choice of the phase of. The choice of is xed by the requirement for the validity of the Born-Oppenheimer approximation, d d. This requires . Since the total phase accumulated by in a stationary state is 2 , we must minimize the variation of along the trajectory by choosing the phase of so thatis an integer, = 2; =n : (17) Thus the spectrum is determined by the eigenvalues of the matrix phase U(2) evaluated for orbits in momen- tum space with radius jnj=q j2eBzj(n+1 2). Note that in the absence of electric elds the Hamiltonian (1) commutes with the guiding center operators X= x+y eBz;Y=yx eBz, thus each Landau eigenstate may be chosen to be a simultaneous eigenstate of X;Y. This leads to the usual degeneracy per unit areajeBzj 2. The quantization condition (15) may be expressed in terms of the energy, J(E) =2mE jeBzj = 2(n(E) +1 2): (18)The left-hand side of (18) is equal to total phase accu- mulated by the wavefunction () in a stationary state, and is therefore equal the classical action integrated over a single period. At a given energy, there exist two orbits, whose radii in momentum space are given by (from (13)) 1 j2eBzjZ 2 d= 2(n(E) +1 2) (19) and may be determined e.g. from magnetic focusing.51 The periods of the two spin trajectories are given by the derivative of the action with respect to the energy, T=2 !=dJ dE=2m jeBzjd dE; (20) regardingEas a continuous variable in the semiclassical limit. While we have performed a detailed derivation in the case of a quadratic dispersion, it is intuitively clear that our argument and results may be rigorously general- ized to the case of non-quadratic dispersions,2 2m! (). In this case the wavefunction (11) takes the form () =ei j2eBzjR 2d+i 2() where() satises the same Schr odinger equation, (12) with the spin-orbit in- teraction(x;y) evaluated for quantized orbits of con- stant energy satisfying the condition1 j2eBzjR 2 nd= 2(n+1 2). The spectrum (16) for non-quadratic dis- persions becomes En;=(n) +!(1 2 2); (21) where the oscillator frequency !must be calculated from the classical equations of motion corresponding to the general dispersion (). III. MAGNETIC OSCILLATIONS According to Onsager's principle, the oscillations in resistivity of a 2D system as function of perpendicular magnetic eld directly measure the semiclassical phase (18) accumulated over an orbit for a particle at the Fermi energy.42We will calculate the oscillating resistivity for a general spin-orbit coupled system in a similar man- ner to the method of Lifshitz and Kosevich52. In the Drude approximation the conductivity is proportional to the density of states A(E) =jeBzj 42TrGR=jeBzj 42ImX n;=1 EEn+i 2 (22) where the retarded Greens function GRis averaged over disorder, and we have included the Landau level de- generacy factorjeBzj 2. We only consider the situation where impurities are short-ranged, so that relaxation is4 described in rst order by a single parameter 1equal to the total scattering cross section at53Bz= 0. Apply- ing the Poisson summation formula to (22), one regards En!E(J) as a function of the continuous variable J(18): A(E) =jeBzj 42Im1X l=0;=Zeil(J) EE(J) +i 2dJ: (23) Performing a change of variables and a contour integra- tion gives the density of states at E=EF A(EF) =jeBzj 21X l=0;=1 !el !cos [lJ(EF)l]; =jeBzj 21X l=0;=1 !el !cosl2EF !(EF) ; (24) where!=!are the frequencies of the spin trajec- tories (20). The spin-dependent phase shift in magnetic oscillations is therefore equal to + ; for orbits eval- uated at the Fermi energy. In the semiclassical regime, n1, the dierence between !+;!may be neglected in the rst approximation (typically oscillations are ob- served up to n40 in electron systems1andn20 in hole systems2,43). Accounting for only the rst harmonic in (24), the resistivity becomes xx(Bz) =xx(0)(1 +e !cos (EF) cos 2EF !1 2 ); (25) and the oscillatory part vanishes when ( EF) =(n+1 2). Since the spin-orbit interaction is generally highly tun- able by experimental parameters1,2, measurement of the envelope cos  over a range of parameters would per- mit the indirect mapping of semiclassical spin dynamics along the cyclotron trajectory (as we shall demonstrate in Section IVC). A. Berry phase In the typical experimental situation reported in magnetotransport measurements in n-type narrow gap systems,1,22{29the spin-orbit interaction is suciently strongj(;)j!, that spin precession is adiabatic, i.e. the spin polarization is locally aligned with the vec- tor(;) along the cyclotron orbit. In this r egime the phase  contains a Berry phase54contribution 'B equal to1 2the solid angle enclosed by the precessing spin polarization on the sphere. While this contribu- tion has been experimentally observed33{36and theoret- ically studied38{41in the context of 2D Dirac materials, measurement of the Berry phase via magnetotransport insemiconductor systems has yet to be reported. Neverthe- less, Eq. (25) demonstrates that the Berry phase should appear as a correction to the phase of the resistivity os- cillations, typically of order for strong spin-orbit inter- action, which may signicantly alter the amplitude of the oscillating resistivity. In the case of a strong Rashba in- teraction, the Berry phase 'B=is a constant shift corresponding to a phase inversion of the oscillations. The phase may be expressed in terms of the spin-split densities measured at zero magnetic eld, J(EF)!4 j2eBzj'B: (26) When the Berry phase is constant as a function of the perpendicular eld, it does not aect the spin-split den- sitieswhich are usually extracted by performing a Fourier transform of the resistivity. In this case, the Berry phase appears only as a constant shift of the oscilla- tions. Explicitly, the Fourier transform of the resistivity with respect to the inverse magnetic eld is given by F(r) =Z eibrxx(b)db ; b =82 j2eBzj(27) and the maxima of the function F(r) occur at r=r+;r; r= 2@ @b=@'B @b: (28) Note that we assume that the spin-orbit interaction is held constant while Bzis varied. While in general, the derivative of the Berry phase appears as a cor- rection to the peaks of the Fourier transform (along- side the zero-eld densities ), in the limit of strong spin-orbit interaction typically encountered in narrow- gap semiconductors1,22{29, the Berry phase is a constant shift and does not contribute to the position of the peaks: a Fourier analysis of the oscillations gives only the zero- eld densities , which corresponds to the rst term in (26). Comparison to the envelope (25) can therefore directly reveal the Berry phase shift. In the general situation, determination of the zero-eld densitiesfrom a Fourier analysis of the oscillations is not straighforward due to the presence of the derivative of the Berry phase in (28). In this case, measuring the oscillations at suciently low elds for which only one species contributes to the resistivity22would allow the Berry phase to be simply extracted from the positions of the maxima of the oscillations (as it is, e.g. in Dirac semimetals33{36). IV. THE CASE OF ROTATING INTERACTIONS: COMPARISON OF EXACT AND SEMICLASSICAL SOLUTIONS In the typical experimental situation, semiconduc- tor heterostructures are subject to the Rashba55and5 Dresselhaus56spin-orbit interactions, in addition to ap- plied magnetic elds. The competition between these in- teractions, which are often of the same order58{60result in complex spin trajectories which are re ected in both the spectrum and magnetic oscillations via the spin evo- lution matrix U(2) (14). Nevertheless, an important situation arises when a single spin-orbit interaction is present which corresponds to a eld which performs an integer number of rotations Waround a circle in momen- tum space, = (kcos(W+);ksin(W+);z). We consider the realisation of this situation in both electron and hole systems. In electron systems, a pure Rashba interaction, HR=(xyyx) corresponds to wind- ing number W= +1, a pure Dresselhaus interaction in zincblende systems conned perpendicular to a cu- bic axis,HD=(xxyy) corresponds to winding numberW=1. In hole systems, the higher angular momentum J=3 2 for holes implies that interactions linear in Jx;Jyhave higher winding numbers than their counterparts in elec- tron systems. The Rashba and Dresselhaus interactions correspond to W= +3 andW= +1 respectively, and an applied in-plane magnetic eld corresponds to W= +2. This statement may be derived from the observation that, for hole systems conned to a two-dimensional plane, the low energy subspace consists of the heavy hole doublet with angular momentum quantized along the perpendic- ular axis,j+i=jJz=3 2i;ji=jJz=3 2i. Inter- actions which are linear in Jx;Jydo not couple states j+i;ji. In order to obtain a coupling between these states, it is necessary to account for the additional in- teraction/2 +J2 +h:c:which appears in the Luttinger Hamiltonian57. In combination with interactions linear inJx;Jy, this contributes a factor 2 +which, after pro- jection onto heavy hole states raises the winding num- ber by 2. In perturbation theory one obtains, for an in-plane magnetic eld a Hamiltonian /B+2 ++h:c: withW= +2; for the Rashba interaction the Hamilto- nian is/i3 +withW= +3, and for the Dresselhaus interaction the Hamiltonian is /f2 +;g+h:c:with W= +1. In this section we present analytical results for these situations, and compare the semiclassical approx- imation to the exact spectra obtained from brute force diagonalization. Let us rst consider the situation for a general winding numberW. The interaction with the eective magnetic eld may be written 
=g(0
)g1withg=eiWz 2 and0= (kcos;ksin;z) constant along the circu- lar trajectory. Performing a transformation to the coro- tating frame, =g0, the Schr odinger equation for spin (12) reads i!d0 d= 0
!+i!g1@g @ 0; = (0+!W ^z 2)
! 0: (29) The eective magnetic eld in the co-rotating frame isstatic, B=0+!W 2^z ; (30) and direct integration gives the evolution operator (in the laboratory frame) U() =eiWz 2ei !B
(31) and the eigenvalues of U(2) are given by ei=eiW2ijBj !: (32) The phase  of the eigenvalues are unambiguously de- ned only up to a multiple of 2 . In order to select the phase , we note that the Born-Oppenheimer approxi- mation is valid only when the spin state is slowly vary- ing. The spin states (in the laboratory frame) are given explicitly by +() =eijBj !i 2(cos 2eiW 2j+i+ sin 2eiW 2+iji); () =eijBj !+i 2(sin 2eiW 2ij+i+ cos 2eiW 2ji) (33) whereis the angle between Band the plane, tan=Bk() Bz(); (34) j+i;jiare spin states with polarization along the z- axis, and0= (kcos;ksin;z). The spin-up and spin-down components of accumulate dierent phases over the orbital trajectory. We may dene  so that the largest spin component of +is constant, with the smaller spin component acquiring a phase of 2 Waround the trajectory. This choice ensures that as the spin- orbit interaction is reduced to zero, the energies Enand quantum states are continuously related to the sim- ply Zeeman-split levels in a uniform magnetic eld. For the caseBz>0, the spin polarization is tilted above the plane and the largest spin component is h+j+i, while forBz<0 the spin polarization is tilted below the plane and the largest component is hj+i, thus:  =2jBj !Wsgn(z+!W 2): (35) Thus the wavefunctions are given by n+() =ein(cosn 2j+i+ sinn 2eiW+iji); n() =ein(sinn 2eiWij+i+ cosn 2ji) (36) forBz=z+!W 2>0, and n+() =ein(cosn 2eiWij+i+ sinn 2ji); n()ein(sinn 2j+i+ cosn 2eiW+iji) (37)6 forBz=z+!W 2<0, andncorrespond to angles  (34) evaluated for values of =n. The spin polarization in the upper spin state +is alongSk+!W ^z 2, which is tilted out of the plane due to the rotation of the eec- tive magnetic eld. This out-of-plane tilting is due to a geometric term ig1@g @in the equation of motion (29). In the adiabatic limit !jj, spin will align along the direction of the eective magnetic eld, Sk, neverthe- less the geometric contribution leads to a correction to  which is equal to the Berry phase discussed in Section IIIA. It follows from (16) that the spectrum is En;=!(n+1 2) jBj!W 2sgn(z+!W 2) =!(n+1 2) "r (z+!W 2)2+2 k!W 2sgn(z+!W 2)# (38) where the eective magnetic eld () is taken along mo- mentum space orbits corresponding to integer values of . We note that, while the choice of phase (35) minimises the error in the semiclassical solution, we may arbitrarilyredene the phase by addition of an integer multiple of 2. After re-labeling of Landau levels (which only aects the ground state), addition of 2 to the phase is equiva- lent to a shift of index in the spin-dependent part of the energy,En;=!(n+1 2+n 2)!!(n+1 2+n+1 2), which leads to an error of the same order in the semiclassical limit, although the numerical error may be larger for al- ternative choices of . We discuss this point further in Appendix A. In the remainder of the section we shall ap- ply these results to specic cases and compare them to the exact solutions. A. Rashba interaction in n-type systems The case of Rashba and Dresselhaus interactions in n- type systems has been extensively discussed in previous literatue.44{47; we will review only the situation in which one of these interactions is present. For the Rashba in- teraction, the Hamiltonian is given by H=2 2m+R(yxxy)gBBz 2z;(39) whereRis the Rashba constant, and the eective magnetic eld (x;y) = (Ry;Rx;gBBz 2) has winding number W= +1. From (38) the semiclassical solution is given by En;=!(n+1 2)"r (! 2gBBz 2)2+2 Rj2eBzj(n+1 2) +! 2sgn(! 2gBBz 2)# : (40) A derivation of the exact solution is presented in Appendix B. We obtain the exact spectrum En;+=8 >>< >>:!(n+1 2) +q (! 2+gBBz 2)2+j2eBzj2 R(n+ 1) +! 2 ;! 2gBBz 2>0; !(n+1 2) +q (! 2+gBBz 2)2+j2eBzj2 Rn! 2 ;! 2gBBz 2<0; En;=8 >>< >>:!(n+1 2)q (! 2+gBBz 2)2+j2eBzj2 Rn+! 2 ;! 2gBBz 2>0; !(n+1 2)q (! 2+gBBz 2)2+j2eBzj2 R(n+ 1)! 2 ;! 2gBBz 2<0(41) The exact wavefunctions are given by n;+() =ein(cosn+1 2j+i+isinn+1 2eiji); n;() =ein(sinn 2eij+i+icosn 2ji) (42)for! 2gBBz 2>0, and n;+() =ein(cosn 2eij+i+isinn 2ji); n;() =ein(sinn+1 2j+i+icosn+1 2eiji) (43) for! 2gBBz 2<0. The angles nare dened in the same way as in the previous section (34), tan n= Rp j2eBzjn ! 2gBBz 2.7 0.01 0.02 0.03 0.04 0.05050100 0 0.5050100 En(meV) EF zB α~ Rn = 40 n = 30 n = 20 n = 10 FIG. 1. ( Color online ). Energies of Landau level states n= 10;20;30;40 inn-type 2D system in the presence of a Rashba interaction, plotted as a function of the dimension- less constant ~ R=RpF EFwhereEF= 73meV is the Fermi energy corresponding to a 2D electron gas with typical ex- perimental density1= 71011cm2(the Fermi energy is indicated by the dashed horizontal line) and band parame- ters corresponding to InAs. The left panel shows energies as a function of BzatR= 0 and the right panel shows energies as a function of ~ RatBz= 0:5T. Red and blue lines indicate states of opposite spin. The dierence between the exact (41) and semiclassical (40) solutions is not visible. FIG. 2. ( Color online. ) Spin precession of Landau eigenstates along the momentum space trajectory due to the Rashba in- teraction in n-type InAs, shown for the highest lled Landau level (n= 29) at experimental density1= 71011cm2. The spin polarization is indicated by red arrows, and the ef- fective magnetic eld is indicated by blue arrows.The error in the semiclassical solution is sin2 8n(En+ En). The Landau level energies for levels n= 10;20;30;40 atBz= 0:5T are plotted in Fig. 1 as a function of the dimensionless parameter ~ R=RpF EFwhereEF= 73meV;pFare the Fermi energy and mo- mentum corresponding to a 2D electron gas at experi- mental density1= 0:61012cm2. The band parame- ters are taken for InAs49,m= 0:0229me;g=14:9. The semiclassical and exact results are both shown, although in this situation they are indistinguishable. For n1 the wavefunctions (42) and (43) reduce to the semiclas- sical expressions (36) and (37) with W= +1 and= 2. The precessing wavefunction is illustrated in Fig. 2 for the highest lled Landau level ( n= 29) at the experi- mental density with the same parameters used in Fig. 1. The spin polarization y() () is indicated by red ar- rows and the eective magnetic eld () is indicated by blue arrows. While the eective magnetic eld is tilted above the plane, the spin polarization is tilted below the plane, illustrating the size of the geometric contribution (30)B0=! 2^zin the experimental parameter r egime. B. Dresselhaus interaction in n-type systems We consider only the linear Dresselhaus interaction in (100)-oriented heterostructures, for which the Hamilto- nian is given by H=2 2m+D(xxyy)gBBz 2z;(44) whereDis the Dresselhaus constant for the heterostruc- ture. The rotating eective magnetic eld, (x;y) = (Dx;Dy;gBBz 2) has winding number W=1, and the semiclassical solution is identical in form to the solution for the Rashba case (41) with the exception that, in the spin-dependent part !is replaced by!due to the opposite winding number: En;=!(n+1 2)"r (! 2gBBz 2)2+2 Dj2eBzjn! 2sgn(! 2gBBz 2)# : (45)8 A derivation of the exact solution is presented in Appendix B. The exact energies are given by En;+=8 >>< >>:!(n+1 2) +q (! 2gBBz 2)2+j2eBzj2 Dn! 2 ;! 2gBBz 2>0; !(n+1 2+q (! 2gBBz 2)2+j2eBzj2 D(n+ 1) +! 2 ;! 2gBBz 2<0:(46) En;=8 >>< >>:!(n+1 2)q (! 2gBBz 2)2+j2eBzj2 D(n+ 1)! 2 ;! 2gBBz 2; !(n+1 2)q (! 2gBBz 2)2+j2eBzj2 Dn+! 2 ;! 2gBBz 2<0: The exact wavefunctions are given by n;+() =ein(cosn 2j+i+ sinn 2eiji); n;() =ein(sinn+1 2eij+i+ cosn+1 2ji) (47) for! 2gBBz 2>0, and n;+() =ein(cosn+1 2eij+i+ sinn+1 2ji); n;() =ein(sinn 2j+i+ cosn 2eiji) (48) for! 2gBBz 2<0. Forn1, the wavefunctions reduce to their semiclassical expressions (36), (37) with W=1 and= 0.C.p-type systems with in-plane magnetic eld The Hamiltonian in case when both in-plane and per- pendicular components of the magnetic eld are present is given by H=2 2mH 2(B+2 ++B2 +)gBBz 2: (49) whereHis a constant which depends on the 2D con- ning potential and the bulk g-factor. The rotating ef- fective magnetic eld is (x;y) =(HBk2 kcos(2+ );HBk2 ksin(2+);gBBz 2) whereB+=Bkeiand has winding number W= 2. Thus the semiclassical spec- trum (38) is given by En;=!(n+1 2)"r (!gBBz 2)2+ (2eHBkBz)2(n+1 2)2!sgn(!gBBz 2)# : (50) A derivation of the exact solution is presented in Appendix C. The exact energies are given by En+=8 >>< >>:!(n+1 2) +q (!gBBz 2)2+ (2eHBzBkn)2! ; !gBBz 2>0; !(n+1 2) +q (!gBBz 2)2+ (2eHBzBkn+2)2+! ; !gBBz 2<0:(51) En=8 >>< >>:!(n+1 2)q (!gBBz 2)2+ (2eHBzBkn+2)2! ; !gBBz 2>0; !(n+1 2)q (!gBBz 2)2+ (2eHBzBkn)2+! ; !gBBz 2<0: where n=p n(n1): (52)The exact wavefunctions are n;+() =ein(cosn 2j+i+ sinn 2e2iji); n;() =ein(sinn+2 2e2ij+i+ cosn+2 2ji) (53)9 0.05 0.1 0.1500.511.5 0 0.200.511.5 En(meV) EF zB yn = 12 n = 10 n = 8 n = 4n = 6 FIG. 3. ( Color online ). Energies of Landau level states n= 4;6;8;10;12 in a 2D GaAs hole gas in the presence of an in-plane magnetic eld Bk, plotted as a function of the dimensionless constant y= 2mHBk. The left panel shows energies as a function of Bzaty= 0 and the right panel shows energies as a function of yatBz= 0:2T. The exact solutions (51) are indicated in solid lines, and the semiclassical approximation (50) is indicated in dashed lines. Red and blue lines indicate states of opposite spin. The Fermi energy EF= 0:89meV corresponding to the typical experimental density43 = 9:31010cm2is indicated by the dashed horizontal line. The arrows indicate possible ESR transitions (discussed in Section V). for!gBBz 2>0, withB+=Bkeiand n;+() =ein(cosn+2 2e2ij+i+ sinn+2 2ji); n;() =ein(sinn 2j+i+ cosn 2e2iji) (54) for!gBBz 2<0, and the angles nare given by tann=2eHBzBkn !gBBz 2: (55) Forn1 we haven!nand the exact wavefunctions reduce to the semiclassical expressions (36), (37) with W= +2. The error in the semiclassical solution is 1 2nsin2n(En+En). The Landau level ener- gies forn= 4;6;8;10;12 are plotted in Fig. 3 as a function of the dimensionless parameter y= 2mHBkfor a GaAs 2D hole system, with eective mass m= 0:25me corresponding to the experimental situation reported in43. We also take a value for the g-factor in GaAs49 g= 6= 7:2. The experimentally measured value of Hin experiment corresponds to y= 0:029 atBk= 1T. The exact solution (51) is shown in solid lines and the semiclassical solution (50) is shown in dashed lines. The horizontal dashed line indicates the Fermi energy at experimental density43= 9:31010cm2. 5 6 7 8 9 10 B (T)z-1Rxx (arb.) 34568104590ooo oooooFIG. 4. ( Color online ). The oscillating resistivity Rxx(Bz) (arbitrary units) as a function of B1 z, with the ratioBz Bx= tantiltkept xed. The values of tiltcorresponding to the individual traces are shown on the right of the gure. The solid lines indicate the oscillations obtained from the exact solution (51) while the dashed lines indicate the semiclassical solution (50). The semiclassical and exact results can only be distinguished for angles tilt= 3;4;5. The oscillating resistivityRxx(Bz) Rxx(Bz=0)is plotted in Fig. 4 for various values of Bx, with the ratioBz Bx= tantilt kept xed. The tilt anglestiltcorresponding to the in- dividual traces are shown on the right side of the gure. The solid line indicates the oscillations obtained from the exact solution (50), while the dashed line indicates the semiclassical solution (49). The semiclassical and exact results can only be distinguished at the lowest angles, tilt= 3;4;5. The precessing wavefunction is illus- trated in Fig. 5 for the highest lled Landau level ( n= 9 atBz= 0:2T) at the experimental density with the same band parameters used in Fig. 3. The in-plane magnetic eldBxcorresponds to a value y= 2mHBx= 0:116. The spin polarization y() () is indicated by red ar- rows and the eective magnetic eld () is indicated by blue arrows. The dierence between the spin polarization and the eective magnetic eld is given by the geometric contribution (30) B0=!^zcorresponding to a rotating eective magnetic eld with winding number W= +2. D. Rashba interaction in p-type systems The Hamiltonian in this case is given by H=2 2m+i0 R 2(3 +3 +) (56)10 FIG. 5. ( Color online. ) Spin precession of a Landau level eigenstate along the momentum space trajectory due to an in-plane magnetic eld Bxcorresponding to y= 2mHBx= 0:116 in a GaAs hole gas, shown for the highest lled Landau level (n= 9,Bz= 0:2T) at experimental density43= 9:3 1010cm2. The spin polarization is indicated by red arrows, and the eective magnetic eld is indicated by blue arrows.where0 Ris the Rashba constant for the heterostruc- ture. The rotating eective magnetic eld, (x;y) = (0 R3 ksin 3;0 R3 kcos 3;gBBz 2) has winding number W= 3. The semiclassical solution is given by (38) En;=!(n+1 2)"r (3! 2gBBz 2)2+ (0 R)2(2eBz)3(n+1 2)33! 2sgn(3! 2gBBz 2)# : (57) A derivation of the exact solution is presented in Appendix C. The exact energies are given by En;+=8 >>< >>:!(n+1 2) +q (3! 2gBBz 2)2+ (0 R)2(2eBzn)33! 2 ;3! 2gBBz 2>0; !(n+1 2) +q (3! 2gBBz 2)2+ (0 R)2(2eBzn+3)3+3! 2 ;3! 2gBBz 2<0(58) En;=8 >>< >>:!(n+1 2)q (3! 2gBBz 2)2+ (0 R)2(2eBzn+3)23! 2 ;3! 2gBBz 2>0; !(n+1 2)q (3! 2gBBz 2)2+ (0 R)2(2eBzn)2+3! 2 ;3! 2gBBz 2<0 where n= (n(n1)(n2))1 3 (59) and the wavefunctions are given by n;+() =ein(cosn 2j+i+isinn 2e3iji); n;() =ein(sinn+3 2e3ij+i+icosn+3 2ji) (60) for3! 2gBBz 2>0, and n;+() =ein(cosn+3 2e3ij+i+isinn+3 2ji); n;() =ein(sinn 2j+i+icosn 2e3iji) (61)for3! 2gBBz 2<0. Here the angles nare given by tann=0 R(2eBzn)3 2 3! 2gBBz 2: (62) Forn1 we haven!nand the exact wavefunctions reduce to the semiclassical expressions (36), (37) with W= +3;= 2. The error in the semiclassical solution is  9 8nsin2n(En+En). The Landau level energies at Bz= 0:5T are plotted in Fig. 6 for n= 4;8;12;16 as a function of the dimensionless parameter ~ 0 R=0 Rp3 F EF where the Fermi energy EF= 2meV (indicated by the dashed horizontal line) corresponds to the experimental density2= 31011cm2. The exact solutions (58) are shown in solid lines, and the semiclassical solutions (57) are shown in dashed lines. The precessing wavefunctions are illustrated in Fig. 6 for the highest lled Landau level ( n= 12) at the exper-11 0.05 0.10.15 0.20.2501234 0 0.501234 En(meV) EF zB α~' Rn = 16 n = 12 n = 8n = 4 FIG. 6. ( Color online ). Energies of Landau level states n= 4;8;12;16 in a 2D GaAs hole gas in the presence of a Rashba interaction, plotted as a function of the dimensionless constant ~0 R=Rp3 F EFwhere the Fermi energy EF= 2meV (indicated by the dashed horizontal line) corresponds to the typical experimental density2= 31011cm2. The left panel shows energies as a function of Bzat ~0 R= 0 and the right panel shows energies as a function of ~ 0 RatBz= 0:5T. The exact solutions (58) are indicated in solid lines, and the semiclassical approximation (57) is indicated in dashed lines. Red and blue lines indicate states of opposite spin.imental density with the same parameters used in Fig. 7. The spin polarization y() () is indicated by red arrows and the eective magnetic eld () is indicated by blue arrows. The dierence between the spin polar- ization and the eective magnetic eld is given by the geometric contribution (30) B0=3! 2^zcorrespond- ing to a rotating eective magnetic eld with winding numberW= +3. E. Dresselhaus interaction in p-type systems The Hamiltonian in the case of a pure Dresselhaus in- teraction is given by H=2 2mgBBz 2z +0 D 4((2 ++2 +)+ (+2 +2 +)+); (63) where0 Dis the Dresselhaus constant for the heterostruc- ture. The rotating eective magnetic eld, (x;y) = (0 D2y;0 D2x;gBBz 2) has winding number W= 1. The semiclassical solution is therefore given by (38) En;=!(n+1 2)"r (! 2gBBz 2)2+ (0 D)2(2eBz)3(n+1 2)3! 2sgn(! 2gBBz 2)# : (64) A derivation of the exact solution is presented in Appendix C. The exact energies are given by En;+=8 >>< >>:!(n+1 2) +q (! 2gBBz 2)2+ (0 D)2(2eBzn)3! 2 ;! 2gBBz 2>0; !(n+1 2) +q (! 2gBBz 2)2+ (0 D)2(2eBz(n+ 1))3+! 2 ;! 2gBBz 2<0 En;=8 >>< >>:!(n+1 2)q (! 2gBBz 2)2+ (0 D)2(2eBz)2(n+ 1)2! 2 ;! 2gBBz 2>0; !(n+1 2)q (! 2gBBz 2)2+ (0 D)2(2eBzn)2+! 2 ;! 2gBBz 2<0(65) and the wavefunctions are given by n;+() =ein(cosn 2j+i+ sinn 2eiji) n;() =ein(sinn+1 2eij+i+ cosn+1 2ji) (66)for! 2gBBz 2>0, and n;+() =ein(cosn+1 2eij+i+ sinn+1 2ji) n;() =ein(sinn 2j+i+ cosn 2eiji) (67) for! 2gBBz 2<0. Here tann=0 D(2eBzn)3 2 ! 2gBBz 2: (68)12 FIG. 7. ( Color online. ) Spin precession of Landau eigenstates along the momentum space trajectory due to the Rashba in- teraction shown for the highest lled Landau level ( n= 12) at the experimental density2= 31011cm2. The spin polar- ization is indicated by red arrows, and the eective magnetic eldis indicated by blue arrows. Forn1 the exact wavefunctions reduce to the semi- classical expressions (36), (37) with W= +1;= 0. V. ELECTRON SPIN RESONANCE In the past, both cyclotron resonance and electron spin resonance (ESR) have been used to study the band pa- rameters of 2D semiconductor systems61{64. In the ab- sence of spin-orbit interaction, ESR occurs when the fre- quency of the applied in-plane magnetic eld coincides with the energy splitting between Zeeman states belong- ing to the orbital level, and therefore simply measures theg-factor. In the presence of spin-orbit coupling, an oscillating in-plane magnetic eld may result in transi- tions between dierent orbital levels, and we expect ESR to be observed in the same range of frequencies as cy- clotron resonance. The ESR probability depends on the angle (34) which describes mixing between j+iandjistates in the pre- cessing Landau level wavefunctions. Thus while mag- netic oscillations oer a sensitive probe of the phase of the eigenvalues of the matrix phase U(2), ESR may pro- vide a complementary measurement in the sense that it probes the spin structure of the Landau level eigenstates. Let us rst consider the situation in electron systems. The probability of transition between dierent levels, n1 may be calculated from the semiclassical wave- functions (36), (37). An oscillating magnetic eld applied in the 2D plane, H/bxx+byygenerates transitions with probability amplitude hn0s0jbxx+byyjnsi=Zh y s0()(bxx+byy)s()i ei(nn0)d 2(69) wheres=nsn0s, sincen;n0are large. There are transitions within the same orbital level ( n+! n), aswell as transitions between dierent spin states in dier- ent orbital levels, ( n+! n+2W;forW! 2gBBz 2>0 and n+! n2W;forW! 2gBBz 2<0). In addition, there exist purely orbital transitions ( n+! nW;+ and n! nW;). The transition probabilities are summarized in Table 1. Let us now consider the case for hole systems. The transition matrix element is given by Hhn0s0jb+2 ++b2 +jnsi= 2eHBznZ y s0() b+e2i+be2i+ s()ei(n0n)d 2: (70) We obtain transitions within the same orbital level only in the caseW= 1 (corresponding, e.g. to the (100) Dres- selhaus interaction). There are transitions between oppo- site spin states ( n+! n2W+2;; n+! n2;for W! 2gBBz 2>0 and n+! n+2W2;; n+! n+2; forW! 2gBBz 2<0), as well as purely orbital tran- sitions ( n+! n+2W;+; n! n+W2;). These results are summarized in Table 1. For the case W= 2, the probability of transition between opposite spin states exhibits a dependence on the direction of the oscillating magnetic eld. Let us consider the case when a static magnetic eld Bk= (Bx;By) is present. The matrix element for the transition is h n2;jHj n;+i/1sin2 2[1 + cos 2(')] (71) where'is the angle between the static and oscillating magnetic eld. These transitions are indicated by the vertical arrows in Fig. 3. VI. CONCLUSION We have obtained a semiclassical expression for the Landau level spectrum in a 2D spin-orbit coupled sys- tem via the one-dimensional Schr odinger equation (12) describing spin evolution around the cyclotron trajectory. In the Born-Oppenheimer approximation, the semiclas- sical quantization condition is strongly modied by spin dynamics and the Landau level problem becomes equiv- alent to that of calculating the SU(2) matrix U() asso- ciated with spin precession around a momentum space orbit of xed radius jj=q j2eBzj(n+1 2). In the semi- classical quantization condition, the eigenvalues of U(2) constitutes the spin-dependent correction to the phase, and in the case of a rotating spin-orbit interaction, con- tains a geometric contribution which is associated with the out-of-plane tilting of the precessing spin wavefunc- tions relative to the driving spin-orbit eld . The impor- tance of the geometric contribution in the experimental regime was illustrated for both nandptype systems in Figs. 2,5,7. We have shown that magnetic oscillations di- rectly probe the phase  of the eigenvalues U(2), while13 Transition jhjHjij2 ElectronsW! 2gBBz 2>0jn;+i!jn;i cos4 2 jn;+i!jn+ 2W;i sin4 2 jn;+i!jnW;+i 1 4sin2jn;+i!jn+W;+i jn;i!jnW;i jn;i!jn+W;i W! 2gBBz 2<0jn;+i!jn;i sin4 2 jn;+i!jn2W;i cos4 2 jn;+i!jn+W;+i 1 4sin2jn;i!jnW;+i jn;i!jn+W;i jn;i!jnW;i HolesW! 2gBBz 2>0jn;+i!jn2;i cos4 2(W6= 2) jn;+i!jn+ 22W;i sin4 2(W6= 2) jn;+i!jn2;i 1sin2 2[1 + cos 2(')] (W= 2) jn;+i!jn+ 2W;+i 1 4sin2jn;+i!jn+W2;+i jn;i!jn+ 2W;i jn;i!jn+W2;i W! 2gBBz 2<0jn;+i!jn+ 2;i sin4 2(W6= 2) jn;+i!jn+ 2W2;i cos4 2(W6= 2) jn;+i!jn2;i 1sin2 2[1 + cos 2(')] (W= 2) jn;+i!jn+ 2W;+i 1 4sin2jn;+i!jn+W2;+i jn;i!jn+ 2W;i jn;i!jn+W2;i TABLE I. ESR matrix elements for transitions between precessing Landau level eigenstates in a spin-orbit eld which rotates about thez-axis with winding number W. The matrix element is given in terms of the angle between spin polarization and thez-axis, and in units (gBjBkj 2)2for electrons and (2 eHBzjBkjn)2for holes. In the case of a static in-plane magnetic eld (W= 2), the ESR matrix element depends on the angle 'between the static and oscillating magnetic elds. in the rotating case ESR measures the polarization of pre- cessing Landau states. When spin dynamics is controlled by variation of external parameters such as the external magnetic eld and gate voltage, this allows mapping of spin precession along the classical orbit. Appendix A: Choice of Landau level labeling In this appendix we present a more detailed derivation of the semiclassical solutions (36), (37) and demonstrate the choice for labeling Landau levels. The semiclassical formalism for the wavefunction () =ein() satis- es Eq. (12) for () i!@ @= [(;n)
!] (A1) where the examples in this work have the form (;n) = [kcos(W+);ksin(W+);z],Wis an integer and kis implicitly ndependent. Using the unitary transfor- mationg() =e1 2iWzand() =e1 2izg()0() wesimplify the interaction term and the equation for 0() (similar to Eq. (29)) ei 2(W+)z(()
)ei 2(W+)z=kx+zz !i!@0() @= kx+ (z+1 2!W )z! 0() (A2) We next rotate around the y axis by the angle (Eq. 34) where cos =k=jBj;sin= (z+1 2!W )=jBjand jBj=q 2 k+ (z+1 2!W )2, so that the equation for 00() =e1 2iy0() becomes i!@(00()) @= [z!]00() !00()ei(1 !jBjz)(A3)14 This latter form multiplies any independent spinor, choosing the states (1 ;0);(0;1) we nd the solutions +() =C+ei(1 !jBj) e1 2i(W+)cos 2 e1 2i(W+)sin 2! () =Cei(1 !jBj) e1 2i(W+)sin 2 e1 2i(W+)cos 2! (A4) where the constants Careindependent. Periodic boundary condition, i.e. uniqueness of wavefunction im- ply integers m, hence two eigenvalues , where (jBj!)1 2W!=m! !!=(jBj+1 2W! )m! (A5) and exponents with1 2W!1 2Ware also integers since the dierence is an integer W. Hence +() =C+eim+ eiWi 2cos1 2 ei 2sin1 2! () =Ceim ei 2sin1 2 eiW+i 2cos1 2! (A6) It is interesting to note that while the full Hamiltonian (1) is not time-reversal invariant, the one-dimensional equation for spin (A1) is invariant under the time-reversal operationT=iyK(whereKis complex conjuation). Hence the solutions () are related by this operator so thatC=C +andm=m+. The integers mcorrespond to relabeling the Landau level index nand within the semiclassical scheme any choice with mnis acceptable. The energies are then given by E n=!(n+1 2) +! =!(nm+1 2)(jB(n)j+1 2W! ) !!(n0+1 2)(jB(n0+m)j+1 2W! ) (A7) where the relabelling n!n0=n+monly aects the lowest lying Landau levels which are not accessible in the semiclassical method. In the semiclassical limit, jB(n0)jjB (n)j; (n0)(n) formn, so the ener- gies and wavefunctions are unchanged under the relabel- ing. For the reasons stated in the text, we have chosen the solutions (36) corresponding to m=W; C=ei 2, and (37) corresponding to m= 0; C=ei 2, which minimise the leading error in the semiclassical scheme and therefore have best agreement with the exact solu- tions presented in Section IV.Appendix B: Exact spectra for n-type systems The Landau level spectrum of systems with a pure Rashba or Dresselhaus interaction may be solved by in- troducing creation and annihilation operators a=p j2eBzj;ay=+p j2eBzj(B1) and diagonalizing the Hamiltonian in the number basis jn;i=jniji whereayajni=njni,zji=ji . 1. Rashba interaction The Hamiltonian is H=!(aya+1 2) +iRp j2eBzj 2(a+ay)gBBz 2z: (B2) The Rashba interaction HR/ia++h:c:couples basis statesjn;iandjn1;+iforn1, withj0;ibeing an eigenstate with energy! 2+gBBz 2. The remaining spectrum may be obtained by diagonalizing the 2 2 Hamiltonian H! !(n+1 2) +gBBz 2iRp j2eBzjn iRp j2eBzjn ! (n1 2)gBBz 2! (B3) in the basis (jn;i;jn1;+i) forn1, which gives energies En;=!nr (! 2+gBBz 2)2+j2eBzj2 Rn : (B4) When! 2+gBBz 2>0, the energy of the eigenstate j0;i coincides with E0;+; in the opposite situation it coincides withE0;. Therefore the complete spectrum is given by En;+; n= 0;1;2;::: En;; n= 1;2;::: (B5) for! 2+gBBz 2>0, and En;+; n= 1;2;::: En;; n= 0;1;2;::: (B6) for! 2+gBBz 2<0. The eigenstates are given by n+= cosn 2jn1;+i+isinn 2jn;i; n=sinn 2jn1;+i+icosn 2jn;i (B7) where tann=Rp j2eBzjn ! 2gBBz 2; (B8)15 andntakes the same values as in the expressions (B5), (B6). The wavefunctions n;() =hj n;iin the- representation may be obtained by use of Eq. (3); we obtain n;+() =ein(cosn 2eij+i+isinn 2ji); n;() =ein(sinn 2eij+i+icosn 2ji):(B9) After a shift of index, En!En+1;, n! n+1;for ! 2+gBBz 2>0 andEn+!En+1;+, n+! n+1;+for ! 2+gBBz 2<0 (so that the spectra in both spin series begin with index n= 0), we obtain the energies (41) and wavefunctions (42), (43) shown in the text. 2. Dresselhaus interaction The Hamiltonian is H=!(aya+1 2) +Dp j2eBzj 2(a+ay+)gBBz 2z: (B10) the Dresselhaus interaction HD/a+h:c:couples ba- sis statesjn;+iandjn1;iwithj0;+ibeing an eigen- state with energy! 2gBBz 2. The remaining spectrum may be obtained by diagonalizing the 2 2 Hamiltonian H! !(n+1 2)gBBz 2Dp j2eBzjn Dp j2eBzjn ! (n1 2) +gBBz 2! (B11) in the basis (jn;+i;jn1;i), which gives energies En;=!nr (! 2gBBz 2)2+j2eBzj2 Dn : (B12) When! 2gBBz 2>0, the energy of the eigenstate j0;+i coincides with E0;+; in the opposite situation it coincides withE0;. Therefore the complete spectrum is given by En;+; n= 0;1;2;::: En;; n= 1;2;::: (B13) for! 2gBBz 2>0, and En;+; n= 1;2;::: En;; n= 0;1;2;::: (B14) for! 2gBBz 2<0. The eigenstates are given by n+= cosn 2jn;+i+ sinn 2jn1;i; n=sinn 2jn;+i+ cosn 2jn1;i (B15)where tann=Dp j2eBzjn ! 2gBBz 2; (B16) andntakes the same values as in the expressions (B13), (B14). The wavefunctions n;() =hj n;iin the- representation may be obtained by use of Eq. (3); we obtain n;+() =ein(cosn 2j+i+ sinn 2eiji); n;() =ein(sinn 2j+i+ cosn 2eiji):(B17) After a shift of index, En!En+1;, n! n+1;for ! 2gBBz 2>0 andEn+!En+1;+, n+! n+1;+for ! 2gBBz 2<0 (so that the spectra in both spin series begin with index n= 0), we obtain the energies (46) and wavefunctions (47), (48) shown in the text. Appendix C: Exact spectra for p-type systems In the text, three situations are discussed: the case of an in-plane magnetic eld, a pure Rashba interaction, and a pure Dresselhaus interaction. In the hole case the creation and annihilation operators are a=+p2eBz;ay=+p2eBz(C1) (note that they are reversed in comparison to the elec- tron case due to the opposite sign of the electric charge). As in the electron case we obtain analytical solutions by diagonalizing the Hamiltonian in the number representa- tion. 1. In-plane magnetic eld The Hamiltonian is H=!(aya+1 2) +eBzHBk(eia2+ei(ay)2+) gBBz 2z: (C2) where the phase is related to the direction of the in- plane magnetic eld via B+=Bkei. The in plane eld, HZ/eia2+h:c: couples basis states jn;+iand jn2;iwithj0;+iandj1;+ibeing eigenstates with energies! 2gBBz 2and3! 2gBBz 2respectively. The re- maining spectrum may be obtained by diagonalizing the 22 Hamiltonian H! !(n+1 2)gBBz 22eHBzBkein 2eHBzBkein!(n3 2) +gBBz 2! ; (C3)16 where n=p n(n1) (C4) in the basis (jn;+i;jn2;i), which gives energies En;=!(n1 2)r (!gBBz 2)2+ (2eHBzBkn)2: (C5) When!gBBz 2>0, the energy of the eigenstates j0;+i andj1;+icoincide with E0;+andE1;+respectively; in the opposite situation they coincide with E0;;E1;. Therefore the complete spectrum is given by En;+; n= 0;1;2;::: En;; n= 2;3;::: (C6) for!gBBz 2>0, and En;+; n= 2;3;::: En;; n= 0;1;2;::: (C7) for!gBBz 2<0. The eigenstates are given by n+= cosn 2jn;+i+ sinn 2eijn2;i; n=sinn 2jn;+i+ cosn 2eijn2;i (C8) where tann=2eHBzBkn !gBBz 2; (C9) andntakes the same values as in the expressions (C6), (C7). The wavefunctions n;() =hj n;iin the- representation may be obtained by use of Eq. (3); we obtain n;+() =ein(cosn 2j+i+ sinn 2e2iji); n;() =ein(sinn 2j+i+ cosn 2e2iji):(C10) After a shift of index, En!En+2;, n! n+2; for!gBBz 2>0 andEn+!En+2;+, n+! n+2;+ for!gBBz 2<0 (so that the spectra in both spin series begin with index n= 0), we obtain the spectrum (51) and wavefunctions (53), (54) shown in the text. 2. Rashba interaction The Hamiltonian is H=!(aya+1 2) +i0 R(2eBz)3 2 2(a3(ay)3+) gBBz 2z: (C11)The Rashba interaction H0 R/a3couples basis states jn;+iandjn3;iforn3. Forn= 0;1;2, the basis statesjn;+iare eigenstates with energy !(n+1 2) gBBz 2. The remaining spectrum may be obtained by diagonalizing the 2 2 Hamiltonian H! !(n+1 2)gBBz 2i0 R(2eBzn)3 i0 R(2eBzn)3!(n5 2) +gBBz 2)! (C12) where n= (n(n1)(n2))1 3 (C13) in the basis (jn;+i;jn3;i), giving energies En;=!(n2)r (3! 2gBBz 2)2+ (0 R)2(2eBzn)3: (C14) When3! 2gBBz 2>0, the energies of the eigenstates jn;+iforn= 0;1;2 coincide with En;+; in the opposite situation they coincide with En;. Therefore the com- plete spectrum is given by En;+; n= 0;1;2;::: En;; n= 3;4;5;::: (C15) for3! 2gBBz 2>0 and En;+; n= 3;4;5;::: En;; n= 0;1;2;::: (C16) for3! 2gBBz 2<0. The eigenstates are given by n;+= cosn 2jn;+i+isinn 2jn3;i n;=sinn 2jn;+i+icosn 2jn3;i (C17) where tann=0 R(2eBzn)3 2 3! 2gBBz 2; (C18) andntakes the same values as in the expressions (C15), (C16). The wavefunctions the -representation are given by projection of the states (C17) onto the basis (3), n;+() =ein(cosn 2j+i+isinn 2e3iji); n;() =ein(sinn 2j+i+icosn 2e3iji): (C19) After a shift of index, En!En+2;, n! n+3; for3! 2gBBz 2>0 andEn;+!En+2;+, n;+! n+3;+ for3! 2gBBz 2<0, we obtain the energies (58) and wavefunctions (60), (61) shown in the text.17 3. Dresselhaus interaction The Hamiltonian is H=!(aya+1 2)gBBz 2z+ 0 D(2eBz)3 2 4((a2ay+aya2)+ ((ay)2a+a(ay)2)) (C20) where0 Dis the Dresselhaus constant for the heterostruc- ture. The Dresselhaus interaction couples basis states jn;+iandjn1;iforn1. The statej0;+iis an eigenstate with energy! 2gBBz 2. The remaining spec- trum is given by diagonalization of the 2 2 Hamiltonian H! !(n+1 2)gBBz 20 D(2eBzn)3 2 0 D(2eBzn)3 2!(n1 2) +gBBz 2! : (C21) in the basis (jn;+i;jn1;i). The energies are given by En+=!nr (! 2gBBz 2)2+ (0 D)2(2eBzn)3: (C22) When! 2gBBz 2>0, the energy of the state j0;+ico- incides with E0;+. In the opposite situation, it coincides withE0;. Thus the complete energy spectrum consists of En;+; n= 0;1;2;::: En;; n= 1;2;::: (C23)for! 2gBBz 2>0, and En;+; n= 1;2;::: En;; n= 0;1;2;::: (C24) for! 2gBBz 2<0. The eigenstates are given by n;+= cosn 2jn;+i+ sinn 2jn1;i n;=sinn 2jn;+i+ cosn 2jn1;i (C25) where tann=0 D(2eBzn)3 2 ! 2gBBz 2; (C26) andntakes the same values as in the expressions (C23), (C24). The wavefunctions in the -representation are given by projection of the states (C25) onto the basis (3), n;+() =ein(cosn 2j+i+ sinn eiji); n;() =ein(sinn 2j+i+ cosn 2eiji):(C27) After a shift of index, En!En+1;, n;! n+1; for! 2gBBz 2>0 andEn;+!En+1;+, n;+! n+1;+ for! 2gBBz 2<0, we obtain the energies (58) and wave- functions (60), (61) shown in the text. 1G. Engels, J. Lange, Th. Sch apers, and H. L uth, Phys. Rev. B 55, 1958 (1997). 2B. Grbi c, et. al. , Phys. Rev. B 77, 125312 (2008). 3S. Datta, B. Das, Appl. Phys. Lett. 56, 665 (1990). 4J. Schliemann, J.C. Egues, D. Loss, Phys. Rev. Lett. 90, 146801 (2003). 5J. Fabian et. al. , Acta Physica Slovaca 57, 474 (2007). 6Zutic I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76 323 7P. Nozi eres and T. Lewiner, Journal de Physique 34, 901 (1973). 8T. Jungwirth, Q. Niu, A.H. MacDonald, Phys. Rev. Lett. 88207208 (2002). 9N. Nagaosa, J. Sinova et. al. , Rev. Mod. Phys. 82, 1539 (2010). 10S. Murakami, N. Nagaosa, S.-C. Zhang, Science 3011348 (2003). 11J. Sinova, et. al. , Phys. Rev. Lett. 92, 126603 (2004). 12B.A. Bernevig, S.-C. Zhang, Phys. Rev. Lett. 95, 016801 (2005). 13J. Schliemann and D. Loss, Phys. Rev. B 71, 085308 (2005).14G. Bergmann, Phys. Rep. 1071 (1984). 15A.G. Aronov and Y.B. Lyanda-Geller, Phys. Rev. Lett. 70, 343 (1993). 16D.P. Arovas and Y. Lyanda-Geller, Phys. Rev. B 57, 12302 (1998). 17J.-B. Yau, E.P. De Poortere, and M. Shayegan, Phys. Rev. Lett. 88, 146801 (2002). 18F. Nagasawa et. al. Nat. Commun. 4, 2526 (2013). 19T. Li and O.P. Sushkov, Phys. Rev. B 87, 165434 (2013). 20R. Winkler, S.J. Papadakis, E.P. De Poortere, and M. Shayegan, Phys. Rev. Lett. 84, 713 (2000). 21S. Keppeler and R. Winkler, Phys. Rev. Lett. 88, 046401 (2002). 22J. P. Eisenstein, H. L. St ormer, V. Narayanamurti, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 53, 2579 (1984). 23J. Luo, H. Munekata, F.F. Fang, P.J. Stiles, Phys. Rev. B 38, 10142R (1988). 24J. Luo, H. Munekata, F.F. Fang, P.J. Stiles, Phys. Rev. B 41, 7685 (1990). 25M. Schultz, F. Heinrichs, U. Merkt, T. Colen, T. Skauli, S. Lovold, Semicond. Sci. Technol. 11, 1168 (1996).18 26P. Ramvall, B. Kowalski, and P. Omling, Phys. Rev. B 55 7160 (1997). 27J.P. Heida, B.J. van Wees, J.J. Kuipers, T.M. Klapwijk, and G. Borghs, Phys. Rev. B 57, 11911 (1998). 28C.-M Hu, et. al. Phys. Rev. B 607736 (1999). 29D. Grundler, Phys. Rev. Lett. 846074 (2000). 30S.I. Dorozhkin, Solid State Commun. 72, 211 (1989). 31T. Matsuyama, R. K ursten, C. Meissner, U. Merkt, Phys. Rev. B 61, 15588 (2000). 32K.S. Novoselov et. al. , Nature 438, 197 (2005). 33Y. Zhang et. al. , Nature 438, 201 (2005). 34J.G. Analytis et. al , Nature Phys. 6, 910 (2010). 35F. Xiu et. al. , Nature Nanophys. 6, 216 (2011). 36J. Xiong et. al. , Physica E 44, 917 (2012). 37B. Sacepe et. al. , Nature Comm. 2, 575 (2011). 38J.N. Fuchs, F. Pi echon, M.O. Goerbig, G. Montambaux, Europhys. J. B 77351 (2010). 39A.R. Wright, R.H. McKenzie, Phys. Rev. B 87085411 (2013). 40A.R. Wright, Phys. Rev. B 87085426 (2013). 41M.O. Goerbig, G. Montambaux, F. Pi echon, Europhys. Lett. 105, 57005 (2014). 42L. Onsager, Philos. Mag. 43, 1006 (2006). 43T.Li, L.Yeoh, A. Srinavasan, O. Klochan, D.A. Ritchie, M.Y. Simmons, O.P. Sushkov, A.R. Hamilton, to be pub- lished . 44M. Zarea and S.E. Ulloa, Phys. Rev. B 72, 085342 (2005). 45D. Zhang, J. Phys. A: Math. Gen. 39, L477 (2006). 46E. Papp and C. Micu, Superlatt. Microstruct. 48, 9 (2010).47S.I. Erlingsson, J.C. Egues, D. Loss, Phys. Rev. B 82, 155456 (2010). 48Hereafter we indicate quantum operators in bold, and clas- sical phase space variables in plain text. 49R. Winkler, Spin-Orbit Coupling Eects in Two- Dimensional Electron and Hole Systems. Springer Tracts in Modern Physics. Vol. 191 (2003). 50For a review of quantization of the harmonic oscillator in action-angle variables, seeP. Carruthers, M.M. Niento, Rev. Mod. Phys. 40, 411 (1968). 51L. P. Rokhinson, V. Larkina, Y. B. Lyanda-Geller, L. N. Pfeier, K. W. West, Phys. Rev. Lett. 93, 146601 (2004). 52I.M. Lifshitz and L.M. Kosevich, Sov. Phys. JETP-USSR, 6, 67 (1958). 53M. Duckheim and D. Loss, Nature Phys. 2, 195 (2006). 54M.V. Berry, Proc. R. Soc. Lond. Ser. A 39245 (1982). 55Y.A. Bychkov and E.I. Rashba, Sov. Phys. -JETP Lett. 39, 78 (1984). 56G. Dresselhaus, Phys. Rev. 100, 580 (1955). 57J. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955). 58S.D. Ganichev, et. al. , Phys. Rev. Lett. 92, 256601 (2004). 59Phys. Rev. B 74, 033305 (2006). 60L. Meier, et. al. Nature Phys. 3, 650 (2007). 61D. Stein, K. von Klitzing, G. Weimann, Phys. Rev. Lett. 51, 130 (1983). 62C.F.O. Grae, M.S. Brandt, M. Stutzmann, M. Holzmann, G. Abstreiter, and F. Sch aer, Phys. Rev. B 59, 13242 (1999). 63Z. Wilamowski, W. Jantsch, H. Malissa, U. R ossler, Phys. Rev. B 66, 195315 (2002). 64J. Matsunami, M. Ooya, T. Okamoto, Phys. Rev. Lett. 97, 066602 (2006).

1003.2735v4.Motion_and_gravitational_wave_forms_of_eccentric_compact_binaries_with_orbital_angular_momentum_aligned_spins_under_next_to_leading_order_in_spin_orbit_and_leading_order_in_spin_1__spin_2__and_spin_squared_couplings.pdf

arXiv:1003.2735v4 [gr-qc] 29 Aug 2013Motion and gravitational wave forms of eccentric compact bi naries with orbital-angular-momentum-aligned spins under next- to-leading order in spin-orbit and leading-order in spin(1)-spin(2) a nd spin-squared coupling M Tessmer, J Hartung, and G Schäfer Theoretisch-Physikalisches Institut, Friedrich-Schill er-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany E-mail:m.tessmer@uni-jena.de Abstract. A quasi-Keplerian parameterisation for the solutions of se cond post-Newtonian (PN) accurate equations of motion for spinning compact binaries is obtain ed including leading order spin-spin and next- to-leading order spin-orbit interactions. Rotational def ormation of the compact objects is incorporated. For arbitrary mass ratios the spin orientations are taken to be parallel or anti-parallel to the orbital angular momentum vector. The emitted gravitational wave forms are g iven in analytic form up to 2PN point particle, 1.5PN spin-orbit and 1PN spin-spin contributions, whereby the spins are assumed to be of 0PN order. PACS numbers: 04.25.Nx, 04.25.-g, 04.20.-q, 04.30.-wMotion and gravitational wave forms of eccentric spinning com pact binaries 2 1. Introduction Inspiralling and merging neutron star (NS) and/or black hol e (BH) binaries are promising sources for continuous gravitational waves (GW). Ground-based laser i nterferometers as e.g. LIGO, VIRGO, and GEO are already searching for those astrophysical sources [1]. For a successful search with the help of matched filtering of the emitted GW signals, one needs a detailed knowledge of t he orbital dynamics. Spin effects of higher order were discussed in [2, 3, 4, 5, 6] for the inspiral of compact binaries were the orbits were assumed to be quasi-circular. A recent publication [7] gave a numeri cal insight into the evolution of binary systems having spins that are parallel to the orbital angular moment um and evolving in quasi-circular orbits. Because there are many physical degrees of freedom involved , it is computationally desirable to have an analytical description, especially for interferometer s working in the early inspiral phase, where numerical relativity currently fails to produce hundreds of orbital c ycles. For non-spinning compact binaries, the post-Newtonian (PN) expansion in the near-zone has been car ried out through 3.5PN order [8] and 3.5PN accurate inspiral templates have been established for circ ular orbits [9, 10]. For numerical performances of these templates see [11, 12]. Observations lead to the assum ption that many astrophysical objects carry a non-negligible spin, such that the effect of spin angular momentum cannot be ignor ed for detailed data analysis. The problem of spins in General Relativity (GR) wa s first discussed in [13, 14, 15] and considerable further developments were made in the 1970s [16, 17, 18, 19], and in recent years as well. Apostolatos [20] showed in his analysis of simple precession for “circular” orbits and spinning self-gravitating sourc es that the form of the GW signal is affected. The amount of the energy radi ated by the binary system with spin has been determined by [21]. Therefore, we want to implement recent breakthroughs in dyn amical relativity of spinning compact binaries into a useful prescription for data analysis appli cations. Our aim is to connect the following items. (i) The “standard” procedure to compute the evolution of ecc entric orbits from the Hamilton equations of motion (EOM). For eccentric orbits, but neglecting spin effe cts, Damour and Deruelle [22] presented a phasing at 1PN employing conchoidal transformations to red uce the structure for the radial motion. Later publications [23, 24, 25] used Hamilton EOM instead of Lagrange ones and employed a more general scheme for a solution to conservative 3PN dynamics w ithout spin. (ii) The 2PN point particle (PP), next-to-leading order spi n-orbit (NLO-SO) and leading-order spin(1)- spin(2) (LO-S 1S2) and spin(1)-spin(1) (LO-S2) interaction contributions. As a starting point, we assume (anti–) aligned spin and orbit al angular momentum vectors for an estimation of the effects. It is interesting to analyse this matter syste m configuration because numerical results of a recent publication indicate that maximum equal-spins alig ned with the orbital angular momentum lead to observable volume of up to ∼30 times larger than the corresponding binaries with the spi ns anti-aligned to the orbital angular momentum [26]. From Figure 10 in [26], one can also find an observable volume of those binaries up to ∼8 times larger compared to non-spinning binaries. These aut hors conclude that those systems are among the most efficient GW sources in the universe . In another recent publication [27] it can be found that in gas-rich environments the spins of two black holes can align with the larger scale accretion disc on a timescale that is short as 1%of the accretion time. Due to the model of those authors, havi ng two black holes interacting independently with an accretion di sc, their spins tend to be aligned with each other and with the orbital angular momentum more or less depending on the model parameters. We work only with the conservative Hamiltonian for the time b eing, and restrict our attention to terms up to 2.5PN order overall, assuming maximally spinning hole s. This means neglecting both the well-known 3PN PP contributions, and the NLO- S1S2[28], as well as the NLO- S2contributions, which have recently been derived for general compact binaries [29]. This latter publ ication came out at a late stage in our calculations, but it should be a straightforward task to include these term s in a future publication. If the objects are slowly rotating, the considered leading- order spin-squared contributions are shifted to 3PN order and, for consistency, the 3PN PP Hamiltonian has to be included. The 3PN PP contributions to the orbital elements are available in the literature [25] an d simply have to be added to what we are going to present in this paper. Anyway, this work is consistently wor ked out to all terms up to 2.5PN, having maximal rotation or not and will list all results in the spins which ar e counted of 0PN order. The paper is organised as follows: Section 2 summarises and d iscusses the Hamiltonian terms we want to include in our prescription. Section 3 investigates the c onservation of initial spin and orbital angularMotion and gravitational wave forms of eccentric spinning com pact binaries 3 momentum alignment conditions. In Section 4, we briefly summ arise the Keplerian parameterisation for Newtonian orbital dynamics and outline the generalisation to higher PN accurate dynamics. The solution of the Hamilton EOM is the subject of Section 5. Section 6 summar ises all important results of our procedure. In Section 7 we give some formulae for the polarisations of th e gravitational waves which are emitted by the system. Calculations were mostly done with Mathematica andxTensor (see [30, 31] and references therein). 2. Spin and orbital dynamics In the following sections, the dynamics of spinning compact binaries is investigated, where the SO contributions are restricted to NLO and the S1S2andS2to LO. The PP contributions are cut off after the 2PN terms. The Hamiltonian associated therewith reads ˆH(ˆx1,ˆx2,ˆp1,ˆp2,ˆS1,ˆS2) =ˆHN PP+ˆH1PN PP+ˆH2PN PP+ˆHLO SO+ˆHNLO SO+ˆHLO S2+ˆHLO S1S2. (1) These are sufficient for maximally rotating black holes up to a nd including 2.5PN. The variables ˆpaandˆxa are the linear canonical momentum and position vectors, res pectively. They commute with the spin vectors ˆSa, where “ a” denotes the particle label, a= 1,2.HPPis the conservative point-particle ADM Hamiltonian known up to 3PN, see, e.g., [32] and [33]. The LO spin dependen t contributions are well-known, see, e.g., [17, 19, 34]. HNLO SOwas recently found in [35, 36] and HNLO S1S2in [36, 28] (the latter was confirmed in [37]). The leading-order S2 1andS2 2Hamiltonians were derived in [18] and [38]. Measuring the GW signal, determinantion of constraints of the equation of state of both extended bodi es is possible in principle. Hamiltonians of cubic and higher order in spin are given in [39, 40], and higher PN or ders linear in spin are tackled in [41, 42]. The four-dimensional model behind the Hamiltonians linear in the single spin variables is given by the Mathisson-Papapetrou equations [14, 13] DSµν a dτ= 2p[µ auν] a, (2) Dpµ a dτ=−1 2Rµ ρβαuρ aSβα a, (3) and the Tulczyjew stress-energy tensor density [43, 44] √−gTµν=/summationdisplay a/integraldisplay dτ/bracketleftbigg u(µ apν) aδ(4)a+(u(µ aSν)α aδ(4)a);α/bracketrightbigg , (4) which can be used as the source of the gravitational field in th e Einstein equations (see [45] and references therein for spin-squared corrections in the stress-energy tensor). Here, the four-dimensional coordinate of the a-th object is denoted by xµ aandpaµis the linear momentum, uµ athe 4-velocity, normalised as uµuµ=−1, τthe proper time parameter, Sµν athe spin tensor, “ ;” denotes the four-dimensional covariant derivative, and δ(4)a=δ(x−xa)with normalisation/integraltextd4xδ(4)a= 1.Rµ ρβαis the four-dimensional Riemann tensor and D/dτthe absolute derivative, which is a derivative in direction of the 4-velocity of the (massive) particle. In order to close the system of equations, one has to impose a spi n supplementary condition (SSC), which is most conveniently taken to be Sµν apaν= 0. (5) To linear order in spin, paµ=mauaµ, wheremais the mass parameter of the a-th object. Notice that the matter variables appearing in the Mathisson-Papapetrou eq uations and the stress-energy tensor are related to the canonical variables appearing in the Hamiltonians by rather complicated redefinitions. We are going to work in the centre-of-mass (COM) frame, where the total linear momentum vector is zero, i.e. ˆp2=−ˆp1=−ˆp. The Hamiltonians taken into account depend on ˆx1andˆx2only in the combinations ˆx1−ˆx2, so they can be re-expressed in terms of n12=−n21=ˆx/ˆ r =x/r, the normalised direction from particle 1 to 2, and ˆ r =|ˆx1−ˆx2|withˆx=ˆx1−ˆx2. We will make use of the following scalings to convert quantit ies with hat to dimensionless ones, H≡ˆH µc2, (6) x≡ˆx/parenleftbiggGm c2/parenrightbigg−1 , (7)Motion and gravitational wave forms of eccentric spinning com pact binaries 4 p≡ˆp(µc)−1, (8) Sa≡ˆSa/parenleftbiggGma c2(mac)/parenrightbigg−1 . (9) Here,m≡m1+m2denotes the total mass and µ≡m1m2/mis the reduced mass. The speed of light is denoted by candGis Newton’s gravitational constant. Additionally, we intr oduce the reduced orbital angular momentum vector h≡rn12×pand its norm h≡ |h|. Explicitly, the contributions to the rescaled version of Eq uation (1) read HN PP=p2 2−1 r, (10) H1PN PP=ǫ2/braceleftbigg1 8(3η−1)/parenleftbig p2/parenrightbig2−1 2/bracketleftBig (3+η)(p2)+η(n12·p)2/bracketrightBig1 r+1 2r2/bracerightbigg , (11) H2PN PP=ǫ4/braceleftBigg 1 16/parenleftbig 1−5η+5η2/parenrightbig/parenleftbig p2/parenrightbig3+1 8/bracketleftbigg/parenleftbig 5−20η−3η2/parenrightbig/parenleftbig p2/parenrightbig2 −2η2(n12·p)2/parenleftbig p2/parenrightbig −3η2(n12·p)4/bracketrightbigg1 r+1 2/bracketleftBig (5+8η)/parenleftbig p2/parenrightbig +3η(n12·p)2/bracketrightBig1 r2 −1 4(1+3η)1 r3/bracerightBigg , (12) HLO SO=ǫ2δαso r3/braceleftBig/parenleftBig 1−η 2+/radicalbig 1−4η/parenrightBig (h·S1)+/parenleftBig 1−η 2−/radicalbig 1−4η/parenrightBig (h·S2)/bracerightBig , (13) HNLO SO=ǫ4δαso 16r4/braceleftbigg (h·S1)/bracketleftbigg 12ηr/parenleftBig 1−η+/radicalbig 1−4η/parenrightBig (n12·p)2 +ηr/parenleftBig 9−6η+19/radicalbig 1−4η/parenrightBig/parenleftbig p2/parenrightbig −16/parenleftBig (η+3)/radicalbig 1−4η+3/parenrightBig/bracketrightbigg −(h·S2)/bracketleftbigg 12ηr/parenleftBig −1+η+/radicalbig 1−4η/parenrightBig (n12·p)2 +ηr/parenleftBig −9+6η+19/radicalbig 1−4η/parenrightBig/parenleftbig p2/parenrightbig −16/parenleftBig (η+3)/radicalbig 1−4η−3/parenrightBig/bracketrightbigg/bracerightbigg , (14) HLO S1S2=ǫ2δ2αs1s2η r3{3(n12·S1)(n12·S2)−(S1·S2)}, (15) HLO S2=ǫ2δ2αs2 2r3/braceleftbigg λ1/parenleftBig −1+2η−/radicalbig 1−4η/parenrightBig/parenleftBig 3(n12·S1)2−(S1·S1)/parenrightBig +λ2/parenleftBig −1+2η+/radicalbig 1−4η/parenrightBig/parenleftBig 3(n12·S2)2−(S2·S2)/parenrightBig/bracerightbigg , (16) whereη≡µ/mis the symmetric mass ratio. Without loss of generality we as sume that m1> m2. Such an assumption is necessary, because the spins are scaled with t he individual masses in a non-symmetric way. We introduced dimensionless “book-keeping” parameters ǫto count the formal 1/corder and δto count the spin order (linear or quadratic). Evaluating all given q uantities, those have to be given the numerical value 1. The parameters αso,αs1s2,αs2distinguish the spin–orbit, spin(1)–spin(2) and the spin- squared contributions and can have values 1or0, depending on whether the reader likes to incorporate the as sociated interactions. The spins are denoted by S1for object 1andS2for object 2. Notice that the S2 1and S2 2Hamiltonians depend on constants λ1andλ2, respectively, which parametrise the quadrupole deformat ion of the objects 1 and 2 due to the spin and take different values for, e.g., blac k holes and neutron stars. For black holes, λa=−1 2and for neutron stars, λacan take continuous values from the interval [−2,−4][34, 46]‡. ‡Note that the definition of the λadepends on the definition of the spin Hamiltonian and, thus, c an be arbitrarily normalised.Motion and gravitational wave forms of eccentric spinning com pact binaries 5 The parallelism condition tells us to set the spins to Sa=χah/h, where−1< χa<1. During our calculations, we insert the condition of maximal rotation ( Sa∼ǫ) to cut off every quantity after 2.5PN, but list our results in formal orders Sa∼ǫ0(for the formal counting, see, e.g., [40] and also Appendix A of [42]). However, for Sa∼ǫ2, many spin contributions are of the order O(ǫ6), i.e. 3PN which is beyond our present 2PN PP dynamics. The reader may insert either Sa∼ǫ(maximal rotation) or Sa∼ǫ2(slow rotation). The next step is to evaluate the EOM due to these Hamiltonians and to find a parametric solution. As stated, we will restrict ourselves to parallel or anti-parallel ang ular momenta and will, finally, only have to take care of the motion in the orbital plane. 3. Conservation of parallelism of handS1andS2 The motion of binaries with arbitrarily-oriented spins is, in general, chaotic as soon as the spin-spin interaction is included [47, 48]. For special configurations, despite th is, it is possible to integrate the EOM analytically, which particularly is the case for aligned spins and orbital angular momentum. The time derivatives of the spins Saand the total angular momentum Jare governed by the Poisson brackets with the total Hamiltonian, given by [S1,H] =δ2ǫ2/braceleftbigg αs1s2η r3(3(S2·n12)(n12×S1)+(S1×S2)) +αs2(n12×S1) r3(S1·n12)3λ1/parenleftBig 2η−1−/radicalbig 1−4η/parenrightBig/bracerightbigg +δ/braceleftbigg αsoǫ2(h×S1) r3/parenleftBig −η 2+/radicalbig 1−4η+1/parenrightBig +αsoǫ4/bracketleftbigg(h×S1) r3/parenleftbigg3 4(p·n12)2η/parenleftBig 1−η+/radicalbig 1−4η/parenrightBig +1 16(p2)η/parenleftBig 9−6η+19/radicalbig 1−4η/parenrightBig/parenrightbigg −(h×S1) r4/parenleftBig 3+(η+3)/radicalbig 1−4η/parenrightBig/bracketrightbigg/bracerightbigg , (17) [S2,H] = [S1,H](1↔2), (18) [J,H] = [h,H]+[S1,H]+[S2,H] = 0. (19) Furthermore, the magnitudes of the spins are conserved, bec ause the spins commute with the linear momentum and the position vector and fulfill the canonical an gular momentum algebra. Note that the operation (1↔2)switches the label indices of the individual particles and g oes along with n12↔n21=−n12. Equation (19) is not displayed completely here like Equatio n (17). If we assume parallel spins and orbital angular momentum at t= 0, all the above Poisson brackets vanish exactly. Anyway, thi s is insufficient to conclude the conservation of parallelism of hand the spins for all times t >0since S1(t) =S1|t=0+[S1,H]|t=0t+1 2[[S1,H],H]|t=0t2+... =∞/summationdisplay n=01 n![S1,H]n|t=0tn, (20) where [S1,H]n=/bracketleftbig [S1,H]n−1,H/bracketrightbig , [S1,H]0=S1. (21) Because the system of variables S1andS2has to be completed with randpto characterise the matter system, one has to give clear information about the full syst em of EOM. It is important that, even with vanishing Poisson brackets of HwithS1andS2,randpdo change due to the orbital revolution. Thus, one has to clarify if this non-stationary subsystem of the EO M is able to cause violation of the parallelism We consistently use the notation mentioned above.Motion and gravitational wave forms of eccentric spinning com pact binaries 6 condition during time evolution. From the stability theory of autonomous ordinary differential equations it is well known that there is a fixed point if alltime derivatives of the system vanish. In the case of a system starting at such a fixed point at t= 0it will not be able to evolve away from this point. The discuss ion of these issues is the main point in the following two subsectio ns. 3.1. Discussion via conservation of constraints One way to show the non-violation of the initial constraint o fh/bardblS1,S2due to the motion of the binary is to argue via the time derivatives of the constraints. These s hould be written as a linear combination of the constraints themselves. Let Ca(x,p,S) = 0, (22) be the initial constraints of the system. Dirac [49, p. 36] ar gued: If one can write ˙Ca=/summationdisplay bDab(x,p,S)Cb, (23) for the time derivatives of the constraints, the constraint s are conserved. That is due to the fact that every time derivative of Equation (23) generates only new time der ivatives of the constraints on the one hand, which can be expressed as a linear combination of constraint s, or time derivatives of the quantities appearing inDabtimes the constraints on the other. In our case the constraints read S1−|S1| hh=S1−˜χ1h= 0, (24) S2−|S2| hh=S2−˜χ2h= 0, (25) with˜χ1and˜χ2denoting the ratios of the spin lengths and the orbital angul ar momentum. In general, the quantities ˜χahave non-vanishing time derivatives, d˜χa dt=−|Sa| h3/parenleftBig ˙h·h/parenrightBig =−˜χa/parenleftBig ˙h·h/parenrightBig h2. (26) Due to the conservation of the total angular momentum J, the derivatives of (24) and (25) can be expressed via§ d dt(Sa−˜χah) =dSa dt+ ˜χa/parenleftBig ˙h·h/parenrightBig h2h−˜χadh dt =dSa dt−˜χa/parenleftbigg 1−h h⊗h h/parenrightbiggdh dt =dSa dt+ ˜χa/parenleftbigg 1−h h⊗h h/parenrightbigg/summationdisplay bdSb dt, (27) where the tensor product 1/h2h⊗his the projector onto the hdirection. Note that the constraint equations only depend on the spin derivatives in linear manner. Hence, it is sufficient to analyse the structure of Equation (17), dS1 dt=D1(S2·n12)(n12×S1)+D2(S1×S2) +D3(S1·n12)(n12×S1)+D4(h×S1). (28) The coefficients Dk(k= 1,...,4) are all scalar functions of the linear momentum p, the separation rand other intrinsic quantities. We are allowed to add vanishing terms to Equation (28), namely dS1 dt=D1[(S2·n12)−˜χ2(h·n12)](n12×S1)+D2(S1×S2) +D3[(S1·n12)−˜χ1(h·n12)](n12×S1)+D4(h×S1). (29) §The reader should be aware that, for a general discussion, sp ins and orbital angular momentum have to be scaled with the sa me quantity to preserve the conservation of the total angular m omentum. In our case, the scaling is different, but the discus sion is only of structural nature and the mass coefficients emerging f rom a differing scaling can be absorbed into the appearing fac tors.Motion and gravitational wave forms of eccentric spinning com pact binaries 7 As well, we can add a term to the D2coefficient and subtract it at the end, getting dS1 dt=D1[(S2·n12)−˜χ2(h·n12)](n12×S1) +D2[(S1×S2)−˜χ2(S1×h)] +D3[(S1·n12)−˜χ1(h·n12)](n12×S1) +(D4−D2˜χ2)(h×S1). (30) Finally, we can insert a vanishing term into the modified last one: dS1 dt=D1[(S2·n12)−˜χ2(h·n12)](n12×S1) +D2[(S1×S2)−˜χ2(S1×h)] +D3[(S1·n12)−˜χ1(h·n12)](n12×S1) +(D4−D2˜χ2)[(h×S1)−˜χ1(h×h)]. (31) We still need to compute the time derivative of S2to obtain the full derivative of the constraints. Therefore , letEkbe the scalar coefficients in dS2/dt(equivalent to the Dkin (31)). Using this, we can rewrite it as dS2 dt=E1[(S1·n12)−˜χ1(h·n12)](n12×S2) +E2[(S1×S2)−˜χ1(h×S2)] +E3[(S2·n12)−˜χ2(h·n12)](n12×S2) +(E4+E2˜χ1)[(h×S2)−˜χ2(h×h)]. (32) Thus, the complete time derivative of e.g. the S1constraint (27) is given by d dt(S1−˜χ1h) =/parenleftbigg (1+ ˜χ1)1−˜χ1h h⊗h h/parenrightbigg ·/braceleftbigg D1[(S2·n12)−˜χ2(h·n12)](n12×S1) +D2[(S1×S2)−˜χ2(S1×h)] +D3[(S1·n12)−˜χ1(h·n12)](n12×S1) +(D4−D2˜χ2)[(h×S1)−˜χ1(h×h)]/bracerightbigg + ˜χ1/parenleftbigg 1−h h⊗h h/parenrightbigg ·/braceleftbigg E1[(S1·n12)−˜χ1(h·n12)](n12×S2) +E2[(S1×S2)−˜χ1(h×S2)] +E3[(S2·n12)−˜χ2(h·n12)](n12×S2) +(E4+E2˜χ1)[(h×S2)−˜χ2(h×h)]/bracerightbigg . (33) In each of the summands of EkandDkin the above equation, one can factor out the constraints lin early. Thus, they do vanish if the constraints are inserted. 3.2. Discussion via symmetry arguments To underline the results of subsection 3.1 we want to show tha t the multi Poisson brackets (21) all vanish if we demand the parallelism of hand the spins, as a complement to the constraint evolution an alysis. During the calculation of the expressions, we truncated the terms t o quadratic order in spin and to 2.5PN order, counting the spin maximally rotating. There is a finite set of terms which are axial vectors and linear or quadratic in spin. Additionally, the spin has to appear in a v ector product. The reason is that the spins commute with the PP Hamiltonians, and Poisson brackets of sp ins with spin Hamiltonians will give crossMotion and gravitational wave forms of eccentric spinning com pact binaries 8 products of spins with angular momentum horS1,S2, respectively. In the Hamiltonians, there are only scalar products of spins with other vectors or with the spins themselves, so that the ǫijkare still remaining after evaluation of the Poisson brackets. The products with the correct symmetry and linear and quadratic in spin are of the form A/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright A×A P×P·S/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright A·A P·PandP/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright A×P·PS/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright A·P whereAstands for axial vectors, Pfor polar vectors, Sfor scalars and PSfor pseudo scalars. The vector Acan beh,S1,S2which are axial vectors and Pcan ben12,pwhich are polar vectors. Now we enumerate all spin products that emerge when we compute the first multi P oisson brackets, omitting scalar factors like e.g. functions of randη: [S1,H] : ( S1×n12)(S1·n12);(S1×n12)(S2·n12); (S1×S2);(h×S1), (34) [[S1,H],H] : ( S1×n12)(p·S1);(S1×n12)(p·S2); (S1×p)(n12·S1);(S1×p)(n12·S2), (35) [S1,H]3: (S1×p)(p·S1);(S1×p)(p·S2); (S1×p)(n12·S1);(S1×p)(n12·S2). (36) Additionally, terms coming from Poisson brackets of S2withHappear and can be computed from above with the operation ( 1↔2). Evaluation of higher multi Poisson brackets will not crea te new terms, but only increase the number of already known factors in the products . These terms vanish identically if we consider the parallelism between hand the spins because of (h·n12) = 0 and(h·p) = 0, by construction. Due to the vanishing Poisson bracket of J=h+S1+S2with the Hamiltonian H(this is true even without demanding the parallelism), the disappearance of all multi Poisson br ackets of S1andS2withHturns out to be sufficient to conclude the constancy of hif the motion starts with h/bardblS1,S2. 4. Kepler Parameterisation In Newtonian dynamics the Keplerian parameterisation of a c ompact binary is a well-known tool for celestial mechanics, see e.g. [50]. After going to spherical coordina tes in the COM, ( r,θ,φ) with the associated orthonormal vectors ( er,eθ,eφ) and restricting to the θ=π/2plane, the Keplerian parameterisation has the following form: r=a(1−ecosu), (37) φ−φ0=v, (38) v= 2 arctan/bracketleftBigg/radicalbigg 1+e 1−etanu 2/bracketrightBigg . (39) Here,ais the semimajor axis, eis the numerical eccentricity, uandvare eccentric and true anomaly, respectively. The time dependency of randφis given by the Kepler equation, ℓ=n(t−t0) =u−esinu, (40) whereℓis the mean anomaly and nthe so-called mean motion, defined as n≡2π PwithPas the orbital period [51]. In these formulae t0andφ0are some initial instant and the associated initial phase. I n terms of the conserved quantities E, which is the scaled energy (see Equation (6)) and numerical ly identical to H, and the orbital angular momentum h, the orbital elements e,aandnsatisfy a=1 2|E|, (41) e2= 1−2h2|E|, (42) n= (2|E|)3/2. (43)Motion and gravitational wave forms of eccentric spinning com pact binaries 9 For higher PN accurate EOM it is possible to get a solution in a perturbative way, having the inverse speed of light as the perturbation parameter. The 1PN accurate Keplerian like (from now on we refer to quasi- Keplerian ) parameterisation was first found in [22] and extended for no n-spinning compact binaries in [23, 25] to 2PN and finally 3PN accuracy. In the recent past a number of efforts has been undertaken to ob tain a solution to the problem of spinning compact binaries via calculating the EOM for spin-related a ngular variables in harmonic gauge. For circular orbits, including radiation reaction (RR), the authors of [ 52] evaluated several contributions to the frequency evolution and the number of accumulated GW cycles up to 2PN, s uch as from the spin, mass quadrupole and the magnetic dipole moment parts. The gravitational wav e form amplitudes as functions of separations and velocities up to and including 1.5PN PP and 1.5PN SO corre ctions are given in [53], discussed for the extreme mass ratio limit in the Lense-Thirring approximati on and later in [54] and [55] for comparable mass binaries. Recently, in [56] a set of independent variables a nd their EOM, characterising the angular momenta, has been provided. For circular orbits with arbitrary spin orientations and le ading-order spin-orbit interactions, the spin and orbital solutions for slightly differing masses were giv en in [57]. Including LO contributions of S2, S1S2and SO as well as the Newtonian and 1PN contributions to the EO M, a certain time-averaged orbital parameterisation was found in [58], for a time scale where th e spin orientations are almost constant, but arbitrary and the radial motion has been determined. Symbol ically, those solutions suggest the following form for the quasi-Keplerian parametrisation including sp in interactions: r =ar(1−ercosu), (44) n(t−t0) =u−etsinu+Fv−u(v−u)+Fvsinv+F2vsin2v+F3vsin3v+... , (45) 2π Φ(φ−φ0) =v+G2vsin2v+G3vsin3v+G4vsin4v+G5vsin5v+... , (46) v = 2 arctan/bracketleftBigg/radicalBigg 1+eφ 1−eφtanu 2/bracketrightBigg . (47) The coefficients F...,G...are PN functions of E,handη. At the end of the calculation for binary dynamics with spin, they will obviously include spin dependencies as well. In case RR is included, the orbital elements are not longer to be regarded as constants. Damour, Gopakumar and Iyer published equations of motion for these e lements for the case that the RR is a small effect and the time derivatives due to RR will not contribute t o the GW expressions explicitly [59, 60]. Spin effects of RR in eccentric orbits were discussed in [61] and re ferences therein. 5. The quasi-Keplerian parameterisation for aligned spinn ing compact binaries Having proven constancy in time of the directions of angular momenta, we can adopt the choice of spherical coordinates with h/bardbleθ(in theθ=π/2plane) and the basis ( n12=er,eφ). Hamilton’s equations of motion dictate ˙r=n12·˙r=n12·∂H ∂p, (48) r˙φ=eφ·˙r=eφ·∂H ∂p, (49) with˙r= dr/dtand˙φ= dφ/dt, as usual. The next standard step is to introduce s≡1/r, such that ˙r=−˙s/s2. Using Equations (48) and (49), we obtain a relation for ˙r2and thus ˙s2and another one for ˙φ/˙s= dφ/ds, where the polynomial of ˙s2is of third degree in s. To obtain a formal 2PN accurate parameterisation /bardbl, we first concentrate on the radial part and search for the two non zero roots of ˙s2= 0, namely s+ands−. The results, to Newtonian order, are s+=1 ar(1−er)=1+/radicalbig 1−2h2|E| h2+O(ǫ2), (50) /bardblWhen we talk about a formal solution at 2PN here, we mean that w e incorporate all terms up to the order ǫ4where the spins are formally counted of order ǫ0.Motion and gravitational wave forms of eccentric spinning com pact binaries 10 s−=1 ar(1+er)=1−/radicalbig 1−2h2|E| h2+O(ǫ2), (51) s−representing periastron and s+as the apastron. Next, we factorise ˙s2with these roots and obtain the following two integrals for the elapsed time tand the total radial period P, P= 2/integraldisplays+ s−P5(τ)dτ τ2/radicalbig (τ−s−)(s+−τ), (52) which is a linear combination of integrals of the type I′ n= 2/integraldisplays+ s−τndτ τ2/radicalbig (τ−s−)(s+−τ). (53) The time elapsed from stos+, t−t0=/integraldisplays+ sP5(τ)dτ τ2/radicalbig (τ−s−)(s+−τ), (54) is a linear combination of integrals of the type In=/integraldisplays+ sτndτ τ2/radicalbig (τ−s−)(s+−τ). (55) Both integrals InandI′ nare given in Appendix A in terms of s+ands−forI′and in terms of ar,er,uand ˜vforI, respectively. The function P5(s)is a fifth order polynomial in sand the factor 2follows from the fact that from s−tos+it is only a half revolution. With the help of the quasi-Keple rian parameterisation r=ar(1−ercosu), (56) wherearanderare some 2PN accurate semi-major axis and radial eccentrici ty, respectively, satisfying ar=1 2s++s− s−s+, (57) er=1 2s+−s− s−+s+, (58) due to (50) and (51), we obtain a 2PN accurate expression for aranderin terms of several intrinsic quantities. With Equation (54), we get a preliminary expres sion for the Kepler Equation, as we express n(t−t0) =2π P(t−t0)in terms of u, and as standard, we introduce an auxiliary variable ˜v≡2arctan/bracketleftbigg/radicalbigg 1+er 1−ertanu 2/bracketrightbigg . (59) At this stage, we have ℓ≡n(t−t0) =u+˜Fusinu+˜F˜v−u(˜v−u)+˜F˜vsin˜v, (60) with˜F...as some 2PN accurate functions of E,h,η,λaandχa. These functions are lengthy and only temporarily needed in the derivation of later results, so we will not provide them. Let us now move on to the angular part. As for the time variable , we factorise the polynomial of dφ/ds with the two roots s−ands+and obtain the elapsed phase at sand the total phase Φfroms−tos+, φ−φ0=/integraldisplays+ sB3(τ)/radicalbig (s−−τ)(τ−s+)dτ , (61) Φ = 2/integraldisplays+ s−B3(τ)/radicalbig (s−−τ)(τ−s+)dτ , (62) where the function B3(τ)is a polynomial of third order in τ, respectively. Using Equation (61) and 62, the elapsed phase scaled by the total phase2π Φ(φ−φ0)in terms of ˜vis computed as 2π Φ(φ−φ0) = ˜v+˜G˜vsin˜v+˜G2˜vsin2˜v+˜G3˜vsin3˜v. (63)Motion and gravitational wave forms of eccentric spinning com pact binaries 11 For the following, we change from the auxiliary variable ˜vto the true anomaly due to Equation (47) with eφ=er(1+ǫ2c1+ǫ4c2), (64) differing from the radial eccentricity by some 1PN and 2PN lev el corrections c1andc2. These corrections are fixed in such a way that the sinvcontribution in2π Φ(φ−φ0)vanishes at each PN order, resulting in a formal lowest order correction with respect to vat 2PN. Therefore, we eliminate uin Equation (59) with the help of (47) and insert the result into (63). For convenience of the reader, we give a 2PN accurate expansion of the expression for ˜vin terms of v: ˜v=v+ǫ2c1er e2r−1sinv +ǫ4/braceleftBigg/parenleftbigg c2−c2 1e2 r e2r−1/parenrightbigger e2r−1sinv+1 4c2 1e2 r (e2r−1)2sin(2v)/bracerightBigg . (65) Having determined the final expression for eφ, Equation (63) takes the form 2π Φ(φ−φ0) =v+G2vsin2v+G3vsin3v. (66) With the help of v, we can re-express the preliminary Kepler equation (60) in t he form of ℓ=n(t−t0) =u−etsinu+Fv−u(v−u)+Fvsinv. (67) Here,etis the time eccentricity and simply represents the sum of all terms with the factor sinuinℓ. All the orbital quantities will be detailed in the next section. 6. Summarising the results We present all the orbital elements ar,er,et,eφ,nand the functions F...andG...of the quasi-Keplerian parameterisation r =ar(1−ercosu), (68) n(t−t0) =u−etsinu+Fv−u(v−u)+Fvsinv, (69) 2π Φ(φ−φ0) =v+G2vsin(2v)+G3vsin(3v), (70) v = 2 arctan/bracketleftBigg/radicalBigg 1+eφ 1−eφtanu 2/bracketrightBigg , (71) in the following list. For δ= 0(remember that δcounts the spin order) one recovers the results from, e.g. [23]. ar=1 2|E| +ǫ2/braceleftbiggη−7 4+δ hαso/bracketleftBig/radicalbig 1−4η(χ1−χ2)+/parenleftBig 1−η 2/parenrightBig (χ1+χ2)/bracketrightBig +δ2 h2/bracketleftbigg (χ1−χ2)2/parenleftbiggαs1s2η 4+αs2/parenleftbigg1 8/radicalbig 1−4η(λ1−λ2)+1 8(1−2η)(λ1+λ2)/parenrightbigg/parenrightbigg +(χ1+χ2)2/parenleftbigg αs2/parenleftbigg1 8/radicalbig 1−4η(λ1−λ2)+1 8(1−2η)(λ1+λ2)/parenrightbigg −αs1s2η 4/parenrightbigg +αs2(χ1+χ2)(χ1−χ2)/parenleftbigg1 4(1−2η)(λ1−λ2)+1 4/radicalbig 1−4η(λ1+λ2)/parenrightbigg/bracketrightbigg/bracerightbigg +ǫ4/braceleftbigg |E|/parenleftbigg1 8/parenleftbig η2+10η+1/parenrightbig +δ hαso/bracketleftbigg1 8/parenleftbig −6η2+19η−8/parenrightbig (χ1+χ2)+1 8/radicalbig 1−4η(5η−8)(χ1−χ2)/bracketrightbigg/parenrightbiggMotion and gravitational wave forms of eccentric spinning com pact binaries 12 +δ h3αso/bracketleftbigg/parenleftbigg η2−39η 4+8/parenrightbigg (χ1+χ2)+1 4(32−9η)/radicalbig 1−4η(χ1−χ2)/bracketrightbigg +1 4h2(11η−17)/bracerightbigg , (72) e2 r= 1−2h2|E| +ǫ2/braceleftbigg h2|E|2/parenleftbigg 5(η−3)+δ hαso/bracketleftBig 8/radicalbig 1−4η(χ1−χ2)+(8−4η)(χ1+χ2)/bracketrightBig +δ2 h2/bracketleftbigg (χ1−χ2)2/parenleftBig 2αs1s2η+αs2/parenleftBig/radicalbig 1−4η(λ1−λ2)−(2η−1)(λ1+λ2)/parenrightBig/parenrightBig +(χ1+χ2)2/parenleftBig αs2/parenleftBig/radicalbig 1−4η(λ1−λ2)−(2η−1)(λ1+λ2)/parenrightBig −2αs1s2η/parenrightBig +αs2(χ1+χ2)(χ1−χ2)/parenleftBig 2/radicalbig 1−4η(λ1+λ2)−2(2η−1)(λ1−λ2)/parenrightBig/bracketrightbigg/parenrightbigg +|E|/parenleftbigg −2(η−6)+δ hαso/bracketleftBig 4(η−2)(χ1+χ2)−8/radicalbig 1−4η(χ1−χ2)/bracketrightBig −δ2 h2/bracketleftbigg (χ1−χ2)2/parenleftBig 2αs1s2η+αs2/parenleftBig/radicalbig 1−4η(λ1−λ2)−(2η−1)(λ1+λ2)/parenrightBig/parenrightBig +(χ1+χ2)2/parenleftBig αs2/parenleftBig/radicalbig 1−4η(λ1−λ2)−(2η−1)(λ1+λ2)/parenrightBig −2αs1s2η/parenrightBig +αs2(χ1+χ2)(χ1−χ2)/parenleftBig 2/radicalbig 1−4η(λ1+λ2)−2(2η−1)(λ1−λ2)/parenrightBig/bracketrightbigg/parenrightbigg/bracerightbigg +ǫ4/braceleftbigg h2|E|3/parenleftbigg −4η2+55η−80 +δ hαso/bracketleftBig/parenleftbig 6η2−49η+80/parenrightbig (χ1+χ2)+(80−19η)/radicalbig 1−4η(χ1−χ2)/bracketrightBig/parenrightbigg +|E| h2/parenleftbigg −22η+34 +δ hαso/bracketleftBig/parenleftbig −8η2+78η−64/parenrightbig (χ1+χ2)+2/radicalbig 1−4η(9η−32)(χ1−χ2)/bracketrightBig/parenrightbigg +|E|2/parenleftbigg η2+η+26 +δ hαso/bracketleftBig/parenleftbig 10η2−70η+4/parenrightbig (χ1+χ2)+4(1−4η)3/2(χ1−χ2)/bracketrightBig/parenrightbigg/bracerightbigg , (73) n= 2√ 2|E|3/2+ǫ2|E|5/2(η−15)√ 2 +ǫ4/braceleftbigg 4|E|3 h/parenleftbigg 6η−15 +αsoδ h/bracketleftBig 2/parenleftbig η2−8η+6/parenrightbig (χ1+χ2)−4/radicalbig 1−4η(η−3)(χ1−χ2)/bracketrightBig/parenrightbigg +|E|7/2 8√ 2/parenleftbig 11η2+30η+555/parenrightbig/bracerightbigg , (74) e2 t= 1−2h2|E|+ǫ2/braceleftbigg |E|/parenleftbig h2|E|(17−7η)+4(η−1)/parenrightbig +δ hαso|E|/bracketleftBig 2(η−2)(χ1+χ2)−4/radicalbig 1−4η(χ1−χ2)/bracketrightBig +δ2|E| h2/bracketleftbigg (χ1−χ2)2/parenleftbigg αs2/parenleftbigg/parenleftbigg η−1 2/parenrightbigg (λ1+λ2)−1 2/radicalbig 1−4η(λ1−λ2)/parenrightbigg −αs1s2η/parenrightbiggMotion and gravitational wave forms of eccentric spinning com pact binaries 13 +(χ1+χ2)2/parenleftbigg αs1s2η+αs2/parenleftbigg/parenleftbigg η−1 2/parenrightbigg (λ1+λ2)−1 2/radicalbig 1−4η(λ1−λ2)/parenrightbigg/parenrightbigg +αs2(χ1+χ2)(χ1−χ2)/parenleftBig (2η−1)(λ1−λ2)−/radicalbig 1−4η(λ1+λ2)/parenrightBig/bracketrightbigg/bracerightbigg +ǫ4/braceleftbigg|E| h2/parenleftbigg −11η+17+h4|E|2/parenleftbig −16η2+47η−112/parenrightbig +12√ 2h3|E|3/2(5−2η) +2h2|E|/parenleftbig 5η2+η+2/parenrightbig +6√ 2h/radicalbig |E|(2η−5)/parenrightbigg +δ hαso|E| 2h2/bracketleftbigg (χ1+χ2)/parenleftbigg −16√ 2h3|E|3/2/parenleftbig η2−8η+6/parenrightbig +h2|E|/parenleftbig 32η2−159η+124/parenrightbig +8√ 2/radicalbig |E|h/parenleftbig η2−8η+6/parenrightbig −8η2+78η−64/parenrightbigg +/radicalbig 1−4η(χ1−χ2)/parenleftbigg 32√ 2h3|E|3/2(η−3)+h2|E|(124−59η) −16√ 2h/radicalbig |E|(η−3)+18η−64/parenrightbigg/bracketrightbigg/bracerightbigg , (75) Fv−u=−ǫ42√ 2|E|3/2 h/braceleftbigg 3/parenleftbigg η−5 2/parenrightbigg +δ hαso/bracketleftBig/parenleftbig η2−8η+6/parenrightbig (χ1+χ2)−2/radicalbig 1−4η(η−3)(χ1−χ2)/bracketrightBig/bracerightbigg , (76) Fv=ǫ4|E|3/2 2√ 2h/radicalbig 1−2h2|E|/braceleftbigg −η(η+4) −δ hαso/bracketleftbigg/radicalbig 1−4η(η+8)(χ1−χ2)−(13η−8)(χ1+χ2)/bracketrightbigg/bracerightbigg , (77) Φ 2π= 1+ǫ21 h2/braceleftbigg 3+δ hαso/bracketleftBig (η−2)(χ1+χ2)−2/radicalbig 1−4η(χ1−χ2)/bracketrightBig +δ2 h2/bracketleftbigg3 8(χ1−χ2)2/parenleftBig −2αs1s2η−αs2/radicalbig 1−4η(λ1−λ2)+αs2(2η−1)(λ1+λ2)/parenrightBig +3 8(χ1+χ2)2/parenleftBig 2αs1s2η−αs2/radicalbig 1−4η(λ1−λ2)+αs2(2η−1)(λ1+λ2)/parenrightBig +3 4αs2(χ1+χ2)(χ1−χ2)/parenleftBig (2η−1)(λ1−λ2)−/radicalbig 1−4η(λ1+λ2)/parenrightBig/bracketrightbigg/bracerightbigg +ǫ4/braceleftbigg|E| h2/parenleftbigg 3η−15 2+δ hαso/bracketleftBig 2/parenleftbig η2−8η+6/parenrightbig (χ1+χ2)−4/radicalbig 1−4η(η−3)(χ1−χ2)/bracketrightBig/parenrightbigg +1 h4/parenleftbiggδ hαso/bracketleftbigg21 4/radicalbig 1−4η(η−8)(χ1−χ2)−3 4/parenleftbig 2η2−49η+56/parenrightbig (χ1+χ2)/bracketrightbigg −15 4(2η−7)/parenrightbigg/bracerightbigg , (78) G2v=ǫ4/parenleftbig 2h2|E|−1/parenrightbig 4h4/braceleftbiggη(3η−1) 2 +3δ hαso/bracketleftBig/radicalbig 1−4ηη(χ1−χ2)−(η−1)η(χ1+χ2)/bracketrightBig/bracerightbigg , (79) G3v=ǫ4/parenleftbig 1−2h2|E|/parenrightbig3/2 8h4/braceleftbigg −3η2 4 +δ hαso/bracketleftBig (η−1)η(χ1+χ2)−/radicalbig 1−4ηη(χ1−χ2)/bracketrightBig/bracerightbigg , (80)Motion and gravitational wave forms of eccentric spinning com pact binaries 14 e2 φ= 1−2h2|E|+ǫ2/braceleftbigg |E|/parenleftbig h2|E|(η−15)+12/parenrightbig +δ hαso|E|/parenleftbig h2|E|−1/parenrightbig/bracketleftBig 8/radicalbig 1−4η(χ1−χ2)−4(η−2)(χ1+χ2)/bracketrightBig +δ2|E| h2/parenleftbig 4h2|E|−3/parenrightbig ×/bracketleftbigg (χ1−χ2)2/parenleftbigg αs1s2η+αs2/parenleftbigg1 2/radicalbig 1−4η(λ1−λ2)−1 2(2η−1)(λ1+λ2)/parenrightbigg/parenrightbigg +(χ1+χ2)2/parenleftbigg −αs1s2η+αs2/parenleftbigg1 2/radicalbig 1−4η(λ1−λ2)−1 2(2η−1)(λ1+λ2)/parenrightbigg/parenrightbigg +αs2(χ1+χ2)(χ1−χ2)/parenleftBig/radicalbig 1−4η(λ1+λ2)−(2η−1)(λ1−λ2)/parenrightBig/bracketrightbigg/bracerightbigg +ǫ4/braceleftbigg|E| 8h2/parenleftbig −4h4|E|2/parenleftbig 3η2−30η+160/parenrightbig +4h2|E|/parenleftbig 9η2+88η−16/parenrightbig −15η2−232η+408/parenrightbig +δ hαso|E| 2h2/bracketleftbigg (χ1+χ2)/parenleftbig 2h4|E|2(80−31η)+8h2|E|/parenleftbig η2−36η+17/parenrightbig −3/parenleftbig η2−71η+64/parenrightbig/parenrightbig +/radicalbig 1−4η(χ1−χ2)/parenleftbig −2h4|E|2(η−80)+4h2|E|(34−15η)+33η−192/parenrightbig/bracketrightbigg/bracerightbigg . (81) For the case that one chooses etinstead of the other eccentricities as the intrinsic parame ter to be searched for in the data analysis investigations, we give the ratios o f the other eccentricities with respect to et: er et= 1+ǫ2/braceleftbigg (8−3η)|E| +αsoδ|E| h/bracketleftBig (η−2)(χ1+χ2)−2/radicalbig 1−4η(χ1−χ2)/bracketrightBig +δ2|E| h2/bracketleftbigg (χ1−χ2)2/parenleftbigg αs2/parenleftbigg1 4(2η−1)(λ1+λ2)−1 4/radicalbig 1−4η(λ1−λ2)/parenrightbigg −αs1s2η 2/parenrightbigg +(χ1+χ2)2/parenleftbiggαs1s2η 2+αs2/parenleftbigg1 4(2η−1)(λ1+λ2)−1 4/radicalbig 1−4η(λ1−λ2)/parenrightbigg/parenrightbigg +αs2(χ1+χ2)(χ1−χ2)/parenleftbigg/parenleftbigg η−1 2/parenrightbigg (λ1−λ2)−1 2/radicalbig 1−4η(λ1+λ2)/parenrightbigg/bracketrightbigg/bracerightbigg +ǫ4/braceleftbigg|E| 2h2/bracketleftBig h2|E|/parenleftbig 6η2−63η+56/parenrightbig −6√ 2h/radicalbig |E|(2η−5)−11η+17/bracketrightBig +αsoδ h|E| 4h2/bracketleftBig/radicalbig 1−4η(χ1−χ2)/parenleftBig h2|E|(23η−84)+16√ 2h/radicalbig |E|(η−3)+18η−64/parenrightBig −(χ1+χ2)/parenleftBig h2|E|/parenleftbig 8η2−55η+84/parenrightbig +8√ 2h/radicalbig |E|/parenleftbig η2−8η+6/parenrightbig +8η2−78η+64/parenrightBig/bracketrightBig/bracerightbigg , (82) eφ et= 1+ǫ2/braceleftbigg −2(η−4)|E| +αsoδ|E| h/bracketleftBig (η−2)(χ1+χ2)−2/radicalbig 1−4η(χ1−χ2)/bracketrightBig +δ2|E| h2/bracketleftbigg (χ1−χ2)2/parenleftbigg αs2/parenleftbigg/parenleftbigg η−1 2/parenrightbigg (λ1+λ2)−1 2/radicalbig 1−4η(λ1−λ2)/parenrightbigg −αs1s2η/parenrightbigg +(χ1+χ2)2/parenleftbigg αs1s2η+αs2/parenleftbigg/parenleftbigg η−1 2/parenrightbigg (λ1+λ2)−1 2/radicalbig 1−4η(λ1−λ2)/parenrightbigg/parenrightbigg +αs2(χ1+χ2)(χ1−χ2)/parenleftBig (2η−1)(λ1−λ2)−/radicalbig 1−4η(λ1+λ2)/parenrightBig/bracketrightbigg/bracerightbiggMotion and gravitational wave forms of eccentric spinning com pact binaries 15 +ǫ4/braceleftbigg|E| 16h2/bracketleftBig 2h2|E|/parenleftbig 11η2−168η+224/parenrightbig −48√ 2h/radicalbig |E|(2η−5)−15η2−144η+272/bracketrightBig +αsoδ h|E| 4h2/bracketleftBig (χ1+χ2)/parenleftBig −3h2|E|/parenleftbig 2η2−15η+28/parenrightbig −8√ 2h/radicalbig |E|/parenleftbig η2−8η+6/parenrightbig +5η2+135η−128/parenrightBig +/radicalbig 1−4η(χ1−χ2)/parenleftBig h2|E|(13η−84)+16√ 2h/radicalbig |E|(η−3)+15η−128/parenrightBig/bracketrightBig/bracerightbigg . (83) 7. Gravitational wave forms The final form of the GW model consists of two ingredients. The first one is the expression for the far-zone radiation field, which will naturally depend on general kine matic quantities describing the binary system. The explicit time evolution of these kinematical quantitie s represents the second one. In the first part of this section, we will compute the GW amplit udesh×andh+. We will require the associated corrections to the desired order, which are avai lable in the literature in harmonic coordinates, see below for references. As well, we will use coordinate transf ormations from ADM to harmonic coordinates to be able to apply the time evolution of the orbital elements in the previous sections, which we have computed in ADM coordinates only. In the second part, we will provide the latter quantities as i mplicit functions of time. 7.1. PN expansion of the gravitational radiation amplitude s The transverse-traceless (TT) projection of the radiation field and thus h×andh+, the two polarisations, strongly depend on the observer’s position relative to the s ource. To obtain h×andh+, it is necessary to give position relations of the orbital plane to the directio n where the detector is situated. An observer– dependent coordinate system will be helpful to give the time domain waveform expressions in terms of the radial separation, orbital angular velocity and the spins. We start the calculation by defining the unit line-of-sight- vectorNas pointing from the source to the observer. Now, let the unit vectors pandqspan the plane of the sky for the observer and complete the orthonormal basis ( p,q,N), p×q=Nand cyclic . (84) Additionally, let us define an invariant reference coordina te system ( ex,ey,ez). Both coordinate systems can be coupled by a special orthogonal matrix. We follow [57] and construct the triad ( p,q,N) by a rotation around the vector exwith some constant inclination angle i0, ex ey ez = 1 0 0 0 cosi0sini0 0−sini0cosi0 p q N . (85) Figure 1 shows shows a representation of what has been done. W e clearly see that the vector pcoincides withex¶. Next, we express the radial separation rin the orbital plane ( ex,ey) and perform the rotation Equation (85) to move to the observer’s triad and calculate randv, r=r(pcosφ+qcosi0sinφ+Nsini0sinφ), (86) v=p/parenleftBig ˙rcosφ−r˙φsinφ/parenrightBig +q/parenleftBig r˙φcosi0cosφ+ ˙rcosi0sinφ/parenrightBig +N/parenleftBig r˙φsini0cosφ+ ˙rsini0sinφ/parenrightBig . (87) This provides the orbital contributions to the field. To comp ute the spin contributions to the radiation field, we also expand the spins in the orbital triad, S1=χ1ez=χ1(Ncosi0−qsini0), (88) S2=χ2ez=χ2(Ncosi0−qsini0). (89) ¶In reference [57] the caption for Figure 2 should be made prec ise. The plane of the sky meets the orbital plane at exfor Υ = 0 only. Generally, at ex=pthe plane of the sky meets the invariable plane.Motion and gravitational wave forms of eccentric spinning com pact binaries 16 PSfrag replacements ex=peyez,h/h n12pqN ϕ φi0i0 hinvariable plane orbital planeplane of sky i Figure 1. The geometry of the binary. We have added the observer relate d frame (p,q,N)(in dashed and dotted lines) with Nas the line–of–sight vector. Npoints from the origin of the invariable frame (ex,ey,ez) to the observer. Note that the orbital angular momentum hlies on the ezaxis and so do the spins. Nis chosen to lie in the ey-ez–plane, and measures a constant angle i0(associated with the rotation around ex) fromez, such that p=ex, and this is the line where the orbital plane meets the plane o f the sky. The angle i0is also found between the vector q, itself positioned in (ey,ez), too, and ey. We also need to know how h×andh+are extracted from the TT part. This is done via following pro jections: h×=1 2(piqj+pjqi)PTTij klhTT kl, (90) h+=1 2(pipj−qjqi)PTTij klhTT kl, (91) wherePTTij klis the usual TT projector onto the line-of-sight vector N, PTTij kl≡(δi k−NiNk)(δj l−NjNl)−1 2(δij−NiNj)(δkl−NkNl), (92) and we define P(×) ij≡1 2(qipj+piqj), (93) P(+) ij≡1 2(pipj−qiqj), (94) which are unaffected by the TT projection operator. The above expressions, Equation (84)-(94), enable us to com pute all the considered contributions to the radiation field polarisations. Following [62] and [63], we l ist the lowest order contributions to the gravitational waveform in harmonic coordinates. These are the PP contribu tions to 2PN, including the NLO-SO and LO-S1S2terms. We also add the terms emerging from the gauge transfor mation from ADM to harmonic coordinates, hTT ij=2η R′/bracketleftBigg ξ(0)PP ij+ǫξ(0.5)PP ij+ǫ2ξ(1)PP ij+ǫ3ξ(1.5)PP ij+ǫ4ξ(2)PP ijMotion and gravitational wave forms of eccentric spinning com pact binaries 17 +ǫ2δαsoξ(1)SO ij+ǫ3δαsoξ(1.5)SO ij+ǫ2δ2αs1s2ξ(1)S1S2 ij +ǫ2δαsogξ(0+1)PP+SO ij +ǫ3δαsogξ(0.5+1)PP+SO ij +ǫ4gξ(0+2)PP+PP ij/bracketrightBigg . (95) Those terms in the last line of the above equation, labeled “ g”, denote corrections coming from the gauge transformation from ADM to harmonic coordinates to the desi red order [35, 64]. Appendix B gives deeper information about how velocities, distances and normal vec tors change within this transformation. We find it convenient to give a hint to their origin by putting the GW m ultipole order and the order/type of the correction in the label, for example “ (0+1)PP+SO ” is the first Taylor correction of the “Newtonian” (PP) quadrupole moment where the coordinates are shifted by a 1PN SO term. According to Equation (93) and Equation (94) one can define th e projected components of the ξvia ξ(order) ×=P(×) ijξ(order) ij, (96) ξ(order) +=P(+) ijξ(order) ij, (97) where the “cross” and “plus” polarisations read ξ(0)PP ×,+= 2/braceleftbigg P(×,+) vv−1 rP(×,+) nn/bracerightbigg , (98) ξ(0.5)PP ×,+=δm m/braceleftbigg 3(N·n12)1 r/bracketleftBig 2P(×,+) nv−˙rP(×,+) nn/bracketrightBig +(N·v)/bracketleftbigg1 rP(×,+) nn−2P(×,+) vv/bracketrightbigg/bracerightbigg , (99) ξ(1)PP ×,+=1 3/braceleftbigg (1−3η)/bracketleftbigg (N·n12)21 r/parenleftbigg/parenleftbigg 3v2−15˙r2+71 r/parenrightbigg P(×,+) nn+30˙rP(×,+) nv−14P(×,+) vv/parenrightbigg +(N·n12)(N·v)1 r/bracketleftBig 12˙rP(×,+) nn−32P(×,+) nv/bracketrightBig +(N·v)2/bracketleftbigg 6P(×,+) vv−21 rP(×,+) nn/bracketrightbigg/bracketrightbigg +/bracketleftbigg 3(1−3η)v2−2(2−3η)1 r/bracketrightbigg P(×,+) vv+41 r˙r(5+3η)P(×,+) nv +1 r/bracketleftbigg 3(1−3η)˙r2−(10+3η)v2+291 r/bracketrightbigg P(×,+) nn/bracerightbigg , (100) ξ(1.5)PP ×,+=δm m/braceleftbigg1 12(1−2η)/braceleftbigg (N·n12)31 r/bracketleftbigg/parenleftbigg 45v2−105˙r2+901 r/parenrightbigg ˙rP(×,+) nn−96˙rP(×,+) vv −/parenleftbigg 42v2−210˙r2+881 r/parenrightbigg P(×,+) nv/bracketrightbigg −(N·n12)2(N·v)1 r/bracketleftbigg/parenleftbigg 27v2−135˙r2+841 r/parenrightbigg P(×,+) nn+336˙rP(×,+) nv−172P(×,+) vv/bracketrightbigg −(N·n12)(N·v)21 r/bracketleftbigg 48˙rP(×,+) nn−184P(×,+) nv/bracketrightbigg +(N·v)3/bracketleftbigg 41 rP(×,+) nn−24P(×,+) vv/bracketrightbigg/bracerightbigg −1 12(N·n12)1 r/braceleftbigg/bracketleftbigg (69−66η)v2−(15−90η)˙r2−(242−24η)1 r/bracketrightbigg ˙rP(×,+) nn −/bracketleftbigg (66−36η)v2+(138+84 η)˙r2 −(256−72η)1 r/bracketrightbigg P(×,+) nv+(192+12 η)˙rP(×,+) vv/bracerightbigg +1 12(N·v)/braceleftbigg/bracketleftbigg (23−10η)v2−(9−18η)˙r2−(104−12η)1 r/bracketrightbigg1 rP(×,+) nn −(88+40η)1 r˙rP(×,+) nv−/bracketleftbigg (12−60η)v2−(20−52η)1 r/bracketrightbigg P(×,+) vv/bracerightbigg/bracerightbigg , (101) ξ(2)PP ×,+=1 120(1−5η+5η2)/braceleftbigg 240 (N·v)4P(×,+) vv−(N·n12)4Motion and gravitational wave forms of eccentric spinning com pact binaries 18 1 r/bracketleftbigg/parenleftbigg 90(v2)2+(3181 r−1260˙r2)v2+3441 r2+1890˙r4 −23101 r˙r2/parenrightbigg P(×,+) nn +/parenleftbigg 1620v2+30001 r−3780˙r2/parenrightbigg ˙rP(×,+) nv−/parenleftbigg 336v2−1680˙r2+6881 r/parenrightbigg P(×,+) vv/bracketrightbigg −(N·n12)3(N·v)1 r/bracketleftbigg/parenleftbigg 1440v2−3360˙r2+36001 r/parenrightbigg ˙rP(×,+) nn −/parenleftbigg 1608v2−8040˙r2+38641 r/parenrightbigg P(×,+) nv−3960˙rP(×,+) vv/bracketrightbigg +120(N·v)3(N·n12)1 r/parenleftbigg 3˙rP(×,+) nn−20P(×,+) nv/parenrightbigg +(N·n12)2(N·v)21 r/bracketleftbigg/parenleftbigg 396v2−1980˙r2+16681 r/parenrightbigg P(×,+) nn+6480˙rP(×,+) nv −3600P(×,+) vv/bracketrightbigg/bracerightbigg −1 30(N·v)2/braceleftbigg/bracketleftbigg (87−315η+145η2)v2−(135−465η+75η2)˙r2 −(289−905η+115η2)1 r/bracketrightbigg1 rP(×,+) nn −/parenleftbigg 240−660η−240η2/parenrightbigg ˙rP(×,+) nv −/bracketleftbigg (30−270η+630η2)v2−60(1−6η+10η2)1 r/bracketrightbigg P(×,+) vv/bracerightbigg +1 30(N·n12)(N·v)1 r/braceleftbigg/bracketleftbigg (270−1140η+1170η2)v2 −(60−450η+900η2)˙r2−(1270−3920η+360η2)1 r/bracketrightbigg ˙rP(×,+) nn −/bracketleftbigg (186−810η+1450η2)v2+(990−2910η−930η2)˙r2 −(1242−4170η+1930η2)1 r/bracketrightbigg P(×,+) nv +/bracketleftbigg 1230−3810η−90η2/bracketrightbigg ˙rP(×,+) vv/bracerightbigg +1 60(N·n12)21 r/braceleftbigg/bracketleftbigg (117−480η+540η2)(v2)2−(630−2850η+4050η2)v2˙r2 −(125−740η+900η2)1 rv2 +(105−1050η+3150η2)˙r4+(2715−8580η+1260η2)1 r˙r2 −(1048−3120η+240η2)1 r2/bracketrightbigg P(×,+) nn +/bracketleftbigg (216−1380η+4320η2)v2+(1260−3300η−3600η2)˙r2 −(3952−12860η+3660η2)1 r/bracketrightbigg ˙rP(×,+) nv −/bracketleftbigg (12−180η+1160η2)v2+(1260−3840η−780η2)˙r2Motion and gravitational wave forms of eccentric spinning com pact binaries 19 −(664−2360η+1700η2)1 r/bracketrightbigg P(×,+) vv/bracerightbigg −1 60/braceleftbigg/bracketleftbigg (66−15η−125η2)(v2)2 +(90−180η−480η2)v2˙r2−(389+1030 η−110η2)1 rv2 +(45−225η+225η2)˙r4+(915−1440η+720η2)1 r˙r2 +(1284+1090 η)1 r2/bracketrightbigg1 rP(×,+) nn −/bracketleftbigg (132+540 η−580η2)v2+(300−1140η+300η2)˙r2 +(856+400 η+700η2)1 r/bracketrightbigg1 r˙rP(×,+) nv −/bracketleftbigg (45−315η+585η2)(v2)2+(354−210η−550η2)1 rv2 −(270−30η+270η2)1 r˙r2 −(638+1400 η−130η2)1 r2/bracketrightbigg P(×,+) vv/bracerightbigg , (102) ξ(1)SO ×,+=−1 r2/braceleftBig/bracketleftBig P(×,+) ij(∆×N)inj 12/bracketrightBig +/radicalbig 1−4η/bracketleftBig P(×,+) ij(S×N)inj 12/bracketrightBig/bracerightBig , (103) ξ(1.5)SO ×,+=1 r2/braceleftBigg /radicalbig 1−4η/bracketleftBigg 6P(×,+) nn[v·(∆×n12)]−6˙r/bracketleftBig P(×,+) ij(∆×n12)inj 12/bracketrightBig +4/bracketleftBig P(×,+) ij(∆×v)inj 12/bracketrightBig/bracketrightBigg +6P(×,+) nn[v·(S×n12)] +η/bracketleftBig P(×,+) ij(S×N)inj 12/bracketrightBig (6˙r(N·n12)−4(N·v)) −6˙r/bracketleftBig P(×,+) ij(S×n12)inj 12/bracketrightBig +η/parenleftBig 4/bracketleftBig P(×,+) ij(S×n12)ivj/bracketrightBig −4(N·n12)/bracketleftBig P(×,+) ij(S×N)ivj/bracketrightBig/parenrightBig +(2η+4)/bracketleftBig P(×,+) ij(S×v)inj 12/bracketrightBig/bracerightBigg , (104) ξ(2)S1S2 ×=−3η r3χ1χ2cos(i0)sin(2φ), (105) ξ(2)S1S2 +=−3η 4r3χ1χ2/parenleftbig cos(2i0)cos(2φ)+2sin2(i0)+3cos(2 φ)/parenrightbig . (106) The remaining contributions are the gauge dependent terms. Explicitly, they read gξ(0+1)PP+SO ×,+ =−η r2/braceleftBigg 3P(×,+) nn[v·(S×n12)]+2/bracketleftBig P(×,+) ij(S×v)inj 12/bracketrightBig +2/bracketleftBig P(×,+) ij(S×n12)ivj/bracketrightBig/bracerightBigg ,(107)Motion and gravitational wave forms of eccentric spinning com pact binaries 20 gξ(0.5+1)PP+SO ×,+ =δm mη 2r2/braceleftBigg P(×,+) nn/bracketleftBig −15˙r(N·n12)[v·(S×n12)]−3˙r[N·(S×v)] +3(N·v)[v·(S×n12)]−1 r[N·(S×n12)]/bracketrightBig +(N·n12)/bracketleftBigg −6˙r/bracketleftBig P(×,+) ij(S×v)inj 12/bracketrightBig +18P(×,+) nv[v·(S×n12)] −6 r/bracketleftBig P(×,+) ij(S×n12)inj 12/bracketrightBig +6/bracketleftBig P(×,+) ij(S×v)ivj/bracketrightBig/bracketrightBigg +(N·v)/bracketleftBigg 2/bracketleftBig P(×,+) ij(S×v)inj 12/bracketrightBig +4/bracketleftBig P(×,+) ij(S×n12)ivj/bracketrightBig/bracketrightBigg +6P(×,+) nv[N·(S×v)]+2P(×,+) vv[N·(S×n12)]/bracerightBigg , (108) gξ(0+2)PP+PP ×,+ =1 r/braceleftBigg P(×,+) nv˙r/bracketleftbigg1 2η/parenleftbig 3˙r2−7v2/parenrightbig −2(5η−1) r/bracketrightbigg +P(×,+) nn/bracketleftBigg 5η/parenleftbig v2−11˙r2/parenrightbig 4r+12η+1 2r2/bracketrightBigg +P(×,+) vv/bracketleftbigg1 2η/parenleftbig 17˙r2−13v2/parenrightbig +21η+1 r/bracketrightbigg/bracerightBigg . (109) Equation (109) shows total agreement with the transformati on term in Equation (A2) of [59]. The next block of equations evaluates the scalar products of vectors and pr ojectors containing the spins. First, we list those with the total spin S=S1+S2. For those terms with ∆=S1−S2instead of S, simply replace ( S→∆) on the left hand side and (χ1+χ2)→(χ1−χ2)on the right. The used abbreviations are given by [S·(n12×v)] = ˙φr(χ1+χ2), (110) [N·(S×n12)] = ( χ1+χ2)sin(i0)cos(φ), (111) [N·(S×v)] =/parenleftbig χ1+χ2/parenrightbig sin/parenleftbig i0/parenrightbig/parenleftbig ˙rcos(φ)−˙φrsin(φ)/parenrightbig , (112) P(×) ijvj(S×n12)i=1 2(χ1+χ2)cos(i0)/braceleftBig ˙rcos(2φ)−˙φrsin(2φ)/bracerightBig , (113) P(+) ijvj(S×n12)i=1 8(χ1+χ2)/braceleftbigg −˙φr(cos(2i0)+3)cos(2 φ)+2˙φrsin2(i0) −˙r(cos(2i0)+3)sin(2 φ)/bracerightbigg , (114) P(×) ijni 12(S×n12)j=1 2(χ1+χ2)cos(i0)cos(2φ), (115) P(+) ijni 12(S×n12)j=−1 8(χ1+χ2){cos(2i0)+3}sin(2φ), (116) P(×) ijvi(S×v)j=−1 2(χ1+χ2)cos(i0)/braceleftBig 2˙φ˙rrsin(2φ)+cos(2 φ)(˙φr−˙r)(˙φr+ ˙r)/bracerightBig , (117) P(+) ijvi(S×v)j=1 8(χ1+χ2)(cos(2i0)+3)/braceleftBig sin(2φ)(˙φr−˙r)(˙φr+ ˙r)−2˙φ˙rrcos(2φ)/bracerightBig , (118) P(×) ijni 12(S×v)j=1 2/parenleftbig χ1+χ2/parenrightbig cos/parenleftbig i0/parenrightbig/braceleftbig ˙rcos(2φ)−˙φrsin(2φ)/bracerightbig , (119) P(+) ijni 12(S×v)j=1 8/parenleftBig χ1+χ2/parenrightBig/braceleftBig −˙φrcos/parenleftbig 2φ/parenrightbig/parenleftbig 3+cos(2 i0)/parenrightbig −2˙φrsin2(i0) −˙rsin/parenleftbig 2φ/parenrightbig/parenleftbig 3+cos(2 i0)/parenrightbig/bracerightBig , (120)Motion and gravitational wave forms of eccentric spinning com pact binaries 21 P(+) ij(S×N)inj 12=−1 2(χ1+χ2)sin(i0)cos(φ), (121) P(×) ij(S×N)inj 12=−1 4(χ1+χ2)sin(2i0)sin(φ), (122) P(+) ij(S×N)ivj=−1 2/parenleftbig χ1+χ2/parenrightbig sin/parenleftbig i0/parenrightbig/parenleftbig ˙rcos(φ)−˙φrsin(φ)/parenrightbig , (123) P(×) ij(S×N)ivj=−1 4/parenleftbig χ1+χ2/parenrightbig sin/parenleftbig 2i0/parenrightbig/parenleftbig˙φrcos(φ)+ ˙rsin(φ)/parenrightbig . (124) The spin-independent projections and the ratio of the differ ence to the sum of the masses read (N·n12)≡Nini 12= sin(i0)sin(φ), (125) (N·v)≡Nivi= sin(i0)/braceleftBig r˙φcos(φ)+ ˙rsin(φ)/bracerightBig , (126) v2≡vivi =r2˙φ2+ ˙r2, (127) P(×) nn≡ P(×) ijni 12nj 12= cos(i0)sin(φ)cos(φ), (128) P(×) vv≡ P(×) ijvivj=1 2cos(i0)/braceleftBig sin(2φ)/parenleftBig ˙r2−˙φ2r2/parenrightBig +2˙φ˙rrcos(2φ)/bracerightBig , (129) P(×) nv≡ P(×) ijni 12vj=1 2cos(i0)/braceleftBig ˙φrcos(2φ)+ ˙rsin(2φ)/bracerightBig , (130) P(+) nn≡ P(+) ijni 12nj 12=1 2/braceleftbig cos2(φ)−cos2(i0)sin2(φ)/bracerightbig , (131) P(+) vv≡ P(+) ijvivj=1 2/braceleftBig (˙rcos(φ)−˙φrsin(φ))2−cos2(i0)(˙φrcos(φ)+ ˙rsin(φ))2/bracerightBig , (132) P(+) nv≡ P(+) ijni 12vj=1 8/braceleftBig −˙φr(cos(2i0)+3)sin(2 φ)−4˙rcos2(i0)sin2(φ)+4˙rcos2(φ)/bracerightBig , (133) δm m≡m1−m2 m=/radicalbig 1−4η. (134) In the expression for the emitted gravitational wave amplit udes, Equation (95), R′is the rescaled distance from the observer to the binary system, R′=RGm c2. (135) We note that it is very important that R′has got the same scaling as rin order to remove the physical dimensions. The common factor c−4ofhTT ijwill be split in c−2for the distance R′andc−2for theξ(...), in order to make allterms dimensionless. Also note that the Equations (98)-(13 3) in our special coordinates are valid only when his constant in time. In the non-aligned case, additional ang ular velocity contributions kick in and the expressions become rather impractical. From reference [6], the reader can extract explicit higher-order spin corrections to the Newtonian quadrupola r field for the case of quasi-circular orbits. 7.2. Dynamical orbital variables as implicit functions of tim e We are now in the position to compute the time domain gravitat ional wave polarisations with the help of our orbital elements, to be expressed in terms of conserved quan tities and the mean anomaly, which is an implicit function of time. Using Equations (68), (69), (70) and (71) o ne can express the quantities r,˙r,φ,˙φ(which are used in the radiation formulas) in terms of the eccentric anomaly u, other orbital elements and several formal 2PN accurate functions. The most compact quantity is rwhich is given by Equation (68), namely r(u) =ar(1−ercosu) (136) =(1−etcosu) 2|E|/bracketleftBigg 1+ǫ2|E| 2(1−etcosu){(1−etcosu)(9−5η)+6η−16} +ǫ4|E|2 (1−etcosu)/braceleftBigg 1 4(e2 t−1)/bracketleftBig (1−etcosu)(η(7η−58)+1)/parenleftbig e2 t−1/parenrightbigMotion and gravitational wave forms of eccentric spinning com pact binaries 22 +2η/parenleftbig −3η/parenleftbig e2 t−1/parenrightbig +34e2 t−56/parenrightbig +68/bracketrightBig −6((1−etcosu)−1)(2η−5)/radicalbig 1−e2 t/bracerightBigg −ǫ2δαso|E|3/2 (1−etcosu)/radicalbig 1−e2 t/braceleftBig 2√ 2/radicalbig 1−4η(χ1−χ2)−√ 2(η−2)(χ1+χ2)/bracerightBig +ǫ2δ2|E|2 (1−etcosu)× /braceleftBigg (χ1−χ2)2/bracketleftBigg αs2/parenleftbig (λ1+λ2)(2η−1)−(λ1−λ2)√1−4η/parenrightbig 2(e2 t−1)−ηαs1s2 e2 t−1/bracketrightBigg +(χ1+χ2)2/bracketleftBigg αs2/parenleftbig (λ1+λ2)(2η−1)−(λ1−λ2)√1−4η/parenrightbig 2(e2 t−1)+ηαs1s2 e2 t−1/bracketrightBigg +αs2(χ1−χ2)(χ1+χ2) e2 t−1/bracketleftBig (λ1−λ2)(2η−1)−(λ1+λ2)/radicalbig 1−4η/bracketrightBig/bracerightBigg +ǫ4δ|E|5/2 2√ 2(1−etcosu)× /braceleftBigg 16((1−etcosu)−1) (e2 t−1)/bracketleftBig ((η−8)η+6)(χ1+χ2)−2(η−3)/radicalbig 1−4η(χ1−χ2)/bracketrightBig +4 /radicalbig 1−e2 t3/bracketleftBig 3(η(2η−15)+12)( χ1+χ2)−(13η−36)/radicalbig 1−4η(χ1−χ2)/bracketrightBig +1/radicalbig 1−e2 t/bracketleftBig (19η−42)/radicalbig 1−4η(χ1−χ2)−(η(13η−50)+42)( χ1+χ2)/bracketrightBig/bracerightBigg/bracketrightBigg . (137) Using expression (136), we calculate the derivative via cha in rule, given by ˙r(u) =dr dudu dt=narersinu× /braceleftbigg 1−etcosu+Fv/radicalBig 1−e2 φ(eφ−cosu) (1−eφcosu)2+Fv−u /radicalBig 1−e2 φ 1−eφcosu−1 /bracerightbigg−1 (138) =et/radicalbig 2|E|sinu 1−etcosu/bracketleftBigg 1+ǫ2|E|3 4(1−3η)+ǫ4|E|2/braceleftbigg1 32(23+η(47η−98)) +6(2η−5)/radicalbig 1−e2 t (e2 t−1)(1−etcosu)+−1 2(η−20)η−30 (1−etcosu)2−η(η+4)/parenleftbig e2 t−1/parenrightbig 2(1−etcosu)3/bracerightbigg +αsoδ× ǫ4|E|5/2/braceleftbigg4√ 2/parenleftbig −2√1−4η(η−3)(χ1−χ2)+((η−8)η+6)(χ1+χ2)/parenrightbig (e2 t−1)(1−etcosu) −/parenleftbig√1−4η(17η−40)(χ1−χ2)+((51−8η)η−40)(χ1+χ2)/parenrightbig /radicalbig 2−2e2 t(1−etcosu)2 +/radicalbig 1−e2 t/parenleftbig√1−4η(η+8)(χ1−χ2)+(8−13η)(χ1+χ2)/parenrightbig √ 2(1−etcosu)3/bracerightbigg/bracketrightBigg . (139) The final expression for φin terms of uis rather complicated. It is convenient to give a short expre ssion and a description how to obtain it. From Equations (70) and (71) o ne can eliminate vto obtain φ(u) =φ0+Φ 2π/braceleftbigg 2arctan/bracketleftBigg/radicalBigg 1+eφ 1−eφtanu 2/bracketrightBigg +G2v2/radicalBig 1−e2 φsin(u)(eφ−cos(u)) (eφcos(u)−1)2Motion and gravitational wave forms of eccentric spinning com pact binaries 23 −G3v/radicalBig 1−e2 φsin(u)/parenleftBig/parenleftBig e2 φ−4/parenrightBig cos(2u)−7e2 φ+12eφcos(u)−2/parenrightBig 2(eφcos(u)−1)3/bracerightbigg . (140) Using the chain rule once more one gets an expression for the a ngular velocity via ˙φ(u) = (dφ/dv)(dv/du)(du/dt), symbolically, ˙φ(u) =Φ P/radicalBig 1−e2 φ (1−eφcosu)× /braceleftBigg 1+G2v/parenleftBig 3e2 φ−4eφcos(u)−(e2 φ−2)cos(2u)/parenrightBig (eφcos(u)−1)2 +G3v/parenleftBig 30e3 φ−45e2 φcos(u)−18/parenleftBig e2 φ−2/parenrightBig eφcos(2u)+3/parenleftBig 3e2 φ−4/parenrightBig cos(3u)/parenrightBig 4(eφcos(u)−1)3/bracerightBigg × /braceleftBigg 1−etcosu+Fv/radicalBig 1−e2 φ(eφ−cosu) (1−eφcosu)2+Fv−u /radicalBig 1−e2 φ 1−eφcosu−1 /bracerightBigg−1 . (141) Again,Pcan easily be computed with the help of the already known defin itionn≡2π/Pand Equation (74). 8. Conclusions In this paper we presented a quasi-Keplerian parameterisat ion for compact binaries with spin and arbitrary mass ratio. We assumed that the spins are aligned or anti-ali gned with the orbital angular momentum and restricted ourselves to the leading-order spin-spin and ne xt-to-leading order spin-orbit, as well as to 2PN point particle contributions. The conservation of alignment for all times holds if the alig nment is assumed at the initial instant of time. It turned out that the effects of the spins do not destroy the polynomial structure of the integrals for both the angular and the radial variables, for which the stan dard routine is valid [23, 59, 65] and enabled us to give a fully analytic prescription for the orbital elemen ts in terms of the binding energy, the mass ratio and the magnitudes of the angular momenta. Furthermore, in contrast to the literature where mostly the emphasis was put on the consistent PN accurate presentation of the phasing, we provided PN extend ed formulae for the radiation polarisations in analytic form as well. These were derived from the results of [62, 59] due to the currently highest available order in spin. We are aware that there is a missing term linear in spin at 2PN o rder in the wave amplitude. Blanchet et al. [3] provided the current and mass multipole moments th at are necessary to compute the far-zone fluxes resulting from the next-to-leading order spin-orbit terms in the acceleration, but the wave amplitude at this order was not given. This missing spin-orbit part at 2PN will be given in a forthcoming publication. We justify this decision by stating that there is a number of rel atively complicated terms of higher order due to the transformation from harmonic to ADM coordinates. To t his order, the coordinate transformation contains next-to-leading order spin-orbit terms which wil l result in lengthy expressions in the radiation field. The difficulty of computing the 2PN amplitude itself becomes c lear when we keep in mind the errata of reference [3]. An outstanding question is the stability of the spin configur ations under purely conservative dynamics. If we assume that the spins have tiny differences in their dire ctions, it is interesting to know if the enclosed angles will grow secularly or will oscillate in an unknown ma nner. This will be task of a further investigation, as well as the inclusion of additional higher order spin Hami ltonians. Aspects of the time evolution of the misalignment of spins due to the radiation reaction were alr eady discussed by Kidder in [62]. Another task to be tackled is the effect of radiation reaction to the orbital elements. It is possible to include the conservative contributions of the spin to the or bital motion into the equations of the far-zoneMotion and gravitational wave forms of eccentric spinning com pact binaries 24 energy and angular momentum flux expressions. The goal is an e quation of motion for the orbital elements to be obtained in an adiabatic approach. Acknowledgments We thank Jan Steinhoff and Steven Hergt for many useful discus sions. This work is partly funded by the Deutsche Forschungsgemeinschaft (DFG) through SFB/TR7 “G ravitationswellenastronomie” and the Research Training School GRK 1523 “Quanten- und Gravitatio nsfelder” and by the Deutsches Zentrum für Luft- und Raumfahrt (DLR) through “LISA Germany”. An anonym ous referee’s helpful comments and suggestions for improvements are thankfully acknowledged . [1] S. Rowan and J. Hough, “Gravitational wave detection by i nterferometry (ground and space),” Living Rev. Relativity 3 (2000) 3. http://www.livingreviews.org/lrr-2000-3 . [2] G. Faye, L. Blanchet, and A. Buonanno, “Higher-order spi n effects in the dynamics of compact binaries. I. Equations of motion,” Phys. Rev. D 74(2006) 104033, arXiv:gr-qc/0605139 . [3] L. Blanchet, A. Buonanno, and G. Faye, “Higher-order spi n effects in the dynamics of compact binaries. II. Radiation field,” Phys. Rev. D 74(2006) 104034, arXiv:gr-qc/0605140 . [4] L. Blanchet, A. Buonanno, and G. Faye, “Erratum: Higher- order spin effects in the dynamics of compact binaries. II. Radiation field,” Phys. Rev. D 75(2007) 049903(E). [5] L. Blanchet, A. Buonanno, and G. Faye, “Erratum: Higher- order spin effects in the dynamics of compact binaries. II. Radiation field,” Phys. Rev. D 81(2010) 089901(E). [6] K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, “Higher- order spin effects in the amplitude and phase of gravitational waveforms emitted by inspiraling compact bi naries: Ready-to-use gravitational waveforms,” Phys. Rev. D 79(2009) 104023, arXiv:0810.5336 [gr-qc] . [7] M. Hannam, S. Husa, B. Brügmann, and A. Gopakumar, “Compa rison between numerical-relativity and post-Newtonian waveforms from spinning binaries: The orbital hang-up case ,”Phys. Rev. D 78(2008) 104007, arXiv:0712.3787 [gr-qc] . [8] P. Jaranowski and G. Schäfer, “Radiative 3.5 post-Newto nian ADM Hamiltonian for many-body point-mass systems,” Phys. Rev. D 55(1997) 4712–4722. [9] L. Blanchet, G. Faye, B. R. Iyer, and B. Joguet, “Gravitat ional-wave inspiral of compact binary systems to 7/2 post-Newtonian order,” Phys. Rev. D 65(2002) 061501(R), arXiv:gr-qc/0105099 . [10] L. Blanchet, G. Faye, B. R. Iyer, and B. Joguet, “Erratum : Gravitational-wave inspiral of compact binary systems to 7/2 post-Newtonian order,” Phys. Rev. D 71(2005) 129902(E). [11] P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, A. M. Sintes, J. T. Whelan, B. Brügmann, P. Diener, N. Dorband, J. Gonzalez, M. Hannam, S. Husa, D. Pollney, L. Re zzolla, L. Santamaría, U. Sperhake, and J. Thornburg, “Template bank for gravitational waveforms f rom coalescing binary black holes: Nonspinning binaries,” Phys. Rev. D 77(2008) 104017, arXiv:0710.2335 [gr-qc] . [12] P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, A. M. Sintes, J. T. Whelan, B. Brügmann, P. Diener, N. Dorband, J. Gonzalez, M. Hannam, S. Husa, D. Pollney, L. Re zzolla, L. Santamaría, U. Sperhake, and J. Thornburg, “Erratum: Template bank for gravitational wa veforms from coalescing binary black holes: Nonspinning binaries,” Phys. Rev. D 79(2009) 129901(E). [13] M. Mathisson, “Neue Mechanik materieller Systeme,” Acta Phys. Pol. 6(1937) 163–200. [14] A. Papapetrou, “Spinning test-particles in general re lativity. I,” Proc. R. Soc. A 209(1951) 248–258. [15] E. Corinaldesi and A. Papapetrou, “Spinning test-part icles in general relativity. II,” Proc. R. Soc. A 209(1951) 259–268. [16] B. M. Barker and R. F. O’Connell, “Derivation of the equa tions of motion of a gyroscope from the quantum theory of gravitation,” Phys. Rev. D 2(1970) 1428–1435. [17] B. M. Barker and R. F. O’Connell, “Gravitational two-bo dy problem with arbitrary masses, spins, and quadrupole moments,” Phys. Rev. D 12(1975) 329–335. [18] P. D. D’Eath, “Interaction of two black holes in the slow -motion limit,” Phys. Rev. D 12(1975) 2183–2199. [19] B. M. Barker and R. F. O’Connell, “The gravitational int eraction: Spin, rotation, and quantum effects—a review,” Gen. Relativ. Gravit. 11(1979) 149–175. [20] T. A. Apostolatos, “Search templates for gravitationa l waves from precessing, inspiraling binaries,” Phys. Rev. D 52(1995) 605–620. [21] L. E. Kidder, C. M. Will, and A. G. Wiseman, “Spin effects i n the inspiral of coalescing compact binaries,” Phys. Rev. D 47(1993) R4183–R4187, arXiv:gr-qc/9211025 . [22] T. Damour and N. Deruelle, “General relativistic celes tial mechanics of binary systems. I. the post-Newtonian mot ion.,” Ann. Inst. H. Poincaré A 43(1985) 107–132. [23] G. Schäfer and N. Wex, “Second post-Newtonian motion of compact binaries,” Phys. Lett. A 174(1993) 196–205. [24] G. Schäfer and N. Wex, “Erratum: Second post-Newtonian motion of compact binaries,” Phys. Lett. A 177(1993) 461(E).Motion and gravitational wave forms of eccentric spinning com pact binaries 25 [25] R.-M. Memmesheimer, A. Gopakumar, and G. Schäfer, “Thi rd post-Newtonian accurate generalized quasi-Keplerian parametrization for compact binaries in eccentric orbits, ”Phys. Rev. D 70(2004) 104011, arXiv:gr-qc/0407049 . [26] C. Reisswig, S. Husa, L. Rezzolla, E. N. Dorband, D. Poll ney, and J. Seiler, “Gravitational-wave detectability of equal-mass black-hole binaries with aligned spins,” Phys. Rev. D 80(2009) 124026, arXiv:0907.0462 [gr-qc] . [27] T. Bogdanović, C. S. Reynolds, and M. C. Miller, “Alignm ent of the spins of supermassive black holes prior to coalescence,” ApJ661(2007) L147–L150, arXiv:gr-qc/0703054 . [28] J. Steinhoff, S. Hergt, and G. Schäfer, “Next-to-leadin g order gravitational spin(1)-spin(2) dynamics in Hamilto nian form,” Phys. Rev. D 77(2008) 081501(R), arXiv:0712.1716 [gr-qc] . [29] S. Hergt, J. Steinhoff, and G. Schäfer, “The reduced Hami ltonian for next-to-leading-order spin-squared dynamics of general compact binaries,” Class. Quant. Grav. 27(2010) 135007, arXiv:1002.2093 [gr-qc] . [30] J. M. Martín-García, “xPerm: fast index canonicalizat ion for tensor computer algebra,” Comp. Phys. Commun. 179(2008) 597–603, arXiv:0803.0862 [cs.SC] . [31] J. M. Martín-García, xAct: Efficient Tensor Computer Algebra .http://www.xact.es/ . [32] P. Jaranowski and G. Schäfer, “Third post-Newtonian hi gher order ADM Hamilton dynamics for two-body point-mass systems,” Phys. Rev. D 57(1998) 7274–7291, arXiv:gr-qc/9712075 . [33] T. Damour, P. Jaranowski, and G. Schäfer, “Dimensional regularization of the gravitational interaction of point m asses,” Phys. Lett. B 513(2001) 147–155, arXiv:gr-qc/0105038 . [34] E. Poisson, “Gravitational waves from inspiraling com pact binaries: The quadrupole-moment term,” Phys. Rev. D 57(1998) 5287–5290, arXiv:gr-qc/9709032 . [35] T. Damour, P. Jaranowski, and G. Schäfer, “Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling,” Phys. Rev. D 77(2008) 064032, arXiv:0711.1048 [gr-qc] . [36] J. Steinhoff, G. Schäfer, and S. Hergt, “ADM canonical fo rmalism for gravitating spinning objects,” Phys. Rev. D 77(2008) 104018, arXiv:0805.3136 [gr-qc] . [37] M. Levi, “Next to leading order gravitational spin1-sp in2 coupling with Kaluza-Klein reduction,” arXiv:0802.1508 [gr-qc] . [38] K. S. Thorne and J. B. Hartle, “Laws of motion and precess ion for black holes and other bodies,” Phys. Rev. D 31(1985) 1815–1837. [39] S. Hergt and G. Schäfer, “Higher-order-in-spin intera ction Hamiltonians for binary black holes from source terms of Kerr geometry in approximate ADM coordinates,” Phys. Rev. D 77(2008) 104001, arXiv:0712.1515 [gr-qc] . [40] S. Hergt and G. Schäfer, “Higher-order-in-spin intera ction Hamiltonians for binary black holes from Poincaré inv ariance,” Phys. Rev. D 78(2008) 124004, arXiv:0809.2208 [gr-qc] . [41] J. Steinhoff and G. Schäfer, “Canonical formulation of s elf-gravitating spinning-object systems,” Europhys. Lett. 87(2009) 50004, arXiv:0907.1967 [gr-qc] . [42] J. Steinhoff and H. Wang, “Canonical formulation of grav itating spinning objects at 3.5 post-Newtonian order,” Phys. Rev. D 81(2010) 024022, arXiv:0910.1008 [gr-qc] . [43] W. M. Tulczyjew, “Motion of multipole particles in gene ral relativity theory,” Acta Phys. Pol. 18(1959) 393–409. [44] W. G. Dixon, “Extended bodies in general relativity: Th eir description and motion,” in Proceedings of the International School of Physics Enrico Fermi LXVII , J. Ehlers, ed., pp. 156–219. North Holland, Amsterdam, 197 9. [45] J. Steinhoff and D. Puetzfeld, “Multipolar equations of motion for extended test bodies in general relativity,” Phys. Rev. D 81(2010) 044019, arXiv:0909.3756 [gr-qc] . [46] W. G. Laarakkers and E. Poisson, “Quadrupole moments of rotating neutron stars,” Astrophys. J. 512(1999) 282–287, arXiv:gr-qc/9709033 . [47] J. Levin, “Gravity waves, chaos, and spinning compact b inaries,” Phys. Rev. Lett. 84(2000) 3515–3518, arXiv:gr-qc/9910040 . [48] C. Sohr, “Chaos in gravitativ-gebundenen Binärsystem en mit Spin.,” diploma thesis, Friedrich-Schiller-Univer sität Jena, 2009. unpublished. [49] P. A. M. Dirac, Lectures on Quantum Mechanics . Yeshiva University Press, New York, 1964. [50] H. Goldstein, Classical Mechanics . Addison-Wiley Publishing Company, 2nd ed., 1981. [51] P. Colwell, Solving Kepler’s equation over three centuries . Willman-Bell, Inc., Richmond, VA 23235, 1993. [52] B. Mikóczi, M. Vasúth, and L. Á. Gergely, “Self-interac tion spin effects in inspiralling compact binaries,” Phys. Rev. D 71(2005) 124043, arXiv:astro-ph/0504538 . [53] J. Majár and M. Vasúth, “Gravitational waveforms from a Lense-Thirring system,” Phys. Rev. D 74(2006) 124007, arXiv:gr-qc/0611105 . [54] M. Vasúth and J. Majár, “Gravitational waveforms for fin ite mass binaries,” Int. J. Mod. Phys. A 22(2007) 2405–2414, arXiv:0705.3481 [gr-qc] . [55] J. Majár and M. Vasúth, “Gravitational waveforms for sp inning compact binaries,” Phys. Rev. D 77(2008) 104005, arXiv:0806.2273 [gr-qc] . [56] L. Á. Gergely, “Spinning compact binary inspiral: Inde pendent variables and dynamically preserved spin configura tions,” arXiv:0912.0459 . [57] M. Tessmer, “Gravitational waveforms from unequal-ma ss binaries with arbitrary spins under leading order spin-o rbit coupling,” Phys. Rev. D 80(2009) 124034, arXiv:0910.5931 [gr-qc] . [58] Z. Keresztes, B. Mikóczi, and L. Á. Gergely, “The Kepler equation for inspiralling compact binaries,” Phys. Rev. D 72(2005) 104022, arXiv:astro-ph/0510602 . [59] T. Damour, A. Gopakumar, and B. R. Iyer, “Phasing of grav itational waves from inspiralling eccentric binaries,” Phys. Rev. D 70(2004) 064028, arXiv:gr-qc/0404128 . [60] C. Königsdörffer and A. Gopakumar, “Phasing of gravitat ional waves from inspiralling eccentric binaries at the third-and-a-half post-Newtonian order,” Phys. Rev. D 73(2006) 124012, arXiv:gr-qc/0603056 .Motion and gravitational wave forms of eccentric spinning com pact binaries 26 [61] L. Á. Gergely, Z. I. Perjés, and M. Vasúth, “Spin effects i n gravitational radiation backreaction. III: Compact bina ries with two spinning components,” Phys. Rev. D 58(1998) 124001, arXiv:gr-qc/9808063 . [62] L. E. Kidder, “Coalescing binary systems of compact obj ects to (post)5/2-Newtonian order. V. Spin effects,” Phys. Rev. D 52(1995) 821–847, arXiv:gr-qc/9506022 . [63] A. Gopakumar and B. R. Iyer, “Gravitational waves from i nspiraling compact binaries: Angular momentum flux, evolution of the orbital elements, and the waveform to the se cond post-newtonian order,” Phys. Rev. D 56(1997) 7708–7731, arXiv:gr-qc/0110100 . [64] T. Damour, P. Jaranowski, and G. Schäfer, “Equivalence between the ADM-Hamiltonian and the harmonic-coordinates approaches to the third post-Newtonian dynamics of compact binaries,” Phys. Rev. D 63(2001) 044021, arXiv:gr-qc/0010040 . [65] C. Königsdörffer and A. Gopakumar, “Post-Newtonian acc urate parametric solution to the dynamics of spinning compa ct binaries in eccentric orbits: The leading order spin-orbit interaction,” Phys. Rev. D 71(2005) 024039, arXiv:gr-qc/0501011 . [66] T. Damour and G. Schäfer, “Lagrangians for n point masse s at the second post-Newtonian approximation of general relativity,” Gen. Relativ. Gravit. 17(1985) 879–905. Appendix A. Integrals For the sake of completeness we give the results of the definit e integrals InandI′ nfor different n: I′ 0=π(s−+s+) (s−s+)3/2, (A.1) I′ 1=2π√s−s+, (A.2) I′ 2= 2π, (A.3) I′ 3=π(s−+s+), (A.4) I′ 4=1 4π/parenleftbig 3s2 −+2s−s++3s2 +/parenrightbig , (A.5) I′ 5=1 8π(s−+s+)/parenleftbig 5s2 −−2s−s++5s2 +/parenrightbig . (A.6) The more complicated integrals with boundary sin terms of u,˜v,erandarare given by I0=a2 r/radicalbig 1−e2r(u−sinu), (A.7) I1=ar/radicalbig 1−e2ru, (A.8) I2= ˜v, (A.9) I3=˜v+ersin(˜v) ar(1−e2r), (A.10) I4=2(2+e2 r)˜v+8ersin˜v+e2 rsin(2˜v) 4a2r(1−e2r)2, (A.11) I5=6/parenleftbig 2+3e2 r/parenrightbig ˜v+9er/parenleftbig 4+e2 r/parenrightbig sin˜v+9e2 rsin(2˜v)+e3 rsin(3˜v) 12a3r(1−e2r)3. (A.12) Appendix B. Coordinate transformation from ADM to harmonic From section IV of [35] and from [64], we collect the contribu tions for the coordinate transformation from ADM to harmonic coordinates for spinning compact binaries, including LO effects of spin-orbit interaction and 2PN PP contributions. Let Yalabel the harmonic position of the a-th particle as a function of the ADM positions xb, momenta pband spins Sb. Then, to 2PN order, the transformation reads in their notation Ya(xb,pb) =xa+ǫ2YSO a(xb,pb,Sb)+ǫ4Y2PN a(xb,pb) (B.1) with YSO a(xb,pb,Sb) =Sa×pa 2m2a, (B.2)Motion and gravitational wave forms of eccentric spinning com pact binaries 27 Y2PN 1(xa,pa) =Gm2/braceleftBigg/bracketleftBigg 5 8p2 2 m2 2−1 8(n12·p2)2 m2 2+Gm1 r12/parenleftbigg7 4+1 4m2 m1/parenrightbigg/bracketrightBigg n12 +1 2(n12·p2) m2p1 m1−7 4(n12·p2) m2p2 m2/bracerightBigg , (B.3) whereY2PN 2(xa,pa)is simply obtained by exchanging the particle indices (1 ↔2). We find it very important to mention some of the rules to obtain the relative separatio n vector with the scaling introduced in this paper. The above equations are notgiven in relative coordinates. Thus, we scale every Sawithm2 a. Next, we subtract Y1fromY2, setting p2=−p1=−pfor the centre-of-mass frame and scale pwithµas in Equation (8) to get a dimensionless momentum. Finally, we di vide the obtained separation vector with Gm and obtain the separation in terms of the linear momentum and the ADM spin momenta. There is an additional transformation at 2PN which relates t he ADM time with the time in harmonic coordinates [66], which reads tADM=th+ǫ4η(n12·v) (B.4) where we removed the scales and used only dimensionless term s. The harmonic velocity is obtained by plugging the harmonic p ositions in the Poisson brackets with the total Hamiltonian and adding the internal derivation of the ADM time with respect to the harmonic time, v=/bracketleftbig x,HADM/bracketrightbig , (B.5) vhcov=/bracketleftbig xhcov,HADM/bracketrightbig/parenleftbiggdtADM dthcov/parenrightbigg , (B.6) The linear momentum pcan then be expressed in terms of the velocity perturbativel y. It is important to expresspin terms of the ADM velocity first and then to plug it into the ex pression for vhcovafterwards. To 2PN order, the radial separation, the velocity and the unit n ormal vector, rharm,vharmandn12harmtransform due to xharm=x+1 2ǫ2δη(S×v)+ǫ4/braceleftbigg12η+1 4rn12−1 8η/parenleftbig n12/parenleftbig ˙r2−5v2/parenrightbig +18˙rv/parenrightbig/bracerightbigg , (B.7) vharm=v−ǫ2δη 2r2(S×n12) 1 8c4/braceleftBigg n12/parenleftbigg η/bracketleftbigg3˙r2−7v2 r−38 r2/bracketrightbigg −4 r2/parenrightbigg ˙r+v/parenleftbigg η/bracketleftbigg9˙r2−5v2 r+34 r2/bracketrightbigg +2 r2/parenrightbigg/bracerightBigg . (B.8) rharm=r−1 2ǫ2δη[S·(n12×v)]+ǫ4/braceleftbigg1 8η/parenleftbig 5v2−19˙r2/parenrightbig +3η+1 4 r/bracerightbigg , (B.9) n12harm=n12+ǫ2δη 2r{n12[S·(n12×v)]+(S×v)}+ǫ49˙rη 4r{˙rn12−v}, (B.10) where every quantity on the right hand side is written in ADM c oordinates. Note that n12[S·(n12×v)]+(S×v) = (1−n12⊗n12)(S×v) (B.11) is the part of (S×v)which is orthogonal to n12.

1306.6727v2.The_role_of_orbital_order_in_the_stabilization_of_the___π_0___ordered_magnetic_state_in_a_minimal_two_band_model_for_iron_pnictides.pdf

arXiv:1306.6727v2 [cond-mat.str-el] 5 Jul 2015The role of orbital order in the stabilization of the ( π,0) ordered magnetic state in a minimal two-band model for iron pnictides Sayandip Ghosh1,a)and Avinash Singh1 Department of Physics, Indian Institute of Technology Kanp ur - 208016 Spin waveexcitations and stability ofthe ( π,0) orderedmagnetic state areinvestigated in a minimal two-band itinerant-electron model for iron pnictides. Presence of hopping a nisotropy generates a strong ferro-orbital order in the dxzanddyzFe orbitals. The orbital order sign is as observed in experiments. Th e induced ferro- orbital order strongly enhances the spin wave energy scale and st abilizes the magnetic state by optimizing the strength of the emergent AF and F spin couplings through optim al band fillings in the two orbitals. The calculated spin-wave dispersion is in quantitative agreement with neu tron scattering measurements. Finite inter-orbital Hund’s coupling is shown to further enhance the spin w ave energies state by coupling the two magnetic sub-systems. A more realistic two-band model with less ho pping anisotropy is also considered which yields not only the circular hole pockets, also correct ferro-orbita l order and emergent F spin coupling. PACS numbers: 75.25.Dk, 74.70.Xa, 75.30.Ds, 71.10.Fd I. INTRODUCTION The iron pnictides exhibit a typical phase diagram1,2 inwhichtheparentcompoundgoesthroughatetragonal- to-orthorhombic structural phase transition (at TS) and spin ordering transition (at TN). Upon doping, both transitions are suppressed and superconductivity emerges. Single crystal neutron scattering experiments show that Fe moments align antiferromagnetically (AF) along the adirection and ferromagnetically(F) along the bdirection3, so that the magnetically ordered state can be viewed as a ( π,0) ordered spin density wave (SDW) state. Inelastic neutron scattering (INS) experiments4–6 yield well-defined spin-wave excitations up to the F zone boundary q=(0,π) on an energy scale ∼200 meV. Several weak coupling models with Fermi surface (FS) nesting have been proposed7–9to account for the ob- served magnetic order. Although these models can explain low-energy magnetic excitations, they fail at higher energies and suggest that spin waves enter the particle-hole continuum at high energies and become over-damped Stoner-type excitations9. The recently ob- served existence6of spin-wave excitations even above TN is also contrary to this weak coupling nesting picture, withinwhichthereisnodifferencebetweenmomentmelt- ing and moment disordering temperatures. In fact, LDA calculations10,11suggest that Fe onsite interaction Uis comparable to Fe 3 dbandwidth (W) indicating that iron pnitides are moderately correlated materials. Apart from the magnetic excitations, angle-resolved photoemission spectroscopy (ARPES)12–14and X-ray linear dichroism (XLD) experiments15have clearly re- vealedtheexistenceoforbitalorderinthesematerials. In the magnetic state, the Fe dyzband is shifted up relative to thedxzband13,15, causing electron density difference between the two orbitals. This type of orbital ordering a)Electronic mail: sayandip@iitk.ac.inwas previously proposed16–19to explain experimentally observed in-plane anisotropic behavior like anisotropy in magnetic exchange coupling4, transport properties20,21, FS structure13, and electronic structure22. Is there a significant correlation between this observed orbitalorderingandstabilityofthe( π,0)magneticstate? In this paper, we will investigate the effect of orbital or- dering on the SDW state stability within a minimal two- band itinerant-electron model. We will study how the variations in orbital disparity affects the induced AF and F spin couplings. This provides a microscopic under- standing of the role of orbital ordering on the stability of the (π,0) orderedSDWstate in the relevant intermediate coupling regime. The minimal two-band model proposed earlier by Raghuet al.23gave Fermi surface structure consistent with LDA calculations at half-filling. Although nesting between hole and electron Fermi pockets yields low crit- ical value of Ufor (π,0) ordering, it has three major shortcomings: (i) no F spin coupling is generated due to nesting as shown by the vanishing spin wave energy at the F zone boundary24,25, whereasINS experiments yield maximum spin waveenergy, (ii) no orbital orderingis ob- tained in this model for electron filling corresponding to nesting condition, and (iii) Fermi surface nesting is rel- evant only in weak coupling limit, whereas pnictides are in intermediate coupling regime ( U∼W). At interme- diate coupling, nesting is not very relevant for magnetic orderingasevidenced bytheobservationofastable( π,0) state in one-band Hubbard model26. Evidently, modifications are required to this two-band model and therefore investigation of models and mech- anisms beyond nesting become relevant in order to re- produce experimentally observed orbital ordering as well as the spin wave dispersion. The present study is an in- vestigation in this direction. As hopping anisotropy has been shown to yield orbital order in the ( π,0) state27, we will first consider the extreme hopping anisotropy case within a minimal two-band model (Section 2) and in- vestigate the consequences on spin couplings and spin2 wave dispersion (Section 3), to bring out in a physically transparent manner how the resulting orbital order en- hances the F and AF spin couplings in stabilizing the (π,0) structure and yielding the observed features of the spin wave dispersion. Guided by the above investiga- tions, Section 4 describes modifications to the two-band model to obtain circular hole pockets and orbital order in agreement with experiments, and also emergence of required F spin coupling. This comprehensive approach of simultaneously keeping account of the Fermi surface as well as emergent spin couplings and spin wave dis- persion provides important physical insight into further extension to a three-band model. II. MINIMAL TWO-BAND MODEL The iron pnictides have a quasi two-dimensional struc- ture with layers of FeAs stacked along the caxis. Among the five Fe 3 dorbitals, only dxzanddyzcontribute to or- bital ordering (due to C4symmetry of others) and we retain only these two in our model. The hybridization of Fe 3dorbitals with themselves as well as through the As 3porbitals lying above and below the square pla- quettes formed by the Fe atoms leads to hopping pa- rameters in our two-orbital model. The hopping am- plitudes are shown in Fig. 1 in which the dxzanddyz orbitals are taken to be extended along x(a) andy(b) direction respectively since the cores of Fe dxzanddyz Wannier Functions extend towards the direction of mag- netic ordering17. Although hybridization between the or- bitals can lead to finite tπ(i.e.πoverlap), for simplicity we consider tσ(i.e.σoverlap) only in our model. How- ever, orbital order in our model persists as long as there is hopping anisotropy i.e. tσis larger than tπ. A fi- nite intra-orbital next-nearest-neighbor (NNN) hopping t′, expected due to presence of Fe-As-Fe path along pla- quette diagonals, is also included. Finite t′plays a very important role in stabilizing the SDW ordering. We start with the two-orbital Hubbard model Hamil- tonian: H=−/summationdisplay /angbracketleftij/angbracketrightµνσtµν ij(a† iµσajνσ+a† jνσaiµσ)+/summationdisplay iµUµniµ↑niµ↓ (1) wherei,jrefertolatticesites; µ,νaretheorbitalindices, tµν ijare the hopping terms as shown in Fig. 1, and Uµ are the intra-orbital Coulomb interactions. The role of inter-orbital density interaction and Hund’s coupling will be discussed later. In this model, the two orbitals are decoupled, with non-magnetic state dispersions εα(k)=−2tcoskx− 4t′coskxcoskyfor the dxzandεβ(k)=−2tcosky− 4t′coskxcoskyfor thedyzorbitals. Although the dis- persions are different, the energy bands are degenerate in the non-magnetic state wherein xandydirections are equivalent. This degeneracy is important to satisfy point-group symmetry conditions. It is noteworthy that FIG. 1. Effective hopping parameters tµν ijin the minimal two- band model involving dxz(filled) and dyz(empty) orbitals, referred to as αandβrespectively. thedxzbandalongΓ −YanddyzalongΓ−Xaredegener- ate in energy, as indeed observed in ARPES studies13on BaFe2As2in the non-magnetic state above the ordering temperature. In the ( π,0) ordered SDW state, the Hartree-Fock (HF) level Hamiltonian matrix is expressed in a com- posite two-orbital ( α β), two-sublattice (A B) basis as: Hσ HF(k) = −σ∆αεx k+εxy k0 0 εx k+εxy kσ∆α 0 0 0 0 −σ∆β+εy kεxy k 0 0 εxy kσ∆β+εy k (2) whereεx(y) k=−2tcoskx(y),εxy k=−4t′coskxcosky,σ=± for the two spins. The self-consistent exchange fields are given by 2∆ µ=Uµmµin terms of the sublattice mag- netizations mµfor the two orbitals µ=α,β. As the intermediate-couplingregime will be consideredthrough- out, the term “SDW state” is used here without any im- plicit weak-coupling connotation. The density terms arising in the HF approximation, Uµnµ/2 = ∆ µ+Uµn↓ µfor the two orbitals, have not been shown explicitly in (1). In the following, we will take identical exchange fields ∆ α= ∆β≡∆, for which we find the relative band shift Uαn↓ α−Uβn↓ βto be quite small, and this will be absorbed in an energy offset to be introduced later. In iron pnictides, each iron ion has six electrons dis- tributed among five 3 dorbitals. Two egorbitalsdx2−y2 andd3z2−r2are completely occupied by four electrons due to large crystal-field splitting between egandt2g states28, and the three t2gorbitals are partially filled by the remaining two electrons. ARPES experiments13 show that the dxyorbital has a finite contribution to the Fermi Surface. Therefore, the expected electron filling in our two band model will be less than half filling, and will correspond to a “hole doped” condition. The partial densities of states for the two orbitals are shown in Fig. 2 for the non-magnetic (∆=0) and mag- netic (∆ /|t|=2) cases. While the two orbitals are de-3 0 0.5 1 1.5 2 2.5 (a)DOS (arb. unit)nα = 0.75α (xz) 0 0.5 1 1.5 2 -3-2-1 0 1 2 3 ε/|t|(b) EFnβ = 0.75β (yz)(c) nα = 1.0Upper Lower -3-2-1 0 1 2 3 ε/|t|(d) EFnβ = 0.5Upper Lower FIG. 2. Calculated partial densities of state (DOS) for αand βorbitals in the non-magnetic [panels (a),(b)] and magnetic [panels (c),(d)] states. Here, t′/t=0.5. The degeneracy be- tweenαandβorbitals is lifted by magnetic ordering. For a typicalEFas shown, nα> nβ. Enhanced nαand reduced nβ due to hopping anisotropy result in stronger AF and F spin couplings. generate in the non-magnetic state, the degeneracy is lifted in the magnetic state since xandydirections are no longer equivalent. The hopping anisotropy together with magnetic ordering anisotropy naturally results in self orbital order in the magnetic state. For hole doping (EF<0), the lower βband becomes partially unoccu- pied whereas the lower αband remains nearly half-filled. For total electron occupation n≈0.75 per orbital (i.e. 25% hole doping), we have nα≈1.0 andnβ≈0.5. This sign of ferro-orbital order ( nα−nβ>0) is in agreement with experiments12–15. The higher value of αDOS than βDOS at Fermi energy naturally leads to more conduc- tivity along AF direction than F direction which agrees with experiments21. III. TRANSVERSE SPIN FLUCTUATIONS Spin-wave excitations in this spontaneously-broken- symmetry SDW state are obtained from the transverse spin fluctuation propagator: [χ−+ RPA(q,ω)] =[χ0(q,ω)] 1−[U][χ0(q,ω)](3) at the RPA level. The interaction matrix [ U] includes Uα andUβas diagonal matrix elements. The bare particle- hole propagator [ χ0(q,ω)] is evaluated in the composite orbital-sublattice basis24by integrating out the fermions in the (π,0) ordered SDW state as: [χ0(q,ω)]ab=i/integraldisplaydω′ 2π/summationdisplay k′[G↑ HF(k′,ω′)]ab [G↓ HF(k′−q,ω′−ω)]ba(4) where [Gσ HF(k,ω)]=[ω1−Hσ HF(k)]−1are the HF level Green’s function in the SDW state and a,bbeing the 0 40 80 120 160 200 240 280 320 ΓM X ΓM' Xωq (meV)α magnet β magnet FIG. 3. Spin-wave dispersions for the two magnet modes α andβalong symmetry directions of the BZ for 25% hole dop- ing. Here t=−200 meV, t′/t=0.5, ∆/|t|=2.0, and the Fermi energyEF/|t| ≈ −2.0. 0 40 80 120 160 200 ωq (meV)(a)δn = 0.2 0 40 80 120 160 ΓM X ΓM' X(b)ωq (meV)δn = 0.3 0 40 80 120 160 200 (c) δn = 0.4 0 40 80 120 160 ΓM X ΓM' X(d)δn = 0.5 FIG. 4. Spin-wave energies with different orbital polarizat ion δn=nα−nβfor 25% hole doping. Crossover from negative to positive energy modes shows strong stabilization of the AF- F state with increasing orbital order. The hopping parameter s and exchange fields are same as in Fig. 3. orbital-sublattice basis indices. The spin-wave energies are obtained from the poles of Eq. (3). In the absenceof anyHund’s coupling, the twoorbitals are decoupled and the magnetic system reduces to two independent magnetic sub-systems αandβinvolving AF ordering in xdirection ( αmagnet) and F ordering in y direction ( βmagnet). The spin-wave dispersions for the two magnet modes αandβare shown in Fig. 3 along (0,0)−→(π,0)−→(π,π)−→(0,π)−→(π,π)[Γ−→M−→ X−→M′−→X]. Fort′=0, the system further reduces to independent AF and F chains, which yield zero spin wave energy for wave vector in the respectively perpen- dicular directions (due to absence of any spin coupling along those directions), implying instabilities of the mag- netic state. Fig. 3 thus highlights the important role of finitet′and the correspondingfinite inter-chain spin cou- plingsinstabilizingthetwo αandβmagnetswithrespect to spin twisting in any direction, as indicated by positive spin wave energies over the entire BZ. To investigate the effect of orbital order on the SDW4 state stability, we plot the spin-wave dispersions for dif- ferent orbital polarization δn=nα−nβin Fig. 4. As magnetic state instability is indicated by the spin-wave energy crossing zero and going negative, we will focus only on the lowest (out of the two αandβmode) ener- gies. Here, we have maintained a constant SDW order parameter ∆ and total electron filling n = 0.75 per or- bital, and the occupations nαandnβof the two orbitals are controlled by introducing an energy offset ∆ αβbe- tween the two orbitals. The orbital occupations are (a) nα= 0.85,nβ= 0.65 (∆ αβ=0.38), (b) nα= 0.9,nβ= 0.6 (∆ αβ=0.29), (c) nα= 0.95,nβ= 0.55 (∆ αβ=0.20) and (d) nα= 1.0,nβ= 0.5 (∆ αβ=0). Figure 4 shows a strong stabilization of the SDW state with orbital or- der, as seen by crossoverfrom negative to positive energy modes. The origin of this orbital-order-induced stabilization is as follows. Effectively, the AF and F spin couplings are optimized by the electron density redistribution associ- ated with orbital order. The increased electron den- sity inαband favors AF coupling (super exchange) in thexdirection (NN) and in the diagonal direction (NNN), whereas the increased hole density in βband fa- vors carrier-mediated F coupling in the ydirection (NN) and AF coupling in the diagonal direction (NNN)29. Thus, all the spin couplings work together and the α andβmagnets both reinforce ( π,0) ordering without any frustration. It is important to note here that the magnetic and orbital orderings effectively stabilize each otherand constitute a composite spin-orbital ordered state with m/negationslash=0,δn/negationslash=0. Our model thus provides a mi- croscopic understanding of the close relation between in-plane anisotropy, orbital order, and the SDW mag- netic order. Ifthe inter-orbitalCoulombinteractionterm Vniαniβisincludedinourmodel, itwillonlyenhancethe orbital offset ∆ αβdue to orbital disparity, and therefore enhance the effect discussed above. Ferro orbital order was reported in a recent study of magnetic excitations in iron pnictides within a degener- ate double-exchange model including antiferromagnetic superexchange interactions27. However, the sign of the ferro orbital order reported in this work ( nyz> nxz) does not agree with experiments. Furthermore, for a re- alistic NN hopping value of 200 meV, their calculated spin wave energy scale of around 30 meV is well below the nearly 200 meV energy scale measured in INS exper- iments. Ferro orbital was also reported in other multi- orbital models due to electron correlation31, anisotropic inter-orbital hopping32and electron-lattice coupling33. However, spin wave excitations and role of orbital order- ing in stabilization of ( π,0) state with respect to trans- verse spin fluctuations was not investigated. Moreover, crystal field splitting due to orthorhombic distortion is necessary to stabilize orbital ordering in these models unlike our model. Now, we investigate the effect of finite inter-orbital 0 50 100 150 200 250 300 350ωq (meV) ΓM X ΓM' XJ/|t| = 0 2 0 250 500 Γ M FIG. 5. Strong enhancement in the spin-wave energies on including the Hund’s coupling. Here, t=−200 meV, t′/t=0.3, ∆/|t|=2.0, and hole doping 25%. Hund’s coupling J, and consider the Hamiltonian: H=−/summationdisplay /angbracketleftij/angbracketrightµνσtµν ij(a† iµσajνσ+a† jνσaiµσ)−/summationdisplay iµνUµνSiµ·Siν (5) where the interaction matrix elements Uµν=Uµforµ=ν andUµν=2Jforµ/negationslash=ν. The self-consistent exchangefields are now given by 2∆ α=Uαmα+Jmβand 2∆ β=Uβmβ+ Jmα. The transverse spin fluctuation propagator now in- cludes both UandJladders at RPA level. Figure 5 shows the spin-wave dispersion with and without Hund’s coupling, displaying a strong enhancement of spin-wave energies with Hund’s coupling, as expected since now the αandβmagnets are coupled. ForJ=0, there are two independent Goldstone modes, as theαandβmagnets are independent. When finite Hund’s coupling is included, the two modes get coupled, leading to a single Goldstone mode corresponding to “in- phase” fluctuations (acoustic mode). The “out-of-phase” modeisnowconvertedtoanopticalbranchwhichrapidly becomes significantly gapped even at small values of J. The spin-waves for the acoustic and optical modes (solid and dotted respectively) are shown in the inset [Fig. 5]. The calculated spin-wave energy scale ( ∼200 meV) agrees well with neutron scattering experiments for |t|=200 meV and Uin the intermediate coupling range (U∼W). The value of U ( ∼1-2 eV) agrees with LDA calculations30. Furthermore, the spin-wave dispersion shows a peak at the F zone boundary which is consis- tent with experiments. For ∆ /|t|=2.0, the SDW state effective gap ∼400 meV is well above the maximum spin wave energy. Thus, contrary to other itinerant models9, spinwaveexcitationsinourmodeldonotrapidlydissolve into the particle-hole continuum, as indeed not observed experimentally up to energies of 200 meV. The spin wave energy scales in the Γ-M (AF) and M- X (F) directions are indicative of the AF and F spin coupling strengths. The hole doping dependence of spin wave dispersion (Fig. 6) effectively shows the evolution of emergent spin couplings. At high hole doping, the5 0 50 100 150 200 250 Γ M X Γωq (meV)x =15 % 20 % 25 % 35 % FIG. 6. Variation of spin wave dispersion with hole doping, showing strong suppression of AF (F) spin couplings at high (low) doping. Here t=−200 meV, t′/t=0.3, and ∆= J=2|t|. 0 50 100 150 200 250 300 ΓM X ΓM' Xωq (meV)tπ = 0.0 0.5 1.0 FIG. 7. Strong enhancement in spin wave energy scale and SDW state stability with increasing hopping anisotropy. He re t=−200 meV, t′/t=0.3, ∆= J=2|t|, and hole doping 25%. AF spin coupling is strongly suppressed due to electron density depletion in the AF band, whereas the F spin coupling is optimized due to enhanced hole doping in the F band. On the other hand, at low hole doping, the AF spin coupling gets saturated as the AF (lower) band is filled, whereas the F spin coupling is strongly weakened. Maximum SDW state stability is seen for about 25% hole doping. If this correspondsto the parent compound, then electron doping (reduction in hole doping from this level) results in crossoverto negative-energymodes, indi- cating destabilization of the SDW state. This is in agree- ment with the observedrapiddecreaseofmagneticorder- ing temperature in iron pnictides with electron doping1. Spin waves were studied earlier24in a two-band model with isotropic hopping and therefore no orbital order. For somewhat larger interaction strength (∆ /|t|)=3) and hole doping ( ∼40%), the spin wave energies are qual- itatively similar as in Fig. 5. However, for lower in- teraction strength and hole doping, increasing hopping anisotropy significantly enhances the spin wave energy scale and SDW state stability, as shown in Fig. 7. This confirms that the emergent AF and F spin couplings be- come stronger due to the electron density redistribution and orbital order in the two orbitals, corresponding to enhanced electron and hole densities in the dxzanddyz Fe orbitals, respectively.ΓX M FIG. 8. Fermi surface in the folded BZ for t1=−1.0,t2=0.5, t3=t4=−0.5, where the hopping parameters are defined in the same way as in23. Here,EF=−0.3|t1|for 25% hole doping. 0 50 100 150 200 250 300 350 ωq (meV) Γ M X Γ M' X FIG. 9. Spin wave dispersion for the same parameters as in Fig. 8 (solid line) and for nesting condition as in23(dot- ted line). Here, ∆=3.0 |t1|,J=2.0|t1|and|t1|=200 meV. The finite spin wave energy at M’ indicates emergent F spin cou- pling. IV. MODIFIED VERSION OF TWO-BAND MODEL OF RAGHU ET AL. With the insight thus obtained, we consider a modified version of the two-band model of Raghu et al.which re- tains most of their essential features, and simultaneously yields not only the two circular hole pockets, but also appropriate orbital order, and also evidence of emergent F spin couplings. The hopping parameters are defined in the same way as in23and their values are taken as t1=−1.0,t2=0.5,t3=t4=−0.5. The corresponding Fermi surface and the spin wave dispersion for 25% hole dop- ing are shown in Fig. 8 and Fig. 9, respectively. The finite spin wave energy at the F zone boundary M’ im- plies the emergence of F spin coupling. Although nesting is no longer obtained due to absence of electron pock- ets, we have nα≃nβ≃0.75 in paramagnetic state, nα≃0.82,nβ≃0.68 in (π,0) state and hence orbital order of correct sign is obtained in this model. This sug- gests that an additional dxyband may need to be in- cluded to simultaneously understand all the important electronic Fermi surface features as observed in ARPES experiments (including the elliptical electron pockets)6 and the spin wave features as observed in INS studies. V. CONCLUSION In conclusion, we have shown that within a mini- mal two-orbital itinerant-electron model with hopping anisotropy, the magnetically anisotropic ( π,0) ordered SDW state is stabilized through the generation of orbital orderwhich optimizes both AF and F spin couplings. We have also shown that with the inclusion of inter-orbital Hund’s coupling, the spin stiffness further increases sub- stantially due to the coupling of the two magnetic sub- systems. The calculated spin-wave energy scale, nature ofspin-wavedispersionandpresenceofferro-orbitalorder are in agreement with experimental observations. Thus, the proclivity of iron arsenides towards ( π,0) magnetic ordering may actively involve the orbital degree of free- dom. We have also considered a two-band model taking hopping parameters similar to Raghu et al.23, but with hopping anisotropy, which yields appropriate orbital or- der, circular hole pockets, as well as emergent F spin couplings. The presence of orbitally ordered state in our model allows the possibility of exploring orbital fluc- tuations which have been seen in recent experiments15. Orbital fluctuations may play an important role on the electron-paring mechanism34and can induce s++ wave superconducting state35, and therefore a combination of orbital and spin fluctuations as a possible mechanism for superconductivity also needs to be explored in these ma- terials. Sayandip Ghosh acknowledges financial support from Council of Scientific and Industrial Research, India. REFERENCES 1J. Zhao, Q. Huang, C. de la Cruz, S. Li, J. W. Lynn, Y. Chen, M. A. Green, G. F. Chen, G. Li, Z. Li, J. L. Luo, N. L. Wang, and P. Dai Nature Mat. 7, 953 (2008). 2S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud’ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, and A. I. Goldman Phys. Rev. Lett. 104, 057006 (2010). 3A .I. Goldman, D. N. Argyriou, B. Ouladdiaf, T. Chatterji, A. Kreyssig, S. Nandi, N. Ni, S. L. Bud’ko, P. C. Canfield, and R. J. McQueeney Phys. Rev. B 78, 100506(R) (2008). 4J. Zhao, D.T. Adroja, D-XYao, R.Bewley, S. Li, X. F.Wang, G. Wu, X. H. Chen, J. Hu, and P. Dai Nature Phys. 5, 555 (2009). 5S. O. Diallo, V. P. Antropov, T. G. Perring, C. Broholm, J. Pulikkotil, N. Ni, S. L. Bud’ko, P. C. Canfield, A. Kreyssig, A . I. Goldman, and R.J. McQueeny Phys. Rev. Lett. 102, 187206 (2009). 6R.A. Ewings, T. G. Perring, J. Gillett, S. D. Das, S. E. Sebast ian, A. E. Taylor, T. Guidi, and A. T. Boothroyd Phys. Rev. B 83, 214519 (2011). 7I. Eremin, and A. V. Chubukov Phys. Rev. B 81, 024511 (2010). 8P. M. R. Brydon, and C. Timm Phys. Rev. B 80, 174401 (2009).9J. Knolle, I. Eremin, A. V. Chubukov, and R. Moessner Phys. Rev. B81140506(R) (2010). 10K. Nakamura, R. Arita, and M. Imada J. Phys. Soc. Jpn. 77, 093711 (2008). 11S. L. Skornyakov, A. V. Efremov, N. A. Skorikov, M. A. Ko- rotin, Y. A. Izyumov, V. I. Anisimov, A. V. Kozhevnikov, and D. Vollhardt Phys. Rev. B 80, 092501 (2009). 12T. Shimojima, K. Ishizaka, Y. Ishida, N. Katayama, K. Ohgush i, T. Kiss, M. Okawa, T. Togashi, X-Y Wang, C-T Chen, S. Watan- abe, R. Kadota, T. Oguchi, A. Chainani, and S. Shin Phys. Rev. Lett.104, 057002 (2010). 13M. Yi, D. Lu, J.-H. Chu, J. G. Analytis, A. P. Sorini, A. F. Kemper, B. Moritz, S.-K. Mo, R. G.Moore, M. Hashimoto, W.- S. Lee, Z. Hussain , T. P. Devereaux, I. R. Fisher, and Z-X Shen Proc. Nat. Ac. Science 108, 6878 (2011). 14M. F. Jensen, V. Brouet, E. Papalazarou, A. Nicolaou, A. Tale b- Ibrahimi, P. Le F` evre, F. Bertran, A. Forget, and D. Colson D Phys. Rev. B 84014509 (2011). 15Y. K. Kim, W. S. Jung, G. R. Han, K.-Y.Choi, K.-H.Kim, C.-C. Chen, T. P. Devereaux, A. Chainani, J. Miyawaki, Y. Takata, Y . Tanaka, M. Oura, S. Shin, A. P. Singh, H. G. Lee , J.-Y. Kim, and C. Kim Phys. Rev. Lett. 111, 217001 (2013). 16W. Lv, J. Wu, and P. Phillips Phys. Rev. B 80, 224506 (2009). 17C.-C. Lee, W.-G. Yin, and W. Ku Phys. Rev. Lett. 103, 267001 (2009). 18C.-C. Chen, J. Maciejko, A. P. Sorini, B. Moritz, R. R. P. Sing h, and T. P. Devereaux Phys. Rev. B 82, 100504(R) (2010). 19W. Lv, and P. Phillips Phys. Rev. B 84174512 (2011). 20M. A. Tanatar, E. C. Blomberg, A. Kreyssig, M. G. Kim, N. Ni, A. Thaler, S. L. Bud’ko, P. C. Canfield, A. I. Goldman, I. I. Mazin, and R. Prozorov Phys. Rev. B 81, 184508 (2010). 21J-H Chu, J. G. Analytis, K. De Greve, P. L. McMahon, Z. Islam, Y. Yamamoto, I. R. Fisher Science 329, 824 (2010). 22T-M Chuang, M. P. Allan, J. Lee, Xie Yang, Ni Ni, S. L. Bud’ko, G. S. Boebinger, P. C. Canfield, and J. C. Davis Science 327, 181 (2010). 23S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C. Zhan g Phys. Rev. B 77, 220503(R) (2008). 24N. Raghuvanshi, and A. Singh J. Phys.: Cond. Mat. 23, 312201 (2011). 25J. Knolle, I. Eremin , and R. Moessner Phys. Rev. B 83, 224503 (2011). 26N. Raghuvanshi, and A. Singh J. Phys.: Cond. Mat. 22, 422202 (2010). 27W. Lv, F. Kr¨ uger, and P. Phillips Phys. Rev. B 82, 045125 (2010). 28F. Kr¨ uger, S. Kumar, J. Zaanen, and J. van den Brink Phys. Rev. B79, 054504 (2009). 29A. Singh, and H. Ghosh Phys. Rev. B 65, 134414 (2002). 30W. L. Yang, A. P. Sorini, C.-C. Chen, B. Moritz, W.-S. Lee, F. Vernay, P. Olalde-Velasco, J. D. Denlinger, B. Delley, J. -H. Chu, J. G. Analytis, I. R. Fisher, Z. A. Ren, J. Yang, W. Lu, Z. X. Zhao, J. van den Brink, Z. Hussain, Z.-X. Shen, and T. P. Devereaux Phys. Rev. B 80, 014508 (2009). 31Y.-M. Quan, L.-J. Zou, D.-Y. Liu, and H.-Q. Lin J. Phys.: Cond . Mat.24, 085603 (2012). 32E. Bascones, M. J. Calderon, and B. Valenzuela Phys. Rev. Let t. 104, 227201 (2010). 33D.-Y. Liu, Y.-M. Quan, D.-M. Chen, L.-J. Zou, and H.-Q. Lin Phys. Rev. B 84, 064435 (2011). 34T. Shimojima, F. Sakaguchi, K. Ishizaka, Y. Ishida, T. Kiss, M. Okawa, T. Togashi, C.-T. Chen, S. Watanabe, M. Arita, K. Shi- mada , H. Namatame, M. Taniguchi, K. Ohgushi, S. Kasahara, T. Terashima, T. Shibauchi , Y. Matsuda, A. Chainani, and S. Shin Science 332, 564 (2011). 35T. Saito, S. Onari, and H. Kontani Phys. Rev. B 82, 144510 (2010).

1511.01109v2.Spin_Orbit_Coupling_Induced_Back_action_Cooling_in_Cavity_Optomechanics_with_a_Bose_Einstein_Condensate.pdf

arXiv:1511.01109v2 [quant-ph] 6 Jan 2017Spin-Orbit Coupling Induced Back-action Cooling in Cavity -Optomechanics with a Bose-Einstein Condensate Kashif Ammar Yasir,1,2,∗Lin Zhuang,3,†and Wu-Ming Liu1,2,‡ 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China. 3School of Physics, Sun Yat-Sen University, Guangzhou 51027 5, P. R. China. We report a spin-orbit coupling induced back-action coolin g in an optomechanical system, com- posedofaspin-orbitcoupledBose-Einstein condensatetra ppedinanoptical cavitywithonemovable end mirror, by suppressing heating effects of quantum noises . The collective density excitations of the spin-orbit coupling mediated hyperfine states – serving as atomic oscillators equally coupled to the cavity field – trigger strongly driven atomic back-actio n. We find that the back-action not only revamps low-temperature dynamics of its own but also provid es an opportunity to cool the mechan- ical mirror to its quantum mechanical ground state. Further , we demonstrate that the strength of spin-orbit coupling also superintends dynamic structur e factor and squeezes nonlinear quantum noises, like thermo-mechanical and photon shot noise, whic h enhances optomechanical features of hybrid cavity beyond the previous investigations. Our findi ngs are testable in a realistic setup and enhance the functionality of cavity-optomechanics with sp in-orbit coupled hyperfine states in the field of quantum optics and quantum computation. PACS numbers: 42.50.Wk, 37.10.De, 71.70.Ej I. INTRODUCTION Cavity-optomechanics provides splendid foundations in utilizing mechanical effects of light to couple optical degree of freedom with mechanical degree of freedom [ 1– 3]. A pivotal paradigm was to cool vibrational modes of mechanical degree of freedom to its quantum mechan- ical ground state which has been attempted to achieve via laser radiations, electronic feedback and dynamical back-action [ 4–12]. The dynamical back-action is the cavity delay induced by the interactions of intra-cavity radiation pressure and the Brownian motion of the mir- ror which leads to cool mirror depending upon detun- ing [13–16]. The demonstration of cavity-optomechanics with other physical objects, like cold atomic gases [ 17] and Bose-Einstein condensate (BEC) [ 18–20], opens up various new aspects to further cool vibrational modes through atomic back-action [ 21–25]. However, thermo- mechanical heating, due to the quantum noises [ 26–30], is a major obstacle in achieving an oscillator with long phonon lifetime in quantum ground state which we in- tend to solve by the inclusion of spin-orbit (SO)-coupled BEC. The SO-coupling, a stunning phenomenon describing interaction between spin of quantum particle and its mo- mentum, has made remarkable progress in the last few years [31–33] and is essential to analyze spin-Hall effect [34,35], topological insulators [ 36–39] and spintronic de- vices [40]. The demonstration of SO-coupling in one- dimensional optical lattices [ 41–52] and optical cavities ∗kayasir@iphy.ac.cn †stszhl@mail.sysu.edu.cn ‡wliu@iphy.ac.cn[53–55] enables us to utilize this phenomenon in optome- chanical environment. The SO-coupling induces signifi- cant variations in cavity radiation pressure by separat- ing atomic mode in hyperfine spin-states which gives rise to self-confinement via dynamical back-action [ 1,56–58]. Further, dynamic structure factor, a phenomenon de- scribing density fluctuations and mean energy of excited quasi-particles, is crucially important in quantum many- body systems [ 59] and provides explanation of quasi- particle evolution under noise effects [ 60–62]. In this paper, wereportthe SO-couplinginduced back- action cooling mechanism in a hybrid optomechanical cavity with SO-coupled BEC and one moving end mir- ror in the presence of quantum noises. We show that the SO-coupling induced features modify intra-cavity atomic back-action which not only leads to maneuver low-temperaturedynamics of atomic mode but alsohelps (a)(b) FIG. 1. (Color online) (a) Schematic diagram of spin-orbit (SO)-coupled87RbBose-Einstein condensate (BEC) trapped inside a high- QFabry-P´ erot cavity with one moving end mir- ror, where ˆ y-axis is the cavity axis and ˆ x-axis is the direction of incident Raman beams. (b) Energy level configuration of SO-coupled BEC.2 in ground state cooling of vibrational modes of the cav- ity mirror. Further, the coupling of mechanical mirror with cavity modifies the eigenenergy spectrum of hyper- fine states via transformation of phonons to atomic de- gree of freedom and provides a way to tune quantum phase transitions in BEC. Furthermore, we compute dy- namic structure factor by manipulating two frequency auto-correlation of photons leaking-out from cavity and observe the influence of SO-coupling on dynamic struc- ture factor. II. CAVITY-OPTOMECHANICS WITH SO-COUPLED BEC The system consists of a high- QFabry-P´ erot cav- ity, with one fixed and one movable mirror, contain- ing SO-coupled BEC illuminated along ˆ y-axis and coher- ently driven by single-mode optical field with frequency ωp=ωR+δωR= 8.8×2πMHz, see Fig. 1(a). To produce SO-coupling, we chose two internal atomic pseudo-spin- states inN= 1.8×105 87Rbbosonic particles having F= 1 electronic ground manifold of 5 S1/2electronic lev- els labeled as | ↑/an}bracketri}ht=|F= 1,mF= 0/an}bracketri}ht(pseudo-spin-up) and| ↓/an}bracketri}ht=|F= 1,mF=−1/an}bracketri}ht(pseudo-spin-down), as shown in Fig. 1(b). The magnetic 10 Gbias fieldB0is applied along cavity axis (ˆ y-axis) to induce Zeeman shift /planckover2pi1ωz, whereωz≈4.8×2πkHz. Two counter-propagating Ramanlasersalong ˆ x-axis,withwavelength λ= 804.1nm anddetuning δ= 1.6ER, interactwith atomicspin-states in opposite direction. The frequencies of these Raman beams areωRandωR+δωR, respectively, with constant frequency difference δωR=ωz+δ//planckover2pi1⋍4.8×2πMHz. kLkLkL=/planckover2pi1ky=√ 2π/planckover2pi1/λandER= (/planckover2pi1ky)2/2ma= 20×2π kHz represent unit-less momentum and energy, respec- tively. The mechanical mirror is coupled to the cavity mode, oscillating with frequency ωc= 4×2πMHzand detuning ∆ c=ωp−ωc=δωR, via radiation pressure force [1,18]. The system Hamiltonian consists of three parts, ˆH= ˆHa+ˆHm+ˆHf. In strong detuning regime and under rotating-wave approximation, the many-body Hamilto- nian for atomic mode ( Ha) is given as [ 42,53,54], ˆHa=/integraldisplay drˆψ†ˆψ†ˆψ†(rrr)/parenleftbigg H0+VLAT/parenrightbigg ˆψˆψˆψ(rrr) +1 2/integraldisplay drrr/summationdisplay σ,´σUσ,´σˆψ† σ(r)ˆψ† ´σ(r)ˆψ´σ(r)ˆψσ(rrr),(1) wheremais the mass of an atom, ˆψˆψˆψ= [ˆψ↑,ˆψ↓]Trep- resents bosonic field operator for pseudo-spin-up and - down atomic states. H0=/planckover2pi12kkk2σ0/2ma+ ˜αkxkxkxσy+ δ 2σy+Ωz 2σzdescribes single-particle Hamiltonian con- taining SO-coupling terms [ 33,63], where ˜α=ER/kLkLkL is the strength of SO-coupling. δ=−gµBBzand Ω z= −gµBByare related to the Zeeman field effects along ˆ z and ˆyaxis, respectively. σx,y,zrepresents 2 ×2 Pauli ma- trices under pseudo-spin rotation and σ0is a unit matrix[42,63].VLAT=/planckover2pi1ˆc†ˆcU0[cos2(kx)+cos2(ky)] is the intra- cavity optical lattice under assumption kx=ky=kand both atomic states are equally coupled to the cavity be- cause of having same motional quantum state [ 54,64]. /planckover2pi1ˆc†ˆcU0is optical potential depth with atom-photon cou- plingU0=g2 0/∆a, whereg0is the vacuum Rabi fre- quencyand∆ aisfar-offdetuningbetweenfield frequency and atomic transition frequency ω0. Here, ˆc(ˆc†) are annihilation (creation) operators for cavity mode. Fi- nally, last term explains many-body intra-species and inter-species interactions for atomic spin-states, where σ,´σ∈ {↑,↓}.Uσ,´σ= 4πa2 σ,´σ/planckover2pi12/maaccounts for strength of atom-atom interactions, where aσ,´σis the s-wave scat- tering length. The Hamiltonian for moving end mirror is ˆHm= /planckover2pi1ωmˆb†ˆb−i/planckover2pi1gm√ 2ˆc†ˆc(ˆb†+ˆb), where first term describes the motionofmechanicalmirrorwith frequency ωmandˆb(ˆb†) are annihilation (creation) operators for mechanical mir- ror with commutation relation [ ˆb†,ˆb] = 1. Second term accommodates mechanical mirror coupling with cavity mode with coupling strength gm=√ 2(ωc/L)x0, where x0=/radicalbig /planckover2pi1/2mωmis zero point motion of mechanical mir- ror having mass m.ˆHc=/planckover2pi1△cˆc†ˆc−i/planckover2pi1η(ˆc−ˆc†), where first term is the strength of cavity mode and second part is associated with its coupling with external pump field with strength |η|=/radicalbig Pκ//planckover2pi1ωp, wherePis the input field power. We substitute plane-wave ansatz ˆψˆψˆψ(r) =eikrˆϕˆϕˆϕ, where ˆϕˆϕˆϕ= [ˆϕ↑,ˆϕ↓]T, in atomic mode Hamiltonian Haby con- sidering homogeneous atomic modes distribution with normalization condition |ˆϕ↑|2+|ˆϕ↓|2=N. We assume that the strengths of intra-species interactions of both spin-states are equal with each other and are defined as, U↑,↑=U↓,↓=U. Similarly, inter-species interactions can be modeled as, U↑,↓=U↓,↑=εU, where parame- terεdepends upon the incident laser configuration [ 33]. Under these considerations, we solve equation Haand compute quantum Langevin equations for the system by usingstandardquantum-noiseoperatorsto include quan- tum noises and dissipations associated with the system [21,50]. The quantum Langevin equation helps us in developing coupled and time dependent set of equations, containing noise operators for optical, mechanical and atomic degrees of freedom, dˆc dt=˙ˆc= (i˜∆+igm√ 2(ˆb+ˆb†)−igaˆϕ†ˆϕˆϕ†ˆϕˆϕ†ˆϕ−κ)ˆc+η +√ 2κain, (2) dˆb dt=˙ˆb=−ωmˆb−gm√ 2ˆc†ˆc−γmˆb+√γmfm, (3) dˆϕˆϕˆϕ dt=˙ˆϕˆϕˆϕ= (/planckover2pi1kkk2σ0 2m+ ˜αkxkxkxσy+δ 2σy+Ωz 2σz−γa+gaˆc†ˆc)ˆϕˆϕˆϕ +1 2Uˆϕˆϕˆϕ†ˆϕˆϕˆϕˆϕˆϕˆϕ+1 2εUˆϕ† σˆϕ´σˆϕσ++√γafa, (4)3 FIG. 2. (Color online) ( a-c) Eigenenergies spectrum ENof spin-orbit (SO)-coupled Bose-Einstein condensate (BEC) a s a function of quasi-momentum kx/kLkx/kLkx/kLfor different Ω z/ωmand δ/ωm, whengm/ωm= 0.1 andα/ωm= 20π. (It should be noted that we consider kykyky=kzkzkz= 0 because SO-coupling is occurringonlyinthedirectionof ˆ x-axis.) Theblackcurverep- resents dispersion at Ω z/ωm= 0while magenta shaded curves (from darkest to lightest) correspond to Ω z/ωm= 2,4,6,8, respectively. ( a), (b) and (c) show the behavior of disper- sionENforδ/ωm=−1,0,1, respectively. ( d) and (e) show dispersion ENversuskx/kLkx/kLkx/kLandgm/ωm, withα/ωm= 30π andα/ωm= 50π, respectively, at Ω z=ωmandδ/ωm= 0. The other parameters used are U/ωm= 5.5,ε/ωm= 0.1, κ/ωm= 0.1,γa/ωm= 0.01,γm/ωm= 0.05 and mechanical mirror frequency ωm≈ωc−ωp. where˜∆ = ∆ c−NU0/2 is the modified detuning of the system and ˆ cinis Markovian input noise operator associated with intra-cavity field, having zero-average /an}bracketle{tˆcin(t)/an}bracketri}ht= 0 and delta-correlation /an}bracketle{tˆcin(t)ˆc† in(´t)/an}bracketri}ht=δ(t−´t) under the condition /planckover2pi1ωc>> kBT. The term γmde- scribes mechanical energy decay rate of the moving end mirror and ˆfmis noise operator (or zero-mean Langevin-force operator) connected with the Brownian motion of mechanical mirror and can be defined by us- ing non-Markovian correlation [ 21,50]/an}bracketle{tˆfm(t)ˆfm(´t)/an}bracketri}ht= γm 2πωm/integraltext dωe−iω(t−´t)[1 +Coth(/planckover2pi1ω 2kBT)]. The external har- monic trapping potential of the condensate, which we have ignored so far because it appeared to be spin in- dependent, cause the damping of the atomic motion. The parameter γarepresents such damping of atomic dressed states motion while ˆfais the associated noise op- erators assumed to be Markovian with delta-correlation /an}bracketle{tˆfa(t)ˆf† a(´t)/an}bracketri}ht=δ(t−´t) under the condition /planckover2pi1Ω>>kBT. Further,σ,´σ∈ {↑,↓}andga=ωc L/radicalbig /planckover2pi1/mbec4ωris the coupling of atomic mode with intra-cavity field, hav- ing effective BEC frequency Ω = 4 ωrand massmbec= ℏω2 c/(L2U2 0ωr), whereωr= 3.8×2πkHz is the recoil fre- quency of atoms and L= 1.25×10−4m is the cavity length. III. CONDENSATE DISPERSION SPECTRUM The eigenenergy spectrum is calculated from time- dependent quantum Langevin equations (for details seeAppendix A). The SO-coupling will create two minima corresponding to lowest energy-levels of atomic spin- states, as illustrated in Fig. 2. At Ω z/ωm= 0, no band- gap appears between lower and upper dispersion states causing the phase mixing of atomic dressed states. How- ever, in presence of Raman coupling, the band-gap be- tween upper EN>0 and lower EN<0 dispersion states appears in the form of Dirac-cone which increases with the increase in Raman coupling. The higher values of Ωz/ωmmerge two minima corresponding to the dressed states into single minima causing quantum phase tran- sitions from mixed phase to separate phase of atomic mode. It can also be seen that the non-zero Raman de- tuningδ/ωm/ne}ationslash= 0 leads to the symmetry-breaking of dis- persion over quasi-momentum. For the small value of Raman coupling (Ω z<4ωm), the dispersion appears in the form of double-well potential in the quasi-momentum which leads to the zero group velocity of atoms [ 65]. The asymmetric behavior indicates rapid population transfer and enhancement in band-gap induced features of hyper- fine states in cavity environment. Moreover, it is noted that because of cavity confinement, the atomic quasi- momentum kx/kLkx/kLkx/kLinteracts with the optical mode along cavity axises. Thus, SO-coupled BEC face an anisotropic potential which leads towards spatial spread of BEC en- ergy spectrum along cavity axis, as can be seen in Fig. 2 The coupling between atomic states and intra-cavity potential is disturbed by the existence ofmechanical mir- ror, and vice-versa, when the atomic modes become res- onant with the optical sideband. At this point, atomic spin states will absorb some phonons emitted by the mechanical mirror via cavity mode and will behave as a phononic-well. Therefore, the increase in mirror-field couplinggivesrisetoatomic-stateenergylevels, asshown in Fig.2(d) and2(e), which provides precise control over the dispersion relation of atomic energy spectrum and quantum phase transitions of BEC. IV. ATOMIC DENSITY-NOISE SPECTRUM We calculate density-noise spectrum (DNS) of atomic spin states by taking two-frequency auto-correlation of the frequency domain solu- tion of quantum Langevin equations, S↑,↓(ω,∆) = 1 4π/integraltext e−i(ω+ω′)t/an}bracketle{tδˆq↑,↓(ω)δˆq↑,↓(ω′)+δˆq↑,↓(ω′)δˆq↑,↓(ω)/an}bracketri}htdω′, whereδˆq↑,↓are dimensionless position quadratures of spin states defined as, δˆq↑,↓=1√ 2(ˆϕ↑,↓+ ˆϕ† ↑,↓). Here the effective system detuning is ∆ = ˜∆−gmqs+gaN, where qsis steady-state position quadratures of mechanical mirror, while Gm=√ 2gm|cs|andGa=√ 2ga|cs|are the effective couplings of intra-cavity field with me- chanical mirror and atomic modes, respectively, tuned by the steady-state cavity mode amplitude cs=η κ+i∆ (for detailed calculations see Appendix BandC). By considering the correlation operators of Markovian and Brownian noises in frequency domain [ 21,22,66,67], we4 (d) (e) (f)(a) (b) (c) FIG. 3. (Color online)( a-c) Illustrate DNS S↑(ω,∆)asafunc- tion of ∆ /ωmandω/ωmforα/ωm= 0π,150πand 250π, re- spectively, when Ω z=ωm,δ/ωm= 1 and Ga/ωm= 28.5. (Note: The color configuration corresponds to the strength of DNS.) Similarly, ( d-f) demonstrate DNS S↓(ω,∆) for α/ωm= 0π,150πand 250π, respectively. Here Gm/ωm= 1.5, Ω/ωm= 70.8,ωm= 3.8×2πkHz and the thermal reservoir temperature is taken as T= 300K. plot the DNS for pseudo spin- ↑and spin- ↓atomic states as shown in Fig. 3. The inclusion of SO-coupling in trapped-atoms mod- ifies atomic back-action generated by the interaction of intra-cavity radiation pressure with BEC. These modifi- cations enhance the cavity induced self-regulatory mech- anism of atomic mode. In the absence of SO-coupling (α/ωm= 0π), bothS↑(ω,∆) andS↓(ω,∆) behave in a similar way as shown in Fig. 3(a) and3(d) [21,22], where α= ˜αkxkxkxis the effective strength of SO-coupling. Both the cooling as well as heating mechanisms are observ- able because area under S↑(ω,∆) describes the effective temperature of atomic mode, as shown in effective tem- perature calculation of mechanical mirror in next sec- tion. One can observe a semi-circular structure appear- ing with the increase in ∆ /ωmcaused by the red-shift in the peak frequency of S↑(ω,∆). Height of the structure initially decreases with increase in ∆ /ωmtowardsω/ωm but shortly again starts rising along the semi-circular structure. The optimal cooling is achieved at ∆ = ωm/2 with a considerable shrink in the area underneath atomic DNS. In the presence of α/ωm,S↑(ω,∆) andS↓(ω,∆) start behaving in a different manner because of the emergence of Zeeman shift among the hyperfine atomic states with SO-coupling. For S↑(ω,∆), the height of semi-circular structure is suppressed due to the energy transforma- tion via intra-cavity potential, as can be seen in Fig. 3(b) and3(c), where α/ωm= 150πand 250π, respectively. The optimal cooling point is now shifted to ∆ /ωm= 1. The existence of SO-coupling not only decreases the area underneath S↑(ω,∆) but also suppresses the radius of that semi-circular structure providing controlled cooling of atomic mode. However, at α/ωm/ne}ationslash= 0π,S↓(ω,∆) be- have differently because of more interaction with quan- tum noise effects, as shown in Fig. 3(e) and3(f). Now the height of semi-circular structure appearing in S↓(ω,∆) is 00.51.01.52.08.0 6.0 4.0 2.0 0.0 Effective Temperature T eff ( mK )(c) (d) Ωz = 0.0 ωm Ωz = 60 ωm(a) (b) FIG. 4. (Color online) ( a) Illustrates Sm(ω,∆) versus nor- malized frequency ω/ωmat ∆/ωm= 1.8,Gm/ωm= 10 andGa/ωm= 20. The black curve is for α/ωm= 0πand magenta curves from dark shade to light shade represent α/ωm= 6π,10π,16π,20πand 30π, respectively. ( b) De- scribesTeff(in units of mK) versus ∆ /ωmatω/ωm= 0.1, Gm/ωm= andGa/ωm= 5. Similarly, the black corre- sponds to α/ωm= 0πwhile shaded curves from darkest to lightest represent α/ωm= 60π,90π,120π,150πand 170π, re- spectively. ( c) and (d) showTeff, as a fucntion of α/ωm andU/ωm, at Ω z/ωm= 0 and 80, respectively. Here, Gm/ωm= 18,Ga/ωm= 16 and α/ωm= 30π. being increased with increase in SO-coupling. The SO- couplinginduced Zeemanfield effectgeneratestheenergy gap between dressed states by increasing and decreasing the ground state energies of pseudo spin- ↓and pseudo spin-↑states, respectively. Therefore, by increasing SO- coupling, spin- ↓state will interact with more noise ef- fects and get heated because of having more ground state energy as compared to spin- ↑state. However, it can be controlled by varying system parameters and the radius of semi-circular structure still appears to be de- creasing with SO-coupling due to cavity mediated self- confinement via back-action. Further, in presence of SO- coupling, atom-atom interactions will effect similarly the low-temperature dynamics of atomic mode, as explained in Appendix E. V. MECHANICAL MIRROR COOLING The effective temperature of mechanical mode ( Teff) is calculated by formula Teff=/an}bracketle{tEm/an}bracketri}ht/kB, where/an}bracketle{tEm/an}bracketri}ht= mω2 m/an}bracketle{tδˆq2/an}bracketri}ht/2 +/an}bracketle{tδˆp2/an}bracketri}ht/2m=mω2 m(neff+ 1/2), corre- sponds to the mean energy which is experimentally mea- sured by calculating area underneath DNS of mechan- ical mirror Sm(ω,∆) =1 4π/integraltext e−i(ω+ω′)t/an}bracketle{tδˆq(ω)δˆq(ω′) + δˆq(ω′)δˆq(ω)/an}bracketri}htdω′, whereδˆqis dimensionless position quadrature of mechanical mirror defined as, δˆq=1√ 2(ˆb+ ˆb†) (for detail see Appendix BandC).neffis the ef- fective phonon number which should be less than one in order to achieve ground state cooling. The position5 and momentum variances are related to DNS, /an}bracketle{tδˆR2/an}bracketri}ht= 1 2π/integraltextSR(ω,∆)dω, where Ris a generic operator rep- resenting position δˆqand momentum δˆpquadrature of mechanical mirror. Here the DNS of mechanical mir- ror in momentum space is defined as Sm(p)(ω,∆) = m2ω2 mSm(ω,∆), where mis the effective mass of me- chanical mirror. The cooling mechanism for mechanical mirror, which can simply be explained by thermodynamic arguments, only occurswhen the intra-cavityoptical sideband is cen- tered atωmwhich is in fact a resolved-sideband regime. Therefore, the BEC should also oscillate at optical side- band frequency in order to absorb excitation energies of the mirror from cavity mode otherwise mirror tem- perature will be unaffected. The implication of SO- coupling splits atomic mode into dressed spin states – acting like two atomic mirrors equally coupled to the cavity – which will modify atomic back-action inside the cavity. This phenomenon enables us to transfer more excitation energies in the form of phonons from mechan- ical mirror to atomic degree of freedom. Fig. 4(a) illus- trates such effects where the suppression of mechanical mirror DNS Sm(ω,∆) can be seen with the increase in SO-coupling. The SO-coupling suppresses the heating effects induced by the Brownian motion of the mirror and enhances back-action cooling of oscillating mirror. Fig.4(a) demonstrates mirror DNS at system detuning ∆/ωm= 1.8. If we change system detuning, it will mod- ify mirror DNS by increasing or decreasing its strength but the effects of SO-couplingon mirror DNS will remain the same, as discussed in Appendix F. Further, the SO- coupling reduces Teffover a wide range of detuning be- cause of energy transformation via modified back-action, as shown in Fig. 4(b), where optimal temperature is de- creasing with the increase in SO-coupling. This implies, like atomic-field coupling and atom-atom interactions(as discussed in Appendix G) [13–16,23–25], SO-coupling significantly alters ground state properties of mechanical mirror. Thus, SO-coupling provides another handle to achieve and sustain ground state cooling which is even beyond the previous back-action cooling mechanism. To further analyze the influence of SO-coupled dressed states on mechanical mirror, we plot Teff, as a function ofU/ωmandα/ωm, in absence(Fig. 4(c)) andin presence (Fig.4(d)) of Raman coupling Ω z/ωm. At Ω z/ωm= 0, the maximum value of Teffappears to be approximately centered at U/ωm≈6 and remains saturated with in- crease inα/ωm. One can state that the maximum value ofTeffshows a kind of localized behavior with SO- coupling which is similar to the results presented in ref- erence [23]. On the other hand, in presence of Raman coupling,Teffshows squeezed and exponential behavior withα/ωm, as illustrated in Fig. 4(d). The SO-coupling in presence of strong Raman coupling, which transforms atomic dispersion spectrum into single minima, absorbs more mirror excitation energies and manipulates atom- atom interactionseffects on mirrortemperature bymodi- fying back-action. Intuitively saying, the higher values of (a) (b) (c) (d) (e) (f)0100200300400500 0.01.02.03.04.05.0 Sout(P,
)6 4 2 0 -2 -4 -60246802468 FIG. 5. (Color online) ( a) and (b) DNS of optical field leaking-out of cavity Sout(P,ω) (in units of W/Hz), versus P/Pcrandω/ωm, forα/ωm= 0πand 80π, respectively. ( c-f) show dynamic structure factor (in units of 1 /Hz) correspond- ing to the out-going optical mode DNS versus ω/ωmfor dif- ferent values of P/Pcr. SO-coupling α/ωm= 0π(80π) for (c) and(d). WhilethestrengthofSO-couplingis α/ωm= 80πfor (e) and (f). HereGm/ωm= 1.5,Ga/ωm= 0.9,U/ωm= 5.5 andδ/ωm= 0. Ramancouplingchangequantumphaseoftrappedatoms causing the alteration in their many-body interactions as well as in SO-coupling effects. Therefore, the suppressed and nonlinear behavior of Teffis caused by the emer- gence of band-gap induced quantum phase transitions of BEC and can be further enhanced by increasing Ω z/ωm providing control over temperature of mechanical mirror [13,14,23]. VI. DYNAMIC STRUCTURE FACTOR We analyze dynamic structure factor by computing Fourier domain auto-correlations of light leaking-out of the cavity. The resultant dynamic structure factor is given as [ 62],SD(k,ω) =4(κ2+∆2) Nη2(1 2πSout(P,ω) + n2 sδ(ω)),wherensis the steady-state photon number andSout(P,ω) is DNS of out-going optical mode (see Appendix D). The frequency ωis referred to the shifted frequency of input field after interacting with the system which causes inelastic photon scattering. In absence of SO-coupling, Sout(P,ω) contains two sidebands at ω <0ωmandω >0ωmcaused by the in- coherent creation and annihilation of quasi-particles [ 62], respectively, see Fig. 5(a). If we increase the input power, both the sidebands tend to move towards ω= 0ωmbe- cause of quantum fluctuations which decrease the spec- tral densities of quasi-particles. Intuitively, it is referred to the scattering of intra-cavity optical mode at Bragg planes in the density-modulated cloud [ 68,69]. Both the sidebands seem to get mixed with each other due to the presenceofanothersecondarystructureapproximatelyat P≈6Pcr. The secondary structure, which is centered at ω= 0ωm, is caused by associated quantum noises [ 62]. In the presence of SO-coupling, secondary structure is6 (a) (b) FIG. 6. (Color online) The dynamic structure factor SD(k,ω) (in units of 1 /Hz) as a function of ω/ωmfor different strengths of α/ωmatU= 5.5ωm(a) and atom-atom interac- tionsU/ωmatα= 10πωm(b). The input field power ratio is fixed to P/Pcr= 4 and the remaining coupling strengths are same as in Fig. 5. In (a), the green shaded curves from darkest to lightest correspond to the SO-coupling α/ωm= 0π,60π,100πand 140π, respectively. Similarly, in ( b), green curves (from dark shade to light shade) carry the influence of atom-atom interactions with strengths U/ωm= 5.5,7.5,9.5 and 11.5, respectively. shifted toP≈7Pcrdue to the modifications in the in- elastic scattering of cavity mode by atomic spin-phase transitions [ 59], as shown in Fig. 5(b). The spectral den- sities of both sidebands as well as secondary structure are also increased due to the addition of quasi-particles excited by the SO-coupling. Dynamic structure factor SD(k,ω) at power ratio P/Pcr= 3 shows two sidebands at ω≈ −2ωmand ω≈2ωmcorresponding to creation and annihilation of quasi-particles, respectively, as shown in Fig. 5(c). An- other, comparatively small, fluctuating structure can be seen atω= 0ωminduced by the quantum-noise effects which verifies the experimental finding of dynamic struc- ture factor in reference [ 62]. If we increase the input field power, the dynamic structure factor will be suppressed by the increase in system fluctuations and the sidebands will move towards ω= 0ωm, as shown in Fig. 5(d). How- ever, the strength of structure appearing at ω= 0ωmis increasedbecause ofquantum noise effects. Interestingly, these effects can be suppressed by SO-coupling because of enhanced intra-cavity atomic back-action, see Fig. 5(e) and5(f), which leads to the enhancement of sideband spectrum. InordertofurtherunderstandinfluenceofSO-coupling as well as cavity mediated long-range atom-atom in- teractions, we plot SD(k,ω) for multiple values of SO-coupling and atom-atom interactions. Fig. 6(a) and6(b) carrySD(k,ω) as a function of normalized frequency for different strengths of α/ωmandU/ωm, respectively, at input field power P/Pcr= 5. It can be clearly seen that by increasing the SO-coupling and atom-atom interactions, the sidebands are enhanced and shifted away from ω= 0ωmdue to the addition of quasi-particles. However, quantum noise fluctuations, appearing at ω≈0, are now being suppressed by increasing α/ωmandU/ωmcausing enhancement in optomechancial applications. Here, it should be noted that the strength of atom-atom interactions definesinter-species as well as intra-species interactions and the frequency of the atomic mode is directly proportional to the strength of interactions√ U[23]. Therefore, like SO-coupling, atom-atom interactions have significant influence on the intra-cavity atomic back-action which leads to the enhancement in inelastic scattering of the cavity mode. Thus, the inclusion of SO-coupling purifies dynamic structure factor by squeezing quantum noises. VII. CONCLUSION We demonstrate SO-coupling induced back-action cooling in cavity-optomechanics with SO-coupled BEC. The SO-coupling modifies dynamical back-action which enhances low-temperature profile of atomic mode by squeezing associated noises. It has been shown that the existence of SO-coupling leads to cool vibrating end mir- ror to its quantum mechanical ground state. Further, by computing dynamic structure factor, we have shown that the SO-coupling enables us to manage and imple- ment noiseless quasi-particles. Likewise, mechanical mir- rorgivesrisetotheeigenenergyspectrumofatomicstates providing control over quantum phase transitions. We chose a particular set of parameters and procedures very close to the present experimental ventures which makes our study experimentally feasible. Our findings con- stitute a significant step towards the utilization of SO- coupled BEC-optomechanics in the field of quantum op- tics and quantum information. ACKNOWLEDGMENT This work was supported by the NKRDP under grants Nos. 2016YFA0301500, NSFC under grants Nos. 11434015, 61227902, 61378017, KZ201610005011, SKLQOQOD under grants No. KF201403, SPRPCAS under grants No. XDB01020300, XDB21030300. We also acknowledge the financial support from CAS-TWAS president’s fellowship programme (2014). Appendix A: Atomic eigenenergies calculation Here we provide some details about energy disper- sion calculation of atomic states. By adopting a mean- field approximation, we consider the intra-cavity field in steady-state and replace the intra-cavity field operator by its expectation value ˆ c→ /an}bracketle{tc/an}bracketri}ht ≡cs. To calculate en- ergy dispersion ENof atomic modes, we define ENas the solution of nonlinear quantum Langevin equations and replace the time derivative id/dtwith eigenenergy EN. After performing some mathematics and applying Pauli matrices, the coupled Langevin equations will take the form [ 33,50,55],7 ns=c† scs=η κ2+(˜∆−gm√ 2(ˆb†+ˆb)+ga(ˆϕ† ↑ˆϕ↑+ ˆϕ† ↓ˆϕ↓))2, (A1) ˆb=gmc† scs√ 2(EN+iωm+γm),ˆb†=gmc† scs√ 2(EN−iωm+γm), (A2) EN/parenleftbiggˆϕ↑ ˆϕ↓/parenrightbigg =/parenleftigg /planckover2pi1kkk2 2m+Ωz 2+gac† scs+1 2UN−γa−i(αkxkxkx+δ 2)+1 2U(ε−1)ˆϕ† ↓ˆϕ↑ i(αkxkxkx+δ 2)+1 2U(ε−1)ˆϕ† ↑ˆϕ↓/planckover2pi1kkk2 2m−Ωz 2+gac† scs+1 2UN−γa/parenrightigg/parenleftbiggˆϕ↑ ˆϕ↓/parenrightbigg , (A3) wherensis the steady-state photon number inside cav- ity. For simplicity, we have ignored quantum noises as- sociated with the system while calculating eigenenergies of atomic mode. Under number conversation condition |ˆϕ↑|2+|ˆϕ↓|2= 1, we substitute steady-state mechanical mirror operators into equ. A1and simplify in term of ns as, n3 s+2L1L2n2 s+Kns=η2(A4) where, L1= ∆c+ga, (A5) L2=g2 m(EN+γm) (E−n+γm)2+ω2m, (A6) K=κ2+L2 1. (A7) Now, by assuming eigenenergies of moving-end mirror and atomic mode independent by keeping mechanical mirror in steady-state, we rewrite equ. A3, for ˆϕ↑and ˆϕ↓, as, EN/parenleftBigg ˆϕ↑ ˆϕ↓/parenrightBigg = /parenleftBigg h1 w+1 2U(ε−1)ˆϕ† ↓ˆϕ↑ w∗+1 2U(ε−1)ˆϕ† ↓ˆϕ↑ h2/parenrightBigg/parenleftBigg ˆϕ↑ ˆϕ↓/parenrightBigg ,(A8) where, h1,2=/planckover2pi1kxkxkx2 2m±Ωz 2+gans+1 2UN−γa,(A9) w=αkxkxkx+δ 2. (A10) By dividing first line of equ. A8with the conjugate of the second line, we obtain, EN−h1 EN−h2|ˆϕ↑|2=|ˆϕ↓|2. (A11) Further, by denoting s=EN−h1 EN−h2and using number con- versation condition, we calculate |ˆϕ↑|and|ˆϕ↓|as, |ˆϕ↑|2=1 s+1, (A12) |ˆϕ↓|2=s s+1, (A13)and by substituting these values in the first line of equ.A8, we obtain, (EN−h1)2−U(ǫ−1)(EN−h2)s s+1+U2(ǫ−1)2s2 4(s+1)2 =sw2. (A14) Finally, by numerically finding roots of steady-state pho- ton number nsfrom equ. A4and substituting them into equ.A14, we plot the roots of eigenenergies versus quasi- momentum kxkxkx, as shownin Fig. 2. (Note that weconsider kykyky=kzkzkz= 0 because SO-coupling is occurring only in the direction of ˆ x-axis.) Appendix B: Langevin equations and frequency domain solutions The coupled Langevin equations of the system contain nonlinear terms in the form of coupling among different degrees of freedom and noises associated with the sys- tem. By considering intense external pump field, these equations can be linearized with the help of quantum fluctuations as, ˆO(t) =Os+δO(t), where Ocan be any operator of the system, Osrepresents steady-state value andδO(t) is the first order quantum fluctuation. During these calculation for simplicity, we assume that both the atomic states, spin- ↑and spin- ↓, have equal amount of particles, i.e ˆ ϕ↑†ˆϕ↑= ˆϕ↓†ˆϕ↓=N/2. Furthermore, we define system quadratures in the form of dimensionless position and momentum quadratures as, ˆqO=1√ 2(ˆO+ˆO†) and ˆpO=i√ 2(ˆO−ˆO†), respectively, (Ois generic operator) having commutation relation [ˆqO,ˆpO] =iwhich reveals the value of scaled Planck’s constant /planckover2pi1= 1. Now the linearized Langevin equation are defined in form of ˙X=KX+F, where vector X= [δqc(t),δpc(t),δq(t),δp(t),δq↑(t),δp↑(t),δq↓(t),δp↓(t)]τ contains position and momentum quadratures of the system (here pandqwith↑and↓indicate atomic states, with cindicates cavity mode and without anything indicate mechanical mirror’s mo- mentum and position quadrature) and vector F= [√ 2κqin c,√ 2κpin c,0,2√γmfm,0,2√γafa,0,2√γafa]τ defines noises associated with the system. The matrix Kcontains dynamical parameters associated with the system,8 K= −κ∆ 0 0 0 0 0 0 ∆−κ−Gm0Ga 0Ga 0 −2Gm0−γmωm0 0 0 0 0 0 −ωm−γm0 0 0 0 2Ga0 0 0 MΩz 2(α−δ 2) 0 0 0 0 0Ωz 2M 0−(α−δ 2) 2Ga0 0 0 ( −α+δ 2) 0 M −Ωz 2 0 0 0 0 0 −(−α+δ 2)−Ωz 2M , whereM=Ω 2+v+UN(1−ε)−γa,v=gansand Ω =/planckover2pi1kkk2/mais the recoil frequency of atomic states. α= ˜αkxkxkxis the effective strength of SO-coupling. The evolution of the system can be analyzed by matrix K which contains multiple crucial parameters such as ef- fective detuning ∆ = ˜∆−gmqs+gaN, whereqsis steady-state quadraturesof mechanicalmirror, and mod- ified coupling of intra-cavity optical mode with mechan- ical mirror Gm=√ 2gm|cs|and atomic modes Ga=√ 2ga|cs|, tuned by the mean intra-cavity field with am- plitudecs=η κ+i∆. The particular interlaced nature of thesesteady-stateparametersprovidesanefficientoppor- tunity to understand nonlinear and bistable dynamics of the system. To make the system accurate and useful, we have to ensure stability of the system and for this purpose, we perform stability analysis of the system. The system can onlybestableiftherootsofthecharacteristicpolynomial of matrix Klie in the left half of the complex plane. For thispurpose, weapplyRouth-HurwitzStabilityCriterion [22] on matrix Kand numerically develop stability condi- tions for the system. These stability conditions are given as,M >κ+γm, (α−δ/2)2+M2>κ2+∆2−ω2 m−Ω2 z, ωm>∆>κ>γ m>0 and ∆G2 a+∆G2 m>M(κ2−Ω2 z). We strictly follow these conditions while performing all numerical calculations in the manuscript. Furthermore, we take Fourier transform of linearized Langevin equations to preform frequency domain analy- sis and solve them for position and momentum quadra- tures of intra-cavity field, δqc(ω) =1 L(ω)/parenleftbigg√ 2κ[∆δpin c+(κ+iω)δqin c] + ∆[Gaδq↑(ω)+Gaδq↓(ω)−Gmδq(ω)]/parenrightbigg ,(B1) δpc(ω) =1 L(ω)/parenleftbigg√ 2κ[∆δqin c+(κ+iω)δpin c]+(κ+iω) [Gaδq↑(ω)+Gaδq↓(ω)−Gmδq(ω)]/parenrightbigg ,(B2) respectively, position quadrature of atomic modes, δq↑,↓(ω) =1 X(ω)/parenleftbigg (B↑,↓(ω)+A↓,↑(ω))C(ω)[∆δpin c+(κ +iω)δqin c]+L1,3(ω)fm+L2,4(ω)fa/parenrightbigg ,(B3)andfinallyforthe positionquadratureofmechanicalmir- ror, δq(ω) =1 Xm(ω)/parenleftbigg Am(ω)[∆δpin c+(κ+iω)δqin c] +Bm(ω)fm+Cm(ω)fa/parenrightbigg . (B4) The parameter L(ω) = (κ+iω)2−∆2con- tains effective detuning of the system, W(ω) = γa+iω−Ω/2−v−UN(1−ε),K(ω) =W2(ω)+(α2−δ/2)2 describes atom-atom interactions and Sm(ω) = (γm+iω)2L(ω)−L(ω)ω2 m+ 2G2 m∆(γm+iω) is related to mirror coupling with intra-cavity field. A↑,↓(ω) = 4W(ω)K(ω)L(ω)Sm(ω)±Ω2 zL(ω)Sm(ω)− 8G2 a∆K(ω)Sm(ω) + 16G2 a∆2G2 m(γm+iω)K(ω) andB↑,↓(ω) = ±Ω2 zL(ω)Sm(ω) + 4( ±α∓ δ/2)K(ω)L(ω)Sm(ω) + 8 G2 a∆K(ω)Sm(ω)− 16G2 a∆2G2 m(γm+iω)K(ω) describes the behavior of atomic mode and its association with moving-end mirror of the system. Bm(ω) = 2Gm/radicalbig (2κ)(γm+iω)X(ω)+ = 2Gm∆Ga(A↑(ω) +A↓(ω) +B↑(ω) +B↓(ω)) rep- resents mechanical mirror behavior and its cou- pling with atomic modes. Further, C(ω) = 8G2 a/radicalbig (2κ)K(ω)Sm(ω) + 16G2 a√ 2κG2 m(γm+iω)K(ω), L1,3(ω) = (B↑,↓(ω)+A↓,↑(ω))8G2 a√γm∆K(ω)L(ω)(γm+ iω),L2,4(ω) = (B↑,↓(ω) +A↓,↑(ω))8G2 a√γaK(ω)Sm(ω), Bm(ω) = 2Gm∆Ga(L1(ω) +L3(ω)) + 2√γmL(ω)X(ω) andCm(ω) = 2Gm∆Ga(L2(ω) +L4(ω)). The term X(ω) =A↑(ω)A↓(ω) +B↑(ω)B↓(ω) represents modified susceptibility atomic states and Xm(ω) =X(ω)Sm(ω) corresponds to the modified susceptibility of mechanical mirror. Appendix C: Density-noise spectrum (DNS) By using frequency domain solutions given above and standard formalism for auto-correlation, SO= 1 2π/integraltext e−i(ω−´ω)/an}bracketle{tO(ω)O(´ω)/an}bracketri}htd´ω, as discussed in main text, whereO(ω) is the generic operator, the DNS for pseudo9 spin-↑and spin- ↓atomic states will be read as, S↑,↓(ω,∆) =1 |X(ω)|2/parenleftbigg 2π|C(ω)|2(|B↑,↓(ω)|2+ |A↓,↑(ω)|2)[∆2+κ2+ω2]+2πL2,4(ω) +L1,3(ω)γmω ωm[1+Coth(/planckover2pi1ω 2kBT)]/parenrightbigg .(C1) Similarly, wecanwriteDNS equationformechanicalmir- ror of the system as, Sm(ω,∆) =1 |Xm(ω)|2/parenleftigg |Am(ω)|2(∆2+κ2+ω2) +2πBm(ω)+Cm(ω)γmω ωm[1 +Coth(/planckover2pi1ω 2kBT)]/parenrightigg . (C2) Appendix D: Spectral density of out-going optical field In order to calculate output optical field of the system, we use input-output field relation, δqout c=√ 2kδqc−δqin c andδpout c=√ 2kδpc−δpin c, wherepin,qinandpout,qout represent input and output field quadratures, respec- tively. By utilizing above relation and intra-cavity field quadrature, we obtain output field relation as, δqout c(ω) =1 L(ω)/parenleftigg /bracketleftbig 2κ∆δpin c+(κ2+ω2+∆2)δqin c/bracketrightbig +√ 2κ∆/bracketleftbig Gaδq↑(ω)+Gaδq↓(ω) −Gmδq(ω)/bracketrightbig/parenrightigg , (D1) δpout c(ω) =1 L(ω)/parenleftigg /bracketleftbig 2κ∆δqin c+(κ2+ω2+∆2)δpin c/bracketrightbig +√ 2κ(κ+iω)/bracketleftbig Gaδq↑(ω)+Gaδq↓(ω) −Gmδq(ω)/bracketrightbig/parenrightigg . (D2) Now, by combining position and momentum quadratures of field, we obtain out-going field operator coutas, δcout(ω) =1 L(ω)/parenleftigg /bracketleftbig 2κ∆δc† in+(κ2+ω2+∆2)δcin/bracketrightbig +√ 2κ∆/bracketleftbig Gaδq↑(ω)+Gaδq↓(ω) −Gmδq(ω)/bracketrightbig/parenrightigg . (D3) Further, to determine the dependence of out-going opti- calmode on the externalpump field power P, we redefinecoupling terms as a function of P, Gm=√ 2CSgm=2ωc L/radicaligg Pκ mωmωp(κ2+∆2)2,(D4) Ga=√ 2CSga=2ωc L/radicaligg Pκ mΩωp(κ2+∆2)2,(D5) and after this, we calculate Density-noise spectrum (DNS) of out-going optical mode by simply using two frequency auto-correlation formula, Sout(P,ω) =2π |L(ω)|2/parenleftigg /bracketleftbig κ2+ω2+∆2+2κ∆/bracketrightbig +4κ∆/bracketleftbig GaS↑(ω,∆)+GaS↓(ω,∆) −GmδSm(ω,∆)/bracketrightbig/parenrightigg . (D6) Appendix E: Influence of atom-atom interactions on atomic density-noise spectrum (a) (b)(c) FIG. 7. (Color online) The dynamics of S↑(ω,∆), as a func- tion of detuning∆ /ωmand frequency ω/ωm, under the effects of many-body interactions U/ωm. (a), (b) and (c) demon- strateS↑(ω,∆) with atom-atom interaction U= 5.5ωm, 7.5ωmand 9.5ωm, respectively. Here, the strength of SO- coupling is kept constant α= 10πωm. One can observe that the strength of atom-atom interactions influences S↑(ω,∆) in a similar way as SO-coupling does. By increasing U/ωm, the area underneath S↑(ω,∆) is decreased which leads to the cooling of atomic mode [ 23]. The remaining parameters are same as in Fig. 2. The atom-atom interactions U/ωmof atomic dressed states show similar influence on S↑(ω,∆) as the influence of SO-coupling on the atomic dressed states, which can be seen in Fig. 7(a-c), where the strength of atom-atom interactions is considered as, U= 5.5ωm,7.5ωm,9.5ωm, respectively. The radius as well as height of the atomic DNS decreases with increase in atom-atom interactions U/ωmof dressed states. (Note: The effects of atom-atom interactions S↓(ω,∆) are not shown here because they will be like-wise as on S↑(ω,∆).) As atom-atom interac- tions are the combinationof inter-speciesas well as intra- species interactions and modifies the coupling atomic states with intra-cavity potential, therefore, by increas- ing interactions, the strength of atomic back-action will be increased leading to more self confinement. Thus, the strength of atom-atom interactions can likely be used to10 control the low-temperature dynamics of atomic dressed states as SO-coupling. Appendix F: Mirror density-noise spectrum under influence of SO-coupling and atom-atom interactions (d) (e) (f)(a) (b) (c) FIG. 8. (Color online) Density-noise spectrum (DNS) Sm(ω,∆)(in unitsof W/Hz)for mechanical mirror ofthe sys- temversusnormalized effectivedetuning∆ /ωmandfrequency ω/ωmunder the influence of spin-orbit (SO) coupling α/ωm of atomic spin-states and normalized atom-atom interactio ns U/ωmamong atomic states. ( a) Demonstrates Sm(ω,∆) in the absence of SO-coupling α= 0πωmwith atom-atom in- teractions U= 5.5ωm. The color configuration corresponds to the strength of mechanical mirror DNS ( Sm(ω,∆)). (b) and (c) illustrate the behavior of Sm(ω,∆) under the influ- ence ofα= 60πωmand 120πωm, respectively. The dynam- ics ofSm(ω,∆) under the effects of many-body interactions U/ωmare illustrated in ( d-f). (d), (e) and (f) demonstrate Sm(ω,∆) as a function of ∆ /ωmandω/ωmwith atom-atom interaction U= 5.5ωm, 7.5ωmand 9.5ωm, respectively, while the strength of SO-coupling is kept constant α= 10πωm. The atom-field coupling is considered as Ga= 4.1ωmwhile the mirror-field coupling is taken as Gm= 1.5ωm. The re- maining parameters, used in numerical calculations, are sa me as in Fig. 2. The dynamics of mechanical mirror will also be influ- enced by the existence of atomic-states as atomic dressed states are influenced by the existence of mechanical mir- ror. Fig.8demonstrates Sm(ω,∆) as a function of ∆ /ωm andω/ωm, under the influence ofSO-coupling and atom- atom interaction. The atom-field coupling is considered asGa= 4.1ωmwhile the mirror-field coupling is taken as Gm= 1.5ωm. The behavior of Sm(ω,∆) in the absence of SO-coupling is shown in Fig. 8(a), which is similar to the behavior of atomic DNS. A semi-circular structure appears with increase in detuning ∆ /ωmtowards fre- quencyω/ωm. The height of Sm(ω,∆) decreasesinitially and achieves optimal cooling point. However, when the system detuning is further increased from ∆ = 1 ωm, the Sm(ω,∆) shows rapid increase in the height of structure giving rise to the temperature of mechanical mirror. The strength of SO-coupling induces similar influence as it does on the atomic DNS. The radiusofthe structure is suppressed by the increased SO-coupling, as shown in Fig.8(b) and8(c), where the strength of SO-coupling is increased to α= 60ωmand 130ωm, respectively. Notonly SO-coupling but also the atom-atom interactions of atomic dressed states will show similar effects on me- chanical DNS as they are inducing in atomic DNS. The increase in atom-atom interactions will also reduce the radius of semi-circular structure, as shown in Fig. 8(d), 8(e) and 8(f), where the strength of atom-atom inter- actions is increased to U= 5.5ωm,7.7ωmand 9.5ωm, respectively. The SO-coupling and atom-atom interac- tions modify the atomic density mode excitation lead- ing to the variation in intra-cavity optical spectrum in the form of modified atomic back-action which will con- sequently lead to the absorption of more mirror excita- tions by spin states. It can also be considered as the atomic and mechanical states are connected with each other through intra-cavity radiation pressure, acting as a spring between these two independent entities, therefore, the modifications produced by SO-coupling and atom- atom interaction will show similar influence on mechani- cal mirror as they are producing on atomic dressed states [23]. Appendix G: Influence of atom-field coupling and atom-atom interactions on mechanical mirror temperature (b) (a) FIG. 9. (Color online) ( a) Illustrates the effective- temperature Teffof mechanical mirror under the influence ofGa/ωm. The SO-coupling strength is now considered as α= 100πωmand many-body interaction is kept as U= 5.5ωm. The black curve represents Ga= 20ωmwhile ma- genta curves from dark shade to light shade are for atom- field coupling Ga= 23ωm,26ωm,29ωm,31ωmand 34ωm, re- spectively. Similarly, ( c) deals with the behavior effective- temperature Teffof mechanical mirror under the influence ofU/ωmatα= 100πωmandGa= 20ωm. Similarly, the black curve represents U= 5ωmwhile magenta curves from dark shade to light shade represent atom-atom interactions U= 6ωm,7ωm,8ωm,9ωmand 10ωm, respectively. The other parameters used in numerical calculation are same as in Fig. 2. If we increase the atomic mode coupling with intra- cavity field, atomic dressed states will absorb more phonons emitted by mechanical mirror of the system which will decrease the thermal excitation of mechan- ical mirror. Fig. 9(a) shows such influence of atom-field couplingonmechanicaloscillatorofthe system wherethe effective-temperatureofmirroris decreasedby increasing atom-field coupling [ 21]. The atom-atom interactions of atomic mode will also influence the mechanical mirror11 similarly as SO-coupling has shown [ 23]. The atom-atom interactions also contribute to control the thermal exci-tation of mechanical mirror which lead to cool oscillating mirrorto its quantum mechanical ground state, as shown in Fig.9(b). [1] T. J. Kippenberg, and K. J. Vahala, Science 321, 1172 (2008); M. Aspelmeyer, T. J. Kippenberg, and F. Mar- quardt,Rev. Mod. Phys. 86, 1391 (2014). [2] T. J. Kippenberg, and K. J. Vahala, Optics Express 17, 20911 (2009). [3] P. Meystre, Annalen der Physik 525, 215 (2013). [4] A. D. O′Connell et al.,Nature464, 697 (2010). [5] J. D. Teufel et al.,Nature475, 359 (2011). [6] J. Chan et al.,Nature478, 89 (2011). [7] M. Yuan et al.,Nat. Commun. 6, 8491 (2015). [8] O. Arcizet et al.,Nature444, 71 (2006). [9] S. Gigan et al.,Nature444, 67 (2006). [10] D. Kleckner, and D. Bouwmeester, Nature 444, 75 (2006). [11] A. Schliesser et al.,Phys. Rev. Lett. 97, 243905 (2006). [12] J. D. Teufel et al.,Phys. Rev. Lett. 101, 197203 (2008). [13] D. J. Wilson et al.,Nature524, 325 (2015). [14] I. Wilson-Rae et al.,Phys. Rev. Lett. 99, 093901 (2007). [15] E. E. Wollman et al.,Science349, 852-855 (2015). [16] A. Nigu´ es, A. Siria, and P. Verlot, Nat. Commun. 6, 8104 (2015). [17] K. W. Murch et al.,Nature Physics 4, 561 (2008). [18] F. Brennecke et al.,Science322, 235 (2008). [19] K. A. Yasir, and W. M. Liu, Sci. Rep. 5, 10612 (2015). [20] K. A. Yasir, and W. M. Liu, Sci. Rep. 6, 22651 (2016). [21] M. Paternostro et al.,New J. Phys. 8, 107 (2006). [22] M. Paternostro, G. D. Chiara, and G. M. Palma, Phys. Rev. Lett. 104, 243602 (2010). [23] S.Mahajan, N.Aggarwal, A.B.Bhattacherjee, andMan- Mohan, J. Phys. B: At. Mol. Opt. Phys. 46, 085301 (2013). [24] Y. C. Liu et al.,Phys. Rev. Lett. 110, 153606 (2013). [25] Florian Elste, S. M. Girwin, and A. A. Clerk, Phys. Rev. Lett.102, 207209 (2009). [26] D. W. C. Brooks et al.,Nature488, 476 (2012). [27] K. Zhang, F. Bariani, and P. Meystre, Phys. Rev. Lett. 112, 150602 (2014). [28] A. H. Safavi-Naeini et al.,New J. Phys. 15, 035007 (2013). [29] Y. Yanay, J. C. Sankey, and A. A. Clerk, Phys. Rev. A 93, 063809 (2016). [30] P. Weber et al.,Nat. Commun. 7, 12496 (2016). [31] V. Galitski, and I. B. Spielman, Nature494, 49 (2013). [32] X.J. Liu et al.,Phys. Rev. Lett. 102, 046402 (2009). [33] Y. J. Lin et al.,Nature462, 628 (2009). [34] Y. K. Kato et al.,Science306, 1910-1913 (2004).[35] M. Konig et al.,Science318, 766-770 (2007). [36] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314, 1757-1761 (2006). [37] D. Hsieh, et al.,Nature452, 970-974 (2008). [38] X. Li, E. Zhao, and W. V. Liu, Nat. Commun. 4, 4023 (2013). [39] K. Sun et al.,Nat. Phys. 8, 67-70 (2012). [40] J. D. Koralek et al.,Nature458, 610-613 (2009). [41] C. Hamner, Phys. Rev. Lett. 114, 070401 (2015). [42] Y. J. Lin, K. Jim´ enez-Garc´ ιa, and I. B. Spielman, Nature 471, 83 (2011). [43] H. Hu et al.,Phys. Rev. Lett. 108, 010402 (2012). [44] L. Jiang et al.,Phys. Rev. A 90, 053606 (2014). [45] X. F. Zhou et al.,Phys. Rev. A 91, 033603 (2015). [46] Z. Cai, X. Zhou, and C. Wu, Phys. Rev. A 85, 061605(R) (2012). [47] S. W. Su et al.,Phys. Rev. A 86, 023601 (2012); S. W. Suet al.,ibid.84, 023601 (2011). [48] Y. Y. Xiang, J. Ye, and W. M. Liu, Phys. Rev. A 90, 053603 (2014); S. S. Zhang et al.,ibid.87, 063623 (2013). [49] N. Goldman et al.,Rep. Prog. Phys. 77, 126401 (2014). [50] J. Dalibard et al.,Rev. Mod. Phys. 83, 1523 (2011). [51] X. Zhou, Y. Li, Z. Cai, and C. Wu, J. Phys. B: At. Mol. Opt. Phys. 48, 249501 (2015). [52] C. Wu, M. S. Ian, X. F. Zhou, Chin. Phys. Lett. 28, 097102 (2011). [53] Y. Deng et al.,Phys. Rev. Lett. 112, 143007 (2014). [54] B. Padhi, and S. Ghosh, Phys. Rev. A 90, 023627 (2014). [55] L. Dong et al.,Phys. Rev. A 89, 011602(R) (2014). [56] L. Dong, C. Zhu, and H. Pu, Atoms3(2), 182-194 (2015). [57] J. S. Bennett et al.,New J. Phys. 18, 053030 (2016). [58] I. Wilson-Rae et al.,New J. Phys. 10, 095007 (2008). [59] M. Punk, D. Chowdhury, and S. Sachdev, Nat. Phys. 10, 289 (2014). [60] H. Miyake et al.,Phys. Rev. Lett. 107, 175302 (2011). [61] S. Sachdev, Science288, 475-480 (2000). [62] R. Landig et al.,Nat. Commun. 6, 7046 (2015). [63] A. L. Fetter, Phys. Rev. A 89, 023629 (2014). [64] C. Maschler, I. B. Mekhov, and H. Ritsch, Eur. Phys. J. D46, 545-560 (2008). [65] J. Higbie, and D. M. Stamper-Kurn, Phys. Rev. A 69, 053605 (2004). [66] S. F¨ ollinget al.,Nature434, 481-484 (2005). [67] J. M. Pirkkalainen et al.,Nat. Commun. 6, 6981 (2015). [68] G. Birkl et al.,Phys. Rev. Lett. 75, 2823 (1995). [69] H. Miyake et al.,Phys. Rev. Lett. 107, 175302 (2011).

1108.4212v1.Vortices_in_spin_orbit_coupled_Bose_Einstein_condensates.pdf

Vortices in spin-orbit-coupled Bose-Einstein condensates J. Radi c,1T. A. Sedrakyan,1I. B. Spielman,1, 2and V. Galitski1 1Joint Quantum Institute, University of Maryland, College Park, Maryland 20742-4111, USA 2National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (Dated: March 2, 2022) Realistic methods to create vortices in spin-orbit-coupled Bose-Einstein condensates are discussed. It is shown that, contrary to common intuition, rotation of the trap containing a spin-orbit con- densate does not lead to an equilibrium state with static vortex structures, but gives rise instead to non-equilibrium behavior described by an intrinsically time-dependent Hamiltonian. We propose here the following alternative methods to induce thermodynamically stable static vortex congura- tions: (1) to rotate both the lasers and the anisotropic trap; and (2) to impose a synthetic Abelian eld on top of synthetic spin-orbit interactions. Eective Hamiltonians for spin-orbit condensates under such perturbations are derived for most currently known realistic laser schemes that induce synthetic spin-orbit couplings. The Gross-Pitaevskii equation is solved for several experimentally relevant regimes. The new interesting eects include spatial separation of left- and right-moving spin-orbit condensates, the appearance of unusual vortex arrangements, and parity eects in vortex nucleation where the topological excitations are predicted to appear in pairs. All these phenomena are shown to be highly non-universal and depend strongly on a specic laser scheme and system parameters. I. INTRODUCTION Spin-orbit-coupled cold atoms represent a very new and quickly growing area of research that promises to host an even richer variety of exotic phenomena than solid-state spintronics [1]. Indeed, within just a few years of experimental research in the eld, a number of exciting phenomena have already been observed [2{5] and there are clearly many more low-hanging fruits awaiting their experimental discovery. The key ideas underlying cold-atom spintronics - that studies particles with a synthetic spin degree of freedom coupled to their motion - grew out of the early theo- retical work by Juzeli unas et al. [6{12], which showed that single-particle physics of atom-laser dressed states, where internal atomic states are coupled by position- dependent laser elds, can be described in terms of a non-Abelian vector potential acting on the dressed ex- citations. Later, it was demonstrated theoretically [13] that specic realizations of such laser congurations, in- cluding the early-proposed tripod scheme, give rise to spin-orbit-coupled Hamiltonians of Rashba-Dresselhaus type, familiar from solid-state semiconductor spintronics and that this \spintronics" description is a convenient al- ternative to the description in terms of the non-Abelian elds. Most importantly, it was quickly realized [14] that contrary to solid-state spintronics, where the underlying particles are bound to be electronic excitations, the syn- thetic spin-1/2 degree of freedom in cold atoms can be carried by dressed spin-orbit-coupled bosons that were predicted to condense into a state, dubbed in Ref. [14] a \spin-orbit coupled Bose-Einstein Condensate (BEC)." It was also shown [14] that multiple peaks in the time- of- ight expansion would be a smoking gun signature of such a new quantum state. Remarkably, this type of behavior was observed experimentally [2] by one of the authors shortly after. The specic laser setup used inRef. [2] - that gives rise to an \Abelian" spin-orbit cou- pling (sometimes referred to as the \persistent-spin-helix symmetry point," [15{21] where the Rashba and Dres- selhaus couplings are equal to each other) - was later analyzed in detail by Ho and collaborators in Ref. [22]. These experimental and theoretical successes have moti- vated other interesting theoretical proposals for realistic experimental schemes that can be used to create spin- orbit-coupled systems [23, 24]. Spin-orbit-coupled BECs have also been studied theoretically in Refs. [25{35] for dierent types of spin-orbit interactions and dierent in- ternal structures of bosons (pseudospin-1/2, spin-1 and spin-2 bosons). Among the obvious questions about the spin-orbit BECs is the physics of topological excitations - vortices - that play a central role in the physics of conventional BECs. This is subject of this paper, where we focus primarily on exploring experimentally-relevant methods that can be used to nucleate static vortex structures in spin-orbit BECs. In contrast to the conventional con- densates, the situation here is shown to be signicantly more complicated as the vortex physics is obscured by the interplay of external perturbations intended to cre- ate them and the hyperne structure underlying the syn- thetic spin-orbit-coupling setup. It is widely known and often taken for granted that rotating a Bose-Einstein condensate gives rise to the for- mation of vortices that arrange themselves into static vortex lattice structures. However, this picture is not in fact an obvious outcome of rotation, which represents a time-dependent perturbation due to a rotating anisotropic trap potential. The many subtleties involved in under- standing the fundamentals of the related phenomena are discussed in detail in the reviews by Leggett [36, 37], but the main conclusion is indeed that the physics of a one- component BEC conned to a spinning anisotropic trap can be mapped onto a statistical-mechanical problem ofarXiv:1108.4212v1 [cond-mat.quant-gas] 21 Aug 20112 the BEC with an eective time-independent Hamiltonian, He=H!r
L, which describes the system in a ro- tating frame of reference (here, Lis the orbital angular momentum operator and !ris the frequency of rotation). A na ve expectation therefore is that to rotate an anisotropic trap would be a straightforward means to create vortex structures in spin-orbit-coupled BECs as well. However, this paper shows that this is generally not so and other, more sophisticated methods have to be involved in order to create static vortex structures. We show that the problem with rotation arises here because atoms are not in uenced by the trapping po- tential only, but also by the lasers which create spin- orbit coupling in the rst place. Therefore, if only the anisotropic potential rotates, it is in general impossible to choose a frame of reference where the Hamiltonian is time-independent, because the \spin-orbit coupling" lasers, stationary in the lab frame, are rotating in the ro- tating frame, generally resulting in non-trivial dynamics in any rotating frame. While there do exist rare degen- erate cases, where a unitary transformation that elimi- nates time-dependence from the non-interacting Hamil- tonian can be explicitly found, the interaction terms gen- erally become time-dependent under the unitary trans- formation, resulting again in a non-equilibrium problem. Hence, we argue that the residual time-dependence ap- pears to be an essential and unwelcome property of a spin-orbit-coupled BEC with a rotating anisotropic po- tential (at least for the realistic laser schemes currently known to us). We believe that while the specics of time- evolution of rotating spin-orbit BECs are sensitive to de- tails of both the laser setup used and interactions, the typical scenario will involve non-universal dynamics that would inevitably lead to heating and destruction of the coherent state in contrast to the conventional BECs. It is therefore desirable to develop other experimentally-relevant methods to create vortices, like rotation or a magnetic eld, for spin-orbit-coupled BECs. Two other ways suggested here and examined in detail are as follows: (i) to rotate both the lasers creating spin-orbit coupling and the trap, if the latter is anisotropic, or just the lasers for an isotropic trap (note that to rotate an isotropic trap has no meaning); (ii) To combine synthetic spin-orbit-couplings with a synthetic Abelian magnetic eld. Theoretically, both methods are shown to give rise to interesting phenomena, including the appearance of sought-after static vortices and vortex lattices, parity eects in vortex nucleation, and real-space splitting of the spin-orbit BEC where the left- and right-moving parts are physically separated (an eect, which bears some similarity to the spin-Hall eect [38, 39] known in condensed matter spintronics). Our paper is structured as follows: Sec. II derives eec- tive Hamiltonians corresponding to a rotating trapping potential and/or rotating \spin-orbit lasers" for various spin-orbit-coupled laser schemes. In Sec. III, we solve the Gross-Pitaevskii equation to describe individual vortices and collective vortex structures for the laser scheme de-scribed in Ref. [4] with a rotating trap and Raman lasers. In Sec.IV, we investigate vortex nucleation and other ef- fects associated with a synthetic magnetic eld that can be imposed on top of the spin-orbit coupled system used in [4] by applying a spatially dependent Zeeman eld. II. ROTATION IN SYSTEMS WITH ENGINEERED SPIN-ORBIT COUPLING In this section, we investigate the eect of rotation of an anisotropic trapping potential and/or spin-orbit lasers in three dierent laser schemes that have been proposed to create eective spin-orbit couplings. To distinguish between the dierent schemes, we will refer to the setup used in Ref. [4] as \M-scheme," the proposal described in Refs. [8, 13] as \tripod-scheme," and the recent proposal of Ref. [24] as \4-level-scheme." A. M-scheme We rst focus on the scheme used in recent experi- ment [4] and investigate the Hamiltonian for the case in which both trap and spin-orbit coupling lasers are ro- tating about the z-axis. The atoms in [4] are under the in uence of three external sources: trapping potential, Raman lasers which create spin-orbit coupling and mag- netic eld which creates Zeeman splitting (aligned along y-direction). If we wanted to get a time-independent Hamiltonian in the rotating frame we would have to ro- tate trapping potential, Raman lasers and magnetic eld. To make things easier it is possible to change direction of the magnetic eld to be along z-axis, which makes rota- tion of magnetic eld about the z-axis unnecessary. If the change of the direction of magnetic eld is accompanied by change in polarization of Raman lasers (the direction of lasers stays the same) the system is described by the same eective equations as in [4]. It is also important to note that, in the case of an isotropic trap, rotation of the trap does not have any eect and in that case rotating only the Raman lasers suces. The stationary system is described by the following Hamiltonian (see methods in [4]): ^H0=" ~2^k2 2m+V(r)# 1 +0 @~(!z+!q) 0 0 0 0 0 0 0 ~!z1 A +p 2 3;xcos(2kLx+
!Lt);(1) where ^k=ir,V(r) is the trapping potential, 1 is the 33 identity matrix,  3;x;y;z are the 33 spin ma- trices,kL=p 2=, is the Raman coupling strength, !zand!qare the linear and quadratic Zeeman shifts, respectively. Here is the wavelength and
!Lis the frequency dierence of the two Raman beams used in the M-scheme. The Hamiltonian is written in the basis of hyperne states fjmF= +1i;jmF= 0i;jmF=1ig3 which are quantized in ^ zdirection (direction of the ex- ternal magnetic eld). When the trap and Raman lasers rotate with a con- stant frequency !rabout the z-axis, the Hamiltonian ^Hrotin the laboratory frame can be obtained from Eq. (1) using the following substitutions: V(x;y;z )!V(x(t);y(t);z) 3;xcos(2kLx+
!Lt)!3;x(t) cos(2kLx(t) +
!Lt); (2) where x(t) =xcos(!rt) +ysin(!rt) y(t) =ycos(!rt)xsin(!rt) 3;x(t) = 3;xcos(!rt) + 3;ysin(!rt):(3) The Hamiltonian ^Hrotcan be also written in a more com- pact form: ^Hrot=ei!rt(^Lz+^Sz)=~^H0ei!rt(^Lz+^Sz)=~; (4) where ^Lis the orbital angular momentum operator and ^Sis the spin operator, and ^Lzand ^Szare theirz- components: ^Lz=~ x^kyy^kx
1,^Sz=~3;z. The Hamiltonian (4) is time-dependent in the labora- tory frame, but we show below that this time-dependence can be eliminated by a unitary transform. Recall that an arbitrary unitary transform, ^U(t), of the Hamiltonian ^H produces a new Hamiltonian, ^H0, as follows ^H0=^U^H^Uyi~^U@^Uy @t: (5) We rst go to the rotating frame of reference (rotating together with both the trap and the lasers) [40]: j RFi= ^U(t)j i, where ^U(t) = exp[i!rt(^Lz+^Sz)=~]. Eq. (5) yields ^HRF=^H0!r(^Lz+^Sz); (6) where ^HRFdenotes the Hamiltonian in the rotating frame. The remaining time-dependence, arising from the oscillating Raman laser elds in ^H0, can be re- moved in the framework of the rotating wave approxi- mation. To obtain an eective description of the system in terms of two internal pseudo-spin states, we follow [4] and choose the quadratic Zeeman shift ~!qto be large enough, so that the state jmz= 1ican be neglected. Us- ing the pseudo-spin-1/2 labels for internal states, we get, j"ijmz= 0i,j#ijmz=1i. The nal Hamiltonian can be expressed in the form used in Ref. [4] (a detailed derivation is much analogous to Ref. [4] and is presented in appendix A) as follows, ^HRF;2=" ~2^k2 2m+V(r)!r^Lz+EL# 1 +~2kL m^kxz+ 2x+~!rkLyz+ 0 0 0~!r ; (7)where 1 is 22 unit matrix,  x;y;z are 22 Pauli matrices and =~(
!L!z) is a detuning from the Raman resonance. Since the resulting Hamiltonian is time-independent in the rotating frame, it leads to the appearance of stationary vortex structures studied below in Sec. III. In the case where only the anisotropic trap is rotating, the Hamiltonian in the laboratory frame is given by (1), withV x;y;z
!V x(t);y(t);z
. Importantly, if we go to the rotating frame and make the rotating wave approximation (exactly as in the above), we are still left with a time-dependence (for details see appendix B): ^H0 RF;2=" ~2^k2 2m+V(r)!r^Lz# 1 +~2kL m^kx(t)z+ 2x+ 2z;(8) where ^kx(t) =^kxcos(!rt)^kysin(!rt). B. Tripod scheme We now concentrate on the proposal described in Refs. [8, 13], which uses a so-called \tripod scheme," that consists of three degenerate ground states of an atom cou- pled to an excited state. The resulting energy spectrum includes two degenerate \dark" states and two \bright" states (one of the bright states is higher and the other is lower in energy with respect to degenerate dark states). In the strong coupling regime and within the adiabatic approximation, the energy dierence between the dark and bright states is very large compared to other char- acteristic energies of the system. In this case, a cou- pling between the dark and bright states is negligible, and consequently if the atoms initially exist within the dark states subspace, they are expected to stay there for a long time. From now on, we use pseudo-spin-1/2 nota- tions for the two degenerate dark states. The eective stationary Hamiltonian (projected onto the dark-state subspace) reads: H=^p2 2m+w(r) 1v0^pxyv1^pyz+0z;(9) where p=i~r,w(r) is a spin-independent part of the trapping potential (see appendix C for details), v0andv1 characterize the strength and type of spin-orbit coupling, and0is the eective Zeeman splitting. 1 is a 22 unit matrix, x;y;z are 22 Pauli matrices. We rst investigate the case with both the trap and the spin-orbit lasers rotating. The derivation, presented in appendix C, leads to the following Hamiltonian in the4 rotating frame ^HRF;2=^p2 2m+w(r)^Lz 1v0^pxyv1^pyz +0z+m~!r(v1xzv0yy) ~!r sin2 sincoscos sincoscoscos2cos2sin2 ;(10) where=mv0x=cos,0= sin2 v0 cos
2+v1 sin2(=2)
2 =2 =2, andis a constant. Let us note here that Ref. [41] previously considered the tripod scheme un- der rotation, but obtained slightly dierent results (the spin angular momentum part ( !r^Sz) was ignored in Ref. [41] ). Our result (10), together with Eq. (7) for the M- scheme, clearly shows that the eect of rotation in sys- tems with synthetic spin-orbit interaction does not re- duce to just adding the !rLzterm for the Hamiltonian in the rotating frame, but also produces other position- dependent terms, which depend on a particular scheme. We now consider the tripod scheme with only the trap rotating. We rst address the following question: if the trapping potential is time-dependent, can we get the ef- fective pseudo-spin Hamiltonian in the laboratory frame just by changing V!V(t) in (9); or in other words, are we still allowed to restrict to the dark-state subspace if the external potential is time dependent? The answer is certainly \yes," if the trapping potential is the same for all three degenerate ground states (which is most often the case for optical trapping), because this kind of time- dependent potential does not couple the dark and bright states. In a general tripod scheme however, the trapping po- tential is not spin-independent ( ^V(r) =P jVj(r)jjihjj, V1=V2=w(r) andV3=w(r) +). To better under- stand this case, let us choose states fj1i;j2i;j3igto be eigenstates of ^Sz(z-component of the total spin opera- tor). Then, the rotation of the trapping potential about the z-axis is described by: V1=V2=w0(r;t) andV3= w0(r;t) +, wherew0(r;t) =ei!rt^Lz=~w(r)ei!rt^Lz=~. We can therefore separate ^V(r) into a stationary spin-dependent term and a time-dependent but spin- independent term: ^V(r;t) =j3ih3j+w0(r;t) j1ih1j+ j2ih2j+j3ih3j
. Therefore the time-dependent part of trapping potential is spin-independent and it will not couple dark and bright states. With this, the tripod sys- tem with a trap rotating about the z-axis is described by ^H=^p2 2m+w0(r;t) 1v0^pxyv1^pyz+0z:(11) We now make the following transformation. ^U(t) =exp[i!rt(^Lz=~+ x=2)], which gives: ^H0=^p2 2m+w(r)!r^Lz 1v0^px(t)y(t) v1^py(t)z(t) +0z(t)~!r 2x;(12) where ^px(t) = ^pxcos(!rt)^pysin(!rt); ^py(t) = ^pycos(!rt) + ^pxsin(!rt); y(t) = ycos(!rt)zsin(!rt); z(t) = zcos(!rt) + ysin(!rt):(13) The Hamiltonian (12) is generally time-dependent. How- ever in the case of Rashba coupling ( v0=v1=v) and 0= 0, this non-interacting part of the Hamiltonian be- comes static and reads, ^H0=^p2 2m+w(r)!r^Lz 1 v(^pxy+ ^pyz)~!r 2x:(14) C. 4-level scheme Here we study the 4-level-scheme [24] with a rotating trap. The stationary eective Hamiltonian (projected to the lowest energy states) is given by [24]: ^H=" ~2^k2 2m+V(r)# 1 +(x^kyy^kx) +(x^ky+ y^kx) +
z 2z;(15) whereanddenote strengths of Rashba and Dres- selhaus couplings respectively (in this scheme, is xed andcan be tuned), and
zis an eective Zeeman eld. Per the same arguments as in the tripod scheme, we are allowed to simply replace V!V(t) in (15) (if an exter- nal potential is time-dependent; note also, that the trap- ping potential here is spin independent). The rotating trap potential reads: V(r;t) =ei!rt^Lz=~V(r)ei!rt^Lz=~. We now make the following transformation: ^U(t) = exp[i!rt(^Lz=~+ z=2)], which gives: ^H0=" ~2^k2 2m+V(r)!r^Lz# 1 +(x^kyy^kx) + ycos(2!rt) + xsin(2!rt)^kx + xcos(2!rt)ysin(2!rt)^ky +
z 2~!r 2 z:(16)5 Again, this non-interacting part of the Hamiltonian is in general time-dependent, however for pure Rashba cou- pling, it becomes time-independent. Note that to get the full Hamiltonian in the rotat- ing frame, we must also include interactions between the bosons and apply to them the same transformations as in the non-interacting part above. If both the trap and spin-orbit lasers rotate, the corresponding unitary oper- ator, ^U(t) = exp[i!rt(^Lz+^Sz)=~], describes a spatial rotation about the z-axis. If the bare interactions are rotationally-invariant, the interaction part of the Hamil- tonian does not change in the rotating frame. In con- trast to this result however, if only the trap is rotating, the interactions will generally acquire time-dependence as well (we have found a few very special cases - with se- rious constraints on the parameters of the system - where a unitary transform can be found that makes both the pure Rashba non-interacting part and interactions time- independent, but whether these degenerate cases can be realized experimentally remains unclear at this stage). III. CREATING VORTICES BY ROTATION In the previous section, we have shown that the Hamil- tonian for the M-scheme in the presence of a rotating trap and Raman lasers becomes time-independent in the ro- tating frame. In analogy with the physics of \ordinary" BEC under rotation, there will be thermal equilibration in the system and vortices will form in the condensate. Let us assume that ~!z(is the chemical po- tential), which gives an eective 2D system, where the motion in z-direction is eectively frozen (this can be achieved by applying a 1D optical lattice in ^ zdirection). We also assume the interaction part of the Hamiltonian to have the form: ^Hint=Z d2r1 2G1^2 "+1 2G2^2 #+G12^"^# ;(17) whereG1,G2andG12are eective 2D interaction strengths and are related to 3D interaction strengths: G1=G3d 1=(p 2lz),G2=G3d 2=(p 2lz) andG12= G3d 12=(p 2lz), wherelz=p ~=(m!z). ^"and ^#are density operators for j"i,j#istates (normal ordering of the corresponding creation/annihilation operators is im- plied). We are interested in nding the ground state con- guration of bosons in a rotating system described by (7,17). First, we have to make an assumption about the ground state and we assume below that (at the mean-eld level) all atoms occupy the same single-particle state de- scribed by the spinor wave-function, "(r); #(r)
(we also call it condensate wave-function). The condensate wave-function satises the Gross-Pitaevskii (GP) equa-tions below:  "= ~2 2mr2i~2kL m@ @x+V(r)!r^Lz~kLy
+NG 1j "j2+NG 12j #j2 "+ 2 #  #= ~2 2mr2+i~2kL m@ @x+V(r)!r^Lz+~kLy~
+NG 2j #j2+NG 12j "j2 #+ 2 "(18) whereNis the total number of particles and is the Lagrange multiplier associated with the constraintR d2r j "j2+j #j2
= 1 (it can be shown that has a physical meaning of chemical potential [36]). We solve the GP equations by using norm-preserving imaginary time propagation method (see for example Ref. [40, 42]). We consider a trapping potential of the following form: V=1 2m!2 x2+ 2y2
, where!and !are trapping frequencies in the ^ xand ^ydirection. It is convenient to measure lengths in the units of the harmonic oscillator length,a0=p ~=(m!) and energy in terms of ~!. We introduce dimensionless position variable r0=r=a0. The corresponding \dimensionless GP equations" reads  "= 1 2r02ik0 L@ @x0+1 2 x02+ 2y02
!0 r^L0 zkLy0
+g1j "j2+g12j #j2 "+ 0 2 #  #= 1 2r02+ik0 L@ @x0+1 2 x02+ 2y02
0 !0 r^L0 z+kLy01
+g2j #j2+g12j "j2 #+ 0 2 "; (19) wherek0 L=kLa0, 0= =(~!),0==(~!),!0 r= !r=!,^Lz=i x0@y0y0@x0
,g1=NG 1=(~!a2 0),g2= NG 2=(~!a2 0) andg12=NG 12=(~!a2 0). In simulations for the rotating system we consider87Rb atoms and we use the experimentally-relevant parame- ters:= 804:1 nm,!= 250 Hz and = 1. These parameters give a0=p ~=m! = 1:52m,k0 L= 8:42. From now on we express length in units of a0(coordinates (x;y) in gures are also given in the units of a0). We per- formed simulations specically for the rotation frequency !r= 0:7!and for three dierent coupling strengths: no coupling ( = 0), weak coupling ( = 2 EL), and strong coupling ( = 10 EL) (EL= 35:4~!). In simulations we chooseg1= 1000,g2= 995,g12= 995. The ratio betweeng1,g2andg12corresponds to interaction coef- cients in87Rb (the interaction coecients for87Rb in statesfjF= 1;m= 0i;jF= 1;m=1igis given in6 Ref. [4]). In absence of rotation and for = 0, = 0 our choice ofg1,g2andg12produces cloud radius of 8 :4m. We also set ~!r= 0. Without rotation and spin-orbit coupling, j"iandj#i components are miscible for our choice of interaction pa- rameters. In the case of rotation and no spin-orbit cou- pling there are several dierent phases depending on !r and ratio of interaction coecients [43]: triangular lat- tice, square lattice, stripe or double-core vortex lattice and vortex sheet. Since our Hamiltonian is almost equiv- alent to the Hamiltonian in Ref. [43] for = 0 and ~!r= 0 (there is a very small dierence in interac- tion coecients; the equivalence of non-interaction part of two systems is clear from (A3)) we reproduced results of Ref. [43]. The results for = 0 are shown in Fig. 1(a), which dis- play the densities of the j"iandj#icomponents forming spatially-separated density stripes with lines of vortices along the minima of the density. As expected, our re- sults reproduce stripe vortex lattice phase described in Ref. 43. Note that for = 0, the Hamiltonian (7) con- serves number of the j"iandj#iparticles separately. We have chosen N"=N#(Ni=R d2rj ij2). A weak spin-orbit coupling ( = 2 EL) (Fig. 1(b)) does not appear to lead to any signicant qualitative changes in the observed behavior: the densities of the j"iand j#icomponents are still spatially separated and there are lines of vortices along the density minima of each com- ponent. A signicant change comes in the strong-coupling regime: see the = 10 ELdata shown in Fig. 1(c). The vortices arrange themselves in a lattice in j"iand j#icomponents and densities of both components are al- most identical. This behavior can be understood from the following part of the Hamiltonian (7): ^H0=~2^k2 x 2m1 +~2kL^kx mz+ 2x: (20) The spectrum of (20) for dierent 's is shown in Fig. 2(a). For large , it consists of two bands with an en- ergy separation much larger than all other characteristic energies of the system. Therefore, our system is \con- ned" to the lower band with a single minimum, which eectively makes it a single-component system. This ex- plains almost identical densities of the two components in Fig. 1(c). IV. CREATING VORTICES BY SPATIALLY-DEPENDENT DETUNING A. The model Vortices in spin-orbit systems like [4] can be created without any rotation, but by imposing an additonal syn- thetic magnetic eld. In [44], it has been shown that a spatially-dependent detuning, , in the M-scheme results (a) (b) (c) FIG. 1: (color online) The density proles for the rotat- ing spin-orbit-coupled BEC are shown. The rst, second and third columns show density of j"icomponent (j "j2), den- sity ofj#icomponent (j #j2) and the total density ( T= j "j2+j #j2), respectively. Figure (a) shows results for = 0 which are characterized by density stripes and lines of vortices in both components. The results for = 2 EL(b) are qual- itatively similar to the = 0 case. Figure (c) shows results for = 10EL; a vortex lattice is formed in both components and densities of the two components are almost identical. in a synthetic magnetic eld, which creates vortices in the strong Raman coupling ( ) regime. Our goal is to investigate the same system for a wide range of (from weak to strong Raman coupling) and to see what kind of vortex structures it yields. The setup is described by the following eective Hamil- tonian (see [4, 44]): ^H= ~2^k2 2m+V! 1 +~2kL^kx mz+ 2x+(y) 2z:(21) We again assume strong connement in the ^ zdirec- tion and describe interactions by equation (17). We are looking for the ground state in the same way as in the rotating case and following the same steps we get the \dimensionless GP equations":  "= 1 2r02ik0 L@ @x0+1 2(x02+ 2y02) +0(y0) 2+g1j "j2+g12j #j2 "+ 0 2 #7  #= ~2 2mr02+ik0 L@ @x0+1 2(x02+ 2y02) 0(y0) 2+g2j #j2+g12j "j2 #+ 0 2 ":(22) Parameters 0,0,k0 L,g1,g2,g12are dened in the same way as in (19). B. Qualitative discussion To get a better understanding of the model, we inves- tigate Hamiltonian (21) in more detail. It is instructive to rst focus on the following part of (21): ^H0=~2^k2 x 2m1 +~2kL^kx mz+ 2x+ 2z:(23) We rst assume that is constant in space. In that case Hamiltonian (23) can be easily diagonalized in the mo- mentum basis: Uy(kx)H0(kx)U(kx) = E+(kx) 0 0E(kx) . The resulting spectrum consists of an upper(+) and lower(-) band, as shown in Fig. 2. The gap separat- ing the bands is large compared to other characteristic energies of the system and it is safe to assume that the condensate occupies only the states in the lower band. In Fig. 2(a), spectra for dierent coupling strengths and= 0 are shown. For <4EL, the spectrum has two minima and BEC will involve states near both left and right minima. At = 4 EL, there is a transition from a spectrum with two minima to a spectrum with one minimum, which changes the structure of the con- densate wave-function. I.e., for >4EL, the BEC is expected to occupy only states with momentum around kx= 0. The eect of detuning in the low- regime is shown in Fig. 2(b). We see that shifts the energies and positions of the left and right minima. In the case of constant , the BEC would occupy only the states around the global minimum (for example, the right minimum in Fig. 2(b)). Those cases have been tested experimentally in [4]. Now, consider a spatially-dependent (y). We will con- sider it to be a linear function of y:(y) =0+y, which is the simplest and the most experimentally relevant regime. The interesting physics is evident from the fol- lowing arguments: for constant detuning, the spectrum around a minimum can be simply described by (we use dimensionless variables, see (19)) ( kxkmin)2=(2me) + Emin, whereme,kmin,Eminare the eective mass, position of the minimum, and the energy at the min- imum, respectively. Note that all these quantities de- pend on. Ifisy-dependent, the values of me, kmin,Eminwill also become spatially-dependent. Hence, the spectrum around the minimum can now be written as: kxkmin(y)2=(2me) +Emin(y) which describes particles moving in an eective gauge eld ( A;) = kmin(y);0;0;Emin(y)
with a spatially-varying eective −2−1012−3−2−101 kx/kLE/EL(a) −2−1012−2−10 kx/kLE/EL (b) −0.5 00.5−8−7.5 kx/kLE/EL (c) FIG. 2: (color online) The energy spectrum of H0. In (a) spectra for dierent (from = 0 to = 6 EL) and= 0 are shown (spectrum for = 0 is at the top while spectrum for = 6ELis at the bottom). The eect of in small regime is shown in (b) ( = 1 EL,= 0:5EL(solid blue line), = 1EL(dashed red line) and = 2EL(dotted black line)). The eect of in large regime is shown in (c) ( = 16 EL, = 0EL(solid blue line), = 1EL(dashed red line) and = 2EL(dotted black line)). changes position and energy of the minimum. massme(y) [44]. The spatially-dependent vector poten- tialAinduces an eective magnetic eld ( Be=rA), which may lead to creation of vortices if strong enough. This approximation provides a good description of the system only if the particles at some point yhave the momentum kxnear the minimum. Our numerical simu- lations presented below indicate that this approximation in fact gives a very good qualitative description in a wide parameter range. We calculate parameters me(y),kmin(y), andEmin(y) by diagonalizing (23) for dierent y's since=(y). The procedure of deriving eective equations for lower band for Hamiltonian (21) in high (single minimum) regime is described in Ref. [44]. Let us note however, that the method we use below to nd the ground state is exact (in particular, we do not limit our system to lower band and we do not simplify interaction terms).8 C. Results In simulations for a system with a spatially-dependent detuningwe use the same experimental parameters as in the simulations of a rotating system, which gives a0=p ~=(m!) = 1:52m andk0 L= 8:42. We choose interaction parameters to be g1= 1600,g2= 1593, g12= 1593 and constant part of detuning 0= 0. The results for = 0, = 4 ~!=a0and = 1 are shown in Fig. 3 and are straightforward to understand. In this case, we may write the Hamiltonian (21) as ^H=H"0 0H#
; whereH"=~2 2m ^k2+ 2kL^kx +V"(r),H#= ~2 2m ^k22kL^kx +V#(r) andV"(r) =V(r) +(y)=2, V#(r) =V(r)(y)=2:We see that motion of j"iandj#i particles is decoupled in ^Hand that they experience dif- ferent potentials V"(r),V#(r). Detuning gradient shifts the minima of V"(r) (V#(r)) fory0==(2m!2 2) in the positive/negative ^ y-direction and therefore, the centers of thej"iandj#idensities are shifted from the origin by y0(the origin is located in the minimum of V(r)), see Fig. 3(b). Also, it is clear from ^Hand Fig. 2(a) that the momentum distribution of j"i(j#i) particles will be cen- tered around k= (kL;0) (k= (kL;0)), see Fig. 3(c). The eect of repulsive interactions between the particles with dierent spins is clearly seen (the overlap between j"iandj#idensities is quite small). If we introduce a nite , the Hamiltonian becomes: ^H= H" =2 =2H# , The corresponding -term creates cou- pling betweenj"iandj#iparticles. If = const = 0 and is small, the states around the left (right) minimum in the spectrum in Fig. 2(a) still consist mainly of the j"i (j#i) particles, but there is also some admixture of the component with the opposite spin, which grows with . It means that "(r) ( #(r)) will mainly consist of states with momentum around the left (right) minimum, but also of states around the right (left) minimum. We can therefore write:  "(r) #(r) = "L(r) #L(r) + "R(r) #R(r) ; (24) where "L(r) and #L(r) consist only of states with mo- menta around left peak, while "R(r) and #R(r) consist only of the states with momenta around right peak of mo- mentum distribution. We therefore call "L(r); #L(r)
and "R(r); #R(r)
left and right wave-function. In the spatially-dependent detuning case it may happen that momentum distribution is separated in two peaks (i.e., there exist \left"- and \right-movers") even for >4EL (see for example Fig. 7(c)). In that case also the notion of left and right wave-function applies. To investigate the eect of , which couples j"iand j#istates, we consider the regime with = 3 ELand (a) (b) (c) FIG. 3: (color online) The gure shows results for = 0, = 4 ~!=a0and = 1. In (a) the total density is shown. The shape of the density is determined by spatially-dependent detuning, which shifts the densities of j"iandj#iparticles (b). Momentum distribution of j"iandj#icomponents is shown in (c). = 8~!=a0(Fig. 4). The total density T(r) is shown in Fig. 4(a) and there is a characteristic series of minima along thex-direction at y= 0, which come from vortices in the "and #wave-functions, see Fig. 4(b), which are positioned along xand neary= 0. We have checked that the phase winding around zero density points of j "j2and9 j #j2is2. Since vortices in j#iandj"icomponents are slightly displaced from y= 0, the density at minima inTare close to, but not exactly equal to zero. To (a) (b) (c) FIG. 4: (color online) The gure shows results for = 3 EL, = 8 ~!=a0and = 1. In (a) the total density is shown. The series of minima at y= 0 comes from vortices in j"iand j#iwavefunctions (b). Momentum distribution of j"iandj#i components is shown in (c). explain the existence of the line of vortices in the j"i andj#icomponents, we examine the left and right wave- functions. Fig. 5(a) displays j "Lj2andj "Rj2(note thatthe amplitude of "Ris considerably smaller than the amplitude of "L:R d2rj "Rj2= 0:05 andR d2rj "Lj2= 0:45). The momentum distribution in Fig. 4(c) shows that the wave-packet, "L, has an average momentum of kleft=0:8kLand "Rhas an average momentum of kright = 0:8kL. Since "is a superposition of the left- and right-movers, "= "L+ "R, the appearance of the line of vortices at overlapping region is expected. The separation of vortices dis then simply given by ( kright kleft)d= 2ord= (2)=(krightkleft). The analytical expression for dts perfectly well to our numerical data. (a) (b) FIG. 5: (color online) The gure (a) shows j "Lj2andj "Rj2 the relative amplitude of which is given byR d2rj "Lj2= 0:45 andR d2rj "Rj2= 0:05 for the parameters = 3 EL, = 8~!=a0and = 1. The superposition of "Land "R, "= "L+ "R, produces vortices in ". The density of left- and right-moving particles ( L=j "Lj2+j #Lj2,R= j "Rj2+j #Rj2) particles is shown in (b). To explain the density prole and momentum dis- tribution, it is useful to consider an eective gauge- eld picture. The eective gauge eld, ( A;) = (kmin(y);0;0;Emin(y)), can be calculated by diagonal- izingH0. As discussed earlier, we may approximate the low-energy band physics by the following Hamilto- nian (we use again the dimensionless variables, where the lengths are measured in terms of a0and the wave-vectors,10 k, in terms of 1 =a0): HEGF=1 2me(y) kxA(y)2+1 2k2 y+(y)+V(r);(25) whereV(r) =1 2(x2+ 2y2). For 4ELthere is a sin- gle local minimum in lower band of the Hamiltonian (23) spectrum for any . For <4ELthe spectrum has two minima for = 0, however when becomes large enough the spectrum has a single local minimum (Fig. 2(b)). The spectrum around each local minimum can be ap- proximated by the form given in (25), and therefore there will beAL(y), L(y),me;L(y) corresponding to the left minimum and AR(y), R(y),me;R(y) corresponding to the right minimum of the spectrum. Left-movers feel the \left gauge eld" ( AL(y);0;0;L(y)) while right-movers feel the \right gauge eld" ( AR(y);0;0;R(y)). To get the eective potential in ^ ydirection acting on left- and right-movers we dene: Ve;L(y) = L(y) + 1 2 2y2,Ve;R(y) = R(y) +1 2 2y2. In Fig. 6 we show L=R(y),Ve;L=R(y),AL=R(y) and 1=me;L=R(y) for = 3ELand= 8 ~!=a0.Ve;L=Rhave minima aty0;R=L =3:2 which explains the total density prole (Fig. 4(a)) which has maxima at y=3:2. The position of two peaks in momentum distribution in Fig. 4(c) can be understood as follows: for particles positioned near the minimum of Ve;Lin Fig. 6(b), it is energetically fa- vorable to have the ^ x-component of momentum approxi- mately equal to A(y0;L) and the ^y-component near zero. Fig. 6(c) shows that A(y0;L)0:79kL, while from Fig. 4 (c), we see that the momentum distribution is centered aroundkx=0:80kL. The same explanation applies for the momentum distribution of right-movers. To investigate the regime with a single minimum in the spectrum ( 4EL) we did calculations for parame- ters: = 5 EL,= 12 ~!=a0and = 1 (Fig. 7). In this \single-minimum" case one might expect momentum dis- tribution to be concentrated around a single point as was observed in Ref. [4]. However, in spatially-dependent- detuning case this is not necessarily true: the momen- tum distribution (Fig. 7(c)) shows two peaks around kx=0:55kL. Also, the total density (Fig. 7(a)) has a characteristic series of minima along y= 0 line which come from vortices in the "and #wave-functions (Fig. 7(b)) created in the overlapping region of left- and right-movers. The results can again be explained by the eective gauge eld. The eective potential in ^ y-direction Ve(y) = (y) +1 2 2y2(Fig. 8(a)) has two minima at y0;R=L =3:4 which explains the density distribution which has maxima at y=3:3. Also, equation (23) tells us it is energetically favourable for particles near the left (right) minimum of Ve(y) to have momentum around A(y0;L) =0:56kL(A(y0;R) = 0:56kL) (Fig. 8(c)) which explains momentum distribution. We also note that in Fig. 8(c) A(y) has a large gradient and therefore magnetic eld ( Be@A=@y ) is strong around y= 0 which may serve as an alternative explanation of line of vortices appearing in Fig. 7(a). We now study the system with strong Raman cou- −10−50510−80−60−40 yΦeff/(¯ hω)(a) −10−50510−60−55−50 yVeff/(¯ hω) (b) −50510−1−0.500.51 yA/kL (c) −5051000.51 y1/meff (d) FIG. 6: (color online) The gure shows the scalar potential (y) (a), the eective trapping potential in y-direction Ve(y) (b), vector potential A(y) (c) and inverse of the eective mass (d) for = 3 EL,= 8~!=a0and = 1. Values correspond- ing to the left minimum of the spectrum are represented by a solid red line while the values corresponding to the left min- imum of the spectrum are represented by a dashed blue line (see the text for details). pling and weak detuning gradient (i.e.is not large enough to produce spatial separation of a cloud along ^yas in previous cases). Results for = 10 EL, = 12 ~!=a0are shown in Fig. 9 and can be explained by the associated eective gauge eld shown in Fig. 10. The total density (Fig. 9(a)) and j "j2,j #j2(Fig. 9(b)) show the existence of a vortex in the centre of the cloud. The vortex appears only for strong enough eective mag- netic eld which is tuned by changing . We dene the eective magnetic eld Be=rA(y) and in our case, ( A= (A(y);0;0)),Be=@A(y) @y^z. The magnetic eld points in the ^ zdirection, depends on y, and is con- stant along x. We also note that since me(y)6= 1 (Fig. 10(d)), the eective equations will dier from those for an ordinary charged particle in a magnetic eld Be(y)^z. The vector potential A(y) and the eective magnetic eld Be(y) are shown in Fig. 10(b,c). It is useful to know the critical eld needed for vor- tex creation and we may get a crude estimate by us- ing the equation for critical magnetic eld of a single- component 2D gas in the Thomas-Fermi limit: Bc= 4(a0=R)2ln 0:888(R=a 0)2
, whereRis the Thomas- Fermi radius of the cloud [45]. We take R= 6:5a0(the size of our cloud), which gives Bc0:35. It is important to notice that larger number of particles or stronger in- teractions increase R, which lowers the critical eld ( Bc decreases with increasing R). To ndBc, we did simu- lations for = 10 EL, = 1 and for dierent values of (which controls the strength of the eective magnetic11 (a) (b) (c) FIG. 7: (color online) The gure shows results for = 5 EL, = 12 ~!=a0and = 1. In (a) the total density is shown. The series of minima at y= 0 comes from vortices in j"iand j#iwavefunctions (b). Momentum distribution of j"iandj#i components is shown in (c). eld). We found that the vortices start to appear for a critical eective magnetic eld Bc0:34, which is very close to our estimate presented above. If the eective eld is strong enough, a vortex \lat- tice" is formed, as shown in Fig. 11, which corresponds to = 10 EL,= 40 ~!=a0and = 1:85. From the gure, we see that vortices are concentrated along the −505−110−100−90−80 yΦeff/(¯ hω)(a) −5 05−90−88−86 yVeff/(¯ hω) (b) −50510−1−0.500.51 yA/kL (c) −5051000.51 y1/meff (d) FIG. 8: (color online) The gure shows the scalar potential (y) (a), the eective trapping potential in y-direction Ve(y) (b), vector potential A(y) (c) and inverse of the eective mass (d) for = 5 EL,= 12 ~!=a0and = 1. x-axis and around y= 0. This is because Be(y) is not homogeneous, i.e. the eld is strongest at y= 0 and it weakens with increasing jyj. We had to increase trap- ping strength in the ^ ydirection ( = 1:85) because scalar potential ( y) separates the clouds (e.g. see Fig. 8(a)) and for a weaker trapping strength, the eective poten- tial would have two minima (it would look like eective potential in Fig. 8(b)). The most interesting regime is the one in which left and right moving phases "L(r), "L(r)
and "R(r), "R(r)
are spatially separated along ^ ydirection and there is a vortex (or vortices) in each phase in addition to a vortex line. This requires double minimum structure of the eective potential in ^ ydirectionVe(y), which sep- arates the phases and strong enough eective magnetic eld in each phase to create additional vortices, which tend to appear in pairs (i.e. the number of vortices is equal in both phases which is a consequence of the fact that in our simulations the eective gauge eld is sym- metric with respect to re ection about y= 0 line and interactions are almost spin-independent). In Fig. 12(a), we show results for != 210 Hz, = 4EL,= 20 ~!=a0(a0=p ~=(m!)). By choosing != 210 Hz, parameter k0 Lin dimensionless GP equations (22) becomes k0 L= 18:83, while interaction coecients stay the same ( g1= 1600,g2= 1593 and g12= 1593). Having larger k0 Lmeans we can create stronger eective magnetic eld. We increased the trapping frequency in they-direction ( = 1:3) to bring two phases closer to y= 0, where the eective eld is stronger (to counter the eective scalar potential ( y), which separates the phases). In Fig. 12(b) we show results for != 210 Hz,12 (a) (b) FIG. 9: (color online) The gure shows results for = 10 EL, = 12 ~!=a0and = 1. In (a) the total density is shown, while (b) and (c) show densities of j"iandj#i(c) components. The vortex in the center appears for strong enough eective magnetic eld. = 10EL,= 150 ~!=a0and = 2:75. Here the left and right phases are completely separated in space and the eective magnetic eld is strong enough to produce multiple vortices in each phase. Also, it is clear that the vortices are not located in centers of two phases, but are positioned closer to y= 0 which is expected because the eld is stronger near y= 0. It is important to discuss the means of experimen- tally observing results we presented. We concentrate on the time-of- ight imaging, which is widely used to probe cold-atoms systems. The time-of- ight picture here will be determined by the underlying momentum distribution of particles. If this momentum distribu- tion consists of two separated peaks, the initial cloud will strongly separate during expansion (see for exam- ple [3, 4]). We note that due to the transformation 0 "(r) = "(r)eikLx, 0 #(r) = #(r)eikLxused when de- riving Hamiltonian (21), the real momentum distribu- tion ofj"iparticles will in fact be shifted by kLwith respect to the momentum distribution shown in gures and the momentum distribution of j#iparticles is shifted bykL(see [4]). In the case of = 0, both j"iand j#iparticles will have zero average momentum, which means both components of the condensate will expand, while the position of centre of mass will be stationary during time-of- ight. For = 3 ELand= 8 ~!=a0, we expect four separated clouds to be seen in the time-of- ight: since the real momentum distributions of j"iand −5 05−180−170−160 yVeff/(¯ hω)(a) −10−50510−0.500.5 yA/kL (b) −50510−0.6−0.4−0.20 yBeff (c) −5051000.51 y1/meff (d) FIG. 10: (color online) The gure shows the eective trapping potential in y-direction Ve(y) (a), vector potential A(y) (b), the eective magnetic eld Be(c) and inverse of the eective mass (d) for = 10 EL,= 12 ~!=a0and = 1. FIG. 11: (color online) The gure shows the total density for = 10EL,= 40 ~!=a0and = 1:85. j#iparticles are shifted by kLandkL, there will be two clouds ofj"iparticles with average momenta of 0 :2kL (larger cloud) and 1 :8kL(smaller cloud) and two clouds ofj#iparticles with average momenta of 0:2kL(larger cloud) and1:8kL(smaller cloud). It is important to notice that the vortex line will not be easily visible in those images, because it exists only due to the the over- lap of the wave-packets with dierent average momenta. During the time-of- ight, two wave-packets ";L; ";R
or #;L; #;R
separate, which means that they do not overlap any more and there is no clear vortex line present. For the case in Fig. 12, the vortices in each phase will be visible since they are not a result of overlapping the left- and right-moving condensates.13 (a) (b) FIG. 12: (color online) Figures show separated left and right phases with vortices in each phase. Trapping frequency is != 210 Hz. Figure (a) shows total density for = 4 EL, = 20 ~!=a0and = 1:3. Figure (b) shows total density for = 10EL,= 150 ~!=a0and = 2:75. V. CONCLUSION In this paper, we have investigated realistic experi- mental methods that can be used to create vortex exci- tations in spin-orbit-coupled Bose-Einstein condensates. The main conclusion of the work is that due to a com- plicated interplay between eects associated with the ap- plied laser elds and rotation, the resulting state of the spin-orbit BEC under additional perturbations is highly non-universal and depends strongly on the system pa- rameters and specic laser schemes. In particular, we argued that a spin-orbit BEC under rotation of the trap alone does not achieve a thermodynamically stable state at all, but acquires a complicated non-equilibrium dy- namics that eventually leads to heating and the destruc- tion of the condensate. We have also suggested two alternative experimen- tal methods to mimic an Abelian \orbital" magnetic eld that involve either rotation of the entire experimen- tal setup, or a spatially-dependent detuning. We per- formed numerical simulations of the resulting thermody- namically stable density distributions, focusing mostly on the M-scheme that has already been realized experimen- tally. This scheme gives rise to an \Abelian" spin-orbit- coupling with a well-understood ground state that we used as a basis of our numerical simulations that showed topological excitations above the ground state. We ex- pect that the predicted vortex congurations, in particu- lar vortices appearing in pairs in the spatially-separated left- and right-moving regions, would be straightforward to observe experimentally, as all necessary ingredients are already experimentally available. Finally, we would like to mention that to this point only an \Abelian" spin-orbit-coupling scheme has been actually realized in experiment [4] and we mostly focused here on vortex topological excitations in such systemswith a well-understood ground state. What remains of great interest of course is an expermiental realization of a truly \non-Abelian" spin-orbit interaction (either of pure Rashba or Dresselhaus type or a non-equal mix- ture of those), which can be achieved using laser schemes described in Secs. IIB, IIC, and Refs. [13, 24]. Note that it was argued theoretically [14] that in the Rashba- Dresselhaus system with single-particle dispersion of the double-well type, a fragmented condensed state [46, 47] can be selected by energetics for repulsive interactions that do not break the underlynig Kramers symmetry. This state arises because repulsive interactions in the real space tend to localize particles in the dual momentum space per the fundamental Heisenberg uncertainty prin- ciple. This robust argument together with the protection provided by Kramers symmetry and momentum conser- vation (modulo nite-size eects due to the trap) suggest that the long-sought-after fragmented BEC (which takes the form of a many-body Schr odinger's cat-state in this case [14]) is more stable in spin-orbit-coupled systems than that in BECs conned to real-space double-well po- tentials and hence can be observed experimentally. Topo- logical excitations above this exotic ground state are ex- pected to also be of exotic nature and may potentially re- alize much of the exciting physics discussed in the context of multi-component superconductors [48, 49]. Finally, the nature of the ground state and topological excita- tions above it in the pure bosonic Rashba model remain of great interest as well. Depending on the interaction pa- rameters, this model with a continuous ring of minima on a circle in momentum space, may potentially host topo- logical BECs, spontaneous symmetry-broken states [33], and exotic Bose-liquid states [50], where strong quantum uctuations prohibit order even at zero temperature. Acknowledgments The authors are grateful to Austen Lamacraft, Egor Babaev, and Sankar Das Sarma for useful discussions. This work was supported by AROs atomtronics MURI (JR and IBS) and US-ARO grant, \Spin-orbit-coupled BECs " (TAS and VG). Note added { After this work was completed, we be- came aware of two papers which study spin-orbit-coupled BECs under rotation [51, 52]. The fundamental assump- tions in these papers are qualitatively dierent from our theory, in that Refs. [51, 52] start with an eective spin- orbit-coupled Hamiltonian and assume that it remains stationary under rotation. This is in contrast to our the- ory, where we consider realistic experimental schemes, where rotation is shown to lead to a dierent descrip- tion.14 Appendix A: M-scheme with rotating trap and spin-orbit lasers The Hamiltonian in the rotating frame (6) is: ^HRF=" ~2^k2 2m+V(r)!r^Lz# 1 +0 @~(!z+!q) 0 0 0 0 0 0 0 ~!z1 A +p 2 3;xcos(2kLx+
!Lt)!r3;z:(A1) The Hamiltonian becomes time-independent if we trans- fer to the rotating-wave frame and if we do the rotating- wave approximation: ^HRF=" ~2^k2 2m+V(r)!r^Lz# 1 +0 @+~!q0 0 0 0 0 0 01 A + p 23;xcos(2kLx) p 23;ysin(2kLx)~!r3;z; (A2) where=~(
!L!z). We set quadratic Zeeman shift ~!qto be much greater than and so we may restrict to the subspace spanned by fjmz= 0i;jmz=1ig: ^HRF;2=" ~2^k2 2m+V(r)!r^Lz# 1 + 2xcos(2kLx) 2ysin(2kLx) + 0 0 0~!r ; (A3) where 1 is 22 unit matrix and  x;y;z are 22 Pauli matrices. Since there are eecively two internal degrees of freedom we introduce pseudospin-1/2 notation, i.e. we denej"ijmz= 0i,j#ijmz=1i. We fol- low the steps in [4] and make transformation: 0 "(r) = "(r)eikLx, 0 #(r) = #(r)eikLx, where "(r); #(r)
is a spinor wavefunction on which Hamiltonian (A3) acts. The Hamiltonian then becomes: ^HRF;2=" ~2^k2 2m+V(r)!r^Lz+EL# 1 +~2kL^kx mz+ 2x+~!rkLyz+ 0 0 0~!r ; (A4) whereEL=~2k2 L=2m. We can drop EL1 term by simply renormalizing the energy.Appendix B: M-scheme with rotating trap The Hamiltonian H0 rotdescribing M-scheme with ro- tating trap in the laboratory frame is: ^H0=" ~2^k2 2m+V x(t);y(t);z
!r^Lz# 1 +0 @~(!z+!q) 0 0 0 0 0 0 0 ~!z1 A +p 2 3;xcos(2kLx+
!Lt);(B1) wherex(t) is dened in (3). After transfering to the rotating frame ( ^U(t) = exp[i!rt(^Lz+^Sz)=~]) and making the rotating wave approximation the Hamiltonian is: ^H0 RF=" ~2^k2 2m+V(r)!r^Lz# 1~!r3;z +0 @3=2 +~!q0 0 0=2 0 0 0=21 A + p 23;xcos(2kLx0(t) +!rt) p 23;ysin(2kLx0(t) +!rt);(B2) wherex0(t) =xcos(!rt)ysin(!rt). We may again neglect statejmz= 1iassuming!q>>. To get the Hamiltonian in a more familiar spin-orbit-coupling form we make the following transformation: 0 "(r) = "(r)eikLx0(t); 0 #(r) = #(r)eikLx0(t)+i!rt, which gives: ^H0 RF;2=" ~2^k2 2m+V(r)!r^Lz+EL# 1 +~2kL m^kx(t)z+ 2x+ 2z;(B3) where ^kx(t) = ^kxcos(!rt)^kysin(!rt). We can drop EL1 term by renormalizing the energy. Appendix C: Tripod scheme with rotating trap and spin-orbit lasers The original Hamiltonian for the tripod scheme (sta- tionary system) is (see [13]): ^H0=~2^k2 2m1 +^V(r) +^Hal; (C1) where ^V(r) =P jVj(r)jjihjjis spin dependent trap- ping potential, atom-laser interaction ^Hal=
j0ih0j 1j0ih1j+ 2j0ih2j+ 3j0ih3j+ H:c:
and1 is 44 unit matrix.
is detuning from resonance and 1;2;315 are Rabi frequencies: 1(r) = sincos(mvax)eimvby, 2(r) = sin sin(mvax)eimvby, 1(r) = cos , where , ,vaandvbare constants (see [13] for details). If we start rotating spin-orbit lasers in the laboratory, atom-laser interaction part of the Hamiltonian becomes ei!rt(^Lz+^Sz)=~^Halei!rt(^Lz+^Sz)=~. If the trap rotates, trapping potential becomes ei!rt(^Lz+^Sz)=~^Vei!rt(^Lz+^Sz)=~. Therefore, we can write the Hamiltonian of the rotating system as: ^Hrot=ei!rt(^Lz+^Sz)=~^H0ei!rt(^Lz+^Sz)=~: (C2) The Hamiltonian in the rotating frame is then: ^HRF= ^H0!r(^Lz+^Sz). Since ^HRFis time-independent we can use exactly the same procedure for getting the eective spin-orbit coupling described in [8, 13], i.e. we project the Hamiltonian to the dark states subspace. Here weassume that three degenerate hyperne groundstates are part of F=1 manifold (for example the ground state of 87Rb) and that they are eigenstates of ^Sz. This gives us the precise form of ^Szoperator. As in [13] we take V1=V2=w(r) andV3=w(r) +. After projecting to dark states we get: ^HRF;2=^p2 2m+w(r)^Lz 1v0^pxyv1^pyz +0z+m~!r(v1xzv0yy) ~!r sin2 sincoscos sincoscoscos2cos2sin2 ; (C3) where0= sin2 v0 cos
2+v1 sin2(=2)
2 =2 =2. [1] I. Zuti c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323-410 (2004). [2] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102, 130401 (2009). [3] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. Porto, and I. B. Spielman, Nature 462, 628-632 (2009). [4] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83-86 (2011). [5] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, W. D. Phillips, J. V. Porto, and I. B. Spielman, Nature Physics 7, 531-534 (2011). [6] G. Juzeli unas and P. Ohberg, Phys. Rev. Lett. 93, 033602 (2004). [7] G. Juzeli unas, P. Ohberg, J. Ruseckas, and A. Klein, Phys. Rev. A 71, 053614 (2005). [8] J. Ruseckas, G. Juzeli unas, P. Ohberg, and M. Fleis- chhauer, Phys. Rev. Lett. 95, 010404 (2005). [9] D. Jaksch and P. Zoller, New. J. Phys. 5, 56 (2003). [10] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewen- stein, Phys. Rev. Lett 95, 010403 (2005). [11] S. Zhu, H. Fu, C. Wu, S. Zhang, and L. Duan, Phys. Rev. Lett97, 240401 (2006). [12] I. I. Satija, D. C. Dakin, and C. W. Clark, Phys. Rev. Lett97, 216401 (2006). [13] T. D. Stanescu, C. Zhang, and V. M. Galitski, Phys. Rev. Lett. 99, 110403 (2007). [14] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A78, 023616 (2008). [15] J. Schliemann, J. C. Egues, and D. Loss, Rev. Rev. Lett. 90, 146801 (2003). [16] O. Chalaev and D. Loss, Rev. Rev. B 71, 245318 (2005). [17] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Rev. Rev. Lett. 97, 236601 (2006). [18] T. D. Stanescu and V. Galitski, Phys. Rev. B 75, 125307 (2007). [19] J. Schliemann, J. C. Egues, and D. Loss, Rev. Rev. B 80, 235327 (2009). [20] J. D. Koralek, C. P. Weber, J.Orenstein, B. A. Bernevig, S.-C. Zhang, S. Mack, and D. Awschalom, Nature 458, 610-613 (2009).[21] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [22] T.-L. Ho and S. Zhang, arXiv:1007.0650 (2010). [23] G. Juzeli unas, J. Ruseckas, and J. Dalibard, Phys. Rev. A81, 053403 (2010). [24] D. L. Campbell, G. Juzeli unas, and I. B. Spielman, arXiv:1102.3945 (2011). [25] J. Larson and E. Sjoqvist, Phys. Rev. A 79, 043627 (2009). [26] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett. 105, 160403 (2010). [27] S.-K. Yip, Phys. Rev. A 83, 043616 (2011). [28] Y. Zhang, L. Mao, and C. Zhang, arXiv:1102.4045 (2011). [29] Z. F. Xu, R. Lu, and L. You, Phys. Rev. A 83, 053602 (2011). [30] T. Kawakami, T. Mizushima, and K. Machida, Phys. Rev. A 84, 011607(R) (2011). [31] C.-M. Jian and H. Zhai, arXiv:1105.5700 (2011). [32] D.-W. Zhang, Z.-Y. Xue, H. Yan, Z. D. Wang, and S.-L. Zhu, arXiv:1104.0444 (2011). [33] S. Gopalakrishnan, A. Lamacraft, and P. M. Goldbart, arXiv:1106.2552 (2011). [34] C. Wu, I. Mondragon-Shem, and X.-F. Zhou, arXiv:0809.3532 (2011). [35] J. Dalibard, G. J. F. Gerbier, and P. Ohberg, arXiv:1008.5378 (2010). [36] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). [37] A. J. Leggett, in Bose-Einstein Condensation: from Atomic Physics to Quantum Fluids , Proceedings of the 13th Physics Summer School, edited by C.M. Savage and M. Das (World Scientic, Singapore) (2000). [38] M. I. Dyakonov and V. I. Perel, Phys. Lett. 35A, 459 (1971); J.E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999); J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, ibid.92, 126603 (2004); S. Mu- rakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003); V. Galitski, A. A. Burkov, and S. Das Sarma, Phys. Rev. B 74, 115331 (2006). [39] Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306,1910 (2004); J. Wunderlich, B.16 Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005). [40] Y. Castin and R. Dum, Eur. Phys. J. D 7, 399-412 (1999). [41] M. Burrello and A. Trombettoni, Phys. Rev. Lett. 105, 125304 (2010). [42] F. Dalfovo and S. Stringari, Phys. Rev. A 53, 2477 (1996). [43] K. Kasamatsu, M. Tsubota, and M. Ueda, Phys. Rev. Lett. 91, 150406 (2003). [44] I. B. Spielman, Phys. Rev. A 79, 063613 (2009). [45] The equation can be derived using expressions given in C.J. Pethick and H. Smith, 2008, Bose-Einstein Con- densation in Dilute Gases (Cambridge University Press, Cambridge, England).[46] A. Ashhab and A. J. Leggett, Rev. Rev. A 68, 063612 (2003). [47] E. J. Mueller, T.-L. Ho, M. Ueda, and G. Baym, Rev. Rev. A 74, 033612 (2006). [48] E. Babaev, A. Sudbo, and N. W. Aschroft, Nature 431, 666-668 (2004). [49] E. Babaev and N. W. Aschroft, Nature Physics 3, 530- 533 (2007). [50] C. N. Varney, K. Sun, V. Galitski, and M. Rigol, Rev. Rev. Lett. 107, 077201 (2011). [51] X. Q. Xu and J. H. Han, arXiv:1107.4845 (2011). [52] X.-F. Zhou, J. Zhou, and C. Wu, arXiv:1108.1238 (2011).

1108.6128v3.Curvature_induced_spin_orbit_coupling_and_spin_relaxation_in_a_chemically_clean_single_layer_graphene.pdf

Curvature-induced spin-orbit coupling and spin relaxation in a chemically clean single-layer graphene Jae-Seung Jeong School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea Jeongkyu Shin and Hyun-Woo Lee Department of Physics and PCTP, Pohang University of Science and Technology, Pohang 790-784, Korea (Dated: October 31, 2018) The study of spin-related phenomena in materials requires knowledge of the precise form of eec- tive spin-orbit coupling for conducting carriers in solid-states systems. We demonstrate theoretically that curvature induced by corrugations or periodic ripples in single-layer graphenes generates two types of eective spin-orbit couplings. In addition to the spin-orbit coupling reported previously that couples with sublattice pseudospin and corresponds to the Rashba-type spin-orbit coupling in a corrugated single-layer graphene, there is an additional spin-orbit coupling that does not cou- ple with the pseudospin, which can not be obtained from the extension of the curvature-induced spin-orbit coupling of carbon nanotubes. Via numerical calculation we show that both types of the curvature-induced spin-orbit coupling make the same order of contribution to spin relaxation in chemically clean single-layer graphene with nanoscale corrugation. The spin relaxation dependence on the corrugation roughness is also studied. I. INTRODUCTION Graphene has attracted much interest due to its un- usual linear energy spectrum and electronic properties.1 It is also a promising material for spin-based applica- tions since spin relaxation (SR) in graphenes is expected to be weak due to the suppression of the hyperne in- teraction of12C and the weak atomic spin-orbit coupling (SOC). Various spintronic devices2and spin qubits3on graphenes have been suggested. Recent experiments, however, measured SR times (SRTs) of the order of 0 :10:5 ns,4which were much shorter than expected and comparable to other materi- als such as aluminum5and copper.6Those experiments stirred up intensive theoretical investigation of the SR in single-layer graphene (SG) to clarify the origin of the short SRT. SG samples in Ref.4have low mobility 2000 cm2=Vs and the SRT was found to increase with increasing carrier density, which is consistent with the Elliot-Yafet (EY) SR mechanism7arising from the inter- play of SOC and momentum scattering. However, it is not clear what kind of SOC is mainly responsible for the SR in SG, and various kinds of SOCs are examined. It was suggested8that adatoms on SG can enhance the local SOC to the order of 10 meV, which results in short SRT comparable to the experimental value4for low mobility SG samples. Thus the adatom-induced SOC provides a possible origin9,11of the short SRT in Ref.4. If this SOC is indeed the main origin, which requires further conr- mation, it implies that the SRT may be enhanced in SGs without adatoms. To nd out how greatly the SRT can be enhanced, various factors limiting the SRT should be examined carefully. In exfoliated SGs, charged impuri- ties or surface phonons in the substrate9can induce the eective SOC of the order of 0 :01 meV. Also, eective SOC of the order of 0 :010:1 meV10can be induced by local curvature eects arising from corrugations12{17and periodic ripples18,19observed in suspended12,17,18 and exfoliated13{16,19SGs. Each substrate-induced and curvature-induced SOC combined with momentum scat- tering o impurities11,20or electron-electron Coulomb scattering,21and substrate-induced or curvature-induced SOC itself22can cause SR. The estimated SRTs are at least of the order of 10 ns, which is about two order of magnitude larger than measured SRTs.4 For a more reliable investigation of spin-related phe- nomena such as SR and spin coherence, it is important to know the precise form of SOC. In this respect, existing studies of the curvature-induced SOC in SGs are not sat- isfactory, since the precise form of this SOC is not known. Previous theoretical studies10,20{22assumed a particular form of the SOC inferred from that of a carbon nan- otube (CNT).10However the SOC-induced energy split- ting measurement23in ultra-clean CNTs revealed that the form of the SOC in Ref.10,24does not provide a sat- isfactory description of the SOC eect in a CNT. Later theoretical works25,26reported that the correct form of the SOC in a CNT has an additional term in addition to the SOC form in Ref.10,24. The additional term does not couple to the pseudospin degrees of freedom in a CNT and thus diers qualitatively from the previously known SOC term,10,24which couples to the pseudospin. It was demonstrated in Ref.26that the interplay of the two SOC terms can explain the SOC-induced energy splitting mea- surement.23,27It is then reasonable to expect that a sim- ilar additional SOC term may exist for SGs as well. Finding the correct curvature-induced SOC can have an implication not only on SR in the current experi- ments,4but also on SR in chemically clean SGs. As technology for the preparation of SGs progresses so as to diminish impurity eects, the fundamental limit of SR would be governed by intrinsic sources. Recalling that atomically at SGs are thermodynamically unstable,28 the curvature-induced SOC is expected to be one of thearXiv:1108.6128v3 [cond-mat.mes-hall] 29 Nov 20112 intrinsic origins determining the upperbound of the SRT in chemically clean SGs.29We believe that this expec- tation is very plausible since SGs and CNTs are very similar chemically and the dominant source of the SOC in ultra-clean CNTs23,27is the curvature-induced SOC. In this paper, we show that in addition to the curvature-induced Rashba-type SOC reported in an ex- isting theory,10which couples to the pseudospin, there exists an additional type of the curvature-induced SOC, which does not couple to the pseudospin and thus ap- pears as the diagonal term in the pseudospin representa- tion. In this respect, this additional SOC term is similar to the additional SOC term in CNTs.25,26However these two additional terms have topologically dierent origins since the periodicity in the circumferential direction aris- ing from the tube topology of CNTs is absent in SGs. Thus the additional SOC term in SGs cannot be obtained from the additional SOC term in CNTs through a trivial extension. This point was not addressed in an earlier at- tempt26to derive the curvature-induced SOC in SGs. In this paper, we also examine the SR due to the curvature in chemically clean SGs and show that the eect of the additional SOC on the SR is comparable in magnitude to the eect of the previously known curvature-induced SOC. Thus the additional curvature-induced SOC term should be included in studies of spin-related phenomena in SGs. We also investigate SR dependence on the frac- tal dimension of the corrugation roughness,14,30and show that corrugation roughness aects the SR. This paper is organized as follows. In Sec. II, we show analytical expressions of two types of the curvature- induced SOCs in SGs, and then demonstrate how to obtain them microscopically using tight-binding (TB) Hamiltonian of the local curvature eect and the atomic SOC of carbon. In Sec. III, we investigate eects of the eective SOC on SR based on the kinetic equation of the spin density operator in corrugated SGs, and calculate numerically SRT. We give a brief summary in Sec. IV II. EFFECTIVE SPIN-ORBIT COUPLING In this section, we present the eective Dirac Hamilto- nian in at SGs, and then, the eective SOC Hamiltonian arising from the local curvature eect combined with the atomic SOC in corrugated SGs. After that, we demon- strate how to evaluate the eective SOC using the TB Hamiltonian via second-order perturbation theory, and why the SOC can not be inferred from the extension of that of CNTs. A. Analytic expression Graphene honeycomb lattice with two sublattices A andB(Fig. 1), has -band consisting of pzorbitals near the Fermi level situated at hexagonal corners in the Brillouin zone. Close to the Dirac point K1andK2=K1,-band states with wave vector k= (kx;ky) [jkjjK1(2)j] relative to the K1(2), can be described by the eective Dirac Hamiltonian HDirac. When the two sublattices AandBof the honeycomb lattices are used as bases, theHDirac is written as31 HDirac =~vF 0eiz(zkxiky) eiz(zkx+iky) 0 (1) wherevFis the Fermi velocity, and z=1 denotes the K1(2). Here, the angle is dened counterclockwise from the^ydirection and the C-C bond vector bcc-direction (Fig. 1). Local curvature eects arising from smooth corru- gations or ripples in SGs combined with atomic SOC of carbon32generate the eective SOC for the -band states.10The corrugated SG can be described as a sur- facez=h(r) with height h(r) as a function of spatial position r= (x;y) in the two-dimensional (2D) refer- ence plane (Fig. 2). Experiments12{15,17suggest that h(r) shows uctuations of the order of 1 nm over scales 10 nm, which can be specied by hihj1 (i;j=x;y) withhibeing the partial derivative along the idirection. When the local structure of the SG surface has two prin- cipal curvatures 1(r) and2(r), we nd that the Hsoc(r) is written as Hsoc(r) = 0z(r)
seiz(r)(zsy+isx) eiz(r)(zsyisx)0z(r)
s (2) where s= (sx;sy;sz) denotes the real spin Pauli ma- trix. Here, and0are material parameters, and (r) and(r) are geometrical parameters containing the lo- cal curvature information. For = 0, the o-diagonal part of Eq. (2) reduces to the Rashba-type SOC / (ysxzxsy) reported in existing theories10,20, where = (x;y;z) denotes the sublattice-pseudospin Pauli matrix. On the other hand, the diagonal part of Eq. (2) is one of main results of this paper, which was not ignored ✓1(r) ✓2(r)xy↵ xyzAB x0(r)y0(r)z0(r)&11&21&311(r)a ↵b0cc(r)bcc 


 FIG. 1. (Color online) Single-layer at graphene honeycomb lattice with two sublattices AandB. The angle is dened counterclockwise from the ^yto the C-C bond vector bcc.a= jbccj1:42A is the carbon-carbon distance.3 in previous studies. Note that this additional SOC term in the diagonal part does not couple to the pseudospin. The material parameters and0are, respectively, given by=aso(ps)(V pp+V pp)=(12V2 sp) and0= asoV pp=(2(V ppV pp)), wheresois the atomic SOC strength of the porbitals,s(p)is the atomic energy for s(p) orbitals and ais the carbon-carbon distance. Here, V spandV() pp represent the coupling strength in the ab- sence of the curvature for the coupling between nearest- neighborsandporbitals and the () coupling between nearest-neighbor porbitals, respectively. The geometrical parameters (r) and(r) are given as follows. (r) is a sum of the two local curvatures, (r) =1(r) +2(r) and(r) is a weighted sum of the two local curvatures, (r) = (x(r);y(r)) =2X j=1j(r) (sin[2j(r)3];cos[2j(r)3]); (3) where1(r) and2(r) are the angles between a local x0(r)-direction and the two principal curvature directions (see Fig. 2 for their precise denitions). Since the two principal curvature directions are mutually orthogonal, 2(r) =1(r)=2. ✓1(r) ✓2(r)xy↵ xyzAB x0(r)y0(r)z0(r)&11&21&311(r)a ↵b0cc(r)bcc x0(r)B1B2B3A 


 FIG. 2. (Color online) Schematic of a partial convex struc- ture in a corrugated SG surface described by z=h(r) where r= (x;y) within the 2D reference xy-plane. The ^z0(r) is the unit vector normal to the local tangential plane and the b0 cc(r) (green arrow) is the projected vector from the local C-C bond onto the tangent plane specied by the (blue) dashed quad- rangle normal to the ^z0(r). The y0(r)-direction in the tangent plane is dened to make the angle [Fig. 1] clockwise from theb0 cc(r) in the tangent plane. Then the x0(r)-direction is determined automatically to be mutually orthogonal to z0(r)- direction and y0(r)-direction. Dotted (blue) lines are cur- vature lines tangent to principal directions along orthogonal (blue) arrows where the direction of arrowheads is arbitrary. The angle1(2)(r) is dened by x0(r)-direction and the princi- pal directions, and 2(r) =1(r)=2. The sublattice Aand its nearest-neighbor sites Bl(l= 1;2;3) with two curvature lines are indicated.Both diagonal and o-diagonal terms in Eq. (2) are invariant under the time reversal symmetry. The o- diagonal term is allowed when the mirror symmetry with respect to the xy-plane is broken.33On top of that, owing to the broken C3symmetry in corrugated SGs, the diago- nal term of the form zsx(y)is allowed as well. Note that the diagonal term has the 1(2)(r)-dependence. However, it can not be obtained from the naive extension of the diagonal SOC of CNTs that depends on the chiral angle through cos(3 ).25,26Quantitatively, both terms have the same order of magnitude. Using so10 meV,32 s=7:3 eV,p= 0 eV,V sp= 4:20 eV,V pp= 5:38 eV, V pp=2:24 eV in Ref.34we obtain00:21 meVnm and0:15 meVnm. Thej(r) and the-dependence ofHsoc(r) combined with the SG morphology have interesting implications. First of all, since 2(r) =1(r)=2, the diagonal term depends on the dierence between two principal curvatures in contrast to the o-diagonal term depend- ing on their sum. As a result, at umbilical points where 1(r) =2(r), the diagonal term disappears while the o-diagonal term remains, which conrms that every tan- gent vector can be a principal direction at the umbilical points.35In contrast with the umbilical points, at sad- dle points where 1(r) =2(r), the diagonal term only appears while the o-diagonal term disappears. In addition, for periodic ripples having unidirectional curvature direction ( 1(r)6= 0 and2(r) = 0), the diago- nal SOC eld direction varies depending on the principal curvature direction and the honeycomb lattice orienta- tion as well. When =q=3 (q2Z) so that the ^xis along the zigzag direction [Fig. 1], for example, the di- agonal term is/1(r)syfor1(r) = 0 and =2, while it is/1(r)sxfor1(r) ==4. On the other hand, when=q=3 +=6 so that the ^xis along the arm- chair direction [Fig. 1], the diagonal term is /1(r)sxfor 1(r) = 0 and=2, while it is/1(r)syfor1(r) ==4. Recent experiments shows that the periodic ripple line direction can be controlled18and the multiple periodic ripple domains with dierent ripple lines can occur,19 which implies that the variation of the diagonal term de- pending on the curvature direction and the honeycomb lattice orientation could be signicant for the analysis of spin transport in SGs with those structural deformation. After all, in order to describe precisely the curvature- induced SOC in SGs both terms should be considered on an equal footing. TheHsoc(r) can be written as an explicit function ofh(r) in the SG surface, z=h(r) wherehihj1 (i;j=x;y). The principal curvature is the eigenvalue of the shape operator S=F1 1F2withF1(2)being the rst (second) fundamental form.35Moreover, since its eigen- vector ^v1(2)(r) is along the principal direction, 1(2)(r) can be determined by the ^v1(2)(r) and ^x0(r), which is as- sumed as ^x0(r)(1;0;hx) in the zeroth order of hihj. F1andF2are, respectively, given by354 F1=1 +h2 xhxhy hxhy1 +h2 y ;F2=1p 1 +jrhj2 hxxhxy hxyhyy ; (4) and then, the 1(2)(r) and ^v1(2)(r) are written as 1(2)(r) hxx+hyy(1)1(2)[(hxxhyy)2+ 4h2 xy]1 2
=2 and^v1(2)(r)[1(2)^x+^y+(1(2)hx+hy)^z]=N, respec- tively, where 1(2)hxxhyy(1)1(2)[(hxxhyy)2+ 4h2 xy]1 2and2hxywith normalization constant N. B. Tight-binding Hamiltonian Now we demonstrate how to derive Eq. (2) microscopi- cally. Using second-order perturbation theory36with the local curvature and the atomic SOC as weak perturba- tion, we obtain Eq. (2). Thus we need to know the TB Hamiltonian of the local curvature Hcand the atomic SOCHsotaking the honeycomb lattice structure into ac- count exactly. Most parts of this subsection are devoted to the TB theory of the -hybridization among sand porbitals. The unoccupied dorbital eects37,38are dis- cussed at the end of this subsection. Firstly,Hcis determined purely by local curvature ef- fect dependent on 1(2)(r) and1(2)(r) at three bonds be- tween an atomic site rAin the sublattice Aand nearest- neighbor sites Bl(l= 1;2;3) in the sublattice B[Fig. 2]. For the smoothly corrugated SGs, we consider solely one of two principal curvature eects assuming that the other principal curvature eect is absent. There are also mu- tual curvature eects between them, but those eects can be ignored for the smooth corrugated surface with xy↵ABbccB1 B1B2B3A 1(rA)&31&21&11 B2B3AB1 &12&22&322(rA) (b)(a) 


 FIG. 3. (Color online) Schematic of a partial convex SG sur- face with sublattices AandBl(l= 1;2;3) indicated in Fig. 2, which is assumed to have unidirectional principal curvature. Although the original surface [Fig. 2] has nite principal cur- vatures1(rA) and2(rA), each curvature eect is evaluated under the assumption that the other principal curvature is zero.1(rA)6= 0 and2(rA) = 0 in (a) and 1(rA) = 0 and 2(rA)6= 0 in (b). Dotted (blue) lines represent curvature lines, and principal direction at rAis tangent to the curva- ture lines. The arc length &l j(j= 1;2) between a solid (green) line passing Aatom normal to the principal direction, and its parallel solid (green) line passing Blatom, is related with !l j by!l j&l jj(r)=2 [Eq. (5)],hihj1 (i;j=x;y) since they are higher order in a=R1(2)(rA) whereR1(2)(rA) =1 1(2)(rA). Then, it is enough to deal with two unidirectional ripple structures whose principal directions are orthogonal as shown in Fig. 3. For the unidirectional ripple structure, each prin- cipal curvature and the angle between the principal direc- tion and the x0(r) at three neighbor C-C bonds around therAare same as 1(2)(rA) and1(2)(rA), respectively. Then, up to rst order in a=R1(2)(rA), theHccan be approximated as HcH1+H2, each of which describes one principal curvature eect in the absence of the other one. Two principal curvatures yield correlation eects between term in Hc, but they are higher order in term of a=R1(2)(r) and thus can be ignored. The Hj(j= 1;2) is written as10,26 Hj=X rA3X l=1X =";#h Sl j csy rAcz0 Bl+cz0y rAcs Bl +X0l j cx0y rAcz0 Blcz0y rAcx0 Bl +Y0l j cy0y rAcz0 Blcz0y rAcy0 Bli + H:c:; (5) wherecx0 r"(#),cy0 r"(#), andcz0 r"(#)represent the annihila- tion operators for px0,py0,pz0orbital states with eigen- spinor"(#)along ^z0at a carbon atom r. Here,Sl j, X0l j, andY0l jrepresent the curvature-induced coupling strength of px0,py0, andsorbitals with a nearest- neighborpz0orbitals. Their expressions are written as Sl j=!l j~Sl j,X0l j=!l j~Xl jcos[j(rA)]!l j~Yl jsin[j(rA)], andY0l j=!l j~Xl jsin[j(rA)]+!l j~Yl jcos[j(rA)], with!1 j a=(2Rj(rA)) sin[j(rA)],!2 ja=(2Rj(rA)) sin[=3 j(rA) +], and!3 ja=(2Rj(rA)) sin[=3 +j(rA)]. Here, ~Sl j,~Xl j, and ~Yl jare, respectively, written as ~S1 j=V spsin [j(rA)]; ~S2 j=V spcosh 6+j(rA)i ; ~S3 j=V spcosh 6j(rA)i ; ~X1 j=V ppV ppcos2[j(rA)]V ppsin2[j(rA)]; ~X2 j=V ppV ppcos2h 3j(rA)i V ppsin2h 3j(rA)i ; ~X3 j=V pp+V ppcos2h 6j(rA)i +V ppsin2h 6j(rA)i ; ~Y1 j=1 2 V ppV pp
sin [2j(rA)]; ~Y2 j=1 2 V ppV pp
sin 2j(rA)2 3 ; ~Y3 j=1 2 V ppV pp
sinh 2j(rA) 3i ; (6) wherej(rA)=j(rA). Note that the local curvature generates interband transition between -band and - band consisting of s,p0 x, andp0 yorbitals.10,26,395 Secondly,Hsois given by Hso=soP rLr
Sr,10,40 where Lr(Sr) is atomic-orbital (spin) angular momentum of an electron at r, and can be expressed by10,26 Hso=so 2X r=rA=B cz0y r#cx0 r"cz0y r"cx0 r#+icz0y r"cy0 r#+icz0y r#cy0 r" +icy0y r"cx0 r"icy0y r#cx0 r# + H:c:: (7) Note that on-site coupling between pz0andpx0(y0)orbitals generates-interband transition that, combined with the curvature-induced interband transition, contributes to theHsoc(r). In CNTs, however, the coupling between pz0orbital andporbital along the CNT axis has no contribution to the eective SOC. For instance, if the Eq. (7) is ex- pressed with eigenspinor along the CNT axis parallel to ^y0(^x0)-direction instead of "(#), the coupling between pz0andpy0(x0)orbitals acquires a phase term eiwith the azimuthal angle along the circumference10,26, and, as a result, its contribution averages out to zero after integration over the circumference.10Hence, the trivial extension from the curvature-induced SOC of the CNT to that of the SG is inevitably incomplete because of their topological dierence in the geometrical structure. Second-order process resulting from two consecutive -interband transitions gives rise to the eective SOC Hamiltonian for the unperturbed -band states of Eq. (1).41For the-band calculation, we use the Slater- Koster parametrization42ofs,px, andpyorbitals. Dur- ing the second-order process, pseudospin-conserving and pseudospin- ipping processes occur; the former and the latter generate, respectively, the diagonal and the o- diagonal SOC in Hsoc(r) [Eq. (2)].26The resulting eec- tive SOC, then, is expressed in local coordinate axes of ^x0(r),^y0(r),^z0(r). However, for the smooth corrugation wherehihj1 (i;j=x;y),^x0(r),^y0(r),^z0(r) in the re- sulting eective SOC may be replaced by ^x,^y,^zto obtain Hsoc(r) in Eq. (2) since the dierence between the two sets of axes [for instance, ^z0(r)
^z= 1 +jrh(r)j2
1 21] generates higher order terms in a=R1(2)(r). Lastly, we remark dorbitals eects on the curvature- induced SOC, which were not considered in our calcula- tion. For the spin-orbit induced gap, the dorbital eects are recently reported to be important since the hybridiza- tion between dorbitals and porbitals generate the gap, which is linear in the atomic SOC strength dofdor- bitals.37,38This linear contribution can be dominant over theporbital contribution, which is quadratic in so. In contrast, the contribution of the dorbitals to the exter- nal eld-induced Rashba SOC is smaller than that of the porbitals.38The coupling between panddorbitals can also lead to the curvature-induced SOC in the rst order ofa=R1(2)(r). Due to the symmetrical reason, however, its form is expected to be same as the form of Hsoc(r) [Eq. (2)] except parameters in and0. The curvature- induced SOC owing to the dorbitals can be generated asfollows. The -band consisting of the pzorbital and the nearest-neighbor dxzanddyzorbitals can couple with the bands through on-site coupling between dorbitals in -band and-band by the atomic SOC. Then, electrons in-band can delocalize to the nearest-neighbor pandd orbitals within bands. Finally, the panddorbitals in -bands can couple with the nearest-neighbor pzorbitals due to the curvature, yielding the curvature-induced SOC that is proportional to a=R1(2)(r),d, and (dp)2. Based on the estimation of the Rashba SOC strength and comparison with the porbital contribution given in Ref.38, we can compare roughly the dorbital contribution of the curvature-induced SOC with the Hsoc(r) [Eq. 2]. Thedis smaller than so, and further, ( dp)2mul- tiplied by hopping parameters associated with dandp orbitals is a small quantity compared with parameters in same dimension in and0. Hence, we believe that the contribution of dorbitals to the curvature-induced SOC is smaller than that of porbitals. In order to get more reliable quantitative estimates, systematic study consid- ering coupling between p,sorbitals and dorbitals is re- quired.38 III. SPIN RELAXATION In this section, we investigate eects of the geometric curvature on SR in chemically clean SGs with nanoscale corrugations. Motivated by recent experiments14about the corrugation roughness of SGs, we study SR depen- dence on the fractal dimension of the corrugation rough- ness within the assumption that both SOC and mo- mentum scattering30arise mainly from the corrugation. Since the substrate-induced SOC eect on SR was ad- dressed in a previous theory,9we focus on the curvature- induced SOC here. A. Kinetic equation of the real spin density opeator In order to calculate SRT, we use the kinetic equa- tion of the real spin density operator .22,43Besides theHsoc(r), we include the strain-induced vector po- tentialHv(r) that occurs in corrugated SGs.44{46Al- thoughHv(r) is independent of the spin, it is still rel- evant for the SRT since it aects the momentum scat- tering rate. Then the local potential Hamiltonian Hp(r) is written asHp(r)H soc(r) +Hv(r) which is consid- ered as weak perturbation in the kinetic equation. We assumehHp(r)i0 whereh


i denotes ensemble aver- age. SinceHp(r) is expected to change slowly over scales larger than the lattice spacing so that the hybridization betweenz=1 andz=1 states can be ignored, we cal- culate SRT for the electron state of the HDirac [Eq. (1)] whenz= 1 and= 0, where eigenenergy "kand eigen- statej kiare, respectively, given by "k=~vFkwith kjkj, andj ki= 1=p 2(ei'k1)Tjkiwith polar angle6 'kof the wavevector k.47Then a 22 real spin density operatorkdiagonal in momentum has a representation in eigenspinors chosen to be along ^zdirection. The ki- netic equation of kis written as22,43 @k @t+i ~[Vkk;k] =Ck; (8) whereVkk0=h kjHp(r)j k0i, andCkis the collision inte- gral describing momentum scattering, which is given by Ck==~P k0[2Vkk0k0Vk0kfk;Vkk0Vk0kg]("k0"k). The commutator in Eq. (8) can be ignored because Hp(r) has no regular contribution. Since the energy scale of Hv(r) is typically larger than that of Hsoc(r), the SRT would be longer than a time scale in which the momen- tum distribution during momentum scattering becomes isotropic.43Thus, thekcan be represented as k=+0 k with the anisotropic part 0 kdue to theHsoc(r), where the bar means averaging over 'k. Inserting the k=+0 kinto Eq. (8), we obtain the equation of as, @ @t=d ~I d'kI d'k0 V0 kk0V0 k0k1 2n ;jV0 kk0j2o ;(9) whered=k=(~vF) is the density of states. Note that collision term consisting of the product of Hsoc(r) and Hv(r) disappears since there is no correlation between them. More specically, since the vector potential is writ- ten as quadratic in hi;j,44,45the collision terms propor- tional tohhihj(r)iorhhihj(r)ivanish by the Wick theorem.30Thus, the random gauge eld does not con- tribute to the SR in chemically clean SGs.48 Since the real spin density P=(Px;Py;Pz) can be eval- uated by P=Tr (s), we can obtain the kinetic equation ofPby tracing Eq. (9) multiplied by sover the real spin space, which is written as @Pi=@tPi=s i(i=x;y;z ) withs ibeing the SRT of spin along the i-direction. Here, s x(y)ands zare evaluated as 1 s x(y)=43d ~Z Z drdr0 02hy(x)(r)y(x)(r0)iJ+2h(r)(r0)iJ0 ; (10) and 1=s z= 1=s x+ 1=s y, whereJandJ0are, respec- tively, dened as JJ2 0(kR)+J2 1(kR), andJ0J2 0(kR) withJ0(1)(x) being the Bessel function of zeroth (rst) order andR=jrr0j. If we assumehx(r)x(r0)i hy(r)y(r0)iandhx(r)y(r0)i0 for the random cor- rugation, we obtain s xs yands x(y)2s z, which means that spins out of plane relax twice as fast as spins in plane.20Note that since the carrier density nis given byn=k2=, the SR vanishes at the Dirac points where n= 0.22 B. Numerical results Figure 4 shows SRTs, hs zithat are calculated nu- merically in corrugated SG surfaces for = 0 and are ensemble-averaged for ten surfaces. Those surfaces are approximated as square lattices, which can be justi- ed since theHsoc(r) contains the curvature eects tak- ing into account the honeycomb lattice. The surface with uctuating h(r) is constructed based on the height- height correlation function g(R) =h(h(r)h(r0))2i= 2 2 1e(R=)2H
, whereHcharacterizes the fractal di- mension,14,30,49represents the correlation length, and =p hh2(r)i. As the fractal dimension Hincreases, the corrugated surface becomes smooth. A recent experiment12in suspended SGs shows that the height of corrugations grows approximately linearly with increasing length, which corresponds to a corru- gated surface for 2 H= 2. For exfoliated SGs on the substrate, on the other hand, the fractal dimension with2H1 was observed.14There is a controversy over the origin of observed corrugation roughness.16Here, we in- Hsoc(r)Hsoc(r) 50100150200(a)2H=12H=2 50100150200 1 2 3n (1012 cm-2 )2H=1(b)withwith the off-diagonal SOC in h⌧szi(ns) 


 FIG. 4. (Color online) Spin relaxation time hs zias a function of the carrier density n=k2=and the fractal dimension H. Thehs ziare ensemble-averaged quantities over ten surfaces for= 0. The corrugated square lattice with the height h(r), the lattice constant of 0 :2 nm, and the side length of 200 nm, are constructed numerically based on the random midpoint displacement method50with height-height correlation func- tiong(R) = 2 2 1e(R=)2H
. Here, we set = 0:3 nm and = 20 nm from recent experiments,14,15and interpolate dis- creteh(r) in order to mimic the continuous SG surface. (a) hs zias a function of nfor 2H= 1 (red solid line) and 2 H= 2 (blue dashed line). (b) hs zias a function of nfor 2H= 1 consideringHsoc(r) (red solid line) and only the o-diagonal SOC inHsoc(r) (black dashed line), respectively.7 vestigate SR in corrugated SGs for 2 H= 1 and 2H= 2 representatively. As shown in Fig. 4 (a), the hs zibecomes shorter with decreasing 2 H, which can be understood in terms of a growing irregularity of Hsoc(r) [Eq. 2] induced by rougher surface corrugation. Also, we check that the hs x(y)iis approximately twice as large as the hs zi(not shown here). Next, in order to investigate the eect of the diagonal SOC inHsoc(r), we compare two SRTs, one calculated considering the entire Hsoc(r) and the other considering only the o-diagonal SOC term in Hsoc(r), respectively, in SGs with 2 H= 1. As shown in Fig. 4 (b), the lat- ter SRT is noticeably larger than the former true SRT. Hence, it is necessary to consider both SOC terms for the precise analysis of SR depending on the corrugation roughness. The calculated SRT as shown in Fig. 4 is at least of the order10 ns, and becomes shorter monotonically as the carrier density nincreases, which is primarily because the density of states dincreases linearly with n1=2[Eq. (10)]. We check that the integrand of Eq. (10) has little depen- dence relatively upon k. This implies that the eective spatial range of the random SOC aecting the SR sub- stantially does not depend signicantly on the variation ofkover the scale10 nm1. Our quantitative and qualitative results are in agreement with a recent the- ory.22As mentioned in the Ref.22, however, those results cannot explain recent spin transport experiments.4 One possibility for that discrepancy is that the ef- fect of charged impurity in the substrate causing mo- mentum scattering51is not considered in our calculation. The momentum scattering combined with the curvature- induced Rashba-type SOC was already investigated the- oretically,20but it cannot explain the measured SRT, ei- ther. However, the inclusion of the additional diagonal SOC inHsoc(r) in addition to the o-diagonal SOC into the EY mechanism could give a possibility to calculate the SRT more precisely. In the EY mechanism,7spin ip scattering occurs at the very moment when the momen- tum scattering takes place since the electron wave func- tions normally have an admixture of the opposite-spin states due to the SOC.IV. CONCLUSION We have demonstrated that the combined action of the curvature and the atomic SOC of carbon gives rise to two types of the eective SOC in SGs with corrugation or pe- riodic ripples. One of them couples with the sublattice pseudospin, which corresponds to the Rashba-type SOC in a corrugated SG reported in previous theories,10,20{22. The other SOC, on the other hand, does not couple with the pseudospin, and was not recognized in previous litera- ture. The additional curvature-induced SOC depends on the principal curvature direction, which is similar to the curvature-induced SOC of CNTs whose diagonal term in pseudospin representation depends on the chiral an- gle of CNTs. However, the curvature-induced SOC in SGs can not be obtained from the trivial extension of the curvature-induced SOC in CNTs due to their dis- tinct topological structure between the SG surface and the CNT cylinder. We have also investigated SR in chemically clean SGs with nanoscale corrugation, and found that the diagonal SOC makes the same order of contribution to SRT as the o-diagonal SOC. The SRT becomes longer as the fractal dimension of the corrugation roughness increases since the irregularity of the SOC decreases in smoother SGs. The calculated SRT in chemically clean SGs, how- ever, can not explain recent experimental results of SR in current exfoliated SG samples. A natural direction for future research would be to calculate SRT in the presence of charged impurities that cause momentum scattering,51considering both the diag- onal and the o-diagonal SOC. In addition to the SR in SGs, we expect that the curvature-induced SOC [ Hsoc(r)] may be relevant for the analysis of other spin-related phe- nomena in SGs. ACKNOWLEDGMENTS We thank D. Gang, T. Takimoto, S. Kettemann, K. Kim, B. H. Kim, J. Han, I. Kim, S. Kim, K. Hashimoto, F. Ziltener, Y.-W. Son, S.-M. Choi, and S. Lee for useful discussions. This work was nancially supported by the NRF (2010-0014109, 2011-0030790) and BK21. jsjeong@kias.re.kr 1A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009); A. K. Geim, Science 324, 1530 (2009). 2Y.-W. Son, M. L. Cohen, and S. G. Louie, Nature (Lon- don) 444, 347 (2006). W. Y. Kim and K. S. Kim, Nat. Nanotechnol. 3, 408 (2008). 3B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Nat. Phys. 3, 192 (2007); P. Recher and B. Trauzettel, Nanotechnology 21, 302001 (2010). 4N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, andB. J. van Wees, Nature (London) 448, 571 (2007); N. Tombros, S. Tanabe, A. Veligura, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Phys. Rev. Lett. 101, 046601 (2008); C. J ozsa, T. Maassen, M. Popinciuc, P. J. Zomer, A. Veligura, H. T. Jonkman, and B. J. van Wees, Phys. Rev. B 80, 241403(R) (2009); W. Han, and R. K. Kawakami, Phys. Rev. Lett. 107, 047207 (2011); S. Jo, D.-K. Ki, D. Jeong, H.-J. Lee, and S. Kettemann, Phys. Rev. B 84, 075453 (2011); W. Han, K. M. McCreary, K. Pi, W. H. Wang, Y. Li, H. Wen, J. R. Chen, and R. K. Kawakami, J. Magn. Magn. Mater. 324, 369 (2012).8 5F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Basel- mans, and B. J. van Wees, Nature (London) 416, 713 (2002). 6F. J. Jedema, M. S. Nijboer, A. T. Filip, and B. J. van Wees, Phys. Rev. B 67, 085319 (2003). 7I.Zuti c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 8A. H. Castro Neto and F. Guinea, Phys. Rev. Lett. 103, 026804 (2009). 9C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B 80, 041405 (R) (2009). 10D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74, 155426 (2006). 11P. Zhang and M. W. Wu, Phys. Rev. B 84, 045304 (2011); H. Ochoa, A. H. Castro Neto, and F. Guinea, arXiv:1107.3386v2 (2011); P. Zhang, and M. W. Wu, arXiv:1108.0283v2 (2011). 12J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, Nature (London), 446, 60 (2007). 13E. Stolyarova, K. T. Rim, S. Ruy, J. Maultzsch, P. Kim, L. E. Brus, T. F. Heinz, M. S. Hybertsen, and G. W. Flynn, Proc. Natl. Acad. Sci. U.S.A. 104, 9209 (2007). 14M. Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D. Williams, Nano Lett. 7, 1643 (2007). 15V. Geringer, M. Liebmann, T. Echtermeyer, S. Runte, M. Schmidt, R. R uckamp, M. C. Lemme, and M. Morgenstern, Phys. Rev. Lett. 102, 076102 (2009). 16W. G. Cullen, M. Yamamoto, K. M. Burson, J. H. Chen, C. Jang, L. Li, M. S. Fuhrer, and E. D. Williams, Phys. Rev. Lett. 105, 215504 (2010). 17K. R. Knox, A. Locatelli, M. B. Yilmaz, D. Cvetko, T. O. Mente s, M. A. Ni~ no, P. Kim, A. Morgante, and R. M. Osgood Jr., Phys. Rev. B 84, 115401 (2011). 18W. Bao, F. Miao, Z. Chen, H. Zhang, W. Jang, C. Dames, and C. N. Lau, Nat. Nanotechnol. 4, 562 (2009). 19J. S. Choi, J.-S. Kim, I.-S. Byun, D. H. Lee, M. J. Lee, B. H. Park, C. Lee, D. Yoon, H. Cheong, K. H. Lee, Y.- W. Son, J. Y. Park, and M. Salmeron, Science, 333, 607 (2011). 20D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. Lett. 103, 146801 (2009). 21Y. Zhou and M. W. Wu, Phys. Rev. B 82, 085304 (2010). 22V. K. Dugaev, E. Ya. Sherman, and J. Barna s, Phys. Rev. B83, 085306 (2011). 23F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen, Nature (London) 452, 448 (2008). 24T. Ando, J. Phys. Soc. Jpn. 69, 1757 (2000). 25W. Izumida, K. Sato, and R. Saito, J. Phys. Soc. Jpn. 78, 074707 (2009). 26J.-S. Jeong and H.-W. Lee, Phys. Rev. B 80, 075409 (2009). 27T. S. Jespersen, K. Grove-Rasmussen, J. Paaske, K. Mu- raki, T. Fujisawa, J. Nyg ard, and K. Flensberg, Nat. Phys. 7, 348 (2011). 28A. Fasolino, J. H. Los, and M. I. Katsnelson, Nature Mater. 6, 858 (2007). 29It was reported recently that the corrugation can be sup- pressed if the SG is exfoliated onto mica surfaces [C. H.Lui, L. Lui, K. F. Mak, G. W. Flynn, and T. F. Heinz, Nature (London), 462, 339 (2009)]. 30M. I. Katsnelson and A. K. Geim, Phil. Trans. R. Soc. A 366, 195 (2008). 31G. W. Semeno, Phys. Rev. Lett. 53, 2449 (1984); D. P. DiVincenzo and E. J. Mele, Phys. Rev. B 29, 1685 (1984). 32J. Serrano, M. Cardona, and T. Ruf, Solid State Commun. 113, 411 (2000). 33C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 34D. Tom anek and M. A. Schluter, Phys. Rev. Lett. 67, 2331 (1991). 35D. J. Struik, Lectures on Classical Dierential Geometry , 2nd ed. (Dover, New York, 1988). 36Leonard I. Schi, Quantum Mechanics (McGraw-Hill, New York, 1968). 37M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, and J. Fabian, Phys. Rev. B 80, 235431 (2009). 38S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B 82, 245412 (2010). 39E.-A. Kim and A. H. Castro Neto, Europhys. Lett. 84, 57007 (2008). 40H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald, Phys. Rev. B 74, 165310 (2006); Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, ibid75, 041401(R) (2007). 41The comprehensive explanation of the degenerate second- order perturbation theory is described well in Ref.10,26. 42J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). 43Optical Orientation , edited by F. Mayer and B. Za- kharchenya (North-Holland, Amsterdam, 1984); N. S. Averkiev, L. E. Golub, and M. Willander, J. Phys.:Condens. Matter 14, R271 (2002); S. A. Tarasenko, JETP Letters 84, 199 (2006). 44T. Ando, J. Phys. Soc. Jpn. 75, 124701 (2006), 45S. V. Morozov, K S. Novoselov, M. I. Katsnelson, F. Schedin, L. A. Ponomarenko, D. Jiang, and A. K. Geim, Phys. Rev. Lett. 97, 016801 (2006). 46F. Guinea, B. Horovitz, and P. Le Doussal, Phys. Rev. B 77, 205421 (2008). 47We evaluate SRT for hole states of Eq. (1) as well, and obtain electron-hole symmetric SRT. 48A strain-induced vector potential arising from in-plane atomic displacement44,45is not correlated with the Hsoc(r) that is dependent on the curvature only. In addition, the scalar potential induced by the curvature eects39is quadratic in local mean curvature. 49B. B. Mandelbrodt, The Fractal Geometry of Nature (Freeman, New York, 1982); G. Palasantzas, Phys. Rev. B48, 14472 (1993). 50A. Fournier, D. Fussel, and L. Carpenter, Communications of the ACM, 25, 371 (1982). 51T. Ando, J. Phys. Soc. Jpn. 75, 074716 (2006); K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, 256602 (2006); K. Nomura and A. H. MacDonald, ibid98, 076602 (2007); E. H. Hwang, S. Adam, and S. Das Sarma, ibid98, 186806 (2007); S. Adam, E. H Hwang, V. Galitski, and S. Das Sarma, Proc. Natl. Acad. Sci. U. S. A. 104, 18392 (2007).

1807.06669v1.Effects_of_spin_orbit_coupling_on_the_optical_response_of_a_material.pdf

arXiv:1807.06669v1 [cond-mat.mtrl-sci] 17 Jul 2018Effects of spin-orbit coupling on the optical response of a ma terial Tae Yun Kim1, Andrea Ferretti2, and Cheol-Hwan Park1∗ 1Department of Physics, Seoul National University, Seoul 08 826, Korea 2Centro S3, CNR-Istituto Nanoscienze, 41125 Modena, Italy (Dated: July 19, 2018) We investigate the effects of spin-orbit coupling on the opti cal response of materials. In particular, we study the effects of the commutator between the spin-orbit coupling part of the potential and the position operator on the optical matrix elements. Using a formalism that separates a fully- relativistic Kleinman-Bylander pseudopotential into the scalar-relativistic and spin-orbit-coupling parts, we calculate the contribution of the commutator aris ing from spin-orbit coupling to the squared optical matrix elements of isolated atoms, monolay er transition metal dichalcogenides, and topological insulators. In the case of isolated atoms from H (Z= 1) to Bi ( Z= 83), the contribution of spin-orbit coupling to the squared matrix elements can be as large as 14 %. On the other hand, in the cases of monolayer transition metal dichalcogenides an d topological insulators, we find that this contribution is less than 1 % and that it is sufficient to calcul ate the optical matrix elements and subsequentphysical quantities without considering the co mmutator arising from spin-orbit coupling. I. INTRODUCTION The electronic structure of materials containing heavy elements can be significantly affected by spin-orbit cou- pling (SOC). Due to the recent advances in the inves- tigation of materials having strong SOC effects such as transition metal dichalcogenides (TMDCs), topological insulators, or Weyl semimetals to name a few, it becomes importanttoaccuratelysimulatethe effectsofSOCusing first-principles density functional theory (DFT) calcula- tions. Because SOC allows the manipulation of the spin degrees of freedom in materials by using light,1–6under- standing the effects of SOC on the optical response of materials is a matter of importance. Consider a system described by an effective Hamilto- nian ˆH=ˆp2 2m+Vloc(ˆr)+ˆVNL, (1) wheremis the mass of an electron, ˆpis the momentum operator, ˆris the position operator, and Vloc(ˆr) andˆVNL are the local and non-local parts of the potential, respec- tively. The optical matrix elements of the system are given by the matrix elements of the velocity operator7 ˆv=ˆp m+i /planckover2pi1/bracketleftBig ˆVNL,ˆr/bracketrightBig . (2) In many DFT calculations, the pseudopotential method is used because of its computational efficiency. Within the non-relativistic and scalar-relativistic pseu- dopotential methods, the effects of the commutator in Eq. (2) on the optical matrix elements and absorption spectra have been investigated for various types of sys- tems such as isolated atoms,8semiconductors,9and met- als.10It was reported that the contribution of the com- mutatorcanbe large,e.g. asin thecasesofcarbonatom8 and bulk copper.10 SOC is proportional to ˆL·ˆS, whereˆLandˆSare the orbital and spin angular momentum operators, respec- tively. Because the orbital angular momentum operatordoes not commute with the position operator, SOC re- sults in an additional contribution to the velocity oper- ator via the commutator and to the optical matrix ele- ments. It has not been well established whether the effects of the commutator arising from the SOC part of the poten- tial are important or not in a system where the influence of SOC on the electronic structure is known to be strong. For example, in some previous studies on the optical re- sponse of Bi 2Se3, a topological insulator that has been extensively investigated, the contribution of the commu- tator arising from SOC was neglected and ˆp/mas an approximation to ˆv[Eq. (2)] was used to calculate the optical matrix elements.5,6,11On the other hand, the au- thors of a recent study12on the circular dichroism of Bi2Te3claimed that the SOC contribution to the veloc- ity operator plays a crucial role in explaining the results of their photoemission experiments. In this study, we investigate the effects of SOC on the optical matrix elements and absorption spectra in vari- oustypesofsystems: isolatedatoms, monolayerTMDCs, and topological insulators. The method used in this study allows for the calculation of the optical matrix el- ements with and without inclusion of the commutator arising from the intrinsic non-locality of SOC while using the same (fully-relativistic) pseudopotential, from which we can directly assess the importance of the effects of SOC in evaluating the optical matrix elements and opti- cal properties of materials. II. METHODS The non-local part of a fully-relativistic pseudopoten- tial in semi-local form can be written as13 ˆVSL=lmax/summationdisplay l=0l+1 2/summationdisplay j=|l−1 2|j/summationdisplay mj=−j|l,j,mj/angbracketrightVl,j(r)/angbracketleftl,j,mj|,(3)2 whereVl,j(r) is the radial potential of ˆVSLfor a given pair of the orbital angular momentum quantum num- berland the total angular momentum jand|l,j,mj/angbracketright is the spin-angular function14satisfying ˆJ2|l,j,mj/angbracketright= j(j+1)/planckover2pi12|l,j,mj/angbracketrightandˆJz|l,j,mj/angbracketright=mj/planckover2pi1|l,j,mj/angbracketright(ˆJ= ˆL+ˆS). The spin-angular function can be explicitly writ- ten in terms of the orbital angular momentum eigen- states|l,ml/angbracketrightsatisfying ˆL2|l,ml/angbracketright=l(l+1)/planckover2pi12|l,ml/angbracketrightand ˆLz|l,ml/angbracketright=ml/planckover2pi1|l,ml/angbracketright(ml=−l,···,l) and the spin an- gular momentum eigenstates |↑/angbracketrightand|↓/angbracketright: forj=l+1/2, |l,j,mj/angbracketright=/radicalBigg l+mj+1 2 2l+1/vextendsingle/vextendsingle/vextendsingle/vextendsinglel,mj−1 2/angbracketrightbigg |↑/angbracketright +/radicalBigg l−mj+1 2 2l+1/vextendsingle/vextendsingle/vextendsingle/vextendsinglel,mj+1 2/angbracketrightbigg |↓/angbracketright,(4) and forj=|l−1/2|, |l,j,mj/angbracketright=/radicalBigg l−mj+1 2 2l+1/vextendsingle/vextendsingle/vextendsingle/vextendsinglel,mj−1 2/angbracketrightbigg |↑/angbracketright −/radicalBigg l+mj+1 2 2l+1/vextendsingle/vextendsingle/vextendsingle/vextendsinglel,mj+1 2/angbracketrightbigg |↓/angbracketright.(5) Using the fact that the spin-angular function is an eigenstate of ˆL·ˆS,ˆVSLcan be rewritten as the sum of the scalar-relativistic and SOC parts:13,15 ˆVSL=lmax/summationdisplay l=0|l/angbracketrightVSR l(r)/angbracketleftl|+lmax/summationdisplay l=1|l/angbracketrightVSO l(r)ˆL·ˆS/angbracketleftl|,(6) where|l/angbracketright/angbracketleftl|is the orbital angular momentum projector for a givenl, which is the sum of |l,ml/angbracketright/angbracketleftl,ml|overallml. In Eq. (6), the radial potentials of the scalar-relativistic and SOC parts of ˆVSLare given as VSR l(r) =l+1 2l+1Vl,l+1 2(r)+l 2l+1Vl,|l−1 2|(r)(7) and VSO l(r) =2 2l+1/bracketleftBig Vl,l+1 2(r)−Vl,|l−1 2|(r)/bracketrightBig ,(8) respectively. The scalar-relativistic potential, VSR l(r), includes the effects of the Darwin term and the mass- velocity term.16Hybertsen and Louie17considered the effects of the SOC potential, VSO l(r), on the spin-orbit splittings in the band structure of semiconductors within first-order perturbation theory and found good agree- ment with experiments. For computational efficiency, pseudopotentials of the fully-separable Kleinman-Bylander (KB) form18–20are commonly used instead of those of the semi-local form.The non-local part of a fully-relativistic KB pseudopo- tential can be written as ˆVKB=lmax/summationdisplay l=0l+1 2/summationdisplay j=|l−1 2|j/summationdisplay mj=−j|l,j,mj/angbracketright|βl,j/angbracketright/angbracketleftβl,j|/angbracketleftl,j,mj|, (9) where the radially non-local projector |βl,j/angbracketright/angbracketleftβl,j|is used instead of the radial potential Vl,j(r). Similarly to the case of the semi-local pseudopotential, a fully-relativistic KB pseudopotential can be rewritten as the sum of the scalar-relativistic and SOC parts: ˆVKB=lmax/summationdisplay l=0|l/angbracketrightˆVSR l/angbracketleftl|+lmax/summationdisplay l=1|l/angbracketrightˆVSO lˆL·ˆS/angbracketleftl|,(10) where the non-local potentials of the scalar-relativistic and SOC parts of ˆVKBare defined as ˆVSR l=l+1 2l+1/vextendsingle/vextendsingle/vextendsingleβl,l+1 2/angbracketrightBig/angbracketleftBig βl,l+1 2/vextendsingle/vextendsingle/vextendsingle +l 2l+1/vextendsingle/vextendsingle/vextendsingleβl,|l−1 2|/angbracketrightBig/angbracketleftBig βl,|l−1 2|/vextendsingle/vextendsingle/vextendsingle(11) and ˆVSO l=2 2l+1/parenleftBig/vextendsingle/vextendsingle/vextendsingleβl,l+1 2/angbracketrightBig/angbracketleftBig βl,l+1 2/vextendsingle/vextendsingle/vextendsingle −/vextendsingle/vextendsingle/vextendsingleβl,|l−1 2|/angbracketrightBig/angbracketleftBig βl,|l−1 2|/vextendsingle/vextendsingle/vextendsingle/parenrightBig ,(12) respectively. The fully-relativistic velocity operatorthat includes all thenon-localeffectsofthefully-relativisticKBpseudopo- tential is written as ˆv(FR)=ˆv(p)+i /planckover2pi1/bracketleftBig ˆVKB,ˆr/bracketrightBig , (13) whereˆv(p)(=ˆp/m) is introduced for notational con- venience. The commutator on the right-hand side of Eq.(13)canbeseparatedintoscalar-relativisticandSOC parts. By using the expressions in Eqs. (10), (11), and (12), we define the scalar-relativistic velocity operator that includes only the effects arising from the scalar- relativistic part of ˆVKB: ˆv(SR)=ˆv(p)+i /planckover2pi1lmax/summationdisplay l=0/bracketleftBig |l/angbracketrightˆVSR l/angbracketleftl|,ˆr/bracketrightBig .(14) Within this formalism, the non-local effects of SOC on the velocity operator arise from the difference between ˆv(FR)andˆv(SR)which can be written as the sum of the commutators arising from ˆV(SO) l: ˆv(SO)≡ˆv(FR)−ˆv(SR)=lmax/summationdisplay l=1ˆv(SO) l(15) where ˆv(SO) l=i//planckover2pi1/bracketleftBig |l/angbracketrightˆVSO lˆL·ˆS/angbracketleftl|,ˆr/bracketrightBig .(16)3 The optical matrix elements of our interest are /angbracketleftf|e·ˆv(FR/SR/p)|i/angbracketright, where|i/angbracketrightand|f/angbracketrightare the initial and final electronic states, respectively, and eis the polariza- tion vector of the incident light. We investigate the dif- ference between the matrix elements of ˆv(FR)andˆv(SR) in several systems having heavy elements such as W and Bi. In the case of an isolated atom, the initial and final states are the eigenstates of angular momentum opera- tors|n,l,j,m j/angbracketrightwherenis the principal quantum num- ber. In periodic systems, the initial and final states are the Bloch states in the valence band |v,k/angbracketrightand those in the conduction band |c,k/angbracketright, respectively, where kis the crystal momentum, and vandcare the indices of the valence and conduction bands, respectively. The imaginary part of the dielectric function is calcu- lated within the independent-particle random-phase ap- proximation: Imε(p/SR/FR)(ω) =4π ω2ΩNk/summationdisplay k/summationdisplay c,v/vextendsingle/vextendsingle/vextendsingle/angbracketleftc,k|e·ˆv(p/SR/FR)|v,k/angbracketright/vextendsingle/vextendsingle/vextendsingle2 ×δ(Ec,k−Ev,k−/planckover2pi1ω),(17) whereωis the frequency of the incident light, Ω is the volume of the unit cell, Nkis the number of kpoints in the Brillouinzone, and Ev,kandEc,karethe Kohn-Sham energy eigenvalues of |v,k/angbracketrightand|c,k/angbracketright, respectively. From Eq. (17), we can see that SOC affects the ab- soprtion spectra of materials in two different ways: ( i) SOC changes the Kohn-Sham energy eigenvalues, Ev,k andEc,k, and eigenstates, |c,k/angbracketrightand|v,k/angbracketright, and (ii) SOC gives an additional contribution, ˆv(SO)in Eq. (15), to the (fully-relativistic) velocity operator. The focus of our work is on the second contribution. In this work, we performed fully-relativistic DFT cal- culations within the generalized gradient approxima- tion21using the Quantum ESPRESSO package.22,23The optical matrix elements and the imaginary part of the dielectric function were calculated by using the Yambo code.24We modified the program so that we can con- structboththescalar-relativisticandfully-relativisticve- locity operators using the same set of fully-relativistic KB pseudopotentials. All the fully-relativistic KB pseu- dopotentials used in this work were generated by using the ONCVPSP code.25The generating parameters for the pseudopotentials were taken from the work of Schlipf and Gygi,26while slight modifications were made to get the fully-relativistic pseudopotential of Bi. III. RESULTS AND DISCUSSION A. Isolated atomic systems We study the effects of SOC on the optical matrix el- ements of isolated W and Bi atoms which are heavy ele- ments and have an electronic structure strongly affectedTABLE I. Pseudopotentials used in this work. The pseudiza- tion radii of the pseudo-wavefunctions with different orbit al angular momenta, rs,rp,rd, andrf, are shown in units of the Bohr radius. Core Valence rsrprdrf S [Ne] 3 s23p42.12 1.51 - - Se [Ar] 3 d104s24p42.60 2.71 3.33 - Mo [Ar] 3 d104s24p65s24d42.00 2.50 2.56 - W(1) [Kr] 4 d104f145s25p66s25d42.12 2.19 1.88 3.03 W(2) [Kr] 4 d104f145s25p66s25d42.02 1.93 1.84 2.73 Bi [Xe] 4 f145d106s26p33.19 3.15 3.00 3.34 by SOC. We calculated the squared optical matrix el- ements of the form |/angbracketleftf|e+·ˆv(p/SR/FR)|i/angbracketright|2, where the initial and final states are chosen to be the total angular momentum eigenstates, |n,l,j,m j/angbracketright, ande+is the polar- ization vector of the left-circularly polarized light propa- gating along the zdirection. By comparing the squared matrix elements of ˆv(FR)andˆv(SR)for a given pair of initial and final states, we calculated the effects of SOC on the individual optical matrix element. Figures 1(a) and 1(b) show the squared matrix ele- ments of ˆv(FR),ˆv(SR), andˆv(p). In the case of a W atom,thedifferencebetweenthesquaredmatrixelements ofˆv(SR)andˆv(p)is not very large. In the case of a Bi atom, however, the squared matrix elements of ˆv(SR) significantly differ from those of ˆv(p), especially for the 5d→6ptransitions. It is known that such non-local ef- fects arising from the scalar-relativistic part of ˆVKBcan be significant if there is a large difference between the l-orbital components of the scalar-relativistic potential, ˆVSR l.24On the other hand, the difference between the squared matrix elements of ˆv(FR)andˆv(SR)is relatively small for all the optical transitions. Figures 1(c) and 1(d) show the difference between the squared matrix elements of ˆv(FR)andˆv(SR). In the case of a W atom, the difference between the squared matrix elements of ˆv(FR)andˆv(SR)can be as large as 4.3 % of the squared matrix elements of ˆv(FR). This difference is even more significant for a Bi atom and can reach 14 % (in the case of 6 s→6ptransitions). Although the effects of SOC on the optical matrix elements strongly depend upon the charactersof the initial and final states, we find that the non-local effects of SOC on the optical matrix elements are not negligible in W and Bi atoms. The relatively large effects of SOC on the optical ma- trixelements in the caseofaBi atomcan be qualitatively understood by looking at the SOC potentials of the W and Bi pseudopotentials in the semi-local form [Eq. (8)]. Figure 2 shows VSO l(r) of the W and Bi pseudopotentials used in our calculations (their generating parameters are shown in Tab. I). Because VSO l(r) is defined as the dif- ference between Vl,l+1/2(r) andVl,|l−1/2|(r),VSO l(r) is localized within the pseudization radii of Vl,l+1/2(r) and Vl,|l−1/2|(r). Because the pseudization radii of the W pseudopotential are smaller than those of the Bi pseu- dopotential (see Tab. I), VSO l(r) of the W pseudopoten-4 −0.03−0.02−0.0100.010.020.03 j mj1.5 1.51.5 0.51.5 −0.51.5 −1.5(5p→5d) Initial state ( j,mj)j mj2.5 −0.52.5 −1.52.5 −2.50.5 −0.5(5d→6p) (6s→6p) Initial state ( j,mj)|/angbracketlefte+·ˆv(p/SR/FR)/angbracketright|2(a.u.) |/angbracketlefte+·ˆv(FR)/angbracketright|2−|/angbracketlefte+·ˆv(SR)/angbracketright|2(a.u.)00.20.40.60.811.2ˆv(p) ˆv(SR) ˆv(FR) (c) W (d) Bi(b) Bi (a) W FIG. 1. (a) and (b) The squared optical matrix elements of W and Bi atoms obtained by using the fully-relativistic velo c- ity operator, the scalar-relativistic velocity operator, and the momentum operator for the optical transitions between the total angular momentum eigenstates. Only the cases of the largest four squared matrix elements of the fully-relativi stic velocity operator among 5 p→5dtransitions in a W atom and those among 5 d→6pand 6s→6ptransitions in a Bi atom are shown. (c) and(d)Thedifference betweenthe squaredop- tical matrix elements obtained by using the fully-relativi stic velocity operator and the scalar-relativistic velocity op erator. The incident light is left-circularly polarized. The matri x el- ements are in Hartree. −0.500.511.522.53 0 1 2 3VSO l(r) (Ry) r(bohr)l= 1 l= 2 l= 3 0 1 2 3 r(bohr)(a) W (b) Bi FIG. 2. The spin-orbit coupling potential [Eq. (8)] of the fully-relativistic pseudopotentialsofW[W(2)pseudopot ential in Tab. I] and Bi atoms.tial are more localized than those of the Bi pseudopoten- tial. We note that for both atoms the p-orbital part of the SOC potential, VSO l=1(r), is much larger than the d- andf-orbital parts, VSO l=2(r) andVSO l=3(r). The matrix elements of ˆv(SO) l[Eq. (16)] can be explic- itly written as /angbracketleftf|ˆv(SO) l|i/angbracketright=i /planckover2pi1/summationdisplay σ,σ′/integraldisplay drVSO l(r)× ψ∗ f(r,σ)/bracketleftBig ˆL·ˆSσσ′|l/angbracketright/angbracketleftl|,r/bracketrightBig ψi(r,σ′),(18) whereσandσ′are the spin indices, ψi(r,σ) andψf(r,σ) are the (r,σ) component of |i/angbracketrightand|f/angbracketright. Because Eq. (18) contains the volume integration of VSO l(r), not only the value ofVSO l(r) near the core region ( r∼0) but also the spatial extent of VSO l(r) is an important factor that affects the magnitude of /angbracketleftf|ˆv(SO) l|i/angbracketright. To sketch the influence of VSO l(r) on the matrix ele- ments of ˆv(SO) l, we evaluate the volume integration of |VSO l(r)|for the atoms in the periodic table from H (Z= 1) to Bi ( Z= 83) except those in the Lanthanide series (Fig. 3). Roughly speaking, the volume integra- tion of|VSO l(r)|increases with the atomic number Z, or with the atomic mass. For example, the volume integra- tion of|VSO l=1(r)|of the Bi (Z= 83) pseudopotential is 0.78 Hartree, while the same quantity of the W ( Z= 74) pseudopotential is 0.3. The result is consistent with our observation in Fig. 1 that the effects of SOC on the opti- cal matrix elements increase with the atomic mass (4.3 % and 14 % of the squared matrix elements of ˆv(FR)for W and Bi atoms, respectively). For most atoms in Fig. 3, we find that the p-orbital component of the SOC potential, |VSO l=1(r)|, is the largest, thed-orbital component, |VSO l=2(r)|, is the second largest, and thef-orbital compoenent, |VSO l=3(r)|, is the small- est. Therefore, it is natural to expect that the contribu- tion ofˆv(SO) l=1to the optical matrix elements is the most important one and that the contribution of ˆv(SO) lbe- comes smaller as lincreases. To investigate the contri- bution of ˆv(SO) lto the squared matrix elements of ˆv(FR), we calculated the squared matrix elements of ˆv(FR)and ˆv(FR)−ˆv(SO) l. Figure 4 shows the difference between the squared ma- trix elements of ˆv(FR)andˆv(FR)−ˆv(SO) lin the casesofW and Bi atoms. In both cases, the contribution of ˆv(SO) l=1to thesquaredmatrixelementsof ˆv(FR)isthelargestamong the contributions of ˆv(SO) l. The contribution of ˆv(SO) l=2is the second largest and that of ˆv(SO) l=3is the smallest and negligible. In the case of a W atom, there is a case where the contribution of ˆv(SO) l=2is almost half of the contribu- tion ofˆv(SO) l=1. In the case of a Bi atom, the contributions ofˆv(SO) l=2andˆv(SO) l=3are very small. Because ˆv(SO) lcontains the orbital angular momentum projector, |l/angbracketright/angbracketleftl|, the matrix elements of ˆv(SO) lare finite5 00.020.040.060.080.10.12 H He Li Be B C N O F NeNaMgAl Si P S Cl Ar K CaSc Ti V CrMnFeCoNiCuZnG aGeAs Se Br1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 atom:Z: l= 1 l= 2 l= 3 00.020.040.060.080.10.12 H He Li Be B C N O F NeNaMgAl Si P S Cl Ar K CaSc Ti V CrMnFeCoNiCuZnG aGeAs Se Br1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 atom:Z: l= 1 l= 2 l= 300.020.040.060.080.10.12 H He Li Be B C N O F NeNaMgAl Si P S Cl Ar K CaSc Ti V CrMnFeCoNiCuZnG aGeAs Se Br1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 atom:Z: l= 1 l= 2 l= 3 00.10.20.30.40.50.60.70.80.9 KrRbSr Y ZrNbMoTcRuRhPdAgCd In Sn Sb Te I XeCsBaLa Hf Ta W ReOs Ir PtAuHgTl Pb Bi36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 72 73 74 7 5 76 77 78 79 80 81 82 83 atom:Z: 00.10.20.30.40.50.60.70.80.9 KrRbSr Y ZrNbMoTcRuRhPdAgCd In Sn Sb Te I XeCsBaLa Hf Ta W ReOs Ir PtAuHgTl Pb Bi36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 72 73 74 7 5 76 77 78 79 80 81 82 83 atom:Z:00.10.20.30.40.50.60.70.80.9 KrRbSr Y ZrNbMoTcRuRhPdAgCd In Sn Sb Te I XeCsBaLa Hf Ta W ReOs Ir PtAuHgTl Pb Bi36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 72 73 74 7 5 76 77 78 79 80 81 82 83 atom:Z:/integraldisplay/vextendsingle/vextendsingleVSO l(r)/vextendsingle/vextendsingler2dr(a.u.) FIG. 3. The volume integration of |VSO l(r)|[see Eq. (8)] of the fully-relativistic pseudopotentials o f the atoms from H ( Z= 1) to Bi (Z= 83) except those in the Lanthanide series. j mj2.5 −0.52.5 −1.52.5 −2.50.5 −0.5(5d→6p) (6s→6p) Initial state ( j,mj)−0.06−0.04−0.0200.020.04 (5p→5d) j mj1.5 1.51.5 0.51.5 −0.51.5 −1.5 Initial state ( j,mj)l= 1 l= 2 l= 3|/angbracketlefte+·ˆv(FR)/angbracketright|2−|/angbracketlefte+·(ˆv(FR)−ˆv(SO) l)/angbracketright|2(a.u.) (a) W (b) Bi FIG. 4. The difference between the squared matrix elements of W and Bi atoms obtained by using the fully-relativistic velocity operator and ˆv(FR)−ˆv(SO) l[see Eqs. (13) and (16)]. The initial and final states are the same as those in Fig. 1. only if the initial state or the final state has l-orbital an- gularmomentumcharacter. Inaddition, accordingtothe opticalselection rule, the matrix elements /angbracketleftf|ˆv(SO) l|i/angbracketrightare finite if the difference between the l’s of|i/angbracketrightand|f/angbracketrightis±1. Therefore, the matrix elements of ˆv(SO) l=3are finite onlyfor the transitions between the states that have d- and f-orbital angular momentum characters. In our calcu- lations, the contribution of ˆv(SO) l=3to the squared matrix elements of ˆv(FR)is usually very small for such transi- tions. It is known that the effects of SOC on the energy levels and wavefunctions of atomic systems are the largest for p-orbitals and become smaller for d- andf-orbitals.27In fact, if we recall the fine structure of a hydrogen atom, we easily see that the energy splittings induced by SOC show the same l-dependent behavior ( p > d > f ).28,29 Our results are in line with these relativistic effects on atomic systems. B. Monolayer transition metal dichalcogenides We calculate the optical matrix elements and absorp- tion spectra of a monolayer of four 2H-type semicon- ducting TMDCs (MoS 2, MoSe 2, WS2, and WSe 2). Fig- ure 5(a) shows the structure of the two-dimensional crys- tal. The TMDC monolayer of 2H-type consists of a tran- sition metal layer (Mo or W) which is sandwiched by two chalcogen layers (S or Se). Figure 5(b) shows the two-dimensional Brillouin zone. In our DFT calculations using fully-relativistic pseu- dopotentials,wesetthekineticenergycutoffoftheplane-6 a1M Γb2 b1 −M(M′)−K(K′)a2KMo, W S, Se(a) (b) −2−10123 Γ /C3 /C5Γ/BX/D2/CT/D6/CV/DD /B4/CT /CE/B5 Γ /C3 /C5ΓΓ /C3 /C5ΓΓ /C3 /C5Γ(c) MoS 2(d) MoSe 2(e) WS 2(f) WSe 2 FIG. 5. (a) and (b) The crystal structure and the Bril- louin zone of a monoayer of 2H-type semiconducting tran- sition metal dichalcogenides, respectively. (c)-(f) The e lec- tronic band structure of a monolayer of four 2H-type transi- tion metal dichalcogenides. wave basis to 80 Ry and sample the Brilluoin zone with a 12×12×1Monkhorst-Pack grid.30In calculations of ab- sorptionspectra,weuseadenser120 ×120×1Monkhorst- Pack grid for k-point summations. Figures 5(c)-(f) show the electronic band structure of TMDC monolayers. At the valence-bandmaximumat K, we see energysplittings between the two highest bands in valence thanks to the large SOC of 4d and 5d transition metals. The magnitude of the SOC-induced energy split- tings varies from 153 to 468 meV (larger for the TMDCs having W atoms). All the results of our calculations are consistent with previous theoretical studies.31,32 To investigate the effects of SOC on the absorption spectra of monolayer TMDCs, we calculated the imagi- nary part of the independent-particle dielectric function usingˆv(FR),ˆv(SR), andˆv(p): Imε(FR)(ω), Imε(SR)(ω), and Imε(p)(ω). Figures 6(a)-(d) show Im ε(FR)(ω) of monolayer TMDCs. The onset energies of Im ε(FR)(ω) correspond to the band gaps of monolayer TMDCs. In the low energy regime ( /planckover2pi1ω<2 eV), we see step-function- like behaviors of Im ε(FR)(ω) which mainly result from the optical transitions between the band-edge states at K and K′, i.e. the two highest-energy valence and the two lowest-energy conduction bands. These band-edge states mostly consist of the dorbitals of transition metal atoms and the porbitals of chalcogen atoms. In the high energy regime ( /planckover2pi1ω >2 eV), additional sharp features arise as the states other than the band-edge states make contributions to Im ε(FR)(ω). Figures 6(e)-(h) show the difference between Imε(FR)(ω) and Im ε(SR)(ω). In the low energy regime (/planckover2pi1ω <2 eV), the difference between Im ε(FR)(ω) and Imε(SR)(ω) is three orders of magnitudes smallerthan Imε(FR)(ω) itself, which indicates that the non- local effects of SOC on Im ε(FR)(ω) are negligibly small in this range of energy. The effects of SOC become larger at higher energies ( /planckover2pi1ω >2 eV). However, the difference between Im ε(FR)(ω) and Imε(SR)(ω) remains smaller than 1 % of Im ε(FR)(ω). In general, the non-local part of a pseudopotential strongly depends on its generating parameters such as valence(and core)configurations,pseudization radii, and the local part of the pseudopotential. Therefore, the ef- fects of the commutator on the optical matrix elements and absorption spectra of a material (including the con- tributions from the scalar-relativistic and SOC parts of the pseudopotential) can changesignificantly by the non- local character of the pseudopotential used in the calcu- lations. In Fig. 7, we plot the imaginary part of the dielectric function ofmonolayerWSe 2obtained byusing twodiffer- ent fully-relativistic KB pseudopotentials of W (for com- parison, we fixed the pseudopotential of Se). We checked that the two different pseudopotentials of W yield al- most the same band structure within the energy range of our interest ( /planckover2pi1ω= 0–4 eV). The absorption spectra in Figs. 7(a) and 7(c) were obtained by using a W pseu- dopotential that includes 4 felectrons in the valence [see W(1) in Tab. I], while those in Fig. 7(b) and 7(d) were obtained by using a W pseudopotential that includes 4 f electrons in the core [see W(2) in Tab. I]. By comparing Im ε(p)(ω) in Figs. 7(a) and 7(b), we find that the absorption spectrum strongly depends on the pseudopotential of W, if we neglect all the effects arising from the non-local part of the pseudopotential and useˆv(p)as the velocity operator. In the case of the W(1) pseudopotential, Im ε(p)(ω) is much smaller than Imε(SR)(ω). The result shows that the commutator aris- ing from the scalar-relativistic part of the W(1) pseu- dopotential affects strongly the absorption spectrum. The large difference between Im ε(SR)(ω) and Imε(p)(ω) isattributedtothepresenceof4 felectronsinthe valence which makes the W(1) pseudopotential strongly non- local. On the other hand, in the case of the W(2) pseu- dopotential, Im ε(p)(ω) is quite similar to Im ε(SR)(ω). The effects of the commutator arising from the scalar- relativistic part of the pseudopotential are much smaller for the W(2) pseudopotential where 4 felectrons are treated as core electrons. We note that in both cases, Imε(SR)(ω) is almost identical to Im ε(FR)(ω). Figures 7(c) and 7(d) show the difference between Imε(FR)(ω) and Imε(SR)(ω). We find that the contri- butions of SOC to Im ε(FR)(ω) for the two different pseu- dopotentials of W are very similar to each other in the whole energy range. The result shows that the effects of the commutator arising from SOC on the absorption spectra do not depend much on the pseudopotential. It is possible that even if the non-local effects of SOC on the individual optical matrix element are large, the effects on the absorption spectrum are small as we sum over the contributions from many optical matrix ele-7 05101520Imε(FR) 0 1 2 3/BX/D2/CT/D6/CV/DD /B4/CT /CE/B50 1 2 3/BX/D2/CT/D6/CV/DD /B4/CT /CE/B50 1 2 3/BX/D2/CT/D6/CV/DD /B4/CT /CE/B5−0.04−0.0200.020.04 0 1 2 3Imε(FR)−Imε(SR)/BX/D2/CT/D6/CV/DD /B4/CT /CE/B5(a) MoS 2 (e) MoS 2(b) MoSe 2 (f) MoSe 2(c) WS 2 (d) WSe 2 (h) WSe 2 (g) WS 2 FIG. 6. (a)-(d) The imaginary part of the dielectric functio n of monolayer transition metal dichalcogenides obtained b y using the fully-relativistic velocity operator. (e)-(h) T he difference between the imaginary part of the dielectric fu nctions of monolayer transition metal dichalcogenides obtained by us ing the fully-relativistic velocity operator and the scala r-relativistic velocity operator. 0 1 2 3 Energy (eV)05101520Imε(p/SR/FR)Imε(FR) Imε(SR) Imε(p) −0.04−0.0200.020.04 0 1 2 3Imε(FR)−Imε(SR) Energy (eV)W(1) pseudopotential W(2) pseudopotential (a) (b) (d) (c) FIG. 7. (a) and (b) The imaginary part of the dielectric function of monolayer WSe 2obtained by using two different pseudopotentials of W, which were generated from two differ- ent valence configurations (see Tab. I for details). Im ε(p)(ω), Imε(SR)(ω), and Im ε(FR)(ω) are the imaginary part of the dielectric function obtained by using the momentum oper- ator, the scalar-relativistic velocity operator, and the f ully- relativistic velocity operator, respectively. (c) and (d) The difference between the imaginary part of the dielectric func - tion obtained by using the fully-relativistic and the scala r- relativistic velocity operators.ments with different momenta and band indices. To check this possibility, we calculated the squared ma- trix elements |/angbracketleftci,k|e+·ˆv(SR/FR)|vj,k/angbracketright|2, whereiand jare 1 or 2, v1andv2are the band indices of the highest-energy and second-highest-energy states in the valence band, respectively, c1andc2are the band indices of the lowest-energy and second-lowest-energy states in the conduction band, respectively, and kis on the path −M→ −K→Γ→K→M [Fig. 5(b)]. Figures 8(a)-(d) show the squared matrix elements of ˆv(FR). Here, we see that the squared matrix elements ofˆv(FR)near K are larger in magnitude than those near−K. Because we assumed the incident light to be left-circularly polarized, the result can be explained by the valley-selective circular dichroism of monolayer TMDCs.33 Figures 8(e)-(h) show the difference between the squared matrix elements of ˆv(FR)andˆv(SR). Although the contribution of the commutator arising from SOC to the squared matrix elements of ˆv(FR)becomes larger in the case of having heavier transition metal atoms (WS 2 and WSe 2), even in those cases the contribution from SOC remains smaller than 1 % of the squared matrix elements of ˆv(FR). If we compare this result with the previous result of an isolated W atom, the influence of the commutator arising from SOC on the optical matrix elements is much suppressed: In the case of a W atom, the effectsofSOC onthesquaredopticalmatrixelements can be as large as 4.3 % of the squared matrix elements ofˆv(FR)[Figs. 1(a) and 1(c)]. Next, we further investigate the dependence of the squared matrix elements of ˆv(FR),ˆv(SR), andˆv(p)on8 00.050.10.150.2|/angbracketlefte+·ˆv(FR)/angbracketright|2(a.u.)v1−c1 v1−c2 v2−c1 v2−c2 −0.0015−0.001−0.000500.00050.0010.00150.002 −M−K Γ K M|/angbracketlefte+·ˆv(FR)/angbracketright|2−|/angbracketlefte+·ˆv(SR)/angbracketright|2(a.u.) −M−K Γ K M −M−K Γ K M −M−K Γ K M(a) MoS 2 (b) MoSe 2 (c) WS 2 (d) WSe 2 (g) WS 2 (f) MoSe 2 (e) MoS 2 (h) WSe 2 FIG. 8. (a)-(d) The squared optical matrix elements obtaine d by using the fully-relativistic velocity operator of mono layer transition metal dichalcogenides for the optical transiti ons involving the highest-energy and second-highest-ener gy states in the valence band, v1andv2, respectively, and the lowest-energy and second-lowest-e nergy states in the conduction band, c1andc2, respectively. The squared optical matrix elements were cal culated along the path in the momentum space, −M→ −K→Γ→ K→M. (e)-(h) The difference between the squared optical matrix elements obtained by using the fully-relativistic velocit y operator and those obtained by using the scalar-relativist ic velocity operator. In all cases, the incident light is lef t-circularly polarized. the initial and final states in the case of monolayer WSe2. We calculated the squared matrix elements at K,|/angbracketleftc,K|e+·ˆv(p/SR/FR)|v,K/angbracketright|2, wherevandcare the band indices of the initial and final states, respectively. The band indices are in increasing order of energy (the state at the valence band maximum is v= 26). We consider three initial states having different orbital char- acters: (i)|3,K/angbracketrightwhich mostly consists of the 5 porbitals of W atoms, ( ii)|18,K/angbracketrightwhich consists of the 5 pand 4d orbitals of W atoms and the 3 porbitals of Se atoms, and (iii)|26,K/angbracketrightwhich consists of the 4 dorbitals of W atoms and the 3porbitals of Se atoms. For each initial state, we considerall the final states satisfying Ec,K−Ev,K<1 Ry. Figures 9(a)-(c) show the squared matrix elements of ˆv(FR),ˆv(SR), andˆv(p)and Figs. 9(d)-(f) show the differ- ence between the squared matrix elements of ˆv(FR)and ˆv(SR). By comparing the results of Figs. 9(d), 9(e), and 9(f), we find that the effects of SOC on the optical ma- trix elements are larger for the optical transitions whose initial state is more localized at W atoms. In the case of|3,K/angbracketright, the effects of SOC on the squared matrix ele- ments of ˆv(FR)can be as large as 6.8 %, while in the case of|26,K/angbracketright, the effects are much smaller, less than 1.1 %. The result of |18,K/angbracketrightfalls somewhere between the resultsof|3,K/angbracketrightand|26,K/angbracketright. Figure 10 show the differences between the squared matrixelementsof ˆv(FR)andˆv(FR)−ˆv(SO) lforthe optical transitionshaving |3,K/angbracketright,|18,K/angbracketright, and|26,K/angbracketrightastheinitial states. In the case of |3,K/angbracketright, we find that among ˆv(SO) l’s ˆv(SO) l=1givesthelargestcontributiontothesquaredmatrix elements of ˆv(FR). Thed-orbital part ˆv(SO) l=2gives the secondlargestcontributionandthe contributionfromthe f-orbital part ˆv(SO) l=3is negligible. This result is similar to the result of an isolated W atom [Fig. 4(a)]. Figures 10(b) and 10(c) show that ˆv(SO) l=1is relatively less important in the cases of |18,K/angbracketrightand|26,K/angbracketrightthan in the case of |3,K/angbracketright. The result can be qualitatively under- stood by looking at the W p- andd-orbital characters of the initial states. As we move from |3,K/angbracketrightto|18,K/angbracketrightand |26,K/angbracketright, the proportion of the W 5 p-orbital component in the initial state decreases while that of the W 4 d-orbital component increases. In the case of |26,K/angbracketright, because the initial state mostly consists of the W 4 dorbitals, the ma- trix elements of ˆv(SO) l=1are finite only for the final states having W 6 p-orbital character (∆ l=±1). Such final states are much more delocalized than the initial state and the matrix elements of ˆv(SO) l=1are small.9 00.050.10.150.20.25|/angbracketlefte+·ˆv(p/SR/FR)/angbracketright|2(a.u.)ˆv(p) ˆv(SR) ˆv(FR) 27 36465868808698 111120Final state band index c−0.00200.0020.0040.006 34384459767792 104134144|/angbracketlefte+·ˆv(FR)/angbracketright|2−|/angbracketlefte+·ˆv(SR)/angbracketright|2(a.u.) Final state band index c 28 354556677997 112127155Final state band index c(b)v= 18 (c)v= 26 (f)v= 26 (e)v= 18 (d)v= 3(a)v= 3 FIG. 9. (a)-(c) The squared optical matrix elements of monol ayer WSe 2obtained by using the fully-relativistic velocity operator, the scalar-relativistic operator, and the momen tum operator for the optical transitions at K in the momentum space. (d)-(f) The difference between the squared optical matrix el ements obtained by using the fully-relativistic velocity o perator and the scalar-relativistic operator. In all cases, left-c ircularly polarized light was considered. −0.006−0.004−0.00200.0020.0040.0060.0080.01 34384459767792 104134144Final state band index cl= 1 l= 2 l= 3 27 36465868808698 111120Final state band index c 28 354556677997 112127155Final state band index c|/angbracketlefte+·ˆv(FR)/angbracketright|2−|/angbracketlefte+·(ˆv(FR)−ˆv(SO) l)/angbracketright|2(a.u.) (c)v= 26 (b)v= 18 (a)v= 3 FIG. 10. (a)-(c) The difference between the squared matrix el ements of monolayer WSe 2obtained by using the fully-relativistic velocity operator and ˆv(FR)−ˆv(SO) l[see Eqs. (13) and (15)] for the optical transitions at K in th e momentum space. In all cases, left-circularly polarized light was considered.10 Γ /C3−1−0.8−0.6−0.4−0.200.20.40.60.8 Γ /C3/BX/D2/CT/D6/CV/DD /B4/CT /CE/B5 Kb1(a) Bi2Se35QL b2 Γ(b) Bi2Te35QL (c) FIG. 11. (a) and (b) The electronic band structure of 5- quintuple-layer slabs of Bi 2Se3and Bi 2Te3. (c) The Brillouin zone of 5-quintuple-layer slabs of Bi 2Se3and Bi 2Te3. C. Bi 2Se3and Bi 2Te3 We investigatethe effects ofSOC on the optical matrix elements of 5-quintuple-layer slabs of Bi 2Se3and Bi 2Te3. Here, we focus on the optical transitions whose initial states are the topological surface states. In DFT calcula- tions, we slightly broke the inversion symmetry to induce small energy splittings between the surface states local- ized at the top and bottom sides of Bi 2Se3and Bi 2Te3 slabs. In this way, we can obtain the surface states |v,k/angbracketright (vis the band index and v= 241 and 391 for Bi 2Se3 andBi 2Te3, respectively)whicharelocalizedwelloneach surface of the slabs. In the calculation of the optical ma- trix elements, we chose the surface state with momentum k′= 0.05ΓK as our initial state [blue dots in Figs. 11(a) and 11(b)]. Also here, we consider all the final states that satisfy Ec,k′−Ev,k′<1 Ry. We set the kinetic en- ergy cutoff of the plane-wave basis to 80 Ry and use a 6×6×1 Monkhorst-Pack grid for k-point sampling. Figures 12(a) and (b) show the squared matrix ele- ments of ˆv(FR),ˆv(SR), andˆv(p). We find that the dif- ference between the squared matrix elements of ˆv(FR) andˆv(SR)is very small, while the difference between the squared matrix elements of ˆv(SR)andˆv(p)is large in some cases of Bi 2Te3. Figures 12(c) and (d) show the differences between the squared matrix elements of ˆv(FR)andˆv(SR)in a different scale. As in the case of thetransitionsfromthevalence-bandmaximumofmono- layer WSe 2[Figs. 9(c) and (f)], the effects of SOC on the optical matrix elements of Bi 2Se3and Bi 2Te3slabs are very small (less than 1 % of the squared matrix elements ofˆv(FR)). Figure 13 shows the difference between the squared matrix elements of ˆv(FR)andˆv(FR)−ˆv(SO) lfor the same optical transitions presented in Fig. 12. We find that the effects of the p-orbital part ˆv(SO) l=1on the optical matrix elements are usually the largest and the effects of the d- andf-orbital parts, ˆv(SO) l=2andˆv(SO) l=3, are much smaller. This is because ( i) thep-orbital component of the SOC00.0050.010.0150.020.025ˆv(p) ˆv(SR) ˆv(FR) −0.0005−0.0004−0.0003−0.0002−0.000100.00010.0002 285288 289296 297299 465 474475478Final state band index c|/angbracketlefte+·ˆv(FR)/angbracketright|2−|/angbracketlefte+·ˆv(SR)/angbracketright|2(a.u.)|/angbracketlefte+·ˆv(p/SR/FR)/angbracketright|2(a.u.) 397404 405409 414458 460 478480628Final state band index c(c)v= 241 (d) v= 391(b)v= 391Bi2Se35QL Bi 2Te35QL (a)v= 241 FIG. 12. (a) and (b) The squared optical matrix ele- ments of 5-quintuple-layer slabs of Bi 2Se3and Bi 2Te3ob- tained by using the fully-relativistic velocity operator, the scalar-relativistic operator, and the momentum operator f or the optical transitions having the topological surface sta te with momentum k= 0.05ΓK as the initial state. (c) and (d) The difference between the squared optical matrix elements obtained by using the fully-relativistic velocity operato r and the scalar-relativistic operator. In all cases, left-circ ularly po- larized light was considered. potentialVSO l=1(r) of Bi is much larger than the d- and f-orbital components VSO l=2,3(r) (see Figs. 2 and 3) and (ii) in particular, the surface states of Bi 2Se3and Bi 2Te3 mostly consist of the 6 porbitals of Bi atoms. The circular dichroism is defined as the relative dif- ference between the squared optical matrix elements for left- and right-circularly-polarized light (see the top of Fig. 14). We calculated the circular dichroism by using ˆv(FR)andˆv(SR)and investigated whether the effects of the commutator arising from SOC change the circular dichroism of Bi 2Se3and Bi 2Te3slabs. In Fig. 14, we seethat the difference between the circu- lardichroisms obtained by using ˆv(FR)andˆv(SR)is negli- gible. Contraryto the argumentsin Ref. 12, the effects of the commutator arising from SOC cannot change the cir- cular dichroism of Bi 2Se3and Bi 2Te3slabs, which means that the methods used in Refs. 5, 6, and 11 will give cor- rect results. Although we did not find the correct final states (satisfying the proper boundary condition) in the calculations of the optical matrix elements, because the effects of SOC on the optical matrix elements are negligi-11397404 405409 414458 460 478480628Final state band index c|/angbracketlefte+·ˆv(FR)/angbracketright|2−|/angbracketlefte+·(ˆv(FR)−ˆv(SO) l)/angbracketright|2(a.u.) −0.0005−0.0004−0.0003−0.0002−0.000100.00010.0002 285288 289296 297299 465 474475478Final state band index cl= 1 l= 2 l= 3(b)v= 391Bi2Se35QL Bi 2Te35QL (a)v= 241 FIG.13. Thedifferencebetweenthesquaredmatrixelements of 5-quintuple-layer slabs of Bi 2Se3and Bi 2Te3obtained by using the fully-relativistic velocity operator and ˆv(FR)−ˆv(SO) l [see Eqs. (13) and (15)] for the optical transitions having t he topological surface state with momentum k= 0.05ΓK as the initial state. In all cases, left-circularly polarized lig ht was considered. −0.500.51 285288 289296 297299 465 474475478Final state band index cˆv(FR) ˆv(SR) 397404 405409 414458 460 478480628Final state band index cC.D.=|/angbracketlefte+·ˆv(FR/SR)/angbracketright|2−|/angbracketlefte−·ˆv(FR/SR)/angbracketright|2 |/angbracketlefte+·ˆv(FR/SR)/angbracketright|2+|/angbracketlefte−·ˆv(FR/SR)/angbracketright|2 (b)v= 391Bi2Te35QL Bi2Se35QL (a)v= 241 FIG. 14. The circular dichroism of 5-quintuple-layer slabs of Bi2Se3and Bi 2Te3for the optical transitions having the topological surface state with momentum k= 0.05ΓK as the initial state. ble over a wide range of energies ( Ec,k′−Ev,k′<1 Ry), it is likely that imposing the correct boundary condition on the final states will not make a significant difference between the results obtained by using ˆv(FR)andˆv(SR). IV. CONCLUSION In this study, we investigated the effects of spin-orbit coupling on the optical responses of isolated atoms,monolayer transition metal dichaocogenides, and the topological surface states of topological insulators using first-principles calculations with fully-relativistic pseu- dopotentials. By using a method that can separate a fully-relativistic Kleinman-Bylander pseudopotential into the scalar-relativistic and spin-orbit coupling parts, we wereableto study the effects of spin-orbitcouplingon the velocity operator and its matrix elements in various systems. In the case of W and Bi atoms, we find that the rel- ative contribution of the commutator arising from SOC to the squared optical matrix elements can be 4.3 % for W and 14 % for Bi. We find that the p-orbital part of the commutator arising from SOC gives the largest con- tribution to the optical matrix elements. The influence of thep-orbital part of the spin-orbit coupling potential is much larger than those of the d- andf-orbital parts. In the case of monolayer transition metal dichalco- genides, the effects of the commutator arising from spin- orbitcouplingaremuchsmallerthaninthecaseofatomic systems, less than 1 % of the squared optical matrix el- ements for the optical transitions from the valence band edge states. In the case of five-quintuple layer slabs of Bi 2Se3and Bi2Te3, the effects of spin-orbit coupling on the optical matrix elements are again very small as in the case of monolayertransitionmetaldichalcogenides. Wefindthat the non-local effects of spin-orbit coupling on the opti- cal matrix elements are so small that the effects do not changethe circulardichroism ofBi 2Se3and Bi 2Te3slabs. In conclusion, we confirm that while the effects of the commutator arising from spin-orbit coupling on the opti- cal matrix elements are not negligible in atomic systems, the effects are much suppressed in the cases of monolayer transition metal dichalcogenides and topological insula- tors where the effects of spin-orbit coupling on the elec- tronic structure are considered to be important. Our calculation results show that in studying the opti- cal response of a material with heavy elements, it is suffi- cient to calculate the optical matrix elements neglecting the commutator arising from spin-orbit coupling in the velocity operator if one has obtained well the electronic structure of the system, i.e. the energy eigenvalues and eigenstates, from fully-relativistic first-principles calcula- tions. T.Y.K. and C.-H.P. were supported by Korean NRF No-2016R1A1A1A05919979 and by the Creative- Pioneering Research Program through Seoul National University. AF acknowledges financial support from the EU Centre of Excellence “MaX - Materials Design at the Exascale”(Horizon2020EINFRA-5, GrantNo. 676598). Computational resourceswere provided by KISTI Super- computing Center (KSC-2018-C2-0002).12 ∗cheolhwan@snu.ac.kr 1C. Jozwiak, C.-H. Park, K. Gotlieb, C. Hwang, D.-H. Lee, S. G. Louie, J. D. Denlinger, C. R. Rotundu, R. J. Birge- neau, Z. Hussain, and A. Lanzara, “Photoelectron spin- flipping and texture manipulation in a topological insula- tor,” Nature Phys. 9, 293 (2013). 2C.-H.ParkandS.GLouie, “Spinpolarization ofphotoelec- trons from topological insulators,” Phys. Rev. Lett. 109, 097601 (2012). 3J. H. Ryooand C.-H. Park, “Spin-conserving and reversing photoemission from the surface states of Bi 2Se3and Au (111),” Phys. Rev. B 93, 085419 (2016). 4J. H. Ryoo and C.-H. Park, “Momentum-dependent spin selection rule in photoemission with glide symmetry,” arXiv:1807.02368. 5Z.-H. Zhu, C. N. Veenstra, G. Levy, A. Ubaldini, P. Syers, N. P. Butch, J. Paglione, M. W. Haverkort, I. S. Elfi- mov, and A. Damascelli, “Layer-by-layer entangled spin- orbital texture of the topological surface state in Bi 2Se3,” Phys. Rev. Lett. 110, 216401 (2013). 6Z.-H. Zhu, C. N. Veenstra, S. Zhdanovich, M. P. Schneider, T. Okuda, K. Miyamoto, S.-Y. Zhu, H. Na- matame, M. Taniguchi, M. W. Haverkort, I. S. Elfimov, and A. Damascelli, “Photoelectron spin- polarization control in the topological insulator Bi 2Se3,” Phys. Rev. Lett. 112, 076802 (2014). 7B. Adolph, V. I. Gavrilenko, K. Tenelsen, F. Bechst- edt, and R. Del Sole, “Nonlocality and many-body effects in the optical properties of semiconductors,” Phys. Rev. B 53, 9797 (1996). 8A. J. Read and R. J. Needs, “Calculation of opti- cal matrix elements with nonlocal pseudopotentials,” Phys. Rev. B 44, 13071 (1991). 9S. Baroni and R. Resta, “ Ab initio calculation of the macroscopic dielectric constant in silicon,” Phys. Rev. B 33, 7017 (1986). 10A. Marini, G. Onida, and R. Del Sole, “Plane-wave DFT- LDA calculation of the electronic structure and absorption spectrum of copper,” Phys. Rev. B 64, 195125 (2001). 11Y. Cao, J. A. Waugh, X.-W. Zhang, J.-W. Luo, Q. Wang, T. J. Reber, S. K. Mo, Z. Xu, A. Yang, J. Schneeloch, G. D. Gu, M. Brahlek, N. Bansal, S. Oh, A. Zunger, and D. S. Dessau, “Mapping the orbital wavefunction of the surface states in three-dimensional topological insulato rs,” Nature Phys. 9, 499 (2013). 12C.-Z. Xu, Y. Liu, R. Yukawa, L.-X. Zhang, I. Mat- suda, T. Miller, and T.-C. Chiang, “Photoemis- sion circular dichroism and spin polarization of the topological surface states in ultrathin Bi 2Te3films,” Phys. Rev. Lett. 115, 016801 (2015). 13L. Kleinman, “Relativistic norm-conserving pseudopoten- tial,” Phys. Rev. B 21, 2630 (1980). 14J. J. Sakurai and J. Napolitano, Modern quantum mechan- ics(Addison-Wesley, 2011). 15G. B. Bachelet, D. R. Hamann, and M. Schl¨ uter, “Pseudopotentials that work: From H to Pu,” Phys. Rev. B 26, 4199 (1982). 16L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968). 17M. S. Hybertsen and S. G. Louie, “Spin-orbit splitting in semiconductorsandinsulators fromthe ab initio pseudopo-tential,” Phys. Rev. B 34, 2920 (1986). 18L. Kleinman and D. M. Bylander, “Effi- cacious form for model pseudopotentials,” Phys. Rev. Lett. 48, 1425 (1982). 19L. A. Hemstreet, C. Y. Fong, and J. S. Nel- son, “First-principles calculations of spin-orbit splitt ings in solids using nonlocal separable pseudopotentials,” Phys. Rev. B 47, 4238 (1993). 20G. Theurich and N. A. Hill, “Self-consistent treat- ment of spin-orbit coupling in solids using rela- tivistic fully separable ab initio pseudopotentials,” Phys. Rev. B 64, 073106 (2001). 21J. P. Perdew, K. Burke, and M. Ernzerhof, “Gen- eralized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865 (1996). 22P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- cioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fab- ris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougous- sis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentz- covitch, “QUANTUM ESPRESSO: a modular and open- source software project for quantum simulations of mate- rials,” J. Phys.: Condens. Matter 21, 395502 (2009). 23P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavaz- zoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carn- imeo, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio Jr., A. Ferretti, A. Floris, G. Fratesi, G. Fu- gallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. K¨ u¸ c¨ ukbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. Otero de-la Roza, L. Paulatto, S. Ponc´ e, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thon- hauser, P. Umari, N. Vast, X. Wu, and S. Baroni, “Ad- vanced capabilities for materials modelling with Quan- tum ESPRESSO,” J. Phys.: Condens. Matter 29, 465901 (2017). 24A. Marini, C. Hogan, M. Gr¨ uning, and Daniele Varsano, “yambo: An ab initio tool for excited state calculations,” Comput. Phys. Commun. 180, 1392 (2009). 25D. R. Hamann, “Optimized norm-conserving Vanderbilt pseudopotentials,” Phys. Rev. B 88, 085117 (2013). 26M. Schlipf and Gygi F., “Optimization algorithm for the generation of ONCV pseudopotentials,” Comput. Phys. Commun. 196, 36 (2015). 27V. M. Burke and I. P. Grant, “The effect of relativity on atomic wave functions,” Proc. Phys. Soc. 90, 297 (1967). 28D. J. Griffiths, Introduction to quantum mechanics (Pear- son Prentice Hall, Upper Saddle River, NJ, 2005). 29See, for example, Fig. 6.9 of Ref. 28. 30H. J. Monkhorst and J. D. Pack, “Spe- cial points for Brillouin-zone integrations,” Phys. Rev. B 13, 5188 (1976). 31Z. Y. Zhu, Y. C. Cheng, and U. Schwingen- schl¨ ogl, “Giant spin-orbit-induced spin splitting in two - dimensional transition-metal dichalcogenide semiconduc - tors,” Phys. Rev. B 84, 153402 (2011).13 32M. Gibertini, F. M. D. Pellegrino, N. Marzari, and M. Polini, “Spin-resolved optical conductivity of two- dimensional group-VIB transition-metal dichalcogenides ,” Phys. Rev. B 90, 245411 (2014).33T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, “Valley-selective circular dichroism of monolayer molybdenum disulphide,” Nature Commun. 3, 887 (2012).

1205.3483v1.Spin_Injection_Spectroscopy_of_a_Spin_Orbit_Coupled_Fermi_Gas.pdf

Spin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi Gas Lawrence W. Cheuk,1Ariel T. Sommer,1Zoran Hadzibabic,1, 2 Tarik Yefsah,1Waseem S. Bakr,1and Martin W. Zwierlein1 1Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA 2Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom The coupling of the spin of electrons to their motional state lies at the heart of recently discovered topological phases of matter [1{3]. Here we create and detect spin-orbit coupling in an atomic Fermi gas, a highly controllable form of quantum degenerate matter [4, 5]. We reveal the spin-orbit gap [6] via spin-injection spectroscopy, which characterizes the energy-momentum dispersion and spin composition of the quantum states. For energies within the spin-orbit gap, the system acts as a spin diode. To fully inhibit transport, we open an additional spin gap, thereby creating a spin-orbit coupled lattice [7] whose spinful band structure we probe. In the presence of s-wave interactions, such systems should display induced p-wave pairing [8], topological super uidity [9], and Majorana edge states [10]. Spin-orbit coupling is responsible for a variety of phe- nomena, from the ne structure of atomic spectra to the spin Hall eect, topological edge states, and the predicted phenomenon of topological superconductivity [3, 11]. In electronic systems, spin-orbit coupling arises from the rel- ativistic transformation of electric elds into magnetic elds in a moving reference frame. In the reference frame of an electron moving with wavevector kin an electric eld, the motional magnetic eld couples to the electron spin through the magnetic dipole interaction. This spin-orbit coupling phenomenon is responsible for lifting the degeneracy of spin states in the excited or- bitals of atoms and solid-state materials such as zinc- blende structures [12]. In a two-dimensional semicon- ductor heterostructure, the electric eld can arise from structure or bulk inversion asymmetry [13], leading to magnetic elds of the form B(R)=(ky;kx;0) or B(D)=(ky;kx;0). The resulting spin-orbit coupling terms in the Hamiltonian are known as the Rashba [14] and Dresselhaus [12] contributions, respectively. In- cluding a possible momentum-independent Zeeman eld B(Z)= (0;B(Z) y;B(Z) z), the Hamiltonian of the electron takes the form: H=~2k2 2mgB ~S
(B(D)+B(R)+B(Z));(1) wheregis the electron g-factor,Bis the Bohr magneton andSis the electron spin. The energy-momentum dispersion and the associated spin texture of the Hamiltonian in Eq. (1) are shown in Figure 1A for B(Z) y= 0 and=. In the absence of a perpendicular Zeeman eld B(Z) z, the spectrum consists of free particle parabolas for the two spin states that are shifted relative to each other in k-space owing to the spin- orbit interaction. For a nite eld B(Z) z, a gap opens in the spectrum. This gap, known as the spin-orbit gap, has been recently observed in one-dimensional quantum wires [6, 15]. The two energy bands are spinful in the sense that the spin of an atom is locked to its momentum.Similar band structures have been used to explain the anomalous quantum Hall eect and predict a saturation of the Hall conductivity for Fermi energies in the gap region [16]. In this work, we engineer the Hamiltonian in Eq. (1) with equal Rashba and Dresselhaus strengths in an opti- cally trapped, degenerate gas of fermionic lithium atoms via Raman dressing of atomic hyperne states [17, 18]. Raman elds have previously been used to generate spin- orbit coupling and gauge elds in pioneering work on Bose-Einstein condensates [19{21], and recently spin- orbit coupling in Fermi gases [22]. Here, we directly measure the spin-orbit band structure of Eq. (1), as well as the rich band structure of a spin-orbit coupled lat- tice. For this, we introduce spin-injection spectroscopy, which is capable of completely characterizing the quan- tum states of spin-orbit coupled fermions, including the energy-momentum dispersion and the associated spin- texture. By tracing the evolution of quantum states in the Brillouin zone, this method is able to directly mea- sure topological invariants, such as the Chern number in a two-dimensional system [3, 11, 23]. In order to directly reveal the single-particle eigen- states of the spin-orbit coupled system, we reduce the in- teractions in our Fermi gas to a negligible strength. This is convenient for studying topological insulators, whose behavior is mostly governed by single-particle physics. On the other hand, a single-component spin-orbit cou- pled Fermi gas is expected to develop eective p-wave interactions mediated by s-wave interactions [8], either in the presence of an s-wave Feshbach resonance, or in the presence of at bands as realized below. This can lead to BCS pairing in a p-wave channel, and in a two-dimensional system with pure Rashba coupling, to px+ipypairing and chiral super uidity [8, 9]. We generate spin-orbit coupling using a pair of laser beams that connect two atomic hyperne levels, la- beledj"iandj#i, via a two-photon Raman transition (Fig. 1B,C). The Raman process imparts momentumarXiv:1205.3483v1 [cond-mat.quant-gas] 15 May 20122 FIG. 1. Realization of spin-orbit coupling in an atomic Fermi gas. (A) Energy bands as a function of quasi-momentum qfor Raman coupling strength of ~ R= 0:25ERand~= 0. The spin composition of the states is indicated by the color. Dashed lines show energy bands for ~ R= 0ERand~= 0ER. (B) Geometry of the Raman beams: A pair of Raman beams at 19 relative to the ^ yaxis couples states j#;kx=qiandj";kx=q+Qi. A bias magnetic eld Bin the ^zdirection provides the quantization axis. (C) The hyperne interaction splits j"iandj#iby~!0, and the relevant polarization components are and +.~is the two-photon detuning. (D) Momentum-dependent Rabi oscillations in the spin texture after sudden switch-on of the Raman beams. Here ~ R= 0:78(2)ERand the detuning ~=0:25(1)ER. (E) A-pulse for the resonant momentum- class of atoms is applied at dierent two-photon detunings ~. The Raman strength is ~ R= 0:035(5)ERin order to retain momentum selectivity. (F and G): Adiabatic loading and unloading of atoms into the upper (lower) band at coupling strength of~ R= 0:53(5)ER. The Raman beams are turned on with =8:5 R, and the detuning is swept linearly to = 0 and back at a rate ofj_j= 0:27(5) 2 R. This loads atoms into the upper (lower) band, as indicated by the diagrams on the right. The spin texture follows the instantaneous value of , indicating adiabaticity. ~Q^xto an atom while changing its spin from j#itoj"i, and momentum~Q^xwhile changing the spin from j"i toj#i. Dening a quasimomentum q=kxQ 2for spin j#iandq=kx+Q 2for spinj"i, one obtains the Hamilto- nian of the form given in Eq. (1) [19]. In this mapping, B(Z) z=~ R=gB, where Ris the two-photon Rabi fre- quency,B(Z) y=~=gB, whereis the two-photon de- tuning, and ==~2Q 2mgB(see Supplemental Material). When the spin-orbit gap is opened suddenly, an atom prepared in the state j#;kx=qQ=2ioscillates between j#;kx=qQ=2iandj";kx=q+Q=2iwith a momen- tum dependent frequency
( q)=h, where
(q) is the en- ergy dierence between the bands at quasimomentum q. Such Rabi oscillations correspond to Larmor preces- sion of the pseudo-spin in the eective magnetic eld B(SO)=B(D)+B(R)+B(Z). We have observed these oscillations by starting with atoms in j#i, pulsing on the Raman eld for a variable duration , and imaging the atoms spin-selectively after time-of- ight expansion from the trap. Time-of- ight maps momentum to real space, allowing direct momentum resolution of the spin popula- tions. As a function of pulse duration, we observe oscil- lations of the pseudospin polarization with momentum- dependent frequencies (Fig. 1D). Our Fermi gas occupiesa large range of momentum states with near-unity occu- pation. Therefore, each image at a given pulse duration  contains information for a large range of momenta q. The observation of momentum-dependent oscillations demon- strates the presence of a spin-orbit gap, and shows that the atomic system is coherent over many cycles. To high- light the momentum selectivity of this process, we pre- pare an equal mixture of atoms in states j"iandj#iand pulse on the Raman elds for a time t== Rfor dif- ferent two-photon detunings . This inverts the spin for atoms with momentum qwhere
(q) is minimal. Since the minimum of
( q) varies linearly with due to the Doppler shift/kxQ, the momentum qat which the spin is inverted depends linearly on (Fig. 1E). Instead of pulsing on the Raman eld and projecting the initial state into a superposition of states in the two bands, one can introduce the spin-orbit gap adiabatically with respect to band populations. This is achieved by starting with a spin-polarized Fermi gas and sweeping the two-photon detuning from an initial value ito a nal detuning f. The magnitude of the initial detun- ingjijis much larger than the two-photon recoil energy ER=~2Q2=2m, so that the eective Zeeman eld is al- most entirely parallel with the spins. Depending on the3 FIG. 2. Spin{injection spectroscopy of a spin-orbit coupled Fermi gas. (A) A radiofrequency (RF) pulse transfers atoms from the reservoir states (shown in black) j"iRandj#iRinto the spin-orbit coupled system (shown in red and blue). Transfer occurs when the RF photon energy equals the energy dierence between the reservoir state and the spin-orbit coupled state at quasi- momentum q. (B,C,D and E) Transfer as a function of RF frequency detuning h
and quasi momentum qat Raman coupling strength of ~ R= 0:43(5)ERand ~= 0:00(3)ER. Note that starting with reservoir j#iR(j"iR), transfer to state j"i(j#i) is entirely due to spin-orbit coupling. Hence the signal is generally much weaker than that for state j#i(j"i) except right in the gap, where their ratio approaches 50% =50%. (B and C) Spin-resolved j#iandj"ispectra, respectively, when transferring out of j"iR. (D and E) Spin-resolved j#iandj"ispectra, respectively, when transferring out of j#iR. (F,G and H) The reconstructed spinful dispersions for ~= 0:00(3)ERand~ R= 0ER,~ R= 0:43(5)ERand~ R= 0:9(1)ER, respectively. direction of the sweep, this loads atoms into either the upper or the lower dressed band. We interrupt the sweep at various times, and image the spin-momentum distri- bution. This reveals that the spin texture follows the eective Zeeman eld. The process is reversible, as we verify by sweeping the detuning back to iand restoring full spin-polarization. (Fig. 1F and G). Having demonstrated the ability to engineer spin-orbit coupling in a Fermi gas, we introduce a general approach to measure the complete eigenstates and energies of fermions at each quasi-momentum qand thus resolve the band structure and associated spin texture of spin-orbit coupled atomic systems. Our approach yields equivalent information to spin and angle-resolved photoemission spectroscopy (spin-ARPES), a powerful technique re- cently developed in condensed matter physics [24]. Spin- ARPES is particularly useful for studying magnetic and quantum spin Hall materials; it has been used, for exam- ple, to directly measure topological quantum numbers in the Bi 1xSbxseries, revealing the presence of topological order and chiral properties [25]. Our spectroscopic technique uses radiofrequency (RF)spin-injection of atoms from a free Fermi gas into an empty spin-orbit coupled system using photons of a known energy (Fig. 2A). After injection, the momentum and spin of the injected atoms are analyzed using time of ight [26] combined with spin-resolved detection. Atoms are initially loaded into one of two free \reservoir" atomic statesj#iRandj"iRwhich can be coupled to the states j#i andj"i, respectively, via the RF spin-injection eld, with- out changing the quasimomentum. The injection occurs when the frequency of the RF pulse matches the energy dierence between the spin-orbit coupled bands and the initial reservoir state (see Fig. 2A). Spin-injection from j#iR(j"iR) populates mostly the region of the spin-orbit coupled bands with a strong admixture of j#i(j"i) states. Thus, the use of two reservoir states allows us to measure both thej#i-rich and thej"i-rich parts of the spin-orbit coupled bands. Following the injection process, the Ra- man beams are switched o, and the atoms are simulta- neously released from the trap. After a suciently long time of free expansion, the density distribution gives ac- cess to the momentum distribution, which we measure using state-selective absorption imaging. By counting4 the number of atoms of a given spin and momentum as a function of injection energy, we determine the disper- sion of the spin-orbit coupled bands along with their spin texture. The topological characteristics of the bands, which are encoded in the eigenstates, can be extracted from the spin and momentum composition. For our spin-orbit sys- tem with= 0, the spin of the eigenstates is conned to they-zplane on the Bloch sphere because the eec- tive magnetic eld has no ^ xcomponent. More general couplings may not restrict the spin to a great circle on the Bloch sphere, in which case at least two spin compo- nents must be measured for a complete characterization of the bands. This can be achieved by rotating the dif- ferent spin components onto the measurement basis with an RF pulse. Applying spin-injection spectroscopy, we have mea- sured the band structure of the equal-part Rashba- Dresselhaus Hamiltonian at = 0 for several R. Fig- ure 2B, C, D and E show spin- and momentum- resolved spin-injection spectra obtained with atoms starting in the j"iRreservoir (top row) and starting in the j#iRreservoir (bottom row), for the case ~ R= 0:43(5)ERand= 0. The (q;")$(q;#) symmetry of the system can be seen in the spectra in Fig. 2. The energy at each quasimo- mentum is found by adding the energy injected into the system by the RF pulse to the initial kinetic energy of the free particle in the reservoir. Figure 2F, G and H show the dispersion and spin texture of the bands obtained from the data. As Ris increased, we observe the open- ing of a spin-orbit gap at q= 0. The spin composition of the bands evolves from purely j"iorj#iaway from the spin-orbit gap to a mixture of the two spin states in the vicinity of the spin-orbit gap, where the spin states are resonantly coupled. The dispersion investigated above is the simplest pos- sible for a spin-orbit coupled system and arises naturally in some condensed matter systems. A Fermi gas with this dispersion has an interesting spinful semi-metallic behavior when the Fermi energy lies within the spin- orbit gap. When the Fermi energy is outside the spin- orbit gap, there is a four-fold degeneracy of states at the Fermi surface. Inside the gap, however, the degeneracy is halved. Furthermore, propagation of spin up parti- cles at the Fermi energy can only occur in the positive qdirection, while spin down fermions can only propa- gate in the opposite way. Particles are thus protected from back-scattering in the absence of magnetic impuri- ties that would rotate their spin. Such a spinful semi- metal can be used to build spin-current diodes, since the material permits ow of polarized spin-currents in one direction only. An even richer band structure involving multiple spin- ful bands separated by fully insulating gaps can arise in the presence of a periodic lattice potential. This has been realized for Bose-Einstein condensates by addingRF coupling between the Raman-coupled states j"iand j#i[7]. Using a similar method, we create a spinful lat- tice for ultracold fermions, and use spin-injection spec- troscopy to probe the resulting spinful band structure. The combined Raman/RF coupling scheme is shown in Fig. 3A. The Raman eld couples the states j#;kx=qi andj";kx=q+Qiwith strength R, whereas the RF eld couples the states j#;kx=qiandj";kx=qiwith strength RF. As a result, the set of coupled states for a given quasimomentum q, shown in the repeated Bril- louin scheme in Fig. 3B, is j;kx=q+nQifor integer nand=";#. The lowest four bands are degenerate at the band center q= 0 when R= RF= 0. The Raman eld splits the degeneracy between the rst and fourth band, leaving the other two degenerate. The remaining degeneracy, which is a Dirac point, is removed with the addition of the RF eld. Thus, when the system is lled up to the top of the second band, it is an insulator. Fur- thermore, when RFis large enough, a band gap also opens between the rst and second bands. Fig. 3D and E show the spin-injection spectra, mea- sured with fermions initially in reservoir state j#iR, which is sucient to reconstruct the full band structure given the (q;")$(q;#) symmetry of the Hamiltonian. The transitions between the reservoir and the spin-orbit cou- pled bands for ~ R= 0:40(5)ERand~ RF= 0:28(2)ER are shown in Fig. 3C. The experimental spectra (Fig. 3D and E) for the same parameters are compared to the corresponding theoretically calculated spectra, shown in Fig. 3F and G. We focus on the features of the j#ichannel of the spectrum, which is stronger because of the better spin-composition overlap with the reservoir state. The spectrum exhibits four prominent features separated by three energy gaps, labeled
1,
2and
3in Fig. 3F and 3G. The gaps giving rise to these features are shown on the band structure in Fig. 3C. The gap
1is opened by the spin-orbit coupling, while
2is opened by a direct RF coupling and
3is opened by a second order process that involves both the RF and Raman elds, explaining its smallness. We have explored the Raman/RF system for a range of coupling strengths as shown in the spectra in Fig. 4B and 4C. The corresponding band structures are shown in Fig 4A. With a careful choice of the Raman/RF coupling strengths, spinful at bands are realized, where interactions should play a dominant role [27]. To illustrate how the energy bands along with the corresponding eigenstates can be extracted, we recon- struct the energy bands along with the spin texture for ~ R= 0:93(7)ERand~ RF= 0:28(2)ER, as shown in Fig. 4D. The energies of the bands are obtained from the resonant frequencies in the spin-injection spectra, while the spin composition is extracted from the relative weights of the signal in the two spin channels (see Supple- mental Material). In general, the eigenvector (n)(q) for thenth energy band at a given quasi-momentum q, can be expanded in terms of free space eigenstates as5 FIG. 3. Creating and probing a spin-orbit coupled lattice. (A) The addition of a radiofrequency eld allows momentum transfer of any multiple of Q. The combined Raman-RF system produces a spinful lattice band structure. (B) The band structure of the Raman-RF system in the repeated zone scheme. The topmost band structure corresponds to ~ RF= 0 and ~ R= 0:25ER, which has a band crossing at quasi-momentum q= 0. The middle band structure corresponds to a larger Raman coupling of ~ R= 0:5ERwith ~ RF= 0. In the bottom-most band structure, ~ R= 0:5ERwhile ~ RFis increased to 0 :25ER. (C) Spin injection from free particle bands to spinful lattice bands, starting from j#iR. Transitions near zero RF detuning ( h
0) that give rise to dominant spectral features are identied. (D and E) Experimental spectrum of the Raman-RF system with ~ R= 0:40(5)ERand ~ RF= 0:28(2)ERin the spinj#iand spinj"ichannels, measured after injection from reservoir j#iR. The dominant features span many Brillouin zones, corresponding to projection of lattice states onto free particle states after time-of- ight. (F and G) The theoretical spectra corresponding to D and E, respectively. The features corresponding to the gaps and transitions identied in C are labeled. j n(q)i=P m;c(n) (kx=q+mQ)j;kx=q+mQi. In spin-injection spectroscopy, the projection of the lattice wavefunctions onto free particle states allows us to not only extract the average spin, but also the magnitude of the coecients c(n) (kx). From the projection coecients c(n) (kx), one can dene the spin ~S(kx) (see Supplemental materials). In Fig. 4E, F and G, we show the extracted value ofSy(kx) andjSz(kx)jfor the bottommost band when ~ R= 0:93(7)ERand ~ RF= 0:28(2)ER. For more general spin-orbit Hamiltonians involving x, one can extract the phase between all three components of ~S(kx) with additional RF pulses, and fully characterize the eigenstate for the corresponding quasimomentum q. The topology of the band, encoded in the evolution of its eigenstates across the Brillouin zone, can thus be mea- sured. In summary, we have created and directly probed a spin-orbit gap in a Fermi gas of ultracold atoms and re- alized a fully gapped band structure allowing for spinful at bands. We introduced spin-injection spectroscopy to characterize the spin-textured energy-momentum dis- persion. Such measurements would reveal the non-trivial topology of the bands in systems with more general spin- orbit couplings [28], opening a path to probing topo- logical insulators with ultracold atoms. Recently de-veloped high numerical aperture imaging techniques can be used for microscopic patterning of lower dimensional Fermi gases into heterostructures with regions charac- terized by dierent topological numbers separated by sharp interfaces [29, 30]. In such systems, spatially resolved spin-injection spectroscopy can directly reveal topologically protected edge states such as Majorana fermions, which have been proposed for topological quan- tum computation[10, 31, 32]. This work was supported by the NSF, a grant from the Army Research Oce with funding from the DARPA OLE program, ARO-MURI on Atomtronics, AFOSR- MURI, ONR YIP, DARPA YFA, an AFOSR PECASE, and the David and Lucile Packard Foundation. Z. H. acknowledges funding from EPSRC under Grant No. EP/I010580/1.6 FIG. 4. Evolution of spin-textured energy bands of a spin-orbit coupled lattice. (A) Theoretical band structures for various combinations of Rand RF. The rst band becomes at while remaining spinful for ~ R= 0:93ERand~ RF= 0:11ERand 0:28ER. (B and C) The corresponding experimental Raman-RF spin-injection spectra for injection from j#iRfor channelsj#i andj"i, respectively. The color map used is the same as Fig. 2B and 2E after rescaling to the maximum intensity (which for 4C is 20% of 4E), except for the top left panel in (C), which is scaled to the maximum intensity of the corresponding panel in (B). Possible interaction eects between j"iwithj#iR(see Fig. S2) makes only the dominant features resolvable in j"i, while ner features are visible in j#i. (D) Experimentally reconstructed band structure for ~ R= 0:93(7)ERand ~ RF= 0:28(2). The spin texture is indicated by the color of the points. (E,F and G) Experimentally measured spin components SyandjSzj as a function of momentum kxfor the lattice wavefunctions corresponding to the bottommost band in D.7 SUPPLEMENTAL MATERIALS System preparation and Raman setup Fermionic6Li in the hyperne state jF= 3=2;mF= 3=2iis sympathetically cooled to degeneracy by23Na atoms in a magnetic trap. The atoms are transferred to a nearly spherical crossed opti- cal dipole trap with mean trapping frequency 150 Hz. Depending on the measurement, the atoms are then transferred into one of the lowest four hyperne states in the ground state manifold via radiofrequency (RF) sweeps. The four lowest hyperne states are states j#iR, j#i,j"i,j"iRin the text. Subsequently, the magnetic eld is ramped to B=B0^z, withB0= 11:6 G. The geometry of the Raman beams is shown in Fig. 1B. The two beams, detuned by 3.96 GHz to the blue of the D 1 line, generate a moving lattice with lattice wavevector Q= 2(1:0m)1and corresponding recoil energy ofER=~2Q2 2m=h32(1) kHz. The calibration of the recoil energy ERis performed using the data in Fig. 1E and relies only on the relation of Qto the traveled distance for a given time-of- ight. To couple statesj"iandj#i, the frequency dierence between the two beams is set near the hyperne splitting of !0= 2207.7 MHz. For the one-photon detuning that we use, R=sc240, where scis the scattering rate. Note that one-body losses from single-photon scattering events do not perturb the measured spin-injection spectra, as they aect all energy states equally and only reduce the number of atoms in each momentum state. Experimental procedure In the presence of the Raman beams, a dierential Stark shift between j"iandj#ican alter the resonant two-photon frequency. The frequency corresponding to = 0 is calibrated using RF spectroscopy on the j"ito j#itransition in the presence of the Raman beams with a large two-photon detuning 21 MHz. The res- onant frequencies for j#iR! j#i andj"iR! j"i are calibrated similarly. The dierential Stark shift for the largest Raman coupling strength is measured to be 4 kHz forj"i ! j#i , and<1 kHz for thej#iR! j#i and j"iR!j"i transitions. For spin-injection spectroscopy, the injection process uses a RF eld that couples the statesj#iR(j"iR) andj"i(j#iR). All spectra are taken with a RF injection pulse duration of 0.5 ms, and an in- jection eld strength corresponding to a maximum trans- fer fraction <0:30. The experimental frequency resolu- tion for spin-injection is 3 kHz 0:1ER, while the mo- mentum resolution is estimated to be 0 :05Q, limited by expansion time and imaging resolution. For all measure- ments, state-selective absorption images are taken aftertime-of- ight of 4 ms. As the atoms are released, the bias magnetic eld is ramped to 300 G, where the reso- nant imaging frequencies for dierent hyperne states are well-resolved, allowing spin-selective absorption imaging. To obtain the spectra shown in Fig. 1 and 2, the time-of- ight images for each spin state are rst integrated along ^y, orthogonal to the spin-orbit direction. For a given quasi-momentum q, the integrated one-dimensional den- sity proles from the two spin channels are then combined to produce the nal spectrum. As an example, we show in Fig. S1 the time-of- ight images and the corresponding integrated density proles, at a specic detuning =ER for the spectrum in Fig. 1E. Hamiltonian for Raman-coupled system The spin of an atom is coupled to its momentum using a pair of laser beams near a Raman transition. Using the rotating wave approximation, the Raman beams generate a spinor potential V(~ r) =~ R 2(xcosQxysinQx) + 2z;(2) where Ris the two-photon Rabi frequency. After a local pseudo-spin rotation about the zaxis with angle Qx, the Hamiltonian becomes [19] H=~2k2 2m+~2Q 2mzq+~ R 2x+ 2z+ER 4;(3) whereqis the quasi-momentum dened in the text. Fol- lowing a global pseudo-spin rotation z!y,y!x andx!z, the Hamiltonian becomes H=~2k2 2m+~2Q 2myq+~ R 2z+ 2y+ER 4;(4) which up to a constant has the same Rashba-Dresselhaus form as Eq. (1), with ==~2Q 2mgB,B(Z) y=~=gB andB(Z) z=~ R=gB. In this convention, the bare hy- perne states, labeled j"iandj#iare eigenstates of y. Reconstructing the spinful dispersion for the Raman-coupled system For the equal Rashba-Dresselhaus system, the two channels in a spectrum from reservoir jiRare rst re- labeled by quasi-momentum q. The ratio of the trans- ferred atoms in each channel at a given qdirectly mea- sures theq-dependent spin composition. The dispersion is then reconstructed by adding the free particle disper- sion0(q) = (qQ=2)2=2mto the spectrum correspond- ing to injection from j"iR(j#iR). The result is shown in Fig. 2F,G and H, where the color denotes the spin tex- ture and the strength of the color is weighed by the total number of atoms at a given q.8 FIG. S1. Example of converting time-of- ight images to a spinful spectrum. Here, we show the conversion process for the spectrum in Fig. 1E for a specic detuning ~=1:88ER. (A and B) Time-of- ight images rescaled in terms of the recoil momentum Qin thej"iandj#ichannels, respectively. (C and D) The time-of- ight images in A and B are integrated along the kydirection to produce 1-dimensional densities n(q=Q). The conversion to quasi-momentum qinvolves adding a spin channel dependent momentum oset. (E) Combining C and D for every quasi-momentum qproduces a slice in the nal spectrum. In the gure, this is the region bounded by the two dashed lines. Repeating the same procedure for other detunings yields the full spectrum. Hamiltonian for the Raman/RF system Raman dressing creates a spin-orbit gap in momentum space. Adding radiofrequency (RF) coupling with zero momentum transfer creates a lattice potential with true band gaps [7]. The RF drive is applied at the same fre- quency as the Raman frequency, with coupling strength RF. An atom can now receive an arbitrary number of units of the recoil momentum ~Qby interacting alter- nately with the Raman eld and the RF eld (see Figure 3(A)). When the two spin states are coupled at dierent momenta using Raman lasers, and additionally at the same momentum using RF, the Hamiltonian becomes H=~2k2 2m+~ R 2(xcosQxysinQx)+~ RF 2x+ 2z: (5) Since the Hamiltonian has discrete translational symme- try along ^x, its eigenstates can be expanded in plane waves as j n(kx)i=X j;=";#c(n) (kx);kx=~kx+jQE ;(6) where ~kxis the quasi-momentum given by to kxrestricted to the rst Brillouin zone, and nis the band index.Spin-injection spectrum for the Raman/RF system Starting from reservoir =";#, the injected popula- tion in band nwith quasi-momentum ~kxat a given RF frequency!,PI;(!;n;m; ~kx), is given by PI;(!;n;l; ~kx)/ 2 I;X jn(~kx+lQ)jc(n) (~kx+jQ)j2 L((~!+0 l(~kx+jQ))(~!0+n(~kx)));(7) where I;is the RF strength coupling the reservoir state toji,n(k) is the trap-averaged momentum distribution for reservoir state jiR,0 l(k) =~2(k+lQ)2 2mis the free par- ticle dispersion of the lth non-interacting band, n(k) is the dispersion for the nth band,!0is the hyperne fre- quency dierence between j"iandj#i, andL(x) is the RF lineshape. After injection, the atoms are released from the trap. After sucient time-of- ight, the momentum distribution is given by the real space atomic density pro- le, which for the spin 0channel is PTOF;0;(!;kx) =X n;lPI;(!;n;l; ~kx)jc(n) 0(kx)j2:(8) The theoretical spectra in Fig. 3F and G are obtained using Eq. (8) and coecients c(n) (kx) found by numeri- cally diagonalizing the Hamiltonian in Eq. (5). The ex- perimental and theoretical spectra for Fig. 4 are shown in Fig. S2.9 FIG. S2. Experimental and theoretical Spin-injection spectra of Raman/RF system for dierent Raman/RF strengths. (A and B) The experimental Raman/RF spin-injection spectra for injection from j#iRfor channelsj#iandj"i, respectively. The color map used is the same as Fig. 2B and 2E after rescaling to the maximum intensity (which for 4C is 20% of 4E), except for the top left panel in (B), which is scaled to the maximum intensity of the corresponding panel in (A). (C and D) Theoretical spectra corresponding to B and C. We do not take into account nite imaging resolution, which aects the sharpness along the momentum axis kx=Q. Scattering due to residual interactions between hyperne states can also lead to blurring of momenta alongkx=Q, and will be most pronounced for the spectra in B, as state j"iscatters more strongly with j#iR(scattering length 450a0) thanj#idoes withj#iR(scattering length 2a0with typical initial 1 =kF3000a0before TOF). Reconstructing the spinful band structure for the Raman/RF System We rst describe a general procedure to reconstruct the band structure for any spinful lattice from spin-injection spectra. One performs spin-injection spectroscopy with reservoir states that are lled up to a Fermi momentum smaller than half the recoil momentum, kF<Q= 2. One then selects a prominent feature on the spectrum andnds the resonant transfer frequencies as a function of kx, over one Brillouin zone. A mask centered on the resonant frequencies and repeated over all kxwith period Qis then created. With the mask for a specic feature applied, the spin channel 0for the spectrum starting from jiRhas10 transfer intensity given by I0;(n;l;kx)/X jn(~kx+lQ)jc(n) (~kx+jQ)j2jc(n) 0(kx)j2: (9) Here, the Fermi momentum kFis less than half the recoil momentum Q=2 and therefore only the l= 0 contributes. DeningN(n;l;~kx) =P j;0I0;(n;l;~kx+jQ), one ob- tains jc(n) 0(kx)j2=I0;(n;l;kx) N(n;l;~kx): (10) Dening~S(kx) =1 2c(n)(kx)yc(n)(kx), where c(n)(kx) = (c(n) "(kx);c(n) #(kx))T, allows to measure Sz(kx). With ad- ditional RF pulses, one can measure Sx(kx) andSy(kx). After the bands that coupled strongly to j;ki;k<Q are measured, one iterates the process with a larger Fermi sea to obtain other bands. In the text, we demonstrate extraction of band struc- ture and spin texture in a spin-orbit coupled lattice. The initial Fermi sea has kF> Q , therefore spectral fea- tures corresponding to transitions out of dierent non- interacting reservoir bands can all appear near zero RF detuning. One can however identify spectral features cor- responding to dierent transitions and apply the above procedure. In addition, since the Raman/RF system is set to= 0, one can invoke the additional symmetry of the band structure about q= 0, meaning that it is sucient to t a certain spectral feature over half of a Brillouin zone. Due to a larger signal in the j#ichannel, features in this channel were used to create the mask. In order to be consistent with the earlier conven- tion for the Rashba-Dresselhaus system, we apply a global spin rotation such that Sy(kx) corresponds to 1 2 jc(n) "(kx)j2jc(n) #(kx)j2 . The experimentally mea- suredSy(kx) for the lowest band for ~ R= 0:93(7)ER and ~ RF= 0:28(2)ERis shown in Fig. 4F. Since the Hamiltonian only has components in the pseudo- spin ^yand ^zdirection, one can also extract jSz(kx)j=q 1 4Sy(kx)2, as shown in Fig. 4E. [1] M. K onig, et al. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science 318, 766 (2007). [2] D. Hsieh, et al. A topological dirac insulator in a quan- tum spin hall phase. Rev. Mod. Phys. 82, 3045 (2010). [3] M. Z. Hasan, C. L. Kane. Colloquium : Topological insu- lators. Rev. Mod. Phys. 82, 3045 (2010). [4] M. Inguscio, W. Ketterle, C. Salomon, eds., Ultracold Fermi gases, Proceedings of the International School of Physics Enrico Fermi, Course CLXIV, Varenna, 20-30 June 2006 (IOS Press, Amsterdam, 2008). [5] I. Bloch, J. Dalibard, W. Zwerger. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008).[6] C. H. L. Quay, et al. Observation of a one-dimensional spin-orbit gap in a quantum wire. Nat. Phys. 6, 336 (2010). [7] K. Jim enez-Garc a, et al . The Peierls substitution in an engineered lattice potential. arXiv:1201.6630v1 [cond- mat.quant-gas] (2012). [8] R. A. Williams, et al. Synthetic partial waves in ultracold atomic collisions. Science 335, 314 (2012). [9] C. Zhang, S. Tewari, R. M. Lutchyn, S. Das Sarma. px+ ipysuper uid from s-wave interactions of fermionic cold atoms. Phys. Rev. Lett. 101, 160401 (2008). [10] M. Sato, Y. Takahashi, S. Fujimoto. Non-abelian topo- logical order in s-wave super uids of ultracold fermionic atoms. Phys. Rev. Lett. 103, 020401 (2009). [11] X.-L. Qi, S.-C. Zhang. Topological insulators and super- conductors. Rev. Mod. Phys. 83, 1057 (2011). [12] G. Dresselhaus. Spin-orbit coupling eects in zinc blende structures. Phys. Rev. 100, 580 (1955). [13] R. Winkler, Spin-Orbit Coupling Eects in Two- Dimensional Electron and Hole Systems , vol. 191 of Springer Tracts in Modern Physics (Springer, Berlin, 2003). [14] Y. A. Bychkov, E. I. Rashba. Oscillatory eects and the magnetic susceptibility of carriers in inversion layers. J. Phys. C 17, 6039 (1984). [15] S. Nadj-Perge, et al. Spectroscopy of spin-orbit quantum bits in indium antimonide nanowires. Phys. Rev. Lett. 108, 166801 (2012). [16] D. Xiao, M.-C. Chang, Q. Niu. Berry phase eects on electronic properties. Rev. Mod. Phys. 82, 1959 (2010). [17] X.-J. Liu, M. F. Borunda, X. Liu, J. Sinova. Eect of in- duced spin-orbit coupling for atoms via laser elds. Phys. Rev. Lett. 102, 046402 (2009). [18] J. Dalibard, F. Gerbier, G. Juzeli unas, P. Ohberg. Col- loquium : Articial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523 (2011). [19] Y.-J. Lin, Jim enez-Garc a, I. B. Spielman. Spin-orbit- coupled Bose-Einstein condensates. Nature 471, 83 (2011). [20] Y.-J. Lin, R. L. Compton, Jim enez-Garc a, J. V. Porto, I. B. Spielman. Synthetic magnetic elds for ultracold neutral atoms. Nature 462, 628 (2009). [21] M. Aidelsburger, et al. Experimental realization of strong eective magnetic elds in an optical lattice. Phys. Rev. Lett.107, 255301 (2011). [22] P. Wang, et al . Spin-Orbit Coupled Degenerate Fermi Gases. arXiv:1204.1887v1 [cond-mat.quant-gas] (2012). [23] E. Zhao, N. Bray-Ali, C. J. Williams, I. B. Spielman, I. I. Satija. Chern numbers hiding in time-of- ight images. Phys. Rev. A 84, 063629 (2011). [24] M. Hoesch, et al. Spin-polarized fermi surface mapping. Journal of Electron Spectroscopy and Related Phenomena 124, 263 (2002). [25] D. Hsieh, et al. Observation of unconventional quantum spin textures in topological insulators. Science 323, 919 (2009). [26] J. T. Stewart, J. P. Gaebler, D. S. Jin. Using photoemis- sion spectroscopy to probe a strongly interacting Fermi gas. Nature 454, 744 (2008). [27] K. Sun, Z. Gu, H. Katsura, S. Das Sarma. Nearly at- bands with nontrivial topology. Phys. Rev. Lett. 106, 236803 (2011). [28] J. D. Sau, R. Sensarma, S. Powell, I. B. Spielman,11 S. Das Sarma. Chiral rashba spin textures in ultracold fermi gases. Phys. Rev. B 83, 140510 (2011). [29] W. S. Bakr, et al. Probing the Super uid-to-Mott Insu- lator Transition at the Single-Atom Level. Science 329, 547 (2010). [30] J. Sherson, et al. Single-atom-resolved uorescence imag- ing of an atomic Mott insulator. Nature 467, 68 (2010).[31] S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, P. Zoller. Quantum computation using vortices and Majorana zero modes of a px+ipysuper uid of fermionic cold atoms. Phys. Rev. Lett. 98, 010506 (2007). [32] L. Jiang, et al. Majorana fermions in equilibrium and in driven cold-atom quantum wires. Phys. Rev. Lett. 106, 220402 (2011).

1801.02887v1.Inverse_engineering_for_fast_transport_and_spin_control_of_spin_orbit_coupled_Bose_Einstein_condensates_in_moving_harmonic_traps.pdf

arXiv:1801.02887v1 [quant-ph] 9 Jan 2018Inverse engineering for fast transport and spin control of s pin-orbit-coupled Bose-Einstein condensates in moving harmonic traps Xi Chen,1,∗Ruan-Lei Jiang,1Jing Li,1Yue Ban,2,3,†and E. Ya. Sherman4,5 1Department of Physics, Shanghai University, 200444 Shangh ai, People’s Republic of China 2Department of Electronic Information Materials, Shanghai University, 200444 Shanghai, People’s Republic o f China 3Instituto de Ciencia de Material de Madrid, CSIC, 28049 Madr id, Spain 4Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080, Bilbao, Spain 5IKERBASQUE Basque Foundation for Science, Bilbao, Spain (Dated: November 8, 2018) We investigate fast transport and spin manipulation of tuna ble spin-orbit-coupled Bose-Einstein condensates in a moving harmonic trap. Motivated by the conc ept of “shortcuts to adiabatic- ity”, we design inversely the time-dependent trap position and spin-orbit coupling strength. By choosing appropriate boundary conditions we obtain fast tr ansport and spin flip simultaneously. The non-adiabatic transport and relevant spin dynamics are illustrated with numerical examples, and compared with the adiabatic transport with constant spi n-orbit-coupling strength and velocity. Moreover, the influence of nonlinearity induced by interato mic interaction is discussed in terms of the Gross-Pitaevskii approach, showing the robustness of t he proposed protocols. With the state- of-the-art experiments, such inverse engineering techniq ue paves the way for coherent control of spin-orbit-coupled Bose-Einstein condensates in harmoni c traps. I. INTRODUCTION Spin-orbit-coupling (SOC), linking a quantum par- ticle’s momentum to its spin, is a fundamental effect in solid-states spintronics [ 1]. In recent years, the ex- perimental breakthrough, realizing a synthetic SOC for (pseudo) spin-1/2bosonic[ 2]andfermions[ 3,4], haspro- vided a platform for quantum simulation of exotic states in condensed matter physics and a flexible tool for ma- nipulatingcoldatoms, seereview[ 5,6]. In particular,the static and dynamical properties, relevant to the SOC ef- fects, have been extensively investigated in such atomic systems [ 8–16], which open new possibility to probe or control quantum spin dynamics such as spin relaxation, Zitterbewegung, spin resonance, and the spin-Hall effect. Cold atoms are comprehensively stored and manipu- lated in traps formed by designed electromagnetic field configurations, with fundamental interest and potential applications in atom interferometry, metrology or quan- tum information processing. Very often, a transport of neutral or ionized cold atoms and Bose-Einstein conden- sates (BECs) to appropriate location without any exci- tation and losses [ 17–21] is demanded. However, most transport processes require long time, satisfying the slow adiabatic criteria, which could be problematic due to the noise and decoherence. An alternative way out is to apply the concept of “shortcuts to adiabaticity” [ 22] to reach the same results at a relatively short time. Among all shortcut techniques, the inverse engineering method, based on Lewis-Riesenfeld invariant and corresponding dynamical modes, can be also applicable to achieve fast ∗xchen@shu.edu.cn †yban@shu.edu.cnnon-adiabatic but reliably controllable transport [ 23–27], or expansion [ 28,29] in harmonic traps and spin control in (effective) two-level systems [ 30]. More interestingly, the spin and motional states can be precisely and simul- taneously controlled by such inverse engineering method in a Morse potential with SOC [ 31]. In such static po- tential trap, the direction and magnitude ofthe synthetic SOC field are chosen as tunable parameters in effective two level system to manipulate spin states, and the po- sition transfer does result from the effect of SOC on the orbital motion, rather than the modulation of the trap center. In this paper, we propose a method for controlling spin dynamics and orbital motion of BECs in moving harmonic traps with Raman-induced SOC. Similarly to electrons in quantum dots [ 32–35], cold atoms are con- fined in harmonic traps, and the spin state and orbital motion can be controllable by time-dependent SOC. The exact wave function of atoms trapped in a moving har- monic trap in presence of time-dependent SOC can be solved analytically [ 34,35], to demonstrate the control- lability of spin state and orbital motion. Instead of (non- adiabatic) cyclic evolution in Ref. [ 35], we apply inverse engineering approach to design the position of moving trap and time-dependent strength of SOC, in order to transport cold atoms, arriving at appropriate location with spin flip simultaneously. The non-adiabatic trans- port and relevant spin dynamics are illustrated with nu- merical examples. As compared with adiabatic transport with constant SOC strength and velocity, we illustrate that inverse engineering provides more flexibility to ma- nipulate the cold atoms and spin-orbit qubits in a fast and robust way. By extension, we also discuss the SOC BECs in the presence of interaction between the atoms and show the stability against the effects of nonlinearity. The results presented below thereby may acquire wide2 Raman laser(a)bias field Raman laserBECxyz |↓ |1,-1› ›= |↑ |1,0› ›= (b) 0 d FIG. 1. (a) Experimental scheme for realizing synthetic SOC BECs and the level diagram. A homogeneous magnetic field prepares the ground state spin-polarized along the x- direction. Two Raman lasers are used to couple the spin-up and spin-down states. (b) Schematic diagram of the BECs transport in an effective one-dimensional harmonic potenti al fromx0= 0 tox0=d. applications. II. GENERAL EQUATION FOR ORBITAL AND SPIN MOTION Our starting point is the BECs in a one-dimensional (alongthex−axis)harmonicpotentialwithSOC,seeFig. 1, where a cloud of ultracold87Rb atoms is strongly con- fined in the y-zplane. Internal hyperfine ground states | ↑/angbracketright=|F= 1,mf= 0/angbracketrightand| ↓/angbracketright=|F= 1,mf=−1/angbracketright coupled by two Raman lasers can be identified here as (pseudo-)spins. The dynamics in a moving harmonic po- tential is governed by the following Hamiltonian H=p2 2m+1 2mω2[x−x0(t)]2+α(t)pσz,(1) wherepandxare the momentum and position opera- tors, respectively, mis the particle mass, ωis the time- independent potential frequency, and σzis the corre- sponding 2 ×2 Pauli matrix. Here x0(t) is the time- dependent trap position, which can be tuned, for in- stance, by changing Gaussian beam waist center [ 21]. The parameter α(t) is a controllable SOC strength, ad- justed by the geometry of two Raman lasers [ 2], see Fig. 1(a). Here we neglect the Zeeman term, ∆ σx, since the magnetic field is supposed to be switched off after the initial spin state is prepared. We present the |Ψ(x,t)/angbracketright ≡[Ψ↑(x,t),Ψ↓(x,t)]Tsolu- tion of the time-dependent Schr¨ odinger equation with Hamiltonian ( 1) in the form: |Ψ(x,t)/angbracketright=e−iEt//planckover2pi1U(t)|ψ(x)/angbracketright|χs/angbracketright, (2) whereEis the eigenvalue, U(t) is a unitary transforma- tion,|χs/angbracketrightis a spinor with spin s,|ψ(x)/angbracketrightstands for theeigenfunction of the stationary harmonic oscillator /bracketleftbiggp2 2m+1 2mω2x2/bracketrightbigg ψ(x) =Eψ(x). (3) Following the approach of Refs. [ 34,35], we introduce the unitary operator, U(t) =Us(t)Uo(t), with the spin and orbit parts, Us(t) =e−iφα(t)e−iφ(t)σze−imac(t)xσz//planckover2pi1e−i˙ac(t)pσz/(/planckover2pi1ω2) Uo(t) =e−iφx0(t)e−ixc(t)p//planckover2pi1eim˙xc(t)x//planckover2pi1, (4) where the upper dot represents time derivative, and we shall seek for the yet unknown phase factors which de- termine their time-dependence. Here two action phase factors: φα(t) =−1 /planckover2pi1/integraldisplayt 0dτLα(τ), (5) φx0(t) =−1 /planckover2pi1/integraldisplayt 0dτLx0(τ), (6) are expressed with the Lagrangians for classical mechan- ics: Lα(t) =1 2ω2m˙a2 c(t)−1 2ma2 c(t)+mac(t)α(t),(7) Lx0(t) =1 2m˙x2 c(t)−1 2mω2[xc(t)−x0(t)]2.(8) To guarantee that Eq. ( 2) is the exact solution, we intro- duce two auxiliary functions, xc(t) andac(t),satisfying equations ¨xc(t)+ω2[xc(t)−x0(t)] = 0, (9) ¨ac(t)+ω2[ac(t)−α(t)] = 0, (10) which describe the center-of-mass position of the BECs and its spin precession, respectively. Also the phase fac- tor, standing for the spin rotation along the z-direction, couples these two parameters as follows φσ(t) =−m /planckover2pi1/integraldisplayt 0˙ac(τ)x0(τ)dτ. (11) Without SOC, α(t) = 0, the second auxiliary Eq. ( 10) becomes trivial, thus the problem is reduced to the previ- oustransportdesign[ 23,24]byusinginverseengineering. As a matter of fact, when the interaction between atoms is negligible, the wave function of in Eq. ( 2) is nothing but the transport modes based on Lewis-Riesenfeld dy- namical invariant, with the eigenvalue En= (n+1/2)/planckover2pi1ω and eigenstate |ψ(x)/angbracketrightof a stationary harmonic trap [ 23], see Eq. ( 3). Additionally, the Hamiltonian ( 1) resembles the one for electron in a moving quantum dot with time- dependent SOC [ 34,35]. Here we shall concentrate on the transport of BECs with tunable SOC. Inwhatfollowsweshalldeveloptheinverseengineering method for transporting BECs rapidly and flipping the spin simultaneously by using Eqs. ( 9)-(11). This is sub- stantially different from the holonomic transformation of3 spin-orbit qubits [ 35], where the controllable parameters are periodically modulated, thus the spin rotation is de- termined by the non-adiabatic Aharonov-Anandanphase for cyclic time evolution. The strategy presented here is that we first consider the position of moving potential x0(t) and the time-dependent SOC strength α(t) as free controllable parameters, and then design them inversely basedonthesolutionsof xc(t)andac(t)satisfyingtheap- propriate boundary conditions. Thus, the spin rotation andorbitalmotioncanbesimultaneouslymanipulatedas we wish. To illustrate the technique, we begin with the linear transport of the BECs with time-dependent SOC, neglecting interatomic interaction. Later, we will check the influence of nonlinearity resulting from interaction between the atoms by a numerical simulation. The de- signed shortcut protocol will be compared with the adia- batic transportwith constantvelocityand SOC strength, showing the advantages of inverse engineering proposed here. III. INVERSE ENGINEERING To design the potential position and SOC strength in- versely, we focus on Eqs. ( 9) and (10) by choosing appro- priate boundary conditions. Suppose that the potential minimum moved from x0(0) = 0 at initial time t= 0 to x0(tf) =dwithin a time interval tf. To guarantee the transport without a final excitation, we set the following boundary conditions [ 23,24]: xc(0) = 0,˙xc(0) = 0,¨xc(0) = 0, (12) xc(tf) =d,˙xc(tf) = 0,¨xc(tf) = 0.(13) These boundary conditions can be satisfied by an infinite set of functions. For simplicity, we choose a flexible poly- nomial ansatz in the form, xc(t) =/summationtext5 i=0biti.Finally, we can obtain the center-of-mass position of the cold atoms xc(t) as: xc(t) =d(10s3−15s4+6s5), (14) withs≡t/tf, which provides the desired x0(t) obtained from Eq. ( 9), namely,x0(t) =xc(t)+ ¨xc(t)/ω2. Now we can fulfill the task of the spin flipping without excitation of the orbital motion. To ensure that the ef- fects ofac(t) and ˙ac(t) in the unitary transformation ( 4) vanish at the initial and final time, we set the boundary conditions: ac(0) =ac(tf) = 0, (15) ˙ac(0) = ˙ac(tf) = 0, (16) ¨ac(0) = ¨ac(tf) = 0. (17) Here the boundary conditions for the second deriva- tive avoid the abrupt changes in the SOC strength at the edges, t= 0 andtf, which could be implemented by switching on/off the Raman laser. In addition, the phase factor of the spin rotation, exp[ −iφσ(t)σz], acts0 0.2 0.4 0.6 0.8 1 t/tf00.20.40.60.81x0/d, xc/d(a) 0 0.2 0.4 0.6 0.8 1 t/tf00.10.20.3α, ac(b) FIG. 2. (a) Dependence of the minimum position x0/d(solid red line) and the center-of-mass xc/d(dashed blue line) on timet/tf. (b) Dependence of the SOC strength α(solid red line) and the parameter ac(dashed blue line) on time t/tf. Parameters: tf= 8/ωandd= 10. on the initial eigenstate of σx([1/√ 2,1/√ 2]T) in theσz- representation. As a result, when φσ(tf) =π/2, the spin state changes to [1 /√ 2,−1/√ 2]T, achieving the spin flip, that is, φσ(tf) =−m /planckover2pi1/integraldisplaytf 0˙ac(τ)x0(τ)dτ=π 2,(18) makes the spin rotate around the z-axis by the π-angle. By combining all the conditions, ( 15)-(18), we solve the polynomial ansatz, ac(t) =/summationtext6 j=0cjtj, and obtain ac(t) =−231π/planckover2pi1 mdω2t2 f 5ω2t2 f−66/parenleftbig s6−3s5+3s4−s3/parenrightbig . (19) Onceac(t) is fixed, we calculate α(t) from Eq. ( 10). In a realistic setup, the SOC was realized for87Rb atoms in an external potential, where the mass of atom ism= 1.443×10−22g and the confining potential fre- quency isω= 2π×250 Hz. To simplify the numeri- cal calculations below, we choose m=/planckover2pi1=ω= 1 and4 (b)|ψ|2 -0.5 0 0.5 1 1.5 x/d FIG. 3. (a) Contour plot of the wave packet propagation during the fast transport designed by the inverse engineeri ng method. (b) Time evolution of the wave packet with spin-up (solid red line) and spin-down (dashed blue line) component s at different times: t= 0,tf/4,tf/2,3tf/4,tf. Parameters are the same as those in Fig. 2. the units of relevant physical parameters are re-scaled byT= 1/ω≈0.637 ms and the characteristic length a0=/radicalbig /planckover2pi1/(mω)≈0.682µm, correspondingly. Figure 2demonstrates the designed position x0(t) and time- dependent SOC strength α(t). IV. FAST TRANSPORT AND SPIN DYNAMICS For the sake of simplicity, we first consider the fast transport and spin flip without interatomic interaction. We assume the initially prepared state, |Ψ(x,0)/angbracketright=1√ 2/parenleftbigg1 1/parenrightbigg ⊗|ψ(x,0)/angbracketright, (20) where |ψ(x,0)/angbracketright=/parenleftbigg1 πa2/parenrightbigg1/4 exp/bracketleftbigg −x2 2a2/bracketrightbigg (21) is the ground state in the harmonic potential, which we consider here without loss of generality. The final wave0 0.2 0.4 0.6 0.8 1 t/tf-1-0.500.51/angbracketleftσi/angbracketright FIG. 4. Time evolution of spin components /angbracketleftσi/angbracketrightduring the fast transport representing /angbracketleftσx/angbracketright(solid black line), /angbracketleftσy/angbracketright(dot- ted red line), and /angbracketleftσz/angbracketright(dashed blue line). Parameters are the same as those in Fig. 2. function has the form |Ψ(x,tf)/angbracketright=1√ 2/parenleftbigg1 −1/parenrightbigg ⊗|ψ(x,tf)/angbracketright,(22) with |ψ(x,tf)/angbracketright=/parenleftbigg1 πa2/parenrightbigg1/4 exp/bracketleftbigg −(x−d)2 2a2/bracketrightbigg .(23) Figure3(a) demonstrates that by using the designed trap position, the BECs is transported from x0= 0 to x0=dwithout anyfinalexcitation. Toillustratethespin motion, the propagation of spin components is displayed inFig.3(b). Sincetheinitialspinparalleltothe x-axisis not an eigenstate of the Hamiltonian ( 1), the spin starts to rotate and the wave packet splits into two components having different velocities. To understand this splitting, we define the velocity operator, taking into account the spin-dependent contribution, v=i /planckover2pi1[H,x] =p m+ασz. (24) As a result, the SOC leads to different velocities for spin- projected components of wave packet. Initially, the spin is parallel to the x-axis, the expectation value of the ve- locity vanishes, /angbracketleftv/angbracketright= 0 att= 0, the two components of the wave packet coincide and start to split. At the mid- dle timet=tf/2, the spin is parallel to the y-direction, and the two components merge again with the same but non-zero expectation value of velocity. At the final time, based on the boundary conditions, ( 15)-(17), the spin is antiparallel to the x-direction, such that the spin compo- nents of the wave packet coincide and /angbracketleftv/angbracketright= 0, see Fig. 3 (b). Next, we discuss the spin evolution in terms of the5 reduced density matrix [ 36] ρ(t) =|Ψ(x,t)/angbracketright/angbracketleftΨ(x,t)|=/bracketleftbigg ρ11(t)ρ12(t) ρ21(t)ρ22(t)/bracketrightbigg ,(25) where ρij(t) =/integraldisplay Ψi(x,t)Ψ∗ j(x,t)dx,(i,j=↑,↓), and tr(ρ) =ρ11(t) +ρ22(t) = 1 because of the nor- malization of the wave function. As a consequence, the three spin components can be defined by /angbracketleftσi/angbracketright= tr(σiρ) (i=x,y,z). Figure 4shows the time evolution of the spin components, where the expectation value of spin polarization at initial time t= 0 is/angbracketleftσx/angbracketright(0) = 1, and /angbracketleftσx/angbracketright(tf) =−1 at the final time t=tf. As a re- sult, the spin flips with rotation around the z-axis, when the BECs is being transported. Additionally, we define the length of the spin vector inside the Bloch sphere asP= (/summationtext3 i=1/angbracketleftσi/angbracketright2)1/2. At the initial and final times P= 1, implying that the spin is in a pure state on the Bloch sphere. During the non-adiabatic transport, spin- dependent excitations of the orbital modes occur, result- ing in a mixed state in the spin subspace with P <1. V. COMPARISON WITH ADIABATIC TRANSPORT WITH CONSTANT SOC Forcomparison, wenowconsiderthe caseofaconstant SOC strength, α, and the adiabatic transportwith a con- stant velocity, d/tf. The adiabatic transport for linear protocol [ 26] requirestad f≫d/radicalbig m/(2/planckover2pi1ω)≈7.07,where we taked= 10 as in Fig. 2. In the adiabatic limit, ne- glect the derivatives ¨ xcand ˙xc, resulting in xc(t) =x(t). Substituting the constant SOC strength α, we solve Eq. (10) and finally obtain ac(t) =α[1−cos(ωt)], (26) with the initial boundary conditions, ac(0) = ˙ac(0) = 0. Obviously, when ωtf= 2πk(k= 1,2,3,...), the boundary conditions ac(tf) = ˙ac(tf) = 0 are fulfilled. On the contrary, the other solution ac=αis neglected, since in this case the spin part of unitary transformation U(t) becomesUs=e−imαxσ z//planckover2pi1, and the initial and final spin states are transformed, due to Us(0) =Us(tf)/negationslash= 1. Furthermore, the phase factor ( 11) is calculated as φσ(tf) =−mαd /planckover2pi1ωtf[sin(ωtf)−ωtfcos(ωtf)],(27) by using ˙ac=αωsin(ωt) andx0(t) =dt/tf. When ωtf= 2πk(k= 1,2,3,...), such phase factor is sim- plified asφσ(tf) =d/λsowithλso=/planckover2pi1/(mα). As a con- sequence, we see from Eq. ( 4) that the spin is rotated by the angle 2 d/λsoaround the z-direction. By further imposingφσ(tf) =π/2,we have the characteristiclength for spin flip dsp=πλso/2, (28)-0.5 0 0.5 1 1.5 x/d00.20.40.6|ψ|2(a) 0 0.2 0.4 0.6 0.8 1 t/tf-1-0.500.51/angbracketleftσi/angbracketright(b) FIG. 5. (a) Profiles of total density |Ψ(x,t)|2(solid black line) and density of spin components, |Ψ↑,↓(x,t)|2(solid red and dashed blue lines, undistinguishable) at the initial ti me t= 0 and at the final time t=tf. (b) Time evolution of spin components /angbracketleftσi/angbracketrightduring the fast transport representing /angbracketleftσx/angbracketright (solid black line), /angbracketleftσy/angbracketright(dotted red line), and /angbracketleftσz/angbracketright(dashed blue line). Parameters: α= 1,tf= 100π/ω,d= 10. and the spin-flip time is tsp= (dsp/d)tf. This is consis- tent with the result of Ref. [ 34]. Figure5(a) illustrates that the BECs can be trans- ported from x0= 0 tox0=dwhentf= 100π/ωis suf- ficiently long to satisfy the adiabatic criteria. When the timetfis anintegermultiple of2 π/ω, there isnofinal ex- citation of the orbital motion, due to ac(tf) = ˙ac(tf) = 0 in Eq. ( 4). The orbital wave function is exactly dis- placed bydin this case. Meanwhile, the spin dynamics is determined by the characteristic length dsp. In other words, the spin flip can be realized if the transported distance is dsp. Whend= 10 andα= 1, we can obtain the periodical time tsp= 0.157tffor spin flip, see Fig. 5 (b). In this case, the final spin state is not the eigenstate ofσx,and the spin cannot be flipped completely, since d/negationslash= (2k−1)dsp(k= 1,2,3,...). However, at the final time the wave functions of the spin components coincide being characterized by the same displacement d, see Fig. 5(a), since the terms on acand ˙acin Eq. (4) vanish at6 t=tf. Therefore, one can transport atoms from x0= 0 tox0= (2k−1)dspadiabaticallywith flipping the spin si- multaneously. Note that in the adiabatic approximation, the characteristic spin rotation length depends only on the SOC strength being independent of the transport ve- locity. As compared to the inversely designed protocols, the adiabatic transport with constant SOC and velocity can achieve the same effect on the orbital and spin mo- tion, but it requiresa much longertime and possible only for a relatively small set of final positions. VI. THE EFFECTS OF INTERATOMIC INTERACTIONS In this section, we briefly present the influence of in- teratomic interaction on the orbital motion and spin dy- namics designed by the inverse engineering method. The Hamiltonian is rewritten as H=p2 2m+1 2mω2[x−x0(t)]2+α(t)pσz+g|Ψ(x,t)|2,(29) where the repulsive interaction characterized by param- eterg= 2as/planckover2pi1ω⊥>0 is involved, with the scatter- ing lengthasand the transverse confinement frequency ω⊥≫ω. The nonlinearity gcan experimentally be ad- justed by Feshbach resonances and the transversal con- finement. In the following numerical calculations, the di- mensionless ˜ g= 2˜as˜ω⊥with/planckover2pi1= 1, where ˜ asis the scat- tering length in the units of a0and ˜ω⊥is the transverse confinement frequency in the units of ω. Here Ψ(x,t) is the wave function of the condensate described by the mean-field Gross-Pitaevskii (GP) equation, and its nor- malization is/integraltext+∞ −∞|Ψ(x,t)|2dx=N, with the number of atomsN. In general, one can transport the ground state of the Gross-Pitaevskii (GP) equation as the initial wave packet [ 36]. Instead, for consistency we assume the following initial Gaussian wave packet |Ψ(x,0)/angbracketright=1√ 2/parenleftbigg1 1/parenrightbigg ⊗/parenleftbiggN2 πa2/parenrightbigg1/4 exp/parenleftbigg −x2 2a2/parenrightbigg .(30) This Gaussian assumption turns out to be an appropri- ate choice, particularly for weak interaction gand small number of atoms N[37,38], otherwise the wave function becomes an inverted parabolic shape at strong repulsive interaction, especially N >600, see [ 39]. The influence of the interatomic interaction and the numbers of atom is illustrated by the fidelity, F= |/angbracketleft˜Ψ(tf)|Ψ(tf)/angbracketright|2, see Fig. 6, where the target state is defined as |˜Ψ(tf)/angbracketright=1√ 2/parenleftbigg1 −1/parenrightbigg ⊗/parenleftbiggN2 πa2/parenrightbigg1/4 exp/bracketleftbigg −(x−d)2 2a2/bracketrightbigg , with the displacement dand spin flip, and the wave func- tion|Ψ(tf)/angbracketrightisthe numericalresult calculatedby the split operator method. Figure 6demonstrates that the Gaus- sian approximation is good to describe such a non-linear0 0.1 0.2 0.3 0.4 0.5 ˜gN0.9920.9940.9960.9981F FIG. 6. Fidelity versus ˜ gN, resulting from interatomic inter- action, where N= 100 and other parameters are the same as those in Fig. 2. system, especially when ˜ gNis reasonably smaller than 1. However, the fidelity becomes worse with increasing the nonlinearity gand the number of atoms N. To un- derstand this effect, we shall analyze the time evolution of two spin components. As mentioned above, due to the SOC the initial wave packet starts to split in two spin components with different velocities. The repulsive interaction helps separation and hinders merging. The spin dynamics and the orbital motion become different from the linear case ( g= 0), and the two spin compo- nents cannot merge at t=tf/2 andt=tf. This causes the final separationofthe spin components and decreases the fidelity. VII. CONCLUSION We have presented a method for achieving the fast transport and spin control of spin-orbit coupled BECs in moving harmonic potentials. The inverse engineer- ing, based on the concept of ”shortcuts to adiabatic- ity“ is applied to design the potential position and the time-dependent strength of SOC, by choosing appropri- ate boundary conditions. The adiabatic transport with a constant SOC has been compared with the developed protocolto illustrate the advantageof the shortcut-based design. Finally, we have discussed the SOC BECs trans- port taking into account the interatomic interaction at the level of the Gross-Pitaevskii equation. The inverse engineering method proposed here is help- ful for manipulating the SOC BECs and controllingspin- orbit coupled qubits by designing the time-dependent SOC and the potential motion. This might have appli- cations in quantum information processing, atom inter- ferometry, and quantum metrology. Several natural ex- tensions of this approach can be done in the near future. For instance, one can hybridize the inverse engineering and the optimal control theory [ 25,27] to optimize the7 shortcuts in the presence of noise and device-related er- rors. Being combined with the variational principle [ 37] and hydrodynamic approach [ 40], the shortcuts can be further designed for soliton dynamics [ 41] or quench dy- namics in the SOC BECs. Last but not least, our system resembles electron confined in parabolic quantum dots or wires [ 32–35], which can be useful for generating spin- dependent coherent and Schr¨ odinger cat states [ 42]. ACKNOWLEDGMENTS We thank Yongping Zhang, Thomas Busch, and Guanzhuo Yang for helpful discussions. This work ispartially supported by the NSFC (11474193, 61404079), the Shuguang (14SG35), and the Program for Profes- sor of Special Appointment (Eastern Scholar). E.Y.S. acknowledges support of the University of the Basque Country UPV/EHU under program UFI 11/55, Spanish MEC/FEDER (FIS2015-67161-P) and Grupos Consoli- dados UPV/EHU del Gobierno Vasco (IT-472-10). Y.B. also acknowledges Juan de la Cierva program. [1] I.ˇZuti´ c, J. Fabian, and S. Das Sarman, Rev. Mod. Phys. 76, 323 (2004). [2] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Na- ture(London) 471, 83 (2011). [3] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [4] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301(2012). [5] V. Galitski and I. B. Spielman, Nature 494, 49 (2013). [6] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015). [7] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett . 108, 225301 (2012). [8] T. Yu and M. W. Wu, Phys. Rev. A 88, 043634 (2013). [9] Y.-C. Zhang, S.-W. Song, C.-F. Liu, and W.-M. Liu, Phys. Rev. A 87, 023612 (2013). [10] C. L. Qu, C. Hamner, M. Gong, C. W. Zhang, and P. Engels, Phys. Rev. A 88, 021604 (2013). [11] A. J. Olson, S.-J. Wang, R. J. Niffenegger, C.-H. Li, C. H. Greene, and Y. P. Chen, Phys. Rev. A 90, 013616 (2014). [12] K. K. Jim´ enez-Garc´ ıa, L. J. LeBlanc, R. A. Williams, M . C. Beeler, C. Qu, M. Gong, C. Zhang, andI.B. Spielman, Phys. Rev. Lett. 114, 125301 (2015). [13] B. Xiong, J.-H. Zheng, and D.-W. Wang, Phys. Rev. A 91, 063602 (2015). [14] D. Sokolovski and E. Ya. Sherman Phys. Rev. A 89, 043614 (2014). [15] Y. Li, C.-L. Qu, Y.-S. Zhang, and C.-W. Zhang, Phys. Rev. A92, 013635 (2015). [16] L. Wen, Q. Sun, Y. Chen, D.-S. Wang, J. Hu, H. Chen, W.-M. Liu, G. Juzeli¯ unas, B. A. Malomed, and A.-C. Ji, Phys. Rev. A 94, 061602 (2016). [17] W. H¨ ansel, P. Hommelhoff, T.W. H¨ ansch, and J. Reichel, Nature(London) 413, 498 (2001). [18] M. Greiner, I. Bloch, T. W. H¨ ansch, and T. Esslinger, Phys. Rev. A 63, 031401(R) (2001). [19] T. L. Gustavson, A. P. Chikkatur, A. E. Leanhardt, A. G¨ orlitz, S. Gupta, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 88, 020401 (2001). [20] A. Couvert, T. Kawalec, G. Reinaudi, and D. Gu´ ery- Odelin, Europhys. Lett. 83, 13001 (2008). [21] J. L´ eonard, M. Lee, A. Morales, T. M Karg, T. Esslinger, and T. Donner, New J. Phys. 16, 093028 (2014).[22] E. Torrontegui, S. Ib´ a˜ nez, S. Mart´ ınez-Garaot, M. M od- ugno, A. del Campo, D. Gu´ ey-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, Adv. At. Mol. Opt. Phys. 62, 117 (2013). [23] E. Torrontegui, S. Ib´ a˜ nez, X. Chen, A. Ruschhaupt, D. Gu´ ery-Odelin, and J. G. Muga, Phys. Rev. A 83, 013415 (2011). [24] E. Torrontegui, X. Chen, M. Modugno, S. Schmidt, A. Ruschaupt, and J. G. Muga, New J. Phys. 14, 013031 (2012). [25] X.-J. Lu, J. G. Muga, X. Chen, U. G. Poschinger, F. Schmidt-Kaler, and A. Ruschhaupt, Phys. Rev. A 89, 063414 (2014). [26] Q. Zhang, X. Chen, and D. Gu´ ery-Odelin, Phys. Rev. A 92, 043410 (2015). [27] Q. Zhang, J. G. Muga, D. Gu´ ery-Odelin, J. Phys. B: At. Mol. Opt. Phys. 49, 125503 (2016). [28] J. G. Muga, X. Chen, A. Ruschhaupt, and D. Gu´ ery- Odelin, J. Phys. B: At. Mol. Opt. Phys. 42, 241001 (2009). [29] X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Gu´ ery-Odelin, and J. G. Muga, Phys. Rev. Lett. 104, 063002 (2010). [30] Y. Ban, X. Chen, E. Y. Sherman, and J. G. Muga, Phys. Rev. Lett. 109, 206602 (2012). [31] Y. Ban, X. Chen, J. G. Muga, and E. Y. Sherman, Phys. Rev. A91, 023604 (2015). [32] C. Echeverr´ ıa-Arrondo and E. Ya. Sherman, Phys. Rev. B88, 155328 (2013). [33] R. Li, J. Q. You, C. P. Sun, and F. Nori, Phys. Rev. Lett. 111, 086805 (2013). [34] T.ˇCadeˇ z, J. H. Jefferson, and A. Ramˇ sak, New J. Phys. 15, 013029 (2013). [35] T.ˇCadeˇ z, J. H. Jefferson, and A. Ramˇ sak, Phys. Rev. Lett.112, 150402 (2014). [36] Sh. Mardonov, M. Modugno, and E. Ya. Sherman, J. Phys. B: At. Mol. Opt. Phys. 48, 115302 (2015). [37] V. M. P´ erez-Garc´ ıa, H. Michinel, J. I. Cirac, M. Lewen - stein, and P. Zoller, Phys. Rev. Lett. 77, 5320 (1996). [38] E. QuinnandM. Haque, Phys.Rev.A 90, 053609 (2014). [39] R. R. Sakhel and A. R. Sakhel, J. Phys. Opt. Phys. 50, 105301 (2017). [40] C.-L. Qu, L. P. Pitaevskii, and S. Stringari, New J. Phys . 19, 085006 (2017).8 [41] J. Li, K. Sun, and X. Chen, Sci. Rep. 6, 38258 (2016). [42] J. Pawlowski, P. Szumniak, and S. Bednar ek, Phys. Rev. B94, 155407 (2016).

1810.03720v1.Zeeman_spin_orbit_coupling_in_antiferromagnetic_conductors.pdf

Zeeman spin-orbit coupling in antiferromagnetic conductors Revaz Ramazashvili Laboratoire de Physique Th eorique, Universit e de Toulouse, CNRS, UPS, France (Dated: October 10, 2018) This article is a brief review of Zeeman spin-orbit coupling, arising in a low-carrier commensurate N eel antiferromagnet subject to magnetic eld. The eld tends to lift the degeneracy of the electron spectrum. However, a hidden symmetry protects double degeneracy of Bloch eigenstates at special momenta in the Brillouin zone. The eective transverse g-factor vanishes at such points, thus acquiring a substan- tial momentum dependence, which turns a textbook Zeeman term into a spin-orbit coupling. After describing the symmetry underpinnings of the Zeeman spin-orbit coupling, I compare it with its intrinsic counterparts such as Rashba coupling, and then show how Zeeman spin-orbit coupling may survive in the presence of intrinsic spin-orbit coupling. Finally, I outline some of the likely experimental manifestations of Zeeman spin-orbit coupling, and compare it with similar phenomena in other settings such as semiconducting quantum wells.arXiv:1810.03720v1 [cond-mat.str-el] 8 Oct 20182 I. INTRODUCTION Spin-orbit coupling, the central character of this Special Issue, appears in non-relativistic quantum mechanics as only a vestige of relativity, in the form of the Pauli term HPin the Schr odinger Hamiltonian1 HP=~ 4m2 0c2
prV(r); (1) where ~is the Planck constant, cthe speed of light, m0is the free electron mass, pis the electron momentum, its spin, and V(r) its potential energy as a function of the electron coordinate r. In the low-energy expansion, the HParrives as a second-order term in the expansion in weak coupling constant =e2=~c1=137. In this sense, spin-orbit coupling is indeed small. Yet in solids, spin-orbit coupling is responsible for a variety of fundamental phenomena. In magnets, it may induce magnetocrystalline anisotropy, whereby spontaneous magnetization acquires a preferred set of directions with respect to the crystal axes.2,3In antiferromagnets, spin-orbit coupling may give rise to \weak" ferromagnetism.2,4{6In transition metal com- pounds, coupling of spin, orbital and structural degrees of freedom leads to a multitude of unusual phases.7In Mott insulators, spin-orbit coupling may produce interesting eects such as realization of an eective Heisenberg-Kitaev model.8Last but not the least, in an innocu- ous band insulator spin-orbit coupling may bring to life topologically non-trivial electron states, that have been a subject of much attention.9,10 In all of these cases, spin-orbit coupling is intrinsic : it acts in the absence of any exter- nal perturbation applied to the crystal. By contrast, the present article is devoted to the Zeeman spin-orbit coupling, that may appear in a low-carrier N eel antiferromagnet, subject to magnetic eld. Zeeman spin-orbit coupling is thus a particular example of the Zeeman eect. At the same time, it entangles orbital motion of an electron with its spin, and hence represents a true spin-orbit coupling. Zeeman spin-orbit coupling is proportional to the applied magnetic eld, and thus is inherently tunable. This distinguishes it from intrinsic spin-orbit coupling, that acts in the absence of any external eld, and that most articles in this Topical Issue focus on. By virtue of the Zeeman spin-orbit coupling, magnetic eld splits a single doubly- degenerate electron band into two bands that are non-degenerate almost everywhere in the Brillouin zone. While this is indeed just aform of the Zeeman eect, there is also a simi-3 larity here to how, in the absence of inversion symmetry, intrinsic spin-orbit coupling lifts double degeneracy of an electron band in a non-magnetic crystal. This similarity provides a useful perspective, hence the article opens with Section II, that presents a basic overview of degeneracies in a non-magnetic crystal, with and without external magnetic eld. Section III does the same for a commensurate collinear N eel antiferromagnet, and elucidates the analogy with non-magnetic case. Most of the Section III is a pedagogical presentation of the degeneracies expected in the absence of intrinsic spin-orbit coupling, where the electron spin is entirely decoupled from its orbital motion.11,12The subsection III.c then analyses spectral degeneracies that may appear in transverse magnetic eld in the presence of in- trinsic spin-orbit coupling. Section IV outlines the implications in the linear order in the eld, introduces the Zeeman spin-orbit coupling and recapitulates some of its properties. Section V addresses some of the likely experimental manifestations of the Zeeman spin-orbit coupling, while Section VI oers a brief comparison with other settings, and an outlook. The review covers early studies of the Zeeman spin-orbit coupling in antiferromagnets as well as a case that remains largely unexplored, where antiferromagnetism coexists with substantial intrinsic spin-orbit coupling. I attempted to make the presentation pedagogical and coherent, and hope the reader will nd such a review useful. II. SPECTRAL SYMMETRIES IN ZERO FIELD A. A non-magnetic crystal Key insight into spectral degeneracies can be gained by simple symmetry arguments. Consider a non-magnetic crystal, that is one symmetric under time reversal . Kramers theorem1tells us, that every single-electron Bloch eigenstate jpiat momentum phas a degenerate orthogonal partner jpiat momentump.13 But are there degenerate states at a given momentum? For a generic momentum pin the Brillouin zone, time reversal symmetry alone does not protect such a degeneracy { with an important exception of special momenta p, that are equivalent to their opposite up to a reciprocal lattice vector Q, so thatUp=p+Q, whereUis a point symmetry of the lattice. For such a p, the stateUjpiresides at momentum Up=p+Q. The states Ujpiandjpiare degenerate and orthogonal. In the nomenclature of the Brillouin zone,4 thepandUp=p+Qare one and the same momentum { which, therefore, hosts two degenerate orthogonal eigenstates. This can be most simply illustrated in one dimension, whereUis the identity, and p= 0 (the -point) and p=are the only such special momenta. At all other momenta, the eigenstates are non-degenerate, as shown in the central panel of the Fig. 1. Attentive reader will instantly recognize that these special momenta pare nothing but the \time reversal-invariant momenta" (TRIM), that have played crucial role in the analysis of topological insulators.14However, the role TRIM may play in the appearance of spectral degeneracies was appreciated long before the advent of topological electron systems.15{17 If, in addition to time reversal, the crystal is symmetric with respect to inversion I, then Bloch eigenstates are doubly degenerate in the entire Brillouin zone: at anymomentum p, the statesjpiandIjpireside at the very same momentum pand are orthogonal.18Put otherwise, if the crystal is symmetric with respect to both the time reversal and inversion I, then, by applying I,andIto a Bloch eigenstate jpiat an arbitrary momentum p, one obtains a quartet of degenerate mutually orthogonal states, of which Ijpiandjpireside at momentum p, whileIjpiandjpireside atp. This is schematically shown in the rightmost panel of the Fig. 1.19 Breaking time reversal symmetry (for instance, by applying magnetic eld) lifts the double degeneracy { generally, at all momenta in the Brillouin zone. However, an arbitrary Bloch eigenstatejpiat momentum pstill has a degenerate orthogonal partner state Ijpi at momentump, as long as pandpare dierent in the Brillouin zone: if p=p+Q, thenjpiandIjpiare one and the same state, as illustrated by the leftmost panel of the Fig. 1. As we saw above, breaking the inversion symmetry while retaining time reversal has a similar yet weaker eect: the double degeneracy at a given momentum is lifted everywhere except for those special momenta, that are equal to their opposite in the Brillouin zone. At the same time, a Bloch eigenstate jpiat an arbitrary momentum pstill has a degenerate orthogonal partner jpiat momentump. These simple arguments show that, in a non-magnetic crystal, it is the inversion asym- metry that lifts the double degeneracy of a given electron band almost everywhere in the Brillouin zone. The eect can be encapsulated in an intrinsic spin-orbit coupling term HSO of the form HSO=dp
; (2)5 -π πpε(p) -π πpε(p) -π πpε(p) p〉 ℐp〉θp〉p〉 p〉 ℐθp〉 ℐp〉θp〉 FIG. 1. (color online). Typical electron spectra "(p) in a one-dimensional non-magnetic crystal. Left panel: Only the inversion symmetry Iis present, but no time reversal . A Bloch eigenstate jpiat momentum pis degenerate with its partner state Ijpiat momentump, the two states denoted by black dots. Central panel: Only the time reversal symmetry is present, but no inversion symmetry. A Bloch eigenstate jpiat momentum pis degenerate with its partner state jpiat momentump, the two states denoted by black dots. The only two symmetry-protected degeneracies at a given momentum, also marked by black dots, reside at p= 0 and p=.Right panel: Both theandIare symmetries. In this case, the two degenerate orthogonal states jpi andIjpiat an arbitrary momentum pform a quartet with their degenerate partner states Ijpi andjpiat momentump. where dpis a real pseudovector, that is an odd function of momentum p, and is the electron spin. Depending on the crystal structure and chemical composition, spin-orbit coupling may take a multitude of forms, whose review would go far beyond the subject of the present article. A concise analysis of symmetry properties of spin-orbit coupling can be found in Refs.20,21. An early example of such a coupling was introduced by E. I. Rashba.22,23 B. Rashba spin-orbit coupling Rashba spin-orbit coupling22{25is a form of theHSOof the Eq. (2) in semiconductor quantum wells and heterostructures, where band mismatch at the interface produces electron states that are localized in the transverse direction, but behave as Bloch states in the two dimensions along the interface. For such states, at low momenta, the HSOin the Eq. (2) reduces to HR=R(
p^n); (3) whereRis a material-specic constant, pandare the electron momentum and spin operators, and ^nis the normal to the interface.6 What may be the order of magnitude of the R? For a na ve estimate, let us turn to the textbook Pauli spin-orbit coupling term of the Eq. (1). If this very term were responsible for the Rashba coupling, what value of Rwould it give rise to? In a quantum well, formed in an elemental material with atomic number Znear a at interface with vacuum, the average rV0(r) points transversely to the boundary, and is of the order of the Coulomb energy Ze2=aB, divided by the Bohr radius aB=~2 Zme2:hrV0(r)iZe2 a2 B^n. Properly averaged26, this yields an estimate RZ2e2 ~h e2 ~ci2 vB
Z 1372, withvB=e2 ~being the hydrogen Bohr velocity. Of course, this is not more than a crude order-of-magnitude estimate: a reliable evaluation of spin-orbit coupling parameters such as Rfrom rst principles is a problem in its own right, as is extracting Rfrom experimental data.24,25,27,28In most materials of experimental interest, the ratio R=c(cbeing the speed of light) falls in the range R=c 104102. The example of Rashba coupling will prove useful below, as a reference point for the strength of the Zeeman spin-orbit coupling. III. SPECTRAL SYMMETRIES OF A COLLINEAR N EEL ANTIFERROMAGNET A. Collinear N eel antiferromagnet in zero eld Before turning to the Zeeman spin-orbit coupling, let us dene the class of materials, where it is expected to appear { a collinear commensurate N eel antiferromagnet. At each point rin the sample, such an antiferromagnet is characterized by spontaneous local mag- netization density M(r), that changes sign upon translation by a period aof the crystal lattice: M(r+a) =M(r). The adjective \collinear" implies that, everywhere in the sample, M(r) points along or opposite a single direction, one and the same throughout the sample. In the following, M(r) will be treated as static; both thermal and quantum uctuations of magnetic order are thus entirely neglected. This will restrict us to the bulk of the anti- ferromagnetic phase, far from any phase transition, thermal or quantum. In particular, this requires temperatures far below both the N eel temperature and the electron bandwidth. In practice, this also implies the ordered moment, noticeable on the scale of the Bohr magneton. Some of potentially relevant materials were discussed in the Ref.12.7 First, it is helpful to understand the spectral degeneracies of such a magnet in zero eld, and there is a perfect analogy here with what happens in a non-magnetic crystal, whose spectral symmetries were recapitulated in the Subsection II.a. The local magnetization density M(r) couples to the electron spin via the N eel exchange coupling HN=
r
, where the
ris proportional to M(r) and thus inherits its transformation properties:
r changes sign upon lattice translation Taas well as upon time reversal . Thus, while neither Tanoris a symmetry of the N eel state, their product Tais indeed a symmetry. In the presence of inversion symmetry, action of Iand ofTaon an exact Bloch eigenstate jpiat an arbitrary momentum pgenerates a quartet of mutually orthogonal degenerate eigenstates: the doublet of jpiandITajpiat momentum p, and the doublet of Ijpiand Tajpiatp, as shown in the left panel of Fig. 3. Notice the analogy with the quartet of mutually orthogonal degenerate eigenstates in a centrosymmetric non-magnetic crystal, with a doublet of jpiandIjpiat momentum pand the doublet of Ijpiandjpiatp.29 In a non-centrosymmetric antiferromagnetic crystal, there is no single universal symmetry to protect a degeneracy of Bloch eigenstates at an arbitrary momentum. Moreover, in an antiferromagnet, the Kramers orthogonality relation hpjjpi=hpjjpifor a non-magnetic crystal is replaced by a less stringent condition hpjTajpi=e2ip
ahpjTajpi (4) for the scalar product of a given Bloch eigenstate jpiat momentum pand its symmetry partnerTajpiat momentump. The momenta psuch that p=p+Q(where Qis anantiferromagnetic reciprocal lattice vector) now fall into two classes.30,31The rst class are those p, wheree2ip
a= 1 (such as the relevant points at the non-magnetic Brillouin zone boundary, and the point at the center of the Brillouin zone); they host a doublet of degenerate states. By contrast, for those p, wheree2ip
a=1, theTaprotects no degeneracy. Up to the distinction outlined in the preceding paragraph, the analogy between a collinear commensurate N eel antiferromagnet and a paramagnet is complete. In a centrosymmetric material, all bands are doubly degenerate at all momenta in the Brillouin zone, be it an antiferromagnet or a non-magnetic crystal. In the absence of inversion symmetry, the elec- tron bands are non-degenerate, with the exception of special momenta in the Brillouin zone, where the time reversal (in a paramagnet) or Ta(in an antiferromagnet) protect a double8 degeneracy. Near such a momentum, the two split sub-bands can be described by an eective Hamiltonian with an intrinsic spin-orbit coupling term (2) with dp= 0 at the degeneracy point. Described above, symmetry-protected degeneracies in a commensurate N eel antiferro- magnet were understood over half a century ago.32However, symmetry in the presence of magnetic eld was not explored until much later. B. Collinear centrosymmetric N eel antiferromagnet in magnetic eld A fundamental dierence between a non-magnetic crystal and an antiferromagnet resides in how the double degeneracy of Bloch eigenstates at a given momentum is lifted by magnetic eld. In a non-magnetic crystal, the Zeeman term HZ= (H
) tends to lift the double degeneracy everywhere in the Brillouin zone. In the presence of inversion symmetry, this is illustrated in the leftmost panel of the Fig. 1. In an antiferromagnet, the situation is more interesting. Here, the eld is applied to a system that already has a preferred direction, dened by the staggered exchange eld
r. Thus, the lifting of the double degeneracy by magnetic eld Hmay be sensitive to how H is oriented relative to
r. It turns out, that a longitudinal eld Hk
rlifts the degeneracy everywhere in the Brillouin zone, as illustrated in the leftmost panel of the Fig. 1. This is hardly surprising. Much more interestingly, in a transverse eld H?
r, the system retains enough symmetry to protect double degeneracy of Bloch eigenstates at a special set of momenta in the Brillouin zone. This symmetry becomes apparent as soon as one pictures a collinear centrosymmetric N eel state in a transverse eld H, as sketched in the Fig. 2. In a lattice model, the two N eel sublattices simply tilt towards H, as do the
rin the upper left corner of the gure and the
r+ain its lower right corner. Magnetic moment along the eld is invariant upon translation bya, while the moment along the initial direction of the staggered magnetization changes sign. More generally, if the physics is not restricted to lattice sites, local magnetization is a function of continuous coordinate rand, in a transverse eld,
ris no longer collinear. However, exactly as in a lattice model, the component
k ralong the initial direction of
r inherits the N eel translation antisymmetry:
k r+a=
k r. By contrast, the component
? r along the applied eld represents the eld-induced magnetic moment and is translationally9 symmetric:
? r+a=
? r. π∆ ∆ θπ∆θπ∆=∆Hr+a rnU ( )n r rU ( ) r+a r naT U ( ) r U ( )n FIG. 2. (color online). Two points in space, randr+a, separated by half a period aof the N eel order { and the exchange eld
rand
r+aat these two points, upon applying magnetic eld H transversely to the initial direction nof the staggered magnetization. The gure also shows how the exchange eld changes upon various transformations, such as (i) Un() { spin rotation by  around n, (ii)Un() { combination of Un() with time reversal , and (iii)Un() combined with translation Ta. To simplify notations, the transformations are shown as if they were applied directly to
rrather than to
r
. Notice that, in a nite transverse eld H, the triple product TaUn() is a symmetry of the tilted N eel state. As we saw above, in zero eld the N eel exchange term HN=
r
was symmetric under Ta. Due to collinearity of
r, theHNwas also symmetric under spin rotation Un() around the N eel axis nby an arbitrary angle . The Un() is thus a symmetry of the zero-eld Hamiltonian { but only in the absence of spin-orbit coupling HSO=dp
, since the latter obviously varies under Un() due to non-collinearity of the dp. So, both Un() andTaare symmetries of the longitudinal part
k r
of the exchange term. Yet, both of these symmetries are broken by the Zeeman term HZ= (H
) and by the eld-induced term
? r
, since both change sign under Ta. Remarkably, this can be undone by a single uniform rotation Un(), a symmetry of the zero-eld state.33As a result, the combination TaUn() is a symmetry of the N eel state in a nite transverse eld. Its action on a Bloch eigenstatejpiproduces a degenerate partner eigenstate at momentum p. Does this symmetry lead to a degeneracy at a given momentum? It does. Anti-unitarity ofTaUn() leads to an analogue of the Kramers orthogonality relation11,12 hpjTaUn()jpi=e2ip
ahpjTaUn()jpi (5)10 between the Bloch eigenstate jpiat momentum pand its symmetry partner state at p. Those pthat are equivalent to their opposite modulo a reciprocal lattice vector Qof the antiferromagnetic state ( p=p+Q), and for whom the exponent in the right-hand side of the Eq. (5) is dierent from unity, host a doublet of degenerate states. The equation p=p+Qimplies that such momenta plie at the antiferromagnetic Brillouin zone boundary. However, for those pthat also belong to the non-magnetic Bril- louin zone boundary,34the exponent e2ip
ain the r.h.s. of the Eq. (5) equals unity, and thus such a pdoes not host a symmetry-protected degeneracy. The precise geometry of the set of pdepends on the conspiracy between the crystal symmetry and the periodicity of the antiferromagnetic order. A number of possible examples were discussed in the Ref.12. In a one-dimensional doubly-commensurate antiferromagnet, the degeneracy is guaranteed atp==2, as illustrated in the Fig. 3. -π -π 2π 2πpϵ(p) -π -π 2π 2πpϵ(p) p〉 ℐθTap〉 ℐp〉θTap〉θTaUn(π)p〉 p〉 FIG. 3. (color online). Typical electron spectra "(p) in a centrosymmetric doubly-commensurate one-dimensional N eel antiferromagnet. Left panel: In zero eld, for any momentum pin the Brillouin zone, the symmetries IandTagenerate a quartet of mutually orthogonal degenerate Bloch eigenstates: a doublet jpiandITajpiat momentum p, and a doubletIjpiandTajpiat momentump. Each doublet is denoted by black dot. Right panel: In a transverse magnetic eldH, the surviving symmetry TaUn() protects double degeneracy at the magnetic Brillouin zone boundary p==2. The degenerate Bloch states jpiandTaUn()jpiare denoted by black dots. C. N eel insulator with intrinsic spin-orbit coupling The arguments above demonstrated how, at a special set of momenta in the Brillouin zone, symmetry protects double degeneracy of the electron spectrum in a centrosymmetric11 N eel antiferromagnet, subject to a nite transverse magnetic eld. However, these arguments tacitly implied both the N eel order and the electron spin to be decoupled from the underlying crystal lattice. In other words, the intrinsic spin-orbit coupling HSOof the Eq. (2) was neglected altogether. Indeed, as a (formally) small relativistic correction, the HSOmay often be ignored. At the same time, the opposite limit of a signicant spin-orbit coupling is of considerable interest: rstly, intrinsic spin-orbit coupling grows rapidly with the atomic number Z. In many antiferromagnets, this alone may rule out the possibility of neglecting HSO. Secondly, spin- orbit coupling plays a key role in topological properties of condensed matter and, recently, a body of work has been devoted to topological properties of antiferromagnetic insulators { see Ref.35and the subsequent work by several groups of authors. Topologically non-trivial electron properties are intimately related to degeneracies in the electron spectrum. In a N eel antiferromagnet without intrinsic spin-orbit coupling, such degeneracies are symmetry-protected even in a nite transverse magnetic eld11,12. But could symmetry-protected degeneracies be present in an antiferromagnet with substantial intrinsic spin-orbit coupling?36 At rst sight, this is unlikely. As we saw in Subsections II.a and III.a, intrinsic spin-orbit coupling tends to lift the spectral degeneracy everywhere, except for a special set of points in the Brillouin zone. It seems that magnetic eld could only lift any remaining degeneracy. Yet, this is not necessarily the case, as shown below. To advance further, we must analyze the symmetries of a Hamiltonian that involves, in addition to a common \non-magnetic" part, three key terms. Firstly, as outlined above, antiferromagnetic order couples to the electron spin via the N eel exchange coupling HN=
r
with collinear eld
r, such that
r+a=
r. Secondly, an intrinsic spin-orbit coupling of the form HSO= (dp
) involves the eld dpthat is, generally, non-collinear, and transforms as a vector representation of the crystal point group. Finally, magnetic eld Hgives rise to the Zeeman term HZ= (H
). Without magnetic eld, the double degeneracy at a given momentum is protected by the very same symmetry Taas in an antiferromagnet without intrinsic spin-orbit coupling: the HSO= (dp
) respects both andTaseparately, while the N eel coupling HN=
r
 is symmetric only under the product Ta, which protects the double degeneracy at those momenta pin the Brillouin zone, that are equivalent to their opposite ( p=p+Q) in the12 antiferromagnetic Brillouin zone. While the set of such momenta in the antiferromagnetic state is dierent from its paramagnetic counterpart, the protecting symmetry Taremains the same. This simple picture changes once magnetic eld is turned on. The reason can be traced back to a N eel antiferromagnet without intrinsic spin-orbit coupling, where Kramers de- generacy in a transverse magnetic eld hinges on the combined symmetry TaUn(), with Un() being the spin rotation by around the unit vector nof the N eel magnetization. Generally, intrinsic spin-orbit coupling HSO= (dp
) is not invariant under Un(), let alone Un() with an arbitrary rotation angle . Thus, with theHSOpresent,TaUn() is no longer a symmetry of the problem. To begin with, in the presence of HSOwe must specify the orientation of the collinear eld
rwith respect to the dp; the latter is generally non-collinear and realizes a vector repre- sentation of the crystal point group. My goal here is not to provide a complete classication, but to demonstrate an interesting possibility which is, at the same time, general enough. Not surprisingly, such a possibility appears in a symmetric conguration, where the
r points along a high-symmetry direction of the dp. For the Bernevig-Hughes-Zhang (BHZ) model37withdBHZ p/(px;py;dz[p2 x+p2 y]), the high-symmetry direction is the z-axis. Below, I consider the case of the
rpointing along the z-axis of the dBHZ p. In fact, this choice was implicitly made already in prior work on antiferromagnetic topological insulators,35where the
rwas chosen to point along the symmetry axis ^ zof the Bernevig-Hughes-Zhang (BHZ) model37in a two-dimensional crystal of square symmetry, with dBHZ p/(px;py;dz[p2 x+p2 y]) near the point of the Brillouin zone. What is the unitary symmetry of the BHZ spin-orbit coupling HBHZ SO=dBHZ p
? Upon spin rotation by angle , the dBHZ p rotates by the same angle. To make a symmetry, this spin rotation must be compensated by orbital rotation Rn() by the opposite angle. The HBHZ SOis thus symmetric under the product Un()Rn() with an angle , respecting the point symmetry of the dBHZ p. Now, what happens in a eld H?, transverse with respect to n? WithoutHBHZ SO, the productTaUn() was a symmetry in such a transverse eld. The HSOis symmetric under both theandTa, but not under Un(): as we saw in the previous paragraph, to make a symmetry, Un() had to be combined with the orbital rotation Rn() around the same axis. As a result, a generic Hamiltonian, involving the HBHZ SO, the N eel exchange HN=
r
13 and the transverse eld H?
, is invariant under the combination TaUn()Rn(). Does this symmetry induce a degeneracy? It does, if [TaUn()Rn()]26= 1. Which, in turn, depends on the mutual orientation of aandn. Ifa?n, then [ TaRn()]2= 1, hence [ TaUn()Rn()]2= 1, and the symmetry TaUn()Rn() induces no degeneracy.38In two dimensions, a?nis the only possibil- ity. Thus, in a two-dimensional N eel antiferromagnet with intrinsic spin-orbit coupling, transverse magnetic eld lifts the Kramers degeneracy. In three dimensions, the aandnmay be parallel; then Tacommutes withRn(), and [TaRn()]2=T2a. As a result, when acting on a Bloch eigenstate jpi, [TaUn()Rn()]2= T2a=e2ip
a. Also, in contrast to the case a?n, foraknthe stateTaUn()Rn()jpi resides at the momentum p.39In the antiferromagnetic Brillouin zone, the momentum planes p
a==2 are equivalent, and for such momenta [ TaUn()Rn()]2=T2a=ei= 1. Thus, at such momenta, the symmetry TaUn()Rn() protects double degeneracy in a transverse magnetic eld. This is illustrated in the Fig. 4. It is convenient to illustrate the arguments above by the following scheme, showing the symmetries of the various terms, and how they are broken as new terms are included in the Hamiltonian. The arrows show how the broken symmetries combine to form a surviving symmetry. From left to right, the columns represent the N eel exchange HN=
r
, the intrinsic spin-orbit coupling HSO=dp
, and the combination of the two with the transverse Zeeman term HZ=H?
.
r
 dp
 (dp+
r)
 (dp+
r+H?)
 T2a Ta T2a T2a Ta  Ta Un() Un()Rn() Un()Rn()TaUn()Rn() Rn( )14 FIG. 4. (color online). The Brillouin zone of a three-dimensional N eel antiferromagnet with intrinsic spin-orbit coupling. The momentum plane p
a=marks the paramagnetic Brillouin zone boundary in the zdirection, and is shown in darker gray. As explained in the main text, both the N eel half-period aand the staggered magnetization
rpoint along the high-symmetry axis n of the spin-orbit coupling, that is along the zaxis of the Brillouin zone, shown by the red arrow. The blue arrow shows the magnetic eld, normal to z. The momentum plane p
a==2, shown in lighter gray, marks the antiferromagnetic Brillouin zone boundary in the zdirection, and hosts doubly degenerate Bloch states. IV. ZEEMAN SPIN-ORBIT COUPLING The arguments of the preceding section have demonstrated a rather peculiar phenomenon: in a commensurate N eel antiferromagnet subject to transverse magnetic eld, Bloch eigen- states remain doubly degenerate at a set of special momenta in the Brillouin zone. This degeneracy is protected by symmetry even in a nite eld { and, therefore, holds to any order in the eld. A fundamental consequence of this degeneracy appears already in the rst order in the eld, that is in the form of the eective Zeeman termHeff Z. By its very nature, antiferro- magnet has a special direction, set by the staggered magnetization. Of course, other special directions may exist { dened, for instance, by the crystal structure, but I will rst describe the limit where there are none, that is in the absence of any intrinsic spin-orbit coupling. In the latter case, the only anisotropy of Heff Zis set by the orientation of the magnetic15 eldHrelative to the unit vector nof the staggered magnetization: Heff Z=B 2 gk(Hk
) +g?(H?
) : Here,Bis the Bohr magneton, the Hkn(H
n) and H?HHkare the longitudinal and transverse components of the eld with respect to the unit vector nof the staggered magnetization, and the gkandg?are the longitudinal and transverse g-factors, respectively. Now, in a transverse eld, double degeneracy of Bloch eigenstates at certain special momenta p=pin the Brillouin zone means that the g?must vanish at such momenta. Not being identically equal to zero, the g?must, therefore, substantially depend on momentum p, and theHeff Zshall be re-written as Heff Z=B 2 gk(Hk
) +g?(p)(H?
) : (6) Momentum dependence of the second term in the r.h.s. above, along with the presence of electron spin , turns the textbook Zeeman term into a veritable spin-orbit coupling that is the subject of the present article. Thegkis a constant, while general properties of the g?(p) are as follows.12In the absence of intrinsic spin-orbit coupling, the g?(p) vanishes on a manifold, dened by the equation g?(p) = 0 in the d-dimensional Brillouin zone. This \degeneracy manifold" is ( d1)- dimensional. In one dimension, it comprises a set of special points. In two dimensions { a set of special lines.40,41In three dimensions, it forms a set of special surfaces. Nevertheless, over most of the Brillouin zone, the g?(p) is close to1, and diers from these two values only within a momentum range of the order of~ ~ aaround the degeneracy manifold. Here is of the order of the antiferromagnetic coherence length12, and is large on the scale of the lattice spacing a. Variation of g?(p) is thus limited to a momentum range that is small compared with the size of the Brillouin zone. Within a momentum range of about~ around the manifold g?(p) = 0, theg?(p) varies linearly with momentum: g?(p)p=~, wherep is measured along the local normal to the degeneracy manifold. Such a variation of g?(p) is illustrated in the Fig. 5. This simplest form of the Zeeman spin-orbit coupling (6) is linear in momentum, as is the intrinsic Rashba spin-orbit coupling (3). Thus it is instructive to compare the two. Firstly, the Zeeman spin-orbit coupling eld dZSO(p) =B 2H?p ~points along the H?. By contrast, the spin-orbit eld dR(p) =Rp^ nof the Rashba coupling is manifestly non-collinear. Secondly, let us compare the two coecients of the linear momentum dependence.16 FIG. 5. (color online). Typical variation of g?(p) as a function of momentum along the local normal to the degeneracy manifold g?(p) = 0. Within a momentum range of the order of ~=, theg?(p) varies essentially linearly: g?(p)p=~. Beyond this range, the g?(p) becomes nearly constant. For the Rashba coupling, the coecient is given by the R. For an elemental material with atomic number Z, the na ve order-of-magnitude estimate of the Section II.b gave Re2 ~
Z 1372. Relative to the speed of light c, this yields R=c1 137h Ze2 ~ci2 1 137Z 1372. In practice, R=c104102. By contrast, the Zeeman spin-orbit coecient relative to cis given by ZSO=c BH? ~cvF cBH?
e2 ~cBH?
1 137BH?
, where the Fermi velocity vFand the antifer- romagnetic coherence length were estimated as per vFe2 ~and~
=vF, with
being the antiferromagnetic gap in the electron spectrum. Notice that ZSO=ccarries only a single weak coupling constant 1=137 compared with 3in the estimate of R=c. The ratio of the two couplings is thus ZSO=R~c Ze22BH?
137 Z2BH?
. The present theory applies only in the limitBH?
1. However, the factor137 Z2is generally large. While this is, indeed, only a hand-waving argument, it indicates that even in a relatively weak eldBH?
1, Zeeman spin-orbit coupling may be comparable to a rather substantial Rashba coupling. Near higher-symmetry points in the Brillouin zone, the degeneracy manifold may cross itself; in this case the g?(p) shows a more interesting behavior. For example, in a two- dimensional antiferromagnet on a square lattice, the degeneracy lines coincide with the magnetic Brillouin zone boundary px+py=andpx+py=. Near the corner point (0;) and (;0) of the magnetic Brillouin zone (point X), one nds g?(p)/p2 xp2 y. At such a point, a degenerate isotropic band extremum (p) =p2=2mis split by transverse eld into two ellipsoid sub-bands (p) = [1H?=
]p2 x=2m+ [1H?=
]p2 y=2m, where the energy scale
characterizes g?(p) near point X. Such a behavior is shown in the17 Fig. 6.40The same gure also illustrates linear band splitting near point  = ( =2;=2), where a band minimum (p) =p2 x=2mx+p2 y=2myis split by transverse eld H?as per (p) =p2 x=2mx+p2 y=2mypy ~(H?
). The degenerate band minimum splits into two identical non-degenerate minima, shifted with respect to each other in momentum space: (p) =p2 x=2mx+h pymy ~(H?
)i2 =2my. FIG. 6. (color online). Zeeman splitting of small carrier pockets, notionally centered at the points Xand  in the rst quadrant of the Brillouin zone of a two-dimensional N eel antiferromagnet on a square lattice. The splitting is induced by magnetic eld, transverse to the staggered magnetization. The dashed line, passing through the points Xand , is the magnetic Brillouin zone boundary, whereg?(p) = 0. The pocket sizes and the splitting are exaggerated. What is the leading term in the momentum expansion of the Zeeman spin-orbit coupling g?(p)(H?
) around the degeneracy planes pz==2, in the presence of an intrinsic spin-orbit coupling, described in the Subsection III.c? The symmetry of g?(p)(H?
) must match that of the Hamiltonian. In particular, it must be invariant under TaUn()Rn(). This implies g?(px;py;pz) =g ?(px;py;pz), while the degeneracy means g?(px;py;=2) = 0. Sinceg?(p)(H?
) is Hermitian, a linear term in pzwith an imaginary coecient is forbidden. The leading term allowed is thus Heff Z/p2 z(H?
). V. EXPERIMENTAL MANIFESTATIONS Zeeman spin-orbit coupling may manifest itself in a number of ways, that all stem from eld-induced entanglement of the electron spin with its orbital motion. As a result, the18 Landau level spectrum and its Zeeman splitting acquire an unusual dependence on the eld orientation42{44that, in turn, has a number of experimental consequences. A. Magnetic quantum oscillations as a diagnostic tool One such consequence is that, in a purely transverse eld, Landau levels undergo no Zeeman splitting42{44{ if the carrier pocket is centered at a symmetry-protected degener- acy point such as those shown in the Fig. 6.40,41In magnetic quantum oscillations, Zeeman splitting of the Landau levels produces the so-called `spin zeros'45, that is special eld orien- tations, where the oscillation amplitude vanishes. In an antiferromagnetic insulator, absence of spin zeros may thus constrain the precise position of the band extremum in the Brillouin zone, or even pinpoint it exactly.40,41This could be particularly useful in antiferromagnets of complex structure and chemical composition, where the location of the band extrema in the Brillouin zone may be less than obvious. B. Zeeman electric-dipole resonance In addition to thermodynamic measurements such as magnetic quantum oscillations, the Landau level spectrum can be studied by resonant spectroscopy. According to textbook, AC electric eld excites spin-conserving cyclotron resonance (CR) transitions between adjacent Landau levels, at the Larmor frequency, whereas AC magnetic eld excites spin ip (spin resonance, ESR) transitions at the Zeeman frequency, at a xed Landau level. Intrinsic spin-orbit coupling enriches this simple picture rather dramatically: it allows one to induce ESR transitions by an AC electric rather than magnetic eld { and, more generally, to induce transitions at combined frequencies. Predicted by E. I. Rashba over fty years ago,20 such transitions are called \electric-dipole spin resonance" or \combined resonance". In case of Zeeman spin-orbit coupling, such transitions may be called Zeeman electric-dipole resonance; their theory was developed in the Ref.44. The term `electric-dipole resonance' implies that the resonance arises from a dipole term eE
rin the Hamiltonian, where Eis the external electric eld, and ris the displacement. The resonance matrix elements are thus determined by the characteristic scale of r. For the textbook ESR, this length scale is given by the Compton length C=~ mc0:4 pm.19 By contrast, for the Zeeman electric-dipole resonance, the length scale in question is the antiferromagnetic coherence length =~vF=
,44where
is the antiferromagnetic gap in the electron spectrum. Thus, matrix elements of Zeeman electric-dipole resonance exceed those of ESR by about ~c e2
F
137
F
, or at least by two orders of magnitude. Resonance absorption is proportional to the square of the transition matrix element; thus the absorption due to electric excitation of spin transitions exceeds that of ESR at least by four orders of magnitude. Last but not the least, Zeeman electric-dipole resonance absorption shows a non-trivial dependence on the orientation of the AC electric eld with respect to the crystal axes, and on the orientation of the DC magnetic eld with respect to the staggered magnetization. VI. SUMMARY AND OUTLOOK As argued above, Zeeman spin-orbit coupling in a N eel antiferromagnet is induced by magnetic eld and arises due to a hidden symmetry, that protects double degeneracy of electron eigenstates at special momenta in the Brillouin zone. These special momenta form a degeneracy manifold, whose dimensionality is reduced against that of the Brillouin zone. Limited to special momenta, the degeneracy means that the transverse g-factor acquires a substantial momentum dependence which is, however, limited to a relatively small part of the Brillouin zone. Therefore, observation of Zeeman spin-orbit coupling requires that carriers be limited to this small part of the Brillouin zone. Which, in turn, suggests low- carrier antiferromagnetic conductors as a likely system with Zeeman spin-orbit coupling. An extended discussion of relevant materials and experimental constraints may be found in the Refs.12,44. A. Zeeman spin-orbit coupling in semiconducting quantum wells To gain a better perspective, it is instructive to look for Zeeman spin-orbit coupling in materials other than antiferromagnets. A welcome example is provided by III-V direct band gap semiconductors of zinc-blende structure, such as GaAs and InAs. Before turning to details, let us recall that, in the simplest case of well-separated conduction and valence bands, it is the intrinsic spin-orbit coupling that may render the g-tensor momentum-dependent.20 In a centrosymmetric system, spin-orbit coupling has no intraband matrix elements20,21, and thus the corrections it induces to the g-tensor are small at least in the measure of
SO=E01, where
SOis the interband matrix element of the intrinsic spin-orbit coupling, andE0is the band gap.46The same small ratio limits the relative variation of the g-tensor across the Brillouin zone. This picture becomes more involved for touching bands, and also if other bands are present nearby, which is the case at the -point of bulk GaAs and InAs (see Fig. 7). Here, the conduction band is made mostly of s-orbital states, whereas the holes are of p-orbital nature. At the -point, the six hole p-states split into a J= 3=2 quartet and a J= 1=2 doublet, separated from the quartet by the `spin-orbit gap'
0. Upon leaving the -point, the quartet splits into two doubly degenerate bands: the predominantly Jz=3=2 `heavy hole' (HH) band and its predominantly Jz=1=2 `light hole' (LH) counterpart. Yet, in bulk GaAs, the electron g-factor remains isotropic, with g0:44, whereas in bulk InAs g15; in both materials relative variation of gacross the Brillouin zone is of the order of 0:04.47 E(k) kzz LH:J =+1/2J=3/2 J=1/2HH:J =+3/2 SOvalence band (p)conduction band (s) ∆0E0 FIG. 7. (color online). Band structure of GaAs, sketched near fundamental gap E0in the vicinity of the -point. As explained in the main text, the conduction band is formed by s-orbitals while the valence band is formed by p-orbitals. The valence band ( J= 3=2) splits into Jz=3=2 heavy hole (HH) band and Jz=1=2 light hole (LH) band, touching at the -point. The HH and LH bands are separated from the J= 1=2 band by the spin-orbit gap
0. Eective anisotropy can be enhanced by crossing over from a bulk sample to nearly two- dimensional semiconducting quantum well. Due to a nite size of the well, hole motion in the growth direction becomes quantized, and interplay of the reduced dimensionality with the21 p-state anisotropy of holes brings about many interesting eects, that have been a subject of active ongoing study.27,47,48Here, we will be interested only in the properties of the hole g-tensor and, for simplicity, will consider only eld along the plane of the quantum well. Level separation due to nite size of the well in the growth direction will be assumed to greatly exceed all the other relevant energy scales such as the spin-orbit gap
0and the hole cyclotron frequencies. Zeeman splitting of both the `heavy holes' HH ( Jz=3=2 at momentum k= 0) and `light holes' LH ( Jz=1=2 atk= 0) in magnetic eld Harises27from the following two terms: H=2B (J
H) +q(J3 xHx+J3 yHy+J3 zHz) ; (7) whereBis the Bohr magneton, vector J= (Jx;Jy;Jz) is made of the 4 4 matrices of the angular momentum J= 3=2, andandqare parameters of the Luttinger Hamiltonian.49 Notice that the rst term above is spherically symmetric, whereas the second term arises because the bulk symmetry is cubic rather than spherical. Accordingly, the coecient is of the order of unity, whereas qtends to be small: in GaAs, 1:2 andq0:04.47For light holes at k= 0, the rst term in the Eq. (7) produces non-zero splitting proportional to ; the light hole g-factor is thus of the order of unity.50By contrast, for heavy holes at the -pointJz= 3, and the rst term of the Eq. (7) does not contribute to the Zeeman eect. Zeeman splitting of heavy holes arises from the second term in the Eq. (7) and is small in the measure of q0:04.50At non-zero k, admixture of Jz=1=2 components to the heavy holes produces contributions to the gfactor, proportional to k2,k4and so on: g(k) =g0+ (ka)2g2() + (ka)4g4() +:::; (8) whereais mostly dened by the characteristic width of the well in the growth direction, g0/qis a constant, and g2() andg4() are functions of the momentum direction in the plane of the well.27,51{54Hole carriers in a GaAs quantum well are thus subject to Zeeman spin-orbit coupling. Let us compare the latter with its counterpart in a N eel antiferromagnet: Firstly, in semiconducting quantum wells Zeeman spin-orbit coupling appears due to anisotropy, enhanced by reduced dimensionality of the well. By contrast, in a N eel an- tiferromagnet, Zeeman spin-orbit coupling emerges due to conspiracy of the anti-unitary symmetry of the N eel order with the symmetry of the crystal lattice. As shown in Section22 III.c, in an antiferromagnet intrinsic spin-orbit coupling and reduced dimensionality may play a subsidiary role. Secondly, in a quantum well Zeeman spin-orbit coupling appears on the background of a non-zero constant g-factor: one may divide the in-plane components of the g-tensor into a momentum-independent term g06= 0 and a momentum-dependent part g(k), so that g(k) =g0+g(k), withg(0) = 0. The functional form of g(k) and its scale relative to g0depend on the width of the well and its growth direction. In GaAs quantum wells, some studies have found g(k)g0,51, while others27,52,55found the scales of g(k) andg0to be comparable for most growth directions { and numerically small, with typical measured values ofgin the plane being of the order of 102.27,51,52,55Notionally reducing g(k) at a constantg0(for instance, by increasing the width of the quantum well), this picture can be continuously tuned to the textbook case of the constant momentum-independent g-tensor, and Zeeman spin-orbit coupling can thus be \switched o". In a N eel antiferromagnet, symmetry-protected zeros of the transverse g-factor are qualitatively dierent: here g0= 0 while the characteristic scale of g(k) is unity, and Zeeman spin-orbit coupling can be \switched o" only by destroying the N eel state. Quantum wells grown in the high-symmetry direction [111] are an exception to the above. Such wells are symmetric under 2 =3 rotation around the growth axis, whereas the second term in the Eq. (7) is not, and thus its Zeeman matrix elements vanish: in such a well, g0= 0 is protected by the symmetry of growth direction, just as in an antiferromagnet g0= 0 is protected by a hidden anti-unitary symmetry.56However, in contrast to an antiferromagnet, the momentum-dependent part g(k) in GaAs is still small compared with unity. Much greater bulk g-factor (g15) makes InAs quantum wells promising in this regard.57 Thirdly, III-V semiconductors of zinc-blende structure lack inversion center. Therefore, in a generic quantum well, both HH and LH bands are split by Dresselhaus and Rashba spin-orbit couplings.58,59This splitting may be signicant, thus obscuring the Zeeman eect in general and Zeeman spin-orbit coupling in particular. By contrast, in a centrosymmetric N eel antiferromagnet such eects are absent. Moreover, as we saw in Section III.c, in an antiferromagnet Zeeman spin-orbit coupling may be symmetry-protected even in the presence of intrinsic spin-orbit coupling. To summarize, both weakly-doped antiferromagnetic insulators and III-V semiconduct- ing quantum wells have their stronger and weaker points for the study of Zeeman spin-orbit23 coupling. GaAs quantum wells have been studied in great detail. At the same time, typi- cal values of in-plane hole g-factor in this material are small. Semiconductors with greater g-factor, such as InAs, hold promise in this regard. By contrast, in a low-carrier N eel antifer- romagnet, Zeeman spin-orbit coupling is protected by symmetry. At the same time, many such materials have not yet been as well-characterized as III-V semiconductors. Clearly, more studies are called for. B. Other possibilities One may ask whether analogues of the Zeeman spin-orbit coupling may arise in a dierent physical context. Indeed, one such possibility is known in the cold-atom physics, where synthetic spin-orbit coupling may be generated and tuned in an experiment, involving laser- induced Raman transitions between two internal states of an atom playing the role of its `pseudospin' (for a review, see60and references therein). Synthetic spin-orbit coupling can be presented as an articial gauge eld. Just as Zeeman spin-orbit coupling, synthetic spin- orbit coupling in cold-atom setups is tunable, which holds great promise in spite of numerous experimental challenges60. Another closely related and very interesting development involves non-symmorphic mag- netic crystals61,62, that may give rise to degeneracies akin to those described above { and, therefore, to some of the similar eects. So far explored very little, this direction may bring some positive surprises. C. Acknowledgements It is my pleasure to thank Prof. P. Pujol for inspiring conversations on the subject, and Profs. M. M. Glazov, Y. B. Lyanda-Geller and E. Ya. Sherman for helpful discussions of the physics of quantum wells.24 VII. APPENDIX: ORTHOGONALITY RELATION AND KRAMERS THEOREM This Appendix proves the relation hj[O]+j[O]j i=h ji (9) and points out some of its consequences. Here jiandj iare arbitrary states, Ois an arbitrary unitary operator, and is time reversal. In the main text, this relation is used for ji=Oj i; in this case, when read right to left, Eqn. (9) yields h jOj i=h j[(O)+]2j(O)j i: (10) Wheneverj iis an eigenvector of the linear operator [ O]2with an eigenvalue dierent from unity, the Eqn. (10) proves orthogonality of j iandOj i. The proof of Eqn. (9) is based on the obvious relation ( C;C ) = ( ;) for arbitrary complex vectors and , where ( ;)P i  iidenotes scalar product, and Cis complex conjugation. Hence, for an arbitrary unitary operator O, one nds (OC;OC ) = ( ;), due to invariance of scalar product under unitary transformation. Time reversal can be presented as a product of Cand a unitary operator63:=VC, thusC=V1and, therefore, (O;O ) = ( ;). As a result, for arbitrary states j iandji, one nds hj[O]+j[O]j i=h ji, which indeed amounts to (9). Electron being a spin-1 =2 particle leads one to dene time reversal for a single-electron wave function as per =iyC, where the Pauli matrix yacts on the electron spinor, and Cis complex conjugation. This implies 2=1. Ifis a symmetry and j iis an eigenstate, thenj iandj iare degenerate. But are they linearly independent? For O= 1, the equality 2=1 makes the Eqn. (10) read h jj i=h jj i= 0, demonstrating orthogonality of thej iandj iand thus proving the Kramers theorem: in a time reversal-invariant system, every single-electron level is at least doubly degenerate. The Kramers theorem above relies on the 2=1 property of spin-1 =2 particles. By contrast, in a N eel antiferromagnet, some of the relevant anti-unitary symmetry operators such asTa,TaUn() andTaUn()Rn() do notsquare to a C-number { but, rather, to an operator: for instance [ Ta]2=T2a. However, the Bloch states in question are eigen- states of such squared operators, with the eigenvalue dierent from unity. This guarantees25 degeneracy, as mentioned immediately below the Eq. (10). 1C. Kittel, Quantum Theory of Solids , John Wiley & Sons, Inc., New York { London (1963). 2L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media , Volume VIII of Course in Theoretical Physics, Elsevier (2008). 3B. D. Cullity and C. D. Graham, Introduction to Magnetic Materials , John Wiley & Sons, Inc., Hoboken, New Jersey (2009). 4I. E. Dzialoshinskii, Sov. Phys. JETP 5, 1259 (1957). 5I. E. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). 6T. Moriya, Phys. Rev. 120, 91 (1960). 7K. I. Kugel' and D. I. Khomski, Sov. Phys. Usp. 25, 231 (1982). 8G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). 9M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 10J. E. Moore, Nature 464, 194 (2010). 11R. Ramazashvili, Phys. Rev. Lett. 101, 137202 (2008). 12R. Ramazashvili, Phys. Rev. 79, 184432 (2009). 13The fact that the state jpiresides at momentum pcan be demonstrated by acting on jpiby an arbitrary lattice translation Tb: Tbjpi=Tbjpi=exp [ip
b]jpi= exp [ip
b]jpi: Here the rst equality used the commutativity of Tband, the second used the Bloch theorem (Tbjki= exp [ik
b]jki), and the third one used the anti-unitarity of . The orthogonality relationhpjjpi= 0 follows from the Eq. (10) of the Appendix with O= 1 { and from the equality2=1, stemming from the spin-1 =2 nature of the electron. 14L. Fu, C. L. Kane, E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007). 15C. Herring, Phys. Rev. 52, 361 (1937). 16J. O. Dimmock and R. G. Wheeler, J. Phys. Chem. Solids 23, 729 (1962). 17J. O. Dimmock and R. G. Wheeler, Phys. Rev. 127, 391 (1962). 18The fact that the state Ijpihas momentum label pcan be veried by acting on Ijpiby a26 translation symmetry operator Tb: TbIjpi=ITbjpi=ITbjpi=Ieib
pjpi=eib
pIjpi; whereas the orthogonality relation hpjIjpi= 0 follows from the Eq. (10) of the Appendix and from the equality [ I]2=1. Notice that this result depends neither on the realization of the system nor on the choice of its eigenstates. 19Notice that the four states of the quartet are mutually orthogonal only when the momenta p andpare dierent in the Brillouin zone. At p= 0 and p=, only two of the four states remain linearly independent, as illustrated in the rightmost panel of the Fig. 1 20E. I. Rashba, Sov. Physics Uspekhi, 7, 823 (1965) 21K.V. Samokhin, Annals of Physics 324, 2385 (2009) 22E. I. Rashba, V. I. Sheka, Fizika Tverdogo Tela; Collected Papers (Moscow and Leningrad: Academy of Sciences of the USSR) Vol. 2, p. 162 (1959). 23E. I. Rashba, Proc. Int. Conf. Semiconductor Physics (Prague) 1960 (Prague: Czechoslovakian Academy of Sciences Publishing House), p. 45 (1961). 24Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984). 25Yu. A. Bychkov and E. I. Rashba, J. Phys. C: Solid State Phys., 17, 6039 (1984). 26L. D. Landau and E. M. Lifshitz, Quantum Mechanics , Volume III of Course in Theoretical Physics, Pergamon Press (1991). 27R. Winkler, Spin-Orbit Coupling Eects in Two-Dimensional Electron and Hole Systems , Springer Tracts in Modern Physics, Volume 191, Berlin { Heidelberg (2003). 28J. R. Bindel, M. Pezzotta, J. Ulrich, M. Liebmann, E. Ya. Sherman and M. Morgenstern, Nature Physics 12, 920 (2016). 29The momentum labels of the three states Ijpi,TajpiandITajpican be found as it was done for a non-magnetic crystal, by acting on the state with a translation operator Tb, that is a symmetry of the N eel state. As an example, consider the state ITajpi, the others can be treated analogously: TbITajpi=ITaTbjpi=ITaexp [ip
b]jpi= exp [ip
b]ITajpi: The orthogonality of the states TajpiandITajpito the statejpifollows from the Eq. (10) of the Appendix, and from the equalities [ ITa]2=1 and [Ta]2=T2a=exp [2ip
a]. The last equality holds for the action on jpi, since T2ais a translation symmetry of the N eel state.27 The orthogonality of Ijpiandjpican be demonstrated as per hpjIjpi=hpjITaTajpi= hpjTaITajpi= exp [2ip
a]hpjIjpi= 0. As in a non-magnetic crystal, the four states of the quartet are mutually orthogonal only when the momenta pandpare dierent in the Brillouin zone. At p= 0 and p=, only two of these four states are linearly independent, as illustrated in the left panel of Fig. 3 30F. B egue, P. Pujol, R. Ramazashvili, Phys. Letters A 381, 1268 (2017). 31F. B egue, P. Pujol, R. Ramazashvili, JETP 126, 90 (2018). 32C. Herring, in Magnetism , edited by George T. Rado and Harry Suhl (Academic Press, New York and London, 1966), Vol. IV, Chap. XIII. 33Again, only in the absence of HSO=dp
, since the latter varies under Un(). 34By non-magnetic Brillouin zone boundary I imply the zone boundary of the Hamiltonian with the N eel exchange term HN=
r
notionally set to zero. 35R. S. K. Mong, A. M. Essin, J. E. Moore, Phys. Rev. B 81, 245209 (2010). 36R. Ramazashvili, Physica B 407, 1930 (2012). 37B. A. Bernevig, T. L. Hughes, S-C. Zhang, Science 314, 1757 (2006). 38As a side remark, notice that, for a?n, the state TaUn()Rn()jpiresides at the same momentum p, as the statejpi:Tb[TaUn()Rn()]jpi= [TaUn()Rn()]Tbjpi= [TaUn()Rn()]eip
bjpi=eip
b[TaUn()Rn()]jpi. 39Tb[TaUn()Rn()]jpi= [TaUn()Rn()]Tbjpi= [TaUn()Rn()]eip
bjpi= eip
b[TaUn()Rn()]jpi: 40R. Ramazashvili, Phys. Rev. Lett. 105, 216404 (2010). 41M. R. Norman and Jie Lin, Phys. Rev. B 82, 060509(R) (2010). 42V. V. Kabanov and A. S. Alexandrov, Phys. Rev. B 77, 132403 (2008). 43R. R. Ramazashvili, Jr., Zh. Eksp. Teor. Fiz. 100, 915 (1991) [Sov. Phys. JETP 73, 505 (1991)]. 44R. Ramazashvili, Phys. Rev. B 80, 054405 (2009). 45D. Shoenberg, Magnetic Oscillations in Metals (Cambridge University Press, Cambridge, Eng- land, 1984). 46L. M. Roth, B. Lax and S. Zwerdling, Phys. Rev. 114, 90 (1959). 47E. L. Ivchenko, G. Pikus, Superlattices and Other Heterostructures , Springer Series in Solid- State Sciences, Volume 110, Berlin { Heidelberg (1997). 48M. I. Dyakonov (Editor) Spin Physics in Semiconductors , Springer Series in Solid-State Sciences,28 Volume 157, Berlin { Heidelberg (2017). 49J. M. Luttinger, Phys. Rev. 102, 1030 (1956). 50A. A. Kiselev, K. W. Kim, and E. Yablonovitch, Phys. Rev. B 64, 125303 (2001). 51X. Marie, T. Amand, P. Le Jeune, M. Paillard, P. Renucci, L. E. Golub, V. D. Dymnikov, E. L. Ivchenko, Phys. Rev. B 60, 5811 (1999) 52R. Winkler, S. J. Papadakis, E. P. De Poortere, M. Shayegan, Phys. Rev. Lett. 85, 4574 (2000). 53D. S. Miserev, O. P. Sushkov, Phys. Rev. B 95, 085431 (2017). 54D.S. Miserev, A. Srinivasan, O.A. Tkachenko, V.A. Tkachenko, I. Farrer, D.A. Ritchie, A.R. Hamilton, O.P. Sushkov, Phys. Rev. Lett. 119, 116803 (2017). 55C. Gradl et al. , Phys. Rev. X 8, 021068 (2018). 56Notice that for a [001] well a similar argument induces no constraint: a =2 rotation around the growth axis [001] transforms ( Jx;Jy;Jz) and (Hx;Hy;Hz) into (Jy;Jx;Jz) and (Hy;Hx;Hz), thus leaving the second term in the Eq. (7) invariant. 57S. Matsuo, H. Kamata, S. Baba, R. S. Deacon, J. Shabani, C. J. Palmstrm, and S. Tarucha, Phys. Rev. B 96, 201404(R) (2017). 58E. I. Rashba and E. Ya. Sherman, Phys. Letters A 129, 175 (1988). 59M. V. Durnev, M. M. Glazov, and E. L. Ivchenko, Phys. Rev. B 89, 075430 (2014). 60V. Galitski and I. B. Spielman, Nature 494, 49 (2013). 61C. Fang and L. Fu, Phys. Rev. B 91, 161105(R) (2015). 62W. Brzezicki and M. Cuoco, Phys. Rev. B 95, 155108 (2017). 63E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra , (Academic Press 1971).

1708.07574v1.Magnetization_in_pristine_graphene_with_Zeeman_splitting_and_variable_spin_orbit_coupling.pdf

arXiv:1708.07574v1 [cond-mat.str-el] 24 Aug 2017Magnetization in pristine graphene with Zeeman splitting a nd variable spin-orbit coupling F. Escudero†, L. Sourrouille†, J.S. Ardenghi†∗and P. Jasen† †IFISUR, Departamento de F´ ısica (UNS-CONICET) Avenida Alem 1253, Bah´ ıa Blanca, Buenos Aires, Argentina August 28, 2017 Abstract The aim of this work is to describe the spin magnetization of g raphene with Rashba spin- orbit coupling and Zeeman effect. It is shown that the magneti zation depends critically on the spin-orbit coupling λthat is controlled with an external electric field. In turn, b y manipulating the density of charge carriers, it is shown that spin up and do wn Landau levels mix introducing jumps in the spin magnetization. Two magnetic oscillations phases are described that can be tunable through the applied external fields. The maximum and minimum of the oscillations can be alternated by taking into account how the energy levels ar e filled when the Rashba-spin-orbit coupling is turned on. The results obtained are of importanc e to design superlattices with variable spin-orbit coupling with different configurations in which s pin oscillations and spin filters can be developed. 1 Introduction Graphene, a one-atom-thick allotrope, has become one of the mos t significant topics in solid state physics due to its two-dimensional structure as well as from its uniq ue electronic properties ([1],[2],[3], [4], [5]). The carbon atoms form a honey-comb lattice made of two inte rpenetrating triangular sub- lattices,AandBthat create specific electronic band structure at the Fermi level: electrons move with a constant velocity about c/300. In turn, the electronic properties are dictated by the πandπ′ bands that form conical valleys touching at the two independent hig h symmetry points at the corner of the Brillouin zone, the so called valley pseudospin [8]. In the absenc e of defects, electrons near these symmetry points behave as massless relativistic Dirac fermions with a n effective Dirac-Weyl Hamil- tonian [4] which allows to consider graphene as a solid-state toy for r elativistic quantum mechanics. When the interaction between the orbital electron motion and spin d egrees of freedom is taken into account, the spin-orbit coupling (SOC) induces a gap in the spectru m. SOC is the most important interaction affecting electronic spin transport in nonmagnetic mate rials. The use of graphene in spin- tronics ([9] and [10]) would require detailed knowledge of graphene’s s pin-orbit coupling effects, as well as discovering ways of increasing and controlling them. The SOC in graphene consists of intrinsic and extrinsic components ([11] and [12]). The intrinsic component of SOC is weak in the graphene sheet, although considerable magnitudes can be obtained in nanosc aled graphene ([13] and [14]). The extrinsic component, known as the Rashba spin-orbit coupling (RSO C) [15] can be larger than the in- trinsic component [16]. The RSOC may be controlled by the application o f external electric fields ([17] and [18]). On the other side, when a magnetic field is applied perpendicu lar to the graphene sheet, a discretization of the energy levels is obtained, the so called Landau levels [19]. These quantized energy levels still appear also for relativistic electrons, just their d ependence on field and quantization ∗email: jsardenghi@gmail.com, fax number: +54-291-459514 2 1parameter is different. In a conventional non-relativistic electron gas, Landau quantization produces equidistant energy levels, which is due to the parabolic dispersion law o f free electrons. In graphene, the electrons have relativistic dispersion law, which strongly modifies the Landau quantization of the energy and the position of the levels. In particular, these levels are not equidistant as occurs in a conventional non-relativistic electron gas in a magnetic field. This lar ge gap allows one to observe the quantum Hall effect in graphene, even at room temperature [20]. In turn, when magnetic fields are applied to solids an important effect called the de Haas-vanAlphen (dH vA) [21] appears asoscillations of magnetization as a function of inverse magnetic field. This effect is a purely quantum mechanical phenomenon and is a powerful toof for maping the Fermi surface, i.e. the electronic states at the Fermi energy ([22],[23]) and because gives important information on t he energy spectrum, it is one of the most important tasks in condensed matter physics. The diffe rent frequencies involved in the oscillations are related to the closed orbits that electrons perform on the Fermi surface. It has been predicted in graphene that magnetization oscillates periodically in a sa wtooth pattern, in agreement with the old Peierls prediction [24], although the basic aspects of the b ehavior of the magnetic oscilla- tions for quasi-2D materials remains yet unclear [25]. In contrast to 2D conventional semiconductors, where the oscillating center of the magnetization Mremains exactly at zero, in graphene the oscil- lating center has a positive value because the diamagnetic contribut ion is half reduced with that in the conventional semiconductor [7]. From an experimental point of view, carbon-based materials are more promising because the available samples already allows one to obs erve the Shubnikov-de Haas effect ([26] and [27]) and then may be easier to interpret than quant um oscillations in its transport properties. Because the dHvA signal in 2D systems are free of the kzsmearing, it should be easier to obtain much rich information about the electron processes. In add ition, the SOC, which is consider- ably large in graphene [28] plays an important role in the determination of the magnetic oscillations because of the fundamental difference with conventional semicon ductor 2DEG. Motivated by this phenonema, in the present paper we study the dH vA oscillations in the spin magnetization in graphene by taking into account the Rashba spin-o rbit interaction modulated by a perpendicular external electric field and the Zeeman effect modulat ed by a constant magnetic field perpendicular to the graphene sheet. It is well known that each ele ctron contributes with µBto the magnetization density if the spin is parallel to the applied magnetic field and−µBif it is antiparallel. Hence, ifN±is the number of electrons per area with spin parallel (+) or antipara llel (−), the magnetization density will be M=µB(N+−N−). As it was said before, the introduction of a constant magnetic field introduces the Landau levels that are splitt ed by the Zeeman effect. The filling of these levels is not trivial in graphene due to its square root de pendence in the Landau index. The degeneracy, that depends linearly on B, defines the number of states that are completely filled, but these levels may not be sorted in ascending order with intercalat ed spins. In fact, whenever the n+ 1 Landau level with spin up is lower than the nLandau level with spin down, an enhancement of the spin magnetization will be obtained. This behavior can be alter d rastically with the RSOC, because in this case, the ordering of the energy levels is not trivial. I n this sense and since both the magnetic field and RSOC are externallycontrollable, in what follows the roleof each ofthe parameters involved, as the electron density and the electric and magnetic field s trength are discussed in relation to the maximum and minimum of the oscillations. The spin-orbit effects a re important, besides the fundamental electronic and band structure and its topology, to u nderstand spin relaxation, spin Hall effect and other effects such as weak (antilocalization). There are two main routes to implement spintronics devices: the giant magneto resistance effect (GMR) [29 ] and the spin effect field transistor (spin-FET) [30]. Both devices consists in a sandwich structure made of two ferromagnetic materials separated by a non-magnetic later in GMR and two dimensional electr on gas in spin-FET. In this device, the spin-orbit coupling causes the electron spin to precess with a precession length determined by the strength of the spin-orbit coupling, which through a gate vo ltage becomes tunable. A detailed knowledge of the interplay of the the different parameters enterin g in the Hamiltonian is vital to fundamentally understand the spin-dependent phenomena in grap hene.1This work will be organized 1For a good review of spintronics in graphene see [31]. 2as follow: In section II, pristine graphene under a constant magne tic field with spin-orbit coupling and Zeeman splitting is introduced and the magnetic oscillations are dis cussed. In section III, N+ andN−populations as a function of B, the electron density and RSOC parameter is studied and the oscillations are computed and discussed. The principal findings of th is paper are highlighted in the conclusion. 2 Magnetic oscillations with Zeeman splitting For a self-contained lecture of this paper, a brief introduction of t he quantum mechanics of graphene in a constant magnetic field in the long wavelength approximation will be introduced (see [5]). The Hamiltonian in one of the two inequivalent corners of the Brillouin zones can be put in a compact notation as (see [6] and [7]) H=vF(σ·π)+1 2∆R(σ×s)−∆Zsz (1) whereσare the Pauli matrices acting on the pseudospin space and sare the Pauli matrices acting on the spin space. ∆ Ris the extrinsic spin-orbit coupling that arise when an external elect ric field is applied perpendicular to the graphene sheet or from a gate voltag e or charged impurities in the substrate ([32], [33] and [34]).2∆Z=gµB 2Bis the Zeeman energy that depends on the magnetic field strength. In the following we will write the extrinsic Rashba spin orbit coupling as ∆ R=/planckover2pi1ωy. The quasiparticle momentum must be replaced by π=p−eA, whereeis the electron charge and Ais the vector potential which in the Landau gauge reads A=(−By,0,0). For the Kvalley, the third term can be written as −i∆R(σ+s+−σ−s−) whereσ±=σx±iσyands±=sx±isy. This term describes a coupling between pseudospin and spin states, that is, a spin-flip pr ocess can be achieved by hopping an electron in the Asublattice with spin up to the Bsublattice with spin down. The eigenvalues of the Hamiltonian of eq.(1) has been computed in [6] and [7] without the Z eeman term. By introducing this term the eigenvalues reads3 En,s,l=l/planckover2pi1√ 2/radicalbigg 2ω2 Z+2nω2 L+ω2y−s/radicalBig 16nω2 Zω2 L+4nω2 Lω2y+ω4y (2) wherel= +1(−1) is for the conduction (valence) band, n= 0,1,2,...is the Landau level index and s=±1 is the spin. The number of conduction electrons will be given by N=neAwhereneis the electron density and Ais the area of the graphene sheet. For simplicity, we can consider on ly conduction electrons, which implies that the added electrons are du e to a gate voltage applied to the graphene sheet so that necan be varied as a function of VG. When a magnetic field is applied, the energy is discretized and each level is degenerated with degene racyD=BA/φ, whereφis the magnetic unit flux. There is a critical magnetic field in which the degene racyDequals the number of electronsN BC=neφ (3) If we would consider only valence electrons, then BC∼80×103T, which is an unfeasible experimen- tally. Nevertheless, BCdefined as in last equation will serve as an upper bound for the numer ical calculations. Let us consider the first case under study: graphene under a mag netic field with Zeeman effect. In this case, the energy levels can be found and reads En,s,l=−s/planckover2pi1ωz+l/planckover2pi1ωL√n (4) 2Intrinsic spin-orbit coupling ∆ intthat contains contributions from the porbitals and dorbitals (see eq.(10) of [11])is neglected. 3In [35] it is discussed the intrinsic and extrinsic spin-orb it couplings and the quasi-classical limit for higher Landa u levels. 30.01 0.02 0.03 0.04 0.05 0.06B BC0.010.020.030.040.050.06M NΜB Figure 1: Dimensionless magnetization per number of electrons in the region where spin up and down levels do not mix. beings=±1 for spin up and down respectively, n= 0,1,2,...is the Landau level index and l= 1 for the conduction band and l=−1 for the valence band. For the conduction band, electrons will sta rt filling the lowest levels, so in the case in which |/planckover2pi1ωz|<1 2|/planckover2pi1ωL|, that is, when the Zeeman splitting is lower than half the separation between consecutive Landau levels , then the filling will be done considering the Landau index first and then the spin index. B0 20 40 60 80 100 120 140 160 180 200M/NµB 00.10.20.30.40.50.60.70.80.91 y=0 B0 20 40 60 80 100 120 140 160 180 200Number of spin up and down electrons/N 00.10.20.30.40.50.60.70.80.91 Figure 2: Left. Spin magnetization as a function of the magnetic field strength, where ne= 0.1nm2 andA= 10nm2without spin-orbit coupling. Right: Number ofspin up and down electr onsnormalized with respect the number of electrons N. This implies that if we consider the energy levels in ascending order, th ere will be no consecutive Landau levels with same spin filled. This assumption is fulfilled for normal systems, where ωLis cyclotron frequency ωL=eB mand the Landau levels are /planckover2pi1ωC(n+1/2). In this case, because ωZis proportional to B, then the condition |/planckover2pi1ωz|<1 2|/planckover2pi1ωL|implies that gµB 2</planckover2pi1e m. For graphene, due to the relativistic dispersion relation, ωLis proportional to√ Band the 4B0 20 40 60 80 100 120 140 160 180 200M/NµB 00.10.20.30.40.50.60.70.80.91 y=0.01 Figure 3: Spin magnetization for y= 0.01eV.B0 20 40 60 80 100 120 140 160 180 200M/NµB 00.10.20.30.40.50.60.70.80.91 y=0.05 Figure 4: Spin magnetization for y= 0.05eV. B0 20 40 60 80 100 120 140 160 180 200M/NµB 00.10.20.30.40.50.60.70.80.91 y=0.1 Figure 5: Spin magnetization for y= 0.1eV.B0 20 40 60 80 100 120 140 160 180 200M/NµB 00.10.20.30.40.50.60.70.80.91 y=0.2 Figure 6: Spin magnetization for y= 0.2eV. Landau levels increase as√n. Then if we write /planckover2pi1ωz=γZBand/planckover2pi1ωL=γL√ B, then the condition for no consecutive spin up or down filling is given by B </parenleftbiggγL 2γZ/parenrightbigg2/parenleftbig√ n+1−√n/parenrightbig2(5) For grapheneγL 2γZ∼36.29 2·0.12∼151.2 (see [36] and [37]).Under this regime, the ordered energy levels can be written as Ep=−(−1)p/planckover2pi1ωZ+/planckover2pi1ωc/radicalbigg p 2−1 4(1−(−1)p) (6) forpeven we obtain the spin up levels and for podd the spin down levels. The filling factor v=BC/B can be written as v=q+θ, whereq=/bracketleftbigBC B/bracketrightbig , where [x] is the floor function defined as the largest integer less than or equal to xandθ=ν−q. The spin Pauli paramagnetism is given by the remaining term for the Landau filling which is proportional to λ M=µB(N↑−N↓) =NµBB BC/bracketleftbigg1 2/parenleftBig 1−(−1)[BC B]/parenrightBig +(−1)/bracketleftBigBC B/bracketrightBig/parenleftbiggBC B−/bracketleftbiggBC B/bracketrightbigg/parenrightbigg/bracketrightbigg where the difference ( N↑−N↓) is proportional to θand the factor ( −1)qselects the majority of spin up or down states in the last Landau level partially filled. In figure 1, t he dimensionless magnetization per electron M/Nµ Bis plotted against x=B BC. 5B0 20 40 60 80 100 120 140 160 180 200M/NµB 00.10.20.30.40.50.60.70.80.91 y=2 B0 20 40 60 80 100 120 140 160 180 200Number of spin up and down electrons/N 00.10.20.30.40.50.60.70.80.91 Figure7: Left. Spinmagnetizationasafunctionofthemagneticfield strength,where ne= 0.1nm2and A= 10nm2. The Rashba spin-orbit coupling is largerthan the critical value wher e spin magnetization saturates. Right: Number of spin up and down electrons normalized with respect the number of electronsN. It must be stressed that condition of eq.(5) depends on nand because√n+1−√n<1 forn>1, then for higher Landau levels the upper bound for Bdecreases and we must take into account the lowest Landau level completely filled n=q=/bracketleftbigBC B/bracketrightbig . This gives the condition for B B </parenleftbiggγL 2γZ/parenrightbigg2/parenleftBigg/radicalBigg/bracketleftbiggBC B/bracketrightbigg +1−/radicalBigg/bracketleftbiggBC B/bracketrightbigg/parenrightBigg2 (7) The jumps in the magnetization are well understood in terms of the fi lling of the Landau levels up to the Fermi level. Because the Fermi level indicates the lowest Landa u level filled but the degeneration of each level is not a integer number, then when a level is completely fi lled, the Fermi level jumps and in consequence the magnetization. In the second regime, where B≥/parenleftBig γL 2γZ/parenrightBig2/parenleftbig√n+1−√n/parenrightbig2, the energy levels will show consecutive Landau levels with the same spin fille d. In this case, it is necessary to use numerical methods to sort the energy levels in ascending ord er taking into account if the level belongs to a spin up or down state. In figure 2, a sawtooth like behav ior for the spin magnetization is obtained which is given by the spin up and down population, where we ha veused that ne= 0.1. When ν= 2, thenD=N/2 which implies that the fundamental and first excited levels are comp letely filled. For this particular value of B, the first excited level are filled with spin down states. This behavior is shown in figure 2, where both population are identical for B=BC/2. Between this value and B=BC(ν= 1), the spin up population decreases linearly. The same behavior is e xpected for v=l wherel= 1,2,... For those values of B, the spin up and down population are identical. The spin magnetization peaks follows a linear behavior for high magnetic fields M=B neφ0. ForB >n eφ0the spin magnetization reaches a plateau given by M=µBN. 2.1 Spin-orbit coupling When an external electric field is applied perpendicular to the graphe ne sheet, the Byckhov-Rashba effect appears (see [11]). The parameter ythat describes this interaction depends linearly with the external electric field strength. In figures 3, 4, 5 and 6, a sequen ce of plots for different values of yare 6Figure 8: Rashba spin-orbit coupling yas a function of the electron density ne. Blue regions between curves indicates those values of ywhere spin magnetization changes with respect the y= 0 case. Figure 9: Rashba spin-orbit coupling yas a function of the electron density ne. Each curve indicates the values of ywhere the spin magnetization jumps from 0 to M=µBN/2jwherej= 1,2,3.... 7shown, where ne= 0.1 andA= 10nm−2(N= 1).4As it can be seen in the supplementary material, there are set of values of ywhere there are no changes in the magnetization. These set of valu es are given when the spin up Landau energy level nis lower than the spin down energy level n−1. For simplicity, let us consider that ν= 2. Then only the two first Landau levels are completely filled wheny= 0. In this case, because these two levels correspond to the spin u p and down n= 0 Landau levels, then the spin magnetization vanishes because N+=N−. Wheny>0, there is a critical value y(2) cwhereE1,↑< E0,↓which implies that the two first Landau levels are spin up states. Then the magnetization jumps to a constant value M=µBN. The condition E1,↑<E0,↓implies that /radicalBig 2γ2 ZB2+2y2≥/radicalbigg 2γ2 ZB2+2γ2 LB+y2−/radicalBig 4γ2 ZB2(4Bγ2 L+y2)+y4 (8) and the solutions reads y2≥γ2 L 2B−2γ2 ZB2(9) whichgivestwosolutions, onefor positivevalues of ywhich is the caseofinterest. In turn, when ν= 3, thenD=N/3, then only the first three Landau levels are completely filled. For y= 0, there is no spin mixing and therefore the magnetization is positive and maximum. W heny>0 and in particular wheny > y(3) cthenE1,↑< E0,↓, but in this case, when we increase the magnetic field strength, the degeneration increases, which implies that only the two first Landau levels are filled completely, which correspond to spin up states. Then, the magnetization increase a sBdoes until we reach B=BC 2. These considerations imply that there is a critical value y(3) c< y < y(2) cdefined from the equation E1,↑=E0,↓whenB=neφ0 2fory(2)and whenB=neφ0 3fory(3) cwhere the magnetization changes with respect the case y= 0. The regionsfor y(j+1) c<y<y(j) candnewhere the magnetization changes can be obtained from the condition Ej+1,↑=Ej,↓whenB=neφ0 jfory(j) candB=neφ0 j+1fory(j+1) c. In figure 8 these set of values are shown. For small values of ythe regions tends to a continuum when j→ ∞. In turn, in figures 7, the spin magnetization as a function of Bis shown when y>y(1) c, where the magnetization saturates. The set of magnetic fields Bj=neφ0 2jare particularly important. For these values and y(j) c=/radicaltp/radicalvertex/radicalvertex/radicalbtneφ0 2j(γ2 L−4γ2 Zneφ0 2j)2−16(j−1)γ2 Zγ2 L(neφ0 2j)2 4(j−1)γ2 L+2γ2 L−8γ2 Zneφ0 2j(10) which is found by computing Ej+1,↑<Ej,↓and replacing Bj=neφ0 2j, the magnetization jumps from M= 0 toMj=µBN/2j(see figure 9). There are infinite mixes of Landau spin up and down st ates that are defined through the inequality En+r,↑≤Er,↓which implies that y2≥16nγ2 Zγ2 L−(rγ2 L−4γ2 Z)2 8γ2 Z−2(2n+r)γ2 L(11) These set of values are located between the regions described in fig ure 8. These results are of importance to develop two-dimensional super tlattices structures by using grapheneorsilicene,whichsupportschargecarriersbehavingasm asslessDiracfermionswithgraphene- like electronic band structure ([38] and [39]). Spin-orbit interaction in silicene can be 1000times larger that of graphene, which implies that quantum spin Hall effect in silicene is experimentally accesible. Considering two semi-infinite graphene ribbons, that are employed a s source and drain, and a drain voltage that introduces charge carriers with a concentration neit is possible to develop a superlattice ofdifferent graphenesamples with specific values of yin series. By fixing the magneticfield strength in any of the critical values Bj=neφ0 2j, the possible values of ycan be located over the curves in figure 9 4A supplementary video has been uploaded where the spin magne tization is plotted for increasing values of yfor the particular case ne= 0.1nm−2. 8in such a way to increase the spin polarized density of states across the superlattice. Different configu- rations can be obtained by developing superlattice with alternating c onstant electric fields in order to obtainspin flipping, which can be used asa spin down orup filter [40]. By applyingLandauer-B¨ uttiker formalism, the electronic behavior of Dirac fermions in the superlatt ice can be studied. The conduc- tance will shows resonant tunneling behavior depending on the numb er of barriers and barrier width [41]. In turn, the spin polarization lifetime can be controlled by the app lied electric field that controls Rashbaspin-orbit coupling [42] and [43]. It has been shown the angula rrange ofthe spin-inversioncan be efficiently controlled by the number of barriers [44]. Magnons can be obtained in the superlattice structure by combining alternating applied electric fields, where the wavelength can be accomodated by the width of the middle graphene samples [45]. It is source of futur e works to develop graphene superlattice with different configurations of Rashba spin-orbit cou plings and external magnetic fields. Finally, to further explore the magnetic activation by changing the R SOC, we can consider the following setup: N= 2, that is, two conduction electrons only, B=Bc 2, whereD= 1 which implies that we have to consider the first two Landau levels only. When y= 0, one electron occupy the ground state with spin up and the secondelectron the ground state with sp in downwhich implies that the spin magnetizationis zero. Suppose that we consider the action ofa fas t sudden perturbation in the system where the RSOC changes to y=y(1) c, where as we said before, y(1) cis such that E1,↑=E0,↓which is given in eq.(9). Then, because the first excited spin up level is below the ground state with spin down, the first two Landau levels filled are with spin up, which implies tha tMjumps toM= 2µB. The transition amplitude is given by the inner product of both eigenst ates. To obtain this value we can consider the anti-symmetric state for y= 0 |ψM=0/angbracketright=1√ 2/bracketleftBig/vextendsingle/vextendsingle/vextendsingleϕy=0 0,↑/angbracketrightBig ⊗/vextendsingle/vextendsingle/vextendsingleϕy=0 0,↓/angbracketrightBig −/vextendsingle/vextendsingle/vextendsingleϕy=0 0,↓/angbracketrightBig ⊗/vextendsingle/vextendsingle/vextendsingleϕy=0 0,↑/angbracketrightBig/bracketrightBig (12) and the anti-symmetric state for y=y(1) c |ψM=2µB/angbracketright=1√ 2/bracketleftBig/vextendsingle/vextendsingle/vextendsingleϕy=y(1) c 0,↑/angbracketrightBig ⊗/vextendsingle/vextendsingle/vextendsingleϕy=y(1) c 1,↑/angbracketrightBig −/vextendsingle/vextendsingle/vextendsingleϕy=y(1) c 1,↑/angbracketrightBig ⊗/vextendsingle/vextendsingle/vextendsingleϕy=y(1) c 0,↑/angbracketrightBig/bracketrightBig (13) where we can note that the state for the second electron is spin up . where we are considering that both electrons are not interacting. Each vector/vextendsingle/vextendsingleϕy=0 n,s/angbracketrightbig can be computed by considering the diagonalization of the Hamiltonian of eq.(1). This eigenfunctions are c omputed in [46] and [47] and we can write/vextendsingle/vextendsingle/vextendsingleϕy=0 0,↑/angbracketrightBig =1√ 2L/parenleftbigφ0φ00 0/parenrightbig ,/vextendsingle/vextendsingle/vextendsingleϕy=0 0,↓/angbracketrightBig =1√ 2L/parenleftbig0 0φ0φ0/parenrightbig ,/vextendsingle/vextendsingle/vextendsingleϕy=y(1) c 0,↑/angbracketrightBig = 1√ 2L√ |β1|2+1/parenleftbigβ1φ00 0φ0/parenrightbig and/vextendsingle/vextendsingle/vextendsingleϕy=y(1) c 1,↑/angbracketrightBig =1√ 2L√ |α1|2+|α2|2+|α3|2+1/parenleftbigα1φ1α2φ0α3φ1φ0/parenrightbig , whereφn(ξ) =π−1/4 √ 2nn!e−1 2ξ2Hn(ξ) whereHnare the Hermite polynomials of order nandξ=ξ= y lB−lBk, wherelB=/radicalbig 1/eBis the magnetic length and kis the wavevector in the xdirection. 2 L is the total length of the graphene sample in the xdirection. The coefficients β1,α1,α2andα3are obtained through the diagonalization of the Hamiltonian and reads β1=i/parenleftBig√ 2γZB+/radicalbig γ2 LB−2γ2 ZB2/parenrightBig /radicalbig γ2 LB−4γ2 ZB2(14) α1=i/parenleftBig γ2 LB+2√ 2γZB/radicalbig γ2 LB−2γ2 ZB2/parenrightBig /radicalbig γ2 LB−2γ2 ZB2/parenleftBig −√ 2γZB+/radicalbig γ2 LB−2γ2 ZB2/parenrightBig (15) α2=2i√ 2γ2 LB/radicalbig γ2 LB−4γ2 ZB2 2γ2 LB−4√ 2γZB/radicalbig γ2 LB−2γ2 ZB2(16) α3=γ2 LB 2−γZB+/radicalbig γ2 LB+2γ2 ZB2(17) 920 40 60 80 100ne0.10.20.30.40.5/LBracketBar1/LAngleBracket1ΨM/Equal2ΜB/RBracketBar1ΨM/Equal0/RAngleBracket1/RBracketBar12 0.00.20.40.60.81.00.0200.0220.0240.0260.0280.0300.0320.034 Figure 10: Transition amplitude for magnetic activation by a sudden p erturbation y= 0→y=y(1) c as a function of electron density ne. wherewehavetoreplace B=Bc 2andy=y(1) c. Infigure10, theprobabilityamplitude |/angbracketleftψM=0|ψM=2µB/angbracketright|2 is plotted as a function of the electron density ne. It is shown that the transition tends to zero as ne→0 and a non-trivial zero for ne∼21nm−2. The curve in this region follows the behavior of the curvey=y(1) cas it can be seen in figure 9, although it is not exactly a semicircle. For h igher values of ne, the probabilityamplitude tends to zero. Then it ispossible to increas ethe spin magnetic transition for the specific value, where the probablity amplitude is maximum for ne. In this model we have not consider more than two electrons because in this case we should tak e into account the interactions between electrons in the same state, which can be done by consider ing the Laughlin wavefunctions [48] and the subtleties introduced by the fractional quantum Hall e ffect. 3 Conclusions In this work we have examined the spin magnetization in pristine graph ene with spin-orbit coupling and Zeeman splitting. The magnetization has been found as a functio n of the electron density, the magneticfield strength and the Rashba spin-orbitcoupling paramet er. We havederivedand compared the maximum and minimum of the magnetization with and without spin-or bit coupling showing that for certain regions of the spin-orbit coupling parameter yand certain values of the electron density ne, the magnetization jumps from zero to a constant value given Mj=µBN/2jbeingNthe number of electrons and j= 1,2,3...being an integer number that controls when a spin up state with Land au levelj+ 1 is lower than the spin down state with Landau level j. Morever, these regions for y follows a complex relation with nethat numerically shows certain values of ywhere magnetization is not altered by y. These results show that it is possible to obtain spin filters or spin osc illations by considering different superlattice configurations, where the inner graphene samples can alternate the applied electric field in such a way to fix the magnetic field strength in th ose values where the Landau spin up and down states mix. Finally, we have studied the probability am plitude for a sudden change of the Rashba spin-orbit coupling from the non-magnetic state M= 0 toM= 2µBwhen there are two electrons, B=neφ0 2andy=y(1) c, showing that the transition has a a peak for ne>0 and a non-trivial zero. 104 Acknowledgment This paper was partially supported by grants of CONICET (Argentin a National Research Council) and Universidad Nacional del Sur (UNS) and by ANPCyT through PIC T 1770, and PIP-CONICET Nos. 114-200901-00272 and 114-200901-00068 research gran ts, as well as by SGCyT-UNS., J. S. A. and L. S. are members of CONICET., F. E. is a fellow researcher at th is institution. 5 Author contributions All authors contributed equally to all aspects of this work. References [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsn elson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature,438, 197 (2005). [2] A.K. Geim and K. S. Novoselov, Nature Materials, 6, 183 (2007). [3] Y. B. Zhang, Y.W. Tan, H. L. Stormer and P. Kim, Nature,438, 201 (2005). [4] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselovan d A. K. Geim, Rev. Mod. Phys. , 81, 109 (2009). [5] M. O. Goerbig, Rev. Mod. Phys., 83, 4, (2011). [6] E. I. Rashba, Phys. Rev. B 79, 161409, (2009). [7] F. Zhen-Guo, W. Zhi-Gang, L. Shu-Shen and Z. Ping, Chin. Phys. B ,20, 5 058103 (2011). [8] J. McClure, Phys. Rev. ,104, 666, (1956). [9] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). [10] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic ,Acta Phys. Slov. 57, 565 (2007). [11] S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B ,82, 245412 (2010). [12] J. Balakrishnan, G.K. Koon, M. Jaiswal, A.H.C. Neto, B. Ozyilmaz, Nat. Phys. 101038 (2013). [13] C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95226801 (2005). [14] C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95146802 (2005). [15] Y.A. Bychkov, E.I. Rashba, JETP Lett. 3978 (1984). [16] R. Decker, Y. Wang, V.W. Brar, W. Regan, H.Z. Tsai, Q. Wu, W. Ga nnett, A. Zettl, M.F. Crommie, Nano Lett. 112291 (2011). [17] C. Ertler, S. Konschuh, M. Gmitra, J. Fabian, Phys. Rev. B, 80041405 (2009). [18] D.A. Abanin, P.A. Lee, L.S. Levitov, Phys. Rev. Lett., 96176803 (2006). [19] S. Kuru, J. Negro and L. M. Nieto, J. Phys.: Condens. Matter ,21,455305 (2009). [20] Y. Zheng and T. Ando, Phys. Rev. B ,65,245420 (2002). [21] W. de Haas and P. van Alphen, Proceedings of the Academy of Science of Amsterdam ,33, 1106– 1118 (1930). 11[22] I. Meinel, T. Hengstmann, D. Grundler, D. Heitmann, W. Wegsch eider and M. Bichler , Phys. Rev. Lett. ,82, 819 (1999). [23] M. P. Schwarz, M. A. Wilde, S. Groth, D. Grundler, C. Heyn and D . Heitmann, Phys. Rev. B , 65, 245315 (2002). [24] S. G. Sharapov, V. P. Gusynin and H. Beck, Phys. Rev. B ,69, 075104 (2004). [25] T. Champel and V.P. Mineev, Philos. Mag. B ,81, 55 (2001). [26] S. Uji, J.S. Brooks, and Y. Iye, Physica B ,299246-247 (1998). [27] Z.M. Wang, Q.Y. Xu, G. Ni, and Y.W. Du, Phys. Lett. A, 314, 328 (2003). [28] A. Varykhalov, J. Sanchez-Barriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O. Rader , Phys. Rev. Lett .101157601 (2008). [29] P. Gr¨ unberg, R. Schreiber, Y. Pang, M. B. Brodsky and H. So wers,Phys. Rev. Lett., 57(19):2442, (1986). [30] S. Datta and B. Das, Appl. Phys. Lett. ,56:665, (1990). [31] N. Kheirabadi, A. Shafiekhani and M. Fathipour, Superlattices and Microstructures ,74, 123-145 (2014). [32] G.Y. Wu, N.-Y. Lue, Phys. Rev. B ,86, 045456 (2012) . [33] A. Manjavacas, S. Thongrattanasiri, D.E. Chang, F.J.G. deAba jo,New J. Phys. ,14, 123020 (2012). [34] G. Burkard, Nat. Nanotech. ,7, 617 (2012). [35] V. Yu. Tsaran and S. G. Sharapov, Phys. Rev. B, 90, 205417 (2014). [36] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod. Phys. ,83, 407 (2011). [37] M.O. Goerbig, R. Moessner, B. Doucot, Phys. Rev. B ,74, 161407 (2006). [38] C. C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. ,107, 076802 (2011). [39] S. Cahangirov,M. Topsakal, E.Akturk, H. Sahin, S. Ciradi, Phys. Rev. Lett. ,102, 236804(2009). [40] N. PournaghaviM.Esmaeilzadeh,S. Ahmadi, M.Farokhnezhad ,Solid State Commun ,226, 33-58 (2016). [41] X-S. Wang, M. Shen, X-T. An and J. J. Liu, Phys. Rev. Lett. ,380, 1663-1667 (2016). [42] X-T. An, Phys. Lett. A ,379, 723-727 (2015). [43] D. Wang, G. Jin, Phys. Lett. A ,378, 2557-2560, (2014). [44] F. Sattari, . Faizabadi, Superlattices and microstructures ,81, 80-87 (2015). [45] Y. Hajati and Z. Rashidian, Superlattices and microstructures ,92, 264-277 (2016). [46] J. S. Ardenghi, P. Bechthold, E. Gonzalez, P. Jasen and A. Jua n,Eur. Phys. J. B ,88: 47 (2015). [47] J. S. Ardenghi, P. Bechthold, E. Gonzalez, P. Jasen and A. Jua n,Physica B ,433,28-36 (2014). [48] R. B. Laughlin, Phys. Rev. Lett., 50, 1395 (1983). 12

2207.06501v1.Jahn_Teller_states_mixed_by_spin_orbit_coupling_in_an_electromagnetic_field.pdf

Jahn-Teller states mixed by spin-orbit coupling in an electromagnetic field A. S. Mi˜ narro, G. Herranz Institut de Ci` encia de Materials de Barcelona (ICMAB-CSIC), Campus UAB, 08193 Bellaterra, Catalonia, Spain Spin-orbit coupling plays a pivotal role in condensed matter physics. For instance, spin-orbit interactions affect the magnetization and transport dynamics in solids, while spins and momenta are locked in topological matter. Alternatively, spin-orbit entanglement may play an important role in exotic phenomena, like quantum spin liquids in 4d and 5d systems. An interesting question is how electronic states mixed by spin orbit coupling interact with electromagnetic fields, which may hold potential to tune their properties and reveal interesting physics. Motivated by our recent discovery of large gyrotropic signals in some Jahn-Teller manganites, here we explore the interaction of light with spin-mixed t2g−egstates in a d4transition metal. We show that spin-orbit mixing enables electronic transitions that are sensitive to circularly polarized light, giving rise to a gyrotropic response that increases with spin-orbit coupling. Interestingly, photoexcited transitions that involve spin reversal are behind such gyrotropic resonances. Additionally, we find that the interaction with the electromagnetic field depends strongly on the relative orientation of the propagation of light with respect to Jahn-Teller distortions and spin quantization. We suggest that such interactions offer the opportunity to use electromagnetic waves at optical wavelengths to entangle orbital and spin degrees of freedom. Our approach, which includes a group-theoretical treatment of spin-orbit coupling, has wide applicability and provides a versatile tool to explore the interaction of electromagnetic fields with electronic states in transition metals with arbitrary spin-orbit coupling strength and point- group symmetries. I. INTRODUCTION Orbital degrees of freedom are an essential ingredi- ent of the physics and chemistry of transition metal compounds [2–5]. The coupling of orbitals to spin, charge or lattice determines many properties of solids and molecules. In the presence of orbital degeneracy, the symmetry of non-linear molecules and solid state systems is broken spontaneously through the Jahn-Teller effect [6–8]. This phenomenon has far-reaching consequences in spectroscopic and chemical properties [9–11], and is also responsible for the emergence of nontrivial quan- tum effects, associated with the appearance of rotational quantization of vibronic states and geometric phases [12– 14]. On the other hand, spin orbit interactions are key to new developments in classical and quantum computa- tion and lie beneath new discoveries in condensed matter physics related to topological matter, such as the quan- tum spin Hall effect or the realization of topological insu- lators and Weyl semimetals [15–17] and Kitaev physics in quantum spin-liquids [18–21]. At the same time, there is a highly nontrivial interplay between spin-orbit coupling and the Jahn-Teller effect when t2gstates are partially filled, where entangled quantum spin-orbital states may emerge [22–24]. An interesting question is how electromagnetic fields interact with spin-orbit mixed states, which could pave the ground to explore quantum physics in these systems. Motivated by our recent finding of large gyrotropic sig- nals in La 2/3Ca1/3MnO 3(originated by the different opti- cal response to light of opposite handedness in the pres- ence of Jahn-Teller distortions) [25], we present here a group-theoretical analysis to study this problem. Our formalism has general applicability and provides a usefulroute to extend the analysis to heavy transition metals in arbitrary point-group symmetries. In the following, we describe in great detail the interaction with an electro- magnetic field of spin-orbit mixed t2g−egstates in a 3 d metal, which provides the clues to its generalization to other transition metals. We first note that when dealing with spin-orbit physics in light transition metals, the mixing between t2gandeg orbitals is usually neglected, since crystal-field splitting and exchange energies are much larger than spin-orbit coupling [22]. However, this approximation breaks down under particular conditions. To illustrate this point, we consider the Tanabe-Sugano diagram for the case of an ion with d4configuration in Ohsymmetry [1, 26]. We see that for values of the crystal field 10 Dqand the Racah parameter Bthat fulfill the condition ( Dq/B )c≤2.7 the ground state term is5Eg, whereas for large enough Dq/B the ground state is3T1g(Fig. 1a). In both limits, a good approximation is that spin-orbit interactions act only on the t2gmanifold (the orbital moment is quenched foregstates), and the spin-orbit mixing of egandt2g states (and, therefore, between5Egand3T1g) can be ignored, at least to first order in spin-orbit coupling. This results in the so-called T−Pequivalence, where the spin- orbit physics of the t2gmanifold can be described with an effective angular momentum L= 1, like for porbitals [1]. However, this approximation breaks down for specific situations. For instance, in 4 d/5dtransition metals, where several energy scales (including spin-orbit coupling and crystal-field) are comparable [27, 28], the t2g−eg mixing can not be generally ignored [29, 30], which is relevant to predict magnetic excitations in heavy transi- tion metal oxides. Alternatively, the T−Pequivalence2 0 1 2 3 4 50246810 5E3T1 5T2 Dq/BE/B (a) d3T1g 5EgEFT/2 FE/2 +GE3A2g 3Eg 5A1g 5B1g3 2FT 2(FE+GE)B1g, B2g Eg A1g, A2gξSO/2 ξSO/2 ∆Oh D4h SOC (b) FIG. 1. a) Tanabe-Sugano diagram for a d4configuration, representing the lowest energy terms5T2,3T1and5EinOhpoint symmetry [1]. Along the ordinate axis, the energy Eis shown relative to the Racah parameter B. The abscissa displays the ratio between the ”differential of quanta” DqandB(for octahedral complexes the crystal field energy is 10 Dq). b) Splitting of the spectroscopic lines in Oh,D4hand spin-orbit coupling (SOC) point symmetries. In this work, we consider that the reduction to D4hsymmetry is driven by Jahn-Teller interactions. fails when the difference in energies between spectro- scopic terms becomes comparable with spin-orbital cou- pling. This may happen for ratios ( Dq/B )c≈2.7 in the Tanabe-Sugano diagram [1], which may occur in 3 d- transition metals (Fig. 1(a)). Alternatively, a reduction from OhtoD4hsymmetry (e.g., induced by a Jahn-Teller instability, as we discuss below), may lead to spin-orbital mixing between spectroscopic terms5A1gand3Egaway from ( Dq/B )c≈2.7, see Fig. 1(b). As we show be- low, spin-orbital mixing between t2gandegstates –which may be also induced by electronic correlations or struc- tural distortions– enables optical transitions that can be probed by circularly polarized light. The sensitivity to circular polarization stems from terms of different spin multiplicity mixed by spin-orbit coupling (e.g., S=1 for 3EgandS=2 for5A1ginD4hpoint symmetry), which allows optical transitions between states with different spin projections, which, otherwise, are absent in the ab- sence of spin-orbit mixing. For the sake of conciseness, we restrict our discussion to 3 d4ions in which spin-orbital mixing is induced by symmetry reduction to D4hinduced by Jahn-Teller instabilities (Fig. 1b). At the end we discuss briefly how the group-theoretical analysis can be extended to other transition metals with arbitrary spin- orbit coupling strength and their interaction with elec- tromagnetic radiation. To describe the physics of spin-orbital mixing we con-sider a Hamiltonian that has the following form: H=HS+W=X i X ψiEψ|ψi⟩⟨ψi|+ X ψi,ϕiVψϕ|ψi⟩⟨ϕi| +X i̸=j ψi,ψjαij|ψi⟩⟨ψj|(1) where HSstands for on-site interactions and Wrepre- sents the interaction of electrons with an electromag- netic field. We consider that the interaction with light induces the hopping with amplitude αijof the fourth electron in the d4ion to neighboring d3sites i,jin the lattice. The on-site Hamiltonian HScontains diagonal terms denoted by Eψ|ψi⟩⟨ψi|and off-diagonal Vψϕ|ψi⟩⟨ϕi| contributions, coming from vibronic couplings and spin- orbit interactions. The Hamiltonian can be formally ex- pressed in terms of irreducible representations |ψi⟩,|ϕi⟩ ∈ {3A2g, [B1g+B2g],Eg, [A1g+A2g],5A1g,5B1g}. The terms of this basis are expressed as linear combina- tions of Slater determinants of monoelectronic orbitals t2g∈(|xy⟩,|yz⟩,|xz⟩) and eg∈(|x2−y2⟩,|z2⟩) that re- spect the Pauli exclusion principle and the point-group symmetries in orbital ( D4h, due to Jahn-Teller instabili- ties) and spin spaces (in Appendix A we give a detailed derivation of this basis and the development of the corre- sponding Slater determinants). We note that ( B1g, B2g) and ( A1g, A2g) are degenerate (possibly this accidental degeneracy is broken if we consider higher-order relativis- tic corrections) and they are lumped together in the ba-3 sis|ψi⟩. Additionally, the presence of magnetic fields can lift the 2 S+ 1-fold degeneracy of terms3A2g(S= 1), 5A1g(S= 2) and5B1g(S= 2), giving a total of 19 states for the full dimensionality of the |ψi⟩basis. The on-site Hamiltonian HScan be decomposed as: HS=H0+HJT+HSO (2) where H0includes the energy splitting between Ohpoint- group terms5Egand3T1gdue to crystal field and ex- change interactions (see Fig. 1b), while HJTtakes into account interactions with Jahn-Teller modes and HSOis the spin-orbit coupling contribution. The paper is organized as follows. In Sec. II A, we describe the spontaneous breaking of orbital degeneracy driven by Jahn-Teller instabilities in a d4ion under Oh symmetry. In this situation,5E2gand3T1gelectronic states interact with doubly degenerate EgJahn-Teller modes, resulting in E⊗eandT⊗evibronic interac- tions. As a result, the point-group symmetry is reduced toD4hand the terms split into3A2g,3Eg,5A1gand 5B1gstates (see Fig. 1b). In Sec. II B we study the point symmetries in orbital and spin spaces related to the spin-orbit operator. The combination of Jahn-Teller and spin-orbit interactions split further these terms into irreducible representations |ψi⟩ ∈ {3A2g, [B1g+B2g], Eg, [A1g+A2g],5A1g,5B1g}, see Fig. 1b and Appendix A. In Sec. III, we describe the interaction of electrons with electromagnetic fields. As a result of this interac- tion, we assume that electrons hop between neighboring sites. We focus on the problem of a lattice in which the fourth electron of an isolated Jahn-Teller d4ion hops tod3nearest-neighbours. We assume that the solid is a transition metal oxide with perovskite structure with ABO 3chemical formulation, in which generally Ais a rare-earth element, Ba transition metal and Ois oxygen. These systems form a large family of materials that in- cludes La 2/3Ca1/3MnO 3, with a broad diversity of phys- ical properties, including magnetism, ferroelectricity or superconductivity [31–34]. The undisturbed perovskite is formed by octahedral unit cells with the metal Bsit- ting at the center of an octahedron formed by six ligand oxygen anions with Ohpoint symmetry [35, 36], see Fig. 2(a). Using the formalism of two-center Slater-Koster integrals, we derive analytic expressions for the light- induced hopping amplitudes between lattice sites. The perturbative analysis discussed in Sec. IV demonstrates that spin-orbit coupling and intraatomic t2g−egmixing are essential to the appearance of gyrotropic responses, and that the latter involve photoexcitations in which one of the spins is inverted. Remarkably, this observation opens the possibility of using electromagnetic fields to manipulate spins via the mechanism described here. Sub- sequently, in Sec. V, we analyze the electronic response to circularly polarized electromagnetic waves. For that purpose, we analyze the density of5B1gstates from the imaginary part of quantum propagators of the different electronic orbitals and obtain expressions for their spec- tral functions. In Sec. VI we analyze these spectral func-tions in circularly polarized electromagnetic fields as a function of the relative strength of Jahn-Teller and spin- orbit interactions. From this analysis, we extract infor- mation about the gyrotropic responses, by which the po- larization of light is changed as a result of the interactions with t2g−egspin-orbit mixed states. Finally, in Sec. VII, we summarize the main results and discuss perspectives of further work, especially the possibility of entangling spin and orbital degrees of freedom using electromagnetic waves, which could be relevant in the framework of non- trivial quantum states in other systems, including heavy 4d-5dtransition metals. II. THEORY: ON-SITE INTERACTIONS A. Jahn-Teller interactions We first derive the Hamiltonian terms for the interac- tion of electron orbitals with Jahn-Teller modes. Under Ohpoint-group symmetry they can interact with two de- generate representations for EgJahn-Teller vibrational modes corresponding to tetragonal modes QEgu=Q3= 2∆z−∆x−∆yand orthorhombic modes QEgv=Q2=√ 3(∆x−∆y), respectively (see Fig. 2). We thus need to solve the E⊗eandT⊗eproblems to derive analytic ex- pressions for the corresponding vibronic interactions be- tween Jahn-Teller Egmodes and doubly degenerate Eg and triply degenerate T1gelectronic states in Ohsymme- try [37]. A convenient way to derive these expressions is to write the Jahn-Teller modes in terms of an angle ϑas Q3= cos ϑandQ2= sin ϑ[38, 39]. Using Pauli matri- cesυiin the pseudospin space of [5A1g,5B1g] states in D4hsymmetry, the E⊗eJahn-Teller interaction can be expressed as: HE⊗e JT=FE+ 2GE 2υ0+ (FE+GE)υz+ + (FE−2GE)δϑυx(3) where FEandGEare linear and quadratic vibronic con- stants, and δϑrepresents perturbative Q2orthorhombic distortions that will be described below. The dependence of Jahn-Teller modes on ϑdefines a potential energy sur- face, which, in the case of harmonic approximation (i.e., GE= 0), defines a ”Mexican hat” [40]. In solids, how- ever, anharmonic contributions are usually relevant and quadratic constants (so that GE̸= 0) must be included inEq.(3). As a result, the surface potential warps pro- ducing three minima at ϑn= 2nπ/3, which correspond to tetragonal elongations of the octahedral cell along z, yandxaxes for n= 0,2,1, which stabilize the occu- pation of dz2,dy2anddx2orbitals, respectively [41–43] (see also Appendix B for a detailed description of the Jahn-Teller Hamiltonian and the vibronic interactions). Actually, the stabilization of tetragonal Jahn-Teller dis- tortions in solids has been confirmed experimentally in a large number of compounds [44], including manganites [45–48].4 (a) (b) (c) (d) FIG. 2. a) Depiction of a transition metal in an octahedral co- ordination with six oxygen ligands. b) Tetragonal Jahn-Teller distortion corresponding to the mode Q=Q3. c) Orthorhom- bic Jahn-Teller distortion corresponding to the mode Q=Q2. d) Superposition of the two distortion modes corresponding toQ=Q3cosϑ+Q2sinϑ. Now we derive the analytic expressions for vibronic interactions involving T1gelectronic states, which can in- teract with EgandT2grepresentations of Jahn-Teller modes [37]. Consequently, such derivation requires solv- ing the T⊗eandT⊗tproblems [41]. We assume, how- ever, that the T⊗tcontribution is negligible, since in our case lattice deformations are predominantly driven by Jahn-Teller instabilities of electrons in egstates. There-fore, we only consider the contribution of T⊗eto the Hamiltonian as follows: HT⊗e JT=1 2FTh λ0−√ 3λ8−δϑλ3i (4) where FTis the vibronic coupling constant for T⊗eand λ2andλ8are Gell-Mann matrices, defined as λ3= 1 0 0 0−1 0 0 0 0 (5a) λ8=1√ 3 1 0 0 0 1 0 0 0−2 (5b) As mentioned above, anharmonic lattice contributions stabilize tetragonal elongations along the three main axes of the ABO 3octahedral cell units, denoted as ϑn= 2nπ/3. This entails breaking the degeneracy of5E2g electronic states in Ohinto5A1gand5B1gterms in D4h symmetry, being the latter lower in energy for the elon- gated tetragonal distortions (Fig. 1b). In this work we also consider small orthorhombic Jahn-Teller distortions corresponding to Q2modes, which modify perturbatively the elongated tetragonal distortions, so that the angle inQ2−Q3space is mapped to ϑn→ϑn+δϑn. As can be inferred from the E⊗eHamiltonian ((2)), these orthorhombic perturbations induce non-diagonal transi- tions between5A1gand5B1gterms. On the other hand, for the t2gsector, the reduction to D4hsymmetry splits the3T1gterm into single-degenerate3A2gand doubly degenerate3Egterms, being the latter lower in energy. In this case, it can be shown that, to first-order, ϑn+δϑn perturbations do not mix3A2gand3Egterms. B. Spin-orbit coupling In order to compute the matrix elements of spin-orbit coupling in D4hsymmetry we use the operator equivalent method [1, 49]. In this approach, the spin-orbit opera- tor is defined by linear combinations VΛ λq=TΛ λS1 q, where TΛ λcorresponds to irreducible representations Λ in the or- bital space with basis λandS1 qcorresponds to irreducible representations D(1) qin the spin-rotation group. Since the T⊗eHamiltonian term in Equation (3) is relatively small –i.e., FTδϑ⪅ξSO, where ξSOis the spin-orbit coupling constant– we can study the orbital space TΛ λin the D4h point-group. Therefore, while TΛ λtransforms as T1ginOh symmetry, the reduction to D4himplies that TΛ λtrans- forms according to irreducible representations A2gwith spatial symmetry νandEgwith spatial symmetries κ, µ (see Appendix A 2 b for a definition of these symmetries). On the other hand, the spin part S1 qis expressed using spherical coordinates, with quantum numbers q= 0,±1. Taking this into account, it can be demonstrated that the spin-orbit coupling Hamiltonian can be expressed in5 terms of operator equivalent matrices VEg κ±1,VEg µ±1and VA2g ν0as follows: HSO=ξSO⃗L·⃗S=ξSO −1√ 2 VEg κ1−VEg κ˘1 + +ı√ 2 VEg µ1+VEg µ˘1 +VA2g ν0 (6) where the sign of qis denoted by a breve symbol, i.e., ˘q=−q. Then we apply the Wigner-Eckart theorem toVΛ λq, which implies working with reduced matrices ⟨ΓS||VΛ||Γ′S′⟩[50]. In our case, these are 4 ×4 matri- ces defined in terms of the irreducible representations of theD4hpoint-group {3Eg,3A2g,5A1g,5B1g}, which are expressed as follows (see Appendix C for the details of this derivation): VA2g=ı √ 3 0 0 0 0 0√ 10 0 0√ 10 0 0 0 0 0 0 (7a) VEg=ı 0√ 3√ 5−√ 15√ 3 0 0 0√ 5 0 0 0 −√ 15 0 0 0 (7b) Once the reduced matrices are computed, the spin- orbit elements can be found using Clebsch-Gordan coef-ficients as follows: ⟨3EgκM′|⃗L·⃗S|3EgµM⟩=−ı 2δM M′ (8a) ⟨3A2gνM′|⃗L·⃗S|3EgκM⟩=1 2√ 2h δM+1 M′−δM−1 M′i (8b) ⟨3A2gνM′|⃗L·⃗S|3EgµM⟩=−ı 2√ 2h δM+1 M′+δM−1 M′i (8c) ⟨3A2gνM′|⃗L·⃗S|5A1guM⟩=−ır 4− |M| 3δM M′ (8d) ⟨3EgκM′|⃗L·⃗S|5A1guM⟩=ı 4s M2+ 3|M|+ 2 6× ×h δM+1 M′−δM−1 M′i(8e) ⟨3EgµM′|⃗L·⃗S|5A1guM⟩=1 4s M2+ 3|M|+ 2 6× ×h δM+1 M′+δM−1 M′i(8f) ⟨3EgκM′|⃗L·⃗S|5B1gvM⟩=ı 4s M2+ 3|M|+ 2 2× ×h δM+1 M′−δM−1 M′i(8g) ⟨3EgµM′|⃗L·⃗S|5B1gvM⟩=−1 4s M2+ 3|M|+ 2 2× ×h δM+1 M′+δM−1 M′i(8h) With these relations, and taking the basis {|3Egκ1⟩, |3Egκ0⟩,|3Egκ˘1⟩,|3Egµ1⟩,|3Egµ0⟩,|3Egµ˘1⟩, |3A2gν1⟩,|3A2gν0⟩,|3A2gν˘1⟩,|5A1gu2⟩,|5A1gu1⟩, |5A1gu0⟩,|5A1gu˘1⟩,|5A1gu˘2⟩,|5B1gv2⟩,|5B1gv1⟩, |5B1gv0⟩,|5B1gv˘1⟩,|5B1gv˘2⟩}, where again the breve symbol denotes the sign of spin quantum numbers, one can write the full 19x19-dimensional spin-orbit matrix as: HSO=ξSO 4√ 3× × 0 0 0 −ı2√ 3 0 0 0−√ 6 0−ı√ 6 0 ı 0 0−ı3√ 20 ı√ 3 0 0 0 0 0 0 0 0√ 6 0−√ 6 0−ı√ 3 0 ı√ 3 0 0−ı3 0 ı3 0 0 0 0 0 0 ı2√ 3 0√ 6 0 0 0 −ı 0 ı√ 6 0 0−ı√ 30ı3√ 2 ı2√ 3 0 0 0 0 0 0 ı√ 6 0√ 6 0 1 0 0−3√ 2 0−√ 30 0 0 0 0 0 0 0 ı√ 6 0 ı√ 6 0√ 3 0√ 3 0 0 −3 0−3 0 0 0−ı2√ 3 0 0 0 0 ı√ 6 0 0 0 1 0√ 6 0 0−√ 30−3√ 2 0√ 6 0 0−ı√ 6 0 0−ı4√ 30 0 0 −√ 6 0√ 6−ı√ 6 0−ı√ 6 0 0−ı8 0 0 0−√ 6 0 0−ı√ 6 0 0 0 0−ı4√ 30 ı√ 6 0 0√ 6 0 0 0 0 0 0 ı√ 3 0 0√ 3 0 ı4√ 3 0 0 −ı 0 ı 1 0 1 0 ı8 0 0−ı√ 3 0 0√ 3 0 0 0ı4√ 3 0 0−ı√ 6 0 0√ 6 0 0 0 ı2√ 3 0 0 −3√ 2 0 0 0 ı3 0 0 −3 0 −ı√ 3 0 ı√ 3−√ 3 0−√ 3 0 −ı3 0 0 −3 0 0 0−ı2√ 3 0 0−2√ 30 0 0 0 0 0 0 (9) In this expression, solid lines separate the ma- trix elements corresponding to spectroscopic terms {3Eg,3A2g,5A1g,5B1g}ordered from left to right columns. On the other hand, dotted lines separate the orbital angular momentum components ( γ=κ, µ) for the 3Egterm. Finally the spin projections Mof the differ-ent elements are displayed in decreasing order from left to right. We note that matrix Eq. (9) is represented for the quantization of ⃗Land⃗Salong the same axis. However, in general, the quantum spin axis can be oriented along6 arbitrary directions with respect to ⃗L. Therefore, it is convenient to apply appropriate rotations Rin the spin space to orient the spin quantization axis along arbitrary directions defined by ˆ nas follows: S′ z=RSzR†= ˆn·⃗S (10) This rotation is characterized by an axis ˆt= (ˆz׈n)/|ˆz׈n|and a rotation angle θ= arccos ˆ z·ˆn. R= e−ıθˆt·⃗S(11) If we define ˆ n= (sin θcosϕ,sinθsinϕ, cosθ ), we have ˆt= (−sinϕ,cosϕ). Then, we obtain the following spin- rotation matrices for the cases S= 1 and S= 2: R(S=1)=1 2 1 + cos θ√ 2eıϕsinθe2ıϕ(1−cosθ) −√ 2e−ıϕsinθ 2 cos θ√ 2eıϕsinθ e−2ıϕ(1−cosθ)−√ 2e−ıϕsinθ 1 + cos θ (12a) R(S=2)=1 8 2(1 + cosθ)24eıϕsinθ(1 + cos θ) 2√ 6e2ıϕ(sinθ)2 −4e−ıϕsinθ(1 + cos θ) [(1 + 4 cos θ)2−9]/2 4√ 6eıϕsinθcosθ 2√ 6e−2ıϕ(sinθ)2−4√ 6e−ıϕsinθcosθ 4[3(cos θ)2−1] −4e−3ıϕsinθ(1−cosθ)−e−2ıϕ[(1−4 cos θ)2−9]/2−4√ 6e−ıϕsinθcosθ 2e−4ıϕ(1−cosθ)2−4e−3ıϕsinθ(1−cosθ) 2√ 6e−2ıϕ(sinθ)2 4e3ıϕsinθ(1−cosθ) 2e4ıϕ(1−cosθ)2 −e2ıϕ[(1−4 cos θ)2−9]/2 4e3ıϕsinθ(1−cosθ) 4√ 6eıϕsinθcosθ 2√ 6e−2ıϕ(sinθ)2 [(1 + 4 cos θ)2−9]/2 4eıϕsinθ(1 + cos θ) −4e−ıϕsinθ(1 + cos θ) 2(1 + cosθ)2 (12b) We use these matrices to compute the spin-orbit ele- ments of Eq. (8) for arbitrary directions of the quantized spin axis. III. THEORY: INTERACTION WITH ELECTROMAGNETIC FIELDS A. Light-induced electron transfer between lattice sites εd εpεd′ (1) (2) FIG. 3. Diagram of the electron transfer between neighbour- ing sites mediated by oxygen ions, induced by the interaction with an electromagnetic field. In the process, there is a first transition from oxygen p-orbital to neighbouring manganese d’-orbital. A second transition involves a transfer from a man- ganese d-orbital to an oxygen p-orbital. So far we have considered on-site interactions of elec- tronic orbitals with the crystal field, Jahn-Teller vibra-tional modes and atomic spin-orbit coupling. In the fol- lowing, we describe their interaction with electromag- netic fields, which we assume induce electron transfer between neighboring sites in the perovskite lattice. The idea of light-induced electron transfer has been proposed, e.g., in some manganites, where optical energy excita- tions have been associated to polaronic transport due to intersite eg−egphotoinduced transitions [25, 51, 52]. Since the cation separation in perovskites ( ≈4˚A) is large for significant direct overlap [53–55], we consider the elec- tron transfer dominated by hopping through porbitals of oxygen. As depicted in Fig. 3, we consider the transfer between neighbouring d4andd3ions, which can be de- scribed as: d4p6d3→d4p5d4→d3p6d4(13) In order to describe this transfer, we define the fol- lowing many-electronic wavefunctions for the two neigh- bouring dions and the oxygen ligands: |Ψ⟩=|2S+1ΓγM⟩|1S⟩|4A2gN⟩ (14a) |Φpw⟩=|2S+1ΓγM⟩|2Pw±1 2⟩|2S+1Γ′γ′M⟩(14b) |Ψ′⟩=|4A2gN⟩|1S⟩|2S+1Γ′γ′M⟩ (14c) Eq. (14a) corresponds to the initial configuration d4p6d3, where the d4ion is described by some of the2S+1Γ repre- sentations discussed in Sec. I, whereas4A2gis the ground state for the d3ion, according to the Tanabe-Sugano di- agram [1]. On the other hand, the ligand orbitals, which7 have filled shells, are described by1Sand Eq. (14c) describes the final state of the transfer, where the spin part of the wavefunction is unchanged since light can- not interact directly with spins. Finally, Eq. (14b) is the intermediate state where the two transition metals have 4 d-electrons and there is a vacancy in the ligand in apworbital ( w∈ {x, y, z}). This intermediate state re- quires an energy equivalent to the charge transfer energy, ∆CT≈4eV for Mn3+[44]. Therefore, in the presence of an electromagnetic field, the orbitals panddare cou- pled, so that the states |Ψ⟩and|Ψ′⟩are perturbed by the intermediate states |Φpw⟩as follows: |˜Ψ⟩ ≈ |Ψ⟩ −ıtpd 2∆CTˆϵ·X w⟨Φpw|⃗∇|Ψ⟩|Φpw⟩(15a) |˜Ψ′⟩ ≈ |Ψ′⟩ −ıtpd 2∆CTˆϵ·X w⟨Φpw|⃗∇|Ψ′⟩|Φpw⟩(15b) where tpd/2 is the p−dhopping amplitude induced by the electromagnetic field, which allows nonzero matrix elements between states |˜Ψ⟩and|˜Ψ′⟩. We treat the interaction with light to first order, so that the amplitude of the light-induced transfer requires the computation of electromagnetic matrix elements that involve two-center integrals including the vector potential −ı⃗∇(defined in the Coulomb gauge): Pψ qˆϵw=1 ıaZ ψ(⃗ r)⃗∇ϕpw(⃗ r±aˆeq)d⃗ r ·ˆϵ (16) where ˆ eqindicates the hopping direction in the lattice, ˆ ϵis the unit vector along the orientation of the vector poten- tial,ais the lattice parameter and ψ,ϕpwdescribe mono- electronic orbitals in the transition metal and oxygen, re- spectively, that are involved in the photoinduced transfer. We note that although the spectroscopic terms are given as combinations of Slater determinants, the vector poten- tial in Eq. (16) is a one-body operator that acts only on the monoelectronic orbital where the transferred electron resides (see Appendix D for a detailed discussion of how one-body operators act on the many-electron wavefunc- tions). The matrix elements shown in Eq. (16) are there- fore expressed in terms of monoelectronic functions ψand ϕpw. This derives from the properties of the one-body po- tential, whereby matrix elements such as −ı⟨Φpw|⃗∇|Ψ⟩, where |Φpw⟩,|Ψ⟩are many-electron functions described by Eq. (14), can be rewritten as −ı⟨ψ|⃗∇|ϕpw⟩, where |ϕpw⟩,|ψ⟩describe monoelectronic orbitals. On the other hand, while the expression in Eq. (16) corresponds to a transfer from a pto a dorbital, the dtoptransition is described by its complex conjugate (Pψ qˆϵw)∗. Interestingly, it can be shown that expressions like∂ˆϵφpw(with ˆ ϵalong ˆ x, ˆy, or ˆz) appearing in Eq. (16) can be expressed as linear combinations of Slater-Koster coefficients (see Chapter 7 in Ref. [31] for a derivation). For instance, for the vector potential along ˆ ϵ||ˆx, we makeuse of the following coefficients: (sdσ)≡1 aZ ψz2(⃗ r)¯ψs(⃗ r±aˆez)d⃗ r (17) (ddσ)≡1 aZ ψz2(⃗ r)¯ψz2(⃗ r±aˆez)d⃗ r (18) (ddπ)≡1 aZ ψxy(⃗ r)¯ψxy(⃗ r±aˆex)d⃗ r (19) (ddδ)≡1 aZ ψxy(⃗ r)¯ψxy(⃗ r±aˆez)d⃗ r (20) where ψz2,ψxyare wavefunctions for the monoelectronic states |z2⟩and|xy⟩, and ¯ψs,¯ψz2and ¯ψxyare effective wavefunctions which have the same symmetries as s,dz2 anddxyorbitals (see Ref. [31]). Table I displays all nonzero matrix elements for the vector potential along the three directions in space in terms of the coefficients (sdσ), (ddσ), (ddπ) and ( ddδ). The hopping amplitudes αψiϕjqbetween ψiandϕjor- bitals located at neighboring sites ( i, jsuch that ⃗ ri−⃗ rj∥ ˆeq) are calculated perturbatively, taking into account the p−dhopping tpdand the charge transfer energy ∆ CT between panddorbitals [44]: αψiϕj ˆϵq=t2 pd ∆CTX wPϕj qˆϵw Pψi qˆϵw∗ (21) Since the electromagnetic field cannot interact directly with spins, the matrix elements of the electromagnetic operator Wˆϵhave the following form: ⟨iΓγSM|Wˆϵ|jΓ′γ′S′M′⟩=αψiϕj ˆϵqδS′ SδM′ M (22) where i,jrefer to neighboring locations in the lattice. In the next section we explain how the hopping ampli- tudes depend on the light polarization, which is described by the unit polarization vector ˆ ϵalong an arbitrary di- rection. B. Cooperative Jahn-Teller effects Although we address the dynamics of electron transfer from isolated Jahn-Teller ions, we incorporate coopera- tive effects, known to be relevant in solids [39, 44, 56–61]. The reason is that the dynamics of ions is much slower than the electronic transfer rates, so that we assume that cooperative effects restrict the possible Jahn-Teller defor- mations of the neighbouring sites where the transferred electron can jump into (see Fig. 4). As discussed in Sec. II A, we consider Jahn-Teller modes of the d4ion described by angles ϑn= 2nπ/3 +δϑ, n = 0,1,2 and δϑ≪2π/3. In consequence, there are three possible ori- entations for the transfer across the six oxygen anions surrounding the initial d-site, namely along ±ˆx,±ˆyor ±ˆz. Then, cooperative effects are incorporated by impos- ing restrictions on the hopping from an initial d4ion with8 χw q=x q=y q=z ux −1 2sdσ−1 2√ 3ddσ −1 2sdσ+1 4√ 3ddσ+√ 3 8ddδ sdσ−1 2√ 3ddσ vx√ 3 2sdσ+1 2ddσ −√ 3 2sdσ+1 4ddσ−1 8ddδ −1 4ddδ ηz1 2ddπ1 2ddδ1 2ddπ τy1 2ddπ1 2ddπ1 2ddδ uy−1 2sdσ+1 4√ 3ddσ+√ 3 8ddδ −1 2sdσ−1 2√ 3ddσ sdσ−1 2√ 3ddσ vy√ 3 2sdσ−1 4ddσ+1 8ddδ −√ 3 2sdσ−1 2ddσ1 4ddδ ζz1 2ddδ1 2ddπ1 2ddπ τx1 2ddπ1 2ddπ1 2ddδ uz−1 2sdσ+1 4√ 3ddσ−√ 3 8ddδ −1 2sdσ+1 4√ 3ddσ−√ 3 8ddδ sdσ+1√ 3ddσ vz√ 3 2sdσ−1 4ddσ−1 8ddδ−√ 3 2sdσ+1 4ddσ+1 8ddδ 0 ζy1 2ddδ1 2ddπ1 2ddπ ηx1 2ddπ1 2ddδ1 2ddπˆϵ= ˆx ˆϵ= ˆy ˆϵ= ˆz TABLE I. Nonzero matrix elements ıPψ qˆϵwfor the vector potential along the three directions in space ˆ ϵ= ˆx,ˆy,ˆzin terms of Slater-Koster coefficients. tetragonal distortion along ˆ z(Jahn-Teller mode with an- gleϑ0) to neighboring sites along the three directions (see Fig. 4). For instance, when δϑ > 0 there is a contraction along yaxis, forcing neighbours on the xyplane to be dis- torted along the ydirection with δϑ < 0. On the other hand, when a site distorted along zhasδϑ < 0, there is a slight contraction along x, so their neighbours are distorted along xwith δϑ > 0. Our model considers the orbital ordering with maximum entropy, which consists in having half of the octahedra distorted along a partic- ular direction (chosen to be z) and the rest is equallydistributed among elongations along xandy[62]. The resulting orbital ordering is depicted in Fig. 4.9 xyz (1) (2) FIG. 4. Graphical representation of the cooperative distor- tions taking place during the light-induced transfer of elec- trons across the lattice. One of such transfers is illustrated by labels ”1” and ”2”, where the electron jumps through an intervening oxygen. Each vertex represents a transition metal, around which the octahedron elongates along the solid lines. The electron is initially located at d4sites whose distortion is along z. Due to cooperative effects, the d3sites around the initial d4site are elongated along directions perpendicular to z. C. Hopping amplitudes for circularly polarized light 1. Left- and right- handed basis for circularly polarized light We describe the polarization of light by a complex vec- tor ˆϵ∈C3normalized to ˆ ϵ·ˆϵ∗= 1: ˆϵ=1q E2 0x+E2 0y+E2 0z E0x E0ye−ı(ϕy−ϕx) E0ze−ı(ϕz−ϕx) (23) Here E0iis the amplitude of the i-th component of the electric field and ϕithe correspondent phase, defined in this expression to keep ϵxreal. Since light is a transverse wave, we need only two com- plex vectors to define a basis for the polarization, both orthogonal to propagation direction ˆk. For circularly po- larized light we use left-handed and right-handed polar- izations, which, in the case of wave propagation along ˆk= ˆzare defined as: ˆςL=1√ 2 1 ı 0 ˆςR=1√ 2 1 −ı 0 (24) For arbitrary orientations of the propagation of light, we use the rotation matrix Rto find the new basis ˆϵL,R=RˆςL,Rfor the polarization. This rotation is char- acterized by an axis ˆ u= (ˆz׈k)/sinαand an angle ofrotation cos φ= ˆz·ˆk. Since every unit vector ˆkcan be described using polar and azimuth angles, α, β, then ˆk= (sin αcosβ,sinαsinβ,cosα), ˆu= (−sinβ,cosβ,0) andφ=α. Then, the rotation matrix for arbitrary wave propagation can be defined as: R= cα+s2 β(1−cα)−sβcβ(1−cα)sαcβ −sβcβ(1−cα)cα+c2 β(1−cα)sαcβ −sαcβ −sαsβ cα (25) where a contracted notation for trigonometric functions is used, namely, sx= sin xandcx= cos x. Finally the polarization vector for arbitrary wavevector orientation has the following expression: ˆϵL,R(ˆk) =1√ 2 cα∓ısβ(1−cα)e±ıβ ±ı[cα+cβ(1−cα)e±ıβ] −sαe±ıβ (26) 2. Electromagnetic response and time-reversal symmetry Left- and right-handed polarizations are related to each other by complex conjugation, ˆ ϵ∗ L= ˆϵR. With this rela- tion we can deduce that ( Pψ qLw)∗=−Pψ qRw(see Eq. (16)) ifψ(⃗ r)∈R,∀⃗ r∈R3, in other words: ψhas real spatial symmetry, which is the case of the basis used here. Time reversal involves complex conjugation (since light acts only on the orbital angular momentum) and the inter- change of the initial and final orbitals in the hopping, so that we have (αψϕ qL)∗=αϕψ qL=αψϕ qR (27) which means that KW LK†=WR, being Kthe complex conjugation operator and WL,Rthe electromagnetic op- erator for left- and right- handed light. We note that the Hamiltonian terms (see Eq. (2)) H0andHJTare expressed as real matrices, while HSOis complex. Thus, in the absence of spin-orbit coupling ( ξSO= 0), we have KHLK†=HRand, as a consequence, the gyrotropic sig- nal is zero. A nonzero gyrotropic response (a different re- sponse to electromagnetic waves of opposite handedness) arises only when ξSO̸= 0, which implies KHRK†̸=HL. This conclusion does not depend on the basis, since it holds even when the wavefunctions are not real, for in- stance, when they are expressed in spherical basis. In- deed, transforming from spherical to a real basis involves a unitary transformation U, so that the relation between WLandWRis U†KUW LU†KU=˜KW L˜K†=WR (28) where ˜Kis also an antiunitary transformation that keeps invariant H0andHJT. This confirms that a change of basis does not break the time-reversal invariant relation between HLandHRwhen ξSO= 0.10 5B1g5A1g3Eg3A2g tpdEigenvalues FIG. 5. Schematic depiction of the dependence on the tpd hopping of the eigenvalues of the wavefunctions correspond- ing to the irreducible representations5A1g,5B1g,3Egand 3A2g. The shadowed area corresponds to the parameter space where the spin-orbit mixing between t2gandegstates and, consequently, the gyrotropic response, are both strong. 3. Orbital-selective gyrotropic responses in broken time-reversal symmetry The origin of the gyrotropic responses can be traced back to the transfer induced by light between specific orbitals in the dmanifold. To shed light on this issue, it is convenient to express the polarization as ˆ ϵL= (a, b, c ) and ˆϵR= (a∗, b∗, c∗). With this, the transfer amplitudes tψϕ qˆϵin Eq. (27) can be expanded as products of Pψ qˆϵˆw integrals (see Eq. (16)) as αψϕ qL=−X w |a|2Pψ qxwPϕ qxw+|b|2Pψ qywPϕ qyw+ +|c|2Pψ qzwPϕ qzw+a∗bPψ qxwPϕ qyw+ +a∗cPψ qxwPϕ qzw+ab∗Pψ qywPϕ qxw+ +b∗cPψ qywPϕ qzw+ac∗Pψ qzwPϕ qxw+ +bc∗Pψ qzwPϕ qyw(29) By inspection of Eq. (22) one realizes that any transfer involving hopping between egandt2gorbitals at neigh- boring sites is forbidden, since S= 2 and S′= 1 (or viceversa). Then, by taking into account Eq. (29) and the nonzero transfer integrals Pψ qˆϵwdisplayed in Table I, one can verify that all hopping amplitudes involving hopping between neighboring eg−egorbitals are real and are consequently time-reversal invariant. Therefore, the transfer between eg−egorbitals cannot give a gy- rotropic response, at least to first order perturbation in the electromagnetic field. On the other hand, the light- induced transfer between neighboring t2g−t2gorbitalshas complex amplitude and breaks time-reversal symme- try, causing distinct electromagnetic responses for light of opposite handedness. As a consequence, both the pres- ence of spin-orbit coupling and intersite t2g−t2gtransfer are key ingredients to have a gyrotropic signal. In the light of the previous discussion, one expects a strong influence of the tpdhopping integral on the gy- rotropic signal. Since the overlapping integrals ( sdσ) and (ddσ) are significantly larger than ( ddπ) and ( ddδ), the energy of t3 2ge1 gstates (corresponding to5A1gand5B1g representations) is influenced much more strongly by tpd than t4 2gstates with3Egand3A2grepresentations (see Fig. 5). It is then expected that as the value of tpd grows, the eigenvalues of t4 2gstates will cross eventually the eigenvalues of t3 2ge1 gstates, producing a strong spin- orbit mixing and enhancing the gyrotropic signal (shad- owed area in Fig. 5). In the present problem, we have verified numerically that this condition is fulfilled for val- ues in the range t2 pd/∆CT∼0.3eV−1eV. IV. PERTURBATION ANALYSIS OF THE GYROTROPIC RESPONSE As discussed previously, a gyrotropic signal requires photoinduced transfer between adjacent t2g−t2gorbitals. In consequence, the unperturbed ground state of the iso- lated Jahn Teller ion5B1g(t3 2ge1 g) has to be excited to a t4 2gconfiguration to activate this transfer channel. Here we derive a perturbation analysis in spin-orbit coupling and orthorhombic modes to understand the electronic transitions that contribute to the gyrotropic signal. First of all, we introduce the notation |i2S+1ΓγM⟩, which in- dicates the irreducible representation of the wavefunction at the i-th site in the lattice. The introduction of inter- site hopping by interaction with the electromagnetic field breaks the degeneracy between the same state at different sites, originating 0-th order eigenstates denoted by: |α2S+1ΓγM⟩=X icα i|i2S+1ΓγM⟩ (30) To continue with the perturbed states we have to un- derstand how the Hamiltonian acts on those 0-th order eigenstates. Recalling that the Hamiltonian has on-site HSand inter-site electromagnetic Wterms, we have: ⟨α′2S′+1Γ′γ′M′|HS|α2S+1ΓγM⟩= =X i(c′α′ i)∗cα i⟨i2S′+1Γ′γ′M′|HS|i2S+1ΓγM⟩(31a) ⟨α′2S′+1Γ′γ′M′|W|α2S+1ΓγM⟩= =X i̸=j(c′α′ i)∗cα j⟨i2S′+1Γ′γ′M′|W|j2S+1ΓγM⟩(31b) A difficulty arises to compute such matrix elements due to cooperative Jahn-Teller effects. In particular, a given11 irreducible representation may contain different wave- functions at adjacent sites in the lattice. For instance, the 3A2gterm corresponds to a |ζητ¯τ|Slater determinant if the distortion is along the zdirection, but if this distor- tion is along yit corresponds to |ζητ¯η|. Thus, the compu- tation of matrix elements described by Eq. (31), neces- sary to determine the perturbed eigenstates, is challeng- ing. We sort out this difficulty by approximating the ma- trix elements, for instance, ⟨α′3Egtq′M′|H|α3EgtqM⟩ ∼ t2 pd/∆CT (ddπ)2+ξSO+FT. In this example, which can be generalized to arbitrary elements, every term is not determined exactly, but it gives a reasonable esti- mate of the contributions coming from intersite hopping, spin-orbit coupling and Jahn-Teller interactions.Now we develop the perturbative analysis. Since we consider the dynamics of an electron initially located in a tetragonally elongated site, we are therefore interested in calculating the transition rates between5B1g(cor- responding to a t3 2ge1 gconfiguration) and3Egor3A2g terms (both corresponding to a t4 2gconfiguration). By the effect of spin-orbit coupling and orthorhombic dis- tortions, the non-perturbed wavefunctions |α3A2gνM⟩ and|α3EgtqM⟩become, respectively, |α3A2gνM⟩•and |α3EgtqM⟩•, where spherical harmonics are used to de- scribe the orbital components tqof the wavefunctions (see Eq. (A10) for the definition of tq). The matrix elements are then approximated to first order in spin-orbit cou- pling and orthorhombic modes as follows: ⟨5B1gv(±1−q)|3Egtq(±1)⟩•∼ξSO EEB(32a) ⟨5B1gv(±1 +q)|3Egtq(±1)⟩•∼δϑ EEB" FT−ξSO EEA FE−2GE±t2 pd ∆CT(sdσ)2 δϑ!# (32b) ⟨5B1gv(±1)|3Egκ0⟩•∼ξSO EEB" 1 +δϑ EEA FE−2GE±t2 pd ∆CT(sdσ)2 δϑ!# (32c) ⟨5B1gv(±1)|3Egµ0⟩•∼ξSO EEB" 1−δϑ EEA FE−2GE±t2 pd ∆CT(sdσ)2 δϑ!# (32d) ⟨5B1gvM|3A2gνM⟩•∼ξSOδϑ EABEAA FE−2GE±t2 pd ∆CT(sdσ)2 δϑ! (32e) where EEBandEEAare, respectively, the energy gaps between3Egand5B1gand between3Egand5A1g, while EABandEAAare the analogous gaps correspond- ing to3A2ginstead of3Eg. These energy gaps determine the degree of orbital mixing between egandt2gstates. The vibronic constants ( FE, GE, FT), spin-orbit coupling (ξSO) and intersite hopping ( tpd) are also included in the expressions above. According to this perturbational analysis, the different transitions contributing to the gy- rotropic signal are sketched in Fig. 6. We first note that spin-orbit corrections connect5B1gwith3Eg, giv- ing rise to the matrix element of Eq. (32a). On the other hand, Eq. (32b) stems from inter-site hoppings and or- thorhombic corrections connecting5B1gand5A1gfol- lowed by spin-orbit mixing of5A1gwith3Eg, while Eq. (32e) takes account of inter-site hoppings and orthorhom- bic corrections connecting5B1gand5A1gplus spin-orbit coupling between5A1gand3A2g. Finally, Eq. (32c) and Eq. (32d) come from spin-orbit interactions within the3Egsubspace with M= 0, where the degeneracy of the wave-functions, which is preserved by spin-orbit coupling, lifts under the action of orthorhombic modes. An inspection of these expressions allows to extract thefollowing conclusions: •A relevant gyrotropic signal appears when the eg−t2gspin-orbit mixing is large. According to Fig. 5, this happens when the gap EEAbetween 5A1gand3Egis reduced by the effect of light in- duced transfer through the tpdhopping integral. In this case, the strong reduction of EEAentails an en- hancement of contributions described by Eqs. Eq. (32b)-Eq. (32e). •All amplitudes involving5B1g,3Egand3A2gde- scribed by Eq. (32) imply transitions between t3 2ge1 g andt4 2gconfigurations, where one of the spins is in- verted during the transition. The only exception is the transition described by Eq. (32e), which is a second-order correction in ξSOδϑ, i.e., it re- quires the simultaneous action of spin-orbit and orthorhombic interactions. Since δϑis small, and considering typical values for the vibronic constants (FT, FE, GE. see section Sec. VI), the contribution of this term is negligible. Therefore, we conclude that the observation of a large gyrotropic signal is fundamentally contributed by transitions that in-12 volve a spin reversal. •The perturbative influence of orthorhombic Jahn- Teller modes is described by the parameter δϑ. For small values of the5A1g−3Eggap, i.e., EEA⪅ξSO, the predominant transition contributing to the gy- rotropic signal is given by Eq. (32b). In this case, in addition to spin-orbit coupling, the hopping be- tween neighboring t2g−t2gstates and orthorhom- bic modes enhance the gyrotropic signal. However, since the inter-site hopping is far larger than the energy of the orthorhombic distortions, the depen- dence of the gyrotropic response on δϑis very weak. On the other hand, for large enough values of the gapEEA> ξSO, the transition described by Eq. (32a) becomes predominant, but its amplitude is significantly smaller than for the case EEA⪅ξSO. We can then conclude that the role of orthorhom- bic perturbations is minor, at least in the regime where FTδϑ⪅ξSOand, therefore, the gyrotropic response is dominated by transitions between5A1g and3Eg, where the wavefunctions are perturbed by spin-orbit coupling. We end this section by discussing the effects of the geometry on the gyrotropic signal, stemming from the relative orientations of light propagation and spin quan- tization, taking ˆ zas the orientation along the tetragonal distortion. An inspection of Eq. (29) reveals that for light propagating along ˆk= ˆz, namely, when light prop- agates along the distortion, the allowed gyrotropic hop- ping channel is mediated by ζ−ηorbitals. In contrast, when light propagates perpendicular to the Jahn-Teller distortion the allowed gyrotropic hopping is η−τfor ˆk= ˆx, while for propagation along ˆk= ˆythe gyrotropic hopping is mediated by τ−ζorbitals (see Eq. (A3c), Eq. (A3c) and Eq. (A3e) for a definition of the t2gorbitals ζ,ηandτ). As a result, the magnitude of the gyrotropic signal strongly depends on both spin axis and light prop- agation. The reason is as follows. In general, for a given couple of t2gorbitals in the hopping channel, the matrix elements of the angular momentum are nonzero only if the direction of the momentum component is contained in both spatial symmetries of the orbitals. For instance, forτ∼xyandη∼xzorbitals, the only non-vanishing element is ⟨τ|lx|η⟩ ̸= 0. In addition, for a given pair of coupled orbitals in the hopping channel, it can be shown that the spin axis has to be oriented along the component of the nonzero matrix element to have a gyrotropic signal. For instance, for light propagating along the Jahn-Teller distortion, i.e., for ˆk= ˆz, the only gyrotropic channel is ζ−η, Therefore, if the spin is quantized along x, then ⟨ζ|lx|η⟩= 0 and the gyrotropic signal is completely extin- guished. Numerical calculations, discussed below, have been performed to study systematically the effect of ge- ometry on the response to circularly polarized light. In the following, we introduce the formalism to evalu- ate the gyrotropic response (section Sec. V), which we use to perform numerical calculations based on the exactdiagonalization of the full Hamiltonian. In Sec. VI we an- alyze the influence of spin-orbit coupling and orthorhom- bic Jahn-Teller modes on the gyrotropic response, which confirms the general tendencies discussed in this section. 5B1g5A1g3Eg3A2g |tqq⟩ |tq˘q⟩|µ0⟩ |κ0⟩ FIG. 6. Sketch of the transitions allowed by intersite hopping induced by light (green solid lines), spin-orbit coupling (brown dashed lines) and Jahn-Teller orthorhombic distortions (blue dashed-dotted lines). Thicker arrows indicate stronger inter- actions, corresponding to transitions between5B1gand5A1g mediated by intersite hopping and between5A1gand3Eg mediated by spin-orbit coupling. V. RESPONSE TO CIRCULARLY POLARIZED ELECTROMAGNETIC WAVES We consider the excitation of an electron located ini- tially in a d4site distorted along z, see Fig. 4. In the presence of an electromagnetic field, this electron is trans- ferred to any of the six nearest neighboring d3sites in the lattice. As explained in section Sec. III A, we assume that cooperative effects induce orbital ordering around the initial d4site, so that the site that receives the trans- ferred electron can only deform along particular orienta- tions, as shown in Fig. 4. In the calculations, the orbital ordering is extended periodically throughout the solid. To compute the dynamics, we suppose that the system is prepared in an ensemble ϱ=X ψpψ|ψ⟩⟨ψ| (33) where ψrefers to state5B1gin the central site, which has the lowest energy (see Fig. 1b). Here, pψis the relative weight assigned to each spin projection allowed by the irreducible representations of the corresponding many-electron wavefunctions. The values of pψare in- deed obtained for each specific case after diagonalization of the full Hamiltonian. The ensemble in Eq. (33) evolves in time as ϱ(t) = e−ıHtϱeıHt(34)13 which allows us to compute the quantum propagator [63, 64] G(t) =−ıΘ(t)⟨[ϱ(t), ϱ]⟩ (35) where the Heaviside function Θ( t) accounts for causality and⟨O⟩is the thermal average of the operator O. ⟨O⟩= tr [ ρO] (36) Here ρis the density matrix for a thermal bath ρ=1 ZX ke−βH|k⟩⟨k|=1 ZX ke−βωk|k⟩⟨k| (37) where Zis the partition function and the second equality holds if {|k⟩}is an eigenbasis of the Hamiltonian. We can then express the quantum propagator in the following way G(t) =−ıΘ(t)1 ZX ψ k,mp2 ψ e−βωk−e−βωm
× ×eı(ωm−ωk)t|⟨k|ψ⟩|2|⟨m|ψ⟩|2(38) Here we have introduced the identity using the Hamil- tonian eigenbasis labeled with m. Defining Ω km= |ωk−ωm|and approximating e−βω≈1 for ω < β−1 and e−βω≈0 for ω > β−1we can rewrite the propagator as follows preserving only the terms such that ωk≫ωm: G(t) =−ıΘ(t)1 ZX ψp2 ψX k,me−ıΩkmt× × |⟨k|ψ⟩|2|⟨m|ψ⟩|2+c.c.(39) The first term in Eq. (39) corresponds to the retarded propagator, while the complex conjugate term is the ad- vanced propagator. In order to compute the spectral response, we use the Heaviside function in the frequency domain Θ(t) =−1 2πılim η→0+Z∞ −∞dωe−ıωt ω+ıη(40) Using the expression Eq. (40) we can write the spectral representation of the propagator in frequency domain Gr(ω) =1 ZX ψ,mp2 ψ|⟨m|ψ⟩|2X k|⟨k|ψ⟩|2 ω−Ωkm+ıη(41) where Ω kmdenotes the frequency eigenvalues of the full Hamiltonian. In the limit η→0+, we have ς(ω) =−1 πℑ[Gr(ω)] =1 ZX ψ,mp2 ψ|⟨m|ψ⟩|2× ×X k|⟨k|ψ⟩|2δ(ω−Ωkm)(42)which has the form of a density of states, which we denote asς(ω), while the parameter ηis related to the lifetime of the excited states. After some algebra, the spectral function can be rewritten as ς(ω) =η ZπX ψ,mp2 ψ|⟨m|ψ⟩|2X k|⟨k|ψ⟩|2 (ω−Ωkm)2+η2(43) In the next section, we use this function to evaluate the gyrotropic response when time-reversal symmetry is broken. VI. NUMERICAL SIMULATIONS A. Calculation of the spectral functions for circularly polarized light For the calculation of the spectral functions defined by Eq. (43) we have solved the full Hamiltonian (Eq. (1)) to compute the eigenvalues. The spectral functions have been obtained for left- ( ςL) and right- ( ςR) circularly po- larized light, by calculating the hopping amplitudes as described in Sec. III C. From these functions, we have built non-gyrotropic ( ςng) and gyrotropic ( ςgy) spectral functions in frequency space, which give account, respec- tively, of the dynamic responses that are insensitive and sensitive to the handedness of the polarization of light. These functions are defined as follows: ςng(ω) =ςL(ω) +ςR(ω) 2(44a) ςgy(ω) =ςL(ω)−ςR(ω) 2(44b) Finally, we define a function that integrates the gy- rotropic signal over the analyzed spectral range (0eV − 3.5eV): Ngy=Z∞ 0ςgy(ω)dω (45) The numerical calculations were carried out by setting the vibronic constants to FE= 450meV, FT= 130meV andGE= 20meV. These values are in agreement with the Jahn-Teller splitting observed for egandt2gelec- trons in 3d elements [44]. On the other hand, the charge transfer gap has been set to ∆ CT= 4eV [44], the damp- ing factor to η= 180meV (see Eq. (43)) and the p−d hopping to tpd= 1.2eV (Eq. (21)). Finally, the Slater- Koster coefficients were set to ( sdσ) = 1, ( ddσ) = 0 .82, (ddπ) = 0 .29 and ( ddδ) = 0 .07. We studied different geometries by varying the rela- tive orientation of the light propagation, Jahn-Teller dis- tortions and spin quantization. By way of illustration, the spectral functions ςngandςgydisplayed in Fig. 7 were calculated for three different geometries, which are sketched in the top panels of each column. The spec- tral functions were computed for four different values of14 xz JT ˆk ˆs ∆ = 12 .5meV∆ = 0 .44eV∆ = 0 .89eV∆ = 1 .36eV (a) xy JTˆk ˆs (b) xz JT ˆk ˆs (c) FIG. 7. Nongyrotropic ( ςng) (black solid lines) and gyrotropic ( ςgy) (red dashed lines) spectral functions calculated for different energy gaps ∆ defined between5A1gand3Egterms (see Fig. 1b). The spectra computed for ∆ = 1.36eV ,0.89eV ,0.44eV ,12.5meV are displayed in descending order for each column (labelled a), b) and c)). The different functions have been computed in different geometric configurations, as sketched on the top of each column. The visible part of the spectrum has been shadowed and divided in two parts, below and above 550nm. This division enables an easier comparison with the experimental magneto-optical spectra reported in Ref. [25]. Those experiments show two absorption peaks centered, respectively, at wavelengths <550nm and >550nm, where only the latter gives rise to a gyrotropic response. The spectra computed in column b) for a gap ∆ = 1 .19eV are in agreement with the experimental spectra reported in [25]. the energy gap ∆ (as indicated in the panels of Fig. 7), while the spin-orbit coupling was set to ξSO= 20meV. The gap ∆ is defined as the energy difference between the unperturbed5A1gand3Egterms (see Fig. 1b), which gives an estimation of the degree of t2g−egmixing before the introduction of the electromagnetic field. All param- eters, including ∆, were chosen to work in a region of the Tanabe-Sugano diagram appropriate for manganese ions, for which the crystal field is 10 Dq≈2eVand the Racah parameter is B≈0.11−0.13eV[26, 65, 66]. On the other hand, in Fig. 8, the integrated gyrotropic signal described by Ngy(Eq. (45)) is mapped as a function of spin-orbit coupling ξSOand orthorhombic perturbations δθfor each value of ∆. Panels in Fig. 8 are organized in the same way as in Fig. 7, i.e., each column correspondsto each of the geometries sketched on the top. B. Discussion of gyrotropic and nongyrotropic responses We first discuss the nongyrotropic spectra described by functions ςngshown in Fig. 7. First of all, we ob- serve that the structure of resonances remains virtually unchanged, as long as the geometry is fixed, regardless of the values of the other parameters. In addition, a comparison between the ςngspectra displayed in Fig. 7 (a) and (c) shows that the nongyrotropic response does not depend on the direction of the spin quantization ˆ s, provided that the relative orientations of light propaga-15 xz JT ˆk ˆs (a) xy JTˆk ˆs (b) xz JT ˆk ˆs ∆ = 12 .5meV∆ = 0 .44eV∆ = 0 .89eV∆ = 1 .36eV (c) FIG. 8. Maps of the integrated spectral function Ngydefined by Eq. (45), as a function of the spin-orbit coupling ( ξSO) and orthorhombic perturbations ( δϑ). The maps were computed for the different geometries sketched in the top of each column, labelled as a), b) and c). For each column, the integrated spectral functions Ngywere calculated for different values of the energy gap ∆. tion and distortions remain the same. This is an indica- tion that the nongyrotropic spectra are contributed es- sentially by transitions between5B1gand5A1gstates. This observation is supported by the fact that the cor- responding spin-orbit elements are zero for these states (see Eq. (9)), which explains why the ςngspectra re- main unchanged as the axis of spin quantization changes. Therefore, the structure of resonances observed in ςng arises basically from the interactions between5B1gand 5A1gstates, mostly through intersite hopping induced by light (Fig. 5). More specifically, such transitions con- nect5B1gand5A1gbonding/antibonding states emerg- ing from light-induced hybrization, which explains the structure of the peaks in the spectra (Fig. 7). The res- onances located at lower energy, below the visible range (shadowed areas in Fig. 7) are mostly contributed by transitions between hybridized5B1gorbitals, while the resonances located in the visible region correspond to transitions involving5A1gstates. We turn now our attention to the gyrotropic ςgyspec- tra. As observed in Fig. 7, their structure is much sim- pler, with a main resonance located in the visible or near- infrared, depending on the geometry and value of ∆. Thecontributions to this resonance come mainly from tran- sitions between3Egand3A2gstates that are perturbed by spin-orbit coupling (see section Sec. IV). The spectral weight of these transitions is too small to be observed in the nongyrotropic spectra. Nonetheless, their effect on the gyrotropic response is crucial, via the orbital mixing induced by spin orbit coupling. One way to evaluate this mixing is by varying the energy gap ∆ defined above. In particular, we observe in Fig. 7 that the gyrotropic sig- nal is the smallest for the largest value of ∆, as expected from the smaller orbital mixing in this case. Let us now discuss the effect of orbital hybridization induced by the coupling to the electromagnetic field. As discussed in Sec. IV, the coupling to light induces p−d hybridization, which causes an evolution of the eigen- states as a function of the overlapping between oxygen and transition metal states, as sketched in Fig. 5. In our numerical calculations we observe that for values tpd≈1.2eV and ∆ ⪅0.89eV there is a strong orbital mixing. Indeed, for this choice of values, the calculated spectra are in agreement with the experimental spectra reported in Ref. [25]. In particular, in the calculated spectra we observe two nongyrotropic resonances in the16 red and blue parts of the visible range, respectively, while a main gyrotropic resonance is seen in the blue region, in agreement with the experiments [25]. Next we discuss the effects of the geometry on the gy- rotropic response. The data shown in Fig. 7 and Fig. 8 reveals the strong dependence of the spectral functions on geometric factors. The effect is particularly evident for spectra shown in panels (c) of both Figures, which show that the gyrotropic signal is completely extinguished for this particular geometry. As discussed in Sec. IV, the reason for this extinction is that the matrix elements of the angular momentum that connect the t2gorbitals in adjacent sites are null in this case, because the direc- tion of the momentum component is not contained in the space symmetries of the t2gorbitals involved in the transfer. On the other hand, the gyrotropic signal for the geometry sketched in panels (b) of both Figures is sig- nificantly larger than for the spectra and maps shown in panels (a). The reason is that for the geometry of spec- tra and mappings shown in Fig. 7(a) and Fig. 8(a), the photoinduced transfer that contributes to the gyrotropic response happens only between adjacent η−τorbitals. In contrast, for the geometry studied in Fig. 7(b) and Fig. 8(b) both ζ−ηandζ−τhopping channels are allowed, increasing the gyrotropic signal. Finally, the data shown in the maps of Fig. 8 re- veals the dependence of the gyrotropic response on spin- orbit coupling. Indeed, when the latter tends to zero, the gyrotropic signal becomes vanishingly small, while it becomes progressively more intense as the spin-orbit coupling increases. On the other hand, we see that or- thorhombic perturbations barely modify the gyrotropic spectra, as expected from the fact that these perturba- tions are much smaller than the intersite hopping induced by coupling to the electromagnetic field. Both observa- tions are in agreement with the conclusions drawn from the perturbative analysis discussed in Sec. IV. VII. CONCLUSIONS AND PERSPECTIVES We used a group theoretical approach to study the in- teraction of transition metals with electromagnetic fields. For that purpose, we described the relevant electronic states by irreducible representations of pertinent point- group symmetries, which were constructed from many- electron wavefunctions based on Slater determinants. The energetics of the problem was established to com- ply with Tanabe-Sugano diagrams corresponding to the particular ion under study [1]. Starting from an initial Ohsymmetry, we analyzed the effect of symmetry reduc- tion due to Jahn-Teller interactions and spin-orbit cou- pling. The interaction with electromagnetic fields was assumed to produce photoinduced transfer of electrons across the lattice. The model predicts an electronic re- sponse that depends on the handedness of circular po- larization of light. Key ingredients to this gyrotropic response are spin-orbit coupling and intraatomic eg−t2gorbital mixing. Remarkably, the gyrotropic resonances are related to photoexcitations where one of the spins is inverted, enabling the use of electromagnetic fields to manipulate spins. We also analyzed the dependence of the gyrotropic response on the relative orientation of Jahn-Teller distortions, light propagation and spin quan- tization. In particular, we found specific conditions for which the gyrotropic response is largely reduced or even extinguished. We focused our analysis on 3 dions, with the aim of understanding our recent observation of a gy- rotropic response associated with Jahn-Teller polarons in La2/3Ca1/3MnO 3[25]. Using standard values for Jahn- Teller constants, spin-orbit coupling and charge transfer our model replicates a gyrotropic response in the blue region of the visible range, in agreement with the exper- iments [25]. Beyond this particular case, a similar approach may be generalized to study the interaction with electromag- netic fields of transition metals with arbitrary point sym- metries and spin-orbit couplings. One perspective is the entanglement of spin and orbital degrees of freedom us- ing light at optical wavelengths. One could think, for in- stance, of studying quantum tunneling of E⊗eorT⊗e Jahn-Teller vibronic states [37]. In particular, the inter- action with light may drive photoexcited states, whose ground state is formed by coherent superpositions of those vibronic states through quantum tunneling, which may be detected with polarized light. Such excitations could form the basis for quantum states of interest for quantum technologies [22, 67]. Another prospect may be the study of 4d-5d transition metals hosting (quan- tum) spin liquids [18–21]. In this case, magnetic inter- actions would compete with the coupling to the elec- tromagnetic field, which could lead to a rich diagram of quantum phases as a function of the wavelength of the electromagnetic radiation. To solve this problem, a group-theoretical approach would require working in ap- propriate regions of Tanabe-Sugano diagrams [1, 49]. For instance, for heavy metal d4ions the ground state is3T1g instead of5Eg, due to the larger crystal field that leads to the condition ( Dq/B )c>2.7, see Figure 1a. Addi- tionally, since in this case Jahn-Teller interactions in the t2gmanifold are typically smaller than spin-orbit cou- pling [20, 37, 44], the group-theoretical analysis should consider lowering the point symmetry by spin-orbit in- teractions in the first place. We note that our model Hamiltonian considers elec- trons that are subject to Jahn-Teller instabilities. In general, these may coexist with electrons in delocalized bands. This is indeed the case of many oxides, includ- ingLa2/3Ca1/3MnO 3, where both Jahn-Teller polarons and delocalized electrons participate in transport [68–70]. We stress that our model captures the essential physics of electrons that are affected by Jahn-Teller interactions, neglecting contributions from delocalized electrons. As we demonstrate here, this is enough to describe the spe- cific contribution of Jahn-Teller vibronic states to the gyrotropic response, which is experimentally distinguish-17 able from the conventional response arising from delocal- ized bands [25]. On the other hand, although the analysis based on spectral functions gives a fundamental under- standing of light-matter interactions in these solids, fur- ther developments can address linear response theory to obtain responses like optical conductivity and permittiv- ity that can be matched with experiments. We also note that our assumption of photoinduced electron transfer implies an enhanced conductivity at resonant frequencies, which, eventually could be tested experimentally by measuring electronic transport un- der illumination at relevant wavelengths. These ex- periments could be done in La2/3Ca1/3MnO 3, but other candidates would also comprise materials like (PrxLa1−x)2/3Ca1/3MnO 3[71] or magnetite [72], where optical signatures of Jahn-Teller polarons have been ob- served [73]. Generally, materials prone to Jahn-Teller in- stabilities, including colossal magnetoresistance mangan- ites [68–70], could be worth exploring in search for gy- rotropic responses arising from spin-orbital mixing. On the theoretical side, other models can extend the anal- ysis to the optical responses of clusters of Jahn-Teller ions rather than isolated ions. Alternative models may also explore these responses in the absence of photo- transfer, e.g., the photoexcitation of Jahn-Teller states in molecules [37, 74], in which the group-theoretical ap- proach should be applied at the level of molecular or- bitals. Finally, in the present model, the electromagnetic radiation and the lattice modes are treated as classical fields. Further extensions would require a full quantum approach to describe these fields, especially relevant for the application of the aforementioned ideas to concepts like cavity quantum electrodynamics [67, 75]. ACKNOWLEDGMENTS We acknowledge financial support from PID2020- 118479RBI00 and Severo Ochoa FUNFUTURE (CEX2019-000917-S) projects of the Spanish Ministry of Science and Innovation (Grant No. MCIN/AEI/10.13039/501100011033). Appendix A: Wavefunctions of many-electron states 1. Notation We use group theory to construct many-electron wave- functions. We work in a product basis between orbital and spin momenta, so that wavefunctions are defined by kets like: |(G)2S+1Γ(γ)M⟩ (A1) Here Sis the spin magnitude, so that 2 S+ 1 is the spin degeneracy, and Γ is the irreducible representation in or- bital space of group G, expressed in the basis γ(which isomitted for unidimensional representations). Finally M is referred to the spin quantum number, M=−S, ..., S . In the description of spin-orbit coupling, wavefunctions are denoted by |Γγ⟩since orbital and spin angular mo- menta are coupled and double group representations are best suited to take into account the electron spin. The wavefunctions describe multi-electronic states, so they are linear combinations of Slater determinants: |ψ1···ψN|=1√ N!ψ1(1)···ψN(1) ......... ψ1(N)···ψN(N)(A2) Here ψi=ϕiχiis the i-th mono-electronic wavefunction, where the orbital angular momentum part is described by ϕi=ζ, η, τ, u, v (labels are defined in EqEq. (A3)) and χi=α, βis the spinor part. Being ˆ nthe quantization axis for the spin, then ˆ n·ˆsα= +1 /2 and ˆ n·ˆsβ=−1/2. For brevity, the notation inside the Slater determinants is written as ϕα→ϕandϕβ→¯ϕ, which is taken from Ref. [1]. Finally, the sign in bras and kets for wavefunctions is denoted by a breve symbol, i.e., ˘M=−M. 2. Construction of the wavefunctions a.Ohpoint symmetry In the case of one electron in a dshell in a cubic crys- tal field the ten-fold degeneracy of the free ion is bro- ken into a egshell (with four-fold degeneracy) and a t2g shell (six-fold degeneracy). The basis angular functions of these two shells can be expressed as linear combina- tions of spherical harmonics Ym lto get real spatial sym- metries: u=Y0 2∼1 2(3z2−r2) (A3a) v=1√ 2 Y+2 2+Y−2 2 ∼√ 3 2(x2−y2) (A3b) ζ=ı√ 2 Y+1 2+Y−1 2 ∼√ 3yz (A3c) η=−1√ 2 Y+1 2−Y−1 2 ∼√ 3xz (A3d) τ=−ı√ 2 Y+2 2−Y−2 2 ∼√ 3xy (A3e) uandvare the basis for the egshell and ζ,ηandτare the basis for the t2gshell. When there is more than one electron in the dshell we construct many-electron wavefunctions using Slater determinants. Since the spin-orbit interaction is small compared to exchange interactions, the many-electron wavefunctions are built by coupling separately the or- bital and spin momenta, following the Russell-Saunders coupling scheme and using Clebsh-Gordan coefficients. For the orbital part, the coefficients are adapted for the18 Ohpoint-group symmetry. The many-electron wave- functions are expressed as linear combinations of Slater determinants, which take all possible permutations of electrons sitting on the different monoelectronic orbitals and having all possible spin orientations [1]. In the d3configuration the ground state for all values of the Racah parameter ∆ /Bin the Sugano-Tanabe di- agrams is4A2g. It can be shown that the wavefunction that describes this four-fold degenerated term is built from Slater determinants with the three monoelectronic orbitals of t2gas follows [1]: |4A2g3 2⟩=−|ζητ| (A4a) |4A2g1 2⟩=−1√ 3 |ζη¯τ|+|ζ¯ητ|+|¯ζητ| (A4b) |4A2g˘1 2⟩=−1√ 3 |¯ζη¯τ|+|ζ¯η¯τ|+|¯ζ¯ητ| (A4c) |4A2g˘3 2⟩=−|¯ζ¯η¯τ| (A4d) When a fourth electron is added to the t2gshell, the many-electron wavefunction transforms as a T1grepre- sentation with total spin S= 1 and is built from nine degenerate Slater determinants as follows: |3T1gκ1⟩=|ζητ¯ζ| (A5a) |3T1gκ0⟩=1√ 2 |ζ¯ητ¯ζ|+|ζη¯τ¯ζ| (A5b) |3T1gκ˘1⟩=|ζ¯η¯τ¯ζ| (A5c) The orbital basis functions for T1gareκ, µ, ν which have the same relation under rotations as the Cartesian coordinates x, y, z but being even under parity. Eq. (A5) give us three determinants. The remaining six are obtained for the µandνbasis of the3T1grepresentation. The latter are obtained from κby rotating in the orbital space by ±2π/3 around the [111] axis. On the other hand, if the fourth electron is in the eg shell the many-electron wavefunction corresponds to an Egrepresentation. In this case, the spin number is S= 2 and the many-electron wavefunctions can be expressed as ten linear combinations of determinants as follows: |5Egγ2⟩=±|ζητγ′| (A6a) |5Egγ1⟩=±1 2 |ζητ¯γ′|+|¯ζητγ′|+|ζ¯ητγ′|+ +|ζη¯τγ′|](A6b) |5Egγ0⟩=±1√ 6 |¯ζητ¯γ′|+|ζ¯ητ¯γ′|+|ζη¯τ¯γ′|+ +|ζ¯η¯τγ′|+|¯ζη¯τγ′|+|¯ζ¯ητγ′|(A6c) |5Egγ˘1⟩=±1 2 |¯ζ¯η¯τγ′|+|ζ¯η¯τ¯γ′|+|¯ζη¯τ¯γ′|+ +|¯ζ¯ητ¯γ′|(A6d) |5Egγ˘2⟩=±|¯ζ¯η¯τ¯γ′| (A6e)Here the bases for the irreducible representation5Eg areγ, γ′=u, vbeing γ̸=γ′and the positive (+) signs corresponding to γ=uand the negative ( −) signs corresponding to γ=v. b.D4hpoint symmetry When the cell is tetragonally distorted by a Jahn-Teller instability (reducing the symmetry to D4h), the T1grep- resentation is broken into a representation A2gwith or- bital symmetry γ=ν, and Egwith symmetries γ=κ, µ. On the other hand, the reduction to D4hsymmetry splits theEgrepresentation into A1gwith basis γ=uandB1g with basis γ=v. Therefore, in D4hpoint symmetry the terms are split in the following representations (we show only the terms with maximum spin quantum number): |3A2gν1⟩=|ζητ¯τ| (A7a) |3Egκ1⟩=|ζητ¯ζ| (A7b) |3Egµ1⟩=|ζητ¯η| (A7c) |5A1gu2⟩=|ζητv| (A7d) |5B1gv2⟩=|ζητu| (A7e) We note that under a tetragonal elongated distortion the term5B1gis lowest in energy. Although in the mono-electronic picture the fourth electron that drives the Jahn-Teller instability sits in an orbital with u symmetry, the many-electron wave-function of the3B1g term has vsymmetry. We also note that we introduce a global phase −1 in the term5B1gto eliminate a minus sign. c. Spin-orbit coupling Finally, we analyze how spin-orbit coupling splits the representations expressed in Eq. (A7). We remind that under spin-orbit coupling, wavefunctions are expressed as double group representations |Γγ⟩. In Eq. (A8) and Eq. (A9), double group representations (on the left side) are expressed in terms of the irreducible representations inD4hpoint symmetry (right side). First of all, note that the only term that splits under spin-orbit coupling is3Eg. This can be understood by observing the reduced matrices in Eq. (7a) of the main text: only the reduced matrix VA2ghas one non-zero diagonal element corre- sponding to this term. The spin part of3Egcannot be described by a j= 1 representation in continuous rota- tion group because spins interact with the orbital space, which, in this case, is described by the Egrepresentation inD4h. In this symmetry, the continuous spin rotation group D(S=1)decomposes into A2g+Egrepresentations –which are gerade , since spinors are even under parity inversion [76]–, with representations in spherical basis19 A2gwith q= 0, and Egwith q=±1. Therefore, we need to obtain the representation of the composite prod- uct (A2g+Eg)⊗Eg=A2g⊗Eg+Eg⊗Eg. Thus, for A2g⊗Eg, i.e., when the orbital component Egcouples to the A2grepresentation in the spin space, it generates two functions that transform as Egrepresentations: |Egκ⟩=|3Egµ0⟩ (A8a) |Egµ⟩=−|3Egκ0⟩ (A8b) On the other hand, the Egrepresentation of the or- bital part combines with the Egrepresentation of the spin part, giving the following irreducible representations Eg⊗Eg=A1g+A2g+B1g+B2g, which are expressed as follows. |A1g⟩=−1 2 |3Egκ1⟩ − |3Egκ˘1⟩ −ı|3Egµ1⟩ −ı|3Egµ˘1⟩ (A9a) |A2g⟩=ı 2 |3Egκ1⟩+|3Egκ˘1⟩ −ı|3Egµ1⟩+ı|3Egµ˘1⟩ (A9b) |B1g⟩=1 2 |3Egκ1⟩ − |3Egκ˘1⟩+ı|3Egµ1⟩+ı|3Egµ˘1⟩ (A9c) |B2g⟩=−ı 2 |3Egκ1⟩+|3Egκ˘1⟩+ı|3Egµ1⟩ −ı|3Egµ˘1⟩ (A9d) Note that we used the Clebsch-Gordan coefficients dis- played in Table II in Appendix C to obtain the expres- sions in Eq. (A8) and Eq. (A9). On the other hand, for some calculations it may be convenient to express Eq. (A8) and Eq. (A9) in spherical basis for the orbital an- gular momentum of the3Egterm, which contains com- ponents with quantum numbers ML=±1 labeled as t±, andt0. We can then rewrite the corresponding terms as: |3Egt±M⟩=∓1√ 2 |3EgκM⟩ ±ı|3EgµM⟩ (A10) |A1g⟩=−1√ 2 |3Egt+˘1⟩+|3Egt−1⟩ (A11a) |A2g⟩=−ı√ 2 |3Egt+˘1⟩ − |3Egt−1⟩ (A11b) |Egκ⟩=ı√ 2 |3Egt+0⟩ − |3Egt−0⟩ (A11c) |Egµ⟩=−1√ 2 |3Egt+0⟩+|3Egt−0⟩ (A11d) |B1g⟩=−1√ 2 |3Egt+1⟩+|3Egt−˘1⟩ (A11e) |B2g⟩=ı√ 2 |3Egt+1⟩ − |3Egt−˘1⟩ (A11f) Finally, we discuss how spin-orbit coupling splits the representation3EginD4hsymmetry, see (Fig. 1b). For that purpose, we use the Wigner-Eckart theorem to com- pute the eigenenergies of the spin-orbit matrix elements. First, since ⟨Egκ|EgγEgκ⟩=⟨Egµ|EgγEgµ⟩= 0 for γ=κ, µ, ν , the spin-orbit eigenenergies of the Egterms (see Eq. (A8)) are 0. On the other hand, by virtue of the expressions in Eq. (A11), the terms in Eq. (A9) can be expressed in the following way:|¯Γ⟩=X γ,qcγq|3Egγq⟩ (A12) where cγqcan be obtained from Eq. (A11). The Clebsh-Gordan coefficients necessary to apply Wigner- Eckart are ⟨Egγ|A2gνEgγ′⟩= δκ γ−δµ γ
 1−δγ′ γ and ⟨1q|101q′⟩=δq′ qq/√ 2, so that the matrix elements can be computed as: ⟨¯Γ|⃗L·⃗S|¯Γ⟩=ıq√ 2X γ,q γ′,q′c∗ γqcγ′q′ δκ γ−δµ γ
× × 1−δγ′ γ δq′ q(A13) This gives matrix elements ⟨A1g|⃗L·⃗S|A1g⟩=⟨A2g|⃗L· ⃗S|A2g⟩=−1/2 and ⟨B1g|⃗L·⃗S|B1g⟩=⟨B2g|⃗L·⃗S|B2g⟩= 1/2. As a result, spin-orbit coupling does not change the energy of the doubly degenerated Egspin-orbit term, while symmetric ( A1g, A2g) and antisymmetric (B1g, B2g) representations split by ∓ξSO/2 with respect to the Egterm (Fig. 1b). We also note that the ac- cidental degeneracy of ( A1g, A2g) and ( B1g, B2g) terms may be eventually lifted if one considers developments beyond first-order relativistic contributions. Appendix B: Jahn-Teller Hamiltonian Atoms or ions in a molecule or a unit cell have a posi- tion where the energy of the system is minimized. Suffi- ciently small deviations from these equilibrium positions can be described through a force constant: KΓ¯Γ= ∂2EΓ ∂Q¯Γ2! 0(B1) Here Q¯Γare the vibronic coordinates that transform un- der irreducible representation ¯Γ, which can be described in the frame of group theory being linear combinations of the displacements of the atoms in Cartesian coordinates ∆Xn,∆Yn,∆Zn, and EΓis the energy of the system which depends on the irreducible representation Γ of the electronic wavefunction. In the presence of orbital degen- eracy, the equilibrium positions change spontaneously, reducing the symmetry through the Jahn-Teller theorem [6]. This situation can be described by the addition of an- other term in the energy of the system that includes the potential energy of the nuclei in the field of the electrons in the state defined by the representation Γ and basis γ, i.e., the adiabatic potential energy surface (APES) εΓ γ(⃗Q) [9]: ε(⃗Q) =X Γ,¯Γ1 2KΓ¯ΓQ2+εΓ γ(⃗Q) (B2)20 where εΓ γ(⃗Q) is obtained by solving the secular equation for the vibronic coupling matrix operator Wwhich, to second order, is defined as: W(r, Q) =X Γγ∂V ∂QΓγ 0QΓ γ+ +1 2X Γ′γ′Γ′′γ′′ ∂2V ∂QΓ′ γ′∂QΓ′′ γ′′! 0QΓ′ γ′QΓ′′ γ′′(B3) where Vrefers to the electron-ion interaction potential. We can thus define first-order Eq. (B4a) and second- order Eq. (B4b) vibronic coupling terms as follows [9]: XΓ γ=∂V ∂QΓγ 0(B4a) XΓ1Γ2 γ1γ2=∂2V ∂QΓ1γ1∂QΓ2γ2 0(B4b) These operators transform as the representation of the group corresponding to the lattice distortions [9], so for the computation of the matrix elements we can use the Wigner-Eckart theorem. For the E⊗eproblem, the following matrix elements can be derived using the functions defined in Eq. (A3): FE=⟨v|XEg u|v⟩ (B5) GE=⟨u|XEgEg vv|u⟩ (B6) By using the Wigner-Eckart theorem we can develop the corresponding Hamiltonian as: HE⊗e JT=1 2KEρ2υ0+  FEρcosϑ+GEρ2cos(2 ϑ) υz+ + FEρsinϑ−GEρ2sin(2ϑ) υx(B7) where υiare the Pauli matrices in the pseudospin space of {v, u}and the vibronic coordinates have been normalized asQ2=ρsinϑandQ3=ρcosϑ. The eigenstates of this Hamiltonian are: εE(ρ, ϑ) =1 2KEρ2± ±ρq F2 E+G2 Eρ2+ 2FEGEρcos(3 ϑ)(B8) with the following eigenstates: w+=1√ 2 vcosΩ 2+usinΩ 2 (B9a) w−=1√ 2 ucosΩ 2−vsinΩ 2 (B9b) The energy minima are found when ϑ= 2nπ/3,n= 0,1,2 –which corresponds to tetragonal elongations along the three axes– and ρ=FE/(KE−2GE). In order tosimplify the computation, the radial variable is normal- ized to ρ= 1. Around the tetragonal elongations, the parameter Ω in Eq. (B9), which is defined as: tan Ω =FEsinϑ+|GE|sin(2ϑ) FEcosϑ− |GE|cos(2 ϑ)(B10) can be approximated as Ω ≈ϑ. In this situation, w− has a u-like symmetry and w+has a v-like symmetry. Since we consider tetragonal elongations with orthorhom- bic perturbations ( ϑ= 2nπ/3±δϑ), we can work with the following basis: ˇv=1√ 2 vcosϑ 2+usinϑ 2 (B11a) ˇu=1√ 2 ucosϑ 2−vsinϑ 2 (B11b) With this basis, and taking into account δϑ, the expres- sion Eq. (B7) is transformed as: HE⊗e JT=FE+ 2GE 2υ0+ (FE+GE)υz+ + (FE−2GE)δϑυx(B12) As argued in the main text, in the t2gshell we only consider the T⊗eproblem. For this problem we also ne- glect second order vibronic constants. The T⊗evibronic constant is defined as: FT=⟨τ|QEg u|τ⟩ (B13) Using again the Wigner-Eckart theorem, the Hamilto- nian of the T⊗eJahn-Teller interaction is derived as a function of Gell-Mann matrices, λk, in the basis {ζ, η, τ}: HT⊗e JT=1 2KTρ2λ0−1 2FTρh√ 3λ8cosϑ+ +λ3sinϑ](B14) The energy minima correspond again to the tetragonal elongations with ρ=FT/KT. Since the nuclei motion is much slower than the electronic transitions, we can make the assumption that these minima are the same as the ones for the E⊗eproblem, so we normalize again ρ= 1, so that FT=KT. These minima correspond to the basis defined before. We can generalize the expres- sion Eq. (B14) to the local basis at each value ϑn, de- noted as {ˇζ,ˇη,ˇτ}, which is defined by rotations around the [111] axis in the orbital space, i.e., ˇ τ=ˆRn 3(xyz)τ, where ˆRn k(x...) defines a rotation of k-th order (angle 2π/k) executed ntimes along the axis defined by the coordinates in the parentheses. Then, the orthorhombic distortions in T⊗eare described by: HT⊗e JT=1 2FTh λ0−√ 3λ8−δϑλ3i (B15) The description so far is done with monoelectronic or- bitals. We can generalize these results to many-electron21 wavefunctions. The vibronic constants are calculated us- ing a one-body potential, so that for the non-diagonal matrix elements we only need to check the orbitals that are different (see also Appendix D). All the wavefunc- tions described by Eq. (A5) and Eq. (A6) are defined in the orbital part in terms of determinants of the type |ζητγ|. Since there is just one different orbital in each Slater determinant, the off-diagonal elements are not modified. Then, for the first order vibronic constants, since the sums for ⟨t|XEgu|t⟩and⟨t|XEgv|t⟩fort=ζ, η, τ are null, the results for many-electron wavefunctions are the same as for the monoelectronic orbitals. We note that the same arguments apply for the T⊗eproblem, since second order vibronic constants are neglected. Appendix C: Reduced matrix elements of the spin-orbit coupling operator For the computation of the matrix elements of the spin- orbit coupling, we use the Wigner-Eckart theorem ap- plied to the spin-orbit operator VΛ λqdefined in Sec. II B, see also Ref. [1]. This operator transforms according to irreducible representations Λ in the orbital space with basis λandS1 qcorresponds to irreducible representations in the spin-rotation group. To calculate a given matrix element, we apply the Wigner-Eckart theorem as follows: ⟨ΓγSM|VΛ λq|Γ′γ′S′M′⟩=(−1)1−gΛ p gΓ(2S+ 1)× × ⟨ΓS||VΛ||Γ′S′⟩⟨Γγ|ΛλΓ′γ′⟩× × ⟨SM|1qS′M′⟩(C1) where |ΓγSM⟩and|Γ′γ′S′M′⟩correspond to wavefunc- tions that transform as irreducible representations Γ, Γ′ in bases γ,γ′with spin S,S′and spin quantum num- bers M,M′, while gΛandgΓare the dimensionality of representations Λ and Γ. The application of the Wigner- Eckart theorem requires the computation of the reduced matrices ⟨ΓS||VΛ||Γ′S′⟩. The latter have to be hermitic, which, as will be shown below, is guaranteed by the fol- lowing expression [1]: ⟨Γ′γ′S′M′|VΛ λq|ΓγSM⟩ =−(−1)q⟨ΓγSM|VΛ λ˘q|Γ′γ′S′M′⟩(C2) where, since ⃗Sis expressed in spherical coordinates, we have q= +1 ,0,−1. One has to consider also the fol- A1A2B1B2 EEuνv τ κ1√ 20−1√ 20 µ0−1√ 20−1√ 2 κ01√ 20−1√ 2 µ1√ 201√ 20κ µ TABLE II. Clebsh-Gordan coefficients for E⊗EinD4hgrouplowing relations between the Clebsh-Gordan coefficients: ⟨SM|1qS′M′⟩= (−1)S−S′+qr 2S+ 1 2S′+ 1× × ⟨S′M′|1˘qSM⟩(C3a) ⟨SM|1qS′M′⟩= (−1)1+S′−S⟨SM|S′M′1q⟩(C3b) ⟨Γγ|ΛλΓ′γ′⟩=rgΓ gΓ′ϵ(ΓΛΓ′)⟨Γ′γ′|ΛλΓγ⟩ (C3c) ⟨Γγ|ΛλΓ′γ′⟩=χ(ΓΛΓ′)⟨Γγ|Γ′γ′Λλ⟩ (C3d) The factors ϵ(ΓΛΓ′) and χ(ΓΛΓ′) depend on the phase convention [50]. We have fixed this convention by impos- ing⟨Γγ|A1guΓγ⟩= 1 for any representation Γ and basis γ. Assuming this convention, we have: ϵ(A1gEgEg) =ϵ(B1gEgEg) = 1 (C4a) ϵ(A2gEgEg) =ϵ(B2gEgEg) =−1 (C4b) ϵ(A2gA2gA1g) = 1 (C4c) χ(EgEgA1g) =χ(EgEgB1g) = 1 (C4d) χ(EgEgA2g) =χ(EgEgB2g) =−1 (C4e) χ(A2gA2gA1g) = 1 (C4f) One can verify that Eq. (C3) and Eq. (C4) imply that ⟨ΓS||VΛ||Γ′S′⟩=⟨Γ′S′||VΛ||ΓS⟩, which, as mentioned above, guarantees the expected hermiticity of the spin- orbit operator. As shown in Sec. II B, the reduced matrices for the spin-orbit operator are expressed through irreducible rep- resentations VA2gandVEg. To derive the matrix ele- ments of these matrices, we use the Clebsh-Gordan co- efficients expressed in Table II and the ladder operators defined as: J±|m⟩=p j(j+ 1)−m(m±1)|m±1⟩ (C5) which are related to the spherical components of the an- gular momentum operators through: J±=∓1√ 2J±1 (C6) We first derive the matrix elements correspond- ing to the representation VA2g. We remind that the reduced matrices are expressed in the basis {3Eg,3A2g,5A1g,5B1g}(see discussion in Sec. II B). We first note that the direct product A2g⊗A2g=A1g implies that the spin-orbit operator in representation A2g has nonzero matrix elements connecting5A1gand3A2g. Let us find such elements by applying the operators of angular and spin momenta to the wavefunctions of rep- resentation3A2g. We choose a spin-orbit operator in representation q= 1 (corresponding to spin operator s+) and|3A2gν1⟩expressed in terms of the corresponding Slater determinants (Eq. (A7a)). By applying the oper- ators directly to the wavefunctions it follows that:22 ⟨5A1gu2|VA2g ν1|3A2gν1⟩= =−1√ 2⟨v|lz|τ⟩⟨+1 2|s+| −1 2⟩=ı√ 2(C7) On the other hand, by applying the Wigner-Eckart the- orem (Eq. (C1)), we obtain: ⟨5A1gu2|VA2g ν1|3A2gν1⟩= =1√ 5⟨5A1g||VA2g||3A2g⟩(C8) Combining Eq. (C7) and Eq. (C8) we obtain the re- duced matrix element ⟨5A1g||VA2g||3A2g⟩=ı√ 10. Next, we note that the direct product Eg⊗A2g=Eg implies that Egwavefunctions can be connected through the spin-orbit operator only to wavefunctions of the same representation. This gives a diagonal element in the VA2g matrix. To find such element we apply, as before, the operators of angular and spin momenta to wavefunctions 3Eg(Eq. (A7b) and Eq. (A7c)) and choose a repre- sentation q= 0, involving the spin operator s0. The application of the operators to the wavefunctions gives: ⟨3Egκ1|VA2g ν0|3Egµ1⟩= =⟨ζ|lz|η⟩⟨−1 2|s0| −1 2⟩=−ı 2(C9) On the other hand, the application of the Wigner- Eckart theorem gives: ⟨3Egκ1|VA2g ν0|3Egµ1⟩= =−1√ 12⟨3Eg||VA2g||3Eg⟩(C10) We therefore obtain ⟨3Eg||VA2g||3Eg⟩=ı√ 3. Using similar arguments, it can be shown that the rest of matrix elements of VA2gare zero, resulting in the reduced matrix described by Eq. (7a). We derive now the matrix elements corresponding to the representation VEg. We first note that the direct product Eg⊗Eg=A1g⊕A2g⊕B1g⊕B2gmeans that the term3Egcan be connected by the spin-orbit oper- ator to all other representations. As done for VA2g, we combine the application of the operators of angular and spin momenta to the pertinent wavefunctions with the application of the Wigner-Eckart theorem. We obtain the following expressions: ⟨3A2gν1|VEg κ0|3Egx1⟩= =⟨τ|lx|ζ⟩⟨−1 2|s0| −1 2⟩=ı 2 =1√ 12⟨3A2g||VEg||3Eg⟩(C11)⟨5A1gu2|VEg κ1|3Egκ1⟩= =⟨v|lx|ζ⟩⟨+1 2|s+| −1 2⟩=−ı√ 2 =−1√ 10⟨5A1g||VEg||3Eg⟩(C12) ⟨5B1gv2|VEg κ1|3Egκ1⟩= =⟨u|lx|ζ⟩⟨+1 2|s+| −1 2⟩=−ır 3 2 =1√ 10⟨5B1g||VEg||3Eg⟩(C13) which allow us to obtain all the matrix elements for the VEgmatrix as follows: ⟨3A2g||VEg||3Eg⟩= ı√ 3,⟨5A1g||VEg||3Eg⟩=ı√ 5 and ⟨5B1g||VEg||3Eg⟩= −ı√ 15. These elements give the reduced matrix VEgexpressed in Eq. (7b). Appendix D: One-body operators In this work, Jahn-Teller and spin-orbit Hamiltonians contain one-body operators. In the same way, light- induced transfer requires also one-body operators in the electromagnetic Hamiltonian. In the following, we ex- plain how one-body operators act in the formalism of the many-electron wavefunctions defined in Appendix A. For that purpose, we recall that to comply with Pauli exclusion principle we need to define a multielectronic wavefunction Ψ through the antisymmetrization opera- torAacting on the product of monoelectronic states oc- cupied by electrons: Ψ =√ N!AY iψi(i) =1√ N!X σ∈P(−1)σY iψσ(i)(i) (D1) where σis an element in the permutation group P,N is the number of fermions of the system and in ( −1)σ represents the parity of the permutation. This results in the formation of Slater determinants. We can define a one-body operator Oas the sum of operators okacting over the k-th fermion as follows: O=X kok (D2) To find the matrix elements, we take into account the following properties of the antisymetrization operator: •Applying Ato a Slater determinant returns the23 same Slater determinant, so that A2=A. AΨ =1√ N!X σ∈P(−1)σAY iψσ(i)(i) =1 (N!)3/2X σ,τ∈P(−1)σ+τY iψτ(σ(i))(i) =1√ N!X κ∈P(−1)κY iψκ(i)(i)(D3) We see that the composition of the two antisym-metrization operators defines another permutation inP,κ(i) =τ(σ(i)) with parity κ=τ+σ, with N! possible different compositions τσthat return κ. •SinceAis a real operator A†=A. •Since any one-body operator is even under permu- tations it always commute with A, [O,A] = 0. Consequently, the matrix elements involving one-body operators between many-electron wavefunctions can be found as follows: ⟨Φ|O|Ψ⟩=N!⟨Y jϕj(j)|A†OA|Y iψi(i)⟩=X kX σ∈P(−1)σ⟨Y jϕj(j)|ok|Y iψσ(i)(i)⟩ =X kX σ∈P(−1)σ⟨ϕk(k)|ok|ψσ(k)(k)⟩Y i⟨ϕi(i)|ψσ(i)(i)⟩=X k⟨ϕk(k)|ok|ψk(k)⟩Y i⟨ϕi(i)|ψi(i)⟩(D4) By orthogonality, the only permutation that does not vanish is the identity. Note that diagonal elements (Φ = Ψ) do not vanish: ⟨Ψ|O|Ψ⟩=X k⟨ψk(k)|ok|ψk(k)⟩ (D5)On the other hand, off-diagonal elements (Φ ̸= Ψ) can be nonzero only if the many-electron wavefunctions differ only by one one-body wavefunction. Otherwise, if they differ by more than one one-body wavefunction, the inner product vanishes. ⟨Φ|O|Ψ⟩=⟨ϕ|o|ψ⟩ (D6) [1] S. Sugano, Y. Tanabe, and H. Kamimura, Miltiplets of Transition-Metal Ions in Crystals , 1st ed. (Academic Press, 111 Fifth Avenue, New York, 1970). [2] Y. Tokura and N. Nagosa, Science 288, 462 (2000). [3] G. Khaliullin, Progress of Theoretical Physics Supple- ment 160, 155 (2005). [4] S. V. Streltsov and D. I. Khomskii, Physics-Uspekhi 60, 1121 (2017). [5] H. Wu, Computational Materials Science 112, 459 (2016), computational Materials Science in China. [6] H. A. Jahn and E. Teller, Proceedings of The Royal So- ciety A 161, 220 (1937). [7] H. Jahn, Proc. Roy. Soc. A 164, 117 (1938). [8] M. C. M. O’Brien and C. C. Chancey, American Journal of Physics 61, 688 (1993). [9] I. B. Bersuker and V. Z. Polinger, Vibronic Interactions in Molecules and Crystals , 1st ed. (Springer, Heidelberg, Berlin, 1989). [10] W. Domcke, D. Yarkony, and H. K¨ oppel, Conical inter- sections: electronic structure, dynamics & spectroscopy , Vol. 15 (World Scientific, 2004). [11] A. Jasper, B. Kendrick, C. A. Mead, and D. Truhlar, Yang, X , 329 (2004). [12] H. C. Longuet-Higgins, U. ¨Opik, M. H. Lecorney Pryce, and R. A. Sack, Proceedings of The Royal Society A 244,1 (1958). [13] H. C. Longuet-Higgins, Proceedings of The Royal Society A344, 147 (1975). [14] G. Herzberg and H. C. Longuet-Higgins, Discussions of the Faraday Society 35, 77 (1963). [15] N. P. Armitage, E. J. Mele, and A. Vishwanath, Reviews of Modern Physics 90, 10.1103/revmodphys.90.015001 (2018). [16] B. Keimer and J. E. Moore, Nature Physics 13, 1045 (2017). [17] J. Wang and S.-C. Zhang, Nature Physics 16, 1062 (2017). [18] L. Clark and A. H. Abdeldaim, Annual Review of Mate- rial Research 51, 495 (2021). [19] H. Takagi, T. Takayama, G. Jackeli, et al. , Nature Re- views Physics 1, 264 (2019). [20] D. I. Komskii and S. V. Streltsov, Chemical Reviews 121, 2992 (2019). [21] K. Kitagawa, T. Takayama, Y. Matsumoto, et al., Nature 554, 341 (2018). [22] S. V. Streltsov and D. I. Khomskii, Phys. Rev. X 10, 31 (2020). [23] T. Takayama, J. Chaloupka, A. Smerald, et al. , Jour- nal of the Physical Society of Japan 90, 062001 (2021), https://doi.org/10.7566/JPSJ.90.062001.24 [24] S. Bhowal and I. Dasgupta, Journal of Physics: Con- densed Matter 33, 453001 (2021). [25] B. Casals, R. Cichelero, P. Garc´ ıa Fern´ andez, et al., Phys. Rev. Lett. 117, 026401 (2016). [26] Y. Tanabe and S. Sugano, Journal of the Physical Society of Japan 9, 766 (1954). [27] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annual Review of Condensed Matter Physics 7, 195 (2016), https://doi.org/10.1146/annurev-conmatphys-031115- 011319. [28] R. Schaffer, E. K.-H. Lee, B.-J. Yang, and Y. B. Kim, Reports on Progress in Physics 79, 094504 (2016). [29] G. L. Stamokostas and G. A. Fiete, Phys. Rev. B 97, 085150 (2018). [30] A. Paramekanti, D. J. Singh, B. Yuan, et al. , Phys. Rev. B97, 235119 (2018). [31] T. Wolfram and S ¸. Ellialtio˘ glu, Electronic and Optical properties of d-band Perovskites , 1st ed. (Cambridge Uni- versity Press, The Edinburgh Building, Cambridge CB2 8RU, 2006). [32] M. R. Filip and F. Giustino, Proceedings of the National Academy of Sciences 115, 5397 (2018). [33] C. Li, K. C. K. Soh, and P. Wu, Journal of Alloys and Compounds 372, 40 (2004). [34] P. Zubko, S. Gariglio, M. Gabay, et al. , Annual Review of Condensed Matter Physics 2, 141 (2011). [35] M. Coll, J. Fontcuberta, M. Althammer, et al. , Applied Surface Science 482, 1 (2019), roadmap for Oxide Elec- tronics, pushed forward by the TO-BE EU Cost action. [36] C. N. R. Rao, Annual Review of Physical Chemistry 40, 291 (1989). [37] I. B. Bersuker and I. Borisovich, The Jahn-Teller effect (Cambridge University Press, Cambridge, UK, 2006). [38] K. I. Kugel 'and D. I. Khomski˘ ı, Soviet Physics Uspekhi 25, 231 (1982). [39] J. Kanamori, Journal of Applied Physics 31, S14 (1960). [40] J. H. Van Vleck, The Journal of Chemical Physics 7, 72 (1939). [41] U. ¨Opik and M. H. L. Pryce, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 238, 425 (1957). [42] P. Garc´ ıa-Fern´ andez, I. B. Bersuker, J. A. Aramburu, et al. , Phys. Rev. B 71, 184117 (2005). [43] J. van den Brink and D. Khomskii, Physical Review B 63, 140416 (2001). [44] D. I. Komskii, Transition Metal Compounds , 1st ed. (Cambridge University Press, Cambridge CB2 8BS, UK, 2014). [45] D. Louca, T. Egami, E. L. Brosha, et al. , Phys. Rev. B 56, R8475 (1997). [46] T. Proffen, R. G. DiFrancesco, S. J. L. Billinge, et al. , Phys. Rev. B 60, 9973 (1999). [47] A. Sartbaeva, S. A. Wells, M. F. Thorpe, et al. , Phys. Rev. Lett. 97, 065501 (2006). [48] W.-T. Chen, C.-W. Wang, C.-C. Cheng, et al. , Nature Communications 12, 6319 (2021).[49] B. Tsukerblat, Group Theory in Chemistry and Spec- troscopy: A Simple Guide to Advanced Usage , Dover Books on Chemistry (Dover Publications, 2006). [50] S. L. Altmann and P. Herzig, Point-Group Theory Tables , 2nd ed. (Clarendon Press, Wien, 2011). [51] S. G. Kaplan, M. Quijada, H. D. Drew, et al. , Phys. Rev. Lett. 77, 2081 (1996). [52] M. Quijada, J. ˇCerne, J. R. Simpson, et al. , Phys. Rev. B58, 16093 (1998). [53] M. L. Munzarov´ a, P. Kub´ aˇ cek, and M. Kaupp, Journal of the American Chemical Society 122, 11900 (2000). [54] R. Thiele, S.-K. Son, B. Ziaja, and R. Santra, Phys. Rev. A86, 033411 (2012). [55] N. Jiang, W. H. E. Schwarz, and J. Li, Inorganic Chem- istry54, 7171 (2015), pMID: 26161576. [56] B. Halperin and R. Englman, Phys. Rev. B 3, 1698 (1971). [57] A. E. Hughes, Acta Crystallographica Section A 29, 108 (1973). [58] G. A. Gehring and K. A. Gehring, Reports on Progress in Physics 38, 1 (1975). [59] I. Bersuker, Physics Letters 20, 589 (1966). [60] M. M. Schmitt, Y. Zhang, A. Mercy, and P. Ghosez, Phys. Rev. B 101, 214304 (2020). [61] S. Margadonna and G. Karotsis, Journal of the American Chemical Society 128, 16436 (2006), pMID: 17177357. [62] K. H. H¨ ock, G. Schr¨ oder, and H. Thomas, Zeitschrift f¨ ur Physik B Condensed Matter 30, 403 (1978). [63] G. D. Mahan, Many-particle physics (Springer Science & Business Media, 2013). [64] R. D. Mattuck, A guide to Feynman diagrams in the many-body problem (Courier Corporation, 1992). [65] A. Bocquet, T. Mizokawa, T. Saitoh, et al. , Physical Re- view B 46, 3771 (1992). [66] A. Weiße and H. Fehske, New Journal of Physics 6, 158 (2004). [67] A. Laucht, F. Hohls, N. Ubbelohde, et al. , Nanotechnol- ogy32, 162003 (2021). [68] E. Nagaev, Physics Reports 346, 387 (2001). [69] E. Dagotto, Nanoscale phase separation and colossal magnetoresistance: the physics of manganites and related compounds (Springer Science & Business Media, 2003). [70] A. P. Ramirez, Journal of Physics: Condensed Matter 9, 8171 (1997). [71] J. M. Caicedo, M. C. Dekker, K. D¨ orr, et al. , Phys. Rev. B82, 140410 (2010). [72] J. M. Caicedo, S. K. Arora, R. Ramos, et al., New Journal of Physics 12, 103023 (2010). [73] J. M. Caicedo, J. Fontcuberta, and G. Herranz, Phys. Rev. B 89, 045121 (2014). [74] I. B. Bersuker, Electronic structure and properties of transition metal compounds: introduction to the theory (John Wiley & Sons, 2010). [75] J. M. Fink, M. G¨ oppl, M. Baur, et al. , Nature 454, 315 (2008). [76] S. L. Altmann, Rotations, Quaternions, and Double Groups (Oxford: Clarendon Press., 1986).

1606.05220v1.Magnetic_Properties_from_the_Viewpoints_of_Electronic_Hamiltonian__Spin_Exchange_Parameters__Spin_Orientation_and_Spin_Half_Misconception.pdf

1 Magnetic Properties from the Vie wpoints of Electronic Hamiltoni an: Spin Exchange Parameters, Spin Orientation and Spin-Half Misconception Myung-Hwan Whangbo1,* and Hongjun Xiang2,3 1 Department of Chemistry, North Carolina State University, Rale igh, North Carolina 27695-8204, USA 2 Key Laboratory of Computational Physical Sciences (Ministry of Education), State Key Laboratory of Surface Physics, a nd Department of Physics, Fudan University, Shanghai 200433, P. R. China 3 Collaborative Innovation Center of Advanced Microstructures, N anjing 210093, P. R. China E-mails: mike_whangbo@ncsu.edu 2 Abstract In this chapter we review the quantitative and qualitative asp ects of describing the properties of magnetic solids on the basis of electronic Hamilt onian, which describes the energy states of a magnetic syste m using both orbital and spin degrees of freedom. To quantitatively discuss a magnetic property of a given magnetic system, one needs to generate the spectrum of its energy states and subsequently ave rage the properties of these states with each state weighted by its Boltzmann distribu tion factor. Currently, this is an impossible task to achieve on the basis of an electronic Hamiltonian, so it is necessary to resort to a simple model Hamiltonian, i.e., a spin Hamiltonian that describes the energy states of a magnetic system using only the spin degr ee of freedom. We show that a spin Hamiltonian approach becomes consistent with an ele ctronic Hamiltonian approach if the spin lattice a nd its associated spin exchange p arameters, to be used for the spin Hamiltonian, are determined by the energy-mapping analysis based on DFT calculations. The preferred spin orientation (i.e., the magneti c anisotropy) of a magnetic ion is not predicted by a spin Hamiltonian because it does not include the orbital degree of freedom explicitly. In contrast, the magnetic anisotropy is readily predicted by electronic structure theories e mploying both orbital and spin d egrees of freedom, if one takes into consideration the spin-orbit coupling (SOC), LˆSˆ , of a magnetic ion where S ˆ and Lˆ are respectively the spin and orbital operators, and the SOC constant. It was shown that the preferred spin orientation of a magnetic ion can be predicted and understood in terms of the HOMO-LUMO interactions of the magnet ic ion by taking SOC, L ˆSˆ , as perturbation. A spin Hamiltonian gives rise to the spin-ha lf misconception, namely, the blind belief that spin-half magnetic ions do not possess 3 magnetic anisotropy that arise from SOC. This misconception con tradicts not only experimental observations on spin -half ions but also theoretica l results based on DFT calculations and perturbation theo ry analyses based on an elect ronic Hamiltonian. This misconception is a direct consequence from the limitedness of a spin Hamiltonian that it lacks the orbital degree of fr eedom. We show that the magnetic properties of 5d ion oxides are better explained by the LS-coupling than by the jj-c oupling scheme of SOC, that the spin-orbital entanglem ent of 5d ions is not as strong as has been assumed. 4 1. Introduction In this chapter we examine how to think about and describe the magnetic properties of crystalline solids, which arise from their transi tion-metal magnetic ions, from the perspectives of an electronic Hamiltonian. The latter represents the energy states of a magnetic system using both orbital and spin degrees of fre edom, that is, the angular property of a magnetic ion is described by a set of orbital/spi n states z zS,SL,L . Compared with the strength of chemical bonding (of the order of several eV), the unpaired electrons of a magnetic ion interact very weakly with those of neighboring magnetic ions so that the energy scale involved in magnetic sta tes is very small, and the states responsible for the magnetic properties are closely pack ed in energy ( Fig. 1 ). (For example, at the magnetic field H of 1 Tesla, µ BH = 5.810–2 meV = 0.67 K in k B units. Other energy scales for discussing magnetic properties are 1 me V = 11.6 K = 8.06 cm-1, and 1 cm-1 = 1.44 K.) To quantitatively describe the magnetic properties of such a system at any given temperature, it is necessary to obtain the spectru m of the energy states and subsequently Boltzmann-average the properties of these states. Since solving this problem on the basis of an electronic Hamiltonian is very diffi cult, one employs a spin Hamiltonian spinHˆ, which represents each magnetic ion using only a set of spin s tates zS,S . This toy Hamiltonian allows one to generate the energy states without self- consistent-field calculations thereby greatly simplifying calcu lations, because it is specified by a few spin exchange interactions j iijSˆSˆJ between certain spin sites i and j, j iijji spin SˆSˆ J Hˆ ( 1 ) 5 where the constants J ij (i.e., spin exchange parameters) are the numerical parameters to be determined. The repeat pattern of the chosen spin exchange path s i-j forms the spin lattice (e.g., an isolated dimer, a uniform chain, an alternating chain , a two-leg ladder, etc.) of the magnetic ions ( Fig. 2 ). Once a spin lattice is selected, this model Hamiltonian grea tly simplifies the generation of its energy states as a function of the numerical parameters J ij, which are fixed as those that best simulate the experimental ma gnetic data (e.g., magnetic susceptibility, specific heat, a nd spin wave dispersion relatio ns). The purpose of using such a toy Hamiltonian is to capture the essential physics of o bserved magnetic properties with a minimal number of adjustable parameters J ij. A general problem facing such a toy Hamiltonian analysis is th at more than one spin lattice may equally well simulate the available experiment al data. Since the novelty of a chosen spin lattice presen ts an opportunity to discover a new physics, the practitioners of spin Hamiltonian analyses tend to favor the in terpretation of experimental data using a novel spin lattice without checking if the chosen spin lattice is consistent with the electronic structure of a magnetic system under examin ation. Not infrequently, therefore, a chosen spin lattice turns out to be irrelevant for the system under examination, thus generating “an answer in search of a problem”. A bright si de of such a regrettable situation would be that the ge nerated physics can stimulate exp erimental interests to search for a system that fits the “predicted” physics. These da ys one can readily determine what spin exchanges paths are relevant for any given magnetic system by performing energy-mapping analysis1-4 on the basis of DFT electronic structure calculations. This theoretical/c omputational tool makes it poss ible to interpret experimental data in terms o f the relevant spin lattice. 6 An implicit assumption behind using a spin Hamiltonian is that one can correctly describe all magnetic phenomena in terms of the energy states i t generates. The strength of a spin Hamiltonian analysis is to simplify complex calculati ons as a result of using only the spin degree of freedom, but this strength is also the very cause for its failure at the fundamental level; a spin Hamiltonian description leads to the conceptual impasse recently termed the spin-half syndrome.5,6 A classic example showing this spin-half misconception is the study of CuCl 2·2H 2O by Moriya and Yoshida more than six decades ago;7 a s t h e c a u s e f o r t h e o b s e r v e d s p i n o r i e n t a t i o n o f t h e S = 1 / 2 ion Cu2+, they dismissed outright the possibility that the S = 1/2 ion has mag netic anisotropy induced by SOC, LˆSˆ , which is a single-spin site interaction (i.e., a local intera ction), and then proceeded to explain the observed spin orientation in terms of nonlocal interactions (e.g., anisotropic spin exchange and magnetic dipole-dipole interactio ns). Over the years the spin-half misconception has been p erpetuated in monographs and textbooks on magnetism.8 However, this misconception contradicts not only the experimen tal observations that spin-half ions (e.g., Cu2+, V4+, Ir4+) exhibit magnetic anisotropies,5,6,9 but also theoretical results based on electronic Hamiltonians i n which the energy states of a magnetic system are described by using both orbital and spin degrees of freedom.5,6,9 A transition-metal magnetic ion of any spin (S = 1/2 – 5/2) has magnetic anisotropy as a consequence of SOC, LˆSˆ , because the latter induces interactions among its crystal-field split d-states and because the energy-lowering associated with these interactions depend on the spin orientation.2,5,6,9,10 In an electronic Hamiltonian approach the energy states of a magnetic system are discussed i n terms of its magnetic orbitals (i.e., its singly occupied orbitals). Each magnetic or bital represents either the up-7 spin state 21 21, or the down-spin state 21 21, , s o t h e o v e r a l l s p i n S o f a magnetic ion is related to how many magnetic orbitals it genera tes. Thus, each magnetic ion of a magnetic orbital in spin state zS,S (= or ) is described by the orbital/spin state z zS,SL,L . The magnetic states are modified by SOC, LˆSˆ , due to the associated intermixing between them, but this intermixing d oes not occur in the spin part zS,S , but in the orbital part zL,L , of each state. For example, when there is no degeneracy in the magnetic orbitals, a given magnetic orbital z zS,SL,L is modified by the intermixing as z z z z2 2S,S L,L L,L L,L) 1( , (2) where a n d are the mixing coefficients (see Section 7 for more details). This SOC- induced orbital mixing is independent of whether the overall sp in S of the magnetic ion is 1/2 or greater because this mixing occurs in each individual ma gnetic orbital and hence does not depend on how many magnetic orbitals a magnetic ion ge nerates. This is why magnetic anisotropy is predicted for S = 1/2 ions on an equal f ooting to S > 1/2 ions in an electronic Hamiltonian approach. This fundamental result is not described by a spin Hamiltonian simply because it lacks the orbital degree of freed om; having completely suppressed the orbital zL,L of a magnetic ion, a spin Hamiltonian does not allow one to discuss the SOC, LˆSˆ , and hence is unable to describe the preferred spin orientatio n of any magnetic ion. The spin-half misconception is a direct co nsequence from this deficiency of a spin Hamiltonian. Anyone who attempted to publish the finding that the spin-half misconception is erroneous would have experienced eye-opening discourses with it s proponents (mostly, 8 practitioners of spin Hamiltonian analyses), to learn that they t r e a t t h e a t t e m p t a s a n affront to their work and do th eir utmost to suppress its publi cation. For those schooled in the electronic structure descrip tion, it is only natural to des cribe the energy states of a magnetic system by using both orbital and spin degrees of freed om, because unpaired electrons responsible for magne tic properties must be accommoda ted in certain orbitals, and hence have no problem in finding that a spin Hamiltonian is a theoretically limited tool. However, most of those schooled in doing physics with spi n Hamiltonian do not appear to realize that this toy Hamiltonian was born out of the necessity to simplify calculations. They tend to believe that the correct energy stat es of a magnetic system are those generated by using only the spin degree of freedom, and i nsist that an electronic Hamiltonian description should pr oduce the same conclusion as d oes a spin Hamiltonian description even if it is an e rroneous one resulting from its d eficiency. To help break this conceptual impasse, it is necessary to expose the origin of the spin-half misconception by discussing how the properties o f solid state magnetic materials are described from the perspectives of an electronic Hamiltonian. Analysis of magnetic properties on the basis of an electronic H amiltonian deals with two competing issues; one i s to produce accurate quantitat ive predictions, and the other is to provide qualitative pictures with which to organize and think about. These two subjects are discussed by organizing our work as follows: In Se ction 2 we first discuss the angular properties of the atomic orbitals and then the crystal-field split d-states of magnetic ions. Section 3 covers the energy-mapping analysis tha t allows one to relate the spin Hamiltonian analysis of a given magnetic system to its ele ctronic structure by evaluating the spin exchange parameters this toy Hamiltonian ne eds. In Section 4 we 9 discuss the qualitative features of spin exchange interactions i n t e r m s o f o r b i t a l interactions involving magnetic orbitals. In Section 5 we descr ibe indirect ways of incorporating SOC into a spin Hamiltonian and the associated en ergy-mapping analysis as well as the origin of the spin-half misconception. The condi tion leading to uniaxial magnetism is discussed in Section 6 to prepare for our discussi on of magnetic anisotropy. Section 7 describes the qualitative rules that allow one to pre dict the preferred spin orientations of magnetic ions on the basis of the perturbation theory in which SOC is taken as perturbation with the crystal-field split d-states as the unperturbed state. In Section 8 we discuss several i ssues concerning the magnetic pro perties of 5d magnetic ions. Our concluding remarks ar e summarized in Section 9. 2. Atomic orbitals and magnetic orbitals 2.1. Angular properties of atomic orbitals The angular properties of atomic orbitals are specified by the spherical harmonics, zL,L , defined in terms of two quantum numbers; the orbital quantum number L (= 0, 1, 2, … ) and its z-axis component L z = L, L+1, … , L 1, L for a given L. The angular behaviors of the atomic p- and d-orbitals are summarized in Table 1 . I n t e r m s o f t h e magnetic quantum numbers L z, the d-orbitals are grouped into three sets: 2 2 zz2 2 z y xxy, for 2 Lyz,xz for 1 Lr 3z for 0 L Similarly, the p-orbitals are expressed as linear combinations of the spherical harmonics zL,L , where L =1, and L z = 1, 0, 1. Thus, the p-orbitals are grouped into two sets: 10 yx, for 1 L z for 0 L zz Consequently, as depicted in Fig. 3 , the minimum difference | Lz| i n t h e m a g n e t i c quantum numbers between different atomic orbitals is summarized as follows: 0 L z between y and xy xandxy yz and xz 2 2 1 L z between y} ,x{ and z}y x{xy, and yz} ,xz{yz} ,xz{ and rz3 2 22 2 2 L z between 3z2-r2 and {xy, x2-y2} These |Lz| values play a crucial role in understanding the preferred spi n orientations of magnetic ions on the basis of the SOC-induced HOMO-LUMO interac tions of their crystal-field split d-states (see Section 7). In quantum mechanics the orbital angular momentum L is replaced by the orbital angular momentum operator Lˆ, which has three components xLˆ,yLˆ and zLˆ in a Cartesian coordinate system. Most calculations associated with orbital angular momentum make use of zLˆ, Lˆ and Lˆ, where Lˆ and Lˆ are the ladder operators defined by y xy x Lˆi Lˆ Lˆ Lˆi Lˆ Lˆ The orbitals zL,L are affected by the operators zLˆ, Lˆ and Lˆ as follows: 11 1L,L )1L(L)1L(L L,LLˆ1L,L )1L(L)1L(L L,LLˆL,LL L,LLˆ z z z zz z z zz z z z ( 3 ) Here we use the atomic unit in which the unit of angular moment um, ħ, is equal to 1. The Lˆ raises the L z of zL,L by 1 as long as L z + 1 L, while Lˆ lowers the L z of zL,L by 1 as long as L z 1 L. In our later discussion, we n eed to evaluate the integrals jLˆix, jLˆiy and jLˆiz involving atomic p-orbitals )z,y,xj,i( as well as those involving atomic d-orbitals (i, j = 3z2r2, xz, yz, x2 – y2, xy). By using Eq. 3 and the expressions of the atomic orbital s listed in Table 1 , we obtain the nonzero integrals listed in Table 2 .9 2.2. Crystal-field split d-states In most cases we are concerned with systems containing transit ion-metal magnetic ions M in magnetic oxides. The preferred orientations of their spin moments are determined by their d-states split by their surrounding ligands L. It depends on the symmetry and composition of the ML n (typically, n = 4 6) polyhedron how the d-states of the ion M split. In a descrip tion of electronic structures u sing an effective one-electron Hamiltonian effHˆ, each split d-level of a ML n polyhedron does not change its energy and shape regardless of whether it is occupied by one or two electr ons, because the presence of electron-electron repulsion in a doubly-occupied level is ig nored. We discuss this simple picture first and then consider how to modify these one- electron levels by electron correlation. 12 2.2.1. One-electron states witho ut electron correlation How strongly the d-orbitals of the transition metal M interact with the p-orbitals of the ligands L depends on the nature of the d-orbitals and the shape of the ML n polyhedron.11 In the split d-states that result from these interactions, the ligand p-orbitals are combined out-of-phase to the metal d-orbitals. Therefore, a given d-state lies high in energy if the M-L antibonding is strong in the state. Let us start from the d-states of an ML 6 octahedron ( Fig. 4a ), which are split into the triply-degenerate t 2g state lying below the doubly-degenerate e g state ( Fig. 4b ). The three components of the t 2g state are each described by M-L -antibonding, and the two components of the e g state by M-L - antibonding ( Fig. 4c ). Some ML n (typically, n = 4 – 6) polyhedra can be regarded as derived from the ML 6 octahedron by lengthening and/ or removing a few M-L bonds. The split d-states of such polyhedra can be readily predicted by consideri ng how the extent of the -antibonding and/or -antibonding of the M-L bonds varies under the geometrical changes ( Fig. 5 ). For an axially-elongated ML 6 octahedron with the z-axis taken along the elongated M-L bonds, the d-states are split as depicted in Fig. 5b ; the 3z2r2 s t a t e (commonly, referred to as the z2 state, for simplicity) is significantly lowered in energy because the -antibonding is reduced, while the xz and yz states are slightl y lowered in energy because the -antibonding is reduced. For a square-planar ML 4 w i t h t h e z - a x i s taken perpendicular to the plane, the d-states are split as shown in Fig. 5c ; the 3z2r2 state is lowered to become the lowest in energy because the -antibonding along the z-13 direction is totally absent while that in the xy-plane is furth er reduced because the girdle of the 3z2r2 state is diminished in size by the second-order orbital mixing of the upper s- orbital of M.11 In addition, the xz and yz states of the ML 4 square plane are lower than those of the axially-elongated ML 6 octahedron because the -antibonding is absent along the z-direction. For a linear ML 2 with the z-axis taken along the M-L bonds, the d-states are split as depicted in Fig. 5d ; the xy and x2y2 states are lowered more in energy than are the xz and yz states because -antibonding is absent in the xy and x2y2 states while it is present in the xz and yz states. In discussing the t 2g and e g states of an ML 6 octahedron, there occur cases when it is more convenient to take the z-axis along one 3-fold rotation al axis of the octahedron (Fig. 6a ) 12 rather than along one M-L bond (i.e., along one 4-fold rotational axis) ( Fig. 5a). Then their orbital character changes as summarized in Table 3 ; the 3z2r2 state becomes one of the t 2g set, while the (xy, x2y2) degenerate set mixes with the (xz, yz) degenerate set to give the (1e x, 1e y) and (2e x, 2e y) sets ( Fig. 6b ). The (xy, x2y2) set has a larger contribution than does the (xz, yz) set in the (1e x, 1e y) set, and the opposite is the case in the (2e x, 2e y) set ( Table 3 ). Such orbital representations as described by Fig. 6 and Table 3 will be employed in Section 7. 2.2.2. One-electron states with electron correlation The essence of electron correlation is that, when a given ener gy state is doubly occupied, its energy is raised by electron-electron repulsion. The latter is partly reduced in spin-polarized electronic structure calculations, in which u p-spin states are allowed to differ in energy and shape from t heir down-spin counterparts. F or strongly correlated 14 systems, the energy split arisi ng from spin-polarized electroni c structure calculations is not strong enough to generate singly-occupied states needed to describe their magnetic insulating states. In spin-polarized DFT calculations, this def iciency is corrected by adding the on-site repulsion U on magnetic ions to force a larg e split between their up- spin and down-spin states ( Fig. 7 ).13 Such calculations are referred to as DFT+U calculations. An important consequence of spi n polarized DFT+U calculations is found when two adjacent spin sites interact.2 If the two equivalent spin sites have a ferromagnetic (FM) arrangement ( Fig. 8a ), the up-spin states of the two sites are degenerate, and so a re the down-spin states of the two s ites. However, if the two equi valent spin sites have an antiferromagnetic (AFM) arrangement ( Fig. 8b ), the up -spin states of the two sites are nondegenerate, and so are the down-spin states of the two sites . In general, orbital interactions between degenerate states are stronger than those between nondegenerate states.11 Since orbital interactions between states require that their s pins be identical, the AFM arrangement leads to a weaker orbital interaction between a djacent spin sites than does the FM arrangement.2 From the viewpoint of the split d-states obtained from an effective one-electron Hamiltonian, the qualitative feat ures of DFT+U calculations can be simulated by splitting the up-spin d-states from those of the down-spin d-states approximately by the amount of U, as illustrated in Fig. 9 , for a high-spin (S = 2) d6 ion forming a square planar site forming a FeL 4 square plane. For simplicity, the separation between the up-sp in and down-spin d-states is exaggerated in Fig. 9 . What is important to note is that the HOMO and the LUMO levels occur within the down-spin states if the d-shell is more than half-15 filled, but within the up-spin states if the d-shell less than half-filled. (This is due to the convention in which the majority and minority spin states are r egarded as up-spin and down-spin states, respectively.) Only when the d-shell is half-filled in a high-spin manner, the HOMO and the LUMO levels o ccur between the up-spin and down -spin states. An alternative way of correcting the deficiency of spin-polariz ed DFT calculations is the hybrid functional method,14 in which the exchange-correlation functional needed for calculations is obtained by mixing some a mount, (typically, 0.2), of the Hartree-Fock exchange pot ential into the DFT functional. The on-site repulsion U is an empirical parameter in DFT+U calculations, and so is the mixing parameter in DFT+hybrid calculations. In general, DFT+U calculations are muc h less time-consuming than are DFT+hybrid calculations. It should be emphasized that density functional calculations are first principles calculations only after the v alue of U is fixed in DFT+U calculations, and only after the value of is fixed in DFT+hybrid calculations. Given computing resources, DFT calculations with or without in cluding SOC effects15 can be readily carried out by us ing user-friendly DFT program packages such as VASP,16 which considers only valence el ectrons using the frozen-core p rojector augmented waves, and WIEN2k,17 which considers all electrons. As the exchange- correlation functional needed for DFT calculations, the general ized gradient approximation 18 is commonly used for studying solid state materials. In unders tanding results of DFT, DFT+U and DFT+U+SOC calculations or predicting results prior to calculations, the concept of orbital interaction analysis,11 developed on the basis of one electron picture, is useful (see below). 16 3. Energy mapping analysis For two spins 1Sˆ and 2Sˆ at spin sites 1 and 2, respectively, the dot product 2 1SˆSˆ has three Cartesian components, i.e., z2z1 y2y1 x2x1 2 1 SˆSˆ SˆSˆ SˆSˆ SˆSˆ . Thus a general expression for the spin exchange interaction energy between the two spin sites can be written as z2z1z y2y1y x2x1x spin SˆSˆJ SˆSˆJ SˆSˆJ Hˆ , where J x, Jy and J z are anisotropic spin exchange s along the x-, y- and z-directio ns, respectively. If J x = J y = J z = J , n a m e l y , i f t h e s p i n e x c h ange is isotropic, the above expression is simplified as )SˆSˆ SˆSˆ SˆSˆ(J Hˆ z2z1 y2y1 x2x1 spin , which represents a Heisenberg spi n Hamiltonian. Another extreme case is given by J x = Jy = 0, for which we obtain an Ising spin Hamiltonian z2z1z spin SˆSˆJ Hˆ . This Hamiltonian describes a mag netic system made up of uniaxia l magnetic ions (namely, those ions with a nonzero moment only in one direction , see Section 6). The deviation of spin exchange from the isotropic character is a co nsequence of SOC. In this section we focus on how to deter mine isotropic spin exchanges, which are often referred to as Heisenberg or symmetric spi n exchanges. The evaluation of anisotropic spin exchanges will be discussed in Section 5.2. 3.1. Use of eigenstates for an isolated spin dimer1,19 17 To gain insight into the meaning of the spin exchange interacti on, we consider a spin dimer consisting of two equivalent spin-1/2 spin sites, 1 and 2, with one electron at each spin site ( Fig. 10 ). The energy of the spin dimer arising from the spin exchange interaction between the spins 1Sˆ and 2Sˆ is given by the spin Hamiltonian 2 1 spin SˆSˆ J Hˆ , ( 4 a ) where J is the spin exchange parameter. If the spins are regard ed as vectors 1S and 2S , then the Hamiltonian is written as 2 1 spin SS J Hˆ ( 4 b ) In the present work, we will use the operator and vector repres entations of spin interchangeably. Note the absence of the negative sign in this expression. With this definition, the AFM and FM spin exchange interactions are given by J > 0 and J < 0, respectively. Given the dot product between 1S and 2S , the lowest energy for J > 0 occurs when the angle between the two spins is 180 (i.e., the spins are AFM), but that for J < 0 when = 0 (i.e., the spins are FM). In e ither case, the spin Hamiltonian leads to a collinear spin arrangement. In principle, the spin at site i (= 1, 2) of the spin dimer ca n have either up-spin or down-spin state. For a single spin S = 1/2 and S z = 1/2 so that, in terms of the zSS, notations, these states are given by 21 2121 21 ,, . These states obey the follow ing general relationships: 18 zz zz zz z z zz z zSS S SS S SS S S S1 SS 1 S S 1SS S S S1 SS 1S S 1 ˆ ,, ˆ ,( ) ( ) , ˆ ,( ) ( ) , ( 5 ) where the ladder opera tors are given by y xy x Sˆ iSˆ SˆSˆ iSˆ Sˆ Using these ladder operator s, Eq. 4a is rewritten as 2/)SˆSˆ SˆSˆ( J SˆSˆ J Hˆ 2 1 2 1 z2z1 spin ( 4 c ) The eigenstates of spinHˆ allowed for the spin dimer are the singlet state S and triplet state T, which are given by 2/T2/ S 2 1 2 12 12 12 1 2 1 Note that the broken-symme try (or Néel) states, 2 1 and 2 1 , interact through spinHˆ to give the symmetry-adapted states S and T. We evaluate the energies of T and S by using Eq. 5 to find Espin(T) = T HˆTspin = J/4 Espin(S) = S HˆSspin = 3J/4. 19 Thus, the energy difference betw een the two states is given by Espin = E spin(T) Espin ( S ) = J , ( 6 ) so the spin exchange constant J represents the energy differenc e between the singlet and triplet spin states of the spin dimer. The singlet state is low er in energy than the triplet state if the spin exchange J is AFM (i.e., J > 0), and the oppo site is the case if the spin exchange J is FM (i.e., J < 0). We now examine how the triplet and singlet states of the spin d imer are described in terms of electronic structure calculations. The electronic H amiltonian elecHˆ for this two-electron system can be written as elec 12Hh 1 h 2 1 rˆˆ ˆ () ( ) / , ( 7 ) where hiˆ() (i = 1, 2) is the one-electron energy (i.e., the kinetic and t he electron-nuclear attraction energies) of the electron i (= 1, 2), and r 12 is the distance between electrons 1 and 2. Assume that the unpaired electrons at sites 1 and 2 are accommodated in the orbitals 1 and 2, respectively, in the absence of interaction between them. Suc h singly- occupied orbitals are referred to as magnetic orbitals. The wea k interaction between 1 and 2 leads to the two levels 1 and 2 of the dimer separated by a small energy gap e (Fig. 11 ), which are approximated by 2/) (2/) ( 2 1 22 1 1 . As depicted in Fig. 12 , one of the three triplet-state wave functions is represented by the electron configuration T. When e is very small (compared with that expected for 20 chemical bonding), the singlet sta te electron configurations 1 and 2 are very close in energy, and interact strongly under elecHˆ to give 1e l e c 2 1 2ˆHK , where K 12 is the exchange repulsion between 1 and 2. 12 1 2 12 2 1K( 1 ) ( 2 ) 1 / r ( 1 ) ( 2 ) , which is the self-repulsion resu lting from the overlap density 12. Thus the true singlet state S is described by the lower-ener gy state of the configuration-in teraction (CI) wave functions i (i = 1, 2), i1 i 1 2 i 2CC (i = 1, 2), namely, S = 1. The energies of S and T, ECI(S) and E CI(T), respectively, can be evaluated in terms of elecHˆ by using the dimer orbitals 1 and 2 determined from the calculations for the triplet state T. Then, after some manipulations, the electronic energy difference between the singlet and triplet state is written as1,19 ECI = E CI(S) ECI(T) = U)e(K22 12 . ( 8 ) The effective on-site repulsion U is given by 12 11J JU , where J 11 and J 12 are the Coulomb repulsions 11 1 1 12 1 1J( 1 ) ( 2 ) 1 / r ( 1 ) ( 2 ) 12 1 2 12 1 2J( 1 ) ( 2 ) 1 / r ( 1 ) ( 2 ) . Then, by mapping the energy spectrum of spinHˆ onto that of elecHˆ, namely, 21 Espin = ECI, we obtain J = ECI = U)e(K22 12 ( 9 ) It is important to note the qualitative aspect of the spin exch ange J on the basis of the above expression. Since the repulsion terms K 12 and U are always positive, the spin exchange J is divided into the FM and AFM components J F (< 0) and J AF ( > 0 ) , respectively. That is, J = J F + J AF, where J F = 2K12 ( 1 0 a ) J AF = U)e(2 ( 10b) The FM term J F term becomes stronger with increasing the exchange integral K 12, which in turn increases with increasing the overlap density, 12. The AFM term J AF becomes stronger with increasing e, which in turn becomes larger with increasing the overlap integral, 12 . In addition, the J AF term becomes weaker with increasing the on-site repulsion, U. 3.2. Use of broken-symmetry states for an isolated spin dimer For a general magnetic system, it is practically impossible to determine the eigenvalue spectrum of either elecˆH or spinˆH . However, for broken-symmetry states, 22 which are not eigenstates of elecˆH and spinˆH , their relative energies can be readily determined in terms of both elecˆH and spinˆH . With DFT calculations, the energy-mapping for a spin dimer between the energy spectra of elecˆH and spinˆH is carried out by using high-spin and broke n-symmetry states ( HS and BS, respectively).1-5,20,21 For example, let us reconsider the spin dimer shown in Fig. 10 , for which the pure-spin HS and BS states are given by 2 1 2 12 1 2 1 or BS or HS Here the HS state is an eigenstate of the spin Hamiltonian spinˆH in Eq. 3a, but the BS state is not. In terms of this Hamiltonian, the energies of the collinear-spin states HS and BS are given by E spin(HS) = spinˆ HS H HS = J/4 E spin(BS) = spinˆBS H BS = J/4, Thus, Espin = E spin(HS) Espin(BS) = J/2. In terms of DFT calculations, the electronic structures of the HS and BS states are readily evaluated to determine their energies, E DFT(HS) and E DFT(HS), respectively, and hence obtain the energy difference EDFT = E DFT(HS) EDFT(BS). 23 Consequently, by mapping Espin onto EDFT, we obtain J / 2 = EDFT. ( 1 1 ) 3.3. Use of broken-symmetry states for general magnetic solids The energy-mapping analysis based on DFT calculations employs the broken- symmetry state that is not an eigenstate of the spin Hamiltonia n . F o r a g e n e r a l s p i n Hamiltonian defined in terms of several spin exchange parameter s (Eq. 1), it is impossible to determine its eigenstates analytically in terms o f the spin exchange parameters to be determined and is also difficult to determine them numerically even when their values are known. For any realistic magnetic system requiring a spin Hamiltonian defined in terms of various spin exchange parameter s, the energy-mapping analysis based on DFT greatly f acilitates the quantitative eval uation of the spin exchange parameters because it does not rely on the eigenstates but on t he broken-symmetry states of the spin Hamiltonian. For broke n-symmetry states, the energy expressions of the spin Hamiltonian can be readily written down (see below) and the cor responding electronic energies can be readily determined by DFT calculations as well. In general, the magnetic energy levels of a magnetic system are described by employing a spin Hamiltonian spinˆH defined in terms of several different spin exchange parameters (Eq. 1). This model Hamiltonian generates a set of m agnetic energy levels as the sum of pair-wise interactions ij i jˆˆJS S. It is interesting that the sum of such “two-body interactions” can reasonably well describe the magnetic energy spectrum. This is due to the fact that spin exchange inte ractions are determined primari ly by the tails of magnetic 24 orbitals (see Section 4).1,2 The spin exchange constants J ij of a given magnetic system can be evaluated by employing the en ergy-mapping method as describe d below.2 (a) Select a set of N spin exchange paths J ij (= J 1, J2, … , J N) for a given magnetic system on the basis of inspecting the geometrical arrangement of its m agnetic ions and also considering the nature of its M-L -M and M-L…L-M exchange paths. (b) Construct N+1 ordered spin states (i.e., br oken-symmetry states) i = 1, 2, … , N+1, in which all spins are collinear so that any given pair of spins h as either FM or AFM arrangement. For a general spin dimer whose spin sites i and j possess N i and N j unpaired spins (hence, spins S i = N i/2 and S j = N j/2), respectively, the spin exchange energies of the FM and AFM arrangements (E FM and E AFM, respectively) are given by3 E FM = +N iNjJij/4 = +S iSjJij, E AFM = NiNjJij/4 = SiSjJij, ( 1 2 ) where J ij (= J 1, J2, … , J N) is the spin exchange parameter for the spin exchange path ij = 1, 2, … , N. Thus, the total spin exchange energy of an ordered spin arran gement is readily obtained by summing up a ll pair-wise interactions to fi nd the energy expression E spin(i) (i = 1, 2, … , N+1) in terms of the parameters to be determined and hence the N relative energies Espin(i – 1) = E spin(i) Espin(1) (i = 2, 3, … , N+1) (c) Determine the electronic energies E DFT(i) of N+1 ordered spin states i = 1, 2, … , N+1 by DFT calculations to obtain the N relative energies 25 EDFT (i – 1) = E DFT(i) EDFT(1) (j = 2, 3, … , N+1) As already mentioned, DFT calcu lations for a magnetic insulator t e n d t o g i v e a metallic electronic structure because the electron correlation of a magnetic ion leading to spin polarization is not well described. Since we de a l w i t h t h e e n e r g y spectrum of a magnetic insulator, it is necessary that the elec tronic structure of each ordered spin state obtained fr om DFT calculations be magnetic i nsulating. To ensure this aspect, it is necessary to perform DFT+U calculations 13 by adding on-site repulsion U eff = U – J with on-site repulsion U and on-site exchange interact ion J on magnetic ions. Furthermore, as can be seen from Eq. 10b, the AF M component of a spin exchange decreases with increasing U eff so that the magnitude and sign of a spin exchange constant may be affected by U eff. It is therefore necessary to carry out DFT+U calculations with several different U eff values. (d) Finally, determine the values of J 1, J2, … , J N by mapping the N relative energies EDFT onto the N relative energies Espin, EDFT (i – 1) = Espin(i – 1) (i = 2 – N+1) (13) In determining N spin exchanges J 1, J2, … , J N, one may employ more than N+1 ordered spin states, hence obtaining more than N relative energies EDFT and Espin for the mapping. In this case, the N parameters J 1, J2, … , J N can be determined by performing least-squares fitting analysis. 3.4. Energy-mapping based on four ordered spin states4 26 For our calculations, we regard the spin operators iSˆ and jSˆ as the classical vectors of iS and jS , respectively. Then, the spin Hamiltonian can be written as j i ji ij j i ji ij spin SS J SˆSˆ J Hˆ ( 1 4 ) Without loss of generality, the spin pair i-j will be regarded as 1-2. For simplicity, all spin sites are assumed to have an identical spin S. We carry out DFT +U calculations for the following four ordered spin states: State Spin 1 Spin 2 Other spin sites 1 (0, 0, S) (0, 0, S) Either (0, 0, S) or (0, 0, -S) according to the experimental (or a low-energy) spin state. Keep the same for the four spin states 2 (0, 0, S) (0, 0, -S) 3 (0, 0, -S) (0, 0, S) 4 (0, 0, -S) (0, 0, -S) where the notations (0, 0, S) and (0, 0, -S), for example, mean that the spin vectors are pointed along the positive and ne gative z-directions, respectiv ely. We represent the energies of the spin states 1 – 4 as E 1 – E 4, respectively. Then, according to Eq. 14, the energy difference, E 1 + E 4 – E 2 – E 3, is related to the spin exchange J as 23 2 4 1 12S4E E EEJ ( 1 5 ) Once the energies E 1 – E 4 are obtained from DFT+U+SOC calculations, we can readily determine J 12. 3.5. General features of spin exch anges numerically extracted Common DFT functionals suffer from the self-interaction error, i.e., a single electron interacts with itself, which is unphysical. This error results in a spurious 27 delocalization of orbitals including magnetic orbitals. Consequ ently, spin exchange interactions are overestimated by the usual DFT methods. This s elf-interaction error can be reduced by using the DFT+U method, in which the on-site Coul omb interaction is taken into consideration. This on-site interaction is parameter ized by the effective on-site Coulomb interaction U eff = U – J. By adding such Hartree-Fock-like terms, the DFT+U method makes the magnetic orbitals more localized and decreases the overlap between magnetic orbitals hence reducing the magnitudes of spin exchang e interactions. Currently, there is no reliable way of determining the U and J parameters needed for DFT+U calculations. A practical way of probing the magnetic propertie s of a given system is to carry out DFT+U calculations for several different U eff values, which provide several sets of the J 1, J2, … , J N values. It is important to find trends common to these sets. W hat matters in finding a spin lattice are the relative magnitudes o f the spin exchanges. As already pointed out, the purpose of using a spin Hamiltonian is to quantitatively describe the observed experimental data with a minimal set of J ij values hence capturing the essence of the chemistry and physics involved. Experimentally, such a set of J ij values for a given magnetic system is dedu ced first by choosing a few spin exchange paths J ij that one considers as important for the system and then by evaluatin g their signs and magnitudes such that the energy spectrum of the resulting spin Hamiltonian best simulates the observed experimental data. The numerical values of J ij deduced from this fitting analysis depends on what spin lattice model one employs for the fitting, and hence more than one spin lattice may fit the experimental data equall y well. This non- uniqueness of the fitting analysis has been the source of contr oversies in the literature over the years. Ultimately, the s pin lattice of a magnetic syst em deduced from 28 experimental fitting analysis should be consistent with the one determined from the energy-mapping analysis based on DFT calculations, because the observed magnetic properties are a consequence of t he electronic structure of the magnetic system. 4. Orbital interactions controlling spin exchanges For a given magnetic system, one can determine the values of it s various spin exchanges using the energy mapping analysis based on DFT+U calc ulations and hence ultimately find the spin lattice appropriate for it. What the e nergy-mapping analysis cannot tell us is why a certain spin exchange is strong or weak . To answer this question, it is necessary to understand how the strength of a given spin exchange interaction between two magnetic ions is related to the orbital interaction b e t w e e n t h e m a g n e t i c orbitals representing the magne tic ions. In this section, we co nsider the qualitative aspects of the orbital interactions controlling spin exchange interacti ons. Given a magnetic solid made up of ML n polyhedra containing a magnetic transition cation Mx+ (x = oxidation state), there may occur two types of spin excha nge paths, namely, M-L-M exchange and/or M-L…L-M exchange paths. The qualitative factors governing the signs and magnitudes of M-L-M exchanges were well established many decades ago.22,23 However, the importance of M-L…L-M exchange paths has been realized much later.1,2 In leading to AFM interactions, M-L…L-M exchanges can be much stronger than M-L-M exchanges. What was not realized in the early studies of M-L-M exchanges is the importance of the magnetic orbitals of ML n polyhedra, in which the M d- orbitals are combined out-of-phase with the L p-orbitals. In M-L…L-M spin exchanges the magnetic orbitals of the two metal sites can interact stron gly as long as their L p -29 orbital tails can interact through the L…L contact.1 In what follows we examine qualitatively the through-space and through-bond orbital intera ctions2 that govern M- L…L-M spin exchanges. As a representative example capturing the essence of spin excha nge interactions, let us examine those of LiCuVO 4 24-26 in which the CuO 2 ribbon chains, made up of edge- sharing CuO 4 square planes running along the b-direction are interconnected along the a- direction by sharing corners with VO 4 tetrahedra. This is shown in Fig. 13 . In LiCuVO 4 the Cu2+(S = 1/2, d9) ions are magnetic, but the V5+ (d0) ions are nonmagnetic. As for the spin exchange paths of LiCuVO 4, we consider the nearest ne ighbor (nn) and next-nearest- neighbor (nnn) intrachain spin exchanges, J nn and J nnn, respectively, in each CuO 2 ribbon chain as well as the interchain spin exchange J a along the a-direction ( Fig. 13 ). The magnetic orbital of the Cu2+ (S = 1/2, d9) ion is given by the x2-y2 - antibonding orbital contained in the CuO 4 square plane ( Fig. 14a ), in which the Cu 3d x2- y2 orbital is combined out-of-phase with the 2p orbitals of the f our surrounding O ligands. As already emphasized,1,2 it is not the “head” part (the Cu 3d x2-y2 orbital) but the “tail” part (the O 2p orbitals) of the magnetic orbital that controls the magnitudes and signs of these spin exchange interactions. Let us first consider the Cu- O-Cu exchange J nn. When the x2-y2 magnetic orbitals 1 and 2 of the two spin sites are brought together to form the Cu-O-Cu bridges, the O 2p orbital tails at the bridging O atoms make a nearly orthogonal arrangement ( Fig. 14b ). Thus, the overlap integral 12 between the two magnetic orbitals is almost zero, which leads to J AF 0. In contrast, the overlap density 12 of the magnetic orbitals is substantial, which leads to nonzero J F. A s a c o n s e q u e n c e , t h e J nn exchange becomes FM.25,26 30 For the intra-chain Cu-O…O-Cu exchange J nnn (Fig. 14c ), the O 2p orbital tails of the magnetic orbitals 1 and 2 at the terminal O atoms are well separated by the O…O contacts. Thus, the overlap density 12 of the magnetic orbitals is negligible leading to J F 0. However, the overlap integral 12 is nonzero because the O 2p tails of 1 and 2 overlap through the O…O contact s. This through-space interactio n between 1 and 2 produces a large energy split e between + and -, which are in-phase and out-of-phase combinations of 1 a n d 2, respectively ( Fig. 15a ), thereby leading to nonzero J AF. Consequently, the J nnn exchange becomes AFM.25,26 In the interchain spin exchange path J a, the two CuO 4 square planes are corner- shared with VO 4 tetrahedra. In the Cu-O…V5+…O-Cu exchange paths, the empty V 3d orbitals should interact in a bonding manner with the Cu x2-y2 orbitals. In the absence of the V 3d orbitals, the energy split e between + and - arising from the through-space interaction between 1 and 2 would be substantial, as expected from the intrachain exchange J nnn, so that one might expect a str ong AFM exchange for the interc hain exchange J a. However, in the Cu-O…V5+…O-Cu exchange paths, the bridging VO 4 units provides a through-bond interaction between the empty V 3d orbitals and the O 2p tails of the magnetic orbitals on the O…O contacts. By symmetry, this through-bond interaction is possible only with - (Fig. 15b ,c). The V 3d orbital being empty, the O 2p tails of - on the O…O contacts interact in-phase with the empty V 3d orbital hence lowering the - level, whereas + is unaffected by the V 3d orbital, thereby reducing the energy split e between + and - o f t h e C u - O … V5+…O-Cu exchange paths and consequently weakening the interchain spin exchange J a.25,26 As a consequence, the 31 magnetic properties are dominated by the one-dimensional charac ter of the CuO 2 ribbon chain. It is important to observe the corollary of the above observati on for general M- L…Ay+…L-M spin exchange, where the cation Ay+ provides through-bond interactions. If the e between + and - is negligible in terms of the through-space interaction, then the effect of the through-bond interaction would make e large leading to a strong AFM interaction.2 When the ML n polyhedra containing M cations are condensed together by sharing a corner, an edge or a face, they give rise to M-L-M exchanges, which are the subject of the Goodenough rules.22 When these polyhedra are not condensed, they give rise to M- L…L-M and M-L…Ay+…L-M exchanges,1,2 where Ay+ (y = oxidation state) refers to the intervening d0 metal cation. The importance of the latter spin exchanges, not covered by the Goodenough rules, was recognized1,2 only after realizing that the magnetic orbitals of an M ion include both the M d-orbitals and the L p-orbitals of the ML n polyhedron, and that the L p-orbital tails of the magnetic orbitals control the magnitudes and signs of such spin exchange interactions.1,2 Concerning the M-L…L-M exchanges, there are several important consequences of this observation:1,2 (a) The strength of a given M-L…L-M spin exchange is not determined by the shortness of the M…M distance, but rather by that of the L…L distance; it is strong when the L…L distance is in the vicinity of the van der Waals radii sum or shorter.1 (b) In a given magnetic system consisting of both M-L-M and M-L…L-M spin exchanges, the M-L…L-M spin exchanges are very often stronger than the M-L-M spin exchanges. 32 (c) The strength of an M-L…L-M spin exchange determined by through-space interaction between the L np tails on the L…L contact can be significantly modified when the L…L contact has a through-bond inte raction with the intervening d0 metal cation Ay+ (y = oxidation state)25,27 or even the p0 metal cation (e.g., Cs+ as found for Cs 2CuCl 4 28). Such an M-L…Ay+…L-M spin exchange becomes strong if the corresponding M- L…L-M through-space exchange is weak, but becomes weaker if the corr esponding M-L…L-M through-space exchange is strong. This is so because the empty d orbital of Ay+ interacts only with the - orbital of the M-L…L-M exchange. In general, the empty d orbital has a much stronger th rough-bond effect than does the empty p orbital. 5. Incorporating the effect of S OC indirectly into spin Hamilto nian When a magnetic ion is present in molecules and solids to form a ML n polyhedron with surrounding ligands L, its orbital momentum L is mostly quenched with a small momentum L remaining unquenched.10 Exceptional cases occur when the ML n polyhedron has n-fold ( n 3) rotational symmetry so that it has doubly-degenerate d- states and when the d-electron count of ML n is such that a degenerate d-state is unevenly occupied. In this case, the orbital momentum L is not quenched so that the effect of the SOC, LˆSˆ , becomes strong often leading to uniaxial magnetism (see Secti on 6). In this section, we consider the cases when the orbital quenching is no t complete so a small orbital momentum L remains at each magnetic ion. In the past this situation has b een discussed on the basis of the effective spin approximation,10,29 in which the need to explicitly describe the unquenche d orbital momentum is circumve nted by treating the 33 system as a spin-only system. In this approximation the effect of SOC arising from L is absorbed into the coefficient for certain terms made up of only spin operators. This approximation deals with both si ngle-spin site and two-spin sit e problems. The former includes the single-ion anisotropy, while the latter include th e asymmetric spin exchange and the Dzyaloshinskii-Moriya (DM) exchange.30,31 The DM exchange is often referred to as antisymmetric exchange. 5.1. SOC effect on a single-spin sit e and spin-half misconcepti on For a magnetic ion with nondegenerate magnetic orbital (e.g., C u2+) , t h e S O C Hamiltonian LˆSˆ Hˆ SO is transformed into the zero-field spin Hamiltonian zfHˆ10 )SˆSˆ SˆSˆ(E)Sˆ Sˆ(D )Sˆ Sˆ(E)Sˆ Sˆ(D Hˆ 21 2 31 2 z2 y2 x2 31 2 z zf ( 1 6 ) where the constants D and E orig inate from the SOC associated w ith the remnant orbital momentum L , that is, D 2(L|| L) E 2(Lx Ly) where L|| and L are the the ||z- and z-components of L , respectively, while Lx and Ly are the x- and y-components of L, respectively. For S > 1/2 ions, Eq. 16 predicts magnetic anisotropy. For ins tance, a S = 1 ion is described by three spin states, 1,1, 0 ,1 and 1,1. Thus, 34 1,1D 1,1)]1S(S Sˆ[D1,1)Sˆ Sˆ(D0 ,1D 0 ,1)]1S(S Sˆ[D0 ,1)Sˆ Sˆ(D1,1D 1,1)]1S(S Sˆ[D1,1)Sˆ Sˆ(D 31 31 2 z2 31 2 z32 31 2 z2 31 2 z31 31 2 z2 31 2 z and 1,1E1,1)SˆSˆ SˆSˆ(E00 ,1)SˆSˆ SˆSˆ(E1,1E1,1)SˆSˆ SˆSˆ(E This shows that the 1,1 states are separated in energy from the 0 ,1 state by |D|. In addition, the 1,1 and 1,1 states interact and become split in energy by |E|. Due to this energy split, the thermal populations of the three states differ, hence leading to magnetic anisotropy. A similar conclusion is reached for S > 1 ions. For example, a S = 3/2 ion is described by the four states, 23 23,, 21 23,, 21 23, and 23 23,. Therefore, D ,)]1S(S Sˆ[D ,)Sˆ Sˆ(D0 ,)]1S(S Sˆ[D ,)Sˆ Sˆ(D0 ,)]1S(S Sˆ[D ,)Sˆ Sˆ(DD ,)]1S(S Sˆ[D ,)Sˆ Sˆ(D 23 23 31 2 z 23 23 2 31 2 z21 23 31 2 z 21 23 2 31 2 z21 23 31 2 z 21 23 2 31 2 z23 23 31 2 z 23 23 2 31 2 z and 21 23 23 2323 23 21 2323 23 21 2321 23 23 23 ,E ,)SˆSˆ SˆSˆ(E,E ,)SˆSˆ SˆSˆ(E,E ,)SˆSˆ SˆSˆ(E,E ,)SˆSˆ SˆSˆ(E Thus, the 23 23, states are separated in energy from the 21 23, states by |D|. Without loss of generality, it can be assumed that the 23 23, states lie higher than the 21 23, 35 states. The 23 23, and 21 23, states interact with interaction energy E, and so are the states 23 23, and 21 23,. Then, according to perturbation theory, the 23 23, states are raised in energy by E2/|D|, and the 21 23, states are lowered in energy by E2/|D|. Consequently, the 23 23, states become separated in energy from the 21 23, states by |D| + 2E2/|D|. The aforementioned energy split for S > 1/2 ions, and the asso ciated magnetic anisotropy, is a consequence of S OC albeit indirectly through t h e c o n s t a n t s D a n d E . Since the information about the orbital zL,L of the magnetic ion is completely hidden in these constants, it is not possible to predict the preferred spin orientation of a S > 1/2 ion on the basis of Eq. 16, although one can infer that such an i o n h a s m a g n e t i c anisotropy as described above. A rather different situation occurs for a S = 1/2 ion, which i s described by two spin states, 21 21, and 21 21, . We note that 0 ,)]1S(S Sˆ[D ,)Sˆ Sˆ(D0 ,)]1S(S Sˆ[D ,)Sˆ Sˆ(D 21 21 31 2 z 21 21 2 31 2 z21 21 31 2 z 21 21 2 31 2 z and 0 ,)SˆSˆ SˆSˆ(E0 ,)SˆSˆ SˆSˆ(E 21 2121 21 Consequently, the up-spin and dow n-spin states do not interact under zfˆH, so their degeneracy is not split. (This r esult obeys the Kramers degener acy theorem,32 w h i c h states that the degeneracy of an odd-spin system should not be split in the absence of an 36 external magnetic field.) This is so even though the constants D and E are nonzero, that is, even though SOC effects are ta ken into consideration though ind irectly. Thus, the thermal populations of the two states and are identical, hence leading to the conclusion that an S = 1/2 ion has no magnetic anisotropy that arise from SOC. This is the origin of the spin-half misconception. Note that LˆSˆ Hˆ SO and zfHˆ are local (i.e., single-spin site) operators, and do not describe interactions between different spin sites. The SOC -induced magnetic anisotropy for S > 1/2 ions is commonly referred to as the sing le-ion anisotropy, to which practitioners of spin Hamiltonian analysis have no objection. H owever, most of them deny strenuously that S = 1/2 ions have single-ion anisotropy a nd suggest the use of the term “magneto-crystalline anisotropy” to describe the experimen tally observed magnetic anisotropy of S = 1/2 ions. In the vernacular this term is a re d herring, because it means that the observed anisotropy is not caused by the single-spin s ite effect (i.e., SOC) but rather by nonlocal effects (i.e., anything other than SOC, e.g. , asymmetric spin exchange and magnetic dipole-dipole inter actions), just as Moriya and Yo shida argued for the S = 1/2 system CuCl 22H2O more than six decades ago.7 However, as recently shown 5,6,9 for various magnetic solids of S = 1/2 ions (see Section 7), the sp in-half misconception is erroneous. Unfortunately, this misconception remains unabated b ecause it is perpetuated in monographs and textbooks on magnetism.8 In defense of the spin-half misconception, one might argue that the true magnetic energy states are not th ose generated by an electronic Hamiltonian, but those generated by a spin Hamiltoni an. However, this argument is even more fallacious than the spin-half misconcepti on, because it amounts to arguing that there exists no orbi tal momentum. The magnetic pro perties of a magnetic ion 37 are ultimately related to its moment , which is the derivative of its total electronic energy with respect to an applied magnetic field (see Section 6 ).10 T h e m o m e n t consists of both orbital and spin components, i.e., S L, and these components are related to the orbital and spin momenta as LB L and S2B S , where B is the Bohr magneton. Consequently, the magnetic energy states become identical to those generated by a spin Hamiltonian, only if 0L , that is, only if the quenching of orbital momentum is complete. The latter condition is hardly met for al l magnetic ions in molecules and solids. It is satisfied for all magnetic ions in a spin Hamiltonian analysis by definition. In short, S = 1/2 ions do possess single-ion ani sotropy, but a spin Hamiltonian analysis predicts erroneously that they do not. 5.2. SOC effect on spin exchange: Mapping analysis for anisotro pic spin exchange33 In some cases the spin exchange between two spin sites may not be isotropic. This is an indirect consequence of SOC because a spin at a given sit e has a preferred orientation due to SOC and because this orientation preference can influence the strength of the spin exchange. Given two spin sites, say, 1 and 2, one m ay take the z-axis along the exchange paths between 1 and 2. As already mentioned in Sec tion 3, the anisotropic spin exchange interaction betw een two sites 1 and 2 is written as z2z1z y2y1y x2x1x spin SˆSˆJ SˆSˆJ SˆSˆJ Hˆ , ( 1 7 a ) To evaluate J x, Jy a n d J z, we perform energy-mapping analysis by determining the energies of appropriate broken-sy mmetry spin states on the basi s of DFT+U+SOC calculations. To determine the J x component, we consider the following four ordered spin states, 38 State Spin 1 Spin 2 Other spin site 1 (S, 0, 0) (S, 0, 0) Either (0, 0, S) or (0, 0, -S) according to the experimental (or a low-energy) spin state. Keep the same for the four spin states 2 (S, 0, 0) (-S, 0, 0) 3 (-S, 0, 0) (S, 0, 0) 4 (-S, 0, 0) (-S, 0, 0) Then, the energy difference, 3 2 4 1 E E EE , of the four states is related to the spin exchange J x as, 23 2 4 1 xS4E E EEJ Then, on the basis of DFT+U+SOC calculations for the four spin states, the value of J x is readily determined. The values of J y and J z are obtained in a similar manner. To obtain J y, we do DFT+U+SOC calculations f or the following states: State Spin 1 Spin 2 Other spin site 1 (0, S, 0) (0, S, 0) Either (0, 0, S) or (0, 0, -S) according to the experimental (or a low-energy) spin state. Keep the same for the four spin states 2 (0, S, 0) (0, -S, 0) 3 (0, -S, 0) (0, S, 0) 4 (0, -S, 0) (0, -S, 0) Then, we find 23 2 4 1 yS4E E EEJ To determine J z, we perform DFT+U+SOC calculati ons for the following states: 39 State Spin 1 Spin 2 Other spin sites 1 (0, 0, S) (0, 0, S) Either (S, 0, 0) or (-S, 0, 0) according to the experimental (or a low-energy) spin state. Keep the same for the four spin states 2 (0, 0, S) (0, 0, -S) 3 (0, 0, -S) (0, 0, S) 4 (0, 0, -S) (0, 0, -S) Then, we find 23 2 4 1 zS4E E EEJ 5.3. SOC effect on two adjacent spin sites Another important consequence of SOC is the Dzyaloshinskii-Mor iya (DM) interaction between two adjacent spin sites. Consider the SOC i n a spin dimer made up of two spin sites 1 and 2, for which the SOC Hamiltonian is given by10 SO 1 2 1 2 1 1 2 2ˆ ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ HL S ( L L ) ( S S ) ( L S L S ) . (18) where the last equality follows from the fact that the SOC is a local interaction. It is important to note that, although SO C describes a single-spin si te interaction, the two spin sites can interact indirectly hence influencing their relative spin orientations.2,31 A s illustrated in Fig. 16 , we suppose that an occupied orbital i interacts with an unoccupied orbital j at spin site 1 via SOC, and that the i and j of site 1 interact with an occupied orbital k of site 2 via orbital interact ion. The orbital mixing between i and k introduces the spin character of site 2 into i of site 1, while that between j and k introduces the spin character of site 2 into j of site 1. Namely, 2 2(1 ) (1 )ii i k jj j k , 40 where refers to a small mixing coefficient. Then, the SOC between su ch modified 'i and 'j at site 1 ind irectly introduces the SOC-in duced in teraction b e tween the spins at sites 1 and 2. For a spin dimer, t here can be a number of inter actions like the one depicted in Fig. 16 at both spin sites, so summing up all such contributions gives rise to the DM interaction energy E DM between spin sites 1 and 2. Suppose that 1L and 2L are the remnant orbital angular momenta at sites 1 and 2, respectively. Then, use of the SOˆH (Eq. 18) as perturbation leads to the DM interaction energy E DM,10,31 E DM = 1 2 1 2 12 1 212[J(L L ) ] ( S S ) D ( S S ) In this expression, the DM vector 12D is related to the difference in the unquenched orbital angular momenta on the two magnetic sites 1 and 2, name ly, 12 12 1 2DJ ( L L ) . F o r a s p i n d i m e r w i t h s p i n e x c h a n g e J 12, the strength of its DM exchange 12D is discussed by considering the ratio D12/J12, which is often approximated by D12/J12 g/g, where g is the contribution of the orbital moment to the g-factor g i n the effective spin approximation. In general, the g/g value is at most 0.1, so that the D12/J12 ratio is often expected to be 0.1 at most. However, it is important to recogni ze an implicit assumption behind this reasoning, namely, that the spin sites 1 and 2 have an identical chemical environment. When the two spin sites have different chemical en vironments, the D12/J12 ratio can be very large as found for a particular Mn(2)3+-O-Mn(3)4+ spin exchange path of 41 CaMn 7O12 (i.e., D12/J12 = 0.54).34 As depicted in Fig. 16 , the magnitude of a DM vector D12 is determined by the three matrix elements, tSO = iS O jˆH , tik = eff ikˆH, and tjk =eff jkˆH. When tSO, tik and tjk are all strong, the magnitude of the DM vector D 12 can be unusually large.35 5.4. Mapping analysis for the DM v ector of an isolated spin dim er2 Let us consider how to determine the DM vector of an isolated s pin dimer. So far, a spin dimer made up of spin sites 1 and 2 has been described b y the spin Hamiltonian, spin 12 1 2ˆˆ ˆHJ S S , composed of only a Heisenberg spin exchange. This Hamiltonian leads to a collinear spin arrangement (either FM or AFM), as already mentioned. To allow for a canting of the spins 1S and 2S from the collinear arrangeme nt (typically from the AFM arrangement), which is experimentally observed, it is necessary to include the DM exchange interaction 12 1 2ˆˆ D( SS ) into the spin Hamiltonian. That is, spin 12 1 2 12 1 2ˆˆ ˆ ˆ ˆHJ S S D ( S S ) . ( 1 9 ) The 12ˆˆSS term, being proportional to sin, where is the angle between the two spin vectors 1S and 2S , is nonzero only if the two spins are not collinear. Thus, the D M interaction 12 1 2ˆˆ D( SS ) induces spin canting. Even when a model Hamiltonian consists of only Heisenberg spin exchanges, a magnetic system with more than two spin sites can have a non-collinear spin arrangement so as to reduce the exten t of spin frustration if there exists substant ial spin frustration. 42 As discussed in Section 3; the spin exchange J 12 of Eq. 19 can be evaluated on the basis of energy-mapping analysis by considering two collinear s pin states HS and BS (i.e., FM and AFM spin arrangements, respectively) because the DM exchange 12 1 2ˆˆ D( SS ) is zero for such collinear spin states. To evaluate the DM vec tor 12D , we carry out energy-mapping analysis on the basis of DFT+U+SOC cal culations. In terms of its Cartesian components, 12D is expressed as xy z 12 12 12 12ˆˆ ˆD i D j D k D Therefore, the DM interaction energy 12 1 2ˆˆ D( SS ) is rewritten as z 2y 2x 2z 1y 1x 1z 12y 12x 12 2 1 12 Sˆ Sˆ SˆSˆ Sˆ Sˆkˆjˆiˆ )Dkˆ Djˆ Diˆ( )SˆSˆ( D )SˆSˆ SˆSˆ(D)SˆSˆ SˆSˆ(D)SˆSˆ SˆSˆ(D x 2y 1y 2x 1z 12x 2z 1z 2x 1y 12y 2z 1z 2y 1x 12 (20) To determine the z 12D component, we consider the following two orthogonally ordered spin states, State Spin 1 Spin 2 1 (S, 0, 0) (0, S, 0) 2 (S, 0, 0,) (0, -S, 0) For these states, 12SS 0 and 2 12SS S so that, according to Eq. 20, the energies of the two states are given by E 1 = S2 z 12D, and E 2 = -S2 z 12D. Consequently, 43 22 1 z 12S2EED . ( 2 1 a ) Thus, the z 12D is determined by evaluating the energies E 1 and E 2 on the basis of DFT+U+SOC calculations. The y 12D and x 12D components are determined in a similar manner. Using the following two orthogonal spin states, State Spin 1 Spin 2 3 (S, 0, 0) (0, 0, S) 4 (S, 0, 0,) (0, 0, -S) the y 12D component is obtained as 24 3 y 12S2E ED ( 21b) In terms of the following t wo orthogonal spin states, State Spin 1 Spin 2 5 (0, S, 0) (0, 0, S) 6 (0, S, 0,) (0, 0, -S) the x 12D term is given by 26 5 z 12S2E ED ( 2 1 c ) 5.4. Mapping analysis for the DM vectors using the four-state m ethod for a general magnetic solid 4 44 For a given pair of spins in a general magnetic solid, the x 12D, y 12D and z 12D components can be similarly extracted by performing DFT+U+SOC c alculations for four non-collinearly ordered spin state s in which all spin exchange interactions associated with the spin sites 1 and 2 vanish.4 In such a case the relative energies of the four states are related only to the energy differences in their DM interact ions. To calculate the z- component of 12D, i.e., z 12D, we carry out DFT+U+SOC cal culations for the following four ordered spin states: State Spin 1 Spin 2 Other spin sites 1 (S, 0, 0) (0, S, 0) Either (0, 0, S) or (0, 0, -S) according to the experimental (or a low-energy) spin state. Keep the same for the four spin states 2 (S, 0, 0) (0, -S, 0) 3 (-S, 0, 0) (0, S, 0) 4 (-S, 0, 0) (0, -S, 0) Then, we obtain 23 2 4 1 z 12S4E E EED ( 2 2 a ) To determine the y-component of 12D, i.e., y 12D, we perform DFT+U+SOC calculations for the following four or dered spin states: State Spin 1 Spin 2 Other spin site 1 (S, 0, 0) (0, 0, S) Either (0, S, 0) or (0, -S, 0) according to the experimental (or a low-energy) spin state. Keep the same for the four spin states 2 (S, 0, 0) (0, 0, -S) 3 (-S, 0, 0) (0, 0, S) 4 (-S, 0, 0) (0, 0, -S) Then, 45 23 2 4 1 y 12S4E E EED ( 22b) To determine the x-component of 12D, i.e., x 12D, we carry out DFT+U+SOC calculations for the following four or dered spin states: State Spin 1 Spin 2 Other spin site 1 (0, S, 0) (0, 0, S) Either (S, 0, 0) or (-S, 0, 0) according to the experimental (or a low-energy) spin state. Keep the same for the four spin states 2 (0, S, 0) (0, 0, -S) 3 (0, -S, 0) (0, 0, S) 4 (0, -S, 0) (0, 0, -S) Then, 23 2 4 1 x 12S4E E EED ( 2 2 c ) 6. Uniaxial magnetism10,36 In classical mechanics, the magnetic moment of a system refers to the change of its energy E with respect t o the applied magnetic field H , E H. ( 2 3 ) A uniaxial magnetic ion has a nonzero magnetic moment only in o ne direction in coordinate space, while an isotropic magnetic ion has a nonzero moment in all directions with equal magnitude. An anisotropic magnetic ion, lying betwee n these two cases, has a moment with magnitude depending on the spin direction. When a t ransition-metal 46 magnetic ion is located at a coordination site with 3-fold or h igher rotational symmetry, its d-states have doubly-degenerate levels, namely, {xz, yz} and {xy, x2-y2}, if the z-axis is taken along the rotational axis. In terms of t he {L z, Lz} set of magnetic quantum numbers, the {xz, yz} and {xy, x2-y2} sets are equivalent to {xz, yz} {1, 1} {xy, x2-y2} {2, 2} An uneven filling of such a degenerate level leading to configu rations such as (L z, Lz)1 and (L z, Lz)3 generates an unquenched orbital angular momentum of magnitude L (in units of ). Thus, an uneven filling of the {1, 1} set leads to L = 1, and that of the {2, 2} set to L = 2. Such an electron filling generates a Jahn-Tell er (JT) instability, but the unquenched orbital momentum remains if the associated JT-distor tion is prevented by steric congestion around the magnetic ions. The orbital momentu m L couples with the spin momentum S by the SOC, LS , leading to the total angular momentum SLJ . The resulting total angular momentum states zJ,J are doublets specified by the two quantum numbers J and J z = J, i.e., { J,J, J,J}.36 In identifying the ground doublet state, it is important to notice10 that < 0 for an ion with more than half-filled d-shell > 0 for an ion with less th an half-filled d-shell. If < 0, the lowest-energy doublet state of the LS term results when S and L are in the same direction. If > 0, however, it results when S and L have the opposite 47 directions. Consequently, for a magnetic ion with L and S, the total angular quantum number J for the spin-orbit coupled ground state is given by 0 fi SL0 fi SLJ : doublet Ground. For < 0, the energy of the J-state increases as J decreases. Howev er, the opposite is the case for > 0.36 In quantum mechanical description, the moment is related to an energy split of a degenerate level by an applied magnetic field. The Zeeman inter action under magnetic field is given by36 ZBˆ ˆ ˆ H( L 2 S ) H ( 2 4 ) If we take the z-axis along the rotational axis responsible for the degeneracy of the doublet state { J,J, J,J}, the Zeeman interaction for the field along the z-direction, H||, is written as Z|| B z z ||ˆ ˆˆH( L 2 S ) H . ( 2 5 a ) This Hamiltonian always lifts the degeneracy of { J,J, J,J}, because 0J,JHˆJ,JJ,JHˆJ,J J,JHˆJ,JH)S2L(J,JHˆJ,J |||| |||| B || Therefore, the energy split ||JE is given by || B ||J H)S2L(2 E ( 25b) and the associated g-factor g || by 48 )S2L(2 H/E g|| B ||J || The Zeeman interaction for the f ield perpendicular to the z-dir ection, H , is written as 1 ZB 2ˆˆ ˆˆ ˆH[ ( L L ) ( S S ) ] H , ( 2 6 a ) for which we find H J,J)Sˆ Sˆ()Lˆ Lˆ(J,J J,JHˆJ,J0J,JHˆJ,J J,JHˆJ,J B 21 B. Then, the associated energy split JE is given by ||H2 EB J . ( 26b) The J,J and J,J states differ in their J z values by 2J, so JE = 0 unless J = 1/2 because J,J state cannot become J,J by the ladder operator Lˆ or Sˆ in such a case. Thus, for magnetic ions with unquenched orbital momentum L , we find uniaxial magnetism if J > 1/2.36 It should be noted that a spin Hamiltonian does not allow one to predict whether or not a given magnetic ion in molecules and solids will exhibi t uniaxial magnetism because it cannot describe SOC, LˆSˆ , explicitly due to the lack of the orbital degree of freedom. Nevertheless, once a magnetic system is known to exhib it uniaxial magnetism, one might use an Ising spin Hamiltonian (Section 3) to discuss its magnetic property. 7. Describing SOC effects with both orbital and spin degrees of freedom: Magnetic anisotropy5 49 In this section we probe the effect of SOC by explicitly consi dering the orbital and spin degrees of freedom. This enables one to quantitatively determine the preferred spin orientation of a magnetic ion M with any spin (i.e., S = 1 /2 – 5/2) by performing DFT+U+SOC calculations and qualitatively predict it on the basi s of analyzing the HOMO-LUMO interactions of the ML n polyhedron induced by SOC, LˆSˆ . For this purpose, the states of a magnetic ion are described by z zS,SL,L instead of approximating it with zS,S . If a coordinate (x , y, z) is employed for the spin Sˆ, and (x, y, z) for the orbital Lˆ, the z direction is the preferred spin orientation by convention. The latter is specified with respect to the (x, y, z) coordinate by defining the polar angles and as depicted in Fig. 17 . In evaluating whether or not the SOC-induced interactions between different electronic st ates vanish, one needs to recall that the orbital states zL,L are orthonormal, and so are the spin states zS,S. That is, therwiseo ,0SS if ,1S,SS,S therwiseo ,0L L if ,1L,LL,L z z z zz z z z 7.1. Selection rules for pref erred spin-orientation Using the (x, y, z) and (x , y, z) coordinates for Lˆ and Sˆ, respectively, the SOC Hamiltonian LˆSˆ Hˆ is rewritten as SO0 SOHˆ Hˆ Hˆ ,2,10,37,38 where sineLˆ 21sineLˆ 21cosLˆSˆ Hˆi i z z0 SO ( 2 7 a ) ) sin sinLˆ cos sinLˆ cosLˆ(Sˆ y x z z . (27b) 50 sin cosLˆ cos cosLˆ sinLˆ)Sˆ Sˆ(2Hˆ y x z SO (28) We now consider if the preferred spin orientation is parallel t o the local z-direction (||z) (of the ML n under consideration) or perpendicular to it ( z). The SOC-induced interaction between two d-states, i and j, involves the interaction energy j SO iHˆ . For our discussion, it is necessa ry to know whether this integr al is zero or not. Since the angular part of a d- or p-orbital is expressed in terms of products zL,LzS,S, the evaluation of j SO iHˆ involves the spin integrals z 'z z S,SSˆS,S and z z S,SSˆS,S as well as the orbital integrals z z z L,LLˆL,L and z z L,LLˆL,L . The SOC Hamiltonian 0 SOHˆ allows interactions only betw een identical spin states, because zSˆ and zSˆ are nonzero. For two states, i and j, of identical spin, we consider the cases when | Lz| = 0 or 1. Then, we find 1 L if ,sin0 L if , cos Hˆ zz j0 SO i . ( 2 9 a ) For the 0 L z case, j0 SO iHˆ is maximum at = 0, i.e., when the spin has the ||z orientation. For the 1 L z case, j0 SO iHˆ becomes maximum at = 9 0, i.e., when the spin has the z orientation. Under SOC i and j do not interact when 1 Lz , because 0 Hˆ j0 SO i in such a case. 51 The total energy of ML n is lowered under SOC by the interactions of the filled d- states with the empty ones. Since the strength of SOC is very w eak, these interactions can be described in terms of perturbation theory in which the SOC H amiltonian is taken as perturbation with the split d-states of ML n as unperturbed states. T hen, the most important interaction of the occupied d-states with the unoccupied d-states is the one between the HOMO and the LUMO (with energies e HO a n d e LU, respectively), and the associated energy stabilization E is given by5 LU HO LU HO20 SOLU HO0 SO e e if ,e eLU Hˆ HOe e if ,LU Hˆ HO E (29b) Thus, we obtain the predictions f or the preferred spin orientat ion as summarized in Table 4. In general, the effect of a degenerate interaction is stronge r than that of a nondegenerate interaction. A syste m with degenerate HOMO and LU MO has JT instability, and the degeneracy would be lifted if the associat ed JT-distortion were to take place.39 According to Eq. 29, the preferr ed spin orientation is either ||z or z . F o r t h e preferred spin orientation t o lie in between the ||z and z directions, therefore, there must be two “HOMO-LUMO” interactions t hat predict different spin ori entations (one for ||z, and the other for z). Such a situation occurs for Na 2IrO 3, as will be discussed below. 7.2. Degenerate perturbation and uniaxial magnetism 52 For a certain metal ion M, the electron configuration of ML n has unevenly-filled degenerate level. For example, the hexagonal perovskites Ca 3CoMnO 6 40 consist of CoMnO 6 chains in which CoO 6 trigonal prisms containing high-spin Co2+ (S = 3/2, d7) ions alternate with MnO 6 octahedra containing high-spin Mn4+ (S = 3/2, d3) ions by sharing their triangular faces ( Fig. 18a ). The d-states of the high-spin Co2+ (S = 3/2, d7) ion in each CoO 6 trigonal prism ( Fig. 18b ) can be described by the electron configuration, (z2)2 < (xy, x2y2)3 < (xz, yz)2, in the one-electron picture.36,39 Thus, the spin-polarized d- states of the high-spin Co2+ is written as, ( z2)1 < (xy, x2y2)2 < (xz, yz)2 < (z2)1 < (xy, x2y2)1 < (xz, yz)0. Due to the half-filled configuration (xy , x2y2)1, the HOMO and LUMO are degenerate with | Lz| = 0, so the preferred spin orientation is ||z, i.e., along th e three-fold rotational axis of the trigonal prism. Furthermore, the configu ration (xy, x2y2)1 leads to an unquenched orbital mome ntum for L = 2. Since the d-shell of the high-spin Co2+ (d7, S = 3/2) ion is more than half filled, < 0, so that J = L + S = 2 + 3/2 = 7/2 for the ground doublet state. Since J > 1/ 2, this ion has uniaxial magn etism, that is, it has a nonzero magnetic moment only along the 3-fold rotational axis of the CoO 6 trigonal prism. E a c h h i g h - s p i n F e2+ ( S = 2 , d6) ion of Fe[C(Si(CH 3)3)3]2 is located at a linear coordinate site ( Fig. 5d ),36,41 so that its down-spin d-states are filled as depicted in Fig. 19 leading to the configuration (xy , x2-y2)1. Thus, with L = 2 and S = 2, the spin-orbit coupled ground doublet state is described by J = L + S = 4 with Jz = 4. Since J > 1/2, this ion has uniaxial magnetism; it has a nonzero magnetic mome nt only along the C-53 Fe-C axis (i.e., along the C -rotational axis), and hence this Fe2+ ion has uniaxial magnetism We now examine the uniaxial magnetism that arises from metal i ons at octahedral sites by considering the FeO 6 octahedra with high-spin Fe2+ (d6, S = 2) ions present in the oxide BaFe 2(PO 4)2, the honeycomb layers of which are made up of edge-sharing FeO 6 octahedra. This oxide exhibits a uniaxial magnetism.42 For our analysis of this observation, it is convenient to take the z-axis along one thre e-fold rotational axis of an ML 6 octahedron ( Fig. 6a ).12 The high-spin Fe2+ ion has the (t 2g)4(eg)2 configuration, the (t2g)4 configuration of which can be described by 1,Fe or 2,Fe shown below 1 y x2 y x1 3 y x1 1,Fe )e1 ,e1()e1 ,e1()a1()e1 ,e1()a1( 1 2 y x1 2 y x2 2,Fe )a1()e1 ,e1()a1()e1 ,e1()a1( The occupancy of the down-spin d-states for 1,Fe and 2,Fe are presented in Fig. 20a and 20b, respectively. An energy-low ering through SOC is strong for 1,Fe because it has an unevenly filled degenerate configuration 1 y x )e1 ,e1( , but not by 2,Fe because the latter has an evenly filled degenerate configuration 2 y x )e1 ,e1( . According to Table 3 , the down-spin configuration 1 y x )e1 ,e1( of 1,Fe is expressed as 1 31 1 2 2 32 1 y x )yz,xz( )y x,xy( )e1 ,e1( . (30) The orbital-unquenched state 1 2 2)yx,xy( leads to L = 2, but the state 1)yz,xz( to L = 1. The SOC constant < 0 for the 1,Fe configuration of Fe2+ (S = 2, d6) so that the ground doublet is J = L + S = 4 from the component 1 2 2)yx,xy( (L = 2), and J = 3 from 1)yz,xz( (L = 1). In terms of the notation {J z, Jz} representing a spin-orbit 54 coupled doublet set, the doublet 4,4 is more stable than 3,3 because < 0, so the 3 y x )e1 ,e1( configuration of Fe2+ is expressed as 1 2 1 y x2 y x3 y x23,34,4 )e1 ,e1()e1 ,e1()e1 ,e1( :eF With J = 3 for the singly-filled doublet, uniaxial magnetism is predicted for the high-spin Fe2+ ion at an octahedral site with | |z spin orientation. In suppor t of this analysis, DFT calculations show the orbital moment of the Fe2+ ion to be 1 B (i.e., L 1).43 Note that the 2,Fe configuration ( Fig. 20b ) leads to | Lz| = 1 and hence the preference for the z spin orientation. 7.3. Nondegenerate perturbation and weak magnetic anisotropy We now examine the preferred spin orientations of magnetic ion s with nondegenerate HOMO and LUMO. The layered compound SrFeO 2 consists of FeO 2 layers made up of corner-sharing FeO 4 square planes containing high-spin Fe2+ (d6, S = 2) ions.44 Corner-sharing FeO 4 square planes are also found in Sr 3Fe2O5, in which they form two-leg ladder chains.45 The d-states of a FeO 4 square plane are split as in Fig. 5c ,46,47 so that the down-spin d-states have only the 3z2r2 level filled, with the empty {xz , yz} set lying immediately above ( Fig. 9 ). Thus, between these HOMO and LUMO, with | Lz| = 1 so the preferred spin direction is z, i.e., parallel to the FeO 4 plane.46,47 A regular MnO 6 octahedron containing a high-spin Mn3+ (d4, S = 2) ion has JT instability and hence adopts an axially-elongated MnO 6 octahedron ( Fig. 5b ). Such JT- distorted MnO 6 octahedra are found in TbMnO 3 48 a n d A g 2MnO 2.49,50 The neutron diffraction studies show tha t the spins of the Mn3+ ions are aligned along the elongated 55 Mn-O bonds.48,50 With four unpaired electrons to fill the split d-states, the LUMO is the x2y2 and the HOMO is the 3z2r2 (Fig. 21 ). Between these two states, | Lz| = 2 so that they do not interact under SO C. The closest-lying filled d-state that can interact with the LUMO is the xy . Now, |Lz| = 0 between the x2y2 and xy states, the preferred spin orientation is ||z, i.e., p arallel to the elongated Mn-O b onds.50,51 T h e N i O 6 trigonal prisms containing Ni2+ (d8, S = 1) ions are found in the NiPtO 6 chains of Sr 3NiPtO 6,52 which is isostructural with Ca 3CoMnO 6. Each NiPtO 6 c h a i n consists of face-sharing NiO 6 trigonal prisms and PtO 6 octahedra. The Pt4+ ( d6, S = 0) ions are nonmagnetic. As depicted in Fig. 22 for the down-spin d-states of Ni2+ (d8, S = 1), |Lz| = 1 between the HOMO and LUMO. Consequently, the preferred sp in orientation of the Ni2+ (d8, S = 1) ion is z, i.e., perpendicular to the NiPtO 6 chain. This in agreement with DFT calculations.6 7.4. Magnetic anisotropy of S = 1/2 systems and spin-half misco nception In this section we examine the experimentally observed magneti c anisotropies of v ario u s S = 1 /2 io n s M. These observations are correctly reproduced by DFT+U+SOC calculations and also correctly e xplained by the SOC-induced HO MO-LUMO interactions of their ML n polyhedra. The experimental and theoretical evidence against the spin-half misconception is overwhelming to say the least. F i r s t , w e c o n s i d e r t h e m a g n e t i c i o n s w i t h S = 1 / 2 i n w h i c h t h e HOMO and LUMO of the crystal-field d-states are not degenerate. An axially-elongated IrO 6 octahedra containing low-spin Ir4+ (d5, S = 1/2) ions are found in the layered compound Sr2IrO 4, in which the corner-sharing of the IrO 6 octahedra using the equatorial oxygen 56 atoms forms the IrO 4 layers with the elongated Ir-O bonds perpendicular to the laye r.53-55 The neutron diffraction studies of Sr 2IrO 4 show that the Ir4+ spins are parallel to the IrO 4 layer.54,55 With the z-axis chosen along the elongated Ir-O bond, the t 2g level of the IrO 6 octahedron is split into {xz, yz} < xy. With five d-electrons to fill the three levels, the down-spin states xz and yz are filled while the xy state is empty, as depicted in Fig. 23a. Consequently, | Lz| = 1 between the HOMO and LUMO , so that the preferred spin orientation is z. This is in agreement with e xperiment and DFT calculations (S ee Section 8.1 for further discussion).6,56 N a 2IrO 3 consists of honeycomb layers made up of edge-sharing IrO 6 octahedra,57,58 which are substantially compressed along the direction perpend icular to the layer (lying in the ab-plane) , i.e., the c*-direction. Stri ctly speaking, each IrO 6 octahedron of Na 2IrO 3 has no 3-fold-rotational symmetry but has a pseudo 3-fold rotational axis along the c*-direction, whi ch we take as the lo cal z-axis. As for the preferred spin orientation of the Ir4+ ions of Na 2IrO 3, experimental studies have not been unequivocal, nor have been DFT studies, but it has become clear that the preferred spin orientation has components along the c*- and a-directions (name ly, ||z and z components).6,59,60 Due to the compression of the IrO 6 octahedron along this axis, its t 2g state is split into 1a < (1e x, 1e y), where 1e x and 1e y are approximately degenerate, so that the down-spin d-states would be occupied as depicted in Fig. 23b . For the Ir4+ ion of Na2IrO 3, therefore, the HOMO and LUMO o ccur from the down-spin electro n configuration close to (1a )1(1e x, 1e y)1, so the preferred spin orientation would be the ||z direction (namely, the c*-direction) because 0 L z . The electron configuration (1a)1(1e x, 1e y)1, deduced from an isolated IrO 6 octahedron, explains the c*-axis 57 component, but cannot explain the presence of the a-axis compon ent in the observed spin moment.59-61 The perturbation theory analysis requires the split d-states o f an IrO 6 octahedron present in Na 2IrO 3, not an isolated IrO 6 octahedron. The former have the effect of the intersite interactions, but the latter do not. An alysis of the intersite interaction showed 6 that they effectively reduce the energy split between 1a and (1e x, 1ey), so the (1a )0(1e x, 1e y)2 configuration also partici pates substantially in controlling the spin orientation thereby giving rise to the a-a xis component (See Section 8.1). C u C l 2·2H 2O is a molecular crystal made up of CuCl 2(OH 2)2 complexes containing Cu2+ (d9, S = 1/2) ions, in which the linear O-Cu-O unit is perpendicul ar to the linear Cl-Cu-Cl unit ( Fig. 24a ).62 The spins of the Cu2+ ions are aligned along the Cu-O direction,63 namely, the Cu2+ ions have easy-plane anisot ropy. The split down-spin d- states of CuCl 2·2H 2O show that the LUMO, x2y2 has the smallest energy gap with the HOMO, xz (Fig. 24b ).9 Since |Lz| = 1, the preferred spin orientation is z. To see if the spin prefers the x- or y-direction in the xy-plane, we use Eq. 27b. The matrix elements j iLˆ of the angular momentum operators )z,y,x(Lˆ are nonzero only for the following {i, j} sets (see Table 2 ):9 For zLˆ: {xz, yz}, {xy, x2y2} For xLˆ: {yz, 3z2r2}, {yz, x2y2}, {xz, xy} For yLˆ: {xz, 3z2r2}, {xz, x2y2}, {yz, xy} The only nonzero interac tion between the LUMO x2y2 and the HOMO xz under SOC i s t h e t e r m xzLˆy xy2 2 involving yLˆ. Eq. 27b shows that this term comes with angular dependency of sinsin , which is maximized when = 90 and = 90. Thus, 58 the preferred spin orientation of CuCl 2(OH 2)2 is along the y-direction, namely, along the Cu-O bonds.9 In CuCl 2,64,65 CuBr 2 66 and LiCuVO 4,24 the square planar CuL 4 units (L = Cl, Br, O) share their opposite edges to form CuL 2 ribbon chains ( Fig. 25a ). The split d-states in the CuL 2 ribbon chains of CuCl 2, CuBr 2 and LiCuVO 4 can be deduced by examining their projected density of states (PDOS) plots. Analyses of the s e p l o t s c a n b e b e s t described by the effective sequence of the down-spin d-states shown in Eq. 31a.9 ( 3 z2r2)1(xy)1(xz, yz)2(x2y2)0 for a CuL 4 of a CuL 2 ribbon chain (31a) ( 3 z2r2)1(xz, yz)2(xy)1(x2y2)0 for an isolated CuL 4 square plane (31b) Consequently, the interaction of the LUMO x2-y2 with the HOMO (xz , yz) will lead to the z spin orientation for the Cu2+ ions of the CuL 2 ribbon chains.9 This down-spin d- state sequence is different fro m the corresponding one expected f o r a n i s o l a t e d C u L 4 square plane (shown in Eq. 31b). This is due to the orbital int eractions between adjacent CuL 4 s q u a r e p l a n e s i n t h e C u L 2 ribbon chain, in particular , the direct metal-metal interactions involving the xy or bitals through the shared edges between adjacent CuL 4 square planes. Now we consider the magnetic ions with S = 1/2 whose HOMO and LUMO are degenerate. Sr 3NiIrO 6 67 is isostructural with Ca 3CoMnO 6, and its NiIrO 6 chains are made up of face-sharing IrO 6 octahedra and NiO 6 trigonal prisms. Each NiO 6 trigonal prism has a Ni2+ ( d8, S = 1) ion, and each IrO 6 octahedron a low-spin Ir4+ ( d5, S = 1/2) ion. Magnetic susceptibility and magnetization measurements68,69 indicate that Sr 3NiIrO 6 has uniaxial magnetism with the spins of both Ni2+ and Ir4+ ions aligned along the chain direction. Neutron diffraction measurements show that in each c hain the spins of adjacent 59 Ni2+ and Ir4+ ions are antiferromagnetically coupled.68 The low-spin Ir4+(d5, S = 1/2) ion has the configuration (t 2g)5, which can be represented by 1,Ir or 2,Ir 4 y x1 2,Ir3 y x2 1,Ir )e1 ,e1()a1()e1 ,e1()a1( The occupancies of the down-spin d-states for 1,Ir and 2,Ir are given as depicted in Fig. 26a and 26b, respectively. It is 1,Ir, not 2,Ir, that can lower energy strongly under SOC. The down-spin part 1 y x )e1 ,e1( of the configuration 3 y x )e1 ,e1( in 1,Ir can be rewritten as in Eq. 30 so that L = 2. For the low-spin Ir4+, < 0, because the t 2g-shell is more than half-filled.10 With S = 1/2, we have J = L + S = 5/2 from (xy, x2y2)3, and 3/2 from (xz, yz)3. Thus, the 3 y x )e1 ,e1( configuration of Ir4+ is expressed as 1 2 1 y x2 y x3 y x42/3,2/32/5,2/5 )e1 ,e1()e1 ,e1()e1 ,e1( :rI The singly-filled doublet has J = 3/2, so uniaxial magnetism is predicted with the spin orientation along the ||z direc tion. This explains why the S = 1/2 ion Ir4+ ion exhibits a strong magnetic anisotropy with t he preferred spin direction al ong the z-axis. In contrast to the case of Sr 3NiPtO 6, the Ni2+ ions of Sr 3NiIrO 6 have the ||z spin orientation. This is due to the combined effect of the uniaxial magnetism of the Ir4+ ions and the strong AFM spin exchange between adjacent Ir4+ and Ni2+ ions in each NiIrO 6 chain, which overrides the weak preference for the z spin orientation for the Ni2+ ion in an “isolated NiO 6” trigonal prism (See Section 8.1 for further discussions).5,6 Let us consider the spin orientation of the S = 1/2 ions V4+ ( d1) in the VO 6 octahedra of R 2V2O7 (R = rare earth),70 in which each VO 6 octahedron is axially compressed along the direction of its local three-fold rotation al axis ( Fig. 27a ) so that its 60 t2g state is split into the 1a < 1e pattern ( Fig. 27b ). With the local z-axis along the three- fold rotational axis of VO 6, the HOMO is the 1a state, which is represented by 3z2r2, which interacts with the LUMO 1e = (1e x, 1e y) states under SOC through their (xz , yz) components. Consequently, | Lz| = 1 and the preferred spin orientation would be z. However, the observed spin orientation is ||z,71 which has also been confirmed by DFT calculations.72 This finding is explained if the V4+ ion has some uniaxial magnetic character despite that the HOMO and LUMO are not degenerate. Fo r the latter to be true, the true ground state of each V4+ i o n i n R 2V2O7 should be a “contaminated state” 1a , which has some contributions of the 1e and 2e character of its isolated VO 6 octahedron, namely, e2 e1 a1 a1 where a n d are small mixing coefficients. This is possible because each V O6 octahedron present in R 2V2O7 h a s a l o w e r s y m m e t r y t h a n d o e s a n i s o l a t e d V O 6 octahedron. The VO 6 octahedra are corner-shared to form a tetrahedral cluster ( Fig. 27c ), and such tetrahedral clusters further share their corners to fo rm a pyroclore lattice ( Fig. 27d). Indeed, the PDOS plots for the up-spin d-states of the V4+ ions in R 2V2O7 show the presence of slight contributions of the 1e and 2e states to the occupied 1a state.5,72 As reviewed above, both experimental and theoretical studies r eveal that S = 1/2 ions do have magnetic anisotropy induced by SOC. The spin-half misconception is in clear contradiction to these e xperimental and theoretical obser vations. Due to the simplification it introduces for doing complex calculations, sp in Hamiltonian has been a practical tool of choice in doing physics on magnetism and will remain so for some time to come. Nevertheless, this success does not justify the perpet uation of the spin-half 61 misconception. This failure of a spin Hamiltonian should be con sidered as a small price to pay for the enormous gain it provides. 7.5. Ligand-controlled spin orientation For the CuBr 4 square planes of CuBr 2 ribbon chain,66 the CuBr 5 square pyramids of (C 5H12N)CuBr 3,73,74 and the CrI 6 octahedra of the layered compound CrI 3,75,76 t h e ligand L is heavier than M, so the SOC between two d-states of ML n results more from the SOC-induced interactions between the p-orbitals of the ligands L rather than from those between the d-orbitals of M. We clarify this point by c onsidering a square planar ML 4 using the coordinate system of Fig. 25a . The metal and ligand contributions in the yz, xy and x2y2 states of ML 4 are shown in Fig. 25b-d , respectively. The SOC-induced interaction between different d-states can occur by the SOC of M, and also by that of each ligand L. The interaction between the z and {x, y} orbitals at each L has |Lz| = 1, leading to the z spin orientation. In contrast, the interaction between the x and y orbitals at each L has | Lz| = 0, leading to the ||z spin orientation ( Table 2 ). When the ligand L is much heavier than the metal M, the SOC constant o f L is greater than that of M. Furthermore, such ligands L possess diffuse and high-lying p-orbitals, which makes the magnetic orbitals of ML n dominated by the ligand p-orbitals and also makes the d-states of ML n weakly split. This makes the SOC effect in ML n dominated by the ligands. 7.6. High-spin d5 systems High-spin d5 transition-metal ions with S = 5/2 possess a small nonzero orb ital momentum L and exhibit weakly preferred spin orientations. For such a mag netic ion, 62 the SOC-induced HOMO-LUMO inter action should be based on the SOHˆ term (Eq. 28), because the HOMO and LUMO occur from different spin states. The comparison of Eq. 27b with Eq. 28 reveals that the predictions concerning the ||z v s . z spin orientation from the term SOHˆ are exactly opposite to those from the term 0 SOHˆ. A similar situation occurs for a d3 magnetic ion at octahedral sites, as found for the Os5+ ions in Ca 2ScOsO 6 77 and the Ir6+ ions in Sr 2CuIrO 6,78 because such an ion has the (t 2g)3 configuration and because the t 2g states are well separated in energy from the e g states. Thus, the occupied up-spin t 2g states, t 2g, become the HOMO, and the unoccupied down-spin t 2g states, t 2g, the LUMO. It is known79 that the orbital momentum of such a cation can be discussed by usi ng the pseudo-orbital states zL,L with L = 1 and zL = 1, 0, –1. To a first approximation, therefore, the orbital mome ntum of such a d3 magnetic ion is zero. However, the quenching of the orbital momentum is not complete so that a (t2g)3 ion has a small nonzero orbital momentum L . Thus the preferred spin orientation of (t 2g)3 ions is governed by the SOC-i nduced HOMO-LUMO interaction base d on the SOHˆ term (Eq. 28).80 8. Magnetic properties of 5d ion oxides6 T h e d orbitals of 5d ions are more diffuse than those of 3d ions, so that electron correlation is much weaker for 5d ions than for 3d ions. For a given MOn polyhedron, the M 3d and O 2 p orbitals do not differ strongly in their contractedness so tha t the associated crystal-field splitting of an isolated MOn polyhedron is strong. However, the M 5d orbitals are much more diffuse than O 2 p o r b i t a l s s o t h a t t h e 5 d-state splitting of an 63 isolated MOn polyhedron is weak. In addition, the interactions between adja cent metal ions M through the M-O-M bridges are stronger for 5d ions than for 3d ions. Thus, for 5 d ion oxides, the relative ordering of their split d-states deduced from an isolated MOn polyhedron might change by the interactions between adjacent me tal ions (i.e., the intersite interactions). Furthermore, each of the crystal-field s p l i t d-states can be split further by SOC,81 and this effect is much stronger for 5d ion oxides than for 3d ion oxides because the strength of SOC is much stronger for 5d ions than f or 3d ions. The weak electron correlation and strong SOC in 5d ion oxides have impor tant consequences, as discussed below. 8.1. Spin-orbit Mott insulating s tate and Madelung potential The combination of strong SOC and weak electr on correlation cr eates a magnetic insulating state, as first reported for Ba 2NaOsO 6 containing Os7+ ( d1) ions.81 T h i s phenomenon, quite common in 5d ion oxides, was considered as a consequence of strong spin-orbital entanglement,82 and the resulting magnetic insulating state is described as a SOC-induced Mott insulating state83 or spin-orbit Mott insulating state.84 Both Sr 3NiIrO 6 and Sr 2IrO 4 are magnetic insulators, namely, they have a band gap at all t emperature.85-90 Na2IrO 3 has been thought to be a magnetic insulator,91,92 but a recent DFT study suggested that it might be a Slater insulator.93 T h e l a t t e r r e f e r s t o a s y s t e m w i t h a partially-filled bands and weak e lectron correlation that opens a band gap when it undergoes a metal-insulator tran sition at a temperature below w hich an AFM ordering sets in.94 In addition to the local factor s affecting electron localizati on such as the oxidation state and the SOC constant of a metal ion M, the extent of electron 64 localization is influenced by th e Madelung potential acting at the M, which is a non-local factor.6 The Madelung potentials acting on the Ir4+ sites less negative (i.e., less attractive) for of Na 2IrO 3 than for Sr 3NiIrO 6 and Sr 2IrO 4, namely, the 5d electrons of an Ir4+ i o n would be less strongly bound (i.e ., less strongly localized) to the ion.6 8.2. Influence of intersite intera ctions on crystal field-split d-states6 In predicting the preferred spin orientations of magnetic ions M in magnetic oxides on the basis of the SOC-i nduced HOMO-LUMO interactions, the split d-states of their local MOn polyhedra are needed. As pointed out above, for oxides of 5d i ons, the relative ordering of their split d-states deduced from an isolated MOn polyhedron might change by the intersite interacti on. In the following we examin e how the intersite interactions affect the split 5 d-states of the Ir4+ ions in Sr 3NiIrO 6, Sr 2IrO 4 and Na 2IrO 3 and explore their consequences. The ESR study of Sr 2IrO 4 showed 95 that the g-factors of the Ir4+ ion along the ||c and c directions are explained if the t 2g-states are split as xy < (xz, yz) rather than as (xz, yz) < xy (discussed in Section 7.4). This finding, puzzling fro m the viewpoints of the split t 2g s t a t e s o f a n i s o l a t e d I r O 6 octahedron, reflects6 that the split d-state patterns of Sr2IrO 4 differ from those of an isolated IrO 6 octahedron due to the intersite interactions. In each IrO 4 layer of Sr 2IrO 4 the Ir-O-Ir linkages in the ab- plane are bent as shown in Fig. 28a. This bending of the Ir-O-Ir linkages does not weaken the -antibonding interactions between adjacent xz/yz orbitals ( Fig. 28b ), but does weaken those between adjacent xy orbitals ( Fig. 28c ). Namely, the -type interactions between a djacent xz/yz orbitals are stronger than those between adj acent xy orbitals. The split d-states of an IrO 6 octahedron 65 embedded in Sr 2IrO 4 and hence having the intersite interactions can be approximate d by those of a dimer made up of two adjacent corner-sharing IrO 6 octahedra. Then, the interactions between two adjacent Ir4+ sites alter the crystal-field split t 2g states as depicted in Fig. 28d , so that the HOMO has the xy character, and the LUMO the xz/yz character. This picture e x p l a i n s t h e P D O S p l o t s o f S r 2IrO 4 shown in Fig. 28e , and predicts the c spin orientation as does the crystal-field split t 2g states of an isolated IrO 6 octahedron ( Fig. 23a ). In addition, this explains why the ESR results95 o f S r 2IrO 4 a r e explained by the d-state ordering xy < (xz, yz), despite that it consists of axia lly- elongated IrO 6 octahedra. I n N a 2IrO 3, edge-sharing IrO 6 octahedra form honeycomb layers ( Fig. 29a ), and such layers are stacked a long the c-direction ( Fig. 29b ). DFT+U+SOC calculations reveal that the preferred spin orientation of the Ir4+ ions in Na 2IrO 3 h a s b o t h | | c * a n d | | a components.6,96 To examine the cause for this observation, we consider how the intersite interaction affects relative ord ering of the down-spin 1a and 1 e states of an Ir4+ ion ( Fig. 23b). Consider a dimer made up of two adjacent Ir4+ ions and recall that the d-orbital component of the 1a state is the 3z2-r2 orbital, while those of the 1e state are the (xy, x2-y2) and (xz, yz) orbitals ( Table 3 ). As depicted in Fig. 29c , the intersite interaction between the two 1a states leads to the 1a + and 1a - states, and that between the 1e states to the 1e + and 1e - states. The split between 1a + and 1a - states is weak because the lateral extension of the 3z2-r2 orbitals within the plane of the honeycomb layer is small. In contrast, the split between the 1e + and 1e - states is large because the lateral extension of the (xy, x2-y2) orbitals is large and because so is that of the (xz, yz) orbita ls. With four down-spin electrons in the dimer, the 1e - states are empty while the rem aining states are filled. The 66 |Lz| = 1 interactions between the 1a +/1a- and 1e - states predict the z spin orientation. The interactions between the 1e + and 1e - s t a t e s g i v e r i s e t o t h e | Lz| = 0 interactions, between their (xz, yz) sets and between their (xy, x2-y2) sets, predicting the ||z spin orientation. Consequently, if the 1a + and 1a - states are close in energy to the 1e + states, then the preferred spin orientation of the Ir4+ ion would be the ( z + ||z) direction. In essence, the ||a component of the Ir4+ spin orientation in Na 2IrO 3 is a consequence of the intersite interactions, because only the ||c* direction is pred icted in their absence. The magnetic insulating state of Sr 3NiIrO 6 is reproduced by DFT+U+SOC calculations only when adjacent Ni2+ and Ir4+ spins have an AFM coupling in each NiIrO 6 chain.6,97,98 It is known experimentally85,86 that the preferred orientation of the Ir4+ spins is the ||c-direction. DFT+U+SOC calculations showed that the pr eferred orientation of the Ir4+ spins is the ||c-direction if the Ni2+ and Ir4+ spins have an AFM coupling,6 but it is the c-direction if they have an FM coupling.6,99 In each NiIrO 6 chain the nearest-neighbor Ir…Ni distance is short due to the face-sharing between the IrO 6 and NiO 6 polyhedra so that the overlap between the Ir and Ni 3z2-r2 orbitals can be strong. As illustrated in Fig. 30a and 30b, the Ni 3z2-r2 orbital is closer in energy to the Ir 3z2-r2 orbital when adjacent Ni2+ and Ir4+ spins have an FM coupling than when they have an AFM coupling (see Section 2.2.2 and Fig. 8 ). The latter makes the intera ction between the Ir and Ni 3z2-r2 states stronger for the FM than for the AFM spin arrangement.2,10,39b As a consequence, the resulting antibonding state (3z2-r2) is unoccupied for the FM spin arrangement, but it is occupied for the AFM spin arrangement ( Fig. 30a and 30b), as found by DFT+U calculations for Sr 3NiIrO 6;6 the PDOS plots for the FM and AFM arrangements, presented in Fig. 30c and 30d, respectively, reveal that the AFM arrangement is 67 consistent with the local e lectron configuration (1a )1(1e x, 1e y)1 (Fig. 26a ), predicting the ||z spin orientation, while the FM arrangement is consisten t with the local configuration (1a )0(1e x, 1e y)1 (Fig. 26b ), predicting the z spin orientation. 8.3. Perturbation theory analysis of preferred spin orientation6 The energy stabilization E associated with the SOC-i nduced interaction between the HOMO and the LUMO (with energies e HO and e LU, respectively) is given by Eq. 29b. For the Ir4+ (low-spin d5) ion systems Sr 3NiIrO 6, Sr 2IrO 4 and Na 2IrO 3, the overall widths of the t 2g-block bandwidths are of the order of 2 eV (i.e., 1.7, 2.6 and 2.4 eV, respectively from our DFT+U calculations) an d the HOMO-LUMO energy differenc es LU HOe e values are of the order of 0.2 eV (0.2, 0.2 and 0.3 eV, respect ively.6 The SOC constant of Ir4+ is of the order of 0.5 eV 8e so that 2 is comparable in magnitude to LU HOe e for the case of e HO < e LU. In such a case, use of perturbation theory does not lead to a n accurate estimation of E. However, this does not affect our qualitative predictions of the preferred spin orientations, because the latter do not require a quantitative evaluation of E. 8.4. LS vs jj coupling scheme of SOC6 The effects of SOC are discussed in terms of either the LS or the jj coupling scheme depending on the strength of SOC. In the LS (or Russel-S aunders) scheme the electron spin momenta are summed up to find the total spin mome ntum i S s , and the orbital momenta of individual el ectrons to find the total orbit al momentum i L l . 68 Then, the SOC is included to couple S and L to obtain the total angular momentum J , leading to the SOC Hamiltonian, LS Hˆ SO . The LS-coupling scheme is typically employed for elements with weak SOC (e.g., 3d- and 4d-elements) . In this scheme the crystal-field split d-states of a MOn polyhedron are closely related to the orbital states zL,L of M in the up-spin or down-spin state magnetic orbitals of MOn. As found for Sr 3NiIrO 6, Sr 2IrO 4 and Na 2IrO 3 6 and for Ba 2NaOsO 6,81 our analyses based on the LS-coupling scheme explain th e spin-orbit Mott insulating s tates of these 5d oxides as well as their observed magnetic anisotropies. T h e j j -coupling scheme, appropriate for elements with strong SOC (e.g. , 4f and 5f elements), has recently become popular in discussing the spin-o rbit Mott insulating states of 5d oxides.82 In this scheme, the spin and orbital momenta are added to obta in the total angular momentum i i i slj for each electron of a magnetic ion M, and the ij ’s of the individual electrons are added to find the total angular momentum, i J j , of M. In this approach, it is not readily obvious how to relate the J states to the crystal-field split d-states of MOn unless the corresponding analysis is done by using the LS-coup ling scheme, because the crystal-field split d-states of MOn are determined by the interactions of the orbital states zL,L of M with the 2p orbitals of the surrounding O ligands and because the information about the orbital states zL,L of M is completely hidden in the jj-coupling scheme. As a consequence, use of the jj scheme make s it difficult to predict such fundamental magnetic properties as the preferred spin orie ntation and the uniaxial magnetism of a magnetic ion M. The latter are readily predicted by the LS coupling scheme. As found for the Ir4+ ion of Sr 3NiIrO 6, the need to employ “J-states” in the LS 69 scheme arises only when a magne tic ion has an unevenly-filled d egenerate d-state, leading to an unquenched orbital momentum L that combines with S to form LSJ . In the LS scheme, use of J-states is inappropriate for Sr 2IrO 4 and Na 2IrO 3 because they possess no unquenched orbital momentum L to combine with S . Studies on Sr 3NiIrO 6, Sr 2IrO 4 a n d N a 2IrO 3 6 and on Ba 2NaOsO 6 81 strongly suggest that the magnetic properties of the 5d oxides are bette r explained by the LS scheme than by the jj scheme. The latter implies that the spin- orbital entanglement in 5d elements is not as strong as has been assumed.82 These conclusions are consistent with the view that SOC for 5d element s lies in between the LS- and j j-coupling schemes, but is closer to the LS scheme.100 9. Concluding remarks In this chapter we have reviewed how to think about magnetic pr operties of solid state materials from the perspectives of an electronic Hamilton ian. On the quantitative level, use of this Hamiltonian enables one (a) to determine the relative stabilities of various spin arran gements on the basis of DFT+U or DFT+U+SOC calculations, (b) to evaluate the spin exchange and DM exchange parameters th at a spin Hamiltonian requires by performing energy-mapping analysis based on DFT+U o r DFT+U+SOC calculations, and (c) to characterize the magnetic anisotropy of a magnetic ion b y performing DFT+U+SOC calculations. On the qualitative level, use of an electronic Hamiltonian allo ws one 70 (a) to examine spin lattices in terms of M-L-M as well as M-L…L -M spin exchanges, (b) to discuss how the strengths of M-L…L-M spin exchanges are modified by through- space and through-bond interactions, and (c) to predict/rationalize the preferred spin orientation of a magnetic ion on the basis of its SOC-induced HOMO-LUMO interactions. The qualitative concepts governin g these structure-property cor relations help one organize/think about known experimental/theoretical observation s, design new experiments to do and new calcula tions to perform, and predict/ rationalize the outcomes of the new studies. In the past, a spin lattice required for spin Hamiltonian anal ysis used to be chosen by inspecting the pattern of magnetic ion arrangement and emplo ying the Goodenough rules, 22 which cover only M-L-M spin exchanges. Use of Goodenough rules often led to spin lattices that are inconsistent with the electronic structu res of the magnetic systems they are supposed to describe, to find that Goodenough rules ar e not adequate enough. The reason for this observation is that M-L-M spin exchanges are frequently much weaker than those spin exchanges not covered by Goodenough rule s, namely, M-L…L-M and/or M-L…Ay+…L-M spin exchanges. This is understandable, because Goodenough rules were formulated in the mid 1950’s, when the magnetic orbi tals of M i o n s w e r e regarded as their singly-occupied pure d-orbitals of M. The importance of M-L…L-M and/or M-L…Ay+…L-M spin exchanges were recognized only in the late 1990’s and the early 2000’s, when it was realized1,2 that the strengths of spin exchanges are not governed by the metal d-orbital components, but by the ligand p-orbital components, of the magnetic orbitals of ML n. Quantitative evaluations of M-L-M , M-L…L-M and M-71 L…Ay+…L-M spin exchanges became possibl e by the energy-mapping analysis1-4 based on DFT+U calculations developed in the early 2000’s. This quant itative analysis helps one find, for any magnetic system , the spin lattice consistent with its electronic structure. The spin-orbit Mott insulati ng states of the 5d oxides Sr 3NiIrO 6, Sr 2IrO 4 and Na2IrO 3 as well as Ba 2NaOsO 6 are well explained by analyses based on the LS-coupling scheme of SOC. Furthermore, their observed magnetic anisotropie s are better explained by the LS scheme rather than by the jj scheme. Consequently, th e spin-orbital entanglement invoked for 5d elements is not as strong as has be en put forward.82 These observations are in support of the view that SOC for 5d element s lies in between the LS- and jj -coupling schemes, but is closer to the LS-coupling scheme.100 A magnetic ion has a preferred spin orientation because SOC ind uces interactions among its crystal-field split d-states and because the associ ated energy lowering depends on the spin orientation. The pre ferred spin orientation of a ma gnetic ion is readily predicted on the basis of the se lection rule involving the SOC- induced HOMO-LUMO interaction. In the electronic st ructure description of a magne tic ion, each of its states has both orbital and spin components, that is, each state is repres ented by a set of orbital/spin states z zS,SL,L . The states of a magnetic ion are modified by SOC, LˆSˆ , because it induces intermixing between them, but this intermixing takes pl ace only in the orbital component zL,L of each state. This explains why a magnetic ion has magnetic anisotropy regardless of whether its spin is 1/2 or not. A spin Hamiltonian analysis fails to explain this fundamental result because it represents each m agnetic state in terms of only spin states zS,S . The effects of SOC, LˆSˆ , can be included into a spin Hamiltonian only indirectly by using the zero-field Hamiltonian zfHˆ (Eq. 16). This 72 Hamiltonian does not allow one to predict the preferred spin or ientation for S > 1/2 ions, although it shows the presence of m agnetic anisotropy arising f rom SOC for such ions in agreement with experiment. As for the S = 1/2 ions, however, th is Hamiltonian is downright incorrect because it predicts the absence of magnetic anisotropy induced by SOC, LˆSˆ , not to mention that it cannot p redict their preferred spin or ientation. It is high time for the proponents of the spin-half misconcepti on to recognize this shortcoming of a spin Hamiltonian analysis. Nevertheless, we ar e not unaware of the astute observation by Max Planck : “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rat her because its opponents eventually die and a new genera tion grows up that is familiar w ith it.”101 This observation is more explicitly paraphrased a s “Death is an essential elemen t in the progress of science, since it takes care of conservati ve scientists of a previous ge neration reluctant to let go of an old, fallacious theory and embrace a new and accurate one.”102 The debate on the spin- half misconception, which has just begun,5,6,9 is certainly not as gran d and epochal as that on the earth- vs. sun-centered model of the planetary motion, t he single- vs. multi-galaxy universe, or the classical vs. quantum theory in the past, but unmistakable parallels exist between them. It is our hope that the readers of this chapter w ill have an open-minded view on magnetism and avoid falling into such a conceptual trap as the spin-half misconception. Acknowledgments This research used resources of the National Energy Research S cientific Computing Center, a DOE Office of Science User Facility support ed by the Office of 73 Science of the U.S. Department of Energy under Contract No. DE- AC02-05CH11231. Work at Fudan was supported by NSFC (11374056), the Special Fun ds for Major State Basic Research (2012CB921400, 2015 CB921700), Program for Profes sor of Special Appointment (Eastern Scholar), Qi ng Nian Bo Jian Program, and F ok Ying Tung Education Foundation. MHW would lik e to thank Dr. Reinhard K. K remer, Prof. Hyun- Joo Koo, Dr. Changhoon Lee and Elijah E. Gordon for their inval uable discussions over the years. References 1. M.-H. Whangbo, H.-J. Koo and D. Dai, J. Solid State Chem . 176, 417 (2003). 2. H. J. Xiang, C. Lee, H.-J. Koo , X. G. Gong and M.-H. Whangbo , Dalton Trans . 42, 823 (2013). 3. (a) D. Dai and M.-H. Whangbo, J. Chem. Phys. 114, 2887 (2001). (b) D. Dai and M.-H. Whangbo, J. Chem. Phys. 118, 29 (2003). 4. H. J. Xiang, E. J. Kan, S.-H. W ei, M.-H. Whangbo and X. G. G ong, Phys. Rev. B 84, 224429 (2011). 5. M.-H. Whangbo, E. E. Gordon, H. J. Xiang, H.-J. Koo and C. L ee, Whangbo, Acc. Chem. Res. 48, 3080 (2015). 6. E. E. Gordon, H. J. Xiang, J. Köhler and M.-H. Whangbo, J. Chem. Phys. 144, 114706 (2016). 7. T. Moriya and K. Yoshida, Prog. Theoret. Phys . 9, 663 (1953). 8. (a) K. Yoshida, “Theory of Magnetism”, Springer, New York, 1 996, pp. 34 37. (b) R. Skomski, “Simple Models of Magnetism”, Oxford University Pre ss, New York, 74 2008, pp. 84 104. (c) R. M. White, “Quantum Theory of Magnetism, Third Ed.”, Springer, Berlin, 2007, pp. 211 212. (d) N. Majlis, “The Quantum Theory of Magnetism, Second Ed.”, World Scientific, New Jersey, 2007, pp. 2324. (e) D. I. Khomskii, “Transition Metal Compounds”, Cambridge University Pr ess, Cambridge, 2014, Section 5.3.1. 9. J. Liu, H.-J. Koo, H. J. Xiang , R. K. Kremer and M.-H. Whang bo, J. Chem. Phys . 141, 124113 (2014). 10. D. Dai, H. J. Xiang and M.-H. Whangbo, J. Comput. Chem . 29, 2187 (2008). 11. T. A. Albright, J. K. Burdett a nd M.-H. Whangbo, “Orbital I nteractions in Chemistry, 2nd Edition”, Wiley, New York, 2013. 12. L. E. Orgel, “An Introduction to Transition Metal Chemistry ”, Wiley, New York, 1969, p 174. 13. (a) S. L. Dudarev, G. A. Bott on, S. Y. Savrasov, C. J. Hump hreys and A. P. Sutton, Phys. Rev. B 57, 1505 (1998). (b) A. I. Liechtens tein, V. I. Anisimov and J. Z aanen, Phys. Rev. B 52, 5467 (1995). 14. J. Heyd, G. E. Scuseri a and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003). 15. J. Kuneš, P. Novák, R. Schmi d, P. Blaha and K. Schwarz, Phys. Rev. B 64, 153102 (2001). 16. (a) G. Kresse and J . Hafner, Phys. Rev. B 47, 558 (1993); (b) G. Kresse and J. Furthmüller, J. Comput. Mater. Sci. 6, 15 (1996); (c) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). 17. P. Blaha, K. Schwarz, G. K. H. M adsen, D. Kvasnicka and J. Luitz, WIEN2k (Vienna University of Technol ogy, 2001) (ISBN 3-9501031-1-2). 75 18. J. P. Perdew, K. Bur ke and M. Ernzerhof, Phys. Rev. Lett ., 1996, 77, 3865-3868. 19. P. J. Hay, J. C. Thibeault and R. Hoffmann, J. Am. Chem. Soc. 97, 4884 (1975). 20. L. Noodleman, J. Chem. Phys . 74, 5737 (1981). 21. L. Noodleman and E. R. Davidson, Chem. Phys . 109, 131 (1986). 22. J. B. Goodenough, Magnetism and the Chemical Bond , Interscience-Wiley, New York, 1963. 23. P. W. Anderson, Phys. Rev . 79, 350 (1950). 24. B. J. Gibson, R. K. Kremer, A . V. Prokofiev, W. Assmus and G. J. McIntyre, Physica B 350, e253 (2004). 25. D. Dai, H.-J. Koo and M.-H. Whangbo, Inorg. Chem . 43, 4026 (2004). 26. H.-J. Koo, C. Lee, M.-H. Whangbo, G. J. McIntyre and R. K. Kremer, Inorg. Chem ., 50, 3582 (2011). 27. H.-J. Koo and M.-H. Whangbo, Inorg. Chem . 45, 4440 (2006). 28. C. Lee, J. Kang, K. H. Lee and M.-H. Whangbo, Inorg. Chem . 48, 4185 (2009). 29. M. H. L. Pryce, Proc. Phys. Soc (London) A 63, 25 (1950). 30. I. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958). 31. T. Moriya, Phys. Rev. Lett . 4, 228 (1960). 32. H. A. Kramers, Proc. Amsterdam Acad . 33, 959 (1930). 33. Z. L. Li, J. H. Yang, G. H. C hen, M.-H. Whangbo, H. J. Xian g and X. G. Gong, Phys. Rev. B , 85, 054426 (2012). 34. X. Z. Lu, M.-H. Whangbo, S. A . Dong, X. G. Gong and H. J. X iang, Phys. Rev. Lett., 108, 187204 (2012). 76 35. J. H. Yang , Z. L. Li , X. Z. Lu , M.-H. Whangbo , S.-H. Wei , X. G. Gong , H. J. Xiang , Phys. Rev. Lett . 109, 107203 (2012). 36. D. Dai and M.-H. Whangbo, Inorg. Chem ., 44, 4407 (2005). 37. X. Wang, R. Wu, D.-S. Wang and A. J. Freeman, Phys. Rev. B 54, 61 (1996). 38. H. H. Kim, I. H. Yu, H . S. Kim, H.-J. Koo and M.-H. Whangbo , Inorg. Chem . 54, 4966 (2015). 39. (a) Y. Zhang, H. J. Xiang and M.-H. Whangbo, Phys. Rev. B 79, 054432 (2009). (b) Y. Zhang, E. J. Kan, H. J. Xiang, A. Villesuzanne, M.-H. Whangb o, Inorg. Chem . 50, 1758 (2011). 40. (a) V. G. Zubkov, G. V. Bazuev, A. P. Tyutyunnik and I. F. Berger, J. Solid State Chem . 160, 293 (2001). (b) H. Wu, T. Burnus, Z. Hu, C. Martin, A. Maigna n, J. C. Cezar, A. Tanaka, N. B. Brookes, D. I. Khomskii and L. H. Tjeng , Phys. Rev. Lett . 102, 026404 (2009). 41. W. M. Reiff, A. M. LaPointe and E. H.Witten, J. Am. Chem. Soc . 126, 10206 (2004). 42. H. Kabbour, R. David, A. Pautr at, H.-J. Koo, M.-H. Whangbo, G. André and O. Mentré, Angew. Chem., Int. Ed . 51, 11745 (2012). 43. Y.-J. Song, K.-W. Lee and W. E. Pickett, Phys. Rev. B 92, 125109 (2015). 44. Y. Sujimoto, C. Tassel, N. H ayashi, T. Watanabe, H. Kageyam a, K. Yoshimura, M. Takano, M. Ceretti, C. Ritter and W. Paulus, Nature , 450, 1062 (2007). 45. H. Kageyama, T. Watanabe, Y. Tsujimoto, A. Kitada, Y. Sumid a, K. Kanamori, K. Yoshimura, N. Hayashi, S. Muranaka, M. Takano, M. Ceretti, W. P aulus, C. Ritter and A. Gilles, Angew. Chem. Int. Ed . 47, 5740 (2008). 77 46. H. J. Xiang, S.-H. Wei and M.-H. Whangbo, Phys. Rev. Lett . 100, 167207 (2008). 47. H.-J. Koo, H. J. Xiang, C. Lee and M.-H. Whangbo, Inorg. Chem . 48, 9051 (2009). 48. J. Blasco, C. Ritter, J. Garcia, J. M. de Teresa, J. Perez- Cacho and M. R. Ibarra, Phys. Rev. B 62, 5609 (2000). 49. F. M. Chang and M. Jansen, Anorg. Allg. Chem . 507, 59 (1983). 50. S. Ji, E. J. Kan, M.-H. Whangbo , J.-H. Kim, Y. Qiu, M. Mats uda, H. Yoshida, Z. Hiroi, M. A. Green, T. Ziman and S.-H. Lee, Phys. Rev. B 81, 094421 (2010). 51. H. J. Xiang, S.-H. Wei, M.-H. Whangbo and J. L. F. Da Silva , Phys. Rev. Lett . 101, 037209 (2008). 52. J. B. Claridge, R. C. Layland, W. H. Henley and H.-C. zur L oye, Chem. Mater . 11, 1376 (1999). 53. T. Shimora, A. Inaguma, T. Nak amura, M. Itoh and Y. Morii, Phys. Rev. B 52, 9143 (1995). 54. F. Ye, S. X. Chi, B. C. Cha koumakos, J. A. Fernandez-Baca, T. F. Qi and G. Cao, G. Phys. Rev. B 87, 140406 (R) (2013). 55. S. W. Lovesey and D. D. Khalyavin, J. Phys.: Condens. Matter 26, 322201 (2014). 56. P. Liu, S. Khmelevskyi, B. Kim, M. Marsman, D. Li, X.-Q. Ch en, D. D. Sarma, G. Kresse and C. Franchini, Phys. Rev. B 92, 054428 (2015). 57. S. K. Choi, R. Coldea, A. N. K olmogorov, T. Lancaster, I. I . Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P. Geg enwart, K. R. Choi, S.-W. Cheon g, P. J. Baker, C. Stock and J. Taylor, Phys. Rev. Lett . 108, 127204 (2012). 58. F. Ye, S. Chi, H. C ao, B. C. Chakoumakos, J. A. Fernandez-B aca, R. Custelcean, T. F. Qi, O. B. Korneta and G. C ao, Phys. Rev. B 85, 180403(R) (20 12). 78 59. H.-J. Kim, J.-H . Lee and J.-H. Cho, Sci. Rep . 4, 5253 (2014). 60. S. H. Chun, J.-W. Kim, J. Kim , H. Zheng, C. C. Stoumpos, C. D. Malliakas, J. F. Mitchell, K. Mehlawat , Y. Singh, Y. Choi , T. Gog, A. Al-Zein, M . M. Sala, M. Krisch, J. Chaloupka, G. Jackeli , G. Khaliullin and B. J. Kim, Nat. Phys . 11, 3322 (2015). 61. K. Hu, F. Wang, and J. Feng , Phys. Rev. Lett . 115, 167204 (2015). 62. S. Brownstein, N. F. Han, E . J. Gabe and Y. le Page, Kristallogr . 189, 13 (1989). 63. N. J. Poulis and G. E. G. Haderman, Physica 18, 201 (1952). 64. M. G. Banks, R. K. Kremer, C. Hoch, A. Simon, B. Ouladdiaf, J.-M. Broto, H. Rakoto, C. Lee and M.-H. Whangbo, Phys. Rev. B 80, 024404 (2009). 65. L. Zhao, T.-L. Hung, C.-C. Li, Y.-Y. Chen, M.-K. Wu, R. K. Kremer, M. G. Banks, A. Simon, M.-H. Whangbo, C. Lee, J. S. Kim, I. G. Kim and K. H. K i m , Adv. Mater . 24, 2469 (2012). 66. H.-J. Koo, C. Lee, M.-H. Whangbo, G. J. McIntyre and R. K. Kremer, Inorg. Chem . 50, 3582 (2011). 67. T. N. Nguyen and H.-C. zur Loye, J. Solid State Chem . 117, 300 (1995). 68. E. Lefrançois, L. C. Chapon, V. Simonet, P. Lejay, D. Khaly avin, S. Rayaprol, E. V. Sampathkumaran, R. Ballou and D. T. Adroja, Phys. Rev. B 90, 014408 (2014). 69. J. W. Kim and S.-W. Cheong, unpublished results (private co mmunication). 70. A. A. Haghighirad, F. Ritter and W. Assmus, Crystal Growth and Design , 8, 1961 (2008). 71. G. T. Knoke, A. Niazi, J. M . Hill and D. C. Johnston, Phys. Rev. B 76, 054439 (2007). 79 72. H. J. Xiang, E. J. Kan, M.-H. W hangbo, C. Lee, S.-H. Wei an d X. G. Gong, Phys. Rev. B 83, 174402 (2011). 73. B. Pan, Y. Wang, L. Zhang and S. Li, Inorg. Chem. 53, 3606 (2014). 74. C. Lee, J. Hong, W. J. Son, E. J. Kan, J. H. Shim and M.-H. Whangbo, RSC Adv . 6, 22722 (2016). 75. M. A. McGuire, H. Dixit, V . R. Cooper, B. C. Sales, Chem. Mater . 27, 612 (2015). 76. H.-J. Koo, J. Köhler and M.-H . Whangbo, to be published. 77. S. Vasala, H. Yamauch i and M. Karppinen, J. Solid State Chem . 220, 28 (2014). 78. D. D. Russell, A. J. Neer, B. C . Melot and S. Derakhshan, Inorg. Chem . 55, 2240 (2016). 79. M. E. Lines, Phys. Rev . 131, 546 (1963). 80. C. Lee, H.-J. Koo and M.-H. Whangbo, to be published. 81. H. J. Xiang and M.-H. Whangbo, Phys. Rev. B , 75, 052407 (2007). 82. (a) B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C . S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J . Oh, J.-H. Park, V. Durairaj, G. Cao and E. Rotenberg, Phys. Rev. Lett. 101, 076402 (2008). (b) B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi and T. Arima, Science , 323, 1329 (2009). 83. G. R. Zhang, X. L. Zhang, T. Jia, Z. Zeng and H. Q. Lin, J. Appl. Phys . 107, 09E120 (2010). 84. Y.-J. Song, K.-W. Lee and W. E. Pickett, Phys. Rev. B , 92, 125109 (2015). 85. E. Lefrançois, L. C. Chapon, V. Simonet, P. Lejay, D. Khaly avin, S. Rayaprol, E. V. Sampathkumaran, R. Ballou and D. T. Adroja, Phys. Rev. B 90, 014408 (2014). 80 86. W. Wu , D. T. Adroja , S. Toth , S. Rayaprol and E. V. Sampathkumaran , arXiv: 1501.05735 [cond-mat.str-el]. 87. J. W. Kim and S.-W. Cheong, unpublished work (2015). 88. F. Ye, S. X. Chi, B. C. Chakoumakos, J. A. Fernandez-Baca, T. F. Qi and G. Cao, Phys. Rev. B 87, 140406 (R) (2013). 89. S. W. Lovesey and D. D. Khalyavin, J. Phys.: Condens. Matter 26 322201 (2014). 90. P. Liu, S. Khmelevskyi, B. Kim, M. Marsman, D. Li, X.-Q. Ch en, D. D. Sarma, G. Kresse and C. Franchini, Phys. Rev. B 92, 054428 (2015). 91. R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veenstra, J. A. Rosen, Yogesh Singh, P. Gegenwart, D. Stricker, J. N. Hancock, D. van der Mar el, I. S. Elfimov, and A. Damascelli, Phys. Rev. Lett . 109, 266406 (2012). 92. H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y. Singh, S. Manni, P. Gegenwart, J. Kim, A. H. Said, D. Casa, T. Gog, M. H. Upton, H. -S. Kim, J. Yu, V. M. Katukuri, L. Hozoi, J. van den Brink, and Y.-J. Kim, Phys. Rev. Lett . 110, 076402 (2013). 93. H.-J. Kim, J.-H . Lee and J.-H. Cho, Sci. Rep . 4, 5253 (2014). 94. J. C. Slater, Phys. Rev . 82, 538 (1951). 95. N. A. Bogdanov, V. M. Katukuri , J. Romhányi, V. Yushankhai, V. Kataev, B. Büchner, J. van den Brink and L. Hozoi, Nat. Commun . 6, 7306 (2015). 96. K. Hu, F. Wang, and J. Feng , Phys. Rev. Lett . 115, 167204 (2015). 97. G. R. Zhang, X. L. Zhang, T. Jia, Z. Zeng and H. Q. Lin, J. Appl. Phys . 107, 09E120 (2010). 98. X. D. Ou and H. Wu, Sci. Rep . 4, 4609 (2014). 81 99. S. Sarkar, S. Kanungo and T. Saha-Dasgupta, Phys. Rev. B 82, 235122 (2010). 100. D. I. Khomskii, “Transition Metal Compounds”, Cambridge Un iversity Press, Cambridge, 2014, p. 34. 101. M. K. Planck, “Scientific A utobiography and Other Papers”, Philosophical Library, New York, 1950, p. 33. 102. S. Singh, “Big Bang”, HaperCollins Publishers, New York, 2 004, p. 75. 82 Table 1. Angular properties of atomic p- and d-orbitals x 2/1,11,1 y 2/1,11,1i z 0,1 3z2r2 0,2 xz 2/1,2 1,2 yz 2/1,2 1,2i xy 2/2,2 2,2i x2y2 2/2,2 2,2 83 Table 2. Nonzero integrals of t he angular momentum operators, jLˆix, jLˆiy and jLˆiz, where )z,y,xj,i( or (i, j = 3z2r2, xz, yz, x2 – y2, xy). zLˆ ixLˆyz xLˆ iyLˆzx yLˆ izLˆxy zLˆ i2 y xLˆxy2 2 z i yzLˆxzz xLˆ 3i yzLˆrz3x2 2 i yzLˆy xx2 2 i xyLˆxzx yLˆ 3i xzLˆrz3y2 2 i xzLˆy xy2 2 i xyLˆyzy 84 Table 3. Orbital character of the d-states of an ML 6 octahedron in two different settings of the Cartesian coordinates z-axis direction Along one M-L bond (Fig. 4a) Along one C 3-rotational axis (Fig. 6a) t2g xy 2 2rz3a1 xz xz xy e131 32 x yz yz )yx( e131 2 2 32 y eg x2y2 xz xy e232 31 x 2 2rz3 yz )y x( e232 2 2 31 y 85 Table 4. The preferred spin or ientations of magnetic ions predi cted using the Lz values associated with the SOC-indu ced HOMO-LUMO interactions Spin orientation Requirement Interactions between ||z 0 Lz xz and yz xy and x2y2 x and y z 1 Lz {3z2r2} and {xz, yz} {xz, yz} and {xy, x2y2} z and {x, y} 86 Figure captions Figure 1. Close-packed energy states of a magnetic system, whic h arise from weak interactions among the unpaired electrons of its magnetic ions. Figure 2. Examples of simple spin lattices: an isolated spin di mer and a uniform chain requiring one spin excha nge constant, and an alternating chain and a two-leg ladder requiring two spin exchange constants. Figure 3. Minimum difference in the magnetic quantum numbers, | Lz|, between pairs of (a) d-orbitals and (b) p-orbitals. Figure 4. (a) An ideal ML 6 octahedron with the local z-axis taken along one M-O bond, i.e., one 4-fold rotational ax is. (b) The orbital composi tions of the t 2g and e g states. (c) The -antibonding in the xy, xz and yz components of the t 2g state, and the -antibonding in the x2y2 and the 3z2r2 components of the e g state. Figure 5. The split d-states of (a) an ideal ML 6 octahedron, (b) an axially-elongated ML 6 octahedron, (c) a square planar ML 4, and (d) a linear ML 2. Figure 6. (a) An ideal ML 6 octahedron with the local z -axis taken along one 3-fold rotational axis. (b) The orbital compositions of the t 2g and e g states as listed in Table 3. 87 Figure 7. The split of the up-spin and down-spin states by an o n-site repulsion U. These states are degenerate in the non- spin-polarized descript ion (left), but are split in the spin-polarized description (right). Figure 8. The orbital interactio ns between two equivalent spin sites for cases when they have (a) a FM arrangement and (b) an AFM arrangement. Figure 9. The simulation of the split d-states obtained from DFT+U calculations in terms of those obtained from an effective one-electron Hamilton ian for a high-spin (S = 2) d6 ion at a square planar site forming a FeL 4 square plane. Figure 10. A spin dimer made up of two equivalent spin sites wi th an unpaired electron at each site. The unpaired electrons at the sites 1 an d 2 are accommodated in the orbitals 1 and 2, respectively, and the spin exchange constant J describes the strength and sign of the int eraction between the two unpair ed electrons. Figure 11. The interaction be tween the magnetic orbitals 1 and 2 of a spin dimer leading to the bonding and antibonding molecular orbitals 1 and 2 of the dimer, respectively, which are split by the energy e. Figure 12. The occupation of the molecular orbitals 1 and 2 of the dimer with two electrons leading to the triplet configuration T as well as two singlet configurations 1 and 2. 88 Figure 13. Two CuO 2 ribbon chains of LiCuVO 4 interconnected by VO 4 tetrahedra, where grey circle = Cu, cyan circle = V, and white circle = O. The intrachain spin exchange paths J nn a n d J nnn as well as the interchain spin exchange path J a a r e indicated by the legends “nn” , “nnn” and “a”, respectively. Figure 14. (a) The x 2-y2 magnetic orbital of a CuO 4 square plane. (b) The Cu-O-Cu spin exchange interaction between nearest-neighbor CuO 4 square planes in a CuO 2 ribbon chain. (c) The Cu-O…O-Cu s pin exchange interaction betwe en next-nearest- neighbor CuO 4 square planes in a CuO 2 ribbon chain. Figure 15. The through-space (TS) and the through-bond (TB) int eractions between the two x 2-y2 magnetic orbitals in the Cu-O…V5+…O-Cu interchain spin exchange Ja in LiCuVO 4: (a) The energy split between + and - due to the TS interaction. (b) The bonding interaction of the V d orbital with the O 2p tails of - i n t h e O…V5+…O bridge. (c) The energy split between + and - due to the through- space (TS) and through-bond (TB) interactions. Figure 16. Three interactions c ontrolling the strength of a DM interaction. Figure 17. Polar angles θ and ϕ defining the preferred or ientation of the spin (i.e., the z-axis) with respect to the (x, y , z) coordinate used to describ e the orbital. 89 Figure 18. (a) A schematic view of an isolated CoMnO 6 chain of Ca 3CoMnO 6, which is made up of the CoO 6 trigonal prisms containing high-spin Co2+ (d7, S = 3/2) ions and the MnO 6 octahedra containing high-spin Mn4+ ( d3, S = 3/2) ions. (b) The occupancy of the down-spin d-states for a high-spin Co2+ ion in an isolated CoO 6 trigonal prism. Figure 19. The down-spin electron configuration of a high-spin Fe 2+ (d6, S = 2) at a linear coordination site that induces uniaxial magnetism. Figure 20. The down-spin electr on configurations of a high-spin Fe 2+ (d6, S = 2) at an octahedral site that induce (a) uniaxial magnetism and (b) no u niaxial magnetism. Figure 21. The high-spin configuration of a Mn 3+ ( d4) ion in an axially-elongated MnO 6 octahedron with the z-axis ta ken along the elongated Mn-O bond s. Figure 22. The down-spin elect ron configuration of a Ni 2+ (d8, S = 1) ion at a trigonal prism site. Figure 23. The down-spin states of the low-spin Ir 4+ (S = 1/2, d5) i o n i n ( a ) t h e axially-elongated IrO 6 octahedron along the 4-fold rotational axis in Sr 2IrO 4 and (b) the axially-compressed IrO 6 octahedron along the 3-fo ld rotational axis in Na 2IrO 3. 90 Figure 24. (a) The structure and the down-spin d-states of a CuCl 2(OH 2)2 complex: blue circle = Cu, green circle = Cl, medium white circle = O, a nd small white circle = H. (b) The down-spin elect ron configuration of a Cu2+ (d9, S = 1/2) ion. Figure 25. (a) The CuL 2 ribbon chain made up of edge-sharing CuL 4 square planes. The contributions of the metal d- and the ligand p-orbitals in the (b) yz, (c) xy and (d) x2y2 states of a CuL 4 square plane. Figure 26. The down-spin electron configurations of a low-spin Ir4+ (d5, S = 1/2) ion at an octahedral site that induce (a) uniaxial magnetism and (b ) no uniaxial magnetism. Figure 27. (a) An axially-compressed VO 6 octahedron of R 2V2O7 ( R = r a r e e a r t h ) along the local z-direction taken along a 3-fold rotational axi s. (b) The split t 2g state of a V4+ (d1, S = 1/2) ion at each VO 6 octahedron. (c) A tetrahedral cluster made up of four VO 6 octahedra. The local z-axes of the four VO 6 octahedra are all pointed to the center of the V 4 tetrahedron. (d) The pyrochlore lattice of the V4+ ions in R2V2O7. Figure 28. (a) A view of an isolated Sr 2IrO 4 layer made up of corner-sharing axially- elongated IrO 6 octahedra approximately along th e c-direction. (b) The interac tion between adjacent xz orbitals (or adjacent yz orbitals) through the O 2 p orbitals through each bent Ir-O eq-O bridge. (c) The interactio n between adjacent xy orbitals 91 through the O 2 p orbitals through each bent Ir-O eq-O bridge. (d) The split d-states of a dimer made up of two adjacent Ir4+ ions after incorporating the effect of the intersite interactions for the cases of the axially-elongated I rO6 octahedra. (e) The PDOS plots for the d-states of Ir4+ in Sr 2IrO 4 in case when the IrO 6 octahedra are axially elongated, where the legends (2, -2), (1, -1), and 0 in dicate the sets of orbitals (xy, x2-y2), (xz, yz) and 3z2-r2, respectively. Figure 29. (a) A projection view of a NaIrO 3 honeycomb layer made up of edge- sharing IrO 6 octahedra with Na (light blue circle) at the center of each Ir 6 hexagon. (b) A perspective view o f how the honeycomb NaIrO 3 layers repeat along the c- direction in Na 2IrO 3, where the layer of Na atoms lying in between the NaIrO 3 honeycomb layers is not shown for simplicity. (c) The split d-states of a dimer made up of two adjacent Ir4+ ions after incorporating the effect of the inter-site interactions. Figure 30. (a, b) Interactions between the Ir and Ni 3z 2-r2 states in each NiIrO 6 chain of Sr 3NiIrO 6 when the spins of adjacent Ir4+ and Ni2+ ions have a FM coupling in (a), and an AFM coupling in (b). (c, d) The PDOS plots for the down- spin d-states of Ir4+ in Sr 3NiIrO 6 in cases when adjacent Ir4+ and Ni2+ ions in each NiIrO 6 chain have a FM coupling in (c), and an AFM coupling in (d). The legends (2, -2), (1, -1) and 0 refer respectively to the (xy, x2-y2), (xz, yz) and 3z2-r2 sets. 92 Fig. 1 Fig. 2 93 Fig. 3 z y x Lz=1 Lz=1 Lz=0 (b) 3z2‐r2 xy,x2‐y2 xz,yz Lz=2 Lz=1Lz=1 Lz=0 Lz=0 (a) 94 Fig. 4 *:xy,xz,yz *: x2y2 *: 3z2r2 95 Fig. 5 96 Fig. 6. Fig. 7 1a 1ey 1ex2ey 2ex t2geg (a) 97 Fig. 8. (a) (b) 98 Fig. 9. 99 Fig. 10 Fig. 11 Fig. 12 100 Fig. 13 Fig. 14 a b nn nnn a 101 Fig. 15 Fig. 16 102 Fig. 17 Figure 18 Lz=0zc(a) (b) Mn Co 103 Fig. 19 Fig. 20 Lz=0z (a) (b) Lz=1z Lz=0z 104 Fig. 21 Fig. 22 Lz = 1 z Lz=0z Lz=2noeffect 105 Fig. 23 (b) 1a1e |Lz| = 0 ||z z (a) |Lz| = 1 z z106 Fig. 24 (b) Lz=1y(a) x y 107 Fig. 25 (a) (b) yz (c) xy (d) x2y2 y x108 Fig. 26 Fig. 27 (a) (b) (c) Lz=1z (d) (a) (b) Lz=0z Lz=1z 109 Fig. 28 xz, yz xy |Lz| = 1 (c) (d) (b) (a) (e) 110 Fig. 29 (a) |Lz| = 0 |Lz| = 1 1a1e1e- 1e+ 1a+1a-(c) (b) c a c* 111 Fig. 30 (a) (b) (c) (d)

1204.5597v1.Spin_freezing_by_Anderson_localization_in_one_dimensional_semiconductors.pdf

arXiv:1204.5597v1 [cond-mat.mes-hall] 25 Apr 2012Spin freezing by Anderson localization in one-dimensional semiconductors C. Echeverr´ ıa-Arrondo Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV/EHU, 48080 Bilbao, Spain E. Ya. Sherman Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV/EHU, 48080 Bilbao, Spain and IKERBASQUE Basque Foundation for Science, 48011 Bibao, Biz kaia, Spain (Dated: July 13, 2018) One-dimensional quantum wires are considered as prospecti ve elements for spin transport and manipulation in spintronics. We study spin dynamics in semi conductor GaAs-like nanowires with disorder and spin-orbit interaction by using a rotation in t he spin subspace gauging away the spin- orbit field. At a strong enough disorder spin density, after a relatively fast relaxation stage, reaches a plateau, which remains a constant for long time. This effect is a manifestation of the Anderson localization and depends in a universal way on the disorder a nd the spin-orbit coupling strength. As a result, at a given disorder, semiconductor nanowires ca n permit a long-term spin polarization tunable with the spin-orbit interactions. PACS numbers: 72.25.Rb,72.70.+m,78.47.-p I. INTRODUCTION The main idea of spintronics - the design and applica- tion of devices controlling not only the charge dynam- ics but also the electron spin evolution - can be use- ful for information storage, transfer, and manipulation technologies.1–3Possible realizations of spintronics de- vices can be based on semiconductor nanowires4–10for quasi-ballistic electron transport, coherent transmission of information, and spin control. These systems attract a great deal of attention due to a clear interplay of trans- port and spin-orbit (SO) coupling physics.11–16 This control faces the problem of inevitable spin relax- ation due to the coupling of electron spin to environment through SO coupling. As a result, the factors determin- ingthespinrelaxationratebecomeofcrucialimportance. Twolimiting casesofspin relaxationarewellunderstood. For the itinerant electrons spin relaxation in mainly de- termined by the Dyakonov-Perel’ mechanism, that is by random precession of electron spin due to the random in time electron momentum. A different approach should be applied for elec- trons localized in a regular external potential form- ing quantum dots promising for quantum information applications.17Here momentum is not a well-defined quantity, and the momentum-dependent splitting re- quired for the Dyakonov-Perel mechanism vanishes. As a result, spin relaxation through SO interaction requires phonon-induced coupling of different orbital states of the localized electron and nonzero external magnetic field.18 In the absence of magnetic field and spin-orbit coupling, spin relaxationcanoccurdue to the hyperfinecouplingof electronspintospinsoflatticenuclei.19Inbothcases,the initial spin polarization goes asymptotically to zero. The characteristic timescale of spin relaxation of electrons lo- calized in quantum dots is expected to be several orders of magnitude longer than that of itinerant electrons. While these two limits of free and strongly localizedFIG. 1: (Color online) Semiconductor nanowire with random impurities shown as filled circles. Although we consider a one-dimensional electron motion, impurities can be random ly distributed over the cross-section of the wire. electrons are well understood, the interplay of disorder- induced localization and spin relaxation of itinerant elec- trons remains an open question although some aspects of the problem have been addressed.20–23The questions here are (i) how the localization forms the spin relax- ation, and (ii) whether the initially prepared spin density relaxes to zero. As a nontrivial example of this interplay we mention that weak localization of two-dimensional electrons leads to a long power-like rather than exponen- tial spin relaxation.24,25Here we analyze this problem for the one-dimensional system, providing, on one hand, the basic example of localization physics in a random potential,26,27and, on the other hand, an example of a system, where spin-orbit coupling can be gauged away by a SU(2) transformation. This paper is organized as follows. In Sec. II, we show how to treat spin relaxation in one-dimensional systems with the gauge transformation and introduce the tight- binding Hamiltonian for the model. The spin dynamics will be analyzed by a numerically exact calculation in Sec. III, where we show that spin density does not re- lax to zero, in contrast to what expected. In addition, in thisSec. III westudyhowasymptoticvalueofspinpolar- ization depends on the disorder and spin-orbit coupling. Conclusions summarize the results in Sec. IV.2 II. MODEL A. Hamiltonian and gauge transformation The investigated structure is a quantum wire extended along thexaxis, as shown in Fig. 1. The total Hamilto- nian has the form ˆH=¯h2 2m(kx−Ax)2+U(x)−mα2 2¯h2,(1) whereAx=−mασy/¯h2standsfortheRashbacoupling28 with the strength α,σyis the Pauli matrix, kxis the elec- tron wavevector, and mis the effective mass. The Dres- selhaus coupling29is obtained with Ax=−mβσx/¯h2, whereβis the coupling constant. Without loss of gener- ality, we concentrate here on the Rashba coupling, which can be changed on demand by applying external electric field across the structure.30 The SO interaction can be removed from ˆHin Eq.(1) through a gauge transformation31,32with a SU(2) spin rotation: ˆS= exp(−ixσy/2ξ),whereξ= ¯h2/2mαis the spin-precession length. After this transformation the system Hamiltonian has the form:ˆ/tildewideH= ¯h2k2 x/2m+U(x). Since for the Hamiltonian (1), σyis the integral of mo- tion, the spin density component along the y-axis is time independent. A nontrivial dynamics of the transformed spin occurs for the γ= (x,z) spin components /an}bracketle{t/tildewidesγ(x,t)/an}bracketri}ht and can be expressed in terms of the spin diffusion /an}bracketle{t/tildewidesγ(x,t)/an}bracketri}ht=/integraldisplay Dγβ(x−x′,t)/an}bracketle{t/tildewidesβ(x′,0)/an}bracketri}htdx′,(2) whereDγβ(x,t) is the exact disorder-dependent one- dimensional spin diffusion Green’s function. In a non- magnetic system without SO coupling Dγβ(x,t) = δγβD(x,t) is diagonalin the spin subspace. As a result of the gauge transformation, the uniform density dynamics is determined by only the Fourier component25 D(q,t) =/integraldisplay∞ −∞dxe−iqxD(x,t) (3) withq= 1/2ξandEq. (2) simplifies forthe physicalmea- surable spins as /an}bracketle{tsγ(t)/an}bracketri}ht=/an}bracketle{tsγ(0)/an}bracketri}htD(1/2ξ,t). Here we will use a similar, however, somewhat different approach based on numerically exact analysis of the direct time evolution of the initial spin-polarized states. It will be shown that the resulting spin dynamics has unexpected features, including a long-time plateau in the spin polar- ization. The eigenfunctions ofˆ/tildewideHcan be taken in the form ψ(x) =ψ(x)|1/an}bracketri}htandψ(x) =ψ(x)|−1/an}bracketri}ht, where|±1/an}bracketri}htare the eigenstates of σzwith the corresponding eigenvalues. The eigenstates of ˆH,φ(x) can be obtained by spin ro- tation of the ψ(x)|σ/an}bracketri}htstates. For example, with spin-up initial state ψ(x)|1/an}bracketri}htone obtains: φ(x) =ψ(x)/bracketleftbigg cos/parenleftbiggx 2ξ/parenrightbigg |1/an}bracketri}ht+sin/parenleftbiggx 2ξ/parenrightbigg |−1/an}bracketri}ht/bracketrightbigg .(4) FIG. 2: (Color online) Site dependent components of φN/4(xn) for a qualitative description of entanglement in- duced by the gauge transformation for (a) α= 0, (b)α= 0.125×10−6meVcm, and (c) α= 10−6meVcm (U0= 55 meV). The solid and dashed lines represent |1/angbracketrightand|−1/angbracketrightcom- ponents, respectively. The spin dynamics and spin relaxation in the system, as it will be shown below, is solely due to the entanglement of spin and coordinate in Eq.(4). B. Tight-binding model and disorder We perform numerical analysis using the tight-binding model, employing the approach similar to Refs.[16,33]. The one-dimensional electron gas is sampled with N= 213(8192) grid points xn=nl, where 1 ≤n≤Nand lis the effective lattice constant with periodic boundary conditions.34The effective hopping matrix element be- tween two nearest neighbors is chosen as t= 50 meV, and the kinetic energy is E(kx) = 2t(1−cos(kxl)). As a3 result, the eigenenergies span the range of [0 ,200] meVs. The distance between two neighbor grid points becomes l= ¯h/√ 2mt= 3.37 nm to satisfy the electron effective massm= 0.067m0in GaAs semiconductor with m0 being the free electron mass. The random potential Un=U(xn) uniformly spans the range [ −U0/2,U0/2] with the white noise correla- tor/an}bracketle{tU(xn1)U(xn2)/an}bracketri}ht=/an}bracketle{tU2/an}bracketri}htδn1,n2, where/an}bracketle{tU2/an}bracketri}ht=U2 0/12. The effects of disorder can be approximately charac- terized through the energy-dependent momentum relax- ation timeτE, which wedefine as¯ h/τE=/an}bracketle{tU2/an}bracketri}htlνE, where νE=√m/π¯h√ 2Eis the density of states per spin com- ponent. The resulting mean free path ℓE=vEτE, where vE=/radicalbig 2E/mis the electron speed and the correspond- ing diffusion coefficient DE=v2 EτE. In this representation the eigenstates ofˆ/tildewideHandˆHform basis sets, {ψi}and{φi}respectively, where 1 ≤i≤2N. Forthesame i, these twosetsarerelatedbythe localspin rotation ˆS.Weassumethat ψi=ψi(xn)|1/an}bracketri}htfor1≤i≤N andψi=ψi−N(xn)|−1/an}bracketri}htforN <i≤2N. III. SPIN DYNAMICS We study dynamics of initial ψistates with 1 ≤i≤N, corresponding to the evolution upon instant switching of the SO coupling. The time dependence can be expressed with the spectral decomposition as: ψso j(t) =/summationdisplay i=1,2Naijφie−itεi/¯h, (5) whereaij=/an}bracketle{tφi|ψj/an}bracketri}ht, andεiare the corresponding eigenenergies. The spin component expectation value /an}bracketle{tσz(t)/an}bracketri}htj=/angbracketleftBig ψso j(t)/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleψso j(t)/angbracketrightBig isdetermined by the spec- trum and eigenstates of the system. In order to give an idea of the entanglement induced by SO coupling, we present in Fig. 2 the evolution of φN/4(xn) state with the increase in the spin-orbit cou- pling. Atα= 0, we obtain a product state φN/4(xn) = ψN/4(xn)|1/an}bracketri}ht, and with the increase in αentangled states areformed. Theoverlapof φi(xn) andψj(xn)eigenstates is characterized by two sets of matrix elements aij; for example, for 1 ≤j≤N: aij=/summationdisplay ncos/parenleftbiggxn 2ξ/parenrightbigg ψi(xn)ψj(xn),1≤i≤N(6) aij=/summationdisplay nsin/parenleftbiggxn 2ξ/parenrightbigg ψi(xn)ψj(xn), N <i ≤2N. The behavior of aijpresented Fig. 3 demonstrates that for givenjit has nonnegligible values only in a certain, rather narrow, range of i. To illustrate the role of the random potential, we con- sider as examples weak ( U0= 15 meV, U0≪t) and FIG. 3: (Color online) Absolute values of aijaround the ini- tial spin-up state ψN/4; hereα= 10−6meVcm (strong SO coupling) and U0=55 meV (strong disorder). FIG. 4: (Color online) Inverseparticipation ratio ζfor the low part of the energy spectrum; gray (red) circles denote stron g disorder (U0=55 meV) and black circles denote weak disorder (U0=15 meV). Since even for U0=55 meV we obtain ζ≪1, the localized states are distributed over many lattice site s, confirming applicability of the tight-binding Hamiltonian for the localization problem. strong (U0= 55 meV, U0> t) disorder. For free elec- trons in state j=N/4 andE= 31 meV, the result- ing ¯h/τEis about 0.1 and 1 meV, respectively. For a free electron with the energy E≈20 meV the velocity vE≈3.5×107cm/s, the mean free path ℓE∼2.5×10−5 cm (¯h/τE= 1 meV), and the corresponding diffusion co- efficientDE∼103cm2/s. These parameters provide an effective integral characteristic of the disorder and corre- spond to realistic parameters of the wires, which, how- ever, can strongly vary from sample to sample and from experiment to experiment. The effect of localization by disorder is seen in the in- verse participation ratio35(IPR)ζi=/summationtext n/vextendsingle/vextendsingleψ4 i(xn)/vextendsingle/vextendsingle. The4 FIG. 5: (Color online) Time-dependent polarization in the weak-disorder regime ( U0=15 meV). The initial bins are cen- tered at the states (a) N/4 (bin width 6.8 meV), (b) N/8 (bin width 3.7 meV), and (c) and N/16 (bin width 2.1 meV) with energies decreasing in the same order. The curves for SO couplings 0 .125×10−6meVcm, 0.5×10−6meVcm, and 2×10−6meVcm are drawn with circles, triangles, and squares, respectively. Note that after the relaxation stag e the spin density remains a finite constant. IPR calculated for the low-energy spectrum is presented inFig.4. Asexpected,thedegreeoflocalizationincreases withU0and this effect is more pronounced for the elec- trons with lowest energies. In contrast to the results of Ref.[23], the IPR in this system does not depend on the SO coupling. We now study the effects of disorder and spin-orbit coupling on the averagespin dynamics of a bin of 256 initial spin-up states and 8 realizations of the ran- dom potential. The statistical error of this approach is, therefore1/√ 2048=2.2%, makingthe resultsstatistically representative. We take three example bins with three different de- grees of localization. The bins are centered around the spin-up states ψN/4,ψN/8, andψN/16, whose IPR val- ues increase in the same order (energies decreasing, see Fig. 4). The calculated bin- and potential realization- averaged spin dynamics is shown in Figs. 5 and 6, re- vealing strong influence of the disorder-induced spatial localization of states. Physically, collisions of electrons with impurities force electron spin to frequently reverse the precession direction. In the classical picture, this leads to a long Dyakonov-Perel’ spin relaxation. If the FIG. 6: (Color online) Time-dependent spin polarization in the strong-disorder regime ( U0=55 meV) with the same no- tations as in Fig. 5. (a) N/4 (bin width 7.2 meV), (b) N/8 (bin width 4.5 meV), and (c) N/16 (bin width 3.7 meV). Note that forα= 0.125×10−6meVcm the spin is almost constant in time, thus suitable for spin-based operations. quantum effects of localization are important, the result- ing effect is the “freezing” of the electronic spin. As one can see in Figs. 5 and 6, the electron spin density re- laxes for ≃5 ps and then remains constant in time for infinitely long (beyond 0.2 ns in our computation). As expected, the spin polarization plateau is higher (i) for localized states and (ii) for weak SO interaction. Al- most time-independent spin states are achieved e.g., at U0= 55 meV and α= 0.125×10−6meVcm. To gain insight into the problem, we study the depen- dence of asymptotic spin density on SO coupling and the localization of electrons in more detail. The long-term densities are plotted in Fig. 7 against parameter ξ/an}bracketle{tζ/an}bracketri}ht. This parameter combines the two factors determining the spin dynamics, SO coupling and spatial localization, where/an}bracketle{tζ/an}bracketri}htis averaged over 256 bin states and 8 realiza- tion of the random potential. The given values follow a universal dependence indicating a unique trend for long- term spins against SO coupling and localization through disorder. This trend corresponds to a fast increase in the asymptoticsteadypolarizationfor ξ/an}bracketle{tζ/an}bracketri}ht<1andasmooth increaseandsaturationfor ξ/an}bracketle{tζ/an}bracketri}ht>1. Theseresultscanbe understood as follows. To show an efficient spin dynam- ics, the electron should move the distance of the order ofπξ. Therefore, the spatial spread of the correspond-5 FIG. 7: (Color online) Long-term relative polarization as a function of ξ/angbracketleftζ/angbracketrightfor three different degrees of localization. Pa- rameterξis modified by changing the coupling constant α. ing states should be larger than πξ. With a stronger localization, the spread and the overlap decrease leading to the universal behavior shown in Fig.7. Qualitatively, in the “clean” ξ/an}bracketle{tζ/an}bracketri}ht ≪1 regime the spin relaxation has the Dyakonov-Perel’ mechanism either purely exponen- tial for Ω EτE≪1 or a combination of oscillations and exponential decay if Ω EτE≥1, where the spin preces- sion rate Ω E= 2α√ 2mE/¯hcorresponds to the electron momentum at given energy E. IV. CONCLUSION To summarize, localization effects of disorder and SO coupling in semiconductor nanowires determine the dy- namics of electronic spins. Our tight-binding model cal-culations show that a prepared spin density relaxes un- til reaching a plateau, directly related to the disorder and strength of SO interaction. In contrast to the ex- pected decay to zero, a long-time constant polarization plateau survives to infinite time. The asymptotic spin density has a universal dependence on the product of the inverse participation ratio and the spin precession length. In the absence of magnetic field, the hyperfine coupling to the spins of nuclei will lead to spin relaxation on timescales at least two orders of magnitude longer than the timescale of the plateau formation of the or- der of 10 ps.19As the experiments on spin transport did not reveal electron-electron interaction effects,8here we have neglected them. Furthermore, whether there exists a range of parameters where the Coulomb forces can be strong enough to modify our results for localized states, remains to be investigated. An immediate consequence of this result is the ability, by choosing the desired Rashba SO parameter for a given wire, to produce and destroy steady spin states, which are of interest for spin-based operations. These results suggest that semiconductor nanowires can be used for coherenttransmissionandstorageofinformation, manip- ulated by spatially and temporally modulated spin-orbit coupling. V. ACKNOWLEDGMENTS We thank G. Japaridze, J. Siewert, and L.A. Wu for helpful discussions. This workwassupportedbythe MCI of Spain grant FIS2009-12773-C02-01and ”Grupos Con- solidados UPV/EHU del Gobierno Vasco” grant IT-472- 10. 1I. Zuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004); J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, Acta Physica Slovaca 57, 565 (2007). 2A. Fert, Rev. Mod. Phys. 80, 1517 (2008). 3M. W. Wu, J. H. Jiang, and M. Q. Weng, Physics Reports 493, 61 (2010). 4X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, Nature409, 66 (2001). 5S.Nadj-Perge, S.Frolov, E.Bakkers, andL.Kouwenhoven, Nature468, 1084 (2010). 6S. Pramanik, S. Bandyopadhyay, and M. Cahay, Phys. Rev. B68, 075313 (2003). 7A. A. Kiselev and K. W. Kim, Phys. Rev. B 61, 13115 (2000). 8C. H. L. Quay, T. L. Hughes, J. A. Sulpizio, L. N. Pfeiffer, K. W. Baldwin, K. W. West, D. Goldhaber-Gordon, and R. de Picciotto, Nat. Phys 6, 336 (2010). 9A. Bringer and T. Sch¨ apers, Phys. Rev. B 83, 115305 (2011). 10M. Governale and U. Z¨ ulicke, Phys. Rev. B 66, 073311 (2002).11Y. V. Pershin, J. A. Nesteroff, and V. Privman, Phys. Rev. B69, 121306 (2004); V. A. Slipko, I. Savran, and Y. V. Pershin, Phys. Rev. B 83, 193302 (2011). 12A. W. Holleitner, V. Sih, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 97, 036805 (2006). 13M. M. Gelabert, L. Serra, D. S´ anchez, and R. L´ opez Phys. Rev. B81, 165317 (2010); D. S´ anchez, L. Serra, and M.-S. Choi Phys. Rev. B 77, 035315 (2008). 14C. L¨ u, U. Z¨ ulicke, and M. W. Wu, Phys. Rev.B 78, 165321 (2008) 15C. L. Romano, P. I. Tamborenea, and S. E. Ulloa, Phys. Rev. B74, 155433 (2006) 16G. I. Japaridze, H. Johannesson, and A. Ferraz, Phys. Rev. B80, 041308 (2009); M. Malard, I. Grusha, G. I. Japaridze, and H. Johannesson, Phys. Rev. B 84, 075466 (2011). 17G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999); L.-A. Wu and D. A. Lidar, Phys. Rev. Lett.91, 097904 (2003); L.-A. Wu, D. A. Lidar and M. Friesen, Phys. Rev. Lett. 93, 030501 (2004). 18P.StanoandJ.Fabian, Phys.Rev.Lett. 96, 186602(2006);6 P. Stano and J. Fabian, Phys. Rev. B 74, 045320 (2006). 19I. A. Merkulov, Al. L. Efros, and M. Rosen, Phys. Rev. B 65, 205309 (2002). 20T. P. Pareek and P. Bruno, Phys. Rev. B 65, 241305 (2002). 21B. I. Shklovskii, Phys. Rev. B 73, 193201 (2006). 22T. Kaneko, M. Koshino, and T. Ando, Physica E 40, 383 (2007). 23G. A. Intronati, P. I. Tamborenea, D. Weinmann, and R. A. Jalabert, arXiv:1102.4753. 24I. S. Lyubinskiy and V. Yu. Kachorovskii, Phys. Rev. B 70, 205335 (2004). 25I. V.Tokatly andE. Ya.Sherman, Phys.Rev.B 82, 161305 (2010). 26P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, Phys. Rev. B 22, 3519 (1980). 27I. M. Lifshitz, S. A. Gredeskul, and L. A. Pastur, In- troduction to the Theory of Disordered Systems, Wiley, NY, (1988); A. L. Efros and B. I. Shklovskii, Electronic Properties of Doped Semiconductors, Springer, Heidelberg, (1989). 28Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39, 79 (1984). 29G. Dresselhaus, Phys. Rev. 100, 580 (1955). 30D. Grundler, Phys. Rev. Lett. 84, 6074 (2000); O. Z. Ka- rimov, G. H. John, R. T. Harley, W. H. Lau, M. E. Flatte,M. Henini, and R. Airey, Phys. Rev. Lett. 91, 246601 (2003), A. Balocchi, Q. H. Duong, P. Renucci, B. L. Liu, C. Fontaine, T. Amand, D. Lagarde, and X. Marie, Phys. Rev. Lett. 107, 136604 (2011). 31L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324 (2003). 32I. Tokatly and E. Ya. Sherman, Annals of Phys. 325, 1104 (2010). 33E. M. Hankiewicz, L. W. Molenkamp, T. Jungwirth, and J. Sinova, Phys. Rev. B 70, 241301 (2004). 34Since we are studying the spin dynamics in the lower part of the spectrum, the matrix Hamiltonian is only partially diagonalized up to 2800 eigenenergies and eigenstates. We use for this purpose the Lanczos method as implemented in the PARPACK package, a collec- tion of Fortran routines for parallel computing. See www.caam.rice.edu/software/ARPACK/index.html for more information about the parallel version of ARPACK. 35F. Evers and A. D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000). 36V. N. Gridnev, Pis’ma Zh. Eksp. Teor. Fiz. 74, 417 (2001) [JETP Lett. 74, 380 (2001)]; C. Grimaldi, Phys. Rev. B 72, 075307 (2005); M. M. Glazov, Solid State Commun. 142, 531 (2007); X. Liu, X.-J. Liu, and J. Sinova, Phys. Rev. B84, 035318 (2011).

1206.6726v2.Spin_orbit_coupled_transport_and_spin_torque_in_a_ferromagnetic_heterostructure.pdf

arXiv:1206.6726v2 [cond-mat.mes-hall] 5 Feb 2014Spin-orbit coupled transport and spin torque in a ferromagn etic heterostructure Xuhui Wang,∗Christian Ortiz Pauyac, and Aur´ elien Manchon† King Abdullah University of Science and Technology (KAUST) , Physical Science and Engineering Division, Thuwal 23955-6 900, Saudi Arabia Ferromagnetic heterostructures provide an ideal platform to explore the nature of spin-orbit torques arising from the interplay mediated by itinerant el ectrons between a Rashba-type spin- orbit coupling and a ferromagnetic exchange interaction. F or such a prototypic system, we develop a set of coupled diffusion equations to describe the diffusive spin dynamics and spin-orbit torques. We characterize the spin torque and its two prominent– out-of-plane andin-plane–components for a wide range of relative strength between the Rashba couplin g and ferromagnetic exchange. The symmetry and angular dependence of the spin torque emerging from our simple Rashba model is in an agreement with experiments. The spin diffusion equatio n can be generalized to incorporate dynamic effect such as spin pumping and magnetic damping.2 I. INTRODUCTION Spin-orbit coupling is a key mechanism in many prominent physical phen omena ranging from the electrically generated bulk spin polarization,1,2to the dissipationless spin current in bulk semiconductors,3to the spin-Hall effect in metals4and two-dimensional electron gas.5,6In searching for an efficient mechanism for magnetization switching, interplay between spin-orbit coupling and magnetism7,8has brought a new member to the spin-transfer torque community,9,10i.e., the spin-orbit torque. Diluted magnetic semiconductors provide an ideal platform to theoretically11and experimentally12–14study the current-driven magnetization dynamics induced by spin- orbit torques. Simply speaking, spin-orbit torque operates through th e competition between the exchange and spin-orbit fields in polarizing the itinerant electrons (or holes) and gives rise to a torque on the ferromagnetic order parameter. When it comes to magnetization switching, the advantage of spin-or bit torque over the conventional one is clear: there is no need to employ a separate ferromagnet as polarizer. Recent experiments on current-driven magnetization dynamics pe rformed in multilayer systems15–19have also achieved current-induced switching in a singleferromagnet film sandwiched between a heavy metal and metal oxid e, which indicates the presence of spin-orbit coupling and therefore s pin-orbit torque as a potential driving force. These systems–mostly consisting of conducting interfaces between fer romagnetic metal films and heavy metals (or metal oxides)–are nowadays often referred to as ferromagnetic hete rostructures. A viable candidate believed to exist in such structures is the Rashba-type spin-orbit interaction due to inver sion symmetry breaking.20Theoretical efforts are made to uncover dominant components of the spin-orbit torque ind uced by the Rashba coupling.21–25They usually treat the coexistence of ferromagnetism and spin-orbit coupling a s an intrinsic property. The underlying physics is fairly simple and intuitive: when a charge current is applied in the struc ture, the Rashba spin-orbit coupling creates an effective magnetic field (coined as the Rashba field BR); so long as BRpolarizes charge carriers to the direction that is misaligned with the magnetization direction m, a spin torque emerges to act on the magnetization and to induce switching. This torque, named as Rashba spin-orbit torque o r Rashba torque, has gained much attention from academia as well as industries and is exactly the central topic o f this paper. Both theories and experiments have shown that the Rashba torque shall, in general, comprise two m ajor components, i.e. a fieldlike torque (or in- plane component) and a damping-like torque (or out-of-plane comp onent). More recently, the symmetry of spin-orbit torque has been scrutinized experimentally. The experiments by Ga relloet alrevealan intriguing yet complex angular dependence on the magnetization direction.26This observation challenges the commonly accepted form described by in-plane and out-of-plane components. In this paper, we provide a systematic theoretical study on the sp in-orbit torque and spin dynamics in a ferro- magnetic ultrathin film without structure inversion symmetry. We co nstruct a simple two-dimensional model that accommodates both a Rashba spin-orbit coupling and an exchange in teraction. For the Rashba torque, we propose a general form that not only contains the in-plane and out-of-plan e components but also possesses symmetry and complex angular dependence supported by experiments. In Sec. I I, we employ the quantum kinetic equation to derive coupled diffusion equations for the charge and spin densities. We acc ount for the fact that Rashba coupling not only produces an effective magnetic field but also induces spin relaxation t hrough the D’yakonov-Perel mechanism27that is dominant in a quasi-two-dimensional system. In the absence of ma gnetism, analytical and numerical solutions in Sec. IIIA are able to describe the spin-galvanic and spin-Hall effect s. We demonstrate in Sec. IIIB that the diffusion equation provides a coherent framework to describe the spin dyna mics in a ferromagnetic metal. Section IV employs the spin diffusion equation given in Sec. II to analyze the general sym metry properties and angular dependence of the Rashba torque in the limits of both a weak and strong spin-orbit coup ling. We are able to provide for the spin-orbit torque an angular dependence that agrees well with recent exper iments. In Sec. V, we evaluate the spin density and Rashba torque numerically for a wide range of relative strength bet ween the Rashba coupling and exchange splitting. In Sec. VI, we further show the formulation proposed in Sec. II ca n be generalized to describe spin pumping and magnetic damping by an inclusion of temporal and spatial variations o f ferromagnetic order parameter. We find agreement with earlier results approached using other methods. S ection VII discusses the validity of the Rashba model and outlines a brief comparative study between the Rashba t orque and spin-Hall effect-induced torque. Section VIII concludes the article. Nevertheless, we emphasize that we make no attempt to argue tha t the Rashba model provides the ultimate answer to the spin-orbit torque in ferromagnetic heterostructures. Sp in-Hall effect4in the nonmagnetic metal layer provides an alternative explanation to several experiments.28However, we must admit, as Haney et al.have pointed out, both explanations have their strength and weakness.29Despite the limitations, the results presented here and their agreement with experiments lead us to believe such a simple model doe s shed light on the nature of the spin-orbit torque in ferromagnetic thin films.3 z y xPt Oxides Co FIG. 1. (Color online) Schematic view of the cross section of a typical ferromagnetic heterostructure that accommodate s both a Rashba spin-orbit coupling and an exchange interaction. A n ultrathin ferromagnetic metal film (e.g., Co) is sandwiche d between an oxide (e.g., AlO x) and a heavy metal layer (e.g., Pt or Ta). A charge current is i njected into the ferromagnetic layer along the ˆxdirection. The dashed red arrow points to the direction of th e effective Rashba field. We shall note that the system is not isolated but connected to an external source an d drain. II. FROM HAMILTONIAN TO DIFFUSION EQUATION InFig. 1, wesketchaschematicviewofacross-sectionofatypical ferromagneticheterostructureunderinvestigation: a ferromagnetic ultrathin metal film (rolled out in the x-yplane) is sandwiched by a heavy metal layer and an oxide; two asymmetric interfaces provide a weakconfinement in the zdirection, along which the inversion symmetry is broken. The potential gradient across the interface generate s a Rashba spin-orbit coupling.20Without loss of generality, our starting point is therefore a simplified quasi-two-dim ensional single-particle Hamiltonian ( /planckover2pi1= 1 is assumed throughout), ˆH=ˆk2 2m+αˆσ·(ˆk׈z)+1 2∆xcˆσ·m+ˆHi, (1) for an electron with momentum ˆk. In Eq. (1), ˆσis the Pauli matrix, mthe effective mass, and mthe magnetization direction. The ferromagnetic exchange splitting is given by ∆ xcandαrepresents the Rashba constant (parameter). The Hamiltonian ˆHi=/summationtextN j=1V(r−rj) accounts for all nonmagnetic impurity scattering potentials V(r) localized at rj. Throughout the following discussion, we assume that the exchang e interaction and spin-orbit splitting are smaller than the Fermi energy, while leaving the ratio of the spin-orbit coup ling to the exchange interaction arbitrary. Before we proceed to detailed discussion, we clarify the validity of su ch a quasi two-dimensional model. In principle, carrier transport in the system under consideration is a three-dim ensional phenomenon in which size effect may arise. Here, our quasi two-dimensional Rashba model assumes a d irect coupling between the exchange and effective Rashba field and is thus requiring ultrathin layers in which diffusive motio n normal to the thin film plane can be neglected. Haney et alhave recently conducted a thorough discussion using a three-dime nsional semiclassical Boltzmann description in a bilayer structure augmented by an interf acial Rashba spin-orbit coupling.29Their results are consistent with the ones obtained from quasi-two-dimensional transport modeled in Refs. [21–25, and 30]. To derive a diffusion equation for the nonequilibrium charge and spin de nsities, we apply the Keldysh formalism.31 We use the Dyson equation, in a two-by-two spin space, to obtain a k inetic equation that assembles the retarded (advanced) Green’s function ˆGR(ˆGA), the Keldysh component of the Green function ˆGK, and the self-energy ˆΣK, i.e., [ˆGR]−1ˆGK−ˆGK[ˆGA]−1=ˆΣKˆGA−ˆGRˆΣK, (2) where all Green’s functions are full functions with interactions tak en care of by the self-energies ˆΣR,A,K. The retarded (advanced) Green’s function in momentum and energy space is ˆGR(A)(k,ǫ) =1 ǫ−ǫk−ˆσ·b(k)−ˆΣR(A)(k,ǫ), (3) whereǫk=k2/(2m) is the single-particle energy. The impurity scattering has been tak en into account by the self- energy, as to be shown below. We have introduced a k-dependent total effective field b(k) = ∆xcm/2+α(k׈z) with magnitude bk=|∆xcm/2+α(k׈z)|and direction ˆb=b(k)/bk. We neglect localization effects and electron-electron interactions a nd assume a short-range δ-function type impu- rity scattering potential. At a low impurity concentration and a weak coupling to electrons, a second-order Born approximation is justified,31i.e., the self-energy due to impurity scattering is32 ˆΣK,R,A(r,r′) =δ(r,r′) mτˆGK,R,A(r,r), (4)4 where the momentum relaxation time is given by 1 τ≈2π/integraldisplayd2k′ (2π)2|V(k−k′)|2δ(ǫk−ǫk′). (5) V(k) is the Fourier transform of the scattering potential and the mag nitude of kandk′is evaluated at Fermi vector kF. The quasiclassical distribution function ˆ g≡ˆgk,ǫ(T,R), defined as the Wigner transform of the Keldysh function ˆGK(r,t;r′,t′), is obtained by integrating out the relative spatial-temporal coor dinates while retaining the center-of- mass ones R= (r+r′)/2 andT= (t+t′)/2. The spatialprofile of the quasiclassicaldistribution function is co nsidered smooth on the scale of Fermi wavelength, we may thus apply the gra dient expansion technique on Eq. (2),33which gives us a transport equation for macroscopic quantities. Under t he gradient expansion, the left-hand side of Eq. (2) becomes [ˆGR]−1ˆGK−ˆGK[ˆGA]−1 ≈[ˆg,ˆσ·b(k)]+i τˆg+i∂ˆg ∂T+i 2/braceleftbiggk m+α(ˆz׈σ),∇Rˆg/bracerightbigg , (6) where{·,·}is the anticommutator. The relaxation time approximation renders t he right-hand side of Eq. (2) as ˆΣKˆGA−ˆGRˆΣK≈1 τ/bracketleftBig ˆρ(ǫ,T,R)ˆGA(k,ǫ)−ˆGR(k,ǫ)ˆρ(ǫ,T,R)/bracketrightBig (7) where we have introduced the density matrix by integrating out k′in ˆg, i.e., ˆρ(ǫ,T,R) =1 2πN/integraldisplayd2k′ (2π)2ˆgk′,ǫ(T,R), (8) whereNis the density of states for one spin specie. For the convenience of discussion, the time variable is changed from Ttot. At this stage, we have a kinetic equation depending on ˆ ρand ˆg i[ˆσ·b(k),ˆg] +1 τˆg+∂ˆg ∂t+1 2/braceleftbiggk m+α(ˆz׈σ),∇Rˆg/bracerightbigg =i τ/bracketleftBig ˆGR(k,ǫ)ˆρ(ǫ)−ˆρ(ǫ)ˆGA(k,ǫ)/bracketrightBig . (9) We perform a Fourier transformation on temporal variable to the f requency domain ω, which leads to Ωˆg−bk[ˆUk,ˆg] =iˆK, (10) where Ω = ω+i/τand the operator ˆUk≡ˆσ·ˆbsatisfies ˆUkˆUk= 1. The right-hand side of Eq. (10) is partitioned according to ˆK=i τ/bracketleftBig ˆGR(k,ǫ)ˆρ(ǫ)−ˆρ(ǫ)ˆGA(k,ǫ)/bracketrightBig /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright ˆK(0) +−1 2/braceleftbiggk m+α(ˆz׈σ),∇Rˆg/bracerightbigg /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright ˆK(1), (11) whereˆK(0)contributes to the lowest-order solution to ˆ gand the gradient correction ˆK(1)is treated as a perturbation. Both functions ˆ gand ˆρare in the frequency domain. Equation (10) is solved formally to give a solution to ˆ g: ˆg=i(2b2 k−Ω2)ˆK+2b2 kˆUkˆKˆUk−Ωbk[ˆUk,ˆK] Ω(4b2 k−Ω2)≡L[ˆK]. (12) An iteration procedure to solve Eq. (12) has been outlined in Ref.[32]. We adopt the procedures here: according to the partition scheme of ˆK, we use ˆK(0)to obtain the zeroth order approximation given by ˆ g(0)≡L[ˆK(0)(ˆρ)] which5 replaces ˆ ginˆK(1)to generate a correction due to the gradient term, i.e., ˆK(1)(ˆg(0)); we further insert ˆK(1)(ˆg(0)) back to Eq. (12) to obtain a correction L[ˆK(1)(ˆg(0))]; then we have the first order approximation to the quasiclassical distribution function, ˆg(1)= ˆg(0)+L[ˆK(1)(ˆg(0))]. (13) The above procedure is repeated to a desired order using ˆg(n)= ˆg(n−1)+L[ˆK(1)(ˆg(n−1))]. (14) In this paper, the second orderapproximation is sufficient. The full expression of the second order approximationfor ˆ g is tedious thus to be excluded in the following. The diffusion equation is d erived by an angle averaging in momentum space, which allows all terms that are of odd order in ki(i=x,y) to vanish while the combinations such as kikj contribute to the averaging by a factor k2 Fδij.33 A Fourier transform from frequency domain back to the real time b rings us a diffusionlike equation for the density matrix, ∂ ∂tˆρ(t) =D∇2ˆρ−1 τdpˆρ+1 2τdp(ˆz׈σ)·ˆρ(ˆz׈σ) +iC[ˆz׈σ,∇ˆρ]−B{ˆz׈σ,∇ˆρ} +Γ[(m×∇)zˆρ−ˆσ·m∇ˆρ·(ˆz׈σ)−(ˆz׈σ)·∇ˆρˆσ·m] +1 2τϕ(ˆσ·mˆρˆσ·m−ˆρ)−i˜∆xc[ˆσ·m,ˆρ] −2R{ˆσ·m,(m×∇)zˆρ}, (15) where ˆρassumes an energy dependence ˆ ρ≡ˆρ(ǫ). The subscript is omitted for the brevity of notation. In a two- dimensionalsystem, the diffusionconstant D=τv2 F/2isgivenin termsofFermivelocity vFandmomentum relaxation timeτ. The renormalized exchange splitting reads ˜∆xc= (∆xc/2)/(4ς2+ 1), where ς2= (∆2 xc/4 +α2k2 F)τ2. The other parameters are given by C=αkFvFτ (4ς2+1)2,Γ =α∆xcvFkFτ2 2(4ς2+1)2,R=α∆2 xcτ2 2(4ς2+1), B=2α3k2 Fτ2 4ς2+1, 1 τdp=2α2k2 Fτ 4ς2+1,1 τϕ=∆2 xcτ 4ς2+1. τdpis the relaxation time due to the D’yakonov-Perelmechanism1andτ∆≡1/∆xcsets the time scale for the coherent precession of the spin density around the magnetization. Equation (15) is valid in the dirty limit ς≪1, which enables the approximation 1+4 ς2≈1. The charge density nand spin density Sare introduced by a vector decomposition of the density matrix ˆ ρǫ=nǫ/2+Sǫ·ˆσ. In real experiments,15,18,19spin transport in a ferromagnetic film experiences random magnetic scatterers, for which we introduce phenomenolo gically an isotropic spin-flip relaxation S/τsf. After an integration over energy ǫ, i.e.,n=N/integraltext dǫ nǫandS=N/integraltext dǫSǫ, we obtain a set of diffusion equations for the charge and spin densities, i.e., ∂n ∂t=D∇2n+B∇z·S+Γ∇z·mn+R∇z·m(S·m), (16) where∇z≡ˆz×∇and ∂S ∂t=D∇2S−S τsf−S+Szz τdp−1 τ∆S×m−m×(S×m) τϕ +B∇zn+2C∇z×S+2R(m·∇zn)m +Γ[m×(∇z×S)+∇z×(m×S)]. (17) The anisotropy in spin relaxation is embedded naturally in our model: th e spin density components SxˆxandSyˆyare relaxed at a rate 1 /τdp+1/τsf, whileSzˆzis submitted to a higher rate 2 /τdp+1/τsf. Equations (16) and (17) comprise one of the most important result s in this paper. For a broad range of the relative strength between the spin-orbit coupling and exchange splitting αkF/∆xc, Eqs. (16) and (17) not only describe the spin dynamics in a ferromagnetic film but also capture the symmetry o f the spin-orbit torque. When the magnetism vanishes ∆ xc= 0, the Bterm behaves as a source that generates spin density electrically.2,32On the other hand,6 when the Rashba spin-orbit coupling is absent ( α= 0), the first two lines in Eq. (17) describe a diffusive motion of spin density in a ferromagnetic metal, which, to be shown in the next s ection, agrees with early results.34TheCterm describes the coherent precession of the spin density around the effective Rashba field. The spin density induced by the Rashba field precesses around the exchange field, which is desc ribed by the Γ term, and is thus at a higher order than the Cterm in the dirty limit, for Γ = ∆ xcτC/2. TheRterm contributes to a magnetization renormalization. We shall assign a proper physical meaning to the transverse spin dephasing time τϕdefined in this paper. Here, the dephasing time τϕis different from the transverse spin scattering time in, for example , Eq. (34) in Ref.[35] that describes the disorder contribution to the transverse spin scatt ering.τϕrather contributes to the transverse spin conductivity σtr∝n mτϕand it plays the same role to the transverse component of spin curr ent as the momentum relaxation time τdoes in the ordinary Drude conductivity. In fact, τϕagrees with the calculation in Ref.[35] when the weak ferromagnet limit is taken, i.e., µ↑≈µ↓≈ǫFandν↑≈ν↓≈ N. III. SPIN TRANSPORT A. Edelstein effect and spin-Hall effect: vanishing magnetis m An electrically generated nonequilibrium spin density due to spin-orbit coupling2can be extracted from Eq. (17) by setting the exchange interaction to zero ∆ xc= 0. If we keep D’yakonov-Perel as the only spin relaxation mechanis m and letτsf=∞, Eq. (17) reads D∇2S−S+Szˆz τdp+2C∇z×S+B∇zn= 0, (18) which also describes the spin-Hall effect in the diffusive regime.32,36,37Besides the spin relaxation, the second term in Eq. (18), the spin dynamics is controlled by two competing effects: the spin precession around the Rashba field (third term) and the electrical spin generation first pointed out by Edelstein.2In an infinite medium where a charge current is flowing along the ˆxdirection, Eq. (18) leads to a solution S=eEτdpBn ǫFˆy=eEζ πvFˆy, (19) where only the linear term in electric field has been retained. On the rig ht-hand side, we have used the charge density in a two-dimensional system n=k2 F/(2π) and introduced the parameter ζ=αkFτas used in Ref.[32]. In the presence of a weak spin-orbit coupling, only the spin precession term survives ; the electrical spin generation dominates when the coupling is strong. B. Spin diffusion in a ferromagnet Spin diffusion in a ferromagnet has been discussed actively in the field o f spintronics34,35,38,39. In this section we show explicitly that, by suppressing the Rashba spin-orbit coupling, Eq. (17) is able to describe spin diffusion in a ferromagnetic metal. A vanishing Rashba spin-orbit coupling means α= 0 and Eq. (17) reduces to ∂S ∂t=D∇2S+m×S τ∆−S τsf−m×(S×m) τϕ, (20) This equation only differs from the result of Ref. [38] by a dephasing t erm of the transverse component of the spin density. In a ferromagnetic metal, we may divide the spin density into a longitudinal component that follows the magneti- zation direction adiabatically, and a deviation that is perpendicular to the magnetization, i.e., S=s0m+δSwhere s0is the local equilibrium spin density. Such a partition, after restoring the electric field by ∇→∇+(e/ǫF)E, gives rise to ∂ ∂tδS+∂ ∂ts0m=s0D∇2m+D∇2δS+DePFNFE·∇m −δS τsf−s0m τsf−δS τϕ+1 τ∆m×δS, (21)7 where the magnetic order parameter is allowed to be spatial depend entm=m(r,t). We introduce PFthe spin polarization and NFthe density of state; both are at Fermi energy ǫF. In a smooth magnetic texture, the characteristic length scale of t he magnetic profile is much larger than the length scale of electron transport; we discard the contribution D∇2δS.34The diffusion of the equilibrium spin density follows s0D∇2m≈s0m/τsf. Inthispaper, weretainonlytermsthatareatfirstorderintempo ralderivative, whichsimplifies Eq. (21) to ξδS τ∆−m×δS τ∆=DePFNFE·∇m−s0∂m ∂t. (22) The last equation can be solved exactly to show δS=τ∆ 1+ξ2/bracketleftbiggPF em×(je·∇)m+ξPF e(je·∇)m −s0m×∂m ∂t−ξs0∂m ∂t/bracketrightbigg (23) whereξ=τ∆(1/τsf+ 1/τϕ) and the electric current je=e2nτE/mis given in terms of electron density n. Apart from the transverse dephasing time absorbed in parameter ξ, the nonequilibrium spin density Eq. (23) agrees with Eq. (8) in Ref.[34]. Given the knowledge of the nonequilibrium spin densit y, the spin torque, defined as T=−1 τ∆m×δS+1 τϕδS, (24) is given by T=−(1−ξ˜β)s0∂m ∂t+˜βs0m×∂m ∂t +(1−ξ˜β)PF e(je·∇)m−˜βPF em×(je·∇)m (25) whereβ=τ∆/τsfand˜β=β/(1 +ξ2). By assuming a long dephasing time of the transverse component τϕ→ ∞ thenξ≈β, Eq. (25) reproduces Eq. (9) in Ref.[34]. On the other hand, a shor t spin dephasing time τϕ→0 yields ˜β→0 which results in a pure adiabatic torque, i.e., the torque reduces to the first and third terms in Eq. (25). IV. RASHBA SPIN TORQUE The primary focus of this article is the Rashba torque originating fro m the coexistence of magnetism and Rashba spin-orbit coupling. In this section, we apply Eqs. (16) and (17) to s tudy the properties of this torque and concentrate on possible analytical aspects in the bulk system or an infinite medium. Analytical results provide a better under- standing of the physical processes behind the Rashba torque and a more transparent view on the structure of the diffusion equations derived in Sec. II. To serve this purpose, we firs t derive a formula that characterizes the general symmetry and angular dependence of the Rashba torque. Then fo r two limiting cases at weak and strong spin-orbit couplings, we are able to directly compare our results to experiment s. A. General symmetry and angular dependence Recent studies showed that the spin-orbit torque in a ferromagne tic heterostructure possesses peculiar symmetries with respect to magnetization inversion and a complex angular depen dence.26To be more specific, the angular dependence discussed here refers to the experimental observa tion that the torque amplitudes vary as functions of magnetization direction. We demonstrate in the following that such s ymmetries and angular dependence are encoded coherently in our simple model. Needless to say, finding a general analytical solution to Eq. (17) wit h boundary conditions is by no means an easy task. But, such solutions to the spin density and spin torque do exis t in an infinite medium and the behavior featured by these solutions, as the numerical solutions suggest, persists in to a finite system.40We reorganize Eq. (17) as S τsf+S+Szˆz τdp+1 τ∆S×m+1 τϕm×(S×m) =X, (26)8 where the right-hand side combines the time and spatial derivatives of the spin and charge densities X≡ −∂S ∂t+D∇2S+B∇zn+2C∇z×S+2R(m·∇zn)m +Γ[m×(∇z×S)+∇z×(m×S)]. (27) A stationary state solution defined by ∂S/∂t= 0 is of our current interest, whereas, in the next section, we will s ee that this term induces a correction contributing to the spin and cha rge pumping effects. In an infinite medium with an applied electric field E, we again replace the spatial gradient ∇by (e/ǫF)EandXreduces to X≈e ǫF[Bnˆz×E+2C(ˆz×E)×S+2Rn(m·(ˆz×E))m +Γm×((ˆz×E)×S)+Γ(ˆz×E)×(m×S)] (28) where we have discarded ∇2Sthat is quadratic in E. For a general expression Eq. (28), we may solve Eq. (26) using the partition S=S/bardblm+δS. A lengthy algebra leads to S/bardbl=τdpγθχθX·/bracketleftBig m(1+ξχ˜βsin2θ) −χ˜βmz(ξm׈z×m+ˆz×m)/bracketrightBig , (29) δS=τ∆γθ 1+ξ2[m×X+ξm×X×m−ξχθmz(X·m)m׈z×m +χθ(mzX·m+βX·(ˆz×m))ˆz×m] (30) whereθis defined as the azimuthal angle between mandˆzand χ=τsf τsf+τdp, χθ=χ 1+χcos2θ, γθ=1 1+ξ˜βχθsin2θ. (31) The results of spin density in Eq. (30) give rise to the spin torque defi ned by Eq. (24). In an infinite medium, the spin torque reads T=γθ/bracketleftBig (1−ξ˜β)m×X×m+˜βm×X +˜βχθ[(ξ−β)X·(ˆz×m)−mz(X·m)]ˆz×m −χθ[(1−ξ˜β)mzX·m+˜βX·(ˆz×m)]m׈z×m/bracketrightBig , (32) which clearly exhibits two outstanding features. First, it is possible t o divide the torque into two components that are either odd or even with respect to inversion of magnetization direct ion. Second, every component has a pronounced angular dependence. The formulation of Eq. (32) motivates us to attempt an interpreta tion ofXas asourceterm, which allows us to extend the applicability of Eq. (32) to include other driving mechan isms due to temporal variation or magnetic texture. We may further simplify Eq. (28) by observing that all non equilibrium spin and charge densities shall–in the lowest order–be linear in E. As we are only interested in the linear response regime, we can appr oximate Sby an equilibrium value nPFmand the source term becomes X≈eNF[Bˆz×E+2CPF(ˆz×E)×m+2R(m·(ˆz×E))m +ΓPFm×((ˆz×E)×m)], (33) which serves as a starting point of the following discussions on spin to rques in two major limits. B. Weak spin-orbit coupling In our system, a weak Rashba spin-orbit coupling implies a low D’yakono v-Perel relaxation rate 1 /τdp∝α2such thatτdp≫τsf,τ∆, indicating the spin relaxation is dominated by random magnetic impurit ies. In this regime, spin precession about the total field dominates the electrical spin gene ration; we may retain only Cand Γ terms in Eq. (33) and discard BandRterms that are of higher order in α. Therefore, when an electric field is applied along the ˆxdirection, Eq. (33) becomes X≈eNFPFE[2Cˆy×m+Γm×(ˆy×m)] (34)9 and the torque given in Eq. (32) reduces to a commonly accepted fo rm T=T⊥ˆy×m+T/bardblm×(ˆy×m), (35) consisting of both out-of-plane ( T⊥) and in-plane ( T/bardbl) components with magnitudes determined by T⊥=eEPFNF/bracketleftBig 2(1−ξ˜β)C+˜βΓ/bracketrightBig , (36) T/bardbl=eEPFNF/bracketleftBig (1−ξ˜β)Γ−2˜βC/bracketrightBig . (37) Note that the in-plane torque in Eq. (37) may experience a sign flip, d epending on the competition between spin relaxation and precession. To compare directly with the results in Ref.[21], we allow τsf→ ∞andτϕ→ ∞, thenβ≈0. Under these assumptions, we have T⊥≈2eEPFNFCandT/bardbl≈eEPFNFΓ. In the dirty limit, Γ ≪Cdue to ∆ xcτ≪1. By making use of the relation for the polarization PF= ∆xc/ǫFand the Drude relation je=e2nτE/m, we obtain the out-of-plane torque T= 2αm∆xc eǫFjeˆy×m, (38) which agrees with the spin torque in an infinite system in the correspo nding limit as derived in Ref.[21]. C. Strong spin-orbit coupling In the presence of a strong spin-orbit coupling, two effects are do minating: electric generation of spin density2and D’yakonov-Perel spin relaxation mechanism.27As the electric field is aligned along the ˆxdirection, Xis simplified to be X≈eNFE[Bˆy+2CPFˆy×m+2Rmym+ΓPFm×(ˆy×m)], (39) and the corresponding spin torque is T=γθ(T0 ⊥ˆy×m+T0 /bardblm׈y×m) +γθχθ(Tx ⊥mx+Tyz ⊥mymz)ˆz×m +γθχθ(Tx /bardblmx+Tyz /bardblmymz)m×(ˆz×m), (40) where the torque amplitude parameters are defined as T0 /bardbl=eENF[2(1−ξ˜β)CPF+˜β(B+ΓPF)], T0 ⊥=eENF[−2˜βCPF+(1−ξ˜β)(B+ΓPF)], Tx /bardbl=−eENF˜β(B+ΓPF), Tyz /bardbl=eENF[2˜βCPF−(1−ξ˜β)(B+2R)], Tx ⊥=−eENF˜β(ξ−β)(B+ΓPF), Tyz ⊥=−eENF[2(ξ−β)CPF+(1−ξ˜β)(B+2R)]. (41) Equation (40) comprises another major result of this paper. The fi rst term, an out-of-plane torque, can be un- derstood as a fieldlike torque produced through the ferromagnet and the spin density generated by the inverse spin- galvanic effect. The second term, an in-plane torque, originates fr om the Slonczewski-Berger-typespin-transfer torque that requires spin dephasing of the transverse component of spin density. The last two terms ˆz×mandm×(ˆz×m) are governed by the anisotropy in spin relaxation which allows the gen eration of spin density components to be per- pendicular to both mand the effective Rashba field. In general, the relative magnitude of these different terms are material dependent. In fact, the symmetry reflected in Eq. (40) compares favorably t o the spin torque formula proposed based on experiments by Garello et al.26More interestingly, if we allow the anisotropy in spin relaxation time to v anish by taking τdp≫τsf, Eq. (40) reduces to the form of Eq. (35) consisting only of the in- plane and out-of-plane components, whereas the complex angular dependence diminishes a ccordingly. This is a strong indication that this angular dependence discovered in our model arises from the anisot ropic spin relaxation. Meanwhile, such an angular dependence obtained here in an infinite medium persists into a realistic experimental setup with boundaries and it is insensitive to the change in sample size.4010 −25 −15 −5 5 15 25 −2 −1 0 y (nm) −25 −15 −5 5 15 25 135 y (nm) Nonequilibrium Spin Density: S y / (P Fje/eD) (nm) −0.8−0.40Nonequilibrium Spin Density: S z / (P Fje/eD) (nm) 0.51.52.5 α = 0.005 eV nm 0.01 0.03 0.05α = 0.0005 eV nm 0.001 0.003 0.005a) c) b) d) FIG. 2. (Color online) Spatial profile of the nonequilibrium spin density Sz(a),(c) and Sy(b),(d) for various values of the Rashba constant. The width of the wire is L= 50 nm. The magnetization direction is along the ˆxaxis. Other parameters are momentum relaxation time τ= 10−15s, exchange splitting τ∆= 10−14s, spin relaxation time τsf= 10−12s, and the Fermi vectorkF= 4.3 nm−1. V. NUMERICAL RESULTS In previous sections, analytical results for the spin density and Ra shba torque were obtained in various limits with respect to the relative magnitude between the spin-orbit coupling a nd exchange field. In this section, we numerically solve Eqs. (16) and (17) to demonstrate that they provide a cohe rent framework to describe the spin dynamics as well as spin torques in the diffusive regime for a wider range of parameter s. Here, we consider an in-plane magnetization that lies along the ˆxdirection and another case where the magnetization is perpendicula r to the thin film plane is reported elsewhere.23For such a two-dimensional electron system, we adopt the following boundary conditions. First, we enforce a vanishing spin accumulation at the edges along the tran sverse direction, i.e., S(y= 0,L) = 0. This condition implies a strong spin-flip scattering at the edges, which is co nsistent with the experimental observations in spin-Hall effect.6Second, an electric field is applied along the ˆxdirection; therefore, we set the charge densities at two ends of the propagation direction to be constant nL=nR=nF. The second boundary condition sets the charge density at the Fermi level. Equivalently, one can apply a voltage drop along the transport direction instead of an explicit inclusion of an electric field. The numerical results of the spin densities are summarized in Fig. 2. F rom the top panels [(a),(b)] to the lower ones [(c),(d)], for a fixed exchange splitting, the system transitions fro m a weak (spin-orbit-) coupling regime to a strong coupling regime. To illustrate this transition, the Szcomponent of the spin density evolves from a symmetric spatial distribution in the weak spin-orbit-coupling regime, with α= 5×10−4eV nm in Fig. 1(a), to an antisymmetric spatial distribution in the strong coupling regime, with α= 5×10−2eV nm in Fig. 1(c). Note that throughout this transition, the in-plane spin density Syis robust yet roughly constant in the bulk. This change in symmetry and the emergence of peaks close to the bo undaries are resulting from the competition between the Rashba and exchange fields. In the weak coupling regim e, the total field is dominated by the exchange field pointing at the ˆxdirection, about which the spin density profile is symmetric in space. A s the spin-orbit coupling increases, the total field is tilted towards the ˆyaxis; then the spin projections along + yand−yare no longer symmetric, as indicted by curves with intermediate αvalues in Figs. 2(a) and 2(b). In the strong coupling regime, when the Rashba coupling overrules the exchange field, the antisym metric profile of Szand the symmetric one of Sy follow naturally from the spin-Hall effect induced by the spin-orbit int eraction. The out-of-plane and in-plane torques are plotted in Fig. 3 with resp ect to the Rashba constant αfor various exchange splittings. The transition regions are of particular intere st. During the transition from a weak to strong coupling, see Fig. 3(a), the magnitude of the out-of-plane torque T⊥first reaches a plateau, then rises again as α increases. In the large αlimit, though the magnitude of the torque increases with α, the torque efficiency defined as dT⊥/dαis actually smaller than it is in the weak coupling. This picture is consisten t with the semiclassical Boltzmann equation description in Ref. [21]. This behavior is caused by the differe nt processes generating the Rashba torque in both regimes. As discussed in Sec. IVB and Sec. IVC, in the weak co upling regime, the torque is dominated by the spin precession around the Rashba field, whereas in the strong coupling, the electrical generation of spin density11 0.01 0.03 0.05048x 1013 α (eV nm)T⊥ / (P F j e / e D) (nm s −1 ) 0.01 0.03 0.05012x 1013 α (eV nm) T|| / (P F j e / e D) (nm s −1 ) ∆xc = 0.004 eV 0.007 0.01 0.02a) b) FIG. 3. (Color online) Magnitude of the out-of-plane torque T⊥(a) and in-plane torque T/bardbl(b) as a function of Rashba constant for various exchange splitting. Other parameters are the sa me as in Fig. 2. dominates. These two distinct processes show different efficiencies . The in-plane torque T/bardblbehaves differently. In the strong coupling limit, T/bardblis proportional to 1 /αdue to the large D’yakonov-Perel spin relaxation rate that is of order α2. A stronger spin-orbit coupling therefore means a decrease in the torque magnitude. The transition suggests that the optimal m agnitude of the in-plane torque is achieved when the exchange energy is about the same order of magnitude as the R ashba splitting αkF. VI. DYNAMICS Our focus in the previous sections has always been on a stationary s tate with a homogeneous magnetization and the temporal and spatial variations of the ferromagnetic order p arameter are neglected entirely. In this section, we demonstrate that the formulation outlined in Eq. (17) is able to addr ess dynamic effects such as spin pumping and magnetization damping. We shall consider only the adiabatic limit where the frequency of magnetization motion is much lower than that of any electronic processes. Without the loss of generality, the anisotropy in spin relaxation is suppressed, for we are keen to provide a qualitative picture rathe r than pinpointing subtleties. A. Spin and charge pumping Now we consider a homogeneous single-domain ferromagnet with a mo ving magnetization in the absence of external electricfield. Intheadiabaticlimit, whiletreatingthespin-orbitcouplin gasaperturbation,thelowest-ordercorrection to the spin dynamics is to let the source term X≈ −∂tS. (42) The magnetization motion brings the system out of equilibrium and indu ces a nonequilibrium spin density. We can no longer naively assume that the spin density is alwaysfollowing the magn etization direction. To get the nonequilibrium part, we perform the usual decomposition S=s0m+δSas in Sec. IIIB, where δSis referring to the nonequilibrium part induced by the magnetization motion. Here, we neglect terms lik eD∇2δSand a simple algebra leads to δS=τ∆s0 1+ξ2/parenleftbigg m×dm dt+ξdm dt/parenrightbigg . (43) Equation (43) is a formal analogy to the conventional spin pumping t heory developed in magnetic multilayers using the scattering matrix approach.39,41Two components exist in the pumping-induced spin density and both o f them are perpendicular to the magnetization direction. In the absence of sp in-flip scattering τsf→ ∞thusξ≪1 (in the dirty limit considered here), the first term m×˙mdominates. In the conventional spin pumping theory, this contribu tion is governed by the real part of the spin-mixing conductance that is u sually much larger than its imaginary counterpart associated with ˙m. Equation (43) seems to suggest a similar trend. On the other hand , a strong spin-flip scattering is expected to be detrimental to the nonequilibrium spin density, whic h is also encoded in Eq. (43): the magnitude ofδSdecreases when the spin-flip relaxation rate 1 /τsfincreases.12 Furthermore, the spin density induced by the magnetization motion generates a charge current via the spin galvanic effect,42which can be estimated qualitatively to be Jc∝ατ∆s0 1+ξ2ˆz×/parenleftbigg m×dm dt+ξdm dt/parenrightbigg (44) The magnitude of the charge current is proportional to the frequ ency of the magnetization precession. B. Magnetic damping Whenamagnetizationmovesinaseaofitinerantelectrons,thecoup lingbetweenthelocalizedanditinerantelectrons induces a frictionto this motion. This friction has been described in terms of the recipr ocal of the spin pumping in a magnetic texture.43The dynamical motion of the magnetic texture pumps a spin current that contributes to a magnetic damping when reabsorbed by the texture. In the presen t case, we show that the pumping of a spin-polarized current studied above can also contribute to the magnetic damping following the same process. In order to describe a magnetic texture, we allow the magnetization direction to assume a spatial dependence, i.e., m=m(r). We limit ourselves to a weak Rashba spin-orbit coupling in order to avoid the c omplexity due to anisotropy in spin relaxation. For the present purpose, we keep the following sourceterm: X=−∂tS+D∇2S+2C(z×∇)×S. (45) To be more specific, we identify two sources for spatially dependent magnetic damping. One comes from the interplay between the diffusive spin dynamics and the magnetization m otion, i.e., the second term in Eq. (45). The other apparently attributes to the Rashba torque, i.e., the third t erm in Eq. (45). We consider here an adiabatic magnetization dynamics, meaning that the electronic spin process, characterized by a time scale τ∆, is the fastest whereas the magnetization motion, with a time scale τM, is the slowest. Without loss of generality, we allow the spin dephasing time to sit in between, i.e., τ∆≪τϕ≪τM. Under these assumptions, the nonequilibrium spin density pumped by the magnetization motion reads δS≈ −s0τ∆m×∂tmand the spatial dependent damping torques are given by TD=−s0τ∆Dm×[m×∇2(m×∂tm)], (46) TR= 2s0τ∆Cm×{m×[(z×∇)×(m×∂tm)]}. (47) It is worth pointing out the symmetry properties of the last two dam ping torques. The damping torque due to spin diffusion, TD, is second order in the spatial gradient and is thus invariant under s patial inversion ∇→ −∇. In fact, Eq.(46) has the same symmetry as the damping torque obtained by Z hang and Zhang.43The other damping torque that arises from the Rashba spin-orbit coupling, TR, is only linearin spatial gradient and is therefore referred to as chiral. In other words, in contrast to TD, the magnetic damping due to Rashba spin-orbit coupling is antisymme tric upon spatial inversion. Equation (47) is in agreement with the dampin g formula derived by Kim et al.44Moreover, a more complex angular dependence of the damping coefficient emerg es when the D’yakonov-Perel anisotropic spin relaxation is taken into account. VII. DISCUSSION Current-induced magnetization dynamics in a single ferromagnetic la yer has been observed in various structures that involve interfaces between transition metal ferromagnets, heavy metals, and/or metal-oxide insulators. Existing experimental systems are Pt/Co/AlO x,15,16,18,19Ta/CoFeB/MgO,17and Pt/NiFe and Pt/Co bilayers,28as well as dilute magnetic semiconductors such as (Ga,Mn)As.12,13Besides the structural complexity in such systems, an unclear picture of spin-orbit coupling in the bulk as well as at interfaces place s a challenge to unravel the nature of spin-orbit torque. A. Validity of Rashba model in realistic interfaces The well-knownRashba-typeeffective interfacial spin-orbit Hamilto nian was pioneeredby E. I. Rashba to model the influence of asymmetric interfaces in semiconducting two-dimension al electron gas:20a sharp potential drop, emerging at the interface (say, in the x-yplane) between two materials, gives rise to a potential gradient ∇Vthat is normal to13 the interface, i.e., ∇V≈ξso(r)ˆz. In case a rotational symmetry exists in-plane, a spherical Fermi surface assumption allows the spin-orbit interaction Hamiltonian to have the form ˆHR=αˆσ·(p׈z), where α≈ ∝an}b∇acketle{tξ∝an}b∇acket∇i}ht/4m2c2. As a matter of fact, in semiconducting interfaces where the transport is desc ribed by a limited number of bands around a high symmetry point, the Rashba form can be recovered by k·ptheory.45 However, one can properly question the validity of the simple Rashba spin-orbit-coupling model for interfaces involving heavy metals and ferromagnets, where the band structu re and Fermi surfaces are much more complex than low doped semiconducting two-dimensional electron gases for which it was initially proposed. Nonetheless, the existence ofa symmetrybreaking-inducedspin splitting ofthe Rash batype has been well established by angle-resolved photoemission spectroscopy in systems consisting of a wide variety of metallic surfaces,46–48quantum wells,49and even oxide heterointerfaces.50Several important works published in the past few years on spin spir als induced by Dzyaloshinskii-Moriya interaction at W/Fe and W/Mn interfaces51also argue in favor of the presence of a sizable Rashba-type spin-orbit coupling. Besides the aforementioned experimental investigations perform ed on clean and epitaxially grown systems, efforts in numerical calculations have been made to the identification of an as ymmetric spin splitting in the band structure of conventional metallic interfaces and surfaces. It is rather intrigu ing to observe that, in spite of the complexity of the band structure arising from complex hybridization among s,p, anddorbitals, first principle calculations do observe such ak-antisymmetric spin splitting in the energy dispersion of interfacial s tates.52–54Although this spin splitting is more subtle that the simple Rashba model depicted in Eq. (1), it ten ds to confirm the phenomenological intuition of Rashba20at metallic interfaces. B. Comparison between SHE torque and Rashba torque At this stage, it is interesting to compare the parameter dependen ce of the in-plane torque T/bardbl[in Eq. (37)] and the torque generated by a spin-Hall effect4in the bulk of a heavy metal material such as Pt. In the latter case, the torque T(SH)exerted on the normal metal/ferromagnet interface is obtained b y projecting out the spin current ( j(SH)due to spin-Hall effect) that is transverse to the magnetization direct ion.28In the bulk, the spin current can be estimated using the ratio between spin-Hall ( σSH) and longitudinal ( σxx) conductivities (the so-called spin-Hall angle), i.e., j(SH)=σSH σxxje. (48) A perturbation calculation using the second-order Born approxima tion gives rise to a spin current; thus the torque with a magnitude given by T(SH)=ηsomγ 2eτ0 trje (49) where, in general, γ >1 is a dimensionless parameter taking into account both side-jump ( γ= 1) and skew-scattering (γ >1) contributions to the spin-Hall effect.55ηsois the spin-orbit-coupling parameter and τ0 tris the transport relaxation time due to bulk impurities, the same definitions as in Ref.[55] except here the definition of spin current differs by a unit 1 /(2e). Meanwhile, the magnitude of the Rashba-induced in-plane torque , i.e., Eq. (37), can be simplified to, since ∆ xcτ≪1, T/bardbl≈4αm ǫFeτsfje. (50) The spin-orbit-coupling parameter αin our definition in Eq. (1) has the unit of energy. Equations (50) and (49) actually show that the in-plane Rashba torque and the spin-Hall tor que have a very similar parameter dependence. Meanwhile, a diffusive description of the bilayer system, consisting of ferromagnet/heavy metal, has shown that both SHE torque and Rashba torque adopt a similar form, T=T/bardblm×(ˆy×m)+T⊥ˆy×m. (51) The similarity in the geometrical form of the two torques implies that, in principle, they are able to induce the same type of magnetic excitation.56 The complexity of the underlying physics of spin-orbit torque and th e geometrical similarity between spin-Hall- induced and Rashba torque make it a challenge to distinguish between two possible origins. Recent progress has been made towards a plausible distinction between the bulk and interfacial origin of different torque components by varying the bilayer thickness,57decoupling the heavy metal from the ferromagnet58or dusting the interface with impurities,59 which has revealed additional complex behaviors that question curr ent models including the Rashba model.14 VIII. CONCLUSION Using Keldysh technique, in the presence of both magnetism and a Ra shba spin-orbit coupling, we derive a spin diffusion equation that provides a coherent description to the diffus ive spin dynamics. In particular, we have derived a general analytical expression for the Rashba torque in the bulk o f a ferromagnetic metal layer in both weak and strong Rashba limits. We find that the spin-orbit torque in general c onsists of not only in-plane and out-of-plane components but also a complex angular dependence, which we attrib ute to the anisotropic spin relaxation induced by the D’yakonov-Perel mechanism. In the presenceofmagnetizationdynamics, wehavedemonstrate dthat ourspin diffusion equationisableto describe a wealth of phenomena including spin pumping and magnetic damping. In particular, these results are in agreement with the earlier ones derived using other methods. We have discusse d the common features shared by the Rashba and SHE torques. We also expect that further investigation involving st ructural modification of the system shall provide a deeper knowledge on the interfacial spin-orbit interaction as well as the current-induced magnetization switching in a single ferromagnet. ACKNOWLEDGMENTS We thank G. E. W. Bauer, H. -W. Lee, K. -J. Lee, J. Sinova, M. D. St iles, X. Waintal, and S. Zhang for numerous stimulating discussions. ∗xuhui.wang@kaust.edu.sa †aurelien.manchon@kaust.edu.sa 1M. I. D’yakonov and V. I. Perel, JETP Lett. 13, 467 (1971). 2V. M. Edelstein, Solid State Commun. 73, 233 (1990). 3S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301, 1348 (2003). 4J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999); S. Zhang, Phys. Rev. Lett. 85, 393 (2000). 5J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, an d A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004). 6Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Sci ence306, 1910 (2004). 7A. S. N´ u˜ nez and A. H. MacDonald, Solid State Commun. 139, 31 (2006). 8K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008). 9J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996). 10M. D. Stiles and J. Miltat, Top. Appl. Phys. 101, 225 (2006); D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater .320, 1190 (2008); J. Z. Sun and D. C. Ralph, J. Magn. Magn. Mater. 320, 1227 (2008). 11I. Garate and A. H. MacDonald, Phys. Rev. B 80, 134403 (2009); P. M. Haney and M. D. Stiles, Phys. Rev. Lett. 105, 126602 (2010); H. Li, X. Wang, F. Dogan, and A. Manchon, Appl. Phys. Lett. 102, 192411 (2013). 12A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y. Lyanda-Ge ller, and L. P. Rokhinson, Nature Phys. 5, 656 (2009). 13D. Fang, H. Kurebayashi, J. Wunderlich, K. V´ yborn´ y, L. P. Z ˆ arbo, R. P. Campion, A. Casiraghi, B. L. Gallagher, T. Jungwirth, and A. J. Ferguson, Nature Nanotechnol. 6, 413 (2011). 14M. Endo, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 97, 222501 (2010). 15I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pi zzini, J. Vogel, and P. Gambardella, Nature Mater. 9, 230 (2010). 16U. H. Pi, K. W. Kim, J. Y. Bae, S. C. Lee, Y. J. Cho, K. S. Kim, and S . Seo, Appl. Phys. Lett. 97, 162507 (2010). 17T. Suzuki, S. Fukami, N. Ishiwata, M. Yamanouchi, S. Ikeda, N . Kasai, and H. Ohno, Appl. Phys. Lett. 98, 142505 (2011). 18I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu, S . Auffret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonfim, A. Schuhl, and G. Gaudin, Nature Mater. 10, 419 (2011). 19I. M. Miron, K. Garello, G. Gaudin, P. J. Zermatten, M. V. Cost ache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature (London) 476, 189 (2011). 20Yu. A. Bychkov and E. I. Rashba, J. Phys. C: Solid State Phys. 17, 6039 (1984). 21A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008); A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009). 22A. Matos-Abiague and R. L. Rodriguez-Suarez, Phys. Rev. B 80094424 (2009). 23X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201 (2012). In Fig. 3(a) of that paper, a typo misplaces t he diffusion constant Din the expression of the vertical axis. 24D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86, 014416 (2012). 25E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406 (2012). 26K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov , S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Nature Nanotechnol. 8, 587 (2013). 27M. I. D’yakonov and V. I. Perel, Sov. Phys. Solid State 13, 3023 (1972).15 28L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett.106, 036601 (2011); L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012); L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, a nd R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). 29P. M. Haney, H. W. Lee, K. -J. Lee, A. Manchon, and M. D. Stiles, Phys. Rev. B 87, 174411 (2013). 30K. W. Kim, S. M. Seo, J. Ryu, K. J. Lee, and H. W. Lee, Phys. Rev. B 85, 180404(R) (2011); S. M. Seo, K. W. Kim, J. Ryu, H. W. Lee, and K. J. Lee, Appl. Phys. Lett. 101, 022405 (2012). 31J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). 32E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett.93, 226602 (2004). 33J. Rammer, Quantum Field Theory of Non-equilibrium States (Cambridge University Press, Cambridge, UK, 2007). 34S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 35Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale, Phys. Rev. B79, 094415 (2009). 36A. A. Burkov, A. V. N´ u˜ nez, and A. H. MacDonald, Phys. Rev. B 70, 155308 (2004). 37˙I. Adagideli and G. E. W. Bauer, Phys. Rev. Lett. 95, 256602 (2005). 38S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002). 39Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin , Rev. Mod. Phys. 77, 1375 (2005). 40C. Ortiz-Pauyac, X. Wang, M. Chshiev, and A. Manchon, Appl. P hys. Lett. 102, 252403 (2013). 41Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Let t.88, 117601 (2002). 42S. D. Ganichev, E. L. Ivchenko, V. V. Belkov, S. A. Tarasenko, M. Sollinger, D. Weiss, W. Wegscheider, and W. Prettl, Nature (London) 417, 6885 (2002). 43S. Zhang and S. S. -L. Zhang, Phys. Rev. Lett. 102, 086601 (2009). 44K. W. Kim, J. H. Moon, K. J. Lee, and H. W. Lee, Phys. Rev. Lett. 108, 217202 (2012). 45R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, Berlin, 2003). 46S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev. Lett. 77, 3419 (1996); F. Reinert, G. Nicolay, S. Schmidt, D. Ehm, and S. H¨ ufner, Phys. Rev. B 63, 115415 (2001); Nicolay G, F. Reinert, S. Hefner, and P. Blah a, Phys. Rev. B 65, 033407 (2001); M. Hoesch, M. Muntwiler, V. N. Petrov, M. Heng sberger, L. Patthey, M. Shi, M. Falub, T. Greber, and J. Osterwalder, Phys. Rev. B 69, 241401(R) (2004). 47O. Krupin, G. Bihlmayer, K. Starke, S. Gorovikov, J. E. Priet o, K. D¨ obrich, S. Bl¨ ugel, and G. Kaindl, Phys. Rev. B 71, 201403(R) (2005); O. Krupin, G. Bihlmayer, K. M. D¨ obrich, J . E. Prieto, K. Starke, S. Gorovikov, S. Bl¨ ugel, S. Kevan, an d G. Kaindl, New. J. Phys. 11, 013035 (2009). 48Ph. Hofmann, Prog. Surf. Sci. 81, 191 (2006); T. Hirahara, K. Miyamoto, I. Matsuda, T. Kadono , A. Kimura, T. Nagao, G. Bihlmayer, E. V. Chulkov, S. Qiao, K. Shimada, H. Namatame, M . Taniguchi, and S. Hasegawa, Phys. Rev. B 76, 153305 (2007); H. Mirhosseini, J. Henk, A. Ernst, S. Ostanin, C. T. C hiang, P. Yu, A. Winkelmann, and J. Kirschner, Phys. Rev. B 79, 245428 (2009); A. Takayama, T. Sato, S. Souma, and T. Takaha shi, Phys. Rev. Lett. 106, 166401 (2011); K. Ishizaka, M. S. Bahramy, H. Murakawa, M. Sakano, T. Shimojima, T. Sonob e, K. Koizumi, S. Shin, H. Miyahara, A. Kimura, K. Miyamoto, T. Okuda, H. Namatame, M. Taniguchi, R. Arita, N. N agaosa, K. Kobayashi, Y. Murakami, R. Kumai, Y. Kaneko, Y. Onose, and Y. Tokura, Nature Mater. 10, 521 (2011). 49A. Varykhalov, J. S´ anchez-Barriga, A. M. Shikin, W. Gudat, W. Eberhardt, and O. Rader, Phys. Rev. Lett. 101, 256601 (2008); J. Hugo Dil, F. Meier, J. Lobo-Checa, L. Patthey, G. B ihlmayer, and J. Osterwalder, Phys. Rev. Lett. 101, 266802 (2008); A. G. Rybkin, A. M. Shikin, V. K. Adamchuk, D. Marchen ko, C. Biswas, A. Varykhalov, and O. Rader, Phys. Rev. B82, 233403 (2010). 50A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cancelli eri, and J. M. Triscone, Phys. Rev. Lett. 104, 126803 (2010). 51P. Ferriani, K. von Bergmann, E. Y. Vedmedenko, S. Heinze, M. Bode, M. Heide, G. Bihlmayer, S. Bl¨ ugel, and R. Wiesen- danger, Phys. Rev. Lett. 101, 027201 (2008). 52A. N. Chantis, K. D. Belashchenko, E. Y. Tsymbal, and M. van Sc hilfgaarde, Phys. Rev. Lett. 98, 046601 (2007). 53G. Bihlmayer, S. Bl¨ ugel, and E. V. Chulkov, 2007 Phys. Rev. B75, 195414 (2007); G. Bihlmayer, Yu. M. Koroteev, P. M. Echenique, E. V. Chulkov, and S. Bl¨ ugel, Surf. Sci. 600, 3888 (2006); M. Nagano, A. Kodama, T. Shishidou, and T. Oguc hi, J. Phys.: Condens. Mater. 21, 064239 (2008). 54S. R. Park, C. H. Kim, J. Yu, J. H. Han, and C. Kim, Phys. Rev. Let t.107, 156803 (2011); J. H. Park, C. H. Kim, H. W. Lee, and J. H. Han, Phys. Rev. B 87, 041301 (2013). 55S. Takahashi and S. Maekawa, J. Phys. Soc. Jpn. 77, 031009 (2008). 56A. Manchon, arXiv:1204.4869. 57J. Kim, J. Sinha, H. Hayashi, M. Yamanouchi, S. Fukami, T. Suz uki, S. Mitani, and S. Ohno, Nature Mater. 12, 240 (2012). 58X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, and J. Q. Xiao, Nat ure Commun. 4, 1799 (2013). 59K. S. Ryu, L. Thomas, S. H. Yang, and S. Parkin, Nature Nanotec hnol.8, 527 (2013).

2106.07762v1.Semi_Implicit_finite_difference_methods_to_study_the_spin_orbit_and_coherently_coupled_spinor_Bose_Einstein_condensates.pdf

Semi-Implicit nite-dierence methods to study the spin-orbit and coherently coupled spinor Bose-Einstein condensates Paramjeet Banger∗, Pardeep Kaur†, Sandeep Gautam‡ June 16, 2021 Abstract We develop time-splitting nite dierence methods, using implicit Backward- Euler and semi-implicit Crank-Nicolson discretization schemes, to study the spin-orbit coupled spinor Bose Einstein condensates with coherent coupling in quasi-one and quasi-two-dimensional traps. The split equa- tions involving kinetic energy and spin-orbit coupling operators are solved using either time implicit Backward-Euler or semi-implicit Crank-Nicolson methods. We explicitly develop the method for pseudospin-1/2, spin- 1 and spin-2 condensates. The results for ground states obtained with time-splitting Backward-Euler and Crank-Nicolson methods are in excel- lent agreement with time-splitting Fourier spectral method which is one of the popular methods to solve the mean-eld models for spin-orbit cou- pled spinor condensates. We conrm the emergence of dierent phases in spin-orbit coupled pseudospin-1/2, spin-1 and spin-2 condensates with coherent coupling. 1 Introduction With experimental realization of optical traps [1], all the hyperne spin states of spin-fultracold bosonic atoms could be trapped and that led to the discovery of 2f+ 1 component Bose Einstein condensates (BECs) termed as spinor BECs [2]. A spinor condensate can be described by a 2 f+ 1 component order param- eter that can vary over space and time [3, 4]. Till date, spinor condensates in ultracold gases of spin-1/287Rb [5], spin-123Na [1], spin-187Rb [6], spin-223Na [7], spin-287Rb [6] and spin-352Cr atoms [8] have been experimentally realized. In later experiments [9], spin-orbit coupling (SOC) was also engineered in neu- tral quantum gases like spinor BECs by controlling the atom light interaction that led to the generation of articial gauge potentials coupled to the atoms [10, 11, 12]. SOC was rst realised experimentally in a BEC of87Rb [9] by dressing two of its internal spin states from within the ground-state manifold by employing pair of Raman lasers that can create a momentum sensitive coupling between two internal atomic states resulting in an eective Zeeman shift. The ∗2018phz0003@iitrpr.ac.in †2018phz0004@iitrpr.ac.in ‡sandeep@iitrpr.ac.in 1arXiv:2106.07762v1 [cond-mat.quant-gas] 14 Jun 2021strength of SOC can be tuned by Raman laser wavelength, whereas the coherent coupling can be tuned by the laser intensity [13]. SOC and spin-dependent in- teractions provide a new platform to explore the novel phases in spin-orbit (SO) coupled spinor BECs [14, 15]. In the mean-eld approximation, a spin- fBEC in the presence of SO and coherent couplings can be well described by a set of 2f+ 1 coupled nonlinear Gross-Pitaevskii equations (CGPEs) [3, 16, 17, 18]. A wide range of numerical techniques have been employed in literature to study the scalar BEC [19, 20, 21, 22] and spinor BECs [23, 24, 25, 26]. In our earlier works, we also provided sets of Fortran 90/95 codes to solve the mean-eld model of SO coupled f= 1 [27] and f= 2 [28] spinor BECs with Rashba SO-coupling using time-splitting Fourier spectral (TSFS) method. In the present work, we describe time-splitting nite-dierence methods to solve the 2 f+ 1 CGPEs of spin-f(f= 1=2;1;2) spinor BECs in quasi-one-dimensional (q1D), quasi-two- dimensional (q2D) traps with SO and coherent couplings. The method can be easily extended to three-dimensional traps and higher spin system (say spin-3 BEC) if needed. We use the time-splitting Backward-Euler (TSBE) or time- splitting Crank-Nicolson (TSCN) nite-dierence methods to solve the split equations corresponding to kinetic energy and spin-orbit coupling operators of spin-fBEC. These discretizaton schemes are employed with periodic boundary conditions and result in 2 f+1 decoupled sets of linear circulant systems of equa- tions for each spatial dimension. The key property of a circulant matrix is that its columns (rows) can be written in terms of powers of the shift matrix times the rst column (row), which allows it to be diagonalized using the discrete Fourier transform [31]. The implementation of TSBE and TSCN is discussed in all its detail for an SO-coupled pseudospin-1/2 condensate, and then extended to higher spin condensates. The rest of this paper is organized as follows: in section 2, we introduce a generic mean-eld model suitable to describe the prop- erties SO and coherently coupled pseudospin-1/2, spin-1 and spin-2 BECs. In section 3, we discuss the TSBE and TSCN schemes to numerically solve the CGPEs, i.e. the mean-eld model. In section 4, we present the results for en- ergies and component densities corresponding to the stationary states of these spinor BECs having f= 1/2, 1 and 2. We also compare the results of the nite dierence methods with the Fourier spectral method. 2 Spinor condensates with spin-orbit and coher- ent coupling A generic spinfcondensate with Rashba SO coupling can be modelled at temperatures well below the critical temperature with a matrix equation of form [3, 16] @ @t= (Hp+Hcoh+Hd+Hnd) ; (1) where is a 2 f+ 1 component order parameter, and =p1. In this work, we consider f= 1=2;1;2 corresponding, respectively, to pseudospin-1/2, spin-1 and spin-2 condensates. In Eq. (1), HpandHcohare 2f+ 12f+ 1 matrix 2operators dened as Hp=1^p2 x+ ^p2 y+ ^p2 z 2+ (Sx^pySy^px); (2) Hcoh = 2Sx; (3) where 1represents a 2 f+12f+1 identity matrix, and are the strengths of SO and coherent couplings, respectively, and ^ p=@=@ with=x;y;z .Sx andSyare the irreducible representations of the xandycomponents of angular momentum operators for spin- fmatrix, respectively. The ( m0;m)thelement of these 2f+ 12f+ 1 matrices are (Sx)m0;m= 2p (f(f+ 1)m0m)m0;m+1+p (f(f+ 1)m0m)m0+1;m ;(4) (Sy)m0;m= 2ip (f(f+ 1)m0m)m0;m+1p (f(f+ 1)m0m)m0+1;m ;(5) herem0andmvary fromf,f1;. . .;f. In Eqs. (4)-(5) = 2 forf= 1=2 and = 1 forf= 1;2. The interatomic interactions in the spinor condensate are accounted by diagonal matrix Hdand non-diagonal matrix Hnd. The trapping potential also enters into the Hdmatrix. In the present work, we consider the harmonic trapping potential for all the spinor condensates. These matrices for a pseudospin-1/2 condensate are [16] Hd=V+P2 l=1g1lj lj20 0 V+P2 l=1g2lj lj2 ; H nd= 0; (6) where V=1 2X 2 2; gll=4Nall aosc; gl;3l=4Nal;3l aosc; wheregllandgl;3lwithl= 1;2 are intra- and inter-species interaction strengths, respectively, aoscis the oscillator length chosen as a unit of length, Nis the total number of particles in the condensate, =!=!xwith=x;y;z is the ratio of conning-potential frequencies along th direction to xdirection. The intraspecies interaction strengths are dened in terms of s-wave scattering lengths,a11anda22, whereas interspecies interaction strength is dened in terms of interspecies s-wave scattering length a12=a21. Similarly, these matrices for spin-1 condensate are [3, 17, 29] Hd=0 @V+c0+c1(0+) 0 0 0 V+c0+c1+ 0 0 0 V+c0+c1(0)1 A; (7a) Hnd=c10 @0 0  1 0  0 1 0  0 1 0 0  1 01 A; (7b) herel=j lj2withl= 0;1,=P ll, and=+11and c0=4N(a0+ 2a2) 3aosc; c 1=4N(a2a0) 3aosc: (8) 3The interaction strengths c0andc1are dened in terms of s-wave scattering lengthsa0anda2. The subscript 0 or 2 in the scattering length characterises the total spin of the allowed scattering channel. Lastly, HdandHndfor a spin-2 condensate are [3, 18] Hd= diag (h+2;h+1;h0;h1;h2); (9a) Hnd=0 BBBB@0h12h13 0 0 h 12 0h23 0 0 h 13h 23 0h34h35 0 0h 34 0h45 0 0h 35h 45 01 CCCCA; (9b) where h2=V+c02c1Fz+2 5c2j 2j2; h 0=V+c0+1 5c2j 0j2; h1=V+c0c1Fz+2 5c2j 1j2; h 12=c1F2 5c2 1  2; h13=1 5c2 0  2; h 23=p 6 2c1F1 5c2 0  1; h34=p 6 2c1F1 5c2 1  0; h 35=1 5c2 2  0; h 45=c1F2 5c2 2  1; and Fz=2X l=2lj lj2; F=F += 2  2 1+p 6  1 0+p 6  0 1+ 2 2  1; (10a) c0=4N(4a2+ 3a4) 7aosc; c1=4N(a4a2) 7aosc; c2=4N(7a010a2+ 3a4) 7aosc: (10b) In Eq. (10b), c0;c1;andc2are three interaction parameters, and a0;a2;a4are thes-wave scattering lengths in the permitted scattering channels. The order parameter for three spin systems is normalized to unity as ZX lj l(x;t)j2dx=X lNl= 1: (11) The order parameter's norm along with the energy of these SO coupled spinor condensate, which is dened as E=Z2 4X l;m  l(Hp+Hcoh+Hd+Hnd)lm m3 5dx; (12) wherel;m run over species' labels, are the two conserved quantities for an SO-coupled condensate. In the present work, the species' labels are 1 ;2 for pseudospin-1/2, 1 ;0;1 for spin-1 and 2 ;1;0;1;2 for spin-2 BECs. The species' labels 1 and 2 for pseudospin-1/2 BEC are equivalents of labels 1 =2 and 1=2, respectively, used in this work. For the sake of the compactness of the notations, the explicit functional dependence of Vonxand lonxandthas been suppressed. 43 Time-splitting Finite dierence methods We describe the (semi)-implicit nite-dierence schemes to numerically solve the coupled Gross{Pitaevskii equations (CGPEs) for SO-coupled spinor con- densates. We use time-splitting Backward-Euler (TSBE) and time-splitting Crank-Nicolson (TSCN) methods to solve the coupled sets of non-linear partial dierential equations describing SO-coupled pseudospin-1/2, spin-1 and spin- 2 BECs. The implementation is explained in all its detail for an SO-coupled pseudospin-1/2 condensate, and then extended to higher spin condensates. The results obtained with these nite dierence schemes are compared with results from Fourier spectral method. The latter method has been used by us to solve CGPEs for SO-coupled spin-1 [27] and spin-2 condensates [28]. 3.1 SO-coupled Pseudospin-1/2 Condensate 3.1.1 Quasi-one-dimensional pseudospin-1/2 BEC We consider a two-component pseudospin-1/2 BEC conned by a harmonic trapping potential with Rashba SO and coherent couplings. We rst elaborate the method for solving one-dimensional CGPEs which describe an SO-coupled pseudospin-1/2 BEC trapped by a q1D trapping potential. In such a trap, the yandzcoordinates can be integrated out and after a rotation by =2 about z-axis in spin-space which changes SytoSx, the resultant matrix operator Hp is Hp=1^p2 x 2 Sy^px1^p2 x 2+ Sx^px; (13) where 1is a 22 identity matrix, and SxandSyare Pauli spin matrices. The form ofHcoh,Hd, andHndremain same as in Eqs. (3) and (6) with the caveat that x=x; V =1 22 xx2; gll=2Nallpyz aosc; gl;3l=2Nal;3lpyz aosc; where the terms have the same meanings as described in the previous section. The time evolution of an SO-coupled spinor condensate as per Eq. (1) is ap- proximated by a rst order operator splitting, wherein one is required to solve the following equations successively over the same period @ @t=Hp ; (14a) @ @t=Hcoh ; (14b) @ @t=Hd ; (14c) where ( x;t) = [ 1(x;t); 2(x;t)]TwithTdenoting the transpose. The matrix Eq. (14a) in terms of coupled component equations is @ l(x;t) @t=1 2@2 l(x;t) @x2 @ 3l(x;t) @x; (15a) wherel= 1;2 is species' label. The spatial domain x2[Lx=2;Lx=2) is discretized via Nxuniformly spaced points with a spacing of
x. The resulting 5one-dimensional space grid is xi=Lx=2 + (i1)
xwherei= 1;2;:::;Nx. Using
tas the time-step to discretize time, the discrete analogue of l(x;t) isn (i;l)which represents the value of lth component of the order parameter at spatial coordinate xiat timen
t. The discretizaton scheme employs the periodic boundary conditions by ensuring that n (1;l)=n (Nx+1;l); n (0;l)=n (Nx;l): (16) In the present work, indices landmare exclusively used for species' labels, indicesiandjare used to denote only space-grid point, nis the index used for time, and=x;y;z . The discrete analogue of Eq. (15a) using Backward-Euler or Crank-Nicolson discretization schemes is n+1 (i;l)n (i;l)=
t 4
x2h  n+1 (i+1;l)2n+1 (i;l)+n+1 (i1;l) + n (i+1;l)2n (i;l) +n (i1;l)i
t 4
xh  n+1 (i+1;3l)n+1 (i1;3l) + n (i+1;3l) n (i1;3l)i ; (17) where= 2;= 0 for Backward-Euler discretization, and == 1 for Crank- Nicolson discretization. The local truncation error incurred in Backward-Euler and Crank-Nicolson discretizations are, respectively, of the order O(
x2+
t) andO(
x2+
t2) [30]. Considering Backward-Euler discretization rst, Eq. (17) is n+1 (i;l)n (i;l)
t=n+1 (i+1;l)2n+1 (i;l)+n+1 (i1;l) 2
x2 n+1 (i+1;3l)n+1 (i1;3l) 2
x: Forl= 1;2, the time evolution as per Backward-Euler is equivalent to " n+1 (i;1) n+1 (i;2)# = (1+Hp
t)1" n (i;1) n (i;2)# ; (18) where Hp" n+1 (i;1) n+1 (i;2)# =2 4n+1 (i+1;1)2n+1 (i;1)+n+1 (i1;1) 2
x2 xn+1 (i+1;2)n+1 (i1;2) 2
x n+1 (i+1;2)2n+1 (i;2)+n+1 (i1;2) 2
x2 xn+1 (i+1;1)n+1 (i1;1) 2
x3 5: (19) AsHpis an Hermitian operator, time evolution operator ( 1+Hp
t)1in Backward-Euler discretization is not unitary leading to the norm being not conserved. In contrast to this, the time evolution as per Crank-Nicolson is equivalent to" n+1 (i;1) n+1 (i;2)# =1Hp
t 1+Hp
t" n (i;1) n (i;2)# ; (20) corresponding to a unitary operator ( 1Hp
t)=(1+Hp
t). The Backward- Euler method is therefore not suitable for realtime evolution in contrast to Crank-Nicolson method. Nonetheless, in imaginary time evolution, a non- unitary time evolution, used to obtain the stationary state solutions both Backward- 6Euler or Crank-Nicolson methods can be used. Rewriting Eq. (17) as 
t 4
x2h n+1 (i1;l)+n+1 (i+1;l)i + 1 +
t 2
x2 n+1 (i;l)+ 
t 4
x n+1 (i+1;3l)n+1 (i1;3l) =
t 4
x2h n (i1;l)+n (i+1;l)i + 1
t 2
x2 n (i;l) 
t 4
x n (i+1;3l)n (i1;3l) : (21) Using Eq. (16) in Eq. (21) with i= 1;2;:::;Nxandl= 1;2, the resulting set of 2Nxcoupled linear algebraic equations can be written in matrix form as An+1 l+Bn+1 3l=Dl; (22) where A, B are circulant NxNxmatrices and n+1 l,DlareNx1 matrices. These matrices can be expressed as A(i;:) = 1 +
t 2
x2;
t 4
x2;0;


;0;
t 4
x2 (Ci1)T; (23a) B(i;:) = 0;
t 4
x;0;


;0;
t 4
x (Ci1)T; (23b) n+1 l= n+1 (1;l); n+1 (2;l); n+1 (3;l);


n+1 (Nx;l)T ; (23c) dl(i) =
t 4
x2n n (i1;l)+n (i+1;l)o + 1
t 2
x2 n (i;l) 
t 4
x n (i+1;3l)n (i1;3l) ; (23d) whereA(i;:) andB(i;:) are theith rows of A and B, respectively, dl(i) is the ith element of column matrix Dl, andCis dened as C=2 66640 0::: 1 1 0::: 0 ......... 0::: 1 03 7775: (24) Forl= 1;2, Eq. (22) represents two coupled matrix equations which can be decoupled to yield (B2A2)n+1 l=BD3lADl; (25) which forl= 1 and 2 represents two decoupled sets of linear circulant system of equations. Now, B2A2being a circulant matrix, it can be diagonalised using Fourier matrix as [31] B2A2=F1F; where (26a) Fi;j=1pNxexp 2 Nx(i1)(j1) ;and (26b)  = diag[p NxFfB2(:;1)A2(:;1)g]: (26c) 7Now, the product of the Fourier matrix ( F) with a one-dimensional array is equal to the discrete Fourier transform of the array, and hence the solution to Eq. (25) using Eqs. (26a)-(26c) is [31] n+1 l= IDFT DFT(BD3lADl):=DFT(B2(:;1)A2(:;1))
; (27) where DFFT and IDFT stand for discrete forward Fourier and inverse discrete Fourier transforms, respectively, A2(:;1) andB2(:;1) denote the rst columns ofA2andB2, and:=indicates the element wise division. Now, Eq. (14b) is evolved in time from tn=n
ttotn+1= (n+ 1)
tconsidering Eq. (27) as the solution at tn. The exact analytic solution to Eq. (14b) is (x;tn+1) = exp[Hcoh
t] (x;tn) = 1cos
t 2 Sxsin
t 2 (x;tn): (28) The last step involves solving Eq. (14c) over the same period treating the solu- tion in Eq. (28) as the solution at tn=n
t. The exact solution to Eq. (14c) is (x;tn+1) = exp[Hd
t] (x;tn): (29) Quasi-two-dimensional pseudospin-1/2 BEC In a quasi-two-dimensional trap with tight connement along zaxis, the form of matrix operator Hpafter integrating out the zcoordinate becomes Hp=1^p2 x+ ^p2 y 2+ (Sx^pySy^px); (30) whereas the form Hcoh,Hd,Hndagain remain unchanged from those in Eqs. (3) and (6) with a caveat that x(x;y); V =1 2(2 xx2+2 yy2); glm=2Nalmp2z aosc: (31) Using the time-splitting, the time evolution of the condensate from tntotn+1is approximated by successive solutions to the following equations over the same period @ @t=Hpx ; (32a) @ @t=Hpy ; (32b) @ @t=Hcoh ; (32c) @ @t=Hd ; (32d) whereHpxandHpyare dened as Hpx=1^p2 x 2 Sy^px; Hpy=1^p2 y 2+ Sx^py: (33) Here, we consider a two-dimensional spatial grid dened as i=L=2 + (i 1)
, wherei= 1;2;:::;N,=x;y, and
is spatial step size. The discrete 8analogue of component wavefunction is n (i;j;l)which is equal to value of the lth wavefunction at space point ( xi;yj) attntime. Similar to quasi-one-dimensional condensates, nite dierence equivalents of each of Eq. (32a) and Eq. (32b) can be simplied to two decoupled matrix equations (Bx2+Ax2)Xn+1 l=AxDx l+ (1)lBxDx 3l; (34a) (By2Ay2)Yn+1 l=ByDy 3lAyDy l; (34b) whereA,B(with=x;y),Xn+1 l;Yn+1 l;D lare dened A(i;:) = 1 +
t 2
2;
t 4
2;0;


;0;
t 4
2 (Ci1)T; (35a) Bx(i;:) = 0;
t 4
x;0;


;0;
t 4
x (Ci1)T; (35b) By(i;:) = 0;
t 4
y;0;


;0;
t 4
y (Ci1)T; (35c) Xn+1 l= n+1 (1;j;l)n+1 (2;j;l)n+1 (3;j;l)


n+1 (Nx;j;l)T ; (35d) Yn+1 l= n+1 (i;1;l); n+1 (i;2;l); n+1 (i;3;l);


n+1 (i;Ny;l)T ; (35e) dx l(i) =
t 4
x2n n (i1;j;l)+n (i+1;j;l)o + 1
t 2
x2 n (i;j;l) +(1)l 
t 4
x n (i+1;j;3l)n (i1;j;3l) ; (35f) dy l(i) =
t 4
y2n n (i;j1;l)+n (i;j+1;l)o + 1
t 2
y2 n (i;j;l) 
t 4
y n (i;j+1;3l)n (i;j1;3l) ; (35g) whereA(i;:) andB(i;:) are theith row ofAandB, respectively, d l(i) is theith element of column matrix D l, andCis dened in Eq. (24). For a xed value ofj(y-index) and l(species index), Eqs. (34a) is a linear circulant system of equations which can be solved by the same procedure as discussed to solve Eq. (25). The solution to Eq. (32a) is obtained by solving Eq. (34a) for all j andlvalues following exactly the same procedure as discussed Sec. 3.1.1. This solution, then, is considered as an input solution at tnwhile solving another set of linear circulant system of Eqs. (34b) over the same period from tnto tn+
t. The solutions to Eqs. (32c)-(32d) are again given as in Eqs. (28)-(29) with ( x;tn) = [ 1(x;y;tn); 2(x;y;tn)]TwithTstanding for transpose. 3.2 SO-coupled spin-1 condensate 3.2.1 Quasi-one-dimensional spin-1 BEC In quasi-one-dimensional trap, Hpfor an SO-coupled spin-1 BEC takes the form Hp=1^p2 x 2+ Sx^px; (36) 9where 1is a 33 identity matrix, and Sxis the 33 spin-1 matrix. The form ofHcoh,Hd, andHndin Eqs. (3), (7a), (7b) remain unchanged, provided x=x; V =1 22 xx2; c 0=pyz2N(a0+ 2a2) 3aosc; c 1=pyz2N(a2a0) 3aosc: Using the rst order time-splitting, the solution of the Eq. (1) is equivalent to solving following equations successively @ @t=Hp ; (37a) @ @t= (Hnd+Hcoh) =Hnd+ ; (37b) @ @t=Hd : (37c) whereHnd+=Hnd+Hcoh, and ( x;t) = [ 1(x;t); 0(x;t); 1(x;t)]T. We solve Eq. (37a) using nite dierence schemes described in detail for pseudospin- 1/2 BEC. Using Backward-Euler (and/or Crank-Nicolson) discretization schemes along with periodic boundary conditions, viz. Eq. (16), Eq. (37a) reduces to three coupled matrix equations An+1 1+Bn+1 0 =D1; (38a) An+1 0+B(n+1 1+ n+1 1) =D0: (38b) Eqs. (38a)-(38b), can be decoupled into following three independent matrix equations, (2B2AA3)n+1 1= (B2A2)D1+ABD 0B2D1; (39a) (A22B2)n+1 0 =AD0B(D1+D1); (39b) whereAand n+1 lwithl= 1;0;1 are same as in Eq. (23a) and Eq. (23c), respectively, whereas rows of Band elements of Djare now dened as B(i;:) = 0;
t 4p 2
x;0;


;0;
t 4p 2
x (Ci1)T(40a) d1(i) =
t 4
x2n n (i1;1)+n (i+1;1)o + 1
t 2
x2 n (i;1) 
t 4p 2
x n (i+1;0)n (i1;0) (40b) d0(i) =
t 4
x2n n (i1;0)+n (i+1;0)o + 1
t 2
x2 n (i;0) 
t 4p 2
x n (i+1;1)n (i1;1)+n (i+1;1)n (i1;1) : (40c) The decoupled matrix Eqs. (39a) -(39b) are linear circulant system of equations which can be solved by using the method described for pseudospin-1/2 BEC. The analytic solution to Eq. (37b) is [27] (x;tn+1) 1+cos1 2
t2H2 nd+sin 
tHnd+ (x;tn); (41) 10where=
tq jc1 0  1+ 2p 2j2+jc1 0  1+ 2p 2j2. Finally, the solution to Eq. (37c) is again given as in Eq. (29) with the caveat that the various quantities are identied as those corresponding to spin-1 BEC. Quasi-two-dimensional spin-1 BEC Here the form of matrix operator Hpis same as in Eq. (30) with 1representing a 33 identity matrix, and Swith=x;ydenoting the spin-1 matrices. Also, the form of Hcoh,Hd, andHndin Eqs. (3), (7a), (7b), respectively, remain unchanged, provided x(x;y); V =X =x;y2 2 2; (42) c0=p 2z2N(a0+ 2a2) 3aosc; c 1=p 2z2N(a2a0) 3aosc: (43) The CGPEs of a quasi-2D spin-1 BEC with Rashba SO coupling can be split into following set of equations, and these has to be solved successively over the same period. @ @t=Hpx ; (44a) @ @t=Hpy ; (44b) @ @t= (Hnd+Hcoh) =Hnd+ ; (44c) @ @t=Hd : (44d) whereHpxandHpyare dened in Eq. (33) with 1, andSbeing identied as 33 identity and spin-1 matrices, respectively. Similar to quasi-two-dimensional pseudospin-1/2 BEC, each of Eq. (44a) and Eq. (44b) can be discretized into three decoupled matrix equations, such as (A3 x+ 2AxB2 x)Xn+1 1= (A2 x+B2 x)Dx 1AxBxDx 0+B2 xDx 1;(45a) (A2 x+ 2B2 x)Xn+1 0 =AxDx 0+Bx(Dx 1Dx 1) (45b) for Eq. (44a), and (2B2 yAyA3 y)Yn+1 1 = (B2 yA2 y)Dy 1+AyByDy 0B2 yDy 1;(46a) (A2 y2B2 y)Yn+1 0 =AyDy 0By(Dy 1+Dy 1); (46b) 11for Eq. (44b). Here, A(with=x;y),Xn+1 l,Yn+1 l, are dened as in Eq. (35a), Eq. (35d) and Eq. (35e) respectively, whereas B;D lare now dened as Bx(i;:) = 0;
t 4p 2
x;0;


;0;
t 4p 2
x (Ci1)T(47a) By(i;:) = 0;
t 4p 2
y;0;


;0;
t 4p 2
y (Ci1)T(47b) dx 1(i) =
t 4
x2n n (i1;j;1)+n (i+1;j;1)o + 1
t 2
x2 n (i;j;1)  
t 4p 2
x n (i+1;j;0)n (i1;j;0) (47c) dx 0(i) =
t 4
x2n n (i1;j;0)+n (i+1;j;0)o + 1
t 2
x2 n (i;j;0) +
t 4p 2
x n (i+1;j;1)n (i1;j;1)(n (i1;j;1)n (i+1;j;1)) :(47d) dy 1(i) =
t 4
y2n n (i;j1;1)+n (i;j+1;1)o + 1
t 2
y2 n (i;j;1) 
t 4p 2
y n (i;j+1;0)n (i;j1;0) (47e) dy 0(i) =
t 4
y2n n (i;j1;0)+n (i;j+1;0)o + 1
t 2
y2 n (i;j;0) 
t 4p 2
y n (i;j+1;1)n (i;j1;1)+n (i;j+1;1)n (i;j1;1) :(47f) Eqs. (45a)-(45b) and (46a)-(46b) are linear circulant system of equations, and thus can be solved as described for pseudospin-1/2 condensates in Sec. 3.1.1. The solution to Eqs. (44c)-(44d) is similar as described for quasi-one- dimensional spin-1 condensates. 3.3 SO-coupled spin-2 condensate 3.3.1 Quasi-one-dimensional spin-2 BEC Similar to quasi-one-dimensional pseudospin-1/2 and spin-1 BECs, form of Hp is1^p2 x=2+ Sx^pxwhereSxdenotes the spin-2 matrix and forms of Hcoh,Hdand Hndremain the same as in Eqs. (3), (9a), and (9b), respectively. The trapping potential and interaction parameters are V(x) =2 xx2 2; c 0=pyz2N(4a2+ 3a4) 7aosc; (48a) c1=pyz2N(a4a2) 7aosc; c 2=pyz2N(7a010a2+ 3a4) 7aosc(48b) 12Using the rst order time-splitting, the solution of the Eq. (1) is equivalent to solving following equations successively @ @t=Hp ; (49a) @ @t= (Hnd+Hcoh) =Hnd+ ; (49b) @ @t=Hd ; (49c) whereHnd+=Hnd+Hcoh, and ( x;t) = [ 2(x;t); 1(x;t); 0(x;t); 1(x;t); 2(x;t)]T. Similar to pseudospin-1/2 and spin-1 condensates, nite dierence discretization of Eq. (49a) along with periodic boundary conditions, viz. Eq. (16), reduces it to ve decoupled matrix equations A(A2B2)(A34AB2)n+1 2=r 3 2(B4A2B2)D03 2AB3D1 +3 2B4D2+5 2AB3BA3 D1 +3 2B4+A44A2B2 D2; (50a) (A2B2)(A34AB2)n+1 1=r 3 2(B3A2B)D03 2B3D2+ 3 2AB2D1+ A35 2AB2 D1 +5 2B3A2B D2; (50b) (A34AB2)n+1 0=(A2B2)D0r 3 2AB(D1+D1) +r 3 2B2(D2+D2); (50c) whereA,B, and n+1 lare same as in Eq. (23a), (23b) and (23c) respectively, whereas the elements of column matrices Dlwithl= 2;1;0;1;2 are d2(i) =
t 4
x2n n (i1;2)+n (i+1;2)o + 1
t 2
x2 n (i;2) 
t 4
x n (i+1;1)n (i1;1) ; (51a) d1(i) =
t 4
x2n n (i1;1)+n (i+1;1)o + 1
t 2
x2 n (i;1) 
t 4
x  n (i+1;2)n (i1;2) r 3 2 
t 4
x n (i+1;0)n (i1;0)# ;(51b) d0(i) =" 
t 4
x2n n (i1;0)+n (i+1;1)o + 1
t 2
x2 n (i;0)r 3 2 
t 4
x  n (i+1;1)n (i1;1) r 3 2 
t 4
x n (i+1;1)n (i1;1)# :(51c) 13The ve decoupled sets of linear circulant system of Eqs. (50a)-(50c) can be solved as discussed in Sec. 3.1.1. The detailed procedure to solve Eq. (49b) is discussed in the appendix, and the exact solution to Eq. (49c) is same as in Eq. (29). 3.4 Quasi-two-dimensional spin-2 BEC Here the form of matrix operator Hpis same as in Eq. (30) with 1representing a 55 identity matrix, Sare spin-2 matrices and the forms of Hcoh,Hd, and Hndin Eqs. (3), (9a), (9b), respectively, remain unchanged, with x(x;y); V =X =x;y2 2 2;c0=p 2z2N(4a2+ 3a4) 7aosc; (52) c1=p 2z2N(a4a2) 7aosc; c 2=p 2z2N(7a010a2+ 3a4) 7aosc: (53) Here also, similar to quasi-two-dimensional spin-1 BEC, each of Eq. (44a) and (44b) can be discretized into ve decoupled matrix equations, such as (4AxB2 x+A3 x)(B2 x+A2 x)AxXn+1 2=r 3 2(B4 x+A2 xB2 x)Dx 03 2AxB3 xDx 1 +3 2B4 xDx 2(5 2AxB3 x+BxA3 x)Dx 1 + (3 2B4 x+A4 x+ 4A2 xB2 x)Dx 2; (54a) (4AxB2 x+A3 x)(B2 x+A2 x)Xn+1 1=r 3 2(B3 x+A2 xBx)Dx 03 2B3 xDx 2 (5 2B3 x+A2 xBx)Dx 2+3 2AxB2 xDx 1 + (5 2AxB2 x+A3 x)Dx 1; (54b) (4AxB2 x+Ax3)Xn+1 0= (B2 x+A2 x)Dx 0+r 3 2AxBx(Dx 1Dx 1) +r 3 2B2 x(Dx 2+Dx 2) (54c) 14for Eq. (44a), and Ay(A2 yB2 y)(A3 y4AyB2 y)Yn+1 2=r 3 2(B4 yA2 yB2 y)Dy 03 2AyB3 yDy 1 +3 2B4 yDy 2+ (5 2AyB3 yByA3 y)Dy 1 + (3 2B4 y+A4 y4A2 yB2 y)Dy 2; (55a) (A2 yB2 y)(A3 y4AyB2 y)Yn+1 1=r 3 2(B3 yA2 yBy)Dy 0+3 2AyB2 yDy 1 3 2B3 yDy 2+ (A3 y5 2AyB2 y)Dy 1 + (5 2B3 yA2 yBy)Dy 2; (55b) (A3 y4AyB2 y)Yn+1 0= (A2 yB2 y)Dy 0r 3 2AyBy(Dy 1+Dy 1) +r 3 2B2 y(Dy 2+Dy 2) (55c) for Eq. (44b). A(with=x;y);Xn+1 l, andYn+1 lare dened as in Eq. (35a), Eq. (35d) and Eq. (35e) respectively, and B;D lare now dened as Bx(i;:) = 0;
t 4
x;0;


;0;
t 4
x (Ci1)T; (56a) By(i;:) = 0;
t 4
y;0;


;0;
t 4
y (Ci1)T; (56b) dx 2(i) =
t 4
x2n n (i1;j;2)+n (i+1;j;2)o + 1
t 2
x2 n (i;j;2)  
t 4
x n (i+1;j;1)n (i1;j;1) ; (56c) dx 1(i) =
t 4
x2n n (i1;j;1)+n (i+1;j;1)o + 1
t 2
x2 n (i;j;1)  
t 4
x n (i+1;j;2)n (i1;j;2) r 3 2 
t 4
x n (i+1;j;0)n (i1;j;0)# ; (56d) dx 0(i) =
t 4
x2n n (i1;j;0)+n (i+1;j;1)o + 1
t 2
x2 n (i;j;0) +r 3 2 
t 4
x n (i+1;j;1)n (i1;j;1) r 3 2 
t 4
x n (i+1;j;1)n (i1;j;1)# ; (56e) 15dy 2(i) =
t 4
y2n n (i;j1;2)+n (i;j+1;2)o + 1
t 2
y2 n (i;2) 
t 4
y n (i;j+1;1)n (i;j1;1) ; (57a) dy 1(i) =
t 4
y2n n (i;j1;1)+n (i;j+1;1)o + 1
t 2
y2 n (i;j;1) 
t 4
y  n (i;j+1;2)n (i;j1;2) r 3 2 
t 4
y n (i;j+1;0)n (i;j1;0)# ; (57b) dy 0(i) =" 
t 4
y2n n (i;j1;0)+n (i;j+1;1)o + 1
t 2
y2 n (i;j;0)r 3 2 
t 4
y  n (i;j+1;1)n (i;j1;1) r 3 2 
t 4
y n (i;j+1;1)n (i;j1;1)# : (57c) whereB(i;:) is theith row ofBandd l(i) is theith element of column matrix D l. Eqs. (54a)-(54c) and (55a)-(55c) are linear circulant system of equations, and thus can be solved in a similar manner as described for pseudospin-1/2 and spin-1 condensates. The solution to Eqs. (44c)-(44d) is on similar lines as described for quasi-one-dimensional spin-2 condensates. 4 Numerical Results Here, we present the numerical results with TSBE and TSCN methods for the pseudospin-1/2, spin-1, and spin-2 in the presence as well as absence of coher- ent coupling. Both TSBE and TSCN can be used to obtain the ground state solutions of an SO and coherently coupled spinor BEC. This can be achieved by considering an initial guess solution to the CGPEs and replacing tbyt=~tto solve CGPEs. The resultant imaginary time evolution is not norm preserving, and hence total norm needs to xed to unity after each time iteration. The quantity= maxjn+1 (i;j;l)n (i;j;l)j=
~tserves as the convergence criterion to quantify convergence in imaginary time propagation. The stationary state so- lutions reported in this section has been obtained with = 106. In contrast to imaginary time evolution, realtime dynamics of the spinor BECs can be studied with TSCN and not with TSBE as the later does not conserve norm as was discussed in Sec. 3.1.1. 4.1 Pseudospin-1/2 For pseudospin-1/2 case, we choose an experimentally realizable87Rb pseudospinor- 1/2 BEC with scattering length a11= 101:8aB, interaction strengths g12= 1:1g11,g22= 0:9g11andg12=g21, whereaBis the Bohr radius. We con- sider 5000 atoms trapped in q1D trapping potential with !x= 220Hz, 16!y= 2400Hz and !z= 2400Hz. The interaction strengths in dimen- sionless units are given as (g11; g22; g12) = (446:95;402:26;491:65); (58) withg12=g21. For q2D BEC, we consider 5000 atoms of87Rb in a trap with trapping frequencies !x=!y= 220Hz,!z= 2400Hz. For this case, the interaction strengths g22= 0:9g11,g12= 1:1g11, andg12=g21fora11= 101:8aB are given as (g11; g22; g12) = (250;225;275); (59) withg12=g21. In both these cases, we compare the results from TSFS, TSBE and TSCN in the presence as well as absence of coherent coupling and nd an excellent agreement. The comparison of the ground state energies obtained with three methods for dierent values of are given in Table-1 for = 0 and Table- 2 for = 0 :5. The results with TSBE and TSCN are in very good agreement with those from TSFS. Table 1: Comparison of ground state energies of pseudospin-1/2 BEC of87Rb obtained with TSFS, TSBE and TSCN for dierent values of in the absence of coherent coupling . The interaction strength parameters are g11= 446:95, g22= 402:26 andg12=g21= 491:65 for q1D BEC, whereas the same for q2D BEC areg11= 250:52,g22= 225:47 andg12=g21= 275:57.
x= 0:1,
~t= 0:01
x= 0:1,
~t= 0:005 TSFS TSBE TSCN TSFS TSBE TSCN q1D 0.5 21.4357 21.4357 21.4357 21.4357 21.4357 21.4357 1.0 21.4186 21.4186 21.4186 21.4186 21.4186 21.4186 1.5 21.3333 21.3334 21.3333 21.3324 21.3324 21.3324 2.0 20.7018 20.7035 20.7022 20.7001 20.7014 20.7011 q2D 0.5 5.7201 5.7201 5.7201 5.7201 5.7201 5.7201 1.0 5.4707 5.4707 5.4707 5.4707 5.4707 5.4707 1.5 4.8520 4.8520 4.8520 4.8518 4.8518 4.8518 2.0 3.9783 3.9786 3.9787 3.9883 3.9786 3.9786 The component densities corresponding to ground state solutions obtained with TSBE and TSCN methods for q1D87Rb BEC are shown in Fig. 1. The densities obtained with two methods are in an excellent agreement. Similarly, the component densities, obtained with TSCN method, for q2D87Rb BEC for dierent values of and are shown in Fig. 2. We also study the variation of the convergence criterion as a function of ~tin imaginary-time propagation with TSBE, TSCN, and TSFS to obtain the ground state solution. As an example, in the imaginary-time propagation to obtain the ground state of q1D pseudospin- 1/2 BEC of87Rb starting with normalized Gaussian initial guess wavefunctions for the two components, the variation of as a function of ~t, obtained with three methods, is shown in Fig. 3(a) for
x= 0:1 and
~t= 0:01 and in Fig. 3(b) for
x= 0:2 and
~t= 0:02. It is evident that TSCN shows faster convergence than TSBE. As discussed in the Sec. 3.1.1, the TSBE does not lead to a unitary time evolution in contrast to TSCN. In order to conrm this, we consider the 17Table 2: Comparison of ground state energies of pseudospin-1/2 BEC obtained with TSFS, TSBE and TSCN for = 0 :5 and dierent values of . The results have been obtained with
x= 0:1 and
~t= 0:01. The interaction strength parameters considered for q1D BEC are g11= 446:95,g22= 402:26 andg12=g21= 491:65, whereas the same for q2D BEC are g11= 250:52, g22= 225:47 andg12=g21= 275:57. TSFS TSBE TSCN q1D 0.5 21.4231 21.4231 21.4231 1.0 21.4002 21.4002 21.4002 1.5 21.3033 21.3034 21.3033 2.0 20.6711 20.6727 20.6715 q2D 0.5 5.6457 5.6457 5.6457 1.0 5.3339 5.3339 5.3339 1.5 4.7181 4.7181 4.7181 2.0 3.8434 3.8438 3.8438 0.020.040.060.08 γ= 0.5 (a)Ω = 0.5ρ1 ρ2ρj(x) γ= 1.0 (b)Ω = 0.5ρ1 ρ2 γ= 2.0 (c)Ω = 0.5ρ1 ρ2 -10 -5 0 5 10 x0.020.040.060.08 γ= 0.5 (d)Ω = 2.0ρ1 ρ2ρj(x) -10 -5 0 5 10 x γ= 1.0 (e)Ω = 2.0ρ1 ρ2 -10 -5 0 5 10 x γ= 2.0 (f)Ω = 2.0ρ1 ρ2 1 Figure 1: (Color online) (a)-(c) are the component densities for an SO-coupled q1D87Rb pseudospin-1 2BEC with = 0 :5 and = 0:5;1;2, respectively. The same for = 2 are shown in (d)-(f), respectively. The lines and points correspond to the results from TSBE and TSCN, respectively. The interaction strengths considered in (a)-(f) are g22= 0:9g11andg12= 1:1g11withg11= 446:95. real-time evolution of the ground state solution of the q1D87Rb shown in Fig. 1(a) with TSBE and TSCN. For this we consider the ground state solution corresponding to interaction parameters in Eq. (58) with = = 0:5 as the 18Figure 2: (Color online) (a1)-(a2) are the component densities obtained with TSCN method for an SO-coupled q2D87Rb pseudospin-1 2BEC with g11= 250; g22= 225,g12=g21= 275, = 0:5 and = 0. The same for ( ; ) = (2;0);(0:5;0;5), and (2;0:5) are shown in (b1)-(b2), (c1)-(c2), and (d1)-(d2), respectively. Figure 3: (Color online) The variation of convergence criterion during imaginary-time propagation to calculate the ground state of q1D87Rb. In (a), we have chosen
x= 0:1 and
~t= 0:01, whereas for (b)
x= 0:2 and
~t= 0:02. initial solution at t= 0 in real-time evolution. The variation of total norm and energy as a function of time obtained using TSFS, TSBE, and TSCN are shown in Fig. 4(a)-(b), respectively. The non-conservation of norm and hence energy in TSBE makes the method unsuitable to study any realtime dynamics. The dynamics of the ground state, a stationary state, is trivial in the sense that besides norm and energy the expectation values of various operators are also conserved. 190 40 80 120 160 200 t0.9870.9900.9930.9960.999N (a)TSFS TSBE TSCN 0 40 80 120 160 200 t21.021.221.4E (b)TSFS TSBE TSCN 0 40 80 120 160 200 t00.20.40.60.81Nl (c)N1 N2 NFigure 4: (Color online) (a) Norm Nas a function of time and (b) energy E as a function of time for the ground state solution of pseudospin-1/2 BEC of 87Rb with = = 0:5. (c) NormNandNlas a function of time in realtime obtained for non-stationary initial solution using TSCN. The real-time evolution of initial solution is obtained using TSFS, TSBE and TSCN with
x= 0:1 and
t= 0:005. Next, we consider the dynamics of non-stationary state using TSCN. We rst obtain a non-stationary state by solving CGPEs for q1D87Rb with interaction strengths as dened in Eq. (58) and = = 0:5 under the constraint of zero polarization. The solution thus obtained is non-stationary, and is then evolved in realtime (without any additional constraint) using TSCN. The variation of component norms as a function of time is shown in Fig. 4(c). 4.2 Spin-1 We consider (1)23Na and (2)87Rb spin-1 BECs corresponding to antiferro- magnetic and ferromagnetic phases in the absence of coupling. The scattering lengths corresponding to system (1) and (2) are a0= 50:00aB,a1= 55:01aB[32] anda0= 101:8aB,a1= 100:4aB[33], respectively. We consider 10000 atoms trapped in q1D trapping potential with !x= 220Hz,!y=!z= 2400Hz. The interaction strengths c0andc2in dimensionless units are given as (1) (c0;c2) = (240 :83;7:54); (60a) (2) (c0;c2) = (885 :72;4:09): (60b) The same number of atoms trapped in q2D trapping potential with !x=!y= 220Hz,!z= 2400Hz leads to following interaction strengths (1) (c0;c2) = (134 :98;4:22); (61a) (2) (c0;c2) = (248 :22;1:15); (61b) for23Na and87Rb spin-1 BECs, respectively. The oscillator lengths for system (1) and (2) are 4 :69m and 2:41m, respectively. For these two cases, the comparison of ground state energies obtained from TSFS, TSBE and TSCN shows an excellent agreement as reported in Table-(3). 20Table 3: Comparison of ground state energies of SO and coherently coupled spin- 1 BECs using TSFS, TSBE, and TSCN methods with
x= 0:1 and
~t= 0:005. The energies correspond to dierent values . The coherent coupling used for q1D and q2D systems are 0 :5 and 0:1, respectively. 23Na83Rb TSFS TSBE TSCN TSFS TSBE TSCN 0.5 15.0623 15.0623 15.0623 35.7812 35.7812 35.7812 q1D 1.0 14.6873 14.6873 14.6873 35.4062 35.4062 35.4062 1.5 14.0623 14.0623 14.0623 34.7812 34.7812 34.7812 2.0 13.1873 13.1876 13.1876 34.9062 33.9062 33.9065 0.5 4.3797 4.3797 4.3797 8.2638 8.2638 8.2638 q2D 1.0 3.9602 3.9602 3.9601 7.8747 7.8747 7.8747 1.5 3.3303 3.3303 3.3303 7.2435 7.2435 7.2435 2.0 2.4486 2.4489 2.4489 6.3658 6.3661 6.3661 The numerically obtained component densities in the ground states of har- monically trapped q1D23Na and87Rb spin-1 BECs with dierent values of and are shown in Fig. 5. The component densities obtained using TSBE 0.020.040.060.080.10 γ= 0.5 (a)Ω = 0.5ρ+1 ρ0 ρ−1ρj(x)[23Na] γ= 1.0 (b)Ω = 0.5ρ+1 ρ0 ρ−1 γ= 1.5 (c)Ω = 0.5ρ+1 ρ0 ρ−1 -10 -5 0 510 x0.010.020.03 γ= 0.5 (d)Ω = 0.5ρ+1 ρ0 ρ−1ρj(x)[87Rb] -10 -5 0 5 10 x γ= 1.0 (e)Ω = 0.5ρ+1 ρ0 ρ−1 -10 -5 0 5 10 x γ= 1.5 (f)Ω = 0.5ρ+1 ρ0 ρ−1 1 Figure 5: (Color online) (a)-(c) are the ground state component densities for an SO-coupled q1D23Na spin-1 BEC for dierent values and = 0:5. The same for87Rb spin-1 BEC are shown in (d)-(f). In (a)-(f), lines and points correspond to the results from TSBE and TSCN, respectively. and TSCN are in an excellent agreement. Similarly, in Fig. (6) we have shown some distinct ground state density proles for q2D23Na and87Rb spin-1 BECs obtained using TSCN. 21Figure 6: (Color online) (a1)-(a3) are the ground-state component densities for an SO-coupled q2D23Na spin-1 BEC with = 0:2 and = 0, whereas the same for87Rb are in (b1)-(b3). (c1)-(c3) and (d1)-(d3) are the ground-state component densities for23Na and87Rb BECs, respectively, with = 2 and = 0:1. 4.3 Spin-2 We consider (1)83Rb, (2)23Na, and (3)87Rb spin-2 BECs corresponding to ferromagnetic, anti-ferromagnetic and cyclic phases. The three sets of scattering 22length corresponding to these systems are [18, 34] (1)a0= 83:0aB; a 2= 82:0aB; a 4= 81:0aB; (62) (2)a0= 34:9aB; a 2= 45:8aB; a 4= 64:5aB; (63) (3)a0= 87:93aB; a 2= 91:28aB; a 4= 99:18aB; (64) respectively. We consider 10000 atoms of each of these systems trapped in q1D trapping potential with !x= 220Hz,!y=!z= 2400Hz. The interaction strengthsc0,c1andc2in dimensionless units are given as (1) (c0;c1;c2) = (699 :62;1:23;4:90); (2) (c0;c1;c2) = (242 :97;12:06;13:03); (3) (c0;c1;c2) = (831 :26;9:91;0:31): Similarly, we consider 10000 atoms of each of three systems trapped in a q2D trapping potential with !x=!y= 220Hz,!z= 2400Hz. The resultant interaction strengths for q2D83Rb,23Na,87Rb spin-2 BECs are (1) (c0;c1;c2) = (392 :14;0:67;2:74); (2) (c0;c1;c2) = (136 :18;6:76;7:30); (3) (c0;c1;c2) = (465 :92;5:55;0:18); respectively. The oscillator lengths corresponding to three systems (1), (2) and (3) are 2:47m, 4:69m and 2:41m, respectively. For these set of parameters, the ground state energies obtained with TSFS, TSBE, and TSCN are reported in Table 4. The agreement between the results with three methods is very good. Table 4: Comparison of ground state energies of q1D and q2D spin-2 BECs of 83Rb,23Na and87Rb obtained with TSFS, TSBE, and TSCN using
x= 0:1 and
~t= 0:001 for dierent values of and . 83Rb23Na87Rb ( ; ) TSFS TSBE TSCN TSFS TSBE TSCN TSFS TSBE TSCN q1D (0.5,0.5) 29.8496 29.8496 29.8496 14.6877 14.6877 14.6877 34.2036 34.2036 34.2036 (1.0,0.5) 28.3496 28.3499 28.3499 13.1877 13.1881 13.1881 32.7036 32.7039 32.7039 q2D (0.5,0.1) 7.0875 7.0875 7.0875 3.9648 3.9645 3.9645 7.6850 7.6850 7.6850 Similarly, the ground-state component densities of q1D spin-283Rb,23Na, and87Rb BECs with dierent values of and calculated using TSBE and TSCN are in very good agreement as shown in Fig. (7). In q2D spin-2 BECs also, ground-state component densities calculated using three methods are in a very good agreement. Here we illustrate some qualitatively distinct ground-state density proles obtained with TSCN. The component densities in the ground state of q2D83Rb,23Na, and87Rb spin-2 BECs with = 0:5 and = 0 :1 are shown in Fig. 8. The ground state of83Rb and23Na spin-2 BECs have vortices of winding number 1;0;+1;+2;+3 and2;1;0;+1;+2 associated 230.010.020.030.040.05 γ= 0.25 (a)Ω = 0.583Rb ρ+2 ρ+1 ρ0 ρ−1 ρ−2ρj(x) 0.020.040.060.08 γ= 0.25 (b)Ω = 0.523Na ρ+2 ρ1 ρ0 ρ−1 ρ−2 0.020.040.06 γ= 0.25 (c)Ω = 0.587Rb ρ+2 ρ1 ρ0 ρ−1 ρ−2 -10 -5 0 5 10 x0.010.020.030.040.05 γ= 0.5 (d)Ω = 0.5ρ+2 ρ+1 ρ0 ρ−1 ρ−2ρj(x) -10 -5 0 510 x0.020.040.060.08 γ= 0.5 (e)Ω = 0.5ρ+2 ρ1 ρ0 ρ−1 ρ−2 -10 -5 0510 x0.020.040.06 γ= 0.5 (f)Ω = 0.5ρ+2 ρ1 ρ0 ρ−1 ρ−2 1Figure 7: (Color online) (a)-(c) are the ground-state component densities for SO- coupled quasi-1D83Rb,23Na and87Rb spin-2 BEC with = 0:25 and = 0 :5 respectively, whereas (d)-(f) are the same for = 0:5 and = 0 :5. In (a)-(f), lines and points correspond to densities with TSBE and TSCN, respectively. with thel= 2;1;0;1;2 components, respectively. The ground state of q2D 87Rb spin-2 BEC has stripe pattern in component densities for = 0:5 and = 0:1. For q2D87Rb spin-2 BEC, the ground state component densities with = 1; = 0 and = 2; = 0 are also illustrated in Figs. 9(a1)-(a5) and Figs. 9(b1)- (b5), respectively. The ground-state component densities have triangular lattice pattern for = 1 and stripe density pattern for = 2. 24Figure 8: (Color online) (a1)-(a5) are the component densities for an SO-coupled quasi-2D83Rb, (b1)-(b5) are for23Na and (c1)-(c5) are for87Rb spin-2 BEC with = 0:1, and = 0:5 respectively. Figure 9: (Color online) (a1)-(a5) are the component densities for an SO-coupled quasi-2D87Rb with = 1:0 whereas (b1)-(b5) are for same system with = 2:0 respectively. 25Summary We have discussed time-splitting Backward-Euler and Crank-Nicolson methods to study the SO-coupled spinor BECs with coherent coupling. We have devel- oped the methods for pseudospin-1/2, spin-1 and spin-2 BEC in q1D and q2D traps. We have considered Rashba SO coupling in the present work, one can also consider Dresselhaus coupling or a combination of both within the framework of same numerical schemes. We have compared the results obtained with these nite dierence methods with the time-splitting Fourier spectral method. The numerical results for stationary states obtained with the three methods are in very good agreement. We have provided the comparison of ground state energies and component density proles calculated using three methods for several illus- trative cases. In imaginary-time propagation, TSCN shows faster convergence as compared to TSBE. Moreover, the time evolution as per TSCN is unitary time evolution consistent with the underlying Hermitian Hamiltonian. This is not the case with TSBE which results in non-unitary time evolution and thus rendering the method not suitable to the study any real-time dynamics. The nite dierence methods developed in the present work can be easily extended to higher spin system like spin-3 BEC. Appendix The split equation for Hnd+is i@ @t=Hnd+ (65) where Hnd+=0 BBBB@0h12h13 0 0 h 12 0h23 0 0 h 13h 23 0h34h35 0 0h 34 0h45 0 0h 35h 45 01 CCCCA; (66) Eq. (66) can split into two operator H1andH2 H1=0 BBBB@0h12 0 0 0 h 12 0h23 0 0 0h 23 0h34 0 0 0h 34 0h45 0 0 0 h 45 01 CCCCA; H 2=0 BBBB@0 0h130 0 0 0 0 0 0 h 130 0 0 h35 0 0 0 0 0 0 0h 350 01 CCCCA;(67) where h12=c1F2 5c2 1  2+ 2; h 23=p 6 2c1F1 5c2 0  1+p 6 4; h34=p 6 2c1F1 5c2 1  0+p 6 4; h 45=c1F2 5c2 2  1+ 2; h13=1 5c2 0  2; h 35=1 5c2 2  0: 26The approximate solution of Eq. (65) is given by (x;t+t) = exp (iHnd+dt) (x;t); exp (iH2dt) exp (iH1dt) (x;t); = exp(itPA 2P1) exp(itSA 1S1) (x;t); =Pexp(itA 2)P1Sexp(itA 1)S1 (x;t); (68) where 55 matrix S=u1; u 2; u 3; u 4u5
: The (u1;u2;u3;u4;u5) are normalised eigen vectors which can be obtained from un-normalised eigen vectors ( v1;v2;v3;v4;v5), dened as v1=" h23h45 h 12h 34;0;h45 h 34;0;1#T ; v2=" h12 2+2+ 2jh12j2+ 2jh23j22jh34j22jh45j2
4h 23h 34h 45;  22+ 2jh12j22jh23j2+ 2jh34j2+ 2jh45j2
4p 2h 23h 34h 45; 2+2+ 2jh12j2+ 2jh23j2+ 2jh34j22jh45j2
4h 34h 45;p 2h 45;1#T ; v3=" h12 2+2+ 2jh12j2+ 2jh23j22jh34j22jh45j2
4h 23h 34h 45;  22+ 2jh12j22jh23j2+ 2jh34j2+ 2jh45j2
4p 2h 23h 34h 45; 2+2+ 2jh12j2+ 2jh23j2+ 2jh34j22jh45j2
4h 34h 45;p 2h 45;1#T ; v4=" h12 22+ 2jh12j2+ 2jh23j22jh34j22jh45j2
4h 23h 34h 45;  2+2+ 2jh12j22jh23j2+ 2jh34j2+ 2jh45j2
4p 2h 23h 34h 45; 22+ 2jh12j2+ 2jh23j2+ 2jh34j22jh45j2
4h 34h 45;p 2h 45;1#T ; v5=" h12 22+ 2jh12j2+ 2jh23j22jh34j22jh45j2
4h 23h 34h 45;  2+2+ 2jh12j22jh23j2+ 2jh34j2+ 2jh45j2
4p 2h 23h 34h 45; 22+ 2jh12j2+ 2jh23j2+ 2jh34j22jh45j2
4h 34h 45;p 2h 45;1#T ;(69) 27by using Gram-Schmidt orthogonalization. The matrix A1= diag 0;p 2;p 2;p 2;p 2 ; (70) where 2=p (jh12j2+jh23j2+jh34j2+jh45j2)24(jh12j2jh34j2+jh45j2(jh12j2+jh23j2)) +jh12j2+jh23j2+jh34j2+jh45j2; 2=p (jh12j2+jh23j2+jh34j2+jh45j2)24(jh12j2jh34j2+jh45j2(jh12j2+jh23j2)) +jh12j2+jh23j2+jh34j2+jh45j2: Similarly, 55 matrix P=w1; w 2; w 3; w 4w5
; where w1=" h35jh13j h 13p jh13j2+jh35j2;0;0;0;1q h35h 35 jh13j2+ 1T ; w2= [0;0;0;1;0]T; w 3= [0;1;0;0;0]T; w4=" h13jh35jp 2h 35p jh13j2+jh35j2;0;jh35jp 2h 35;0;jh35jp 2p jh13j2+jh35j2T ; w5=" h13jh35jp 2h 35p jh13j2+jh35j2;0;jh35jp 2h 35;0;jh35jp 2p jh13j2+jh35j2T ;(71) and 55 matrix A2= diag 0;0;0;p jh13j2+jh35j2;p jh13j2+jh35j2 : (72) References [1] D.M. Stamper-Kurn, M.R. Andrews, A.P. Chikkatur, S. Inouye, H.-J. Mies- ner, J. Stenger, W. Ketterle, Phys. Rev. Lett. 80(1998) 2027. [2] J. Stenger, S. Inouye, D.M. Stamper-Kurn, H.J. Miesner, A.P. Chikkatur, W. Ketterle, Nature, 396(1998) 345. [3] Y. Kawaguchi, M. Ueda, Physics Reports 520(2012) 253. [4] D.M. Stamper-Kurn, M. Ueda, Rev. Mod. Phys. 85(2013) 1191. [5] C.J. Myatt, E.A. Burt, R.W. Ghrist, E.A. Cornell, C.E. Wieman, Phys. Rev. Lett. 78(1997) 586. [6] M.S. Chang, C.D. Hamley, M.D. Barrett, J.A. Sauer, K.M. Fortier, W. Zhang, L. You, and M.S. Chapman, Phys. Rev. Lett. 92(2004) 140403. 28[7] A. G orlitz, T.L. Gustavson, A.E. Leanhardt, R. L ow, A.P. Chikkatur, S. Gupta, S. Inouye, D.E. Pritchard and W. Ketterle Phys. Rev. Lett. 90 (2003) 090401. [8] B. Pasquiou, E. Mar echal, G. Bismut, P. Pedri, L. Vernac, O. Gorceix, B. Laburthe-Tolra, Phys. Rev. Lett. 106(2011) 255303. [9] Y.-J. Lin, K. Jim enez-Garc a, I.B. Spielman, Nature 471(2011) 83. [10] K. Osterloh, M. Baig, L. Santos, P. Zoller, M. Lewenstein, Phys. Rev. Lett. 95(2005) 010403. [11] J. Ruseckas, G. Juzeliunas, P. Ohberg, M. Fleischhauer, Phys. Rev. Lett. 95(2005) 010404. [12] N. Goldman, G. Juzeliunas, P. Ohberg, I.B. Spielman, Rep. Prog. Phys. 77(2014) 126401. [13] V. Galitski, I.B. Spielman, Nature, 494(2013) 49. [14] C. Wang, C. Gao, C.-M. Jian, H. Zhai, Phys. Rev. Lett. 105(2010) 160403. [15] Y. Li, L.P. Pitaevskii, S. Stringari, Phys. Rev.Lett. 108225301 (2012). [16] H. Zhai, Physics Reports 78(2015) 026001. [17] T. L. Ho, Phys. Rev. Lett. 81(1998) 742. [18] C.V. Ciobanu, S. Yip, T. Ho, Phys. Rev. A 61(2000) 033607. [19] S.-M. Chang, W.-W. Lin, and S.-F. Shieh, J. Comp. Phys. 202(2005) 367. [20] W. Bao and J. Shen, SIAM Journal on Scientic Computing 26(2005) 2010. [21] W. Bao, Multiscale Model. Simul., 2(2004) 210236. [22] S. Ashhab and C. Lobo, Phys. Rev. A, 66(2002) 013609. [23] H. Wang, Int. J. Comp. Math. 84(2007) 925. [24] W. Bao and F.Y. Lim, F. Y., SIAM Journal on Scientic Computing 30 (2008) 1925. [25] W. Bao, I.-L. Chern, and Y. Zhang, J. Comp. Phys. 253(2013) 189. [26] H. Wang and Z. Xu, Comp. Phys. Comm. 185 (2014) 2803; H. Wang, J. Comp. Phys. 274(2014) 473. [27] P. Kaur, A. Roy, and S. Gautam, Comp. Phys. Comm. 259(2021) 107671. [28] P. Banger, P. Kaur, A. Roy, and S. Gautam, arXiv preprint arXiv:2011.08892 (2020). [29] T. Ohmi, K. Machida, J. Phys. Soc. Japan 67(1998) 1822. [30] R. L. Burden and J. D. Faires, \Numerical Analysis," 9th Edition, Brooks/Cole, Pacic Grove, 2011. 29[31] M. Rezghi, E. Elden, Linear Algebra and its Application, 435 (2011) 422447. [32] A. Crubellier, O. Dulieu, F. Masnou-Seeuws, M. Elbs, H. Kn ockel, E. Tie- mann, Eur. Phys. J. D 6(1999) 211. [33] E.G.M. van Kempen, S.J.J.M.F. Kokkelmans, D.J. Heinzen, B.J. Verhaar, Phys. Rev. Lett. 88(2002) 093201. [34] A. Widera, F. Gerbier, S. F olling, T. Gericke, O. Tatjana and I. Bloch, New Journal of Physics 8(2006) 152. 30

1112.0394v1.The_contribution_of_spin_torque_to_spin_Hall_coefficient_and_spin_motive_force_in_spin_orbit_coupling_system.pdf

arXiv:1112.0394v1 [cond-mat.str-el] 2 Dec 2011The contribution of spin torque to spin Hall coefficient and spin motive force in spin-orbit coupling system Yong-Ping Fu1,2, Dong Wang1,2, F. J. Huang1,2, Y. D. Li2, W. M. Liu1 1Beijing National Laboratory for Condensed Matter Physics, Inst itute of Physics, Chinese Academy of Sciences, Beijing 100080, China 2School of Physics, Yunnan’s University, Kunming 650091, China E-mail:ynufyp@sina.cn; fuyongping44@yahoo.com.cn Abstract. We derive rigorously the relativistic angular momentum conservation equation by means of quantum electrodynamics. The novel nonrela tivistic spin current and torque in the spin-orbit coupling system, up to the order of 1 /c4, are exactly investigated by using Foldy-Wouthuysen transformation. We find a perfect spin Hall coefficient including the contribution of spin torque dipole. A novel sp in motive force, analogueto the Lorentz force, is also obtained for understanding of the spin Hall effect.2 1. Introduction Spintronics has become a fast developing field since it developed. The transport concerned aspect of the carriers’ spin degree and the spin Hall eff ect [1-3] were paid a lot of attention recently. In order to describe the spin transpor t properly, the definition of spin current was discussed and various theories of spin current have been established [4, 5]. In a traditional review, the spin current was pres ented in terms of an anticommutator of the velocity and the spin, (1 /2)ϕ+{v,s}ϕ. However, under such a definition one of problem is that there is not a conjugate spin f orce to link the spin current. Therefore, the Onsager relation can not be establis hed [6]. Furthermore, becausethespinhasitsowndynamicsinitsHilbertspace, thecurren twithbothspinand spatial degree is not conserved due to the spin-orbit coupling. With the consideration of a spin torque, a source in the spin continuity equation can be achie ved. Previous investigations in the spin torque depend on the spin relaxation time [7- 12]. To our knowledge, an explicit torque beyond of approximation of spin relaxa tion time has not been established yet. In the studies of spin Hall effect, the experiments and theories foc us on the spin Hall coefficient σSH[4, 5, 13-31]. In comparison of Ohm’s law in electronics responded to the applied electric field a spin current jkl sis generated, jkl s=σSHεlkmEm[4]. Recent studies shew that the spin Hall coefficient σSHnot only includes the contribution of the conventional spin current, but also the torque dipoles which are contained in semiconductor models with the effect of disorder [6, 32]. However, those contributions from the torque dipoles have not been clearly found yet. Basedontheaboveconsiderations, theconsistencyofquantume lectrodynamicsand Noether’stheoreminthederivationoftheexactconservationequ ationfortherelativistic angular momentum was suggested [33, 34]. It is found that the spin c urrent including a correction is different from the traditional definition. In the applic ation the spin Hall conductivity σSHinvolved the correction can be obtained. Under the requirement of the Onsager relation the spin force is found to relate to the spin Hall coe fficient, therefore, relate the topological aspect of systems with the spin-orbit couplin g. 2. Spin continuity equation Let us firstly consider the relativistic Lagrangian with Dirac fields Ψ an d¯Ψ coupled to an electromagnetic field Aµ,L=LD+Lem+Lint, whereLD=¯Ψ(i¯hcγµ∂µ−mc2)Ψ describes the free Dirac fields of spin 1 /2,Lem=−(1/4)FµνFµνis the Lagrangian of electromagnetic field, where Fµν=∂µAν−∂νAµ, the interaction between Dirac fields and electromagnetic field is given by Lint=−e¯ΨγµAµΨ, and the four-vector γµis represented as γµ= (γ0,γ) in terms of Pauli matrices σ. The energy-momentum tensor of a gauge invariant form is found to beθµν= θµν D+θµν em+θµν int, whereθµν D=¯Ψi¯hcγµ∂νΨ−gµνLD,θµν em=−Fµσ∂νAσ−gµνLem, and θµν int=−gµνLint. Heregµν=gµνis the metric tensor with g00= 1,gii=−1 (i= 1,2,3)3 ✑✑✑✑✑ ✑✑✑✑✑ ✑✑✑✑✑s s✑✑ ✸✲ ❄✑ ✰✲✻ slJkl s,fk Em Jkl s,fkEmsl ✑✑✑✑ ✑✑✑✑s✑✑ ✰✻Jkl s fksl ✑✑ ✸ ❄sJkl s fk sl✲Em ✑✑✑✑✑ ✑✑✑✑✑ ✑✑✑✑✑ s✑ ✰✲✻sl Jkl s,fkEm✻Bla c b Figure 1. The spin current Jkl sand the spin motive force fkvia the spin sand the electric field E, whereJkl srepresents the current of the lcomponent slof the spin along the direction k. (a) the spin current Jkl sand the spin motive force fkin the spin-orbit coupling system without an external magnetic field, wher eEm,slandJkl s (orfk) satisfy the right-hand rule; (b) the spin current and the spin mot ive force in the spin-orbit coupling system under an external magnetic field Balong the ldirection; (c) the spin current and the spin motive force in the two-dimensional Ra shba spin-orbit coupling system. andgµν= 0 (µ,ν= 0,1,2,3,µ/ne}ationslash=ν). This energy-momentum tensor satisfies the conservation law, i.e., ∂µθµν= 0. With the tensor the angular momentum tensor can be written in the form of Mαµν=sαµν+lαµν. Here lαµν=xµθαν−xνθαµ is the orbital angular momentum tensor and sαµν=sαµν D+sαµν emis spin angular momentum tensor, where sαµν D= (∂L/∂∂αΨ)Iµν DΨ andsαµν em= (∂L/∂∂αAσ)(Iµν em)σρAρ. Considering the notations Iµν D=−iσµν/2 and (Iµν em)σρ=gµ σgν ρ−gµ ρgν σ, it is found sαµν D= i(¯hc/4)¯Ψγα[γµ,γν]Ψ andsαµν em=AµFαν−AνFαµ. The corresponding conservation law for the total angular momentum is ∂αMαµν= 0. In order to obtain the nonrelativistic form of the conservation law, the Foldy- Wouthuysen transformation is used in the following calculations up to 1/c4. The nonrelativistic wave function is written in terms of a transformation on the relativistic wave function Ψ, Ψ′′= exp[is′(α)]exp[is(α)]Ψ, where the operators in the exponential are is(α)≡(β/2mc)α·πandis′(α)≡(i¯he/4m2c3)α·E. Here Eis the electric field intensity. Correspondingly the wave function is wr itten in the form as Ψ′′= (ϕ′′,χ′′)T, where ϕ′′=/bracketleftBig 1−s(σ)2/2β2/bracketrightBig ϕandχ′′=/bracketleftBig is′(σ)−i(E−eφ)s(σ)/2mc2β−is(σ)3/3β3/bracketrightBig ϕ. Introducing a notion η=i¯heσ·E− (E−eφ)σ·π−(σ·π)3/6m,χ′′is presented as χ′′= (η/4m2c3)ϕ. With the help of formulaeis(α)ˆOe−is(α)=ˆO+[is,ˆO]+[is,[is,ˆO]]/2+···+[is,[is,···,[is,ˆO]···]]/n!+··· and letˆObeM0ijandMkij, the continuity equation for the nonrelativistic electronic spin can be obtained. The nonrelatistivic form of angular momentum c onservation law reads ∂ ∂tρl s+∇kjkl s=Tl, (1) wherejkl s= (¯h/4m)ϕ†/braceleftBig πk,σl/bracerightBig ϕis the traditional spin current, which represents the4 current of the lcomponent of the spin along the direction k. Here we have written the wave function ϕ′′asϕfor the convenient. The spin density ρl sis obtained as ρl s=¯h 2ϕ+σlϕ+¯h 4m2c2ϕ+(πlσ·π−π2σl)ϕ +¯h2e 8m2c3ϕ+(3Bl−σlσ·B)ϕ+i¯h 8m3c4ϕ+[(σ×π)lη−η+(σ×π)l]ϕ,(2) where magnetic field Bis written out evidently. The first term in Eq. (2) is nothing but a traditional spin density. The second term can be written as (¯ h/4m2c2)ϕ+π×(π×σ)ϕ, which indicates its generation from the spin-orbit coupling. The inter action between the intrinsic magnetic moment and the external magnetic field is given by t he third term. The last term gives a small correction in the order of 1 /c4. Now let us analysis the right hand of Eq. (1), named the spin torque d ensityTl. Up to the same order of the nonrelativistic approximation, it is found Tl=∇k{i¯h 2mϕ+σk(σ×π)lϕ}+i¯he 2mcϕ+[σlσ·B−Bl]ϕ −¯he 4m2c2ϕ+{¯h[∇(E·σ)×σ]l+2σlπ·E−2σ·π El}ϕ +¯h2e 4m2c2∇k{ϕ+[σk(σ×E)l+(σ×E)kσl]ϕ} −1 32m4c4ϕ+(η+{σ·π,{σ·π,(π×σ)l}}+{σ·π,{σ·π,(π×σ)l}}η)ϕ +¯h 64m4c4∇k[ϕ+(η+{σ·π,{σ·π,σkσl}}+{σ·π,{σ·π,σkσl}}η)ϕ].(3) Besides of the relativistic correction up to the order of 1 /c4, the contributions from the spin-orbit coupling and its nonrelativistic correction are presen ted by the first and the forth terms. The second term corresponds to the interactio n of intrinsic magnetic moment and external magnetic field. The effect from the couplings a mong the orbit and the spin to the electric field is given in the third term. Previous discussion of spin Hall effect was given in the case of absenc e of the magnetic field B. In general, to extend the cases for the ferromagnet or the sys tem under the external magnetic field, the magnetic field is remained in th e follows and demonstrated theeffect ofmagneticfieldonthespinHalleffect. Co nsidering anexternal magnetic field along the direction of the spin, one state of the spin po larization is left and all spin transport processes in the presence of both the elect ric fieldEmand the magnetic field Blare shown in Fig. 1(b). The corresponding the spin motive force and the spin Hall coefficient can be obtained. It is worth to point out that the previous spin current does not contain the contribution of spin torque dipole [6]. W hen the torque density is written in the form of a divergence of a torque dipole Tl=−∇kPkl T, where Pkl T=/integraltext vTldxkis integrable, the spin current is found Jkl s=jkl s+Pkl T, (4) which includes the traditional current and a correction of the spin t orque dipole. Eq. (4) can be written as a response equation Jkl s=σscεlkmEmin which σscis the spin Hall5 coefficient. Obviously, the spin current Jkl sis vertical to the direction of the spin sland the electric field Em.Em,sl, andJkl ssatisfy the right-hand rule, as shown in Fig. 1(a). Now the spin continuity equation (1) can be written as ∂ ∂tρl s+∇kJkl s= 0. (5) It implies that the spin current has a natural conjugate spin force . Therefore, the Onsager relation σmk sc=−σkm cscan be established under the time reversal symmetry to link the spin transport with other transport phenomena, such as t he charge transport, whereσmk scandσkm csare the spin-charge and charge-spin conductivity tensors. 3. Spin Hall coefficient and spin motive force We consider the divergence of the spin torque dipole as a product of a electric field and a coefficient χlm(q),−iqkPkl T(q) =χlm(q)Em(q), withqbeing a finite wave vector. The more explicit form of the coefficient can be represented as follow χlm=−¯h 2εlm′mqm′σe e+i¯he 2mcϕ+(q)[(σlσ·B−Bl)/Em]ϕ(q) −¯he 4m2c2ϕ+(q)(i¯hqm′ σmσn′ εlm′n′ +2σlπm)ϕ(q), (6) whereσeis the electric conductivity. The spin Hall coefficient σsccorresponding to our new spin current Jkl scan be written as σsc=σ0 SH+σT SH, (7) whereσ0 SHis the conventional spin Hall conductivity [4, 12], corresponding to the traditional spin current, σT SHis the contribution of the spin torque dipole Pkl T, and σT SH=Re{i∂χlm(q)/∂qk}q=0. In some semiconductors with disorder the spin Hall coefficient is extremely different from the conventional one. We can evaluate the spin Hall coefficient σT SHin the GaAs sample as follows: at room temperature, the carrier density of GaAs is n∼1017cm−3, the mobility of carriers is µ∼350cm2/Vs, the conventional spin Hall coefficient is σ0 SH∼16Ω−1cm−1,σT SH∼5.6Ω−1cm−1. For lower carrier density case, n∼1016cm−3,µ∼400cm2/Vs,σ0 SH∼7.3Ω−1cm−1,σT SH is estimated as σT SH∼0.64Ω−1cm−1. As a kind of correction, σT SHis one order smaller than the conventional spin Hall coefficient σ0 SH. The general spin Hall coefficient σsc should include the conventional one σ0 SHand the correction σT SH. Now the Onsager relation and spin Hall coefficient have been found. T he so-called spin force Fscan be calculated as Fs= (Jc−σccE)/σcs, whereσccis the charge-charge conductivity tensor, and Jcis charge current [6]. Particularly, in Ref. [12], the spin force has a simple form as Fm s=Jk c/σkm csin the two-dimensional electron gas. From Onsager relation, σkm cs=−σmk sc, the charge-spin tensor σkm cscan be obtained, and the intrinsic Hall current Jk cin thekdirection can be detected by experiments. However, the spin force can not be interpreted as a motive force of electron like the Lorentz force in Hall effect, and it has the same direction with the electric field Em.6 To interpret the spin Hall effect, we try to find a spin motive force fkwhich has an analogy to the Lorentz force in the Hall effect. Here the spin motive force is vertical to the direction of the electric field and the spin, i.e., Em,slandfksatisfy the right-hand rule, as shown in Fig. 1(a). The discussion is based on the spin torque . The torque densityTlcan be written as the form Tl=εlmkrmfk=χlmEm. After calculation, we obtainfkas fk=σ1 fEm+σ2 fχlm, (8) where the spin motive force coefficients σ1 fandσ2 fare expressed as σ1 f=1 2Re{εlmk∇mχlm(r)} (9) and σ2 f=1 2Re{εlmk∇mEm(r)}. (10) In the case of the electric field being constant, σ2 fis zero. Here we have obtained the evident formula χlm, and the electric field Emcan be detected in experiments. Thus the spin motive force fkis found. Assuming the mobility of the carriers in the GaAs sample with disorder is µ∼103cm2/Vs and the electric field is E∼10mV/µm, we find the order-of-magnitude of the spin motive force fk∼10−20eV/µm. Obviously, this is an extremely weak quantity. 4. Application in the two-dimensional electron gas We will discuss the properties of the spin motive force in the two-dime nsional electron gas. The Dirac Hamiltonian of relativistic electron is H=cα·P+βmc2. Using the F-W transformation, the nonrelativistic limit of the Dirac Hamilton ian isH= β(mc2+π2/2m−π4/8m3c2)+eφ−(¯he/2mc)βσ·B−/parenleftBig ¯h2e/8m2c2/parenrightBig ∇·E−i/parenleftBig ¯h2e/8m2c2/parenrightBig σ· (∇ ×E)−(¯he/4m2c2)σ·(E×P), where φis the electric potential [34]. In the two- dimensional electron gas, E= (0,0,Em),σ= (σl,σk,σm),P= (Pl,Pk,0), andB= 0, the nonrelativistic Hamiltonian can be written as H=P2/2m−λ(Pkσl−Plσk), this is the Rashba Hamiltonian, where the coupling parameter λ= (¯he/4m2c2)Em[35]. In the two-dimensional electron gas, the formula χlmhas a simple form, χlm= i(¯h/2e)εlm′m∇m′ σe−(¯he/4m2c2)ϕ+(¯h∇m′σmσn′εlm′n′+ 2σlπm)ϕ. Thus the spin motive force can be represented as fk=/parenleftBig εlkm¯he/8m2c2/parenrightBig ∇m[ϕ+(¯h∇m′σmσn′εlm′n′+ 2σlπm)ϕ]Em. The spin motive force fkis nonzero, and it induces the spin current, so the spin Hall effect can be observed in experiments in the two-dimens ional electron gas. In this case, fkshould be vertical to the spin sland electric field Em, as shown in Fig. 1(c). In Ref. [36], the author introduced a spin transverse force which is perpendicular to the spin current. Onthe contrary, our spin motive forceis para llel to the spin current. So it can be used to better understand the mechanism of the spin Ha ll effect. In conclusion, we induce the spin continuity equation from the angula r momentum conservation law with spin-orbit coupling. Our results naturally includ e a correction to7 the traditional spin current. The correction could be considered a s a spin torque dipole, so there is a conjugate force linking the spin current, and the Onsa ger relation can be established. A perfect spin Hall coefficient corresponding to the ne w spin current is conformed. Furthermore, the magnitude of the spin Hall coefficien t is evaluated. From the explicit spin torque, we introduce a spin motive force having the s ame direction with the spin current to better understand the spin Hall effect. We find a novel right- hand rule among the electric field, the spin and spin current (or spin m otive force) in spintronics. We are grateful to Z. S. Ma and Y. G. Yao for helpful discussions. T his work was supported by NSF of China under grant 10347001, 90403034, 90406017, 60525417, 10665003, and by NKBRSF of China under 2005CB724508 and 2006C B921400. References [1]ˇZuti´ c I, Fabian J and Sarma S D 2004 Rev. Mod. Phys. 76323 [2] Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, von Molnar S, R oukes M L, Chtchelkanova A Y, and Treger D M 2001 Science2941488 [3] Prinz G A 1998 Science2821660 [4] Murakami S, Nagaosa N and Zhang s c 2003 Science3011348 [5] Sinova J, Culcer D, Niu Q, Sinitsyn N A, Jungwirth T, and MacDonald A H 2004Phys. Rev. Lett. 92126603 [6] Shi J, Zhang P, Xiao D, and Niu Q 2006 Phys. Rev. Lett. 96076604 [7] Sun Q F and Xie X C 2005 Phys. Rev. B 72245305 [8] Wang Y, Xia K, Su Z B, and Ma Z 2006 Phys. Rev. Lett. 96066601 [9] Zhang S and Yang Z 2005 Phys. Rev. Lett. 94066602 [10] Culcer D, Sinova J, Sinitsyn N A, Jungwirth T, MacDonald A H and Niu Q 2004Phys. Rev. Lett. 93046602 [11] Shen R, Chen Y, Wang Z D, Xing D Y 2006 Phys. Rev. B 74125313 [12] Zhang P and Niu Q cond-mat/0406436 [13] Dyakonov M I and Perel V I 1971 Sov. Phys. JETP 13467 [14] Hirsch J E 1999 Phys. Rev. Lett. 831834 [15] Rashba E I 2004 Phys. Rev. B 70161201 [16] Hu J P, Bernevig B A, and Wu C J 2003 Int. J. Mod. Phys. B 175991 [17] Sinitsyn N A, Hankiewicz E M, Teizer W and Sinova J 2004 Phys. Rev. B 70081312 [18] Bernevig B A 2005 Phys. Rev. B 71073201 [19] Shen S Q, Ma M, Xie X C, and Zhang F C 2004 Phys. Rev. Lett. 92256603 [20] Guo G Y, Yao Y and Niu Q 2005 Phys. Rev. Lett. 94226601 [21] Inoue J I, Bauer G E W and Molenkamp L W 2004 Phys. Rev. B 70041303(R) [22] Mishchenko E G, Shytov A V, and Halperin B I 2004 Phys. Rev. Lett. 93226602 [23] Dimitrova O V 2005 Phys. Rev. B 71245327 [24] Chalaev O and Loss D 2005 Phys. Rev. B 71245318 [25] Bernevig B A and Zhang S C 2005 Phys. Rev. Lett. 95016801 [26] Rashba E I 2003 Phys. Rev. B 68241315(R) [27] Jin P Q, Li Y Q and Zhang F C 2006 J. Phys. A 397115 [28] Wang J, Wang B G, Ren W and Guo H cond-mat/0507159 [29] Kato Y K, Myers R C, Gossard A C and Awschalom D D 2004 Science3061910 [30] Wunderlich J, Kaestner B, Sinova J and Jungwirth T 2005 Phys. Rev. Lett. 94047204 [31] Liu S Y and Lei X L 2005 Phys. Rev. B 72155314 [32] Sugimoto N, Onoda S, Murakami S and Nagaosa N cond-mat/050 34758 [33] Foldy L L and Wouthuysen S A 1950 Phys. Rev. 7829 [34] Bjorken J D and Drell S D 1964 Relativistic Quantum Mechanics (New York: Mc Graw-Hill) [35] Bychkov Y A and Rashba E I 1984 J. Phys. C 176039 [36] Shen S. Q. 2005 Phys. Rev. Lett. 95, 187203

2302.05101v2.Patterning_by_dynamically_unstable_spin_orbit_coupled_Bose_Einstein_condensates.pdf

Patterning by dynamically unstable spin-orbit-coupled Bose-Einstein condensates Yunjia Zhai and Yongping Zhang∗ Department of Physics, Shanghai University, Shanghai 200444, China In a two-dimensional atomic Bose-Einstein condensate, we demonstrate Rashba spin-orbit cou- pling can always introduce dynamical instability into specific zero-quasimomentum states in all pa- rameter regimes. During the evolution of the zero-quasimomentum states, such spin-orbit-coupling- induced instability can fragment the states and lead to a dynamically patterning process. The features of formed patterns are identified from the symmetries of the Bogoliubov-de Gennes Hamil- tonian. We show that spin-orbit-coupled Bose-Einstein condensates provide an interesting platform for the investigation of pattern formations. I. INTRODUCTION Atomic two-component Bose-Einstein condensates (BECs) are a veritable platform to explore pattern for- mation. A key mechanism of patterning is the insta- bility of spatially uniform states against small perturba- tions [1, 2]. The instability triggers fast growth of mode- selected perturbations dynamically giving rise to complex spatial structures [3]. When the interactions between components dominate over these of intra-components, a uniform two-component BEC presents phase separa- tion instability [4–6]. Experimentally tunable interac- tions provide a controllable approach to manipulate the phase separation [7]. The uniform two-component BEC with the instability spontaneously breaks to the spatial density patterns of complex alternating domains due to the immiscible characteristic [8, 9]. Such pattern forma- tions have been experimentally observed in different two- component setups [10, 11]. A linear coupling between two components causes Rabi oscillations between them and can modify the critical condition for phase separation in- stability. Coupling-induced pattern dynamics has been observed in experiment [12]. Furthermore, the coupling- induced pattern formation has been proposed to test the critical phenomena relating to topological defect forma- tion [13–15]. If the coupling-caused Rabi oscillations are spatially inhomogeneous, a stable moving pattern with antiferromagnetic properties can be generated [16]. Manipulating two-component BECs with the purpose to introduce other instability mechanisms is often stud- ied for pattern formations. In spatially segregated two- component BECs, the Rayleigh-Taylor instability [17– 20] and the Kelvin-Helmholtz instability [21] are em- ployed to produce complex patterns along interfaces. The snaking instability of a two-dimensional ring dark soliton leads to different symmetrical patterns with the help of the periodic modulation of the inter-component interac- tions [22]. The dynamical instability of a linear-coupled two-component polariton condensate is demonstrated to induce complex spatiotemporal patterns with phase dis- locations and vortices [23]. The periodic modulation of the transverse confinement of a two-component BEC can ∗yongping11@t.shu.edu.cnbe used to trigger the Faraday instability [24–26]. The formed Faraday patterns, having two different types of density and spin, are observed in a recent two-component experiment [27]. The emergent Faraday patterns in a two-component BEC with parametrically driven dipo- lar interactions [28] and in a two-component Fermi-Bose mixtures with driven interactions [29] are investigated. An interesting study shows the linear coupling between two components can also lead to the Faraday instabil- ity and excite Faraday patterns without the parametric driving [30]. Such an idea of Faraday pattern formations is generalized to the Raman-induced spin-orbit coupling instead of the linear coupling in a very recent work [31]. On the other hand, a two-component BEC is an ideal platform to study spin-orbit-coupled physics. Spin-orbit coupling can be artificially introduced into BECs using Raman lasers [36]. Such a Raman-induced spin-orbit cou- pling is one-dimensional [37]. While Rashba spin-orbit coupling is two-dimensional. It has been successfully synthesized into two-component BECs [32, 33] and de- generate Fermi gases [34, 35]. These experimental ad- vances and abundant spin-orbit-coupled physics revealed by early studies in [38–43] stimulate the wide investi- gation of spin-orbit-coupled BECs [44–50]. Dynamical instability of a spin-orbit-coupled BEC is a fundamental issue and attracts considerable research interest. Zhu, Zhang and Wu check dynamical instability of all states in the lower band of a Rashba-coupled BEC and delin- eate unstable parameter regimes [51]. Ozawa, Pitaevskii and Stringari examine dynamical instability of a Raman- induced spin-orbit-coupled BEC and find that states with the negative effective mass are dynamically unstable [52]. Dynamical instability of a spin-orbit-coupled BEC load- ing into a moving optical lattice is analyzed theoreti- cally [53] and experimentally [54]. For various moving ve- locities of the optical lattice, the instability is experimen- tally measured by observation of the atom loss in BECs. Unstable behaviors relate to breakdown of Galilean in- variance due to spin-orbit coupling [54]. Mardonov et al.study instability of a spin-orbit-coupled BEC with attractive interactions and find that spin-orbit coupling can control instability-induced collapse [55]. Much at- tention has been paid to analyze dynamical instabil- ity of a particular state which has a zero quasimomen- tum [38, 56–64]. The zero-quasimomentum state is ofarXiv:2302.05101v2 [cond-mat.quant-gas] 11 Jul 20232 interest. In the absence of spin-orbit coupling, atoms condensate in this state. Due to the zero quasimomen- tum, spin-orbit coupling itself has no effect on its exis- tence. But, it does make the state dynamically unsta- ble. The spin-orbit-coupling-induced dynamical instabil- ity in a zero-quasimomentum state is firstly revealed by Wang et al. [38]. It has been analyzed in detail for one- dimensional spin-orbit coupling [56–58]. Further relevant studies involve more novel physical environments, such as in the presence of an exotically one-dimensional spin- orbit coupling [59–62], spin-1 spin-orbit coupling [63], and Lee-Huang-Yang interactions [64]. In this paper, we reveal that patterns can be formed by the mechanism of spin-orbit-coupling-induced dynamical instability in a two-dimensional BEC. We first show that spin-orbit coupling always brings dynamical instability to specific zero-quasimomentum states in all parameter regimes. This is so-called spin-orbit-coupling-induced in- stability. There are four different zero-quasimomentum states. We classify them basing on whether they carry current or not. Two of them are purely originated from nonlinearity and are unique since spin-orbit coupling is ir- relevant to their existence but gives them current. We are interested in the four states since they are always dynami- cally unstable in the presence of spin-orbit coupling. Pre- vious studies [38, 56–64] have already shown the instabil- ity of a no-current-carrying zero-quasimomentum state. We uncover spin-orbit-coupling-induced dynamical insta- bility for these four states by analyzing Bogoliubov-de Gennes (BdG) equations. We then demonstrate that the spin-orbit-coupling-induced instability can trigger pat- terning processes for all four states. The current-carrying and no-current-carrying states have different formed pat- terns. The geometry of formed patterns is relevant to the symmetry of BdG Hamiltonian. We further reveal that for an anisotropic spin-orbit coupling BdG Hamil- tonian of all four states has the same symmetry. So similar patterning processes are found for four states with the anisotropic spin-orbit coupling. A tunable spin- orbit coupling can be experimentally synthesized into two-component BECs. Our study demonstrates that a spin-orbit-coupled BEC is an ideal platform for the in- vestigation of pattern formations. The paper is organized as follows. In Sec. II, a theoretical frame to analyze spin-orbit-coupling-induced dynamical instability is provided. It includes Gross- Pitaevskii equations and Bogoliubov-de Gennes analysis. In Sec. III, we prove Rashba coupling can induce dy- namical instability to four different zero-quasimomentum states, and show that dynamical instability can trigger patterning for all states. The features of formed patterns relate to the symmetry of BdG Hamiltonian. We further- more study the case of an anisotropic spin-orbit coupling. Finally, in Sec. IV we summarize our results.II. THEORETICAL MODEL The system is a two-dimensional two-component BEC with spin-orbit coupling. It is described by the following Gross-Pitaevskii equations (GPEs), i∂Ψ ∂t=p2 x+p2 y 2Ψ + ( δσz+γxpxσy−γypyσx) Ψ + HnΨ. (1) The spinor is Ψ( x, y, t ) = [Ψ 1(x, y, t ),Ψ2(x, y, t )]Twith the first component wave function Ψ 1and the second component Ψ 2.px=−i∂/∂x andpy=−i∂/∂y are the momentum operators along the xandydirections respec- tively. ( σx, σy, σz) are standard Pauli spin-1/2 matrices. The term δσzrepresents a Zeeman field along the zdi- rection [32]. The two-dimensional spin-orbit coupling is γxpxσy−γypyσxwith the anisotropic coefficients γxand γy. Ifγx=γy, the coupling becomes Rashba type. In experiment, the parameters δ,γxandγyare tunable [32]. In the GPEs, the mean-field interactions are described by Hn. Hn= g|Ψ1|2+g12|Ψ2|20 0 g12|Ψ1|2+g|Ψ2|2 ,(2) with gandg12being the intra-component and inter- component interaction coefficients respectively. They are proportional to the s-wave scattering lengths. The GPEs in Eq. (1) are dimensionless and we use the units of length, energy and time as 1 /k0,ℏ2k2 0/mandm/(ℏk2 0) respectively, here k0is the wave number of the lasers that are employed to generate spin-orbit coupling [32]. Since the system is spatially homogeneous, stationary solutions of the GPEs are plane waves Ψ(x, y, t ) =e−iµt+ikxx+ikyy ψ1 ψ2 . (3) Here, µis the chemical potential, kxandkyare the quasi- momenta along the xandydirections. The spin popu- lation, which is spatially independent, satisfies |ψ1|2+ |ψ2|2= 1. The nonlinear dispersion relation µ(kx, ky) and the wave functions ( ψ1, ψ2) can be derived after sub- stituting the plane-wave solutions into Eq. (1). Dynamical instability of these plane-wave solutions can be examined from Bogoliubov-de Gennes (BdG) equa- tions. After adding perturbations into the plane-wave solutions in Eq. (3), general wave functions become, Ψ(x, y, t ) =e−iµt+ikxx+ikyy(4) × ψ1+U1eiqxx+iqyy−iωt+V∗ 1e−iqxx−iqyy+iω∗t ψ2+U2eiqxx+iqyy−iωt+V∗ 2e−iqxx−iqyy+iω∗t , where U1,2andV1,2are the perturbation amplitudes, ωis the perturbation energy, and qx, qyare the quasimomenta of perturbation along the x, ydirections. Substituting the general wave functions into Eq. (1) and keeping linear terms relating to the perturbation amplitudes, we get the3 following BdG equations, ω U1 V1 U2 V2 =HBdG U1 V1 U2 V2 . (5) The BdG Hamiltonian in above is, HBdG= A[ψ1, ψ2] +δσzM[ψ1, ψ2] +Tsoc M[ψ2, ψ1] +T∗ socA[ψ2, ψ1]−δσz ,(6) with, A[ϕ1, ϕ2] = q2 x+q2 y 2−µ+ 2g|ϕ1|2+g12|ϕ2|2! σz, M[ϕ1, ϕ2] = g12ϕ1ϕ∗ 2g12ϕ1ϕ2 −g12ϕ∗ 1ϕ∗ 2−g12ϕ∗ 1ϕ2 , and Tsoc=−iγxqxσz−γyqyI. The BdG Hamiltonian is non-Hermitian, which allows for the existence of complex eigenvalues. For a given state in Eq. (3), if ωin BdG equations have complex modes, the state is dynamically unstable. In the presence of complex modes, perturbations in Eq. (4) shall grow up exponentially, which destroys the state. We focus on the states with zero quasimomentum kx= ky= 0. We reveal that they are dynamically unstable by calculating BdG equations. The consequence of their dynamical instability is the formation of density patterns. We show pattern formation by evolving GPEs with initial states as zero-quasimomentum states plus a randomly distributed noise. The time evolution is implemented by the standard split-step Fourier method. The window of two-dimensional space is chosen as ( x, y)∈[−51.2,51.2] and is discretized into a 256 ×256 mesh grid, and the periodic boundary condition is used for time evolution. III. RESULTS AND ANALYSIS The zero-quasimomentum states in plane-wave solu- tions Eq. (3) are of particular interest. Their existence does not depend on spin-orbit coupling. Substituting the zero-quasimomentum solutions ( kx=ky= 0) in Eq. (3) into GPEs, we have µψ1= (g|ψ1|2+g12|ψ2|2+δ)ψ1, µψ2= (g|ψ2|2+g12|ψ1|2−δ)ψ2. (7) Solving above nonlinear equations together with the nor- malization condition |ψ1|2+|ψ2|2= 1, we get four differ- ent zero-quasimomentum states. They are ψ1 ψ2 µ = 1 0 g+δ ; 0 1 g−δ ; ±q 1 2−δ g−g12 q 1 2+δ g−g12 1 2(g+g12) .(8) FIG. 1. Nonlinear bands and the Rashba spin-orbit-coupling- induced dynamical instability of the zero-quasimomentum states with the parameter regime of g−g12>2δ, and γx= γy= 1. The left panel is the nonlinear bands µ(kx= 0, ky) as a function of the quasimomentum ky.g= 1, g12= 0.2 andδ= 0.1, with these parameters the chemical poten- tial of the current-carrying states labeled by ‘c’ and ‘d’ is lower than the no-current-carrying states labeled by ‘a’ and ‘b’. The distributions of unstable modes |Imag[ ω]|(calcu- lated from the BdG equations in Eq. (5)) for the four zero- quasimomentum states demonstrated in the perturbation- quasimomentum space ( qx, qy) in the right panel. It is noted that the four zero-quasimomentum states do not depend on spin-orbit coupling. Therefore, their ex- istence is irrelevant to the detail form of spin-orbit cou- pling. They can also exist if spin-orbit coupling is one dimensional. The velocity operator of GPEs is calculated as [65] ˆv= (px+γxσy)ˆex+ (py−γyσx)ˆey. (9) Current is defined as J=Tr{ˆρˆv}with pure state density operator ˆ ρ=|Ψ⟩⟨Ψ|. The four states are real-valued, so the current becomes J=Tr{ˆρˆv}=⟨Ψ|ˆv|Ψ⟩ =−γyˆey⟨Ψ|σx|Ψ⟩ (10) =−2γyψ1ψ2ˆey. The current possibly happens along the ydirection. The former two solutions in Eq. (8) do not carry current, J= 0. In previous studies on dynamical instability in the presence of a one dimensional spin-orbit coupling, only one of these two has been analyzed [51, 52, 56–64]. The latter two solutions only exist when |g−g12| ≥2δ, which indicates that they completely originate from non- linearity. They have the same chemical potential and carry opposite currents, J=∓2γyq 1 4−δ2 (g−g2)2ˆey. Even though spin-orbit coupling does not affect the existence of these two nonlinear solutions, it gives them the current which is proportional to γy. A. The Rashba spin-orbit coupling γx=γyand g−g12>2δ We show spin-orbit-coupling-induced dynamical insta- bility for all four states in all parameter regimes. We4 FIG. 2. The evolution of the zero-quasimomentum states with 1% uniformly distributed random noise for the Rashba spin-orbit coupling and g−g12>2δ.g= 1, g12= 0.2,δ= 0.1 and γx=γy= 1. (a) and (b) are for the no-current-carrying states, (c) and (d) are for the current-carrying states. In each plot, the upper panel shows the snapshots of the coordinate-space total density, |Ψ(x, y, t )|2=|Ψ1(x, y, t )|2+|Ψ2(x, y, t )|2. The lower panel demonstrates the snapshot of the logarithm of the quasimomentum-space total density, ln |˜Ψ(kx, ky, t)|2= ln˜Ψ1(kx, ky, t)2 +˜Ψ2(kx, ky, t)2 . first analyze Rashba coupling, γx=γy. The chemical potential of the states depends on nonlinear coefficients and the Zeeman field δ. When g−g12>2δ, the current- carrying states have a lower chemical potential than that of no-current-carrying states. In order to show the lo- cation of these four states, we calculate full plane-wave solutions with kx= 0 and an arbitrary ky. The calcu- lated nonlinear bands µ(kx= 0, ky) are demonstrated in the left panel of Fig. 1 with g−g12>2δ. There are two bands and a loop structure adhering to the lower band appears. Nonlinear bands are symmetrical with respect toky= 0. The loop is a pure nonlinear effect, and its appearance requests |g−g12| ≥2δ[66]. The no-current- carrying states are two higher states at ky= 0, which are labeled by ‘a’ and ‘b’ in the figure. The current-carrying states locate in the lower parts at ky= 0, which are la- beled by ‘c’ and ‘d’. These two states have opposite group velocities, ∂µ/∂k y̸= 0, which is a further indication of current carrying. We substitute the four zero-quasimomentum states into BdG equations, and diagonalize the resultant BdGHamiltonian. Dynamical instability is identified if imag- inary parts of ωare not zero. In the right panel of Fig. 1, we demonstrate the absolute value of imaginary parts ofω,|Imag[ ω]|, for the four zero-quasimomentum states in the perturbation-quasimomentum ( qx, qy) plane. The no-current-carrying states [shown in Figs. 1(a) and 1(b)] have a two-ring structure and |Imag[ ω]|is azimuthally symmetrical. The reason of the azimuthal symmetry can be understood in this way. We set qx=qcos(θ) and qy=qsin(θ) with qbeing the magnitude of quasimo- mentum and θbeing the azimuthal angle. For the no- current-carrying states, terms relating to the overlap of two components, such as ψ1ψ2andψ∗ 1ψ2, disappear in BdG Hamiltonian, i.e., M= 0. A rescaling of perturba- tion amplitudes, U1 V1 U2 V2 → e−iθ 2U1 e−iθ 2V1 eiθ 2U2 e−i3θ 2V2 , (11)5 can gauge away the azimuthal angle θin BdG equations. Therefore, ωdoes not dependent on θ, giving rise to the azimuthal symmetry in |Imag[ ω]|shown in Figs. 1(a) and 1(b). Physically, Rashba spin-orbit-coupling γx=γyhas a rotational symmetry eiϕJzwith ϕbeing an arbitrary angle and Jz=−i∂ ∂θ+σz 2. Here θis the azimuthal angle defined from px=pcos(θ) and py=psin(θ) with pbeing the amplitude. [ eiϕJz, γ(pxσy−pyσx)] = 0. BdG Hamil- tonian inherits the symmetry leading to the azimuthal symmetry. Nevertheless, when terms ψ1ψ2in BdG Hamiltonian are non-zero, M ̸= 0 breaks the rotational symmetry, and there does not exist any rescaling to gauge away the azimuthal angle. Consequently, the current-carrying states lose the azimuthal symmetry in |Imag[ ω]|. The cal- culated results |Imag[ ω]|are demonstrated in Figs. 1(c) and 1(d) for the current-carrying states. These two states have the same distribution which possesses a πrota- tional symmetry. The symmetry is because that the wave functions of these two states are real-valued so H∗ BdG(qx, qy) =HBdG(−qx, qy) is satisfied, which leads to|Imag[ ω(qx, qy)]|=|Imag[ ω(−qx, qy)]|. We have revealed that Rashba spin-orbit coupling can induce dynamical instability into the four zero- quasimomentum states. The unstable modes have inter- esting distributions in the perturbation-quasimomentum space. We shall uncover that the dynamical instability can result in the fragmentation of spatially uniform zero- quasimomentum states and leads to pattern formation. The time evolution of GPEs is implemented by using the initial states as Ψ 1,2(x, y, t = 0) = ψ1,2(1 + 0 .01R), here ψ1,2are the zero-quasimomentum states and 1% uniformly distributed random noise Ris added. The role of initial noise is to serve as seeds for boosting unstable perturbation modes. The detailed kind of noise does not affect finally formed patterns but a large amplitude shortens patterning time scale. In Fig. 2, evolution of the coordinate-space total density, |Ψ(x, y, t )|2= |Ψ1(x, y, t )|2+|Ψ2(x, y, t )|2, and the logarithm of the quasimomentum-space total density, ln |˜Ψ(kx, ky, t)|2= ln˜Ψ1(kx, ky, t)2 +˜Ψ2(kx, ky, t)2 , are demon- FIG. 3. Nonlinear bands and the Rashba spin-orbit-coupling- induced dynamical instability of the zero-quasimomentum states with the parameter regime of g−g12<−2δ, and γx=γy= 1. g= 1, g12= 1.8 and δ= 0.1. The plots show same quantities as in Fig. 1.strated in the upper and lower panels of each subplot respectively, here ˜Ψ1,2(kx, ky, t) =R dxdy Ψ1,2(x, y, t )e−ikxx−ikyyare wave functions in the quasimomentum space. Figs. 2(a) and 2(b) are evolution snapshots for the no-current-carrying states [‘a’ and ‘b’ in Fig. 1]. At t= 0, ln |˜Ψ(kx, ky, t)|2shows a uniformly random distribution since we consider the initial random noise. Around t= 12.7, fragmentation of coordinate-space density |Ψ(x, y, t )|2leads to a clear pat- tern. The quasimomentum-space density demonstrates the same structure as unstable modes shown in Figs. 1(a) and 1(b). This indicates that all the unstable modes are selected from the background noise to grow up. Around t= 17.7, the unstable modes shown in ln |˜Ψ(kx, ky, t)|2 completely dominate. The pattern of the coordinate- space density is fully established. Since unstable modes have the azimuthal symmetry, the formed patterns are isotropic in the coordinate space. In Fig. 2(b), the occupation of unstable modes in the inner ring is slightly smaller than that in the outer ring. The situation is opposite in Fig. 2(a). Such difference leads to the distances between patterned density spots in Fig. 2(a) are larger than these in Fig. 2(b). Therefore, the pattern is more dense in Fig. 2(b). Figs. 2(c) and 2(d) demon- strate evolution snapshots for the current-carry states [‘c’ and ‘d’ in Fig. 1]. These two states show a same patterning process. At t= 8.5, all unstable modes begin to take shape as shown in ln |˜Ψ(kx, ky, t)|2. Meanwhile, the coordinate-space density is patterning. At t= 13, patterns are well established. The growing modes shown in ln|˜Ψ(kx, ky, t)|2match with the calculated unstable modes demonstrated in Figs. 1(c) and 1(d). It is noted that the formed patterns for the current-carry states are very different from the no-current-carry states. Since the unstable modes around finite qxandqy= 0 dominate [see Figs. 1(c) and 1(d)], the patterns in Figs. 2(c) and 2(d) become fracted stripes along the xdirection, which also reflects the πrotational symmetry of unstable modes in Figs. 1(c) and 1(d). B. The Rashba spin-orbit coupling γx=γyand g−g12<−2δ We have shown that Rashba-coupling-induced unsta- ble perturbation modes have different symmetries for the no-current-carry and current-carrying states when g−g12>2δ. These unstable modes can grow up from the noisy background, leading to patterned structures. The no-current-carrying states have a different patterned ge- ometry from the current-carrying states. In this section, we study the parameter regime of g−g12<−2δ, where chemical potential of the current-carrying states is higher than that of the no-current-carrying states. Typical non- linear bands µ(kx= 0, ky) are demonstrated in the left panel of Fig. 3. The loop structure adheres to the up- per band, therefore the current-carry states labeled as ‘c’ and ‘d’ have a large chemical potential. The unstable6 FIG. 4. The evolution of the zero-quasimomentum states with 1% uniformly distributed random noise for the Rashba spin-orbit coupling and g−g12<−2δ.g= 1, g12= 1.8,δ= 0.1 and γx=γy= 1. (a) and (b) are for the no-current-carrying states, (c) and (d) are for the current-carrying states. In each plot, the upper panel shows the snapshots of the coordinate-space total density |Ψ(x, y, t )|2. The lower panel demonstrates the snapshot of the logarithm of the quasimomentum-space total density ln|˜Ψ(kx, ky, t)|2. FIG. 5. Nonlinear bands and the Rashba spin-orbit-coupling- induced dynamical instability of the zero-quasimomentum states with the parameter regime of g−g12<2δ, and γx=γy= 1. g= 1, g12= 0.2 and δ= 0.9. There are only the no-current-carrying states labeled as ‘a’ and ‘b’. The plots show same quantities as in Fig. 1. modes |Imag[ ω]|are calculated from BdG equations, and are shown in Fig. 3. They keep the azimuthal symmetry for the no-current-carrying states and form a disc with a little hole in the middle [see Figs. 3(a) and 3(b)]. For the current-carrying states, unstable modes still have theπrotational symmetry, as shown in Figs. 3(c) and 3(d). However, the distribution is very different from that in Figs. 1(c) and 1(d) in the previous section. During the time evolution of these zero- quasimomentum states, the unstable modes shown in Fig. 3 can be spontaneously selected to grow up. Figs. 4(a) and 4(b) describe the evolution of the no- current-carrying states. At t= 10 .4 a density pattern FIG. 6. Nonlinear bands and an anisotropic spin- orbit-coupling-induced dynamical instability of the zero- quasimomentum states with the parameter regime of g−g12< −2δ, and γx= 1, γy= 2. g= 1, g12= 1.8 and δ= 0.1. The plots show same quantities as in Fig. 1.7 FIG. 7. The evolution of the zero-quasimomentum states with 1% uniformly distributed random noise for the anisotropic spin-orbit coupling and g−g12<−2δ.g= 1, g12= 1.8,δ= 0.1 and γx= 1, γy= 2. (a) and (b) are for the no-current-carrying states, (c) and (d) are for the current-carrying states. In each plot, the upper panel shows the snapshots of the coordinate- space total density |Ψ(x, y, t )|2. The lower panel demonstrates the snapshot of the logarithm of the quasimomentum-space total density ln |˜Ψ(kx, ky, t)|2. becomes obvious in Fig. 4(a), while in Fig. 4(b), it is att= 9.8. The different time scale is because that magnitudes of the unstable modes are different [see Fig. 3(a) and 3(b)]. For the ‘b’ state, |Imag[ ω]|have larger magnitudes, which results in a faster growth of the unstable modes. The well established patterns att= 15 .5 in Fig. 4(a) and at t= 13 .6 in Fig. 4(b) distribute isotropically. For the current-carrying states shown in Figs. 4(c) and 4(d), the developed pattern is anisotropic. C. The Rashba spin-orbit coupling γx=γyand |g−g12|<2δ When |g−g12|<2δ, the current-carrying states in Eq. (8) cannot exist. The zero-quasimomentum states are just two no-current-carrying solutions. They are la- beled as ‘a’ and ‘b’ in the full nonlinear bands in Fig. 5. As shown in the left panel, when |g−g12|<2δ,µ(kx= 0, ky) have two nonlinear bands, however, there is no loop structure. The current-carrying states are relevant tothe loop, so they do not exist in Fig. 5. Rashba spin- orbit coupling introduces dynamical instability to the no-current-carrying states. The distributions of unsta- ble modes |Imag[ ω]|in the perturbation-quasimomentum space are demonstrated in Figs. 5(a) and 5(b). For the ‘a’ state, unstable modes take a disc geometry with a small hole in the middle. For the ‘b’ state, they appear as two rings. The time evolution of these two states sup- ports patterning process. The pattern formed from the ‘a’ state is similar to that shown in Figs. 4(a) and 4(b), and formed from the ‘b’ state is similar to that shown in Figs. 2(c) and 2(d). D. The anisotropic spin-orbit coupling γx̸=γy An anisotropic spin-orbit coupling γx̸=γyloses the rotational symmetry eiϕJz. It is expected that distri- butions of unstable modes for an anisotropic spin-orbit coupling do not have the azimuthal symmetry. We study a typically anisotropic case γx= 1 and γy= 2 in the presence of four zero-quasimomentum states. The non-8 linear bands µ(kx= 0, ky) are demonstrated in the left panel of Fig. 6. From the nonlinear bands, we identify lo- cations of the zero-quasimomentum states. Since in this case, g−g12<−2δ, the current-carrying states labeled as ‘c’ and ‘d’ in the plot have a larger chemical potential, and the loop adheres to the upper band. Distributions of unstable modes for the four states are demonstrated in Figs. 6(a)-6(d). The particular outstanding is that the distributions for the no-current-carrying states are not azimuthally symmetrical [see Figs. 6(a) and 6(b)]. They are like a digital number eight. While, the distributions for the current-carrying states are also dominated by a digital number eight [see Figs. 6(c) and 6(d)]. Its size is smaller than that of the no-current-carrying states. For all four states, distributions of unstable modes always have a πrotational symmetry since if the wave functions (ψ1, ψ2) are real-valued, H∗ BdG(qx, qy) =HBdG(−qx, qy) can be satisfied, which gives rise to |Imag[ ω(qx, qy)]|= |Imag[ ω(−qx, qy)]|. The patterning by these four states is demonstrated in Fig. 7. The two no-current-carrying states show a simi- lar patterning process [see Figs. 7(a) and 7(b)]. The two current-carrying states present the exact same process [see Figs. 7(c) and 7(d)]. From the quasimomentum- space distributions in each plot, we can know that the selected unstable modes distribute as a digital number eight for all four states. The differences between the states are the size and magnitudes of the number eight, which gives patterns different length and time scales. The length scale for the current-carrying states is larger than that for the no-current-carry states. The patterning in Fig. 7 reflects the same symmetry of BdG Hamiltonian for the four states. In above, we show that spin-orbit-coupling-induced dynamical instability always exists in the four zero- quasimomentum states for all parameter regimes and the instability leads to patterning processes. Excellence of the zero-quasimomentum states lies in their easily ex- perimental implementations. Experiments start from a quasi-two-dimensional BEC. The zero-quasimomentum states can be realized by precisely controlling popula- tion of two components. Spin-orbit coupling is then sud- denly switched on by shining Raman lasers. The diabatic quench of spin-orbit coupling does not excite the zero-quasimomentum states, since they are also eigenstates of the spin-orbit-coupled system. After holding the system for a certain period during which dynamical instability excites unstable modes to grow up, a time-of-flight mea- surement is performed to measure momentum-space dis- tributions, from which the coordinate-space patterns can be revealed. IV. CONCLUSION Rashba-coupling-induced dynamical instability has been studied in a two-dimensional BEC. There exist four different zero-quasimomentum states. The two of them carry current, and the other two do not. The appear- ance of these four states does not depend on spin-orbit coupling. However, we show that the coupling indeed gives them dynamical instability. From BdG equations, we calculate unstable perturbation modes that make the states dynamically unstable. The momentum-space dis- tributions of unstable modes have different symmetries for the four states. For the no-current-carrying states, the distribution becomes azimuthally symmetrical. While it takes a πrotational symmetry for the current-carrying states. Dynamical instability triggers fast growth of un- stable modes from a noisy background, leading to frag- mentation of the four homogeneous states. We show that the fragmentation is accompanied with a patterning pro- cess. Affected by the symmetries of unstable modes, the formed coordinate-space density patterns have different features for the four states. Patterns in the no-current- carrying states are isotropic and they have a specific ori- entation in the current-carrying states. An anisotropic spin-orbit coupling can also generate dynamical instabil- ity into these four states. Unstable modes for all states have the same symmetry in the momentum space. We re- veal that the four states have a similar patterning process but with different length and time scales. V. ACKNOWLEDGES This work was supported by National Natural Sci- ence Foundation of China with Grants No.11974235 and 11774219. [1] M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851 (1993). [2] J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys. 71, S396 (1999). [3] Z. Wang, C. Navarrete-Benlloch, and Z. Cai, Pattern Formation and Exotic Order in Driven-Dissipative Bose- Hubbard Systems, Phys. Rev. Lett. 125, 115301 (2020). [4] C. K. Law, H. Pu, N. P. Bigelow, and J. H. Eberly, “Sta- bility Signature” in Two-Species Dilute Bose-EinsteinCondensates, Phys. Rev. Lett. 79, 3105 (1997). [5] E. Timmermans, Phase Separation of Bose-Einstein Con- densates, Phys. Rev. Lett. 81, 5718 (1998). [6] V. Penna and A. Richaud, Two-species boson mixture on a ring: A group-theoretic approach to the quantum dynamics of low-energy excitations, Phys. Rev. A 96, 053631 (2017). [7] G. Thalhammer, G. Barontini, L. De Sarlo, J. Catani, F. Minardi, and M. Inguscio, Double Species Bose- Einstein Condensate with Tunable Interspecies Interac-9 tions, Phys. Rev. Lett. 100, 210402 (2008). [8] K. Kasamatsu and M. Tsubota, Multiple Do- main Formation Induced by Modulation Instabil- ity in Two-Component Bose-Einstein Condensates, Phys. Rev. Lett. 93, 100402 (2004). [9] S. Ronen, J. L. Bohn, L. E. Halmo, and M. Edwards, Dynamical pattern formation during growth of a dual- species Bose-Einstein condensate, Phys. Rev. A 78, 053613 (2008). [10] H.-J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. In- ouye, A. P. Chikkatur, and W. Ketterle, Observation of Metastable States in Spinor Bose-Einstein Condensates, Phys. Rev. Lett. 82, 2228 (1999). [11] S. B. Papp, J. M. Pino, and C. E. Wieman, Tunable Miscibility in a Dual-Species Bose-Einstein Condensate, Phys. Rev. Lett. 101, 040402 (2008). [12] E. Nicklas, H. Strobel, T. Zibold, C. Gross, B. A. Mal- omed, P. G. Kevrekidis, and M. K. Oberthaler, Rabi Flopping Induces Spatial Demixing Dynamics, Phys. Rev. Lett. 107, 193001 (2011). [13] C. Lee, Universality and Anomalous Mean-Field Breakdown of Symmetry-Breaking Transitions in a Coupled Two-Component Bose-Einstein Condensate, Phys. Rev. Lett. 102, 070401 (2009). [14] J. Sabbatini, W. H. Zurek, and M. J. Davis, Phase Sepa- ration and Pattern Formation in a Binary Bose-Einstein Condensate, Phys. Rev. Lett. 107, 230402 (2011). [15] S. Wu, Y. Ke, J. Huang, and C. Lee, Kibble- Zurek scalings of continuous magnetic phase transitions in spin-1 spin-orbit-coupled Bose-Einstein condensates, Phys. Rev. A 95, 063606 (2017). [16] C. Hamner, Y. Zhang, J. J. Chang, C. Zhang, and P. En- gels, Phase Winding a Two-Component Bose-Einstein Condensate in an Elongated Trap: Experimental Ob- servation of Moving Magnetic Orders and Dark-Bright Solitons, Phys. Rev. Lett. 111, 264101 (2013). [17] K. Sasaki, N. Suzuki, D. Akamatsu, and H. Saito, Rayleigh-Taylor instability and mushroom-pattern for- mation in a two-component Bose-Einstein condensate, Phys. Rev. A 80, 063611 (2009). [18] S. Gautam and D. Angom, Rayleigh-Taylor instability in binary condensates, Phys. Rev. A 81, 053616 (2010). [19] T. Kadokura, T. Aioi, K. Sasaki, T. Kishimoto, and H. Saito, Rayleigh-Taylor instability in a two-component Bose-Einstein condensate with rotational symmetry, Phys. Rev. A 85, 013602 (2012). [20] K.-T. Xi, T. Byrnes, and H. Saito, Fingering instabilities and pattern formation in a two-component dipolar Bose- Einstein condensate, Phys. Rev. A 97, 023625 (2018). [21] H. Kokubo, K. Kasamatsu, and H. Takeuchi, Pattern formation of quantum Kelvin-Helmholtz instability in bi- nary superfluids, Phys. Rev. A 104, 023312 (2021). [22] Z.-M. He, L. Wen, Y.-J. Wang, G. P. Chen, R.-B. Tan, C.-Q. Dai, and X.-F. Zhang, Dynamics and pattern for- mation of ring dark solitons in a two-dimensional bi- nary Bose-Einstein condensate with tunable interactions, Phys. Rev. E 99, 062216 (2019). [23] T. C. H. Liew, O. A. Egorov, M. Matuszewski, O. Kyri- ienko, X. Ma, and E. A. Ostrovskaya, Instability-induced formation and nonequilibrium dynamics of phase de- fects in polariton condensates, Phys. Rev. B 91, 085413 (2015). [24] K. Staliunas, S. Longhi, and G. J. de Valc´ arcel, Faraday Patterns in Bose-Einstein Condensates,Phys. Rev. Lett. 89, 210406 (2002). [25] P. Engels, C. Atherton, and M. A. Hoefer, Observa- tion of Faraday Waves in a Bose-Einstein Condensate, Phys. Rev. Lett. 98, 095301 (2007). [26] A. Bala˘ z and A. I. Nicolin, Faraday waves in binary non- miscible Bose-Einstein condensates, Phys. Rev. A 85, 023613 (2012). [27] R. Cominotti, A. Berti, A. Farolfi, A. Zenesini, G. Lam- poresi, I. Carusotto, A. Recati, and G. Ferrari, Obser- vation of Massless and Massive Collective Excitations with Faraday Patterns in a Two-Component Superfluid, Phys. Rev. Lett. 128, 210401 (2022). [28] K. Lakomy, R. Nath, and L. Santos, Faraday pat- terns in coupled one-dimensional dipolar condensates, Phys. Rev. A 8686, 023620 (2012). [29] F. Kh. Abdullaev, M. ¨Ogren, and M. P. Sørensen, Fara- day waves in quasi-one-dimensional superfluid Fermi- Bose mixtures, Phys. Rev. A 87, 023616 (2013). [30] T. Chen, K. Shibata, Y. Eto, T. Hirano, and H. Saito, Faraday patterns generated by Rabi oscillation in a binary Bose-Einstein condensate, Phys. Rev. A 100, 063610 (2019). [31] H. Zhang, S. Liu, and Y.-S. Zhang, Faraday pat- terns in spin-orbit-coupled Bose-Einstein condensates, Phys. Rev. A 105, 063319 (2022). [32] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates, Science 354, 83 (2016). [33] W. Sun, B.-Z. Wang, X.-T. Xu, C.-R. Yi, L. Zhang, Z. Wu, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Highly Controllable and Robust 2D Spin-Orbit Coupling for Quantum Gases, Phys. Rev. Lett. 121, 150401 (2018). [34] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental realization of two-dimensional synthetic spin–orbit cou- pling in ultracold Fermi gases, Nat. Phys. 12, 540 (2016). [35] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu, C. Zhang, P. Wang, and J. Zhang, Experimental Ob- servation of a Topological Band Gap Opening in Ultra- cold Fermi Gases with Two-Dimensional Spin-Orbit Cou- pling, Phys. Rev. Lett. 117, 235304 (2016). [36] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spin- orbit-coupled Bose–Einstein condensates, Nature 471, 83 (2011). [37] Y. Zhang, M. E. Mossman, T. Busch, P. Engels, and C. Zhang, Properties of spin-orbit-coupled Bose-Einstein condensates, Frontiers of Physics. 11, 1 (2016). [38] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Spin-orbit coupled spinor Bose-Einstein condensates, Phys. Rev. Lett. 105, 160403 (2010). [39] C. Wu, I. Mondragon-Shem, and X.-F. Zhou, Uncon- ventional Bose-Einstein Condensations from Spin-Orbit Coupling, Chin. Phys. Lett. 28, 097102 (2011). [40] T.-L. Ho and S. Zhang, Bose-Einstein Condensates with Spin-Orbit Interaction, Phys. Rev. Lett. 107, 150403 (2011). [41] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Spin- orbit coupled weakly interacting Bose-Einstein conden- sates in harmonic traps, Phys. Rev. Lett. 108, 010402 (2012). [42] Y. Zhang, L. Mao, and C. Zhang, Mean-field dy- namics of spin-orbit coupled Bose-Einstein condensates,10 Phys. Rev. Lett. 108, 035302 (2012). [43] Y. Li, L. P. Pitaevskii, and S. Stringari, Quantum Tri- criticality and Phase Transitions in Spin-Orbit Coupled Bose-Einstein Condensates, Phys. Rev. Lett. 108, 225301 (2012). [44] N. Goldman, G. Juzeli¯ unas, P. ¨Ohberg, and I. B. Spiel- man, Light-induced gauge fields for ultracold atoms, Rep. Prog. Phys. 77, 126401 (2014). [45] H. Zhai, Degenerate quantum gases with spin-orbit cou- pling: a review, Rep. Prog. Phys. 78, 026001 (2015). [46] J. Zhang, H. Hu, and X. -J. Liu, Fermi gases with syn- thetic spin-orbit coupling, Ann. Rev. Cold At. Mol. 2, 81 (2014). [47] Y. Li, G. I. Martone, S. Stringari, Bose- Einstein condensation with spin-orbit coupling, Ann. Rev. Cold At. Mol. 3, 201 (2015). [48] S. Zhang, W. S. Cole, A. Paramekanti, and N. Trivedi, Spin-orbit coupling in optical lattices, Ann. Rev. Cold At. Mol. 3, 135 (2015). [49] Y. Xu and C. Zhang, Topological Fulde-Ferrell superfluids of a spin-orbit coupled Fermi gas, Int. J. Mod. Phys. B 29, 1530001 (2015). [50] S. Zhang and G.-B. Jo, Recent advances in spin-orbit coupled quantum gases, Journal of Physics and Chem- istry of Solids 128, 75 (2019). [51] Q. Zhu, C. Zhang and B. Wu, Exotic superfluidity in spin-orbit coupled Bose-Einstein condensates, Euro- phys. Lett. 100, 50003 (2012). [52] T. Ozawa, L. P. Pitaevskii, and S. Stringari, Supercurrent and dynamical instability of spin-orbit-coupled ultracold Bose gases, Phys. Rev. A 87, 063610 (2013). [53] Y. Zhang and C. Zhang, Bose-Einstein condensates in spin-orbit-coupled optical lattices: Flat bands and su- perfluidity, Phys. Rev. A 87, 023611 (2013). [54] C. Hamner, Y. Zhang, M. A. Khamehchi, M. J. Davis, and P. Engels, Spin-Orbit-Coupled Bose-Einstein Condensates in a One-Dimensional Optical Lattice, Phys. Rev. Lett. 114, 070401 (2015). [55] Sh. Mardonov, E. Ya. Sherman, J. G. Muga, H.- W. Wang, Y. Ban, and X. Chen, Collapse of spin-orbit- coupled Bose-Einstein condensates, Phys. Rev. A 91,043604 (2015). [56] I. A. Bhat, T. Mithun, B. Malomed, and K. Porsezian, Modulational instability in binary spin-orbit- coupled Bose-Einstein condensates, Phys. Rev. A 92, 063606 (2015). [57] S. Bhuvaneswari, K. Nithyanandan, P. Muruganandam and K. Porsezian, Modulation instability in quasi-two- dimensional spin-orbit coupled Bose-Einstein conden- sates J. Phys. B: At. Mol. Opt. Phys. 49, 245301 (2016). [58] T. Mithun and K. Kasamatsu, Modulation instability as- sociated nonlinear dynamics of spin-orbit coupled Bose- Einstein condensates, J. Phys. B: At. Mol. Opt. Phys. 52, 045301 (2019). [59] X.-X. Li, R.-J. Cheng, A.-X. Zhang, and J.-K. Xue, Mod- ulational instability of Bose-Einstein condensates with helicoidal spin-orbit coupling, Phys. Rev. E 100, 032220 (2019). [60] C. B. Tabi, S. Veni, and T. C. Kofan´ e, Generation of matter waves in Bose-Bose mixtures with helicoidal spin- orbit coupling, Phys. Rev. A 104, 033325 (2021). [61] P. Otlaadisa, C. B. Tabi, and T. C. Kofan´ e, Modulation instability in helicoidal spin-orbit coupled open Bose- Einstein condensates, Phys. Rev. E 103, 052206 (2021). [62] C. B. Tabi, P. Otlaadisaa, T. C. Kofan´ e, Modulation instability of two-dimensional Bose-Einstein condensates with helicoidal and a mixture of Rashba-Dresselhaus spin-orbit couplings, Phys. Lett. A 449, 128334 (2022). [63] G.-Q. Li, G.-D. Chen, P. Peng, Z. Li, and X.- D. Bai, Modulation instability of a spin-1 Bose- Einstein condensate with spin-orbit coupling, J. Phys. B: At. Mol. Opt. Phys. 50, 235302 (2017). [64] D. Singh, M. K. Parit, T. S. Raju, and P. K. Panigrahi, Modulational instability in a one- dimensional spin-orbit coupled Bose-Bose mixture, J. Phys. B: At. Mol. Opt. Phys. 53, 245001 (2020). [65] Sh. Mardonov, M. Modugno, and E. Ya. Sherman, Dy- namics of Spin-Orbit Coupled Bose-Einstein Conden- sates in a Random Potential, Phys. Rev. Lett. 115, 180402 (2015). [66] Y. Zhang, Z. Gui, and Y. Chen, Nonlinear dynam- ics of a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A 99, 023616 (2019).

1204.3728v1.Compensation_effect_in_carbon_nanotube_quantum_dots_coupled_to_polarized_electrodes_in_the_presence_of_spin_orbit_coupling.pdf

arXiv:1204.3728v1 [cond-mat.str-el] 17 Apr 2012Compensation effect in carbon nanotube quantum dots coupled to polarized electrodes in the presence of spin-orbit coupling Lin Li,1Yang-Yang Ni,1Tie-Feng Fang,1and Hong-Gang Luo1,2 1Center of Interdisciplinary Studies and Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University , Lanzhou 730000, China 2Beijing Computational Science Research Center, Beijing 10 0084, China (Dated: November 26, 2018) We study theoretically the Kondo effect in carbon nanotube qu antum dot attached to polarized electrodes. Since both spin and orbit degrees of freedom are involved in such a system, the electrode polarization contains the spin- and orbit-polarizations a s well as the Kramers polarization in the presence of the spin-orbit coupling. In this paper we focus o n the compensation effect of the effective fields induced by different polarizations by applying magnet ic field. The main results are i) while the effective fields induced by the spin- and orbit-polarizat ions remove the degeneracy in the Kondo effect, the effective field induced by the Kramers polarizatio n enhances the degeneracy through sup- pressing the spin-orbit coupling; ii) while the effective fie ld induced by the spin-polarization can not be compensated by applying magnetic field, the effective field induced by the orbit-polarization can be compensated; and iii) the presence of the spin-orbit coup ling does not change the compensation behavior observed in the case without the spin-orbit coupli ng. These results are observable in an ul- traclean carbon-nanotube quantum dot attached to ferromag netic contacts under a parallel applied magnetic field along the tube axis and it would deepen our unde rstanding on the Kondo physics of the carbon nanotube quantum dot. I. INTRODUCTION The experimental and theoretical studies of the Kondo effect1in artificial confined systems has attracted much attentionsince1998,thefirstexperimentalobservationof the Kondo effect in semiconductor quantum dot.2–4The advantage of the quantum dot as the platform studying the Kondo effect is its tunability, namely, one can tune readily the voltages of various electrodes of the quantum dot to control the relevant model parameters in describ- ing the Kondo effect, as a result, one can study in a deep way various aspects of the transport property in such a system. For example, by tuning the gate volt- age one can observe the Coulomb blockade effect,5,6the Kondo effect2–4and its unitary limit,7,8and even from the Kondo regime to the mixed-valence regime.9The non-equilibrium Kondo effect has also been studied by tuning the source-drainbias10,11and the couplingsto the leads has been tuned to observe the offset of the Kondo resonance,10and so on. Recently, due to the development of the spintron- ics the influence of the polarized electrodes attached to the quantum dot has also been intensively investigated experimentally12–15and theoretically.16–24It was found that the effect of the polarized electrodes is equivalent to an effective exchange field and can be compensated by applying an external magnetic field.12,13,18–22 The semiconductor quantum dot involved only one level is a simple system to study the Kondo physics since in such a system only the spin degree of free- dom is involved. A slightly complicated system involved degree of freedom other than the spin is the carbon nanotube (CNT) quantum dot, which includes the or- bital degree of freedom,25–30and the spin-orbit coupling could be stronger than the believed earlier, as predictedtheoretically31,32and confirmed by the experiment.33 The presence of the spin-orbit coupling has a significant influence to the transport behaviors in the CNT quan- tum dot, for example, the SU(4) symmetry discussed in the CNT quantum dot28,29is no longer valid and corre- spondingly the Kondo resonance shows some interesting splitting effects.34,35This motivates us to further study a question, namely, how does the compensation effect observed in the semiconductor quantum dot behave in the CNT quantum dot in the presence of strongly spin- orbit coupling? In Ref. [36], the Kramers polarization due to the presence of the spin-orbit coupling and its in- fluence on the Kondo peak splitting have been studied in detail by using scaling analysis and the slave-boson technique.21,37Thecompensationeffectofthe SU(2)spin and the orbital Kondo effects were discussed in the ab- sence of the spin-orbit coupling. In the present work, we study systematically the compensation effect with and without the spin-orbit coupling and the results show many novel features as presented later. Experimentally, the spin polarized transport through CNT quantum dots attached to ferromagnetic leads has beenreportedintheliterature.38Whentheexternalmag- netic field is applied perpendicularly, Hauptmann et. al. found that the exchange field can be compensated by the external field applied, which is consistent with that ob- served in the semiconductor quantum dot. The possible reason is that in the perpendicular case the spin pro- jection along the CNT axis is no longer a good quantum number and as a result the response of the single-particle energyspectrum to the applied field is approximatelythe Zeeman effect-like.39,40This is because that the perpen- dicular field only couples to the spin, not to the orbital degrees of freedom. In the present work we focus on the parallel magnetic field case, in which the presence of the2 strong spin-orbital coupling should play a significant role in the transport properties with the polarized electrodes. There are three kinds of possible polarizations of con- duction electrons in CNT, namely, the spin-polarization, the orbit-polarizationand the Kramerspolarization.36In the absence of the spin-orbit coupling, the Kramers po- larization is also absent. Thus we discuss the effects of the spin- and orbit-polarizations and compare them with those in the semiconductor quantum dots. In the pres- ence of the spin-orbit coupling, the Kramers polariza- tion is found to have a different feature in comparison to the spin- and orbit-polarizations. When both spin- and orbit-polarizations remove the degeneracy, the Kramers polarization can enhance the degeneracy. By applying magnetic field, the compensation behavior obtained is quite different to that found in the semiconductor quan- tum dot and in the perpendicular field case. While the effective exchange field induced by the orbit-polarization can be compensated, the effective exchange field induced by the spin-polarization can not be compensated. Due to the interplay between the spin-orbit coupling, the polar- ized electrodes (spin-, orbit- and Kramers polarizations) and the magnetic field applied, the Kondo peaks show complicated splitting behaviors. We analyze in detail the correspondences between these sub-peaks and their microscopic tunneling processes. This paper is organized as follows. In Sec. II we use the Anderson model to describe the CNT quantum dot and use the Green’s function formalism to study the po- larized transport behaviors. In Sec. III we present the explicit numerical results and discuss microscopic tun- neling processes. Finally, Sec. IV is devoted to a brief summary. II. MODEL AND GREEN’S FUNCTION FORMALISM The CNT quantum dot can be described by the An- derson impurity model41 H=/summationdisplay kmαǫkmαc† kmαckmα+/summationdisplay mεmd† mdm+U 2/summationdisplay m/ne}ationslash=m′nm′nm +/summationdisplay kmα/parenleftbig Vαd† mckmα+h.c./parenrightbig , (1) whered† m(dm) andc† kmα(ckmα) represent the creation (annihilation) operators of an electron in the dot and the left ( α=L) and the right ( α=R) leads, respec- tively. Here m={σ,τ}describes the configuration of electrons where σ=↑or↓andτ=±denote the spin and orbital quantum numbers. ǫkmαis the single-particle energy spectrum in the leads with the configuration m andεmis the dot level related to the spin-orbit coupling, which will be given below. nm=d† mdmis the occupa- tion operator, Uis the on-site interaction and Vαis the tunneling amplitude between the dot and leads. Here weassume that the configuration mof an electron is con- served during tunneling between the dot and leads. The electronic structure of the dot affects directly its transport properties obtained from the current through the dot within the framework of the Keldysh formalism42,43 I=ie /planckover2pi1ΓLΓR ΓL+ΓR/summationdisplay m/integraldisplay dωρd,m(ω)(fL(ω)−fR(ω)), (2) wherefα(ω) is the Fermi distribution of the lead αand Γα=/summationtext mΓα,mwith Γ α,m=πρ0 α,m|Vα|2. Hereρ0 α,m denotes the density of states of the polarized electrodes with the configuration m, which is related to the spin polarization Ps α, the orbitalpolarization Po αas well as the Kramerspolarization Pk αin the presence ofstronglyspin- orbit coupling.36Thus the coupling matrix of different configurations can be expressed as follows Γα,{↑,+}=Γα 4(1+Ps α+Po α+Pk α) (3) Γα,{↑,−}=Γα 4(1+Ps α−Po α−Pk α) (4) Γα,{↓,+}=Γα 4(1−Ps α+Po α−Pk α) (5) Γα,{↓,−}=Γα 4(1−Ps α−Po α+Pk α) (6) In Eq. (2) ρd,m(ω) =−1 πImGr d,m(ω) is the density of states in the quantum dot, where the retarded Green’s function Gr d,m(ω) can be obtained by using the equation ofmotionapproach,44inwhichthehierarchyofequations has to be truncated at certain level. Here we consider the Lacroix approximation,45which is enough to capture the compensation effect we are interested and the higher- order effects46are neglected here. Before the Green’s function Gr d,m(ω) is derived we first discuss the quantum dot level εm, which is given in the presence of the parallel magnetic field34 εm≡ε{σ,τ}=ε0 d+1 2στ∆so+τµB+σB,(7) whereε0 dis bared level depending on the geometric pa- rameters of the dot and the gate voltage. The second term is due to the spin-orbit coupling.31–33,35The third term is the orbital Zeeman splitting, where B=gµBB is the renomalized magnetic field, µ= 2µorb/(gµB) is the ratio between the orbital magnetic moment µorband the Bohr magneton µBandgis the Land´ e g-factor. It was found that µorbis usually 10 ∼20 times larger than µB.27,33The last term is the spin zeeman splitting in the presence of magnetic fields. Due to the presence of the polarized electrodes, the dot level (7) would be further modified. In the semicon- ductor quantum dots, the ferromagnetic electrodes in- duceaneffectiveexchangefieldduetothespin-dependent charge fluctuations.12,18–20,23This behavior can be well understood by Haldane’s scaling theory.47In the CNT3 quantum dots the orbit degree of freedom comes into play. The spin-, orbit-polarizations as well as possible Kramers-polarization in the presence the spin-orbit cou- pling can also induce effective exchange fields on the dot levels. By the same way one can obtain modified dot levels ˜εm=εm+δεm(notem={σ,τ}), where δε{σ,τ}=/summationdisplay α/integraldisplaydε π/parenleftbiggΓα,στ(1−fα(ε)) εστ−ε+Γα,σ¯τfα(ε) ε−U−εσ¯τ +Γα,¯στfα(ε) ε−U−ε¯στ+Γα,¯σ¯τfα(ε) ε−U−ε¯σ¯τ/parenrightbigg .(8) Here ¯σ(¯τ) =−σ(τ). The first term in Eq. (8) corre-sponds to the charge fluctuations between a single oc- cupied state and empty state in the dot levels. The re- mainingtermsreflect the chargefluctuationsbetween the single occupied state and double occupied states with on- site Coulomb repulsion U. In the semiconductor quan- tum dot, the modified dot levels can be attributed to an effective exchange field and the field can be compen- sated by external magnetic field applied.12,18–20,23Here we explore the possible compensation effects in the CNT quantum dot. With the effective dot levels at hand, one can derive the Green’s function Gr d,m(ω), which reads Gr d,m(ω) =1−/summationtext m′/ne}ationslash=m/an}bracketle{tnm′/an}bracketri}ht−/summationtext m′/ne}ationslash=mAmm′(ω) ω−˜εm−∆m(ω)+∆m(ω)/summationtext m′/ne}ationslash=mAmm′(ω)−/summationtext m′/ne}ationslash=mBmm′(ω). (9) Here the Lacroix’s approximation45is used and for sim- plify, we also consider the limit of U→ ∞. In this case, only the first term survives in Eq. (8). In Eq. (9),/an}bracketle{tnm/an}bracketri}ht=/integraltext ρd,m(ω)fm(ω)dωis the average occupa- tion number of the configuration min the dot with fm(ω) =1 Γm/summationtext αΓα,mfα(ω) and Γ m=/summationtext αΓα,m. The other notations introduced are ∆m(ω) =/summationdisplay kα|Vα|2 ω−ǫkmα, (10) Amm′(ω) =/summationdisplay kαVα/an}bracketle{td† m′ckm′α/an}bracketri}ht ω−˜εm+ ˜εm′−ǫkm′α,(11) Bmm′(ω) =/summationdisplay kk′αα′VαV∗ α′/an}bracketle{tc† k′m′α′ckm′α/an}bracketri}ht ω−˜εm+ ˜εm′−ǫkm′α.(12) By performing the summations of kin Eqs. (10), one has ∆m(ω) =Γm π/integraldisplay dω′P1 ω−ω′−iΓm≈ −iΓm,(13) where “P” denotes the principal part integration and the lastapproximationisobtainedbyneglectingtherealpart of ∆m(ω) for simplify. Similarly, Amm′(ω) andBmm′(ω) can be written as Amm′(ω) =Γm′ π/integraldisplay dω′fm′(ω′)/parenleftBig Gr d,m′(ω′)/parenrightBig∗ ω+iη−˜εm+ ˜εm′−ω′,(14) Bmm′(ω) =Γm′ π/integraldisplay dω′ fm′(ω′) ω+iη−˜εm+ ˜εm′−ω′ +i˜Γm′ π/integraldisplay dω′˜fm′(ω′)/parenleftBig Gr d,m′(ω′)/parenrightBig∗ ω+iη−˜εm+ ˜εm′−ω′,(15) where ˜Γm′= 2Γ L,m′ΓR,m′and˜fm′(ω) = 1 ˜Γm′/summationtext αΓα,m′Γ¯α,m′fα(ω)(1−f¯α(ω)) and here α= (L,R) and ¯α= (R,L). As a result, the Green’s functionGr d,m(ω) can be calculated self-consistently. It is well known that while the Lacroix’s decoupling can capture correctly the Kondo effect, the height of the Kondo peak can not be obtained accurately. The reason is that in this scheme the occupation number can not be calculated accurately. However, in the present work we focus on the compensation effect, which is only related to the positions of the poles of the dot Green’s function. From Eqs. (9), (14) and (15) one notices that the inaccuracy of the occupation number δnwill lead to an error of the position of the poles in the order of δn/ε0 d, which can be safely neglected in the Kondo regime we are interested in. III. RESULTS AND DISCUSSION In this section, we discuss in detail the various po- larization effects by solving numerically the dot Green’s function Eq. (9). While the interesting physics is in- volved only in the parallel configuration of polarization, the spin- and orbit-dependent charge fluctuations in the anti-parallel configuration cancel out each other due to the anti-orientation polarization of the leads.12,18,19,23 Therefore, we only consider the parallel configuration. In addition, since we are interested in the compensation behaviors of the spin- and orbit-polarizations as well as theKramerspolarizationin the presenceofthe spin-orbit coupling, we neglect the possible correlation between the different polarizations36and allow independent change of different polarizations. In the following calculations we take Γ L= ΓR= Γ as the units of energy and fix ε0 d=−7.5Γ. Due to the presence of the spin and orbit degrees of freedom in the system studied, the Kondoresonancescan originate from different cotunneling processes, as shown in Fig. 1. After some careful analysis, it is found that4 there are twelve possible cotunneling processes. As a re- sult, the position of the Kondo resonance obtained from each individual process can be determined by the en- ergy difference between the final and initial configura- tions, namely, eVsd= ˜εm′−˜εm. For convenience, we mark these different contunneling processes by numbers. Explicitly, we use “1” and “4” to denote the tunnelings with both spin and orbit flip, ( | ↑+/an}bracketri}ht ↔ | ↓ −/an}bracketri}ht ) and (| ↑ −/an}bracketri}ht ↔ | ↓ +/an}bracketri}ht), respectively and “2” and “5” to rep- resent the tunnelings with only spin-flip and the orbit remains, i.e., ( | ↑+/an}bracketri}ht ↔ | ↓+/an}bracketri}ht) and (| ↑ −/an}bracketri}ht ↔ | ↓ −/an}bracketri}ht ), re- spectivelyandfinally “3”and“6”mean ( | ↑+/an}bracketri}ht ↔ | ↑ −/an}bracketri}ht ) and (| ↓ −/an}bracketri}ht ↔ | ↓ +/an}bracketri}ht), respectively, namely, the spin keeps unchanged but the orbit flips. These six cotunnel- ing processes will lead to twelve Kondo peaks at most when all degeneracy is lifted by interactions involving spin and orbit degrees of freedom. In the following we explicitly discuss various cases. We study possible com- pensation effects by applying magnetic field and compare them with the results in the SU(2) case.12,13,18–22For completeness, we firstly discuss the case that the spin- orbit coupling is neglected. A. Polarization effects without the spin-orbit coupling 1. Spin-polarization effect In this case, the Kondo physics in the CNT quantum dot involves spin and orbit degrees of freedom but they are independent of each other, thus the system has the SU(4) symmetry.23There are two polarization effects to be considered, namely, spin- and orbit-polarizations. In Fig. 2 we firstly show the spin-polarization effect and its possible compensation by applying external magnetic field. When there is no any polarization in the elec- trodes, all energy levels in the dot are degenerate, so the Kondo resonance shows a single peak, as shown in Fig. 2 (a) for Ps= 0. When the spin polarization is switched on, the Kondo peak splits into three sub-peaks, /s40/s98/s41 /s109/s109/s39 /s109/s39/s101/s86 /s115/s100/s109/s40/s97/s41 FIG. 1. Schematic diagram of the cotunneling process. (a) The configuration before the tunneling and (b) the configu- ration after the tunneling. There are six possible tunnelin g processes involving different spin and orbit flips in carbon nanotube quantum dot ( m⇔m′).where the processes “3” and “6” are unaffected by the spin-polarization but the remaining processes are related to the spin-polarization. This is because that the for- mer does not involve the spin flip but the other processes do involve the spin flip. Moreover, the more stronger the spin-polarization, the more larger is the distance be- tween these three sub-peaks. Furthermore, these three sub-peaks can not be compensated by applying external magnetic field, as shown in Fig. 2 (b). This is in contrast to the statement that the spin-polarization can be com- pensated by applying an external magnetic field.36On the contrary, with increasing magnetic field, these three sub-peaks split further due to orbit- and spin-Zeeman ef- fects. It is noted that the processes “2” and “5” do not change with increasing magnetic field since in these two processes the orbital degree of freedom is not changed. Below we consider the orbit polarization effect. /s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54 /s45/s48/s46/s53/s48 /s45/s48/s46/s50/s53 /s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48/s48/s46/s48/s56/s48/s46/s49/s50/s80/s115 /s32/s61/s32/s48/s46/s53/s80/s115 /s32/s61/s32/s48/s46/s50/s53/s80/s115 /s32/s61/s32/s48 /s52/s49/s40/s50/s44/s53/s41 /s32/s32/s100/s73/s47/s100/s86 /s115/s100/s66/s32/s61/s32/s48/s40/s97/s41 /s40/s51/s44/s54/s41 /s40/s49/s44/s52/s59/s32/s50/s44/s53/s41 /s40/s51/s44/s54/s41 /s80/s115 /s32/s61/s32/s48/s46/s53 /s32/s100/s73/s47/s100/s86 /s115/s100 /s101/s86 /s115/s100/s47/s40/s98/s41 /s66 FIG. 2. The spin-polarization effect and its compensation behavior in the absence of the spin-orbit coupling. (a) The Kondo peak splitting due to the spin-polarization and (b) the Kondo sub-peaks evolution under the external magnetic field applied. For clarity, here and hereafter the curves hav e been shifted perpendicularly to show the change of the sub- peak position with the polarization in (a) and the field in (b) . The parameters used are Po=Pk= 0 and ∆ so= 0. In (b) the magnetic fields applied are 0 ,0.0025Γ,0.005Γ and 0 .0075Γ (from top to bottom).5 2. Orbit-polarization effect When the orbit-polarizationis switched on, the orbital degeneracy is removed. As a result, Fig. 3 (a) shows how the Kondo peaks split with increasing of the orbit polar- ization. It is observed that the processes “2” and “5” do not change with the change of the orbit polarization but the degeneracy between the processes “1” and “4” is re- moved. To compare Fig. 3(a) to Fig. 2 (b), the evolution ofthe Kondosub-peaksunder the external magneticfield without the orbit polarization is the same as that with the orbit polarization with zero field. This means that the effective exchange field induced by the orbit polar- ization can be completely compensated by the external magnetic field. This is true, asshown in Fig. 3 (b) (dash- dotted line). With increasing field, the processes “1” and “4” separated by the orbit polarization merge again at B= 0.006Γ. This is in contrast to the spin-polarization case. The compensation effect of the orbit polarization observed here has been not reported in the literature. The physical reason of this compensation effect is that the orbit Zeeman effect dominates in such a system and the spin Zeeman is very weak. In the following we con- sider the case that the spin-orbit coupling is present. B. Polarization effect with the spin-orbit coupling In the presence of spin-orbit coupling, the spin- and orbit-degrees of freedom are no longer independent of each other and as a result, the SU(4) Kondo physics breaks down.34In this case, even there is no magnetic field, the Kondo resonance shows three sub-peaks, as shown in Fig. 4(a) for Ps= 0. The peak located at the center is due to the processes “1” and “4”, the side- peaks originate from the remaining processes, as pointed out in the previous work.34This is in contrast to the single peak observed in Fig. 2(a). Below we consider the polarization effects in the electrodes. As in the last section, we firstly consider the spin-polarization effect. 1. Spin-polarization effect With increasing the spin-polarization, the processes “3” and “6” keep unchanged, which is the same as Fig. 2 without the spin-orbit coupling. The reason is the same there. However, the remaining processes is completely different. While the degeneracy between (1,4) and (2,5) is removed by the spin-orbit coupling, the processes (1,4) keep untouched by the spin-polarization, the processes (2,5) are sensitive to the spin-polarization. This is be- causethatintheprocesses(1,4)theeffectivefieldinduced by the spin-polarization does not change the symmetry of the spin and orbit since the orbit flips together with the spin. This is not true for the processes (2,5) in whichthe orbital keeps unchanged when the spin flips. As a re- sult, the degeneracy between the processes “2” and “5” is quickly removed with increasing the spin-polarization. Interestingly, by applying the magnetic field, the pro- cesses (2,5) keep unchanged, as shown in Fig. 4(b), the degenerates between the processes “1” and “4” and the processes “3” and “6” are removed. In this case, all de- generacy in such a system has been lifted and the Kondo resonance shows twelve sub-peak structure, as observed in Fig. 4(b). Here the magnetic field does not play a compensation role, on the contrary, it removes all possi- ble degeneracy. 2. Orbit-polarization effect Nowwediscusstheorbit-polarizationeffectinthepres- ence of spin-orbit coupling. When increasing the orbit polarization, one notes that the processes “2” and “5” do not change since in these two processes the orbital degree of freedom does not flip. However, the degener- acy between the processes “1” and “4” is lifted. The processes “3” and “6” so do. As a consequence, all de- /s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50 /s45/s48/s46/s53/s48 /s45/s48/s46/s50/s53 /s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50/s48/s46/s49/s52 /s80/s111 /s61/s48/s46/s51/s40/s98/s41/s49/s40/s50/s44/s53/s41 /s52 /s40/s51/s44/s54/s41/s80/s111 /s32/s61/s32/s48 /s32/s32/s100/s73/s47/s100/s86 /s115/s100/s40/s51/s44/s54/s41 /s52 /s49/s40/s50/s44/s53/s41/s66/s61/s48 /s80/s111 /s32/s61/s32/s48/s46/s49/s53 /s80/s111 /s32/s61/s32/s48/s46/s51/s40/s97/s41 /s66 /s32/s100/s73/s47/s100/s86 /s115/s100 /s101/s86 /s115/s100/s47 FIG. 3. The orbit-polarization effect and its compensation behavior. (a) The Kondo sub-peaks splitting due to the orbit polarization and (b) the Kondo sub-peaks evolution under the external magnetic field applied. The parameters used are Pk= 0 and ∆ so= 0 and Ps= 0.5. In (b) the magnetic fields applied are 0 ,0.0025Γ,0.005Γ,0.006Γ,0.0075Γ (from top to bottom). The dotted line denotes the complete compensation effect at B= 0.006Γ.6 generacy in such a system is removed by the presence of spin- and orbit-polarizations. This can be seen from Fig. 5 (a), in which there are twelve sub-peaks for Po= 0.3. By applying magnetic field, one finds that the splittings of (1,4) and (3,6) observed in Fig. 5(a) can be partly removed, as shown by the dotted line in Fig. 5(b), which is roughly consistent with the curve of Po= 0 in Fig. 5(a). This is exactly the compensation effect induced by the orbit polarization. 3. Kramers-polarization effect In the presence of the spin-orbit coupling, a novel po- larization effect can be proposed, namely, the Kramers polarization. As discussed above, the presences of the spin- and orbit-polarizations can remove all degeneracy in carbon nanotube quantum dot and the Kondo reso- nance can show all possible twelve sub-peak structure. However, the additional Kramers polarization plays an inverse role, namely, it leads to the degeneracy in such a system, as shown in Fig. 6 (a). While the processes (1,4) /s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50/s48/s46/s49/s52 /s54/s51/s53/s50/s40/s49/s44/s52/s41 /s80/s115 /s32/s61/s32/s48/s46/s53 /s40/s98/s41/s80/s115 /s32/s61/s32/s48 /s32/s32/s100/s73/s47/s100/s86 /s115/s100 /s40/s51/s44/s54/s41/s66/s61/s48 /s80/s115 /s32/s61/s32/s48/s46/s50/s53 /s80/s115 /s32/s61/s32/s48/s46/s53/s40/s97/s41 /s52/s49 /s53/s50/s66 /s32/s100/s73/s47/s100/s86 /s115/s100 /s101/s86 /s115/s100/s47 FIG. 4. The spin-polarization effect and its compensation behavior in the presence of the spin-orbit coupling. (a) The Kondo peak splitting due to the spin-polarization at zero fie ld and (b) the Kondo sub-peaks evolution under the external magnetic field applied when Ps= 0.5. The parameters used arePo=Pk= 0 and ∆ so= 0.0075Γ. In (b) the magnetic fields applied are 0 ,0.0025Γ,0.005Γ,0.0075Γ (from top to bot- tom).are not be affected by the Kramers polarization since in such processes the spin and orbit flip together, the non- degeneracyof the processes“2” and “5” is removedwhen Pk= 0.2. The same effect is also seen in the processes “3” and “6”. Therefore, the Kramers polarization plays an effective field to remove the non-degeneracy leaded by the spin-orbit coupling. This effective field can not be compensated by applying magnetic field, as observed in Fig. 6(b). With increasing magnetic field, the processes (2,5) keep untouched, and remaining processes are found to depend strongly on the applied magnetic field. Physi- cally, in the processes (2,5) the orbit does not flip but in the remaining processes the orbit degree of freedom flips. The three peaks structure observedin Fig. 6(b) (the dot- ted line) corresponds to the spin-polarization splitting of theSU(4) Kondo peak since the spin-orbit coupling is suppressed completely by the Kramers polarization and the effective field induced by the orbit polarization are compensated by the external magnetic fields. This indi- cates that the effective field induced by the orbital polar- ization can always be compensated by applying external /s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50 /s45/s48/s46/s53/s48 /s45/s48/s46/s50/s53 /s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50/s48/s46/s49/s52 /s50/s53 /s80/s111 /s32/s61/s32/s48/s46/s51/s40/s98/s41/s80/s111 /s32/s61/s32/s48 /s32/s32/s100/s73/s47/s100/s86 /s115/s100/s40/s51/s44/s54/s41 /s40/s49/s44/s52/s41 /s50/s66/s61/s48 /s80/s111 /s32/s61/s32/s48/s46/s49/s53 /s80/s111 /s32/s61/s32/s48/s46/s51/s40/s97/s41 /s53/s49 /s54/s52/s51 /s66 /s32/s100/s73/s47/s100/s86 /s115/s100 /s101/s86 /s115/s100/s47 FIG. 5. The orbit-polarization effect in the presence of the spin-orbit coupling and its compensation behavior. (a) The Kondo sub-peaks splitting due to the orbit polarization at zero field and (b) the Kondo sub-peaks evolution under the external magnetic fields applied when Po= 0.3. The parame- ters used are Pk= 0 and ∆ so= 0.0075Γ. In (b) the magnetic fieldsappliedare0 ,0.0025Γ,0.005Γ,0.006Γ,0.0075Γ (fromtop to bottom). The dotted line denotes the compensation effect atB= 0.006Γ.7 magnetic field, which can also be seen from Fig. 6(b), in spite of the presence of the spin-orbit coupling. IV. SUMMARY In this paper we study the compensation effect in CNT quantum dot involvingspin and orbit degreesof freedom. In comparison to the semiconductor quantum dot, the polarizedelectrodesattachedtothe dotcontainthree po- larizationeffects,namely, thespin-andorbit-polarization as well as the Kramers polarization in the presence of the spin-orbit coupling. In the semiconductor quantum dot, the effective field induced by the spin-polarization can be completely compensated by applying magnetic field. However, here in the CNT quantum dot, one finds that only the effective field induced by the orbit polarization can be compensated by applying magnetic field and that inducedbythespin-polarizationcannotbecompensated. This can be due to that the orbital magnetic moment is much larger than the spin magnetic moment. This is/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50 /s45/s48/s46/s53/s48 /s45/s48/s46/s50/s53 /s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50/s48/s46/s49/s52/s53/s50/s51 /s54 /s80/s107 /s32/s61/s32/s48/s46/s50/s40/s98/s41/s49/s40/s50/s44/s53/s41 /s52 /s40/s51/s44/s54/s41/s80/s107 /s32/s61/s32/s48 /s32/s32/s100/s73/s47/s100/s86 /s115/s100/s52 /s49/s66/s61/s48 /s80/s107 /s32/s61/s32/s48/s46/s49 /s80/s107 /s32/s61/s32/s48/s46/s50/s40/s97/s41 /s66 /s32/s100/s73/s47/s100/s86 /s115/s100 /s101/s86 /s115/s100/s47 FIG. 6. The Kramers polarization effect in the presence of the spin-orbit coupling and its compensation behavior. (a) The Kondo sub-peaks merging due to the Kramers polariza- tion at zero field and (b) the Kond sub-peaks evolution under the external magnetic fields applied with Pk= 0.2. The pa- rameter used are ∆ so= 0.0075Γ. In (b) the magnetic fields applied are 0 ,0.0025Γ,0.005Γ,0.006Γ,0.0075Γ (from top to bottom). The dotted line denotes the compensation effect at B= 0.006Γ. different from the statement that the SU(2) spin Kondo effects can be compensated by applying an external mag- netic field.36In addition, we also find that the effective fields induced by the spin- and orbit-polarizations re- move the degeneracy but the effective field induced by the Kramers polarization can enhance the degeneracy through suppressing the spin-orbit coupling. Further- more, the presence of the spin-orbit coupling does not change the compensation effect on the effective field on the orbit degree of freedom induced by any polarization. In such a system, the compensation effect of the effective field induced by the orbit polarization has not been re- ported in the literature and could be tested in the future experiments. ACKNOWLEDGMENTS SupportfromCMMMofLanzhouUniversity,theNSF- China, the national program for basic research and the fundamental research funds for the central universities of China is acknowledged.8 1J. Kondo, Prog. Theor. Phys. 32, 37 (1964). 2D. Goldhaber-Gordon, Hadas Shtrikman, D. Mahalu, David Abusch-Magder, U. Meirav, and M. A. Kastner, Na- ture (London) 391, 156 (1998). 3S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen- hoven, Science 281, 540 (1998). 4J. Schmid et al., Physica (Amsterdam) 256BC258B, 182 (1998). 5U. Meirav, M. A. Kastner, and S. J. Wind, Phys. Rev. Lett.65, 771 (1990). 6Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. 66, 3048 (1991). 7W. G. van der Wiel, S. De Franceschi, T. Fujisawa, J. M. Elzerman, S. Tarucha and L. P. Kouwenhoven, Science 289, 2105 (2000). 8J. von Delft, Science 289, 2064 (2000). 9D. Goldhaber-Gordon, J. G¨ ores, M. A. Kastner, H. Shtrik- man, D. Mahalu, and U. Meirav, Phys. Rev. Lett. 81, 5225 (1998). 10F. Simmel, R. H. Blick, J. P. Kotthaus, W. Wegscheider, and M. Bichler, Phys. Rev. Lett. 83, 804 (1999). 11S. De Franceschi, R. Hanson, W. G. van der Wiel, J. M. Elzerman, J. J. Wijpkema, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Phys. Rev. Lett. 89, 156801 (2002). 12A. N. Pasupathy, R. C. Bialczak, J. Martinek, J. E. Grose, L. A. K. Donev, P. L. McEuen, D. C. Ralph, Science 306, 86 (2004). 13K. Hamaya, M. Kitabatake, K. Shibata, M. Jung, M. Kawamura, K. Hirakawa, T. Machida, T. Taniyama, S. Ishida, and Y. Arakawa, Appl. Phys. Lett. 91, 232105 (2007). 14S. Sahoo, T. Kontos, J. Furer, C. Hoffmann, M. Gr¨ aber, A. Cottet and C. Sch¨ onenberger, Nature Phys. 1, 99 (2005). 15L. Hofstetter, A. Geresdi, M. Aagesen, J. Nygard, C. Sch¨ onenberger and S. Csonka, Phys. Rev. Lett. 104, 246804 (2010). 16P. Zhang, Q. -K. Xue, Y. P. Wang, and X. C. Xie, Phys. Rev. Lett. 89, 286803 (2002). 17N. Sergueev, Q. F. Sun, H. Guo, B. G. Wang, and J. Wang, Phys. Rev. B. 65,165303 (2002). 18J. Martinek, Y. Utsumi, H. Imamura, J. Barnas, S. Maekawa, J. Konig, and G. Schon, Phys. Rev. Lett. 91, 127203 (2003) 19J. Martinek, M. Sindel, L. Borda, J. Barnas, J. Konig, G. Schon, and J. von Delft, Phys. Rev. Lett. 91, 247202 (2003) 20M. Sindel, L. Borda, J. Martinek, R. Bulla, J. Konig, G. Schon, S. Maekawa, and J. von Delft, Phys. Rev. B. 76, 045321 (2007). 21M. Krawiec J. Phys.: Condens. Matter 19, 346234 (2007).22J. S. Lim, M. Crisan, D. Sanchez, R. Lopez, and I. Grosu, Phys. Rev. B 81, 235309 (2010). 23M. S. Choi, D. Sanchez, and R. Lopez, Phys. Rev. Lett. 92, 056601 (2004). 24R. Swirkowicz, M. Wilczynski, and J. Barnas J. Phys.: Condens. Matter 18, 2291-2304 (2006). 25T. Ando, J. Phys. Soc. Jpn. 69, 1757 (2000). 26J. Nygard, D. Henry Cobden, and P. E. Lindelof, Nature 408, 342 (2000). 27E. D. Minot, Y. Yaish, V. Sazonova, and P. L. McEuen, Nature428, 536-539 (2004). 28P. Jarillo-Herrero, J. Kong, H. S. J. van der Zant, C. Dekker, L. P. Kouwenhoven and S. D. Franceschi, Na- ture434, 484 (2005); P. Jarillo-Herrero, J. Kong, H. S. J. van der Zant, C. Dekker, L. P. Kouwenhoven and S. DeFranceschi, Phys. Rev. Lett. 94, 156802 (2005). 29M. S. Choi, R. Lopez, and R. Aguado, Phys. Rev. Lett. 95, 067204 (2005). 30A. Makarovski, A. Zhukov, J. Liu, and G. Finkelstein, Phys. Rev. B 75, 241407 (2007). 31D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B74, 155426 (2006). 32D. V. Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B 77, 235301 (2008). 33F. Kuemmeth, S. llani, D. C. Ralph, and P. L. McEuen, Nature452, 448-452 (2008). 34T.-F. Fang, W. Zuo, and H.-G. Luo, Phys. Rev. Lett. 101, 246805 (2008). 35M. R. Galpin, F. W. Jayatilaka, D. E. Logan, and F. B. Anders, Phys. Rev. B 81, 075437 (2010). 36J. S. Lim, R. Lopez, G. L. Giorgi, and D. Sanchez, Phys. Rev. B83, 155325 (2011). 37J. C. Le Guillou and E. Ragoucy, Phys. Rev. B 52, 2403 (1995). 38J. R.Hauptmann, J. Paaske, andP.E.Lindelof, Nat.Phys. 4, 373 (2008). 39D.E.LoganandM.R.Galpin, J.Chem.Phys. 130, 224503 (2009). 40T.-F. Fang, W. Zuo, and H.-G. Luo, Phys. Rev. Lett. 104, 169902(E) (2010). 41P. W. Anderson, Phys. Rev. 124, 41 (1961). 42Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. 70, 2601 (1993). 43A. P. Jauho, N. S. Wingreen and Y. Meir, Rev. B 50, 5528 (1994). 44D. N. Zubarev, Usp. Fiz. Nauk 71, 71 (1960) [Sov. Phys. Usp.3, 320 (1960)]. 45C. Lacroix, J. Phys. F 11, 2389 (1981). 46H.-G. Luo, Z.-J. Ying and S.-J. Wang, Phys. Rev. B 59, 9710 (1999). 47F. D. M. Haldane, Phys. Rev. Lett. 40, 416 (1978).

1906.04851v1.Spin_imbalance_of_charge_carriers_induced_by_an_electric_current.pdf

Spin imbalan ce of charge carriers induced by an electric current Antonio Hernando,a,b , F. Guineab,c and Miguel A. García d aInstituto de Magnetismo Aplicado, UCM -CSIC -ADIF, P. O. Box 155, 28230 -Las Rozas. Madrid, Spain. bIMDEA , Nanociencia, Faraday 9, 28049 Madrid , Spain. cSchool of Physics and Astronomy, University of Manchester, Manchester, M13 9PY, UK . dInstituto de Cerámica y Vidrio, CSIC, C/Kelsen 5, 28049 -Madrid, Spain. We analyze the contribution of the inhomogeneous magnetic field induced by an electrical current to the spin Hall effect in metals. The Zeeman coupling between the field and the electron spin leads to a spin dependent force, and to spin accumulation at the edges. We compare the effect of this relativistic correction to the electr on dynamics to the features induced by the spin -orbit interaction. The effect of current induced magnetic fields on t he spin Hall effect can be comparable to the extrinsic contribution from the spin -orbit interaction, although it does not require the prese nce of heavy elements with a strong spin -orbit interaction. The induced spin s are oriented normal to the metal slab. Introduction. The passage of an electric current in a metallic sheet leads to the accumulation of spin at its edges, the Spin Hall Effect (SHE). This phenomenon, related to the accumulation of c harge when a current is induced in a ferromagnet, the Anomalous Hall Effect (AHE) has opened new ways of manipulating spins in metals. A natural explanation of the SHE and AHE can be given in terms of the spin -orbit interaction1. For the SHE, either spin d ependent scattering due to impurities2, or the action of an intrinsic interaction on accelerating carriers3,4 can lead to the deflection of spin currents, and to spin accumulation at the edges, see Fig.[1]. The nature of the SHE and the AHE are discussed in detail in refs.4,5. In the following, we analyze an additional contribution to the SHE, arising from the existence of non -homogeneous magnetic fields within the metallic sheet, see Fig.[2]. These fields give rise to a spatially dependent Zeeman energy, and to lateral forces on the carriers, whose sign depends on the spin. As a result, a gradient in the spin density within the sample is induced. It will be shown that this mechanism gives rise to spin imbalances comparable to those derived from the extrin sic and intrinsic mechanisms mentioned earlier. Our analysis does not rely explicitly on the existence of a strong spin orbit coupling, so that it can be generalized to metals made from light elements (note, however, that the Zeeman energy can be conside red a relativistic effect which arises from the Dirac equation which describes the electrons). The main features of the model are described next. Then, semi quantitative estimates of the spin accumulation induced by inhomogeneous magnetic fields are made, and compared to the results derived from the contributions to the SHE from extrinsic and intrinsic spin -orbit interactions. Finally, the main conclusions and possible extensions will be presented. We use CGS units throughout the manuscript. This choice hel ps us to highlight the way in which relativistic corrections enter in the analysis. Figure 1. Sketch of the spin currents induced by the magnetic field created by an electrical current in a Hall bar. Inhomogeneous magnetic fields in current carrying metall ic sheets. We consider a thin metallic sheet, of dimensions ℓ𝑥 ,ℓ𝑦 and ℓ𝑧, with ℓ𝑧≪ℓ𝑥 ,ℓ𝑦, see Fig.[1] . We assume a constant current density per unit area, 𝑗𝑦 along the 𝑦 direction. This current induces at the plane of the sheet, a magnetic field along the 𝑧 direction. In the limit ℓ𝑦≫ ℓ𝑥,ℓ𝑧 and ℓ𝑧≪ℓ𝑥, the field is6 (see also the Supplementary Information) 𝐵𝑧𝐼(𝑥)=2𝑗𝑦ℓ𝑧 𝑐log(ℓ𝑥2⁄+𝑥 ℓ𝑥2⁄−𝑥) (1) The label 𝑗 stands for the current density, 𝑗𝑦=𝐼(ℓ𝑥ℓ𝑧) ⁄ , where 𝐼 is the total current . Throughout most of the sheet the field can be well approximated by function proportional to the 𝑥 coordinate6, 𝐵𝑧𝐼(𝑥)≈8𝑗𝑦𝑥 𝑐ℓ𝑧 ℓ𝑥 (2) yxz lylxlzjyelectronThis is the field measured in a frame of refere nce at rest. In the frame of reference where the current carriers are at rest a positive current, associated with the ions, can be defined. This current induces an opposite magnetic field, which is felt by the carriers. The Zeeman energy associated to this field is 𝐸𝑍𝐼=∫𝑑3𝑟⃗𝜇𝐵𝐵𝑧(𝑟⃗)[𝑛↑(𝑟⃗)−𝑛↓(𝑟⃗)] (3) Where 𝜇𝐵=(𝑒ℏ)(2𝑚𝑐) ⁄ is Bohr’s magneton. As a result, a force acting on the spin of the carriers arises 𝐹𝐼≈±8𝜇𝐵𝑗𝑦 𝑐ℓ𝑧 ℓ𝑥=±8𝜇𝐵𝐼 𝑐ℓ𝑥2 (4) This force is sketched in Fig.[2]. Figure 2. Sketch of the inhomogeneous magnetic field created by a current, and the associated Zeeman forces. Numerical estimates. Magnetic field. The magnetic field induced by the cu rrent, given in eq.(1), is maximum at the edges of the sample, 𝑥≈ℓ𝑥2⁄, 𝐵𝑚𝑎𝑥 ≈8𝑗𝑦ℓ𝑧 𝑐 (5) For typical current densities, 𝑗𝑦≈107 A×m−2 and ℓ𝑧≈100 nm we obtain 𝐵𝑚𝑎𝑥 ≈10−2G. This field is much lower th an typical fields used in experiments. We assume that the orbital polarization induced by a magnetic field of this magnitude is negligible. As mentioned before, we focus on the spin dependent forces induced by the gradient of this field. Numerical estimat es. Hall angle. The spin Hall angle, obtained from eq.(4), is described in the Supplementary Information. An approximation based on a three dimensional nearly free electron model for the carriers gives, see eq.(A5): 𝛼≈8 1372𝑘𝐹𝑎𝐵 3𝜋2𝑘𝐹 ℓℓ𝑧 ℓ𝑥 (6) Where 𝛼 is the Hall angle, 𝑘𝐹 is the Fermi wavevector, 𝑎𝐵≈0.53 Å is the Bohr radius, and ℓ is the mean free path. The factor 1372 in the denominator of the r.h.s. in eq.(6) contains the fine structure constant, and it highlights the relativistic origin of the effect. Eq.(6) applies to metallic samples with light elements, where only impurities with heavy elements can lead to a spin Hall effect. Experiments with Al samples w ere reported in6. We assume that 𝑘𝐹𝐴𝑙≈1.7 Å−1 , 𝑘𝐹𝐴𝑙 ℓ≈102 and ℓ𝑧ℓ𝑥⁄ ≈15⁄. We obtain 𝛼≈2.3×10−4. This value is of the same order of magnitude as the results reported in8,9 (see also10,11). The estimate in eq.(6) is valid for nearly free electron systems , where the effective mass is approximately equal to the free electron mass, 𝑚𝑒𝑓𝑓≈𝑚𝑒. The value of 𝛼 in eq.(6) is enhanced by a factor 𝑚𝑒𝑓𝑓 𝑚𝑒⁄ if the two masses differ significantly, as in transition metals. xz l x / 2 -l x / 2 F FF FB B electronA similar analysis to the one leading to eq.(6), but for a quasi -two dimensional electron gas where only one subband is occupied gives, see eq.(A6) 𝛼2𝐷≈8 13721 2𝜋𝑘𝐹 ℓ𝑎𝐵 ℓ𝑥 (7) The dependence of the angle on the fine structure constant implies that , 𝛼∝𝑐−2, where 𝑐 is the velocity of light. This is the same as the one expected for the extrinsic contribution from skew scattering effects, see below. This may complicate the experimental separation of the contribution from the field gradient discussed here from that of skew scattering . A plot of the depend ence of 𝛼 on elastic scattering and on the aspect ratio of the sample, using. Eq.(6), is shown in Fig.[3]. Ref.[2] suggested the measurement of the Spin Hall Effect through the voltage induced in a strip connecting the edges of the sample. This setup does not nec essarily work for the detecting the contribution discussed here, as the current induced magnetic field can also affect the electrons in the strip , see discussion in the Supplementa ry Information . Figure 3. Spin Hall angle, 𝛼, due to the current induced magnetic field, as function of the product of the Fermi wavelength and the mean free path, 𝑘𝐹ℓ and the aspect ratio of the sample, ℓ𝑧ℓ𝑥⁄. See equation (6). Numerical estimates. Spin accumulation at the edges. The net spin at the edges, if the spin diffusion length is smaller than the width of the sample, ℓ𝑥, is determined by the d ifference in the chemical potential for the spin up and spin down electrons, 𝜇↑−𝜇↓≈2𝜇𝐵𝐵𝑧(𝑥≈±ℓ𝑥 2) . The expression for the field is given in eq.(1). This expression has been obtained in the limit ℓ𝑧≪ℓ𝑥 and it breaks close to the edges, when |ℓ𝑥 2−|𝑥||≤ℓ𝑧. We take the average of 𝐵𝑧 over distances fr om the edge comparable to ℓ𝑧, and obtain 〈𝐵𝑧〉≈2𝑗𝑦ℓ𝑧 𝑐log(ℓ𝑥 ℓ𝑧). We finally obtain that the spin accumulation at the edges , in units of electrons per area is 𝑛↑−𝑛↓≈4𝜇𝐵𝑗𝑦ℓ𝑧 𝑐log(ℓ𝑥 ℓ𝑧)×𝑛(𝜖𝐹) (8) Where 𝑛(𝜖𝐹) is the density of states of the metal at the Fermi energy. We assume ℓ𝑥=1𝜇m, ℓ𝑧=0.1𝜇m, 𝑗𝑦= 103Amm2⁄ and 𝑛(𝜖𝐹)=1eV−1Å−3. Inserting these parameters in to eq.(8) we obtai n 𝑛↑−𝑛↓≈1018cm−3. Comparison to the skew scattering contribution to the Spin Hall Effect. We first est imate the combined effect of the spin scattering interaction and defect scattering following ref.[2]. We assume that the effect of the impurities is roughly equivalent to the effect of a magnetic field equal to the magnetization due to the carriers with a given spin, 𝐵𝑒𝑓𝑓≈𝑒ℏ 𝑚𝑐𝑛↑ where 𝑛↑ is the density per unit volume of carriers with spin up. The associated force is 𝐹𝑒𝑥𝑡𝑆𝐻≈v𝑒𝐵𝑒𝑓𝑓 𝑐≈v𝑒2ℏ 𝑚𝑐2𝑛↑ (9) Where v is the carrier velocity. Using 𝐼≈𝑒v(𝑛↑+𝑛↓)ℓ𝑥ℓ𝑧 , we obtain 𝐹𝑒𝑥𝑡𝑟𝑆𝐻≈±ℏ𝑒𝐼 𝑚𝑐2ℓ𝑥ℓ𝑧≈±𝜇𝐵𝐼 𝑐×1 ℓ𝑥ℓ𝑧 (10) Which leads to a spin Hall angle 𝛼𝑒𝑥𝑡𝑟 ≈1 1372𝑘𝐹𝑎𝐵 3𝜋2𝑘𝐹ℓ (11) So that, using eq.(6), we obtain 𝛼𝐼 𝛼𝑒𝑥𝑡𝑟≈8ℓ𝑧 ℓ𝑥 (12) Note that this estimate of 𝐹𝑒𝑥𝑡𝑆𝐻 in eq.( 10) gives an upper bound to its actual value, as the analysis assumes that the spin dependent scattering induced by impurities leads to an effective field comparable to the one generated by the total spin of the carriers in the metal . A general discussion of skew scattering effects, following the original ideas12,13,14, can be found in15. A different estimate of 𝐹𝑒𝑥𝑡𝑆𝐻 based on a two dimensional nearly free electron model a nd the Boltzmann equation is given in the supplementary information, see eq.(A17 ), and it gives 𝛼𝑒𝑥𝑡𝑟2𝐷 ≈(𝑘𝐹𝑎𝐵)2 1372 ℓ ℓ𝑆𝑂𝐼 (13) Where ℓ𝑆𝑂𝐼 is a mean free path associated to the impurities which ind uce the skew scattering. Using this estimate, and comparing to eq.(7), we find 𝛼𝐼2𝐷 𝛼𝑒𝑥𝑡𝑟2𝐷≈8 𝑘𝐹𝑎𝐵ℓ𝑆𝑂𝐼 ℓ𝑥 (14) It is finally interesting to note that three dimensional scattering by impurities with spin -orbit c oupling, described generically by terms such as 𝑉𝑖𝑚𝑝(𝑟⃗)≈𝑉0𝐿⃗⃗𝑠⃗𝛿(𝑟⃗), do not favor a particular spin orientation. This scattering is expected to dominate in metallic systems where many subbands are occupied, such that ℓ𝑧≫𝑘𝐹−1. In these samples, there is no preferred spin orientation which spontaneously will accumulate at the edges (note this in not the case for experiments where a spin polarized current is injected). On the other hand, the magnetic field induced by the current considered here selects a spin orientation, and always leads to a spin accumulation at the edges. Other contributions to the Spin Hall Effect. A similar analytical semi -quantitative comparison between the current induced SHE discussed here and the intrinsi c contribution of the spin orbit interaction is not possible. Simple estimates based on the Rashba interaction give vanishing effects16, once leading effects due to impurities are taken into account4,5,6. More involved calculations using the Berry curvatur e in complex, spin dependent, models for the bands of transition metals suggest the the SHE conductivity can be written as17,18 𝜎𝑥𝑦𝑧≈𝑒 4𝑎×〈𝑙⃗ 𝑠⃗〉𝐹𝑆 ℏ2 (15) Here, 𝑎 is the lattice constant, and 〈𝑙⃗ 𝑠⃗〉𝐹𝑆 is the average over the Fermi surface of the product of the orbital angular momentum and the spin, which depends on the strength of the spin orbit interaction. The value of 〈𝑙⃗ 𝑠⃗〉𝐹𝑆 is determined by the intrinsic spin orbit interaction in the mater ial, which typically scales as 𝑐−2. Hence, the current contrib ution to the spin Hall voltage, the contribution from skew scattering, and the intrinsic contribution all arise from relativistic effects, and show the same dependence on the vel ocity of light . Note that the intrinsic contribution changes the conductivity , see eq.(15). There is finally a contribution to the spin Hall effect from side jump scattering. To our knowledge, there are not reliable techniques to extract simple estimates of the magnitude of these pro cesses. We cannot compare the role of the current induced field considered here to that of side jump scattering. Conclusions. We have analyzed the contribution to the spin Hall effect of the magnetic field associated to the current flowing through a metal lic sample. The spin accumulation at the edges arises from the Zeeman coupling to the inhomogeneous magnetic field which exists within the sample. The gradient of the Zeeman coupling acts like a force which accelerates the carriers in the direction perpend icular to the current, in opposite way for spin up and spin down electrons, where the spins are ori ented along the magnetic field. This effect is relativistic in nature, and its contribution to the spin Hall angle depends on the value of the velocity of li ght as 𝛼∝𝑐−2, similarly to the (extrinsic) contribution from skew scattering processes. The current induced spin accumulation is linearly proportional to the scattering time or the transport mean free path, also like the contribution from skew scatte ring. Hence, it should dominate in clean samples, with 𝑘𝐹ℓ≫1. Estimates from simple models suggest that the effect of the current induced magnetic field should be at least comparable to that of skew scattering. The current induced spin Hall effect dep ends on the aspect ratio of the sample, through the ratio ℓ𝑧ℓ𝑥⁄, where ℓ𝑧 is the thickness, and ℓ𝑥 is the width. Hence, it is not a unique property of the material. This can explain the broad distribution of Hall angles for the same mater ial reported in the literature. In contrast with the effect of skew scattering, the current induced spin Hall effect does not require the existence of a strong spin -orbit interaction, either due to heavy ion impurities or intrinsic to the material. Hence, it should be present in any sample, and play a significant role in light metal materials, such as Al. Samples which are far from being quasi -two dimensional, 𝑘𝐹ℓ𝑧≫1, typically do not have a preferred spin orientation, so that transverse spin curre nts due to skew scattering will average to zero. The current induced magnetic field provides a preferred spin direction, enhancing the robustness of the spin Hall effect. Acknowledgements This work was supported by the Spanish Ministry of Innovation, Scien ce and Technology and Spanish Ministry of Economy and Competitiveness through Researc h Projects MAT2015 -67557 -C2-1-P, MAT2017 -86450 -C4-1-R, S2013/MIT -2850 NANOFRONTMAG and by the European Commission AMPHIBIAN (H2020 -NMBP -2016 - 720853). References 1 M. I. Dya konov, and V. I. Perel, JETP Lett. 13 467 (1971) . 2 J. E. Hirsch , Phys. Rev. Lett. 83, 1834 (1999). 3 R. Karplus, and J. M. Luttinger, Phys. Rev. 95, 1154 (1954). 4 J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004). 5 J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T. Jungwirth , Rev. Mod. Phys . 87, 1213 (2015). 6 M. Lin iers, V. Madurga, M. Vázquez and A. Hernando, Phys. Rev. B 31, 4425 (1985). 7 S. O. Valenzuela, M. Tinkham, Nature 442, 176 (2006). 8 Y. Kato , R. C. Myers , A. C. Gossard , D. D. Awschalom, Science 306 1910 (2004) . 9 K. Ando, and E. Saitoh, Nature Comm. 3, 629 (2012). 10 H.L. Wang, C.H. Du, Y. Pu, R. Adur, P.C. Hammel, and F.Y. Yang Phys. Rev. Lett. 112, 197201 (2014). 11 A. Hoffmann, IEEE Trans. On Magnetics , 49, 5172 (2013). 12 N.F. Mott Proc. Roy. Soc. A 124 425 (1929) . 13 J. Smit, Physica 21, 877 (1955). 14 J. Smit, Physica 24, 39 (1958). 15 N. A. Sinitsyn, Journal of Phys.: Cond. Matt. 20, 023201 (2008). 16 J. Ionue, G. E. W. Bauer, and L. W. Mol enkamp, Phys. Rev. B 70, 041303 (2004). 17 H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Phys. Rev. Lett. 102, 016601 (2009). 18 E. Saitoh, M. Ueda, H. Miyajima, G. Tatara. Appl. Phys. Lett. 88 182509 (2006). Supplementary Information. Calculation of the magnetic field in a thin slab. We analyze a slab which is infinite in the 𝑦 direction, and with dimensions ℓ𝑥,ℓ𝑧 in the 𝑥,𝑧 directions. Then, the field at a position (𝑥,𝑦,𝑧) inside the slab, −ℓ𝑥2⁄≤𝑥≤ ℓ𝑥2⁄ ,−ℓ𝑧2⁄≤𝑧≤ℓ𝑧2⁄ can be written, using the Biot Savart law, as 𝐵 ⃗⃗⃗⃗(𝑥,𝑦,𝑧)=𝑗𝑦 𝑐∫ 𝑑𝑦′∫ 𝑑𝑥′ℓ𝑥2⁄ −ℓ𝑥2⁄∞ −∞∫ 𝑑𝑧′(𝑧−𝑧′)𝑥̂−(𝑥−𝑥′)𝑧̂ [(𝑥−𝑥′)2+(𝑦−𝑦′)2+(𝑧−𝑧′)2]32⁄ℓ𝑧2⁄ −ℓ𝑧2⁄ (A1) Where 𝑥̂ and 𝑧̂ are unit vectors along the 𝑥 and 𝑧 directions. Integrating over 𝑥′ and 𝑦′, we obtain 𝐵𝑥(𝑥,𝑦,𝑧)=2𝑗𝑦 𝑐∫ 𝑑𝑧′tan(𝑥−𝑥′ 𝑧−𝑧′)| 𝑥′=−ℓ𝑥2⁄𝑥′=ℓ𝑥2⁄−ℓ𝑧2⁄ −ℓ𝑧2⁄≈2𝜋𝑗𝑦 𝑐∫ 𝑑𝑧′sign (𝑧−𝑧′)≈±2𝜋𝑗𝑦ℓ𝑧 𝑐−ℓ𝑧2⁄ −ℓ𝑧2⁄ (A2) 𝐵𝑧(𝑥,𝑦,𝑧)=−2𝑗𝑦 𝑐∫ 𝑑𝑧′1 2log[(𝑥−𝑥′)2+(𝑧−𝑧′)2]|𝑥′=−ℓ𝑥2⁄𝑥′=ℓ𝑥2⁄ −ℓ𝑧2⁄ −ℓ𝑧2⁄≈2𝜋𝑗𝑦ℓ𝑧 𝑐log|𝑥+ℓ𝑥2⁄ 𝑥−ℓ𝑥2⁄| (A3) Where, in ord er to obtain the last result , the assumption ℓ𝑧≪ℓ𝑥 has been made. The two signs in the expression for 𝐵𝑥 refer to the top and bottom surfaces of the slab. Note that the integral of the field along a contour defined as a section of the stripe of width 2𝑥 is given by Φ(𝑥)≈2∫𝑑𝑥𝑥 −𝑥𝐵𝑥(𝑥,𝑦,𝑧)≈8𝜋𝑗𝑦𝑥ℓ𝑧 𝑐 (A4) Spin Hall angle due to the current induced magnetic field. The spin Hall angle can be written as 𝛼≈𝐹𝐼 𝑒𝐸∥ (𝐴5) Where 𝐹𝐼 is given in eq.( 4) of the main text, and 𝐸∥ is the field which induces the longitudinal current 𝑗∥=𝑛𝑣 ≈𝑛𝑒𝐸∥𝜏 𝑚=𝑛𝑒𝐸∥ℓ 𝑚𝑣𝐹 (𝐴6) Where 𝑛 is the carrier density, 𝑣 is the drift velocity, 𝜏 is the scattering time, 𝑣𝐹 is the Fermi velocity, and ℓ is the mean free path. For a three dimensional metal and using a nearly free electron description, we have 𝑛=𝑘𝐹3 3𝜋2 𝑣𝐹=ℏ𝑘𝐹 𝑚 (𝐴7) Where 𝑘𝐹 is the Fermi wavevector. From eqs.(A5), (A6), and (A7) we obtain 𝛼≈8𝑒2 𝑚𝑐2𝑘𝐹2 3𝜋2ℓℓ𝑧 ℓ𝑥≈8 1372𝑘𝐹𝑎𝐵 3𝜋2𝑘𝐹 ℓℓ𝑧 ℓ𝑥 (𝐴8) Where 𝑎𝐵≈0.53 Å is the Bohr radius. For a two dimensional system where only one subband is occupied, the carrier density is 𝑛=𝑘𝐹2 2𝜋ℓ𝑧 (𝐴9) The spin Hall angle becomes 𝛼2𝐷≈8 13721 2𝜋𝑘𝐹 ℓ𝑎𝐵 ℓ𝑥 (𝐴10) Skew scattering and mean free path . The contribution from skew scattering to the spin Hall conductivity arises from the spin dependence of the scattering rates in the presence of the spin -orbit interaction3,4,5. We use a nearly free electron approximation to the electron band, and assume that the impurity induced matrix elements between scattering states can be written as 〈𝑘⃗⃗′,𝑠′|𝑉|𝑘⃗⃗,𝑠〉=𝑉𝑘.𝑘′[𝛿𝑠,𝑠′+𝑖ℏ2 4𝑚2𝑐2(⟨𝑠′|𝜎⃗|𝑠⟩×𝑘⃗⃗′)∙𝑘⃗⃗] (𝐴11) We analyze a quasi -two dimensional system, wh ere the Fermi surface can be approximated by a circle of radius 𝑘𝐹. The momenta 𝑘⃗⃗ and 𝑘⃗⃗′ lie in the 𝑥−𝑦 plane. The transition rates near the Fermi surface can be written as function of the angle 𝜃 between 𝑘⃗⃗ and 𝑘⃗⃗′. Using eq.(A11 ), and considering higher order scattering processes, we approximate the transition rates by 𝑊𝜃𝑠𝑧≈1 𝜏+1 𝜏𝑆𝑂𝐼ℏ2𝑘𝐹2𝑠𝑧 𝑚2𝑐2sin(𝜃) (𝐴12) where 𝑠𝑧=±1 2 describes the spin of the electron, and we have distinguished two scattering processes described by two scattering rates, 𝜏, which describe symmetric scattering, and 𝜏𝑆𝑂𝐼 which describes skew scattering induced by impurities with spin -orbit coupling. The transport properties can be compute d from the Boltzmann equation, which can be written as 𝜕𝑓𝜃𝑠𝑧 𝜕𝑡=0=𝑒𝐸⃗⃗𝑣⃗𝜕𝑓0(𝜖) 𝜕𝜖+∑𝑊𝜃−𝜃′𝑠𝑧(𝑓𝜃𝑠𝑧−𝑓𝜃′𝑠𝑧) 𝜃′ (𝐴13) where 𝑓0(𝜖) is the unperturbed Fermi -Dirac distribution and 𝑣⃗=ℏ𝑘⃗⃗ 𝑚. There are two independent equations which describe th e distribution of electrons with 𝑠𝑧=±1 2. We split 𝑓𝜃 into a 𝑠𝑧 independent term, 𝑓̅𝜃, and a term which describes the skew scattering, ±𝛿𝑓𝜃, where the sign refers to the spin. We assume that |𝑓̅𝜃̅|≫|𝛿𝑓𝜃| and obtain 𝑓̅𝜃≈−ℏ𝑘𝐹𝑒|𝐸⃗⃗|𝜏cos(𝜃) 𝑚𝜕𝑓0(𝜖) 𝜕𝜖 (𝐴14) The shift in the carriers’ velocity is 𝑣∥≈𝑒|𝐸⃗⃗|𝜏 𝑚 (𝐴15) Inserting this expression into the Boltzmann equation, we obtain 0=ℏ𝑘𝐹𝑒|𝐸⃗⃗|cos(𝜃) 𝑚𝜕𝑓0(𝜖) 𝜕𝜖+1 𝜏(𝑓̅𝜃±𝛿𝑓𝜃)±ℏ2𝑘𝐹2 2𝑚2𝑐2𝜏 𝜏𝑆𝑂𝐼ℏ𝑘𝐹𝑒|𝐸⃗⃗|sin(𝜃) 𝑚𝜕𝑓0(𝜖) 𝜕𝜖 (𝐴16) This equation can be understood as if the function 𝑓̅𝜃±𝛿𝑓𝜃 is induced by a field which is rotated with respect to the applied field by an angle 𝛼≈±ℏ2𝑘𝐹2 2𝑚2𝑐2 𝜏 𝜏𝑆𝑂𝐼 ≈±(𝑘𝐹𝑎𝐵)2 1372 ℓ ℓ𝑆𝑂𝐼 (𝐴17) Where 𝑎𝐵 is the Bohr radius, and ℓ𝑆𝑂𝐼 is a mean free path related to the impurities which induce the skew scattering. Note that the estimate in eq.(A17 ) is valid only when one two dimensional subband is occupied. In the limit ℓ𝑧−1≪𝑘𝐹 the sample must be approximated using a three dime nsional Fermi surface. In that case, the ansatz for the spin -orbit matrix element s defined in eq.(A5 ) does not favor a particular spin direction, when scattering processes are averaged over the Fermi surface. As a result, an unpolarized current will not l ead to a net spin polarization at the edges of the sample. Effect of Lorentz force due to the field produced by the primary current j x. The field associated with the current 𝐵𝑧(𝑥)exerts a force on the carriers that move along the x axis. This force is given by 𝐹𝑥(𝑥)=𝑒𝑣𝐵𝑧(𝑥)𝑐⁄ and induces a perturbation of the uniform density of carriers. However the value of the maximum Lorentz force for the density currents considered in this article (Hall effect with an applied field of ~1 G) is four orders of magnitude smaller than 𝐹𝑥(𝑥)=𝜇𝐵𝜕𝑥𝐵𝑧(𝑥). Therefore, the charge imbalance generated by the magnetic field induced by the current can be neglected. Experimental techniques capable of measuring the contribution of the magnetic field induced by the current to the spin Hall effect. Ref.(2) in the main text argues that the magnetic field induced by a current will not generate a voltage on a metallic bridge which contacts both edges of a Hall bar. This argument is correct, and it is base d on the fact that, when the bridge is close to the Hall bar, the magnetic field induced by the current will thread both the bar and the bridge, preventing any spin current to flow along the bar, and to induce a voltage across it. On the other hand, the co mbination of a spin gradient and spin dependent scattering by impurities will lead to the voltage discussed in ref.(2). This situation changes if the magnetic field threading the bridge is different from the field threading the bar. This can be achieved by changing the distance between the bridge and the Hall bar. Any technique which does not require a contact between the edges of the bar will be sensitive to the spin accumulation induced by the field associated to the current. Possible experimental techniq ues are. I) injection of a spin polarized current using a ferromagnet, refs.(7), and (9) of the main text, ii) spin pumping from a magnetic insulator, ref.(10) of the main text, iii) measurement of the Kerr rotation by optical means, ref. (8) of the main t ext. A general overview of techniques used to measure the spin Hall effect can be found in the reviews cited in the main text, refs.(5) and (11).

1408.4123v2.Charge_noise__spin_orbit_coupling__and_coherence_of_single_spin_qubits.pdf

Charge noise, spin-orbit coupling and dephasing of single-spin qubits Adam Bermeister, Daniel Keith, and Dimitrie Culcer School of Physics, The University of New South Wales, Sydney 2052, Australia (Dated: October 31, 2021) Quantum dot quantum computing architectures rely on systems in which inversion symmetry is broken, and spin-orbit coupling is present, causing even single-spin qubits to be susceptible to charge noise. We derive an eective Hamiltonian for the combined action of noise and spin-orbit coupling on a single-spin qubit, identify the mechanisms behind dephasing, and estimate the free induction decay dephasing times T 2for common materials such as Si and GaAs. Dephasing is driven by noise matrix elements that cause relative uctuations between orbital levels, which are dominated by screened whole charge defects and unscreened dipole defects in the substrate. Dephasing times T 2 dier markedly between materials, and can be enhanced by increasing gate elds, choosing materials with weak spin-orbit, making dots narrower, or using accumulation dots. Developments in quantum computing hold consider- able promise in the progress of modern information pro- cessing, and this has spurred a large experimental and theoretical eort investigating two-level systems that can be used as quantum bits (qubits). The need for scalabil- ity and long coherence times has led naturally to solid state spin-based devices, such as quantum dot spin sys- tems, as ideal candidates for scalable qubits. The focus has been on single-spin [1] and singlet-triplet qubits. [2] While GaAs quantum dots have been studied for many years, a substantial eort is also underway researching Si spin quantum computing architectures, [3{5] motivated by their compatibility with Si microelectronics and long coherence times. [6{12] Recently, much eort has also been devoted to quantum dot systems with spin-orbit interactions, [13{15] where spin manipulation could in principle be achieved entirely by electrical means. [16{ 18] The coherence of a solid-state spin qubit is quantied by the relaxation time T1and the dephasing time T 2, both of which are determined by mechanisms that cou- ple up spins with down spins. This coupling can either be direct, through the hyperne interaction [19{22] and uctuations in the g-factor, [23] or indirect, through the joint eect of hyperne or spin-orbit coupling and uctu- ating electric elds, such as those due to phonons [24{30] or charge noise. [31{33] Inversion symmetry breaking near an interface makes spin-orbit coupling unavoidable, even in materials such as Si in which it is weak. [34] Hyperne eects typically occur on long time scales, the nuclear bath is relatively well known and can be con- trolled through feedback mechanisms [35] while in ma- terials such as Si hyperne coupling can be eliminated altogether through isotopic purication. [36, 37] The spin relaxation rate due to phonons is proportional to the fth power of the magnetic eld in zinc-blende mate- rials, in which piezoelectric electron-phonon coupling is often dominant, and to the seventh power of the magnetic eld in Si, in which there is no piezoelectric coupling.[28] Hence phonon eects become less pronounced at low magnetic elds. They also become weaker at low tem- peratures. [25] Noise is a well-known source of dephasing in chargequbits. [38{41] Experiments on quantum dots and point contacts have shown noise to be strong even at dilution refrigerator temperatures. [38{46] Noise sources include Pbcenters, which may act as traps that charge and dis- charge, and tunneling two-level systems, which can be modeled as uctuating charge dipoles. [31, 47{52] Noise and spin-orbit coupling give rise to nontrivial physics in 2D and 1D structures. [53{55] In quantum dot spin qubits, Ref. 33 has already shown that spin-orbit and noise lead to spin relaxation, and that noise and phonon eects in general become comparable at low-enough mag- netic elds. Hence, at dilution refrigerator temperatures the interplay of spin-orbit and noise may set the dening bound on spin qubit coherence. In this paper we build on previous decoherence work [56{62] and devise a theory of dephasing due to the com- bined eect of charge noise and spin-orbit interactions, with two aims in mind. The rst is to understand concep- tually how spin-orbit and noise cause dephasing. For ex- ample, noise can give relative uctuations between levels, virtual transitions between levels, as well as uctuations in spin-orbit constants. We wish to isolate the terms that are responsible for dephasing. The second aim is to study the sensitivity to spin-orbit coupling across common ma- terials with similar noise proles. We study a sample qubit with the same specications in dierent materials, we determine sample T 2s due to common noise sources, discuss the variation in T 2across materials, and seek methods to improve T 2generally. We consider a single-spin qubit implemented in a sym- metric, gate-dened quantum dot, located at a sharp at interface (Fig. 1) in a dilution refrigerator at 100mK. The qubit is described by the Hamiltonian H=HQD+HZ+ HSO+HN. The kinetic energy and connement term HQD=~2 2m@2 @x2+@2 @y2 +~2 2ma4(x2+y2);(1) whereais the eective dot radius and mthe eec- tive mass. The eigenstates of HQDare the Fock-DarwinarXiv:1408.4123v2 [cond-mat.mes-hall] 13 Nov 20142 states, with the ground and rst excited states given by 0(x;y) =1 ape x2+y2 2a2 (x;y) =1 a2p(xiy)e x2+y2 2a2 :(2) These have energies "0=~2=2ma2for the orbital ground state and "1= 3~2=2ma2for the twofold degen- erate rst orbital excited state. The orbital level splitting is assumed to be the dominant scale, so that only the ground and rst excited states are considered. The Zee- man Hamiltonian HZ=1 2gB
B, withthe vector of Pauli spin matrices. Since Bis constant, the orbital ef- fect ofBcan be absorbed into the eective dot radius a. We have also not taken into account multi valley eects in Si. For a certain interaction to couple valley states appreciably, it must be suciently sharp in real space. Neither the spin-orbit coupling due to the interface eld nor the electric eld of the defect satisfy this require- ment { even though these interactions are important in relaxation in particular around hot spots. [63] The spin-orbit term HSO=HR1+HD1. The Rashba termHR1=(t)
(k^z), stems from structure in- version asymmetry, where k=irhere is an operator in real space, ^zis the unit vector perpendicular to the interface, and is determined by a material specic pa- rameter as well as the interface electric eld Ez. [64] Thusis also sensitive to stray electric elds and uc- tuates in time, thus we let (t) =[1 +(t)] where 1. For a quantum dot on a (001) surface the lin- ear Dresselhaus term HD1=(ykyxkx) is usually the dominant bulk inversion asymmetry contribution,[64] where=3(=w)2, with3a material-specic param- eter andwthe width of the z-connement perpendicular to the interface. Since HD1can be obtained from HR1by a=2 spin rotation, they give rise to qualitatively similar physics. In Si = 0 due to inversion symmetry, whereas Rashba spin-orbit coupling is expected generally in a 2D electron gas near an interface, and should be present in all gate-dened dots. In zincblende structures HR1and HD1may comparable in magnitude in certain parame- ter regimes, though for Ez107Vm1(0.1 V/10 nm), the Rashba term is expected to be the dominant spin-orbit contribution. The noise Hamiltonian HN(t) is a random function of time. We do not include gate noise in our model, and we rst consider random telegraph noise (RTN). In the simplest case, in which the qubit is only sensitive to one defect,HNrepresents a uctuating Coulomb potential, screened by the nearby 2D electron gas. The 2D screened Coulomb potential Uscris written in terms of its Fourier transform, which is a function of momentum q[65] Uscr(r) =e2 20rZ2kF 0d2q (2)2eiq
r q+qTF; (3) withrthe relative permittivity, qTFthe Tomas-Fermi wave vector, and kFthe Fermi wave vector (the con- Figure 1: Defect locations with respect to the gate-dened quantum dot projected onto the xz-plane, with ^znormal to the interface. In general a top gate is also present (not shown). The red area represents the region of the quantum dot. tribution from q > 2kFis negligible [62]). The ma- trix elements entering HNarev0=h0jUscrj0i,v1= hjUscrji,v2=h0jUscrjih jUscrj0iand v3=hjUscrjih jUscrji. For RTN we can writevi(t) =vi(1)N(t)fori= 0;1;2, andN(t) = 0;1 is a Poisson random variable with switching time . [73] Additional (extrinsic) spin-orbit coupling arises from the electric eld of the defect itself. Yet for a charge de- fect located 40 nm away from the dot this eld is several orders of magnitude smaller than the interface electric eldEz. Because the matrix element involved is second order invi, the contribution this makes to dephasing is many orders of magnitude smaller than the Rashba in- teraction due to Ez, and will not be considered further. In the basisf0";0#;+";+#;";#g, with";# representing up and down spins, the Hamiltonian reads H=0 BBBBBBBBBBB@"+ 0 0v2sRv2isD 0" 0isDv2sRv2 v2isD"+ 10v30 sRv2 0" 10v3 v2sRv30"+ 1 0 isDv2 0v30" 11 CCCCCCCCCCCA(4) where" 0(t) ="0+v0(t)"Zand" 1(t) ="1+v1(t)"Z are the Zeeman-split orbital levels including the noise terms, the Zeeman energy "Z=1 2gBB, and the spin- orbit terms sD==a andsR(t) =sR[1 +(t)], with sR==a(not a function of time). The qubit subspace is simply the Zeeman-split orbital ground statef0";0#g, which has been singled out in the top left hand corner of Eq. 4. These two states are coupled by HNto spin-aligned orbital excited states and byHSOto orbital excited states with anti-aligned spin. By projecting Honto this subspace we encapsulate the combined eect of spin-orbit coupling and noise in an eective qubit Hamiltonian Hqbt. To achieve this, we3 carry out a Schrieer-Wol transformation, eliminating higher orbital excited states. [25, 64, 67, 68] Keeping terms up to the second order in this transformation, Hqbt(t) =HZ2"Zfv2(t)[sR(t)x+sDy] + [sR(t)2+s2 D]zg ["+v(t)]2 (5) where"="0"1(not a function of time) and v(t) = v0(t)v1(t). We retain only terms of rst order in "Z andv. Equation (5) implies that, in addition to HZ, there exists an eective Zeeman term1 2
V(t), where V(t) represents an eective uctuating eective magnetic eld due to the combined action of spin-orbit and noise. For convenience Vhas units of energy and, for RTN, V(t) =V(1)N(t). We will also use V(t) =jV(t)jfor the magnitude of V. Since the Rashba and Dresselhaus contributions are added in quadrature, there is no sweet spot for dephasing. The noise matrix elements appearing in Hqbtmay be divided into two categories. The diagonal elements v0(t);v1(t) cause dierent orbital levels to uctuate by dierent amounts, while the o-diagonal element v2(t) causes transitions between dierent orbital levels. If the qubit is initialized in an o-diagonal state, the diago- nal elements ( z) inHqbtgive dephasing. These terms involve the intraband matrix elements v0(t);v1(t) of the defect potentials. An additional contribution comes from uctuations in , which lead to uctuations in sRit- self. These uctuations can be interpreted as a modu- lation of the g-factor, and are expected to come from defects in the substrate right above the dot, which mod- ifyEz. Since the dot region is depleted, whole charge defects cannot uctuate, except in the very special case in which the defect lies right above the dot. Hence de- fects contributing to Ezare expected to be mostly charge dipoles, stemming for example from passivated traps. Al- though these are weaker than whole charge defects, they are unscreened, leading to a subtle competition. Thus, generally, dephasing stems from noise matrix elements that cause relative uctuations between orbital levels. In contrast, if the qubit is initialized in the spin-up state, the o-diagonal elements ( x) inHqbtgive relaxation ( T1 processes), which was studied in detail in Ref. 33. These elements are of rst order in and involve the interband defect matrix element v2(t). [74] In order to study dephasing further and obtain quan- titative estimates of T 2, we focus on a single-spin qubit described by a spin density matrix (t). The spin density matrix satises the quantum Liouville equation d dt+i ~[Hqbt;] = 0: (6) The spin density matrix (t) =1 2
S(t). Any spin component Sican be found as Si(t) = tr [i(t)], with tr the matrix trace. We restrict our attention to RTN for the time being. Using the time evolution operator e(i=~)Rt 0Hqbt(t0)dt0, we obtain the general time evolutionof the spin as S(t) =S0cosh(S0^h) sinh+^h(^h
S0)(1cosh);(7) where we have dened S0S(t= 0) and, for RTN, h(t) = (V=~)Rt 0(1)N(t0)dt0, withh(t) =jh(t)j. The two components of hhave exactly the same time evo- lution. SincejBj  jV, ifS0=S0x^xis initialised, Sx(t)S0xcos [h(t)]. Averaging over noise realisations [56, 59, 62] hcos [h(t)]ii=et=sinh t + cosh t ; (8) where  =p (~=)2V2=~. All systems of interest in this work satisfy V2(~=)2, in which case we may ap- proximateq~ 
2V2~  1V22 2~2 . When the de- nominator of the sinh is expanded in ( V=~)2, only the leading term in the expansion may be retained. Physi- cally, in this case, the time dynamics of h(t) are a random walk in time, and the spread in cos h(t) leads to motional narrowing. As a result, the initial spin decays exponen- tially asSx(t)S0xet=T 2, where 1 T 2 RTN=V2 2~2: (9) For whole charge defects, where dephasing is dominated by uctuations in the orbital energy, we may set (t) = 0 and retainVwh(t) = 8 (s2 R+s2 D)"Zv(t)=(")3. For dipole charge defects we have Vdip(t) = 8 (s2 R+s2 D)"Z(t)=(")2. We turn our attention next to 1 =fnoise. In semicon- ductors 1=fnoise is Gaussian [70] and is fully described by its spectral density S(tt0) =hHN(t)HN(t0)i. The Fourier transform of this spectral density has the form S(!) = kBT !, where is a parameter typically inferred from experiment. Based on our estimates for RTN above we expect whole charge defects to dominate dephasing. Hence, for the eective uctuating magnetic eld V(t) acting in the qubit subspace, we may write approximately SV(!)8(s2 R+s2 D)"Z (")32 S(!). To study dephasing, we writeSx(t) =S0xe(t), where (t) =2 kBT ~28(s2 R+s2 D)"Z (")32Z1 !0d!sin2!t=2 !3: (10) The low-frequency cut-o !0is usually taken to be the inverse of the measurement time. At times t1=!0 such as we consider here, we can approximate (t)t T 22 ln1 !0t; (11) where the dephasing time is estimated by 1 T 2 1=fr kBT 2~28(s2 R+s2 D)"Z (")3 : (12)4 Table I: Sample T 2for a quantum dot with a= 20 nm,= 4104,= 1s and the defect distance is 40 nm (for RTN), Ez= 20 MV/m, "Z= 60eV, T = 0.1 K, from Refs. 8, 64, from Ref. 64 and S(!) for 1=fnoise estimated from Refs. 38, 42. Following Ref. 8, the connement perpendicular to the interface (k^z) is represented by a square well of width 15 nm. For Si the valley splitting is assumed large. (peV m)(peV m) (T 2)RTN wh (T 2)RTN dip (T 2)1=f wh Si/SiGe 0 :02 0 3 ms 18 s 20 s GaAs 1 :0 0:12 60 ns 280 s 20 ns InAs 23 0 :12 40 ps 65 ns 900 ps InSb 105 3 :4 1 ps 1 ns 200 ps a=20nmH1fL a=10nmH1fL a=20nmHRTN L a=10nmHRTN L 050100150200250300tHmsL 0.00.20.40.60.81.0SxHtLSxH0L Figure 2: Time evolution of the initated spin for dierent dot radiiain Si/SiGe, !0=1 s and other values as in Table I. Since this denition of T 2is approximate, we plot the full time evolution of Sx(t) in Fig. 2. We consider a sample dot with radius a= 20 nm lo- cated atx=y= 0, andas calculated in Refs. 8, 64. For a defect in the plane of the dot with x= 40 nm, v0= 23eV,v1= 71eV andv2= 31eV. Next we es- timate the change in due to a dipole defect right above the dot (x=y= 0) andz= 3 nm away from it. [31, 47] The potential of an unscreened charge dipole located a distanceRDaway from the dot, is Udip(RD) =p
^RD 4"0"rR2 D. The charge dipole has dipole moment p=el, where l= (lx;ly;lz). We take the expectation value of Udip(RD) using  0, and compare it with the matrix element of eEzz, yielding= 4104. We use this gure in all our estimates since "rfor all materials considered are of very similar magnitudes. For 1 =fnoise we extract from experiment. For Si/SiGe we use Ref. 42, and for GaAs Ref. 38, while for InAs and InSb, in the absence of experimental data, we use the same S(!) as for GaAs.The results are listed in Table I, which is the central result of this paper. For all materials, whole charge de- fects dominate dephasing. Table I shows that terms of second-order in spin-orbit are eective in causing dephas- ing, and the dependence on 2causes vast dierences in dephasing times T 2between materials. Hence, using ma- terials with a small such as Si can improve coherence enormously. If spin-orbit coupling is needed for electric dipole spin resonance, increasing Ezwill align the charge dipoles. Although that increases and with it dephas- ing, it also reduces the gate time by an equal amount. Moreover, for 1 =fnoise,T 2/a4, so by halving the dot radius the dephasing time can be increased by an order of magnitude (Fig. 2; for RTN, T 2/a8). One can also use pulse sequences, [43] lower the temperature to reduce S(!), use accumulation dots, in which there is no nearby 2DEG, or focus on reducing charge noise. [42, 44, 45] Following existing calculations of , [8] we have taken the^z-connement in the form of a square well, whereas semiconductor interfaces are more accurately described by a triangular well. Nevertheless, since the form of HR1 andHD1is dictated by symmetry, they will be identi- cal in structure for triangular connement, thus we may simply treat andas phenomenological parameters. Finally, uctuations in waect. Although this eect, likewise driven by uctuating dipoles, can be calculated in the same way as the renormalization of 0by(t), we expect its contribution to be minor, in exact analogy withHR1. In summary, we have shown that spin-orbit coupling and charge noise are an eective source of dephasing in single-spin qubits even in materials such as GaAs in which spin-orbit coupling is weak. Based on realistic ex- perimental parameters vast dierences in spin dephasing times exist between common materials. In the future we will devise a full model of 1 =fnoise [71] as an ensemble of incoherent RTNs, [46] where qubit dynamics is nontriv- ial. [72] Dephasing of hole spin qubits, in which spin-orbit interactions are also strong but the heavy hole-light hole coupling cannot be ignored, will likewise be studied in a future publication. Acknowledgments We thank R. Winkler, Sven Rogge, Joe Sal, Andrea Morello, K. Takeda, Amir Yacoby, L. Vandersypen, Neil Zimmerman, S. Das Sarma, Alex Hamilton, Xuedong Hu, Guido Burkard, Mark Friesen, Andrew Dzurak, Menno Veldhorst, Floris Zwanenburg, J. R. Petta, and Matt House for enlightening discussions. [1] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). [2] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 309, 2180 (2005). [3] J. Morton, D. McCamey, M. Eriksson, and S. Lyon,5 Nature 479, 345 (2011). [4] F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Sim- mons, L. Hollenberg, G. Klimeck, S. Rogge, S. Copper- smith, and M. Eriksson, Rev. Mod. Phys. 85, 961 (2013). [5] X. Hao, R. Ruskov, M. Xiao, C. Tahan, and H. Jiang, Nature Comm. 5, 3860 (2014). [6] G. Feher, Phys. Rev. 114, 1219 (1959). [7] E. Abe, K. M. Itoh, J. Isoya, and S. Yamasaki, Phys. Rev. B 70, 033204 (2004). [8] C. Tahan and R. Joynt, Phys. Rev. B 71, 075315 (2005). [9] A. M. Tyryshkin, J. J. L. Morton, S. C. Benjamin, A. Ar- davan, G. A. D. Briggs, J. W. Ager, and S. A. Lyon, J. Phys. Condens. Matter 18, S783 (2006). [10] L. Wang, K. Shen, B. Y. Sun, and M. W. Wu, Phys. Rev. B 81, 235326 (2010). [11] M. Raith, P. Stano, and J. Fabian, Phys. Rev. B 83, 195318 (2011). [12] J. Muhonen, J. Dehollain, A. Laucht, F. Hudson, T. Sekiguchi, K. Itoh, D. Jamieson, J. McCallum, A. Dzurak, and A. Morello, arXiv:1402.7140 (2014). [13] M. Friesen, C. Tahan, R. Joynt, and M. A. Eriksson, Phys. Rev. Lett. 92, 037901 (2004). [14] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Nature 468, 1084 (2010). [15] A. P alyi, P. R. Struck, M. Rudner, K. Flensberg, and G. Burkard, Phys. Rev. Lett. 108, 206811 (2012). [16] D. Bulaev and D. Loss, Phys. Rev. Lett. 98, 97202 (2007). [17] P. Szumniak, S. Bednarek, B. Partoens, and F. M. Peeters, Phys. Rev. Lett. 109, 107201 (2012). [18] J. C. Budich, D. G. Rothe, E. M. Hankiewicz, and B. Trauzettel, Phys. Rev. B 85, 205425 (2012). [19] I. A. Merkulov, A. I. Efros, and M. Rosen, Phys. Rev. B65, 205309 (2002). [20] A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. B 67, 195329 (2003). [21] C. Deng and X. Hu, Phys. Rev. B 73, 241303 (2006). [22] W. A. Coish and J. Baugh, Phys. Stat. Sol. B 246, 2203 (2009). [23] E. Ivchenko, A. Kiselev, and M. Willander, Solid State Comm. 102, 375 (1997). [24] S. I. Erlingsson and Y. V. Nazarov, Phys. Rev. B 66, 155327 (2002). [25] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 (2004). [26] D. Bulaev and D. Loss, Phys. Rev. Lett. 95, 076805 (2005). [27] P. San-Jose, G. Zarand, A. Shnirman, and G. Sch on, Phys. Rev. Lett. 97, 076803 (2006). [28] M. Prada, R. H. Blick, and R. Joynt, Phys. Rev. B 77, 115438 (2008). [29] X. Hu, Phys. Rev. B 83, 165322 (2011). [30] J. I. Climente, C. Segarra, and J. Planelles, New. J. Phys. 15, 093009 (2013). [31] D. M. Fleetwood, S. T. Pantelides, and R. D. Schrimpf, Defects in Microelectronic Materials and Devices (CRC Press, Boca Raton, FL, 2008). [32] S. W. Jung, T. Fujisawa, Y. Hirayama, and Y. H. Jeong, Appl. Phys. Lett. 85, 768 (2004). [33] P. Huang and X.Hu, Phys. Rev. B 89, 195302 (2014). [34] Z. Wilamowski, W. Jantsch, H. Malissa, and U. R ossler, Phys. Rev. B 66, 195315 (2002). [35] H. Bluhm, S. Foletti, D. Mahalu, V. Umansky, and A. Yacoby, Phys. Rev. Lett. 105, 216803 (2010).[36] W. M. Witzel, X. Hu, and S. Das Sarma, Phys. Rev. B 76, 035212 (2007). [37] K. M. Itoh and H. Watanabe, MRS Communications (to be published). [38] K. D. Petersson, J. R. Petta, H. Lu, and A. C. Gossard, Phys. Rev. Lett. 105, 246804 (2010). [39] E. Dupont-Ferrier, B. Roche, B. Voisin, X. Jehl, R. Wac- quez, M. Vinet, M. Sanquer, and S. D. Franceschi, Phys. Rev. Lett. 110, 136802 (2013). [40] O. E. Dial, M. D. Shulman, S. P. Harvey, H. Bluhm, V. Umansky, and A. Yacoby, Phys. Rev. Lett. 110, 146804 (2013). [41] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt- shuler, Rev. Mod. Phys. 361, 86 (2014). [42] K. Takeda, T. Obata, Y. Fukuoka, W. M. Akhtar, J. Kamioka, T. Kodera, S. Oda, and S. Tarucha, Appl. Phys. Lett. 102, 123113 (2013). [43] H. Ribeiro, G. Burkard, J. R. Petta, H. Lu, and A. C. Gossard, Phys. Rev. Lett. 110, 086804 (2013). [44] C. Buizert, F. Koppens, M. Pioro-Ladrire, H.-P. Tranitz, I. T. Vink, S. Tarucha, W. Wegscheider, and L. Vander- sypen, Phys. Rev. Lett. 101, 226603 (2008). [45] K. Hitachi, T. Ota, and K. Muraki, Appl. Phys. Lett. 102, 192104 (2013). [46] J. M uller, S. von Moln ar, Y. Ohno, and H. Ohno, Phys. Rev. Lett. 96, 186601 (2006). [47] S. M. Sze and K. K. Ng, Physics of Semiconductor De- vices (Wiley, New York, NY, 2006). [48] J. Zimmermann and G. Weber, Phys. Rev. Lett. 46, 661 (1981). [49] J. Jang, K. Lee, and C. Lee, J. Electrochem. Soc. 129, 2770 (1982). [50] J. Reinisch and A. Heuer, J. Phys. Chem B 110, 19044 (2006). [51] R. Biswas and Y.-P. Li, Phys. Rev. Lett. 82, 2512 (2006). [52] N. M. Zimmerman, W. H. Huber, B. Simonds, E. Hour- dakis, A. Fujiwara, Y. Ono, Y. Takahashi, H. Inokawa, M. Furlan, and M. W. Keller, JAP 104, 033710 (2008). [53] M. M. Glazov, E. Y. Sherman, and V. K.Dugaev, Phys- ica E 42, 2157 (2010). [54] M. Glazov and E. Sherman, Phys. Rev. Lett. 107, 156602 (2011). [55] F. Li, Y. V. Pershin, V. A. Slipko, and N. A. Sinitsyn, Phys. Rev. Lett. 111, 067201 (2013). [56] R. de Sousa and S. Das Sarma, Phys. Rev. B 68, 115322 (2003). [57] X. Hu and S. Das Sarma, Phys. Rev. Lett. 96, 100501 (2006). [58] L. Chirolli and G. Burkard, Adv. Phys. 225, 57 (2008). [59] D. Culcer, X. Hu, and S. Das Sarma, Appl. Phys. Lett. 95, 073102 (2009). [60] R. de Sousa, in Electron Spin Resonance and Related Phenomena in Low-Dimensional Structures , edited by M. Fanciulli (Springer, 2009). [61] G. Ramon and X. Hu, Phys. Rev. B 81, 045304 (2010). [62] D. Culcer and N. M. Zimmerman, Appl. Phys. Lett. 102, 232108 (2013). [63] P. Huang and X. Hu, arXiv:1408.1666 (2014). [64] R. Winkler, Spin-orbit eects in two-dimensional electron and hole systems (Springer, 2003). [65] J. H. Davies, The Physics of Low-Dimensional Semicon- ductors: An Introduction (Cambridge University Press, 1998). [66] The eect of uctuators with 1s can be eliminated6 in experiment through dynamical decoupling. Hence, on physical grounds, we impose 1 s as a cuto for the switching time. [67] I. L. Aleiner and V. I. Falko, Phys. Rev. Lett. 87, 256801 (2001). [68] P. Stano and J. Fabian, Phys. Rev. Lett. 96, 186602 (2006). [69] We expect whole charge defect potentials to be dominant in relaxation since they are much stronger than dipole potentials [62]. [70] S. Kogan, Electronic noise and uctuations in solids (Cambridge University Press, New York, 2008).[71] I. Martin and Y. M. Galperin, Phys. Rev. B 73, 180201 (2006). [72] G. Burkard, Phys. Rev. B 79, 125317 (2009). [73] The eect of uctuators with 1s can be eliminated in experiment through dynamical decoupling. Hence, on physical grounds, we impose 1 s as a cuto for the switching time. [74] We expect whole charge defect potentials to be dominant in relaxation since they are much stronger than dipole potentials [62].

0705.0277v3.Charge_current_driven_by_spin_dynamics_in_disordered_Rashba_spin_orbit_system.pdf

arXiv:0705.0277v3 [cond-mat.mes-hall] 28 Jan 2008Charge current driven by spin dynamics in disordered Rashba spin-orbit system Jun-ichiro Ohe∗ I. Institut f¨ ur Theoretische Physik, Universtit¨ at Hambu rg, Jungiusstrasse 9, 20355 Hamburg, Germany Akihito Takeuchi Department of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan Gen Tatara Department of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Jap an (Dated: August 23, 2021) Pumping of charge current by spin dynamics in the presence of the Rashba spin-orbit interaction is theoretically studied. Considering disordered electro n, the exchange coupling and spin-orbit interactions are treated perturbatively. It is found that d ominant current induced by the spin dynamics is interpreted as a consequence of the conversion f rom spin current via the inverse spin Halleffect. Wealso foundthatthecurrenthasanadditionalc omponentfrom afictitiousconservative field. Results are applied to the case of moving domain wall. Recent spintronics studies aim at manipulation both of charge and sp in degrees of freedom [1]. Central roles are played by the spin-orbit interaction and the exchange interact ion between the conduction electrons and local spins [2, 3, 4, 5, 6]. It has been shown that the exchange coupling is u seful for electrical control of magnetization dynamics via spin transfer torque [2, 3]. It can also be used to pump s pin current from the precession of the magnetization [7, 8, 9]. The spin-orbit interaction has been recently found to induce a magnetism by the application of electric voltage (the spin Hall effect) [10, 11, 12]. By combining the exchange and the spin-orbit interactions, various phenomena are expected, and the subject discussed in this paper is one of them; pumping of charge current by dynamical magnetization. The idea is to convert the pumped spin current into a charge current by using of the spin- orbit interaction as proposed by Saitoh et al. [13]. This currentdue tothe inversespin Hall effect wasindeed observed [13, 14, 15] in metallicsystems wherethe spin-orbit interaction is induced by Pt atom. Theoretically, generation of the electric field or voltage due to the d ynamical spin structure was discussed by Stern [16]. The field is not a real electric field, but an effective one due to a spin Berry phase acting only on charge degrees of freedom with spin (like the electron). The mechanism is sim ilar to the Faraday’s law, but a magnetic flux is replaced by a fictitious field from the spin Berry phase. The induced field is described by ∇×Eeff=−˙b, wherebis field of spin Berry phase, and current is divergenceless; ∇·j= 0. The theory was recently applied to a domain wall by Barnes and Maekawa [17], and the effect of the spin relaxation was studied by Duine [18], where the relaxation was introduced by a phenomenological term ( β-term). All these studies have been done in the adiabatic limit, where the exchange coupling between the local spin and the conduction ele ctrons is strong. Chargecurrent as a result of inversespin Hall effect was theoretic allystudied by Zhang and Niu [19] and Hankiewicz et al. [20] (the effect was called reciprocal spin Hall effect). The calc ulation was done as a response to applied spin- dependent chemical potential, which would not be easy to control e xperimentally. In contrast, our study tries to derive direct relation between physically accessible quantities, curr ent and magnetization (local spin). Another type of a voltage generated at an interface of a ferroma gnet-nonmagnet contact was predicted by Wang et al. [21] in order to explain the experimental observation [22]. They ha ve pointed out that a net charging is due to a back flow of spin current at the interface, and the spin accumulatio n at the interface is essential. In this letter, we theoretically predict another mechanism of a spin- induced charge battery realized in disordered conductors. We consider the perturbative regime of the exchang e coupling, that is in the opposite limit of adiabatic cases [16, 17, 18]. Proposed mechanism does not rely on the interfa ce, and the pumped current arises simply when the exchange interaction and the spin-orbit interaction exist simult aneously. Weconsiderthetwo-dimensionalelectrongassystem(2DEGs)with theRashbaspin-orbitinteractionthatoriginates from the lack of the inversion symmetry [5, 6, 23]. The conduction ele ctrons couples to the local magnetic moment via theexchangecoupling. Suchasystemcanbeachievedbythe2DEGs attachedtotheferromagneticcontact[24, 25, 26], or the 2DEGs in magnetic semiconductors (e.g., CdTe/CdMnTe) [27, 2 8]. Let us consider the current representation by the simple argumen t. The current jµ, proportional to the average tr∝angb∇acketleftkµ∝angb∇acket∇ightof the electron wave vector with respect to electron states (tr is trace over spin indices), vanishes if the system is spatially symmetric. The Rashba interaction, proportiona l to (k׈σ)z, breaks the spatial symmetry, but2 jS s-o FIG. 1: Diagrammatic representation of currents at the first order in Jexand the lowest order in Ω, which turns out to vanish. Dotted lines and wavy lines denote local spins Sand the Rashba interaction, respectively, and thick line re presents the diffusion ladder,Dq. charge current does not arise since tr ∝angb∇acketleftkµ(k׈σ)z∝angb∇acket∇ight= 0 (only spin current arises [7]). Charge current appears when we introduce the exchange coupling, proportional to S·ˆσwhereSis a local spin. We would have the current jµ∝tr∝angb∇acketleftkµ(k׈σ)zS·ˆσ∝angb∇acket∇ight ∝ǫµνzSνto the first order exchange interaction, and jµ∝ǫµνz(S×S′)νat the second oder interaction with different spins, SandS′. In order to obtain the precise expression of the current, we perf orm analytical calculations by using a diagrammatic technique. Although the argument in the previous paragraph gives the qualitative idea of pumping charge current, it turns out that the linear term in Svanishes identically in the Rashba case, and the second order term n eeds a dynamical part as S×˙S. We will demonstrate that the pumped current has two component s. One describes the inverse spin Hall effect [13], and the other is a conservative curren t which written as a divergence of a scalar potential. We will show that the current due to the inverse spin Hall effect is dom inant in various cases, and the conservative current is relatively small. However, the conservative one is also impo rtant from the view of the fundamental physics, because it provides the fictitious scalar potential which acts only on the particle having both charge and spin. We consider a disordered 2DEGs with the Rashba spin-orbit interact ion. The 2DEGs also interact with the local spin,Sx(t), via the exchange coupling. The local spin is treated as classical, an d slowly varying in space and time. The system is represented by a Hamiltonian H=H0+Hex(t) +Hso+Himp, whereH0≡/summationtext kεkc† kckdescribes free electrons with εk=/planckover2pi12k2/2m(mbeing the effective mass). The exchange and the Rashba spin-orbit interactions are given by Hex(t) =−Jex/summationdisplay k,q,ΩSq(Ω)eiΩtc† k+qˆσck, Hso=−α/summationdisplay kǫµνzkµ(c† kˆσνck), (1) whereJexis the exchange coupling constant, Sq(Ω) denotes the Fourier transform of the local spin structure andαis the strength of the Rashba spin-orbit interaction. Spin-indepen dent disorder is represented by Himp=/summationtextni i=1/summationtext k,k′u Vei(k−k′)·ric† k′ck, which gives rise an elastic electron lifetime τ= (2πρniu2/V)−1, whereρis the density of states, niis the number of the impurities, uis the strength of the impurity scattering and Vis the volume of the system. The charge current density of this system is given by jµ(x,t) =ie V/summationdisplay k,k′ei(k−k′)·xtr/bracketleftbigg/parenleftbigg(k+k′)µ 2m−αǫµνzˆσν/parenrightbigg G< k,k′(t,t)/bracketrightbigg . (2) G< k,k′(t,t′) is a lesser Green function which is a 2 ×2 matrix in spin space with components G< kσ,k′σ′(t,t′) = i∝angb∇acketleftc† k′σ′(t′)ckσ(t)∝angb∇acket∇ight(σ,σ′=±), where ∝angb∇acketleft···∝angb∇acket∇ightis the expectation value estimated by the total Hamiltonian H. WecalculatethecurrentbytreatingbothRashba(tothefirstord er)andexchangeinteractionsperturbatively,which is valid if Jexτ≪1 andαkFτ≪1[29]. Successiveimpurity scatteringsare denoted by ladderappro ximation, resulting inadiffusionpropagatoratsmallmomentumtransfer( q),Dq≡(Dq2τ)−1, whereD≡k2 Fτ/2m2isadiffusionconstant. Dominant contributions are from diagrams that include a maximal num ber of diffusion propagators. Contributions from the first order in Jexare shown in Fig. 1, that turn out to vanish identically. The leading con tribution coming from the second order in Jexare shown in Fig. 2.3 (a) (b) FIG. 2: Leading contribution from the second order in Jex. Contributions from (a) and (b) are denoted by jφ µandjISH µin Eq. (4), respectively. Contributions from other diagrams c ancel out or are smaller by O(1/εFτ). After straightforward calculations, the current in the slowly vary ing limit (Ω τ≪1) is obtained as jµ(x,t) =4eαJex2 iπmVǫνηz(niu2 V)/summationdisplay q,Qe−iQ·xDq(SQ−q×˙Sq)η ×/bracketleftBig (niu2 V)DqDQAµ QBν qCqQ+DqEµ qQBν q−DQAµ QFν qQ/bracketrightBig , (3) whereAµ Q≡/summationtext kkµgr k−Q 2ga k+Q 2,Bν q≡/summationtext kkν(gr k)2ga k+q,CqQ≡Re/summationtext kgr kga k+qga k+Q,Eµ qQ≡ iIm/summationtext k/parenleftBig k+Q 2/parenrightBig µgr kga k+qga k+QandFν qQ≡Re/summationtext k(k+q)νgr k(ga k+q)2ga k+Q(gr k= (ga k)∗= (εF−εk+i 2τ)−1(εFbeing the Fermi energy)). The second and third terms in Eq. (3) (propo rtionalto Eµ qQandFν qQ, respectively) are the second leading term with a long range limit ( q,Q→0). It is, however, physically the most essential term as we see belo w. For a spatially smooth structure of spins, i.e., q,Q≪ℓ−1(ℓis the electron mean free path), we can approximate Aµ Q∼2πiρτmDQ µ,Bν q∼4πρτ3εFqν,CqQ∼ −ρ/2ε2 F,Eµ qQ∼ −πiρτ2qµandFν qQ∼πρτ3Qν. Then, the current is obtained as [30] jµ(x,t) =jISH µ(x,t)+jφ µ(x,t), where jISH µ(x,t) =−3emαJex2τ2 πǫµηz/integraldisplayd2x1 a2Dx−x1(Sx×˙Sx1)η, jφ µ(x,t) =2eαJex2τ3 π2ǫνηz∂ ∂xµ/integraldisplayd2x1 a2/integraldisplayd2x2 a2Dx−x1∂D(2) x1−x2 ∂x1ν(Sx1×˙Sx2)η. (4) Here,Dx≡a2 V/summationtext qe−iq·xDq,D(2) x≡a2 V/summationtext qe−iq·x(Dq)2andais the lattice constant. (Note that singular behavior at q→0 is cut off at q∼L−1, whereLis a system size.) The first term in Eq. (4), jISH, describes a current whose direction is correlated with the magnet ization direction, perpendicular to S×˙S, and represents inverse spin Hall effect [13]. In order to make clear the physical meaning of this current, we compare with the spin currents pumped by magnet ization. In the absence of spin-orbit interaction, we obtain the pumped spin current at the lowest order in Jexas jsµ(x,t) =JexεFτ2 2π∂ ∂xµ/integraldisplayd2x1 a2Dx−x1/bracketleftbigg ˙Sx1−2Jexτ/integraldisplayd2x2 a2Dx1−x2(Sx1×˙Sx2)/bracketrightbigg . (5) The polarizationofthe spin currentisin both directions, ∝ ∝angb∇acketleft˙S∝angb∇acket∇ightand∝angb∇acketleftS×˙S∝angb∇acket∇ight, where∝angb∇acketleft···∝angb∇acket∇ightdenotes averageoverdiffusive electron motion. This spin current is a gradient of a certain spin pote ntial,jsµ=−∇µφs. The result of Eq. (5) is consistent with the observation by Tserkovnyak et al. [8], where t he spin current appears at the interface between ferromagnetandnormalmetalsassociatedwiththephenomenolog icalparameterofspin-mixingconductance. (Wenote that there is also an equilibrium component of spin current [31]. This co mponent, js(eq) µ(x) =Jex2 24π2ε2 Fτ(∇µSx)×Sx, is free from diffusion poles. Hence, it is local and therefore small comp ared with dynamical contributions.) The meaning of pumped spin current is understood by taking a divergence: ∇·js(x,t) =−mJex 2π/bracketleftbigg ˙Sx−2Jexτ/integraldisplayd2x1 a2Dx−x1(Sx×˙Sx1)/bracketrightbigg . (6)4 FIG. 3: (Color online) Two typical geometries with ferromag nets attached to two-dimensional electron system. Left: Da tta- Das geometry [24] and Right: perpendicular geometry like in Ref. [13]. In both cases, inverse spin Hall current is given b y jISH µ∝ǫµνz(S×˙S)ν(µ=x,y). Comparing the second term ( ∇ ·js(2)ν) tojISH, we see that jISH µ=γISHǫµνz(∇ ·js(2)ν), where γISH=−3eατ. This expression is the Rashba-version of inverse spin Hall effect, j∝js׈σ, proposed in Ref. [13]. (Note that the spin current considered in Ref. [13] is the one flowing through the interf ace that enables the spin current to enter without divergence.) Eq. (6) representsa conservationlawofspin, and co rrectlydescribesa fact that the Gilbert-type damping (∝angb∇acketleftS×˙S∝angb∇acket∇ight) results in a flow of spin current or ˙S. In contrast, the second term in Eq. (4), jφ, is a gradient of a scalar quantity. It can be interpreted as a curre nt arisingfrom apotential or aconservedforce. The fictitious electr ic field, defined by Eind=jφ/σ0, whereσ0=e2nτ/m is Boltzmann conductivity, is written as Eind=−∇φind. The scalar potential is obtained as φind(x,t) =−2αJex2τ2 πeεFǫνηz/integraldisplayd2x1 a2/integraldisplayd2x2 a2Dx−x1∂D(2) x1−x2 ∂x1ν(Sx1×˙Sx2)η. (7) (Note that these scalar potential and field are fictitious ones, act ing only on charge having spin degrees of freedom.) The current jφis in the direction where magnetization changes. It contributes to t he perpendicular current in the Datta-Das spin transistor geometry [24]. However, it is not in-plane current in the layer geometry [13] as shown in Fig. 3. It does not contribute to the case of moving domain walls as we will show below. Let us apply our results to a case of a moving domain wall (as would be r ealized by using magnetic semiconduc- tors [27, 28]). We define polar angles ( θ,φ) with respect to the easy and hard axis of spin as cos θ=Seasy/S,tanφ= Shard/Smid, whereSeasy,ShardandSmiddenote spin components in easy-, hard- and medium-anisotropy dir ections. Rigid and one-dimensional (in the x-direction) domain wall solution is represented by two dynamical var iables,X(t) andφ(t) [32], as cos θ(x,t) = tanhx−X(t) λand sinθ(x,t) = [coshx−X(t) λ]−1, whereλis wall thickness. By assuming a dirty case λ≫ℓ(ℓbeing the mean free path), and noting that Sx×˙Sx′vanishes if |x−x′| ≫λ, we can approx- imateSx×˙Sx′by local value as Sx×˙Sx∼sinθ(x,t)(˙φeθ−˙X λeφ). Here, eθ= (cosθcosφ,cosθsinφ,−sinθ) and eφ= (−sinφ,cosφ,0), indicating that the damping ( S×˙S) on the translational motion ( ˙X) is in the φ-direction, while it is within the wall plane when φvaries. The pumped inverse spin Hall current, jISH µ∼ −j0ǫµηz∝angb∇acketleftS×˙S∝angb∇acket∇ightη, where j0≡3 πemαJex2τ2D0(D0≡L a2/integraltextλ ℓdxDx) depends much on the wall geometry. We consider three types of t he domain wall as shown in Fig. 4, a Neel wall (N) and two Bloch walls ((B-i) and (B- ii)). By estimating ∝angb∇acketleftS×˙S∝angb∇acket∇ightinside the wall (by using ∝angb∇acketleftcosθ∝angb∇acket∇ight=∝angb∇acketlefttanhx λ∝angb∇acket∇ight= 0 etc.), we obtain jISH x(N)∼ −j0˙X Lsinφ, jISH x(B−i)∼j0˙X Lcosφ, (8) which are driven by translational motion, and jISH x(B−ii)∼j0λ L˙φ, (9) which is driven by tilt of the wall. In the case of the Neel and the out-o f-plane Bloch wall, the current is a constant (if wall velocity is constant) at small speed and shows an oscillation (s inφor cosφ) when the domain wall is above Walker’s break down. In contrast, no current is induced for the in- plane Bloch wall at small velocity and finite but steady current arises above breakdown (as long as ˙φis more or less constant). The gradient part of the current, jφ, on the other hand, is averaged out for any type of the domain wall. Let us briefly see the magnitude of current (Eqs. (8)(9)). We use εF= 10meV, Jex= 10meV, a= 10nm and α= 0.3×10−11eVm [6]. Using D0∼(L3/a2ℓ2λ)ln(L/λ) (our diffusive result depends much on sample size L), the current is estimated as j∼4×10−14[C/m]×L2λ2 a4lnL λ×Ω[Hz]. If we choose Ω = 100MHz, L= 1µm andλ∼10a, we obtainj∼10[A/m], i.e., current is I≡jL∼10µA, which would be detectable experimentally.5 FIG. 4: (Color online) Three domain walls in the xy-plane, Neel, Bloch-i and Bloch-ii. We have considered a case of an uniform Rashba interaction. If we a llowαto be position dependent, α(x), we have other contributions. For instance, charge current linear in Sarises proportional to ∝angb∇acketleft(∇α)˙S∝angb∇acket∇ight. Thus, various currents are expected in the case of finite-size Rashba system (finite size is a lways the case in experiments). Inconclusion, wehavetheoreticallyshownthatchargecurrentisp umpedbymagnetizationdynamicsinthepresence of the Rashba spin-orbit interaction. The dominant part was found to be due to the inverse spin Hall effect, i.e., conversion of spin current into charge current by spin-orbit inter action. In addition to the inverse spin Hall current, we found a conservative current flowing basically along the gradient of the magnetization damping. This current is rotation free, and should be distinguished from the inverse spin Ha ll current and from the divergenceless current predicted by Stern and others [16, 17, 18]. It would be extremely int eresting if one could experimentally determine the type of the pumped current. The authors are grateful to E. Saitoh, B. Kramer, R. Raimondi, M. Yamamoto, S. Kettemann, J. Shibata, H. Kohno, S. Murakami, and T. Ohtsuki for valuable discussions. This w ork has been supported by the Deutsche Forschungsgemeinschaft via SFBs 508 and 668 of the Universit¨ at Hamburg. Noted added in proof. —After finishing the manuscript, we found optically induced inverse s pin Hall effect was observed in GaAs[33]. ∗Present address: Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan; Electronic address: johe@imr.tohoku.ac.jp [1] S.A. Wolf et al., Science 294, 1488 (2001). [2] L. Berger, Phys. Rev. B 54, 9353 (1996). [3] J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [4] H. Ohno, Science 281, 951 (1998). [5] E.I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). [6] J. Nitta et al., Phys. Rev. Lett. 78, 1335 (1997). [7] A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Le tt.84, 2481 (2000). [8] Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. Lett.88, 117601 (2002). [9] Y. Tserkovnyak et al., Rev. Mod. Phys. 77, 1375 (2005). [10] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003). [11] J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004). [12] Y.K. Kato et al., Science 306, 1910 (2004). [13] E. Saitoh et al., Appl. Phys. Lett. 88, 182509 (2006). [14] S.O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). [15] T. Kimura et al., Phys. Rev. Lett. 98, 156601 (2007). [16] A. Stern, Phys. Rev. Lett. 68, 1022 (1992). [17] S.E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601 (2007). [18] R.A. Duine, cond-mat/0706.3160. [19] P. Zhang and Q. Niu, cond-mat/0406436. [20] E.M. Hankiewicz et al., Phys. Rev. B 72, 155305 (2005). [21] X. Wang et al., Phys. Rev. Lett. 97, 216602 (2006). [22] M.V. Costache et al., Phys. Rev. Lett. 97, 216603 (2006). [23] Y. Yanase and M. Sigrist, J. Phys. Soc. Jpn. 76, 043712 (2007). [24] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). [25] A.T. Hanbicki et al., Appl. Phys. Lett. 80, 1240 (2002). [26] T. Matsuyama et al., Phys. Rev. B 65, 155322 (2002). [27] S. Scholl et al., Appl. Phys. Lett. 62, 3010 (1993). [28] F. Takano et al., Physica B 298, 407 (2001). [29] J. I. Inoue, G.E.W. Bauer, and L.W. Molenkamp, Phys. Rev . B70, 041303(R) (2004). [30] Correctly speaking, even without spin-orbit interact ion, small charge current arises at the second order of diffus ion pole. This current, proportional to ∝angbracketleftSx·˙Sx′∝angbracketright, is of non-magnetic origin, i.e., due to time-dependent pot ential, and is of minor6 importance, besides being very small when Sis slowly varying in space, S·˙S∼0. [31] G. Tatara and H. Kohno, Phys. Rev. B 67, 113316 (2003). [32] J.C. Slonczewski, Int. J. Magn. 2, 85 (1972). [33] H. Zhao et al., Phys. Rev. Lett. 96, 246601 (2006).

1408.6700v1.Interplay_of_spin_orbit_and_hyperfine_interactions_in_dynamical_nuclear_polarization_in_semiconductor_quantum_dots.pdf

Interplay of spin-orbit and hyperne interactions in dynamical nuclear polarization in semiconductor quantum dots Marko J. Ran ci c and Guido Burkard Department of Physics, University of Konstanz, D-78457 Konstanz, Germany (Dated: August 28, 2021) We theoretically study the interplay of spin-orbit and hyperne interactions in dynamical nuclear polarization in two-electron semiconductor double quantum dots near the singlet ( S) - triplet (T+) anticrossing. The goal of the scheme under study is to extend the singlet ( S) - triplet ( T0) qubit decoherence time T 2by dynamically transferring the polarization from the electron spins to the nuclear spins. This polarization transfer is achieved by cycling the electron spins over the S T+anticrossing. Here, we investigate, both quantitatively and qualitatively, how this hyperne mediated dynamical polarization transfer is in uenced by the Rashba and Dresselhaus spin-orbit interaction. In addition to T 2, we determine the singlet return probability Ps, a quantity that can be measured in experiments. Our results suggest that the spin-orbit interaction establishes a mechanism that can polarize the nuclear spins in the opposite direction compared to hyperne mediated nuclear spin polarization. In materials with relatively strong spin-orbit coupling, this interplay of spin-orbit and hyperne mediated nuclear spin polarizations prevents any notable increase of the ST0qubit decoherence time T 2. I. INTRODUCTION Electron spins in semiconductor quantum dots are con- sidered to be excellent candidates for qubits [1]. In or- der for a full scale quantum computer to be produced, a successful fulllment of the DiVincenzo criteria [2] is necessary. Accurate qubit manipulation [3, 4] and reli- able state preparation [5] are some of the requirements that have been satised in the past years. Techniques for qubit identication and fast readout are also known, e.g., the spin readout for a two-electron double quan- tum dot is most commonly done in the regime of Pauli spin blockade [6] using spin to charge conversion mea- surements [7]. Still, one challenge remains - suciently isolating the qubit from the corruptive eects of its sur- roundings. Due to the in uence of its surroundings, a qubit will irreversibly lose information. Dierent types of informa- tion losses happen on dierent time scales. The time in which a qubit relaxes to a state of thermal equilibrium is the relaxation time T1, whereas the time in which a qubit loses coherence due to the collective eects of its surroundings is the decoherence time T 2. Although ex- perimental and theoretical solutions for overcoming these information losses have been steadily developed for years, [8-13] overcoming qubit decoherence caused by a uctu- ating nuclear spin bath is still an ongoing task. Silicon [14] and graphene [15] have stable isotopes with a zero nuclear spin. Therefore, they can be isotopi- cally puried leaving only spin zero nuclei which do not contribute to the electron spin qubit decoherence. On the other hand, III-IV semiconductors, and particularly InxGa1xAs structures only have stable isotopes with a non-zero nuclear spin. An electron conned in a typi- cal InxGa1xAs quantum dot interacts with 104106 nuclear spins, which contribute strongly to electron spin qubit decoherence. Optically [16-18] or electrically polar- izing the nuclear spins can prolong the coherence timesof electron spins. Such a polarization of nuclear spins is achieved by transferring spin from the electron spins to the nuclear spins in a procedure called dynamical nuclear polarization (DNP) [19]. A suitable system for conducting DNP is a gate dened double quantum dot loaded with two electrons. There has been a variety of proposals [20, 3] to use DQDs as qubits, e.g., by focusing on the singlet jSi= 1=p 2(j"ij#i j#ij"i ) and tripletjT0i= 1=p 2(j"ij#i +j#ij"i ) logi- cal subspace [21], where the generated nuclear dierence eld and the exchange interaction are used to perform universal control of the qubit on the Bloch sphere. Other than the already mentioned DNP, the eects of dephas- ing caused by a nuclear spin bath, can be canceled by applying a Hahn echo sequence [22], or the more elabo- rate CPMG sequences [21]. The generation of a nuclear gradient eld, required to control the ST0qubit [21], can be achieved by cy- cling the electron spins over the anticrossing between the singletjSi= 1=p 2(j"ij#ij#ij"i ) and triplet jT+i=j"ij"i states. During such a ST+cycle, the electron spins transfer polarization to the nuclear spins [23], and a nuclear dierence eld is generated. Further- more, a higher degree of nuclear spin polarization causes a longer spin coherence time of the ST0qubit. In mate- rials with sizable spin-orbit interaction, the spin-orbit in- teraction induces electron spin ips, and this mechanism competes with the hyperne mediated electron spin ips required for DNP. In such materials, we theoretically ex- plore the interplay of spin-orbit and hyperne eects on nuclear spin preparation schemes, in the vicinity of the ST+anticrossing. We assume that the dots are embedded in the semicon- ductor material In xGa1xAs with 0x1. We model 150 nuclear spins per dot fully quantum mechanically, keeping track of how the probabilities and coherences of all nuclear states change in time. As compared to our model, recent models treating more [23] or fewer [24] nu-arXiv:1408.6700v1 [cond-mat.mes-hall] 28 Aug 20142 clear spins fully quantum mechanically, do not take into account the spin-orbit interaction. Although there has been some work on the interplay of spin-orbit and nu- clear eects in GaAs double quantum dots [25-28], to our best knowledge none of these theoretical frameworks treat the nuclear spin dynamics fully quantum mechan- ically, nor investigate the nuclear spin dynamics when subjected to a large number ( 300) of DNP cycles. On the other hand, again to our best knowledge, there has been no theoretical work to describe the ST+DNP in materials having strong spin-orbit interaction, e.g., InAs. Experiments in InAs have been carried out with a single electron spin in a single quantum dot [29], or in a double quantum dot, by using a dierent, more elaborate puls- ing sequence [30]. As a consequence of our fully quantum treatment we can give precise estimations of T 2, compare them to known experiments in GaAs [31], and calculate a value forT 2in InxGa1xAs. Our results can also be be extrapolated to materials with even stronger spin-orbit coupling as compared to InAs such as, e.g., InSb. This paper is organized as follows. In Section II we describe our model, in Section III we discuss the total nuclear spin angular momentum basis which signicantly reduces the dimension of our Hilbert space. In Section IV we study the time evolution during the DNP cycle, in Section V we present results on In 0:2Ga0:8As, a material with an intermediate strength of spin-orbit interaction, and in Section VI we compare results for dierent abun- dances of indium in In xGa1xAs. We conclude in Section VII. II. MODEL The connement in a quantum dot is modeled with a quadratic potential and the electronic wave functions are calculated according to the Hund-Mulliken theory [32]. Our approach is a good approximation in the regime where half of the interdot separation ais larger thanthe eective Bohr radius, a>aB=p h=m!0. Here, !0is the circular frequency of the conning potential, which we later assume to be  h!0= 3:0 meV, and m is the eective electron mass ( m= 0:067m0for GaAs andm= 0:023m0for InAs). The interdot separation 2aneeds to be chosen suciently large, due to the fact that the Hund-Mulliken theory is valid in the regime of weakly interacting quantum dots. On the other hand, the extended tunneling matrix element tHneeds to be nonvanishing, so that our DNP sequence is still possi- ble. Therefore, for In 0:2Ga0:8As, which is the material we study in Section V, we want tH0:01U, whereU is the Coulomb energy of the electrons. This is why we seta= 46:3 nm. A magnetic eld of B= 110 mT is applied perpendicular to the plane spanned by the [110] and [ 110] crystallographic axes, see Fig. 1. The specic value of the magnetic eld is chosen so that the ST+ anticrossing is located at "3U=2, where"is the energy dierence between the quantum dots, Fig. 2. All stable isotopes of gallium and arsenide have a nu- clear spinj= 3=2, while stable isotopes of indium have a nuclear spin j= 9=2. Here we discuss a simplied model in which all of the nuclear spins are assumed to be j= 1=2 [33]. Also, spin-orbit eects depend strongly on the homogeneity of the distribution of In and Ga atoms in InxGa1xAs. Here, we assume a completely homogenous distribution of In and Ga. For numerical convenience we model a geometry in which the [110], [ 110] crystal- lographic axes and the interdot connection axis plie in plane (Fig. 1). We develop a numerical method for mod- eling up to N= 150 nuclear spins per dot, a constraint imposed by our current computational resources. The total Hamiltonian describing the electronic and nuclear degrees of freedom is H=H0(") +HHF+HSO: (1) HereH0(") is the non-relativistic Hamiltonian of two electrons in a QD [32], H0(") =0 BBBBBB@U" Xp 2tH 0 0 0 X U +"p 2tH 0 0 0 p 2tHp 2tHV+ 0 0 0 0 0 0 V+gBBz 0 0 0 0 V 0 0 0 0 0 0 VgBBz1 CCCCCCA; (2) in the basis offS(2;0);S(0;2);S(1;1);T+(1;1);T0(1;1); T(1;1)g. The letter Sdenotes the singlet state, and T+, T,T0are triplet states with the total spin projections ms= +1,ms=1,ms= 0. The numbers in the paren- theses indicate the charge state. More specically, (2 ;0) denotes a state where the left dot is occupied with two electrons and the right dot is empty, (0 ;2) denotes a statewhere the right dot is being occupied with two electrons and the left dot is empty, and (1 ;1) stands for each dot being occupied with one electron. The Hamiltonian [Eq. (2)] acquires time dependence through the bias energy ". To describe the DNP process, the bias energy "will be assumed to be a linear function of time "=rt;where we setr= 2U=, and where = 50 ns is the duration of the3 Figure 1. (Color online) Geometry of the problem. The strength of spin-orbit interaction is tuned by varying the an- glebetween the [110] crystallographic axis and the interdot connection axis p. Spin-orbit interaction generates an eec- tive magnetic eld along theyaxis. The external magnetic eld is perpendicular to the [110] - pplane. bias sweep. The value of ris chosen so that "= 2Uat the beginning of the sweep ( t= 0),"= 0 at the and of the sweep ( t=), as in the experiment by Petta et al. [5]. The quantities in H0are the on-site Coulomb energy U1 meV, the coordinated hopping from one dot to the otherX0:1eV, the doubly occupied singlet and triplet matrix elements, V+; V10eV, and the ex- tended hopping parameter, tH0:01U[32]. The Zee- man energy is given as gBBz, wheregis the electron gfactor (g=0:44 for GaAs, g=14:7 for InAs), the Bohr magneton is B= 5:79105eV/T and Bz= 110 mT is the magnetic eld. For an electron con- ned in an GaAs QD the Zeeman energy at this eld is Ez= 2:8106eV. Due to the fact that we are inter- ested in the ST+transition, we focus our attention on the energy subspace spanned by the states fS(2;0), S(1;1),T+(1;1)g. The singlet S(0;2) is high in energy with respect to the other two singlets [cf. Fig. 2] (for positive values of the detuning ") whereas the remain- ing two singlets S(2;0) andS(1;1) are close in energy. The triplet states T0(1;1), andT(1;1) are split o from theT+(1;1) by the Zeeman energy. It should be men- tioned that we treat the Hamiltonian [Eq. (2)] using the adiabatic approximation, meaning that the system will remain in its instantaneous eigenstates. This allows us to obtain the eigenenergies by diagonalizing the Hamilto- nianH0in the subspace of fS(2;0);S(1;1)g. As a result of the diagonalization we obtain the two hybridized sin- gletsjS+i,jSi[32, 34] with energies ES=U"+V+ 2r (U"+V+)2 4+ 2t2 H;(3) -10123 0 0.5 1 1.5 2E/U ε/UT−(1,1) T0(1,1) T+(1,1) S−S(0,2) S+ Figure 2. (Color online) Two-electron spectrum of a DQD in InAs as a function of the interdot bias ", obtained by di- agonalizing the Hamiltonian H0[Eq. (2)]. The energy E and the detuning "are expressed in units of the Coulomb en- ergyU. The parameters of the plot are the magnetic eld B= 1 T, the Coulomb energy U= 4:86 meV, the extended tunneling hopping tH= 0:11 meV, the triplet matrix element V+= 2:16eV, the doubly occupied singlet matrix element V= 0:42eV, half of the interdot separation a= 73:6 nm. Including hyperne interaction and/or spin-orbit interaction opens up an avoided crossing
[34] (upper inset). The mag- netic eld is chosen large, as compared to the value in the re- mainder of the paper, for visualization purposes. The S(2;0) andS(0;2) are singly occupied singlets, S(1;1) is the doubly occupied singlet. T+,T0andTare triplet states correspond- ing toms= 1,ms= 0 andms=1. TheSandS+are the lower and the upper hybridized singlet [see Eq. (4) and Eq. (5)]. and eigenvectors jSi=c(")jS(1;1)i+p 1c(")2jS(2;0)i; (4) jS+i=p 1c(")2jS(1;1)ic(")jS(2;0)i: (5) Withc(") = cos we denote the charge admixture coef- cient which can be expressed with the charge admixture angle , where cos 2 =UV+"p (UV+")2+ 8t2 H: (6) We only take into account the transitions between the lower hybridized singlet jSiand tripletjT+ibecause the upper hybridized singlet jS+iis higher in energy, and therefore can be neglected, as shown in Fig. 2 . The spin-orbit term HSOin the Hamiltonian is a func- tion of the angle [cf. Fig 1] between the [110] crystal- lographic axis and the interdot connection axis p[34], HSO=i 2 ()
X s;t=";#(cy LsstcRth:c:); (7)4 where () is the spin-orbit eective magnetic eld de- ned by i () =hLj^pjRi(() cose[110]+(+) sine[110]): (8) Hereandare the Rashba [35] and Dresselhaus [36] coecients, the cy r;soperator creates an electron with spin s=";#, in the right or left dot, r=R;L. Further, s;t is the vector of Pauli matrices and  L;Rare the spatial parts of the wavefunctions corresponding to the left and the right dot respectively [34] and ^ pis the component of the momentum operator along the interdot connection axis. For computational simplicity, we choose our coordinate system such that the matrix elements of the spin-orbit part of the Hamiltonian [Eq. (7)] are always real. This is achieved by setting the eyaxis of our coordinate system parallel with [34], as shown in Fig. 1. When the spin- orbit interaction is excluded, our xandyaxes are parallel to the crystallographic axes. Finally, the hyperne part of the Hamiltonian is given by [23] HHF=S1
h1+S2
h2=1 22X i=1(2Sz ihz i+S+ ih i+S ih+ i); (9) whereS() iare theith electron spin ladder operators, Sz iandhz iare thezcomponents of the ith electron spin operator and Overhauser eld operator. Furthermore, h i=hx iihy iare the ladder operators of the Overhauser eld, hi=n(i)X k=1Ak iIk i; (10) where Ik iare the nuclear spin operators for the kth nu- clear spin in contact with the ith electron spin. The strength of the hyperne coupling between the ith elec- tron and the kth nuclear spin is labeled Ak i. In general Ak i can have a dierent value for every nuclear spin, but we simplify this by assuming a constant hyperne coupling Ak i=Atot=N[24]. Performing a diagonalization in the singlet subspace spanned byfS(2;0),S(1;1)g, we nd that the singlet eigenfunctions are bias dependent and therefore time de- pendent [Eq. (4) and Eq. (5)]. This implies that the cou- pling between the lower hybridized singlet jSiand the jT+itriplet state is time dependent as compared to time independent coupling between the jS(1;1)iandjS(2;0)i singlets and the jT+itriplet. The time dependence of the coupling originates on the fact that the coupling de- pends on the charge state of the hybridized singlet [Eq. (4) and Eq. (5)]. The state S(2;0) couples to T+only via the spin-orbit interaction and S(1;1) couples to T+ only by means of the hyperne interaction. By using thewavefunctions of the lower hybridized singlet (see Eq. (4) we can calculate the matrix element of the Hamiltonian between the lower hybridized singlet jSiand the triplet jT+i hSjHjT+i=c(")hS(1;1)jHHFjT+i +p 1c(")2hS(2;0)jHSOjT+i:(11) It should be mentioned that due to time dependent interactions, the model discussed here must go beyond the Landau-Zener model [37-39]. III. THE BASIS OF TOTAL ANGULAR MOMENTUM In our model, all nuclear spins are treated as having spinj= 1=2. This means that the total number of nu- clear spin states is dim( H) = 2N, whereNis the num- ber of nuclear spins in a quantum dot. Because the total number of nuclear spin states scales exponentially with N it would be impossible to treat a large number ( N= 150) of nuclear spins with the computational power at our dis- posal. In order to make the problem treatable we rst make a basis change from the product basis f";#g, to the basis of total angular momentum fjj;mig. Herejis the total nuclear spin quantum number, 0 jN=2, andmis the total nuclear spin projection along the z axis,jmj. Now the total number of states can be written as dim(H) =N=2X j=0X perm(2j+ 1) = 2N: (12) The inner sum runs over all permutation symmetries for a given value of j. The basis of total angular momentum still scales as dim( H) = 2N, but now certain states in the inner sum in Eq. (12) do not need to be taken into account, and states with higher jin the outer sum in Eq. (12) can be neglected due to the low probability of their occurrence. In the remainder of this section we will describe in more detail how we reduce the number of nuclear spin states from dim( H) = 2Nto dim(H0)2N. Neither the hyperne nor the spin-orbit interaction mix states with dierent j, and thus the matrix represent- ing our Hamiltonian is block diagonal with every block corresponding to a value of j=j0; j0+ 1; :::N= 2. The value ofj0depends on the parity of N, for an even N, j0= 0 and for an odd N,j0= 1=2. The probability distribution of nuclear spin states, with respect to the quantum number jis a Gaussian (in the limit N!1 ) with its maximum located at p N=2, Fig. 3. From now on we will refer to this value of jas its most likely value,jmlp N=2. The nuclear spin probability distri- bution, with respect to the number of nuclear spins per dotNand quantum number jis given by the following formula [40]5 00.020.040.060.080.1 0 10 20 30 40 50 60 70p(j, N ) j Gaussian fit included states neglected states Figure 3. (Color online) Initial nuclear spin probability distri- bution with respect to the quantum number jforN= 150 nu- clear spins 1 =2, wherejml=p N=2 andjmax= 18. Through- out our calculations we only consider the states 0 jjmax (blue diamonds) and do not consider the states j > j max (black circles). p(N;j) =(2j+ 1)2N! (N=2 +j+ 1)!(N=2j)!2N: (13) Thejandmquantum numbers are generally not suf- cient to describe all possible nuclear spin states. Other thanjandm, the nuclear spin states are described by their permutation symmetries. For example, for three nuclear spins dened by quantum numbers j= 1=2 and m= 1=2, there are two states j1=2;1=2iandj1=2;1=2i0 with distinct permutation symmetries. These two states are not mixed by homogenous hyperne or by spin-orbit interactions. Furthermore, they remain equally probable as the matrix elements of the Hamiltonian only depend onjandmand not on the symmetry properties. There- fore, by evaluating our system for a certain symmetry j1=2;1=2iwe would also know the behavior of the state with a dierent permutation symmetry j1=2;1=2i0. By generalizing this simple example to N-spin systems we can signicantly reduce the number of the states we con- sider. For every value of jwe need to evaluate only one state of symmetry in Eq. (12), and therefore for each value ofjthe inner sum in Eq. (12) can be replaced by one representing term. We can reduce the number of states further by choosing the maximum value of jwe take into consideration, jmax in a manner thatp N=2jmaxN=2. The omission of all states with j > j maxis justied because these states occur with a very low probability (see Fig. 3 and Eq. (13)). Now the total number of the states we consider scales with jmaxas dim(H0) =jmaxX j=0(2j+ 1)=(jmax+ 1)22N:(14)Due to the fact that the states with dierent jdo not mix by any interaction we consider, we can analyze our system for one value of jat a time and nally average over all included values of j. By doing so, we average over close to (but not exactly) 100% of all possible states. In our case,N= 150 nuclear spins per dot and 0 j75. Constraining ourselves to 0 jjmax= 18, we average over 97:8% of all possible nuclear spin congurations, as shown in Fig. 3. The eciency of our approach can be illustrated best if we calculate the number of states in the f",#gbasis and in the fjj;migbasis after we consider only one symmetry state for every jand consider only 0jjmax. ForN= 150, Eq. (12) yields dim( H) 1:41045and forjmax= 18, Eq. (14) yields dim( H0) = 361. IV. TIME EVOLUTION DURING DNP We now describe a single step in the DNP proce- dure. The system is initialized in a singlet state S(2;0), where both electrons are occupying the same dot. Af- terwards, the electronic system is driven with a nite velocity through the ST+anticrossing (see Fig. 2) by varying the voltage bias ". The electronic state is then measured, and nally the system is reset quickly to the initial state S(2;0) [23]. Accordingly, we propagate the density matrix of the system according to the update rule (i+1)=MSU(i)UyMS+MTU(i)UyMT: (15) Here(i)and(i+1)are the total density matrices be- fore and after the i-th DNP step, Uis the unitary time evolution operator and MSandMTare the singlet and triplet projection operators [41]. They satisfy the rela- tionsMS+MT=I;andMSMT= 0: After the evolution of the system, a measurement of the electronic state takes place. This measurement pro- cedure has two outcomes: either a singlet Sor a triplet T+is detected. The nuclear density matrix is updated accordingly, n=PSS n+PTT n; (16) wherenis the nuclear density matrix and PS= Tr[MSU(i)UyMS] andPT= Tr[MTU(i)UyMT] are the singlet and the triplet outcome probabilities. The superscripts SandTstand for a nuclear density matrix related to the singlet and the triplet measurement outcome. For a certain value of jwe calculate the singlet return probability PS, and the standard deviation of the nuclear dierence eld, (z)=p h(hz)2ihhzi2[13]. After averaging over all included j, we use the stan- dard deviation of the nuclear dierence eld to evaluate theST0spin qubit decoherence time, T 2= h=(z)[13]. We compute the propagator Uby discretizing the time interval (0;). Our model describes the passage through6 the anticrossing with q= 100 equally spaced, step-like time increments. The procedure of computing the prop- agator is the following: For every discrete point in time tiwe compute the Hamiltonian H(ti). We approximate the propagator for the xed time point ti, Uti=eiH(ti)
t=h; (17) with
t==q. By repeating the procedure for every discrete step we obtain the total time evolution operator U=UtqUtq1:::Ut1: (18) Tuning the system across the ST+point and measuring the electronic state after every forward sweep changes the probabilities and coherences of the electronic and the nu- clear states. The qualitative picture is simpler if we rst disregard the spin-orbit interaction. When the spin-orbit interaction is excluded, both the electronic spin singlet and the triplet outcomes increase the probability for nu- clear spins to be in the spin down state [23], correspond- ing to generating negative values of nuclear spin polariza- tionP= (n"n#)=(n"+n#), wherePis the nuclear spin polarization, n"is the number of nuclear spins pointing up andn#is the number of nuclear spins pointing down [cf. Figs. 4(a-d)]. There is one more possible process, involving spin-orbit interaction, which is not shown in Fig. 4. After cycling the electronic system across the ST+anticrossing the system can end up in a virtual T+state due to spin-orbit interaction, but is instantaneously transferred to a singlet state due to hyperne interaction, accompanied by a ip of the nuclear spin from down to up, thus changing the nuclear spin polarization closer to positive values. This is a process that, along with the process visualized on Fig. 4(d), competes with the hyperne-mediated gener- ation of negative polarization of the nuclear spins (down pumping). These two processes combined compensate the down pumping in systems with strong spin-orbit in- teraction. To make an eective comparison between In xGa1xAs systems with dierent indium content xwe keep the same values forBzandd=a=aB= 2:186. This implies that the single particle tunneling and the overlap between the quantum dots would remain the same for every value of x (see Ref. [32]). For a comparison between dierent ma- terials, the relative strength of the spin-orbit interaction can be quantied by the ratio of  = 4 a=SO, where  SO is the spin-orbit length dened by 1 SO=m hq cos2()2+ sin2(+)2:(19) Here,mis the eective electron mass, andare the Rashba and Dresselhaus constants and is the angle be- tween the [110] crystallographic axis and the interdot connection axis p[cf. Fig.4]. The spin-orbit length is the distance which an electron needs to travel in order to have its spin ipped due to spin-orbit interaction. If the electrons are initialized in (a) (b) (c) (d) Figure 4. (Color online) System initialization and measure- ment outcomes. (a) Initially, the quantum dots have an en- ergy bias"and the two electrons rest in a singlet (2 ;0) state on the left dot. (b) After slowly tuning "to zero, and measur- ing a singlet outcome, due to the weak measurement the spin of the nuclear bath decreases. (c) In the case of a spin triplet outcome an electron spin ips and the spin of the nuclear bath is changed accordingly. (d) The electronic spin can also be ipped due to spin-orbit, and the spin of the nuclear bath is pumped in the opposing direction (up) due to the weak measurement. With "we denote the voltage bias, is the angle between the [110] crystallographic axis and the interdot connection axis p, is the spin-orbit eective magnetic eld. a singlet state the probability for ipping the tunneling electron due to spin-orbit interaction is P ip= 1=2 at7 00.20.4 -15 -10 -5 0 5 10 15p(m) m(a) left dot -8 -6 -4 -2 0 2 4 6 8 m(b) right dot Figure 5. (Color online) (a) Probability distribution in the left quantum dot with respect to the nuclear spin projection quantum number mforjL= 14. Blue circles represent the initial probability distribution, black triangles represent the probability distribution after 300 cycles with spin-orbit in- teraction excluded, and red squares represent the probability distribution after 300 cycles with spin-orbit interaction in- cluded. (b) Probability distribution in the right quantum dot with respect to the nuclear spin projection quantum number mforjR= 7. Red pentagons present the initial probability distribution, green triangles represent the probability distri- bution after 300 cycles with spin-orbit interaction excluded and black diamonds represent the probability distribution af- ter 300 cycles when spin-orbit interaction is included corre- sponding to ==2. Here,is the angle between the [110] crystallographic axis and the interdot connection axis p. The number of nuclear spins per quantum dot is N= 150. 2a=  SO=2. This further implies that if  <1, the system is more probable to remain in a singlet state. If  = 1 the SandT+outcomes due to spin-orbit cou- pling are equally probable and nally if 1 <<2 a T+outcome due to spin-orbit is more probable, because the probability that the tunneling electron has ipped its spin is greater than P ip>0:5. In our study  SO=22a which implies  1, thus singlet outcomes due to spin-orbit interaction are always more probable even in pure InAs with the strongest possible value of spin-orbit (==2). In pure InAs, with ==2, 0:63 for d=a=aB= 2:186. V. RESULTS FOR In 0:2Ga0:8As Our attention is now focused on In 0:2Ga0:8As, a ma- terial with an intermediate strength of spin-orbit cou- pling, as compared to the relatively weak spin-orbit cou- pling in GaAs and relatively strong spin-orbit coupling in InAs. We have evaluated the system of N= 150 nuclear spins per dot, for dierent values of the angle and with jmax= 18. States with j >j maxwould further lower the T 2andPsand increase (z). Therefore, we point out that our results provide an upper bound for T 2(includ- ing states with j >jmax= 18 could lower T 2for at most 2:2%, see Fig. 3 and Eq. (13)) and Psand a lower bound for(z). We study the eect of 300 DNP cycles on the nuclear spin state. We nd that the spin-orbit interac- 0.50.60.70.80.91 0 50 100 150 200 250 300Ps Number of cycles spin-orbit excluded θ= 0 θ=π/12 θ=π/6 θ=π/4 θ=π/3 θ=π/2Figure 6. (Color online) The singlet return probability PS as a function of the number of cycles across the ST+an- ticrossing in In 0:2Ga0:8As. Here,is the angle between the [110] crystallographic axis and the interdot connection axis p. tion has a notable eect on nuclear state preparation. In Fig. 5, we plot the probabilities of nuclear spin states for a case with a given value of jL;Rin the left and the right dot. ForjL= 14 andjR= 7 the pumping procedure has al- tered the nuclear probability distribution from a uniform distribution (with respect to the quantum number m) to a probability distribution where states with negative m are more likely. In the case without spin-orbit interac- tion, two processes contribute to this negative pumping of the nuclear spin [23] - the singlet detection accompa- nied by a weak measurement of the nuclear spin state and theT+detection, which ips the nuclear spin down to conserve the total spin of the system [cf. Fig. 4(b) and Fig. 4(c)]. Although including spin-orbit interaction [cf. Fig. 5(a), Fig. 5(b)], changes the nal distribution of nuclear spin states only slightly, spin-orbit eects still have a notable eect on the singlet return probability PS= Tr[MSU(i)UyMS]. In Fig. 6, we plot PSas a function of the number of cycles across the ST+anti- crossing for In 0:2Ga0:8As. Here, we tune the strength of the spin-orbit interaction by varying the angle between the [110] crystallographic axis and the interdot connec- tion axisp. As shown in Fig. 6 (solid red line), re- peatedly cycling the system across the anticrossing point polarizes the nuclear spins, which leads to Ps= 1 af- ter 300 cycles [23]. The situation changes dramatically when we include the spin-orbit interaction, which com- petes with the hyperne mediated down pumping of the nuclear spin. By theoretically varying the strength of the spin-orbit interaction, we nd that when the spin-orbit interaction has the largest possible value for ==2, it signicantly aects the singlet return probability Ps0:72 (Fig. 6).8 405060708090100 0 50 100 150 200 250 300σ(z)[neV] Number of cycles spin-orbit excluded θ= 0 θ=π/6 θ=π/3 θ=π/2 Figure 7. (Color online) Standard deviation of the nuclear dierence eld (z)with respect to the number of DNP cycles across theST+anticrossing and dierent values of angle in In 0:2Ga0:8As. Here, is the angle between the [110] crystallographic axis and the interdot connection axis p. Including spin-orbit interaction generates a mechanism which polarizes nuclear spins in the up direction (see Sec- tion IV and Fig. 5). As a consequence of this behavior, the nuclear preparation mechanism is not ecient when spin-orbit eects are strong. The interplay of the hyper- ne and spin-orbit interactions on nuclear state prepa- ration can be observed better if we plot the standard deviation of the nuclear dierence eld (z)(Fig. 7). We notice that the spin-orbit interaction has prevented the reduction of the standard deviation of the nuclear dierence eld (0 =2, see Fig. 7). Spin-orbit interactions aect the eorts to increase the spin ST0 qubit decoherence time T 2, see Fig. 8. The strongest spin-orbit coupling, corresponding to ==2, slightly lowers the resulting decoherence time from T 215 ns (red line) to T 213 ns (black dashed line with black x symbols). Without the spin-orbit interaction our theory predicts that the ratio of the nal decoherence time (after the cy- cling is complete) T 2;fand initial decoherence time (be- fore the cycling starts) T 2;iisT 2;f=T 2;i2:28 [cf. Fig. 9]. The situation changes when we include spin-orbit in- teraction. For = 0 we nd a value of T 2;f=T 2;i2:20, while for==2 the ratio is T 2;f=T 2;i2:04. After the inclusion of the spin-orbit interaction the ra- tioT 2;f=T 2;idecreases with . Our results suggest that theST+dynamical nuclear polarization is not as eec- tive in materials with intermediate strength of spin-orbit interaction, as compared to those without spin-orbit cou- pling. Nevertheless, the DNP still provides a notable en- hancement of the ST0qubit decoherence time T 2. We work in the so called "giant spin model" and we model the behavior of 104106nuclear spins with signicantly fewer spins,102103. In general (z) i/Ak i, which 68101214 0 50 100 150 200 250 300T∗ 2[ns] Number of cycles spin-orbit excluded θ= 0 θ=π/6 θ=π/3 θ=π/2Figure 8. (Color online) ST0qubit decoherence time T 2as a function of the number of DNP cycles across the ST+anticrossing and strength of spin-orbit interaction in In0:2Ga0:8As. Here,is the angle between the [110] crystal- lographic axis and the interdot connection axis p. 22.052.12.152.22.252.3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6T∗ 2,f/T∗ 2,i θ[rad] spin-orbit excluded spin-orbit included Figure 9. (Color online) The ratio of the nal T 2;fand initial T 2;idecoherence times in In 0:2Ga0:8As, for dierent values of the anglebetween the [110] crystallographic axis and the interdot connection axis p. would give rise to a much higher standard deviation of the nuclear dierence eld than expected. Therefore, we rescale the hyperne constant, such that (z) ihas the same value for N106, andN= 150,jml=p N=2. The predicted decoherence time before the start of the DNP isT 26:2 ns while measurements yield T 210 ns for pure GaAs [5] (where excluding spin-orbit eects is a good approximation). Since (z) i/p N, and(z) fdoes not depend on Nbut on dierent parameters, we can estimate that T 2;f=T 2;ip Nfor our case of N= 150 and the realistic case N= 106(for an electrically de- ned quantum dot in In xGa1xAs). Therefore, we can9 68101214 0 50 100 150 200 250 300T∗ 2[ns] Number of cycles x= 0% x= 20% x= 40% x= 60% x= 80% x= 100% Figure 10. (Color online) ST0electron spin coherence time T 2as a function of the number of DNP cycles across the ST+anticrossing, for dierent abundances of indium xin InxGa1xAs and for ==2. Here,is the angle between the [110] crystallographic axis and the interdot connection axisp. estimate the maximum possible ratio of initial and - nal decoherence times for the realistic case of N= 106 spins and spin-orbit interaction excluded and included to beT 2;f=T 2;i175 without spin-orbit interaction, compared to T 2;f=T 2;i94 for GaAs in reference [23], T 2;f=T 2;i174 for= 0,T 2;f=T 2;i163 for==2. VI. RESULTS FOR In xGa1xAs In this section we will compare the T 2results for InxGa1xAs with varying In content x. We vary the concentration of indium xin the sample between 0 and 1 with a 0:2 increment. For the sake of computational eciency, and the fact that we are interested in a mere comparison between materials with dierent percentages of indium, our computational method is slightly simpli- ed now. Instead of averaging over all possible states ranging from jminjmaxwe setjL=jR=jml=p N=2 for the left and the right quantum dot. This eectively means that we are simulating a situation where an exper- iment is performed only once with the most likely nuclear spin conguration. From Fig. 10 we conclude that raising the concentra- tion of indium in a In xGa1xAs sample has a detrimental eect on the eciency of the ST+DNP scheme. By doping the system with indium, the Rashba spin-orbit coupling is strengthened, thus reducing the overall  SO [Eq. (19)], which as a consequence has more virtual and realT+outcomes due to the spin-orbit interaction. The virtualT+will relax to S, quickly ipping a nuclear spin from down to up in the process. The real spin-orbit me- diatedT+outcomes will also pump the nuclear spin to- 68101214 0 50 100 150 200 250 300T∗ 2[ns] Number of cycles GaAs InAsFigure 11. (Color online) ST0electron spin coherence time T 2for GaAs and InAs as a function of the number of DNP cycles across the ST+anticrossing, for = 0, i.e. the case where the [110] crystallographic axis and the interdot connection axis pare aligned. wards the positive values of the polarization (up). This process can completely vain eorts to increase T 2, even at intermediate concentrations of 40% In (Fig. 10). At higher indium concentrations, DNP is totally suppressed for all values of [cf. Fig. 11]. VII. CONCLUSIONS AND FINAL REMARKS Our results show that pure InAs is a not a suitable can- didate forST+DNP, due to the fact that the enhance- ment ofT 2is strongly suppressed even for the smallest possible strength of the spin-orbit interaction correspond- ing to= 0. Dynamical nuclear polarization in InAs could still be achieved by using single spin single quan- tum dot systems [29] or by using a more elaborate pulsing sequence [30]. A similar behavior could be expected in materials with even stronger spin-orbit as compared to InAs and that is, e.g., InSb. To conclude, we have discussed a nuclear polarization scheme in In xGa1xAs double quantum dots with spin- orbit interaction included. In the presence of spin-orbit interaction a suppression of the enhancement of T 2is predicted. Our conclusions are also valid for materials with fewer nuclear spins. We underline that the ST+ DNP sequence is highly sensitive to the strength of the spin-orbit coupling, and therefore the eciency of the ST+DNP sequence will depend on the angle and the In content xin InxGa1xAs. A stronger spin-orbit interaction will establish a process that will quickly neu- tralize any eorts to prolong T 2. The cases of unequally coupled and/or sized dots, and dierent shapes of the bias [21] are in general treatable by our numerics and will be the subject of our future studies. Charge noise10 [42-44] is neglected in the current model. Investigating the signicance of charge coherence requires an extension of the numerical tools we use [43], and is planned as a forthcoming investigation.ACKNOWLEDGMENTS We thank Hugo Ribeiro for useful discussions and the EU S3NANO Marie Curie ITN and Deutsche Forschungs- gemeinschaft (DFG) within the SFB 767 for nancial support. 1D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 2D. P. DiVincenzo, arXiv:cond-mat/9612126v2. 3J. M. Taylor, H. A. Engel, W. D ur, A. Yacoby, C. M. Mar- cus, P. Zoller, and M. D. Lukin, Nature Phys. 1, 177 (2005). 4K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarov, and L. M. K. Vandersypen, Science 318, 5855 (2007). 5J. R. Petta , A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 309, 5744 (2005). 6K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Science 297, 5585, (2002). 7A. C. Johnson, J. R. Petta, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. B 72, 165308 (2005). 8W. A. Coish and D. Loss, Phys. Rev. B 72, 125337 (2005). 9A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. 88, 186802 (2002). 10E. A. Chekhovich, M. N. Makhonin, A. I. Tartakovskii, A. Yacoby, H. Bluhm, K. C. Nowack, and L. M. K. Van- dersypen, Nature Mater. 12, 494-504 (2013). 11 L. Cywi nski, W. M. Witzel, and S. Das Sarma, Phys. Rev. B79, 245314 (2009). 12 L. Cywi nski, W. M. Witzel, and S. Das Sarma, Phys. Rev. Lett. 102, 057601 (2009). 13R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Rev. Mod. Phys. 791217 (2007). 14B. E. Kane, Nature 393, 133-137 (1998). 15B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Nature Physics 3, 192 - 196 (2007). 16E. A. Chekhovich, M. N. Makhonin, K. V. Kavokin, A. B. Krysa, M. S. Skolnick, and A. I. Tartakovskii, Phys. Rev. Lett. 104, 066804 (2010). 17A. H ogele, M. Kroner, C. Latta, M. Claassen, I. Caru- sotto, C. Bulutay, and A. Imamoglu, Phys. Rev. Lett. 108, 197403 (2012). 18C. Latta, A. H ogele, Y. Zhao, A. N. Vamivakas, P. Maletinsky, M. Kroner, J. Dreiser, I. Carusotto, A. Badolato, D. Schuh, W. Wegscheider, M. Atature, and A. Imamoglu, Nature Physics 5, 758 - 763 (2009). 19M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 110 086601 (2013). 20J. Levy, Phys. Rev. Lett. 89, 147902 (2002). 21H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Umansky, and A. Yacoby, Nature Physics 7, 109-113(2011). 22E. L. Hahn, Phys. Rev. 80, 580 (1950). 23H. Ribeiro and G. Burkard, Phys. Rev. Lett. 102, 216802 (2009). 24G. Ramon and X. Hu, Phys. Rev. B 75, 161301(R) (2007). 25M. S. Rudner and L. S. Levitov, Phys. Rev. B 82, 155418 (2010). 26I. Neder, M. S. Rudner, and B. I. Halperin, arXiv:1309. 3027v1 (2013). 27M. S. Rudner, I. Neder, L. S. Levitov, and B. I. Halperin, Phys. Rev. B. 82, 041311(R) (2010). 28A. Brataas and E. I. Rashba, Phys. Rev. B 84, 045301 (2011). 29B. Eble, O. Krebs, A. Lematre, K. Kowalik, A. Kudelski, P. Voisin, B. Urbaszek, X. Marie, and T. Amand, Phys. Rev. B 74, 081306(R) (2006). 30S. M. Frolov, J. Danon, S. Nadj-Perge, K. Zuo, J. W. W. van Tilburg, V. S. Pribiag, J. W. G. van den Berg, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Phys. Rev. Lett. 109, 236805 (2012). 31D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 321, 817 (2008). 32G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 3 (1999). 33J. M. Taylor, A. Imamoglu, and M. D. Lukin, Phys. Rev. Lett. 91, 246802 (2003). 34D. Stepanenko, M. Rudner, B. Halperin, and D. Loss, Phys. Rev. B 85, 075416 (2012). 35E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). 36G. Dresselhaus, Phys. Rev. 100(2) , 580-586 (1955). 37L. D. Landau, Phys. Z Sowjetunion 2, 46 (1932). 38C. Zener, Proc. R. Soc. A 137, 696 (1932). 39E. C. G. St uckelberg, Helv. Phys. Acta 5, 369 (1932). 40J. Hildmann, E. Kavousanaki, G. Burkard, and H. Ribeiro, Phys. Rev. B 89, 205302 (2014). 41M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, 2000), page 102. 42O. E. Dial, M. D. Shulman, S. P. Harvey, H. Bluhm, V. Umansky, and A. Yacoby, Phys. Rev. Lett. 110, 146804 (2013). 43H. Ribeiro, G. Burkard, J. R. Petta, H. Lu, and A. C. Gos- sard, Phys. Rev. Lett. 110, 086804 (2013). 44X. Hu and S. Das Sarma, Phys. Rev. Lett. 96, 100501 (2006).

0802.1351v1.Coupling_of_Spin_and_Orbital_Motion_of_Electrons_in_Carbon_Nanotubes.pdf

1 Coupling of Spin and Orbital Motion of Electrons in Carbon Nanotubes F. Kuemmeth*, S. Ilani*, D. C. Ralph and P. L. McEuen Laboratory of Atomic and Solid State Physics, De partment of Physics, Cornell University, Ithaca NY 14853 * These authors contributed equally to this work Electrons in atoms possess both spin and orbi tal degrees of freedom. In non-relativistic quantum mechanics, these are independent, res ulting in large degeneracies in atomic spectra. However, relativistic ef fects couple the spin and orbita l motion leading to the well- known fine structure in their spectra. The electronic states in defect-free carbon nanotubes (NTs) are widely believed to be four-fold degenerate1-10, due to independent spin and orbital symmetries, and to also possess electron-hole symmetry11. Here we report measurements demonstrating that in clean NTs the spin and orbital motion of electrons are coupled, thereby breaking all of these symmetri es. This spin-orbit coupling is directly observed as a splitting of the four-fold dege neracy of a single electron in ultra-clean quantum dots. The coupling favours parallel a lignment of the orbital and spin magnetic moments for electrons and anti-parallel al ignment for holes. Our measurements are consistent with recent theories12,13 that predict the existence of spin-orbit coupling in curved graphene and describe it as a spin -dependent topological phase in NTs. Our findings have important implications for spin-b ased applications in carbon-based systems, entailing new design principles for the rea lization of qubits in NTs and providing a mechanism for all-electri cal control of spins14 in NTs. Carbon-based systems are promising candidates fo r spin based applications such as spin- qubits14-19 and spintronics20-23 as they are believed to have exceptionally long spin coherence times due to weak spin-orbit interactions and the absence of nuclear spin in the 12C atom. Carbon NTs may play a particularly interesting role in th is context because in addition to spin they offer a unique two-fold orbital degree of freedom that can also be used for quantum manipulation. The latter arises from the two equi valent dispersion cones (K and K’) in graphene, which lead to doubly-degenerate electronic orbits that encircle the nanotube circumference in a clockwise (CW) and counter-clockwise (CCW) fashion24 (Fig 1a). Together, the two-fold spin degeneracy and two-fold orbital degeneracy are generally assumed to yield a four-fold-degenerate electronic energy spectrum in clean NTs. Understanding the f undamental symmetries of this spectrum is at the heart of successful manipulation of these quantum degrees of freedom. A powerful way to probe the symmetries is by confining the carriers to a quantum dot (QD) and applying a magnetic fiel d parallel to the tube axis, ||B4,5,8,10,24,25. The confinement creates bound states and the field interrogates thei r nature by coupling indepe ndently to their spin and orbital moments. In the absence of spin-o rbit coupling, such a measurement should yield for a defect-free NT the energy sp ectrum shown in figure 1b. At 0||=B the NT spectrum should be four-fold degenerate . With increasing ||B the spectrum splits into pairs of CCW and CW states 2 (going down and up in energy respectively), each pa ir having a smaller internal spin splitting. Indications of approximate four-f old degeneracy have been observ ed in high-field measurements of electron addition spectra2-10 and inelastic cotunneling4,10 in nanotube QDs. However, in previous experiments disorder-i nduced splitting of th e orbital degeneracy and electron-electron interactions in multi-electron QDs have masked the intrinsic symmetries at low energies. In this work we directly measure the in trinsic electronic spect rum by studying a single charge carrier, an electron or a hole, in an u ltra-clean carbon nanotube QD. Remarkably, we find that the expected four-fold symmetry and electr on-hole symmetry are broken by spin-orbit (SO) coupling, demonstrating that the spin and orbital motion in NTs are not independent degrees of freedom. The observed SO coupling further determines the filling order in the many-electron ground states, giving states quite different from models ba sed purely on electron-electron interactions. The geometry of our devices is shown in Fi g. 1c. A single small-bandgap NT is contacted by source and drain electrodes, and is gated fr om below by two gates (see methods). When biased, these gates shift the local Fermi energy in the NT thereby accumulating electrons or holes. In this work we use two independent gates to create a QD that is lo calized either above the left or above the right gate electrode. This is achieved by choosing appropriate combinations of gate voltages that pin the Fermi energy inside the gap on one side of the device while adding carriers to the other side (Fig 1c). Measurement of the linear conductance, SDdVdIG / = , through such a dot (Fig. 1e) shows Coulomb blockade pe aks that correspond to th e addition of individual carriers to the dot, and allows us to identify the first electron and first hole in the dot (see supplementary information for details). Having a single carrier in the do t enables us to study single-particle levels in the ab sence of electron-elec tron interactions, and thus to unambiguously identify the presence of spin-orbit coupling. The results reported here were observed in two independent devices and below we present data from one of them. We probe the quantum states of the NT us ing tunnelling spectroscopy. The differential conductance through the dot, SDdVdIG /= , is measured as a f unction of gate voltage, gV, and source-drain bias, sdV, as the first electron is added to the dot. Figure 2a shows a typical measurement taken at mT 300||=B . The transition between the C oulomb blockade regions of zero and one electron features distinct resonances that correspond to the ground state ( α) as well as the excited states ( β, γ, δ) of the first electron. Their energies can be obtained from a line cut at constant sdV (Fig 2b), by converting the gate volta ges into energies (see methods). The magnetic field dependence of the one-electron states γβα ,, and δ is measured by taking gV traces such as in Fig. 2b for different values of ||B. This is shown in Fig 2c, where we plot SDdVdI/ as a function of gV and ||B. The energies of the states α and β decrease with increasing ||B, hence we identify them as CC W orbital states. The states γ and δ increase in energy and are thus identified as CW orbital states. From the slopes of these resonances with respect to magnetic field we extract an orbital moment of µorb=1.55 meV/T and estimate the NT diameter to be nm5≈d24. 3 A striking difference is observed when we comp are the measured excitation spectrum with the one predicted in Fig. 1b: At zero magnetic field the four stat es in our measurement are not degenerate but rather split into two pairs. To iden tify the nature of this splitting we note that with increasing magnetic field the energy difference between the states α and β increases while the difference between states γ and δ decreases, and both differences are consistent with a g-factor of an electron spin (Figure 2d). This observation allows us to id entify unambiguously the spin and orbital composition of each energy level, as shown in the inset of Fig. 2c. At 0||=B the four-fold degeneracy is split into two Kramer doublets – the lower-energy doublet i nvolves states with parallel alignment of orbital and spin magnetic moments, whereas the higher-energy doublet has states with anti-parallel alignment. The zero-field splitting is therefore identified as a spin-orbit splitting, with a value of meV 02.0 37.0 ± =∆SO (extracted from Fig. 2d). At low fields (Figure 2e) the intersections of states with opposite spin directions (e.g. α and γ) show simple crossing, whereas st ates with parallel spin (e.g. β and γ) show avoided crossing, a signature of di sorder-induced mixing between CCW and CW orbits ('KK∆). In previous experiments, the di sorder-induced mixing was signifi cantly larger, presumably obscuring the effects of SO coupling. In our measurements, the mixing is small, SO KK ∆<< ≈ ∆ eVµ 65' , probably due to smooth electronic c onfinement, enabling the observation of SO effects. We further demonstrate the intrin sic nature of the effect by measuring identical excitation spectra for QDs formed at different locations along the same NT (Supp. Fig S1). Next, we show that SO coupling significa ntly affects the many-body ground states of multiple electrons in a QD. Figure 3a shows the magnetic field dependence of the addition energies for the N-electron ground states ( N=-2 to +4), obtained by measuring the linear conductance as a function of gV and ||B. Near zero magneti c field the sign of ||/dB dVg changes every time an electron is added (or removed), indicating that CCW and CW states are filled alternately. Similar addition sequences were explained in the past by repulsive electron-electron interactions driving electrons to occupy different orbits2-7,9,26 (Fig. 3b). However, in our nanotubes the underlying mechanism is entirely different. Compar ing the one-electron excitation spectrum with the two electron-grou nd state (Fig. 3c), we see that the latter follows exactly the first excited state of the one-electron QD. Specifically, both start with a CW slope at low fields and flip to a CCW slope at the fiel d associated with the SO splitting, mT125||≈B . Thus the two-electron ground state is explained entirely by SO coupling (Fig. 3d). Note that below mT125||≈B SO favours each of the two electrons to pos sess parallel orbital and spin moments, forcing them into two different orbital states. Th erefore, the two-electron ground state is neither the spin-triplet state predicted by the electron-in teraction-based models nor a spin singlet, but rather a Slater determinant of tw o single-electron states each of which have parallel orbital and spin magnetic moments. SO effects are commonly assumed to be neg ligible in carbon-base d systems due to the weak atomic SO splitting in carbon ∆at=E(2P3/2)−E(2P1/2) ~ 8 meV ( )27 and its almost perfect suppression in flat graphene13. However, recent theories have argued that SO coupling can nevertheless be significant in carbon NTs due to their curvat ure and cylindrical topology12,13. The 4 predicted effect is illustrated in Figure 4a. C onsider an electron with a spin moment pointing along the NT axis and orbiting around the NT circumference. The electron occupies the zp orbitals of the carbon atoms, wh ich are pointing perpendicular to the NT surface. In the rest frame of the electron the underlying zp orbital revolves around th e spin exactly once every rotation, independent of the details of the elec tron trajectory. In the pr esence of atomic SO coupling a constant phase accumulates during each rotation, which can therefore be described by a spin-dependent topological flux, SOSφ|| passing through the NT cross section ( 1 /1|| −+=S for spin moment parallel/antiparallel to the NT ax is). This flux modifies the quantization condition of the wavefunction around the circumference: 0 || / 2 φ φπ π πSOS dk dk − →⊥ ⊥ , (1) where ⊥k is the electron’s wave-vector in the circum ferential direction as measured from the K and K’ points, d is the tube diameter and 0φ is the flux quantum. Accord ing to the theory in Ref [12] the flux is given by: 03 010 3512φ φεφπσ πσ−≈⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ +∆= pppp at SOVV, (2) where πσε is the energy splitting of the π and σ bands in graphene and σ ppV, π ppV are the hopping elements within these bands. This flux doe s not depend on the geometrical properties of the NT such as its diameter or the shape of its cross section, signifying its topological origin. Figure 4b illustrates the consequences of the modified quantization conditions for a small- bandgap tube at 0||=B . Near each Dirac cone (K and K’) there are two quantization lines for the two spin directions (dashed lines). Combining Eq . 1 with the linear disp ersion, and including the Aharonov-Bohm flux induced by ||B, 4/2 ||dBAB π φ= , and the Zeeman spin coupling, the energies are: 0|| 00, ||||2 2 ||2 2,2 φφ φφµSO AB B FdSdk k BSgk kv E + + = − + ±=⊥ ⊥ ⊥ h (3) Here Fv is the Fermi velocity, ||k is the wave vector parallel to the NT axis, and F gap v E k h2/0,±=⊥ accounts for the small bandgap, gapE, at zero magnetic field (the opposite signs are for the K’ and K points). The resul ting energy spectrum is schematically shown in Figure 4c, and is in agreement w ith our measurements. From Eq. (3) we see that the SO energy splitting 04 φφSO F SOdvh=∆ (assuming 0||=k ) is inversely proportional to d. Using the estimated diameter of our NT, nm5≈d , and the measured splitting (Fig. 2d) we obtain the value ]nm[/ meV 9.1 dSO≈∆ , in agreement with the predicted13 value of ] nm[/ meV 6.1 dSO≈∆ . 5 An interesting prediction of the theory12,13,18 is the breaking of elect ron-hole symmetry. In the absence of SO intera ctions the low-energy spectrum of a NT exhibits electron-hole symmetry such that each allowed state has a matching stat e with opposite energy; i.e., the spectrum is symmetric upon reflection about the line 0=E . In the presence of SO interactions and an applied magnetic field, Eq. (3) predicts that this symmetry is broken, as is evident from the absence of mirror symmetry around 0=E in the spectrum in Fig. 4c. For 0 >SOφ the theory predicts that in the one-electr on ground state the orbital and spin magnetic moments are parallel, whereas in the one-hole ground state they are anti-parallel. This result allows us to test the breaking of electron-hole symmetry experimentally. The measured excitation spectra for the first hole in the QD (Fig. 4d) clearly shows a SO splitting at 0||=B , and a spin g-factor equal to that of the one-electron QD (Fig. 4e). However, in contrast to the one-electr on case, here the ground state ( α) and the first excited state ( β) converge with increasing ||B, implying that the orbital and spin moments are aligned anti-parallel in the one-hole ground state, opposite to the one-electron case. Th is observation qualitatively confirms the scheme in Figure 4c. We note, however, that the SO splitting observed for the hole () meV 01.021.0± =∆SO is somewhat smaller than that of the electron, a difference that is not accounted for by current theory. Th is might result from different confinement lengths (different ||k in Eq. 3) or different el ectric fields (i.e. different gV) for electrons and holes, but current theories predict an effect that is too small to explain this observation. The existence of SO coup ling in carbon nanotubes i nvalidates several common assumptions about the nature of th e electronic states in this system , such as four-fold degeneracy and electron-hole symmetry, and further leads to the existence of entangled spin and orbital multi-electron ground states. Currently, carbon-based systems are considered to be excellent candidates for spin-based applications in part du e to the belief that they have weak spin-orbit interactions. Here we showed th at this hypothesis is wrong for NTs. Nevertheless, rather than excluding spin-based devices in NT s, our findings may actually prom ote their feasibility, as long as new design principles are adopted for qubits a nd spintronic devices, which make use of the strong spin-orbit coupling. This coupling can pr ovide a valuable capability, so far missing in carbon systems: the ability to us e electrical gates to manipulat e the spin degree of freedom, through its coupling to the orbi tal electronic wavefunction14. Methods Devices were fabricated from degenerately doped silicon-on-insulator wafers, with a 1.5µm thick device layer on top of a 2 µm buried oxide. Two electrically isolated gate electrodes (Fig. 1d) were patterned from the devi ce layer using dry etching and thermal oxidation (thickness 100 nm). Gate contacts (2/50 nm Ti/Pt), source and drain electrodes (5/25 nm Cr/Pt) as well as catalyst pads were patterned using e-beam lithogra phy. NTs were grown in the last step in order to produce clean devices8. Measurements were performed in a He He/4 3 dilution refrigerator at base temperature ( mK 30T= ), using standard lock-in tec hniques with small excitations (typically V 104 µ − ). The electron temperature extracte d from Coulomb peak widths was 6 100−200 m K. The conversion from ga te voltage to energy is obtained from the bias dependence of the tunnelling resonances , such as in Fig. 2a, and is α=0.57 for the first electron (Fig. 2) and α=0.58 for the first hole (Fig. 4). Acknowledgements We thank E. Altman, Y. Gefen, C. L. Henley, Y. Meir, E. Muelle r, Y. Oreg, E. I. Rashba, A. Stern and B. Trauzettel for discussions. This work was supported by the NSF through the Center for Nanoscale systems, and by the MARCO Focuse d Research Center on Ma terials, Structures and Devices. Sample fabrication was perfor med at the Cornell node of the National Nanofabrication Users Network, funded by NSF. References 1. Kane, C. L. and Mele, E. J., Size, shape, and low energy electronic structure of carbon nanotubes. Phys. Rev. Lett. 78, 1932-1935 (1997). 2. Cobden, D. H. and Nygard, J., Shell fill ing in closed single- wall carbon nanotube quantum dots. Phys. Rev. Lett. 89, 046803 (2002). 3. Liang, W. J., Bockrath, M., and Park, H., Shell filling and exchange coupling in metallic single-walled carbon nanotubes. Phys. Rev. Lett. 88, 126801 (2002). 4. Jarillo-Herrero, P. et al. , Electronic transport spectroscopy of carbon nanotubes in a magnetic field. Phys. Rev. Lett. 94, 156802 (2005). 5. Jarillo-Herrero, P. et al. , Orbital Kondo effect in carbon nanotubes. Nature 434, 484-488 (2005). 6. Moriyama, S. et al. , Four-electron shell structures and an interacting two-electron system in carbon-nanotube quantum dots. Phys. Rev. Lett. 94, 186806 (2005). 7. Sapmaz, S. et al. , Electronic excitation spectrum of metallic carbon nanotubes. Phys. Rev. B 71, 153402 (2005). 8. Cao, J., Wang, Q., and Dai, H., Electron transport in very clean, as-grown suspended carbon nanotubes. Nature Mat. 4, 745-749 (2005). 9. Makarovski, A., An, L., Liu, J., and Finkels tein, G., Persistent orbital degeneracy in carbon nanotubes. Phys. Rev. B 74, 155431 (2006). 10. Makarovski, A., Zhukov, A., Liu, J., and Finkelstein, G., SU(2) and SU(4) Kondo effects in carbon nanotube quantum dots. Phys. Rev. B 75, 241407 (2007). 11. Jarillo-Herrero, P. et al. , Electron-hole symmetry in a semiconducting carbon nanotube quantum dot. Nature 429, 389-392 (2004). 12. Ando, T., Spin-orbit inte raction in carbon nanotubes. J. Phys. Soc. Jpn. 69, 1757-1763 (2000). 13. Huertas-Hernando, D., Guinea, F., and Br ataas, A., Spin-orbit coupling in curved graphene, fullerenes, nano tubes, and nanotube caps. Phys. Rev. B 74, 155426 (2006). 14. Nowack, K. C., Koppens, F. H. L., Nazarov, Yu V., and Vandersypen, L. M. K., Coherent Control of a Single Elec tron Spin with Electric Fields. Science 318, 1430-1433 (2007). 7 15. Loss, D. and DiVincenzo, D. P., Qu antum computation with quantum dots. Phys. Rev. A 57, 120-126 (1998). 16. Elzerman, J. M. et al. , Single-shot read-out of an indi vidual electron spin in a quantum dot. Nature 430, 431-435 (2004). 17. Petta, J. R. et al. , Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309, 2180-2184 (2005). 18. Bulaev, D. V., Trauzettel, B., and Loss, D., arXiv:0712.3767. 19. Trauzettel, B., Bulaev, D. V., Loss, D. and Burkard, G., Spin qubits in graphene quantum dots. Nature Phys. 3, 192-196 (2007). 20. Awschalom, D. D. and Flatte, M. E., Challenges for semiconductor spintronics. Nature Phys. 3, 153-159 (2007). 21. Sahoo, S. et al. , Electric field contro l of spin transport. Nature Phys. 1, 99-102 (2005). 22. Tombros, N., van der Molen, S. J., and van Wees, B. J., Separa ting spin and charge transport in single-wall carbon nanotubes. Phys. Rev. B 73, 233403 (2006). 23. Tombros, N. et al. , Electronic spin transport and spin precession in single graphene layers at room temperature. Nature 448, 571-574 (2007). 24. Minot, E. D., Yaish, Y., Sazonova, V., and McEuen, P. L., Determination of electron orbital magnetic moments in carbon nanotubes. Nature 428, 536-539 (2004). 25. Cobden, D. H. et al. , Spin splitting and even-odd effects in carbon nanotubes. Phys. Rev. Lett. 81, 681-684 (1998). 26. Oreg, Y., Byczuk, K., and Halperin, B. I., Spin configurations of a carbon nanotube in a nonuniform external potential. Phys. Rev. Lett. 85, 365-368 (2000). 27. Ralchenko, Yu., et al. , Atomic Spectra Database (versi on 3.1.3), National Institute of Standards and Technology, Gaithersburg, MD. http://physics.nist.gov /asd3 (accessed 14 Nov. 2007). 8 Figures Figure 1: Few-electron carbon nano tube (NT) quantum dot devices. a, Electrons confined in a NT segment have quan tized energy levels, each four-fold degenerate in the absence of spin-orbit coupling and defect scattering. The purple arro w at the left (right) illustrates the current and magnetic moment arisi ng from clockwise (counter-clockwise) orbital motion around the NT. The green arrows i ndicate positive moments due to spin. b, Expected energy splitting for a defect-f ree NT in a magnetic field B|| parallel to the NT axis in the absence of spin-orbit coupling: At T0||=B all four states are de generate. With increasing B|| each state shifts according to its orbital and spin magne tic moments, as indicated by purple and green arrows respectively. c, Device schematic. A single NT is conn ected to source and drain contacts, separated by 500 nm, and gated from below by tw o gate electrodes. The two gate voltages ( Vgl, Vgr) are used to create a quantum dot localized above the right or left gate electrodes. The energy band diagram is shown for the first case. d, Scanning electron microgr aph of the device. e, The measured linear conductance, sddVdIG /= , as function of gate voltage, gV, for a dot localized above the right gate ( B||=6 T, T=30 mK). The number of electr ons or holes in the dot is indicated. The conductance of the to p two peaks is scaled by 1/10. 9 Figure 2: Excited-state spectroscopy of a single electron in a NT dot . a, Differential conductance, sddVdIG /= , measured as function of gate voltage, gV, and source- drain bias, sdV, at B||=300 mT, displaying transitions from ze ro to one electron in the dot. b, A line cut at Vsd=−1.9 mV reveals four energy levels α, β, γ and δ as well as another peak w corresponding to the edge of th e one-electron Coulomb diamond. c, sddVdIG /= as a function of gV and ||B at a constant bias Vsd=−2 mV. The resonances α, β, γ , δ and w are indicated. The energy scale on the right is determined by scaling ∆gV with the lever arm 57.0=α extracted from the slopes in a. Inset: Orbital and spin magnetic mome nts assigned to the observed states. d, Extracted energy splitting between the states α and β as a function of ||B (dots). The linear fit (red line) gives a Zeeman splitting with 1 .0 14.2 ± =g , and a zero-field splitting of 02.0 37.0 ± =∆SO meV. (s.d. error bars) e, Magnified view of panel c showing the zero-field splitting due to SO interaction ∆SO() as well as finite-field anti-crossing due to K-K’ mixing ()'KK∆ . Dashed lines show the calculated spectrum using eV65' µ= ∆KK . 10 Figure 3: The many-electron ground states and th eir explanation by spin-orbit interaction. a, sddVdIG /= , measured as a function of gate voltage, gV, and magnetic field, ||B, showing Coulomb blockade peaks (carrier addition spectra) for the first four electrons and the first two holes (data are offset in gV for clarity). b, Incorrect interpretati on of the addition spectrum shown in a using a model employing exchange interac tions between electrons. Dashed/solid lines represent addition of down/up spin moment s . The two-electron ground state at low fields, indicated at the left, is a spin triplet. c, Comparison of the measured two-electron addition energy from a with the one-electron excita tion spectrum from Fig. 2 e. d, Schematic explanation of the data in a using electronic states with spin-orbit coupling: The two-electron ground state at low fields, indicated on the left, is neither a spin-singlet nor a spin-triplet state. 11 Figure 4: Theoretical model for SO intera ction in nanotubes and the energy level spectroscopy of a single hole . a, Schematic of an electron with spin parallel to the NT axis revolving around the NT circumference. The carbon zp orbitals (red) are perpendicular to the surface. In the rest frame of the electron, the zp orbital rotates around the spin. b, Allowed electron a nd hole energies (red and blue circles) at 0||=B for a small-bandgap NT with SO inte raction. The states are derived by cutting the Dirac cones ( K and K’ ) with spin dependent quantization lines (dashed lines). The allowed k⊥-vectors differ for up and down electron spin moments. c, Calculated energy levels for an electron (red lines) and a hole (blue lines) as a function of ||B. The four distinct slopes arise from the orbital and spin Zeeman shifts. d, sddVdIG /= as a function of gV and ||B at a constant bias Vsd=−2 mV. The resonances labelled α, β, γ, δ and w arise from tunnelling of holes onto the dot and therefore the energy scale points opposite to gV. The ground state ( α) and first excited state ( β) cross at B||≈1.5 T. e, Extracted energy splitting between the states α and β as a function of ||B (dots). The linear fit (blue lin e) gives a Zeeman splitting with 1.0 14.2 ± =g , and a zero-field splitting of 01 .021.0± =∆SO meV. (s.d. error bars) f, Magnified view of the level crossings in d and a model calculation using eVm1.0'= ∆KK (dashed lines). 12 SUPPLEMENTARY INFORMATION Identification of the first electron and the first hole In this paper we measure the energy spectrum of a single charge carrier, an electron or a hole, in a nanotube (NT) quantum dot (QD). Having a singl e carrier is central to our experiment as it allows us to avoid electronic interactions between carriers a nd to unambiguously identify the effects of spin-orbit interactions. Here we explain in more detail how we determine that there is a single carrier in the dot. We start by identifying the transition from elect rons to holes in the addition spectrum with the Coulomb valley labeled “0” in figure 1e, based on two observati ons: First, this valley is significantly larger than all other valleys, reflecting the added contribution of the bandgap to the addition energy (quantitativ e analysis below). Second, electrons and holes can be distinguished by their response to magnetic fiel d (Fig. 3a). At large fields, such that the orbital coupling dominates over the level spacing in the dot, electrons and holes rotate in opposite directions around the NT circumference. This leads to oppos ite signs of ||/dB dVg for electrons and holes (figure 3a, B||>200 mT). Note that this also means th at the energy gap decreases with increasing magnetic field. To confirm that the first Coulomb valleys on the electron and hole sides correspond to the first electron and first hole in the dot it is enough to show that we observe all the charge states in the transport measuremen ts and that there are no non-conducting charge states within the 0 th Coulomb valley. By applying a high ma gnetic field we can reduce the size of the 0th Coulomb valley such that it doe sn’t allow for even a single additional charge state in the gap, thus confirming that the first Coulomb peak s in the electron and hole sides correspond to the addition of the first electron and first hole to the QD. To analyze this quantitatively, we determine al l the parameters of the QD directly from non- linear transport data, similar to those in fig. 2a. Specifically, the char ging energies and level spacings between particle-in-a- box longitudinal modes for the fi rst electron and first hole are meV19=eU , meV 25=hU and meV 8=∆e , ∆h=11 meV (see more details below). We estimate the band gap at 6 Tesla by subtractin g the average charging energy and average level spacing for electrons and holes from the 0th Coulomb valley (55 meV), and obtain meV2 24± =gapE at 6 Tesla. At the highest field in our measurements (9 Tesla) the size of the energy gap is smaller than the charging energies of either the electron or the hole dot, excluding the possibility of hidden char ge states inside the 0th Coulomb valley. Effects of higher particle-i n-a-box longitudinal modes The one-electron and one-hole excitation spectra presented in this paper correspond to the lowest quantized longitudinal mode. Quantized st ates of other longitudinal modes do not appear in the data presented in Figure 2 and Figure 4 becau se of their higher energy. We verified this by measuring excitation spectra at source drain voltage s larger than in Figure 2a, and identifying longitudinal modes by their dependence on the le ngth of the quantum dot. The level spacing extracted for a dot that is exte nded over both right and left gate electrodes is ~4 meV, and it increases continuously to >8 meV as the dot beco mes localized either above the right or above the left gate electrode (using a ppropriate gate voltages). The la tter corresponds to a confinement length smaller then 200 nm and is consistent with the lithographic dimensi ons given in Figure 1c. Therefore, higher longitudinal m odes were ignored in the discu ssion of the one-electron and one- hole quantum dot. 13 The situation is different for the top two Coulomb peaks presented in Figure 3a, which involve three and four electrons in the quantum dot. Because of the increased size of the quantum dot at those charge states, the level spacing is re duced and higher modes become occupied already at ~200 mT. At B||>200 mT it is favorable for all electrons to orbit in counterclockwise direction, ther eby aligning their orbital magne tic moments parallel to the external magnetic field. This explains why the 3e and 4e addition spectra shown in Fig. 3a deviate for B||>200 mT from the 3rd and 4th excitations of the one-electron QD. Comparison between the QD above the left and right gate electrodes The ability to localize the QD on two physically different segments of the same nanotube (Figure S1) helps us determine whether the exci tations observed at low energies depend on local properties in the nanotube such as localized disord er or specific properties of the source or drain electrodes. Figure S1c shows th e non-linear conductance measured for a QD localized above the right gate electrode. The measurement is at finite magnetic field ( B||=300 mT) allowing to resolve the four quantum states in the dot. The individual states are visible at negative bias and are absent at positive bias, indicating that the coupling of the QD to the source is much weaker than to the drain, as expected from its location. Accordingly, the capacitance between the QD and the source is smaller than the cap acitance between the QD and the drain ( Cs = 1/1.8 Cd, extracted from the slopes of the resonances in pane l c). The energies of the quantum states in the dot extracted from this measurement are 0, 0. 40, 0.98 and 1.33 meV (±5%). Figure S1d shows a similar measurement for a QD localized above the left gate electrode. As expected, here the tunnel coupling to the drain is weaker than that to the source, and hence the quantum states are probed by tunneling-in from the drain electrode (i.e. resonances appear at Vsd>0). The capacitance ratio ( Cs = 2.2 Cd) is reversed compared to that in panel c. Most importantly, the excitations in panel c and d have identical energi es up to an experimental uncertainty of ±5%. Similar measurements at other magnetic fields have also shown that the QD excitation energies do not depend on whether it is loca lized above the right or left ga te electrode, demonstrating that the observed excitations are an in trinsic property of the nanotube. Spin-orbit coupling vs. K-K’ scattering The four-fold degeneracy in NTs can be broken by extrinsic sources such as disorder, or by the intrinsic spin-orbit coupling. Disorder breaks the orbital symmetry of NTs in a trivial way and leaves doubly-degenerate spin states as in any other confined system with low symmetry. Spin-orbit coupling, on the othe r hand, breaks the degeneracy by coupling the orbital and spin degrees of freedom in parallel or anti-parallel configurati ons. Figure S2 shows how the theoretical four-fold energy spectrum for a single electron (Fig. 1b) changes in the presence of disorder (Fig. S2a) or spin-orbit coupling (Fig. S2b). In both cases the spectrum is split at 0 ||=B into two Kramer doublets, but the nature of the ne w eigenstates is entirely different. In the case of disorder, the splitting result s from mixing wavefunctions whic h revolve in opposite directions around the NT circumference, and hence the new eigenstates lack a definite sense of rotation around the circumference. Orbital angular moment um ceases to be a good quantum number for these states as is apparent from the fact that they have no coupling to the field ( 0 /||=dBdE at zero field, ignoring the spin Zeeman coupling) . In the presence of sp in-orbit coupling, the components of angular momentum and spin para llel to the NT axis remain good quantum numbers. This is readily seen from the fact that the slopes in magnetic field, ||/dBdE , remain 14 finite even at zero magnetic field. SO interactions thus create non- trivial states in which the spin and orbital degrees of freedom are tied together. Figure S1: Independence of the one-electron excitation energies on the QD location. a, Schematic band diagram for a one-electron QD formed above th e right gate electrode. Here the longer barrier on the left side leads to a weaker tunnel coupling to the source electrode. b, Same for a QD formed above the left gate electrode ( c) Differential conductance, sddVdIG / = , measured as function of gate voltage, gV, and source-drain bias, sdV, at B||=300 mT for the transition from zero to one electron for a dot localized above the right gate electrode. The energies of the one-ele ctron excitations that appear at negative sdV are labeled. Also shown are the ratios of the capacitances between the QD and source (sC), drain (dC) and gate (gC) electrodes, extracted from th e slopes of the resonances. d, Same for a QD localized above the left gate electrode. Figure S2: Breaking of four-fold degeneracy : spin-orbit coupling vs. KK’ scattering. a, The calculated one-electron spectrum as a functi on of parallel magnetic field in the presence of disorder-induced K-K’ scattering and the abse nce of spin-orbit coupling. Dashed and solid lines correspond to spin moment down and up. b, Same, but with spin-o rbit coupling and without disorder-induced K-K’ scattering.

2008.07308v2.Terahertz_spin_dynamics_driven_by_an_optical_spin_orbit_torque.pdf

Terahertz spin dynamics driven by an optical spin-orbit torque Ritwik Mondal,1,Andreas Donges,1and Ulrich Nowak1 1Fachbereich Physik, Universität Konstanz, DE-78457 Konstanz, Germany Spin torques are at the heart of spin manipulations in spintronic devices. Here, we examine the existence of an optical spin-orbit torque, a relativistic spin torque originating from the spin-orbit coupling of an oscillating applied field with the spins. We compare the effect of the nonrelativistic Zeeman torque with the relativistic optical spin-orbit torque for ferromagnetic systems excited by a circularly polarised laser pulse. The latter torque depends on the helicity of the light and scales with the intensity, while being inversely proportional to the frequency. Our results show that the optical spin-orbit torque can provide a torque on the spins, which is quantitatively equivalent to the Zeeman torque. Moreover, temperature dependent calculations show that the effect of optical spin-orbit torque decreases with increasing temperature. However, the effect does not vanish in a ferromagnetic system, even above its Curie temperature. I. INTRODUCTION Interest in controlling spins by means of circularly po- larised pulses has grown immensely due to its potential ap- plications in spin-based memory technologies [1–3]. Apart from the heat-assisted spin manipulations, the spins can alsobe controlled using the inverse Faraday effect (IFE) [4–6], the magnetic field of the terahertz pulses [7–10] and an optical spin-orbit torque (OSOT) that does not impart angular momentum into the spin system [11, 12]. To be able to explain such effects theoretically, one has to simu- late spin dynamics including several nonrelativistic and rel- ativistic effects that might appear at ultrashort timescales [8–12]. The Landau-Lifshitz-Gilbert (LLG) equation of motion, consisting of precession of a spin moment around an effec- tive field and a transverse relaxation, has been used ex- tensively in the past to simulate such spin dynamics [13– 15]. However, for a spin system excited by ultrashort laser pulses, a stochastic LLG equation of motion with atomistic resolutionisrequiredduetothestrongthermalfluctuations in order to study the dynamical processes [16–20]. In these equations, a stochastic field is added to the effective field, in order to quantify the thermal fluctuations in the spin system. Nonetheless, at ultrashort timescales, the exact form of the LLG equation of motion has to be questioned as several other relativistic phenomena can occur. There- fore, we seek for an equation of motion that can capture all the possible interactions during ultrashort laser excitations of a spin system. In a previous work, starting from the relativistic Dirac Hamiltonian yet including the magnetic exchange interac- tions, a rigorous derivation of the LLG equation of motion has been provided [21]. To treat the action of a laser pulse and the corresponding interactions, the Dirac-Kohn-Sham equation with external magnetic vector potential was con- sidered [22–24]. To this end, a semirelativistic expansion of the Dirac-Kohn-Sham Hamiltonian that includes several nonrelativistic and relativistic spin-laser coupling terms was derived [25]. Having these coupling terms, an extended equation of motion that includes not only the spin preces- sion and damping, but also other relativistic torque terms ritwik.mondal@uni-konstanz.dewas obtained [26]. One of these torque terms is the field- derivative torque which appears due to the time-dependent field excitation, e.g., in case of THz pulse excitation [27]. Another new torque term is the OSOT, which stems from the spin-orbit interaction of the applied field with the elec- tron spins i.e., it imparts spin-angular momentum of the applied field to the spins. The new equation of motion for the reduced magnetization vector mi(t)including OSOT cast in the form of an LLG equation is @mi @t= mi Be i+BOSOT
+mi D
@mi @t : (1) We define the gyromagnetic ratio as andD, representing the damping parameter, is in general a tensor. For sim- plification, we consider only a scalar damping parameter  that can be expressed as =1 3Tr(D)[21]. The effective field is represented as Be i, which is the derivative of the total magnetic energy without the relativistic light-spin in- teraction with respect to mias will be specified later on in detail. The additional field BOSOTdescribes the opto- magnetic field induced by the laser pulse due to the OSOT phenomenon. Inthisarticle, weinvestigatetheeffectoftheOSOTterm within atomistic spin dynamics simulations. First, we re- visit the derivation of the OSOT from a general spin-orbit coupling Hamiltonian that has been derived within a rel- ativistic formalism [21]. We find that the OSOT depends on the helicity, frequency and intensity of the light pulse. If the intensity is high and frequency is low, we expect the OSOT terms to show the most significant effects. We sim- ulate the spin dynamics for a spin model, representative for bcc Fe, with the OSOT and find that the OSOT can pro- vide significant contributions at the THz regime. Studying the temperature dependence, we find that the OSOT ef- fects is robust against thermal fluctuations, i.e., we observe no significant reduction of the strength of the OSOT even up to the critical temperature. II. OPTICAL SPIN-ORBIT TORQUE The generalized spin-orbit coupling (SOC) Hamiltonian, as derived within the fully relativistic Dirac framework, canarXiv:2008.07308v2 [cond-mat.mes-hall] 15 Feb 20212 Figure 1. (Color Online) ZT and OSOT for a single spin (big red arrow) excited by an elliptically polarised laser pulse. The fields and torques are represented by yellow and blue arrows, respectively. Due to the presence of elliptical polarisation, the ZF and ZT are drawn as blurry. be written as [21, 23, 26] HSOC=e~ 8m2c2
[Etot(peA)(peA)Etot]; (2) where the electric fields are represented as Etot=Eint+ EextandEext=@A @trwith(A;)asmagneticvector and scalar potentials. The physical constants have their usual meanings and denotes the electron’s spin through Pauli spin matrices and pis the electron momentum. Note that the SOC can occur in several ways, as de- scribed in the following: (i) the angular momentum of an electron couples to the spin of the electron — this can be expressed as
(Eintp), (ii) the first-order magnetic vec- tor potential of the electromagnetic (EM) field couples to the spins — this can be expressed as 
(EintA), (iii) the spin angular momentum of the EM field couples to the spin — this can be expressed as 
(EextA). In a spherically symmetric potential, the first type SOC can be written as traditional l
scoupling, where lands represent the orbital and spin angular momentum respec- tively. It provides explanations to several relativistic effects in magnetism e.g., magnetic Gilbert damping [21, 28, 29] and many others [30, 31]. Such a SOC exists even without an external field. In contrary, the latter two types of SOC depend explicitly on the external field and can be written as H0 SOC=e2~ 4m2c2
[(Eint+Eext)A]:(3) The total internal field depends on the intrinsic field E0 int that even exist without the applied field and the applied field itself. Within the linear response theory we write Eint=E0 int+Eext, wheredefines the coupling strength of the applied EM field relative to the intrinsic one. We thus consider the optical spin-orbit coupling as HOSOC =e2~(1 +) 4m2c2
(EextA): (4) Using the definition of Zeeman coupling of s
BOSOT, the optical spin-orbit couplingcan be shown to induce a field (see Appendix A for de- tails) BOSOT =e2~(1 +) 4m2sB2 0 !sin^ex; (5) whereB0(=E0=c),and!are the envelope of the oscillat- ingZeemanfield, helicityandangularfrequencyoftheellip- tically polarised light, respectively. sdefines the magnetic moment. Note, that due to the OSOT, the induced field points along the direction of the energy flux of the prop- agating wave. Additionally, note that the field BOSOTis largest for circularly polarised light ( ==2). However, if the coupling strength is not the same for right and left circularly polarised light, their combined effect could lead to nonzeroBOSOTfor linearly polarised light. The param- eterdepends on the electron density and absorption of the light that can be different for right and left circularly polarised light, leading to a magnetic circular dichroism (MCD). In the following, we simulate the Zeeman effect from the electromagnetic THz field and the optical spin- orbit coupling effects simultaneously, and make a compar- ison between these two effects in a ferromagnetic system. III. ATOMISTIC SPIN SIMULATIONS Inordertocomputethespindynamics, itisconvinientto transformtheimplicitformofourequationofmotiontothe explicit Landau-Lifshitz (LL) form. For a scalar damping parameters, , the LLG equation (1) can be recast as @mi(t) @t= (1 +2)mi Be i+BOSOT
 (1 +2)mi mi Be i+BOSOT
 :(6) The effective field, Be i, is the derivative of total en- ergy with respect to the magnetic moment, Be i= 1 is@H=@mi. The LLG equation, Eq. (6), hereby consists of the so called field-like (see also Fig. 1) and the weaker damping-like torque which is proportional to the Gilbert damping coefficient = 0:01and describes the coupling of the spins to a heat bath. In the following we consider a spin model for bcc Fe. The total Hamiltonian of the system (without the relativistic spin-light coupling term) Hcan be expressed as H=X i<jJijmi
mjX idz(mz i)2 sB(t)
X imi: (7) The first term describes the traditional Heisenberg ex- change energy, considering the exchange constants Jijup to the third nearest neighbour. The second term rep- resents the uniaxial anisotropy with energy constant dz and the last term is the Zeeman coupling with a time- dependent field B(t) =Rh B0exp t2 22i!ti . The el- lipticity of the applied pulse is taken into account through B0=B0p 2 ^y+ ei^z
, which considers the major and mi- nor axes of the ellipse and defines the pulse duration. In3 the following we use the abbreviations Zeeman field (ZF), Zeeman torque (ZT), and optical spin-orbit field (OSOF). A striking difference between ZT and OSOT is that the ZT does not depend on the frequency of the pulse and is proportional to the amplitude of the applied field pulse. However, the OSOT depends inversely on the frequency and is proportional to the square of the field amplitude, i.e. the intensity. To quantify the OSOT effects, one can estimateitsamplitudebycomputingthecharacteristicfield constant of the OSOF ~B(!;) =4m2 es! e2~(1 +): (8) Considering a central frequency of f= 1 THz, we find that ~B(2f;0) = 157 T . Hence an electromagnetic field ampli- tudeofabout 10 TcouldinduceaOSOFofaround1T,even for the low limit of the coupling strength, !0. Needless to mention that the OSOT effects might exceed the ZT if  is higher. However, the results shown here were computed for the limit !0. Such ultra-intense THz fields are cur- rently only achievable with linear polarisation, with recent experiments pointing towards achieving circular polarized laser pulses in the terahertz frequency regime [32, 33]. IV. APPLICATION TO FERROMAGNETS In our following simulations, we consider bcc Fe as a fer- romagnetic sample. For the zero-temperature simulations it is sufficient to solve the LLG equation for a single spin, due to the homogeneity of the THz pulse excitation. The choice of exchange parameters is only relevant at elevated temperatures. We shine an electromagnetic pulse having 1 pspulse duration along the ^x-direction with yandzfield components as shown in the Fig. 1. We will mainly discuss the limitdz0, since the uniaxial anisotropy does not have a significant impact on the ultrafast spin dynamics (see Appendix B for details). A. Zero temperature simulations In particular we compare the effect of ZT and OSOT at zero temperature. As the optical pulse is applied along the ^x-direction (k=jkj^x), we calculate the change in mag- netization for yandzcomponents in Fig. 2(a). For the case of only ZT the induced OSOT remains obviously zero. For a maximum of 10 Tapplied ZF, the change in magne- tization is about 50% for the y-component and 5% for the z-component. The reason for the triggered magnetisation dynamics is the following: the ZF from the optical pulse has two components ByandBz. TheBycomponent exerts a torque along^xon the equilibrium spin direction:
m/m0^zBy^y=m0By^x; (9) withequilibriummagnetisation m0. Ontheotherhand, the Bzcomponent of the EM field does not exert any torque on the Fe spins at first as the spins are initially aligned along ^z. However, as soon as the above mentioned torque induced some magnetisation
mx, theBzcomponent ofthe EM pulse, one quarter period later, exerts a torque along ^ydirection:
m/
mx^xBz^z=m0ByBz^y:(10) Therefore, there exists a superposition between two torques for the ZF. Note that the change in
mymagnetization is antisymmetric in the helicity of light, which suggests that the effect is similar to IFE [34]. Further note that the ab initio calculations of the IFE were calculated using a nonrelativistic theory [35, 36]. However, present theory is based on a relativistic interaction Hamiltonian. According to Eq. (5), the induced field diverges in the limit !!0, which can be explained by the previous theories of IFE [37– 39], and is in accordance with the ab initio calculations of IFE [35, 36, 40]. We would also like to mention that these ab initio calculations showed the IFE being asymmetric in helicity for ferromagnets which includes the absorption of light [36]. The OSOF, on the other hand, has only one component Bxalong ^xdirection, seeFig.2(b). Thus, itexertsatorque along ^y-direction
m/m0^zBx^x/m0B2 0^y: (11) and changes the magnetization by about 50 %as shown in Fig. 2(b). According to our theory in Eq. (5), the right and left circular polarization will induce an OSOF along the+^xand^xdirections respectively. Therefore, theright circular pulse exerts a torque along ^y(viz. ^z^x=^y) and similarly, the left circular pulse exerts a torque along ^y. The change in magnetization is also antisymmetric in the helicity of the light pulse, similar to ZT effects in
my. Note that we have assumed that the material dependent parameteris zero, which dictates that the antisymmetric behavior is not universal. In fact, the parameter could be different for right and left circular pulse, meaning that the antisymmetry of the plots above would be broken [36]. The effective contributions of the combined ZT and OSOT have been computed in Fig. 2(c). We note that the magnetization change in myis opposite in helicity for ZT and OSOT (e.g., see fifth row plots in Figs. 2(a) and 2(b)). Therefore, the final effective contribution becomes only about 5% change in
my. For thez-component of magnetization, the effective
mzis negligibly small, even though individual changes are about 5% due to ZT and OSOT. To quantitatively understand the ZT and OSOT effects, we compute the field dependent contributions in terms of the maximum change of each magnetization compo- nents
mias a function of the applied field B0for left circular polarised light, shown in Fig. 3. From a sim- ple scaling argument we would expect the spin excitation
m?=
mx^x+
my^yscaling with the magnitude of BEMandBOSOF, respectively. For the EM field, this is simply the amplitude BEM/B0, whereas for the OSOF, that isBOSOF/B2 0according to Eq. (5). As already described above, Eq. (9), the spin excitation
mxfor the EM field is induced by the field-like torque from itsBycomponent, thus, Fig. 3 (a) shows a linear scal- ing with the EM field amplitude B0. The
mycomponent in Fig. 3 (b), however, is induced in a second order process,4 0.5 0.00.5BOSOF,x [T](a) Effect of only ZT Left Right(b) Effect of only OSOT Left Right(c) Effect of ZT + OSOT Left Right 5 05BZF,y [T] 5 05BZF,z [T] 0.2 0.00.2mx [s] 0.2 0.00.2my [s] 4 2 0 2 4 6 Time [ps]0.981.00mz [s] 4 2 0 2 4 6 Time [ps]4 2 0 2 4 6 Time [ps] Figure 2. (Color Online) The dynamical effects of ZT and OSOT for Fe at an maximum applied field of 10 T. The calculations include (a) onlyZT, (b) onlyOSOT and (c) both the ZT and OSOT. The first row shows the applied ZF which has yandz components, the second row represents the induced OSOF which has only an x-component. The other three rows show the change in magnetization along x,yandzcomponents respectively. described in Eq. (10) and thus scales quadratically with the EM field amplitude B0, i.e.,
my/B2 0. A deviation of this scaling law is observed at lower field amplitudes of B0.100 mT, where the damping-like torque of the EM field with
my/B0is taking over. This effect can also be seen by comparing the torque amplitudes, which differ by a factor of
my=
mxin this regime. The
mz component, i.e., the deviation of the equilibrium compo- nent then follows from conservation of magnetization am- plitude:
mz=m0p m0
m2 ?. At low excitation the dominant contribution here is the
mxcomponent and one can simplify
mz
m2 x=2m0/B2 0. In contrast to the EM field, there is only one field-like torque acting along the +^ydirection due to the OSOF in ^xdirection. This torque can be seen in Fig. 3 (b), and shows the quadratic scaling
my/Bx/B2 0implied by Eq. (11). The
mxexcitation in Fig. 3 (a) on the other hand is due to the damping-like OSOT and thus following the same power law, but a factor of smaller compared to the field-like excitation in Fig. 3 (b). Finally, the
mzexcitation of the OSOT follows again from conservation of magnetization amplitude, leading to
mz
m2 y=2m0 and implying
mz/B4 0as shown in Fig. 3 (c). These scaling observations are once more summarized in Tab. I where we display the scaling exponents obtained by fitting our simulation data. An interesting observation here is that although the characteristic field of the OSOT is on the order of 100 Tin this frequency regime, due to the different torque symmetry compared to the ZT, the OSOT is by no means negligible. For the
myexcitation we find that the strength of ZT and OSOT are comparable, though opposite in sign, at fields on the order of 102mT—a value that is in much closer reach experimentally. Altogehter, the OSOT effects can thus provide an equivalent torque compared to the Zeeman effect. Therefore, the OSOT ef- fects cannot be neglected when a circularly polarised light acts on a magnetic system even at the weak coupling limit !0. Up to now we did not take into account the role of a fi- nite anisotropy. In order to investigate this, we performed5 102 101 100101 B0[T]106 105 104 103 102 101 mi, max [s] mx ZT OSOT 102 101 100101 B0[T] damping torque2ndorder field torque my ZT OSOT 102 101 100101 B0[T] mz ZT OSOT Δmy<0Δmy>0(a) Effect ofonlyZT (b) Effect ofonlyOSOT (c) Effect of ZT + OSOT Figure 3. (Color Online) Maximum of the magnetization change as a function of applied Zeeman field for the application of circular polarised THz pulses (absolute values). Lines are fits to the data according to a single (double) power law, see Tab. I for coefficients. Table I. Fit parameters to a scaling function fN(B0) =PN n=1ay;n(B0=T)i;nwhereai;nis given in units of s. For the ZF fit of
mya double power law N= 2has been used, whereas for the remaining fits a monomial N= 1was sufficient. ZF OSOF n a i;n i;n ai;n i;n
mx1 1:981021:00 9:901052:00
my1 8:801040:96 1:981032:00 2 1:741032:09 - -
mz1 2:031042:03 1:961064:00 the same calculation depicted in Fig. 2 with the anisotropy dz= 7:66µeVfor Fe included. We found that the main ef- fect of the anisotropy field is the precession of the induced magnetization
m?around the z-axis over time (see Ap- pendix B for details). On the other hand, no direct effect of the anisotropy can be observed on the ultrafast time scales, i.e., on the time scale of the pulse duration. B. Finite temperature simulations For the finite temperature simulations, the exchange pa- rameters of Pajda et al.[41] are used, where the first two nearest neighbors are strongly ferromagnetically, however, the third nearest neighbor is weakly antiferromagnetically coupled. For these simulations, we also use the uniaxial anisotropy of dz= 7:66µeValong thez-axis, in order to align the magnetization. Calculating the temperature de- pendence of the OSOT for ferromagnetic Fe, we use a sim- ulation grid of 483spins and add a stochastic field to the effective field in Eq. (6), in order to treat the thermal fluc- tuations [17]. The calculated Curie temperature of this sys- tem isTC= 1368 K and thus slightly higher than the true value of Fe. However, since the Curie temperature is the only temperature scale in our simulations, we can basically treat it as a free scaling parameter. We compare in Fig. 4 the OSOT effects at T= 0and T= 0:73TC(0K and 1000K). We find in Fig. 5 that though the OSOT effect in
miappears to decrease withincreasing temperature, this reduction is only related to the thermal reduction of magnetic order meq(T)at finite temperature. In other words the rotation angle of the nor- malized magnetization is not sensitive to the temperature. 0.5 0.00.5Bx OSOF (T)Left Right 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Time (ps)0.2 0.1 0.00.10.2my (s) T=0 T=0.73×TC Figure 4. (Color Online) OSOT effects at finite temperature. Top panel: the OSOT-induced field. Bottom panel: the cor- responding spin dynamics for the change of myatT= 0and T= 0:73TC(0K and 1000K). To illustrate this further, we have systematically calcu- lated the temperature dependence of the OSOT by cal- culating the difference between maximum spin excitation for right and left circular pulses in Fig. 5. For each tem- perature, we performed ten simulations for each circular pulses, and took the average to determine max [
mR;y] max [
mL;y]as a function of temperature, which ensures thatthethermalfluctuationsareminimised. Farawayfrom TCthe spin excitation amplitude is proportional to the equilibrium magnetization curve. Only in the close vicinity of the critical temperature, we find an increase of the net spin excitation, relative to the equilibrium magnetisation. This is simply due to the large amplitude of both ZF and OSOF which induce a transient magnetic order.6 Figure 5. (Color Online) Temperature dependence of OSOT, by the difference between the maximum changes of mydue to right and left polarised light pulses at maximum Zeeman field amplitude of 10 T. For comparison, the right axis shows the equilibrium magnetization. Therefore, OSOT effects should be observed for bcc Fe even at elevated temperatures i.e., at realistic conditions for ultra-intense spin excitations. V. CONCLUSIONS To conclude, we incorporated a new torque into the LLG equation that should appear in ultrafast spin dynamics, namely OSOT, and investigated this effect via computer simulations of an atomistic spin model, representative for bcc Fe. The OSOT originates from the spin-orbit coupling of the electron spins to an external EM field. The strength of this OSOT, unlike the first-order ZT, depends on the intensity of the ultrafast light pulse, as well as on the fre- quency. Inaddition, theOSOTdependsontheellipticityof thepulseanditprovidesmaximumtorqueforcircularlypo- larised light pulses. Throughout the simulations presented here, we considered the weak coupling limit !0. How- ever, the coupling depends on the electronic configurations of the system and could potentially further increase the strength of the OSOT. Although the OSOT is a higher or- der contribution in the external field, we found that the ZT andtheOSOTprovidequantitativelyequivalenttorqueson the spins for circularly polarised laser pulses in the mag- netization component perpendicular to the k-vector and equilibrium magnetisation m0. The effect of the OSOT resembles an already known effect, namely the IFE and it can be considered as a relativistic contributions to the IFE. The temperature dependence study of OSOT shows that the OSOT effect is present at elevated temperatures, even up to the Curie temperature. ACKNOWLEDGMENTS We thank László Szunyogh for valuable discussions and acknowledge financial support from the Alexander vonHumboldt-Stiftung, Zukunftskolleg at Universität Kon- stanz via grant No. P82963319 and the Deutsche Forschungsgemeinschaft via NO 290/5-1. Appendix A: Derivation of optical spin-orbit torque Following Eq. (4), the induced field can be written as BOSOT =e2~(1 +) 4m2c2s(EextA):(A1) Weusethetimedependentfieldas Eext=R E0ei(k
r!t)
and the amplitude as the elliptically polarised light E0= E0p 2 ^y+ ei^z
, withas the ellipticity of the light. There- fore,theelectricfieldcanbewrittenas(whenonlythetime- dependent part is taken) Eext=@A @t)A=R Eextdt that can be calculated as follows A=R E0Z ei!tdt =R iE0ei!t ! :(A2) Therefore, the induced field can be taken from Eq. (A1) as BOSOT =e2~(1 +) 4m2c2s!R[i(E0E? 0)] =e2~(1 +) 4m2c2s!E2 0sin^x =e2~(1 +) 4m2s!B2 0sin^x (A3) In the last step of the calculation, we used the relation E0=cB0. In our simulations, we apply time-dependent Zeeman fields along yandz-directions and the correspond- ing induced optical spin-orbit field acts along x-direction. Appendix B: Effect of anisotropy Here, we compute the influence of uniaxial magnetic anisotropy on the spin dynamics induced by the ZT and OSOT. Fig. 6 shows the magnetization dynamics taking into account the uniaxial anisotropy for bcc Fe. The main effect of anisotropy can be noticed by comparing with the zero-anisotropy simulations in Fig. 2. A small increase in
mxis observed in the case of only ZF or OSOF after the pulse has passed. This is due to the slow precession of the induced
mycomponentwhichstartstoprecessaroundthe anisotropy field along ^zaxis. In case of the superposition of ZF and OSOF the excitation of
myis mostly compen- sated and therefore no
mxemerges either. Additionally, we mention that the anisotropy energies do not affect the
myand
mzon the ultrafast time scale and the net ex- citation remains the same irrespective of the anisotropy — at least for the typically weak magnetic anisotropies of the 3d ferromagnets.7 0.5 0.00.5BOSOF,x [T](a) Effect of only ZT Left Right(b) Effect of only OSOT Left Right(c) Effect of ZT + OSOT Left Right 5 05BZF,y [T] 5 05BZF,z [T] 0.2 0.00.2mx [s] 0.2 0.00.2my [s] 4 2 0 2 4 6 Time [ps]0.981.00mz [s] 4 2 0 2 4 6 Time [ps]4 2 0 2 4 6 Time [ps] Figure 6. (Color Online) The dynamical effects of ZT and OSOT have been calculated for Fe including the uniaxial anisotropy at an applied field of 10 T. The calculations have been performed accounting (a) onlyZeeman effect, (b) onlyOSOT effect and (c) both the Zeeman and OSOT effects. In all the plots, the first row represents the induced spin-orbit field which has only x-component, and the rows two and three show the applied ZF which has yandzcomponents. The other three rows show the change in magnetization alongx,yandzcomponents respectively. The orange and blue colors represent the action of right and left circular pulses. [1] A. V. Kimel, A. Kirilyuk, and Th. Rasing, Laser Photon. Rev.1, 275 (2007). [2] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett. 99, 047601 (2007). [3] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M. Balbashov, and Th. Rasing, Nature 435, 655 (2005). [4] R. John, M. Berritta, D. Hinzke, C. Müller, T. San- tos, H. Ulrichs, P. Nieves, J. Walowski, R. Mondal, O. Chubykalo-Fesenko, J. McCord, P. M. Oppeneer, U. Nowak, and M. Münzenberg, Sci. Rep. 7, 4114 (2017). [5] S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V. Uh- líř, L. Pang, M. Hehn, S. Alebrand, M. Cinchetti, G. Mali- nowski, Y. Fainman, M. Aeschlimann, and E. E. Fullerton, Nat. Mater. 13, 286 (2014). [6] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad, Y. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski, K. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fuller- ton, Science 345, 1403 (2014). [7] M. Shalaby, A. Donges, K. Carva, R. Allenspach, P. M. Oppeneer, U. Nowak, and C. P. Hauri, Phys. Rev. B 98, 014405 (2018). [8] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. Mährlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, andR. Huber, Nat. Photon. 5, 31 (2011). [9] T. Kampfrath, K. Tanaka, and K. A. Nelson, Nat. Photon. 7, 680 (2013). [10] S. Wienholdt, D. Hinzke, and U. Nowak, Phys. Rev. Lett. 108, 247207 (2012). [11] N. Tesarová, P. Nemec, E. Rozkotová, J. Zemen, T. Janda, D. Butkovicová, F. Trojánek, K. Olejnik, V. Novák, P. Maly, and T. Jungwirth, Nat. Photon. 7, 492 (2013). [12] G.-M. Choi, J. H. Oh, D.-K. Lee, S.-W. Lee, K. W. Kim, M. Lim, B.-C. Min, K.-J. Lee, and H.-W. Lee, Nat. Comm. 11, 1482 (2020). [13] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 101 (1935). [14] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [15] H. Kronmüller and M. Fähnle, Micromagnetism and the Microstructure of Ferromagnetic Solids (Cambridge Uni- versity Press, 2003). [16] N. Kazantseva, U. Nowak, R. W. Chantrell, J. Hohlfeld, and A. Rebei, Europhys. Lett. 81, 27004 (2007). [17] U. Nowak, “Handbook of Magnetism and Advanced Mag- netic Materials,” (2007). [18] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A. Ellis, and R. W. Chantrell, J. Phys.: Condens. Matter26, 103202 (2014).8 [19] S. Wienholdt, D. Hinzke, K. Carva, P. M. Oppeneer, and U. Nowak, Phys. Rev. B 88, 020406 (2013). [20] B. Frietsch, A. Donges, R. Carley, M. Teichmann, J. Bowlan, K. Döbrich, K. Carva, D. Legut, P. M. Op- peneer, U. Nowak, and M. Weinelt, 6(2020), 10.1126/sci- adv.abb1601. [21] R. Mondal, M. Berritta, and P. M. Oppeneer, Phys. Rev. B94, 144419 (2016). [22] R. Mondal, M. Berritta, K. Carva, and P. M. Oppeneer, Phys. Rev. B 91, 174415 (2015). [23] R. Mondal, M. Berritta, C. Paillard, S. Singh, B. Dkhil, P. M. Oppeneer, and L. Bellaiche, Phys. Rev. B 92, 100402(R) (2015). [24] A. Crépieux and P. Bruno, Phys. Rev. B 64, 094434 (2001). [25] R. Mondal, M. Berritta, and P. M. Oppeneer, J. Phys. : Condens. Matter 29, 194002 (2017). [26] R. Mondal, M. Berritta, and P. M. Oppeneer, Phys. Rev. B98, 214429 (2018). [27] R. Mondal, A. Donges, U. Ritzmann, P. M. Oppeneer, and U. Nowak, Phys. Rev. B 100, 060409(R) (2019). [28] K. Gilmore, Ph.D. thesis, Montana State University, 2007. [29] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett.99, 027204 (2007). [30] T. Moriya, Phys. Rev. 120, 91 (1960).[31] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). [32] M.Jia, Z.Wang, H.Li, X.Wang, W.Luo, S.Sun, Y.Zhang, Q. He, and L. Zhou, Light: Science & Applications 8, 16 (2019). [33] E. V. Jasper, T. T. Mai, M. T. Warren, R. K. Smith, D. M. Heligman, E. McCormick, Y. S. Ou, M. Sheffield, and R. Valdés Aguilar, Phys. Rev. Materials 4, 013803 (2020). [34] P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom, Phys. Rev. 143, 574 (1966). [35] M. Battiato, G. Barbalinardo, and P. M. Oppeneer, Phys. Rev. B89, 014413 (2014). [36] M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer, Phys. Rev. Lett. 117, 137203 (2016). [37] A. Qaiumzadeh, G. E. W. Bauer, and A. Brataas, Phys. Rev. B88, 064416 (2013). [38] D. Popova, A. Bringer, and S. Blügel, Phys. Rev. B 84, 214421 (2011). [39] R. Hertel, Journal of Magnetism and Magnetic Materials 303, L1 (2006). [40] F. Freimuth, S. Blügel, and Y. Mokrousov, Phys. Rev. B 94, 144432 (2016). [41] M. Pajda, J. Kudrnovský, I. Turek, V. Drchal, and P. Bruno, Phys. Rev. B 64, 174402 (2001).

1706.06068v1.Chiral_and_Topological_Orbital_Magnetism_of_Spin_Textures.pdf

Chiral and Topological Orbital Magnetism of Spin Textures Fabian R. Lux,Frank Freimuth, Stefan Bl ¨ugel, and Yuriy Mokrousov Peter Gr ¨unberg Institut and Institute for Advanced Simulation, Forschungszentrum J ¨ulich and JARA, 52425 J ¨ulich, Germany (Dated: June 20, 2017) Using a semiclassical Green’s function formalism, we discover the emergence of chiral and topological orbital magnetism in two-dimensional chiral spin textures by explicitly finding the corrections to the orbital magneti- zation, proportional to the powers of the gradients of the texture. We show that in the absence of spin-orbit coupling, the resulting orbital moment can be understood as the electronic response to the emergent magnetic field associated with the real-space Berry curvature. By referring to the Rashba model, we demonstrate that by tuning the parameters of surface systems the engineering of emergent orbital magnetism in spin textures can pave the way to novel concepts in orbitronics. Introduction. The importance of chiral magnetic structures such as domain walls and skyrmions is ever growing due to their role in the formulation of advanced concepts in spin- tronics, accompanied by frequent discoveries of novel ef- fects hinging on the finite chirality of these particle-like tex- tures [1]. In recent years significant advances have been made in the reliable detection of chiral textures and in their effi- cient manipulation by external perturbations [2]. For the case of skyrmions, the palette of emergent topological phenomena they give rise to [3] was shown to root predominantly in the “emergent” magnetic field, intrinsically generated by the non- trivial real-space distribution of the orientation of the magne- tization, ^n(x;y): Bz eff=~ 2e^n
@^n @x@^n @y : (1) The integrated flux of this field gives rise to a quantized in- teger topological charge of a skyrmion, Nsk, affecting its dy- namical properties and resulting e.g. in an enhanced robust- ness with respect to scattering and fluctuations [4]. Within an intuitively appealing scenario, the emergent mag- netic field in chiral systems couples directly to the orbital de- gree of freedom and provides an alternative mechanism for Hall effects, such as the topological Hall effect of skyrmions or the anomalous Hall effect in non-collinear magnets [5–7]. Besides these well-known effects, the orbital response to Bz eff gives rise to a phenomenon which was coined topological or- bital magnetization (TOM) [5], and which has not yet been explored to the same extent. Indeed, the emergence of or- bital magnetism that does not rely on spin-orbit interaction in strongly frustrated systems and small-size skyrmions has been shown from first-principles and tight-binding calculations [8– 11]. The novel chirality-driven “topological” channel for the orbital magnetism is a remarkable, attractive effect, since it provides an additional control on the handedness of the un- derlying texture, may exhibit enhanced robustness [11], and generally paves the way for non-trivial spin textures into the realm of orbitronics with the vision of addressing and oper- ating with orbital degrees of freedom of electrons rather than their spin [9]. However, at the current stage, the physics of the TOM is very poorly understood and the rigorous framework for accessing orbital magnetism in chiral spin textures, imper-ative for our ability to engineer and utilize the orbital degree of freedom of chiral systems for the purposes of orbitronics, has been missing so far. In this Letter, by refering to the Green’s function perturba- tion theory [12], we put the orbital magnetism in chiral sys- tems on firm quantum-mechanical ground presenting a rigor- ous theory for the emergence of orbital magnetism in non- collinear systems. By systematically tracing the orders of perturbation theory for chiral magnetic textures we distin- guish corrections to the out-of-plane orbital magnetization Mom=M(^n)ezof a locally ferromagnetic system, appear- ing in higher orders of the gradients of the magnetization: Mcom=M i(^n)(@in)ez (2) Mtom=M ij(^n)(@in)(@jn)ez; (3) where@i=@=@xi. Here, and in the following discussion, summation over repeated indices is implied, with greek in- dices;2fx;y;zgand latin indices i;j;k2fx;yg. In addition to the effect of TOM, appearing at the second order (Eq. (3) and Fig. 1(b)), we thereby propose a novel con- tribution to the orbital magnetization, which is linear in the chirality of the underlying texture (Eq. (2) and Fig. 1(a)), and which we thus call the chiral orbital magnetization (COM). By explicitly referring to the 2D Rashba model, we numer- ically evaluate the magnitude and real-space behavior of the TOM and COM, finding that by tuning the parameters of sur- face and interfacial systems the orbital magnetism of domain walls and chiral skyrmions can be engineered in a desired way. Our findings open new vistas for exploiting the orbital mag- netism in chiral magnetic systems, thereby launching the field of chiral orbitronics. The expansion in magnetic field gradients is naturally achieved within the phase-space formulation of quantum me- chanics, the Wigner representation [12, 13]. The key quantity in this approach is the retarded single-particle Green’s func- tionGR, implicitly given by the Hamiltonian Hvia the Dyson equation H(X;) +i0+
?GR(X;) =id; (4) whereX= (t;X)and= (;)are the four-vectors of position and canonical momentum, respectively. The lat- ter of the two, in terms of the elementary charge e > 0arXiv:1706.06068v1 [cond-mat.mtrl-sci] 19 Jun 20172 FIG. 1. In one-dimensional and two dimensional chiral topologi- cal spin textures, such as spin-spirals (a) or skyrmions (b), the non- trivial distribution of the magnetization ^ngives rise to (a) chiral or- bital magnetization (COM), linear in the gradients of ^n, Eq. (2), and to (b) topological orbital magnetization (TOM), which is second or- der in the gradients of ^n, Eq. (3). Both COM and TOM can exhibit a complex distribution in real space, as illustrated in (a) and (b) respec- tively by explicit calculations for systems discussed in the text (with parameters
xc= 0:9eV ,~R= 2:0eV˚A1,kBT= 0:1eV). and the electromagnetic vector potential A, is related to the zero-field momentum pby the relation (X;p) =p+ eA(X):The?-product, formally defined by the operator ?exp i~ 2 [ @x;! @]eF @! @ of left- and right-acting derivatives$ @, allows for an expansion of GR in powers of ~, gradients of ^nand external electromagnetic fields, captured in a covariant way by the field tensor F= @xA@xA[12]. In this work, we are after the orbital magnetization (OM) in z-direction. Given the grand canonical potential , the surface density of the orbital moment is given by [14, 15] M(x) =@Bh (x)i; (5) which requires an expansion of up to at least first order in the magnetic field B=Bezin the collinear case. In the limit ofT!0, the grand potential is asymptotically related to the Green’s function GRvia h i1 =Zdp (2~)2f()()trGR(x;p); (6) where=denotes the imaginary part, the integral measure is defined as d p=dd2p,f()represents the Fermi function f() = (e()+ 1)1,is the chemical potential and 1=kBT. In our approach, deviations from the collinear theory enter the formalism as gradients of ^nand can be traced systematically in GRand inh i, finally leading to Eq. (2) and (3). For details concerning the analytical and numerical strat- egy of devising this expansion in a diagrammatic way we refer to the supplement.While our approach is very general, for the purposes of il- lustration and feasible numerical estimates, we restrict our fur- ther analysis to the two-dimensional Rashba model H(X;) =2 2me+R()z+
xc
^n(X); (7) wheremeis the electron’s (effective) mass, denotes the vec- tor of Pauli matrices, Ris the Rashba spin-orbit coupling constant, and
xcis the strength of the local exchange field. This model has been proven to be extremely fruitful in unrav- elling various phenomena in surface magnetism [16]. Chiral Orbital Magnetization. When in Eq. (6) is ex- panded in orders of Band in orders of the exchange field gra- dients, the first correction to OM appears at the second order in these perturbations. It is the COM, Eq. (2), and it is lin- ear in the exchange field gradients. To get a first insight into this novel effect, we consider an instructive example of a spin- spiral solution with wave-vector qpropagating in x-direction according to the form ^n= (sin(qx);0;cos(qx))T, as depicted in Fig. 1(a). For this N ´eel-type texture, one finds that up to O(R)the localx-dependent orbital moment is given by Mcom cos=eq R 48sgn(
xc) 132
2xc (j
xcjjj);(8) where=qxis the angle between ^nand thez-axis, and is the Heaviside step-function. In the vicinity of R= 0, the magnitude of COM is thus proportional to the strength of spin-orbit interaction and vanishes in the limit of zero R. It is rewarding to understand this behavior in the language of gauge fields. To linear order in R, the Rashba Hamiltonian can be expressed as a perturbative correction to the canonical momentum !+eAR, withAR=m Rijzeij=e. For jRj  j
xcjthe spin polarization of the wavefunctions is only weakly altered away from ^nand we can use the SU(2) gauge field, defined by Uy(
^n)U=z, to rotate our Hamil- tonian towards the local axis specified by ^n: H!H0=(+eA(X))2 2me+
xcz; (9) where the potential Acomprises the mixing of two gauge fields:A=UyARU+Axc, with the additional contri- butionAxc=i~UyrU=e. The real-space Berry curva- ture corresponding to this vector potential can be recast as an effective, chirality- and spin-orbit-driven magnetic field Bz eff=sgn(
xc)h#j@xAy@yAxj#iaccompanying the “fer- romagnetic” system [17–19]: Bz eff spiral=m Rq esgn(
xc) cos (10) Thus, in the limit of jRjj
xcj, the emergence of chiral orbital magnetization can be understood as the coupling of a mixedSU(2)gauge field to the diamagnetic Landau-Peierls susceptibility "+# LP=e2=(12m e), i.e., one can show that Mcom=1 2"+# LPBz effsgn(
xc); (11)3 FIG. 2. Phase diagram of mtom, Eq. (3), evaluated at the core of a N´eel-type skyrmion ( m= 1,c= 0:9nm,w= 1:2nm) as a function of the parameters
xcandRof the Rashba Hamiltonian (7) with= 0. Inset: phase diagram of Mcom, Eq. (2), evaluated at the position of a spin-spiral ( q= 2:86nm1) with an out-of-plane magnetization as a function of
xcandR. In an intermediate regime of
xc.Rorbital magnetism is strongly enhanced. in the vicinity of the band extrema. The behavior of COM becomes complicated and deviates remarkably from that given by Eq. (8) as the Rashba param- eter increases. To demonstrate this, we numerically calculate the value ofMcomfor a spin-spiral with q=2:86nm1[20] at the position in real space with an out-of-plane magnetiza- tion, in a wide range of parameters
xcandRof the Rashba Hamiltonian (7) with = 0, presenting the results in the inset of Fig. 2. In this plot, we observe that while the gauge field picture is valid in the limit of
xc=R!1 , there exists a pronounced region in the ( R,
xc)-phase-space where COM exhibits a strong non-linear enhancement. In this region, the real-space behavior of Mcomcan be very non-trivial, deviat- ing strongly from the cos-like behavior of Eq. (8), see for ex- ample Fig. 1(a) for the distribution of COM along the spin- spiral for specific values of
xc= 0:9eV ,~R= 2:0eV˚A1, kBT= 0:1eV . Such behavior can be related to the strongly anisotropic properties of the Rashba model, as discussed in detail below. Topological Orbital Magnetization. The TOM appears as the correction to the OM which is second order in the gra- dients of the texture, Eq. (3), and while it vanishes for one- dimensional spin-textures discussed above, we show that it is finite for 2D textures such as skyrmions. Remarkably, in con- trast to COM, the TOM is non-vanishing even without spin- orbit interaction. To investigate this, we set Rto zero, reduc- ing the effective vector potential to A=Axcand with the emergent field turning into Bz eff, Eq. (1) [19]. The gradient expansion now reveals that Mtom=1 4"+# LPBz effsgn(
xc) 132
2xc ; (12)ifjj<j
xcj, and zero otherwise. In the vicinity of the band edges, where jjj
xcj, this again confirms the gauge- theoretical expectation, characterizing TOM as the electronic response to the emergent magnetic field (details on how the scalar spin chirality is entering this equation can be found in the supplement). Remarkably, the similarity between Eqs. (11) and (12) underlines the common origin of the COM and TOM in the “effective” magnetic field in the system, gen- erated by a combination of a gradient of ^nalongxwith spin- orbit interaction (in case of COM), and by a combination of the gradients of ^nalongxandy(in case of TOM). To explore the behavior of TOM in the presence of spin- orbit interaction, R6= 0, we numerically compute the value of TOM at the center of a N ´eel skyrmion of core size c= 0:9nm, with the domain wall width w= 1:2nm and the topo- logical charge Nsk=1(see Fig. 1 and the supplement de- scribing the exact modelling of the skyrmion shape as put for- ward in [21]), as function of
xcandR(at= 0). The corresponding phase diagram, presented in Fig. 2, displays two notable features. The first one is the relative stability of Eq. (12) against a perturbation by a spin-orbit field in the limit ofj
xcjjRj. The second one is the significant enhance- ment of TOM in the regime where jRj>j
xcj, similar to COM (albeit over a larger part of the parameter space). As exemplified in Fig. 1(b), the “local” TOM in this regime of en- hancement deviates strongly in its real-space distribution from the uniform behavior described by Eq. (12) and is therefore not well described by Bz eff. This effect, again in direct analogy to COM, can partially be attributed to the enhanced diamag- netic susceptibility of the Rashba model [22]. And although replacingLPin Eq. (12) with the exact orbital magnetic sus- ceptibility (as e.g. given by the Fukuyama’s result [23]) can account for the behavior of the TOM around ^n=ez, it fails to reproduce the strongly anisotropic feature near ^n?ez, prominent in Fig. 1(b). This anisotropy is a direct consequence of the well-known non-trivial anisotropic k-space topology of the 2D Rashba model [24], which we can analyze by evaluating Mtomas func- tion of the chemical potential . The results shown in Fig. 3(a) for^n=ezreveal the sensitivity of Mtomto theR-induced deformation of the purely parabolic free-electron bands sepa- rated by
xc. The magnitude of TOM is largest and it exhibits pronounced oscillations in a narrow energy interval around the band edges. When we turn ^ninto the in-plane direction, ak!kasymmetric R-driven band crossing occurs along k?^n, eventually pushing the peaks of Mtomthrough the chemical potential which explains the anisotropy of TOM. Be- sides a strong enhancement of TOM, the effective real-space magnetic field picture, as given by Eq. (12), fails to reproduce the strong anisotropy of Mtom, although an appropriate more general geometric theory, accounting for these features and formulated in terms of mixed Berry curvatures in real- and momentum-space, could exist [13]. At this point, we turn to the discussion of the total inte- grated values of the orbital moments in chiral spin textures. As concerning the total value of the COM-driven orbital moment4 FIG. 3. (a) TOM density (thick blue line) at two different positions (indicated by the red arrow) within the Nsk=1N´eel Skyrmion (c= 0:9nm,w= 1:2nm,
xc= 0:9eV ,R= 2:0eV˚A1) as a function of the chemical potential across the band structure n (thin black lines) of the Rashba Hamiltonian (7). The arrows mark the values of used in (b). (b) Integrated TOM (i.e., the topological orbital moment, Mtom) of the skyrmion with parameters from (a) as a function of Rfor different values of . in one-dimensional 360or 180chiral domain walls it al- ways vanishes identically by arguments of symmetry. In sharp contrast, the TOM-driven total orbital moment of skyrmions generally does not vanish. This can be first shown in the limit when the gauge-field approach is valid (i.e.,
xcR). In this case, as follows from Eq. (12), the integrated value of Mtomover the skyrmion, i.e., the topological orbital moment mtom, isquantized to a universal value of BNsk=12, at= 0, independent of the parameters of the electronic structure. In this limit the skyrmion of Nsk6= 0 thus behaves as an en- semble ofNskeffective particles which occupy a macroscopic atomic orbital with associated universal value of the orbital angular momentum of B=12. In the other limit of R>
xcthe magnitude of Mtom can be enhanced drastically with respect to this value. To show this, we calculate Mtomfor N ´eel-type skyrmions with Nsk=1at a fixed value of
xc= 0:9eV while varying R for three different values of , see Fig. 3(b). The presented data reveals an increase in mtomup to as much as 1 Bupon increasingR, which can be attributed to the enhanced val- ues ofMtomin the vicinity of the band extrema and its strong anisotropy in the considered part of the phase diagram, Fig. 2. Another remarkable consequence of the non-trivial behavior ofMtomin energy, Fig. 3(a), is the dependence of the sign of Mtomon the value of . A fundamental result of our analy- sis is an almost complete independence of the values of mtom, discussed above, with respect to the parameters which deter- mine the shape of the skyrmion, including its radius. Given the observed remarkable stability of mtom, we can truly callthis moment topologically-protected in the sense of its robust- ness with respect to diverse perturbations of the underlying spin texture which gives rise to it. On a fundamental level, COM and TOM arise as a con- sequence of the changes in the local electronic structure in response to an emergent field. This opens a way to experi- mentally access MtomandMcomby such techniques as off- axis electron holography [25] (sensitive to local distribution of magnetic moments), or scanning tunneling spectroscopy (sensitive to the local electronic structure) in terms of B-field induced changes in the dI=dU ord2I=dU2spectra [26]. Fur- ther, the emergence of COM and TOM can give a thrust to the field of electron vortex beam microscopy [27] where a beam of incident electrons intrinsically carries orbital angular momentum interacting with the magnetic system into the realm of chiral magnetic systems. For example, we speculate that at sufficient intensities, electron vortex beams can imprint skyrmionic textures possibly by partially transforming its or- bital angular momentum into TOM. At the end, we conclude that the phenomenon of chiral and topological orbital magnetism marks outstanding prospects for “chiral” spintronics and orbitronics. While the magnitude and details of the “local” TOM and COM can be tuned by electronic structure engineering, the topological orbital mo- ment is a new type of property in the physics of skyrmions. This observable, in analogy to the topological charge, ex- hibits either a quantization or a strong protection against de- formations of the underlying spin structure, and thus provides unique means for skyrmion detection, manipulation and uti- lization. In particular, the topological orbital moment can be envisaged to mediate the interaction between skyrmions and circularly-polarized light, giving an opportunity for opti- cal detection and control of the skyrmion topological charge. On the other hand, since the topological orbital moment is di- rectly proportional to the topological charge of the skyrmions, we suggest that the interaction of TOM with external magnetic fields could be used to trigger the formation of skyrmions with large topological charge. Ultimately, the currents of skyrmions can be employed for “lossless” transport of the as- sociated topological orbital momenta over large distances in skyrmionic devices, opening new perspectives in orbitronics. We thank J.-P. Hanke, M.d.S. Dias and S. Lounis for fruit- ful discussions, and gratefully acknowledge computing time on the supercomputers JUQUEEN and JURECA at J ¨ulich Su- percomputing Center, and at the JARA-HPC cluster of RWTH Aachen. We acknowledge funding under SPP 1538 and project MO 1731/5-1 of Deutsche Forschungsgemeinschaft (DFG) and the European Union’s Horizon 2020 research and innovation programme under grant agreement number 665095 (FET-Open project MAGicSky). f.lux@fz-juelich.de [1] A. Fert, V . Cros, and J. Sampaio, Nat. Nanotechnol. 8, 1525 (2013). [2] W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y . Fradin, J. E. Pearson, Y . Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, Science 349, 283 (2015). [3] N. Nagaosa and Y . Tokura, Nat. Nanotechnol. 8, 899 (2013). [4] J. Sampaio, V . Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotechnol. 8, 839 (2013). [5] M. Hoffmann, J. Weischenberg, B. Dup ´e, F. Freimuth, P. Fer- riani, Y . Mokrousov, and S. Heinze, Phys. Rev. B 92, 020401 (2015). [6] J. K ¨ubler and C. Felser, EPL 108, 67001 (2014). [7] P. Bruno, V . K. Dugaev, and M. Taillefumier, Phys. Rev. Lett. 93, 096806 (2004). [8] J.-P. Hanke, F. Freimuth, A. K. Nandy, H. Zhang, S. Bl ¨ugel, and Y . Mokrousov, Phys. Rev. B 94, 121114 (2016). [9] J.-P. Hanke, F. Freimuth, S. Bl ¨ugel, and Y . Mokrousov, Sci. Rep.7, 41078 (2017). [10] R. Shindou and N. Nagaosa, Phys. Rev. Lett. 87, 116801 (2001). [11] M. d. S. Dias, J. Bouaziz, M. Bouhassoune, S. Bl ¨ugel, and S. Lounis, Nat. Commun. 7, 13613 (2016). [12] S. Onoda, N. Sugimoto, and N. Nagaosa, Prog. Theor. Phys. 116, 61 (2006). [13] F. Freimuth, R. Bamler, Y . Mokrousov, and A. Rosch, Phys. Rev. B 88, 214409 (2013).[14] G. Zhu, S. A. Yang, C. Fang, W. M. Liu, and Y . Yao, Phys. Rev. B86, 214415 (2012). [15] J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett. 99, 197202 (2007). [16] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. a. Duine, Nat. Mater. 14, 871 (2015). [17] K. Bliokh and Y . Bliokh, Ann. Phys 319, 13 (2005). [18] C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov, Phys. Rev. B 82, 195316 (2010). [19] T. Fujita, M. B. A. Jalil, S. G. Tan, and S. Murakami, J. Appl. Phys. , J. Appl. Phys 110, 121301 (2011). [20] P. Ferriani, K. von Bergmann, E. Y . Vedmedenko, S. Heinze, M. Bode, M. Heide, G. Bihlmayer, S. Bl ¨ugel, and R. Wiesen- danger, Phys. Rev. Lett. 101, 027201 (2008). [21] N. Romming, A. Kubetzka, C. Hanneken, K. von Bergmann, and R. Wiesendanger, Phys. Rev. Lett. 114, 177203 (2015). [22] G. A. H. Schober, H. Murakawa, M. S. Bahramy, R. Arita, Y . Kaneko, Y . Tokura, and N. Nagaosa, Phys. Rev. Lett. 108, 247208 (2012). [23] H. Fukuyama, Prog. Theor. Phys. 45, 704 (1971). [24] S.-Q. Shen, Phys. Rev. B 70, 081311 (2004). [25] K. Shibata, A. Kov ´acs, N. Kiselev, N. Kanazawa, R. Dunin- Borkowski, and Y . Tokura, Phys. Rev. Lett. 118, 087202 (2017). [26] A. Kubetzka, C. Hanneken, R. Wiesendanger, and K. von Bergmann, Phys. Rev. B 95, 104433 (2017). [27] H. Fujita and M. Sato, Phys. Rev. B 95, 054421 (2017).

1505.03826v1.Spatiotemporal_spin_fluctuations_caused_by_spin_orbit_coupled_Brownian_motion.pdf

Spatiotemporal spin uctuations caused by spin-orbit-coupled Brownian motion A. V. Poshakinskiy1,and S. A. Tarasenko1, 2 1Ioe Institute, St. Petersburg 194021, Russia 2St. Petersburg State Polytechnic University, St. Petersburg 195251, Russia We develop a theory of thermal uctuations of spin density emerging in a two-dimensional electron gas. The spin uctuations probed at spatially separated spots of the sample are correlated due to Brownian motion of electrons and spin-obit coupling. We calculate the spatiotemporal correlation functions of the spin density for both ballistic and diusive transport of electrons and analyze them for dierent types of spin-orbit interaction including the isotropic Rashba model and persistent spin helix regime. The measurement of spatial spin uctuations provides direct access to the parameters of spin-orbit coupling and spin transport in conditions close to the thermal equilibrium. PACS numbers: 72.25.-b, 73.50.Td, 05.40.-a Thermal and quantum uctuations of observables are canonical examples of stochastic processes which are in- herent to physical systems. Since the discovery of the random motion of pollen grains suspended in water by R. Brown followed by the theoretical works by A. Ein- stein [1] and M. Smoluchowski [2], the study of uctua- tions has become central to statistical physics and kinet- ics. The advances in optical spectroscopy have triggered the research of uctuations of electron spins in atomic systems [3, 4] and, more recently, in semiconductor struc- tures including bulk materials [5{7], quantum wells [8{ 10], quantum wires [11{13], and quantum dots [14{18]. Owing to the fundamental connection between uctua- tions and dissipation processes, such a spin noise spec- troscopy is becoming a powerful tool for studying the spin dynamics in conditions close to thermal equilibrium and for determining the spin relaxation times, g-factors, parameters of exchange and hyperne interactions, etc., see recent review papers [19, 20]. The study of spin noise in semiconductors has been focused so far on the evolution of spin uctuations in time (or the spectral density of uctuations) [5{12, 14{ 18]. However, already in the rst experiments on spin noise in an electron gas it was found that the dynam- ics of spin uctuations is sensitive to the spatial size of the probed area of the sample due to diusion of elec- trons out of this area [8]. Moreover, the general analysis shows that the temporal and spatial correlations of spin uctuations emerging in an electron gas are coupled due to Brownian motion of electrons and spin-orbit interac- tion. Therefore, a natural and consistent description of the spin noise in an electron gas is spatiotemporal. Mean- while, such a study of the spin noise has been limited up to now to one-dimensional systems [13] in which the mo- tion of an electron in the real space and its spin rotation are locked and the spin dynamics is rather obvious. In systems of higher dimensions, the spin dynamics is com- plicated by the diversity of electron trajectories and the one-to-one correspondence between the real-space shift of electrons and the spin rotation angle vanishes. Here, we develop a theory of propagating spin uctuations in atwo-dimensional electron gas conned in a quantum well (QW). We calculate the spatiotemporal correlation func- tions of the spin density for both diusive and ballistic regimes and show that the correlation functions provide a direct information on the spin-orbit coupling in an elec- tron gas and parameters of the spin transport. We show that the correlations of spin uctuations drastically in- crease in the regime of persistent spin helix [21], which suggests a noninvasive method for probing this intrigu- ing phenomenon. Since the access to electron spin den- sity with high spatial and temporal resolutions is exper- imentally available nowadays [22{24], the measurements of uctuations will enable the study of spin waves and spin diusion processes in conditions close to the ther- mal equilibrium. The Gedankenexperiment we consider is illustrated in Fig. 1. A two-dimensional electron gas conned in a semi- conductor quantum well is in thermal equilibrium. The gas is spin unpolarized, however there are incessant uc- tuations of the spin density. The spin uctuations prop- agate in the QW plane due to Brownian motion of elec- trons and precess in the eective magnetic eld caused probe 1probe 2 FIG. 1. Probe-probe measurements of the spatial correlations of spin uctuations in an electron gas. Fluctuations of the spin density at the points r1andr2are correlated due to Brownian motion of electrons. The sign of the correlation function, positive or negative, depends on the distance r1r2and the frequency (p) of spin precession in the eective magnetic eld which acts upon the electron spins when electrons walk.arXiv:1505.03826v1 [cond-mat.mes-hall] 14 May 20152 by spin-orbit coupling. In quantum wells, the frequency of spin precession depends linearly on the electron mo- mentumpwhich leads to a relationship between the elec- tron trajectory in real space and spin rotation angle. As a result, the Brownian motion of electrons leads to cor- relations of the spin density probed at dierent spots of the sample. Whether the spin uctuations at the points r1andr2are positively or negatively correlated depends on the average spin rotation angle when electrons walk betweenr1andr2points which depends on the distance r=r1r2and the eective magnetic eld strength. To characterize the uctuations we introduce the corre- lation functions of the spin density S(r1;t1) andS(r2;t2) at dierent points of the sample. For spatially and tem- porally homogenous systems, the correlation functions depend on the time dierence and coordinate dierence and are dened by K(r1r2;t1t2) =Dn ^S(r1;t1);^S(r2;t2)oE ;(1) where ^S(r1;t1) =^ y(r1;t1)^ (r1;t1) is the spin den- sity operator, ^ (r1;t1) is the electron-eld operator,  are the Pauli matrices, fA;Bg= (AB+BA)=2 is the symmetrized product of the operators, and the angular brackets denote averaging with the quantum-mechanical density matrix corresponding to thermodynamic equilib- rium. The necessity to use the symmetrized product of ^S(r1;t1) and ^S(r2;t2) is caused by the fact that dif- ferent components of the spin density operator do not commute with each other. The correlation functions sat- isfy the relations K(r;t) =K (r;t),K(r;t) = K(r;t), andK(r;t) =K(r;t); the latter is due to time inversion symmetry in the absence of an ex- ternal magnetic eld. We calculate the correlation functions by using the uctuation-dissipation theorem [25] that relates the Fourier components of the correlation functions K(q;!) =RR K(r;t) ei!tiq
rdtdrto the Fourier components of the spin susceptibility (q;!), K(q;!) =(q;!) (q;!) 2i~coth~! 2T: (2) Here,(q;!) is dened as the linear response, S(q;!) =(q;!)F(q;!), of the spin density S(q;!) to the \force" F(q;!) whose action upon the system is described by the Hamiltonian perturbation ^V=R^S(r;t)
F(r;t)dr, andTis the temperature. The susceptibility (q;!) is readily expressed via retarded and advanced electron Green's functions GR;A "(r;r0) as follows (q;!) =m 4~2i 4~!Z 0d" 2Z drZ dr0(3) Tr GR "(r;r0)GA "~!(r0;r) eiq(r0r);wheremis the eective mass, is the Kronecker delta, the angular brackets denote averaging over the dis- order, and we assume that the electron gas is degenerate, ~qpFand~!"F, withpFand"Fbeing the Fermi momentum and the Fermi energy, respectively. Averaging over the positions of impurities leads to the sum of the ladder diagrams. For the case of short-range scattering, the sum of the ladder diagrams has the form (q;!) =m 4~2i 4~!Z 0d" 2Tr"X pGR p;"GA p~q;"~! +~3 mX pp0GR p;"GR p0;"GA p0~q;"~!GA p~q;"~!+:::3 5; (4) whereGR;A p;"are impurity-averaged Green's functions, GR;A p;"=1 "F+"p2=2m(~=2) (p)
i~=2; (5) is the relaxation time, (p) is the Larmor frequency corresponding to the eective magnetic eld. Straightforward calculation of the series (4) yields (q;!) =m 4~2[+ i!T(q;!)]; (6) whereT(q;!) is the spin lifetime tensor, T(q;!) =C(q;!)[1C(q;!)]1; (7) C(q;!) is the matrix is given by C(q;!) =Zd'p 2 +  2= 2+ 22;(8) (p) = 1i!+ i(=m)q
p, is the Levi-Civita tensor,'pis the polar angle of the momentum p, and jpj=pF. We note that the tensor T(q;!) describes also the spin density S(q;!) emerging in the sample when the spin is generated at the Fermi level at the rate G(q;!), S=TG. Calculation of the spin lifetime tensor T in the framework of the Boltzmann equation in station- ary and spatially homogenous case for QWs of dierent symmetry and arbitrary is described in Refs. [26, 27]. Combining Eqs. (2) and (6) we obtain the correlation functions at T~! K(q;!) =mT 4~2[T(q;!) +T (q;!)]:(9) Equation (9) describes the spatial and temporal cor- relations of spin density uctuations emerging in two- dimensional electron gas for arbitrary form of the eec- tive magnetic eld and arbitrary parameter . Below, we discuss the correlations for the ballistic ( 1) and3 FIG. 2. Spin density correlation functions K(r;t) for the Rashba spin-orbit interaction and coordinate dierence rkx between the points where the spin uctuations are probed. Map are calculated for R= 5 which is close to ballistic transport of electrons between the points. diusive ( 1) regimes of spin dynamics. We con- sider (001)-oriented QWs where the p-linear spin-orbit coupling is described by the eective magnetic eld (p) = [( D+ R)py=pF;( D R)px=pF;0];(10) Dand Rare the Dresselhaus and Rashba eld strengths at the Fermi level, respectively, xk[110] and yk[110] are the crystal-structure-enforced eigen axes in the QW plane, and zis the QW normal [28]. Figure 2 shows the coordinate and time dependence of the correlation functions K(r;t) for ballistic transport of electrons between the spots where the spin uctuations are probed. The maps are calculated for the Rashba ef- fective magnetic eld. The pronounced correlations of the spin uctuations probed at the spots separated by the distanceremerge at the time delay t= (m=pF)rthat follows from the ballistic behavior of electron transport. The correlation function contains both the diagonal K and o-diagonal K(6=) components. The latter originate from the precession of electron spins in the ef- fective magnetic eld when electrons propagate between the spots. The precession is also responsible for the os- cillatory behavior of the correlation functions. Figure 2 is plotted for rkx. Accordingly, the Rashba eld for electrons propagating between the spots points along the yaxis and couples the xandzcomponents of the spin density. The frequency of spin precession is given by R which results in the oscillations of the correlation func- tions in real space with the wave vector q= Rm=pF. In the case of arbitrary eective eld (p), the corre- lation function in the ballistic limit has the form K(r;t) =mT 82~2rR mr t t (rvFjtj);(11) whereR[] is the matrix of rotation by the angle and vF=pF=mis the Fermi velocity. In particular, the Kzz component for (001)-oriented QWs assumes the form Kzz=mT 82~2rcos tp 1sin 2cos 2'r (rvFjtj); (12) FIG. 3. Spin density correlation functions K(r;t) for the Rashba spin-orbit interaction and coordinate dierence rkx between the points where the spin uctuations are probed. Maps are calculated for R= 0:2 which corresponds to dif- fusive transport of electrons between the points. where =p 2 D+ 2 R,= arctan( R= D), and'r= arctan(y=x) is the polar angle of the vector r. Now we turn to diusive transport of electrons. The calculated dependence of the correlation functions K(r;t) on coordinate and time for this regime is shown in Fig. 3. The uctuations of spin density at the spots separated by the distance rget correlated at the time delayt&r2=D, whereD=v2 F=2 is the diusion coe- cient. Despite the diusive transport, for which electrons can travel along many dierent trajectories, the correla- tion functions do contain oscillations as a function of dis- tance. This can be attributed to the fact that the major contribution to correlations is given by the diusive tra- jectories which are close to the straight line connecting the spots. An analytical equation for the correlation function in the diusive regime can be derived from Eqs. (7)-(9) by considering the case 1 and, hence, !;Dq21. Such a procedure yields K(q;!) =mT 4~21 i!++Dq2+ 2iD(q)+ H:c: ; (13) where  =R ( 2  )d'p=(2) is the D'yakonov{Perel' spin-relaxation-rate tensor for homo- geneous spin distribution [29] and  (q) =R   q
pd'p=(2m). The tensor  (q) describes the preces- sion of electron spin at diusion. At large delay times, the correlations are determined by the spin excitations with the longest lifetime. For pure Rashba or Dressel- haus spin-orbit coupling, the Kzzcomponent of the cor- relation function at t1= 2assumes the form Kzz(r;t) =3m T 32vF~2e(7=32) 2jtj p 4DjtjJ0 r 15 16 r vF! ; (14) whereJ0is the Bessel function. The emergence of long-lived spin polarization waves with the wave vector q=p 15=16 =vFunder inhomogeneous spin pumping was theoretically predicted by V.A. Froltsov [30] and ob-4 FIG. 4. (a)-(d) Correlation function of the out-of-plane spin uctuations Kzzin real space in (001)-grown QW. Maps are calculated for the delay time t= 20=( 2 R), R= 0:2, and dierent ratio of the Dresselhaus and Rashba terms. (e) Dis- tributions of the eective magnetic eld on the Fermi circle for dierent ratio of the Dresselhaus and Rashba terms. served by means of transient spin-grating spectroscopy by C.P. Weber et al. [23]. Finally, we discuss the spin uctuations in the pres- ence of spin splitting anisotropy, caused by interference of the Rashba and Dresselhaus terms, which can be very pronounced in asymmetric (001)-oriented QWs [31]. Fig- ure 4 shows the dependence of the correlation function Kzzon the relative position of the probed spots in the QW plane. The maps are calculated for dierent ratio of the Dresselhaus and Rashba elds. The anisotropy of spin splitting in the momentum space leads to an anisotropy of the correlation function in real space which turns out to be strong even for small ratio D= R, cf. Figs. 4a-d and 4e. Another striking consequence of the interplay of the Rashba and Dresselhaus elds is a dras- tic increase in the lifetime and amplitude of spin corre- lations, see Fig. 4. This is due to the fact that, at equal strengths of the Rashba and Dresselhaus elds, the SU(2) spin rotation symmetry emerges in the system leading to an appearance of spin density waves with innite life- time (persistent spin helix) [21, 32{35]. We note that the emergence of long-lived waves with the wave vector along theyaxis can be clearly seen already at D= R= 0:5, Fig. 4d. An advantage of the spin noise spectroscopy forthe study of spin helix as compared to the pump-probe technique is that no photoexcited carriers and, hence, no additional mechanisms of spin dephasing are introduced. At large delay times, the correlation function is deter- mined by the spin density waves with the longest lifetime. Such waves are directed along the xoryaxis depending on the sign of the product R D. At R D>0, the correlation function Kzzoscillates along the yaxis and Eq. (13) yields its asymptotic behavior Kzz(r;t)/Te~ tcos(~qy) t; (15) where the wave vector ~ qand the decay rate ~ are given by ~q= R+ D vFr 11 16tan4  4 ; (16) ~ =9( R D)2 32 12 91 1 + sin 2 : While the decay rate of the correlation function is mostly determined by the dierence of the strengths of the Rashba and Dresselhau elds, the period of oscillations in real space is determined by the sum of the eld strengths. To summarize, we have developed a theory of spa- tiotemporal uctuations of spin density emerging in a two-dimensional electron gas with spin-orbit coupling. We have calculated the correlation functions of spin den- sity for both ballistic and diusive regimes of electron transport and analyzed them for dierent types of spin- orbit coupling that can be realized in quantum wells. The correlations of spin uctuations at large delay times are determined by the long-lived waves of spin density and drastically increas in the regime of persistent spin helix. Financial support by the RFBR, RF President Grants No. MD-3098.2014.2 and No. NSh-1085.2014.2, EU project SPANGL4Q, and the \Dynasty" Foundation is gratefully acknowledged. poshakinskiy@mail.ioe.ru [1] A. Einstein, \ Uber die von der molekularkinetischen Theorie der W arme geforderte Bewegung von in ruhen- den Fl ussigkeiten suspendierten Teilchen," Annalen der Physik 322, 549{560 (1905). [2] M. Smoluchowski, \Zur kinetischen Theorie der Brown- schen Molekularbewegung und der Suspensionen," An- nalen der Physik 21, 756{780 (1906). [3] E. B. Aleksandrov and V. S. Zapassky, \Magnetic reso- nance in the Faraday-rotation noise spectrum," Zh. Eksp. Teor. Fiz. 81, 132{138 (1981), [JETP 54, 64 (2011)]. [4] S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D.L. Smith, \Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance," Nature (London) 431, 49{52 (2004).5 [5] M. Oestreich, M. R omer, R. J. Haug, and D. H agele, \Spin Noise Spectroscopy in GaAs," Phys. Rev. Lett. 95, 216603 (2005). [6] S. A. Crooker, L. Cheng, and D. L. Smith, \Spin noise of conduction electrons in n-type bulk GaAs," Phys. Rev. B79, 035208 (2009). [7] M. R omer, H. Bernien, G. M uller, D. Schuh, J. H ubner, and M. Oestreich, \Electron-spin relaxation in bulk GaAs for doping densities close to the metal-to-insulator tran- sition," Phys. Rev. B 81, 075216 (2010). [8] G. M. M uller, M. R omer, D. Schuh, W. Wegscheider, J. H ubner, and M. Oestreich, \Spin Noise Spectroscopy in GaAs (110) Quantum Wells: Access to Intrinsic Spin Lifetimes and Equilibrium Electron Dynamics," Phys. Rev. Lett. 101, 206601 (2008). [9] Fuxiang Li, Yuriy V. Pershin, Valeriy A. Slipko, and N. A. Sinitsyn, \Nonequilibrium spin noise spec- troscopy," Phys. Rev. Lett. 111, 067201 (2013). [10] S. V. Poltavtsev, I. I. Ryzhov, M. M. Glazov, G. G. Ko- zlov, V. S. Zapasskii, A. V. Kavokin, P. G. Lagoudakis, D. S. Smirnov, and E. L. Ivchenko, \Spin noise spec- troscopy of a single quantum well microcavity," Phys. Rev. B 89, 081304 (2014). [11] M. M. Glazov and E. Ya. Sherman, \Theory of Spin Noise in Nanowires," Phys. Rev. Lett. 107, 156602 (2011). [12] P. Agnihotri and S. Bandyopadhyay, \Spin dynamics and spin noise in the presence of randomly varying spin-orbit interaction in a semiconductor quantum wire," Journal of Physics: Condensed Matter 24, 215302 (2012). [13] Yu. V. Pershin, V. A. Slipko, D. Roy, and N. A. Sinitsyn, \Two-beam spin noise spectroscopy," Appl. Phys. Lett. 102, 202405 (2013), 10.1063/1.4807011. [14] S. A. Crooker, J. Brandt, C. Sandfort, A. Greilich, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, \Spin Noise of Electrons and Holes in Self-Assembled Quantum Dots," Phys. Rev. Lett. 104, 036601 (2010). [15] Y. Li, N. Sinitsyn, D. L. Smith, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, \Intrinsic Spin Fluctuations Reveal the Dynamical Response Func- tion of Holes Coupled to Nuclear Spin Baths in (In,Ga)As Quantum Dots," Phys. Rev. Lett. 108, 186603 (2012). [16] R. Dahbashi, J. H ubner, F. Berski, J. Wiegand, X. Marie, K. Pierz, H. W. Schumacher, and M. Oestreich, \Mea- surement of heavy-hole spin dephasing in (InGa)As quantum dots," Appl. Phys. Lett. 100, 031906 (2012), 10.1063/1.3678182. [17] V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, \Optical Spectroscopy of Spin Noise," Phys. Rev. Lett. 110, 176601 (2013). [18] D. S. Smirnov, M. M. Glazov, and E. L. Ivchenko, \Eect of exchange interaction on the spin uctuations of local- ized electrons," Phys. Solid State 56, 254{262 (2014). [19] J. H ubner, F. Berski, R. Dahbashi, and M. Oestreich, \The rise of spin noise spectroscopy in semiconductors: From acoustic to GHz frequencies," physica status solidi (b)251, 1824{1838 (2014). [20] V. S. Zapasskii, \Spin-noise spectroscopy: from proof of principle to applications," Adv. Opt. Photon. 5, 131{168(2013). [21] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, \Exact SU(2) Symmetry and Persistent Spin Helix in a Spin- Orbit Coupled System," Phys. Rev. Lett. 97, 236601 (2006). [22] A. R. Cameron, P. Riblet, and A. Miller, \Spin Gratings and the Measurement of Electron Drift Mobility in Mul- tiple Quantum Well Semiconductors," Phys. Rev. Lett. 76, 4793{4796 (1996). [23] C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, J. Stephens, and D. D. Awschalom, \Nondiusive Spin Dynamics in a Two-Dimensional Electron Gas," Phys. Rev. Lett. 98, 076604 (2007). [24] R. V olkl, M. Griesbeck, S. A. Tarasenko, D. Schuh, W. Wegscheider, C. Sch uller, and T. Korn, \Spin dephasing and photoinduced spin diusion in a high- mobility two-dimensional electron system embedded in a GaAs-(Al,Ga)As quantum well grown in the [110] di- rection," Phys. Rev. B 83, 241306 (2011). [25] L. D. Landau and E. M. Lifshitz, Course of The- oretical Physics, Vol. 5 Statistical Physics (3rd ed.) (Butterworth-Heinemann, Oxford, 1980). [26] A. V. Poshakinskiy and S. A. Tarasenko, \Spin-orbit Hanle eect in high-mobility quantum wells," Phys. Rev. B84, 073301 (2011). [27] A. V. Poshakinskiy and S. A. Tarasenko, \Eect of Dresselhaus spin-orbit coupling on spin dephasing in asymmetric and macroscopically symmetric (110)-grown quantum wells," Phys. Rev. B 87, 235301 (2013). [28] E. L. Ivchenko, Optical spectroscopy of semiconductor nanostructures (Alpha Science International, Harrow, UK, 2005). [29] M. I. D'yakonov and V. Yu. Kachorovskii, \Spin re- laxation of conduction electrons in noncentrosymmet- ric semiconductors," Fiz. Tekh. Poluprovodn. 20, 178 (1986), [Sov. Phys. Semicond. 20, 110 (1986)]. [30] V. A. Froltsov, \Diusion of inhomogeneous spin distri- bution in a magnetic eld parallel to interfaces of a III-V semiconductor quantum well," Phys. Rev. B 64, 045311 (2001). [31] S. D. Ganichev and L. E. Golub, \Interplay of Rashba/Dresselhaus spin splittings probed by photogal- vanic spectroscopy | A review," physica status solidi (b) 251, 1801{1823 (2014). [32] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, S. Mack, and D. D. Awschalom, \Emer- gence of the persistent spin helix in semiconductor quan- tum wells," Nature (London) 458, 610{613 (2009). [33] M. P. Walser, C. Reichl, W. Wegscheider, and G. Salis, \Direct mapping of the formation of a persistent spin helix," Nat. Physics 8, 757{762 (2012). [34] A. Sasaki, S. Nonaka, Y. Kunihashi, M. Kohda, T. Bauernfeind, T. Dollinger, K. Richter, and J. Nitta, \Direct determination of spin-orbit interaction coe- cients and realization of the persistent spin helix sym- metry," Nat. Nanotechnology 9, 703{709 (2014). [35] J. Ishihara, Y. Ohno, and H. Ohno, \Direct imag- ing of gate-controlled persistent spin helix state in a modulation-doped GaAs/AlGaAs quantum well," Appl. Phys. Express 7, 013001 (2014).

1008.1600v2.Gate_dependent_spin_orbit_coupling_in_multi_electron_carbon_nanotubes.pdf

arXiv:1008.1600v2 [cond-mat.mes-hall] 12 Aug 2010Gate-dependent spin-orbit coupling in multi-electron car bon nanotubes T. S. Jespersen†,1K. Grove-Rasmussen†,1,2J. Paaske,1K. Muraki,2T. Fujisawa,3J. Nyg˚ ard,1and K. Flensberg1 1Niels Bohr Institute & Nano-Science Center, University of C openhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark 2NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan 3Research Center for Low Temperature Physics, Tokyo Institu te of Technology, Ookayama, Meguro, Tokyo 152-8551, Japan Understanding how the orbital motion of electrons is couple d to the spin degree of freedom in nanoscale systems is central for applications in spin-base d electronics and quantum computation. We demonstrate this coupling of spin and orbit in a carbon nan otube quantum dot in the general multi-electron regime in presence of finite disorder. Furth er, we find a strong systematic depen- dence of the spin-orbit coupling on the electron occupation of the quantum dot. This dependence, which even includes a sign change is not demonstrated in any o ther system and follows from the curvature-induced spin-orbit split Dirac-spectrum of the underlying graphene lattice. Our findings unambiguously show that the spin-orbit coupling is a genera l property of nanotube quantum dots which provide a unique platform for the study of spin-orbit e ffects and their applications. Note: Manuscript with high resolution figures and supplement at www.fys.ku.dk/~tsand/TSJ_KGR2010.pdf The interaction of the spin of electrons with their orbital motion has become a focus of attention in quantum dot research. On the one hand, this spin-orbit interaction (SOI) provides a route for spin decoherence, which is unwanted for purposes of quantum computation[1–3]. On the other hand, if properly controlled, the SOI can be utilized as a means of electrically manipulating the spin degree of freedom[4–7]. In this context, carbon nanotubes provide a number of attractive features, including large confinement en- ergies, nearly nuclear-spin-free environment, and, most importantly, the details of the energy level structure is theoretically well understood and modeled, as well as experimentally highly reproducible. Remarkably, the SOI in nanotubes was largely overlooked in the first two decades of nanotube research and was only recently demonstrated by Kuemmeth et al.for the special case of a single carrier in ultra-clean CNT quantum dots[2, 8, 9]. Except for these reports, the SOI in nanotubes is experimentally unexplored. Theoretically, the focus has exclusively been on the SOI-modified band structure of disorder-free nanotubes[10–14]. Therefore, two important questions remain: how the effective SOI depends on electron filling and how it appears in the general case of quantum dots subject to disorder. Here we answer these two questions. Firstly, by low-temperature electron transport we demonstrate the presence of a significant SOI in a disordered CNT quantum dot holding hundreds of electrons. We identify and analyze the role of SOI in the energy spectrum for one, two, and three electrons in the four-fold degenerate CNT electronic shells, thus describing shells at any electron filling. By rotating the sample, we present for the first time spectroscopy of the same charge-states for magnetic fields both parallel and perpendicular to the nanotube axis, thus controlling the coupling to the orbital magnetic moment. Remarkably,a single-electron model taking into account both SOI and disorder quantitatively describes all essential details of the multi-electron quantum dot spectra. Secondly, by changing the electron occupancy we are able to tune the effective SOI and even reverse its sign in accordance with the expected curvature-induced spin-orbit splitting of the underlying graphene Dirac spectrum[1, 11–14]. Such systematic dependence has not been demon- strated in any other material system and may enable a new range of spin-orbit related applications. This microscopic understanding and detailed modeling is in stark contrasts to situations encountered in alternative strong-SOI quantum-dot materials, such as InAs or InSb nanowires, where the effective SOI arises from bulk crystal effects combined with unknown contributions from surface effects, strain and crystal defects[15]. These systems often exhibit semi-random fluctuations of, e.g., the g-factor as single electrons are added[16, 17]. Thus, beyond fundamental interest and the prospect of realizing recent proposals of SOI-induced spin con- trol in CNTs[1, 18], our findings pave the way for new designsofexperimentsutilizingtheSOIinquantumdots. Zero-field splitting of the four-fold degeneracy Our experimental setup is presented in Fig. 1a. We fabricate devices of single-wall CNTs on highly doped Si substrates capped with an insulating layer of SiO 2 (see Methods section). The size of the quantum dots is defined by the contact separation (400nm) and the electrical properties are investigated in a voltage biased two-terminal configuration applying a voltage Vsdbe- tween source-drain contacts and measuring the resulting currentI. The differential conductance dI/dV sdis mea- sured by standard lock-in techniques. When biased with a voltage Vg, the Si substrate acts as an electrostatic gate controlling the electron occupancy of the dot. The devices are measured at T= 100mK in a3He/4He2 dilution refrigerator, fitted with a piezo-rotator allowing in-plane rotations of the device in magnetic fields up to 9T. Figure 1b shows a typical measurement of dI/dV sd vs.VsdandVgin the multi-electron regime of a small- band-gap semiconducting nanotube. The pattern of diamond-shaped regions of low conductance is expected for a quantum dot in the Coulomb blockade regime and within each diamond the quantum dot hosts a fixed number of electrons N, increasing one-by-one with increasing Vg. The energy Eadd, required for adding a single electron appears as the diamond heights and has been extracted in Fig. 1c. The four-electron periodicity clearly observed in Fig. 1b and c reflects the near four- fold degeneracyin the nanotube energyspectrum[19, 20]; one factor of 2 from the intrinsic spin ( ↑,↓) and one factor of 2 from the so-called isospin ( K,K′) that stems from the rotationalsymmetry of the nanotube - electrons orbit the CNT in a clockwise or anticlockwise direction. As is generally observed[19–23], the addition energy for the second electron in each quartet (yellow in Fig. 1c) exceeds those for one and three. This was previously interpreted as a result of disorder-induced coupling ∆KK′of the clockwise and anticlockwise states[21, 24] that splits the spectrum into two spin-degenerate pairs of bonding/antibonding states separated by ∆ KK′. As mentioned, Kuemmeth et al.showed recently that for the first electron in an ultra-clean suspended nanotube quantum dot, the splitting was instead dominated by the spin-orbit coupling. The first question we address here is whether SOI also appears in the many-electron regime and how it may be modified or masked by disorder. Modeling spin-orbit coupling and disorder Performing level spectroscopy with a magnetic field B applied either parallel ( B/bardbl) or perpendicular ( B⊥) to the nanotube axis proves to be a powerful tool to analyze the separate contributions from disorder and spin-orbit coupling. This is illustrated in Figs. 2a-d, which show calculated single-particle energy level spectra for four limiting combinations of ∆ KK′and the effective spin- orbit coupling ∆ SO(all limits are relevant for nanotube devices depending on the degree of disorder and CNT structure[13, 14]; details of the model are provided in the Supplementary Information). In all cases a parallel field separates the four states into pairs of increasing ( K-like states) and decreasing ( K′-like states) energies. The magnitude of the shift is given by the orbital g-factor gorbreflecting the coupling of B/bardblto the orbital magnetic moment caused by motion around the CNT[25]. Further, each pair exhibits a smaller internal splitting due to the Zeeman effect. Figure 2b shows the disorder induced coupling of KandK′states resulting in an avoided crossing at B/bardbl= 0 and the zero-field splitting discussed above. In the opposite limit with SOI only (Fig. 2c), thezero-field spectrum is also split into two doublets, but the field dependence is markedlydifferent and no avoided crossing appears. In the simplest picture, this behavior originates from coupling of the electron spin to an effective magnetic field BSO=−(v×E)/c2experienced by the electron as it moves with velocity vin an electric fieldE. Here the speed of light, c, reflects the relativistic origin of the effect. In nanotubes, the curvature of the graphene lattice generates an effective radial electric field, and since the velocity is mainly circumferential (and opposite for KandK′),BSOpolarizes the spins along the nanotube axis and favors parallel alignment of the spin and orbital angular momentum. Thus, even in the absence of disorder, the spectrum splits into two Kramers doublets ( K↓,K′↑) and (K↑,K′↓) separated by ∆SO. Interestingly, since a perpendicular field does not couple KandK′the doublets do not split along B⊥on the figure. As a consequence, the g-factor, when measured in a perpendicular field, will vary from zero when ∆ SO≫∆KK′(Fig. 2c) to 2 in the opposite limit (Fig. 2b). The final case, including both disorder and SOI, is of particular importance for the present study, and the calculated spectrum is displayed in Fig. 2d for ∆KK′>∆SO. Importantly, the effects of SOI are not masked despite the dominating disorder: For parallel field, SOI remains responsible for an asymmetric split- ting of the Kramers doublets ( α,β) vs. (δ,γ), and the appearance of an additional degeneracy in the spectrum at finite field ( δandγstates). In a perpendicular field, the effect of SOI is to suppress the Zeeman splitting of the two doublets and since the eigenstates of the SOI have spins along the nanotube axis it couples the states with spins polarized along B⊥resulting in the avoided crossing indicated on the figure. Spin-orbit interaction revealed by spectroscopy With this in mind we now focus on the quartet with 4N0≈180 electrons highlighted in Fig. 1b and expanded in Fig. 3a. In order to investigate the level structure we perform cotunneling spectroscopy, as illustrated in the schematic Fig. 3b[26]: In Coulomb blockade, whenever e|Vsd|matches the energyof a transition from the ground stateαto an excited state ( β,γ,δ), inelastic cotunnel processes, that leave the quantum dot in the excited state, become available for transport. This significantly increases the current and gives rise to steps in the conductance. These appear as gate-independent features in Fig. 3a (arrows) and are clearly seen in the inset showing a trace through the center of the one-electron (4N0+1) diamond along the dashed line. Thus following the magnetic field dependence of this trace, as shown in Fig. 3c, maps out the level structure. The energies of the excitations are given by the inflection points of the curve ( i.e.peaks/dips of d2I/dV2 sd)[27] and the level3 evolution is therefore directly evident in Fig. 3d-i, which show color maps of the second derivative vs. Vsdand B⊥,B/bardblforVgpositioned in the center of the one, two, and three electron charge states. As explained below, the SOI is clearly expressed in all three spectra. Consider first the one-electron case: In a parallel field (Fig. 3d), the asymmetric splitting of the two doublets is evident (black vs. green arrows) and applying the field perpendicularly (Panel g), the SOI is directly expressed as the avoided crossing indicated on the figure. The measurement is in near-perfect agreement with the single-particle excitation spectrum calculated by subtracting the energies of Fig. 3b and shown by the solid lines. The calculation depends on only three parameters: ∆ SO= 0.15meV set directly by the avoided crossing, ∆ KK′= 0.45meV determined from the zero-field splitting of the doublets (see Fig. 2d), and gorb= 5.7 set by the slopes of the excitation lines from αtoγ,δin Panel d. Consider now the role of SOI for the doubly occupied CNT quartet. This situation is of particular importance for quantum computation as a paradigm for preparation of entangled states and a fundamental part of Pauli blockade in double quantum dots[28]. Figures 3e,h show the measured spectra in parallel and perpendicular fields. The model perfectly describes the measurement and now contains no free parameters since these are fixed by one-electron measurement. Six states are expected: the ground state singlet-like state ˜S0formed by the two electrons occupying the low-energy Kramers doublet[29], three triplet-like ˜T−,˜T0,˜T+and a singlet- like state ˜S1, which all use one state from each doublet and the singlet-like ˜S2with both electrons occupying the high-energy doublet. The ground state ˜S0does not appear directly in the measurement, but sets the origin for the cotunneling excitations. The excitation to the high-energy ˜S2(dashed line) is absent in the experiment, since it cannot be reached by promoting only a single electron from ˜S0. In Fig. 3h excitations to ˜T−and˜T+are clearly observed, while ˜S1and˜T0merge into a single high-intensity peak showing that any ex- change splitting Jis below the spectroscopic line width ≈100µeV[23]. In other quartets an exchange splitting is indeed observed (see Supplementary Information). In the two-electron spectra the SOI is directly expressed as the avoided crossing at B⊥≈4.5T accompanying the ˜S0↔˜T−ground state transition[15, 30]. In quartets of yet stronger tunnel-coupling it is replaced by a singlet-triplet Kondo resonance[31] (see Supplementary Information). Finally, the spectrum of three electrons in the four- electron shell is equivalent to that of a single hole in a full shell; at low fields the δ-state becomes the ground-state and γthe first excited state, while αand βthen constitute the excited doublet. As seen by comparing Fig. 2b and 2d SOI breaks the intra-shell electron-hole symmetry of the nanotube spectrum. This is evident in the experiment when comparing Fig. 3dand 3f: In 3f, increasing B/bardbl, the lowest excited state γ, barely separates from the ground state δand at B/bardbl= 1.1T they cross again, causing a ground state transition. At the crossing point, the spin-degenerate ground-state results in a zero-bias Kondo peak (see inset)[32]. Interestingly, this degeneracy also forms the qubit proposed in Ref. 1. For the B⊥-dependence the one- and three-electron cases remain identical and Fig. 3i exhibits again the SOI-induced avoided crossings. Gate-dependent spin-orbit coupling Having established the presence of SOI in the general many-electron disordered quantum dot, we now focus on the dependence of ∆ SOon the quantum dot occupation. To this end, we have repeated the spectroscopy of Fig. 3 for a large number of CNT quartets and in each case extracted ∆ SOby fitting to the single-particle model (all underlying data are presented in the Supplementary In- formation). Figure 4a shows the result as a function of Vg. An overall decrease of ∆ SOis observed as electrons are added to the conduction band and, interestingly, a negative value is found in the valence band ( i.e.SOI favoring anti-parallel rather than parallel spin and or- bital angular momentum, thus effectively interchanging the one and three-electron spectra). The magnitude of ∆ SOis given by the spin-orbit split- ting of the underlying graphene band structure as we will now discuss. For flat graphene this splitting is very weak (∆graphene SO∼1µeV)[12] as it is second-order in the already weak atomic SOI of carbon ∆C SO∼8meV. In nanotubes, however, the curvature induces a coupling between the π- andσ-bands and generates a curvature- induced spin-orbit splitting, which is first order in the atomic SOI and thus greatly enhances ∆ SO. Around a Dirac-point of the Brillouin zone ( e.g.K) the graphene band structure appears as on Fig. 4b[11–14]: The spin- up and spin-down Dirac cones are split by SOI both in energy and along k⊥, the momentum in the circumferen- tial direction of the CNT. The schematic also highlights the CNT band structure (Fig. 4a upper inset) obtained by imposing periodic boundary conditions on k⊥. In a finite-lengthCNTquantumdotalsothewavevectoralong thenanotubeaxis, k/bardbl, isquantized, andletting ǫNdenote the energyofthe Nthlongitudinalmode the effectiveSOI for a small-gap CNT becomes ∆SO,±=EK ↑−EK ↓ (1) = 2∆0 SO±/radicalBig (∆g+∆1 SO)2+ǫ2 N ∓/radicalBig (∆g−∆1 SO)2+ǫ2 N. Here the upper(lower) sign refers to the conduc- tion(valence) band, ∆ gis the curvature induced en- ergy gap[33, 34], and the two terms ∆0 SOand ∆1 SO are the band structure spin-orbit parameters due to4 curvature[11–14]. The separate contributions of the two terms are illustrated in the lower inset to Fig. 4a. ∆1 SO was found already in the work of Ando[10] and accounts for thek⊥-separation of the Dirac-cones in Fig. 4b. For the CNT band structure this term acts equivalently to an Aharonov-Bohm flux from a parallel spin-dependent magneticfield, whichchangesthequantizationconditions in thek⊥-direction. Characteristically, its contribution to ∆SOdecreases with the number of electrons in the dot (ǫN) and reverses sign for the valence band. This contrasts the ǫN-independent contribution from the re- centlypredicted∆0 SO-term[12–14], whichactsasaneffec- tive valley-dependent Zeeman term and accounts for the energy splitting of the Dirac-cones in Fig. 4b. For the nanotube studied in Kuemmeth et al.[9] ∆SOhas the same sign for electrons and holes, i.e.,|∆0 SO|>|∆1 SO|. In our case, the negative values measured in the valence band demonstrate the opposite limit |∆0 SO|<|∆1 SO|. Thus, the measured gate-dependence of ∆ SOagrees with the spin-orbit splitting of the graphene Dirac spectrum caused by the curvature of the nanotube, and fitting to Eq.1(Fig.4a, dashedline) yields∆0 SO= 10±10µeVand ∆1 SO= 220±25µeV[35]. Band structure models relate these parameters to the structure of the nanotube[12– 14]: ∆0 SO=λ0∆C SO∆gDand ∆1 SO=λ1∆C SO/D, where Dis the nanotube diameter and λ1,2constants that de- pend on the CNT class (semiconducting, small-band- gap). Typically CVD-grown single wall CNTs have di- ameters in the range 1-3nm, while obtaining Dfrom the measured values of gorb[36] gives D≈6nm. Thus taking D= 1-6nmwe estimate λ1= 0.03-0.17nmand λ0= 0.7- 4·10−6(nm·meV)−1. While λ1is consistent with the value quoted in the theoretical literature (0 .095nm)[13], the calculated λ0value−4·10−3(nm·meV)−1does not match the experiment and similar deviations appear[13] when comparing the theory to the SOI values measured in Ref. 8. The origin of this discrepancy remains un- known, and further work on SOI in nanotubes with known chirality is needed to make further progress.Methods The devices are made on a highly doped Silicon wafer terminated by 500nm of SiO 2. Alignment marks (Cr, 70nm) are defined by electron beam lithography prior to deposition of catalyst islands made of Iron nitrate (Fe(NO 3)3), Molybdenum acetate and Alumina support particles[37]. Thesampleisthen transferredtoafurnace, where single wall carbon nanotubes are grown by chem- ical vapor deposition at 850-900◦C in an atmosphere of hydrogen, argon and methane gases. Pairs of electrodes consisting of Au/Pd (40/10nm) spaced by 400nm are fabricatedalongsidethecatalystislandsbystandardelec- tron beam lithography techniques. Finally, bonding pads (Au/Cr 150/10nm) are made by optical lithography and the devices are screened by room- and low-temperature measurements. We measured the sample in an Oxford dilution refrig- erator fitted with an Attocube ANRv51 piezo rotator which allows high precision in-plane rotation of the sam- ple in large magnetic fields. The rotator provides resis- tive feedback of the actual position measured by lock-in techniques. For electrical filtering, room-temperature π- filters and low-temperature Thermocoax are used. The base temperature of the modified refrigerator is around 100mK. The CNT measurement setup consists of a Na- tional Instrument digital to analog card, custom made optically coupled amplifiers, a DL Instruments 1211 cur- rent to voltage amplifier and a Princeton Applied Re- search 5210 Lock-in amplifier. Standard dc and lock-in techniques have been used to measure current and differ- ential conductance dI/dV sdwhiled2I/dV2 sdis obtained numerically.5 [1] Bulaev, D., Trauzettel, B. & Loss, D. Spin-orbit interac - tion and anomalous spin relaxation in carbon nanotube quantum dots. Phys. Rev. B 77, 235301 (2008). [2] Churchill, H. et al.Electron-nuclear interaction in c-13 nanotube double quantum dots. Nat. Phys. 5, 321 – 326 (2009). [3] Fischer, J. & Loss, D. Dealing with decoherence. Science 324, 1277–1278 (2009). [4] Flindt, C., Sørensen, A. & Flensberg, K. Spin-orbit me- diated control of spin qubits. Phys. Rev. Lett. 97, 240501 (2006). [5] Nowack, K., Koppens, F., Nazarov, Y. & Vandersypen, L. Coherent control of a single electron spin with electric fields.Science318, 1430 – 1433 (2007). [6] Pfund, A., Shorubalko, I., Ensslin, K. & Leturcq, R. Sup- pression of spin relaxation in an inas nanowire double quantum dot. Phys. Rev. Lett. 99, 036801 (2007). [7] Trif, M., Golovach, V. & Loss, D. Spin-spin coupling in electrostatically coupled quantumdots. Phys. Rev. B 75, 085307 (2007). [8] Kuemmeth, F., Churchill, H., Herring, P. & Marcus, C. Carbon nanotubes for coherent spintronics. Mat. Today 13, 18–26 (2010). [9] Kuemmeth, F., Ilani, S., Ralph, D. & McEuen, P. Cou- pling of spin and orbital motion of electrons in carbon nanotubes. Nature452, 448 – 452 (2008). [10] Ando, T. Spin-orbit interaction in carbon nanotubes. J. Phys. Soc. Jpn. 69, 1757 – 1763 (2000). [11] Chico, L., Lopez-Sancho, M. & Munoz, M. Spin splitting induced by spin-orbit interaction in chiral nanotubes. Phys. Rev. Lett. 93, 176402 (2004). [12] Huertas-Hernando, D., Guinea, F. & Brataas, A. Spin- orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps. Phys. Rev. B 74, 155426 (2006). [13] Izumida, W., Sato, K. & Saito, R. Spin-orbit interac- tion in single wall carbon nanotubes: Symmetry adapted tight-binding calculation and effective model analysis. J. Phys. Soc. Jpn. 78, 074707 (2009). [14] Jeong, J. & Lee, H. Curvature-enhanced spin-orbit cou- pling in a carbon nanotube. Phys. Rev. B 80, 075409 (2009). [15] Fasth, C., Fuhrer, A., Samuelson, L., Golovach, V. & Loss, D. Direct measurement of the spin-orbit inter- action in a two-electron inas nanowire quantum dot. Phys. Rev. Lett. 98, 266801 (2007). [16] Csonka, S. et al.Giant fluctuations and gate control of the g-factor in inas nanowire quantum dots. Nano Lett. 8, 3932–3935 (2008). [17] Nilsson, H. et al.Giant, level-dependent g factors in insb nanowire quantum dots. Nano Lett. 9, 3151–3156 (2009). [18] Flensberg, K. & Marcus, C. Bends in nanotubes allow electric spin control and coupling. Phys. Rev. B 81, 195418 (2010). [19] Cobden, D. & Nygard, J. Shell filling in closed single- wall carbon nanotube quantum dots. Phys. Rev. Lett. 89, 046803 (2002). [20] Liang, W., Bockrath, M. & Park, H. Shell filling and exchange coupling in metallic single-walled carbon nan- otubes.Phys. Rev. Lett. 88, 126801 (2002). [21] Jarillo-Herrero, P. et al. Electronic transport spec- troscopy of carbon nanotubes in a magnetic field.Phys. Rev. Lett. 94, 156802 (2005). [22] Makarovski, A., An, L., Liu, J. & Finkelstein, G. Persistent orbital degeneracy in carbon nanotubes. Phys. Rev. B 74, 155431 (2006). [23] Moriyama, S., Fuse, T., Suzuki, M., Aoyagi, Y. & Ishibashi, K. Four-electron shell structures and an inter- acting two-electron system in carbon-nanotube quantum dots.Phys. Rev. Lett. 94, 186806 (2005). [24] Oreg, Y., Byczuk, K. & Halperin, B. Spin configurations of a carbon nanotube in a nonuniform externalpotential. Phys. Rev. Lett. 85, 365–368 (2000). [25] Minot, E., Yaish, Y., Sazonova, V. & Mceuen, P. Deter- mination of electron orbital magnetic moments in carbon nanotubes. Nature428, 536 – 539 (2004). [26] De Franceschi, S. et al.Electron cotunneling in a semi- conductor quantum dot. Phys. Rev. Lett. 86, 878–881 (2001). [27] Paaske, J. et al.Non-equilibrium singlet-triplet kondo effect in carbon nanotubes. Nat. Phys. 2, 460–464 (2006). [28] Hanson, R., Kouwenhoven, L., Petta, J., Tarucha, S. & Vandersypen, L. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007). [29] The states are not the conventional spin singlets and triplets as the they are modified by SOI as emphasized by the tildes. [30] From the high-field ground state ˜T−, excitations to ˜S2 are actually allowed, however, the ˜S0↔˜T−and˜S2↔˜T+ avoided crossings occur simultaneously and as ˜T−to˜T+ excitations are forbidden the dashed line remains unseen in the experiment. [31] Nygard, J., Cobden, D. & Lindelof, P. Kondo physics in carbon nanotubes. Nature408, 342–346 (2000). [32] Galpin, M., Jayatilaka, F., Logan, D. & Anders, F. In- terplay between kondo physics and spin-orbit coupling in carbon nanotubequantumdots. Phys. Rev. B 81, 075437 (2010). [33] Kane, C. & Mele, E. Size, shape, and low energy elec- tronic structure of carbon nanotubes. Phys. Rev. Lett. 78, 1932–1935 (1997). [34] Kleiner, A. & Eggert, S. Band gaps of primary metallic carbon nanotubes. Phys. Rev. B 63, 073408 (2001). [35] The band-gap of the device Eg≈30meV is measured directly as a large Coulomb diamond at Vg≈1V and ǫN≈25meV/V×Vgis estimated from the level spacing ∆E≈3meV and ≈8shells/V. [36] Within the present theory the orbital g-factor depends on electron filling gorb≈(evFD/2µB)//radicalbig 1+(ǫn/∆g)2 in agreement with the measurements (see SOM). [37] Kong, J., Soh, H., Cassell, A., Quate, C. & Dai, H. Syn- thesis of individual single-walled carbon nanotubes on patterned silicon wafers. Nature395, 878–881 (1998). Acknowledgements We thank P. E. Lindelof, J. Mygind, H.I. Jørgensen, C.M. Marcus, and F. Kuemmeth for discussions and ex- perimental support. T.S.J. acknowledges the Carlsberg Foundation and Lundbeck Foundation for financial sup-6 port. K.G.R. K.F. J.N. acknowledges The Danish Re- search Council and University of Copenhagen Center of Excellence. Author contributions T.S.J. and K.G.R. performed the measurements, ana- lyzed the data and wrote the paper. T.S.J. designed therotating sample stage. K.G.R. made the sample. K.M., T.F. and J.N. participated in discussions and writing the paper. J.P. and K.F. developed the theory and guided the experiment.7 FIG. 1: Four-fold periodic nanotube spectrum. a, Schematic illustration of the device and setup. CNT quantu m dots are measured at T= 100mK in a standard two-terminal configuration in a cryosta t modified to enable measurements in a high magnetic field at arbitrary in-plane angles θto the CNT axis. b, Typical measurement of the differential conductance dI/dV sd vs. source-drain bias Vsdand gate voltage Vgfor a multi-electron CNT quantum dot. c, Addition energy as a function of Vg. Inbandcthe characteristic filling of four-electron shells is clear ly seen. FIG. 2: Role of spin-orbit interaction and disorder for the n anotube energy spectrum. Calculated single-particle ener gy spectrum as a function of magnetic field applied perpendicul ar (B⊥) and parallel ( B/bardbl) to the CNT axis in the limiting cases of neither SOI nor disorder a, disorder alone b, SOI alone c, and the two combined ∆ KK′>∆SO>0d. Depending on the CNT type, electron filling and degree of disorder, all four situa tions can occur.8 FIG. 3: Spin-orbit interaction in a disordered multi-elect ron nanotube quantum dot. a, Measurement of dI/dV sdvs.VsdandVg corresponding to the consecutive addition of four electron s to an empty shell (indicated on Fig. 1b). A strong tunnel cou pling results in significant cotunneling which is evident as horiz ontal lines truncating the diamonds (arrows). The black tra ce shows a cut along the dashed line. b, Schematic illustration of the relevant inelastic cotunne ling processes. c, Traces along the dashed line in afor various B/bardbl(red:B= 0, scale-bar: 0 .1e2/h).d-f, Thesecond derivative d2I/dV2 sdalong the center of the N0+1,N0+ 2 and N0+3 diamonds, respectively, as a function of a parallel magne tic field. Peaks/dips appear at inflection points of the differential conductance and thus correspond t o the energy difference between ground and excited states. In fthe inset shows dI/dV sdvs.−0.3< Vsd<0.3mV and B/bardbl= 0;0.55;1.1;1.65T (arrows) illustrating the splitting and SOI-induced reappearance of a zero-bias Kondo resonance. g-i, Asd-fbut measured as a function of B⊥. The effective spin-orbit coupling appears directly as the avoided crossings indicated by ∆ SO. Ind-ithe black lines results from the single-particle model with parameters ∆ SO= 0.15meV,∆KK′= 0.45meV, and gorb= 11.4. The dashed lines in e,hcorrespond to the excitations to the two-electron singlet-like ˜S2state which cannot be reached by promoting a single electron from the ground state ( ˜S0) and therefore expected to be absent in the measurement.9 FIG. 4: Tuning∆ SOinaccordance with thecurvature-inducedspin-orbitsplit tingofthenanotubeDirac-spectrum. a, Measured effective spin-orbit coupling strength as a function of Vgextracted from spectroscopy measurements like in Fig. 3, re peated for multiple shells. The dashed line is a fit to the theory. Lower i nset: Expected dependence of ∆ SOonǫNhighlighting the two SOI-contributions ∆0 SOand ∆1 SO.b, Graphene dispersion-cones around one K-point of the graphene Brillouin zone. Due to SOI the spin-up (blue) and spin-down (red) Dirac cones are sp lit in both the vertical ( E) andk⊥-direction. The cut shows the resulting CNT band structure also shown in the upper inset in a.

1501.01362v1.Superfluidity_of_pure_spin_current_in_ultracold_Bose_gases.pdf

arXiv:1501.01362v1 [cond-mat.quant-gas] 7 Jan 2015Superfluidity of pure spin current in ultracold Bose gases Qizhong Zhu (朱起忠),1Qing-feng Sun (孙庆丰),1,2and Biao Wu (吴飙)1,2,3 1International Center for Quantum Materials, School of Phys ics, Peking University, Beijing 100871, China 2Collaborative Innovation Center of Quantum Matter, Beijin g 100871, China 3Wilczek Quantum Center, College of Science, Zhejiang Univer sity of Technology, Hangzhou 310014, China (Dated: October 13, 2018) We study the superfluidity of a pure spin current that is a spin current without mass current. We examine two types of pure spin currents, planar and circular , in spin-1 Bose gas. For the planar current, it is usually unstable, but can be stabilized by the quadratic Zeeman effect. The circular current can be generated with spin-orbit coupling. When the spin-orbit coupling strength is weak, we find that the circular pure spin current is the ground state of the system and thus a super-flow. We discuss the experimental schemes to realize and detect a p ure spin current. PACS numbers: 05.30.Jp, 03.75.Mn, 03.75.Kk, 71.70.Ej I. INTRODUCTION Since the experimental realization of Bose-Einstein condensation in optical traps, much effort has been de- voted to the study of spinor superfluid [1–7]. With the degree of freedom of spin, spinor superfluid has much richer phases than scalar superfluid as it has both su- perfluid order and spin textures. However, most spinor superfluids studied till now carry both spin current and mass current. It will be interesting to see whether these two currents can decouple and further whether a pure spin current which has no mass current can flow friction- lessly. Our motivation also originates from condensed matter physics, in which the concept of spin supercon- ductor [8, 9], formed by the Cooper-likepairs of electrons and holes and carrying spin super-current, is proposed. It is interesting to have these ideas realized in the field of cold atoms. In this work we focus on the unpolarized spin-1 Bose- Einstein condensate (BEC), where the pure spin current can be generated by applying a small magnetic gradient. It was found in Ref. [10] that a planar pure spin current insuchasystemisalwaysunstableasthe m= 1,−1com- ponents can collide into the m= 0 component destroying the spin current. We find that the pure spin current can be stabilized with the quadratic Zeeman effect and be- come a super-flow. Furthermore, we propose a scheme to create pure spin current at the ground state, thus free fromtheissueofinstability. Ourschemeutilizesthe spin- orbit coupling. Specifically, we study a spin-1 BEC with Rashbaspin-orbitcouplingconfinedin atwo-dimensional harmonic trap, and numerically find the ground state of the system. For antiferromagnetic interactions, opposite vortices appear in the m= 1,−1 components with equal amplitude when the spin-orbit coupling is weak. Such a state carries pure spin current and no mass current. This spin current is a super-flow as it is the ground state and must be stable. We note that there hasbeen alot oftheoreticaland ex- perimental work on the counterflow of two species BEC [10–21]. For two miscible BECs with counterflow, it isfound that there is a critical relative speed between the two species, beyond which the state is dynamically un- stable [11–19]. It is shown that the instability can lead to proliferation of solitons [14, 15] and quantum turbulence [16]. This kind of counterflow is very similar to a spin current but it is not for two reasons: (1) Theoretically, if weregardthe twospecies astwocomponentsofapseudo- spin, this pseudo-spin has no SU(2) rotational symmetry as the number of bosons in each species is conserved. (2) Experimentally, it is hard to control the number of bosons in each component to create a spin current that has no mass current. The paper is organized as follows. In Sec. II, we first study the stability of a spin-1 planar counterflow. We identify the mechanisms associated with the instabilities, and find that the quadratic Zeeman effect can stabilize suchaplanarflow. Wethen studythe similarsituationin the circular geometry in Sec. III. The pure spin current consists of a vortex and anti-vortex in the m=−1,1 components. The experimental schemes to realize the stable pure spin current is discussed in Sec. IV. Finally, we briefly summarize our main results in Sec. V. II. PLANAR FLOW Thedynamicsofaspin-1BECinfreespaceisgoverned by the mean field Gross-Pitaevskii (GP) equation [7], i/planckover2pi1∂ ∂tψm=−/planckover2pi12∇2 2Mψm+c0ρψm+c21/summationdisplay n=−1s·Smnψn,(1) whereψm(m= 1,0,−1) are the components of the macroscopic wave function. ρ=/summationtext1 m=−1|ψm|2is the total density, si=/summationtext mnψ∗ m(Si)mnψnis the spin density vector and S= (Sx,Sy,Sz) is the spin operator vector withSi(i=x,y,z) being the three Pauli matrices in the spin-1 representation. The collisional interactions in- clude a spin-independent part c0= 4π/planckover2pi12(a0+ 2a2)/3M and a spin-dependent part c2= 4π/planckover2pi12(a2−a0)/3M, with af(f= 0,2) being the s-wave scattering length for spin-1 atoms in the symmetric channel of total spin f.2 We considera spin currentstate of the aboveGP equa- tion with the form ψ=/radicalbiggn 2 eik1·r 0 eik2·r , (2) wherenis the density of the uniform BEC. The require- ment of equal chemical potential leads to |k1|=|k2|. In the case where k1=−k2, this state carries a pure spin current: the total mass current is zero as it has equal mass counterflow while the spin current is nonzero. It is instructive to first consider the special case when there is no counterflow, i.e., k1=k2= 0. The exci- tation spectra are found to be ǫ0=/radicalBig 2c2nǫq+ǫ2qand ǫ±1 1=/radicalBig 2c0nǫq+ǫ2q,ǫ±1 2=/radicalBig 2c2nǫq+ǫ2q, respectively, withǫq=/planckover2pi12q2/2M. So for antiferromagnetic interac- tion (c0>0,c2>0), all branches of the spectra are real and there is a double degeneracy in one branch of the spectra. The phonon excitations give two sound veloci- ties,/radicalbig nci/M(i= 0,2), corresponding to the speeds of density wave and spin wave, respectively. However, the existence of phonon excitation does not mean that the pure spin current ( k1=k2/ne}ationslash= 0) is a super-flow as we can not obtain the current with k1=k2/ne}ationslash= 0 from the state withk1=k2= 0 by a Galilean transformation. The stability of the spin current has been studied in Ref. [10] for the case k1=−k2/ne}ationslash= 0. It is found that, for the antiferromagnetic interaction case ( c0>0,c2>0), the excitation spectrum of the m= 0 component always has nonzero imaginary part in the long wavelength limit as long as there is counterflow between the two compo- nents, and the imaginary excitations in the m= 1,−1 components only appear for a large enough relative ve- locityv1= 2/radicalbig nc2/M. For the ferromagnetic interac- tion case (c0>0,c2<0), both excitation spectra of the m= 0 andm= 1,−1 components have nonzero imagi- nary parts for any relative velocity. This means that the pure spin current cannot be stable in any cases. For the general non-collinear case ( k=k1+k2 2/ne}ationslash= 0) and antiferromagnetic interaction, the excitation spec- trum for the m= 0 component is found to be ǫ0=/radicalBigg/parenleftbigg ǫq+/planckover2pi12 2M(|k|2−|k1|2)+c2n/parenrightbigg2 −c2 2n2.(3) We see here that as long as the momenta of the two com- ponents are not exactly parallel, i.e., k1is not exactly equal to k2, then|k|<|k1|, and there is always dynam- ical instability for the long wavelength excitations. Therefore, the spin current in Eq. (2) is generally un- stable and not a super-flow. This instability originates from the interaction process described by ψ† 0ψ† 0ψ1ψ−1in the second quantized Hamiltonian. This energetically fa- vored process converts two particles in the m= 1,−1 components, respectively, into two stationary particles in them= 0 component. To suppress such a process andachieve a stable pure spin current, one can utilize the quadratic Zeeman effect. With the quadratic Zeeman ef- fect of negative coefficient, the Hamiltonian adopts an additional term λm2(λ <0 andm= 1,0,−1). This term does not change the energy of the m= 0 compo- nent, but lowers the energy of the other two components m= 1,−1. As a result, there arises a barrier for two atoms in the m= 1,−1 components scattering to the m= 0 component, and the scattering process is thus suppressed. The above intuitive argument can be made more rigor- ous and quantitative. Consider the case k1=−k2. With the quadratic Zeeman term, the excitation spectrum for them= 0 component changes to ǫ0=/radicalBigg/parenleftbigg ǫq−/planckover2pi12|k1|2 2M+c2n−λ/parenrightbigg2 −c2 2n2.(4) Soaslongas −λ−/planckover2pi12|k1|2/2M >0, longwavelengthexci- tationswill be stableforthe m= 0component. Fromthe excitationofthe m= 0component, onecanobtainacrit- icalrelativevelocityofthespincurrent, v0= 2/radicalbig −2λ/M. Thereis anothernonzerocriticalvelocity v1= 2/radicalbig nc2/M determined by the excitations of the m= 1,−1 compo- nents. The overall critical velocity of the system is the smaller one of v0andv1. Therefore, below the critical relative velocity vc= min{v0,v1}, the pure spin current is stable and a super-flow. The experimental scheme to realize such a Zeeman effect will be discussed in Sec. IV. III. CIRCULAR FLOW In the cylindrical geometry, we consider a pure spin current formed by two vortices with opposite circulation in them= 1,−1 components. From similar arguments, one can expect that interaction will make such a cur- rent unstable. Inspired by the quadratic Zeeman effect method above, we propose to use spin-orbit coupling to stabilize it. The spin-orbit coupling can be viewed as a momentum-dependent effective magnetic field that ex- erts only on the m= 1,−1 components. Therefore, it is possible that spin-orbit coupling lowers the energy of m= 1,−1 components, and consequently suppresses the interaction process leading to the instability. The model of spin-1 BEC subject to Rashba spin-orbit coupling can be described by the following energy func- tional, E[ψα] =/integraldisplay dr/braceleftBigg/summationdisplay α/planckover2pi12|∇ψα|2 2M+ρV(r)+c0 2ρ2+c2 2s2 +γ/an}b∇acketle{tSxpy−Sypx/an}b∇acket∇i}ht/bracerightBigg , (5) whereρis the density, V(r) =1 2Mω2(x2+y2) is the trap- ping potential, and γis the strength of spin-orbit cou- pling./an}b∇acketle{t···/an}b∇acket∇i}htistheexpectationvaluetakenwithrespectto3 the three component wave function ψ= (ψ1,ψ0,ψ−1)T. The strength of the spin-orbit coupling γdefines a char- acteristic length asoc=/planckover2pi1/Mγ, and can be rescaled to be dimensionless with respect to the harmonic oscillator lengthah=/radicalbig /planckover2pi1/Mω. Then we characterize the strength of spin-orbit coupling with the dimensionless quantity κ=ah/asoc=γ/radicalbig M//planckover2pi1ω. The spin-orbit coupling of Rashbatypeherecanbegeneratedinvariousways,which will be discussed in the next section. The above model can describe a spin-1 BEC of23Na confined in a two-dimensional harmonic trap. Assume the atom number is about 106. Using the estimate of scattering lengths a0= 50aB,a2= 55aB[22], withaB being the Bohr radius, the ground state of spin-123Na should be antiferromagnetic because c0>0,c2>0 [5]. Previous studies of spin-1 BEC with Rashba spin-orbit coupling mostly focus on the strong spin-orbit coupling regime, where the ground state is found to be the plane wave phase or the stripe phase, for ferromagnetic in- teraction and antiferromagnetic interaction, respectively [23]. Here we are interested in the antiferromagnetic in- teraction case and the weak spin-orbit coupling regime (κ≪1), and calculate the ground state wave function of the energy functional with the method of imaginary time evolution. Wefindthatwhenthespin-orbitcouplingisweak( κ≪ 1), the ground state wave function has the form ψ= χ1(r)e−iφ χ0(r) χ−1(r)eiφ , (6) withχ1(r) =−χ−1(r) and allχireal. The ground state is shown in Fig. 1. Such a ground state consists of an anti-vortex in the m= 1 component and a vortex in the m=−1 component. The m= 0 component does not carry angular momentum. Since |ψ1|=|ψ−1|, the net mass current vanishes. The wave function in Eq. (6) can be understood in the single particle level. In terms of the ladder operators of spin and angular momentum, the spin-orbit coupling term reads Hsoc=γ√ M/planckover2pi1ω 2/bracketleftBig S+/parenleftBig ˆaR−ˆa† L/parenrightBig +S−/parenleftBig ˆa† R−ˆaL/parenrightBig/bracketrightBig , (7) whereS±is the ladder operator of spin, and ˆ a† L(R)is the creation operator of the left (right) circular quanta [24]. When the spin-orbit coupling is very weak ( κ≪1), its effect can be accounted for in a perturbative way. From the ground state Ψ(0)=|0,0/an}b∇acket∇i}ht, the first order correction to the wave function for small γis given by Ψ(1)=γ√ M/planckover2pi1ω 2/planckover2pi1ω/parenleftBig −S+ˆa† L+S−ˆa† R/parenrightBig |0,0/an}b∇acket∇i}ht =κ 2(−|1,−1/an}b∇acket∇i}ht+|−1,1/an}b∇acket∇i}ht), (8) where|ms,mo/an}b∇acket∇i}htdenotes a state with spin quantum num- bermsand orbital magnetic quantum number mo. One FIG. 1: (color online) Amplitudes (a1,b1,c1) and phase an- gles (a2,b2,c2) of the three component wave function ψ= (ψ1,ψ0,ψ−1)Tat the ground state of Hamiltonian (5) for a BEC of23Na confined in a 2D harmonic trap. The particle numberis 106, the frequency of the trap is 2 π×42 Hz, and the dimensionless spin-orbit coupling strength is κ= 0.04. The units of the XandYaxes areah. immediately sees that ψ1has angularmomentum −/planckover2pi1and ψ−1has angular momentum /planckover2pi1. Besides, the amplitudes of bothψ1andψ−1are proportional to κ. There exits a continuity equation for spin density and spin current, which is d dt/parenleftbig ψ†Sµψ/parenrightbig +∇·Js µ= 0. (9) The spin current density tensor Js µ(µ=x,y,zdenotes the spin component) is defined as [25, 26] Js µ=1 2/braceleftbig ψ†Sµvψ+c.c./bracerightbig =1 2 /summationdisplay m,n,lψ∗ m(Sµ)mnvnlψl+c.c. ,(10) where vnl=p M+γ(ˆz×Snl), (11)4 −2−1 012−2−1012 XY FIG. 2: (color online) Distribution of the spin current dens i- tiesJs x(blue arrow), Js y(red arrow) and Js z(black arrow) of the ground state shown in Fig. 1. The length of the arrows represents the strength of the spin current. The arrow lengt h of different colors is not to scale. κ= 0.04. The units of the XandYaxes areah. and c.c. means the complex conjugate. The second part invnlis induced by the spin-orbit coupling. By the definition in Eq. (10), the spin current density carried by the ground state (6) is Js x=γsin2φ|ψ1|2ˆx+γ/parenleftbig |ψ0|2+2|ψ1|2sin2φ/parenrightbig ˆy, Js y=−γ/parenleftbig |ψ0|2+2|ψ1|2cos2φ/parenrightbig ˆx−γsin2φ|ψ1|2ˆy, Js z=/parenleftBig −2/planckover2pi1|ψ1|2 Mr+√ 2γ|ψ1ψ0|/parenrightBig ˆφ. (12) From both analytical and numerical results of the wave function, |ψ1| ≪ |ψ0|, soJs xroughly points in the ydi- rection, while Js yalmost points in the −xdirection. Js z represents a flow whose amplitude has rotational symme- try. From the numerical results shown in Fig. 2, we see thatJs zis a counter-clockwise flow. The amplitudes of Js xandJs yare of the same order, both proportional to κ, while that of Js z, proportional to κ2, is much smaller. It is evident that the state in Eq. (6) carries no mass cur- rent and only pure spin current. Since the spin current is in the ground state, it must be stable. In this way, we have realized a superfluid of pure spin current, or a pure spin super-current. IV. EXPERIMENTAL SCHEMES In this section, we propose the experimental schemes to generate and detect the pure spin currents discussed in Sec. II and Sec. III.The planar pure spin current can be easily generated. By applying a magnetic field gradient, the two compo- nentsm= 1,−1 will be accelerated in opposite direc- tions and a pure spin current is generated as done in Refs. [14, 15]. To stabilize this spin current, one needs to generate the quadratic Zeeman effect. We apply an oscil- lating magnetic field Bsinωtwith the frequency ωbeing much larger than the characteristic frequency of the con- densate, e.g., the chemical potential µ. The time averag- ing removes the linear Zeeman effect; only the quadratic Zeeman effect remains. The coefficient of the quadratic Zeeman effect from the second-orderperturbation theory is given by λ= (gµBB)2/∆Ehf, wheregis the Land´ e g- factoroftheatom, µBisthe Bohrmagneton, and∆ Ehfis the hyperfine energy splitting [27]. For the F= 2 mani- fold of87Rb, ∆Ehf<0, sothe coefficient ofthe quadratic Zeeman effect is negative. The circular flow in Sec. III may find prospective real- izations in two different systems: cold atoms and exciton BEC. In cold atoms, we consider a system consisting of a BEC of23Na confined in a pancake trap, where the con- finement in the zdirection is so tight that one can treat the system effectively as two dimensional. The spin-orbit couplingcanbe inducedbytwodifferentmethods. Oneis by the exertion of a strong external electric field Ein the zdirection. Due to the relativistic effect, the magnetic moment of the atom will experience a weak spin-orbit coupling, where the strength γ=gµB|E|/Mc2. HereM is the atomic mass and cis the speed of light. For weak spin-orbit coupling (small γ), the fraction ofatoms in the m= 1,−1 components is proportional to γ2. For an ex- perimentally observable fraction of atoms, e.g., 0 .1% of 106atoms, using the typical parameters of23Na BEC, the estimated electric field is of the same order of mag- nitude as the vacuum breakdown field. For atoms with smaller mass or larger magnetic moment, the required electricfield canbe lowered. Another method ofrealizing spin-orbit coupling is to exploit the atom laser interac- tion, where strong spin-orbit coupling can be created in principle [28]. In exciton BEC systems, as the effective mass of exciton is much smaller than that of atom, the required electric field is four to five orders of magnitude smaller, which is quite feasible in experiments [29–32]. The vortex and anti-vortex in the m= 1,−1 compo- nents can be detected by the method of time of flight. First one can split the three spin components with the Stern-Gerlach effect. The appearance of vortex or anti- vortex in the m= 1,−1 components is signaled by a ring structure in the time of flight image. After a suffi- ciently long time of expansion, the ring structure should be clearly visible. V. CONCLUSION In summary, we have studied the stability of a pure spin current of a spin-1 BEC. In the planar flow, the system always suffers from dynamical instability. The5 origin of the instability is the interaction process that converts two particles in the m= 1,−1 components into them= 0 component. Based on this, we propose a method to stabilize the pure spin current by utilizing the quadratic Zeeman effect. In the circular flow, we have proposedtouse spin-orbitcouplingtomakethepurespin current stable. For weak spin-orbit coupling, we have found that the ground state of the system is a superfluid ofpurespincurrent. Theexperimentalschemestorealizeand detect these pure spin currents have been discussed. Acknowledgements This work is supported by the NBRP of China (2013CB921903,2012CB921300) and the NSF of China (11274024,11274364,11334001,11429402). [1] D. Hall, M. Matthews, J. Ensher, C. Wieman, and E. Cornell, Phys. Rev. Lett. 81, 4531 (1998). [2] D. Hall, M. Matthews, C. Wieman, and E. Cornell, Phys. Rev. Lett. 81, 1543 (1998). [3] M. Matthews et. al., Phys. Rev. Lett. 81, 243 (1998). [4] J. Stenger et. al., Nature (London) 396, 345 (1998). [5] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998). [6] T. Ohmi, and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998). [7] Seefor example, D. M. Stamper-Kurnand M. Ueda, Rev. Mod. Phys. 85, 1191 (2013) and references therein. [8] Q.-F. Sun, Z.-T. Jiang, Y. Yu, and X. C. Xie, Phys. Rev. B84, 214501 (2011). [9] Z.-Q. Bao, X. C. Xie, and Q.-F. Sun, Nat. Commun. 4, 2951 (2013). [10] K. Fujimoto and M. Tsubota, Phys. Rev. A 85, 033642 (2012). [11] C. K. Law, C. M. Chan, P. T. Leung, and M.-C. Chu, Phys. Rev. A 63, 063612 (2001). [12] A. B. Kuklov and B. V. Svistunov, Phys. Rev. Lett. 90, 100401 (2003). [13] V. I. Yukalov and E. P. Yukalova, Laser Phys. Lett. 1, 50 (2004). [14] M. A. Hoefer, J. J. Chang, C. Hamner, and P. Engels, Phys. Rev. A 84, 041605(R) (2011). [15] C. Hamner, J. J. Chang, P. Engels, and M. A. Hoefer, Phys. Rev. Lett. 106, 065302 (2011). [16] H. Takeuchi, S. Ishino, and M. Tsubota, Phys. Rev. Lett. 105, 205301 (2010).[17] S. Ishino, M. Tsubota, and H. Takeuchi, Phys. Rev. A 83, 063602 (2011). [18] L. Y. Kravchenko and D. V. Fil, J. Low Temp. Phys. 155, 219 (2009). [19] M. Abad, A. Sartori, S. Finazzi, and A. Recati, Phys. Rev. A89, 053602 (2014). [20] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M. Stamper-Kurn, Phys. Rev. Lett. 100, 170403 (2008). [21] R. W. Cherng, V. Gritsev, D. M. Stamper-Kurn, and E. Demler, Phys. Rev. Lett. 100, 180404 (2008). [22] A. Crubellier et. al., Eur. Phys. J. D 6, 211 (1999). [23] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett.105, 160403 (2010). [24] C. Cohen-Tannoudji, B.Diu, andF. Laloe, Quantum Me- chanics, Vol. 1, Wiley, 1991. [25] Q.-F. Sun and X. C. Xie, Phys. Rev. B 72245303 (2005). [26] Q.-F. Sun, X. C. Xie, and J. Wang, Phys. Rev. B 77 035327 (2008). [27] M. Ueda, Fundamentals and New Frontiers of Bose- Einstein Condensation , World Scientific, 2010. [28] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg, Rev. Mod. Phys. 83, 1523 (2011). [29] C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys. Rev. Lett. 69, 3314 (1992). [30] J. Kasprzak et al., Nature 443, 409 (2006). [31] K.G. Lagoudakis et al., Nat. Phys. 4, 706 (2008). [32] A. Amo et al., Nat. Phys. 5, 805 (2009).

1703.06234v1.Spin_orbit_scattering_visualized_in_quasiparticle_interference.pdf

Spin-orbit scattering visualized in quasiparticle interference Y. Kohsaka,1,T. Machida,1K. Iwaya,1M. Kanou,2T. Hanaguri,1and T. Sasagawa2 1RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan. 2Laboratory for Materials and Structures, Tokyo Institute of Technology, Yokohama, Kanagawa 226-8503, Japan. (Dated: March 21, 2017) Abstract In the presence of spin-orbit coupling, electron scattering o impurities depends on both spin and orbital angular momentum of electrons { spin-orbit scattering. Although some transport prop- erties are subject to spin-orbit scattering, experimental techniques directly accessible to this eect are limited. Here we show that a signature of spin-orbit scattering manifests itself in quasiparticle interference (QPI) imaged by spectroscopic-imaging scanning tunneling microscopy. The experi- mental data of a polar semiconductor BiTeI are well reproduced by numerical simulations with the T-matrix formalism that include not only scalar scattering normally adopted but also spin-orbit scattering stronger than scalar scattering. To accelerate the simulations, we extend the standard ecient method of QPI calculation for momentum-independent scattering to be applicable even for spin-orbit scattering. We further identify a selection rule that makes spin-orbit scattering visible in the QPI pattern. These results demonstrate that spin-orbit scattering can exert predominant in uence on QPI patterns and thus suggest that QPI measurement is available to detect spin-orbit scattering. kohsaka@riken.jp 1arXiv:1703.06234v1 [cond-mat.mes-hall] 18 Mar 2017I. INTRODUCTION Spin-dependent scattering has played important roles in many elds of physics for a long time. Spin-dependent asymmetric scattering of electron beams in vacuum provided a foun- dation of relativistic quantum mechanics [1, 2]. In condensed matter physics, spin-orbit scattering of electrons propagating in solids contributes to some transport phenomena [3{5]. For example, spin-dependent impurity scattering caused by spin-orbit scattering is among the origins of anomalous Hall eect and (extrinsic) spin Hall eect [4, 5]. Another direct consequence of spin-orbit scattering is rotation of electron spin, changing interference be- tween wave functions of electrons around an impurity. This eect on quantum interference is known as the origin of weak anti-localization [3]. Interference of wave functions results in a periodic modulation of the local density of states (LDOS). This modulation, known as quasiparticle interference (QPI), has been im- aged by spectroscopic imaging scanning tunneling microscopy (SI-STM) in a wide variety of materials [6{14]. QPI has been studied mostly to acquire momentum-resolved information of electronic states from its characteristic periodicity based on an assumption that the scatter- ing center is a scalar one. This assumption is widely used even for strong spin-orbit coupling systems [9{11, 13, 14] where spin-orbit scattering is also likely to be strong. However, the role of spin-orbit scattering in QPI is obscure. A primal dierence between spin-orbit and scalar scattering is that the former depends on both spin and momentum of electrons whereas the latter does not. Consequently, spin-orbit scattering can cause additional enhancement or suppression of QPI that is unanticipated for scalar scattering. Lee et al. theoretically indicate that spin-orbit scattering enhances new scattering channels for the surface states of topological insulator Bi 2Te3[15]. In the experiments, however, such enhancement and resultant multiple branches of QPI have never been observed [9, 10]. In this paper, exploiting atomic-resolution SI-STM and numerical simulations, we reveal that spin-orbit scattering is predominant for QPI of the quasi-two-dimensional states of a polar semiconductor BiTeI. We performed a detailed analysis of QPI using the standard T- matrix formalism with an extended technique to accelerate calculations including not only momentum-independent scalar scattering but also momentum-dependent spin-orbit scatter- ing. All the components of QPI observed by the experiments are successfully reproduced 2when spin-orbit scattering is considered. The key ingredient is the selection rule due to spin- orbit scattering, which selectively enhances one of the main scattering channels contributing QPI of BiTeI. Our nding demonstrates that spin-orbit scattering is actually observed in QPI and provides a foothold to get better insight into the electronic states through QPI. II. OVERVIEW OF EXPERMENTAL RESULTS OF BiTeI BiTeI has a polar crystal structure with layered stacking of triple layers composed of Te, Bi, and I layers, and hosts giant Rashba-type spin splitting in bulk bands as well as at the surface as observed by angle-resolved photoemission spectroscopy (ARPES) [16{20]. A domain structure composed of opposite stacking orders is found in this material, resulting in two kinds of termination at a single surface [21{24]. Especially in the Te-terminated areas, dierential conductance ( dI=dV ) images show clear QPI of quasi-two-dimensional states split o from the bulk valence band by the spontaneous electric polarization [24]. To dene points to be focused, we brie y summarize QPI of BiTeI. (Experimental details are described in Ref. 24.) Figure 1(a) shows a typical image of QPI in the Te-terminated area. The Fourier-transformed image shown in Fig. 1(b) reveals that the QPI patterns consist of three major components: the hexagonal ring surrounding the point, the strong peaks in the -M direction, and the outermost humps in the -M direction. These features are commonly observed in all samples studied. By comparing QPI dispersions with the ARPES results, [16{19] the hexagonal ring and the strong peaks are assigned to interband scattering between the spin-split bands and intraband scattering between the corners of the hexagonally warped outer branch, respectively, as depicted in Fig. 1(c). The outermost humps change their locations depending on bias voltages for stabilizing the scanning tip, and therefore are attributed to the so-called setpoint eect discussed later. The near- feature, which varies from one sample to another, originates from nanometer-scale inhomogeneity due to random distribution of defects. Although the positions of the hexagonal ring and the strong -M peaks are understood as described above, there remains a puzzle in their intensities. The intraband scattering is mostly forbidden because the backscattering from ktokis suppressed due to the anti- parallel spin orientations. The -M peaks are nevertheless allowed because of deviation from the backscattering. Meanwhile, the interband scattering giving the hexagonal ring is always 30.650.650.65 0.600.600.600.750.750.75(nS)(nS)(nS) 0.700.700.70 Setpoint (“hump”)qinterqinterqinterqintraqintraqintra qinterqintra MK qinter qintra MГ M K MqinterqintraIntra-band (“peak”) Inter-band(“ring”)(b) (a)(a)(a) BiTeI, Te-surface, d I/dV BiTeI, Te-surface, dI/dV BiTeI, Te-surface, d I/dV 9 nm9 nm9 nm 0.6 Å-1MK (c)FIG. 1. (color online) (a) A 45 45 nm2dI=dV image taken at the Te-terminated surface of BiTeI. The image was taken at 10 mV with a lock-in modulation voltage of 5 mV rmsand a setup tunneling current of 0.2 nA at a setup bias voltage of 0.2 V. (b) Fourier transform of (a). (c) Schematic gures of the band structure of quasi-two-dimensional states at the Te-terminated surface observed by ARPES [16{18]. From left to right, a three-dimensional illustration of the band structure, the band dispersion in the -M direction, and a constant energy contour. The double-headed arrows denote dominant scattering channels producing the QPI. Spin directions are depicted by the arrows and markers colored in orange and blue; in-plane components are denoted by the arrows and out-of-plane components are denoted by the markers. allowed because the spin orientations are almost parallel. That is, the spin texture of the band structure is more benecial for the hexagonal ring than for the -M peaks, although the former is actually weaker than the latter. This inverted intensity is a robust signature of spin-orbit scattering, as revealed below. III. THE MODEL AND T-MATRIX FORMALISM To solve the puzzle of intensity, we numerically simulate QPI patterns. To model the quasi-two-dimensional state originating from the bulk valence band predominated by Bi 6 pz 4orbitals, [25] we employ an extended Rashba Hamiltonian, H0(kx;ky) = E0+k2 2mE(k) I+V(k)(kxykyx) + (k)(3k2 xk2 y)kyz; (1) whereIandi(i=x;y;z ) are the identity matrix and the Pauli matrices, respectively, with k=pk2 x+k2 y. IfE(k) andV(k) are constant and ( k) = 0, Eq. (1) gives the Bychkov- Rashba Hamiltonian [26]. The last term of Eq. (1) re ects C 3vsymmetry of BiTeI [16, 27]. We extend H0up tok6terms,E(k) = 1 +4k2+6k4,V(k) =v(1 +3k2+5k4), and (k) =(1 + 5k2) so that it is invariant under a three-fold rotation along the zdirection, mirror operation about the xzplane (xis along the -M direction), and the time-reversal operation. The higher terms up to k6are required for QPI calculations performed in the whole surface Brillouin zone whereas k3terms are enough to reproduce the ARPES results near the point [20]. We choose parameters as m= 0:0168 eV1A2,4=2:03A2, 6= 87:5A4,v= 3:13 eV A1,3=2:01A2,5= 323 A4,=41:7 eV A3, 5= 2:43A2, andE0=0:352 eV by tting experimental data (Fig. 2). QPI patterns have been calculated with the standard T-matrix formalism for a single local impurity. In fact, there are many defects in the eld of view of Fig. 1(a). However, the three major features of QPI are independent of details of defect distribution as evidenced by the experimental fact that they are observed in all samples. Therefore, we postulate that multiple impurities work on overall intensity in a statistical manner [12] and a single impurity is a good starting point to discuss QPI patterns. The LDOS is written by the retarded Green's function ^Gin momentum space, (q;!) =1 2iX kTrn ^G(k;kq;!)^G(k;k+q;!)o ; (2) where(q;!) is the Fourier transform of the LDOS, (q;!) =R (r;!)eiq
rdr. Here we consider the Green's function in matrix form to include spin. In the presence of a scattering center the potential of which in momentum space is ^Vk;k0, ^G(k;k0;!) =^G0(k;!)k;k0+^G0(k;!)^Tk;k0(!)^G0(k0;!); (3) where theTmatrix satises ^Tk;k0(!) =^Vk;k0+X p^Vk;p^G0(p;!)^Tp;k0(!): (4) 5-0.4-0.20.0E (eV) K Г M0 eV -0.20.00.2ky (Å-1) -0.2 0.0 0.2 kx (Å-1) -0.2 eV 0.2 eV0.2 eV-0.2 eV 0 eV-0.20.00.2ky (Å-1) -0.2 0.0 0.2 kx (Å-1)-0.3-0.2-0.10.0E (eV) -0.2 0.0 0.2 kx (Å-1)-0.20.00.2V (V) 0.5 0.4 0.3 qx (Å-1)(a) (b) (c) (d) (e)FIG. 2. (color online) Fitting results to determine the parameters of Eq. (1): (a) the Fermi surface, [20] (b) band dispersion in the -M direction, [20] and (c) QPI dispersion the in the -M direction [24]. Open circles and solid curves are experimental data and tting results, respectively. Since there is an energy oset
E0between ARPES and QPI dispersions, E0is given by the sum of two tting parameters, E0=EARPES 0 +
E0, whereEARPES 0 =0:179 eV and
E0=0:173 eV. The eigenvalues of the model Hamiltonian are shown in (d) constant energy contours and (e) dispersion along M--K. Here ^G0is the bare Green's function, ^G0(k;!) =P ngn(k;!) n(k) n(k),gn(k;!) = (!+in(k))1, wheren(k) and n(k) are thenth eigenvalue and the nth eigenstate of the bare Hamiltonian, respectively, with being a small broadening factor (10 meV for our simulations). QPI patterns can be computed in principle with these equations. For momentum- independent scattering (e.g., scalar scattering), a direct calculation of the ksummation requiresO(N4) operations for a single QPI image, where NNis the number of grid points. The amount of this calculation can be reduced to O(N2log2N) by using fast Fourier transform (FFT) [12, 28]. For momentum-dependent scattering (e.g., spin-orbit scattering), however, the FFT-based technique has not been applied and consequently the ksummation requiringO(N6) operations has been directly calculated. The enormous amount of calcula- tion has hindered precise and comprehensive analysis of QPI; calculations have often been done only in a narrow range and at a low resolution of energy and momentum [15]. We nd that the FFT-based technique is still available for momentum-dependent scattering satisfy- 6ing a certain condition. Because of this method, the amount of calculation for spin-orbit scattering can be reduced to O(N2log2N) that greatly accelerates our simulations. Details of the method are described in Appendix A. IV. NUMERICAL RESULTS We begin our simulations with a scalar impurity. In this case, the scattering potential is independent of momentum, ^V=V0I, whereV0is strength of scattering ( V0= 0:1 eV for all simulations). The Tmatrix is also simplied to a momentum-independent form. This simplication makes the calculation greatly easy and is why a scalar impurity is widely assumed as scattering center. The simulation result successfully reproduces the hexagonal ring and the -M peaks as shown in Fig. 3(a). The hexagonal ring appears at the location of interband scattering between the spin-split bands and the -M peaks lie slightly outside of intraband scattering of the outer branch, corroborating the peak assignment described above. No prominent feature appears near the point, being consistent with the experiments. (See Appendix B for details.) Although these basic features are reproduced, the ring is stronger than the peaks, replicating the puzzle. This discrepancy in intensities is robust as long as the band parameters are in a reasonable range, suggesting that the scattering is not a simple scalar one. We then consider two kinds of scattering, magnetic scattering and spin-orbit scattering. Since both rotate electron spin, they may change the situation that is benecial for the hexagonal ring. Figure 3(b) shows the calculation result with magnetic scattering ^V=V0z corresponding to a classical magnetic moment pointing in the zdirection. The -M peaks are still weaker than the hexagonal ring and the QPI pattern is strongly suppressed overall. In contrast, noteworthy results are found for spin-orbit scattering, ^Vk;k0=V0fI+ic(kk0)
g; (5) wherecis the eective spin-orbit coupling parameter [5] and denotes strength of spin-orbit scattering relative to that of scalar scattering. The -M peaks are selectively enhanced as c increases and become stronger than the hexagonal ring as shown in Fig. 3(c). The remaining feature, the humps in the -M direction, can be calculated by taking data acquisition procedures of SI-STM (the setpoint eect) into account. Even if a dI=dV 74.0 3.0 2.0 1.00.3 Å-1 0.03 0.01 0.050.3 Å-1 2.0 1.5 1.0 0.50.3 Å-1(b) magnetic (c) scaler + spin-orbit (a) scalarFIG. 3. (color online) Fourier-transformed images of QPI patterns at 10 meV,(q;!= 10 meV), calculated with dierent scattering centers. (a) Scalar scattering. (b) Magnetic scat- tering. (c) Scalar and spin-orbit scattering with c= 60 A2. The blue dashed lines in (a) depict scattering vectors expected from constant energy contours of the band dispersion. The inner and outer lines denote interband scattering between the spin-split bands and intraband scattering of the outer branch, respectively. spectrum is proportional to LDOS at each location as generally assumed, the proportional constant is not generally uniform but has a spatial structure re ecting variation of the tip height. The height of a scanning tip is adjusted at each location such that the tunneling current is a set value. The current is determined by the LDOS integrated up to a given bias voltageVset. Consequently, a dI=dV image observed by SI-STM depends on Vsetas well as the LDOS, dI dV(r;V;V set)/(r;eV)ZeVset 0(r;EF+)d; (6) whereEFis the Fermi energy [29]. The denominator of Eq. (6) represents the setpoint eect. This eect has been known in the experiments but neglected in the calculations of QPI. Full simulation including spin-orbit scattering and the setpoint eect is shown in Fig. 4. All of the hexagonal ring, the strong -M peaks, and the -M humps are well reproduced. The peak intensities agree with the experiment as shown in Fig. 4(e) with c= 80 A2or a dimensionless parameter c2=a02= 40 (a0= 4.34 A,a-axis length), indicating predominance of spin-orbit scattering over scalar scattering. 83 2 1 0Intensity (a. u.) 0.8 0.6 0.4 0.2 experiment simulation qГM (Å-1) MK Г experiment calculation (full) 0.15 0.00 -0.15 E (eV) 0.6 0.4 0.2 qГM (Å-1)calculation (no setpoint effect) 0.15 0.00 -0.15 E (eV)calculation (full) experiment0.15 0.00 -0.15 V (V)(a) (e)(b) (c) (d)FIG. 4. (color online) Full simulation including spin-orbit scattering with c= 80 A2and the setpoint eect with Vset= 0:2 V. (a) A side-by-side comparison between the experiment (left) and the full simulation (right) at 10 meV. (b, c, d) Energy dependence (dispersion) in the -M direction. (e) Line proles of (a) in the -M direction. An exponential background is subtracted from the experimental data. V. DISCUSSION The above results of numerical simulations clearly show that spin-orbit scattering is the crucial ingredient to explain the QPI intensities of BiTeI. The contrasting results of magnetic and spin-orbit scattering can be understood as follows. Electrons with spin-up and spin- down feel potentials of opposite signs for magnetic scattering [30]. In addition, electrons scattered to the right and the left do as well for spin-orbit scattering. The opposite signs result in suppression of QPI for magnetic scattering whereas the same sign, as a result of a combination of the two eects, is cooperative for QPI in the case of spin-orbit scattering. An essential point of this mechanism is the scattering amplitude. To the rst order of ^V, a contribution to QPI from a scattering process ( k0!k) and its time-reversal counterpart 9(k!k0) are written as k;k0(!) =1 X m;nIm gn(k;!)gm(k0;!) n(k)^Vk;k0 m(k0) m(k0) n(k) ; (7) k0;k(!) =1 X m;nIm gn(k;!)gm(k0;!) n(k)^Vk;k01 m(k0) m(k0) n(k) ; (8) where  is the time-reversal operator. (Derivation of these formulas is written in Ap- pendix C.) The dierence between the two processes is found to be the potential in the scattering amplitude. For scalar scattering ^Vk;k0=V0I, Eqs. (7) and (8) are the same, k;k0=k0;k. For magnetic scattering ^Vk;k0=V0i, the scattering amplitude changes sign under time reversal because  i1=i. Time-reversal processes thus always cancel with each other, k;k0=k0;k, leading to the strong suppression of QPI for magnetic scattering. (This explains why QPI patterns are unchanged even in the presence of magnetic impurities [10, 31].) For spin-orbit scattering ^Vk;k0=icV0(kk0)
, the scattering amplitude of time-reversal processes has the same sign because  ^Vk;k01= (i)cV0(kk0)
() = ^Vk;k0. Therefore, as for spin-orbit scattering, the two scattering processes cooperatively con- tribute to QPI without being canceled as in the case of scalar scattering. The preferential enhancement of the -M peaks (Fig. 3(c)) is attributed to the directional and spin-dependent nature of spin-orbit scattering. For electrons in two dimensions, spin- orbit scattering is written as ^Vk;k0=icV0kk0sink;k0z, wherek;k0is the angle from ktok0. Since the hexagonal ring mainly consists of k;k0withk;k0andh n(k)jzj m(k0)i 0, spin-orbit scattering does not contribute to the hexagonal ring whereas it does con- tribute to the -M peaks because k;k02=3 andh n(k)jzj n(k0)i6= 0. (Whether the scattering amplitude is zero or not is easily estimated as written in Appendix D.) This selective suppression is a selection rule originating from the scattering amplitude h n(k)j^Vk;k0j m(k0)i= 0, which is distinct from a selection rule stemming from orthogo- nal wave functions h n(k)j m(k0)i= 0 [10, 31{34]. Spin-orbit scattering exists in principle in any materials and grows with spin-orbit cou- pling. However, its appearance in QPI depends on details of relevant electronic states. QPI of surface states of Au(111) [6] is insensitive to spin-orbit scattering due to the selection rule,k;k0andh n(k)jzj m(k0)i0, being the same as the hexagonal ring of BiTeI. As for topological surface states of Bi 2Te3, [9, 10] we presume that spin-orbit scattering 10enhances the -M peaks so it does for BiTeI, but its in uence remains to be claried. In this sense, the quasi-two-dimensional states of BiTeI with two scattering channels, one of which is sensitive to spin-orbit scattering and the other insensitive, are suited to investigate eects of spin-orbit scattering. QPI arising from electronic states near the Brillouin-zone boundary may be subject to spin-orbit scattering because of large momenta and k;k0=2 [15]. Such candidates are found in topological crystalline insulators and Weyl semimetals [11, 13, 14]. Including other factors aecting QPI intensities would be conducive to better quantifying strength of spin-orbit scattering. A delta-function scattering potential is used for simplicity in our simulations. In a more realistic case, V0in Eq. (5) is changed from a constant to V(jqj) for a spherical scattering potential. Since V(jqj) usually decreases monotonically with increasingjqj, QPI intensities are prone to be suppressed at large jqj, where the amplitude of spin-orbit scattering is large. Finite sharpness of a scanning tip also causes a similar eect. Strength of spin-orbit scattering therefore may be underestimated due to these factors. Nevertheless, the strength of spin-orbit scattering obtained for BiTeI (80 A2) is much larger than a theoretical value for n-GaAs (5.3 A2), [5] being consistent with strong spin-orbit coupling in BiTeI. We note that the obtained value is averaged over many defects of multiple kinds. If defects are separated enough, spin-orbit scattering can be probed at individual defects and may be available for designing and optimizing materials of spin Hall eect. VI. CONCLUSIONS In conclusion, using atomic-resolution SI-STM and numerical simulations with the T- matrix formalism, we identify a signature of spin-orbit scattering in the QPI patterns of BiTeI. Spin-orbit scattering manifests itself in QPI through a selection rule originating from the scattering amplitude. Our results highlight the importance of the scattering process beyond featureless scalar scattering and, more importantly, suggest a potential capability of QPI measurement as a local, direct (unaected by scattering time), and quantitative probe of spin-orbit scattering detected heretofore by transport measurements. We believe that including spin-orbit scattering into QPI analysis, which is now readily possible as demonstrated in our simulations, leads to a deeper understanding of electronic states of and more functionality from strong spin-orbit coupling systems. 11ACKNOWLEDGMENTS We thank M. S. Bahramy and Wei-Cheng Lee for fruitful discussions. This work is supported by the Murata Science Foundation. Appendix A: Fast calculation of QPI patterns Theksummation in Eq. (2) with the integral equation about the T-matrix Eq. (4) re- quiresO(N4) operations for single q, resulting inO(N6) operations in total to calculate a QPI image. In the case of momentum-independent scattering, the total amount of calcula- tion is reduced to O(N4) and further reduced to O(N2log2N) by using FFT. Here we show that, even in the case of momentum-dependent scattering, the amount of calculation can be reduced toO(N2log2N) fromO(N6) of theksummation when the scattering satises a certain condition. To introduce our approach to reduce the amount of calculation, we begin with momentum- independent scattering. Since the Tmatrix is also independent of k, Eqs.(2)-(4) are simpli- ed to (q;!)1 2iX kTrn ^G0(k;!)^T(!)^G0(kq;!)^G 0(k;!)^T(!)^G 0(k+q;!)o ;(A1) where ^T(!) = I^VP k^G0(k;!)1^V. Only the inhomogeneous part of LDOS is shown in Eq. (A1) for brevity because it gives spatial modulations of QPI patterns. Each matrix element of the ksummation in Eq. (A1) is X kTrn ^G0(k;!)^T(!)^G0(k+q;!)o =X m;nt(mn)(!)X k;jg(jm) 0(k;!)g(nj) 0(k+q;!);(A2) whereg(ij) 0andt(ij)are matrix elements of ^G0and ^T, respectively. The right-hand side of Eq. (A2) is the cross-correlation between g(jm) 0andg(nj) 0, and thus can be expressed with Fourier transform, X kg(jm) 0(k;!)g(nj) 0(k+q;!) =X rg(jm) 0(r;!)g(nj) 0(r;!)eiq
r; (A3) whereg(ij) 0(r;!) =P kg(ij) 0(k;!)eik
r. Since the right-hand side of Eq. (A3) is the inverse Fourier transform of g(jm) 0(r;!)g(nj) 0(r;!), the left-hand side of Eq. (A3) can be calculated 12via FFT without taking the ksummation [12, 28]. The amount of calculation is thus reduced fromO(N4) of the direct calculation to O(N2log2N) of FFT. FFT, the essential point to reduce the amount of calculation, is available as in Eq. (A3) because Eq. (A1) is expressed virtually as a product of two matrices; one is a function of k and the other is a function of k+q. At a glance, this condition is not satised for general scatterers because the Tmatrix depends on momentum. However, when ^Vk;k0is expressed as a sum of products between k- andk0-dependent matrices ^Vk;k0=X j^uj(k)^vj(k0) =tu(k)v(k0); (A4) whereu(k) =t(^u1(k) ^u2(k)


) andv(k) =t(^v1(k) ^v2(k)


), FFT is available to calcu- late QPI patterns as in the case of momentum-independent scattering. We rewrite Eq. (4) in a form of a recurrence formula ^T(n) k;k0(!) =8 >< >:^Vk;k0 (n= 1); ^Vk;k0+P p^Vk;p^G0(p;!)^T(n1) p;k0(!) (n2):(A5) By multiplying v(k)^G0(k;!) from the left and taking a sum with respect to k, we obtain Kn(k0;!) =8 >< >:M(!)v(k0) ( n= 1); M(!)v(k0) +M(!)Kn1(k0;!) (n2);(A6) whereKn(k0;!) =P pv(p)^G0(p;!)^T(n) p;k0(!) andM(!) =P pv(p)^G0(p;!)tu(p). Since these equations are summarized as Kn(k0;!) =Pn j=1fM(!)gjv(k0), Eq. (A5) is also sum- marized as ^T(n) k;k0(!) =tu(k)n1X j=0fM(!)gjv(k0); (A7) wherefM(!)g0is the identity matrix,tu(k)fM(!)g0v(k0) =tu(k)v(k0). Now the Tmatrix and thus the second term of Eq. (3) are expressed as a product of kandk0terms, and FFT is available to calculate QPI patterns as described above. We stress that the integral equation of theTmatrix [Eq. (4)] is reduced virtually to a sum of fM(!)gj, which is independent ofk. Therefore, the calculation size is almost the same as that of momentum-independent scatterers. Namely, the total amount of QPI calculation is drastically reduced from O(N6) toO(N2log2N). 13(b) (c) (a) (e) (d) ~ seconds ~ hours ~ days ~ months ~ yearsFIG. 5. (color online) Simulations with the same parameters as Fig. 3(c) but at lower resolutions. Since the calculation of Fig. 3(c) took 4 s, calculations with direct ksummation of (a){(e) are estimated to take 4 s, 1 h, 1 day, 1 month, and 7 years, respectively. (e) The same as Fig. 3(c) shown for comparison. Spin-orbit scattering, ^Vk;k0=V0fI+ic(kk0)
g, is written in the form of Eq. (A4). In two dimension, it is written as ^Vk;k0=V0 I+ic(kxk0 ykyk0 x)z =tu(k)v(k0), where tu(k) = (I kxI kyI),v(k) =Uu(k), and U=0 BBB@V0I 0 0 0 0 icV0z 0icV0z01 CCCA: (A8) TheMmatrix is M(!) =UX k0 BBB@^G0(k;!)kx^G0(k;!)ky^G0(k;!) kx^G0(k;!)k2 x^G0(k;!)kxky^G0(k;!) ky^G0(k;!)kxky^G0(k;!)k2 y^G0(k;!)1 CCCA: (A9) Note that each element of Mis a 22 matrix independent of kand thusPMjis readily calculable. Calculation of Fig. 3(c) with this method took only several seconds with a desktop computer. This means direct ksummation at the same resolution is estimated to take about several years. Even if the direct calculations were done at lower resolutions, it may be dicult to nd a reasonable compromise between resolution and time as shown in Fig. 5, highlighting drastic reduction of calculation costs with our method. Appendix B: Limitations of the joint density of states approach All QPI patterns observed in the experiments show large intensities near q= 0 as shown in Fig. 1(b). Since these near- features vary from one eld of view to another, and one sample 140.3 Å-10.3 Å-1(b) SJDOS (a) JDOSFIG. 6. (color online) Fourier-transformed images of QPI patterns at 10 meV, calculated by (a) JDOS and (b) spin-dependent JDOS. to another, they originate from the nanoscale inhomogeneity due to random distribution of defects. The varying near- feature means that a QPI pattern near q= 0, if any, is small and masked by the nanoscale inhomogeneity. Therefore, no prominent feature near q= 0 in Fig. 3 is consistent with the experiments. Meanwhile, from the viewpoint of the so-called joint density of states (JDOS), one may expect a large QPI intensity near q= 0; if DOS is large at a given k, large DOS is also found neark, resulting in a large QPI intensity near q= 0. However, as revealed by JDOS calculations shown below, such a JDOS-derived pattern near q= 0 has never been observed in BiTeI. In the JDOS approach, QPI patterns are interpreted to be proportional to JDOS, JDOS(q;!)/X k0(k;!)0(kq;!); (B1) where0(k;!) is the DOS at k,0(k;!) =Imh Tr G0(k;!) i . To include a spin eect, spin-dependent JDOS is considered, SJDOS(q;!)/X i=0;x;y;zX ki(k;!)i(kq;!) (B2) wherei(k;!) =Imh Tr iG0(k;!) i . The JDOS approach always predicts a large QPI intensity near q= 0 because the JDOS is the auto-correlation of i(k;!). Actually, as shown in Fig. 6, calculations with the JDOS and spin-dependent JDOS approaches show an asterisk-like pattern centered at q= 0, sticking out in the -K direction, and extending close to the hexagonal ring of intraband scattering. However, such a salient pattern has never been observed in the experiments. This discrepancy between the JDOS calculations and the observed QPI patterns highlights limitations of the JDOS approach. 15The limitations of the JDOS approach derive from a dierence between QPI and the JDOS. The spin-dependent JDOS of Eq. (B2) can be written as SJDOS(q;!)/X kX m;nIm gn(k;!) Im gm(kq;!)  m(kq) n(k)2 : (B3) A QPI pattern calculated by the T-matrix formalism for scalar scattering is written in a similar form. Equation (7) with ^Vk;k0=V0Igives (q;!) =X kk;kq(!)/X kX m;nIm gn(k;!)gm(kq;!)  m(kq) n(k)2 :(B4) The product of Im gin Eq. (B3) is not included in Eq. (B4); in other words, that QPI is not directly related to JDOS as discussed in Ref. 28. Appendix C: Contribution to QPI from k!k0scattering process We denek;k0(!) such that its summation with respect to kgives(kk0;!), k;k0(!)1 2iTrn ^G(k;k0;!)^G(k0;k;!)o : (C1) Here we consider an approximation to the rst order of ^Vk;k0for simplicity. Since the T matrix is ^Tk;k0^Vk;k0, k;k0(!) =1 2iTrh ^G0(k;!)^Vk;k0^G0(k0;!)n ^G0(k0;!)^Vk0;k^G0(k;!)oi : (C2) The rst term is Trn ^G0(k;!)^Vk;k0^G0(k0;!)o =X m;ngn(k;!)gm(k0;!)Trn n(k) n(k)^Vk;k0 m(k0) m(k0)o (C3) =X m;ngn(k;!)gm(k0;!) n(k)^Vk;k0 m(k0) m(k0) n(k) : (C4) Similarly, the second term is Trn ^G0(k0;!)^Vk0;k^G0(k;!)o =X m;n gn(k;!)gm(k0;!)  n(k)^Vk0;ky m(k0) m(k0) n(k) : (C5) 16Given ^Vk0;ky=^Vk;k0as^Vis Hermitian, Eqs. (C4) and (C5) are summarized to k;k0(!) =1 X m;nIm gn(k;!)gm(k0;!) n(k)^Vk;k0 m(k0) m(k0) n(k) :(7) Replacingk(k0) withk0(k) gives the time-reversal counterpart, k0;k(!) =1 X m;nIm gn(k;!)gm(k0;!) m(k0)^Vk0;k n(k) n(k) m(k0) (C6) =1 X m;nIm gn(k;!)gm(k0;!) n(k)^Vk0;ky1 m(k0) m(k0) n(k) : (C7) The second follows from three identities; gn(k;!) =gn(k;!) asH0is time invariant, and L = 0Ly10 and  = 00 , where0 =  ,0 =  , andLis a linear operator [35]. The last two hold because  is antiunitary. Given ^Vk0;ky=^Vk;k0 as^Vis Hermitian, we obtain k0;k(!) =1 X m;nIm gn(k;!)gm(k0;!) n(k)^Vk;k01 m(k0) m(k0) n(k) : (8) Appendix D: Relation between the scattering amplitude and spin orientation LetSn(k) be the expectation value of the Pauli matrices, Sn(k) = n(k) n(k) , then the following relation holds Sn(k)
Sm(k0) = 2 n(k) m(k0)2 1: (D1) The-rotation operator along the zdirection for a spin-1/2 state is given as eiz=2=iz. Therefore, by rotating Sm(k0) and m(k0) byalong thezdirection, we obtain Sn(k)
RzSm(k0) = 2 n(k)z m(k0)2 1; (D2) whereRzdenotes-rotation of spin orientation along the zdirection. Equation (D2) means n(k)z m(k0) = 0 for two states with the spin orientations parallel and lying in the xy plane. [1] N. F. Mott, \The Scattering of Fast Electrons by Atomic Nuclei," Proc. R. Soc. A 124, 425{442 (1929). 17[2] C. G. Shull, C. T. Chase, and F. E. Myers, \Electron Polarization," Phys. Rev. 63, 29{37 (1943). [3] Shinobu Hikami, Anatoly I. Larkin, and Yosuke Nagaoka, \Spin-Orbit Interaction and Mag- netoresistance in the Two Dimensional Random System," Prog. Theor. Phys. 63, 707{710 (1980). [4] Naoto Nagaosa, Jairo Sinova, Shigeki Onoda, A. H. MacDonald, and N. P. Ong, \Anomalous Hall eect," Rev. Mod. Phys. 82, 1539{1592 (2010). [5] Jairo Sinova, Sergio O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, \Spin Hall eects," Rev. Mod. Phys. 87, 1213{1260 (2015). [6] Y. Hasegawa and P. Avouris, \Direct observation of standing wave formation at surface steps using scanning tunneling spectroscopy," Phys. Rev. Lett. 71, 1071{1074 (1993). [7] M. F. Crommie, C. P. Lutz, and D. M. Eigler, \Imaging standing waves in a two-dimensional electron gas," Nature 363, 524{527 (1993). [8] J. E. Homan, K. McElroy, D.-H. Lee, K. M Lang, H. Eisaki, S. Uchida, and J. C. Davis, \Imaging Quasiparticle Interference in Bi 2Sr2CaCu 2O8+," Science 297, 1148{1151 (2002). [9] Tong Zhang, Peng Cheng, Xi Chen, Jin-Feng Jia, Xucun Ma, Ke He, Lili Wang, Haijun Zhang, Xi Dai, Zhong Fang, Xincheng Xie, and Qi-Kun Xue, \Experimental Demonstration of Topological Surface States Protected by Time-Reversal Symmetry," Phys. Rev. Lett. 103, 266803 (2009). [10] Haim Beidenkopf, Pedram Roushan, Jungpil Seo, Lindsay Gorman, Ilya Drozdov, Yew San Hor, R. J. Cava, and Ali Yazdani, \Spatial uctuations of helical Dirac fermions on the surface of topological insulators," Nat. Phys. 7, 939{943 (2011). [11] Ilija Zeljkovic, Yoshinori Okada, Cheng-Yi Huang, R. Sankar, Daniel Walkup, Wenwen Zhou, Maksym Serbyn, Fangcheng Chou, Wei-Feng Tsai, Hsin Lin, Arun Bansil, Liang Fu, M. Zahid Hasan, and Vidya Madhavan, \Mapping the unconventional orbital texture in topological crystalline insulators," Nat. Phys. 10, 572{577 (2014). [12] C. J. Arguello, E. P. Rosenthal, E. F. Andrade, W. Jin, P. C. Yeh, N. Zaki, S. Jia, R. J. Cava, R. M. Fernandes, A. J. Millis, T. Valla, R. M. Osgood, and A. N. Pasupathy, \Quasipar- ticle Interference, Quasiparticle Interactions, and the Origin of the Charge Density Wave in 2H-NbSe 2," Phys. Rev. Lett. 114, 037001 (2015). 18[13] Hao Zheng, Su-Yang Xu, Guang Bian, Cheng Guo, Guoqing Chang, Daniel S. Sanchez, Ilya Belopolski, Chi-Cheng Lee, Shin-Ming Huang, Xiao Zhang, Raman Sankar, Nasser Alidoust, Tay-Rong Chang, Fan Wu, Titus Neupert, Fangcheng Chou, Horng-Tay Jeng, Nan Yao, Arun Bansil, Shuang Jia, Hsin Lin, and M. Zahid Hasan, \Atomic-Scale Visualization of Quantum Interference on a Weyl Semimetal Surface by Scanning Tunneling Microscopy," ACS Nano 10, 1378{1385 (2016). [14] Hiroyuki Inoue, Andr as Gyenis, Zhijun Wang, Jian Li, Seong Woo Oh, Shan Jiang, Ni Ni, B. Andrei Bernevig, and Ali Yazdani, \Quasiparticle interference of the Fermi arcs and surface-bulk connectivity of a Weyl semimetal," Science 351, 1184{1187 (2016). [15] Wei-Cheng Lee, Congjun Wu, Daniel P. Arovas, and Shou-Cheng Zhang, \Quasiparticle Interference on the Surface of the Topological Insulator Bi 2Te3," Phys. Rev. B 80, 245439 (2009). [16] K. Ishizaka, M. S. Bahramy, H. Murakawa, M. Sakano, T. Shimojima, T. Sonobe, K. Koizumi, S. Shin, H. Miyahara, A. Kimura, K. Miyamoto, T. Okuda, H. Namatame, M. Taniguchi, R. Arita, N. Nagaosa, K. Kobayashi, Y. Murakami, R. Kumai, Y. Kaneko, Y. Onose, and Y. Tokura, \Giant Rashba-type spin splitting in bulk BiTeI," Nat. Mater. 10, 521{526 (2011). [17] A. Crepaldi, L. Moreschini, G. Aut es, C. Tournier-Colletta, S. Moser, N. Virk, H. Berger, Ph. Bugnon, Y. J. Chang, K. Kern, A. Bostwick, E. Rotenberg, O. V. Yazyev, and M. Grioni, \Giant Ambipolar Rashba Eect in the Semiconductor BiTeI," Phys. Rev. Lett. 109, 096803 (2012). [18] Gabriel Landolt, Sergey V. Eremeev, Yury M. Koroteev, Bartosz Slomski, Stefan Mu, Titus Neupert, Masaki Kobayashi, Vladimir N. Strocov, Thorsten Schmitt, Ziya S. Aliev, Maham- mad B. Babanly, Imamaddin R. Amiraslanov, Evgueni V. Chulkov, J urg Osterwalder, and J. Hugo Dil, \Disentanglement of Surface and Bulk Rashba Spin Splittings in Noncentrosym- metric BiTeI," Phys. Rev. Lett. 109, 116403 (2012). [19] M. Sakano, J. Miyawaki, A. Chainani, Y. Takata, T. Sonobe, T. Shimojima, M. Oura, S. Shin, M. S. Bahramy, R. Arita, N. Nagaosa, H. Murakawa, Y. Kaneko, Y. Tokura, and K. Ishizaka, \Three-dimensional bulk band dispersion in polar BiTeI with giant Rashba-type spin split- ting," Phys. Rev. B 86, 085204 (2012). [20] M. Sakano, M. S. Bahramy, A. Katayama, T. Shimojima, H. Murakawa, Y. Kaneko, W. Malaeb, S. Shin, K. Ono, H. Kumigashira, R. Arita, N. Nagaosa, H. Y. Hwang, Y. Tokura, 19and K. Ishizaka, \Strongly Spin-Orbit Coupled Two-Dimensional Electron Gas Emerging near the Surface of Polar Semiconductors," Phys. Rev. Lett. 110, 107204 (2013). [21] C. Tournier-Colletta, G. Aut es, B. Kierren, Ph. Bugnon, H. Berger, Y. Fagot-Revurat, O. V. Yazyev, M. Grioni, and D. Malterre, \Atomic and electronic structure of a Rashba p-n junction at the BiTeI surface," Phys. Rev. B 89, 085402 (2014). [22] Christopher John Butler, Hung-Hsiang Yang, Jhen-Yong Hong, Shih-Hao Hsu, Raman Sankar, Chun-I Lu, Hsin-Yu Lu, Kui-Hon Ou Yang, Hung-Wei Shiu, Chia-Hao Chen, Chao-Cheng Kaun, Guo-Jiun Shu, Fang-Cheng Chou, and Minn-Tsong Lin, \Mapping polarization induced surface band bending on the Rashba semiconductor BiTeI," Nat. commun. 5, 4066 (2014). [23] Sebastian Fiedler, Lydia El-Kareh, Sergey V. Eremeev, Oleg E. Tereshchenko, Christoph Seibel, Peter Lutz, Konstantin A. Kokh, Evgueni V. Chulkov, Tatyana V. Kuznetsova, Vladimir I. Grebennikov, Hendrik Bentmann, Matthias Bode, and Friedrich Reinert, \Defect and structural imperfection eects on the electronic properties of BiTeI surfaces," New J. Phys. 16, 075013 (2014). [24] Y. Kohsaka, M. Kanou, H. Takagi, T. Hanaguri, and T. Sasagawa, \Imaging ambipolar two- dimensional carriers induced by the spontaneous electric polarization of a polar semiconductor BiTeI," Phys. Rev. B 91, 245312 (2015). [25] M. S. Bahramy, R. Arita, and N. Nagaosa, \Origin of giant bulk Rashba splitting: Application to BiTeI," Phys. Rev. B 84, 041202(R) (2011). [26] Yu A. Bychkov and E. I. Rashba, \Properties of a 2D electron gas with lifted spectral degen- eracy," JETP Lett. 39, 78{81 (1984). [27] M. S. Bahramy, B. J. Yang, R. Arita, and N. Nagaosa, \Emergence of non-centrosymmetric topological insulating phase in BiTeI under pressure," Nat. Commun. 3, 679 (2012). [28] Philip G. Derry, Andrew K. Mitchell, and David E. Logan, \Quasiparticle interference from magnetic impurities," Phys. Rev. B 92, 035126 (2015). [29] Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, \An intrinsic bond-centered electronic glass with unidirectional domains in underdoped cuprates." Science 315, 1380{1385 (2007), Eq. (6) is derived in supporting online text 1. [30] Xiaoting Zhou, Chen Fang, Wei-Feng Tsai, and Jiangping Hu, \Theory of quasiparticle scattering in a two-dimensional system of helical Dirac fermions: Surface band structure of a 20three-dimensional topological insulator," Phys. Rev. B 80, 245317 (2009). [31] Anna Str o_ zecka, Asier Eiguren, and Jose Ignacio Pascual, \Quasiparticle Interference around a Magnetic Impurity on a Surface with Strong Spin-Orbit Coupling," Phys. Rev. Lett. 107, 186805 (2011). [32] L. Petersen and P. Hedeg ard, \A simple tight-binding model of spin orbit splitting of sp- derived surface states," Surf. Sci. 459, 49{56 (2000). [33] J. I. Pascual, G. Bihlmayer, Yu. M. Koroteev, H.-P. Rust, G. Ceballos, M. Hansmann, K. Horn, E. V. Chulkov, S. Bl ugel, P. M. Echenique, and Ph. Hofmann, \Role of Spin in Quasiparticle Interference," Phys. Rev. Lett. 93, 196802 (2004). [34] I. Brihuega, P. Mallet, C. Bena, S. Bose, C. Michaelis, L. Vitali, F. Varchon, L. Magaud, K. Kern, and J. Y. Veuillen, \Quasiparticle chirality in epitaxial graphene probed at the nanometer scale," Phys. Rev. Lett. 101, 206802 (2008). [35] J. J. Sakurai, Modern Quantum Mechanics, Revised Edition (Addison-Wesley, Boston, 1993). 21

0912.3517v2.An_improved_effective_one_body_Hamiltonian_for_spinning_black_hole_binaries.pdf

arXiv:0912.3517v2 [gr-qc] 26 Feb 2010An improved effective-one-body Hamiltonian for spinning bl ack-hole binaries Enrico Barausse1and Alessandra Buonanno1 1Maryland Center for Fundamental Physics, Department of Phy sics, University of Maryland, College Park, MD 20742 (Dated: September 3, 2018) Building on a recent paper in which we computed the canonical Hamiltonian of a spinning test particle in curved spacetime, at linear order in the particl e’s spin, we work out an improved effective- one-body (EOB) Hamiltonian for spinning black-hole binari es. As in previous descriptions, we endow the effective particle not only with a mass µ, but also with a spin S∗. Thus, the effective particle interacts with the effective Kerr background (havi ng spinSKerr) through a geodesic-type interaction and an additional spin-dependent interaction proportional to S∗. When expanded in post-Newtonian (PN) orders, the EOB Hamiltonian reproduce s the leading order spin-spin coupling and the spin-orbit coupling through 2.5PN order, for any mas s-ratio. Also, it reproduces allspin- orbitcouplings inthe test-particle limit. Similarly toth etest-particle limit case, when we restrict the EOBdynamicstospins aligned or antialigned with theorbita l angular momentum, for whichcircular orbits exist, the EOBdynamics has several interesting feat ures, suchas the existence of an innermost stable circular orbit, a photon circular orbit, and a maximu m in the orbital frequency during the plunge subsequent to the inspiral. These properties are cru cial for reproducing the dynamics and gravitational-wave emission of spinning black-hole binar ies, as calculated in numerical relativity simulations. PACS numbers: 04.25.D-, 04.25.dg, 04.25.Nx, 04.30.-w I. INTRODUCTION Coalescing black-hole binaries are among the most promising sources for the current and future laser- interferometer gravitational-wave detectors, such as the ground-based detectors LIGO and Virgo [1, 2] and the space-based detector LISA [3]. The search for gravitational waves from coalescing bi- naries and the extraction of the binary’s physical param- eters are based on the matched filtering technique, which requires accurate knowledge of the waveform of the in- coming signal. Because black holes in general relativity are uniquely defined by their masses and spins, the wave- forms for black-hole binaries on a quasi-circular orbits depend on eight parameters, namely the masses m1and m2and the spin vectors S1andS2. Due to the large pa- rameter space, eventually tens of thousands of waveform templates may be needed to extract the gravitational- wave signal from the noise, an impossible demand for numerical-relativity alone. Fortunately, recent work at the interface between analytical and numerical relativity has demonstratedthe possibility ofmodeling analytically the dynamics and the gravitational-waveemission of coa- lescingnon-spinning black holes, thus providing data an- alysts with analytical template families [4–7] to be used for the searches (see also Ref. [8], which considers the cases of extreme mass-ratio inspirals). The next impor- tant step is to extend those studies to spinning precessing black holes. Sofar, theanalyticalmodelingoftheinspiral,plunge1, 1We refer to plunge as the dynamical phase starting soon after the two-body system passes the last stable orbit. During the plu ngemerger2, and ringdown has been obtained within either the effective-one-body (EOB) formalism [4, 6, 7, 9–17] or in Taylor-expanded PN models [13], both calibrated to numerical-relativity simulations, or in phenomenological approaches [5, 18] where the numerical-relativity wave- forms are fitted to templates which resemble the PN ex- pansion, but in which the coefficients predicted by PN theory are replaced by many arbitrary coefficients. Con- sidering the success of the EOB formalism in under- standing the physics of the coalescence of non-spinning black holes and modeling their gravitational-wave emis- sion with a small number of adjustable parameters, in this paper we will use that technique, adapting it to the case of spinning black-hole binaries. The first EOB Hamiltonian which included spin effects was computed in Ref. [19]. In Ref. [20], the authors used the non-spinning EOB Hamiltonian augmented with PN spin terms to carry out the first exploratory study of the dynamics and gravitational radiation of spinning black- holebinariesduringinspiral, mergerandringdown. More recently, Ref. [21] extended the model of Ref. [19] to in- clude the next-to-leading-orderspin-orbitcouplings. The EOBformalismdeveloped in Refs. [19, 21] highlightssev- eral features of the spinning two-body dynamics and was recently compared to numerical-relativity simulations of spinning non-precessing black holes in Ref. [22]. In this paper we build on Refs. [19, 21] and also on Ref. [23], in which we (in collaboration with Etienne Racine) derived the canonical Hamiltonian for a spinning test-particle in curved spacetime, at linear order in the particle’s spin, the motion is driven mostly by the conservative dynamics. 2We refer to merger as the dynamical phase in which the two- body system is described by a single black hole.2 andworkoutanimprovedEOBHamiltonianforspinning black-hole binaries. In particular, our EOB Hamiltonian reproduces the leading order spin-spin coupling and the spin-orbit coupling through 2.5PN order, for any mass- ratio. Also, it resums allthe test-particle limit spin-orbit terms. Moreover, when restricted to the case of spins aligned or antialigned with the orbital angular momen- tum, it presents several important features, such as the existence of an innermost stable circular orbit, a photon circular orbit, and a maximum in the orbital frequency during the plunge subsequent to the inspiral. All of these features are crucial for reproducing the dynamics and gravitational-wave emission of spinning coalescing black holes, as calculated in numerical relativity simulations. This paper is organized as follows. After presenting our notation (Sec. II), in Sec. III we build on Ref. [23] and derive the Hamiltonian for a spinning test particle in axisymmetric stationary spacetimes. In Sec. IV, we spe- cializetheaxisymmetricstationaryspacetimetothe Kerr spacetime in Boyer-Lindquist coordinates. In Sec. V we work out the EOB Hamiltonian of two spinning precess- ing black holes. In Sec. VI we restrict the dynamics to spins aligned or antialigned with the orbital angular mo- mentum and determine several properties of the circular- orbit dynamics. Section VII summarizes our main con- clusions. More details on how the spin-spin sector of the EOB Hamiltonian is constructed are eventually given in Appendix A. II. NOTATION Throughout this paper, we use the signature (−,+,+,+) for the metric. Spacetime tensor indices (ranging from 0 to 3) are denoted with Greek letters, while spatial tensor indices (ranging from 1 to 3) are denoted with lowercase Latin letters. Unless stated oth- erwise, we use geometric units ( G=c= 1), although we restore the factors of cwhen expanding in PN orders. We define a tetrad field as a set consisting of a timelike future-oriented vector ˜ eµ Tand three spacelike vectors ˜ eµ I (I= 1,...,3) — collectively denoted as ˜ eµ A(A= 0,...,3) — satisfying ˜eµ A˜eν Bgµν=ηAB, (2.1) whereηTT=−1,ηTI= 0,ηIJ=δIJ(δIJbeing the Kronecker symbol). Internal tetrad indices denoted with the uppercase Latin letters A,B,CandDalwaysrunfrom 0to3, while internal tetrad indices with the uppercase Latin letters I,J,KandL, associated with the spacelike tetrad vec- tors, run from 1 to 3 only. The timelike tetrad index is denoted by T. Tetrad indices are raised and lowered with the metric ηAB[e.g., ˜eµ A=ηAB(˜eB)µ]. We denote the projections of a vector Vonto the tetrad with VA≡Vµ˜eA µ, and similarly for tensors of higher rank. Partial derivativeswill be denoted with a comma or with ∂, and covariant derivatives with a semicolon. III. HAMILTONIAN FOR A SPINNING TEST-PARTICLE IN AXISYMMETRIC STATIONARY SPACETIMES Following Ref. [24], we write a generic axisymmetric stationary metric in quasi-isotropic coordinates as ds2=−e2νdt2+R2sin2θB2e−2ν(dφ−ωdt)2 +e2µ/parenleftbig dR2+R2dθ2/parenrightbig , (3.1) whereν,µ,Bandωare functions of the coordinates R andθ. Introducing the cartesian quasi-isotropic coordi- nates X=Rsinθcosφ, (3.2a) Y=Rsinθsinφ, (3.2b) Z=Rcosθ, (3.2c) we can write Eq. (3.1) as ds2=e−2ν/bracketleftbig B2ω2/parenleftbig X2+Y2/parenrightbig −e4ν/bracketrightbig dt2 +2B2e−2νω(Y dX−XdY)dt −2/parenleftbig B2e−2ν−e2µ/parenrightbig XY X2+Y2dXdY +e2µX2+B2e−2νY2 X2+Y2dX2 +B2e−2νX2+e2µY2 X2+Y2dY2+e2µdZ2. (3.3) It is straightforward to see that in the flat-spacetime limit (ω=ν=µ= 0,B= 1) Eq. (3.3) reduces to the Minkowski metric. Reference[23] computed the Hamiltonian ofaspinning test-particle in curved spacetime at linear order in the particle’s spin, and showed that it can be written as H=HNS+HS, (3.4) whereHNSis the Hamiltonian for a non-spinning test particle of mass m, given by HNS=βiPi+α/radicalBig m2+γijPiPj,(3.5) with α=1/radicalbig −gtt, (3.6) βi=gti gtt, (3.7) γij=gij−gtigtj gtt, (3.8)3 and HS=−/parenleftBigg βiFK i+FK t+αγijPiFK j/radicalbig m2+γijPiPj/parenrightBigg SK, where the coefficients FI µcan be expressed in terms of a reference tetrad field ˜ eAas FK µ=/parenleftbigg 2EµTI¯ωJ ¯ωT+EµIJ/parenrightbigg ǫIJK,(3.9) Eλµν≡1 2ηAB˜eA µ˜eB ν;λ, (3.10) with ¯ωµ=¯Pµ−m˜eT µ, (3.11) ¯Pi=Pi, (3.12) ¯Pt=−βiPi−α/radicalBig m2+γijPiPj,(3.13) ¯ωT= ¯ωµ˜eµ T=¯Pµ˜eµ T−m, (3.14) ¯ωI= ¯ωµ˜eµ I=¯Pµ˜eµ I. (3.15) Reference [23] also showed that in order to obtain a Hamiltonian giving the usual leading-order spin-orbitcoupling without gauge effects (or, equivalently, HS= 0 in flat spacetime), the reference tetrad field must become cartesian in the flat-spacetime limit. We find that the following choice for the reference tetrad ˜eT α=δt α(−gtt)−1/2=eνδt α, (3.16a) ˜eα 1=Be−µX2+eνY2 B(X2+Y2)δα X+(Be−µ−eν)XY B(X2+Y2)δα Y, (3.16b) ˜eα 2=(Be−µ−eν)XY B(X2+Y2)δα X+eνX2+Be−µY2 B(X2+Y2)δα Y, (3.16c) ˜eα 3=e−µδα Z, (3.16d) indeed reduces to the cartesian tetrad ˜ eT α= 1, ˜eα I=δα I in the flat-spacetime limit. We can then use the tetrad defined by Eqs. (3.16a)– (3.16d) to calculate the coefficients FK µin Eq. (3.9), and obtain HS=HSO+HSS, (3.17) with HSO=e2ν−µ(eµ+ν−B)(ˆP·ξR)SZ B2√QR2ξ2+eν−2µ B2/parenleftbig√Q+1/parenrightbig√QR2ξ2/braceleftBigg Bcosθeµ+ν(ˆP·ξR)/parenleftBig/radicalbig Q+1/parenrightBig (S·N)ξ2 +R(S·ξ)/bracketleftBig µR(ˆP·VR)/parenleftBig/radicalbig Q+1/parenrightBig −µcosθ(ˆP·N)ξ2−/radicalbig Q(νR(ˆP·VR)+(µcosθ−νcosθ)(ˆP·N)ξ2)/bracketrightBig B2 +eµ+ν(ˆP·ξR)/parenleftBig 2/radicalbig Q+1/parenrightBig/bracketleftBig νRR(S·V)−νcosθ(S·N)ξ2/bracketrightBig B−BReµ+ν(ˆP·ξR)/parenleftBig/radicalbig Q+1/parenrightBig R(S·V)/bracerightBigg , (3.18) HSS=ωSZ+e−3µ−νωR 2B/parenleftbig√Q+1/parenrightbig√QRξ2/braceleftBigg −eµ+ν(ˆP·VR)(ˆP·ξR)(S·ξ)B+e2(µ+ν)(ˆP·ξR)2(S·V) +e2µ/parenleftBig 1+/radicalbig Q/parenrightBig/radicalbig QR2(S·V)ξ2B2+(ˆP·N)R/bracketleftBig (ˆP·VR)(S·N)−(ˆP·N)R(S·V)/bracketrightBig ξ2B2/bracerightBigg +e−3µ−νωcosθ 2B/parenleftbig√Q+1/parenrightbig√QR2/braceleftBigg eµ+ν(ˆP·N)(ˆP·ξR)R(S·ξ)B−e2(µ+ν)(ˆP·ξR)2(S·N) +/bracketleftBig (S·N)(ˆP·VR)2−(ˆP·N)R(S·V)(ˆP·VR)−e2µ/parenleftBig 1+/radicalbig Q/parenrightBig/radicalbig QR2(S·N)ξ2/bracketrightBig B2/bracerightBigg ,(3.19) Q= 1+γijˆPiˆPj= 1+e−2µ(ˆP·N)2+e−2µ(ˆP·VR)2 R2ξ2+e2ν(ˆP·ξR)2 B2R2ξ2, (3.20) where we denote ˆP=P m, (3.21) N=X R, (3.22) ξ=eZ×N=−YeX+XeY R,(3.23) V=N×ξ, (3.24)and fR≡∂f(R,cosθ) ∂R, (3.25) fcosθ≡∂f(R,cosθ) ∂(cosθ). (3.26)4 Here, the generic function fcan stand for B,ω,µorν. Note that because ωis proportional to gtφ[see Eq. (3.1)] and thus to the spin of the spacetime, HSS(which is pro- portional to ωand its derivatives) gives the leading-order coupling between the particle’s spin and the spin of the background spacetime (together with other higher order terms). Also, because ˆP·ξR=ˆPφin spherical coordi- nates,HSOis the part ofthe Hamiltonian which givesthe leading-order spin orbit coupling (again, together with other higher order terms). Moreover, note that HS= 0 in a flat spacetime, thus confirming the absence of gauge effects in the leading order spin-orbit coupling. As a consistency test, we specialize to the case of a spherically symmetric spacetime in quasi-isotropic coor- dinates, which was considered in Ref. [23] (see Sec. V A therein). Because the metric for such a spacetime is given by ds2=−f(R)dt2+h(R)(dX2+dY2+dZ2),(3.27) a comparison with Eq. (3.1) immediately reveals that B=/radicalbig f(R)h(R), (3.28) ω= 0, (3.29) ν=1 2log[f(R)], (3.30) µ=1 2log[h(R)]. (3.31) Inserting Eqs.(3.28)–(3.31) in Eqs.(3.17)–(3.20), we find HS=L·S 2mR/radicalbig f(R)h(R)2√Q(1+√Q)× /braceleftBig/radicalbig Q[f′(R)h(R)−f(R)h′(R)]−f(R)h′(R)/bracerightBig , (3.32) where Q= 1+1 hˆP2, (3.33) L=X×P, (3.34) in agreement with Eq. (5.7) in Ref. [23].Let us now investigate how the Hamiltonian (3.17) is affectedby achangeofthe radialcoordinate R. Denoting the new radial coordinate by r=|x|and defining J−1≡dR dr, (3.35) the radial derivatives of the metric potentials can be re- expressed as fR=frJ, (3.36) where again f=B,ω, ν,µ . The spin S, the derivatives of the metric potentials with respect to cos θ, and the quantities N=n=x r, (3.37) ξ=eZ×N=ez×n, (3.38) V=v=n×ξ (3.39) are not affected by the coordinate change. The same applies to the quantities ˆP·VRandˆP·ξRappearing in Eqs. (3.18), (3.19) and (3.20). In fact, in spherical coordinates,wehave ˆP·VR=−ˆPθsinθandˆP·ξR=ˆPφ, hence ˆP·VR=ˆp·vr, (3.40) ˆP·ξR=ˆp·ξr, (3.41) whereˆp=p/mandpis the conjugate momentum in the new coordinate system, i.e., pi=∂Xj/∂xiPj. On the contrary, since ˆP·N=ˆPR, we have ˆP·N= (ˆp·n)J. (3.42) It is therefore straightforward to compute HSin a coor- dinate system related to quasi-isotropic coordinates by a rescaling of the radius. We have HS=HSO+HSS, (3.43) where HSO=e2ν−µ(eµ+ν−B) (ˆp·ξr)Sz B2√QR2ξ2+eν−2µ B2/parenleftbig√Q+1/parenrightbig√QR2ξ2/braceleftBigg Bcosθeµ+ν(ˆp·ξr)/parenleftBig/radicalbig Q+1/parenrightBig (S·n)ξ2 +R(S·ξ)J/bracketleftBig µr(ˆp·vr)/parenleftBig/radicalbig Q+1/parenrightBig −µcosθ(ˆp·n)ξ2−/radicalbig Q(νr(ˆp·vr)+(µcosθ−νcosθ)(ˆp·n)ξ2)/bracketrightBig B2 +eµ+ν(ˆp·ξr)/parenleftBig 2/radicalbig Q+1/parenrightBig/bracketleftBig JνrR(S·v)−νcosθ(S·n)ξ2/bracketrightBig B−JBreµ+ν(ˆp·ξr)/parenleftBig/radicalbig Q+1/parenrightBig R(S·v)/bracerightBigg , (3.44)5 HSS=ωSz+e−3µ−νJωr 2B/parenleftbig√Q+1/parenrightbig√QRξ2/braceleftBigg −eµ+ν(ˆp·vr)(ˆp·ξr)(S·ξ)B+e2(µ+ν)(ˆp·ξr)2(S·v) +e2µ/parenleftBig 1+/radicalbig Q/parenrightBig/radicalbig QR2(S·v)ξ2B2+J(ˆp·n)R[(ˆp·vr)(S·n)−J(ˆp·n)R(S·v)]ξ2B2/bracerightBigg +e−3µ−νωcosθ 2B/parenleftbig√Q+1/parenrightbig√QR2/braceleftBigg −e2(µ+ν)(ˆp·ξr)2(S·n) +eµ+νJ(ˆp·n)(ˆp·ξr)R(S·ξ)B +/bracketleftBig (S·n)(ˆp·vr)2−J(ˆp·n)R(S·v)(ˆp·vr)−e2µ/parenleftBig 1+/radicalbig Q/parenrightBig/radicalbig QR2(S·n)ξ2/bracketrightBig B2/bracerightBigg ,(3.45) Q= 1+γijˆpiˆpj= 1+e−2µ(ˆp·n)2J2+e−2µ(ˆp·vr)2 R2ξ2+e2ν(ˆp·ξr)2 B2R2ξ2, (3.46) and whereRmust of course be expressed in terms of the new radial coordinate r. IV. HAMILTONIAN FOR A SPINNING TEST-PARTICLE IN KERR SPACETIME IN BOYER-LINDQUIST COORDINATES In this section, we will specialize the Hamiltonian de- rivedintheprevioussectiontothe caseofKerrspacetime in Boyer-Lindquist coordinates. We start from the metric potentials appearing in Eq. (3.1), which in the case of a Kerr spacetime take the form [25] B=√ ∆ R, (4.1) ω=2aMr Λ, (4.2) e2ν=∆Σ Λ, (4.3) e2µ=Σ R2, (4.4) with Σ =r2+a2cos2θ, (4.5) ∆ =r2+a2−2Mr, (4.6) ̟2=r2+a2, (4.7) Λ =̟4−a2∆sin2θ, (4.8) where the parameter a, which has the dimensions of a length, is related to the spin vector SKerrof the Kerr black hole by a=|SKerr| M. (4.9) The Boyer-Lindquist coordinate ris related to the quasi- isotropic coordinate Rby r=R+M+R2 H R, (4.10)whereRH=√ M2−a2/2isthehorizon’sradiusinquasi- isotropic coordinates. Note that the inverse of this trans- formation is given, outside the horizon, by R=1 2/parenleftbig r−M+√ ∆/parenrightbig . (4.11) We then obtain that the derivatives of the metric poten- tials take the form Br=r−M−√ ∆ R√ ∆, (4.12a) ωr=2aM[Σ̟2−2r2(Σ+̟2)] Λ2,(4.12b) νr=r−M ∆+r Σ −2r̟2−a2(r−M) sin2θ Λ,(4.12c) µr=r Σ−1√ ∆, (4.12d) Bcosθ= 0, (4.12e) ωcosθ=−4a3Mr∆ cosθ (∆Σ+2Mr̟2)2, (4.12f) νcosθ=2a2Mr̟2cosθ (∆Σ+2Mr̟2)Σ, (4.12g) µcosθ=a2cosθ Σ, (4.12h) and we also have J−1=dR dr=R√ ∆. (4.13) Inserting Eqs. (4.12a)–(4.12h) and Eq. (4.13) into Eqs. (3.43)–(3.46), we find that Rcancels out both in Q, that is Q= 1+∆(ˆp·n)2 Σ+(ˆp·ξr)2Σ Λ sin2θ+(ˆp·vr)2 Σ sin2θ,(4.14)6 and in the Hamiltonian HS. In conclusion, the Hamil- tonian of a spinning test-particle in Kerr spacetime in Boyer-Lindquist coordinates is H=HNS+HS, (4.15) with HNS=βipi+α/radicalBig m2+γijpipj, (4.16)whereα,βiandγijare given in Eqs. (3.6)–(3.8) and need to be computed using the Kerr metric coefficients (4.1)–(4.8), and with HS=HSO+HSS, (4.17) where HSO=e2ν−˜µ/parenleftBig e˜µ+ν−˜B/parenrightBig (ˆp·ξr)(S·ˆSKerr) ˜B2√Qξ2+eν−2˜µ ˜B2/parenleftbig√Q+1/parenrightbig√Qξ2/braceleftBigg (S·ξ)˜J/bracketleftBig µr(ˆp·vr)/parenleftBig/radicalbig Q+1/parenrightBig −µcosθ(ˆp·n)ξ2 −/radicalbig Q(νr(ˆp·vr)+(µcosθ−νcosθ)(ˆp·n)ξ2)/bracketrightBig ˜B2+e˜µ+ν(ˆp·ξr)/parenleftBig 2/radicalbig Q+1/parenrightBig/bracketleftBig ˜Jνr(S·v)−νcosθ(S·n)ξ2/bracketrightBig ˜B −˜J˜Bre˜µ+ν(ˆp·ξr)/parenleftBig/radicalbig Q+1/parenrightBig (S·v)/bracerightBigg , (4.18) and HSS=ω(S·ˆSKerr)+e−3˜µ−ν˜Jωr 2˜B/parenleftbig√Q+1/parenrightbig√Qξ2/braceleftBigg −e˜µ+ν(ˆp·vr)(ˆp·ξr)(S·ξ)˜B+e2(˜µ+ν)(ˆp·ξr)2(S·v) +e2˜µ/parenleftBig 1+/radicalbig Q/parenrightBig/radicalbig Q(S·v)ξ2˜B2+˜J(ˆp·n)/bracketleftBig (ˆp·vr)(S·n)−˜J(ˆp·n)(S·v)/bracketrightBig ξ2˜B2/bracerightBigg +e−3˜µ−νωcosθ 2˜B/parenleftbig√Q+1/parenrightbig√Q/braceleftBigg −e2(˜µ+ν)(ˆp·ξr)2(S·n)+e˜µ+ν˜J(ˆp·n)(ˆp·ξr)(S·ξ)˜B +/bracketleftBig (S·n)(ˆp·vr)2−˜J(ˆp·n)(S·v)(ˆp·vr)−e2˜µ/parenleftBig 1+/radicalbig Q/parenrightBig/radicalbig Q(S·n)ξ2/bracketrightBig ˜B2/bracerightBigg , (4.19) where we define ˜B=BR=√ ∆, (4.20) ˜Br=BrR=r−M−√ ∆√ ∆,(4.21) e2˜µ=e2µR2= Σ, (4.22) ˜J=JR=√ ∆, (4.23) ˆSKerr=SKerr |SKerr|(4.24) and we recall that ξ2= sin2θ. We stress that because thisHamiltonianisexpressedintermsofquantitieswhich are scalar under spatial rotations, we can express it in a cartesian coordinate system in which the spin of the Kerr black hole is not directed along the z-axis. For that purpose, it is sufficient to replace rwith (x2+y2+ z2)1/2, cosθwithˆSKerr·n,ezwithˆSKerrin Eq. (3.38), and express the vectors appearing in Eqs. (4.16)–(4.19) in terms of their cartesian components. As a consistency check, we can compute the Hamilto-nian for a spinning test-particle in a Schwarzschildspace- time in Schwarzschild spherical coordinates by setting a= 0, and compare the result to the expression com- puted in Ref. [23] [see Eq. (5.12) therein]. We find HS=ψ6 R3√Q(1+√Q)× /bracketleftbigg 1−M 2R+2/parenleftbigg 1−M 4R/parenrightbigg/radicalbig Q/bracketrightbigg (L·S∗),(4.25) where S∗=M mS, (4.26) ψ=/parenleftbigg 1+M 2R/parenrightbigg−1 , (4.27) R=1 2/parenleftBig r−M+/radicalbig r2−2Mr/parenrightBig ,(4.28) and Q= 1+(ˆp·n)2/parenleftbigg 1−2M r/parenrightbigg +(ˆp·v)2+(ˆp·ξ)2 sin2θ,(4.29)7 in agreement with Ref. [23]. Also, it is worth noting that the Hamiltonian (4.25) is the same as the quasi- isotropic Schwarzschild Hamiltonian (3.32), expressed in terms of the Schwarzschild coordinate r. This is because the scalar product L·Sis unaffected by a change of the radial coordinate. V. EFFECTIVE-ONE-BODY HAMILTONIAN FOR TWO SPINNING BLACK HOLES The EOB approach was originally introduced in Refs. [9–11, 19] to provide us with an improved (re- summed) Hamiltonian that could be used to evolve a binary system not only during the long inspiral, but also during the plunge, and that could supply a natural mo- mentat which to switch from the two body description to the one-body description, in which the system is rep- resented by a superposition of quasi-normal modes of the remnant black hole. A crucial ingredient of the EOB approach is the real PN-expanded Arnowitt-Deser-Misner (ADM) Hamilto- nian (or realHamiltonian) describing two black holes of massesm1,m2and spins S1,S2. The real Hamiltonian isthencanonicallytransformedandsubsequently mapped to aneffective Hamiltonian Heffdescribing a test-particle of massµ=m1m2/(m1+m2) and suitable spin S∗, moving in a deformed Kerr metric of mass M=m1+m2 and suitable spin SKerr. The parameter regulating the deformation is the symmetric mass ratio of the binary, η=µ/M, which ensuresthat the deformation disappears in the case of extreme mass-ratio binaries. The resulting improved EOB Hamiltonian then takes the form Himproved real=M/radicalBigg 1+2η/parenleftbiggHeff µ−1/parenrightbigg .(5.1) The computation of the improved EOB Hamiltonian consists of several stages. For this reason, we briefly re- view here the main steps and the underpinning logic that we will follow in the rest of this section: (i) We apply a canonical transformation to the PN- expanded ADM Hamiltonian using a generating function which is compatible with the one used in previous EOB work, obtaining the PN-expanded Hamiltonian in EOB canonical coordinates (see Sec. VA); (ii) We compute the effective Hamiltonian correspond- ing to the canonically transformed PN-expanded ADM Hamiltonian (see Sec. VB); (iii) We deform the Hamiltonian of a spinning test- particle in Kerr derived in Sec. IV by deforming the Kerr metric (see Sec. VC) , and expand this deformed Hamiltonian in PN orders (see Sec. VD); (iv) Comparing (iii) and (iii), we work out the mapping between the spin variables in the real and effectivedescriptions, and write the improved EOB Hamil- tonian (see Sec. VE). A. The ADM Hamiltonian canonically transformed to EOB coordinates WedenotetheADM canonicalvariablesinthebinary’s center-of-mass frame with r′andp′. It is convenient to introduce the following spin combinations: σ=S1+S2, (5.2) σ∗=S1m2 m1+S2m1 m2, (5.3) σ0=σ+σ∗. (5.4) Moreover, in order to consistently keep track of the PN orders, we will restore the speed of light cand rescale the spins variables as σ∗→σ∗candσ→σc.3The canoni- cal ADM Hamiltonian is known through 3PN order [26– 30] and partially at higher PN orders [31, 32]. In par- ticular, the spin-orbit and spin-spin coupling terms agree with those computed via effective-field-theory techniques at 1.5PN, 2PN and 3PN order [33–36]. In this paper, we use the spin-independent part of the ADM Hamiltonian through 3PN order, but we only use its spin-dependent part through 2.5 PN order, i.e., we consider the leading- order (1.5 PN) and the next-to-leading order (2.5PN) spin-orbit couplings, but only the leading order (2PN) spin-spin coupling. The expressions for these couplings are [19, 27] HADM SO(r′,p′,σ∗,σ) =1 c3L′ r′3·(gADM σσ+gADM σ∗σ∗), (5.5) HADM SS(r′,p′,σ∗,σ) =1 c4η 2r′3/bracketleftbig 3(n′·σ0)2−σ2 0/bracketrightbig , (5.6) withL′=r′×p′,n′=r′/r′, and gADM σ= 2+1 c2/bracketleftbigg19 8ηˆp′2+3 2η(n′·ˆp′)2 −(6+2η)M r′/bracketrightbigg , (5.7a) gADM σ∗=3 2+1 c2/bracketleftbigg/parenleftbigg −5 8+2η/parenrightbigg ˆp′2+3 4η(n′·ˆp′)2 −(5+2η)M r′/bracketrightbigg , (5.7b) 3This is appropriate for black holes or a rapidly rotating com - pact stars. In the black-hole case, S=χM2/c, withχrang- ing from 0 to 1. In the rapidly spinning star case one has S=Mvrotr∼Mcrs∼M2/c(where we have assumed that the rotational velocity vrotis comparable to cand that the stellar radiusris of the order of the Schwarzschild radius rs∼M/c2).8 wherewehaveintroducedthe rescaledconjugatemomen- tumˆp′=p′/µ. We now perform a canonical transformation from the ADM canonicalvariables r′andp′to the EOB canonical variables randp. Let us first consider the purely orbital generating function G(r′,p) =r′·p+GNS(r′,p), (5.8) GNS(r′,p) =GNS1PN(r′,p) +GNS2PN(r′,p)+GNS3PN(r′,p),(5.9) wherethe1PN-accurategeneratingfunction GNS1PNwas derived in Ref. [10], GNS1PN(r′,p) =1 c2r′·p/bracketleftbigg −1 2ηˆp2+M r′/parenleftbigg 1+1 2η/parenrightbigg/bracketrightbigg , (5.10) while the 2PN and 3PN accurate generating functions, GNS2PNandGNS3PN, were derived in Refs. [10] and [11], respectively. From the definition of generating function, it followsthat the transformationofthe phase-spacevari- ables is implicitly given by xi=x′i+∂GNS(x′,p) ∂pi, (5.11) pi=p′ i−∂GNS(x′,p) ∂x′i, (5.12) while the Hamiltonian transforms as H(r,p) = HADM(r′,p′). At linear order, which is enough for our purposes, Eqs. (5.11) and (5.12) can be written as y=y′−{GNS,y′}, where{...}are the Poisson brackets and where ystands for either xorp. The transforma- tion of the Hamiltonian, again at linear order, is then H(y) =HADM(y) +{GNS,HADM}(y) [21]. Similarly, if one considers a generating function which depends not only on the orbital variables, but also on the spins, G(r′,p,σ∗,σ) =r′·p +GNS(r′,p)+GS(r′,p,σ∗,σ) (5.13) the Hamiltonian will again transform as H(y) = HADM(y) +{GNS,HADM}(y) +{GS,HADM}(y), where now the Poisson brackets in the term {GS,HADM}will involve also the spin variables [21]. In particular, let us consider a spin-dependent generating function GS(r′,p,σ∗,σ) =GS2PN(r′,p,σ) +GS2.5PN(r′,p,σ∗,σ) (5.14) +GSSS2.5PN(r′,p,σ∗,σ). where the 2PN-accuratespin-dependent generating func-tionGS2PNwas implicitly4used in Ref. [19], GS2PN(r′,p,σ) =−1 2c4M2r′2/braceleftBig [σ2−(σ·n′)2](r′·p) +(σ·n′)(r′×p)·(σ×n′)/bracerightBig ; (5.15) the 2.5PN-accurategeneratingfunction GS2.5PNlinearin the spin variables was introduced in Ref. [21], GS2.5PN(r′,p,σ∗,σ) =1 µr′3c5(r′·p)(r′×p)· [a(η)σ+b(η)σ∗],(5.16) a(η) andb(η) being arbitrary gauge functions; also, for reasons which will become clear in Sec. VD, we include the following 2.5PN-accurate generating function, cubic in the spins, GSSS2.5PN(r′,p,σ∗,σ) =µ 2M3r′4c5(σ·r′)[σ∗·(σ×r′)]. (5.17) When applying the generating function (5.13) to the ADM 2PN spin-spin Hamiltonian (5.6), we obtain HSS2PN(r,p,σ∗,σ) =HADM SS2PN(r,p,σ∗,σ) +{GS2PN,HNewt}(r,p,σ), (5.18) with HNewt=−Mµ r+p2 2µ, (5.19) {GS2PN,HNewt}(r,p,σ) =−1 c4η 2r3/bracketleftbig (n·σ)2−σ2/bracketrightbig +1 2µM2r2c4/braceleftBig −[p2−2(p·n)2]σ2 +[(p−2(p·n)n)·σ]p·σ/bracerightBig .(5.20) Similarly, if we apply the same generating function to the ADM spin-orbit Hamiltonian (5.5), the 1.5PN order term remains unaltered [21], while the 2.5PN order term transforms as [21] HSO2.5PN(r,p,σ∗,σ) =HADM SO2.5PN(r,p,σ∗,σ) +{G2.5PN,HNewt}(r,p,σ∗,σ), +{GNS1PN,HADM SO1.5PN}(r,p,σ∗,σ) (5.21) where G2.5PN=GSS2.5PN+GSSS2.5PN, (5.22) 4See discussion in Sec II D of Ref. [19]. The need for this gener - ating function will become apparent with Eq. (5.55) in Sec. V D.9 {G2.5PN,HNewt}(r,p,σ∗,σ) = 1 r3c5L·[b(η)σ∗+a(η)σ]/bracketleftbigg −M r+ˆp2−3(ˆp·n)2/bracketrightbigg +[σ∗·(σ×n)][σ·(p−2(p·n)n)] M3r3c5 +(L·σ∗)σ2−(L·σ)(σ∗·σ) 2M3r4c5, (5.23) and {GNS1PN,HADM SO1.5PN}(r,p,σ∗,σ) = −3L 2r3c5·/parenleftbigg3 2σ∗+2σ/parenrightbigg/braceleftbigg −M r(2+η)+η/bracketleftbig ˆp2+2(ˆp·n)2/bracketrightbig/bracerightbigg . (5.24) Therefore, the complete real Hamiltonian in the EOB canonical coordinates is H(r,p,σ∗,σ) =Hnospin(r,p,σ∗,σ) +HADM SO(r,p,σ∗,σ) +HADM SS(r,p,σ∗,σ) +{G2.5PN,HNewt}(r,p,σ∗,σ) +{GNS1PN,HADM SO1.5PN}(r,p,σ∗,σ) +{GS2PN,HNewt}(r,p,σ), (5.25) whereHnospinis the 3PN ADM Hamiltonian for non- spinning black holes, canonically transformed to EOB coordinates, which can be obtained from Ref. [11]. B. Spin couplings in the effective Hamiltonian Following Refs. [9, 11, 19], we map the effective and real two-body Hamiltonians as Heff µc2=H2 real−m2 1c4−m2 2c4 2m1m2c4,(5.26) whereHrealis the real two-body Hamiltonian containing alsotherest-masscontribution Mc2. Wedenotethenon- relativistic part of the real Hamiltonian by HNR, i.e., HNR≡Hreal−Mc2. Identifying HNRwithHasgivenin Eq. (5.25), and expanding Eq. (5.26) in powers of 1 /c, we find that the 1.5PNand 2.5PNorderspin-orbit couplings of the effective Hamiltonian are Heff SO(r,p,σ∗,σ) =1 c3L r3·/parenleftbig geff σσ+geff σ∗σ∗/parenrightbig +[σ∗·(σ×n)][σ·(p−2(p·n)n)] M3r3c5 +(L·σ∗)σ2−(L·σ)(σ∗·σ) 2M3r4c5, (5.27)where [21] geff σ= 2+1 c2/braceleftbigg/bracketleftbigg3 8η+a(η)/bracketrightbigg ˆp2 −/bracketleftbigg9 2η+3a(η)/bracketrightbigg (ˆp·n)2 −M r[η+a(η)]/bracerightbigg , (5.28a) geff σ∗=3 2+1 c2/braceleftbigg/bracketleftbigg −5 8+1 2η+b(η)/bracketrightbigg ˆp2 −/bracketleftbigg15 4η+3b(η)/bracketrightbigg (ˆp·n)2 −M r/bracketleftbigg1 2+5 4η+b(η)/bracketrightbigg/bracerightbigg ,(5.28b) and the 2PN order spin-spin coupling is Heff SS(r,p,σ∗,σ) =1 c4η 2r3(3ninj−δij)σi 0σj 0 −1 c4η 2r3/bracketleftbig (n·σ)2−σ2/bracketrightbig +1 2µM2r2c4/braceleftBig −[p2−2(p·n)2]σ2 +[(p−2(p·n)n)·σ]p·σ/bracerightBig .(5.29) C. The Hamiltonian of a spinning test-particle in a deformed Kerr spacetime We now deform the Hamiltonian of a spinning test- particle in a Kerr spacetime computed in Sec. IV [see Eqs. (4.16), (4.17), (4.18) and (4.19)] by deforming the Kerr metric. The deformation that we introduce is reg- ulated by the parameter η=µ/M, and therefore dis- appears in the test-particle limit. Also, the deformed Hamiltonianwillbesuchastoreproduce,whenexpanded in PN orders, the spin couplings of the effective Hamil- tonian given in Sec. VB. WhenthespinoftheKerrblackholeiszero,thatis a= 0, we require the metric to coincide with the deformed- Schwarzschildmetricused in the EOBformalismfor non- spinning black-hole binaries [10, 11]. That deformation simply amounts to changing the components gttandgrr of the metric. In the spinning case, following Ref. [21], we seek an extension of this deformation by changing the potential ∆ appearing in the Kerr potentials (4.1)–(4.4). It is worth noting, however, that we are not allowed to deform the Kerr metric in an arbitrary way. We re- call indeed that the Hamiltonian that we have derived in Sec. IV is only valid for a stationary axisymmetric metric, and in coordinates which are related to quasi- isotropic coordinates by a redefinition of the radius. In other words, it must be possible for our deformed metric to be put in the form (3.1) by a coordinate change of the typeR=R(r). For this reason we cannot deform the metric exactly in the same way as in Ref. [21]. Here we10 propose to deform the metric potentials in the following manner B=√∆t R, (5.30) ω=/tildewideωfd Λt, (5.31) e2ν=∆tΣ Λt, (5.32) e2µ=Σ R2, (5.33) and J−1=dR dr=R√∆r, (5.34) where the relation between randRcan be found by integrating Eq. (5.34): R= exp/parenleftbigg/integraldisplaydr√∆r/parenrightbigg . (5.35) The deformed metric therefore takes the form gtt=−Λt ∆tΣ, (5.36a) grr=∆r Σ, (5.36b) gθθ=1 Σ, (5.36c) gφφ=1 Λt/parenleftbigg −/tildewideω2 fd ∆tΣ+Σ sin2θ/parenrightbigg ,(5.36d) gtφ=−/tildewideωfd ∆tΣ, (5.36e) which does not depend on R. Therefore, as we will show explicitly later in this section, we do notneed to com- pute the integral (5.35) to write the Hamiltonian. The quantities ∆ t, ∆r, Λtand/tildewideωfdin Eqs. (5.36a)–(5.36e) are given by ∆t=r2/bracketleftbigg A(u)+a2 M2u2/bracketrightbigg , (5.37) ∆r= ∆tD−1(u), (5.38) Λt=̟4−a2∆tsin2θ, (5.39) /tildewideωfd= 2aMr+ωfd 1ηaM3 r+ωfd 2ηMa3 r,(5.40) whereu=M/r,ωfd 1andωfd 2are adjustable parameters which regulate the strength of the frame-dragging, and through 3PN order [9, 11] A(u) = 1−2u+2ηu3+η/parenleftbigg94 3−41 32π2/parenrightbigg u4,(5.41) D−1(u) = 1+6ηu2+2(26−3η)ηu3. (5.42)We find that our deformed metric is the same as the de- formed metric of Ref. [21], except for gφφandgtφ.5As we prove below, the differences between our deformation and the deformation of Ref. [21] appear in the Hamilto- nian at PN orders higher than 3PN. To obtain the total Hamiltonian (4.15), that is H= HNS+HS, we first compute the Hamiltonian HNSfor a non-spinningparticleinthedeformed-Kerrmetric. Using Eq. (4.16) and Ref. [11], we have HNS=βipi+α/radicalBig m2+γijpipj+Q4(p),(5.43) whereQ4(p) is a term which is quartic in the space mo- mentapiand which was introduced in Ref. [11], and α=1/radicalbig −gtt, (5.44) βi=gti gtt, (5.45) γij=gij−gtigtj gtt. (5.46) In Eqs. (5.44)–(5.46) the metric components have to be replaced with those of the deformed-Kerrmetric (5.36a)– (5.36e). When expanded in PN orders, Eq. (5.43) co- incides, through 3PN order, with the Hamiltonian of a non-spinning test particle in the deformed-Kerr metric given by Ref. [21]. Second, to calculate HSgiven by Eqs. (3.43), (3.44) and (3.45), we need to compute the derivatives of the metric potentials. We obtain Br=√∆r∆t′−2∆t 2√∆r∆tR, (5.47a) ωr=−Λ′ t/tildewideωfd+Λt/tildewideω′ fd Λ2 t, (5.47b) νr=r Σ+̟2/parenleftbig ̟2∆t′−4r∆t/parenrightbig 2Λt∆t,(5.47c) µr=r Σ−1√∆r, (5.47d) Bcosθ= 0, (5.47e) ωcosθ=−2a2cosθ∆t/tildewideωfd Λ2 t, (5.47f) νcosθ=a2̟2cosθ(̟2−∆t) ΛtΣ,(5.47g) µcosθ=a2cosθ Σ, (5.47h) where the prime denotes derivatives with respect to r. As already stressed, although the metric potentials B, 5Ref. [21] chooses gφφ= (−a2sin2θ+ ∆t)/(∆tΣsin2θ) and gtφ=a(∆t−̟2)/(∆tΣ), which are different from our expres- sions (5.36d) and (5.36e) even for ωfd 1=ωfd 2= 0.11 ω,νandµdepend on R, the factors Rcancel out in the deformed-Kerr metric. Therefore, those factors must cancel out also in HS. This happens because the ref- erence tetrad field ˜ eAwhich, together with the metric, completely determines the Hamiltonian [see Eq. (3.9)], can be defined independently of R. Indeed, this turns out to be the case, and if we introduce the rescaled po- tentials ˜B=BR=/radicalbig ∆t, (5.48) ˜Br=BrR=√∆r∆t′−2∆t 2√∆r∆t,(5.49) e2˜µ=e2µR2= Σ, (5.50) ˜J=JR=/radicalbig ∆r (5.51) and define Q= 1+∆r(ˆp·n)2 Σ+(ˆp·ξr)2Σ Λtsin2θ+(ˆp·vr)2 Σ sin2θ,(5.52) the Hamiltonian HSfor the deformed-Kerr metric takes exactlythe sameform asin the Kerrcase[seeEqs.(4.17), (4.18) and (4.19)], where we recall that ξ2= sin2θand wherenowωanditsderivatives, νanditsderivatives,and the derivatives of µare given by Eqs. (5.31), (5.32), and Eqs. (5.47a)–(5.47h). Also, as we have already stressed, in order to express the Hamiltonian HSin a cartesian coordinatesysteminwhichthespinofthe deformed-Kerr black hole is not directed along the z-axis, it is sufficient to replacerwith (x2+y2+z2)1/2, cosθwithˆSKerr·n, ezwithˆSKerrin Eq. (3.38), and to express the vectors appearing in the Hamiltonian in terms of their cartesian components. D. PN expansion of the deformed Hamiltonian We nowexpandthe deformed Hamiltonian H=HNS+ HSderived in the previous section into PN orders. We will denote the spin of the deformed-Kerr metric with SKerr, while for the test particle’s spin we introduce the rescaled spin vector S∗=SM/m,Sbeing the physical, unrescaled spin. Also, we rescale the spins as SKerr→ SKerrcandS∗→S∗c, so as to keep track of the PN orders correctly. Moreover, we set SKerr=χKerrM2, χKerrbeing the dimensionless spin of the deformed-Kerr black hole, with norm |χKerr|ranging from 0 to 1. As already mentioned, the part of the Hamiltonian which does not depend on the test particle’s spin, HNS, agrees through 3PN order with the corresponding HNS computed in Ref. [21]. Moreover, although the metric (5.36a)–(5.36e) only coincides with the Kerr metric for η= 0, the dependence on ηappears neither in the 2PN order coupling of the deformed-Kerr black hole’s spin with itself, nor in its 1.5PN and 2.5PN order spin-orbit couplings. Those couplings are therefore the same as inthe case of the Kerr metric, and they are given by HNS SO1.5PN=1 c32 r3L·SKerr, (5.53) HNS SO2.5PN= 0, (5.54) HNS SS2PN=1 c4m 2Mr3(3ninj−δij)Si KerrSj Kerr −1 c4m 2Mr3/bracketleftbig (n·SKerr)2−S2 Kerr/bracketrightbig +1 2m(Mr)2c4/braceleftBig −[p2−2(p·n)2]S2 Kerr +[(p−2(p·n)n)·SKerr]p·SKerr/bracerightBig . (5.55) ExpandingtheninPNordersthe partoftheHamiltonian that depends on the test particle’s spin, that is HS, we find HS SO1.5PN=3 2r3c3L·S∗, (5.56) HS SO2.5PN=1 r3c5/bracketleftbigg −M r/parenleftbigg1 2+3η/parenrightbigg −5 8ˆp2/bracketrightbigg L·S∗ +[S∗·(SKerr×n)][SKerr·(p−2(p·n)n)] M3r3c5 +(L·S∗)S2 Kerr−(L·SKerr)(S∗·SKerr) 2M3r4c5. (5.57) HS SS2PN=m Mr3c4(3ninj−δij)Si KerrSj ∗.(5.58) We recall that the Hamiltonian for a spinning test parti- cle in curved spacetimefrom which we started the deriva- tion of our novel EOB model [see Eq. (3.9)] is only valid at linear order in the particle’s spin. Therefore, the same restriction applies to the Hamiltonian derived in Sec. VC. In particular, that Hamiltonian does not in- clude the couplings of the particle’s spin with itself. We introduce those couplings by hand, at least at the leading order (2PN), by adding a quadrupole deformation [19] hµν, quadratic in the particle’s spin, to the deformed- Kerr metric in Sec. VC [see Eqs. (5.36a)–(5.36e)]. The expression for hµνand the details of the above proce- dure — together with a way in which it can in principle be extended to reproduce also the next-to-leading order coupling of the particle’s spin with itself — are given in Appendix A. For the purpose of the present discus- sion, however, it is sufficient to mention that the addition of this quadrupole deformation to the metric (5.36a)– (5.36e) augments Eq. (5.55) by the term m 2Mr3c4(3ninj−δij)Si ∗Sj ∗.(5.59) Therefore, the total leading order spin-spin Hamiltonian12 is HSS2PN=HS SS2PN+HNS SS2PN +m 2Mr3c4(3ninj−δij)Si ∗Sj ∗ =m 2Mr3c4(3ninj−δij)Si 0Sj 0 −1 c4m 2Mr3/bracketleftbig (n·SKerr)2−S2 Kerr/bracketrightbig +1 2mM2r2c4/braceleftBig −[p2−2(p·n)2]S2 Kerr +[(p−2(p·n)n)·SKerr]p·SKerr/bracerightBig ,(5.60) withSi 0=Si Kerr+Si ∗. As we will show in Sec. VE, a proper choice of the vectorsSKerrandS∗in terms of the vectors σandσ∗, defined in Eqs. (5.2) and (5.3), allows us to reproduce the PN-expanded effective Hamiltonian [see Eqs. (5.27)– (5.29)] using the PN-expanded deformed-Kerr Hamilto- nian that we have just derived. Finally, it is worth noting that the presence of terms quadratic in the deformed-Kerr black hole’s spin in Eq. (5.57) explains why we introduced the 2.5PN- accurate canonical transformation (5.17). Indeed, the latter produces exactly the same terms in the PN- expanded effective Hamiltonian (5.27) at 2.5PN order. Quite interestingly, the terms quadratic in SKerrappear- ing in Eq. (5.57) could also be eliminated with a suitable choice of the reference tetrad ˜ eA. In fact, as stressed in Sec. III and in Ref. [23], a choice of the reference tetrad field corresponds to choosing a particular gauge for the particle’s spin. In agreement with this interpretation, we find that the terms of Eq. (5.57) which are quadratic inSKerrdisappear if the initial tetrad (3.16a)–(3.16d) is changed to a different tetrad ˜e′ Arelated to the original one by the following purely-spatial rotation: ˜e′T=˜eT,˜e′ I=RIJ˜eJ, (5.61) where the rotation matrix RIJis given by R=RY/bracketleftbigg −a2XZ 2R4/bracketrightbigg RX/bracketleftbigg −a2Y Z 2R4/bracketrightbigg ,(5.62) RX[ψ] andRY[φ] being rotations of angles ψandφ around the axis XandY, respectively. As a consistency check, we have verified that this new tetrad is the same as that used in Ref. [23] when com- putingtheHamiltonianinADMcoordinates,wherethose termsquadraticin SKerrdo notappear. Wehavechecked this by transforming the new tetrad (5.61) from quasi- isotropic to ADM coordinates [which are related by the coordinate transformation(49) in Ref. [31]], and compar- ingittothetetradgiveninEqs.(6.9a)–(6.9b)ofRef.[23], and find that the two tetrads agree through order 1 /c8.E. The effective-one-body Hamiltonian In this section we first find the mapping between the masses µ,Mand the spins σandσ∗of the ef- fective Hamiltonian derived in Sec VB, and those of the deformed-Kerr Hamiltonian derived in Secs. VC and VD, that is m,M,SKerrandS∗. Then, we derive the improved (resummed) EOB Hamiltonian. As shown in Ref. [9], matching the non-spinning parts HNSof these Hamiltonians forces us to identify the total massMofthe twoblackholes in the PNdescription with the deformed-Kerr mass Mof the test-particle descrip- tion, thus justifying our choice of using the same symbol for these two a priori distinct quantities. Similarly, we find thatm=µ[9]. Assuming this mapping between the masses and imposing that the PN-expanded deformed- Kerr Hamiltonian given by Eqs. (5.53)–(5.60) coincides with the effective Hamiltonian given by Eqs. (5.27)– (5.29), we obtain the following mapping between the spins S∗=σ∗+1 c2∆σ∗, (5.63) SKerr=σ+1 c2∆σ, (5.64) where we have set for simplicity a(η) = 0 andb(η) = 0 and where ∆σ=−1 16/braceleftBigg 12∆σ∗+η/bracketleftbigg2M r(4σ−7σ∗) +6(ˆp·n)2(6σ+5σ∗)−ˆp2(3σ+4σ∗)/bracketrightbigg/bracerightBigg . (5.65) Here,∆σ∗is an arbitrary function going to zero at least linearly in ηwhenη→0, so as to get the correct test-particle limit. In fact, if ∆σ∗satisfies this condi- tion and if we assume, as appropriate for black holes, S1,2=χ1,2m2 1,2(with|χ1,2| ≤1 and constant)6, when m2∼0 we have SKerr=S1+O(m2). Similarly, for m2∼0 the physical unrescaled spin of the effective par- ticle isS=S∗m/M=S2+O(m2)2. The equations of motion of our initial Hamiltonian (3.9) coincide with the Papapetrou equations [23], which describe the motion of a spinning test-particle in a curved spacetime [38, 39]. Assuming the canonical commutation relations between xi,pj,S1andS2, we obtain that the Hamilton equa- tions for the effective deformed-Kerr Hamiltonian are ˙y= ˙yP+O(m2). Here, the dot denotes a time deriva- tive,yis a generic phase-space variable ( xi,pj,S1or 6As noted by Ref. [21], a spin mapping such as ours also gives the correct test particle limit if |S1,2|/m1,2= const., but this scaling of the spins with the masses is not appropriate for bl ack holes [37].13 S2), and ˙y= ˙yPare the Papapetrou equations expressed in Hamiltonian form. Therefore, our mapping repro- duces the correct test-particle limit, and the remainders SKerr−S1=O(m2)andS−S2=O(m2)2produceextra- accelerations of order O(m2) or higher. This is compa- rable to the self-force acceleration [40], which appears at the next order in the mass ratio beyond the test-particle limit. Although different choices for the function ∆σ∗are in principle possible, we choose here ∆σ∗=η 12/bracketleftBig2M r(7σ∗−4σ)+ˆp2(3σ+4σ∗) −6(ˆp·n)2(6σ+5σ∗)/bracketrightBig , (5.66) which gives, when inserted into Eq. (5.65), ∆σ= 0. Be- cause this form for ∆σ∗is clearly not covariant under generic coordinate transformations, we choose instead the following form for the mapping of the spins, which is covariant at least as far as the square of the momentum is concerned: ∆σ= 0, (5.67) ∆σ∗=η 12/bracketleftBig2M r(7σ∗−4σ)+(Q−1)(3σ+4σ∗) −6∆r Σ(ˆp·n)2(6σ+5σ∗)/bracketrightBig , (5.68) where we have replaced ˆp2withγijˆpiˆpj=Q−1 [where Qis given in Eq. (5.52)] and ( ˆp·n)2= ˆp2 rwith ∆r(ˆp·n)2/Σ =grrˆp2 r. This form agrees with the pre- vious mapping through order 1 /c2, but differs from it at higher orders. Although neither this form is completely covariant, not even under a rescaling of the radial coor- dinate (as it still features a dependence on the radius r), it proved slightly better as far as the dynamics of the EOB model, analyzed in the next section, is concerned. In particular, the factor grr, which becomes zero at the horizon, quenches the increase of ˆ prat small radii, thus giving a more stable behavior during the plunge subse- quent to the inspiral. (A similar effect was observed in Ref. [22], where the radial momentum was expressed in tortoise coordinates to prevent it from diverging close to the horizon.) Having determined the mass and spin mappings, we can write down the improved (resummed) Hamiltonian (or EOB Hamiltonian) for spinning black holes. To this purpose, it is sufficient to invert the mapping between the real and effective Hamiltonians [Eq. (5.26)]. In units in whichc= 1, we obtain Himproved real=M/radicalBigg 1+2η/parenleftbiggHeff µ−1/parenrightbigg ,(5.69) with Heff=HS+βipi+α/radicalBig µ2+γijpipj+Q4(p) −µ 2Mr3(δij−3ninj)S∗ iS∗ j. (5.70)Here, the −µ/(2Mr3)(δij−3ninj)S∗ iS∗ jterm is the quadrupole deformation introduced in the previous sec- tion to account for the leading order coupling of the par- ticle’s spin with itself (see also Appendix A); βi,αand γijare computed using the deformed-Kerr metric, that is inserting Eqs. (5.36a)–(5.36e) into Eqs. (5.44)–(5.46); HSis obtained by inserting Eqs. (5.31), (5.32), and Eqs. (5.47a)–(5.52) into Eqs. (4.17), (4.18) and (4.19). Lastly, the spin SKerrenters this Hamiltonian through the parameter a=|SKerr|/Mappearingin the deformed- Kerr metric. Before completing this section, we want to discuss the deformation of the Kerr potentials ∆ tand ∆ rgiven in Eqs. (5.37) and (5.38), which play an important role in the EOB Hamiltonian (5.69). It is convenient to re-write the function ∆ tas ∆t=r2∆u(u), (5.71) ∆u(u) =A(u)+a2 M2u2. (5.72) In previous EOB investigations the Pad´ e summation was applied to the function ∆ uto enforce the presence of a zero, correspondingto the EOBhorizon, both in the non- spinning [11] and spinning case [19, 21]. Reference [22] pointed out that when including the 4PN and 5PNterms in the function A(u), the Pad´ e summation generates polesifspinsarepresent. Also, the Pad´ esummationdoes not always ensure the existence of an innermost stable circular orbit (ISCO) for spins aligned and antialigned with the orbital angular momentum and, even when it does, the position of the ISCO does not vary monotoni- cally with the magnitude of the spins. For these reasons, we propose here an alternative way of enforcing the exis- tence of the EOB horizons. Working through 3PN order, we write ∆u(u) =¯∆u(u)/bracketleftbig 1+η∆0+ηlog/parenleftbig 1+∆1u+∆2u2 +∆3u3+∆4u4/parenrightbig/bracketrightbig , (5.73) where ¯∆u(u) =a2 M2/parenleftBigg u−M rEOB H,+/parenrightBigg/parenleftBigg u−M rEOB H,−/parenrightBigg (5.74) =a2u2 M2+2u ηK−1+1 (ηK−1)2,(5.75) rEOB H,±=/parenleftBig M±/radicalbig M2−a2/parenrightBig (1−Kη).(5.76) Here,rEOB H,±are the EOB horizons, which differ from the Kerr horizons when the adjustable parameter Kis differ- ent from zero, and where the log is introduced to quench the divergence of the powers of uat small radii. We could in principle replace the logarithm with any other analytical function with no zeros (e.g., an exponential). However, when studying the dynamics ofthe EOB model (see Sec. VI) the results are more sensible if we choose a function, such as the logarithm, which softens the diver- gence of the truncated PN series.14 The coefficients ∆ 0, ∆1, ∆2, ∆3and ∆ 4can be de- rived by inserting Eq. (5.73) into Eq. (5.71), expanding through3PNorder,andequatingtheresulttoEqs.(5.71)and (5.72), with A(u) given by its PN expansion (5.41). Doing so, we obtain ∆0=K(ηK−2), (5.77) ∆1=−2(ηK−1)(K+∆0), (5.78) ∆2=1 2∆1(−4ηK+∆1+4)−a2 M2(ηK−1)2∆0, (5.79) ∆3=1 3/bracketleftBig −∆3 1+3(ηK−1)∆2 1+3∆2∆1−6(ηK−1)(−ηK+∆2+1)−3a2 M2(ηK−1)2∆1/bracketrightBig ,(5.80) ∆4=1 12/braceleftBig 6a2 M2/parenleftbig ∆2 1−2∆2/parenrightbig (ηK−1)2+3∆4 1−8(ηK−1)∆3 1−12∆2∆2 1+12[2(ηK−1)∆2+∆3] ∆1 +12/parenleftbigg94 3−41 32π2/parenrightbigg (ηK−1)2+6/bracketleftbig ∆2 2−4∆3(ηK−1)/bracketrightbig/bracerightBig . (5.81) By construction, if we expand Eq. (5.73) in PN orders, Kcan only appear at 4PN and higher orders, because we must recover the PN expansion (5.37)–(5.41) through 3PN order. In this sense, Kparameterizes our ignorance of the PN expansion at orders equal or higher than 4PN (i.e.,Kwould not play any role if the PN series were known in its entirety). Similarly, we re-write the poten- tial ∆r[Eq. (5.38)] as ∆r= ∆tD−1(u), (5.82) D−1(u) = 1+log[1+6 ηu2+2(26−3η)ηu3]. (5.83) The coefficients in the above function D−1(u) are such that, when PN expanded, it gives the PN result (5.42), and the logarithmic dependence is once again chosen to quench the divergence of the truncated PN series. Finally, let us stress that if we included PN orders higher than 3PN in the functions A(u) andD(u), we would need to add higher order coefficients ∆ iwithi>4 in Eq. (5.73). VI. EFFECTIVE-ONE-BODY DYNAMICS FOR CIRCULAR, EQUATORIAL ORBITS Inthissectionwestudy thedynamicsofthe novelEOB model that we developed in Sec. VE. We will show that (i) Our EOB model has the correct test-particle limit, for both non-spinning and spinning black holes, for genericorbits and arbitrary spin orientations; (ii) There exist an ISCO when the spins are aligned or antialigned with the orbital angular momentum L; (iii) The radius, energy, total angular momentum, or- bital angular momentum and frequency at theISCO exhibit a smooth dependence on the binary mass-ratio and spins. Also, this dependence looks reasonable based on what we expect from the test- particle limit and from numerical-relativity simula- tions; (iv) The frequency at the ISCO for an extreme mass- ratio non-spinning black-hole binary agrees with the exact result computed by Ref. [41]; (v) During the plunge subsequent to the ISCO, the orbital frequency of black-hole binaries with spins aligned or antialigned with Lgrows and reaches a maximum, after which it decreases. The ra- dius at which the frequency peaks is very close to the radius of the equatorial, circular light ring (or photon orbit). This feature generalizes the non- spinning behavior [9], and it has a clear physical interpretation in terms of frame-dragging. As in the non-spinning case[9], it providesa natural time at which to match the two-body description of the inspiral and plunge to the one-body description of the merger and ringdown. We stress that only (i) applies to generic orbits and spin orientations, while (ii), (iii), (iv) and (v) are true for black-hole binaries with spins aligned or antialigned with L. (It should be noted that circular or spherical orbits, and therefore the ISCO, are not even present for generic orbitsandspin orientations,becausethe systemis notin- tegrable, not even in the test-particle limit [37]). While we will tackle the study of generic orbits and arbitrary spin orientations in a follow-up paper, we argue that the preliminary study presented here is already sufficient to illustrate the potential of the novel EOB model. We re- call [22] that the only existing EOB model for spinning black-hole binaries, proposed in Refs. [19, 21], (i) repro- ducesonlyapproximatelythetestparticlelimit; (ii)when15 including non-spinning terms at 4PN and 5PN order, it does not always present an ISCO for binaries with spins parallel to L, and when it does the spin dependence of quantities evaluated at the ISCO is unusual; (iii) gen- erally, the orbital frequency does not peak during the plunge, making the prediction of the matching time from the two-body to the one-body description quite problem- atic. Let us now go through the points of the list that we presented at the beginning of this section. In order to prove point (i) we first need to observe that the de- formed metric (5.36a)–(5.36e) [with the potentials ∆ r and ∆ tgiven by Eqs. (5.82) and (5.73)] reduces to the Kerr metric as η→0, and the deformation is linear in ηwhenη∼0. Therefore, the acceleration produced by this deformation on the test-particle is comparableto the self-force acceleration, which appears at the next order in the mass ratio beyond the test-particle limit. Sec- ond, as already proved in Sec. VE, the mapping (5.63)–(5.64) of the spins reduces to SKerr=S1+O(m2) and S=S∗m/M=S2+O(m2)2whenm2∼0, where the remainders produce accelerations which are again com- parable to the self-force acceleration. To prove points (ii), (iii), (iv) and (iv), we need to write the effective EOB Hamiltonian (5.70) for equato- rial orbits and for spins parallel to the orbital angular momentum (chosen to be along the z-axis). We obtain Heff=HS+βipi+α/radicalBig µ2+γijpipj+Q4(p) −µ 2Mr3S2 ∗, (6.1) HS=geff SOL·S∗+geff SSS∗, (6.2) where geff SO=e2ν−˜µ/bracketleftBig −√Q∆r(˜Br−2˜Bνr)+/parenleftBig e˜µ+ν−˜B/parenrightBig/parenleftbig√Q+1/parenrightbig +(˜Bνr−˜Br)√∆r/bracketrightBig ˜B2M/parenleftbig√Q+1/parenrightbig√Q, (6.3) geff SS=µ M/bracketleftBigg ω+1 2˜Be−˜µ−νωr/radicalbig ∆r+/parenleftbiggL2 z µ2−˜B2e−2(˜µ+ν)∆rp2 r µ2/parenrightbiggeν−˜µωr√∆r 2˜B/parenleftbig√Q+1/parenrightbig√Q/bracketrightBigg , (6.4) with Q= 1+∆rp2 r µ2r2+L2 zr2 µ2(̟4−a2∆t).(6.5) The above equations can be evaluated explicitly by us- ing Eqs. (5.31), (5.32), (5.47a)–(5.47d), (5.48)–(5.50), (5.71)– (5.82)7. To calculate the radius and the or- bital angular momentum at the ISCO for the EOB model, we insert Eq. (6.1) into the real EOB Hamilto- nian (5.69), and solve numerically the following system of equations [9] ∂Himproved real(r,pr= 0,Lz) ∂r= 0,(6.6) ∂2Himproved real(r,pr= 0,Lz) ∂r2= 0,(6.7) with respect to randLz. Moreover, the frequency for circular orbits is given by Ω =∂Himproved real(r,pr= 0,Lz) ∂Lz, (6.8) 7A Mathematica notebook implementing the Hamiltonian (6.1) – (6.5) is available from the authors upon request.which follows immediately from the Hamilton equations becauseLz=pφ. Finally the binding energy is Ebind= Himproved real−M. Henceforth, we set the adjustable frame-dragging pa- rametersωfd 1=ωfd 2= 0 [see Eq. (5.40)] and write Kin Eq. (5.76) as a polynomial of second order in η, K(η) =K0+K1η+K2η2. (6.9) K0,K1andK2being constants. We find that if we impose K(1/4) =1 2,dK dη(1/4) = 0 (6.10) the functional dependence on ηandχof several physical quantities evaluated at the ISCO is quite smooth and regular. Therefore, imposing these constraints we obtain K(η) =K0(1−4η)2+4(1−2η)η. (6.11) It is worth noting that the values of KanddK/dηatη= 1/4 have a more direct meaning than the coefficients K1 andK2. In fact, current numerical-relativity simulations can evolve binary black holes with η≈0.25 (with only few runs having η∼0.1). Thus, they can determine K(1/4)−1/2, while the value of ( dK/dη)(1/4) can be16 FIG. 1: The frequency at the EOB ISCO for binaries having spins parallel to L, with mass ratio q=m2/m1and with spin-parameter projections onto the direction of Lgiven by χ1=χ2=χ. As expected, the frequency increases with χ for a given mass ratio, while for fixed χit increases with qif χ/lessorsimilar0.9, while it decreases with qifχis almost extremal (see text for details). hopefully determined when more numerical simulations withη= 0.1–0.25 become available. Hereafter, we will use Eq. (6.11) and set K0= 1.4467. The latter is determined by requiring that the ISCO fre- quency for extreme mass-ratio non-spinning black-hole binaries agrees with the exact result of Ref. [41], which computed the shift ofthe ISCOfrequency due to the con- servative part of self-force (see also Ref. [42] where the resultofRef.[41]wascomparedtothenon-spinningEOB prediction which resums the function (5.41) ` a laPad´ e.).8 In Fig. 1 we plot the orbital frequency at the ISCO for binaries with mass ratio q=m2/m1ranging from 10−6 to 1 and spins aligned with L. In particular, denoting byS1,2=χ1,2m2 1,2the projections of the spins along the direction of L, we consider binaries with χ1=χ2=χ. We see that the ISCO frequency increases with the mag- nitude of the spins χif the mass ratio is fixed, as ex- pected from the test-particle case. Also, if the spins are kept fixed and small, the ISCO frequency increases with the mass-ratio, as it should be to reproduce the 8We stress that the most general form of K(η) can include terms depending on a2, witha=|SKerr|/M. In particular, a term not depending on ηand proportional to a2could be determined by a calculation similar to that in Ref. [41], that is by computin g the shiftof the ISCO frequency caused by the conservative parto fthe self-force, for a non-spinning test-particle in a Kerr spac etime.FIG. 2: The final spin parameter χfinas inferred at the EOB ISCO, for binaries having spins parallel to L, with mass ratio q=m2/m1and with spin-parameter projections onto the direction of Lgiven by χ1=χ2=χ. As expected, χfin flattens for large χin the comparable mass case (see text for details). results of numerical-relativity simulations (see, e.g., the non-spinning EOB models of Ref. [6, 7]). However, if the spins are close to χ= 1, the ISCO frequency decreases when the mass ratio increases. This crossover is mir- rored by a similar behavior of other quantities evaluated at the ISCO — such as the energy, the orbital angular momentum, andthecoordinateradius—anditsphysical meaning can be explained as follows. When comparable- mass almost-extremal black holes merge, the resulting black-hole remnant has a spin parameter that is slightly smaller than the spin parameters of the parent black holes. This is a consequence of the cosmic censorship conjecture (see Ref. [46] and references therein) which prevents black holes with spin χ >1 to be formed [47]. Therefore, because in the EOB model the position, and therefore the frequency, of the ISCO (together with the lossofenergyand angularmomentum duringthe plunge) regulate the final spin of the remnant, and because for an isolated black hole the ISCO frequency increases with the spin, any EOB model that satisfies the cosmic cen- sorship conjecture must have an ISCO frequency that slightly decreases with the mass ratio when χ∼1. This interpretation can be confirmed by computing the final spin of the remnant black hole as estimated at the ISCO. We have χfin=S1+S2+LISCO (M+Ebind ISCO)2, (6.12) which is plotted in Fig. 2. Although the final spin gets slightlylargerthan1forhighinitialspins(becauseweare17 FIG. 3: The final spin parameter χfinas inferred at the EOB ISCO for binaries having spins parallel to L, with mass ratio q=m2/m1and with spin-parameter projections onto the di- rection of Lgiven by χ1=χ2=χ, comparedtotheremnant’s final spin parameter predicted by the formula presented in Ref. [43] (“BR09”), which accurately reproduces numerical - relativity results. The EOB model and the BR09 formula agree when the mass ratio is small ( q= 0.1), because the emission during the plunge, merger and ringdown is negligib le in this case. For q= 0.5 andq= 1, there is an offset, because the EOB result, at this stage, neglects the gravitational-w ave emission during the plunge, merger and ringdown (see text for details). neglecting here the energy and angular momentum emit- ted during the plunge, merger and ringdown), the curves are remarkably smooth and monotonic (see the corre- sponding Fig. 5 of Ref. [21]) and they flatten at high ini- tialspins, asexpected. Inparticular,inFig.3wefocuson mass ratios q= 1,q= 0.5 andq= 0.1, and plot the final spinχfinas inferred from the ISCO energy and angular momentum, together with the final spin of the remnant predicted by the formula presented in Ref. [43] which ac- curately reproduce the numerical-relativity results (see also Refs. [45, 49–54] for other formulas for the final spin of the remnant). It is remarkable that in spite of the off- setbetweenthepredictionsoftheformulaofRef.[43]and theEOBresult, whichisduetoneglectingtheenergyand angular momentum emitted during plunge, merger and ringdown, the qualitative behavior of the curves in Fig. 3 is the same. Also, we observe that the difference between corresponding curves decreases with the mass ratio, with the EOB and the numerical-relativity–based results be- ing in very good agreement for q= 0.1. This happens because the energy and angular momentum emitted dur- ing plunge, merger and ringdown become negligible forFIG. 4: The mass loss inferred at the EOB ISCO for binaries having spins parallel to L, with mass ratio q=m2/m1and with spin-parameter projections onto the direction of Lgiven byχ1=χ2=χ, compared to the total mass lost during the inspiral, merger and ringdown, as predicted by the formulas presented in Ref. [44] (“AEI09”) and in Ref. [45] (“RIT09”), which reproduce numerical-relativity results, although w ith different accuracies because of the different parameter regi ons they cover (see the text for details). The EOB model and the AEI09 and RIT09 fits agree when the mass ratio is small (q= 0.1), while there is an offset for q= 0.5 andq= 1. The reason is that the ringdown emission, which is negligibl e for small mass-ratios, is not taken into account by our EOB model at this stage. small mass-ratios.9. Similarly, in Fig. 4 we plot the bind- ing energy at the ISCO for mass ratios q= 1,q= 0.5 and q= 0.1 and compare it with fits to numerical-relativity data for the total mass radiated in gravitational waves during the inspiral, merger and ringdown. In particular, for theq= 1 case we use the fit in Ref. [44] (“AEI09”), while forq= 0.5 andq= 0.1 we use the fit recently pro- posed by Ref. [45] (“RIT09”). While the AEI fit is more accurate than the RIT one for the particular configura- tionconsideredhere(seeFig. 11andrelateddiscussionin Ref. [44]), the AEI fit is only applicable for comparable- mass binaries10, and for this reasonwe resortto the more 9This can be seen by noting that, for a test-particle with mass maround a black holes with mass M, the final plunge lasts a dynamical time ∼M[9], while the inspiral from large radii to the ISCO lasts ∼M2/m. 10The limited applicability of the AEI fit (which is only valid f or equal-mass binaries with spins aligned or anti-aligned) is indeed one reason why it turns out to be more accurate, for the config- uration under consideration, than the RIT fit, which is inste ad applicable to more generic binaries.18 FIG. 5: The frequency at the EOB ISCO for binaries having spins parallel to L, with mass ratio q=m2/m1and with spin-parameter projections onto the direction of Lgiven by χ1=−χ2=χ. As expected, the frequency is constant in the equal-mass case, because the spins of the two black holes cancel out (see text for details). FIG. 6: The final spin parameter χfinas inferred at the EOB ISCO, for binaries having spins parallel to L, with mass ratio q=m2/m1and with spin-parameter projections onto the direction of Lgiven by χ1=−χ2=χ. The results are the same for all equal-mass binaries, for which the spins of the two black holes cancel out (see text for details).FIG. 7: The maximum of the EOB orbital frequency during the plunge, for binaries having spins parallel to L, with mass ratioq=m2/m1and with spin-parameter projections onto the direction of Lgiven by χ1=χ2=χ. As expected, the frequency increases with χfor a given mass ratio, while for fixedχit increases with qifχ/lessorsimilar0.9, while it decreases with qisχis almost extremal (see text for details). generalRIT fitin the q= 0.5andq= 0.1cases. [In Fig.4 we show the predictions of both the AEI and the RIT fit in theq= 1 case. Being the AEI fit more accurate, its difference from the RIT fit gives an idea of the error bars which should be applied to the predictions of the RIT fit forq= 0.5 andq= 0.1.] InFigs.5and6wepresentsimilarresults, fortheISCO frequency and for the final spin estimated at the ISCO, in the case of spins antialigned with the orbital angular momentum. The most apparent feature of these figures is that, in the equal-mass case, the quantities under con- sideration are independent of χ. This happens because in this case the spins S1andS2are equal and opposite, which results in a zero value for the spins SKerrandS∗ entering the EOB Hamiltonian (5.69). As such, in the EOB model, equal-mass binaries with equal and oppo- site spins behave as non-spinning binaries. This feature, which is also shared by the PN-expanded Hamiltonian, until the PN order which is currently known, is also in agreement with the results of numerical simulations. In fact, equal-mass binaries with equal and opposite spins would be indistinguishable with LISA, Virgo and LIGO observations [44, 55]. Except for this feature, and simi- larly to the aligned case discussed above, the behavior of the curves in Figs. 5 and 6 is quite smooth and regular when going from the equal-mass case to the test-particle case. In Fig. 7 we plot the maximum value of the orbital frequency during the plunge subsequent to the inspi-19 FIG. 8: For binaries having spins parallel to L, with mass ratio q=m2/m1= 1 (left panel) and q=m2/m1= 0.1 (right panel) and with spin-parameter projections onto the direction of Lgiven by χ1=χ2=χ, we plot twice the maximum of the EOB orbital frequency during the plunge against the frequencie s of the first 8 overtones of the ℓ= 2,m= 2 quasi-normal mode of a Kerr black hole. The quasi-normal mode frequency is compute d using the final spin and final mass of the remnant. The final spin is estimated by applying the formula of Ref. [43], while for the final mass we use the formula of Ref. [44] (“AEI09”) in theq= 1 case and that of Ref. [45] (“RIT09”) in the q= 0.1 case. As can be seen, in the q= 0.1 case the peak frequencies lie among the high overtones of the ℓ= 2,m= 2 mode, while in the q= 1 case they are generally lower than them. In the q= 1 case we also mark with a square the numerical gravitationa l-wave frequency at the peak of the h22mode when χ= 0. This gravitational-wave frequency coincides with (twice) the maximum of the EOB orbital frequency at the time when the matching of the quasi-normal modes is performed in the non-s pinning EOB model of Ref. [7]. The numerical gravitational- wave frequency is computed from the numerical simulation of Refs . [48] (“Caltech-Cornell”). ral, for binaries with mass ratio q=m2/m1and with χ1=χ2=χ. Moreprecisely, weassumethatthe particle starts off with no radial velocity at the ISCO (thus hav- ing angular momentum LISCOand energy EISCO), and we compute prassuming that the energy and angular momentum are conserved during the plunge. We find that the orbital frequency presents a peak for any value of the spins and any mass ratio, and we denote the value of the frequency at the peak with MΩmax. We note that the behavior of MΩmaxas a function of the mass ratio is similar to that of MΩISCO. In particular, its dependence onηchanges sign when going from small to large spins. The physical interpretation of the peak of the orbital frequency is that the frequency increases as the effec- tive particle spirals in, but when the effective particle gets close to the black hole, the orbital frequency has to decrease because the particle’s motion gets locked to the horizon (this is a well-known phenomenon, see for instance Ref. [56, 57] for some of its effect on the test- particle dynamics). Said in another way, the orbital fre- quency of the effective particle for an observer at infinity goestoaconstant(ortozerointhenon-spinningcase[9]) on the EOB horizon. As a consequence, the peak in the frequency can be used to signal the transition betweentwo regimes [9]: one in which the deformed black hole and the effective particle have different frequencies and oneinwhichthe twobodies basicallymoveandradiateas a single perturbed black hole. For this reason the peak of the frequency provides the EOB approach with the nat- ural point where to switch to the one-body description, i.e., the point where to start describing the gravitational waveforms as a superposition of quasi-normal modes. We find that the values of MΩmaxare roughly those needed to attach the quasi-normal modes used in EOB modelstodescribethemergerandtheringdown[6,7,22]. ThisisshowninFig.8, whereweplottwicethemaximum of the orbital frequency, MΩ22for binaries with q= 1 (left panel) and q= 0.1 (right panel) and with spins χ1=χ2=χ. We compare MΩ22with the frequency of the first 8 overtones of the ℓ= 2,m= 2 quasi-normal mode of a Kerr black hole, computed using the final spin and the final mass of the black-hole remnant [58]. [The final spin parameter is estimated by applying the for- mula of Ref. [43], while for the final mass we use the formula of Ref. [44] (“AEI09”) in the q= 1 case and the one of Ref. [45] (“RIT09”) in the q= 0.1 case.] In theq= 1 case we also mark with a square the numerical gravitational-wavefrequency at the peak ofthe h22mode20 FIG. 9: The maximum of the EOB orbital frequency during the plunge, for binaries having spins parallel to L, with mass ratioq=m2/m1and with spin-parameter projections onto the direction of Lgiven by χ1=−χ2=χ. The results are the same for all equal-mass binaries, for which the spins of the two black holes cancel out (see text for details). whenχ= 0. This gravitational-wavefrequency coincides with (twice) the maximum of the EOB orbital frequency atthetimewhenthematchingofthequasi-normalmodes is performed in the non-spinning EOB model of Ref. [7]. The numerical gravitational-wavefrequency is computed from the numerical simulation of Refs. [48] (“Caltech- Cornell”). As can be seen, while in the q= 0.1 case the peak frequencies lie among the high overtones of the ℓ= 2,m= 2 quasi-normal mode, in the q= 1 case they are generally lower than them. Quite interestingly, we find that the values of MΩ22forχ/greaterorsimilar0.4 can be in- creased up to the frequencies of the quasi-normal modes by assuming ωfd 2∼30–70 in Eq. (5.40). Nevertheless, the frequencies that we obtain are comparable to those used for the matching with the quasi-normal modes in Ref. [7, 22], and we therefore expect such a matching to be possible also in our EOB model. Also, it is interesting to note that the position rmaxof the frequency peak is quite close (to within 8%) to the position of the light ring (or circular photon orbit). This fact, which holds exactly in the non-spinning case [9], further confirms that the potential barrier for massless particles (such as gravitational waves) lies at r∼rmax. Finally, inFig.9weshowthemaximumvalueoftheor- bital frequency during the plunge for binaries with mass ratioq=m2/m1and withχ1=−χ2=χ. As for the ISCO quantities, the dependence on the spins and the massratiosis muchsimplerthan in the alignedcase, with the black-holespins cancelling out in the equal mass-case and thus giving results which are independent of χ. Also,weseeasmooth transitionfrom theequal-massto the ex- trememass-ratiocase,thatourmodelreproducesexactly. Also in this antialigned case, the radius rmaxagrees with the light-ring position to within 4%. VII. CONCLUSIONS In this paper, building on Ref. [23], we computed the Hamiltonian of a spinning test particle, at linear order in the particle’s spin, in an axisymmetric stationary metric and in quasi-isotropic coordinates. Then, by applying a coordinate transformation, we derived the Hamiltonian of a spinning test particle in Kerr spacetime in Boyer- Lindquist coordinates We used those results to construct an improved EOB Hamiltonian for spinning black holes. To achieve this goal, we followed previous studies [19, 21] and mapped the real two-body dynamics into the dynamics of an ef- fective particle with mass µand spin S∗moving in a deformed-Kerr spacetime with spin SKerr, the symmet- ric mass-ratio of the binary, η, acting as the deformation parameter. To derive the improved EOB Hamiltonian, we pro- ceeded as follows. First, we applied a suitable canonical transformationtotherealADM Hamiltonianandworked out the PN-expanded effective Hamiltonian through the relation Heff µ=H2 real−m2 1−m2 2 2m1m2. (7.1) Then, we found an appropriate deformed-Kerr met- ric such that the corresponding Hamiltonian, when ex- panded in PN orders, coincided with the PN-expanded effective Hamiltonian through 3PN order in the non- spinning terms, and 2.5PN order in the spinning terms. The (resummed) improved EOB Hamiltonian is then found by inverting Eq. (7.1), which gives Himproved real=M/radicalBigg 1+2η/parenleftbiggHeff µ−1/parenrightbigg ,(7.2) withHeffgiven by Eq. (5.70), where α,βiandγijare ob- tained by inserting Eqs. (5.36a)–(5.36e) into Eqs. (5.44)– (5.46); where HSis obtained by inserting Eqs. (5.31), (5.32), and Eqs. (5.47a)–(5.52) into Eqs. (4.17), (4.18) and (4.19); and where the effective particle’s spin S∗ and the deformed-Kerr spin SKerr(witha=|SKerr|/M) are expressed in terms of the real spins by means of Eqs. (5.63), (5.64), (5.2) and (5.3). The crucial EOB metric potential for quasi-circular motion is the potential ∆ t(r) (which reduces in the non- spinningcaseto the radialpotential A(r) ofRefs. [9, 10]). To guarantee the presence of an inner and outer horizons in the EOB metric, we proposed to re-write the poten- tial ∆ t(r) in a suitable way [see Eqs. (5.71) and (5.73)],21 introducing the adjustable EOB parameter K(η) regu- lating the higher-order, unknown PN terms. The rea- son why we did not re-write the potential ∆ t(r) using the Pad´ e summation [21] is because Ref. [22] found that when including non-spinning terms at 4PN and 5PN or- der, the Pad´ e summation produces spurious poles, does not always ensure the presence of an ISCO for binaries with spins parallel to Land, even when it does, the spin dependence of physical quantities evaluated at the ISCO is quite unusual. Restricting the study to circular orbits in the equato- rial plane and assuming spins aligned or antialigned with the orbital angular momentum, we investigated several features of our improved EOB Hamiltonian. Using an expression of the EOB adjustable parameter K(η) which reproduces the self-force results in the non-spinning ex- treme mass-ratio limit [41, 42], we computed the orbital frequency at the EOB ISCO, we estimated the final spin fromtheEOBISCO,andthemaximumorbitalfrequency during the plunge. We found that these predictions are quite smooth and regular under a variation of ηand of theblack-holespins. Quiteinterestingly,themaximumof the orbital frequency during the plunge alwaysexists and is close to the light-ring position, as in the non-spinning case [9]. For this reason, as in the non-spinning case [9], the orbital-frequency peak can be used within the EOB to markthe matching time at which the mergerand ring- down start, i.e, the time when, in the EOB formalism, the gravitational waveforms start being described by a superposition of quasi-normal modes. This will be useful in future comparisons of the EOB model with numerical- relativity simulations. The results of Sec. VI are an example of the per- formances that our improved EOB Hamiltonian can achieve. We expect several refinements to be possibly needed when comparing our EOB model with accurate numerical-relativity simulations of binary black holes. We may, forexample, extend ourmodel toreproducealso the next-to-leading order spin-spin couplings, which are known and appear at 3PN order [28–30, 33–36]. Also, we might introduce a different mapping between the black- hole spins S1,S2andS∗,SKerr, a different form of the adjustable parameter K(η), and re-write differently the EOB metric potential ∆ t(r). We could also introduce in it the adjustable parameters a5anda6at 4PN and 5PN order, respectively. Moreover, other choices of the reference tetrad used to work out the Hamiltonian for a spinning test-particle in an axisymmetric stationary spacetime could be in principle used, leading to a differ- ent (canonically related) EOB Hamiltonian. Lastly, the mapping (7.1)-(7.2) could me modified by introducing a dependence on the spin variables. In conclusion, the most remarkable feature of our im- proved EOB Hamiltonian is that in the extreme mass- ratio limit, it exactly reproduces the Hamiltonian of a spinning test particle in a Kerr spacetime, at linear order in the particle’s spin and at all PN orders.Acknowledgments We thank Yi Pan for several useful discussions. E.B. and A.B. acknowledge support from NSF Grants No. PHYS-0603762 and PHY-0903631. A.B. also acknowl- edges support from NASA grant NNX09AI81G. Appendix A: Incorporating spin-spin couplings in the effective-one-body Hamiltonian The Hamiltonian for a spinning particle in a Kerr spacetime that we derived in Sec. IV, and the Hamilto- nian for a spinning particlein a deformed-Kerrspacetime that we derived in Sec. VC are only valid at linear order in the particle’s spin. However, as suggested in Ref. [19], we can introduce the terms that are quadratic in the par- ticle’s spin by modifying the quadrupole moment of the Kerr metric. In particular, we can add a quadrupole which is quadratic in the particle’s spin to the quadrupole of the Kerr metric (which is quadratic in SKerr). The expres- sion for the metric perturbation correspondingto a slight change of the Kerr quadrupole can be extracted from the Hartle-Thorne metric [59, 60], which describes the spacetime of a slowly rotating star. Ref. [61] gives this expression in quasi-Boyer-Lindquist coordinates (i.e., in coordinates that reduce to Boyer-Lindquist coordinates if the quadrupole perturbation is zero, thus reducing the spacetime to pure Kerr). This is exactly what is needed for our purposes, since we work in quasi-Boyer-Lindquist coordinates too. In particular, our procedure for introducing the terms which are quadratic in the particle’s spin into our Hamil- tonian consists of modifying the effective metric (5.36a)– (5.36e) by adding the quadrupole metric hµν=1 M4QijS∗ iS∗ j¯hµν, (A1) where the quadrupole tensor Qijis given by Qij=δij−3ninj, (A2) and¯hµνis given by [61] ¯htt=1 1−2M/rF1(r),¯hti= 0, (A3) ¯hij=−F2(r)/braceleftbigg δij−ninj/bracketleftbigg 1+/parenleftbigg 1−2M r/parenrightbiggF1(r) F2(r)/bracketrightbigg/bracerightbigg . (A4) The functions F1,2(r) in the above equation are derived in Ref. [61] by transforming the Hartle-Thorne metric to22 quasi-Boyer-Lindquist coordinates, and are given by F1(r) =−5(r−M) 8Mr(r−2M)(2M2+6Mr−3r2) −15r(r−2M) 16M2log/parenleftbiggr r−2M/parenrightbigg ,(A5) F2(r) =5 8Mr(2M2−3Mr−3r2)+ 15 16M2(r2−2M2)log/parenleftbiggr r−2M/parenrightbigg .(A6) Because at large radii F2(r)≈ −F1(r)≈(M/r)3, we seethat the deformation hµν, when inserted in the Hamil- tonian (5.43), gives the correct leading-order (2PN) cou- pling of the particle’s spin with itself. Keeping only the leading-order term created by hµν, the effective EOB Hamiltonian therefore becomes H=HS+βipi+α/radicalBig m2+γijpipj+Q4(p) −m 2Mr3QijS∗ iS∗ j. (A7) [1] S. J. Waldman (LIGO Scientific Collaboration), Class. Quantum Grav. 23, S653 (2006). [2] F. Acernese et al. (Virgo Collaboration), Class. Quant. Grav.25, 114045 (2008). [3] B. F. Schutz, Class. Quant. Grav. 26, 094020 (2009). [4] A. Buonanno, Y. Pan, J. G. Baker, J. Centrella, B. J. Kelly, S. T. McWilliams, and J. R. van Meter, Phys. Rev. D76, 104049 (2007). [5] P. Ajith, S. Babak, Y. Chen, M. Hewitson, B. Krishnan, A. M. Sintes, J. T. Whelan, B. Br¨ ugmann, P. Diener, N. Dorband, et al., Phys. Rev. D 77, 104017 (2008). [6] T. Damour and A. Nagar, Phys. Rev. D 79, 081503 (2009). [7] A. Buonanno, Y. Pan, H. P. Pfeiffer, M. A. Scheel, L. T. Buchman, and L. E. Kidder, Phys. Rev. D 79, 124028 (2009). [8] N. Yunes, A. Buonanno, S. A. Hughes, M. Cole- man Miller, and Y. Pan (2009), arXiv:0909.4263 [gr-qc]. [9] A. Buonanno and T. Damour, Phys. Rev. D 62, 064015 (2000). [10] A. Buonanno and T. Damour, Phys. Rev. D 59, 084006 (1999). [11] T. Damour, P. Jaranowski, and G. Sch¨ afer, Phys. Rev. D62, 084011 (2000). [12] A. Buonanno, G. B. Cook, and F. Pretorius, Phys. Rev. D75, 124018 (2007). [13] Y. Pan, A. Buonanno, J. G. Baker, J. Centrella, B. J. Kelly, S. T. McWilliams, F. Pretorius, and J. R. van Meter, Phys. Rev. D 77, 024014 (2008). [14] T. Damour and A. Nagar, Phys. Rev. D 77, 024043 (2008). [15] T. Damour, A. Nagar, E. N. Dorband, D. Pollney, and L. Rezzolla, Phys. Rev. D 77, 084017 (2008). [16] T. Damour, A. Nagar, M. Hannam, S. Husa, and B. Br¨ ugmann, Phys. Rev. D 78, 044039 (2008). [17] M. Boyle, A. Buonanno, L. E. Kidder, A. H. Mrou´ e, Y. Pan, H. P. Pfeiffer, and M. A. Scheel, Phys. Rev. D 78, 104020 (2008). [18] P. Ajith, M. Hannam, S. Husa, Y. Chen, B. Bruegmann, N. Dorband, D. Mueller, F. Ohme, D. Pollney, C. Reiss- wig, et al. (2009), arXiv:0909.2867 [gr-qc]. [19] T. Damour, Phys. Rev. D 64, 124013 (2001). [20] A. Buonanno, Y. Chen, and T. Damour, Phys. Rev. D 74, 104005 (2006). [21] T. Damour, P. Jaranowski, and G. Sch¨ afer, Phys. Rev. D78, 024009 (2008).[22] Y. Pan, A. Buonanno, L. T. Buchman, T. Chu, L. E. Kidder, H. P. Pfeiffer, and M. A. Scheel (2009), arXiv:0912.3466 [gr-qc]. [23] E. Barausse, E. Racine, and A. Buonanno, Phys. Rev. D 80, 104025 (2009). [24] J. M. Bardeen, in Black Holes (Les Astres Occlus) (Gor- don and Breach, New York, 1973), pp. 241–289. [25] G. Cook, Living Reviews in Relativity 3, 5 (2000). [26] T. Damour and G. Sch¨ afer, Nuovo Cimento 101 B, 127 (1988). [27] T. Damour, P. Jaranowski, and G. Schaefer, Phys. Rev. D77, 064032 (2008). [28] J. Steinhoff, S. Hergt, and G. Schaefer, Phys. Rev. D 77, 081501(R) (2008). [29] J. Steinhoff, G. Sch¨ afer, and S. Hergt, Phys. Rev. D 77, 104018 (2008). [30] J. Steinhoff, S. Hergt, and G. Sch¨ afer, Phys. Rev. D 78, 101503 (2008). [31] S. Hergt and G. Schaefer, Phys. Rev. D 77, 104001 (2008). [32] S.HergtandG.Sch¨ afer, Phys.Rev.D 78, 124004(2008). [33] R. A. Porto and I. Z. Rothstein, Phys. Rev. Lett. 97, 021101 (2006). [34] R. A. Porto and I. Z. Rothstein (2007), arXiv:0712.2032 [gr-qc]. [35] R. A.PortoandI.Z.Rothstein, Phys.Rev.D 78, 044012 (2008). [36] R. A.PortoandI.Z.Rothstein, Phys.Rev.D 78, 044013 (2008). [37] M. D. Hartl, Phys. Rev. D67, 024005 (2003). [38] A. Papapetrou, Proc. R. Soc. A 209, 248 (1951). [39] A. Papapetrou, Proc. Phys. Soc. A 64, 57 (1951). [40] E. Poisson, Living Rev. Rel. 7, 6 (2004). [41] L. Barack and N. Sago, Phys. Rev. Lett. 102, 191101 (2009). [42] T. Damour, Phys. Rev. D 81, 024017 (2010), 0910.5533. [43] E. Barausse and L. Rezzolla, Astrophys. J. Lett. 704, L40 (2009). [44] C. Reisswig, S. Husa, L. Rezzolla, E. N. Dorband, D. Pollney, and J. Seiler, Phys. Rev. D 80, 124026 (2009), 0907.0462. [45] C. O. Lousto, M. Campanelli, and Y. Zlochower (2009), arXiv:0904.3541 [gr-qc]. [46] R. Penrose, J. Astrophys. Astron. 20, 233 (1999). [47] S. Dain, C. O. Lousto, and Y. Zlochower, Phys. Rev. D 78, 024039 (2008).23 [48] M. A. Scheel, M. Boyle, T. Chu, L. E. Kidder, K. D. Matthews, and H. P. Pfeiffer, Phys. Rev. D 79, 024003 (2009). [49] L. Rezzolla, E. N. Dorband, C. Reisswig, P. Diener, D. Pollney, E. Schnetter, and B. Szil´ agyi, Astrophys. J. 679, 1422 (2008). [50] L. Rezzolla, P. Diener, E. N. Dorband, D. Pollney, C. Reisswig, E. Schnetter, and J. Seiler, Astrophys. J. Lett.674, L29 (2008). [51] L. Rezzolla, E. Barausse, E. N. Dorband, D. Pollney, C. Reisswig, J. Seiler, and S. Husa, Phys. Rev. D 78, 044002 (2008). [52] E. Barausse (2009), arXiv:0911.1274 [qr-qc]. [53] A. Buonanno, L. E. Kidder, and L. Lehner, Phys. Rev. D77, 026004 (2008).[54] W. Tichy and P. Marronetti, Phys. Rev. D78, 081501 (2008). [55] B. Vaishnav, I. Hinder, F. Herrmann, and D. Shoemaker, Phys. Rev. D 76, 084020 (2007). [56] S. A. Hughes, Phys. Rev. D63, 064016 (2001). [57] E. Barausse, S. A. Hughes, and L. Rezzolla, Phys. Rev. D76, 044007 (2007). [58] E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav.26, 163001 (2009). [59] J. B. Hartle, Astrophys. J. 150, 1005 (1967). [60] J. B. Hartle and K. S. Thorne, Astrophys. J. 153, 807 (1968). [61] K. Glampedakis and S. Babak, Classical and Quantum Gravity 23, 4167 (2006).

1308.3428v1.Spintronics_and_Pseudospintronics_in_Graphene_and_Topological_Insulators.pdf

Spintronics and Pseudospintronics in Graphene and Topological Insulators Dmytro Pesin and Allan H. MacDonald Department of Physics, University of Texas at Austin, Austin TX 78712- 1081, USA ABSTRACT The two- dimensional electron systems in graphene and in topological insulator s are described by massle ss Dirac equations. Although the two systems have similar Hamiltonians , they are polar opposites in terms of spin-orbit coupling strength. We briefly review the status of efforts to achieve long spin relaxation times in graphene with its weak spin- orbit coupling, and to achieve large current -induced spin polarizations in topological - insulator surface states that have strong spin- orbit coupling . We also comment on differences between the magnetic responses and dilute -moment coupling properties of the two systems, and on the pseudospin analog of giant magnetoresistance i n bilayer graphene. I: Introduction --- The central goals of spintronics [1] are to understand mechanisms by which it is possible to achieve efficient electrical control of spin- currents and spin- configurations, and to discover materials in which these mechanisms are prominently exhibited. Because of the obvious relationship to magnetic information storage technologies , the possibility of applications is always in the background of spintronics research topics, and sometimes jumps to the foreground to spectacular effect. Nevertheless, the problems that aris e in this field are often intriguing from a fundamental point of view , and many topics are pursued for their intrinsic interest. Many of the most active themes of spintronics research are reviewed elsewhere in this issue. This Progress A rticle is motivated by recent interest in two new types of electronically two- dimensional (2D) material: graphene layer systems [2][3], and surface -state systems of 3D topological insulator s (materials that act as bulk insulators but have topologically protected surface states) [4][5]. We briefly review recent work that has explored these materials from the spintronics [6] point -of-view, provide a perspective of how graphene and topological insulator systems fit into the broader spintronics context, and speculate on directions for future research. Studies of graphene- based 2D electron systems (2DESs) and of two and three dimensional topological insulators are among the most interesting and active current topics in materials physics. The two systems have closely related, although still distinct, electronic properties. We rest rict our attention primarily to 3D topological insulators in which the low -energy degrees of freedom are surface -state electrons that are described by 2D Dirac equations. The same equations describe low-energy π-band electrons that are confined to a single graphene sheet. Below we refer to the two systems generically as 2D Dirac systems (2DDSs). The spi ntronics field can be organized as summarized in Fig. 1. The most important distinction is between conductors with magnetic order and c onductors without magnetic order. The 2DDSs are in the second category so far, although one can imagine hybrid systems in which spins behave collectively because of proximity effects or because of deliberately introduced dilute moments [7][8][9][10]. Spintronics exists as a topic largely because of the difference ( two or more order of magnitude ) between elec tron velocities in conductors and the speed of light, which almost always makes spin- orbit coupling weak. Spins are therefore usually almost conserved: that is, their relaxation times are normally much longer than other characteristic electronic time scales. In ordered systems , weak spin-orbit coupling leads to magnetic anisotropy energies that determine the energetically favorable magnetization orientations in a crystalline lattice. The anisotropy energies are extremely small compared to magnetic condensat ion energies , which measure how much the energy is lowered by ordering magnetically . For this reason , the spin- densities that can be induced by transport currents [11] are sufficient to induce magnetization switching, even though they are small in a relative sense. On the other hand , the changes in electrical transport properties that can accompany switching are often large, because they reflect the full strength of exchange interactions in the magnetic state. (Thermal transport is also sensitive to magnetic configurations [12] .) Paramagnetic conductors are not spin- polarized in their ground state, but because of spin- orbit coupling transport currents (which break time -reversal invariance) can induce spin- densities. Moreover currents can be weakly spin- polarized even in non- magnetic materials [13]. In spintronics , paramagnetic conductors with especially weak spin- orbit coupling are desirable as 'spin-conservers ' that can transmi t spin -encoded information across a device with high fidelity. On the other hand paramagnetic conductors with strong spin- orbit coupling are desirable because they can be ' spin-generators ' when combined with transport currents. Generally speaking, the surface states of topological insulators have exceptionally strong spin- orbit coupling and could provide an interesting example of a spin-generator system, whereas graphene π-bands have weak spin- orbit coupling and have potential as a spin-conserver system. At the time of writing, fewer potential applications have been identified for the more subtle spintronics effects that occur in non- magnetic conductors, like the materials that we will discuss here , although recent work on magnetization reversal induced by the spin Hall effect near heterojunctions between magnetic and non- magnetic materials [14][15] suggests that their potential has not yet been fully realized. In our view the most intriguing possibilities for 2DDS s are gate -tunable dilute- moment or nanoparticle- array magnetism at topological -insulator or graphene surface s, and layer -pseudospin giant magnetoresistance in graphene bilayers . At the present time, experimental work on graphene- sheet 2DESs is much more advanced than work on topological -insulator surface states. The study of topological in sulators has so far advanced slowly because of materials problems, which include the lack of real ins ulating behavior in many cases. Although these materials issues [16] are critically important, they are not addressed in this Progress Article. Spintronics in Graphene Sheets The states near the Fermi level of a graphene sheet are πelectrons. In neutral graphene , the π- bands are half -filled, and the Fermi level lies at the energy of the Brillouin -zone corner where there is a linear band crossing between π conduction and valence bands - that is, at the Dirac point. For energies close to the Dirac poi nt, the band Hamiltonian 𝐻 is accurately approximated by a two- dimensional Dirac equation: ) (yy xxz k k v H σστ+ = (1) where ),(y xkk is 2D momentum measured from the band- crossing point, 810≈v cm s-1 is the Bloch state velocity at the Dirac point, zτ is a Pauli matrix that acts on the valley degree of freedom that distinguishes the two inequivalent band crossing points, and yx,σ are Pauli matrices that act on the honeycomb lattice's sublattice degree- of-freedom. It is common in the graphene literature to view the sublattice degree- of-freedom as a pseudospin, with the eigenstates of localized on A or B subl attices ( Fig.2.) Using this language , the Ha miltonian is purely off -diagonal in pseudospin ; it represents hopping between sublattices that vanishes at the Dirac point because of destructive interference between the three nearest -neighbor hopping paths. Graphene sheets are ambipolar , - that is, their Fermi energies can be shifted by approximately 3.0± eV relative to the Dirac point by gating. Surface states of topological insulators , discussed in the next section, have similar 2D Dirac bands as explained in Fig. 2 . The spin- conserving potential of graphene is due to the weakness of spin-orbit interactions at energies close to the Dirac point of an intrinsic inversion symmetric sheet. These interactions take the form [17][18][19] zzzIIG SO s H τσ∆=, (2) where zsis a Pauli matrix that acts on the spin degrees -of-freedom. The form of the interaction is determined entirely by symmetry and corresponds to a staggered potential which has opposite signs on opposi te sublattices (the zσfactor), and for each sublattice opposite signs for opposite spins (the zsfactor ). The valley dependent zτ factor indicates that this potential has opposite signs when its dependence on triangular lattice position is Fourier -transformed at the two inequivalent Brillouin -zone- corner Dirac points K and 'K. The magnitude of I∆ is smaller than the characteristic spin -orbit scale of p2 electrons in carbon 6~ξ meV, because [18][19] spin-orbit interactions vanish when projected onto the atomic π bands of a flat sheet . Spin -orbit coupling in a flat graphene sheet is weaker than in the curved graphene s heets of nanotubes [18 ][20][21]. Using a simple tight-binding approach to describe spin- orbit induced mixing between π- and σ- bands leads to the estimate µ1~I∆ eV. Ab initio calculations [22][23] predict values that range from µ1~ eV to µ50~ eV . Although the small value of this coupling constant adds to the difficulty of estimating its value accurately, there is no doubt that it is small compared to characteristic spin- orbit energy scales. Because of its small value, it has also not yet been possible to di rectly measure I∆ in graphene. Experiment has, however, placed an upper bound of µ100~ eV on the value of I∆(ref. [24]). The spin relaxation associated with the intrinsic spin- orbit interaction occurs via the Elliot -Yafet disorder -scattering spin- relaxation mechanism [25][26] . Because band eigenstates do not have a pure spin, spin- reversals can be induced even by a spin- independent disorder potential. In the graphene case, however, the component of spin perpendicular to the surface is not relaxed ( ∞→⊥τ ) because the intrinsic spin- orbit interaction commutes with zs. For the in- plane spin polarization, the spin relaxation rate for electrons with the Fermi energy , Fε, is [25][26] τετ1 1 22 || FI∆≈ (3) where Fε is measured from the band- crossing point. Note that when the Elliot -Yafet mechanism applies , the spin rel axation time, ||τ, is proportional to the momentum relaxation time, τ; this is th e identifying property of the Elliot -Yafet relaxation mechanism. The latter property is quite natural, because in the Elliot -Yafet mechanism spin relaxes only during c ollisions and the spin- and momentum -relaxation rates must therefore be proportional. To obtain a conservative theoretical estimate of the E lliot-Yafet spin-flip time, we adopt the above experimental limit of µ100~ eV as an estimate for the intrinsic spin-orbit str ength, and assume a mobility of 3 ,000 cm2V-1s-1 at the carrier densities of ~1210cm-2 used in most experimental spin- relaxation studies. The resulting spin -relaxation time is 50~ns. If we instead assumed a value of I∆ in the mid- range of theoretical estimates ( µ10~ eV), the spin- relaxation time would exceed a microsecond and the corresponding spin -diffusion length ||τλ Dsd= would be µ300~ m. Here the diffusion constant D is given by 2/2τv . If I∆was closer to the tight -binding estimates , ||τ and sdλwould be even longer. Unfortunately, this simple and happy scenario has not yet been realized; observed spin memory times and lengths are still much shorter than these values. Early Hanle spin- precession studies [27][28][29][30][31][32][33] found that the spin- diffusion length sdλwas indeed proportional to the charge scattering time τ, (Fig. 3) but the measured carrier spin lifetimes in gr aphene sheets were in the 200 100− ps range. Furthermore , the sense of the anisotropy between the in -plane and out - of-plane relaxation rates observed experimentally [34] was that ||ττ<⊥ , opposite to expectations based on Elliot -Yafet relaxation due to intrinsic spin -orbit coupling. These observations indicated that although spins were indeed relaxed by the Elliot -Yafet mechanism, non- intrinsic spin -orbit interaction mechanisms must play an essential role. Motivated by these observations, a variety of different extrinsic spin- relaxation mechanisms were studied theoretically [35][36][37][38]. Recently it was demonstrated [39] that relaxation induced by ferromagnetic contact s can play a role in limiting the apparent spin lifetime in Hanle experiment s performed with transparent contacts. Because this spin -relaxation mechanism can be mitigated by fabricating pinhole- free tunnel barriers between the graphene sheet and spin- injectors and detectors, the discovery is an encouraging one. At present , the largest spin relaxation rates that have been achieved in single layer graphene [40][41] are ~0.5 ns . But because samples prepared in similar ways can show very different spin -relaxation times, the source of the spin- flip scattering that limits the spin-relaxation time in a particular material is usually not known. Longer spin - lifetimes, breaking the 1-ns barrier, have been achieved i n bilayer graphene, possibly owing to enhanced screening or reduced surface exposure in this carbon allotrope [40][41] as summarized in Fig. 3. Importantly, it was observed that in bilayer graphene a regime can be achieved in which the spin relaxation time is inversely proportional to the momentum relaxation time. As we now explain , this observation signals the presence of spin- split bands due to broken inversion symmetry . Quite generally , any material with broken inversion symmetry can exhibit the Dyakonov -Perel spin- relaxation mechanism [42]. This mechanism accounts for changes in spin direction between collisions with impurities. When Bloch states are spin-split, propa gating electrons preces s under the influence of a momentum -dependent effective magnetic field. Dyakonov -Perel relaxation can occur in single and bilayer graphene because of Rashba c oupling due to gating electric fields or ripples. Near the Dirac point, the spin- orbit coupling Hamiltonian for the Rashba interaction is [17] ) (, sy yzx RRG SO s s H στσ− ∆= (4) where R∆is the strength of Rashba spin- orbit coupling. If the effect of ripples is taken into account, R∆acquires a random position- dependent contribution [36]. The dependence on momentum direction enters through the Dirac -Hamiltonian (1) via its dependence on the in- plane components of the lattice pseudospin vector ),(y xσσ . It should be noted that the above Hamiltonian will contribute to Elliot -Yafet -type relaxation in addition to the DP relaxation discussed below. However, the Elliot -Yafet rate is always smaller than the Dyakonov -Perel rate contribution in good conductors , that is, when 1>>τεF , which normally holds in graphene except at small carrier densities. The Dyakonov -Perel relaxation rate can be estimated using simple physical arguments. For weak spin-orbit coupling ( τ/<<∆R ), the Bloch state lifetime τ is not long enough to resolve the small spin-orbit splitting. Spins then relax via a process of random precessional walks on the spin Bloch sphere with step length /τR∆ and step time τ. A spin with a non- equilibrium orientation will relax after a time DPτsuch that .1~τττDP R ∆ (5) These considerations imply that , ~22 ττ RDP∆ (6) and that the spin -diffusio length is . ~ RDP svD∆=τλ (7) For sing le-layer graphene, the Rashba spi -orbit strength is estimated to be tens of microelectronvolts per 1V nm-1 of external electric field. For a density of 10-12 cm-2 we find that R∆ ~0.5µeV and conc lude that gate- induced Rashba spin- orbit coupling is not strong enough to be responsible for the spin- lifetimes observed in single- layer graphene. The random Rashba spin- orbit field due to graphene ripples can be stronger, however [18][36], and st ill stronger effects are possible in bilayer graphene. Because graphene has an exposed 2D surface, it turns out that it can be used not only as a spin- conserver but also as a spin- generator. A very large increase in Rashba spin- orbit coupling strength can be achieved by covalent bonding with absorbates [35][43][44][45][46][47][48]. These findings suggest the possibility of creating lithographi cally defined spintronics devices based on graphene with a designed network including both spin- conserving and spin- generating regions. Spin-generation effects, including the spin Hall effect and current -induced spin- densities, are discussed at greater len gth below in the context of topological insulators, which always have strong spin-orbit interactions as we shall emphasize. Topological Insulator Surface- State Spintronics Strong three -dimensional topological insulators [4][5][49] possess an odd number of 2D Dirac bands localized on the bulk material's surface. The simplest case, that of a single Dirac cone, is realized in the Bi 2Se3 family of topological insulators. Because of its role in the early experimental exploration of topological insulators , Bi2Se3 has been referred to as the hydrogen atom of topological insulators [4]. The surface -state 2DDSs of topological insulators are similar to those present in graphene, but differ in three essential respects as indicated in Fig.2 . First of all , for topological insulators there is only one conduction- and valence- band surface state at each 2D momentum, compared with the four states (two for spin times two for valley) present in the graphene case. Although fundamental, this distinction is often academic because spin and valley coupling effects in graphene are too weak to play a significant role in most observable properties. Second, the position of the Fermi level relative to the Dirac point in these materials is determined by bulk physics, not surface physics; the Fermi level does not generically lie at the Dirac point [50] when the TI is neutral. Pristine neutral graphene is , in contrast , guaranteed to have a Fermi level that intersects the Dirac point. This distinction is also largely academic in practice , as the position of the Fermi level at the surface of a topological insulator can be adjusted with external gate voltages, just as it is in most graphene studies. The third distinction is the most important one, particularly for spintronics. The 2D massless Dirac equation for TI surface states couples spin, not sublattice- pseudospin, to momentum. The potential role of topological insulator surface s tates in spintronics is that of a spin -generator , not a spin- conserver, as we discuss below. The simplest approximate Hamiltonian for a surface state of a strong topological insulator, including both orbital and Zeeman coupling to an external magnetic field is ext B ext TI b g Acepzv H ⋅−−×⋅= σµ σ ) ( (8) In the above Hamiltonian, σ is a P auli matrix that acts on spin (not sublattice pseudospin) degrees of freedom, zis chosen to be the direction normal to the surface of a TI, 𝑝⃗ is the canonical momentum, gis the Lande g -factor for the surface electrons (assumed isotropic for simplicity), 𝜇𝐵 is the electron Bohr magneton, and extA and extb are the external vector potential and magnetic field. When the topological insulator surface is proximity coupled to a ferromagnetically ordered material , extb can include an exchange- field contribution. The orbital term in the Hamiltonian (equat ion (8)) is identical to the Rashba spin- orbit coupling term in the Hamiltonian of a two-dimensional electron gas [1], but here it is the entire Hamiltonian. Spin-orbit interactions are as strong as they could possibly be on any topological -insulator surface. This fact explains why the surface of a topological insulator cannot be used as a spin- conserver. Such a surface is almost always in the regi me in which the spin- orbit field is larger than the elastic scattering rate. The spin-relaxation time is therefore always of the same order as the transport scattering time, which depends on sample quality but is typically much shorter than 10-9 s. At topological -insulator surfaces there is no motivation for identifying the spin- relaxation time as a separate time scale. To compare the spin-generator potential of a topological -insulator surface to other 2DEGs, we take the ratio between the current -induced spin density and the total electron density as a figure of merit. From the Hamiltonian (equation (8)) it follows that the electric current operator is in fact proportional to the spin density operator. (The same property applies in the graphene case when spin is substituted by pseudospin.) The explicit expression is z evj ×=σ (9) Using the Drude formula, m neDrude /2τ σ= with effective mass ,/vpmF= for th e conductivity of a 2DDS on a topological insulator surf ace leads to the following estimate for the spin polarization induced by a transport current [51] , Fy peE evnj ns τ== (10) where Eis the applied electric field, and Fp is the Fermi momentum of surface electrons. The ratio of current -induced spin- density to the full electron density is equal to the ratio of the transport drift velocity to the Fermi velocity. The corresponding figure of merit for, say, Rashba- coupled 2DEG is smaller by an additional factor of (ref. [52]). In this sense a topological -insulator surface is a much more effective spin generator than a regular 2DEG. Larger spin polarizations can be achieved in ferro/normal hybrid systems, but topological -insulator surfaces offer the possibility of gate voltage control. Another way to generate spin polarization in a spin- orbit coupled system is via the spin Hall effect (SHE) [53] in which transverse spin currents appear in response to an applied electric field. These spin currents can then lead to spin accumulation at the boundaries of the system or in adjacent conductors. In general , one expects that both intrinsic mechanis ms [54] related to the spin-orbit coupling in band structure, and helical scattering effects [55] due to spin- orbit impurity potential scattering can contribute to the SHE. For a disordered topological -insulator surface, however, there are two reasons not to expect a large spin Hall signal in spite of the strong spin- orbit in teractions. First, it is known that for a disordered 2D system with Rashba- like spin -orbit coupling that is linear in momentum the d.c. intrinsic SHE vanishes [56][57][58][59][60]. Second, even if cubic in momentum corrections to Hamiltonian (equation (8)), or extrinsic contribution to the SHE are considered, the spin relaxation length coincides with the transport mean free path on a topological - insulator surface, and thus the spin accumulation is restricted to boundary regions with a mean - free-path size. This makes the observation of the resultant spin accumulation a challenge, although not an impossibility (see for example r ef. [61]). These considerations motivated studies of SHE on topological -insulator surfaces to focus on mesoscopic samples in the ballistic regime [62][63]. It was found that the resultant spin polarizations exceed their semiconductor counterparts by at least an order of magnitude. It is important to note that even though the Dirac electrons on t he two opposite surfaces of a topological insulator have opposite chiralities, and thus could partially cancel current -induced spin densities and spin currents, their spatial separation, in princip le, makes it possible to address each surface separately. For instance, a gap could be induced on the side surfaces of the sample, at least in principle, by coating with a magnetic film. (See below on why a magnetic field is able to induce a gap on a surface of a topological insulator.) The experimental observation of these effects awaits t he discovery of techniques that can be used to prepare topological -insulator samples with surfaces of the required quality. Although the dissipationless voltage -induced spin Hall current of topological -insulator surface states is likely small, the correlation between velocity and spin directions implies that large (non- dissipative) spin- currents flow in equilibrium. These spin currents likely have observable consequences only if they can flow from the TI surface to another material with magnetic order or weaker spin- orbit coupling. It seems clear that equilibrium spin currents do not satisfy this condition. The correlation between velocity and spin directions w ill also tend to suppress large- angle scattering, as does the analogous property of graphene sheets, and therefore to support large longitudinal conductivities. These properties of the surface states of 3D topological insulators faintly echo those of the heli cal liquid 1D edge state systems associated with two- dimensional topological insulators [64][65][66]. In the case of the latter edge state , the only possible type of elastic scattering is backscattering between states that are spin and orbit al time -reversed partners , which is absent in the absence of time- reversal -symmetry breaking because of Kramer’s theorem. We now turn to the effects of external perturbations that break time-reversal symmetry. Because the presence of gapless surface states in strong and weak topological insulators is protected by time-reversal invariance, the surface -state system is sensitive to perturbations that break this symmetry. In the Hamiltonian (equation (8)), the external magnetic (or exchange) field is the source of time- reversal invariance breaking. Because of the strictly linear dispersion in the Hamiltonian (equation (8)), an in- plane magnetic field enters the Hamiltonian in precisely the same way as a static, spatially -constant vector -potential. There is therefore no paramagnetic response on a TI surface to uniform static in- plane exchange or magnetic fields. This means that for strictly linear dispersion there is also no transport -current -induced spin polarization perpendicular to the plane . Perpendicular -to-plane spin- polarizations in systems with in- plane spin- orbit effective magnetic fields cannot in any case be explained solely on the basis of Boltzmann transport theory. They were nevertheless discovered [67] experimentally in semiconductor 2DEGs and later explained [68] theoretically in terms of a delicate balance between collision and generation terms in a quantum - kinetic transport theory . Magnetic fields that are applied in the zdirection, on the other hand, do produce large effects. An exchange field applied in the zdirection (see below) opens up a gap at the Dirac point and induces [4][5][69][70] a half -quantized quantum Hall effect when the Fermi level lies in this gap. Non-zero charge anomalous Hall effects [71][72] are present at all carrier densities. An external magnetic field in the z direction opens up a large gap at the Dirac point and smaller gaps at many other filling factors with half -odd-integer quantized Hall conductivities . The Hall conductivity increases as the Fermi level moves further away from the Dirac point, while the corresponding gaps get smaller . A number of interesting magnetoelectric [49][73][74][75], and magneto -optical phenomena [70][76][77][78][79][80][81][82][83] are associated with weak time- reversal -symmetry breaking at the surface of a topological insulator. The properties of topological -insulator surface states in a perpendicular external magnetic field are very similar to those of graphene sheets because the orbital response to a magnetic field, which tends to dominate, is identical in the two cases. The r esponse of such surface states to exchange coupling with magnetic ally ordered systems is, however , entirely different from that of a graphene systems. Two ideas have been suggested for engineering a system in which this exchange coupling is present: i) introducing magnetic impurities [84] that can order due to surface -state mediated interactions, and ii) proximity coupling to evaporated magnetic films [85]. The former possibility was considered theoretically in r efs. [9] and [10]. Spin-momentum locking at the surface leads to an unconventional RKKY interaction. [9][10][86][87] which give s rises to a rich phase diagram [9] [10] that is briefly summarized in Fig. 4. The maximum temperatu re for magnetic impurity ferroma gnetism was estimated to be ~30K, in decent agreement with an experimental observation of 13K [84]. The relatively low transition temperature means that this type of order does not have practical utility in Bi 2Se3 systems, although the same mechanism could lead to much higher transition temper atures in as yet undiscovered topological insulators . A more practical approach in the near term might stem from the demonstration [85] that magnetic proximity effects due to an iron coating ma ke the surface of a strong topological insulator ferromagnetic at room temperature. As a medium for local -moment coupling, graphene sheets might have some advantages over topological -insulator surface states because t heir spin-susceptibility is electrically tunable and can be made large by shifting the Fermi level far from the Dirac point. Local moments can be induced in graphene by defects including vacancies and hydrogen or fluorine adatoms [8]. Unfortunately , the coupling between these defects and graphene's π orbitals seems so far to be weak [88] [89]. Pseudospintronics Any two -component quantum degree of freedom that electrons possess in addi tion to their orbital degrees of freedom can behave in a manner that is mathematically equivalent to spin, and can therefore be viewed as a pseudospin. The analogy between spins and pseudospins is useful only in special circumstances since pseudospins lif etimes are normally short , and the identified degrees of freedom are often not continuous across sample boundaries. A s mentioned earlier, graphene’s sublattice degree of freedom can be viewed as a pseudospin and enters the continuum model Dirac equation i n precisely the same manner that real spin enters the Dirac equation for thr surface states of topological insulators . In the pseudospin language, the Hamiltonian consists only of a pseudospin- orbit coupling term with an effective magnetic field that is l inear in momentum and points in the same direction as momentum. Much of what we have explained about the role of spin in topological -insulator surface states, applies equally well to pseudospin in single- layer graphene. Just as in the spin case , we can expect that charge currents in single- layer graphene wi ll be accompanied by pseudospin currents. It is, however, not obvious how these pseudospin currents could be measured. Another example of a two- valued degree of freedom that is often usefully viewed as a pseudospin is the layer degree of freedom in a bilayer electron system. In the case of bilayer graphene there are four carbon atoms per unit cell , but it turns out [90] that there is also, at least at energies below the interlayer bonding e nergy 3.0~1γ eV, a spin- 1/2 pseudospin degree of freedom which can be interpreted as labeling a layer rat her than a sublattice. The pseudospin effective magnetic field in the balanced bilayer band structure is proportional to momentum squared in magnitude and has a n x-y plane orientation angle that is twice the orientation angle of the momentum. Because the z component of pseudospin in the bilayer case corresponds to layer -polarization, an electric field applied between layers acts like a z-direction pseudospin field. This pseudospin field can readily be tuned experimentally from being near zero to being the largest energy scale in the system. No similar experimental control is available for the sublattice pseudospin of single- layer graphene. Unlike the case of a ferromagnetic metal in which external magnetic fi elds have a negligible influence on the overall density of states, the pseudospin field in bilayer graphene can open up a sizable gap - effectively turning a metal into a semiconductor. This surprising feature of the electronic properties of bilayer graphene seems to off er the best potential [91] for digital electronics based on graphene. In 3D systems , there are two types of spin Hall response associated with a current in the x direct ion: i) current s of y spins in the z direction and ii) current s of z spins in the y direction. For a finite -thickness quasi -2D system, currents in the vertical z direction should lead to spin accumulations of opposite sign near the top and bottom surfaces. In bilayer graphene we can therefore anticipate that a current in the x direction will lead to a z direction pseudospin current that flows in the y direction, and to y direction pseudospin accumulations that have opposite signs in the top and bottom layers. These e ffects have so far not been extensively explored. The large and technologically useful magnetotransport properties of metallic ferromagnets occur even though the coupling between individual charge carriers and a magnetic field is rather weak. Because the collective magnetic orientation coordinate of a ferromagnet balances external fields and current effects against only weak anisotropy field effects, mechanisms that would be weak for individual electrons are strong collectively and sufficient to achieve complete reversal of the magnetization. When the magnetization direction is reversed in a part of a magnetic circuit, the effective potential seen by an electron of a given spin is changed by a large amount - in fact by the size of the exchange spin- splitting in that particular magnet, which can be of the order of electronvolts . We do not expect collective pseudospin order to be present at room temperature in graphene systems, although r ecent evidence [92][93] does suggest that bilayer graphene has collective pseudospin order at low temperatures when the carrier density is very close to the neutrality point. Bilayer graphene does , however , have the unique feature that there is a readily available experimental capability which can vary the z component of the pseudospin field over wide ranges, easily larger than the carrier Fermi energy . This property can be used, for example, to devise [94][95] pseudospin analogs of ferromagnetic metal spin-valve devices, Fig. 5. Looking forward Graphene and topological -insulator surface states have very similar Hamiltonians, but represent two distinct extremes of paramagnetic conductor spintronics. The Dirac equation in graphene captures strong coupling between momentum and the sublattice pseudospin degree of freedom. The coupling between graphene's orbital and spin degrees, on the other hand, is so weak that it is normally neglected in analyses of graphene electronic properties and, in fact, has not yet been measured directly. Spin-orbit coupling is weak in graphene not only because carbon is a relatively light element, but also because of its planar geometry reduc es the coupli ng between the Fermi - level π electrons and σelectrons. This means that the spin -creator spintronic effects, transport - current induced spin- densities and spin- polarized currents, are extremely weak in graphene . Graphene's potential in spintronics is as an unusually effective spin- conserver , which can transmit spin-information over extremely large distances. Spin-lifetimes in initial experiments were limited by hybridization with contact electrodes. At present , spin-lifetimes in graphene are still limited by extrinsic disorder effects that have not been definitively identified. Given the rapid pace of recent progress , we can expect that future efforts will achieve spin- lifetimes that are limited only by intrinsic effects. These experiments should enable a definitive measurement of the intrinsic spin- orbit coupling strength at the Dirac point, whose value still remains quite uncertain. This development would be satisfying, as it might finally allow th e quantum spin Hall effect to be observed in the material for w hich it was originally proposed [17] . An exception to the weak spin - orbit coupling properties of graphene arises when it is hybridized with certain metals. Topological insulator surface state s are strongly spin-orbit coupled, unlike the electron states in graphene sheets. Spin lifetimes will inevitably be close to Bloch- state lifetimes. The interesting spintronic properties of such surface states are characterized by their spin Hall and current -induced spin-density properties. The experimental and theoretical exploration of these effects is still at an early stage, but we can expect that the second effect will be stronger than the first. The ratio of current -induced spin densities to the overal l charge density is usually small by a factor of drift velocity divid ed by Fermi velocity and also by a factor of spin -orbit coupling energy divided by Fermi energy. In the case of topological -insulator surface states , the second small factor is absent. Spin Hall effects, on the other hand, are known to be weak in systems in which the spin- orbit coupling term is approximately linear in momentum. This condition applies to states near the Dirac point , suggesting that spin Hall effects will be weak in spite of the very strong spin -orbit interactions. Because of their electrical tunability, 2DESs are attractive as magnetic stiffness agents in systems containing dilute magnetic moments. In the case of (Ga,Mn)As, the Mn local moments induce too much disorder when inserted into a quantum well and robust 2D ferromagnetism has been elusive. It is interesting to speculate on the potential of 2DDSs in graphene and topological insulators as hosts for diluted- moment ferromagnetism. It is certainly clear that the two materials will behave very differently. In mean -field theories, which are accurate when the moment density is comparable to the carrier density and carriers are able to provide adequate magnetic stiffness, the magnetic transition temperature is proportional to the carrier spin susceptibility. In graphene, the susceptibilit y is nearly perfectly isotropic because of weak spin- orbit interactions. Attractively, the susceptibility is also strongly sensitive to carrier -density, vanishing with the carrier density -of-states at the Dirac point. In the case of topological -insulator surface states, the susceptibility is strongly anisotropic, large only for out -of-plane fields, and reaches a maximum when the Fermi level is placed at the Dirac point. Any magnetic state based on coupling between magnetic moments that is mediated by topological -insulator surface states will have the attractive feature of strong perpendicular anisotropy. Graphene and multilayer graphene π-electron sys tems possess sublattice degrees of freedom at each crystal momentum, which may be viewed as pseudospins. In both single- layer and bilayer graphene, low-energy states have two- component pseudospins that are mathematically equivalent to the electronic spin. The band Hamiltonians in both cases have very strong pseudospin- orbit coupling that is similar to the strong spin -orbit coupling of topological -insulator surface states. Because the sublattice pseudospin is equivalent to layer in the case of bilayer graphene, electric fields perpendicular to the layer couple much more strongly to pseudospin than practical magnetic fields are able to couple to real spins. The unique electronic structure features shared by 2DDS s in graphene and topological -insulator surface state are likely to provide them with a number of roles in spintronics, and t he rapid progress over the past few years, partially chronicled here, points to future surprises. The most intriguing opportunities, in our view, are for gate tunable magnetism in graphene sheets and at topological - insulator surface s (perhaps magnetism involving coupling between nanoparticle arrays) and electrically tunable pseudospin properties in multi layer graphene systems that can be used to create pseudospin analog ues of giant magneto resistance and other e stablished spintronics effects. Acknowledgements We acknowledge financial support from the US Army Research Office (ARO) under award number MURI W911NF -08-1-0364. We thank D. Abanin, I. Grigorie va, J. Sinova, A. Veligura, and B. van Wees for discussions. REFERENCES [1] Igor Žutić, Jaroslav Fabian, and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76, 323 -345 (2004). [2] A.K. Geim and A.H. MacDonald, Explor ing Carbon Flatland, Phys. Today 60, 35 -41 (2007). [3] A.H. Castro Neto, F. Guinea, N. M. R. Torres, K. S. Novoselov, and A. K. Geim , The electronic properties of graphene, Rev. Mod. Phys. 81, 109- 162 (2009). [4] M. Z. Hasan and C. L. Kane, Topological insulators,Rev. Mod. Phys. 82, 3045 -3067 (2010). [5] X.-L. Qi and S. -C. Zhang, Topological Insulators and Superconductors, Rev. Mod. Phys. 83, 1057 -1110 (2011). [6] The elephant in the electronic materials room, silicon, is also interesting from a spintronics point of view. See Ron Jansen, Nature Materials (this issue). [7] A.H. MacDonald, P. Schiffer, and N. Samarth, Ferromagnetic semiconductors: moving beyond (Ga, Mn)As, Nature Materials 4, 195 -202 (2005). [8] For a review of magnetism in carbon nanostructures see Oleg V. Yazyev, Emergence of magnetism in grapheme materials and nanostructures, Rep. Prog. Phys. 73, 056501 --1--16 (2010). [9] D. A. Abanin, and D. A. Pesin, Ordering of Magnetic Impurities and Tunable Electronic Properties of Topological Insulators, P hys. Rev. Lett. 106, 136802 -1-4 (2011). [10] Jia-Ji Zhu, Dao- Xin Yao, Shou- Cheng Zhang, and Kai Chang, Electrically controllable surface magnetism on the surface of topological insulator, Phys. Rev. Lett. 106, 097201- 1-4 (2011). [11] Arne Brataas, Andy Kent and Hideo Ohno, Nature Materials (this issue). [12] Gerrit Bauer, Bart van Wees and Eiji Saitoh, Nature Materials (this issue). [13] Tomas Jungwirth and Joerg Wunderlich, Nature Materials (this issue). [14] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Perpendicular switching of a single ferromagnetic layer induced by in- plane current injection, Natu re 476, 189– 193 (2011). [15] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, R. A. Buhrman, Magnetic switching by spin torque from the spin Hall effect , arXiv:1110.6846v2 (2011) . [16] Guoyu Wang, Lynn Endicott, and Ctirad Uher , Recent Advances in the Growth of Bi -Sb-Te- Se Thin Films , Sci. Adv. Mat. 3, 539 -560 (2011). [17] C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in Graphene, P hys. Rev. Lett. 95, 226801- 1-4 (2005). [18] D. Huertas -Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74, Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps, 155426- 1-15 (2006). [19] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald, Intrinsic and Rashba spin- orbit interactions in graphene sheets, Phys. Rev. B 74, 165310- 1-5 (2006). [20] T. Ando, Spin– orbit interaction in carbon nanotubes, J. Phys. Soc. Jpn. 69, 1757– 1763 (2000). [21] F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen, Coupling of spin and orbital motion of electrons in carbon nanotubes, Nature 452, 448- 452 ( 2008) . [22] Y. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang, Phys. Rev. B 75, Spin-orbit gap of graphene: First- principles calculati ons, 041401- 1-4 (2007); [23] M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch -Draxl, and J. Fabian, Band- structure topologies of graphene: Spin- orbit coupling effects from first principles, Phys. Rev. B 80, 235431- 1- 5 (2009). [24] D. C. Elias, R. V. Gorbachev, A. S . Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim, Dirac cones reshaped by interaction effects in suspended graphene, Nature Phys. 7, 701 -704 (2011). [25] R. J. Elliott, Theory of the Effect of Spin -Orbit Coupling on Magnetic Resonance in Some Semiconductors, Phys. Rev. 96, 266 -279 (1954). [26] Y. Yafet, g -factors and spin -lattice relaxationof conduction electrons, in Solid State Physics , edited by F. Seitz and D. Turnbull (Academic Pre ss Inc., New York, 1963), Vol. 14, p. 1. [27] E. W. Hill, A. K. Geim, K. Novoselov, F. Schedin, P.Blake, Graphene Spin Valve Devices, IEEE Trans. Magn. 42, 2694- 2696 (2006). [28] N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, B. J. van Wees, Electronic spin transport and spin precession in single graphene layers at room temperatu re, Nature 448, 571- 574 (2007). [29] S. Cho, Y. F. Chen, M. S. Fuhrer, Gate-tunable graphene spin valve, Appl. Phys. Lett. 91, 123105- 1-3 (2007). [30] M. Nishioka, A. M. Goldman, Spin trans port through multilayer graphene, Appl. Phys. Lett. 90, 252505 -1-3 (2007). [31] M. Ohishi, M. Shiraishi, R. Nouchi, T. Nozaki, T. Shinjo,Y. Suzuki, Spin Injection into a Graphene Thin Film at Room Temperature, Jpn. J. Appl. Phys., 46, L605 -L607 (2007). [32] W. H . Wang, K. Pi, Y. Li, Y. F. Chiang, P. Wei, J. Shi, and R. K. Kawakami, Magnetotransport properties of mesoscopic graphite spin valves, Phys. Rev. B 77, 020402 -1-4 (2008). [33] C. J'ozsa, T. Maassen, M. Popinciuc, P. J. Zomer, A. Veligura, H. T. Jonkman, and B. J. van Wees, Linear scaling between momentum and spin scattering in graphene, Phys. Rev. B 80, 241403(R) -1-4 (2009). [34] N. Tombros, S. Tanabe, A. Veligura, C. Jozsa1, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Anisotropic Spin Relaxation in Graphene, Phys. Rev. Lett. 101, 046601- 1-4 (2008). [35] A. H. Castro Neto, and F. Guinea, Impurity -Induced Spin- Orbit Coupling in Graphene, Phys. Rev. Lett. 103, 026804- 1-4 (2009). [36] D. Huertas -Hernando, F. Guinea, and Arne Brataas, Spin- Orbit -Mediated Spin Relaxation in Graphene, Phys. Rev. Lett. 103, 146801- 1-4 (2009). [37] C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Electron spin relaxation in graphene: The role of the substrate, Phys. Rev. B 80, 041405 (R)-1-4 (2009). [38] H. Ochoa, A.H. Castro -Neto, and F. Guinea, Elliot -Yafet Mechanism in Graphene, arXiv:1107.3386. [39] W. H an, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami , Tunneling Spin Injec tion into Single Layer Graphene, Phys. Rev. Lett. 105, 167202 -1-4 (2010). [40] W. Han and R. K. Kawakami, Spin Relaxation in Single- Layer and Bilayer Graphene, Phys. Rev. Lett. 107, 047207- 1-4 (2011). [41] T.- Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M. Jaiswal, J. Samm, S. R. Ali, A. Pachoud, M. Zeng, M. Popinciuc, G. Güntherodt, B. Beschoten, and B. Ö zyilmaz, Observation of Long Spin- Relaxation Times in Bilayer Graphene at Room Temperature, Phys. Rev. Lett. 107, 047206- 1-4 (2011). [42] M. I. Dyakonov and V. I. Perel’, S pin relaxation of conduction electrons in noncentrosymmetric semiconducto rs, Sov. Ph ys. Solid State 13, 3023 -3026 (1971). [43] S. Jo, D. -K. Ki, D. Jeong, H. -J. Lee, and S. Kettemann, Spin relaxation properties in graphene due to its linear dispersion, Phys. Rev. B 84, 075453- 1-6 (2011). [44] D. Ma, Z. Li, Z. Yang, Strong spin- orbit splitting in gr aphene with adsorbed Au atoms, CARBON 50, 297- 305 (2012). [45] A. Varykhalov, J. Sa nchez -Barriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O.Rader, Electronic and Magnetic Properties of Q uasifreestanding Graphene on Ni, Phys. Rev. Let t. 101, 157601 (2008). [46] O. Rader, A. Varykhalov, J. Sánchez -Barriga, D. Marchenko, A. Rybkin, and A. M. Shikin, Is There a Rashba Effect in Graphene on 3d Ferromagnets? Phys. Rev.Lett. 102, 057602- 1-4 (2009). [47] Z. Qiao, S. A. Yang, W. Feng, W. -K. Tse, J. Ding, Y. G.Yao, J. Wang, and Q. Niu, Quantum anomalous Hall effect in graphene from Rashba and exchange effects , Phys. Rev. B 82, 161414(R) -1-4 (2010). [48] C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu, Engineering a Robust Quantum Spin Hall State in Graphene via Adatom Deposition, Phys. Rev. X 1, 021001- 1-15 (2011) . [49] Joel E. Moore, The birth of topological insulators, Nature 464, 194 -198 (2010). [50] J. C. Y. Teo, L . Fu, and C. L. Kane, Surface states and topological invariants in three - dimensional topological insulators: Application to Bi_1- xSb_x, Phys. Rev. B 78, 045426- 1-15 (2008). [51] Dimitrie Culcer, E. H. Hwang, Tudor D. Stanescu, and S. Das Sarma, Two-dimensional surface charge transport in topological insulators, Phys. Rev. B 82, 155457- 1-17 (2010). [52] V. M. Edelstein, Spin polarization of conduction electrons induced by electric current in two- dimensional asymmetric electron syst ems, Solid State Comm. 73, 233- 235 (1990). [53] M. I. Dyakonov and A. V. Khaetskii, Spin Hall Effect, in Spin physics in semiconductors , edited by M. Dyakonov. [54] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Experimental Observation of the Spin-Hall Effect in a Two -Dimensional Spin- Orbit Coupled Semiconductor System, Phys. Rev. Lett. 94, 047204 -1-4 (2005). [55] M. I. Dyakonov and V. I. Perel, Current -induced spin orientation of electrons in semiconductors, Phys. Lett. A 35, 459 -460 (1971). [56] J.I. Inoue, G.E.W. Bauer, and L.W. Molenkamp, Suppression of the persistent spin Hall current by defect scattering, Phys.Rev. B 70, 41303(R) -1-4 (2004). [57] E.G. Mishchenko, A.V. Shytov, and B.I. Halperin, Spin Current and Polarization in Impure Two-Dimensional Electron Systems with Spin- Orbit Coupling , Phys. Rev. Lett. 93, 226602- 1-4 (2004). [58] A. Khaetskii, Nonexist ence of Intrinsic Spin Currents, Phys. Rev. Lett. 96, 056602- 1-4 (2006). [59] R. Raimondi and P. Schwab, Spin -Hall effect in a disordered two -dimensional electron system, Phys. Rev. B 71, 33311 -1-4 (2005). [60] O.V. Dimitrova, Spin-Hall conductivity in a two- dimensional Rashba electron gas, Phys. Rev. B 71, 245327 -1-7 (2005). [61] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Room -temperature reversible spin Hall effect, Phys. Rev. Lett. 98, 156601- 1-4 (2007). [62] P.G. Silvestrov, E.G. Mishchenko, Spin -Hall Effect in Chiral Electron Systems: from Semiconductor Heterostruc tures to Topological Insulators, arXiv:0912.4658 (2009). [63] J.-H. Gao, J. Yuan, W. -Q. Chen, Y. Zhou, and F. -C. Zhang, Giant Mesoscopic Spin Hall Effect on the Surface of Topological Insulator, Phys. Rev. Lett. 106, 057205- 1-4 (2011). [64] C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802 -1-4 (2005). [65] B. Andr ei Bernevig, Taylor L. Hughes, and Shou- Cheng Zhang, Quantum Spin Hall Effect and Topological Phase Tr ansition in HgTe Quantum Wells, Science 314, 1757- 1761 (2006). [66] M. König, S . Wiedmann, C . Brüne, A . Roth, H . Buhmann, L. W. Molenkamp, X .-L. Qi, S .-C. Zhang, “Quantum Spin Hall Insulator State in HgTe Quantum Wells,” Science 318, 766- 770 (2007). [67] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Current -Induced Spin Polarization in Strained Semiconductors, Phys. Rev. Lett. 93, 176601 -1-4 (2004). [68] H.-A. Engel, E. I. Rash ba, and B. I. Halperin, Out -of-Plane Spin Polarization from In -Plane Electric and Magnetic Fields, Phys. Rev. Lett. 98, 036602- 1-4 (2007). [69] A.A. Zhuzin and A.A. Burkov, Thin topological insulator film in a perpendicular magnetic field, Phys. Rev. B 83, 195413- 1-4 (2011). [70] W.-K. Tse and A. H. MacDonald, Giant Magneto -Optical Kerr Effect and Universal Faraday Effect in Thin- Film Topological Insulators, Phys. Rev. Lett. 105, 057401 -1-4 (2010). [71] N. A. Sinitsyn, A. H. MacDonald, T. Jungw irth, V. K. Dugaev, and J . Sinova, Anomalous Hall effect in 2D Dirac band: link between Kubo- Streda formula and semiclassical Boltzmann equation approach, Phys. Rev. B 75, 045315 -1-17 (2007). [72] D. Culcer and S. Das Sarma, Anomalous Hall response of topol ogical insulators, Phys. Rev. B 83, 245441 -1-9 (2011). [73] A. M. Essin, J. E. Moore, and D. Vanderbilt, Magnetoelectric Polarizability and Axion Electrodynamics in Crystalline Insulators, Phys. Rev. Lett. 102, 146805- 1-4 (2009). [74] X.-L. Qi, R. Li, J. Zang, and S. -C. Zhang, Inducing a Magnetic Monopole with Topological Surface States, Science 27, 1184 -1187 (2009). [75] X.-L. Qi, T. L. Hughes, and S. -C. Zhang, Topological field theory of tim e-reversal invariant insulators, Phys. Rev. B 78, 195424 -1-43 (2008). [76] W.-K. Tse and A. H. MacDonald, Magneto- optical and magnetoelectric effects of topological insulators in quantizing magnetic fields, Phys. Rev. B 82, 161104(R) -1-4 (2010). [77] A. B. Sushkov, G. S. Jenkins, D. C. Schmadel, N. P. Butch, J. Paglione, and H. D. Drew , Far-infrared cyclotron resonance and Faraday effect in Bi 2Se3, Phys. Rev. B 82, 125110 -1-5 (2010). [78] G. S. Jenkins, A. B. Sushkov, D. C. Schmadel, N. P. Butch, P. Syers, J. Paglione, and H. D. Drew, Terahertz Kerr and reflectivity measurements on t he to pological insulator Bi2Se3, Phys. Rev. B 82, 125120 -1-9 (2010). [79] R. Valdes Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, J. Cerne, A. G. Markelz,S. Oh, and N. P. Armitage, THz response and colossal Kerr rotation from the surface states of the topological insulator Bi 2Se3, arXiv:1105.0237v3. [80] J. N. Hancock, J. L. M. van Mechelen, A. B. Kuzmenko, D. van der Marel, C. Brune, E. G. Novik, G. V. Astakhov, H. Buhmann, L. Molenkamp, Surface State Charge Dynamics of a High - Mobility Three -Dimensional Topological Insulator, Phys. Rev. Lett. 107, 136803- 1-4 (2011) . [81] J. Maciejko, X. -L. Qi, H. D. Drew, and S. -C. Zhang, Topological Quantization in Units of the Fine Structure Constant, Phys. Rev. Lett. 105, 166803 -1-4 (2010). [82] G. Tkachov and E. M. Hankiewicz, Anomalous galvanomagnetism, cyclotron resonance, and microwave spectroscopy of topological insulators, Phys. Rev. B 84, 035405- 1-5 (2011). [83] I. Garate and M. Franz, Inverse Spin- Galvanic Effect in a Topological - Insulator/Ferromagnet Interface, Phys. Rev. Lett. 104, 146802- 1-4 (2010). [84] L. A. Wray, S.- Y. Xu, Y. Xia, D. Hsieh, A . V. Fedorov, Y . S. Hor, R. J. Cava, A. Bansil, H. Lin, and M. Z. Hasan, A topological insulator surface under strong Coulomb, magnetic and disorder perturbations, Nature Phys. 7, 32- 37 (2011). [85] I. Vobornik, U . Manj u, J. Fujii, F . Borgatti, P. Torelli, D. Krizmancic, Y. S. Hor, R. J. Cava, and G. Panaccione, Magnetic Proximity Effect as a Pathway to Spintronic Applications of Topological Insulators, Nano Lett. 11, 4079- 4082 (2011). [86] Q. Liu, C.- X. Liu, C. Xu, X. -L. Qi, and S. -C. Zhang, Magnetic impurities on the Surface of a Topological Insulator, Phys. Rev. Lett. 102, 156603- 1-4 (2009). [87] I. Garate, M. Franz, Magnetoelectric response of the time- reversal invariant helical m etal, Phys. Rev. B 81, 172408 -1-4 (2010). [88] X. Hong, S. -H. Cheng, C. Herding, and J. Zhu, Colossal negative magnetoresistance in dilute fluorinated graphene, Phys. Rev. B 83, 085410- 1-5 (2011). [89] R. R. Nair, M. Sepioni, I -L. Tsai, O. Lehtinen, J. Keinonen, A. V. Krasheninnikov, T. Thomson, A. K. Geim, and I. V. Grigorieva, Spin -half paramagnetism in graphene induced by point defects, Nature Physics (2012) doi:10.1038/nphys2183 [90] E. McCann, V. I. Fal'ko, Landau- Level Degeneracy and Quantum Hall Effect in a Gr aphite Bilayer, Phys. Rev. Lett. 96, 086805 -1-4 (2006) . [91] F. Xia, D . B. Farmer, Y. -m. Lin, and P . Avouris, Graphene field- effect -transistors with high on/off current ratio and large transport band gap at room temperature, Nano Let t. 10, 715- 718 (2010). [92] W. Bao, L. Jing, J. Velasco Jr, Y. Lee, G. Liu, D. Tran, B. Standley, M. Aykol, S. B. Cronin, D. Smirnov, M. Koshino, E. McCann, M. Bockrath, and C. N. Lau, Stacking -dependent band gap and quantum transport in trilayer graphene, Nature Phys. 7, 948– 952 (2011) . [93] J. Velasco Jr., L. Jing, W. Bao, Y. Lee, P. Kratz, V. Aji, M. Bockrath, C.N. Lau, C. Varma, R. Stillwell, D. Smirnov, Fan Zhang, J. Jung, and A.H. MacDonald, Transport Spectroscopy of Symmetry -Broken Insulating States in Bilayer , arXiv:1108.1609.v1. [94] P. San-Jose, E. Prada, E. McCann, H. Schomerus, Pseudospin Valve in Bilayer Graphene: Towards Graphene -Based Pseudospintronics , Phys. Rev. Lett. 102, 247204- 1-4 (2009). [95] M.F.Craciun, S.Russo, M.Yamamoto and S.Tarucha, Tuneable el ectronic properties in graphem e, Nano Today, 6, 42- 60 (2011). Figure 1 | Overview of spintronics. In all panels , the large spins represent overall magnetization and the small spins represent the transport electrons. The field of spintronics is divided in the first place between the study of magnetically ordered conductors (left panels) and the study of paramagnetic metals or semiconductors (right panels). Within each class, one can study effects in which electric fields alter spin configurat ions (bottom panels) and complementary effects (top panels) in which effective magnetic fields due to spin- orbit, exchange, or magnetostatic interactions influence transport properties. The four panels in this figure ( anticlockwise from the top left) schem atically illustrate i) giant magnetoresistance in which variation in magnetization direction increases back scattering and hence resistance; ii) Andreev reflection of spins in non- collinear magnetic systems that leads to spin- transfer torques and current -induced spin- reversal; iii) current - induced spin polarization in paramagnetic conductors ; and iv) the spin Hall effect and spin currents in paramagnetic conductors. Spin- transport effects in paramagnetic conductors always require spin-orbit interactions. This Progress Article concentrates on right -panel phenomena in topological insulators and in monolayer graphene, and left -panel phenomena in bilayer graphene. Figure 2 | Dirac cones in graphene and in topological insulators. Topological -insulator surface states and π-band states in graphene (left panel) are described by a 2D Dirac equation with strong coupling between momentum and spin in the topological -insulator case and between momentum and sublattice pseudospin in the graphene case. Topological -insulator surface states are non - degenerate and are coherent equal -weight linear combinations of two spin components with a momentum -dependent phase difference , 𝜑𝑝. The pseudospin representation of the honeycomb sublattice degree of freedom is illustrated in the right panel. The π-band eigenstates in graphene are coherent equal -weight linear combinations of their two honeycomb sublattice components, denoted by A and B, and have spin- and momentum -space valley degeneracies. Figure 3 | Hanle effect measurements of room -temperature spin- relaxation times in a variety of bilayer graphene samples [41] . Mobilities , 𝜇, of the samples vary by tw o orders of magnitude. Similar results were reported in r ef. [40]. The property that the spin- relaxation time decreases with increasing mobility suggests that the Dyakonov -Perel spin- relaxation mechanism is operative in these systems. Reprinted with permission from ref. [41], © 2011 APS . Figure 4 | Phase diagram for magnetic adatom magnetism mediated by topological insulator surface states. Exchange coupling constants ⊥||,Jcharacterize the coupling between in- plane and out-of-plane components of the spins of the magnetic adatom and topological -insulator surface electrons. Inset: dependence of spin -1/2 adatom disorder -averaged magnetization (solid line) and magnetization fluctuations (dashed line) on the exchange anisotropy, ⊥= JJ/||δ (solid line). The position of the quantum phase transition between the out -of-plane ferromagnet and spin glass phases, 3.1≈cδ , is inferred from the finite- size condition that magnetization has decreased by 50% from its value at .0||=J The position of the transition is further confirmed by a maximum in the magnetization fluctuations at cδδ=. Reprinted with permission from ref. [9], © 2011 APS. Figure 5 | Comparison between a regular ferromagnetic metal spin valve device and a bilayer graphene pseudospin valve device. a. In the case of the usual spin valve, the strong dependence of resistance on the relative orientation of two ferromagnetic elements placed in series in a magnetic circuit is due to the strongly spin- dependent transport of ferromagnets and to interface scatt ering from the exchange potential change at the interface. (See also the upper left panel of Fig. 1). b. In a bilayer graphene pseudospin valve, the relative pseudospin alignment of bilayer segments placed in series is controlled by external gates. The inc rease in device resistance when the gate voltages have opposite sign is due entirely to scattering of injected pseudospin- polarized electrons off the pseudospin texture (indicated by red arrows) created by the gate voltages. Part b reprinted with permission from ref. [94], © 2009 APS.

2111.11054v1.Particle_and_spin_transports_of_spin_orbit_coupled_Fermi_gas_through_a_Quantum_Point_Contact.pdf

arXiv:2111.11054v1 [cond-mat.quant-gas] 22 Nov 2021Particle and spin transports of spin-orbit coupled Fermi ga s through a Quantum Point Contact Xiaoyu Dai1and Boyang Liu1,∗ 1Institute of Theoretical Physics, Faculty of Science, Beijing University of Technology, Beijing, 100124, China (Dated: November 23, 2021) The particle and spin transport through a quantum point cont act between two Fermi gases with Raman-induced spin-orbit coupling are investigated. We sh ow that the particle and spin conduc- tances both demonstrate the structure of plateau due to the m esoscopic scale of the quantum point contact. Compared with the normal Fermi gases the particle c onductance can be significantly en- hanced by the spin-orbit coupling effect. Furthermore, the c onversion of the particle and spin currents can take place in the spin-orbit coupled system, an d we find that it is controlled by the pa- rameter of two-photon detuning. When the parameter of two-p hoton detuning vanishes the particle and spin currents decouple. I. INTRODUCTION Transport measurements are important tools to inves- tigatethefundamentalpropertiesofstatesofmatter. Re- cently,thestudiesoftransportincoldatomshavebecome one of the frontiers in the area. Many experiments have been conducted, including particle transport [1–9], spin transport [10–14] and heat transport [15–17]. Particu- larly, the realization of two-terminal set-up by ETH’s group [1, 2, 4–6, 15, 16] paved a road to extend this paradigmatic tool from condensed matter physics to var- ious unique states in cold atom physics, for instance, the quantized conductance of neutral matter has been ob- served [2], and the anomalous conductance of unitary Fermi gas has been studied both experimentally [5] and theoretically[18–20]. The interplay between charge and spin degrees of free- dom is ubiquitous in physical systems. By utilizing light- atom interactions the synthetic spin-orbit (SO) coupling in neutral atoms has be realized in both bosonic and fermioniccoldatomsystems[21–23]. Theseachievements have stimulated intensive studies in this area [24, 25]. In transport experiments the SO coupling may gener- ate spin and charge currents conversion, which is a key phenomenon in spintronics and can facilitate technologi- cal applications, for instance, it’s possible to control spin signals by manipulating the electric signals. In this work we study the spin and particle transports between two Fermi gases with Raman-induced SO cou- pling. The two reservoirs are connected by a quantum point contact (QPC). We calculate the particle and spin currents using the Keldysh formulism [26]. The chemical potentials of spin-up and spin-down particles are tuned to be different in either of the reservoirs, and then the particle current Ip≡I↑+I↓and spin current Is≡I↑−I↓ can be generated. In the linear response regime they can ∗Electronic address: boyangleo@gmail.combe expressed as /parenleftbigg Ip Is/parenrightbigg =/parenleftbigg σpσo σ′ oσs/parenrightbigg/parenleftbigg ∆µp ∆µs/parenrightbigg , (1) where ∆µp= ¯µR−¯µLand ∆µs=δµR−δµL. ¯µj= (µj↑+µj↓)/2 andδµj=µj↓−µj↑are the average and difference of the chemical potentials of spin-up and spin- down fermions, where j=L,Rdenote the left and right reservoir.σpandσsare the particle and spin conduc- tances.σo, andσ′ odescribe the conversion of the particle and spin currents. We investigate the variations of the elementsσp,σs,σo, andσ′ owith respect to various pa- rameters in system with Raman-induced SO coupling. II. MODEL The whole system, including two reservoirs and the QPC,canbedescribedbythefollowingHamiltonian(set- ting/planckover2pi1= 1) ˆH=ˆHL+ˆHR+ˆHT, (2) whereˆHL(ˆHR) describes the Fermi gas with Raman- induced SO coupling in the left (right) reservoir and is given by ˆHj= Ψ† jG−1 jΨj, (3) where Ψ† j=/parenleftBig ˆψ† j↑ˆψ† j↓/parenrightBig and G−1 j=/parenleftBigg (kx+k0)2+k2 ⊥ 2m+δ 2−µj↑Ω 2 Ω 2(kx−k0)2+k2 ⊥ 2m−δ 2−µj↓/parenrightBigg . (4) The operator ˆψ† jσ(ˆψjσ) describes the creation (annihila- tion) of a fermion atom with spin σ=↑,↓in thej-th reservoir.mis the mass of fermions and k2 ⊥=k2 y+k2 z. Ω/2 is the strength of the Raman coupling and k0is the2 wave vector of the laser. δis the two-photon detuning [25]. With the Pauli matrices the Eq. (4) can be cast as G−1 j=k2 2m+Bk·σ+Er−¯µj, (5) whereBk= (Ω/2,0,kxk0/m+δ/2−δµj/2), andEr= k2 0/2mis the recoil energy. With SO coupling the spin is not a good quantum number anymore. However, the single particle Hamiltonian can be diagonalized to two eigenstates with another good quantum number, the he- licity‘±’, whichdenotebeingspinparalleloranti-parallel to the Zeeman field Bk. With a unitary transformation UjˆHjU† jthe eigenenergies can be obtained as Ek,±=k2 2m±ξ(kx)+Er−/parenleftbig ¯µj∓S(kx)δµj/parenrightbig ,(6) where ξ(kx) =/radicalbigg/parenleftBig kxk0/m+δ/2/parenrightBig2 +Ω2/4, S(kx) =kxk0/m+δ/2 2/radicalbigg/parenleftBig kxk0/m+δ/2/parenrightBig2 +Ω2/4.(7) In this work we study the cases where δµjis small com- pared with the chemical potentials µjσ. Hence, in Eq.(6) and the following work we only keep the terms up to the first order of δµj. The fields ˆψj±corresponding to these two branches can be found by a unitary transformation as the following /parenleftbiggˆψj+(k) ˆψj−(k)/parenrightbigg =/parenleftbigg Dj(kx)−Oj(kx) Oj(kx)Dj(kx)/parenrightbigg/parenleftbiggˆψj↑(k) ˆψj↓(k)/parenrightbigg ,(8) and the elements in above matrix are presented as Dj(kx) =Λ(kx)/radicalbig Λ(kx)2+Ω2/4 +Λ(kx)Ω2/4 2ξ(kx)(Λ(kx)2+Ω2/4)3/2δµj, Oj(kx) =Ω/2/radicalbig Λ(kx)2+Ω2/4 −Λ(kx)2Ω/2 2ξ(kx)(Λ(kx)2+Ω2/4)3/2δµj,(9) where Λ(kx) =kxk0/m+δ/2+ξ(kx). (10) The tunneling between two reservoirs through the QPC is described by ˆHT. In real space it can be written as ˆHT=∞/summationdisplay n=0[T(+) nˆψ† L+(0)ˆψR+(0)+T(−) nˆψ† L−(0)ˆψR−(0)] +h.c.. (11)Here we assume the eigenstates ψR+andψR−are trans- ported through point x= 0 between the two reser- voirs. In the experimental setup the QPC is formed by the confinement in ˆ yand ˆzdirections, which lead to the transport channels with energies of ǫ⊥(ny,nz)≡ (1 2+ny)ωy+ (1 2+nz)ωz+Vg[2, 5]. For simplicity an effective gate potential ¯Vg=Vg+1 2ωy+1 2ωzcan be de- fined and we assume ωz≫ωy, then the several lowest transport channels would be nyωy+¯Vg, and they are non-degenerate. Then the tunneling amplitude T(±) ncan be written as T(±) n(kL,kR) =T/productdisplay j=L,RΘ(ǫ±(kj)−nωy−¯Vg),(12) whereǫ±(k) =k2 2m±ξ(kx)+Eris the single particle en- ergyoffield ˆψj±andΘ(ǫ±(kj)−nωy−¯Vg)istheheaviside step function. Above tunneling amplitude indicates that only the particle with energy ǫk,±> nω+¯Vgcan enter then-th cannel and will come out from the same chan- nel. That is, there is no inter-channel scattering within the QPC region. The current for spin- σis defined as Iσ≡1 2/angbracketleftbigg∂ ∂t(NLσ−NRσ)/angbracketrightbigg , (13) whereNjσ≡/summationtext kσˆψ† jσ(k)ˆψjσ(k). In above expression the averages /an}bracketle{t· · ·/an}bracketri}htis taken over a time-evolving many- body state, which can be calculated using the Keldysh formalism. Please see the appendix A for details. In the linear response regime the particle and spin currents are expressed as the Eq. (1). The elements σp,σs,σo, and σ′ oin Eq. (1) can be calculated as the following σp=α h∞/summationdisplay n=0/braceleftBigg/integraldisplay ǫ+(k1x)<ǫFdk1x/integraldisplay∞ −∞dk2x Θ(ǫ+(k2x)−nωy−¯Vg)F1(k1x,k2x)n+(k2x) +/integraldisplay ǫ−(k1x)<ǫFdk1x/integraldisplay∞ −∞dk2x Θ(ǫ−(k2x)−nωy−¯Vg)F1(k1x,k2x)n−(k2x)/bracerightBigg , (14) σo=α h∞/summationdisplay n=0/braceleftBigg −/integraldisplay ǫ+(k1x)<ǫFdk1x/integraldisplay∞ −∞dk2x Θ(ǫ+(k2x)−nωy−¯Vg)F1(k1x,k2x)S(k2x)n+(k2x) +/integraldisplay ǫ−(k1x)<ǫFdk1x/integraldisplay∞ −∞dk2x Θ(ǫ−(k2x)−nωy−¯Vg)F1(k1x,k2x)S(k2x)n−(k2x)/bracerightBigg , (15)3 σ′ o=α h∞/summationdisplay n=0/braceleftBigg/integraldisplay ǫ+(k1x)<ǫFdk1x/integraldisplay∞ −∞dk2x Θ(ǫ+(k2x)−nωy−¯Vg)F2(k1x,k2x)n+(k2x) +/integraldisplay ǫ−(k1x)<ǫFdk1x/integraldisplay∞ −∞dk2x Θ(ǫ−(k2x)−nωy−¯Vg)F2(k1x,k2x)n−(k2x)/bracerightBigg , (16) σs=α h∞/summationdisplay n=0/braceleftBigg −/integraldisplay ǫ+(k1x)<ǫFdk1x/integraldisplay∞ −∞dk2x Θ(ǫ+(k2x)−nωy−¯Vg)F2(k1x,k2x)S(k2x)n+(k2x) +/integraldisplay ǫ−(k1x)<ǫFdk1x/integraldisplay∞ −∞dk2x Θ(ǫ−(k2x)−nωy−¯Vg)F2(k1x,k2x)S(k2x)n−(k2x)/bracerightBigg , (17) where n±(kx) =1 exp{(ǫ±(kx)−ǫF)/kBT}+1 F1(k1x,k2x) =(Λ(k1x)Λ(k2x)+Ω2/4)2 2(Λ(k1x)2+Ω2/4)(Λ(k2x)2+Ω2/4), F2(k1x,k2x) =Λ(k1x)2Λ(k2x)2−Ω4/16 2(Λ(k1x)2+Ω2/4)(Λ(k2x)2+Ω2/4). (18) In this work we use ¯ µLas the energy scale and define the Fermi energy as ǫF= ¯µL. The so-called transparency is defined as α=|T |2m3ǫF π2. Here we assume perfectly transparent junction and set α= 1. III. PARTICLE AND SPIN CONDUCTANCES FORδ= 0 In this section we investigate the variation of the con- ductance matrix in Eq. (1) for the case of symmetric dispersion, where the two-photo detuning δ= 0. Several properties of the conductances can be found. First, the particle conductance σpdemonstrates the structure of quantized plateaus analogous to the system without SO coupling [2, 27, 28], except that the height of the plateau is not 2 /hany more. For instance, the height of the plateau can be larger than 2 /hfor Ω/ǫF= 0.5 and Er/ǫF= 0.25 as shown in Fig.1 (a). Furthermore, the spin conductance also shows plateau structure as shown in Fig.1 (b). Second, fixing ¯Vg/ǫF= 0.5 the particle and spin con- ductances are at the first plateau, then we can study the variation of the height of the plateau by changing the parameters. In Fig. 2 (a) and (b) we plot σpandσs-1.5-1.0-0.500.51.01.50246810 -Vg/
F p(1/h)(a) Ω/ϵF=0.5,Er/ϵF=0.25 -1.5-1.0-0.500.51.01.50123 -Vg/F s(1/h)(b) FIG. 1: (Color online) The particle conductance σpand spin conductance σsas functions of gate potential ¯Vgfor fixed Ω andEr. as functions of Ω and Er. For the case of small Ω, for instance, the curve of Ω /ǫF= 0.01 in Fig. 2 (a), one observes that σpcan reach 4 /hwhenErapproaches 0, which is double of the height of the plateau in the system without SO coupling [2, 27, 28]. That is, in the region ofEr≪Ω the coupling of spin-up and spin-down fields can enhance the particle transport ability. Then, when Erincreases to the region Ω ≪Er≪ǫFthe particle conductance σpdrops rapidly and can reach the value of 2/h. In this region the spin-up and spin-down fields are roughly decoupled. It’s reduced to a system without SO coupling. Hence, σpreaches the value of 2 /h, which is the conductance of normal gas. When Erincreases further and becomes comparable to ǫFone observes that σpincreases up to 4 /hagain. This can be explained by the graphs of Fig. 3 (a) and (b), in which we plot the transport of particles around the Fermi surface in one- dimensional case. The single particle dispersion of region Er≪ǫFis sketched in Fig. 3 (a). One observes that a particle on the lower branch in the left reservoir can be transported to two states in the right reservoir, since the momentum is not conserved in the tunneling process as show in Eq. (A9) in the appendix A. For the region Er≫ǫFin Fig. 3 (b), we demonstrate that a particle on 0.00.51.01.52.02.53.00123456 Er/ϵFσp(1/h)(a) Ω/ϵF=1Ω/ϵF=0.5Ω/ϵF=0.1Ω/ϵF=0.01 0.00.51.01.52.02.53.00123 Er/ϵFσs(1/h)(b) FIG. 2: (Color online) The particle conductance σpand spin conductance σsas functions of Erfor Ω/ǫF= 0.01, 0.1, 0.5 and 1. In both (a) and (b), we set ¯Vg/ǫF= 0.5 andδ= 0.4 Left reservoir Right reservoir (a) Left reservoir Right reservoir (b) ҧ ߤҧ ߤோ ҧ ߤҧ ߤோ Right reservoir (c) ҧ ߤҧ ߤோ Left reservoir FIG. 3: (Color online) the schematic plot of dispersions and Fermi surfaces for the region of (a) Ω ≪Er≪ǫFand (b) Ω≪ǫF≪Er, the red part of the dispersion represents the lower branch and the blue one represents the upper branch. For the region (a) a particle around the Fermi surface on the lower branch can be transported to two states in the right reservoir. For the region (b) a particle on the lower branch can be transported to four states in the right reservoir. In both (a) and (b) we set δµj= 0. (c), the schematic plot of the tilted Fermi surface due to the non-zero δµj. The red straight line represents the Fermi surface of the lower bran ch, and the blue one is for the upper branch. the lower branch in the left reservoir can be transported to four state in the right reservoir. This explains the in- creasing of the conductance as Erincreases. Fig. 3 (a) and (b) are examples in one-dimensional case. Our sys- tem is three dimensional. The Fermi surface is not a set ofpoints but asphere. The situation is morecomplicated but the logic beneath is the same. Comparing the four curves of Ω /ǫF= 0.01,0.1,0.5,and 1 in Fig. 2 (a) one observe that for any fixed Erthe particle conductance σpincreases as Ω increases. There are two effects that help to increase the particle conductance. One is the the mixing of spin-up and spin-down fields. The other effect is the density of state. The low energy density of state in Raman-induced SO coupled system is larger than the caseofk2 xdispersion[25]. ThecurvesofΩ /ǫF= 0.5and1 demonstrated clear peaks, which correspond to the cases when the Fermi surfaces reach the middle peak of the dispersion of the lower branch, where the density of state is large. Third, in Fig. 2 (b) one observes that the spin conduc- tanceσsincreases as Erincreases and saturates at the value of 2/h. The variation of Ω doesn’t affect σsmuch.0.00   1.01   2.001234 56σp(1/h)(a) δ/ϵF=±  δ/ϵF=±1δ/ϵF=±  δ/ϵF=0 0.0   1.0   2.0-3-2-10123σo(1/h)(b) δ/ϵF=  δ/ϵF=1δ/ϵF=   δ/ϵF=0δ/ϵF=-   δ/ϵF=-1δ/ϵF=-    0.0 ! " 1.0# $ % 2.0-6-4-20246 Er/ϵFσo'(1/h)(c) δ/ϵF=& ' (δ/ϵF=1δ/ϵF= ) * +δ/ϵF=0δ/ϵF=- , - .δ/ϵF=-1δ/ϵF=-/ 2 3 0.04 6 7 1.08 9 : 2.00123 Er/ϵFσs(1/h)(d) δ/ϵF=±; < =δ/ϵF=±1δ/ϵF=±> ? @δ/ϵF=0 FIG. 4: (Color online) σp,σo,σ′ o, andσsas functions of Er for different values of δ. Here we set Ω /ǫF= 0.1. When δ changes sign, σpandσsremain the same, while σoandσ′ o changes sign. Forth, our calculation shows that the off-diagonal termsσoandσ′ oboth vanish in the case of δ= 0. The spin current is driven by the bias of the spin difference ∆µs=δµR−δµL. In Eq. (6) we can see that the spin difference δµjdeforms the Fermi surface. Roughly speaking, it tilts the Fermi surface as sketched in Fig. 3(c). During the spin transport, the Fermi surface of the left reservoir will become more horizontal. The particles on the lower branch with momenta kx<0 in the left reservoir will be transported to the right, while the par- ticles with momenta kx>0 in the right reservoir will be transported to the left. Since the dispersion is sym- metric about the axis kx= 0, the number of particles transported from the left to the right reservoir equals to the one transported from the right to the left reservoir. Hence, the bias of the spin difference ∆ µswill not drive a net particle current. Analogously, ∆ µpwill not drive a spin current. Hence, σoandσ′ oare both zero for δ= 0 case. IV. PARTICLE AND SPIN CONDUCTANCES FORδ/negationslash= 0 Aswediscussedinthe lastsectiontheparticleandspin currents decouple when δvanishes. In this section we in- vestigate how a non-zero δaffect the transport. In Fig. 4 we plotσp,σo,σ′ o, andσsas functions of Erandδfor a typical value of Ω /ǫF= 0.1. One observe that as δin- creasesthe σpandσsareboth enhanced in Fig. 4(a) and (d). Of particular interests are the graph Fig. 4 (b) and5 (c), in which we see that σoandσ′ obecome non-zero for non-zeroδ, that is, the strength of the coupling between particle and spin currents can be controlled by the two photon detuning δ. Furthermore, the signs of σoandσ′ o are also tunable. When δchanges sign, σoandσ′ ochange signs either, which gives more flexibility to manipulate the conversion of the particle and spin currents. V. CONCLUSIONS In summary, the particle and spin transports through a QPC have been studied in a fermioninc system with Raman-induced SO coupling. Due to the mesoscopic scale of the QPC both particle and spin conductance ex- hibit plateau structure. Compared with the system with- out SO couplings the height of the plateau of the particle conductance σpis enlarged by the SO coupling effects. The magnitude of the spin conductance σsmajorly de- pends on the laser wave vector k0. Ask0increasesσsin- creases and saturates at a fixed value. Large two-photon detuningδcan help to enhance σswhenk0is small. Fur- thermore, the conversionofthe particle andspin currents are exclusively controlled by the two-photon detuning δ. Whenδis non-zero, the particle and spin currents couple together. VI. ACKNOWLEDGEMENTS The work is supported by the National Science Foun- dation of China (Grant No. NSFC-11874002), Beijing Natural Science Foundation (Grand No. Z180007). Appendix A: The calculation of current in Keldysh formulism To calculate the particle and spin currents we employ the Keldysh formulism in which the whole system is on a closed time contour. The partition function can be written as Z=1 Z0/integraldisplay D[¯ψj+,ψj+,¯ψj−,ψj−]exp(iS),(A1) whereS=S0+ST+SsandS0,STandSscorrespond to the free, tunneling and sourceterms, respectively. S0can be expressed in the momentum space as the following S0=/integraldisplay dkdω/summationdisplay j=L,R/braceleftBig ¯Ψj+[Gj+]−1Ψj++¯Ψj−[Gj−]−1Ψj−/bracerightBig , (A2) where the fields are given by ¯Ψj+=/parenleftbig¯ψj1+¯ψj2+/parenrightbig ,Ψj+=/parenleftbigg ψj1+ ψj2+/parenrightbigg¯Ψj−=/parenleftbig¯ψj1−¯ψj2−/parenrightbig ,Ψj−=/parenleftbigg ψj1− ψj2−/parenrightbigg .(A3) The fieldψj1±andψj2±are the Keldysh rotation of the fieldsψ+ j±andψ− j±, whicharethefields of ψj±onforward and backward time directions. The Keldysh rotation are given by ψj1±=1√ 2(ψ+ j±+ψ− j±), ψj2±=1√ 2(ψ+ j±−ψ− j±), ¯ψj1±=1√ 2(¯ψ+ j±−¯ψ− j±),¯ψj2±=1√ 2(¯ψ+ j±+¯ψ− j±). (A4) The Green’sfunctions in the Keldyshspaceareexpressed as Gj±=/parenleftbiggGR j±GK j± 0GA j±/parenrightbigg , (A5) The retard(advanced) Green’s function is given by GR j±= 1 ω−k2/2m∓ξ(kx)−Er+(¯µj∓S(kx))δµj)+i0+, GA j±= 1 ω−k2/2m∓ξ(kx)−Er+(¯µj∓S(kx))δµj)−i0+ (A6) and the Keldysh Green function is GK j±= (1−2nF(ω))(GR j±−GA j±).(A7) To write down the action STfor the tunneling Hamil- tonianˆHTwe use the single particle Hamiltonian ˆH+/summationtext j(µj↑ˆNj↑+µj↓ˆNj↓) to construct the time evolution op- erator U(t) =ei[ˆH+/summationtext j(µj↑ˆNj↑+µj↓ˆNj↓)]t(A8) Then the time evolution of the tunneling part is given by HT(t) =U(t)HTU†(t). In momentum space it’s written as ˆHT=∞/summationdisplay n=0/integraldisplaydω 2πd3kL (2π)3d3kR (2π)3 [T(+) nˆψ† L+(ω,kL)ˆψR+(ω−∆µp+S(kRx)∆µs,kR) +T(−) nˆψ† L−(ω,kL)ˆψR−(ω−∆µp−S(kRx)∆µs,kR)] +h.c. (A9) The particle and spin currents are defined in Eq. (1). In momentum space they can be expressed in terms of fieldˆψj±as the following6 I↑(Ω) = −i∞/summationdisplay n=0/integraldisplaydω 2πd3kL (2π)3d3kR (2π)3 /braceleftBig/parenleftbig DLDR+OLOR/parenrightbig/parenleftBig DLDRT(+) n/an}bracketle{tˆψ† L+(ω,kL)ˆψR+(Ω+ω−∆µp+S(kRx)∆µs,kR)/an}bracketri}ht +OLORT(−) n/an}bracketle{tˆψ† L−(ω,kL)ˆψR−(Ω+ω−∆µp−S(kRx)∆µs,kR)/an}bracketri}ht/parenrightBig/bracerightBig +h.c., I↓(Ω) = −i∞/summationdisplay n=0/integraldisplaydω 2πd3kL (2π)3d3kR (2π)3 /braceleftBig/parenleftbig DLDR+OLOR/parenrightbig/parenleftBig OLORT(+) n/an}bracketle{tˆψ† L+(ω,kL)ˆψR+(Ω+ω−∆µp+S(kRx)∆µs,kR)/an}bracketri}ht +DLDRT(−) n/an}bracketle{tˆψ† L−(ω,kL)ˆψR−(Ω+ω−∆µp−S(kRx)∆µs,kR)/an}bracketri}ht/parenrightBig/bracerightBig +h.c. (A10) In above equations the DjandOjare the elements of the unitary matrix in Eq.(8). Then in the Keldysh formulismthe action parts of the tunneling term and source term can be cast as ST=∞/summationdisplay n=0/integraldisplaydω 2πd3kL (2π)3d3kR (2π)3/parenleftBig Jq +(0,ω,kL,kR)+Jq −(0,ω,kL,kR)/parenrightBig +h.c., Ss=−i∞/summationdisplay n=0/integraldisplaydΩ 2πdω 2πd3kL (2π)3d3kR (2π)3/braceleftBig Acl ↑(Ω)[(DLDR+OLOR)(DLDRJq +(Ω,ω,kL,kR)+OLORJq −(Ω,ω,kL,kR))] +Aq ↑(Ω)[(DLDR+OLOR)(DLDRJcl +(Ω,ω,kL,kR)+OLORJcl −(Ω,ω,kL,kR))] +Acl ↓(Ω)[(DLDR+OLOR)(OLORJq +(Ω,ω,kL,kR)+DLDRJq −(Ω,ω,kL,kR))] +Aq ↓(Ω)[(DLDR+OLOR)(OLORJcl +(Ω,ω,kL,kR)+DLDRJcl −(Ω,ω,kL,kR))]/bracerightBig +h.c., (A11) where Jcl +(Ω,ω,kL,kR) =∞/summationdisplay n=0T(+) n /parenleftBig ¯ψL1+(ω,kL)ψR2+(Ω+ω−∆µp+S(kRx)∆µs,kR)+¯ψL2+(ω,kL)ψR1+(Ω+ω−∆µp+S(kRx)∆µs,kR)/parenrightBig , Jcl −(Ω,ω,kL,kR) =∞/summationdisplay n=0T(−) n /parenleftBig ¯ψL1−(ω,kL)ψR2−(Ω+ω−∆µp−S(kRx)∆µs,kR)+¯ψL2−(ω,kL)ψR1−(Ω+ω−∆µp−S(kRx)∆µs,kR)/parenrightBig , Jq +(Ω,ω,kL,kR) =∞/summationdisplay n=0T(+) n /parenleftBig ¯ψL1+(ω,kL)ψR1+(Ω+ω−∆µp+S(kRx)∆µs,kR)+¯ψL2+(ω,kL)ψR2+(Ω+ω−∆µp+S(kRx)∆µs,kR)/parenrightBig , Jq −(Ω,ω,kL,kR) =∞/summationdisplay n=0T(−) n /parenleftBig ¯ψL1−(ω,kL)ψR1−(Ω+ω−∆µp−S(kRx)∆µs,kR)+¯ψL2−(ω,kL)ψR2−(Ω+ω−∆µp−S(kRx)∆µs,kR)/parenrightBig . (A12) The currents of spin-up and spin-down can be calculated by Iσ(t) =/integraldisplayΩ 2πeiΩtIσ(Ω), (A13)7 whereIσ(Ω) is given by Iσ(Ω) =i 2∂Z ∂Aq σ/vextendsingle/vextendsingle/vextendsingle/vextendsingle Aq ↑,Aq ↓,Acl ↑,Acl ↓=0.(A14) [1] D. Stadler, S. Krinner, J. Meineke, J.-P. Brantut, and T, Esslinger, Nature 491, 736 (2012). [2] S. Krinner, D. Stadler, D Husmann, J. -P. Brantut, and T. Esslinger, Nature 517, 64 (2015). [3] G. Valtolina, A.Burchianti, A.Amico, E. Neri, K.Xhani, J. A.Seman, A.Trombettoni, A.Smerzi, M. Zaccanti, M. Inguscio, and G. Roati, Science 350, 1505 (2015). [4] D. Husmann, S. Uchino, S. Krinner, M. Lebrat, T. Gi- amarchi, T. Esslinger, and J.-P. Brantut, Science 350, 1498 (2015). [5] S. Krinner, M. Lebrat, D. Husmann, C. Grenier, J.-P. Brantut, and T. Esslinger, Proc. Natl. Acad. Sci. 113, 8144 (2016). [6] S. H¨ ausler, S. Nakajima, M. Lebrat, D. Husmann, S. Krinner, T.Esslinger, andJ.-P.Brantut, Phys.Rev.Lett. 119, 030403 (2017). [7] A. Burchianti, F. Scazza, A. Amico, G. Valtolina, J. A. Seman, C. Fort, M. Zaccanti, M. Inguscio, and G. Roati, Phys. Rev. Lett. 120, 25302 (2018). [8] W. J. Kwon, G. Del Pace, R. Panza, M. Inguscio, W. Zwerger, M. Zaccanti, F. Scazza, G. Roati, Science 369, 84 (2020). [9] G. Del Pace, W. J. Kwon, M. Zaccanti, G. Roati, and F. Scazza, Phys. Rev. Lett. 126, 055301 (2021). [10] A. Sommer, M. Ku, G. Roati, and M.W. Zwierlein, Na- ture472, 201 (2011). [11] A. B. Bardon, S. Beattie, C. Luciuk, W. Cairncross, D. Fine, N. S. Cheng, G. J. A. Edge, E. Taylor, S. Zhang, S. Trotzky, and J. H. Thywissen, Science 344, 722 (2014). [12] M. Koschorreck, D. Pertot, E. Vogt, M. Kohl, Nat. Phys. 9, 405 (2013). [13] C. Luciuk, S. Smale, F. Bottcher, H. Sharum, B. A. Olsen, S. Trotzky, T. Enss, and J. H. Thywissen, Phys. Rev. Lett. 118, 130405 (2017).[14] G. Valtolina, F. Scazza, A. Amico, A. Burchianti, A. Re- cati, T. Enss, M. Inguscio, M. Zaccanti, and G. Roati, Nat. Phys. 13, 704 (2017). [15] J.-P. Brantut, C. Grenier, J. Meineke, D. Stadler, S. Krinner, C. Kollath, T. Esslinger, A. Georges, Science 342, 713 (2013). [16] D. Husmann, M. Lebrat, S. H¨ ausler, J.-P. Brantut, L. Corman, and T. Esslinger, Proc. Natl. Acad. Sci. USA 115, 8563 (2018). [17] S. H¨ ausler, P. Fabritius, J. Mohan, M. Lebrat, L. Cor- man, and T. Esslinger, Phys. Rev. X 11, 021034 (2021). [18] M. Kan´ asz-Nagy, L. Glazman, T. Esslinger, E. A. Dem- ler, Phys. Rev. Lett. 117, 255302 (2016). [19] B. Liu, H. Zhai, S. Zhang, Phys. Rev. A 95, 013623 (2017). [20] S. Uchino, and M. Ueda, Phys. Rev. Lett. 118, 105303 (2017). [21] Y.-J. Lin, K. Jim´ enez-Garc´ ıa and I. B. Spielman, Natu re 471, 83 (2011). [22] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [23] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [24] N. Goldman, G. Juzeliunas, P. Ohberg, I. B. Spielman, Rep. Prog. Phys. 77, 126401 (2014). [25] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015). [26] A. Kamenev, Field theory of non-equilibrium systems , Cambridge University Press, 2011. [27] R. Landauer, IBM J. Res. Develop. 1, 223 (1957). [28] M. B¨ uttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B31, 6207 (1985).

1506.00911v2.Spin_orbit_coupling_rule_in_bound_fermions_systems.pdf

Spin-orbit coupling rule in bound fermions systems J.-P. Ebran,1E. Khan,2A. Mutschler,2, 3and D. Vretenar4 1CEA,DAM,DIF, F-91297 Arpajon, France 2Institut de Physique Nucl eaire, Universit e Paris-Sud, IN2P3-CNRS, F-91406 Orsay Cedex, France 3Grand Acc el erateur National d'Ions Lourds (GANIL), CEA/DSM-CNRS/IN2P3, Bvd Henri Becquerel, 14076 Caen, France 4Physics Department, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia (Dated: November 4, 2015) Spin-orbit coupling characterizes quantum systems such as atoms, nuclei, hypernuclei, quarkonia, etc., and is essential for understanding their spectroscopic properties. Depending on the system, the eect of spin-orbit coupling on shell structure is large in nuclei, small in quarkonia, perturbative in atoms. In the standard non-relativistic reduction of the single-particle Dirac equation, we derive a universal rule for the relative magnitude of the spin-orbit eect that applies to very dierent quantum systems, regardless of whether the spin-orbit coupling originates from the strong or electromagnetic interaction. It is shown that in nuclei the near equality of the mass of the nucleon and the dierence between the large repulsive and attractive potentials explains the fact that spin-orbit splittings are comparable to the energy spacing between major shells. For a specic ratio between the particle mass and the eective potential whose gradient determines the spin-orbit force, we predict the occurrence of giant spin-orbit energy splittings that dominate the single-particle excitation spectrum. PACS numbers: 21.10.-k,21.60.Jz,21.80.+a I. INTRODUCTION Spin-orbit coupling, known since the early days of quantum mechanics, is among the most studied eects related to the spin of a particle. After nearly a cen- tury, spin-orbit eects continue to provide a basis for a variety of new phenomena in diverse elds of quantum physics, such as spintronics [1], topological insulators [2], cold atomic gases [3], atomic nuclei far from stability [4], etc. In nuclear physics, in particular, the strong cou- pling between the orbital angular momentum and spin of a nucleon accounts for the empirical magic numbers and shell gaps. The energy spacings between spin-orbit partner states can be as large as the gaps between major shells in atomic nuclei, but the spin-orbit splittings can also be considerably reduced in short-lived nuclei with extreme isospin values [4{6]. Starting from the nuclear physics case, we investigate the ratio between the ma- jor energy spacings of levels characterized by principal single-particle quantum numbers and the energy splitting of spin-orbit partner states in systems of bound fermions. This ratio can dier, of course, by orders of magnitude in systems in which the spin-orbit coupling originates from dierent underlying interactions, e.g. the strong eective nuclear force or the electromagnetic interaction. It is, therefore, interesting to try to nd a common character- istic for the spin-orbit splitting in systems as diverse as atoms, nuclei, hypernuclei, quarkonia, etc. II. SPIN-ORBIT COUPLING IN ATOMIC NUCLEI In the nuclear relativistic mean-eld framework [7{9] a nucleus is considered as a system of independent nu-cleons moving in local self-consistent scalar and vector potentials. The single-nucleon dynamics is governed by the Dirac equation: [~ 
~ p+V+(m+S)] i=Ei i (1) where idenotes the Dirac spinor:  i i (2) for the i-th nucleon. For simplicity we only con- sider spherical nuclei and assume time-reversal symme- try (pairwise occupied states with Kramers degeneracy), which ensures that the only non-vanishing components of the vector elds are the time-like ones and thus there is no net contribution from nucleon currents. The lo- cal vectorVand scalarSpotentials are uniquely deter- mined by the actual nucleon density and scalar density of a given nucleus, respectively. In the ground state A nucleons occupy the lowest single-nucleon orbitals, de- termined self-consistently by the iterative solution of the Dirac equation (1). If one expresses the single-nucleon energy asEi=m+"i, wheremis the nucleon mass, and rewrites the Dirac equation as a system of two equa- tions foriandi, then, noticing that for bound states "i<<m , i1 2M(r)(~ 
~ p)i (3) to order"i=m, and M(r)m+1 2(S(r)V(r)): (4) The equation for the upper component iof the Dirac spinor reduces to the Schr odinger-like form [10, 11]  ~ p1 2M(r)~ p+U(r) +VLS(r) i="ii (5)arXiv:1506.00911v2 [nucl-th] 3 Nov 20152 for a nucleon with eective mass M(r) in the potential U(r)V(r) +S(r), and with the spin-orbit potential: VLS1 2M2(r)1 rd dr(V(r)S(r))~l
~ s: (6) The spin-orbit coupling plays a crucial role in nuclear structure, and its inclusion in the eective single-nucleon potential is essential to reproduce the empirical magic numbers. The relativistic mean-eld framework, in par- ticular, naturally includes the nucleon spin degree of free- dom, and the resulting spin-orbit potential emerges au- tomatically with the empirical strength [12]. The nu- clear spin-orbit potential originates from the dierence between two large elds: the vector potential V(short- range repulsion) with typical strength of 350 MeV, and the scalar potential S(medium-range attraction), typically of the order of 400 MeV in nucleonic mat- ter and nite nuclei. In the context of in-medium QCD sum rules [13], the strong scalar and vector mean elds experienced by nucleons can be associated with the lead- ing density dependence of the chiral (quark) condensate, hqqi, and the quark density hqyqi. In the mean-eld phe- nomenology the sum of these two elds V+S50 MeV provides the conning potential that binds the nucleons in a nucleus, whereas the large dierence VS750 MeV determines the pronounced energy spacings be- tween spin-orbit partner states in nite nuclei, of the order of several MeV [7{9, 12, 14, 15]. A puzzling coin- cidence that we would like to explore is that the largest spin-orbit splittings for intruder states are comparable in magnitude to the energy gaps between major shells of the nuclear potential. The aim of this study is to evaluate the typical ratio between the energy spacings of levels characterized by principal single-particle quantum numbers and the en- ergy splitting of spin-orbit partner states (ne structure). Even though the values of this ratio span orders of magni- tude for dierent bound quantum systems, we will show that it is basically governed by two quantities that char- acterize a given system, irrespective whether the binding and spin-orbit potentials originate from the strong (nu- clei) or electromagnetic (atoms) interactions. In the nuclear case the interaction is of short range and the self-consistent potentials display a spatial distri- bution that corresponds to the actual single-nucleon den- sity. The expression for the spin-orbit potential Eq. (6) can, therefore, be rewritten in the following form: VLS'F(r)0(r) 2(r)r~l
~ s; (7) where F(r)V(r)S(r)  m1 2(V(r)S(r))2; (8) and(r) denotes the self-consistent ground-state density of a nucleus with A nucleons. For a typical approxima- tion of the single-nucleon potential, such as the harmonicoscillator, or the more realistic Woods-Saxon potential, one can show [16]: <0(r) 2(r)r>'1 R2 0(9) whereR0=r0A1=3,r01:2 fm and, together with < ~l
~ s >=l=2 forj=l+ 1=2, and<~l
~ s >=1=2(l+ 1) forj=l1=2, the energy spacing between spin-orbit partner states can be approximated by: j
<VLS>jFl~2 R2 0(10) Let us consider the ratio between the major energy spac- ings and the spin-orbit splitting. For the harmonic oscil- lator potential one nds [17] ~!0=~ R0r 2U0 m; (11) where, in our case, the depth of the potential is U0 U(r= 0) =V(0) +S(0). Therefore, x~!0 j
<VLS>j=K1 +1 4; (12) whereK=p2mU0R0=l~, and m VS: (13) K is typically of the order 1 5 for l3 (corresponding to the magic spin-orbit gaps). Since for the nucleon mass m940 MeV and VS750 MeV:= 1:25, it follows from Eq. (12) that for the nuclear system the ratio xis of the order 15, that is, in nuclei the energy splitting between spin-orbit partner states is comparable in magni- tude to the spacings between major oscillator shells. This is because of the near equality of the mass mand the po- tentialVSin nuclei. It should be noted than in the case of the pseudo-spin symmetry [18] (V=-S), Eq. (13) yields =m/2V. An aspect that can be generalised to dierent quantum systems is the specic functional dependence of xon the ratio of the particle mass mand the eective potential whose gradient determines the spin-orbit force. III. OTHER SYSTEMS OF BOUND FERMIONS In the case of atomic systems the binding of an electron is determined by the Coulomb potential V(r) =Z=r , whereZis the charge of the nucleus and the ne- structure constant = 1=137. One can again perform the non-relativistic reduction of the Dirac equation for the electron (Eqs. (1) - (5)), and the resulting spin-orbit potential reads: VLS=1 2M2(r)1 rdV(r) dr~l
~ ; (14)3 with 2M(r)2mV(r). In this case, of course, U(r) in Eq. (5) is just the Coulomb potential. The energy spacing between successive levels with dierent principal quantum number nis proportional to 2, whereas the rst-order spin-orbit splitting is 4. The ratio between the principal energy spacings and the spin-orbit splittings (ne structure) is much larger than in the nuclear case, that is1=2104, known from the early seminal work of Sommerfeld [19]. The interesting fact is that, starting from Eq. (14), this ratio can again be expressed with the same functional dependence on as in Eq. (12). =m=V is now negative and, with m= 0:5 MeV and V(r0) =2:72105MeV for the hydrogen atom and the Bohr's radius r0, it follows that in the atomic case the characteristic value is 1=22104. For large absolute values of , the expression Eq. (12) reduces to x1=2104, in agreement with the empirical value quoted above. The ne structure of atomic spectra thus becomes a limit of the spin-orbit rule (12). It should be noted the validity of the spin-orbit rule in Coulomb-like systems is due to the 1/r behavior of the potential. The spin-orbit rules also applies to the case of ions having Z protons: in these systems the ne structure is known to behave as Z22. In the present approach the Z2factor comes from the Z one of Eq. (13) and the Z one from the r0/Z typical size of the ion. Figure 1 displays the quantity representing the spin- orbit rule: x'1 +1 4(15) as a function of the ratio between the mass of the par- ticle and the eective potential that determines the spin- orbit force in a given quantum system. As shown above, for nucleiis slightly larger than one and, depending on the specic orbital, xlies in the interval 1 5. In atoms is negative and of the order of 104and, therefore, the characteristic value of xis1=2104. The main results of the spin-orbit rule quantatively apply to nuclei and atoms. It may be relevant to test it in a few other quantum systems although a more qual- itative agreement is expected, due to additional eects which would not be included in the present approach. For instance, experimental evidence indicates that in  hypernuclei the -nuclear spin-orbit interaction is very weak compared to the strong spin-orbit interaction in ordinary nuclei [20, 21]. Available data show that the en- ergy spacings between  spin-orbit partner states are of the order of100 keV, although some microscopic mod- els also predict 2 MeV values in some cases [22]. Even though a quantitative study of the smallness of the  spin-orbit interaction necessitates a rather involved anal- ysis based on in-medium chiral SU(3) dynamics [23], one can qualitatively understand the reduction of the spin- orbit splitting in  hypernuclei already at the relativis- tic mean-eld level [24, 25]. A nite-density QCD sum- rule analysis [13, 24] indicates that, when compared the strong scalar and vector mean elds experienced by nu- -5-4 -3 -2 -1012 3 4 5 η012345678910 x 1/x NucleiQuarkonia AtomsAtomsGiant LS Hyper nuclei Nuclei pNuclei pNuclei εj+ > ε j- εj+ < ε j- n in n inp in pNucleipNuclei p inFIG. 1: The ratio between the principal energy spacings and the spin-orbit splittings (ne structure) Eq. (15), as a function of the ratio between the mass of the particle and the eective potential that determines the spin-orbit force in a given quantum system. cleons in ordinary nuclei, the corresponding self-energies of a  hyperon are reduced by a factor 0:40:5, and even smaller if corrections from in-medium conden- sates of higher dimensions are taken into account. With m= 1:12 GeV, this means that in Eq. (13) is of the order 34, leading to larger x value than in nuclei, in qualitative agreement with empirical values of x in such systems. Since the energy spacing between major oscilla- tor shells in  hypernuclei (for instance,13 C,16 O,40 Ca), is of the order of 10 MeV [20], the empirical ratio x ranges from a few units to a few dozens. Our mean-eld estimate is closer to the small values of x but, as shown in Ref. [25], a considerable additional reduction of the splitting between  spin-orbit partner states arises as the eect of the hypernuclear tensor coupling. As explained above, a quantitative explanation of the small spin-orbit splitting in hypernuclei must include beyond-mean-eld eects. An interesting example of quantum systems that are governed by the strong interaction but exhibit negative values of, are quarkonia such as, for instance, char- monium ccand bottomonium bb. The center-of-mass is dierent, compared to the present one-body approach, but this only generates a global scaling factor of 2 on the energy positions, due to the reduced mass. Since we are dealing with order of magnitudes, and most importantly with energies ratios, it is relevant to check if the present approach applies in the case of quarkonia. The use of potential models for these system can be justied by the fact that the bottom and charm masses are large in com- parison to the typical hadronic scale of QCD. Most phe- nomenological approaches to the dynamics of two heavy4 quarks interacting through a potential are variants of the Cornell model [26], which consists of a superposition of the one-gluon-exchange that leads to a Coulomb-like at- tractive vector potential at short distances: V=4 3s(r) r; (16) plus a scalar linear conning potential S=r, with 0:18 GeV2. Therefore, VSis negative in the quarkonia case. The masses of the cquarkbquark are mc1:27 GeV and mb4:2 GeV, respectively. At ra- dial distances that correspond to the mean-square radii of the quarkonia states the Coulomb-like attractive vec- tor potential Eq. (16), with a depth of the order of 1 GeV, dominates over the scalar potential [27]. The mass of the lowest charmonium state is 3 GeV, while that of the lowest bottomonium state is 9:5 GeV. The en- ergy spacing between 1 Sand 2Sstates is of the order of 600 MeV for charmonia, and 560 MeV for bottomonia [28, 29]. The ne splittings between 1 PJstates are less than 100 MeV for the charmonium (
M21= 45:60:2 MeV and
M10= 95:30:4 MeV), and less than 40 MeV for the bottomonium (
M21= 19:40:4 MeV and
M10= 33:30:5 MeV). This means that, in the case of quarkonia, the empirical ratio between the energy spac- ings characterized by the principal quantum numbers and the ne spin-orbit splittings is of the order of 5 to 10, in qualitative agreement with Fig. 1. The particular dependence of this ratio on , shown in Fig. 1, displays interesting features. For positive values of, states for which the orbital angular momentum and spin are aligned are found at lower energy with respect to states for which the orbital angular momentum and spin are anti-aligned (nuclei, hypernuclei), whereas the opposite energy ordering is found for negative (atoms, quarkonia). The ratio x(Eq. (15)) diverges at = 0, that is, in the limit of massless particle. An interesting situation is found in the vicinity of =1/2, for which x= 0. More precisely, values close to 1/2 are in the validity domain of the approximation performed to derive the present spin-orbit rule. This oc- curs when the mass of the particle is close to ( VS)=2, and the energy spacings between states are characterized by very large spin-orbit coupling (giant LS). Such states could be obtained in particular cases for which one would be able to choose the strength of the eective potential whose gradient determines the spin-orbit force. In a hy- pothetical case this limit is approached when the eec- tive potential becomes very deep as it would occur, for instance, for bound antibaryon-nuclear systems which, in addition to ordinary nucleons, contain antibaryons (B= p;;:::) (cf. Figs. 10 and 21 of Ref. [30]). Two cases can be considered. For the antibaryon spec- trum the spin-orbit splitting remains small, of about few hundreds keV [31]. The corresponding V-S potential is smaller than in the nucleon case because V has the op- posite sign in the case of antibaryons [31] and, therefore, the value of becomes larger (Fig. 1). This is both thecase for the antiproton and anti- nuclei [22, 32]: in the last case a good quantitative agreement is found for the relative amplitude of the spin-orbit splitting predicted by the spin-orbit rule (x 40). The case of nucleons in antibaryon-nuclei is closer to the giant LS state: the gradient of the potential obtained from self-consistent calculations increases and the spin- orbit splitting not only gets larger, but displays a pro- nounced increase with respect to the spacing between major shells (Fig. 1). However, such systems have not been observed and this is just an illustrative example, with antibaryon-nucleus potentials subject to large un- certainties [30]. A similar eect, although much weaker, is predicted for single-nucleon spectra in -hypernuclei [33]. Even though the presence of a  induces only a frac- tional change in the central mean-eld potential, through a purely relativistic eect it increases the spin-orbit term in the surface region, providing additional binding for the outermost neutrons. To emphasize this region of gi- ant spin-orbit splittings, in Fig. 1 we also plot the inverse ofx. For negative values of the ratioxdoes not vanish, but again one nds a region of small absolute values of  for which the splitting between spin-orbit partner states is comparable in magnitude to the energy spacing be- tween successive levels with dierent principal quantum number. IV. SUMMARY In conclusion, we have investigated an interesting eect of spin-orbit coupling in systems of bound fermions (elec- trons, quarks, nucleons, hyperons). By performing the usual non-relativistic reduction of the Dirac equation, the single-particle mean-eld equation takes a Schr odinger- like form which, in addition to the conning potential, exhibits the spin-orbit potential explicitly. Starting from the relativistic mean-eld approximation in the nuclear case, we have derived an analytic expression for the ratio between the energy spacings characterized by principal single-particle quantum numbers and the energy split- ting of spin-orbit partner states. This quantity explic- itly depends on the ratio of the particle mass and the eective potential whose gradient determines the spin- orbit force (cf. Eq. (15)). In nuclei the near equality of the mass of the nucleon and the dierence between the large repulsive vector and attractive scalar potentials explains the fact that the spin-orbit splittings are com- parable to the energy spacing between major oscillator shells. The same universal functional form also applies to other bound quantum systems, regardless of whether the spin-orbit potential originates from the strong or elec- tromagnetic interaction, and for which the ratio between the principal energy spacings and the spin-orbit split- tings is orders of magnitude larger than in the nuclear case. For a given particle mass this ratio could be al- tered, in principle, by modifying the potential that gen- erates the spin-orbit coupling eect. When this ratio5 is1=2, our study predicts the occurrence of giant spin-orbit splittings, that is, a single-particle excitation spectrum dominated by large energy spacings between spin-orbit partner states. The extension of the derivation of the spin-orbit rule to exotic nuclei, where the densities are more diuse shall be undertaken in a forthcoming work. This includes halo nuclei involving coupling to the continuum [34{37]. It would also be relevant to derive a pseudo-spin rule, pro-viding the typical magnitude of the related degeneracy raising due to the breaking of the pseudo-spin symmetry [18, 38], with respect to the main shell structure. This work was supported by the Institut Universitaire de France. The authors thank Nguyen Van Giai, Tamara Nik si c, Peter Ring and Olivier Sorlin for reading the manuscript and valuable discussions. [1] I. Zuti c, J. Fabian, and S. Das Sarma, Rev. Mod. phys. 76, 323 (2004). [2] M. Z. Hasan and C. L. Kane, Rev. Mod. phys. 82, 3045 (2010). [3] V. Galitski and Ian B. Spielman, Nature 494, 49 (2013). [4] O. Sorlin and M.-G. Porquet, Prog. Part. Nucl. Phys. 61, 602 (2008). [5] G. A. Lalazissis, D. Vretenar, W. P oschl, and P. Ring, Phys. Lett. B 418, 7 (1998). [6] G. Burgunder et al., Phys. Rev. Lett. 112, 042502 (2014). [7] H.-P. Duerr, Phys. Rev. 103, 469 (1956). [8] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986). [9] B. D. Serot and J. D. Walecka, Int. J. Mod. Phys. E 6, 515 (1997). [10] B. L. Bibrair, L. N. Savushkin, and V.N. Fomenko, Sov. J. Nucl. Phys. 35, 664 (1982). [11] P.-G. Reinhard, Rep. Prog. Phys. 52, 439 (1989). [12] R. J. Furnstahl and B. D. Serot, Comments Nucl. Part. Phys. 2, A23 (2000). [13] T. D. Cohen, R. J. Furnstahl and D. K. Griegel, Phys. Rev. Lett. 67, 961 (1991); T. D. Cohen, R. J. Furnstahl, D. K. Griegel and X. m. Jin, Prog. Part. Nucl. Phys. 35, 221 (1995) 221; E.G. Drukarev and E.M. Levin, Prog. Part. Nucl. Phys. 27, 77 (1991). [14] G. A. Lalazissis, D. Vretenar, and P. Ring, Phys. Rev. C 57, 2294 (1998). [15] D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, and P. Ring, Phys. Rep. 409, 101 (2005). [16] A. Bohr and B. R. Mottelson, Nuclear Structure, W. A. Benjamin, Inc. (1969). [17] P. Ring, P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag (1980). [18] J.N. Ginocchio, Phys. Rev. Lett. 78 (1997) 436. [19] A. Sommerfeld, Annalen der Physik 51, 1 (1916). [20] O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys.57, 564 (2006), and references therein. [21] A. Gal and R.S. Hayano, eds., Special Issue on Strangeness Nuclear Physics , Nucl. Phys. A 804 (2008). [22] Song Chun-Yan, Yao Jiang-Ming, and Meng Jie, Chin. Phys. Lett. 26, 122102 (2009). [23] P. Finelli, N. Kaiser, D. Vretenar and W. Weise, Phys. Lett. B 658, 90 (2007); Nucl. Phys. A 831, 163 (2009). [24] X. Jin, R. J. Furnstahl, Phys. Rev. C 49, 1190 (1994). [25] J. Mare s and B. K. Jennings, Phys. Rev. C 49, 2472 (1994). [26] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. Yan, Phys. Rev. D 17, 3090 (1978). [27] W. Buchm uller and S.-H. H. Tye, Phys. Rev. D 24, 132 (1981). [28] N. Brambilla et al, CERN Yellow Report, CERN-2005- 005, Geneva: CERN, 2005. arXiv:hep-ph/0412158 [29] Particle Data Group, J. Beringer et al, Phys. Rev. D 86, 010001 (2012). [30] I. N. Mishustin, L. M. Satarov, T. J. B urvenich, H. St ocker, and W. Greiner, Phys. Rev. C 71, 035201 (2005). [31] Shan-Gui Zhou, J. Meng, and P. Ring, Phys. Rev. Lett. 91, 262501 (2003). [32] Song Chun-Yan, Yao Jiang-Ming, and Meng Jie, Chin. Phys. Lett. 28, 092101 (2011). [33] D. Vretenar, W. P oschl, G. A. Lalazissis, and P. Ring, Phys. Rev. C 57, R1060 (1998). [34] J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, L.S. Geng, Prog. Part. Nucl. Phys. 57,470 (2006). [35] J. Meng and P. Ring, Phys. Rev. Lett. 77, 3963 (1996). [36] W. P oschl, D. Vretenar, G. A. Lalazissis, and P. Ring, Phys. Rev. Lett. 79, 3841 (1997). [37] J. Meng and P. Ring, Phys. Rev. Lett. 80, 460 (1998). [38] Haozhao Liang, Jie Meng, Shan-Gui Zhou, Phys. Rep. 570, 1 (2015).

1904.01015v1.Spin_charge_coupled_transport_in_van_der_Waals_systems_with_random_tunneling.pdf

Spin-charge coupled transport in van der Waals systems with random tunneling M. Rodriguez-Vega1;2, G. Schwiete3, Enrico Rossi4 1Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA 2Department of Physics, Northeastern University, Boston, MA 02115, USA 3Department of Physics and Astronomy, Center for Materials for Information Technology (MINT), The University of Alabama, Alabama 35487, USA 4Department of Physics, William & Mary, Williamsburg, VA 23187, USA (Dated: July 8, 2021) We study the electron and spin transport in a van der Waals system formed by one layer with strong spin-orbit coupling and a second layer without spin-orbit coupling, in the regime when the interlayer tunneling is random. We find that in the layer without intrinsic spin-orbit coupling spin-charge coupled transport can be induced by two distinct mechanisms. First, the gapless diffusion modes of the two isolated layers hybridize in the presence of tunneling, which constitutes a source of spin-charge coupled transport in the second layer. Second, the random tunneling introduces spin-orbit coupling in the effective disorder-averaged single-particle Hamiltonian of the second layer. This results in non-trivial spin transport and, for sufficiently strong tunneling, in spin- charge coupling. As an example, we consider a van der Waals system formed by a two-dimensional electron gas (2DEG)–such as graphene–and the surface of a topological insulator (TI) and show that the proximity of the TI induces a coupling of the spin and charge transport in the 2DEG. In addition, we show that such coupling can be tuned by varying the doping of the TI’s surface. We then obtain, for a simple geometry, the current-induced non-equilibrium spin accumulation (Edelstein effect) caused in the 2DEG by the coupling of charge and spin transport. In recent years experimentalists have been able to make very novel and high quality heterostructures that allow the re- alization of new effects and states of great fundamental and technological interest [1]. Recently simple heterostructures formed by two graphene layers with a relative twist angle [2– 5] have shown a phase diagram [6, 7] that is remarkably rem- iniscent of the phase diagram of high temperature supercon- ductors. These are just some of the most striking examples that heterostructures can be used to realize novel effects that are not present in the single constituents. Applications of het- erostructure engineering [8] can be found in tunnel junctions [9], plasmonic [10], photoresponsive [11], spintronics [12– 14] and valleytronic [15] devices. One of the essential elements to realize non-trivial topo- logical states and spin-dependent transport phenomena is the presence of spin-orbit coupling (SOC). However, often the presence of spin-orbit coupling is not accompanied by other desirable properties such as high mobility, or superconduct- ing pairing. For this reason heterostructures that combine one constituent with significant SOC and one constituent with no SOC but other distinct properties are very interesting both for fundamental reasons and for their potential for technologi- cal applications. One example, of such heterostructures are graphene–topological-insulator van der Waals systems [16– 24]. So far, the theoretical studies of van der Waals het- erostructures have focused on the regime when the tunneling is not random and a strong hybridization between the elec- tronic states of the isolated systems can be achieved. How- ever, in many situations we can expect the tunneling between the systems forming the heterostructure to be random, due for example to the incommensurate nature of the stacking config- uration and/or the presence of surface roughness. In this work we focus on this situation, and study the elec- tron and spin transport in a two-dimensional van der Waalssystems comprised of one component (layer) with strong SOC and one with no, or negligible, SOC, when the interlayer tun- neling is random. Due to the random nature of the tunneling in most experimental situations the transport will be diffusive even in the absence of disorder. For this reason we consider only the diffusive regime, in which specific details of the sys- tem considered (like the value of the mean free path) do not affect the general expression of the transport equations that, therefore, have a somewhat universal character. We find that in general, if the diffusive transport in the layer with SOC exhibit spin-charge coupling [25–27] such coupling will be present also in the layer without SOC, i.e., in the most com- mon experimental situation. To exemplify this general result we consider the case of a van der Waals system formed by a two-dimensional electron gas (2DEG) placed on the surface of a strong three dimensional topological insulator (TI) [28, 29]. Graphene and the surface of TIs in the tetradymite family such as Bi2Se3have almost commensurate lattices and as a consequence in many graphene-TI heterostructures the K,K0 points of the graphene’s BZ are folded close to the TI’s [19] point. This fact, combined with the random and finite-range nature of the interlayer tunneling, implies that the results that we obtain for a 2DEG-TI van der Waals system are directly relevant to graphene-TI heterostructures, and similar systems. We obtain the diffusive transport equations in the 2DEG layer and show that they describe a transport in which the charge and the spin degrees of freedom are coupled. Finally, we show how the diffusive equations give rise to spin-dependent trans- port effects, analogous to the ones obtained for a 2DEG with Rashba SOC [25] and an isolated TI’s surface [26], that are tunable by simply varying the doping of the TI, and that can be used for possible spintronics applications. The Hamiltonian ^Hfor the heterostructure can be written as ^H=P l=1;2[^Hl+^Vl]+^Twherelis the layer index, ^Hlis thearXiv:1904.01015v1 [cond-mat.mes-hall] 1 Apr 20192 Hamiltonian for layer lin the clean limit, ^Vlis the term due to disorder located in layer l, and ^Tis the term describing inter- layer tunneling. For the 2DEG layer we have ^Hl=^H2d(k) =P kss0^ y 2d;ksH2dss0(k)^ 2d;ks0where, ^ y 2d;ks(^ 2d;ks)is the creation (annihilation) operator for an electron with momen- tumkand spins. Without loss of generality we can lin- earize the 2DEG dispersion around the Fermi surface and as- sumeH2d(k) = (v2djkj2d)0withv2dthe Fermi ve- locity,2dthe chemical potential, and 0the22iden- tity Pauli matrix in spin space. For the TI’s surface we have ^Hl=^HTI=P kss0^ y TI;ksHTIss0(k)^ TI;ks0where y TI;ks ( y TI;ks)creates (annihilates) a surface Dirac fermion with spinsand momentum k,HTI(k) =vTI(k)zTI, vTIbeing the Fermi velocity on the TI’s surface, TIthe TI’s surface chemical potential, and i,i=x;ythe Pauli matrices in spin space. For the disorder potential in layer l,V(D) l(q), we have hV(D) l(r1)V(D) l(r2)i=WD l(r1r2), where the angle brack- ets denote average over disorder realizations, and WD l(r1 r2)is the disorder-averaged spatial correlation. In momen- tum space we have WD l(q) =nl impjU(q)j2wherenl impis the impurity density in layer l, andUl(q)the Fourier trans- form of the potential profile Ul(r)of a single impurity. With- out loss of generality we can assume hV(D) l(r)i= 0. As- suming the tunneling to be spin-conserving we have ^T=P kqsT(q)^ y lks^ lk+qs+h:c: with l6=l. Assuming the tunneling to be random we can characterize it by the spatial average of the tunneling matrix element hT(r1)T(r20)i= Wt(r1r2). In the remainder we assume both the intralayer disorder and interlayer tunneling to be short-range so that Ul(q) = const = Ul,Wt(q) = const = t2. LetGR;A 0l(k;) = (Hl(k)0+)1be the bare re- tarded (advanced) real-time Green’s function for layer l. The total self-energy for layer l,l, has contributions from scat- tering with impurities, 0 l, and random tunneling events t l. We have 0 l(k;) =nl impR qjUl(q)j2Gl(kq;), whereR qR d2q=(2)2. In the self-consistent Born approxima- tion,Glis the disorder-dressed Green’s function for layer l. For the 2DEG, apart from an overall unimportant real con- stant, we have 0 2d=i0 2d0=2, where 0 2d= 1=0 2d= 22dn2d impU2 2d, and2dis the density of states (DOS) at the Fermi energy. For the TI’s surface, due to the fact that the electrons behave as massless Dirac fermions, for UTI(q) = const , we have that the integral in the expression for 0 lhas an ultraviolet divergence [30]. After properly regularizing such divergence [31] one finds that the intralayer disorder, in addition to generating an imaginary part of the self-energy, i0 TI0, with 0 TI= 1=0 TI=TInTI impU2 TIandTI, the TI’s DOS at the Fermi energy, causes a renormalization of the Fermi velocity that we incorporate in the definition of vTI. The same ultraviolet divergence appears for the self-energy correction for the 2DEG due to tunneling events into the TI, t 2d. The proper renormalization of such divergence, consis- tent with the Ward identities, causes t 2dto have a non-trivialreal part so that t 2d(k;) =it 2d0=2 + t2=(4v2 TI)
(k)z:(1) where t 2d= 1=t 2d=TIt2. This result shows that even when the interlayer tunneling processes are random, a spin- orbit coupling term is induced in the 2DEG due to TI’s sur- face proximity. This term of the self energy qualitatively af- fects the diffusive transport in the 2DEG, but it is not nec- essary to induce spin-charge transport in the 2DEG as we will show below. The self-energy correction for the TI due to tunneling events into the 2DEG, t TI, does not require any special care and simply results in an additional broad- ening of the quasiparticles: t TI(k;) =it TI0=2with t TI= 1=t TI= 22dt2. With the self-energy contributions, the dressed 2D system Green’s functions take the form GR=A 2d;(k) =(i2d=2k)0[t2=(4vTI)] (k)z (i2d=2k)2[t4=(4vTI)2]k2; (2) GR=A TI;(k) =(iTI=2)0vTI(k)z (iTI=2)2v2 TIk2; (3) where 2d0 2d+ t 2d,TI0 TI+ t TI. In the diffusive regime, to leading order in 1=(F), the re- tarded dynamical part of the spin-density response function for layerl,dyn lis obtained by summing all ladder vertex corrections to the bare spin-density response. In our case we have two types of ladder diagrams: the ones due to random interlayer tunneling and the ones due to intralayer disorder. In most experimentally relevant situations we expect the scat- tering time due to intralayer disorder to be much smaller than the relaxation time due to the interlayer random tunneling pro- cesses. For this reason in the remainder we assume t0. The main building block for the calculation of dyn 2dis the dif- fusonD2d, which includes both interlayer tunneling and in- tralayer ladder diagrams. It satisfies the self-consistent equa- tion [32, 33] D2d=~D2d+~D2dJTI 2d~DTIJ2d TID2d: (4) In this equation, the auxiliary intralayer diffuson for layer l, ~Dl(l= (2d;TI)) is obtained by taking into account only intralyer disorder and the junctions Jdescribe the transi- tion between the layers. The constant collects disorder- dependent normalizations with 1=n2d impnTI impU2 2dU2 TI. The self-consistency equation (4) is shown diagrammatically in Fig. 1(a). Mathematically, the auxiliary diffuson ~Dlsatisfies the Bethe-Salpeter equation (see Fig. 1(b)) ~Dl(q;!) =nl impU2 l(0 0Pl(q;!))1: (5) Here, the quantum probability Plis defined as Pl(q;!)nl impU2 lZ kGR l;F(k) GA l;F!(kq):(6)3 Dl <latexit sha1_base64="JqJ2p6HAoVeMhWWCi63kgDArko0=">AAAB9HicbVDLSgMxFL3js9ZX1aWbYBFclRkVdFnUhcsK9gHtUO6kaRuayYxJplCGfocbF4q49WPc+Tdm2llo64HA4Zx7uScniAXXxnW/nZXVtfWNzcJWcXtnd2+/dHDY0FGiKKvTSESqFaBmgktWN9wI1ooVwzAQrBmMbjO/OWZK80g+mknM/BAHkvc5RWMlvxOiGVIU6d20K7qlsltxZyDLxMtJGXLUuqWvTi+iScikoQK1bntubPwUleFUsGmxk2gWIx3hgLUtlRgy7aez0FNyapUe6UfKPmnITP29kWKo9SQM7GQWUi96mfif105M/9pPuYwTwySdH+ongpiIZA2QHleMGjGxBKniNiuhQ1RIje2paEvwFr+8TBrnFe+i4j5clqs3eR0FOIYTOAMPrqAK91CDOlB4gmd4hTdn7Lw4787HfHTFyXeO4A+czx/8d5I7</latexit>↵ <latexit sha1_base64="ljfxkPSo6R6GSeRY6vDWjNSNe+E=">AAAB9HicbVBNS8NAEJ34WetX1aOXxSJ4KokKeix68VjBfkAby2S7aZfuJnF3Uyihv8OLB0W8+mO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dJwqyuo0FrFqBaiZ4BGrG24EayWKoQwEawbD26nfHDGleRw9mHHCfIn9iIecorGS39G8L/GxgyIZYLdUdivuDGSZeDkpQ45at/TV6cU0lSwyVKDWbc9NjJ+hMpwKNil2Us0SpEPss7alEUqm/Wx29IScWqVHwljZigyZqb8nMpRaj2VgOyWagV70puJ/Xjs14bWf8ShJDYvofFGYCmJiMk2A9Lhi1IixJUgVt7cSOkCF1NicijYEb/HlZdI4r3gXFff+sly9yeMowDGcwBl4cAVVuIMa1IHCEzzDK7w5I+fFeXc+5q0rTj5zBH/gfP4A5AuSKw==</latexit> <latexit sha1_base64="IhSYI9WFb8ehErzIbIRmqS5XTJc=">AAAB83icbVBNS8NAEJ34WetX1aOXYBE8lUQFPRa9eKxgP6CJZbPdtEt3N2F3IpTQv+HFgyJe/TPe/Ddu2xy09cHA470ZZuZFqeAGPe/bWVldW9/YLG2Vt3d29/YrB4ctk2SasiZNRKI7ETFMcMWayFGwTqoZkZFg7Wh0O/XbT0wbnqgHHKcslGSgeMwpQSsFgeEDSR6DiCHpVapezZvBXSZ+QapQoNGrfAX9hGaSKaSCGNP1vRTDnGjkVLBJOcgMSwkdkQHrWqqIZCbMZzdP3FOr9N040bYUujP190ROpDFjGdlOSXBoFr2p+J/XzTC+DnOu0gyZovNFcSZcTNxpAG6fa0ZRjC0hVHN7q0uHRBOKNqayDcFffHmZtM5r/kXNu7+s1m+KOEpwDCdwBj5cQR3uoAFNoJDCM7zCm5M5L8678zFvXXGKmSP4A+fzBxmWkbc=</latexit>=<latexit sha1_base64="2wsinhV7OEj9020G2B+xBypL2+k=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokKehGKXjy2YD+gDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWM7qZ+6wmV5rF8MOME/YgOJA85o8ZK9ZteqexW3BnIMvFyUoYctV7pq9uPWRqhNExQrTuemxg/o8pwJnBS7KYaE8pGdIAdSyWNUPvZ7NAJObVKn4SxsiUNmam/JzIaaT2OAtsZUTPUi95U/M/rpCa89jMuk9SgZPNFYSqIicn0a9LnCpkRY0soU9zeStiQKsqMzaZoQ/AWX14mzfOKd1Fx65fl6m0eRwGO4QTOwIMrqMI91KABDBCe4RXenEfnxXl3PuatK04+cwR/4Hz+AI13jMM=</latexit>+<latexit sha1_base64="26BDQsRl0AvWjqpXxBRvcak+khY=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSIIQklU0GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+Oyura+sbm4Wt4vbO7t5+6eCwqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3dRvPaHSPJYPZpygH9GB5CFn1Fipft4rld2KOwNZJl5OypCj1it9dfsxSyOUhgmqdcdzE+NnVBnOBE6K3VRjQtmIDrBjqaQRaj+bHTohp1bpkzBWtqQhM/X3REYjrcdRYDsjaoZ60ZuK/3md1IQ3fsZlkhqUbL4oTAUxMZl+TfpcITNibAllittbCRtSRZmx2RRtCN7iy8ukeVHxLitu/apcvc3jKMAxnMAZeHANVbiHGjSAAcIzvMKb8+i8OO/Ox7x1xclnjuAPnM8fci+MsQ==</latexit>=<latexit sha1_base64="2wsinhV7OEj9020G2B+xBypL2+k=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokKehGKXjy2YD+gDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWM7qZ+6wmV5rF8MOME/YgOJA85o8ZK9ZteqexW3BnIMvFyUoYctV7pq9uPWRqhNExQrTuemxg/o8pwJnBS7KYaE8pGdIAdSyWNUPvZ7NAJObVKn4SxsiUNmam/JzIaaT2OAtsZUTPUi95U/M/rpCa89jMuk9SgZPNFYSqIicn0a9LnCpkRY0soU9zeStiQKsqMzaZoQ/AWX14mzfOKd1Fx65fl6m0eRwGO4QTOwIMrqMI91KABDBCe4RXenEfnxXl3PuatK04+cwR/4Hz+AI13jMM=</latexit>+<latexit sha1_base64="26BDQsRl0AvWjqpXxBRvcak+khY=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSIIQklU0GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+Oyura+sbm4Wt4vbO7t5+6eCwqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3dRvPaHSPJYPZpygH9GB5CFn1Fipft4rld2KOwNZJl5OypCj1it9dfsxSyOUhgmqdcdzE+NnVBnOBE6K3VRjQtmIDrBjqaQRaj+bHTohp1bpkzBWtqQhM/X3REYjrcdRYDsjaoZ60ZuK/3md1IQ3fsZlkhqUbL4oTAUxMZl+TfpcITNibAllittbCRtSRZmx2RRtCN7iy8ukeVHxLitu/apcvc3jKMAxnMAZeHANVbiHGjSAAcIzvMKb8+i8OO/Ox7x1xclnjuAPnM8fci+MsQ==</latexit>˜Dl <latexit sha1_base64="9RYtKY8knJs6W170rzf1DpcOb5w=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEVyVRQZdFXbisYG2hCWEyuWmHTh7MTIQSAv6KGxeKuPU73Pk3TtostPXAwOGce7lnjp9yJpVlfRu1peWV1bX6emNjc2t7x9zde5BJJih0acIT0feJBM5i6CqmOPRTASTyOfT88XXp9x5BSJbE92qSghuRYcxCRonSkmceOIrxAHInImpECc9visLjntm0WtYUeJHYFWmiCh3P/HKChGYRxIpyIuXAtlLl5kQoRjkUDSeTkBI6JkMYaBqTCKSbT+MX+FgrAQ4ToV+s8FT9vZGTSMpJ5OvJMqWc90rxP2+QqfDSzVmcZgpiOjsUZhyrBJdd4IAJoIpPNCFUMJ0V0xERhCrdWEOXYM9/eZE8nLbss5Z1d95sX1V11NEhOkInyEYXqI1uUQd1EUU5ekav6M14Ml6Md+NjNlozqp199AfG5w/6qpYi</latexit>Dl <latexit sha1_base64="JqJ2p6HAoVeMhWWCi63kgDArko0=">AAAB9HicbVDLSgMxFL3js9ZX1aWbYBFclRkVdFnUhcsK9gHtUO6kaRuayYxJplCGfocbF4q49WPc+Tdm2llo64HA4Zx7uScniAXXxnW/nZXVtfWNzcJWcXtnd2+/dHDY0FGiKKvTSESqFaBmgktWN9wI1ooVwzAQrBmMbjO/OWZK80g+mknM/BAHkvc5RWMlvxOiGVIU6d20K7qlsltxZyDLxMtJGXLUuqWvTi+iScikoQK1bntubPwUleFUsGmxk2gWIx3hgLUtlRgy7aez0FNyapUe6UfKPmnITP29kWKo9SQM7GQWUi96mfif105M/9pPuYwTwySdH+ongpiIZA2QHleMGjGxBKniNiuhQ1RIje2paEvwFr+8TBrnFe+i4j5clqs3eR0FOIYTOAMPrqAK91CDOlB4gmd4hTdn7Lw4787HfHTFyXeO4A+czx/8d5I7</latexit>˜Dl0 <latexit sha1_base64="rwWadXt/FaW9mUSdqo9ZqXlr4D8=">AAACAXicbVBNS8NAEJ34WetX1IvgJVhETyVRQY9FPXisYD+gCWGz2bZLN5uwuxFKiBf/ihcPinj1X3jz37hpc9DWBwOP92aYmRckjEpl29/GwuLS8spqZa26vrG5tW3u7LZlnApMWjhmsegGSBJGOWkpqhjpJoKgKGCkE4yuC7/zQISkMb9X44R4ERpw2qcYKS355r6rKAtJ5kZIDTFi2U2e+xk7zn2zZtftCax54pSkBiWavvnlhjFOI8IVZkjKnmMnysuQUBQzklfdVJIE4REakJ6mHEVEetnkg9w60kpo9WOhiytrov6eyFAk5TgKdGdxqJz1CvE/r5eq/qWXUZ6kinA8XdRPmaViq4jDCqkgWLGxJggLqm+18BAJhJUOrapDcGZfnift07pzVrfvzmuNqzKOChzAIZyAAxfQgFtoQgswPMIzvMKb8WS8GO/Gx7R1wShn9uAPjM8fOG+XXw==</latexit>˜Dl <latexit sha1_base64="9RYtKY8knJs6W170rzf1DpcOb5w=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEVyVRQZdFXbisYG2hCWEyuWmHTh7MTIQSAv6KGxeKuPU73Pk3TtostPXAwOGce7lnjp9yJpVlfRu1peWV1bX6emNjc2t7x9zde5BJJih0acIT0feJBM5i6CqmOPRTASTyOfT88XXp9x5BSJbE92qSghuRYcxCRonSkmceOIrxAHInImpECc9visLjntm0WtYUeJHYFWmiCh3P/HKChGYRxIpyIuXAtlLl5kQoRjkUDSeTkBI6JkMYaBqTCKSbT+MX+FgrAQ4ToV+s8FT9vZGTSMpJ5OvJMqWc90rxP2+QqfDSzVmcZgpiOjsUZhyrBJdd4IAJoIpPNCFUMJ0V0xERhCrdWEOXYM9/eZE8nLbss5Z1d95sX1V11NEhOkInyEYXqI1uUQd1EUU5ekav6M14Ml6Md+NjNlozqp199AfG5w/6qpYi</latexit>Dl <latexit sha1_base64="JqJ2p6HAoVeMhWWCi63kgDArko0=">AAAB9HicbVDLSgMxFL3js9ZX1aWbYBFclRkVdFnUhcsK9gHtUO6kaRuayYxJplCGfocbF4q49WPc+Tdm2llo64HA4Zx7uScniAXXxnW/nZXVtfWNzcJWcXtnd2+/dHDY0FGiKKvTSESqFaBmgktWN9wI1ooVwzAQrBmMbjO/OWZK80g+mknM/BAHkvc5RWMlvxOiGVIU6d20K7qlsltxZyDLxMtJGXLUuqWvTi+iScikoQK1bntubPwUleFUsGmxk2gWIx3hgLUtlRgy7aez0FNyapUe6UfKPmnITP29kWKo9SQM7GQWUi96mfif105M/9pPuYwTwySdH+ongpiIZA2QHleMGjGxBKniNiuhQ1RIje2paEvwFr+8TBrnFe+i4j5clqs3eR0FOIYTOAMPrqAK91CDOlB4gmd4hTdn7Lw4787HfHTFyXeO4A+czx/8d5I7</latexit>l<latexit sha1_base64="E5Kc1ZKr520j8ga7QDzfGA0mefk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUEP1yxa26c5BV4uWkAjnq/fJXbxCzNEJpmKBadz03MX5GleFM4LTUSzUmlI3pELuWShqh9rP5oVNyZpUBCWNlSxoyV39PZDTSehIFtjOiZqSXvZn4n9dNTXjjZ1wmqUHJFovCVBATk9nXZMAVMiMmllCmuL2VsBFVlBmbTcmG4C2/vEpaF1Xvsuo2riq12zyOIpzAKZyDB9dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/gfP4A1LOM8g==</latexit>l0 <latexit sha1_base64="YQeNZuh1u9q5GjBC/MYKCXfr8No=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lU0GPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSATXxnW/ncLK6tr6RnGztLW9s7tX3j9o6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0O/VbT6g0j+WjGSfoR3QgecgZNVZ6EKe9csWtujOQZeLlpAI56r3yV7cfszRCaZigWnc8NzF+RpXhTOCk1E01JpSN6AA7lkoaofaz2aUTcmKVPgljZUsaMlN/T2Q00nocBbYzomaoF72p+J/XSU147WdcJqlByeaLwlQQE5Pp26TPFTIjxpZQpri9lbAhVZQZG07JhuAtvrxMmudV76Lq3l9Wajd5HEU4gmM4Aw+uoAZ3UIcGMAjhGV7hzRk5L8678zFvLTj5zCH8gfP5AzUujSM=</latexit>l<latexit sha1_base64="E5Kc1ZKr520j8ga7QDzfGA0mefk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUEP1yxa26c5BV4uWkAjnq/fJXbxCzNEJpmKBadz03MX5GleFM4LTUSzUmlI3pELuWShqh9rP5oVNyZpUBCWNlSxoyV39PZDTSehIFtjOiZqSXvZn4n9dNTXjjZ1wmqUHJFovCVBATk9nXZMAVMiMmllCmuL2VsBFVlBmbTcmG4C2/vEpaF1Xvsuo2riq12zyOIpzAKZyDB9dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/gfP4A1LOM8g==</latexit>l0 <latexit sha1_base64="YQeNZuh1u9q5GjBC/MYKCXfr8No=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lU0GPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSATXxnW/ncLK6tr6RnGztLW9s7tX3j9o6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0O/VbT6g0j+WjGSfoR3QgecgZNVZ6EKe9csWtujOQZeLlpAI56r3yV7cfszRCaZigWnc8NzF+RpXhTOCk1E01JpSN6AA7lkoaofaz2aUTcmKVPgljZUsaMlN/T2Q00nocBbYzomaoF72p+J/XSU147WdcJqlByeaLwlQQE5Pp26TPFTIjxpZQpri9lbAhVZQZG07JhuAtvrxMmudV76Lq3l9Wajd5HEU4gmM4Aw+uoAZ3UIcGMAjhGV7hzRk5L8678zFvLTj5zCH8gfP5AzUujSM=</latexit>l0 <latexit sha1_base64="YQeNZuh1u9q5GjBC/MYKCXfr8No=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lU0GPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSATXxnW/ncLK6tr6RnGztLW9s7tX3j9o6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0O/VbT6g0j+WjGSfoR3QgecgZNVZ6EKe9csWtujOQZeLlpAI56r3yV7cfszRCaZigWnc8NzF+RpXhTOCk1E01JpSN6AA7lkoaofaz2aUTcmKVPgljZUsaMlN/T2Q00nocBbYzomaoF72p+J/XSU147WdcJqlByeaLwlQQE5Pp26TPFTIjxpZQpri9lbAhVZQZG07JhuAtvrxMmudV76Lq3l9Wajd5HEU4gmM4Aw+uoAZ3UIcGMAjhGV7hzRk5L8678zFvLTj5zCH8gfP5AzUujSM=</latexit>l0 <latexit sha1_base64="YQeNZuh1u9q5GjBC/MYKCXfr8No=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lU0GPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSATXxnW/ncLK6tr6RnGztLW9s7tX3j9o6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0O/VbT6g0j+WjGSfoR3QgecgZNVZ6EKe9csWtujOQZeLlpAI56r3yV7cfszRCaZigWnc8NzF+RpXhTOCk1E01JpSN6AA7lkoaofaz2aUTcmKVPgljZUsaMlN/T2Q00nocBbYzomaoF72p+J/XSU147WdcJqlByeaLwlQQE5Pp26TPFTIjxpZQpri9lbAhVZQZG07JhuAtvrxMmudV76Lq3l9Wajd5HEU4gmM4Aw+uoAZ3UIcGMAjhGV7hzRk5L8678zFvLTj5zCH8gfP5AzUujSM=</latexit>l<latexit sha1_base64="E5Kc1ZKr520j8ga7QDzfGA0mefk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUEP1yxa26c5BV4uWkAjnq/fJXbxCzNEJpmKBadz03MX5GleFM4LTUSzUmlI3pELuWShqh9rP5oVNyZpUBCWNlSxoyV39PZDTSehIFtjOiZqSXvZn4n9dNTXjjZ1wmqUHJFovCVBATk9nXZMAVMiMmllCmuL2VsBFVlBmbTcmG4C2/vEpaF1Xvsuo2riq12zyOIpzAKZyDB9dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/gfP4A1LOM8g==</latexit>l<latexit sha1_base64="E5Kc1ZKr520j8ga7QDzfGA0mefk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUEP1yxa26c5BV4uWkAjnq/fJXbxCzNEJpmKBadz03MX5GleFM4LTUSzUmlI3pELuWShqh9rP5oVNyZpUBCWNlSxoyV39PZDTSehIFtjOiZqSXvZn4n9dNTXjjZ1wmqUHJFovCVBATk9nXZMAVMiMmllCmuL2VsBFVlBmbTcmG4C2/vEpaF1Xvsuo2riq12zyOIpzAKZyDB9dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/gfP4A1LOM8g==</latexit>˜Dl <latexit sha1_base64="9RYtKY8knJs6W170rzf1DpcOb5w=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEVyVRQZdFXbisYG2hCWEyuWmHTh7MTIQSAv6KGxeKuPU73Pk3TtostPXAwOGce7lnjp9yJpVlfRu1peWV1bX6emNjc2t7x9zde5BJJih0acIT0feJBM5i6CqmOPRTASTyOfT88XXp9x5BSJbE92qSghuRYcxCRonSkmceOIrxAHInImpECc9visLjntm0WtYUeJHYFWmiCh3P/HKChGYRxIpyIuXAtlLl5kQoRjkUDSeTkBI6JkMYaBqTCKSbT+MX+FgrAQ4ToV+s8FT9vZGTSMpJ5OvJMqWc90rxP2+QqfDSzVmcZgpiOjsUZhyrBJdd4IAJoIpPNCFUMJ0V0xERhCrdWEOXYM9/eZE8nLbss5Z1d95sX1V11NEhOkInyEYXqI1uUQd1EUU5ekav6M14Ml6Md+NjNlozqp199AfG5w/6qpYi</latexit>˜Dl <latexit sha1_base64="9RYtKY8knJs6W170rzf1DpcOb5w=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEVyVRQZdFXbisYG2hCWEyuWmHTh7MTIQSAv6KGxeKuPU73Pk3TtostPXAwOGce7lnjp9yJpVlfRu1peWV1bX6emNjc2t7x9zde5BJJih0acIT0feJBM5i6CqmOPRTASTyOfT88XXp9x5BSJbE92qSghuRYcxCRonSkmceOIrxAHInImpECc9visLjntm0WtYUeJHYFWmiCh3P/HKChGYRxIpyIuXAtlLl5kQoRjkUDSeTkBI6JkMYaBqTCKSbT+MX+FgrAQ4ToV+s8FT9vZGTSMpJ5OvJMqWc90rxP2+QqfDSzVmcZgpiOjsUZhyrBJdd4IAJoIpPNCFUMJ0V0xERhCrdWEOXYM9/eZE8nLbss5Z1d95sX1V11NEhOkInyEYXqI1uUQd1EUU5ekav6M14Ml6Md+NjNlozqp199AfG5w/6qpYi</latexit>(a)<latexit sha1_base64="AE0Nq7v1HuaVpS5n46o4fgzmSnw=">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxhoSXRxhKjIAlcyNyyBxv29i67eybkwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRW8epoqxFYxGrToCaCS5Zy3AjWCdRDKNAsMdgfDPzH5+Y0jyWD2aSMD/CoeQhp2isdF/F83654tbcOcgq8XJSgRzNfvmrN4hpGjFpqECtu56bGD9DZTgVbFrqpZolSMc4ZF1LJUZM+9n81Ck5s8qAhLGyJQ2Zq78nMoy0nkSB7YzQjPSyNxP/87qpCa/8jMskNUzSxaIwFcTEZPY3GXDFqBETS5Aqbm8ldIQKqbHplGwI3vLLq6Rdr3kXNfeuXmlc53EU4QROoQoeXEIDbqEJLaAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QOItY1K</latexit>(b)<latexit sha1_base64="GKSjSQBQIgw43TqwKo3Rk5L2ahM=">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxhoSXRxhKjIAlcyN4yBxv29i67eybkwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRW8epYthisYhVJ6AaBZfYMtwI7CQKaRQIfAzGNzP/8QmV5rF8MJME/YgOJQ85o8ZK99XgvF+uuDV3DrJKvJxUIEezX/7qDWKWRigNE1Trrucmxs+oMpwJnJZ6qcaEsjEdYtdSSSPUfjY/dUrOrDIgYaxsSUPm6u+JjEZaT6LAdkbUjPSyNxP/87qpCa/8jMskNSjZYlGYCmJiMvubDLhCZsTEEsoUt7cSNqKKMmPTKdkQvOWXV0m7XvMuau5dvdK4zuMowgmcQhU8uIQG3EITWsBgCM/wCm+OcF6cd+dj0Vpw8plj+APn8weKOo1L</latexit>(c)<latexit sha1_base64="E6bvoD8cCKqyBu1OQ/TffOex1Mg=">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxhoSXRxhKjIAlcyN4yBxv29i67eybkwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRW8epYthisYhVJ6AaBZfYMtwI7CQKaRQIfAzGNzP/8QmV5rF8MJME/YgOJQ85o8ZK91V23i9X3Jo7B1klXk4qkKPZL3/1BjFLI5SGCap113MT42dUGc4ETku9VGNC2ZgOsWuppBFqP5ufOiVnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhFd+xmWSGpRssShMBTExmf1NBlwhM2JiCWWK21sJG1FFmbHplGwI3vLLq6Rdr3kXNfeuXmlc53EU4QROoQoeXEIDbqEJLWAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QOLv41M</latexit> FIG. 1. (a) Illustration of the self-consistency equation for the diffu- son, Eq. (4). Solid and dashed lines symbolize retarded and advanced Green’s functions, respectively. Dotted lines with black circles rep- resent tunneling processes. (b) The Bethe-Salpeter equation for the auxiliary diffuson, Eq. (5). The dotted lines with crosses represent disorder scattering. (c) The diffuson Dlcan be used to calculate the dynamical part of the response function dyn las shown. The junctionsJl0 l=Plt2Pl0account for the tunneling pro- cesses. The expressions of P2dandPTIare given in the SM [34]. For the purpose of finding dyn l, it is convenient to solve Eq. (4) in the spin-charge representation. To this end the dif- fusons, as well as the junctions, are contracted with the Pauli matrices asD 2d=1 2 s1s2Ds1s2;s3s4 2d s3s4where; = (0;x;y;z )correspond to the charge and x;y;z components of the spin, respectively. With the knowledge of Dl, the dynami- cal part of the spin-density response function can be found by introducing charge and spin vertices as illustrated in Fig. 1(c). The full response function is then obtained by adding the static part,l=st l+dyn l, wherest; l/l. For systems with conserved particle number, the density response function 00must satisfy the condition lim!!0(limq!000(q;!)) = 0. In the problem under consideration, electrons can move from one layer to the other. Therefore, a complete description of the time evolution of the charge and spin densities must in- clude the mixed response function ll0withl6=l0, i.e. the response of densities in layer lto perturbations in layer l0.ll0 can be found in analogy to l. In the 2DEG, the charge and spin response to external per- turbations in the form of electric potentials or Zeeman fields may be conveniently cast in the form of coupled transport equations. In the diffusive limit, we find @tn2d=Dr2~n2d+ nslTI(^zr)~s2d@t(V2dVTI) (7) @ts2d= D2dr2t 2d
~s2d+t 2dlTI(^zr)~s2d + t 2dlTI(^zr) [lTI(r~s2d)z+ ~n2d=2] (8) where the effective charge diffusion constant D=t 2dDTI+ t TID2d t 2d+ t TI; t l=1 t l; (9) is a weighted average of the diffusion constants D2d= v2 F0 2d=2, andDTI=v2 TI0 TIin the 2DEG and TI, re- spectively. Moreover, lTI=vTI0 TIis the TI mean free path. The spin-charge coupling in the 2DEG is character- ized by ns= 2t 2dt TI=(t 2d+ t TI). The term containingthe dimensionless constant =F0 2d=(22TIDTI)origi- nates from the induced spin-orbit coupling in the 2DEG. The charge and spin densities ~nand~sappearing on the right hand side of the diffusion equations include external driving po- tentials for the charge, V2d, and spin, h2d, respectively, as ~n2d=n2d+ 22dV2dand~s=s22dh2d. The last term in Eq. (7) accounts for a potential loss of electrons in the 2DEG for a dynamically driven system, with coefficient = 22dt 2d=(t 2d+ t TI). Equations (7)-(9) are the main result of this work. They show that in a 2DEG-TI system charge transport and spin transport are coupled even when the tunneling between the two systems is random. Notice that Eqs. (7)-(9) were ob- tained in the limit in which t l=0 l1, and!1,being the longest relaxation time: = max(t 2d;t TI). Eqs. (7)-(9) can only describe transport over time scales much larger than and therefore are not valid in the limit t= 0 for which !1 . Fort= 0the two systems are decoupled and for the 2DEG the diffusive transport of charge and spin are indepen- dent withD2d=v2 2d0 2d=2. It is instructive to note that there are two mechanisms responsible for the spin-charge and spin-spin coupling in Eqs. (7) and (8). The term with coefficient in Eq. (8) results from the real part of the self-energy in Eq. (1), i.e. from the tunneling-induced spin-orbit coupling in the effective single- particle Hamiltonian of the 2DEG. This term couples in-plane and out-of-plane spin components. The spin-charge coupling in Eqs. (7) and Eqs. (8) has a different origin. The surface of the TI hosts a single gapless diffusion mode in the absence of tunneling, as can be seen by diagonalizing the diffuson [26, 32, 33]. For finite q, this mode has a non-trivial spin structure. By means of the random tunneling, this mode and the gapless modes in the 2DEG hybridize. The hybridization gives rise to spin-charge coupling via the term with coeffi- cient nsin Eq. (7) and the final term in Eq. (8), as well as to anisotropic spin-diffusion encoded in the first term of the second line in Eq. (8). To leading order in tunneling, the two described mechanisms for spin-charge coupling are indepen- dent of each other. As follows from Ref. [25], spin-orbit cou- pling eventually also leads to spin-charge coupled transport at higher orders in the coupling strength. A separate conse- quence of the tunneling in Eq. (8) is that, since spin is not conserved in the coupled system, a gap of size t 2dopens for the spin diffusion modes. Equations (7), (8) show that the strength of the coupling between charge transport and spin transport, and the spin- diffusion anisotropy, are proportional to the ratio t 2d=0 2d. Given that t 2d=t2TI, and thatTIscales linearly with TI, we see that both in the 2DEG both the spin-charge cou- pling and the spin-diffusion anisotropy can be tuned simply by changing the doping of the TI’s surface. We now study the solution of Eqs. (7), (8) for a simple setup, as in Refs. 25 and 26, to highlight some of the transports effects due to the coupling between spin and charge transport described by Eqs. (7), (8), and to highlight some of the main similarities and differences between a 2DEG-TI system, a TI’s surface,4 and a 2DEG with Rashba SOC. We consider a system of size Lalongx,L=2<x<L= 2, and in which all the quantities are uniform along y. In the stationary limit, due to the uni- formity along y, Eqs. (7), (8) separate in two independent sets of equations: one set describing the coupled transport of n andsy, one set describing the coupled transport of sxandsz. Given that we are interested in the coupling between charge and spin transport, we focus on the first set. Due to the as- sumption that all the quantities are homogenous along y, the coupled equations for nandsyfor a 2DEG-TI, a TI, and a 2DEG with Rashba SOC have the same structure: Dn@2 xn+ 2s@xsy= 0; (10) Ds@2 xsysy s+n@xn= 0: (11) whereDn,Ds,n, andsare constants whose expression in terms of the parameters characterizing the system are given in Table I for a 2DEG-TI, a TI, and a 2DEG with Rashba SOC. From charge conservation, using Eq. (7), we find that the charge current takes the form J=Drn2dnslTI(sx 2d^y sy 2d^x);and for the simple case described by Eq. (10), J=J^x, J=Dndn=dx + 2ssy, withDnandsgiven in Table I. Similarly from Eq. (11) we can obtain an expression for the current ofsy. This expression has the term n@xn, however, as pointed out before [35–39], such term describes an equilib- rium spin current and therefore should not be included in the definition of an externally driven spin current. Knowing the expression of Jand of the spin current allows us to write the boundary conditions for Eqs. (10), (11), corresponding to the situation when a charge current Iis injected at x=L=2via a ferromagnetic electrode so that the incoming electrons have a net spin polarization alongsy: Jjx=L 2=I e; Ds@xsyjx=L 2=I e; Ds@xsyjx=L 2= 0; (12) Recalling that the voltage drop
V(x)[40] at position xis given by
V(x) =(1=2e)Rx L=2dx0(dn=dx0), and solv- ing Eqs. (10), (11) with the boundary conditions (12) we find sy(x) =Il eDscosh ((xL=2)=l) sinh (L=l)l2 nI eDnDs(13) and the voltage drop between the leads
V=I 2e2Dn2l2 s Ds nL Dn +L : (14) In Eqs. (13), (14) l2 1=(sDs) + 2ns=(DnDs). Us- ing the expressions given in Table I for Dn,Ds,s,n, and s, Eqs. (13) and (14) for a 2D-TI system become, to leading order in the tunneling amplitude (with lp D2dt 2d) sy(x) =Il eD2dcosh [(xL=2)=l] sinh (L=l)IlTI 2eD; (15)
V=I 2e22dD L+ 2lTIt TI t 2d+ t TI : (16)2D+TI TI Rashba DnDv2 TI0 TI=2v2 R0 R=2 DsD2d+ t 2dl2 TI3Dn=2Dn n1 2t 2dlTIvTI=2(kF0 R)2 s1 2nslTIvTI=2 2n st 2d0 TI 20 R=(2kF0 R)2 TABLE I. Diffusion coefficients for a TI, Rashba 2DEG, and 2D+TI. is the SOC strength in the Rashba 2DEG, and 0 Rthe Rashba scat- tering time. The second term on the r.h.s. of Eq. (15) shows that, as in the case of 2DEG with Rashba SOC [25] and a TI [26], an Edelstein [41] effect is present, i.e., a constant nonequilib- rium spin polarization generated by a charge current I. This effect is present due to the “mirroring” into the 2DEG of the TI’s gapless diffusion mode characterized by the coupling of charge and spin. It is interesting to notice that for a 2DEG- TI system such term, as long as t 2d0 2dto remain in the regime of validity of the diffusion equations (7), (8), is inde- pendent of the interlayer tunneling strength. This is due to the fact that in the 2DEG-TI van der Waals structure, in the 2DEG layer, both the spin relaxation rate, 1=s, and the spin-charge couplingnin Eq. (11) scale as t2. As a consequence we expect that even in the limit of very small ta significant Edel- stein effect should be present in a metallic 2D layer placed in proximity of a system with significant SOC such as a TI’s surface. In addition, we see that for a 2DEG-TI system, con- trary to a TI, the strength of the Edelstein effect can be tuned by varying the doping, and therefore TI, of the TI’s surface. The other important result is that the decay length of syisl that can also be tuned by varying the doping in the TI, and that can be very long in the weak tunneling regime, for wich t 2d0 2d. The last term on the r.h.s. of Eq. (16) is a mag- netoresistance contribution to the voltage drop due to the cou- pling of the charge and spin transport. For a 2DEG-TI system this term is therefore dependent on the relative strength of the disorder in the TI and 2DEG. In conclusion, we have studied the electron and spin trans- port in a van der Waals system formed by one layer with strong spin-orbit coupling and a second layer without spin- orbit coupling, in the regime when the interlayer tunneling is random, and shown that in the layer without intrinsic spin- orbit coupling spin-charge coupled transport can be induced by the hybridization of the diffusion modes of the two iso- lated layers. To exemplify the mechanism we have studied a van der Waals system formed by a 2DEG and TI’s surface and shown how the coupling of the spin and charge transport in the TI is “mirrored” into the 2DEG. In addition, for the specific case of a 2DEG-TI van der Waals system, we show that a spin-orbit coupling term is induced into the 2DEG, and that the induced coupling of spin and charge transport in the 2DEG can be tuned by varying the TI’s doping. Finally we showed how the coupled spin-charge transport described by the diffusive equations that we obtain for the 2DEG leads to5 a current-induced non-equilibrium spin accumulation and a magnetoresistance effect that are also tunable by changing the TI’s doping. We thank Ion Garate for useful discussions. This work was supported in part by the US-Israel Binational Science Foundation grants No. 2014345 (M.R.V .), the NSF CAREER grant DMR-1350663 (M.R.V .), the NSF Materials Research Science and Engineering Center Grant No. DMR-1720595 (M.R.V .), the College of Arts and Sciences at the University of Alabama (G.S.), and the National Science Foundation under Grant No. DMR-1742752 (G.S.). ER acknowledges support from NSF CAREER grant No. DMR-1455233, ONR grant No. N00014-16-1-3158, and ARO grant No. W911NF-18-1- 0290. E.R. thanks the Aspen Center for Physics, which is sup- ported by National Science Foundation grant PHY-1607611, for its hospitality while part of this work was performed. [1] A. K. Geim and I. V . Grigorieva, Nature 499, 419 (2013). [2] J. Lopes dos Santos, N. Peres, and A. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007). [3] E. J. Mele, Phys. Rev. B 81, 161405(R) (2010). [4] R. Bistritzer and A. H. MacDonald, Proc. National Acad. Sci- ences United States Am. 108, 12233 (2011). [5] C.-P. Lu et al. , PNAS 113, 6623 (2016). [6] Y . Cao, V . Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxi- ras, and P. Jarillo-Herrero, Nature 556, 43 (2018). [7] M. Yankowitz, S. Chen, H. Polshyn, Y . Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Science (2019). [8] K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Cas- tro Neto, Science 353(2016). [9] G.-H. Lee, Y .-J. Yu, C. Lee, C. Dean, K. L. Shepard, P. Kim, and J. Hone, Applied Physics Letters 99, 243114 (2011). [10] A. Woessner, M. B. Lundeberg, Y . Gao, A. Principi, P. Alonso- Gonz ´alez, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, Na- ture Materials 14, 421 (2014). [11] K. Roy, M. Padmanabhan, S. Goswami, T. P. Sai, G. Rama- lingam, S. Raghavan, and A. Ghosh, Nature Nanotechnology 8, 826 (2013). [12] I. ˇZuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [13] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). [14] C. Cardoso, D. Soriano, N. A. Garcia-Martinez, and J. Fernandez-Rossier, Phys. Rev. Lett. 121, 067701 (2018). [15] J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, Nature Reviews Materials 1, 16055 (2016). [16] W. Dang, H. Peng, H. Li, P. Wang, and Z. Liu, Nano Letters 10, 2870 (2010). [17] L. Kou, B. Yan, F. Hu, S.-C. Wu, T. O. Wehling, C. Felser, C. Chen, and T. Frauenheim, Nano letters 13, 6251 (2013). [18] K.-H. Jin and S.-H. Jhi, Phys. Rev. B 87, 075442 (2013). [19] J. Zhang, C. Triola, and E. Rossi, Phys. Rev. Lett. 112, 096802 (2014). [20] M. Rodriguez-Vega, J. Fischer, S. Das Sarma, and E. Rossi, Phys. Rev. B 90, 035406 (2014).[21] L. Zhang, Y . Yan, H.-C. Wu, D. Yu, and Z.-M. Liao, ACS Nano 10, 3816 (2016). [22] M. Rodriguez-Vega, G. Schwiete, J. Sinova, and E. Rossi, Phys. Rev. B 96, 235419 (2017). [23] K. Song, D. Soriano, A. W. Cummings, R. Robles, P. Ordej ´on, and S. Roche, Nano Letters 18, 2033 (2018). [24] D. Khokhriakov, A. W. Cummings, K. Song, M. Vila, B. Karpiak, A. Dankert, S. Roche, and S. P. Dash, Science Advances 4(2018). [25] A. A. Burkov, A. S. N ´u˜nez, and A. H. MacDonald, Phys. Rev. B70, 155308 (2004). [26] A. A. Burkov and D. G. Hawthorn, Phys. Rev. Lett. 105, 066802 (2010). [27] K. Shen, R. Raimondi, and G. Vignale, Phys. Rev. B 90, 245302 (2014). [28] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [29] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). [30] A. Sakai and H. Kohno, Phys. Rev. B 89, 165307 (2014). [31] J. Fujimoto, A. Sakai, and H. Kohno, Phys. Rev. B 87, 085437 (2013). [32] I. Garate and L. Glazman, Phys. Rev. B 86, 035422 (2012). [33] H. Velkov, G. N. Bremm, T. Micklitz, and G. Schwiete, Phys. Rev. B 98, 165408 (2018). [34] See supplemental material. [35] E. I. Rashba, Phys. Rev. B 68, 241315(R) (2003). [36] A. G. Mal’shukov, L. Y . Wang, C. S. Chu, and K. A. Chao, Phys. Rev. Lett. 95, 146601 (2005). [37] V . M. Galitski, A. A. Burkov, and S. Das Sarma, Phys. Rev. B 74, 115331 (2006). [38] O. Bleibaum, Phys. Rev. B 73, 035322 (2006). [39] Y . Tserkovnyak, B. I. Halperin, A. A. Kovalev, and A. Brataas, Phys. Rev. B 76, 085319 (2007). [40] Here we make the electric charge explicit, Vn=eV. [41] V . Edelstein, Solid State Communications 73, 233 (1990). [42] G. Bergmann, Phys. Rev. B 39, 11280 (1989).6 SUPPLEMENTAL MATERIAL Expression of quantum probabilities Here, it is convenient to define l= 1=lforl= 0 l+ t land to display formulas for ~Pl= (0 l=l)Pl. In the limit t l=0 l1and0 l=F1, to leading order in !=0 land vFq=0 lwe find: ~P2d(q;!) =~a2d(q;!)0 0+~ba 2d(q;!)0 a + ~ca 2d(q;!)a 0; (17) ~PTI(q;!) =~aTI(q;!)0 0+~ba TI(q;!) 0 a+ a 0
+~dab TI(q;!)a b: (18) For the 2DEG: ~a2d(q;!)1 +i!2d2d~D2dq2(19) ~bx 2d(q;!)2dt 2d~lTIqy=4 =~cx 2d(q;!) (20) ~by 2d(q;!)2dt 2d~lTIqx=4 =~cy 2d(q;!) (21) where ~D2d=v2 2d2d=2and~l2d=v2d2d. For the TI’s surface [26] ~aTI= 1TI~DTIq2+i!TI =2; (22) ~bx TI=i~lTIqy=4; by TI=i~lTIqx=4; (23) ~dxx TI= 1TI~DTI(q2 x+ 3q2 y)=2 +i!TI =4; (24) ~dyy TI= 1TI~DTI(3q2 x+q2 y)=2 +i!TI =4; (25) dxy TI=dyx TI=TI~DTIqxqy=4; (26) where ~DTI=v2 TITI=2and~lTI=vTITI. Spin-charge diffusion equation for TI’s surface To facilitate the comparison between the results that we ob- tain in the main text for a 2DEG-TI system and an isolated TI’s surface we report here the diffusion equations for a TI’s surface, first derived in Ref. 26: @tnTI=DTIr2nTI+vTI(^zr)
~ sTI (27) @tsx TI=DTI 2@2 xsx TI+3DTI 2@2 ysx TIDTI@2 xysy TI sx TI 0 TIvTI 2@ynTI (28) @tsy TI=3DTI 2@2 xsy TI+DTI 2@2 ysy TIDTI@2 xysx TI sy TI 0 TI+vTI 2@xnTI; (29)wherenTIis the carrier density on the TI’s surface, and ~ sTI= (sx TI;sy TI). Notice that the spin densities are damped by scattering with non-magnetic impurities due to spin-orbit coupling. Due to a typo in Ref. 26 the terms with mix deriva- tives have opposite sign compared to Eqs. (28), (28). We can see that the negative sign in front of the terms @2 xysx TI,@2 xysy TI in Eqs. (28), (28) is correct by considering that when nTIis uniform in time and space so that Eq. (27) implies @ysx TI= @xsy TI, Eqs. (28) and (29) lead to @ts TI= ((1=2)DTIr2 1=0 TI)s TI, the expected spin-diffusion equation in this simple limit. Diffusion equations for two coupled 2DEGs In this appendix, we review the density diffusion equation of a 2DEG-2DEG heterostructure. The effect of the coupling in the quantum interference has been studied before [42]. Each layerlposses its own diffusion constant Dland den- sity of states l, wherel=T;B labels the top and bottom 2DEG layer respectively. We obtain @tn(T) 2d=Dr2n(T) 2d; where we have defined D=t TDB+ t BDT t T+ t B;t l=1 t l(30) The renormalized diffusion constant contains corrections pro- portional to the diffusion constant in the bottom layer. The leading corrections to the diffusion constant is given by a term proportional the ratio of the DOS in each layer. Given that there is no spin-orbit coupling, the spin follow analogous dif- fusion equations in each direction.

1404.5150v2.Graphene_with_wedge_disclination_in_the_presence_of_intrinsic_and_Rashba_spin_orbit_couplings.pdf

epl draft Graphene with wedge disclination in the presence of intrinsic and Rashba spin orbit couplings Tarun Choudhari andNivedita Deo Departement of Physics and Astrophysics, University of Delhi.Delhi-110007,India. PACS 72.10.Fk { Scattering by point defects, dislocations, surfaces, and other imperfections PACS 71.10.Pm { Fermions in reduced dimensions PACS 73.22.Pr { Electronic structure of graphene Abstract {In this article, the modied Kane-Mele Hamiltonian is derived for graphene with wedge disclination and spin orbit couplings (intrinsic and Rashba). The wedge disclination changes the at lattice into the conical lattice and hence modies the spin orbit couplings. The Hamiltonian is exactly solved for the intrinsic spin orbit interaction and perturbatively for the Rashba spin orbit interaction. It is shown that there exists the Kramer's degenerate midgap localized spin separated uxon states around the defect. These zero energy spin separated states occur at the external magnetic ux value  
. The external magnetic ux  is introduced to make the wave-function periodic when the electron circulates around the defect. It is found that this separation occurs due to the eect of the conical curvature on the spin orbit coupling. Further, we nd these results are robust to the addition of the Rashba spin orbit interaction which is important for the application to spintronics and nanoelectronics. Introduction. { "Topological Insulator (TI)" like HgTe, Graphene and Fluorinated Stanene [1, 2], are ma- terials which are insulators in bulk and have protected conducting gapless surface (3D) or edge (2D) states for electrons. This denes a new class of quantum materials called quantum spin hall insulators. These topological in- sulators are classied in terms of the symmetry, some of them preserve inversion and time reversal symmetry [3,4], while some of them have broken time reversal symmetry [5, 6]. The TI which preserves time reversal symmetry arises due to the presence of intrinsic spin orbit coupling. The robust properties of surface and edge states of topo- logical insulators are important for applications to quan- tum computers and spintronics devices hence many studies have been made in the past few years to nd new topo- logical materials and structures. The lattice with topolog- ical defects is one of them. It has been naturally [7], as well as experimentally [8,9], and theoretically seen [10{15], that 3D topological insulators (TI) and topological super- conductors (TS) show signicant response to 'Topological Defects' and have gapless states around the defect for an electron circulating it. Recently the bound state of con- ical singularity in graphene with wedge disclination and broken time reversal symmetry have been studied for the spinless electron [16], which shows the existence of local-ized zero energy bound states near the defect. The eect of Coulomb impurities in the presence of a uniform mag- netic eld on the energy spectrum of disclinated graphene has also been studied recently [17]. The present paper deals with the bound state of the electron having spin in graphene with wedge disclination in the presence of curva- ture modied intrinsic and Rashba spin orbit coupling, in order to shed light on how the spin orbit coupling aects the bound state in disclinated graphene. The analysis has been done by using the Kane-Mele Hamiltonian [4] for a spinfull electron on the honeycomb lattice. Although the Kane-Mele model (i.e.quantum spin hall insulator (QSHI) ) is dicult to realize in graphene because the intrinsic spin orbit coupling [24] in pristine graphene is too weak, about 20-50 ev, to have QSHI phase in it. This can be made large enough by having graphene absorbed with hy- drogen ad-atoms [20], heavy elements [21] or by proximity withSb2Te3TI [22]. These methods increases the intrinsic SOC to the order of 20 mev to realize QSHI. In this paper we show that there exists the Kramer's degenerate midgap localized spin uxon states [18, 19] around the defect in graphene with wedge disclination, and these zero energy up and down spin states separate out purely due to the eect of the curvature of the conical honeycomb lattice on the intrinsic spin orbit coupling in the presence of the ex- p-1arXiv:1404.5150v2 [cond-mat.mes-hall] 20 Jan 2015T.Choudhari, N.Deo ternal magnetic ux. So in this work we rst introduce the wedge disclination, derive the curvature modied intrinsic spin orbit coupling Hamiltonian and hence the Kane-Mele Hamiltonian for the conical honeycomb lattice. We then nd the bound state wave function and energy of the low energy electron in the presence of intrinsic and Rashba SOC. Wedge disclination and gauge transformation due to wedge disclination. { The low energy spinfull electron dynamics in at honeycomb lattice is governed by the massive 2D Dirac Hamiltonian, called the Kane-Mele model for honeycomb lattice [4], and is given by H(k) =f~(zxkx+yky) +
sozzsz; (1) where the,andsare the Pauli matrices for valley, sub- lattice and spin space. z=1 for the two valley K(K') in the Brioullin zone, z=1 for the sublattice A(B), sz= 1 for up and down spin of the electron. fis the Fermi velocity and
so= 3p 3t2denes the intrinsic spin orbit coupling parameter in the honeycomb lattice, t2denes the second nearest neighbour hoping amplitude. The to- tal low energy electron wave function for this Hamiltonian is (r) = [(uA";uA#;uB";uB#);(uA0";uA0#;uB0";uB0#)] (r). Whereu(A;B;A0;B0)("#)(r) are the base functions for the sublattice A(B) for momentum K, and for momen- tum K', for up and down spin of the electron. (r) is a 8x1 envelope wave function slowly varying on the lattice. The Kane-Mele Hamiltonian above [eq. (1)] has an in- trinsic spin orbit coupling Hamiltonian as the mass term, which produces a gap in the energy spectrum and is given by,Hso=
sozzsz, this Hamiltonian preserves time re- versal symmetry. In graphene, the wedge disclination is dene by a procedure called the Volterra's cut and glue procedure in which there is a removal or addition of the wedge of angle n=3 from the at lattice followed by the gluing of the edges of a cut to form a continuous lattice by preserving the three fold connectivity of the atoms. The resultant lattice then has a conical shape with apex as a hole in the form of a polygon. The index 'n' denes the type of disclination, for n>0 the wedge is removed (pos- itive curvature) and for n<0 there is an addition of the wedge (negative curvature) in the at honeycomb lattice. So there are four types of disclination in a honeycomb lat- tice which are given by the pentagon defect (n=1), the square defect (n=2), the heptagon defect (n=-1) and the octagon defect (n=-2). Graphene in nano cone form have defects with more than one pentagon and heptagon at the top of the cone. It has been found experimentally [8, 9] that these nano cones can have a single pentagon cell at the top of the cone as well which leads to the existence of the wedge disclination in graphene. To study the bound states of the electron around these defects we have used the continuum model, which introduces the curvature in- duced gauge eld in the low energy electron Dirac Hamil- tonian [eq. (1)] of the honeycomb lattice [13{15]. In the continuum model, the honeycomb lattice with disclination Fig. 1: (color online)Unfolded plane of lattice (Continuum model) with wedge of angle removed from it. is the radius of the defect hole. R() is the curve along which the spinor is rotated. At the centre the red dot represent the ux tube with magnetic ux . ( r;) are the coordinates of a point on the lattice. is considered as an unfolded plane (shown in g. 1) with a cut of angle and the hole of radius ' ' (located in place of the polygon at the top of cone) such that any point on this continuum lattice is dened by the polar coordi- nates (r,). So in the Volterra's cut and glue procedure for the disclinated honeycomb lattice, the wavefunction for the electron with spin must be single valued at the seam if this spinor is rotated around the defect core along the curveR() (g. 1). This leads to the general gauge transformation for the envelope function so as to com- pensate for the mismatch of phases of the base functions uA";uA#;uB";uB#;uA0";uA0#;uB0";uB0#. The Bloch phases are given by != ei2=3and != ei2=3. In g. 2 the distribution of phases for these base functions is shown. By matching the phases for any general disclina- tion (n), [16] the general gauge transformation, [14,15] for electrons with spin is given by (= 2n 3) =ein 6(zzso3yyso) (= 0);(2) where thesois the 2x2 identity matrix for the spin space. The cut and glue procedure transforms the at honey- comb lattice into a conical honeycomb lattice. So if we rescale the angle of unfolded plane to the new angle ==(1n=6) whereis varying from 0 to 2 so as to have conical topology in the unfolded plane then the general gauge transformation for any given disclination dened by index, n=1;2 is (= 2) =ein 6(zzso3yys0) (= 0): (3) This transformation can be broken into two singular trans- formations [16], for the envelope wave function ( ) of the electron with spin so that (= 2) =Us()Vns() (= 0), where Us() =ei 2zzso; Vns() =ein 4yyso: (4) Kane-Mele Hamiltonian for the conical honey- comb lattice. { Now consider an external magnetic ux  applied at the centre of the defect hole using magnetic p-2Bound state in graphene with wedge disclination Fig. 2: (color online)The gure shows the distribution of the phases (!= ei2=3and != ei2=3) of base function uA"and uB0"on the honeycomb lattice.For base functions uA#anduB0# the distribution of phases is same.For the 60oand 120ocut the phases of base functions are matched along the line(OM and OP) and (OM and ON) respectively. The distribution of the phases (!and !) for base function ( uA0#,uB#) and (uA0",uB") is just the complex conjugate of the above gure. ux tube (red dot in g. 1) having a vector potential A given by A= r n0^: (5) with n= (1n=6) and  0=h=ethe magnetic ux quanta. To dene the dynamics for the electron with spin in the honeycomb lattice with disclination we have used the Kane-Mele model [eq. (1)] at low energy in polar co- ordinates (r;) with a external magnetic ux . However, in this Kane-Mele model the intrinsic spin orbit coupling Hamiltonian ( Hso=
sozzsz) is changed into the cur- vature modied intrinsic spin orbit coupling ( H0 so) for the conical honeycomb lattice because the lattice curvature strongly aects the spin orbit coupling [24]. So if we con- sider the honeycomb lattice as a cone having semi-cone angle, then the coordinates of a point on its surface is dened by ( r;;z ). We derived H0 so, using the trans- formation  obtained by transforming the spin basis of the perpendicular component of the spin ( sz) on the at honeycomb lattice to the spin basis of the perpendicular component of the spin ( sn) on the conical lattice [24]. The transformation is given by  =1p 2 ei 2p1 + sin ei 2p1sin ei 2p1sinei 2p1 + sin! ;(6) The Hamiltonian H0 so= yHso is then found to be H0 so=
sozzszf() +
sozzsxcosg()
sozzsysing();(7) whereg() = cosandf() = sin, are the curvature functions and they depend upon the type of wedge discli- nation (n) through the semicone angle , sin= 1jnj=6 [23]. It is interesting to note that this curvature modied intrinsic SOC Hamiltonian has two extra spin ipping in- trinsic Rashba like terms zzsxandzzsywhich occur due to the mixing of pzwithpxandpyorbitals of thecarbon atoms present at the curved surface of the coni- cal graphene. This is in contrast to the the intrinsic SOC Hamiltonian Hsofor the at honeycomb lattice and the cylindrical nano tube SOC but is similar to the spherical honeycomb lattice SOC Hamiltonian [24]. The Kane-Mele Hamiltonian in polar coordinates is then given by HFS(k) =f~(zxkr+yk) + r n0y+H0 so;(8) wherekr=i@=@r andk=i=r n(@=@ ). Now by using the Hamiltonian H0 so[eq. (7)] in the Hamiltonian HFS[eq. (8)] and by applying the gauge transformation Us() andVns() on the Hamiltonian HFS, we nd the Hamiltonian for the spinfull electron in the honeycomb lattice with wedge disclination to be given by HDS(r;) = Uy sVy nsHFSVnsUs, HDS= kri 2r zx+ k+ r n0+n 4r nz y +
so[zzszf() +zzsxg()];(9) where ~=f= 1. We have used H0 sowith the condition = 2because==(1n=6) and for = 2,is always 2. The curvature of conical graphene manifests itself in the generalized Kane-Mele Hamiltonian through the gauge potential like term: i=2randn=4r noccurring due to the gauge transformations Us() andVns() and through the functions f() andg() occurring in the spin orbit coupling mass term (
so). In spite of the presence of Rashba like term in this Hamiltonian an exact solution can be found where as for spherical and spiral honeycomb lattice this is not the case [24], [25]. The bound state wavefunction and energy for the electron around the defect. { The Hamiltonian (HDS), is separable in the coordinates 'r' and ' '. By using the ansatz (r;) =eijX(r) where 'j' is a half inte- ger number and acts as the azimuthal angular momentum quantum number for the electron circulating around the defect core, we nd the modied Kane-Mele Hamiltonian in radial coordinates to be given by H0 DS(r) = kri 2r zx+ ry +
so[zzszf() +zzsxg()] (10) where=for the two emergent valleys and is =j+ 0+n 4 (1n 6): (11) If we solve the eigenvalue equation H0 DSX(r) =X(r), we nd the bound state spinor (r;) for the electron with spin in terms of the modied Bessel functions of the second p-3T.Choudhari, N.Deo kind which decay exponentially for r!1 , (r;) =eij2 6666666666666666666666666664K+1 2(r) (i+
sog()) (+
sof())K+1 2(r) (i
sog()) (+
sof())K++1 2(r) K++1 2(r) (i
sog()) (+
sof())K+1 2(r) K+1 2(r) K1 2(r) (i+
sog()) (+
sof())K1 2(r)3 7777777777777777777777777775: (12) where=p (
so)22. Now the square integrability of does not uniquely dene the quantized state as !0. So to nd this, we have assumed a conning potential V(r <  ) =m0(ozso) in the region r <  in place of the spin orbit coupling mass term in the Kane-Mele model [eq. (1)]. Then by making m0!1 we have found the innite mass boundary condition at r= for the spinfull electron given by (;) =M (;), where Mis a general 8x8 matrix. We have found the matrix Mof innite mass boundary condition for the spinfull electron by using the procedure given in [26] for Dirac current operator, ^J= zx^r+y^and the time reversal operator, T=i(y I sy)C;for the [eq. 1]. The matrix 'M' is given by M=z ^: sz:So the boundary condition becomes (;= 2) =z y sz (;= 2): (13) The boundary condition [eq. (13)] leads to the bound state energy of the up and down spin electron at the K and K' valleys and is given by  z
so=1 +K+ 1 2() K 1 2()s K2 + 1 2() K2  1 2()R+ (Rf2) 1 +K2 + 1 2() K2  1 2() (14) where =1 for upspin ( z=") and down ( z=#) spin state respectively. R= [(g())2+ 1]. These bound state energies depend upon the semi-cone angle , hole radius and the the parameter . Further, if we put f() = 1 andR() = 1 (i.e., g() = 0) in the eq. (14), which means the mass term
sonow becomes independent of the curvature, we nd the same result found by Ruegg and Lin [16], for the bound state energy of the spin-less electron,p
so=p
so+=K1 2()=K+1 2(); where
so=mis the mass term of the Haldane Hamilto- nian for the honeycomb lattice.Addition of Rashba Hamiltonian. { We have also included the Rashba SOC to probe whether the zero en- ergy localized states are robust against the addition of the disorder terms like the Rasbha spin orbit coupling which may arise due to the interaction with a substrate or an ex- ternal electric eld. The Rashba Hamiltonian [24], in the at honeycomb lattice is given by HR=R(zxsyysx). For the conical lattice the curvature modied Rashba Hamiltonian is found to be, H0 R= yHR;given by H0 R=R(zxsycos+zxsxsinyszg()) +R(ysxf() cosysyf() sin):(15) Note that this curvature modied Rashba SOC (which ips spins on hoping) contains a spin preserving term ysz which occurs in this Hamiltonian purely due the curvature of the conical honeycomb lattice. Now we have included this modied Rashba Hamiltonian, H0 R, in our Hamilto- nian,HDS[eq. (9)], for the honeycomb lattice with wedge disclination and the spin full electron after applying the gauge transformation Us() andVns(). The nal Hamil- tonian after addition of the modied Rashba Hamiltonian is given by, HDSR =HDS+H0 R:We observe that this Hamiltonian is not solvable analytically. So we treat the Rashba Hamiltonian H0 Ras a perturbation in the Hamil- tonianHDSbecauseR=
so<1 and used the degenerate perturbation theory to nd the energy spectrum of the bound states for the Hamiltonian HDSR. The change in energy,
due Rashba Hamiltonian is the eigenvalue of the W-matrix [27] and the total bound state energy is given by= z+
;,where the  zis the energy de- ned by eq. (14) and = 1 denes the two energy states of mixed spin states around the defect.
depends upon the curvature of the lattice through the function f() and g(). Results after addition of Rashba SOC are discussed in the later part of the next section. Results and Discussion. { For
so6= 0 and= 0, the energy spectrum for the bound state (  <
so) of the electron for up and down spin states for the pentagon (n=1), square (n=2), heptagon (n=-1) and octagon (n=- 2) defect in the two valleys, K and K' is shown in g. 3(a), which we have plotted using eq. (14) and incorporating sin= 1jnj=6, with respect to general (rst and sec- ond valley). In the g. 3(a) the continuous green curve shows the up spin energy and the dotted red curve shows the down spin state energy for pentagon defect (n=1). We nd that the up and down spin state energy monotonically varies with with a crossing at = 0. The energy spec- trum of the spin-less electron is also shown in g. 3(a) as the grey curve crossing the origin. First, we conclude from the energy spectrum (g. 3(a)), that the zero energy mode, which occurs at = 0 for spin-less electron are also present for up spin and down spin states but are now sep- arated and are at non zero value of =orespectively. Now the magnitude of odepends upon the value of exter- nal magnetic ux () and the type of disclination (n) and p-4Bound state in graphene with wedge disclination /UpArrow /DownArrow Spinless(a) |n| = 1|n| = 2 Τ = ± 1 /MiΝus1.0/MiΝus0.5 0.0 0.5 1.0/MiΝus0.50.00.51.0 ΝΕ/Slash1/CapDΕlΤaso /UpArrow /DownArrow n= 1 Τ = ± 10 .1 /Equal0 .5Ρ/CapDΕlΤaso (b) /MiΝus1.0/MiΝus0.5 0.0 0.5 1.0/MiΝus0.4/MiΝus0.20.00.20.40.60.81.0 ΝΕ/Slash1/CapDΕlΤaso /ShortUpArrow /ShortDownArrown= 1 Τ = + 1(c) /Minus1.0/Minus0.5 0.0 0.5 1.0/Minus0.4/Minus0.20.00.20.40.60.81.0 /CapPhi/Slash1/CapPhioΕ/Slash1/UpTriangleso /ShortUpArrow /ShortDownArrown= 1 Τ = − 1(d) /Minus1.0/Minus0.50.00.51.0/Minus0.4/Minus0.20.00.20.40.60.81.0 /CapPhi/Slash1/CapPhioΕ/Slash1/UpTriangleso /ShortUpArrow /ShortDownArrown= 2 Τ = ± 1(e) /Minus1.0/Minus0.5 0.0 0.5 1.0/Minus0.20.00.20.40.60.81.0 /CapPhi/Slash1/CapPhioΕ/Slash1/CapDΕlΤaso /CapGamma/Equal/Plus1 /CapGamma/Equal/Minus1n= 1 Τ = ± 1(f) /MiΝus1.0/MiΝus0.50.0 0.5 1.0/MiΝus0.4/MiΝus0.20.00.20.40.60.81.0 ΝΕ/Slash1/UpTriangleso Fig. 3: (color online)(a),The general vscurve for upspin(Green,Brown,continuous) ,downspin (Red,Blue,dotted) for all four defects and spinless electron(Grey curve passing through origin) for pentagon defect(n=1) in two K valley for
so6= 0 andR= 0. (b)vscurve for variable 
sorang- ing from 0.1 to 0.5. (c)-(e), vs =oplot for pentgon(n=1) and square (n=2) defect in two K valleys. (f), vsplot for pentgon defect(n=1) with
so6= 0 andR6= 0.All plots are for
so= 0:5. this separation purely occurs due to the eect of curvature on the intrinsic spin orbit coupling mass term
sothrough the curvature functions g() andf(). Second, from this plot we conclude that #=". Then in the expression forgiven by eq. (11) as = (j+ =0+n=4)= n, the occurrence of the term f=n=4 inis due to the wedge disclination and is called the ctitious magnetic ux experienced by the electron when it circulates around the defect core. Then for a given spin z=(";#), z,j zand external ux  the ctitious magnetic ux will be given by f so=so njso=o. Then for  = 0 and j"=j#, we get f "=f #, which implies that the ctitious spin ux (no=4) due to wedge disclinations has opposite signs for the up spin and down spin states. Third, the zero en- ergy up spin and down spin states in the two valleys are localized around the defect as can be seen from the boundstate spinor eq. (12) which decays exponentially with 'r' and the defect acts like a source of spin ux (  zno=4), we recognize these as the zero energy spin uxon states, [18,19]. Fourth, from g. 3(a) we observe that (i) for the jnj= 2 the bound state energy is greater than the energy forjnj= 1 (ii). the bound state energy of the positive curvature (n>0) and the negative curvature ( n<0) are degenerate. The g. 3(a) also shows the Kramer's de- generate pairs for up spin and down spin. The g. 3(b) shows the energy spectrum of the bound state of the spin- full electron for dierent values of the product 
soof the hole radius and spin orbit coupling constant
sofor the pentagon defect (n=1). This shows that at a given value ofthe bound state energy for both up and down spin decrease with increase in the value of 
soranging from 0.1 to 0.5. However, for higher values of the product 
sothe allowed range for to have bound state energy  <
sois increased. This shows that the localized zero energy state around the defect still persists for a minimum size of the defect core ( ) threaded by magnetic ux (). Further, the spectrum shows that the crossing of the up- spin and the down spin state remains xed with a change in the magnitude of 
so. The g. 3(c)-3(e) shows the bound state energy of the up spin and down spin states, at both K valleys, with respect to external magnetic ux () for pentagon (n=1) and square defect(n=2). Now in graphene for the electron moving around the defect core, the spinor picks up a Berry phase of after rotation by angle 2which leads to the antisymmetric boundary con- dition. So to get the periodic boundary condition and zero energy states we have to apply a external magnetic ux of  =h=2e= o=2 in the defect core. But as the defect also acts like the source of ctitious magnetic ux which iso=4 for pentagon defect (n=1) and o=2 for square defect (n=2), so to get zero energy bound state and pe- riodic boundary condition we have to apply an external magnetic ux of  = o=4 and  = 0 for the pentagon and square defect respectively as shown in vs =oplot of [16] for the spin less electron. But for the spinfull elec- tron we nd that the up and down spin zero energy states do not occur exactly at these value of external magnetic ux but they are separated and occur at the external mag- netic ux, 
. For the pentagon defect in the rst valley K, the zero energy states for spin less electron occur at external magnetic ux value,  = 3=2; =2 [16], but for spinfull electron g. 3(c), the pair of up and down spin zero energy states occur at external magnetic ux value 
 ==2=2 and 
 =3=2=2. For the square defect (n=2), g. 3(e) for which the value of the external magnetic ux should be 0 or 2to get spin less zero energy states, we nd the zero energy states oc- cur at a external magnetic ux value  
 = 03=4 and 
 =23=4;23=4. Similarly for the heptagon defect (n=-1) and for the octagon defect (n=-2) (will be reported elsewhere) the
 is 7 =5(=2) and 3=2 respectively. This extra external magnetic ux,
 oc- curs due to the eect of curvature on the intrinsic SOC p-5T.Choudhari, N.Deo through the curvature functions g() andf() because if we make the curvature function g() = 0 and f() = 1 (to remove the eect of spin), we nd that the zero energy states occur at external magnetic ux value . So for all these defects the up spin zero energy state is present at the external magnetic ux value 
 and the down spin zero energy state is present at the external magnetic ux value  +
 which concludes that they are separated. From g. 3(c)-3(e), we also observe the number of up- spin and down-spin states for the pentagon (n=1), square defect(n=2), heptagon (n=-1) and octagon defect(n=-2) change from 2 to 1, 2 to 1, 3 to 2 and 3 to 2 respec- tively at dierent values of magnetic ux. Which means the external magnetic ux applied at the defect core can change the local density of states around the defect. Now for
so;R6= 0 , we have numerically plotted the energy spectrum of the Hamiltonian ( HDSR) for the bound state electron given by =DS+
as shown in g. 3(f) for the rst and second valley with R=
so= 0:3 for the pen- tagon defect. Compared to the energy spectrum shown in the g. 3(a), for which the states of up and down spin cross each other at = 0 in the two K valleys, the states now cross each other at 6= 0. This change in for cross- ing depends upon the magnitude of the ratio of R=
so. Further, the energy spectrum shows that the zero energy states are still present in the spectrum for both the valleys but now move to smaller values of =rfor the two K valleys as compared to g. 3(a) in which they are at =osuch thatr< o. In the plot for the bound state energy vs the external magnetic ux() for all four defects, we observe that the value of
 after adding the Rashba spin orbit coupling does not change and it is still given by
 equal to =2, 3=4, 7=5(=2) and 3=2 for pentagon (n=1), square(n=2), heptagon (n=-1) and oc- tagon defect(n=-2) respectively. Conclusion. { In summary, we have found the bound state of the spin-full electron in graphene with wedge disclination using the modied Kane-Mele model in the presence of curvature modied, intrinsic spin orbit cou- pling (
so) and extrinsic Rashba spin orbit coupling ( R), for the conical graphene lattice. This study leads to the conclusion that for the case
so6= 0 andR= 0, for the electron circulating the defect, there exist Kramer's degenerate pair of zero energy spin bound states local- ized around the defect, acting as the source of spin c- titious ux and these zero energy spin states separates only due to curvature modied SOC of the wedge discli- nated graphene in the presence of external magnetic ux. For the case
so6= 0 andR6= 0 these zero energy spin states still persist and all other results are robust. This concludes that in 2D, disclinated graphene based topolog- ical insulators can act as a source of protected localized zero energy separated spin states around the defect which can be controlled by external magnetic ux. Hence these results can have potential applications in spintronics and nanoelectronics. TC would like to thank the University Grant Commis- sion ( U.G.C ) for Senior research fellowship ( S.R.F ). ND would like to thank the University of Delhi R & D Re- search Grant. REFERENCES [1]Ando Y. ,J. Phys. Soc. Jpn. ,82(2013) 102001. [2]Hasan M. Z. andKane C. L. ,Rev. Mod. Phys. , 82(2010) 3045 [3]Fu L. andKane C. L. ,Phys. Rev. B ,76(2007) 045302. [4]Kane C. L. andMele E. J. ,Phys. Rev. Lett. , 95(2005) 146802. [5]Mong R. ,Phys. Rev. B ,81(2010) 245209. [6]Okada Y. et al. ,Phys. Rev. Lett. ,106 (2011) 206805. [7]Jaszczak J. A. et al. ,Carbon ,41(2003) 2085. [8]Lin C. T. ,Langmuir ,23(2007) 12806. [9]Krishnan A. et al. ,Nature ,388(664) (1997) 451-454. [10]Ulloa P. et al. ,Nanoscale Research Lett ,8 (2013) 384. [11]Ran Y. et al. ,Nature physics ,5(2009) 298. protected zero mode ti and ts [12]Teo J. C. Y. andKane C. L. ,Phys. Rev. B , 82(2010) 115120. [13]Lammert P. E. andCrespi V. H. ,Phys. Rev. Lett.,85(2000) 5190. [14]Vozmediano M.A.H. et al. ,Physics Reports , 496(2010) 109-148. [15]Cortijo A. andVozmediano M. A. H. ,Nu- clear Phys.B ,763[FS] (2007) 293-308. [16]Ruegg A. andLin C. ,Phys. Rev. Lett. ,110 (2013) 046401. [17]de Souza a J.F.O., de Lima Ribeiro b C.A. andFurtado a C. ,Phys. Lett. A ,378 (2014) 2317-2324. [18]Assad F. F. et al. ,Phys. Rev. X ,3(2013) 011015. [19]Ran Y. et al. ,Phys. Rev. Lett. ,101 (2008) 086801. [20]Neto A. H. C. andGuniea F. ,Phys. Rev. Lett.,103(2009) 026804. [21]Weeks C. et al. ,Phys. Rev. X ,1(2011) 021001. [22]Jin K. H. andJhi S. H. ,Phys. Rev. B ,87 (2013) 075442. [23]Fonseca J. M. et al. ,Phys. Lett. A ,374(2010) 4359. [24]Hernando D. H. et al. ,Phys. Rev. B ,74 (2006) 155426, Ando T. ,J. Phys. Soc. Jpn. , 69(2000) 1757, Steele G.A. et al. ,Nat. Com- mun. ,4:1573 (2013) . [25]Avdoshenko S. M. et al. ,Scientic Reports ,3 (2013) 1632. [26]Macann E. andFal'ko V. I. ,J.Phy:Condensed Matter ,16(2004) 2371-2379. [27]Griffiths D. J. ,Introduction to Quantum Mechanics,second edition (Pearson Educa- tion)2008, p. 269. p-6

1712.05678v2.Large_spin_relaxation_anisotropy_and_valley_Zeeman_spin_orbit_coupling_in_WSe2_Gr_hBN_heterostructures.pdf

Large spin relaxation anisotropy and valley-Zeeman spin-orbit coupling in WSe2/Gr/hBN heterostructures Simon Zihlmann,1,∗Aron W. Cummings,2Jose H. Garcia,2M´ at´ e Kedves,3Kenji Watanabe,4Takashi Taniguchi,4Christian Sch¨ onenberger,1and P´ eter Makk1, 3,† 1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland 2Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3Department of Physics, Budapest University of Technology and Economics and Nanoelectronics ’Momentum’ Research Group of the Hungarian Academy of Sciences, Budafoki ut 8, 1111 Budapest, Hungary 4National Institute for Material Science, 1-1 Namiki, Tsukuba, 305-0044, Japan (Dated: December 19, 2017) Large spin-orbital proximity effects have been predicted in graphene interfaced with a transition metal dichalcogenide layer. Whereas clear evidence for an enhanced spin-orbit coupling has been found at large carrier densities, the type of spin-orbit coupling and its relaxation mechanism re- mained unknown. We show for the first time an increased spin-orbit coupling close to the charge neutrality point in graphene, where topological states are expected to appear. Single layer graphene encapsulated between the transition metal dichalcogenide WSe 2and hBN is found to exhibit ex- ceptional quality with mobilities as high as 100 000 cm2V−1s−1. At the same time clear weak anti-localization indicates strong spin-orbit coupling and a large spin relaxation anisotropy due to the presence of a dominating symmetric spin-orbit coupling is found. Doping dependent measure- ments show that the spin relaxation of the in-plane spins is largely dominated by a valley-Zeeman spin-orbit coupling and that the intrinsic spin-orbit coupling plays a minor role in spin relaxation. The strong spin-valley coupling opens new possibilities in exploring spin and valley degree of freedom in graphene with the realization of new concepts in spin manipulation. MOTIVATION/INTRODUCTION In recent years, van der Waals heterostructures (vdW) have gained a huge interest due to their possibility of im- plementing new functionalities in devices by assembling 2D building blocks on demand [1]. It has been shown that the unique band structure of graphene can be en- gineered and enriched with new properties by placing it in proximity to other materials, including the formation of minibands [2–5], magnetic ordering [6, 7], and super- conductivity [8, 9]. Special interest has been paid to the enhancement of spin-orbit coupling (SOC) in graphene since a topological state, a quantum spin Hall phase, was theoretically shown to emerge [10]. First principles cal- culations predicted an intrinsic SOC strength of 12 µeV [11], which is currently not observable even in the clean- est devices. Therefore, several routes were proposed and explored to enhance the SOC in graphene while preserv- ing its high electronic quality [12–14]. One of the most promising approaches is the combination of a transition metal dichalcogenide (TMDC) layer with graphene in a vdW-hetereostructure. TMDCs have very large SOC on the 100 meV–scale in the valence band and large SOC on the order of 10 meV in the conduction band [13]. The realization of topological states is not the only motivation to enhance the SOC in graphene. It has been shown that graphene is an ideal material for spin trans- port [13]. Spin relaxation times on the order of nanosec- onds [15, 16] and relaxation lengths of 24 µm [17] have been observed. However, the presence of only weak SOCin pristine graphene limits the tunability of possible spin- tronics devices made from graphene. The presence of strong SOC would enable fast and efficient spin manip- ulation by electric fields for possible spintronics applica- tions, such as spin-filters [18] or spin-orbit valves [19, 20]. In addition, enhanced SOC leads to large spin-Hall angles [21] that could be used as a source of spin currents or as a detector of spin currents in graphene-based spintronic devices. It was proposed that graphene in contact to a single layer of a TMDC can inherit a substantial SOC from the underlying substrate [14, 22]. The experimental de- tection of clear weak anti-localization (WAL) [23–28] as well as the observation of a beating of Shubnikov de-Haas (SdH) oscillations [24] leave no doubt that the SOC is greatly enhanced in graphene/TMDC heterostructures. First principles calculations of graphene on WSe 2[22] predicted large spin-orbit coupling strength and the for- mation of inverted bands hosting special edge states. At low energy, the band structure can be described in a simple tight-binding model of graphene containing the orbital terms and all the symmetry allowed SOC terms H=H0+H∆+HI+HVZ+HR[22, 29]: H0=/planckover2pi1vF(κkxˆσx+kyˆσy)·ˆs0 H∆= ∆ˆσz·ˆs0 HI=λIκˆσz·ˆsz HVZ=λVZκˆσ0·ˆsz HR=λR(κˆσx·ˆsy−ˆσy·ˆsx).(1) Here, ˆσiare the Pauli matrices acting on the pseudospin,arXiv:1712.05678v2 [cond-mat.mes-hall] 18 Dec 20172 ˆsiare the Pauli matrices acting on the real spin and κ is either±1 and denotes the valley degree of freedom. kxandkyrepresent the k-vector in the graphene plane, /planckover2pi1is the reduced Planck constant, vFis the Fermi ve- locity and λi,∆ are constants. The first term H0is the usual graphene Hamiltonian that describes the lin- ear band structure at low energies. H∆represents an orbital gap that arises from a staggered sublattice poten- tial.HIis the intrinsic SOC term that opens a topolog- ical gap of 2 λI[10].HVZis a valley-Zeeman SOC that couples valley to spin and results from different intrinsic SOC on the two sublattices. This term leads to a Zee- man splitting of 2 λVZthat has opposite sign in the K and K’ valleys and leads to an out of plane spin polar- ization with opposite polarization in each valley. HRis a Rashba SOC arising from the structure inversion asym- metry. This term leads to a spin splitting of the bands with a spin expectation value that lies in the plane and is coupled to the momentum via the pseudospin. At higher energies k-dependent terms, called pseudospin inversion asymmetric (PIA) SOC come into play, which can be neglected at lower doping [29]. Previous studies have estimated the SOC strength from theoretical calculations [23] or extracted only the Rashba SOC at intermediate [27] or at very high dop- ing [25] or gave only a total SOC strength [26]. Fur- ther studies have extracted a combination of Rashba and valley-Zeeman SOC strength form SdH-oscillation beat- ing measurements [24]. Additionally, a very recent study uses the clean limit (precession time) to estimate the SOC strength from diffusive WAL measurements [28]. Here, we give for the first time a clear and comprehen- sive study of SOC at the charge neutrality point (CNP) for WSe 2/Gr/hBN heterostructures. The influence of strong SOC is expected to have the largest impact on the bandstructure close to the CNP. The strength of all possible SOC terms is discussed and we find that the re- laxation times are dominated by the valley-Zeeman SOC. The valley-Zeeman SOC leads to a much faster relaxation of in-plane spins than out-of plane spins. This asym- metry is unique for systems with strong valley-Zeeman SOC and is not present in traditional 2D Rashba sys- tems where the anisotropy is 1/2 [18]. Our study is in contrast to previous WAL measurements [25, 27], but is in good agreement with recent spin-valve measurements reporting a large spin relaxation anisotropy [30, 31]. METHODS WSe 2/Gr/hBN vdW-heterostructures were assembled using a dry pick-up method [32] and Cr/Au 1D-edge con- tacts were fabricated [33]. Obviously a clean interface between high quality WSe 2and graphene is of utmost importance. A short discussion on the influence of the WSe 2quality is given in the Supplemental Material. Af-ter shaping the vdW-heterostructure into a Hall-bar ge- ometry by a reactive ion etching plasma employing SF 6 as the main reactive gas, Ti/Au top gates were fabri- cated with an MgO dielectric layer to prevent it from contacting the exposed graphene at the edge of the vdW- heterostructure. A heavily-doped silicon substrate with 300 nm SiO 2was used as a global back gate. An optical image of a typical device and a cross section is shown in Fig. 1 (a). In total, three different samples with a total of four devices were fabricated. Device A, B and C are pre- sented in the main text and device D is discussed in the Supplemental Material. Standard low frequency lock-in techniques were used to measure two- and four-terminal conductance and resistance. Weak anti-localization was measured at temperatures of 50 mK to 1 .8 K whereas a classical background was measured at sufficiently large temperatures of 30 K to 50 K. RESULTS Device Characterization The two-terminal resistance measured from contact 1 to 2 as a function of applied top and bottom gate is shown in Fig. 1 (b). A pronounced resistance maximum, tun- able by both gates, indicates the CNP of the bulk of the device whereas a fainter line only changing with V BG indicates the CNP from the device areas close to the con- tacts, which are not covered by the top gate. From the four-terminal conductivity, shown in Fig. 1 (c), the field effect mobility µ/similarequal130 000 cm2V−1s−1and the residual dopingn∗= 7×1010cm−2were extracted. The mobil- ity was extracted from a linear fit of the conductivity as a function of density at negative V BG. At positive VBGthe mobility is higher as one can easily see from Fig. 1 (c). At V BG≥25 V, the lever arm of the back gate is greatly reduced since the WSe 2layers gets popu- lated with charge carriers, i.g. the Fermi level is shifted into some trap states in the WSe 2. Although the WSe 2is poorly conducting (low mobility) it can screen potential fluctuations due to disorder and this can lead to a larger mobility in the graphene layer, as similarly observed in graphene on MoS 2[34]. Fig. 1 (d)shows the longitudinal resistance as a func- tion of magnetic field and gate voltage with lines orig- inating from the integer quantum Hall effect. At low fields, the normal single layer spectrum is obtained with plateaus at filling factors ν=±2,±6,±10,±14,..., whereas at larger magnetic fields full degeneracy lift- ing is observed with plateaus at filling factors ν= ±2,±3,±4,±5,±6,.... The presence of symmetry bro- ken states, that are due to electron-electron interactions [35], is indicative of a high device quality. In the ab- sence of interaction driven symmetry breaking, the spin- splitting of the quantum Hall states could be used to3 investigate the SOC strength [36]. The high quality of the devices presented here poses sever limitations on the investigation of the SOC strength using WAL theory. Ballistic transport features (trans- verse magnetic focusing) are observed at densities larger than 8×1011cm−2. Therefore, a true diffusive regime is only obtained close to the CNP, where the charge carriers are quasi-diffusive [37]. 4 2 0 -2 -4VTG (V) 20 0 -20 VBG (V) 1.5x1041.0 0.5R2T (Ω) (b) 2.5x103 2.0 1.5 1.0 0.5 0.0σ4T (e2/h) -4 -2 0 2 4 VTG (V)n* = 7e10 cm-2 130'000 cm2/(Vs)(c) -30 V -15 V -0 V +15 V +30 V -20020VBG (V) 8 6 4 2 0 Bz (T)(d) 43210log(Rxx(Ω)) -10-6-22610Gxy (e2/h) -20 0 20 VBG (V)3 2 1 0Rxx (kΩ)(e) WSe2hBN Cr/Au Cr/AuMgOTi/Au SiO2 Si, p++Bz B||(a) 12 34 56 FIG. 1. Device layout and basic characterization of WSe 2/Gr/hBN vdW-heterostructures. (a) shows an optical image of a device A before the fabrication of the top gate, whose outline is indicated by the white dashed rectan- gle. On the right, a schematic cross section is shown and the directions of the magnetic fields are indicated. The scale bar is 1µm. The data shown in (b)to(e)are from device B. The two terminal resistance measured from lead 1 to 2 is shown as a function of top and back gate voltage. A pronounced re- sistance maximum tunable by both gates indicates the charge neutrality point (CNP) of the bulk device, whereas a fainter line only changing with V BGindicates the CNP from the de- vice area close to the contacts that are not covered by the top gate. Cuts in V TGat different V BGof the conductivity measured in a four-terminal configuration are shown in (c), which are also used to extract field effect mobility (linear fit indicated by black dashed line) and residual doping as indi- cated. The fan plot of longitudinal resistance R xxversus V BG and B zat V TG=−1.42 V is shown in (d)and a cut at B z = 7 T in (e). Clear plateaus are observed at filling factors ν=±2,±3,±4,... and higher, indicating full lifting of the fourfold degeneracy of graphene for magnetic fields >6 T.Magneto conductance In a diffusive conductor, the charge carrier trajectories can form closed loops after several scattering events. The presence of time-reversal symmetry leads to a construc- tive interference of the electronic wave function along these trajectories and therefore to an enhanced back scat- tering probability compared to the classical case. This phenomenon is known as weak localization (WL). Con- sidering the spin degree of freedom of the electrons, this can change. If strong SOC is present the spin can precess between scattering events, leading to destructive interfer- ence and hence to an enhanced forward scattering proba- bility compared to the classical case. This phenomenon is known as weak anti-localization [38]. The quantum cor- rection to the magneto conductivity can therefore reveal the SOC strength. The two-terminal magneto conductivity ∆ σ=σ(B)− σ(B= 0) versus B zand n at T = 0 .25 K and zero perpen- dicular electric field is shown in Fig. 2 (a). A clear feature at B z= 0 mT is visible, as well as large modulations in B z and n due to universal conductance fluctuations (UCFs). UCFs are not averaged out since the device size is on the order of the dephasing length lφ. Therefore, an ensemble average of the magneto conductivity over several densi- ties is performed to reduce the amplitude of the UCFs [23], and curves as in Fig. 2 (b)result. A clear WAL peak is observed at 0 .25 K whereas at 30 K the quan- tum correction is fully suppressed due to a very short phase coherence time and only a classical background in magneto conductivity remains. This high temperature background is then subtracted from the low temperature measurements to extract the real quantum correction to the magneto conductivity [24]. In addition to WL/WAL measurements the phase coherence time can be extracted independently from the autocorrelation function of UCF in magnetic field [39]. UCF as a function of B zwas mea- sured in a range where the WAL did not contribute to the magneto conductivity (e.g. 20 mT to 70 mT) and an average over several densities was performed. The in- flection point in the autocorrelation, determined by the minimum in its derivative, is a robust measure of τφ[40], see Fig. 2 (d). Fitting To extract the spin-orbit scattering times we use the theoretical formula derived by diagrammatic perturba- tion theory [41]. In the case of graphene, the quantum correction to the magneto conductivity ∆ σin the pres-4 -10010Bz (mT) -2-1012 n (1011cm-2)(a) -1.0-0.50.00.5Δσ (e2/h)-2.5e11 < n < +2.5e11 cm-2, E = 0 V/m -0.20-0.15-0.10-0.050.00Δσ (e2/h) -10010 Bz (mT)-0.10-0.050.00Δσ - Δσ30 K (e2/h) -10010 Bz (mT) 0.25 K 30 K 0.25 K 1.8 K 4 K 8 K(b) 0.8 0.4 0.0f(δB) (e4/h2) 1086420 δBz (mT)T = 0.25 K τφ = 8 ps Bip = 1 mT(c) FIG. 2. Magneto conductivity of device A: (a) Magneto conductivity versus B zand n is shown at T = 0 .25 K. A clear feature is observed around B = 0 mT and large modulations due do UCF are observed in B zand n. (b)shows the mag- neto conductivity averaged over all traces at different n. The WAL peak completely disappears at T = 30 K, leaving the classical magneto conductivity as a background. The 30 K trace is offset vertically for clarity. The quantum correction to the magneto conductivity is then obtained by subtracting the high temperature background from the magneto conduc- tivity, see (b)on the right for different temperatures. With increasing temperature the phase coherence time shortens and therefore the WAL peak broadens and reduces in height. (c) shows the autocorrelation of the magneto conductivity in red and its derivative in blue (without scale). The minimum of the derivative indicates the inflection point (B ip) of the auto- correlation, which is a measure of τφ. ence of strong SOC is given by: ∆σ(B) =−e2 2πh/bracketleftBigg F/parenleftBigg τ−1 B τ−1 φ/parenrightBigg −F/parenleftBigg τ−1 B τ−1 φ+ 2τ−1asy/parenrightBigg −2F/parenleftBigg τ−1 B τ−1 φ+τ−1asy+τ−1sym/parenrightBigg/bracketrightBigg ,(2) whereF(x) = ln(x) + Ψ(1/2 + 1/x), with Ψ(x) being the digamma function, τ−1 B= 4eDB/ /planckover2pi1, whereDis the diffusion constant, τφis the phase coherence time, τasy is the spin-orbit scattering time due to SOC terms that are asymmetric upon z/-z inversion ( HR) andτsymisthe spin-orbit scattering time due to SOC terms that are symmetric upon z/-z inversion ( HI,HVZ) [41]. The total spin-orbit scattering time is given by the sum of the asymmetric and symmetric rate τ−1 SO=τ−1 asy+τ−1 sym. In general, Eq. 2 is only valid if the intervalley scattering rateτ−1 ivis much larger than the dephasing rate τ−1 φand the rates due to spin-orbit scattering τ−1 asy,τ−1 sym. In the limit of very weak asymmetric but strong sym- metric SOC ( τasy/greatermuchτφ/greatermuchτsym), Eq. 2 describes re- duced WL since the first two terms cancel and there- fore a positive magneto conductivity results. Contrary to that, in the limit of very weak symmetric but strong asymmetric SOC ( τsym/greatermuchτφ/greatermuchτasy) a clear WAL peak is obtained. If both time scales are shorter than τφ, the ratioτasy/τsymwill determine the quantum correction of the magneto conductivity. In the limit of total weak SOC (τasy,τsym/greatermuchτφ) the normal WL in graphene is obtained [42], as the first two terms cancel and other terms explic- itly involving the inter- and intravalley scattering must be considered (see Supplemental Material). Since the second and the third term can produce very similar dependencies on B zit can be hard to properly distinguish between the influence of τasyandτsymon ∆σ(B), as also previously reported [24, 28]. It is there- fore important to measure and fit the magneto conduc- tivity to sufficiently large fields in order to capture the influence of the second and third term, which only sig- nificantly contribute at larger fields (for strong SOC). However, there is an upper limit of the field scale (the so-called transport field Btr) at which the theory of WAL breaks down. The size of the shortest closed loops that can be formed in a diffusive sample is on the order of l2 mfp, wherelmfpis the mean-free path of the charge carriers. Fields that are larger than Φ 0/l2 mfp, where Φ 0=h/eis the flux quantum, are not meaningful in the framework of diffusive transport. In the most general case there are three different regimes in the presence of strong SOC in graphene: τasy/lessmuchτsym,τasy∼τsymandτasy/greatermuchτsym. Therefore, we fitted the magneto conductivity with initial fit param- eters in these three limits. An example is shown in Fig. 3, where the three different fits are shown as well as the extracted parameters. Obviously, the case τasy/greatermuchτsym (fit1) andτasy∼τsym(fit2) are indistinguishable and fit the data worse than the case τasy/greatermuchτsym(fit3). In addition,τφextracted from the UCF matches best for fit3. Therefore, we can clearly state that the symmetric SOC is stronger than the asymmetric SOC. The flat back- ground as well as the narrow width of the WAL peak can only be reproduced with the third case. A very similar behaviour was found in device C at the CNP. In device B (shown in the Supplemental Material), whose mobility is larger than the one from device A, we cannot clearly dis- tinguish the three limits as the transport field is too low (≈12 mT) and the flat background at larger field cannot be used to disentangle the different parameters from each5 other. However, this does not contradict τasy/greatermuchτsym and the overall strength of the SOC ( τSO/similarequal0.2 ps) is in good agreement with device A shown here. Obviously, the extracted time scales should be taken with care as many things can introduce uncertainties in the extracted time scales. First of all, we are looking at ensemble-averaged quantities and it is clear that this might influence the precision of the extraction of the time scales. In addition, the subtraction of a high temperature background can lead to higher uncertainty of the quan- tum correction. Lastly, the high mobility of the clean devices places severe limitations on the usable range of magnetic field. All these influences lead us to a conser- vative estimation of a 50 % uncertainty for the extracted time scales. Nevertheless, the order of magnitude of the extracted time scales and trends are still robust. -0.14-0.12-0.10-0.08-0.06-0.04-0.020.00Δσ0.25 K - Δσ30 K (e2/h) -15-10-5051015 Bz (mT) sample A fit1 fit2 fit3fit1fit2fit3 τφ4.14.06.6 τasy0.660.936.2 τsym4.11.00.15 τSO0.570.490.15 D = 0.12 m2/s FIG. 3. Fitting of quantum correction to the magneto conductivity of device A The quantum correction to the magneto conductivity is fit using Eq. 2. The results for three different limits are shown and their parameters are indicated (in units of ps). τφis estimated to be 8 ps from the autocor- relation of UCF in magnetic field, see Fig. 2 (d). The presence of a top and a back gate allows us to tune the carrier density and the transverse electric field inde- pendently. The spin-orbit scattering rates were found to be electric field independent at the CNP in the range of−0.05 V nm−1to 0.08 V nm−1within the precision of parameter extraction. Details are given in the Supple- mental Material. Within the investigated electric field rangeτasywas found to be in the range of 5 ps to 10 ps, always close to τφ.τsymon the other hand was found to be around 0 .1 ps to 0.3 ps whileτpwas around 0 .2 ps to 0.3 ps, see Supplemental Material for more details. The lack of electric field tunability of τasyandτsymin the investigated electric field range is not so surprising. The Rashba coupling in this system is expected to change considerably for electric fields on the order of 1 V nm−1, which are much larger than the applied fields here. How- ever, such large electric fields are hard to achieve. Inaddition,τsym, which results from λIandλVZis not ex- pected to change much with electric field as long as the Fermi energy is not shifted into the conduction or va- lence band of the WSe 2[14]. These findings contradict another study [26], which claims an electric field tunabil- ity of both SOC terms. However, there it is not discussed how accurately those parameters were extracted. Density dependence The momentum relaxation time τpcan be tuned by changing the carrier density in graphene. Fig. 4 shows the dependence of τ−1 asyandτ−1 symonτpin a third device C. The lower mobility in device C allowed for WAL mea- surements at higher charge carrier densities not accessi- ble in devices A and B. At the CNP, τ−1 asyandτ−1 symare found to be consistent across all three devices A, B and C. Here,τ−1 symincreases with increasing τpwhereasτ−1 asy is roughly constant with increasing τp. The dependence of the spin-orbit scattering times on the momentum scat- tering time can give useful insights into the dominating spin relaxation mechanisms, as will be discussed later. It is important to note that the extracted τasyis always very close to τφ. Therefore, the extracted τasycould be shorter than what the actual value would be since τφacts as a cutoff. 2.0x1013 1.5 1.0 0.5τsym-1 (s-1)40 30 20 10σ (e2/h) -10x1015-50510 n (m-2)45'000 cm2/(Vs)34'000 cm2/(Vs) 120x109 100 80 60 40τasy-1 (s-1) 350x10-15 300 250 200 τp (s) FIG. 4. Density dependence of device C: The depen- dence of the spin-orbit scattering rates τ−1 symandτ−1 asyas a function of τpare shown for device C. The error bars on the spin-orbit scattering rates are given by a conservative esti- mate of 50 %. The two terminal conductivity is shown in the inset and the extracted mobilities for the n and p side are indicated.6 In-plane magnetic field dependence An in-plane magnetic field (B /bardbl) is expected to lift the influence of SOC on the quantum correction to the mag- neto conductivity at sufficiently large fields. This means that a crossover from WAL to WL for z/-z asymmetric and a crossover from reduced WL to full WL correction for z/-z symmetric spin-orbit coupling is expected at a field where the Zeeman energy is much larger than the SOC strength [41]. The experimental determination of this crossover field allows for an estimate of the SOC strength. The B/bardbldependence of the quantum correction to the magneto conductivity of device A at the CNP and at zero perpendicular electric field was investigated, as shown in Fig. 5. The WAL peak decreases and broadens with increasing B/bardbluntil it completely vanishes at B /bardbl/similarequal3 T. Neither a reappearance of the WAL peak, nor a transition to WL, is observed at higher B /bardblfields (up to 9 T). A qualitatively similar behaviour was observed for device D. Fits with equation 2 allow the extraction of τφand τSO, which are shown in Fig. 5 (b)for B/bardblfields lower than 3 T. A clear decrease of τφis observed while τSO remains constant. The reduction in τφwith increasing B /bardblwas previously attributed to enhanced dephasing due to a random vector potential created by a corrugated graphene layer in an in- plane magnetic field [43]. The clear reduction in τφwith constantτSOand the absence of any appearance of WL at larger B/bardblalso strongly suggests that a similar mechanism is at play here. Therefore, the vanishing WAL peak is due to the loss of phase coherence and not due to the fact that the Zeeman energy ( Ez) is exceeding the SOC strength. Using the range where WAL is still present, we can define a lower bound of the crossover field when τφ drops below 80 % of its initial value, which corresponds to 2 T here. This leads to a lower bound of the SOC strengthλSOC≥Ez∼0.2 meV given a g-factor of 2. DISCUSSION The effect of SOC was investigated in high quality vdW-heterostructures of WSe 2/Gr/hBN at the CNP, as there the effects of SOC are expected to be most impor- tant. The two-terminal conductance measurements are not influenced by contact resistances nor pn-interfaces close to the CNP. At larger doping, the two-terminal con- ductance would need to be considered with care. Phase coherence times around 4 ps to 7 ps were con- sistently found from fits to Eq. 2 and from the autocor- relation of UCF. It is commonly known that the phase coherence time is shorter at the CNP than at larger dop- ing [43, 44]. Moreover, large diffusion coefficients lead to long phase coherence lengths being on the order of the device size ( lφ=/radicalbig Dτφ≈1µm), which in turn leads to 10x10-12 8 6 4 2 0τ (s) 2.5 2.0 1.5 1.0 0.5 0.0 B|| (T) τφ τSO(b)-0.10-0.050.000.05ΔσT=1.8 K - ΔσT=30 K (e2/h) -20 -10 0 10 20 Bz (mT) B|| = 0.0 T B|| = 0.4 T B|| = 1.2 T B|| = 2.0 T B|| = 2.8 T fits(a) B|| = 5 T B|| = 7 T B|| = 9 TFIG. 5. In-plane magnetic field dependence of de- vice A: The quantum correction to the magneto conduc- tivity at the CNP and at zero perpendicular electric field is shown for different in-plane magnetic field strengths B /bardblin (a). Here, n was averaged in the range of −1×1011cm−2to 1×1011cm−2. The WAL peak gradually decreases in height and broadens as B /bardblis increased. The traces at B/bardbl= 5, 7, 9 T are offset by 0 .03 e2/h for clarity. In (b)the extracted phase coherence time τφand the total spin-orbit scattering time τSO are plotted versus B /bardbl.τφclearly reduces, whereas τSOremains roughly constant over the full B /bardblrange investigated. large UCF amplitudes making the analysis harder. In general Eq. 2 is only applicable for short τiv. Since τivis unknown in these devices, only an estimate can be given here. WL measurements of graphene on hBN foundτivon the order of picoseconds [45, 46]. Inter- valley scattering is only possible at sharp scattering cen- tres as it requires a large momentum change. It is a reasonable assumption that the defect density in WSe 2, which is around 1 ×1012cm−2[47], is larger than in the high quality hBN [48]. This leads to shorter τivtimes in graphene placed on top of WSe 2and makes Eq. 2 appli- cable despite the short spin-orbit scattering times found here. In the case of weaker SOC, Eq. 2 cannot be used. Instead, a more complex analysis including τivandτ∗is needed. This was used for device D, and is presented in the Supplemental Material. Spin-orbit scattering rates were successfully extracted at the CNP and τasywas found to be around 4 ps to 7 ps whereas τsymwas found to be much shorter, around 0.1 ps to 0.3 ps. In these systems, if τivis sufficiently short,τasy/2 is predicted to represent the out-of-plane spin relaxation time τ⊥andτsymthen represents the in- plane spin relaxation time τ/bardbl[18]. For the time scales7 stated above, a spin relaxation anisotropy τ⊥/τ/bardbl∼20 is found (see Supplemental Material for detailed calcula- tion). This large anisotropy in spin relaxation is unique for systems with a strong valley-Zeeman SOC. Similar anisotropies have been found recently in spin valves in similar systems [30, 31]. In order to link spin-orbit scattering time scales to SOC strengths, spin relaxation mechanisms have to be consid- ered. The simple definition of /planckover2pi1/τSOas the SOC strength is only valid in the limit where the precession frequency is much larger than the momentum relaxation rate (e.g. full spin precession occurs between scattering events). In the following we concentrate on the parameters from de- vice A that were extracted close to the CNP. The de- pendence on τpin device A can most likely be assumed to be very similar to that observed in device C. Within the investigated density range of −2.5×1011cm−2to 2.5×1011cm−2, including residual doping, an average Fermi energy of 45 meV was estimated. This is based on the density of states of pristine graphene, which should be an adequate assumption for a Fermi energy larger than any SOC strengths. The symmetric spin-orbit scattering time τsymcon- tains contributions from the intrinsic SOC and from the valley-Zeeman SOC. Up to now, only the intrinsic SOC has been considered in the analysis of WAL measure- ments, and the impact of valley-Zeeman SOC has been ignored. However, as we now explain, it is highly unlikely that intrinsic SOC is responsible for the small values of τsym. The intrinsic SOC is expected to relax spin via the Elliott-Yafet (EY) mechanism [49], which is given as τs=/parenleftbigg2EF λI/parenrightbigg2 τp, (3) whereτsis the spin relaxation time, EFis the Fermi energy,λIis the intrinsic SOC strength and τpis the momentum relaxation time [49]. Since the intrinsic SOC does not lead to spin-split bands and hence no spin-orbit fields exist that could lead to spin preces- sion, a relaxation via the Dyakonov-Perel mechanism can be excluded. Therefore, we can estimate λI= 2EF//radicalBig τsymτ−1p∼110 meV using τsym∼0.2 ps, a mean Fermi energy of 45 meV and a momentum relaxation time of 0.3 ps. The extracted value for λIwould correspond to the opening of a topological gap of 220 meV. In the presence of a small residual doping (here 30 meV), such a large topological gap should easily be detectable in trans- port. However, none of our transport measurements con- firm this. In addition, the increase of τ−1 symwithτp, as shown in Fig. 4, does not support the EY mechanism. On the other hand, Cummings et al. have shown that the in-plane spins are also relaxed by the valley-Zeeman term via a Dyakonov-Perel mechanism where τivtakesthe role of the momentum relaxation time [18]: τ−1 s=/parenleftbigg2λVZ /planckover2pi1/parenrightbigg2 τiv. (4) While this equation applies in the motional narrowing regime of spin relaxation, our measurement appears to be near the transition where that regime no longer applies. Taking this into consideration (see Supplemental Mate- rial), we estimate λVZto be in the range of 0 .23 meV to 2.3 meV for a τsymof 0.2 ps and a τivof 0.1 ps to 1 ps. This agrees well with first principles calculations [22]. The large range in λVZcomes from the fact that τivis not exactly known. Obviously, τsymcould still contain parts that are re- lated to the intrinsic SOC ( τ−1 sym =τ−1 sym,I +τ−1 sym,VZ ). As an upper bound of λI, we can give a scale of 15 meV, which corresponds to half the energy scale due to the residual doping in the system. This would lead toτsym,I∼10 ps. Such a slow relaxation rate ( τ−1 sym,I ) is completely masked by the much larger relaxation rate τ−1 sym,VZ coming from the valley-Zeeman term. There- fore, the presence of the valley-Zeeman term makes it very hard to give a reasonable estimate of the intrinsic SOC strength. The asymmetric spin-orbit scattering time τasycon- tains contributions from the Rashba-SOC and from the PIA SOC. Since the PIA SOC scales linearly with the mo- mentum, it can be neglected at the CNP. Here, τasyrep- resents only the spin-orbit scattering time coming from Rashba SOC. It is known that Rashba SOC can relax the spins via the Elliott-Yafet mechanism [49]. In addition, the Rashba SOC leads to a spin splitting of the bands and therefore to a spin-orbit field. This opens a second re- laxation channel via the Dyakonov-Perel mechanism [50]. In principle the dependence on the momentum scatter- ing timeτpallows one to distinguish between these two mechanisms. Here, τ−1 asydoes not monotonically depend onτpas one can see in Fig. 4 and therefore we cannot unambiguously decide between the two mechanisms. Assuming that only the EY mechanism is responsible for spin relaxation, then λR=EF//radicalBig 4τasyτ−1p∼5.0 meV can be estimated, using τasyof 6 ps, a mean Fermi energy of 45 meV and a momentum relaxation time of 0 .3 ps. On the other hand, pure DP-mediated spin relaxation leads toλR=/planckover2pi1//radicalbig2τasyτp∼0.35 meV. The Rashba SOC strength estimated by the EY relaxation mechanism is large compared to first principles calculations [22], which agree much better with the SOC strength estimated by the DP mechanism. This is also in agreement with pre- vious findings [25, 27]. Since there is a finite valley-Zeeman SOC, which is a result of different intrinsic SOC on the A sublattice and B sublattice, a staggered sublattice potential can also be expected. The presence of a staggered potential, meaning that the on-site energy of the A atom is different from8 the B atom on average, leads to the opening of a trivial gap of ∆ at the CNP. Since there is no evidence of an orbital gap, we take the first principles calculations as an estimate of ∆ = 0 .54 meV. Knowing all relevant parameters in Eq. 1, a band structure can be calculated, which is shown in Fig. 6. The bands are spin split mainly due to the presence of strong valley-Zeeman SOC but also due to the weaker Rashba SOC. At very low energies, an inverted band is formed due to the interplay of the valley-Zeeman and Rashba SOC, see Fig. 6 (b). This system was predicted to host helical edge states for zigzag graphene nanorib- bons, demonstrating the quantum spin Hall effect [22]. In the case of stronger intrinsic SOC, which we cannot estimate accurately, a band structure as in Fig. 6 (c)is expected with a topological gap appearing at low ener- gies. We would like to note here, that this system might host a quantum spin Hall phase. However, its detection is still masked by device quality as the minimal Fermi energy is much larger than the topological gap, see also Fig. 6 (a). Our findings are in good agreement with the calcula- tions by Gmitra et al. [22]. However, we have to re- mark that whereas the calculations were performed for single-layer TMDCs, we have used multilayer WSe 2as a substrate. Single-layer TMDCs are direct band-gap semiconductors with the band gap located at the K-point whereas multilayer TMDCs have an indirect band gap. Since the SOC results from the mixing of the graphene orbitals with the WSe 2orbitals, the strength of the in- duced SOC depends on the relative band alignment be- tween the graphene and WSe 2band, which will be differ- ent for single- or multilayer TMDCs. This difference was recently shown by Wakamura et al. [28]. Therefore using single-layer WSe 2to induce SOC might even enhance the coupling found by our studies. Furthermore, the param- eters taken from Ref. [22] for the orbital gap and for the intrinsic SOC therefore have to be taken with care. CONCLUSION In conclusion we measured weak anti-localization in high quality WSe 2/Gr/hBN vdW-heterostructures at the charge neutrality point. The presence of a clear WAL peak reveals a strong SOC with a much faster spin relax- ation of in-plane spins compared to out-of-plane spins. Whereas previous studies have also found a clear WAL signal, we present for the first time a complete interpre- tation of all involved SOC terms considering their relax- ation mechanisms. This includes the finding of a very large spin relaxation anisotropy that is governed by the presence of a valley-Zeeman SOC that couples spin to val- ley. The relaxation mechanism at play here is very special since it relies on intervalley scattering and can only occur in materials where a valley degree of freedom is present -40-2002040E (meV) -100 -50 0 50 100 |k| (106 m-1)(a) ∆ = 0.54 meV λI = -0.06 meV λVZ = 1 meV λR = 0.35 meV -6-4-20246E (meV) -10 -5 0 5 10 |k| (106 m-1)(b) ~ 2 λVZ 8 4 0 -4 -8E (meV) -10 -5 0 5 10 |k| (106 m-1)(c) λI = 5 meVFIG. 6. Possible low energy band structures: (a) and (b)show the band structures using the Hamiltonian of Eq. 1 with the parameters listed in (a). The unknown parameters ∆ andλIwere taken from Ref. [22]. In (a), the band struc- ture is shown in the density range of −2.5×1011cm−2to 2.5×1011cm−2(CNP), which corresponds the the one inves- tigated above. The energy range dominated by charge puddles is indicated by the grey shaded region. (b)shows a zoom in at low energy. In (c),λIof 5 meV is assumed to show the changes due to the unknown λIat low energy. and coupled to spin. This is in excellent good agree- ment with recent spin-valve measurements that found also very large spin relaxation anisotropies in similar sys- tems [30, 31]. In addition, we investigated the influence of an in-plane magnetic field on the WAL signature. Due to the loss of phase coherence, a lower bound of all SOC strengths of 0.2 meV can be given, which is in agreement with the numbers presented above. This approach does not de- pend on accurate fitting of WAL peaks nor on the inter- pretation of spin-orbit scattering rates. The coupling of spin and valley opens new possibili- ties in exploring spin and valley degrees of freedom in graphene. In the case of bilayer graphene in proximity to WSe 2an enormous gate tunability of the SOC strength is predicted since full layer polarization can be achieved by an external electric field [19, 20]. This is just one of many possible routes to investigate in the future. Acknowledgments The authors gratefully acknowledge fruitful discussions on the interpretation of the experimental data with Mar- tin Gmitra and Vladimir Fal’ko. Clevin Handschin is acknowledged for helpful discussions on the sample fab- rication. This work has received funding from the Euro-9 pean Unions Horizon 2020 research and innovation pro- gramme under grant agreement No 696656 (Graphene Flagship), the Swiss National Science Foundation, the Swiss Nanoscience Institute, the Swiss NCCR QSIT and ISpinText FlagERA network OTKA PD-121052 and OTKA FK-123894. P.M. acknowledges support as a Bolyai Fellow. ICN2 is supported by the Severo Ochoa program from Spanish MINECO (Grant No. SEV-2013- 0295) and funded by the CERCA Programme / Gener- alitat de Catalunya. Author contributions S.Z. fabricated and measured the devices with the help of P.M. K.M. contributed to the fabrication of device C. S.Z. analysed the data with help from P.M. and inputs from C.S.. S.Z., P.M., A.W.C., J.H.G and C.S. were involved in the interpretation of the results. S.Z. wrote the manuscript with inputs from P.M., C.S., A.W.C. and J.H.G., K.W. and T.T. provided the hBN crystals used in the devices. ∗Simon.Zihlmann@unibas.ch †peter.makk@mail.bme.hu [1] A. K. Geim and I. V. Grigorieva. Van der Waals het- erostructures. Nature , 499(7459):419–425, July 2013. [2] L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha- Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Falko, and A. K. Geim. Cloning of Dirac fermions in graphene superlattices. Na- ture, 497:594–, May 2013. [3] C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim. Hofstadter’s butterfly and the fractal quantum Hall effect in moir´ e superlattices. Nature , 497:598–, May 2013. [4] B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori. Massive Dirac Fermions and Hofstadter But- terfly in a van der Waals Heterostructure. Science , 340(6139):1427–1430, 2013. [5] Menyoung Lee, John R. Wallbank, Patrick Gal- lagher, Kenji Watanabe, Takashi Taniguchi, Vladimir I. Fal’ko, and David Goldhaber-Gordon. Ballistic mini- band conduction in a graphene superlattice. Science , 353(6307):1526–1529, 2016. [6] Zhiyong Wang, Chi Tang, Raymond Sachs, Yafis Bar- las, and Jing Shi. Proximity-Induced Ferromagnetism in Graphene Revealed by the Anomalous Hall Effect. Phys. Rev. Lett. , 114:016603, Jan 2015. [7] Johannes Christian Leutenantsmeyer, Alexey A Kaverzin, Magdalena Wojtaszek, and Bart J van Wees. Proximity induced room temperature ferromag- netism in graphene probed with spin currents. 2D Materials , 4(1):014001, 2017. [8] D. K. Efetov, L. Wang, C. Handschin, K. B. Efetov, J. Shuang, R. Cava, T. Taniguchi, K. Watanabe, J. Hone,C. R. Dean, and P. Kim. Specular interband Andreev re- flections at van der Waals interfaces between graphene and NbSe2. Nature Physics , 12:328–, December 2015. [9] Landry Bretheau, Joel I-Jan Wang, Riccardo Pisoni, Kenji Watanabe, Takashi Taniguchi, and Pablo Jarillo- Herrero. Tunnelling spectroscopy of Andreev states in graphene. Nature Physics , 13:756–, May 2017. [10] C. L. Kane and E. J. Mele. Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett. , 95:226801, Nov 2005. [11] Sergej Konschuh, Martin Gmitra, and Jaroslav Fabian. Tight-binding theory of the spin-orbit coupling in graphene. Phys. Rev. B , 82:245412, Dec 2010. [12] A. H. Castro Neto and F. Guinea. Impurity-Induced Spin-Orbit Coupling in Graphene. Phys. Rev. Lett. , 103:026804, Jul 2009. [13] Wei Han, Roland K. Kawakami, Martin Gmitra, and Jaroslav Fabian. Graphene spintronics. Nat Nano , 9(10):794–807, October 2014. [14] Martin Gmitra and Jaroslav Fabian. Graphene on transition-metal dichalcogenides: A platform for prox- imity spin-orbit physics and optospintronics. Phys. Rev. B, 92:155403, Oct 2015. [15] Marc Dr¨ ogeler, Frank Volmer, Maik Wolter, Bernat Terr´ es, Kenji Watanabe, Takashi Taniguchi, Gernot G¨ untherodt, Christoph Stampfer, and Bernd Beschoten. Nanosecond Spin Lifetimes in Single- and Few-Layer Graphene-hBN Heterostructures at Room Temperature. Nano Letters , 14(11):6050–6055, 2014. PMID: 25291305. [16] Simranjeet Singh, Jyoti Katoch, Jinsong Xu, Cheng Tan, Tiancong Zhu, Walid Amamou, James Hone, and Roland Kawakami. Nanosecond spin relaxation times in single layer graphene spin valves with hexagonal boron nitride tunnel barriers. Applied Physics Letters , 109(12):122411, 2016. [17] J. Ingla-Ayn´ es, Marcos H. D. Guimar˜ aes, Rick J. Mei- jerink, Paul J. Zomer, and Bart J. van Wees. 24 µm length spin relaxation length in boron nitride encapsu- lated bilayer graphene. arXiv:1506.00472 , 2015. [18] Aron W. Cummings, Jose H. Garcia, Jaroslav Fabian, and Stephan Roche. Giant Spin Lifetime Anisotropy in Graphene Induced by Proximity Effects. Phys. Rev. Lett. , 119:206601, Nov 2017. [19] Martin Gmitra and Jaroslav Fabian. Proximity Effects in Bilayer Graphene on Monolayer wse 2: Field-Effect Spin Valley Locking, Spin-Orbit Valve, and Spin Transistor. Phys. Rev. Lett. , 119:146401, Oct 2017. [20] Jun Yong Khoo, Alberto F. Morpurgo, and Leonid Lev- itov. On-Demand SpinOrbit Interaction from Which- Layer Tunability in Bilayer Graphene. Nano Letters , 17(11):7003–7008, 2017. PMID: 29058917. [21] Jose H. Garcia, Aron W. Cummings, and Stephan Roche. Spin Hall Effect and Weak Antilocalization in Graphene/Transition Metal Dichalcogenide Heterostruc- tures. Nano Letters , 0(0):null, 2017. PMID: 28715194. [22] Martin Gmitra, Denis Kochan, Petra H¨ ogl, and Jaroslav Fabian. Trivial and inverted Dirac bands and the emergence of quantum spin Hall states in graphene on transition-metal dichalcogenides. Phys. Rev. B , 93:155104, Apr 2016. [23] Zhe Wang, DongKeun Ki, Hua Chen, Helmuth Berger, Allan H. MacDonald, and Alberto F. Morpurgo. Strong interface-induced spin-orbit interaction in graphene on WS2. Nature Communications , 6:8339–, September 2015. [24] Zhe Wang, Dong-Keun Ki, Jun Yong Khoo, Diego10 Mauro, Helmuth Berger, Leonid S. Levitov, and Al- berto F. Morpurgo. Origin and Magnitude of ‘De- signer’ Spin-Orbit Interaction in Graphene on Semicon- ducting Transition Metal Dichalcogenides. Phys. Rev. X , 6:041020, Oct 2016. [25] Bowen Yang, Min-Feng Tu, Jeongwoo Kim, Yong Wu, Hui Wang, Jason Alicea, Ruqian Wu, Marc Bockrath, and Jing Shi. Tunable spin-orbit coupling and symmetry- protected edge states in graphene/WS2. 2D Materials , 3(3):031012, 2016. [26] Tobias V¨ olkl, Tobias Rockinger, Martin Drienovsky, Kenji Watanabe, Takashi Taniguchi, Dieter Weiss, and Jonathan Eroms. Magnetotransport in heterostructures of transition metal dichalcogenides and graphene. Phys. Rev. B , 96:125405, Sep 2017. [27] Bowen Yang, Mark Lohmann, David Barroso, Ingrid Liao, Zhisheng Lin, Yawen Liu, Ludwig Bartels, Kenji Watanabe, Takashi Taniguchi, and Jing Shi. Strong electron-hole symmetric Rashba spin-orbit coupling in graphene/monolayer transition metal dichalcogenide het- erostructures. Phys. Rev. B , 96:041409, Jul 2017. [28] Taro Wakamura, Francesco Reale, Pawel Palczynski, So- phie Gu´ eron, Cecilia Mattevi, and H´ el` ene Bouchiat. Strong Spin-Orbit Interaction Induced in Graphene by Monolayer WS 2.arXiv:1710.07483 , 2017. [29] Denis Kochan, Susanne Irmer, and Jaroslav Fabian. Model spin-orbit coupling Hamiltonians for graphene sys- tems. Phys. Rev. B , 95:165415, Apr 2017. [30] Talieh S. Ghiasi, Josep Ingla-Ayn´ es, Alexey A. Kaverzin, and Bart J. van Wees. Large proximity- induced spin lifetime anisotropy in transition-metal dichalcogenide/graphene heterostructures. Nano Letters , 17(12):7528–7532, 2017. PMID: 29172543. [31] L. A. Ben´ ıtez, J. F. Sierra, W. Savero Torres, A. Ar- righi, F. Bonell, M. V. Costache, and S. O. Valenzuela. Strongly anisotropic spin relaxation in graphene/WS 2 van der Waals heterostructures. Nature Physics , 2017. [32] P. J. Zomer, M. H. D. Guimares, J. C. Brant, N. Tombros, and B. J. van Wees. Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride. Applied Physics Letters , 105(1), 2014. [33] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean. One-Dimensional Electrical Contact to a Two- Dimensional Material. Science , 342(6158):614–617, 2013. [34] Lucas Basnzerus, Kenji Watanabe, Takashi Taniguchi, Bern Beschoten, and Christoph Stampfer. Dry transfer of CVD graphene using MoS2-based stamps. Physics Status Solidi RRL , 11:1700136, 2017. [35] A. F. Young, C. R. Dean, L. Wang, H. Ren, P. Cadden- Zimansky, K. Watanabe, T. Taniguchi, J. Hone, K. L. Shepard, and P. Kim. Spin and valley quantum Hall fer- romagnetism in graphene. Nature Physics , 8:550–, May 2012. [36] Tarik P. Cysne, Tatiana G. Rappoport, Jose H. Gar- cia, and Alexandre R. Rocha. Quantum Hall Effect in Graphene with Interface-Induced Spin-Orbit Coupling. arXiv:1711.04811 , 2017. [37] S. Das Sarma, Shaffique Adam, E. H. Hwang, and Enrico Rossi. Electronic transport in two-dimensional graphene. Rev. Mod. Phys. , 83:407–470, May 2011. [38] G. Bergmann. Weak anti-localization - An experimentalproof for the destructive interference of rotated spin 1/2. Solid State Communications , 42(11):815 – 817, 1982. [39] P. A. Lee, A. Douglas Stone, and H. Fukuyama. Univer- sal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field. Phys. Rev. B, 35:1039–1070, Jan 1987. [40] M. B. Lundeberg, J. Renard, and J. A. Folk. Conduc- tance fluctuations in quasi-two-dimensional systems: A practical view. Phys. Rev. B , 86:205413, Nov 2012. [41] Edward McCann and Vladimir I. Fal’ko. z→−zSym- metry of Spin-Orbit Coupling and Weak Localization in Graphene. Phys. Rev. Lett. , 108:166606, Apr 2012. [42] E. McCann, K. Kechedzhi, Vladimir I. Fal’ko, H. Suzuura, T. Ando, and B. L. Altshuler. Weak- Localization Magnetoresistance and Valley Symmetry in Graphene. Phys. Rev. Lett. , 97:146805, Oct 2006. [43] Mark B. Lundeberg and Joshua A. Folk. Rippled Graphene in an In-Plane Magnetic Field: Effects of a Random Vector Potential. Phys. Rev. Lett. , 105:146804, Sep 2010. [44] F. V. Tikhonenko, D. W. Horsell, R. V. Gorbachev, and A. K. Savchenko. Weak Localization in Graphene Flakes. Phys. Rev. Lett. , 100:056802, Feb 2008. [45] WL measuremetns in hBN/Gr/hBN devices revealed in- tervalley scattering times on the order of pico seconds. [46] Nuno J. G. Couto, Davide Costanzo, Stephan Engels, Dong-Keun Ki, Kenji Watanabe, Takashi Taniguchi, Christoph Stampfer, Francisco Guinea, and Alberto F. Morpurgo. Random Strain Fluctuations as Domi- nant Disorder Source for High-Quality On-Substrate Graphene Devices. Phys. Rev. X , 4:041019, Oct 2014. [47] Rafik Addou and Robert M. Wallace. Surface Analysis of WSe2 Crystals: Spatial and Electronic Variability. ACS Applied Materials & Interfaces , 8(39):26400–26406, 2016. PMID: 27599557. [48] T. Taniguchi and K. Watanabe. Synthesis of high-purity boron nitride single crystals under high pressure by using BaBN solvent. Journal of Crystal Growth , 303(2):525 – 529, 2007. [49] H. Ochoa, A. H. Castro Neto, and F. Guinea. Elliot-Yafet Mechanism in Graphene. Phys. Rev. Lett. , 108:206808, May 2012. [50] M. I. Dyakonov and V. I. Perel. Spin Relaxation of Con- duction Electrons in Noncentrosymmetric Semiconduc- tors. Sov. Phys. Solid State , 13(12):3023–3026, 1972.Supplemental Material: Large spin relaxation anisotropy and valley-Zeeman spin-orbit coupling in WSe2/Gr/hBN heterostructures Simon Zihlmann,1,∗Aron W. Cummings,2Jose H. Garcia,2M´ at´ e Kedves,3Kenji Watanabe,4Takashi Taniguchi,4Christian Sch¨ onenberger,1and P´ eter Makk1, 3,† 1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland 2Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3Department of Physics, Budapest University of Technology and Economics and Nanoelectronics ’Momentum’ Research Group of the Hungarian Academy of Sciences, Budafoki ut 8, 1111 Budapest, Hungary 4National Institute for Material Science, 1-1 Namiki, Tsukuba, 305-0044, Japan (Dated: December 19, 2017) FABRICATION AND MEASUREMENT DETAILS Graphene (obtained from natural graphite, NGS), WSe 2(obtained from hQgraphene) and hBN (grown by Taniguchi and Watanabe) were exfoliated with Nitto tape onto Si wafers with 300 nm of SiO 2. The WSe 2/Gr/hBN vdW- heterostructures were assembled using a dry pick-up method developed by Zomer et al. [1]. After the assembly of the vdW-heterostructures, the stacks were annealed in H 2/N2mixture for 100 min at 200◦C to remove polymer residues and to make the stack more homogeneous (merging of bubbles). Higher temperatures were avoided in order not to damage the WSe 2layer. The stacks were shaped into a Hall-bar mesa by standard e-beam lithography and reactive ion etching using a SF 6, O 2and Ar-based plasma. One-dimensional side contacts were then fabricated with e-beam lithography and the evaporation of 10 nm Cr and 50 nm Au. Lift-off was performed in warm acetone. In order to insulate the top gate from the exposed graphene at the edge of the mesa, an insulating MgO layer was evaporated before the Ti/Au of the top gate. Standard low frequency lock-in techniques were used to measure differential conductance and resistance in two- and four-terminal configuration. The samples were measured in a3He system at temperatures down to 0 .25 K and in a variable temperature insert at temperature of 1 .8 K and higher. The magnetic in-plane field was applied using a vector magnet. The small misalignment of the sample plane with the in-plane magnetic field was compensated by a finite offset field in the out-of-plane direction. This offset was found to scale linearly with the applied in-plane field. The back and top gate lever arms ( αBG,αTG) were found from Hall measurements and the charge carrier density in the graphene was calculated using a simple capacitance model, n=αBG/parenleftbig VBG−V0 BG/parenrightbig +αTG/parenleftbig VTG−V0 TG/parenrightbig , (1) whereV0 BGandV0 BGaccount for some offset doping of the graphene. Similarly, the applied electric field (field direction out of plane) was obtained: E=1 dBG/parenleftbig VBG−V0 BG/parenrightbig −1 dTG/parenleftbig VTG−V0 TG/parenrightbig , (2) wheredBGanddTGdenote the thickness of the back and top gate dielectric. The thicknesses of the bottom WSe 2 flake and the top hBN flake were determined by atomic force microscopy. To account for the residual doping, the density was corrected in the following way: ncorr=/radicalbig n2+n2∗. It was the corrected density ncorrthat was used for the calculation of the diffusion constant via the Einstein relation and for the estimation of the Fermi energy. FITTING OF MAGNETO CONDUCTIVITY DATA FROM DEVICE B As mentioned in the main text, a second device B was investigated as well. A gate-gate map of the resistivity of device B is shown in Fig. 1 (a). A field effect mobility of ∼25 000 cm2V−1s−1and a residual doping of ∼7×1010cm−2 were found. The quantum correction to the magneto conductivity was measured at the charge neutrality point for different electric fields. The same analysis was performed as mentioned in the main text. The extracted quantum correction to the magneto conductivity was also fit using Eq. 1 from the main text considering the three different cases as elaborated in the main text. Since the quality of device B is higher than that of device A, the diffusion constant is larger and hence the mean free path lmfpis longer. This leads to a much smaller transport field as thisarXiv:1712.05678v2 [cond-mat.mes-hall] 18 Dec 20172 scales with l−2 mfp. Therefore, the fitting range here was limited to 12 mT, which poses serious limits on the quality of the fit. It is very difficult to independently extract the different spin-orbit scattering times as obviously seen in Fig. 1, where basically all three fits overlap. Only at larger fields would the three fits be distinguishable. However, the time scales extracted here do not contradict the results presented in the main text. The strength of the total SOC, captured in τSO, is roughly the same for all three fits. As can be seen in Fig. 1, the total spin-orbit scattering timeτSOis more robust with respect to different fitting limits. Therefore, we only consider τSOfor device B in the next section. -0.20-0.15-0.10-0.050.00Δσ (e2/h) -15-10-5051015 Bz (mT) sample B fit1 fit2 fit3(b) fit1fit2fit3 τφ3.93.84.2 τasy0.340.521.2 τsym4.60.550.18 τSO0.320.270.16 D= 0.29 m2/s Btr = 12 mT -4-2024VTG (V) -30-20-10 0102030 VBG (V) 4e+07 2e+07 0 -2e+07 4e+12 3e+12 2e+12 1e+12 0 -1e+12 -2e+12 -3e+12 2000 1500 1000 5000ρ (Ω)(a) FIG. 1. Data from device B: (a) shows the resistivity as a function of vTGandVBG. Constant density contours are indicated with red solid lines and constant electric field contours is solid black lines. (b) shows the quantum correction to the magneto conductivity of device B at zero electric field within a density range of −5×1011cm−2to 5×1011cm−2. The same procedure as described in the main text was used. The results for three different limits are shown and their parameters are indicated. The fitting was restricted to the range of the transport field B tr= 12 mT. ELECTRIC FIELD DEPENDENCE OF THE SPIN-ORBIT SCATTERING RATES The presence of a top and a back gate in our devices allows us to tune the carrier density and the transverse electric field independently in devices A and B. In the case of device A, the SOC strength was found to be electric field independent at the CNP in the range of −5×107V/m to 8 ×107V/m as shown in Fig 2. The electric field range was limited by the fact that at large positive gate voltages the Fermi energy was shifted into the conduction band of the WSe 2whereas at large negative gate voltages gate instabilities occurred. Within the investigated electric field rangeτasywas found to be in the range of 5 ps to 10 ps, always close to τφ.τsymon the other hand was found to be around 0.1 ps to 0.3 ps whileτpwas around 0 .2 ps to 0.3 ps for device A. The total spin-orbit scattering time τSOis mostly given by τsym. Device B, where only τφandτSOcould be extracted reliably, shows similar results as device A. Therefore, we conclude that the in the electric field range −5×107V/m to 8 ×107V/m no tuning of the SOC strength with electric field is observed. From first principles calculations, the Rashba SOC is expected to change by 10 % if the electric field is tuned by 1 ×109V m−1and also the intrinsic and valley-Zeeman SOC parameters are expected to change slightly [2]. However, within the resolution of the extraction of the spin-orbit scattering time scales, we cannot establish a clear trend. These findings are in contrast to previous studies that found an electric field tunability of τasyandτSOon a similar electric field scale in graphene/WSe 2devices [3]. However, it is important to note that the changes are small and since no error bars are given, it is hard to tell if the three data points show a clear trend. Another study found a linear tunability of τasyof roughly 10 % on a similar electric field scale in graphene/WS 2devices [4]. There, τsymwas neglected with the argument that it cannot lead to spin relaxation. However, it was shown that τsymcan lead to spin relaxation [5] and therefore it cannot be neglected in the analysis. In our case, it is the dominating spin relaxation mechanism.3 6810-13246810-12246810-11τ (s) 80x106 6040200-20-40-60 E-field (V/m) τφ, B τSO, B τp, B τφ, A τasy, A τsym, A τSO, A τp, A FIG. 2. Electric field dependence of device A and B: The extracted spin-orbit scattering time scales τasy,τsym,τSO andτφwere extracted for different perpendicular electric field around the charge neutrality point. In addition, the momentum scattering time τpextracted from the diffusion constant is also shown. In the case of device B, only the total spin-orbit scattering timeτSOis given, as a reliable extraction of τasyandτsymwas not possible in this device (see discussion above). SPIN RELAXATION ANISOTROPY Cummings et al. have found a giant spin relaxation anisotropy in systems with strong valley-Zeeman SOC that is commonly found in graphene/TMDC heterostructures [5]. They derived the following equation: τ⊥ τ/bardbl=/parenleftbiggλVZ λR/parenrightbigg2τiv τp+ 1/2 (3) whereτ⊥is the out-of-plane spin relaxation time, τ/bardblthe in-plane spin relaxation time, λVZis the SOC strength of the valley-Zeeman SOC, λRis the SOC strength of the Rashba SOC and τivandτprepresent the intervalley and momentum scattering times respectively. If a strong intervalley scattering is assumed, which is a prerequisite for the application of the WAL theory [6], τ⊥is given by τasy/2 andτ/bardblis given by τsym. We therefore get a spin relaxation anisotropy τ⊥/τ/bardbl≈τasy/2τsym≈20, which is much larger than what is expected for usual 2D Rashba systems. Furthermore, assuming a ratio of τiv/τp≈1, which corresponds to very strong intervalley scattering, a ratio ofλVZ/λR≈6 is expected. ESTIMATE OF VALLEY-ZEEMAN SOC STRENGTH For a valley-Zeeman SOC strength λVZ, the spin splitting is 2 λVZand the precession frequency is ω= 2λVZ//planckover2pi1. In the D’yakonov-Perel’ (DP) regime of spin relaxation, when ωτiv<1, the in-plane spin relaxation rate is τ−1 s/bardbl= (2λVZ//planckover2pi1)2τiv. However, if ωτiv>1, then the spin can fully precess before scattering randomizes the spin-orbit field, and the spin lifetime scales with the intervalley time, τs/bardbl= 2τiv. A plot of these two regimes is shown below, where we have taken our derived limits of λVZ= 0.23 and 2.3 meV (see below) as well as the DFT-derived value of 1 .19 meV. Considering this behavior, the condition τs/bardbl≥2τivshould always be satisfied. Meanwhile, our measurements revealedτsym= 0.2 ps andτiv≈0.1−1 ps, which violates this condition for all except the smallest value of τiv. One way to account for this is to consider the impact of spin-orbit disorder on the in-plane spin lifetime. Assuming that theτs/bardblfrom uniform valley-Zeeman SOC is given by 2 τiv, and the rest comes from spin-orbit disorder, we can estimate an upper bound of λVZ=/planckover2pi1//radicalbig 4(2τiv)τiv= 0.23 meV to 2 .3 meV. Another possibility is that since our measurements are right around the transition point ωτiv= 1, we could be extracting the in-plane spin precession frequency; τ−1 sym=ω. Doing so would give λVZ=/planckover2pi1/2τsym= 1.6 meV, which fits in the range derived above. Overall, since the experiments appear to be close to this transition point, all methods of deriving the strength of λVZtend to give similar values, from a few tenths up to a few meV depending on the4 FIG. 3. Dependence of in-plane spin relaxation time τs/bardblon intervalley scattering time τiv. Red and blue lines show the dependence in the DP regime of spin relaxation, for the largest and smallest estimated values of λVZ. The black dashed line show the value derived from DFT [2]. The green line shows the dependence in the coherent spin precession regime. estimate of τiv. We would like to note that it is not fully understood how the spin precession frequency enters into the WAL correction and how the corresponding SOC strength would then be extracted. Therefore, further theoretical work is needed. DATA FROM DEVICE D The third sample with device D is a WSe 2/Gr/hBN stack with a very thin WSe 2(3 nm) as substrate. The gate-gate map of the two terminal resistance is shown in Fig. 4 (a). Due to the very thin bottom WSe 2the mobility in this device is around 50 000 cm2V−1s−1and a residual doping of 2 .5×1011cm−2is found. A typical magneto conductivity trace of this device is shown in Fig. 4. Mostly, positive magneto conductivity is observed with only a very small feature that shows negative magneto conductivity at 30 mK, which was absent at 1 .8 K. The magneto conductivity of device D could not be fitted with the standard WAL formula presented in the main text. However, similar curve shapes could be reproduced by including the influence of τivandτ∗. A complete formula can be derived from equation 9 of Ref. [6]. If all relaxation gaps are included and if disorder SOC is neglected one arrives at the following form: ∆σ(B) =−e2 2πh/bracketleftBigg F/parenleftBigg τ−1 B τ−1 φ/parenrightBigg −F/parenleftBigg τ−1 B τ−1 φ+ 2τ−1asy/parenrightBigg −2F/parenleftBigg τ−1 B τ−1 φ+τ−1asy+τ−1sym/parenrightBigg −F/parenleftBigg τ−1 B τ−1 φ+ 2τ−1 iv/parenrightBigg −2F/parenleftBigg τ−1 B τ−1 φ+τ−1 ∗/parenrightBigg +F/parenleftBigg τ−1 B τ−1 φ+ 2τ−1 iv+ 2τ−1asy/parenrightBigg + 2F/parenleftBigg τ−1 B τ−1 φ+τ−1 ∗+ 2τ−1asy/parenrightBigg +2F/parenleftBigg τ−1 B τ−1 φ+ 2τ−1 iv+τ−1asy+τ−1sym/parenrightBigg + 4F/parenleftBigg τ−1 B τ−1 φ+τ−1 ∗+τ−1asy+τ−1sym/parenrightBigg/bracketrightBigg .(4) However, the addition of two more parameters makes it very hard to unambiguously extract all parameters exactly. Therefore, we do not extract any spin-orbit time scales from this device. The influence of τivandτ∗are much weaker for the data presented in the main text. The long phase coherence time τφ∼25 ps is attributed to the lower temperature (T= 30 mK) at which the mea- surement was performed. At higher temperature (1 .8 K), the phase coherence is significantly shorter ∼4 ps(broader5 4 2 0 -2 -4VTG (V) -40 -20 0 20 40 VBG (V) 3e+08 2e+08 1e+08 0 -1e+08 -2e+08 -3e+08 6e+12 4e+12 2e+12 0 -2e+12 -4e+12 (a) 15 10 5R2terminal (kΩ) 0.5 0.4 0.3 0.2 0.1 0.0Δσ (e2/h) -20-10 01020 Bz (mT)τφ = 2.5e-11 s τasy = 2.8e-11 s τsym = 3.0e-11 s τiv = 1.5e-12 s τ* = 8.0e-13 s D = 0.041 m/s2 30 mK standard WAL complete WAL 1.8 K(b) FIG. 4. Data from device D: (a) shows a gate-gate map of the two-terminal resistance of device D. Constant density (red solid line, in units of cm−2) and electric field (black solid lines, in units of V m−1) lines are superimposed on top of that. (b) shows the quantum quantum correction of the magneto conductivity at zero electric field in the density range of −5×1011cm−2 to 5×1011cm−2. It shows a WL dip with a tiny feature of WAL around zero Bzat a temperature of 30 mK. A possible fit (red) and its parameters, including the influence of τivandτ∗, are indicated. The low magnetic field range can be reasonably well described by the standard WAL formula without τivandτ∗. As a comparison, the magneto conductivity is also shown at 4 K. This trace is vertically offset by −0.06 e2/h for clarity. dip and reduced overall correction) and the influence of the SOC on the magneto conductivity (WAL) is not observed any longer. Bothτasyandτsymseem to be very close to τφin sample D. In particular, τsymis much longer than in the devices presented in the main text. We conclude that even though there is some indication of SOC in sample D, its overall strength must be smaller than in the devices presented in the main text. Certainly the SOC relevant for τsymmust be smaller as this time scale is two orders of magnitude longer than in device A and B. This large difference cannot be explained by the shorter τpthat is roughly a factor of 5 shorter in device D than in device A and B. INFLUENCE OF WSE 2QUALITY In addition to WSe 2obtained from hQ graphene, we also investigated devices with WSe 2obtained from Nanosurf as an alternative source. In general, devices with WSe 2from Nanosurf showed more gate instabilities. Some devices showed mobilities around 20 000 cm2V−1s−1. Magneto conductivity was measured in order to investigate possible enhanced SOC, but in none of the devices we did we find a pronounced WAL signature. Some devices showed signatures of WL, whereas some did not show any clear magneto conductivity. For some devices it was impossible to measure magneto conductivity as the devices were not stable enough. ∗Simon.Zihlmann@unibas.ch †peter.makk@mail.bme.hu [1] P. J. Zomer, M. H. D. Guimares, J. C. Brant, N. Tombros, and B. J. van Wees. Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride. Applied Physics Letters , 105(1), 2014. [2] Martin Gmitra, Denis Kochan, Petra H¨ ogl, and Jaroslav Fabian. Trivial and inverted Dirac bands and the emergence of quantum spin Hall states in graphene on transition-metal dichalcogenides. Phys. Rev. B , 93:155104, Apr 2016. [3] Tobias V¨ olkl, Tobias Rockinger, Martin Drienovsky, Kenji Watanabe, Takashi Taniguchi, Dieter Weiss, and Jonathan Eroms. Magnetotransport in heterostructures of transition metal dichalcogenides and graphene. Phys. Rev. B , 96:125405, Sep 2017.6 [4] Bowen Yang, Min-Feng Tu, Jeongwoo Kim, Yong Wu, Hui Wang, Jason Alicea, Ruqian Wu, Marc Bockrath, and Jing Shi. Tunable spin-orbit coupling and symmetry-protected edge states in graphene/WS2. 2D Materials , 3(3):031012, 2016. [5] Aron W. Cummings, Jose H. Garcia, Jaroslav Fabian, and Stephan Roche. Giant Spin Lifetime Anisotropy in Graphene Induced by Proximity Effects. Phys. Rev. Lett. , 119:206601, Nov 2017. [6] Edward McCann and Vladimir I. Fal’ko. z→−zSymmetry of Spin-Orbit Coupling and Weak Localization in Graphene. Phys. Rev. Lett. , 108:166606, Apr 2012.

1403.6159v1.Optical_spin_injection_in_graphene_with_Rashba_spin_orbit_interaction.pdf

arXiv:1403.6159v1 [cond-mat.mes-hall] 24 Mar 2014Optical spin injection in graphene with Rashba spin-orbit i nteraction M. Inglot,1V. K. Dugaev,1,2E. Ya. Sherman,3,4and J. Barna´ s5,6 1Department of Physics, Rzesz´ ow University of Technology, al. Powsta´ nc´ ow Warszawy 6, 35-959 Rzesz´ ow, Poland 2Departamento de F´ ısica and CFIF, Instituto Superior T´ ecn ico, Universidade de Lisboa, av. Rovisco Pais, 1049-001 Lisbon, Portugal 3Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080 Bilbao, Spain 4IKERBASQUE Basque Foundation for Science, Bilbao, Spain 5Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Pozna´ n, Poland 6Institute of Molecular Physics, Polish Academy of Sciences , ul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland (Dated: October 8, 2018) We calculate the efficiency of infrared optical spin injectio n in single-layer graphene with Rashba spin-orbit coupling and for in-plane magnetic field. The inj ection rate in the photon frequency range corresponding to the Rashba splitting is shown to be proport ional to the ratio of the Zeeman and Rashbasplittings. Asaresult, large spinpolarization can becontrollably achievedfor experimentally available values of the spin-orbit coupling and in magnetic fields below 10 Tesla. PACS numbers: 72.25.Fe, 78.67.Wj, 81.05.ue, 85.75.-d I. INTRODUCTION Graphene–a two-dimensionalhexagonallattice ofcar- bon atoms – was discovered about eight years ago1–3 and is now one of the most promising materials for fu- ture nanoelectronics. The high application potential of this novel material is associated with some peculiari- ties of its electronic and phonon transport properties4 as well as with its outstanding mechanical5,6and optical properties7. Optoelectronic properties of graphene are also very promising for applications8. Moreover, owing to a very long spin relaxation time, which is expected due to a very weak spin-orbit interaction, graphene is also attractive for applications in spin electronics (see, e.g., Ref. [9]). However, to utilize the outstanding properties of graphene for spin-dependent transport, one needs to have a reliable method of controllable spin injection and spin manipulation. The possibility of a relatively strong Rashba spin-orbit coupling has been reported in Refs. [10,11] for graphene deposited on a Ni (or Ni/Au) substrate. Such a strong spin-orbit coupling formally en- ables spin manipulation in graphene. However, even in the absence of a substrate leading to strong spin-orbit coupling, experiments report spin relaxation time on the timescaleoftheorderoforlessthanonenanosecond12–14, which makes applications of graphene in spin electron- ics rather difficult. In all the experiments aimed at the measurements of spin relaxation time, spins are injected en masse from a ferromagnetic contact giving rise to some charge/spin density distribution, which influences its subsequent dynamics. Several theoretical approaches (see for example Refs. [15–19]) have been proposed to describe spin relaxation. However, most of them demon- strated spin relaxation time much longer than that ob- served experimentally. There are several experimental techniques which canbe used to manipulate and control electron spin in graphene. For example, spin current and spin density in graphene nanodisks can be manipulated by varying length of the corresponding zigzag edge20. Quantum pumping of Dirac fermions and spin current in a mono- layer graphene in perpendicular magnetic field, with the gate voltage as a control parameter, has been proposed in Ref. [21]. Furthermore, the method of spin current generation in a monolayer graphene through adiabatic quantum pumping by two oscillating in time potentials has been described in Refs. [22,23] and for the bilayer graphene in Ref. [24]. It is well-known that spin-orbit coupling can lead to a direct optical spin injection - the technique extensively used in the physics of semiconductors25. For graphene, the spin-orbit coupling influences the optical response in the infrared frequency range26. In this paper we con- sider the infrared optical spin injection in a single-layer graphene by linearly polarized light. The graphene is as- sumed to be deposited on a substrate which leads to the Rashba spin-orbit coupling. Due to this interaction, the electronic spectrum of graphene near the Dirac points splits into four bands with parabolic dependence on the electron momentum at small wave vectors and linear dis- persion at large wave vectors27. Splitting of the sub- bands is determined by the spin-orbit coupling strength. We show that optical spin injection becomes allowed in the presence of an external magnetic field, and the injec- tion efficiency is of the order of the ratio of the Zeeman splitting and the spin-orbit coupling matrix element. By modifying the infrared light frequency, one can change the absorption region in the momentum space, and thus control the spin injection. In Sec. 2 we derive some general formula for optical spin injection efficiency in graphene. Numerical results on spin injection rate are presented and described in Sec. 3. Summary and final conclusions are in Sec. 4.2 II. SPIN INJECTION RATE AND EFFICIENCY Weassumeanexternalmagneticfield Borientedinthe graphene plane. Hamiltonian describing the low energy electron excitations near the Dirac point Kin graphene with the Rashba spin-orbit interaction takes then the form28 ˆH=v(τ·k)+g(B·σ)+λ(τxσy−τyσx),(1) whereg=gLµB/2,λ=α/2 withαbeing the cou- pling constant of Rashba spin-orbit interaction,27and gLis the Land´ e factor. The matrices τandσare the Pauli matrices in the sublattice and spin space, respec- tively. The third term of the above Hamiltonian stands for the Rashba spin-orbit coupling induced by the sub- strate. Note, the first term is diagonal in the spin space and forabbreviationthe correspondingunit matrix isnot written explicitly. Similarly, the second term is diagonal in the sublattice spaceand the correspondingunit matrix is not written explicitly, too. Electronic spectrum corresponding to the Hamiltonian (1) consists of four energy bands. In the limit of weak magnetic field, gB/λ→0, this spectrum is described by the formulas E(0) nk=±λ±(λ2+v2k2)1/2, (2) with all possible combination of the + and −signs, and the index nlabeling the bands in the order of increasing energy (see Fig. 1). As in the usual two-dimensional electron gas with Rashba spin-orbit interaction, the expectation value of the spin z-component in eigenstates of Hamiltonian (1) forB= 0 is equal to zero. However, unlike to the two- dimensional electron gas, the expectation value of the in- plane spin depends on the wave vector and is relatively small for low-energy electron states. Indeed, the expec- tation value, /angbracketleftΨ(0) nk|σ|Ψ(0) nk/angbracketright, of electron spin in the state Ψ(0) nkforB= 0 is given by27 s≡ /angbracketleftΨ(0) nk|σ|Ψ(0) nk/angbracketright=ξv(k׈z)√ λ2+v2k2, (3) whereξ=±1 is the band index, ξ= 1 forn= 2,3 and ξ=−1 forn= 1,4 (cf. Ref. [27]). Thus, when vk≪λ, the expectation value of spin is small, |s| ≪1. Moreover, the spins are perpendicular to the wave vectors, similarly as in two-dimensional electron gas. We note that the upper index (0) at the eigenfunctions and eigenenergies indicates they are for B= 0. In the following, we take the in-plane magnetic field B alongthe x-axis and assume it is ratherweak, gB/λ≪1. The former assumptions justifies the absence of Landau quantization, while the latter condition assures that the band dispersion is only weakly perturbed by the static magnetic field, and the resulting spin injection is linear in the applied magnetic field B. The four-band struc- ture is presented in Fig. 1, which also shows the assumed position of the Fermi level µ.FIG. 1: (Color online). Band structure of graphene with a Rashba spin-orbit interaction in the gB/λ≪1 limit. Ar- rows show the k-dependent orientation of electron spins in the eigenstates of the Hamiltonian. Here we consider only the vicinity of the KDirac point, taking into account that the other valley gives exactly the same result for light abso rp- tion and spin injection. Taking into account the first term of Eq. (1), Hamilto- nian describing interaction of graphene with an external periodic electromagnetic field A(t) =A0e−iωtcan be written as ˆHA=−ev /planckover2pi1c(τ·A). (4) This periodic perturbation leads to electron transitions between the bands shown in Fig. 1. The spin states of electrons involved in the transitions are then modified accordingly. Without loss of generality, we assume in the following that the electromagnetic field is oriented along they-axis,A0= (0,A0,0). The total absorption rate of photons can be calculated as the sum of all allowed transitions, I(ω) =/summationdisplay nn′In→n′(ω), (5) whereIn→n′(ω) correspondstothe absorptionassociated with the transitions of electrons from the subband nto the subband n′, which can be calculated from the Fermi golden rule as In→n′(ω) =2π /planckover2pi1/integraldisplayd2k (2π)2/vextendsingle/vextendsingle/vextendsingle/angbracketleftΨnk|ˆHA|Ψn′k/angbracketright/vextendsingle/vextendsingle/vextendsingle2 ×δ(Enk+/planckover2pi1ω−En′k)f(Enk)[1−f(En′k)].(6) Here, Ψ nkandEnkare the eigenfunctions and eigen- values of the total Hamiltonian (1), and f(Enk) is the corresponding Fermi distribution function. It is convenient to introduce an independent of the system parameter I0, defined as I0=ω 4/parenleftBige /planckover2pi1c/parenrightBig2 A2 0, (7)3 and rewrite Eq. (5) as I(ω) =I0/summationdisplay nn′/tildewideIn→n′(ω)≡I0/tildewideI(ω), (8) with the system-dependent functions /tildewideIn→n′(ω). SinceA2 0 in Eq. (7) is related to the incident flux qofy-polarized photons by the formula A2 0= 4π/planckover2pi1cq/ω, Eq. (7) can be presented in the form I0=πe2 /planckover2pi1cq. (9) The ratio I0/q=πe2//planckover2pi1ccorresponds to the absorp- tion coefficient of graphene without Rashba spin-orbit coupling.3,29,30In the limit of large frequency, /planckover2pi1ω≫λ, the absorption rate (8) is constant and does not de- pend on frequency, like in the case of graphene with zero Rashba coupling, I(ω)→I0. Thus,/tildewideI(ω) can be consid- eredasaratioofabsorptioncoefficientsforgraphenewith Rashba spin-orbit interaction and of graphene without Rashba interaction. In other words, /tildewideI(ω) is the absorp- tion coefficient normalized to that for graphene without Rashba interaction. Now, let us define the spin injection rate for the i-th component of the spin density. Following Eq. (6), we write Jn→n′ i(ω) =2π /planckover2pi1/integraldisplayd2k (2π)2/vextendsingle/vextendsingle/vextendsingle/angbracketleftΨnk|ˆHA|Ψn′k/angbracketright/vextendsingle/vextendsingle/vextendsingle2 ×(/angbracketleftΨn′k|σi|Ψn′k/angbracketright−/angbracketleftΨnk|σi|Ψnk/angbracketright) ×δ(Enk+/planckover2pi1ω−En′k)f(Enk)[1−f(En′k)].(10) Similarlytothecaseofabsorption,weintroducethe total spin injection rate as Ji(ω) =/summationtext n,n′Jn→n′ i(ω) and write Ji(ω) =I0/tildewideJi(ω) andJn→n′ i(ω) =I0/tildewideJn→n′ i(ω). Thus, /tildewideJi(ω) and/tildewideJn→n′ i(ω) can be considered as normalized to I0spin injection rates. Before discussing numerical re- sults based on the above formula, let us discuss briefly physical origin of the spin injection. In the absence of magnetic field, symmetry of the ma- trix elements and spin expectation values (as shown in Fig. 1) as well as the independence of energy Enkon the momentum orientationlead to zerospin injection rate, as required by the time-reversal symmetry. In an in-plane magnetic field, in turn, each subband is shifted in en- ergy,Enk−E(0) nk=g/parenleftBig /angbracketleftΨ(0) nk|σ|Ψ(0) nk/angbracketright·B/parenrightBig .As a result, the lines of energy conservation, Enk+/planckover2pi1ω−En′k= 0, for the transitions changing the electron spin, such as 1→3 and 2 →4, are not simple circles anymore and acquire a distortion of the order of ( gB/λ)cosϕ, where ϕis the angle between kand the x-axis. In addition, the 4-component wave functions are modified in the first order perturbation as Ψnk−Ψ(0) nk=g/summationdisplay n′/angbracketleftBig Ψ(0) n′k/vextendsingle/vextendsingle/vextendsingle(σ·B)/vextendsingle/vextendsingle/vextendsingleΨ(0) nk/angbracketrightBig E(0) nk−E(0) n′kΨ(0) n′.(11)Accordingly, expectation values of the spin components /angbracketleftΨnk|σi|Ψnk/angbracketrightand of the interband matrix elements /angbracketleftΨnk|ˆHA|Ψn′k/angbracketrightacquirefirst-ordermodificationintheap- plied magnetic field, which results in a nonzero spin in- jection. III. INFRARED ABSORPTION AND SPIN INJECTION: NUMERICAL RESULTS Now we present some numerical results on the absorp- tion of linearly polarized light and the associated spin injection. In our calculations we assumed the tempera- tureT= 1 K. FIG. 2: (Color online). (a) The normalized absorption coef- ficients corresponding to indicated interband transitions , cal- culated for α= 2λ= 13 meV. (b) The total normalized ab- sorption coefficient for twodifferentvalues oftheRashbaspi n- orbit parameter, as indicated. Both figures are calculated f or magnetic field B= 5 T and for the chemical potential µ= 5 meV. Let us begin with the normalized absorption coeffi- cients presented in Fig. 2 as a function of the frequency ω. Figure 2a shows the normalized absorption coeffi- cients for individual inter-band transitions, while Fig. 2b shows the total normalized absorption coefficient for two different values of the Rashba parameter α. In the lat- ter case, the thin solid line corresponds to the absorption4 coefficient in the absence of Rashba coupling. As one can see in Fig. 2, the spin-orbit coupling strongly modifies the absorption, in agreement with the resultsofRef. [26]. The frequencythresholdforthe inter- band transitions is determined by the Pauli blocking and also depends on the chemical potential µand the spin- orbit splitting 2 λ. In the case considered here, µ <2λ, the transitions of highest frequency occur between the n= 1 and n′= 4 bands and start at the K-point with zero matrix element. At high frequencies, /planckover2pi1ω≫λ, the total absorption coefficient approaches that for a pure single-layer graphene without spin-orbit coupling. FIG. 3: (Color online). (a) The total injection rate for the spin component σxin a magnetic field of B= 5 T parallel to thex-axis and external electromagnetic field polarized along they-axis. The dashed green (solid red) line corresponds to the coupling constant α= 2λ= 13 meV ( α= 2λ= 5 meV). All results are for the chemical potential µ= 5 meV. (b) Spin injection efficiency for spin change per absorbed photon for the same parameters and injection geometry. Let us consider now numerical results on spin injection showninFig.3forthe magneticfield B= 5T,whichcor- responds to the Zeeman splitting 2 gBof approximately 0.6 meV. We show there only the injection rate for the spinx-component, and for clarity we simplified there the notation by omitting the index i,˜Jx(ω)≡˜J(ω). Fig-ure 3a shows the total normalized spin injection rate as a function of frequency for two different values of the Rashba coupling parameter. The dependence of the spin injectionrateonthe lightfrequencyisrathercomplicated due to several interband transitions involved and com- plex dependence of the matrix elements on the transition frequency. For the spin-conserving transitions between the states characterized by the same ξin Eq. (3), such as 1→4 and 2→3, the main contribution to the spin injection comes from changes in the expectation values of/angbracketleftΨnk|σ|Ψnk/angbracketright, while for the other transitions all the changes in the system make comparable contributions. Note, the spin injected is opposite to the magnetic field. The spin injection efficiency can be defined as the av- erage spin injected by a single photon. This efficiency is given by |/tildewideJω)//tildewideI(ω)|, and is shown in Fig. 3b for the same Rashba parameters as in Fig. 3a. As one can see in Fig. 3b, the efficiency can reach 0.2 per incident photon. In general, the injection rate is of the order of gB/λ, and can be manipulated by changing the photon frequency in the range of the order of λ. Since the transitions are rather complicated, the ratio gB/λshould be considered as an order-of-magnitude estimate only. Similarspininjection, thoughconsiderablyweaker,can be obtained for the electric field along the magnetic field, i.e. along the x-axis. IV. SUMMARY AND CONCLUSIONS We have considered theoretically spin injection in single-layergrapheneinthepresenceofRashbaspin-orbit coupling and in-plane external magnetic field. We have found that the spin injection is efficient at frequencies of the order of spin-orbit band splitting, with the efficiency being of the order of the ratio of Zeeman and Rashba splittings. For experimentally achievable parameters of the spin- orbit coupling and magnetic field, the injection efficiency can achieve 0.2 per absorbed photon. This result shows that optical spin injection opens a way for a controllable andefficient method ofspindensity andspin currentgen- eration in graphene, not requiring presence of any ferro- magnetic contact. Acknowledgements This work is partly supported by the National Science Center in Poland as a research project in years 2011 – 2014 and Grants Nos. DEC-2011/01/N/ST3/00394 and DEC-2012/06/M/ST3/00042. TheworkofEYSwassup- ported by the University of Basque Country UPV/EHU under program UFI 11/55, Spanish MEC (FIS2012- 36673-C03-01), and ”Grupos Consolidados UPV/EHU del Gobierno Vasco” (IT-472-10).5 1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). 2Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005). 3M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge Univ. Press, 2012). 4D. L. Nika and A. A. Balandin, J. Phys.: Condens. Matter 24, 233203 (2012). 5A. R. Ranjbartoreh, B. Wang, X. Shen, and G. Wang, J. Appl. Phys. 109, 014306 (2011). 6H. Chen, M. B. Møller, K. J. Gilmore, G. G. Wallace, and D. Li, Adv. Mater. 20, 3557 (2008). 7L. A. Falkovsky, J. Phys.: Conf. Ser. 129, 012004 (2008). 8F. Bonaccorso, Z. Sun, T. Hasan, andA.C. Ferrari, Nature Photonics 4, 611 (2010). 9F. S. M. Guimar˜ aes, A. T. Costa, R. B. Muniz, and M. S. Ferreira, Phys. Rev. B 81, 233402 (2010) 10Y. S. Dedkov, M. Fonin, U. R¨ udiger, and C. Laubschat, Phys. Rev. Lett. 100, 107602 (2008). 11A. Varykhalov, J. S´ anchez-Barriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O. Rader, Phys. Rev. Lett. 101, 157601 (2008). 12N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature 448, 571 (2007). 13Wei Han and R. K. Kawakami, Phys. Rev. Lett. 107, 047207 (2011) 14T. Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M. Jaiswal, J. Samm, S. R. Ali, A. Pachoud, M. Zeng, M. Popinciuc, G. G¨ untherodt, B.Beschoten, andB. ¨Ozyilmaz, Phys. Rev. Lett. 107, 047206 (2011). 15V. K. Dugaev, E. Ya. Sherman, and J. Barna´ s, Phys. Rev. B83, 085306 (2011).16L. Wang and M. W. Wu, Phys. Rev. B 87, 205416 (2013); P. Zhang, Y. Zhou and M. W. Wu, J. Appl. Phys. 112, 073709 (2012). 17D. V. Fedorov, M. Gradhand, S. Ostanin, I. V. Maznichenko, A. Ernst, J. Fabian, and I. Mertig, Phys. Rev. Lett. 110, 156602 (2013), S. Konschuh, M. Gmitra, D.Kochan, andJ. Fabian, Phys.Rev.B 85, 115423 (2012). 18S. Fratini, D. Gos´ albez-Mart´ ınez, P. M. C´ amara, and J. Fern´ andez-Rossier, Phys. Rev. B 88, 115426 (2013). 19H. Ochoa, A. H. Castro Neto, and F. Guinea, Phys. Rev. Lett.108, 206808 (2012). 20M. Ezawa, Physica E 42, 703 (2010). 21R. P. Tiwari and M. Blaauboer, Appl. Phys. Lett. 97, 243112 (2010). 22Q. Zhang, K. S. Chan, and Z. Lin, Appl. Phys. Lett. 98, 032106 (2011). 23D. Greenbaum, S. Das, G. Schwiete, and P. G. Silvestrov, Phys. Rev. Lett. 75, 195437 (2007). 24J. F. Liu and K. S. Chan, Nanotechnology 22, 395201 (2011). 25Spin Physics in Semiconductors (Springer Series in Solid- State Sciences) by M.I. Dyakonov (Berlin, 2008) 26P. Ingenhoven, J. Z. Bern´ ad, U. Z¨ ulicke, and R. Egger, Phys. Rev. B 81, 035421 (2010). 27E. I. Rashba, Phys. Rev. B 79, 161409 (2009). 28C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 29V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006). 30A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008).

1808.08029v1.Residual_spin_susceptibility_in_the_spin_triplet__orbital_singlet_model.pdf

Residual spin susceptibility in the spin-triplet, orbital-singlet model Yue Yu,1Alfred K. C. Cheung,1S. Raghu,1, 2and D. F. Agterberg3 1Department of Physics, Stanford University, Stanford, CA 94305 2SLAC National Accelerator Laboratory, Menlo Park, CA 94025 3Department of Physics, University of Wisconsin, Milwaukee, WI 53201 Nuclear magnetic resonance (NMR) and Knight shift measurements are critical tools in the iden- tication of spin-triplet superconductors. We discuss the eects of spin orbit coupling on the Knight shift and susceptibilities for a variety of spin triplet multi-orbital gap functions with orbital-singlet character and compare their responses to "traditional" single band spin-triplet ( px+ipy) supercon- ductors. We observe a non-negligible residual spin-susceptibility at low temperature. I. INTRODUCTION A working denition for an unconventional supercon- ductor is one whose gap function averaged over the Fermi surface is less than the maximum value of the absolute value of the gap at any point on the Fermi surface. This allows for gaps which do not exhibit isotropic s-wave pairing. In particular, the possibility for pairing in odd parity channels allows for spin-triplet pairing , the most notable purported example being Sr 2RuO 4in which the simplest descriptions involve models with only intra-band pairing. For Sr 2RuO 4, there is still no consensus on the actual form of the superconducting gap with various ex- periments showing con icting results1{5. In light of this, it is useful to consider what other systems might be can- didates for realizing spin-triplet pairing. One recent work proposes that a dierent kind of spin-triplet pairing may be realized in iron-based superconductors that only pos- sess hole pockets6. Based on angle-resolved photoemis- sion spectroscopy and heat capacity measurements, the authors argue in favor of s-wave gaps in these materials and present a new mechanism for its realization. The key ingredients to their proposal are (1) the pres- ence of spin orbit coupling (SOC), and (2) the multi- orbital nature of these systems which introduces the pos- sibility for inter-band paired gap functions7,8. This re- sults in the stabilization of an even parity orbital-singlet, spin-triplet pairing state. Indeed, the s-wave iron-based superconductors are expected to exhibit sizable SOC9 and seem to have the ingredients necessary for the model proposed6,10{12. The same pairing state has also been the focus of study using dynamical mean eld theory8,13{15. These studies all reveal a pairing instability within the strong coupling limit. In order to evaluate the validity of the proposed model for the relevant iron-based supercondcutors, it is crucial to identify the experimental signatures in nuclear mag- netic resonance (NMR) and Knight shift measurements { key experimental testing grounds for unconventional su- perconductors2,16{19. To this end, in this article we com- pare the results in Knight shift between the well-studied single band spin-triplet state ~d(~k) = (0;0;kx+iky) and the inter-band model of Ref. 6. In the absence of SOC, we nd an invariant Knight shift which stays constant into the superconducting phase in every direction for theinter-band model. This distinguishes it from the intra- band (kx+iky)^zstate, where there is a drop in the Knight shift for a eld applied along the zdirection10,20. When including SOC, we observe a substantial decrease in spin susceptibilities, which agrees with previous the- oretical predictions6. However, we observe a non-zero residual spin susceptibility at low temperature, in con- trast with zero residual spin susceptibility for intra-band spin-singlet pairing. Our results on spin susceptibility and Knight shift reveal that the pairing state driven by SOC has both intra-band spin-singlet and inter-band spin-triplet properties. This article is organized as follows. In Section II, we introduce our mean eld theory model involving pairing in the orbital singlet, spin triplet channel. In Section III, we explain how observables in NMR and Knight shift experiments can be calculated within our model. Our results are discussed in Section IV. Concluding remarks appear in Section V. II. THE MODEL We consider a three band model for d-orbital electrons with tetragonal symmetry. The Hamiltonian is given by H=H0+HSOC+HBCS, where: H0=X ~k;a;b;hab 0(~k)cy ~kac~kb; (1) and: hab 0(~k) =2 4yz(~k)V(~k) 0 V(~k)xz(~k) 0 0 0 xy(~k)3 5; (2) is the kinetic energy part of the tight binding model with hybridization between xzandyzorbitals.cy ~ka c~ka
is the creation(annihilation) operator of electrons in orbital a=yz,xz, oryzand spin=";#. We consider a quasi- two dimensional material, where the dispersion in the z-direction is neglected. Here, the form and value of the unhybridized dispersions and the hybridization potentialarXiv:1808.08029v1 [cond-mat.supr-con] 24 Aug 20182 are given in Ref. 21: yz(~k) =02tcosky2t?coskx xz(~k) =02tcoskx2t?cosky xy(~k) =2t0(coskx+ cosky) + 4t00coskxcosky V(~k) =2Vsinkxsinky; (3) which was originally proposed for Sr 2RuO 4. However, we use this model only as a specic example; our analy-sis is not limited to this particular material. We choose our unit of energy to be tin the following analysis. With- out spin orbit coupling (SOC) and superconductivity, the band structure from the diagonalization of H0is given in Ref. 21. The minimal band gap at the Fermi sur- face
band0:05tis between the dxyband and one of the hybridized bands. We add the spin orbit coupling HSOC =~L
~S. The form of the BCS interaction we use is: HBCS=X ~k;~k0;a;b;figVab1234(~k;~k0)c~ka 1c~kb2cy ~k0b3cy ~k0a4;(4) Using a mean eld decomposition, we can calculate the gap function:
ab34(~k0) =X ~k12Vab1234(~k;~k0)hc~ka 1c~kb2i:(5) Thus we obtain the mean eld Hamiltonian, which is now rewritten into Bogoliubov-de-Gennes (BdG) form: y ~k[cy ~kX";cy ~kX#;cy ~kY";cy ~kY#;cy ~kZ";cy ~kZ#] HMF=X ~kjkx>0[ y ~k; T ~k]hBdG(~k) ~k  ~k hBdG(~k) =^h(~k) ^
(~k) ^
(~k)y^hT(~k) ; ^h(~k) =hab 0(~k) 0 +(Lx x+Ly y+Lz z); (6) as a 12 by 12 matrix. Here x;y;z; 0are the Pauli matrices and identity matrix in spin space. For simplicity, the orbital angular momentum operators ( La)bc=iabcare assumed to be the same as for electrons with L= 1. For the case of intra-band pairing, the superconduct- ing order parameter can be rewritten as
( ~k) = (I3) iy~d(~k)
~ , where~ = (x;y;z) andI3is the 3 by 3 identity matrix in orbital space20. Additionally, the multi-band nature of the model introduces the possibil- ity for inter-band coupled gap functions. We consider a BCS gap
BCS = 0:01t, which is of the order of the band gap near EF. Thus, the inter-band pairing cannot be fully neglected. In this calculation, we consider a local model (~k-independent model) of orbital-singlet and spin- triplet pairing, which has the following gap function, andthe corresponding Vab1234(~k;~k0): Vab1234(~k;~k0) =gf(~k)y ba21f(~k0)ab34 ^
ab12(~k) =
fab12(~k) f(~k)Lz (izy); (7) couplingyzandxzorbitals. Since the induced intraband spin-singlet state does not depend on a specic direc- tion, the results on susceptibilities do not show qualita- tive dierences in dierent directions, which will be con- rmed in the next section. Thus, other pairings including (Lx x+Ly y)iywill give similar results. The gap function is even parity, with time reversal symmetry and inversion symmetry. Note that other pairing states (e.g. intra-band pairings or other forms of inter-band pairing) are not considered here. The two electrons comprising the Cooper pair form an orbital singlet using the xzand yzorbitals. In spin space, they form a triplet pair with total spin rotating within the xyplane. The overall Cooper pair is odd under particle exchange. The ~d-vector of this triplet-spin pairing contains only a z-component. If we turn on the SOC, this inter-band pair could de- velop an intra-band spin-singlet component6. The nite intra-band pairing as induced by SOC helps increase the superconducting transition temperature, as we will ex- plain in Sec. IV. III. NMR AND KNIGHT SHIFT We now consider observables in NMR experiments and the Knight shift, which are key experimental techniques in identifying spin triplet superconductors17,19. Under- standing the Knight shift experiment is vital in distin- guishing between dierent types of gap functions20,22,23. We will rst summarize the theoretical background of3 this experiment. Then we will show results for the inter- band paired state under dierent SOC strengths. In atomic physics, the eld induced non-zero spin andorbit angular momentum of the electrons generate a hy- perne eld experienced by the nuclear spin24: ~Bhf=20Bhr3i(~L+L(L+ 1)~S3 2[~L(~L
~S) + (~L
~S)~L]); (8) which leads to the Knight shift in the NMR response. The hyperne eld can be decomposed into orbital and spin contributions. The orbital angular momentum gen- erates a current and hence, a magnetic eld, contribut- ing to the rst term of the hyperne eld in Eq. 8. The dipole-dipole interaction between electron spin and nu- clear spin leads to the remaining two terms in the hyper- ne eld, under the approximation known as the Equiva- lent Operator Method25. Given the atomic wavefunction of an electron with angular momentum l, the strength of the dipole-dipole interaction is = 2=[(2l1)(2l+ 3)]25. In the following calculations, we choose `= 2 which gives = 2=21. The Fermi contact interaction will also con- tribute to the hyperne eld, which is neglected here for L6= 0 systems. Knight shift tensor !Kis then determined by the hy- perne eld through ~Bhf= !K
~B: (9) Here~Bis the external magnetic eld. The spin and or- bital contribution towards the diagonal elements of the Knight shift is directly related to the spin and orbital susceptibilities. Under spin orbit coupling, ~Land~Sare no longer good quantum numbers, and we will take the expectation value of the hyperne eld operator in the simulation. The orbital contribution, which is propor- tional to the orbital magnetic susceptibility, does not change dramatically upon entering the superconducting phase. This is because in the Kubo formula for orbital susceptibility, orbital angular momentum couples states with dierent energy, so the energy shift by superconduc- tivity will not have a strong eect. The orbital contri- bution can be extracted from measurements within the normal state and then substracted in the superconduct- ing Knight shift17,19. The spin contribution has a similar behavior as spin susceptibility and acts as a key feature for distinguishing between spin-singlet and spin-triplet gap functions. In the following simulations, we numer- ically diagonalize the BdG Hamiltonian and obtain the susceptibilities and the diagonal elements of the Knight shift tensor.IV. RESULTS AND DISCUSSION The gap equation can now be solved numerically, and the parameter
in Eq. 7 is obtained self-consistently for a givengand temperature T. After obtaining
, the susceptibilities and Knight shift are calculated from the Kubo formula. We now compare the Knight shift results for the single band ~d(~k) = (0;0;kx+iky) state and the inter-band orbital-singlet, spin-triplet state. In order to shed light on the role of SOC in the formation of intra- band pairing, we consider three SOC regimes: = 0 (i.e. zero SOC) where only inter-band pairing is present, 
BCS when the SOC strength is comparable to the size of the superconducting gap, and 
BCS. The Knight shift and susceptibilities are all normalized to a dimensionless number, for which the Knight shift and susceptibilities in the normal state is unity. If the spin orbit coupling is absent, the critical temper- ature for superconductivity is found to be exponentially small. We present here an unrealistic scenario with a non-zero order parameter
BCS under zero/small SOC, as shown in Fig. 2 and Fig. 3. This setup can be achieved by choosing a relatively large coupling coecient gin Eq. 7, which raises the critical temperature to numer- ically accessible values. This small SOC regime serves only to illustrate the key results, namely the residual spin-susceptibility. Furthermore, to better illustrate the eect of SOC, we choose dierent gparameters for the two curves = 0 and=
0, such that the supercon- ductivity gap at the lowest temperature reached is xed to be
0= 0:01t. For purely inter-band pairing without SOC, we numer- ically obtain the susceptibility and Knight shift. Fig. 2 shows the responses in the presence of an out-of-plane magnetic eld while Fig. 3 is in the presence of an in- plane magnetic eld. In striking contrast to the intra- bandkx+ikymodel2,10,20, the spin and orbital suscepti- bilities and the Knight shift show no decrease in the su- perconducting phase in any direction. We further check the density of states. By redening the Fermi surface to be the states with energy near the chemical potential jEEFj<
BCS, we nd that there is no dramatic change in the density of states on the Fermi surface. Here we provide a qualitative explanation for the above ndings. For intra-band pairing, the Cooper pairs are composed of electrons with the same energies, and the4 Δ"#$% ~Δ"#$%Δ"&'Δ"&' FIG. 1: Schematic representation of the change of band struc- ture for a two band system with inter-band coupling for (left) no superconductivity and (right) with superconductivity. The distance between the BCS gaps and the Fermi surface is of order
Band. If the superconducting order parameter
BCS is much smaller than the energy dierence of the two bands
Band, then the states near the Fermi surface are approxi- mately unchanged. Thus the density of states at the Fermi surface is unaected by superconductivity. 0.4 0.5 0.6 0.7 T/000.51lz =00 =10 0.4 0.5 0.6 0.7 T/000.51sz =00 =10 0.4 0.5 0.6 0.7 T/000.51KSz =00 =10 0.4 0.5 0.6 0.7 T/000.51/0=00 =10 FIG. 2: Orbital susceptibility, spin susceptibility and gap function for triplet pairing
( ~k) =Lz iyzwhen apply- ing an out-of-plane magnetic eld for = 0 and=
BCS. Within the range of the gap function, no drop in susceptibili- ties and Knight shift is observed. Note that the zero temper- ature gap function should be larger than 0 :01t, but we only focus on the range where
BCS
band. contribution to the superconductivity gap is mainly from electrons near the Fermi surface ( jEEFj
BCS). When BCS states are formed, a superconducting gap then opens at the Fermi surface. The density of states at the Fermi surface vanishes, and there is a substantial drop in spin susceptibility. In contrast, the inter-band pairing involves electrons at dierent bands. In a rough approximation, let us assume the pairing only happens at the band crossing (Fig. 1). When Cooper pairs are formed, a superconducting gap will open above and below the Fermi surface. The distance from the Fermi surface is of order
band. If
band
BCS, the Fermi surface is then approximately unchanged. Therefore, the inter- band superconductivity does not exhibit any decrease in 0.4 0.5 0.6 0.7 T/000.51lx =00 =10 0.4 0.5 0.6 0.7 T/000.51sx =00 =10 0.4 0.5 0.6 0.7 T/000.51KSx =00 =10 0.4 0.5 0.6 0.7 T/000.51/0=00 =10FIG. 3: Orbital susceptibility, spin susceptibility, and gap function for triplet pairing
( ~k) =Lz iyzwhen apply- ing a magnetic eld in the in-plane x-direction. The same parameters are applied as in Fig. 2. Knight shift, even though ~dis inz-direction. As we turn on SOC while still keeping it weak ( 
BCS<
Band), the SOC is not yet sucient to gener- ate a considerable intra-band spin-singlet pairing. As a result, the decrease spin-susceptibility and Knight shift remains small (red curves in Fig. 2and Fig. 3). However, we observe a higher critical temperature, which is due to a reduction of the minimal band gap by SOC. 0.1 0.2 0.3 0.4 T/000.51/0=100 =200 =300 =400 =500 0.1 0.2 0.3 0.4 T/000.51KSz=100 =200 =300 =400 =500 0.1 0.2 0.3 0.4 T/000.51lz=100 =200 =300 =400 =500 0.1 0.2 0.3 0.4 T/000.51sz=100 =200 =300 =400 =500 FIG. 4: Orbital susceptibility, spin susceptibility and gap function for triplet pairing
( ~k) =Lz iyzwhen adding z-direction magnetic eld in the large SOC regime. We now consider the regime with large SOC. The in- duced spin-singlet pairing state greatly reduces the spin- susceptibility and the Knight shift in every direction, as shown in Fig. 4. The induced intra-band pairing provides sucient superconducting instability, leading to signif- icant enhancement of the critical temperature, as pre- dicted in the theoretical work6. In Fig. 5 and Fig. 6, we focus on the residual suscepti-5 0 20 40 /000.51lz/lznlz 0 20 40 /000.51sz/sznsz 0 20 40 /000.51KSz/KSznKSz FIG. 5: Residual Knight shift and susceptibilities for an ap- plied magnetic eld in the z-direction, as a function of SOC. The strength of SOC varies from 0 to 50
BCS, i.e. in a regime of strong SOC. 0 20 40 /000.51lx/lxnlx 0 20 40 /000.51sx/sxnsx 0 20 40 /000.51KSx/KSxnKSx FIG. 6: Residual Knight shift and susceptibilities under a magnetic eld applied in the xdirection, as a function of SOC strength. The strength of SOC varies from 0 to 50
BCS. bilities and Knight shift for dierent SOC. We compare the results at two temperature. At the lower tempera- tureT= 0:01
BCS, we add a BCS gap
BCS = 0:01. At the \higher" temperature T=
BCS, we consider the normal state (with zero BCS gap). We take the ratio of the susceptibility in the rst state to its value in the second state as a measure of the residual susceptibility. The aim is to observe the residual susceptibility at low temperature as we tune both the BCS term and SOC strength. The residual spin susceptibility rst exhibits a con-tinuous drop as SOC is increased from zero, due to the formation of intra-band pairing. It reaches a minimum at around= 10
BCS. More importantly, the resid- ual spin susceptibility never goes to zero, even when the orbital susceptibility is approximately unchanged. This feature can be explained by returning to the pedagogical model with = 0 (see Fig. 2 and Fig. 3 and accompa- nying discussion). This is a crucial dierence compared with the well-studied kx+ikymodel, in which the non- zero spin susceptibility under SOC is accompanied by a decreasing orbital susceptibility10,20. The non-zero spin susceptibility persists as the SOC is tuned to very large values. This may be due to mixing between an originally vanishing spin susceptibility and a non-zero orbital susceptibility, similar to the kx+iky model (in which the z-direction susceptibility exhibits similar features) or in a simple single-band spin-singlet model22,23. The system becomes more complex due to band crossings. Therefore, for future work, the analysis for very large SOC could be performed for a model with a large band gap. There is no qualitative dierence between the drops in dierent directions, which conrms the predicted contri- bution from spin-singlet pairing when SOC is present6. Thus, the Knight shift in dierent directions provides a tool in identifying the inter-band state and the SOC strength. V. CONCLUSION In summary, we have calculated the Knight shift for the even-parity orbital-singlet spin-triplet superconduc- tors in a quasi-two dimensional tight binding model, by numerically solving the gap equation. In contrast to the kx+ikyunconventional superconductor, the inter-band model exhibits no decrease in Knight shift in any di- rection. After introducing the required large spin orbit coupling, the predicted intra-band spin singlet state is observed, and there is a drop in spin susceptibility and Knight shift. However, the residual spin susceptibility is non-zero even under large SOC in contrast to spin-singlet intra-band pairing state. Acknowledgments Y.Y., A.C., and S.R. were supported by the DOE Oce of Basic Energy Sciences, contract DEAC02- 76SF00515. D. F. A. was supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant No. GBMF4302. 1T. Scadi, Weak-Coupling Theory of Topological Super- conductivity: The Case of Strontium Ruthenate (2017),ISBN 978-3-319-62866-0. 2K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Mao,6 Y. Mori, and Y. Maeno, Nature 396, 658 (1998). 3J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006), URL https://link.aps.org/doi/10.1103/PhysRevLett.97. 167002 . 4C. W. Hicks, J. R. Kirtley, T. M. Lippman, N. C. Koshnick, M. E. Huber, Y. Maeno, W. M. Yuhasz, M. B. Maple, and K. A. Moler, Phys. Rev. B 81, 214501 (2010), URL https: //link.aps.org/doi/10.1103/PhysRevB.81.214501 . 5A. Steppke, L. Zhao, M. E. Barber, T. Scadi, F. Jerzem- beck, H. Rosner, A. S. Gibbs, Y. Maeno, S. H. Simon, A. P. Mackenzie, et al., Science 355, eaaf9398 (2017). 6O. Vafek and A. V. Chubukov, Phys. Rev. Lett. 118, 087003 (2017), URL https://link.aps.org/doi/ 10.1103/PhysRevLett.118.087003 . 7A. Ramires and M. Sigrist, Phys. Rev. B 94, 104501 (2016), URL https://link.aps.org/doi/10. 1103/PhysRevB.94.104501 . 8S. Hoshino and P. Werner, Phys. Rev. Lett. 115, 247001 (2015), URL https://link.aps.org/doi/10. 1103/PhysRevLett.115.247001 . 9S. V. Borisenko, D. Evtushinsky, Z.-H. Liu, I. Morozov, R. Kappenberger, S. Wurmehl, B. B uchner, A. Yaresko, T. Kim, M. Hoesch, et al., Nature Physics 12, 311 (2016). 10D. Nisson and N. Curro, New Journal of Physics 18, 073041 (2016). 11M. W. Haverkort, I. S. Elmov, L. H. Tjeng, G. A. Sawatzky, and A. Damascelli, Phys. Rev. Lett. 101, 026406 (2008), URL https://link.aps.org/doi/10. 1103/PhysRevLett.101.026406 . 12A. Ramires, D. F. Agterberg, and M. Sigrist, arXivpreprint arXiv:1802.00361 (2018). 13A. Klejnberg and J. Spalek, Journal of Physics: Condensed Matter 11, 6553 (1999). 14J. Spa lek, Phys. Rev. B 63, 104513 (2001), URL https: //link.aps.org/doi/10.1103/PhysRevB.63.104513 . 15J. E. Han, Phys. Rev. B 70, 054513 (2004), URL https: //link.aps.org/doi/10.1103/PhysRevB.70.054513 . 16A. P. Mackenzie and Y. Maeno, Reviews of Modern Physics 75, 657 (2003). 17A. Abragam, The principles of nuclear magnetism , 32 (Ox- ford university press, 1961). 18N. Curro, Reports on Progress in Physics 72, 026502 (2009). 19A. Rigamonti, F. Borsa, and P. Carretta, Reports on Progress in Physics 61, 1367 (1998). 20P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004), URL https://link. aps.org/doi/10.1103/PhysRevLett.92.097001 . 21A. Akbari and P. Thalmeier, Phys. Rev. B 88, 134519 (2013), URL https://link.aps.org/doi/10. 1103/PhysRevB.88.134519 . 22A. Abrikosov and L. Gorkov, Sov. Phys. JETP 15, 752 (1962). 23P. W. Anderson, Phys. Rev. Lett. 3, 325 (1959), URL https://link.aps.org/doi/10.1103/PhysRevLett.3. 325. 24A. Abragam and B. Bleaney, Electron paramagnetic reso- nance of transition ions (OUP Oxford, 2012). 25J. S. Grith, The theory of transition-metal ions (Cam- bridge University Press, 1971).

1205.6950v1.Effect_of_spin_orbit_coupling_on_magnetic_and_orbital_order_in_MgV__2_O__4_.pdf

arXiv:1205.6950v1 [cond-mat.str-el] 31 May 2012Effect of spin-orbit coupling on magnetic and orbital order i n MgV 2O4 Ramandeep Kaur, T. Maitra and T. Nautiyal Department of Physics, Indian Institute of Technology Roor kee, Roorkee- 247667, Uttarakhand, India (Dated: May 30, 2018) Recent measurements on MgV 2O4single crystal have reignited the debate on the role of spin- orbit (SO) coupling in dictating the orbital order in Vanadium spi nel systems. Density functional the- ory calculations were performed using the full-potential l inearized augmented-plane-wave method within the local spin density approximation (LSDA), Coulom b correlated LSDA+U, and with SO interaction (LSDA+U+SO) to study the magnetic and orbital o rdering in low temperature phase of MgV 2O4. It is observed that the spin-orbit coupling in the experime ntally observed antiferro- magnetic phase, affects the orbital order differently in alte rnate V-atom chains along c-axis. This observation is found to be consistent with the experimental prediction that the effect of spin-orbit coupling is intermediate between that in case of ZnV 2O4and MnV 2O4. PACS numbers: 71.20.-b, 75.25.Dk, 71.70.Ej, 71.27.+a I. INTRODUCTION Vanadium spinels AV 2O4(A=Mg, Zn, Cd) are being studied extensively in recent years1–6as they provide a very interesting playground for the study of competing interactions on a frustrated lattice in 3-dimension. The Vanadium (V) ions at the B-sites of the spinel structure form a pyrochlore la ttice, with corner sharing tetrahedra, which is geometrically frustrated. In its 3+ valence state, V ion has two elec trons in the d-shell which, because of a strong Hund’s coupling, alignparalleltoeachothertherebyimpartingahighs pin state( S= 1) tothe ion. Thusin this family of spinels, there is a magnetic ion on a geometrically frustrated lattic e resulting in competing ground states. Things get more involvedwhen the partial occupancy of triply degenerate t2gorbitals by the two d-electronsmakes the orbital degree of freedom unfrozen. As both spin and orbital degrees of freedom remain active, there is a high possibility of spin-orbit (SO) coupling playing important role in the low energy phy sics of this family of systems. Role of this interaction has been a matter of debate recently1,3,4. The manifestation of the interplay of orbital, spin and lattice degrees of freedom in these systems culminates in experiments as a sequence of phase transitions1,7,8. A structural transition, often followed by magnetic transition as the temperatu re is lowered, signifies competing interactions trying to stabilize a particular ground state with gradual lifting of the frus tration. MgV2O4, with a normal spinel structure, has been reported to undergo a structural transition at 51 K from cubic to tetragonalphaseand a magnetictransitionat 42Kfrom non-magn eticto an antiferromagnetic(AFM)phaseconsisting of alternating antiferromagnetic chains of V atoms running parallel toaandbdirections as one goes along c-axis1,17. The high temperature (HT) phase has a cubic spinel structure with F¯43msymmetry where the V ion is surrounded by an almost perfect O6octahedron with all the six V-O bonds having same length. This leads t o a sizable ( ∼2.5 eV) t2g-egcrystal field splitting of the d levels. There is of course a small trigon al distortion also present in this phase. Experimental results further reveal that the structural tran sition to the tetragonal phase at 51 K is accompanied by a compression along c-axis with c/a=0.9941. This lowers the symmetr y to space group I¯4m2. Hence, in addition to thet2g-egsplitting arising from roughly O6octahedral coordination, a further splitting occurs due to the te tragonal compression where the low lying t2gtriplet splits into a lower energy singlet ( dxyorbital) and a higher energy doublet ofdyzanddzxorbitals. The orbital degeneracy is thereby partially lifted with this s tructural distortion. Now out of the two d electrons, one goes to the lower energy dxyorbital while the other still has a choice as it occupies the doubly degenerate dyzanddzxorbitals. This opens up a possibility of orbital order in this system. St ructural transition also partially lifts the frustration of the V−Vbonds in the pyrochlore lattice. This then brings in the second trans ition, at lower temperature of 42 K where a long range antiferromagnetic order sets in1. Thus the presence of any orbital order and the magnetic order observed at low temperatures in all t he Vanadium spinels are interrelated. Severaltheoreticalmodels havebeen proposedin the last few yea rstoexplain the possible orbitalorderin Vanadium spinels so as to be consistent with the observed antiferromagnetic order. Among these, the model proposed by Tsunetsugu and Motome3is based on strong coupling Kugel-Khomskii Hamiltonian and predicts a n orbital order where at each V site, d xyorbital is occupied by one electron and the second electron occupie s either d xzor dyz orbital, alternately, along the c-axis. However, this type of orbita l order was found to be of lower symmetry than that (I41/amd) observed experimentally for ZnV 2O4. In an alternative theoretical model, Tchernychov4considered a dominant SO interaction which then led to the proposal that the se cond electron would occupy a complex orbital of type d xz±idyzat each V site. This orbital order is found to be consistent with the u nderlying crystal symmetry. Also it explains the low magnetic moment per V ion observed in these sys tems as a large negative orbital moment is expected from a strong SO coupling. These findings were also corro borated by electronic structure calculations5for2 ZnV2O4. However, recent measurements on other members of the Vanadiu m spinel family raise doubts about the presence of a strong spin-orbit interaction effect. In fact, there has been a tremendous effort, from both theoreticians and experimentalists working on these systems, to bring out a unified pic ture in terms of the important interactions which underlie the two phase transitions (one structural and the other magnetic). In ZnV 2O4the SO coupling is found to be significant both from theory as well as experiments4,5,10whereas in case of MnV 2O4there seems to be very little or no effect of the SO interaction on the orbital order11,12. Recently Wheeler et al.1performed neutron diffraction measurements on MgV 2O4single crystal and speculated on the basis of their observations th at MgV 2O4might come intermediate between ZnV 2O4and MnV 2O4as far as strength of SO coupling is concerned. Hence it is expected that in MgV 2O4the occupied orbitals, instead of being completely real (Tsunetsug u and Motome model) or completely complex (Tchernychov model), could be a mixture of real and comple x orbitals. In the previous theoretical study on MgV2O413, the issue of impact of SO coupling on orbital order has not been inve stigated. However, as stated above, SO coupling in MgV 2O4is expected to be non-negligible from experimental observations. I n order to investigate thoroughly the effect of SO interaction on magnetic and orbital ord er in MgV 2O4, we have carried out first principle electronic structure calculations incorporating SO coupling. Such a calculation is definitely expected to unfurl the strength of SO coupling in this system, the nature of orbital order (if there is any) and correlation of experimentally observed magnetic order with the orbital order, if present. II. METHODOLOGY We undertake an electronic structure calculation using full-potent ial linearized augmented-plane-wave method with the basis chosen to be linearized augmented plane waves as implement ed in WIEN2K code14. The calculations have been carried out with no shape approximation to the potential and c harge density. These calculations were performed at three levels of sophistication using local spin density approximatio n(LSDA), Coulomb correlatedLSDA+U approx- imation, and with SO interaction i.e. LSDA+U+SO approximation. To rem ove the self-Coulomb and self-exchange- correlation energy included in LSDA approximation, we use self-inter action corrected scheme (LSDA+U(SIC))15, which is appropriate for the strongly correlated systems. The cor rected energy functional is written as15 E=ELSDA−[UN(N−1)/2−JN(N−2)/4]+1/2/summationdisplay m,m′,σUmm′nmσnm′−σ+1/2/summationdisplay m/negationslash=m′,m′,σ(Umm′−Jmm′)nmσnm′σ Here ELSDAis the standard LSDA energy functional, U represents the on-site Coulomb interaction, J is the exchange parameter and n mσare the occupations of the localized orbitals. N is the total number o f localized electrons. In the LSDA+U+SO calculations, SO coupling was considered within the scalar relativistic approximation and the second variational method was employed16. In this method, the eigen value problem is first solved separately fo r spin up and spin down states without inclusion of the SO interaction term ( HSO) in the total Hamiltonian. The resulting eigen values and eigen functions are then used to solve new eigen valu e problem with the H SOterm in the total Hamiltonian. This method is more efficient and computationally less expe nsive than the calculation in which H SOis included in the total Hamiltonian by doubling the dimension of the origina l eigenvalue problem in order to calculate the non-zero matrix elements between spin-up and spin-down stat es. In this method, the calculation of H SOmatrix elements involves much less number of basis functions than in the orig inal basis set. MgV2O4crystallizes in tetragonal structure with symmetry I¯4m2 (space group 119) at low temperatures1. Atomic positions and lattice constants were taken from the experimental data1. The atomic sphere radii were chosen to be 1.96, 1.99, and 1.78 a.u. for Mg, V, and O, respectively. We have used 50kpoints in the irreducible part of the Brillouin zone for the self-consistent calculations. In order to mode l the low temperature magnetic order observed in the experiment, we have constructed a supercell (with 8 inequivale nt Vanadium atoms). The lowering of symmetry of this unit cell arises due to the experimentally observed antiferro magnetic ordering. The network of corner sharing V4O4cubes of low temperature structure is shown in Fig. 1 with the magne tic order. The 8 inequivalent Vanadium atoms considered in the calculation are also marked in the figure with t he corresponding orientation of spins at that particularsite. Onecanseetheantiferromagneticchainsalong a(...V3−V7−V3−V7...)andb(...V6−V2−V6−V2...) axes alternating along c-axis. In each V 4O4cube there are 4 inequivalent V atoms. Due to the presence of coop erative trigonal distortion along c-axis resulting in alternating compression and expansion of cube faces, there is a further symmetry breaking and hence successive cubes along c-axis no long er remain equivalent. Furthermore, the V 4O4 cube containing V1, V5, V2 and V6 does not have the same spin arran gement as that containing V3, V7, V4 and V8. Therefore to model the experimentally observed magnetic order o ne needs to consider 8 inequivalent V atoms in the unit cell.3 FIG. 1: Corner sharing network of V 4O4cubes in the low temperature structure of MgV 2O4showing the experimentally observed magnetic order. The solid and dotted lines joining the V atoms (shown in one cube) represent the shorter V-V FM bonds (2.971 ˚A) and longer V-V AFM bonds (2.98 ˚A) respectively. FIG. 2: Spin polarized (a) total DOS within LSDA, (b) partial DOS for V d-states around the Fermi level within LSDA+U, (c) partial d-DOS within LSDA+U+SO (U-J = 2 eV)in the low temp erature AFM phase. III. RESULTS Our LSDA calculations of experimentally observed antiferromagnet ically ordered phase show that total energy of this phase is indeed lower than that of the corresponding ferromagnet ic (FM) phase by 0.4 eV per formula unit. The density of states (DOS) of this antiferromagnetic state (within LS DA) is shown in Fig. 2(a). It is observed that LSDA givesa metallic state whereasthe system is knownto be a Mott insulat or17. Thus AFM interactionalone is not able to open up the gap. Around the Fermi level, mainly V d-states are seen to be present. In an effort to have the insulating gap as observed experimentally, we included Coulomb correlation in ou r calculations within LSDA+U approximation.4 FIG. 3: Real space electron density at each V site within (a) L SDA+U (b) LSDA+U+SO along two different directions. Isosurface used for both corresponds to 0.5 e/ ˚A3. We performed calculations with U eff(=U-J) values in the range 1 to 4 eV as found to be relevant from the literature on Vanadium spinel systems5,11,13. We present the results for U eff= 2 eV in the following, nevertheless it may be noted that our conclusions remain valid in the whole range of U effvalues considered by us. In Fig. 2(b) we show the partial DOS of five d-orbitals as these are the states present around the Fermi level. As expected, the application of Coulomb correlation Uis able to open up a small gap of 0.12 eV which increases with the increas e inU. The gap originates because of the the splitting of the t 2glevels in addition to the t 2g-egsplitting due to octahedral field. The further splitting of t 2gis primarily caused by the antiferromagnetic interactions which get e nhanced in the presence of Coulomb correlations. Another observation that can clearly be made from the partial DOS of d-orbitals (Fig. 2(b)) is that among the occupied t 2gorbitals, one orbital (i.e. d x2−y2) is more populated while the other two (d xzand dyz) essentially have the same occupancy and seem to be degenerate. The higher occupanc y of dx2−y2orbital is a result of the presence of the tetragonal compression along c-axis at low temperatures. Howev er, closer analysis of occupancies of the apparently degenerate d xzand dyzorbitals at each Vanadium site shows that there is a tendency towar ds orbital ordering. Table I lists the orbital occupancies of d x2−y2, dxzand dyzfor the 8 inequivalent V atoms in the unit cell considered. One clearly observes that the occupancy of d xzand dyzorbitals are different as one moves along the c-axis whereas that of dx2−y2remains the same. The orbital polarization increases on increasing t he value of Uand alternates for the dxzand dyzorbitals (see for example, V1 and V2) in successive Vanadium layers a long c-axis. This is similar to an A-type antiferro-orbital order where the antiferromagnetic V c hains parallel to the ab-plane have ferro-orbital order (e.g. similar orbital occupancies of V1 and V5 ions or that of V2 and V6 ions) whereas along c-axis there is an antiferro-orbital order between d xzand dyzorbitals (see occupancies of V1 and V2 or that of V5 and V6 in Table I). This is consistent with the previous theoretical observation on the same system13. The observed intra-chain ferro- orbital order is also consistent with experimental antiferromagne tic order as per Goodenough-Kanamori-Anderson rules18.The orbital order described above is also revealed in the calculated r eal space electron density at each V site shown in Fig. 3(a). This orbital order was predicted by Tsunetsugu and Motome3for Vanadium spinels from their calculations based on Kugel- Khomskii model in strong coupling limit. As mentioned earlier, the influence of spin-orbit coupling on the magn etic and orbital order in these systems is continuously debated but no conclusion has been reached yet. In order to investigate the effect of spin-orbit interaction in this particular system, we also performed a calculation with spin-orbit coupling within LSDA+U+SO approximation. The solution obtained within LSDA+U+SO has a lower en ergy than that obtained within LSDA+U by 0.095 eV per formula unit for Ueff=2eV. The partial DOS (Fig. 2(c)) clearly shows a non-negligible impac t of SO in general, with an increased energy gap compared to that with LSDA +U. The analysis of orbital occupancies in this case indeed leads to some im portant and interesting observations. The apparent degeneracy of d xzand dyzorbitals observed within LSDA+U is no longer present and there is a co mplete lifting of degeneracy of all the t 2gorbitals (see Table I). Even though, likewise LSDA+U, the antiferro magnetic V chains parallel to the ab-plane are still ferro-orbitally ordered and along c-axis these chains are anti-ferro orbitally ordered, the orbital polarizations in adjacent chains along c-axes are significantly different in the presence of SO interaction. For example, if we compare the occupancies (Table I) f or V1 and V2 with LSDA+U and LSDA+U+SO,5 TABLE I: Orbital occupancies and spin magnetic moment withi n LSDA+U and LSDA+U+SO (U eff=2 eV) orbital occupation spin magnetic dx2−y2dxzdyzmoment ( µB) With LSDA+U V1 (V3) 0.6650.5530.3861.53 (-1.53) V2 (V4) 0.6650.3860.5531.53 (-1.53) V5 (V7) 0.6650.5530.386-1.53 (1.53) V6 (V8) 0.6650.3860.553-1.53 (1.53) With LSDA+U+SO V1 (V3) 0.5950.4510.5911.574 (-1.574) V2 (V4) 0.7590.2140.6651.571 (-1.571) V5 (V7) 0.5950.4510.591-1.574 (1.574) V6 (V8) 0.7590.2140.665-1.571 (1.571) TABLE II: Calculated orbital moments, total magnetic momen t (J) (in µB) and angle of J w.r.t. z-axis within LSDA+U+SO (Ueff= 2 eV). The spin magnetic moment is along z axis and is listed i n Table I µorbital µtotalangle xyzJ V1-0.3550.000-0.4661.1517.79 V2-0.015-0.030-0.5101.051.90 V5-0.3550.0000.466-1.15162.21 V6-0.015-0.0300.510-1.05178.10 we note that orbital occupancies of d x2−y2are no longer same in presence of SO interaction. Furthermore, th e polarization of the d xzand dyzorbitals are also very different (i.e. at V2 the polarization of d yzorbital w.r.t. d xzis much stronger than that at V1). This implies that V chains in success ive layers along c-axis are affected differently by the SO interaction. This is also reflected in the orbital moments of V io ns (listed in Table II and depicted in Fig. 4). The calculated electronic density at each V site is shown in Fig. 3(b) wh ich brings out the impact of SO interaction. In Fig. 4 we show two successive Vanadium chains along c-axis with the calculated electron density at each Vanadium site in the presence of both Coulomb correlation and SO inte raction. We have also marked the direction of orbital and magnetic moments at each site. Effect of SO interact ion is clearly different on the two chains and so is the arrangement of the orbital moments. One chain shows the can ted orbital arrangement and orbital moments are making an angle of 17.790with the c-axis whereas in other chain orbital moment makes an angle of 1.900(almost collinear orbital arrangement) with the c-axis (Table II). On one ch ain (V1-V5-V1-V5) due to canting of orbital FIG. 4: Electron densities for two successive Vanadium chai ns along c-axis within LSDA+U+SO showing the impact of SO interaction on them. The directions of corresponding orbit al and magnetic moments are also shown below each chain.6 moment, the effect of SO interaction reduces whereas in the other chain orbital moments align almost opposite to the magnetic moment implying a substantial SO interaction. The obse rvation that the SO interaction appears to affect alternate V chains along c-axis differently, is interesting. This also substantiates the speculation of Wheeler et al.1that SO interaction in MgV 2O4may not be as large as that in ZnV 2O4or as small as that in MnV 2O4as discussed earlier. The magnitude of orbital moments observed in ou r calculation also corroborates this fact. Thus our results show that a small but non-negligible spin-orbit coupling, along with the significant trigonal distortion present in MgV 2O4structure, has a substantial effect on the orbital order of this s ystem. This observation is consistent with the experimental observations by Wheeler et al.1of antiferromagnetic chains with a strongly reduced moment and the one-dimensional behavior and a single band of excitations proje cted by the inelastic neutron scattering. IV. CONCLUSIONS To conclude, we have studied the effect of spin-orbit interaction on magnetic and orbital order in the low temperature tetragonalphase ofMgV 2O4. We observethat even thoughthe orbitalmoments arerelativelys mall comparedto those of ZnV 2O4, the orbital order in successive Vanadium chains is differently affect ed in the presence of SO interaction. In one chain (V1-V5-V1-V5, parallel to crystallographic aaxis) the three t 2gorbitals are nearly equally populated giving rise to a canted (non-collinear) arrangement of orbital mome nts whereas in the other (V2-V6-V2-V6, parallel tobaxis), the orbitals are highly polarized leading to a collinear arrangeme nt of orbital moments. These results imply that SO interaction in MgV 2O4is non-negligible and has a significant effect on orbital order. Howeve r it is not very strong unlike ZnV 2O4and at the same time not very weak unlike MnV 2O4. V. ACKNOWLEDGEMENT This work is supported by the DST (India) fast track project (gra nt no.: SR/FTP/PS-74/2008). RD acknowledges CSIR (India) for a research fellowship. 1E.M. Wheeler, B. Lake, A.T. M. Nazmul Islam, M. Reehuis, P. St effens, T. Guidi and A. H. Hill, Phys. Rev. B 82, 140406(R) (2010). 2Paolo G Radaelli, New J. Phys. 7, 53 (2005). 3H. Tsunetsugu and Y. Motome, Phys. Rev. B 68060405 (2003); ibid.Prog. Theor. Phys. Suppl. 160, 203 (2005). 4O. Tchernyshyov, Phys. Rev. Lett. 93157206 (2004). 5T. Maitra and R. Valent´ ı; Phys. Rev. Lett. 99, 126401, (2007). 6G. Giovanetti, A. Stroppa,S. Picozzi,D. Baldomir,V. Pardo ,S. Blanco-Canosa,F. Rivadulla,S. Jodlauk,D. Niermann,J . Rohrkamp,T. Lorenz,S. Streltsov,D. I. Khomskii,and J. Hem berger; Phys. Rev. B 83, 060402(R) (2011). 7M. Reehuis, A. Krimmel, N. Bottgen , A. Loidl and A. Prokofiev, Eur. Phys. J. B 35, 311 (2003). 8V. O. Garlea, R. Jin, D. Mandrus, B. Roessli, Q. Huang, M. Mill er, A. J. Schultz, and S. E. Nagler, Phys. Rev. Lett. 100, 066404 (2008). 9H. Mamiya, M. Onoda, T. Furubayashi, J. Tang, and I. Nakatani , J. Appl. Phys. 81, 5289 (1997). 10S. H. Jung, J. Noh1, J. Kim, C. L. Zhang, S. W. Cheong and E. J. Ch oi; J. Phys.: Condens. Matter 20175205 (2008). 11S. Sarkar, T. Maitra, Roser Valent, and T. Saha-Dasgupta; Ph ys. Rev. Lett. 102, 216405 (2009). 12S.-H. Baek, N. J. Curro, K.-Y. Choi, A. P. Reyes, P. L. Kuhns, H . D. Zhou, and C. R. Wiebe, Phys. Rev. B 80, 140406(R) (2009). 13S. K. Pandey; Phys. Rev. B. 84, 094407 (2011). 14P. Blaha, K. Schwartz, G. K. H. Madsen, D. Kvasnicka and J. Lui tz; WIEN2K edited by K. Schwarz (Techn. University Wien, Austria, 2001), ISBN 3-9501031-1-2. 15V.I. Anisimov, I.V. Solovyev, M.A. Korotin, M.T. Czyzyk, an d G.A. Sawatzky, Phys. Rev. B 48, 16929 (1993) 16D.D. Koelling et al., J. Phys. C 10, 3107 (1977); A.H. MacDona ld et al., ibid. 13, 2675 (1980) 17H. Mamiya, M. Onoda, Solid State Communications 95, 217 (1995), 18J. Kanamori, Prog. Theor. Phys. 17177 (1957); J. B. Goodenough, J. Phys. Chem. Solids 6, 287 (1958); P. W. Anderson, Phys. Rev. 79350 (1950).

1812.09270v1.Influence_of_spin_orbit_and_spin_Hall_effects_on_the_spin_Seebeck_current_beyond_linear_response.pdf

arXiv:1812.09270v1 [cond-mat.mes-hall] 21 Dec 2018Influence of spin-orbit and spin-Hall effects on the spin Seeb eck current beyond linear response: A Fokker-Planck approach L. Chotorlishvili1, Z. Toklikishvili2, X.-G. Wang3, V.K. Dugaev4, J. Barna´ s5,6J. Berakdar1 1Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle-W ittenberg, D-06120 Halle/Saale, Germany 2Faculty of Exact and Natural Sciences, Tbilisi State Univer sity, Chavchavadze av.3, 0128 Tbilisi, Georgia 3School of Physics and Electronics, Central South Universit y, Changsha 410083, China 4Department of Physics and Medical Engineering, Rzeszow University of Technology, 35-959 Rzeszow, Poland 5Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Poznan, Poland 6Institute of Molecular Physics, Polish Academy of Sciences , ul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland (Dated: December 24, 2018) We study the spin transport theoretically in heterostructu res consisting of a ferromagnetic metal- lic thin film sandwiched between heavy-metal and oxide layer s. The spin current in the heavy metal layer is generated via the spin Hall effect, while the oxide la yer induces at the interface with the ferromagnetic layer a spin-orbital coupling of the Rashba t ype. Impact of the spin Hall effect and Rashba spin-orbit coupling on the spin Seebeck current is ex plored with a particular emphasis on nonlinear effects. Technically, we employ the Fokker-Planc k approach and contrast the analytical expressions with full numerical micromagnetic simulation s. We show that when an external mag- netic field H0is aligned parallel (antiparallel) to the Rashba field, the s pin-orbit coupling enhances (reduces) the spin pumping current. In turn, the spin Hall eff ect and the Dzyaloshinskii-Moriya interaction are shown to increase the spin pumping current. I. INTRODUCTION In a seminal paper [1], Bychkov and Rashba explored the impact of spin-orbit (SO) interaction on the proper- ties of two-dimensional semiconductor heterostructures. Since then, the basic idea of Bychkov and Rashba was carried over to other research areas of physics. It was shown, for instance, that the SO interaction plays a sig- nificant role in the quantum spin Hall Effect in graphene [2], Bose-Einstein condensates [3], and in the orbital- based electron-spin control [4]. Recent experiments [5, 6] revealed the role of SO interaction in the motion of do- main walls, as well. Combining the SO coupling and thermal effects bring in new insight and phenomena. A thermal bias applied to a ferromagnetic insulator leads to the formation of a thermally assisted magnonic spin current that is proportional to the temperature gradi- ent. This phenomenon falls in the class of spin See- beck effects and may be useful for thermal control of magnetic moments [7–23]. The objective of this paper is to study the impact of SO interaction on the forma- tion and transport of thermally assisted magnonic spin current in spin-active multilayers. We investigate two different heterostructures which include a layer of fer- romagnetic metal sandwiched between heavy metal and oxide materials, see Fig. 1 and Fig. 2. In both cases, an inversion asymmetry is caused by two different inter- faces – heavy-metal/ferromagnet and ferromagnet/oxide ones. A large SO coupling is present in the heavy metal [24–28]. This study is motivated by the experimental work in Ref. [27] with a particular attention to the systems Pt/Co/AlO xand Ta/CoFeB/MgO. Moreover, the torques generated by strong SO coupling are gener- ally different from the Slonczewski’s spin-transfer torque[24, 30, 31], with the prospect for novel physical effects in the heavy-metal/ferromagnetic-metal/oxide heterostruc- tures. An applied an electric voltage (see Fig.1) gener- ates charge current in the ferromagneticand heavy metal layers. This current in the heavy-metal layer leads to spin current due to the spin Hall effect, which is then injected into the thin ferromagnetic layer [32–39] and acts as an extra torque on the localized magnetic mo- ments in the ferromagnet. The induced torque influences the magnetization dynamics, which is the topic of this work. To describe the influence of the spin current on the magnetization dynamics in the ferromagnetic layer we add a relevant term to the Landau-Lifshitz-Gilbert (LLG) equation. In turn, the Rashba SO coupling at the ferromagnet/oxide interface in the presence of the charge current results in a spin polarization at the in- terface, with the exchange coupling exerting a torque on the ferromagnetic layer, as well. Thus, the SHE and the Rashba SO coupling influence the magnetization dynam- ics in the ferromagnetic layer through the Rashba and SHE torques, both incorporated into the LLG equation (the Rashba and SHE fields). The considered setup al- lows to formally investigate the interplay/competition of the torques due to Rashba SO interaction and SH effect. The Rashba SO torque acts field-like, while the torque due to the spin current generated viathe spin Hall effect is predominantlyofdamping/antidumpinglike in nature. We utilizethe Fokker-Planckmethod [40] forthe stochas- tic LLG equation for studying the magnetic dynamics beyond the linear response regime. The influence of the Rashba-type SO coupling on the magnonic spin current was studied in the works [41–43]. In the system shown in Fig.1, the normal metal with temperature TNis attached to the ferromagnet with2 TF> TN. We consider the spin current flowing from the ferromagnetic to a normal metal layer. Magnons from the high-temperature region diffuse to the lower temper- aturepartgivingrisetoamagnonicspincurrentandthus also to the spin Seebeck effect (SSE) [44–46]. Magnonic spin current pumped from the ferromagnet into the nor- mal metal, Isp, increases with the temperature differ- ence,Isp∼TF−TN. However, the spin current in- jected from the ferromagnetic layer to the normal metal is not the only spin current that crosses the normal- metal/ferromagnet interface. The fluctuating spin cur- rentIflis generated in the normal metal and flows to- wards the ferromagnet, i.e. in the direction opposite to the magnonic spin pumping current. The quantity of in- terest is therefore the total spin current, Itot=Isp+Ifl that crossesthe normal-metal/ferromagnetinterface. We show that Itotis drastically influenced by the proximity of the heavy metal (due to spin Hall effect) and the ox- ide (due to Rashba spin-orbit coupling). In the second system (see Fig.2) an additional normal-metal layer is attached to the ferromagnetic one. The paper is organized as follows. In Sec. IIwe in- troduce the model under consideration. In Sec. III andIVwe explore the spin current in two different heterostructures. For the sake of simplicity, we neglect Dzyaloshinskii-Moriya interaction (DMI). Effects of the DMI term and magnetocrystalline anisotropy are ex- plorednumericallybymicromagneticsimulationsandare described in Sec. V. Section VIsummarizesthe findings. Main technical details are deferred to the appendices. II. THEORETICAL MODEL For the heavy-metal/ferromagnetic-metal/oxide sand- wich we choose the ferromagnetic metallic layer to be in direct contact with a nonmagnetic metallic layer, as shown in Fig.1. We also assume that, due to a strong electron-phonon interaction, the local thermal equilib- rium between electrons and phonons in both ferromag- netic and normal-metal layers is established, Tp F=Te F= TFandTp N=Te N=TN. The magnon temperature Tm F in the ferromagneticlayerdiffers in generalfrom the tem- perature of electrons/phonons, Tm F/negationslash=TF[46]. At nonzero temperatures, the thermally activated magnetization dynamics in the ferromagnet gives rise to a spin current flowing into the normal metal. This effect isknownasspin pumping [44, 47–49]. Thecorresponding expression for the spin current density reads [46, 50] Isp(t) =/planckover2pi1 4π[grm(t)×˙m(t)+gi˙m(t)],(1) wheregrandgiare the real and imaginary parts of the dimensionless spin mixing conductance of the ferromagnet/normal-metal ( F|N) interface, while m(t) =M(t)/Msis the dimensionless unit vector along the magnetization orientation (here Msis the saturation magnetization) and ˙m≡dm/dt. The spin current isa tensor describing the spatial distribution of the cur- rent flow and orientation of the flowing spin (magnetic moment). Due to the geometry of the system under con- sideration, the spin current flows along the y-axis, see Fig. 1. In turn, the spin polarization of the current de- pends on the orientation of the magnetic moment and its time derivative. The averagespin depends on the ground state magnetic order which in our case is collinear with the external magnetic field (applied along the y-axis). Therefore, the only nonzero component of the average spin current tensor is Iy sp. Thermal noise in the normal-metal layer activates a fluctuating spin current flowing from the normal metal to the ferromagnet [47], Ifl(t) =−MsV γm(t)×ζ′(t). (2) Here,Vis the total volume of the ferromagnet, γis the gyromagnetic factor, and ζ′(t) =γh′(t) withh′(t) de- noting the random magnetic field. In the classical limit, kBT≫/planckover2pi1ω0, the correlation function /angb∇acketleftζ′ i(t)ζ′ j(t′)/angb∇acket∇ightofζ′(t) reads /angb∇acketleftζ′ i(t)ζ′ j(t′)/angb∇acket∇ight=2α′γkBTN MsVδijδ(t)≡σ′2δijδ(t),(3) where/angb∇acketleft.../angb∇acket∇ightdenotes the ensemble average, and i,j= x,y,z. Furthermore, ω0is the ferromagnetic resonance frequency and α′is the contribution to the damping con- stant due to spin pumping, α′=γ/planckover2pi1gr/4πMsV. We em- phasize that the correlator(Eq.(3)) is proportionalto the temperature TN. The total spin current flowing through the ferromagnet/normal-metal interface is given by the sum ofpumping and fluctuating spin currents, Itot=Isp+Ifl. For clarity of notation, we omit here (and also in the following) the time dependence of spin currents, nor- malized magnetization, random magnetic fields, and their correlators. This dependence will be restored if necessary. According to Eqs. (1) and (2), the total average spin current flowing across the interface can be written in the following form [46]: /angb∇acketleftItot/angb∇acket∇ight=MsV γ[α′/angb∇acketleftm×˙m/angb∇acket∇ight−/angb∇acketleftm×ζ′/angb∇acket∇ight].(4) Now, we assume that a spatially uniform current of densityja=jaixis injected along the x-axis. This cur- rent gives rise to additional torques owing to the spin Hall effect and Rashba spin-orbit interaction. Thus, the magnetizationdynamics is then modified and is governed by the stochastic LLG equation [27]: dm dt=−γm×(Heff+h)+αm×˙m+τSO,(5) whereαis the Gilbert damping constant, his the time- dependentrandommagneticfieldintheferromagnet,and Heffis an effective field. This effective field consists of3 threecontributions: theexchangefield, theexternalmag- netic field oriented along the y-axis, and the field corre- sponding to the DM interaction: Heff=2A µ0Ms∇2m+H0y−1 µ0MsδEDM δm, EDM=D/bracketleftbig mz∇m−/parenleftbig m∇/parenrightbig mz/bracketrightbig . (6) For the sake of simplicity, in the analytical part we take into account only the external magnetic field. In turn, the term τSOin Eq.(5) describes SO torques related to the Rashba SO coupling and the spin Hall effect, τSO=−γm×HR+γηξm×(m×HR) (7) +γm×(m×HSH), whereξis a non-adiabatic parameter, and η= 1 when the torque has Slonczewski-like form, while η= 0 in the opposite case [27]. In the above equation, the DM inter- action enters the effective magnetic field, while the effect of Rashba SO coupling and spin Hall effect are included by means of the extra torqueadded to the LLG equation. As already mentioned above, the charge current flow- ing in the thin ferromagnetic layer leads to spin polar- ization at the ferromagnet/oxide interface. The accumu- lated spin density in vicinity of the interface interacts with the local magnetization by means of the exchange coupling. This effect may be described by an effective Rashba field HR=HRiy[5, 24, 32]: HR=αRP µ0µBMs(iz×ja) =αRPja µ0µBMsiy,(8) whereαRis the Rashba parameter and Pis the degree of spin polarization of conduction electrons [32]. The first terminEq.(7)correspondstotheout-of-planetorqueand isrelatedtotheeffectivefield HR. Thistorqueisoriented perpendicularly to the ( m,HR) plane. The second term in Eq.(7) captures the effects of spin diffusion inside the magnetic layer and the spin current associated with the Rashba interaction at the interface. For more details, we refer to the work [24]. The last term in Eq.(7) corresponds to the spin Hall torque[33,34], expressedbythespinHallfield HSH. The spin current is generateddue to the spin Hall effect in the heavy metal layer and is injected into the ferromagnetic layer. For moredetails, we referto the references [36–39]. The explicit expression for HSHreads: HSH=/planckover2pi1θSHja µ02eMsLziy, (9) whereLzisthethicknessoftheferromagneticlayer,while θSHis the spin Hall angle (defined as the ratio of spin current and charge current densities). As already mentioned above, total random magnetic fieldh(t) has two contributions from different noise sources: the thermal random field h0(t), and the ran- dom field h′(t). Since the random fields are statisti- cally independent, their correlatorsare additive and fully/s122 /s121 /s78/s111/s114/s109/s97/s108 /s77/s101/s116/s97/s108/s79/s120/s105/s100/s101 /s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s105/s99 /s84 /s78/s84 /s70/s73 /s102/s108 /s73 /s83/s80/s72 /s82/s44/s32/s83/s72 /s72/s101/s97/s118/s121 /s77/s101/s116/s97/s108/s106 /s97/s120 FIG. 1. Schematic illustration of the system. A ferromagnet ic metallic layer is sandwiched between the oxide and heavy metal layers. The injected current jaflows in the ferromag- netic and heavy metal layers in the xdirection. The Rashba fieldHRand the spin Hall field are oriented along the yaxis. The normal metal with the temperature TNis attached to the ferromagnetic layer. The temperature of the ferromag- netic layer TFis different from TN. determined by the total (enhanced) magnetic damping α=α0+α′[46] (with α0being the damping parameter of the ferromagnetic material, i.e., without contributions from pumping currents), /angb∇acketleftζi(t)ζj(0)/angb∇acket∇ight=2αγkBTm F MsVδijδ(t) =σ2δijδ(t),(10) whereζ(t) =γh(t), andαTm F=α0TF+α′TN. III. SPIN CURRENT: N/F STRUCTURE Theinjected electricalcurrentcreatesatransversespin current in the heavy-metal layer via the spin Hall effect (or spin accumulation at the boundaries of the sample) [44]. In turn, the Rashba SO interaction in the presence ofchargecurrentgivesrisetoadditionaltorqueasalready described above. In the case under consideration, the Rashba SO field, Eq.(8), and the spin Hall field, Eq.(9), are oriented along the yaxis. When temperature of the ferromagnetic film differs from that of the normal metal, TF/negationslash=TN, the spin Seebeck current emerges in the Fe/N contact. Note, this current also exists in the absence of spin-orbitinteraction and for ja= 0. Below, we calculate the total spin current in the N/F structure, taking into account the Rashba SO field and the spin Hall effect. In order to calculate the spin pumping current, /angb∇acketleftIsp/angb∇acket∇ight= MsV γα′/angb∇acketleftm×˙m/angb∇acket∇ight, we use Eq.(A1) (see Appendix A) and find /angb∇acketleftIsp/angb∇acket∇ight=α′MsV γ(−/angb∇acketleftm×ω2/angb∇acket∇ight−/angb∇acketleftm×m×ω1/angb∇acket∇ight),(11)4 whereω1andω2are defined in the Appendix A, see Eq.(A1). Utilizing Eq.(A2) and Eq.(A15) we find mean values of the magnetization components (see Appendix B): /angb∇acketleftmy/angb∇acket∇ight=−L(βω2),/angb∇acketleftm2 y/angb∇acket∇ight= 1−2L(βω2) βω2, /angb∇acketleftm2 x/angb∇acket∇ight=/angb∇acketleftm2 z/angb∇acket∇ight=L(βω2) βω2,(12) whereβ= 2/σ2, andL(x) = coth x−1 xis the Langevin function. From Eq.(12) we obtain /angb∇acketleftm×ω2/angb∇acket∇ight= 0, and /angb∇acketleft(m×˙m)x/angb∇acket∇ight= 0,/angb∇acketleft(m×˙m)y/angb∇acket∇ight=−ω1(1−/angb∇acketleftm2 y)/angb∇acket∇ight),/angb∇acketleft(m× ˙m)z/angb∇acket∇ight= 0. Thus, the only nonzero component of the spin pumping current is Iy sp, /angb∇acketleftIy sp/angb∇acket∇ight=α′MsV γω1(1−/angb∇acketleftm2 y/angb∇acket∇ight) =α′MsV γ2ω1 βω2L(βω2). (13) For the evaluation of the fluctuating spin current /angb∇acketleftIfl/angb∇acket∇ight=−MsV γ/angb∇acketleftm×ζ′/angb∇acket∇ight, we linearize the LLG equation, Eq.(A1), near to the equilibrium point: /angb∇acketleftmx/angb∇acket∇ight=/angb∇acketleftmz/angb∇acket∇ight= 0,/angb∇acketleftmy/angb∇acket∇ight=−L(βω2): ˙mx=ω1mz+ω2/angb∇acketleftmy/angb∇acket∇ightmx−/angb∇acketleftmy/angb∇acket∇ightζz(t), ˙mz=−ω1mz+ω2/angb∇acketleftmy/angb∇acket∇ightmz+/angb∇acketleftmy/angb∇acket∇ightζx(t).(14) Fourier transforming to the frequency domain (˜ g=/integraltext geiωtdtandg=/integraltext ˜ge−iωtdω/2π), from Eq.(14) we ob- tain ˜mi(ω) =/summationtext jχij(ω)˜ζj(ω), where i,j=x,z, and χij(ω) =/angb∇acketleftmy/angb∇acket∇ight (ω2/angb∇acketleftmy/angb∇acket∇ight+iω)2+ω2 1 ×/parenleftbigg ω1 (ω2/angb∇acketleftmy/angb∇acket∇ight+iω) −(ω2/angb∇acketleftmy/angb∇acket∇ight+iω)ω1/parenrightbigg ,(15) /angb∇acketleftmi(t)ζ′ x(0)/angb∇acket∇ight=σ′2/integraldisplay+∞ −∞χij(ω)e−iωtdω 2π.(16) Equation(16) has nonzero elements: /angb∇acketleftmz(t)ζ′ x(0)/angb∇acket∇ight=−/angb∇acketleftmx(t)ζ′ z(0)/angb∇acket∇ight =−σ′2/angb∇acketleftmy/angb∇acket∇ight/integraldisplay+∞ ∞ω2/angb∇acketleftmy/angb∇acket∇ight+iω (ω2/angb∇acketleftmy/angb∇acket∇ight+iω)2+ω2 1e−iωtdω 2π.(17) Details of calculating the integral in Eq.(17) are pre- sented in Appendix C. Taking into account Eq.(17) one obtains /angb∇acketleftm×ζ′/angb∇acket∇ighty=/angb∇acketleftmzζ′ x−mxζ′ z/angb∇acket∇ight=−σ′2L(βω2).(18) Thefluctuatingspincurrenthasonlyonenonzerocompo- nent, i.e., the ycomponent – similarly as the spin pump- ing current does, /angb∇acketleftIy fl/angb∇acket∇ight=MsV γσ′2L(βω2). (19)We emphasize that when calculating the spin pumping current,wedidnotemployalinearizationprocedure. Ac- cordingly, the expression for the spin pumping current, Eq.(13), is valid for an arbitrary deviation of the magne- tization from the ground state magnetic order, even for thermally assistedmagnetization-reversalinstability pro- cesses, meaning the transversal components mx, mycan be arbitrarily large. On the other hand, the expression for the fluctuation spin current, Eq.(19), was obtained upon a linearization near the equilibrium point, as de- scribed at the beginning of this paragraph. Taking into account the above derived formula Eq.(13) and Eq.(19) for spin pumping and fluctuation currents, respectively, we deduce the following expression of the total spin cur- rent: /angb∇acketleftIy tot/angb∇acket∇ight=MsV γL(βω2)/bracketleftbig α′σ2ω1 ω2+σ′2/bracketrightbig .(20) WhenHeff= (0,H0,0), where H0is the external mag- netic field orientedalongthe yaxis[5], then using Eq.(3), Eq.(10) and Eq.(A2) one obtains from Eq.(20), /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV(αH0+(α−ηξ)HR−HSH) αkBTm F/parenrightbigg ×/parenleftbiggα(H0+HR+αHSH)Tm F αH0+(α−ηξ)HR−HSH−TN/parenrightbigg , (21) whereη= 0,1 and we inspect in the following the η= 0 case. We analyze now in more details Eq.(21) for η= 0 and for several asymptotic cases. Let us begin with the case of a negligible spin Hall effect. Assuming a small HSH, HSH≪αHR, αH0, we derive from Eq.(21) the spin cur- rent in the following two regimes: (i) The low temper- ature regime, MsV(H0+HR)/kBTm F≫1, and (ii) the high temperature regime, MsV(H0+HR)/kBTm F≪1. These two regimes can be equivalently referred to as the high and weak magnetic field limits, respectively. In particular, in the low temperature limit, upon tak- ing into account the property of the Langevin function, L/parenleftbig x/parenrightbig = coth/parenleftbig x/parenrightbig −1/x,L/parenleftbig x≫1/parenrightbig ≈1, we find that the spin current depends neither on the SO coupling nor on the external magnetic field, /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kB(Tm F−TN), and is solely determined by the temperature bias. In the low-temperature regime (strong magnetic field), the magnetic fluctuations are small and the spin current is then linear in the averaged square of these fluctuations. The latter in turn are linear in the relevant temperature. Accordingly, the spin current is proportional to the tem- perature bias. In the high-temperature limit (or equiva- lently a small magnetic field), the magnetic fluctuations are relatively large. Taking into account the asymptotic limit of the Langevin function, L/parenleftbig x≪1/parenrightbig ≈x/3, in the high magnon temperature limit, the spin current is /angb∇acketleftIy tot/angb∇acket∇ight= (2/3)α′MsV(H0+HR)(Tm F−TN)/Tm F. Thus, the spin current is reduced by the factor ( H0+HR)/Tm F, which decreases with increasing magnon temperature or decreasing magnetic field. Note, the spin current is en-5 hanced when the Rashba and the external fields are par- allel and is reduced in the antiparallel case. Remark- ably, the saturation of the spin current is observed in the high magnon temperature limit, Tm F≫TN, where /angb∇acketleftIy tot/angb∇acket∇ight ≈(2/3)α′MsV(H0+HR). Let us assume now a sizable spin-Hall field that can- not be neglected. The first specific case is when HSH≈ α/parenleftbig H0+HR/parenrightbig . Taking into account the asymptotic limit of the Langevin function, L/parenleftbig x≪1/parenrightbig ≈x/3 in the high magnon temperature limit, MsV/parenleftbig αH0+αHR− HSH/parenrightbig /αkBTm F≪1, one finds the following expression for the spin current: /angb∇acketleftIy tot/angb∇acket∇ight=2 3α′MsV/bracketleftbigg/parenleftbig H0+HR+αHSH/parenrightbig −TN Tm F/parenleftbig αH0+αHR−HSH/parenrightbig α/bracketrightbigg . (22) Sinceα/parenleftbig H0+HR/parenrightbig ≈HSH, the second term for any finite TN/Tm Fin Eq.(22) is small and can be neglected. Thus, the saturated spin current is /angb∇acketleftIy tot/angb∇acket∇ight=2 3α′MsV(H0+HR+αHSH).(23) The expression for the saturated spin current, Eq.(23), doesnotdependonthetemperature. However,Eq.(23)is valid only if the magnon temperature Tm Fis high enough. Thus, by tuning the applied external magnetic field H0, a nonzero spin pumping current can be achieved at arbi- trary and even at equal temperatures Tm F=TN. For the opposite external and Rashba fields, H0=−HR, from Eq.(21) follows /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsVHSH αkBTm F/parenrightbigg/parenleftbig α2Tm F+TN/parenrightbig .(24) The obtained result is remarkable as it shows that the net pumping current is finite at arbitrary nonzero tem- peratures Tm FandTNin the absence of the applied tem- perature gradient. Finally we explore the case when the fields are com- parableHSH≈HR≈H0, and since α <1,HSH≫ αHR, αHR. /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV(αH0+αHR−HSH) αkBTm F/parenrightbigg ×/braceleftbiggα/parenleftbig H0+HR+αHSH)Tm F αH0+αHR−HSH)−TN/bracerightbigg . The expression of the total spin current in this case reads /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsVHSH αkBTm F/parenrightbigg ×/braceleftbiggα/parenleftbig H0+HR/parenrightbig Tm F HSH+TN/bracerightbigg . (25) In the low magnon temperature limit we deduce /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kB/braceleftbiggα/parenleftbig H0+HR/parenrightbig Tm F HSH+TN/bracerightbigg ,(26)/s122 /s121 /s78/s111/s114/s109/s97/s108 /s77/s101/s116/s97/s108/s79/s120/s105/s100/s101 /s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s105/s99 /s84 /s78 /s49/s84 /s70/s73 /s102/s108 /s73 /s83/s80/s72 /s82/s44/s32/s83/s72 /s72/s101/s97/s118/s121 /s77/s101/s116/s97/s108/s106 /s97/s120 /s84 /s78 /s50/s78/s111/s114/s109/s97/s108 /s77/s101/s116/s97/s108 /s73 /s83/s80/s73 /s102/s108/s78/s111/s114/s109/s97/s108 /s77/s101/s116/s97/s108 FIG. 2. Schematic illustration of the N/F/N System. The ferromagnetic film is attached to two nonmagnetic layers, N1 on the left and N2on the right side. The temperatures of the layersN1andN2are different. Other notation as in Fig.1. while in the high magnon temperature limit one finds /angb∇acketleftIy tot/angb∇acket∇ight=2 3MsV/braceleftbigg/parenleftbig H0+HR/parenrightbig +HSH αTN TFm/bracerightbigg .(27) As we see from Eq.(26),(27) the role of the field HSHis different. In the lowmagnontemperaturelimit it reduces the spin pumping current, while in the high magnontem- perature limit it enhances the fluctuating spin current. In the analytical calculation, we assumed that temper- atures of the magnon subsystem and normal metal are fixed during the process. However, this is an approxima- tion because the temperatures of the subsystems change slightly during the equilibration process. For illustra- tion, we consider the case when the external and Rashba fields,H0andHR, are parallel and we neglect the spin Hall term. Then from Eq.(21) we deduce: /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV/parenleftbig H0+HR/parenrightbig kBTm F/parenrightbigg/parenleftbig Tm F−TN/parenrightbig .(28) Apparently the total spin current is zero when Tm F=TN. However, the magnon temperature Tm Fthat we used for derivation of the Fokker-Planck equation, is the initial magnon temperature. The electric current due to the Rashba field modifies the magnon density and magnon temperature, leading to a slight difference in effective magnon temperatures Tm F/parenleftbig ja/parenrightbig −Tm F/parenleftbig ja= 0/parenrightbig =δTm F. This correction is beyond the Fokker-Planck equation. Therefore, due to the temperature correction δTm F, in the numerical calculations, we expect to obtain a fi- nite net current even when the initial magnon temper- ature is equal to the temperature of the normal metal, Tm F/parenleftbig ja= 0/parenrightbig =TN. IV. SPIN CURRENT IN THE N/F/N STRUCTURE6 The same method has been utilized to calculate the spin current in the N/F/N system shown schematically in Fig.2. We calculate the spin current defined as the difference of spin currents flowing through the two inter- faces, Itot=Itot1−Itot2 (29) Itot1=Ifl1+Isp1,Itot2=Ifl2+Isp2. HereItot1andItot2is the total spin current in the first and second interfaces. The total spin current includes four terms. Two terms Isp1andIsp2describe the spin pumping currents from the ferromagnetic layer to the leftN1and to the right N2metallic layers, respectively. In turn, the terms Ifl1andIfl2describe the fluctuat- ing spin currents flowing from the left and right metallic layers towards the ferromagnetic layer. We assume that the two metals have different temperatures TN1andTN2. Thespinpumpingcurrentflowingfromtheferromagnetic layer towards metallic layers ( i= 1,2) reads /angb∇acketleftIy sp1/angb∇acket∇ight=α′(TN1)MsV γ2ω1 βω2L(βω2),(30) /angb∇acketleftIy sp2/angb∇acket∇ight=−α′(TN2)MsV γ2ω1 βω2L(βω2). In turn, the fluctuating currents have the components /angb∇acketleftIy fl1/angb∇acket∇ight= 2α′(TN1)kBTN1L(βω2),(31) /angb∇acketleftIy fl2/angb∇acket∇ight=−2α′(TN2)kBTN2L(βω2). As we can see from Eq.(30) and Eq.(31), the difference in the two components of the spin pumping current and fluctuating current is related to the temperature depen- dence of the damping constant α′(TN). For convenience we denote α′(TN1) =α′andα′(TN2) =α′+∆α. If the difference between the temperatures of the metals TN1 andTN2is not too large, the variation of the damping constant ∆ α′is very small |∆α′|/α′<<1 [51, 52]. In such a case /angb∇acketleftIy tot1/angb∇acket∇ight= 2α′kBL(βω2)/parenleftbigg αω1 ω2Tm F+TN1/parenrightbigg ,(32) /angb∇acketleftIy tot2/angb∇acket∇ight=−2α′kBL(βω2)/parenleftbigg αω1 ω2Tm F+TN2/parenrightbigg , and total spin current: /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kBL(βω2)/parenleftbigg 2αω1 ω2Tm F+TN1+TN2/parenrightbigg .(33) WhenHeff= (0,H0,0), then using Eq.(3), Eq.(10) and Eq.(A2) one obtains from Eq.(33): /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV(αH0+(α−ηξ)HR−HSH) αkBTm F/parenrightbigg ×/parenleftbigg 2α(H0+HR+αHSH)Tm F αH0+(α−ηξ)HR−HSH−TN1−TN2/parenrightbigg .(34)Whenη= 0 andHSH≪αH0,αHRfrom Eq.(34) we get: /angb∇acketleftIy tot/angb∇acket∇ight= 2α′kBL/parenleftbiggMsV(H0+HR) kBTm F/parenrightbigg ×(35) /parenleftbig 2Tm F−TN1−TN2/parenrightbig . Again we see that the larger is the difference between magnon and metal temperatures, the larger is the total spin current. V. EFFECT OF DM INTERACTION In order to explore the role of DM interaction, we per- formed micromagnetic simulations for a finite-size N/F system. To be more specific, we study Pt/Co/AlO where the Co layer is 500 nm ×50 nm large with a thickness of 10 nm. The Co layer is sandwiched between Pt and AlO films. The parameters describe the Co layer: the satura- tion magnetization of Ms= 106A/m, and the damping constant α= 0.2. The Rashba field, HR=αRP µBµ0Msja, can be estimated assuming P= 0.5,αR= 10−10eVm, and the spatially uniform current density jaalong the x axis of the order of 1012A/m2. Due to the structure of the Rashba field, an increase in the magnitude of current densityjais formally equivalent to the corresponding in- crease in the SO constant αR. Thus, the dependence of the total spin current on the current density jais equiv- alent to the dependence of the total spin current on the SO constant αR. In Fig. (3), the total spin current is plotted as a func- tion of the electric current density ja(assumed negative), for the case when the external magnetic field H0and the Rashbafield HRareparallel. When ja= 0, the totalspin currentissolelythespinSeebeckcurrentandisabsentfor equal temperatures, Tm F/parenleftbig ja= 0/parenrightbig =TN. However, when |ja|>0, the total spin current is nonzero, as well. As it wasalreadymentioned above,the reasonofanonzeronet spin current is a slight shift of the magnon temperature, Tm F/parenleftbig |ja|>0/parenrightbig −Tm F/parenleftbig ja= 0/parenrightbig =δTm Fand of the magnon density, that occur due to the charge current ja. Ap- parently, in case of antiparallel Rashba HRand external H0magnetic fields, the total net spin current decreases with increasing magnitude of the charge current. This numerical result is consistent with the analytical results obtained in the previous section. As we see, the effect of the DM interaction is diverse: when δTm F>0 and spin current is positive/angbracketleftbig Iy tot/angbracketrightbig =/angbracketleftbig Iy sp/angbracketrightbig +/angbracketleftbig Iy fl/angbracketrightbig >0,/angbracketleftbig Iy fl/angbracketrightbig <0 (i.e. ferromagnetic layer is hotterthan the normal metal layer), the DM interaction enhances the current. How- ever, in the case δTm F<0, when fluctuating spin current is larger than the spin pumping current, and the total net current is negative/angbracketleftbig Iy tot/angbracketrightbig <0, the DM interaction re- duces the spin current. This means that the Rashba HR field alwayshas a positive contributionto the spin pump- ing current. The situation is the same when the spin Hall effect is included, see Fig. (4). As one can see, the spin Hall effect has the opposite effect, it always decreases the7 /s48/s46/s48 /s45/s51/s46/s48/s120/s49/s48/s49/s49 /s45/s54/s46/s48/s120/s49/s48/s49/s49 /s45/s57/s46/s48/s120/s49/s48/s49/s49/s45/s49/s46/s48/s120/s49/s48/s45/s55/s48/s46/s48/s49/s46/s48/s120/s49/s48/s45/s55/s50/s46/s48/s120/s49/s48/s45/s55/s32 /s32/s73/s121 /s116/s111/s116/s40/s74/s47/s109/s50 /s41 /s106 /s97/s32/s40/s65/s47/s109/s50 /s41/s32/s84 /s32/s61/s32/s53/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 /s32/s84 /s32/s61/s32/s49/s48/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 /s32/s84 /s32/s61/s32/s49/s53/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 /s32/s84 /s32/s61/s32/s53/s48/s75/s44/s32 /s68 /s32/s61/s32/s48 /s32/s84 /s32/s61/s32/s49/s48/s48/s75/s44/s32 /s68 /s32/s61/s32/s48 /s32/s84 /s32/s61/s32/s49/s53/s48/s75/s44/s32 /s68 /s32/s61/s32/s48 FIG. 3. Total spin current Iy totin the absence of the spin Hall effect ( θSH= 0), plotted as a function of the electric current density ja. The external field H0and the Rashba fieldHRare parallel. The magnon temperature is T≡Tm F= 50 K (squares line), 100 K (circles line) and 150 K (triangles line). The DMI constant is assumed D= 0 (solid dots) and D=−2.4×10−3J/m2(open dots). The external magnetic fieldH0= 4×105A/m and the normal metal temperature TN= 50 K are assumed. spin pumping current. Therefore for δTm F>0 the total spin current without the spin Hall effect is larger, while forδTm F<0 it is smaller. Finally, we consider the case when the Rashba field HRand the external magnetic H0field are parallel, see Fig. (4). Note that a switching of the direction of the magnetic field alters the ground state magnetic order. Therefore, the spin current changes sign. As we see from Fig. (5), the spin current increases with the electric cur- rent density |ja|. This result is also consistent with the analytical result obtained in the previous section. In order to see the effect of magnetocrystalline anisotropy, we repeated the calculations with the anisotropy term being included. Results of the calcula- tions, plotted in Fig.(6), Fig. (7), and Fig. (8) show that the magnetocrystalline anisotropy has no significant in- fluence on the spin current, so the effects discussed above hold in the presence of the anisotropy, as well. VI. CONCLUSIONS In this paper, we have considered two different het- erostructuresconsistingofathinferromagneticfilmsand- wiched between heavy-metal and oxide layers. Interfac- ing the ferromagnetic layer to the heavy metal may re- sult in spin Hall torque exerted on the magnetic moment, while at the interface of the oxide material a spin-orbit coupling of Rashba type emerges. Both factors (the spin Hall effect and the Rashba spin-orbit coupling) have a significant influence on the magnetic dynamics, and thus also on the spin pumping current. The total spin cur-/s48/s46/s48 /s45/s51/s46/s48/s120/s49/s48/s49/s49 /s45/s54/s46/s48/s120/s49/s48/s49/s49 /s45/s57/s46/s48/s120/s49/s48/s49/s49/s45/s56/s46/s48/s120/s49/s48/s45/s56/s45/s52/s46/s48/s120/s49/s48/s45/s56/s48/s46/s48/s52/s46/s48/s120/s49/s48/s45/s56/s56/s46/s48/s120/s49/s48/s45/s56 /s32/s32/s73/s121 /s116/s111/s116 /s40/s74/s47/s109/s50 /s41 /s106 /s97/s32/s40/s65/s47/s109/s50 /s41/s32 /s83/s72/s68 /s32 /s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 /s32 /s83/s72/s68 /s32 /s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 FIG. 4. The total spin current Iy totwithout thecontributionof the spin Hall effect ( θSH= 0, squares line) and with the spin Halleffect( θSH= 0.08, triangles line), plottedasafunctionof the electric current density ja. The external field H0and the Rashba field HRare parallel. The DM interaction constant is assumed D= 0 (solid dots) and D=−2.4×10−3J/m2 (open dots). The external magnetic field H0= 4×105A/m, the magnon temperature Tm F= 100 K, and the normal metal temperature TN= 50 K are assumed. /s48/s46/s48 /s45/s50/s46/s48/s120/s49/s48/s49/s48 /s45/s52/s46/s48/s120/s49/s48/s49/s48 /s45/s54/s46/s48/s120/s49/s48/s49/s48 /s45/s56/s46/s48/s120/s49/s48/s49/s48/s45/s54/s46/s48/s120/s49/s48/s45/s55/s45/s52/s46/s48/s120/s49/s48/s45/s55/s45/s50/s46/s48/s120/s49/s48/s45/s55/s48/s46/s48 /s32/s32/s73/s121 /s116/s111/s116 /s40/s74/s47/s109/s50 /s41 /s106 /s97/s32/s40/s65/s47/s109/s50 /s41/s32 /s83/s72/s68 /s32/s61/s32 /s32 /s83/s72/s68 /s32/s61/s32 /s32 /s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 FIG. 5. The total spin current Iy totwith the spin Hall ef- fect (θSH= 0.08, circles line) and without spin Hall effect (θSH= 0, squares line), plotted as a function of the electric current density ja. The external field H0and the Rashba fieldHRare antiparallel. The DMI constant is assumed D= 0 (solid dots) and D=−2.4×10−3J/m2(open dots). The external magnetic field is H0=−9×105A/m, the magnon temperature is TF= 100 K, and the temperature of normal metal is TN= 50 K. rent crossing the ferromagnetic/normal-metal interface has two contributions: the spin current pumped from the ferromagnetic metal to the normal one, and the spin fluctuating currentflowingin the opposite direction. The spin Hall effect and the Rashba spin-orbit coupling influ- enceonlyspin pumpingcurrentandthereforeimpactalso8 /s48/s46/s48 /s45/s51/s46/s48/s120/s49/s48/s49/s49 /s45/s54/s46/s48/s120/s49/s48/s49/s49 /s45/s57/s46/s48/s120/s49/s48/s49/s49/s45/s49/s46/s48/s120/s49/s48/s45/s55/s48/s46/s48/s49/s46/s48/s120/s49/s48/s45/s55/s50/s46/s48/s120/s49/s48/s45/s55/s32 /s32/s73/s121 /s116/s111/s116/s40/s74/s47/s109/s50 /s41 /s106 /s97/s32/s40/s65/s47/s109/s50 /s41/s32/s84 /s32/s61/s32/s53/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 /s32/s84 /s32/s61/s32/s49/s48/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 /s32/s84 /s32/s61/s32/s49/s53/s48/s75/s44/s32 /s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 /s32/s84 /s32/s61/s32/s53/s48/s75/s44/s32 /s68 /s32/s61/s32/s48 /s32/s84 /s32/s61/s32/s49/s48/s48/s75/s44/s32 /s68 /s32/s61/s32/s48 /s32/s84 /s32/s61/s32/s49/s53/s48/s75/s44/s32 /s68 /s32/s61/s32/s48 FIG. 6. Total spin current Iy totin the absence of spin Hall contribution ( θSH= 0), plotted as a function of the electric current density ja. The local magnetization and the Rashba field are parallel. The magnon temperature is T≡Tm F= 50 K (squares line), 100 K (circles line) and 150 K (tri- angles line). The DMI constant D= 0 (solid dots) and D=−2.4×10−3J/m2(open dots). The magnetocrystalline anisotropy constant Ky= 3×105J/m3and the normal metal temperature TN= 50 K. The effective anisotropy field is Hani= 2Kymyey/(µ0Ms). the total spin current. We explored the spin Seebeck cur- rent beyond the linear response regime, and found the following interesting features: if the external magnetic fieldH0is parallel to the Rashba SO field HR, then the SO coupling enhances the spin current, in the case of an antiparallel magnetic field H0and a Rashba SO field HR, the SO coupling decreases the spin current. The spin Hall effect and the DM interaction always increase the spin pumping current. The results are confirmed an- alyticallybymeansofthe Fokker-Planckequationand by directmicromagneticnumericalcalculationsforaspecific sample. ACKNOWLEDGMENTS This work is supported by the DFG through the SFB 762 and SFB-TRR 227 and by the National Research Center in Poland as a research project No. DEC- 2017/27/B/ST3/02881. Appendix A: Derivation of the Fokker-Plank equation For the derivation of the Fokker-Plank equation, we follow Ref.[53] and use the functional integration method in order to average the dynamics over all possible real- izations of the random noise field. First we rewrite LLG/s48/s46/s48 /s45/s51/s46/s48/s120/s49/s48/s49/s49 /s45/s54/s46/s48/s120/s49/s48/s49/s49 /s45/s57/s46/s48/s120/s49/s48/s49/s49/s45/s56/s46/s48/s120/s49/s48/s45/s56/s45/s52/s46/s48/s120/s49/s48/s45/s56/s48/s46/s48/s52/s46/s48/s120/s49/s48/s45/s56/s56/s46/s48/s120/s49/s48/s45/s56 /s32/s32/s73/s121 /s116/s111/s116 /s40/s74/s47/s109/s50 /s41 /s106 /s97/s32/s40/s65/s47/s109/s50 /s41/s32 /s83/s72/s68 /s32/s61/s32 /s32 /s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 /s32 /s83/s72/s68 /s32/s61/s32 /s32 /s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 FIG. 7. Total spin current Iy totin the absence of spin Hall effect (θSH= 0, squares line) and with the Hall effect( θSH= 0.08, triangles line), plotted as a function of the electric cu r- rent density ja, for the case when the local magnetization and the Rashba field are parallel. The DM interaction constant D= 0 (solid dots) and D=−2.4×10−3J/m2(open dots). The magnetocrystalline anisotropy constant Ky= 3×105 J/m3, the magnon temperature Tm F= 100 K and the normal metal temperature TN= 50 K. The effective anisotropy field isHani= 2Kymyey/(µ0Ms). equation (Eq.(5)) in the form: dm dt=−m×(ω1+ζ(t))+m×m×ω2,(A1) where ω1=ωeff+ωR+αωSH, ω2=−αωeff−αωR+ηξωR+ωSH,(A2) ωeff=γHeff,ωR=γHR,ωSH=γHSH, andγ→γ/(1 +α2). Here ζ(t) is a random Langevin field with the following correlation properties: /angb∇acketleftζ(t)/angb∇acket∇ight= 0, (A3) /angb∇acketleftζi(t)ζj(t′)/angb∇acket∇ight=σ2δijδ(t−t′). (A4) We introduce the probability distribution function of the random Gaussian noise ζ: F[ζ(t)] =1 Zζexp/bracketleftbigg −1 σ2/integraldisplay+∞ −∞dτζ2(τ)/bracketrightbigg ,(A5) whereZζ=/integraltext DζFis the noise partition function. With the help of Eq.(A5) the average of any noise functional Aζcan be written as /angb∇acketleftA[ζ]/angb∇acket∇ightζ=/integraldisplay DζA[ζ]F[ζ]. (A6) Considering the obvious identity: δζατ δζβ(t)=δαβδ(τ−t), (A7)9 /s48/s46/s48 /s45/s50/s46/s48/s120/s49/s48/s49/s48 /s45/s52/s46/s48/s120/s49/s48/s49/s48 /s45/s54/s46/s48/s120/s49/s48/s49/s48 /s45/s56/s46/s48/s120/s49/s48/s49/s48/s45/s54/s46/s48/s120/s49/s48/s45/s55/s45/s52/s46/s48/s120/s49/s48/s45/s55/s45/s50/s46/s48/s120/s49/s48/s45/s55/s48/s46/s48 /s32/s32/s73/s121 /s116/s111/s116 /s40/s74/s47/s109/s50 /s41 /s106 /s97/s32/s40/s65/s47/s109/s50 /s41/s32 /s83/s72/s68 /s32/s61/s32 /s32 /s83/s72/s68 /s32/s61/s32 /s32 /s83/s72/s68 /s32/s61/s32 /s50/s46/s52/s32/s51 /s32/s74/s47/s109/s50 FIG. 8. The total spin current Iy totwith the spin Hall effect (θSH= 0.08, circles line) and without spin Hall effect ( θSH= 0, squares line), plotted as a function of the electric curre nt densityja. The local magnetization and the Rashba field are antiparallel. DMIconstant D=0(soliddots)and D=−2.4× 10−3J/m2(open dots). The external magnetic field H0= −5×105A/m, the magnetocrystalline anisotropy constant Ky= 3×105J/m3, the magnon temperature Tm F= 100 K, and the normal metal temperature TN= 50 K. The effective anisotropy field is Hani= 2Kymyey/(µ0Ms). we can calculate first and second variations of F[ζ(t)]: δF[ζ] δζα(t)=−1 σ2ζα(t)F[ζ], (A8) δ2F[ζ] δζα(t)δζβ(t′)= [1 σ4ζα(t)ζβ(t′)−1 σ2δαβδ(t−t′)]F[ζ]. (A9) For arbitrary nwe have: /integraldisplay DζδnF[ζ] δζα1(t1)δζα2(t2)...δζαn(tn)= 0.(A10) TakingintoaccountEq.(A8)toEq.(A10), weobtain(A3) and (A4). Now, we introduce the distribution function: f(N,t) =/angb∇acketleftπ([ζ],t)/angb∇acket∇ightζ,π([ζ],t) =δ(N−m(t)),(A11) on the sphere |N|= 1. Taking into account the relation [53]˙π=−∂π ∂N˙m(t) and the equation ofmotion, Eq.(A1), we deduce the Fokker-Plank equation: ∂f ∂t=∂ ∂N[(N×ω1)−(N×N×ω2) +N×/angb∇acketleftζ(t)π([ζ],t)/angb∇acket∇ightζ]. (A12) To calculate /angb∇acketleftζ(t)π([ζ],t)/angb∇acket∇ightζwe use the standard proce- dure, discussed for example in Ref. [53], and obtain /angb∇acketleftζ(t)π([ζ],t)/angb∇acket∇ightζ=−σ2 2N×∂f ∂N.(A13)The Fokker-Plank equation in the final form reads ∂f ∂t=∂ ∂N[(N×ω1) −(N×N×ω2)−σ2 2N×∂f ∂N].(A14) The stationary solution of the Fokker-Plank equation whenω1||ω2has the form f(N) =e−2 σ2/integraltext dN·ω2 /integraltext dNe−2 σ2/integraltext dN·ω2. (A15) Appendix B: Mean values of magnetization Exploiting the parametrization: mx= sinθcosϕ,my= sinθcosϕ,mz= cosθ, 0≤θ≤π,0≤ϕ≤2π, (B1) and taking into account Eq.(A15) and the parametriza- tion Eq.(B1), we can write the probability distribution form: dw(θ,ϕ) =1 Zf(θ,ϕ)dm, (B2) f(θ,ϕ) = exp(−βω2sinθsinϕ), dm= sinθdθdϕ,β =2 σ2. HereZ=4πsin(βω2) βω2is the partition function. From Eq.(B2) we can calculate the mean values of the mag- netization: /angb∇acketleftmx/angb∇acket∇ight=/angb∇acketleftmz/angb∇acket∇ight= 0,/angb∇acketleftmy/angb∇acket∇ight=−L(βω2) (B3) /angb∇acketleftm2 x/angb∇acket∇ight=/angb∇acketleftm2 z/angb∇acket∇ight=L(βω2) βω2,/angb∇acketleftm2 y/angb∇acket∇ight= 1−2L(βω2) βω2 /angb∇acketleftmxmy/angb∇acket∇ight=/angb∇acketleftmzmy/angb∇acket∇ight= 0. Appendix C: Derivation of Eq.(17) To calculate (17) we utilize the Jourdan’s lemma, /integraldisplay+∞ −∞ω2/angb∇acketleftmy/angb∇acket∇ight+iω (ω2/angb∇acketleftmy/angb∇acket∇ight+iω)2+ω2 1e−iωtdω 2π(C1) =/integraldisplay+∞ −∞ω2/angb∇acketleftmy/angb∇acket∇ight−iω (ω2/angb∇acketleftmy/angb∇acket∇ight−iω)2+ω2 1eiωtdω 2π =/braceleftbigg −1 2(e−i(ω1+iω2/angbracketleftmy/angbracketright)t+ei(ω1−iω2/angbracketleftmy/angbracketright)t) ift >0, 0 ift <0. Thisintegralisdiscontinuousat t= 0, thereforethevalue /angb∇acketleftm(t)×ζ′(0)/angb∇acket∇ightyatt= 0 is given by the average of the values at t= 0±. Therefore, from Eq.(C1) we deduce Eq.(18) /angb∇acketleftm(0)×ζ′(0)/angb∇acket∇ighty=σ′2/angb∇acketleftmy/angb∇acket∇ight=−σ′2L(βω2).10 [1] Y. A. Bychkov and E. I. Rasbha, P. Zh. Eksp. Teor. Fiz., 39, 66 (1984). [2] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). [3] Y.J. Lin, K. Jimenez-Garcia, I. B. Spielman, Nature 471, 83 (2011). [4] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91, 126405 (2003). [5] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda- Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonfim, A. Schuhl, and G. Gaudin, Nature Mater. 10, 419 (2011). [6] P. P. J. Haazen, E. Mure, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Nature Mater. 12, 299 (2013). [7] A. A. Kovalev, V. A. Zyuzin, and B. Li, Phys. Rev. B 95, 165106 (2017). [8] U. Ritzmann, D. Hinzke, and U. Nowak, Phys. Rev. B 95, 054411 (2017). [9] P. Yan, G. E. W. Bauer, and H. Zhang, Phys. Rev. B 95, 024417 (2017). [10] J. Barker and G. E. W. Bauer, Phys. Rev. Lett. 117, 217201 (2016). [11] K. I. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M. Ishida, S. Yorozu, S. Maekawa E. Saitoh Proceedings of the IEEE 104, 1946 (2016). [12] E. J. Guo, J. Cramer, A. Kehlberger, C. A. Ferguson, D. A. MacLaren, G. Jakob, and M. Kl¨ aui, Phys. Rev. X 6, 031012 (2016). [13] M. Schreier, F. Kramer, H. Huebl, S. Gepr¨ ags, R. Gross, S. T. B. Goennenwein, T. Noack, T. Langner, A. A. Serga, B. Hillebrands, and V. I. Vasyuchka, Phys. Rev. B93, 224430 (2016). [14] S. R. Boona, S. J. Watzman, and J. P. Heremans, APL Mater.4, 104502 (2016). [15] V. Basso, E. Ferraro, and M. Piazzi, Phys. Rev. B 94, 144422 (2016). [16] G. Lefkidis and S. A. Reyes, Phys. Rev. B 94, 144433 (2016). [17] S. M. Rezende, A. Azevedo, and R. L. Rodriguez-Suarez, J. Phys. D: Appl. Phys. 51, 174004 (2018). [18] M. Bialek, S. Brechet and J. P. Ansermet, J. Phys. D Appl. Phys. 51, 164001 (2018). [19] J. M. Bartell, C. L. Jermain, S. V. Aradhya, J. T. Brang- ham, F. Yang, D. C. Ralph, and G. D. Fuchs, Phys. Rev. Applied 7, 044004 (2017). [20] H. Yua, S. D. Brechetb, J. P. Ansermet, Phys. Lett. A 381, 825 (2017); L. Karwacki, J. Magn. Magn. Mater. 441, 764 (2017). [21] S. R. Etesami, L. Chotorlishvili, A. Sukhov, and J. Be- rakdar Phys. Rev. B 90, 014410 (2014); S. R. Etesami, L. Chotorlishvili, J. Berakdar, Appl. Phys. Lett. 107, 132402 (2015). [22] L. Chotorlishvili, X.-G. Wang, Z. Toklikishvili, and J . Berakdar, Phys. Rev. B 97, 144409 (2018). [23] X.-G. Wang, L. Chotorlishvili, G.-H. Guo, A. Sukhov, V. Dugaev, J. Barna´ s, and J. Berakdar, Phys. Rev. B 94, 104410 (2016); X.-G Wang, L. Chotorlishvili, G.-H Guo, J. Berakdar, J. Phys. D: Appl. Phys. 50, 495005 (2017); X.-G. Wang, L. Chotorlishvili, and J. Berakdar, Front. Mater. 4, 19 (2017); X.-G.Wang, Z.-X. Li, Z.-W.Zhou, Y.-Z. Nie, Q.-L Xia, Z.-M Zeng, L. Chotorlishvili, J. Berakdar, and G.-H. Guo, Phys. Rev. B 95, 020414(R) (2017). [24] X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201 (2012); A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009). [25] B. Gu, I. Sugai, T. Ziman, G. Y. Guo, N. Nagaosa, T. Seki, K. Takanashi, and S. Maekawa, Phys. Rev. Lett. 105, 216401 (2010). [26] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). [27] E. Martinez, S. Emori, and G. S. D. Beach, Appl. Phys. Lett.103, 072406 (2013). [28] K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee, Phys. Rev. B 85, 180404(R) (2012). [29] S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Nature Mater. 12, 611 (2013). [30] R. Lavrijsen, P. P. Haazen, E. Mure, J. H. Franken, J. T. Kohlhepp, H. J. M. Swagten, and B. Koopmans, Appl. Phys. Lett. 100, 262408 (2012). [31] Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984). [32] P. Gambardella and I. M. Miron, Philos. Trans. R. Soc. London, Ser. A 369, 3175 (2011); I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, P. Gambardella, Nature Mater. 9, 230 (2010). [33] M. Dyakonov and V. Perel, JETP Lett. 13, 467 (1971). [34] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). [35] B. Gu, I. Sugai, T. Ziman, G. Y. Guo, N. Nagaosa, T. Seki, K. Takanashi, and S. Maekawa, Phys. Rev. Lett. 105, 216401 (2010). [36] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). [37] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). [38] K. Kondou, H. Sukegawa, S. Mitani, K. Tsukagoshi, and S. Kasai, Appl. Phys. Express 5, 073002 (2012). [39] S.-M. Seo, K.-W. Kim, J. Ryu, H.-W. Lee, and K.-J. Lee, Appl. Phys. Lett. 101, 022405 (2012). [40] Z. Li and S. Zhang, Phys. Rev. B 69, 134416 (2004). [41] N. Okuma and K. Nomura Phys. Rev. B 95, 115403 (2017). [42] F. Mahfouzi and N. Kioussis Phys. Rev. B 95, 184417 (2017) [43] B. K. Nikolic, K. Dolui, Marko Petrovi´ c, P. Plechaˇ c, T . Markussen, K. Stokbro arXiv:1801.05793. [44] E. Saitoh and K. I. Uchida, Spin Seebeck effect, Spin Current, EditedbySadamichiMaekawa, Sergio O.Valen- zuela, Eiji Saitoh, Takashi Kimura, Oxford University press, (2012). [45] K. I. Uchida, S. Takahashi, K. Harii, J. leda, W. Koshibee, K. Ando, S. Maekawa, E. Saitoh, Nature Let- ters,455, 778 (2008). [46] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev B 81, 214418 (2010). [47] J. Foros, A. Brataas, Y. Tserkovnyak and G. E. Bauer, Phys. Rev. Lett. 95, 016601 (2005). [48] Y. Tserkovniak, A. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [49] H. Adachi, K. I. Uchida, E. Saitoh and S. Maekawa, Rep. Prog. Phys. 76, 036501 (2013).11 [50] L. Chotorlishvili, Z. Toklikishvili, V. K. Dugaev, J. Barna´ s, S. Trimper and J. Berakdar, Phys. Rev B 88, 144429, (2013). [51] X. Joyeux, T. Devolder, Joo-von kim, Y. Gomez de la Torre, S. Eimer, C. Chappert, J. Appl. Phys. 110, 063915 (2011).[52] L. Chotorlishvili, Z. Toklikishvili, S. R. Etesami, V. K. Dugaev, J. Barna´ s, J. Berakdar, J. Magn. Magn. Mater. 396, 254 (2015). [53] D. A. Garanin, Phys. Rev. B 55, 3050 (1997).

1801.08349v2.Spin_relaxation_anisotropy_in_a_nanowire_quantum_dot_with_strong_spin_orbit_coupling.pdf

Spin-relaxation anisotropy in a nanowire quantum dot with strong spin-orbit coupling Zhi-Hai Liu ( 刘志海)1and Rui Li ( 李睿)2, 1Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, Beijing 100193, China 2Key Laboratory for Microstructural Material Physics of Hebei Province, School of Science, Yanshan University, Qinhuangdao 066004, China (Dated: June 30, 2021) We study the impacts of the magnetic eld direction on the spin-manipulation and the spin- relaxation in a one-dimensional quantum dot with strong spin-orbit coupling. The energy spectrum and the corresponding eigenfunctions in the quantum dot are obtained exactly. We nd that no matter how large the spin-orbit coupling is, the electric-dipole spin transition rate as a function of the magnetic eld direction always has a periodicity. However, the phonon-induced spin relaxation rate as a function of the magnetic eld direction has a periodicity only in the weak spin-orbit coupling regime, and the periodicity is prolonged to 2 in the strong spin-orbit coupling regime. I. INTRODUCTION In recent decades, the spin-orbit couling (SOC) in III- V semiconductor materials has promoted great advances in the studies of spintronics. For instance, the pseudospin qubit in a spin-orbit coupled quantum dot is control- lable by an external ac electric-eld via electric-dipole spin resonant (EDSR) [1{14]. Furthermore, a spin-orbit coupled nanowire epitaxially covered by superconductors has been proved to be a promising system for search- ing the Majorana quasiparticles [15{18]. Thus, from the viewpoint of both the fundamental science and the prac- tical applications, an accurate understanding of the SOC eect in quantum system becomes important. There are two kinds of SOCs in III-V semiconductor materials: the Dresselhaus SOC generated by the bulk in- version asymmetry and the Rashba SOC induced by the structure inversion asymmetry [19{21]. Moreover, by ex- ploiting the electric-eld dependence of the Rashba SOC in semiconductor nanostructures [22, 23], it provides a promising method for investigating the strong SOC ef- fect in quantum system. In semiconductor quantum dot, in order to observe the nontrivial SOC eect one should rst break the time-reversal symmetry by applying an external mag- netic eld [10, 18]. In the presence of both the mag- netic eld and the SOC, only a few models are exact solvable. For example, an analytic solution for a two- dimensional (2D) quantum dot with hard-wall conning potential was given in Ref. [24{26]. The exact energy spectrum and wavefunctions of a 1D square well quan- tum dot were given in Ref. [27]. In all of the above mod- els, the magnetic eld direction is xed. However, from both the theoretical and the experimental viewpoints, the magnetic eld direction plays an important role for the observable SOC eect in quantum dot [8{11, 28{33]. It is desirable to clarify the in uences of the magnetic eld direction on the spin properties when SOC is strong. ruili@ysu.edu.cn (b)A nanowire quantum dot x 0 -a aV(x) (a) xz oB ϕFIG. 1. (a) The schematic diagram of a nanowire quantum dot with large SOC. (b) The conning potential of the quan- tum dot along the wire axis (x-axis), and an external magnetic eldBapplied on the xzplane. In this paper, we obtain exactly the eigen-energies and -functions of an electron conned in an 1D quantum dot with large SOC. Our special interest is focused on the interplay between the SOC and the magnetic eld di- rection. When the magnetic eld direction is rotated on a plane, we study both the electric-dipole transition rate and the phonon-induced relaxation rate between the lowest Zeeman sublevels. The anisotropy of the eective Land e g-factor is revealed [34]. Here, in order to facilitate the study of the in uence of the magnetic-eld direction on the SOC eects, the original g-factor is assumed to be a constant, i.e., the corresponding bulk value. We nd that no matter how large the SOC is, the Rabi frequency as a function of the magnetic eld direction always has a periodicity. While for the phonon-induced spin relax- ation rate [35], with the increase of the SOC, the period- icity of the relaxation rate changes from in the weak SOC regime to 2 in the strong SOC regime.arXiv:1801.08349v2 [cond-mat.mes-hall] 18 Jul 20182 II. THE MODEL We consider a spin-orbit coupled quasi-1D quantum dot, where an electron conned in an innite square well and subject to an external Zeeman eld [11]. The model under consideration is shown schematically in Fig. 1. The nanowire material can be chosen as those with strong SOC, e.g., InAs and InSb. Note that our approach is also applicable to materials with weak SOC. As illustrated in Fig. 1, the conning potential along the axial direction is modeled by an innite square well V(x) =( 1;jxj>a; 0;jxja;(1) whereais the half-width of the potential well. In the presence of an external magnetic eld applied on the xz planeB=B(cos';sin'), the Hamiltonian describing the nanowire quantum dot reads [28] H=p2 2me+pz+
2(xcos'+zsin') +V(x); (2) wheremeis the electron eective mass, p=i~@=@x is the canonical momentum along the wire, is the Rashba SOC strength,
= gBBcorresponds to the Zeeman splitting (with gandBbeing the Land e factor and the Bohr magneton, respectively [27, 36]), and 'is the mag- netic eld direction. It should be noted that, in the pres- ence of the magnetic eld, there is a vector potential term Ax=(y=2)Bsin'. However, for a quasi-1D quantum dot, we can set y= 0 because the motion of the electron is only allowed in the axial direction [28, 37]. We rst give the boundary condition of our model. Be- cause the conning potential is innite outside the well. Thus, the electron is strictly conned inside the well and the wave function is zero at the boundary sites (a) = 0; (3) where (x) = [ "(x) #(x)]Tis the eigenfunction of the quantum dot, with ";#(x) being its two components. In experiments, the quantum-dot SOC depends mostly on both the material parameters and the external electric eld [19, 22, 23]. As an explicit example, in our following calculations, we have chosen the InSb as our nanowire material [38, 39]. Unless otherwise specied, the model parameters are listed in Table. I. III. THE ENERGY SPECTRUM AND THE WAVE FUNCTIONS Inside the well, the Hamiltonian can be reduced to the following bulk Hamiltonian [ V(x) = 0 in Eq. (2)] Hb=p2 2me+pz+
2(xcos'+zsin'):(4)TABLE I. The relevant parameters of the InSb nanowire quantum dot we are considering. Most of the parameter val- ues are taken from Refs. 38 and 39. me=m0ag a (nm) B(T)xso(nm)' 0:013650:6 50 0 :05 40200 02 D(eV)cl(m/s)(kg=m3)l(kg=m)b 6.6 3690 5774.7 1.8142 1012 am0is the electron mass. bl=r2 0, withr0= 10nm being the radius of the nanowire. The eigenstates of the bulk Hamiltonian can be obtained by solving the bulk Schr odinger equation Hb (x) = E (x). Specically, there are several kinds of bulk wave functions with respect to the energy region [27]. In our following calculations, we focus on the energy region where only bulk plane-wave solutions are allowed. By solving the bulk Schr odinger equation, we nd there are four independent plane-wave solutions 1;2(x) =eik1;2x cos1;2 sin1;2 ; 3;4(x) =eik3;4x sin3;4 cos3;4 ; (5) wherek1;2;3;4is a function of the energy E(the detailed expressions are given in Appendix A) and i=1 2arctan
cos' 2~ki+
sin' : (6) Each independent solution does not satisfy the hard-wall boundary conditions in Eq.(3), i.e., i(a)6= 0. How- ever, a linear combination of all the degenerate bulk wave functions can fulll the boundary condition [24{ 27]. Therefore, the eigenstate of Hamiltonian (2) can be written as (x) =4X i=1ci i(x); (7) whereciare the coecients to be determined. Imposing the hard-wall boundary conditions on ( x), we obtain an equation array for the coecients ci:M
C= 0, whereC= [c1c2c3c4]Tand the detailed expression of Mis given in Appendix A. The matrix Mnow is only a function of E. The condition that there exists nontrivial solution reads Det[M] = 0: (8) Indeed, Eq. (8) indicates an implicit transcendental equa- tion forE, and the roots of this equation give us the energy spectrum of the quantum dot. Once the energy spectrum is obtained, we can obtain the coecients ci by solving M
C= 0, such that the corresponding eigen- functions can be obtained. Let 0(x) and 1(x) be the two lowest eigenstates in the quantum dot, and the corresponding energies are E03 0 � 2� ϕ1.64 1.60 1.56 1.52 1.48() meV E−36 −40 −44 −48 −52 effecttive g-factor(a) (b)E0E1 Magnetic field direction 0.000.060.12()2 0| |xΨ ()2 1| |xΨ↑ ↓ ↑ ↓ −1.0−0.50.00.51.00.000.060.12 /xa FIG. 2. (a) The two lowest energy levels E0andE1as a function of the magnetic eld direction 'for the SOC lengthxso= 50 nm. The eective g-factor is dened as ge(E0E1)=(BB). (b) The probability density distri- butions of the lowest two eigenstates j 0(x)j2andj 1(x)j2 for magnetic direction '==6. The solid lines represent the spin-up components and the dashed lines correspond to the spin-down components. andE1, respectively ( E0< E 1). When the SOC length xso~=(me) is chosen as xso= 50 nm, the lowest two energy levels as a function of the magnetic eld di- rection'are shown in Fig. 2(a). The eective g-factor ge(E1E0)=(BB) as a function of the angle 'is also given. When the Zeeman eld is perpendicular to the spin-orbit eld, i.e., '= 0,, and 2, the eective Zeeman splitting reaches its minimum and gebecomes maximal [28]. When the Zeeman eld is parallel to the spin-orbital eld, i.e., '==2, 3=2, the eective Zee- man splitting reaches its maximum and geequals to the bulk value ( ge=50:6). We also show the probabil- ity density distribution in the quantum dot for the two lowest eigenstates 0(x) and 1(x) [see Fig. 2(b)]. As can be seen from the gure, for a general magnetic eld direction'==6, the eigenfunction contains both the spin-up component and the spin-down component. The spin-up component is dominant in the ground state and the spin-down component is dominant in the rst excited state. 0 � 2�0.01.2 0.6 0.0 2.24.4(b)xso= 200 nm ϕ Magnetic field direction(a) 310 ( )− ↓↑×Ω310 ( )− ↓↑×Ωxso= 40 nmmaxima minimaFIG. 3. The electric-dipole spin transition rate #", in unit of eEa=h, as a function of the magnetic eld direction ', under dierent SOC strengths. Panel (a) show the result for SOC lengthxso= 200 nm, and panel (b) show the result for SOC lengthxso= 40 nm. IV. ELECTRIC-DIPOLE SPIN RESONANCE In the presence of an external magnetic eld, the res- onant electric-dipole spin transition rate in the quan- tum dot was usually calculated using approximated wave functions, either the SOC or the Zeeman eld was treated perturbatively [7, 9{12]. Here, in our exactly solvable model, the dependence of the Rabi frequency on the mag- netic led direction is investigated. When an alternating electric eld is applied along the x-axis, the electric-driving Hamiltonian reads Hed=p2 2me+pz+gB 2B
+V(x) +eExcos!t; (9) whereEand!are the amplitude and frequency of the alternating eld, respectively. Generally, under a small ac electric eld the electric-dipole interaction can be regarded as a perturbation [9, 10], and the resonant electric-dipole transition rate, i.e., the Rabi frequency, can be calculated: ij=eE hZa a y i(x)x j(x)dx; (10) withhbeing the Plank constant. In the rest of this paper, we only consider the electric-dipole transition between the lowest Zeeman sublevels [40]. The spin- ip transition rate #", in unit of eEa=h, as a function of the magnetic direction is shown in Fig. 3. Figure 3(a) shows the result in the weak SOC regime xso>a, and Fig. 3(b) shows the result in the strong SOC regimexso< a[41]. When the Zeeman eld is perpen- dicular to the spin-orbital eld, the spin and the orbital degrees of freedom are hybridized to maximal, such that4 when'= 0,, and 2, the Rabi frequency reaches its maximum. When the Zeeman eld is parallel to the spin- orbit eld, there is no mixing of the spin and the orbital degrees of freedom, i.e., the operator zis a conserved quantity, such that the Rabi frequency becomes zero at the sites'==2 and 3=2. No matter how large the SOC is, we nd that the Rabi frequency as a function of the magnetic eld direction always has a periodicity [see Fig. 3]. V. THE PHONON-INDUCED SPIN RELAXATION On the one hand, the presence of SOC facilitates the manipulation of the electron spin, on the other hand, the SOC also mediates an spin-phonon interaction, which is harmful to the spin lifetime [42{50]. Here we study the dependence of the phonon-induced relaxation rate on the magnetic eld direction in the quantum dot. Due to the high excitation energy of the optical phonons, the phonon-induced spin relaxation in a semi- conductor quantum dot is almost always caused by the acoustic phonons [51{55]. Moreover, for the energetically close two levels, i.e., the lowest Zeeman sublevels, the multi-phonon transition induced by anharmonic phonon terms can also be ignored [56, 57]. Generally, there are two kinds of acoustic electron-phonon (e-ph) interactions: the piezoelectric interaction and the deformation poten- tial interaction [58{62]. For narrow-gap semiconductor materials with strong Rashba SOC and large g-factor, the phonon-induced relaxation is dominated by the de- formation potential phonons [63, 64]. The Hamiltonian describing the e-ph deformational interaction reads [65] Heph=X q~ 2l!qL1=2 eiqxDjqj(bq+by q);(11) wherelis the mass density of the nanowire, Lis the length of the nanowire, Dis the deformation potential coupling strength, b(by) denotes the phonon annihilation (creation) operator, qand!qcorrespond to the wave vec- tor and angular frequency of the acoustic wave. Thus, the total Hamiltonian describing the quantum-dot-phonon system reads Ht=H+Heph+X q~!qby qbq: (12) The phonon-induced relaxation rate between the energy levelsiandjcan be calculated by using the Fermi golden rule [52, 53] ij=D2q2 2l~!qcljWij(q)j2[n(T) + 1](~!q
ij):(13) Here
ij=jEiEijis the energy dierence between the relevant levels, clis the wave velocity, n(T) = 0.500 0.250 0.0000.070 0.035 0.000 0 � 2� ϕ Magnetic field direction() MHz↓↑Γ () MHz↓↑Γ(a) (b)xso= 200 nm xso= 40 nmmaxima minimaFIG. 4. The phonon-induced spin relaxation rate #"as a function of the magnetic eld direction ', under dierent SOC strengths. Panel (a) shows the result in the weak SOC regime withxso= 200 nm; while panel (b) shows the result in the strong SOC regime with xso= 40 nm. [exp(
ij=kBT)1]1is the average phonon number, and the electron-phonon matrix element Wij(q) is given by Wij(q) =Za a y j(x)eiqx i(x)dx: (14) In the case of low temperature, kBT
ij, the aver- age phonon number n(T)0. Because the exact wave functions i(x) and j(x) are already obtained, the tran- sition element Wij(q) can be calculated accurately, and hence the relaxation rate ij. More specically, the spin relaxation rate #"between the two lowest energy levels as a function of the angle ' is shown in Fig. 4. When the magnetic eld is parallel to SOC eld, i.e., '==2 and 3=2, there is no spin relaxation #"= 0 due to the fact that zis a good quantum number. For a relatively weak SOC xso= 200 nm, at the sites '= 0,, and 2, the relaxation rate reaches its maximal value [see Fig. 4(a)]. The magnetic eld dependence in this case is very similar to that of the Rabi frequency shown in Fig. 3. However, when the SOC is strong, i.e., xso= 40 nm, we nd that the sites for the maximal relaxation rate are a little bit deviation from'=and 2[see Fig. 4(b)]. This is actually a strong SOC eect in the quantum dot. Specially, we can expand the electron-phonon operator as follows eiqx= 1 +iqx+1 2(iqx)2+


: (15) When the SOC is weak, the contributions from the high- order terms (/x2or higher orders) to the transition ele- ment (14) are negligible, such that the relaxation rate and the Rabi frequency share the same periodicity. When the SOC becomes strong, the contributions from the high- order terms become important, such that the sites for5 the maximal relaxation rate deviate from the sites of the weak SOC. Therefore, the relaxation rate as a function of the angle 'shows a 2period, in stark contrast to a period in the weak SOC regime [31, 32, 55]. VI. CONCLUSION In this paper, we analytically solve the 1D hard-wall quantum dot problem in the presence of both the strong SOC and the magnetic eld. The EDSR and the phonon- induced spin relaxation are studied in details with specic interest focused on the interplay between the SOC and the magnetic eld direction. In dierent SOC regimes, we nd that the phonon-induced spin relaxation rate showsdierent periodic oscillation over the magnetic direction. The 2periodicity can be served as a signature of the strong SOC eect in quantum dot. The results of our calculations will help clarify the in uence of magnetic eld direction on the spin- manipulation and the spin-relaxation in quantum dot un- der the eect of strong SOC. ACKNOWLEDGEMENTS This work was supported by National Natural Sci- ence Foundation of China (grant No. 11404020) and Postdoctoral Science Foundation of China (grant No. 2014M560039). Appendix A: The detailed expressions of kiand M In this Appendix, the detailed expressions of the wave vectors ki(i= 1;2;3;4) as a function of the energy Eare presented and the detailed form of the matrix Mis also given. Expand the bulk Hamiltonian Hbin Eq. (4) in the spin space j"i;j#i , the bulk Schr odinger equation Hb (x) = E (x) can be rewritten as p2 2me+
2sin'+p
2cos'
2cos'p2 2me
2sin'p! (x) =E (x); (A1) where we have used the identities: zj"i=j"i,zj#i=j#i ,xj"i=j#i, andxj#i=j"i. The eigenstate (x) is assumed to have the form of (x) =eikx 1 2 : (A2) Substituting Eq. (A2) into Eq. (A1), we have ~2k2 2me+
2sin'+~kE
2cos'
2cos'~2k2 2me
2sin'~kE! eikx 1 2 = 0: (A3) Mathematically, the condition that there exists a nontrivial solution to Eq. (A3) reads Det ~2k2 2me+
2sin'+~kE
2cos'
2cos'~2k2 2me
2sin'~kE! = 0: (A4) Essentially, Eq. (A4) implies a quartic equation of k ~4k4 4m2e~2E me+2~2 k2~
sin'k+E2
2 4= 0: (A5) After some tedious algebra, Eq. (A5) can be rewritten as a product of two factor (k2+k0k+)(k2k0k+) = 0; (A6) where = (k3 0+fk0j)=(2k0); = (k3 0+fk0+j)=(2k0): (A7)6 Herek0is a root of the following equation k6 0+ 2fk4 0+ (f24r)k2 0j2= 0; (A8) where the parameters j=4m2 e
sin'=~3; r= 4E2
2
m2 e=~4; f=4 Eme+2m2 e
=~2: (A9) One solution of Eq. (A8) can be written as k0=s 2f2p f2+ 12rcos 3; (A10) where the angle =1 3arccos" 2f3+ 72fr27j2 2p (f2+ 12r)3# : (A11) Then, we can obtain four independent solutions to Eq. (A6) k1=k0+p k2 04 2; k 2=k0p k2 04 2; k3=k0+p k2 04 2; k 4=k0p k2 04 2; (A12) where the complicated dependences of the wave vectors kionEcan be re ected by Eq. (A9). In the following, the detailed expression for the matrix Mis given. Substituting the eigenfunction [given in Eq. (7)] into the boundary condition [see Eq. (3)], we obtain c1eik1acos1+c2eik2acos2+c3eik3asin3+c4eik4asin4= 0; c1eik1asin1+c2eik2asin2c3eik3acos3c4eik4acos4= 0; c1eik1acos1+c2eik2acos2+c3eik3asin3+c4eik4asin4= 0; c1eik1asin1+c2eik2asin2c3eik3acos3c4eik4acos4= 0: (A13) The above equation array can be written as matrix equation M
C= 0, where the matrix Mreads M=0 BB@eik1acos1eik2acos2eik3asin3eik4asin4 eik1asin1eik2asin2eik3acos3eik4acos4 eik1acos1eik2acos2eik3asin3eik4asin4 eik1asin1eik2asin2eik3acos3eik4acos41 CCA: (A14) It should be noted that ialso depends on ki(i= 14) [see Eq. (6)], such that matrix Monly depends on the energyE[see Eqs. (A9) and (A12)]. The condition there exists nontrivial solution for the coecients Creads Det[M] = 0: (A15) Solving this equation, we can obtain the exact energy spectrum of the quantum dot. [1] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L. M. K. Vandersypen, Science 318, 1430 (2007).[2] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Nature 468, 1084 (2010).7 [3] K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung, J. M. Taylor, A. A. Houck, and J. R. Petta, Nature 490, 380 (2012). [4] X. Hu, Y.-X. Liu, and F. Nori, Phys. Rev. B 86, 035314 (2012). [5] E. I. Rashba, Phys. Rev. B 78, 195302 (2008). [6] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys. Rev. B 76, 161308(R)(2007). [7] P. Stano and J. Fabian, Phys. Rev. B 77, 045310 (2008). [8] S. Nadj-Perge, V. S. Pribiag, J. W. G. van den Berg, K. Zuo, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Phys. Rev. Lett. 108, 166801 (2012). [9] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91, 126405 (2003). [10] V. N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B 74, 165319 (2006). [11] R. Li, J. Q. You, C. P. Sun, and F. Nori, Phys. Rev. Lett. 111, 086805 (2013); R. Li, Phys. Scr. 91, 055801 (2016). [12] J. Fan, Y. Chen, G. Chen, L. Xiao, S. Jia, and F. Nori, Sci. Rep. 6, 38851 (2016). [13] Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha, Phys. Rev. Lett. 96, 047202 (2006). [14] D. V. Khomitsky, L. V. Gulyaev, and E. Ya. Sherman, Phys. Rev. B 85, 125312 (2012). [15] M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M. Leijinse, K. Flensberg, J. Nyg ard, P. Krogstrup, and C. M. Marcus, Science 354, 1557 (2016). [16] S. Li, N. Kang, P. Caro, and H. Q. Xu, Phys. Rev. B 95, 014515 (2017). [17] Roman M. Lutchyn, Jay D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010). [18] Jay D. Sau and S. Das Sarma, Nature Commun. 3, 964 (2012). [19] R. Winkler, Spin-orbit Coupling Eects in Two- dimensional Electron and Hole Systems, (Springer, Berlin, 2003). [20] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [21] Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984). [22] D. Liang and X. Gao, Nano Lett. 12(6), 3263-3267 (2012). [23] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997). [24] E. N. Bulgakov and A. F. Sadreev, JETP Lett. 73, 505 (2001). [25] E. Tsitsishvili, G. S. Lozano, and A. O. Gogolin, Phys. Rev. B 70, 115316 (2004). [26] G. A. Intronati, P. I. Tamborenea, D. Weinmann, and R. A. Jalabert, Phys. Rev. B 88, 045303 (2013). [27] R. Li, Phys. Rev. B 97, 085430 (2018); R. Li, Z.-H. Liu, Y. Wu, and C. S. Liu, Sci. Rep. 8, 7400 (2018). [28] M. P. Nowak and B. Szafran, Phys. Rev. B 87, 205436 (2013). [29] F. Dolcini and L. Dell'Anna, Phys. Rev. B 78, 024518 (2008). [30] V. I. Falko, B. L. Altshuler, and O. Tsyplyatyev, Phys. Rev. Lett. 95, 076603 (2005). [31] P. Scarlino, E. Kawakami, P. Stano, M. Shaei, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, Phys. Rev. Lett. 113, 256802 (2014). [32] A. Hofmann, V. F. Maisi, T. Kr ahenmann, C. Reichl, W. Wegscheider, K. Ensslin, and T. Ihn, Phys. Rev. Lett.119, 176807 (2017). [33] J.-T. Hung, E. Marcellina, B. Wang, A. R. Hamilton, and D. Culcer, Phys. Rev. B 95, 195316 (2017). [34] Y. Tomohiro and M. Eto, International Symposium on Access Spaces: 36-40, (2011). [35] M. Borhani and X. Hu, Phys. Rev. B 85, 125132 (2012). [36] Y. Y. Wang and M. W. Wu, Phys. Rev. B 77, 125323 (2008). [37] Z.-H. Liu, R. Li, X. Hu, and J. Q. You, Sci. Rep. 8, 2302 (2018). [38] R. de Sousa and S. Das Sarma, Phys. Rev. B 68, 155330 (2003). [39] C. F. Destefani and S. E. Ulloa, Phys. Rev. B 72, 115326 (2005). [40] J. H. Jiang and M. W. Wu, Phys. Rev. B 75, 035307 (2007). [41] X. Liu, X.-J. Liu, and J. Sinova, Phys. Rev. B 84, 035318 (2011). [42] A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61, 12639 (2000); Phys. Rev. B 64, 125316 (2001). [43] A. Kha, R. Joynt, and D. Culcer, Appl. Phys. Lett.107, 172101 (2015). [44] S. Amasha, K. MacLean, Iuliana P. Radu, D. M. Zumb uhl, M. A. Kastner, M. P. Hanson, and A. C. Gos- sard, Phys. Rev. Lett. 100, 046803 (2008). [45] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys. Rev. Lett. 99, 036801 (2007). [46] T. Meunier, I. T. Vink, L. H. Willems van Beveren, K.-J. Tielrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz, W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van- dersypen, Phys. Rev. Lett. 98, 126601 (2007). [47] A. Bermeister, D. Keith, and D. Culcer, Appl. Phys. Lett.105, 192102 (2014). [48] J. Jing, P. Huang, and X. Hu, Phys. Rev. A 90, 022118 (2014). [49] L. M. Woods, T. L. Reinecke, and Y. Lyanda-Geller, Phys. Rev. B 66, 161318(R) (2002). [50] D. Q. Wang, O. Klochan, J.-T. Hung, D. Culcer, I. Far- rer, D. A. Ritchie, and A. R. Hamilton, Nano Lett. 16(12), 7685-7689 (2016). [51] M. Trif, V. N. Golovach, and D. Loss, Phys. Rev. B 77, 045434 (2008). [52] P. Stano and J. Fabian, Phys. Rev. Lett. 96, 186602 (2006); Phys. Rev. B 74, 045320 (2006). [53] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 (2004). [54] O. Olendski and T. V. Shahbazyan, Phys. Rev. B 75, 041306(R) (2007). [55] C. Segarra, J. Planelles, J. I. Climente, and F. Rajadell, New. J. Phys 17, 033014 (2015). [56] T. Inoshita and H. Sakaki, Solid-State Electronics 37, 1175-1178 (1994). [57] I. A. Dmitriev and R. A. Suris, Phys. Rev. B 90, 155431 (2014). [58] Y. Y. Wang and M. W. Wu, Phys. Rev. B 74, 165312 (2006). [59] J. L. Cheng, M. W. Wu, and C. Lu, Phys. Rev. B 69, 115318 (2004). [60] A. Poudel, L. S. Langsjoen, M. G. Vavilov, and R. Joynt, Phys. Rev. B 87, 045301 (2013). [61] M. Raith, P. Stano, F. Barua, and J. Fabian, Phys. Rev. Lett. 108, 246602 (2012). [62] S. Prabhakar, R. Melnik, and L. L. Bonilla, Phys. Rev. B87, 235202 (2013).8 [63] C. L. Romano, G. E. Marques, L. Sanz and A. M. Al- calde, Phys. Rev. B 77, 033301 (2008). [64] A. M. Alcalde, Q. Fanyao, and G. E. Marques, Physica E (Amsterdam) 20, 228 (2004).[65] A. N. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications, (Springer, Berlin, 2003).

2403.09112v1.Spin_Orbit_Coupled_Insulators_and_Metals_on_the_Verge_of_Kitaev_Spin_Liquids_in_Ilmenite_Heterostructures.pdf

Spin-Orbit Coupled Insulators and Metals on the Verge of Kitaev Spin Liquids in Ilmenite Heterostructures Yi-Feng Zhao,1,∗Seong-Hoon Jang,2and Yukitoshi Motome1,† 1Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan 2Institute for Materials Research, Tohoku University, Aoba, Sendai, 980-8577, Japan Competition and cooperation between electron correlation and relativistic spin-orbit coupling give rise to diverse exotic quantum phenomena in solids. An illustrative example is spin-orbit entan- gled quantum liquids, which exhibit remarkable features such as topological orders and fractional excitations. The Kitaev honeycomb model realizes such interesting states, called the Kitaev spin liquids, but its experimental feasibility is still challenging. Here we theoretically investigate hexag- onal heterostructures including a candidate for the Kitaev magnets, an ilmenite oxide MgIrO 3, to actively manipulate the electronic and magnetic properties toward the realization of the Kitaev spin liquids. For three different structure types of ilmenite bilayers MgIrO 3/ATiO 3with A= Mn, Fe, Co, and Ni, we obtain the optimized lattice structures, the electronic band structures, the stable magnetic orders, and the effective magnetic couplings, by combining ab initio calculations and the effective model approaches. We find that the spin-orbital coupled bands characterized by the pseu- dospin jeff= 1/2, crucially important for the Kitaev-type interactions, are retained in the MgIrO 3 layer for all the heterostructures, but the magnetic state and the band gap depend on the types of heterostructures as well as the Aatoms. In particular, one type becomes metallic irrespective of A, while the other two are mostly insulating. We show that the insulating cases provide spin-orbit coupled Mott insulating states with dominant Kitaev-type interactions, accompanied by different combinations of subdominant interactions depending on the heterostructural type and A, while the metallic cases realize spin-orbit coupled metals with various doping rates. Our results indicate that these hexagonal heterostructures are a good platform for engineering electronic and magnetic prop- erties of the spin-orbital coupled correlated materials, including the possibility of Majorana Fermi surfaces and topological superconductivity. I. INTRODUCTION Strong electron correlations, represented as Coulomb repulsion U, play a pivotal role in 3 dtransition metal compounds and lead to a plethora of intriguing phenom- ena, such as the Mott transition and high-temperature superconductivity [1, 2]. The other key concept of quan- tum materials, the spin-orbit coupling (SOC), repre- sented as λ, is a relativistic effect entangling the spin degree of freedom and the orbital motion of electrons, which is an essential ingredient in the topological insu- lators [3, 4]. Beyond their independent effects, synergy between Uandλhas attracted increasing attention re- cently due to the emergence of new states of matter, such as axion insulators [5, 6] and topological semimetals [7– 9]. In general, it is difficult for the SOC to dramatically influence the electronic properties in 3 dtransition metal compounds since λis much smaller than U. However, when proceeding to 4 dand 5 dsystems, the dorbitals are spatially more spread out, which reduces U, and at the same time, the relativistic effect becomes larger for heavier atoms, enhancing λ. Hence, in these systems, the competition and cooperation between Uandλplay a decisive role in their electronic states and allow us to access the intriguing regime that yields the exotic corre- lated states of matter [10]. ∗zyf@g.ecc.u-tokyo.ac.jp †motome@ap.t.u-tokyo.ac.jpOne of the striking examples is the spin-orbit coupled Mott insulator, typically realized in the iridium oxides with Ir4+valence, e.g., Sr 2IrO4[11, 12]. In each Ir ion located in the center of the IrO 6octahedron, the crystal field energy, which is significantly larger than Uandλ, splits the dorbital manifold into low-energy t2gand high- energy egones. For Sr 2IrO4, five electrons occupy the t2g orbitals and make the system yield the t5 2glow-spin state. Usually, the partially-filled orbital causes a metallic state according to the conventional band theory, but the insu- lating state was observed in experiments [13]. Consider- ing the large SOC, the t2gmanifold continues to split into high-energy doublet characterized with the pesudospin jeff= 1/2 and low-energy quartet with jeff= 3/2; the latter is fully occupied and the former is half filled. Fi- nally, a Mott gap is opened in the half-filled jeff= 1/2 band by U. This accounts for the insulating nature of the system, and the Mott insulating state realized in the spin-orbital coupled bands is called the spin-orbit cou- pled Mott insulator. The quantum spin liquid (QSL), one of the most exotic quantum states in the spin-orbit coupled Mott insulators, has received increasing attention due to the emergence of remarkable properties, e.g., fractional excitations [14] and topological orders [15]. In the QSL, long-range mag- netic ordering is suppressed down to zero temperature due to strong quantum fluctuations, though the localized magnetic moments are quantum entangled [16–19]. The presence of fractional quasiparticles that obey the non- abelian statistics is not only of great fundamental phys-arXiv:2403.09112v1 [cond-mat.str-el] 14 Mar 20242 ical research, but also promising toward quantum com- putation [20]. In general, one route to realizing the QSL depends on geometrical frustration. Indeed, experiments have evidenced several candidates of QSL in antiferro- magnets with lattice structures including triangular unit, where magnetic frustration is the common feature [21– 23]. The other route to the QSL is the so-called exchange frustration caused by the conflicting constraints between anisotrpic exchange interactions [24]. The strong spin- orbital entanglement in the spin-orbit coupled Mott in- sulators, in general, gives rise to spin anisotropy, offering a good playground for the exchange frustration, even on the lattices without geometrical frustration. The Kitaev model is a distinctive quantum spin model realizing the exchange frustration [25]. The model is de- fined for S=1 2local magnetic moments on the two- dimensional (2D) honeycomb lattice with the Ising-type bond-dependent anisotropic interactions, whose Hamil- tonian is given by H=X αX ⟨i,j⟩αKαSα iSα j, (1) where α=x, y, z denote the three bonds of the honey- comb lattice, and Kαis the exchange coupling constant on the αbond; the sum of ⟨i, j⟩αis taken for nearest- neighbor sites of iandjon the αbonds. In this model, the orthogonal anisotropies on the x, y, z bonds provide the exchange frustration. Importantly, the model is ex- actly solvable, and the ground state is a QSL with frac- tional excitations, itinerant Majorana fermions and local- ized Z 2fluxes [25]. Moreover, the anyonic excitations in this model hold promise for applications in quantum com- puting [25, 26]; especially, non-Abelian anyons, which fol- low braiding rules similar to those of conformal blocks for the Ising model, appear under an external magnetic field. The exchange frustration in the Kitaev-type interac- tions can be realized in real materials when two con- ditions are met [27]. First, at each magnetic ion, the spin-orbit coupled Mott insulating state with pseu- dospin jeff= 1/2 should be realized. Second, the pseu- dospins need to interact with each other through the lig- ands shared by neighboring octahedra which forms edge- sharing network. Over the past decade, enormous efforts have been devoted to exploring the candidate materials for the Kitaev QSL that meet these conditions [28–31]. Fortunately, dominant Kitaev-type interactions were in- deed discovered in several materials, such as A2IrO3with A= Li and Na [32–38] and α-RuCl 3[39–46]. Not only the 5dand 4 dcandidates, 3 dtransition metal compounds like Co oxides have also been investigated [47, 48]. In addi- tion to these examples with ferromagnetic (FM) Kitaev interactions, the antiferromagnetic (AFM) Kitaev QSL candidates were also predicted by ab initio calculations, e.g., f-electron based magnets [49, 50] and polar spin- orbit coupled Mott insulators α-RuH 3/2X3/2with X= Cl and Br in the Janus structure [51]. Although a plethora of Kitaev QSL candidates have been investigated, those realizing the Kitaev QSL in theground state are still missing, since a long-range magnetic order due to parasitic interactions such as the Heisenberg interaction hinders the Kitaev QSL. Considerable efforts have been dedicated to suppressing the parasitic inter- actions and/or enhancing the Kitaev-type interaction. One way is to utilize heterostructures that incorporate the Kitaev candidates. For example, the Kitaev interac- tion is promoted more than 50% for the heterostructure composed of 2D monolayers of α-RuCl 3and graphene compared to the pristine α-RuCl 3, predicted by ab initio calculations [52]. The heterostructures between a 2D α- RuCl 3and three-dimensional (3D) topological insulator BiSbTe 1.5Se1.5evidenced the charge transfer phenomena, albeit the magnetic properties were not reported [53]. Within the realm of Kitaev heterostructures, remarkably few studies have been designed for the composite 3D/3D superlattices due to the fabrication challenge [54]. To date, few attempts [55] have been made to investigate the development of the electronic band structure and the magnetic properties, particularly whether the Kitaev in- teraction is still dominant when constructing the 3D/3D heterostructures using Kitaev QSL candidates. In this paper, we theoretically study the electronic and magnetic properties in bilayer heterostructures as an interface of 3D/3D superlattices using a recently- synthesized iridium ilmenite MgIrO 3[56] and other il- menite magnets ATiO 3with A= Mn, Fe, Co, and Ni as the substrate. This material choice is motivated by two key considerations: (i) All of these materials have been successfully synthesized in experiments, which is helpful for the epitaxial growth of multilayer heterostructures, and (ii) the ilmenite MgIrO 3is identified as a good can- didate for Kitaev magnets [57, 58]. We consider three configurations of heterostructures, classified by type-I, II, and III, which are all chemically allowed due to the char- acteristics of the alternative layer stacking in ilmenites, as shown in Fig. 1. The electronic band structures, mag- netic ground states, and the effective exchange interac- tions are systematically investigated by employing the combinatorial of ab initio calculations, construction of the effective tight-binding model, and perturbation ex- pansions. We find that (i) the spin-orbit coupled bands characterized by the effective pseudospin jeff= 1/2, a key demand for Kitaev-type interactions, are still pre- served in the MgIrO 3layer for all types of the het- erostructures, (ii) type-I and III heterostructures realize spin-orbit coupled Mott insulators excluding Mn type- III, whereas type-II ones are spin-orbit coupled metals with doped jeff= 1/2 bands with various carrier concen- trations, and (iii) in almost all of the insulating cases, the Kitaev-type interactions are predominant, whereas the forms and magnitudes of the other parasitic interac- tions depend on the specific types of the heterostructures and the Aatoms. The structure of the remaining article is as follows. In Sec. II, we provide a detailed description of the opti- mized lattice structures of MgIrO 3/ATiO 3heterostruc- tures with A= Mn, Fe, Co, and Ni. In Sec. III, we3 introduce the methods employed in this work, including the means for structural optimization and the ab initio calculations with LDA+SOC+ Uscheme (Sec. III A), the estimates of the effective transfer integral and construc- tion of the multiorbital Hubbard model (Sec. III B), and the second-order perturbation that is used in the estimation of exchange interactions (Sec. III C). In Sec. IV, we systematically display the results of the elec- tronic band structures for three types of heterostruc- ture MgIrO 3/ATiO 3. In Sec. IV A, we present the elec- tronic band structures for the paramagnetic state ob- tained by LDA+SOC, together with the projected den- sity of states (PDOS) derived by the maximally-localized Wannier function (MLWF). In Sec. IV B, we discuss the stable magnetic states within LDA+SOC+ Uand show their band structures and PDOS. In Sec. V, we derive the effective exchange coupling constants for the het- erostructures for which the LDA+SOC+ Ucalculations suggest spin-orbit coupled Mott insulating nature, and show their location in the phase diagram for the K-J-Γ model. In Sec. VI, we discuss the possibility of the real- ization of Majorana Fermi surfaces (Sec. VI A) and exotic superconducting phases (Sec. VI B) in the heterostruc- tures, and the feasibility of these heterostructures in ex- periments (Sec. VI C). Section VII is devoted to the sum- mary and prospects. In Appendix A, we present the de- tails of the energy difference of the magnetic orders and the effective exchange couplings between the Aions. We present additional information on orbital projected band structures for different specific types of heterostructures ofAatoms in Appendix B and the band structures of monolayer MgIrO 3in Appendix C. II. HETEROSTRUCTURES MgIrO 3andATiO 3both belong to ilmenite oxides ABO3with R¯3 space group. The lattice structure con- sists of alternative stacking of honeycomb layers with edge-sharing AO6octahedra and those with BO6octa- hedra. The common stacking layer structures reduce the lattice mismatch to form the heterostructures and also make them feasible to fabricate in experiments. In this study, we consider heterostructures composed of mono- layer of MgIrO 3andATiO 3with the balance chemical formula, to clarify the interface effect on the electronic properties of 3D/3D superlattices. Specifically, we con- struct three types of heterostructures, distinguished by the intersurface atoms and pertinent octahedra in the middle layer, labeled as type-I, II, and III and shown in Fig. 1. For type-I, the top and bottom layers are made of honeycomb networks of IrO 6and TiO 6octa- hedra, respectively, whereas the sandwiching honeycomb layer is formed of alternating MgO 6andAO6octahedra. In the type-II, the bottom layer is replaced by AO6hon- eycomb layer, resulting in a mixture of MgO 6and TiO 6 in the middle layer. The type-III has a similar constitu- tion of top and bottom layers to type-I, while the middle type-III IrMgATi(a) (b) (c)TiMg/AIr AMg/Ti IrIr ATitype-I type-II acbabcFIG. 1. Schematic pictures of crystal structures for three types of the heterostructures MgIrO 3/ATiO 3: (a) type-I, (b) type-II, and (c) type-III with A= Mn, Fe, Co, or Ni. The left and right panels show the side views and the bird’s- eye views, respectively. The type-I is composed of the top honeycomb layer with edge-sharing IrO 6octahedra and the bottom honeycomb layer of TiO 6, sandwiching a honeycomb layer of alternating MgO 6andAO6octahedra. In the type- II, the bottom is replaced by the AO6honeycomb layer, leaving a mixture of MgO 6and TiO 6in the middle, and in the type-III, the middle is replaced by the AO6honey- comb layer. In type-I and II, the chemical formula is com- monly given by Mg 2Ir2O6/A2Ti2O6, but that for type-III is MgAIr2O6/A2Ti2O6. The crystal structures are embodied by VESTA [59]. layer is fully composed of AO6octahedra. We inten- tionally design these structures to balance their chemical valences and prevent the presence of redundant charges. This can be derived from the chemical formula for each type of the heterostructure, such as Mg 2Ir2O6/A2Ti2O6 for type-I and II, and Mg AIr2O6/A2Ti2O6for type-III, respectively. We optimize the lattice structures of the heterostruc- tures by the optimization scheme in Sec. III A. The in- formation of the stable lattice structures, including the in-plane lattice constant, the bond distance between ad- jacent Ir atoms and O atoms, and the angle between the neighboring Ir, O, and Ir atoms, are listed in Table I for three types of the heterostructures with different A atoms. For comparison, the experimental structures of the bulk MgIrO 3are also listed. We find all the in-plane constants are close to the bulk value of 5.158 ˚A, in which4 the maximum and minimum lattice mismatch is 1.2% and 0.1% respectively of type-II for the Fe atom and type-III for the Mn atom. See also the discussion in Sec. VI C. Meanwhile, not only the in-plane constants but also the bond distances of Ir atoms are both enlarged as the in- crease of ionic radii of Aatoms. The heterostructural type can significantly influence the bond distance and an- gle as well. For example, the angle between neighboring Ir atoms and the intermediate O atom θIr−O−Ir= 96.69◦ of type-II for the Ni case largely increases from that of 94.03◦of the bulk case. In terms of the Ir-Ir bond length (dIr−Ir), the length of 2.986 ˚A for the bulk [56] is sub- stantially decreased to 2.930 ˚A of type-II for the Fe case. Meanwhile, the Ir-O bond length dIr−Oof all cases are en- larged compared with that of the bulk system of 1.942 ˚A, in which type-III with Co atoms is maximally influenced. TABLE I. Structural information of optimized heterostruc- tures for MgIrO 3/ATiO 3(A= Mn, Fe, Co, and Ni): ade- notes the in-plane lattice constant, and dandθrepresent the bond distance and the angle between neighboring ions, respec- tively. The experimental information on the bulk MgIrO 3is also shown for comparison. A type a(˚A) dIr−Ir(˚A) dIr−O(˚A) θIr−O−Ir(◦) MnI 5.167 2.986 1.997 95.92 II 5.127 2.962 2.009 95.57 III 5.152 2.977 1.985 95.96 FeI 5.104 2.951 2.012 94.32 II 5.068 2.930 2.004 94.00 III 5.083 2.940 2.008 93.98 CoI 5.116 2.961 1.999 94.76 II 5.113 2.955 1.990 95.26 III 5.115 2.960 2.019 94.31 NiI 5.148 2.977 2.006 95.03 II 5.181 2.994 1.997 96.69 III 5.173 2.994 1.995 94.18 bulk [56] 5.158 2.986 1.942 94.03 III. METHODS A.Ab initio calculations In the ab initio calculations, we use the QUANTUM ESPRESSO [60] based on the density functional the- ory [61]. The exchange-correlation potential is treated as Perdew-Zunger functional by using the projector- augmented-wave method [62, 63]. Under the consider- ation of the SOC effect, the fully relativistic functional is utilized for all the atoms except oxygens [64]. To obtain stable structures for heterostructures, we initially con- struct a bilayer MgIrO 3structure using the experimental structure for the bulk material. Subsequently, we replace the lower half with ATiO 3layer to create three different types of heterostructures in Fig. 1. Then, we perform full optimization for both lattice parameters and the po- sition of each ion until the residual force becomes lessthan 0.0001 Ry/Bohr. During the optimization proce- dure, the structural symmetry is retained as R¯3 space group. The 20 ˚A thick vacuum is adopted to eliminate the interaction between adjacent layers. The 6 ×6×1 and 12×12×1 Monkhorst-Pack k-points meshes are utilized for the structural optimization and self-consistent cal- culations, respectively [65]. The self-consistent conver- gence is set to 10−8Ry and the kinetic energy is chosen to 80 Ry for all the structural configurations, which are respectively small and large enough to guarantee accu- rate results. To simulate the electron correlation effects for 3delectrons of Aatoms and 5 delectrons of Ir atom, we adopt the LDA+SOC+ Ucalculations [66] with the Coulomb repulsions UA= 5.0 eV, 5.3 eV, 4.5 eV, and 6.45 eV with A= Mn, Fe, Co, and Ni atoms, respec- tively, and UIr= 3.0 eV, accompanying with the Hund’s- rule coupling with JH/U= 0.1 according to previous works [67, 68]. Based on the ab initio results, we also obtain the ML- WFs by using the kpoints increased to 18 ×18×1 within the Momkhorst-Pack scheme [65]. We select the t2g, 2p, and 3 dorbitals respectively of Ir, O, and Aatoms to construct the MLWFs by employing the Wannier90 [69]. Herein, we include O 2 pandA3dorbitals due to their significant contribution near the Fermi level, as detailed in Figs. 2-5. By utilizing the MLWFs, we construct the tight-binding models and calculate their band structures for comparison. We also calculate the PDOS of each atomic orbital, including the effective angular momen- tum of Ir atoms jeff, from the tight-binding models. We consider the non-relativistic ab initio calculations and rel- ative MLWFs for the estimation of transfer integrals (see Sec. III B). B. Multiorbital Hubbard model To estimate the effective exchange interactions be- tween the magnetic Ir ions, we need the effective transfer integrals between neighboring Ir t2gorbitals with the as- sociation of O 2 porbitals by constructing MLWFs with LDA calculation in the paramagnetic state. It is notice- able that the effects of relativistic SOC and electron cor- relation are not taken into account in this calculation to circumvent the doublecounting in constructing the ef- fective spin models. Specifically, the effective transfer integral tis estimated as [57] tiu,jv+X ptiu,pt∗ jv,p ∆p−uv. (2) The first term denotes the direct hopping between two adjacent Ir atoms, where tiu,jv represents the transfer integral between orbital uat site iand orbital vat site j. The second term denotes the indirect hopping between the two Ir atoms via the shared O 2 porbitals, where tiu,prepresents the transfer integral between Ir atom u orbital at site iand ligand atom porbital, and ∆ p−uv5 is the harmonic mean of the energy of uandvorbitals measured from that of porbitals. Herein, we consider only hopping processes between the nearest-neighbor Ir atoms. Using the effective transfer integrals, we construct a multiorbital Hubbard model with one hole occupying the t2gorbitals, whose Hamiltonian is given by H=Hhop+Htri+Hsoc+HU. (3) The first term denotes the kinetic energy of the t2gelec- trons as Hhop=−X i,jc† i(ˆTγ ij⊗σ0)cj, (4) where the matrix ˆTγ ijincludes the effective transfer in- tegrals estimated by Eq. (2), γis the x,y, and zbond connected by neighboring sites iandjwhich belong to different honeycomb sublattices, and σ0denotes the 2 ×2 identity matrix; c† i= (c† i,yz,↑,c† i,yz,↓,c† i,zx,↑,c† i,zx,↓,c† i,xy,↑, c† i,xy,↓) denote the creation of one hole in the t2gorbitals (yz,zx, and xy) carrying spin up ( ↑) or down ( ↓) at site i. The second term in Eq. (3) denotes the trigonal crystal splitting as Htri=−X ic† i(ˆTtri⊗σ0)ci, (5) with ˆTtriin the form of ˆTtri= 0 ∆ tri∆tri ∆tri0 ∆ tri ∆tri∆tri0 . (6) The third term denotes the SOC as Hsoc=−λ 2X ic† i 0 iσz−iσy −iσz0 iσx iσy−iσx0 ci,(7) where σ{x,y,z}are Pauli matrices, and λis the SOC co- efficient; for instance, λof Ir atom is estimated at about 0.4 eV [70, 71]. The last term denotes the onsite Coulomb interactions as [72, 73] HU=X iUniu↑niu↓ +X i,u<v,σ[U′niuσniv¯σ+ (U′−JH)niuσnivσ] +X i,u̸=vJH(c† iu↑c† iv↓ciu↓civ↑+c† iu↑c† iu↓civ↓civ↑),(8) with niuσ=c† iuσciuσ; ¯σ=↓(↑) for σ=↑(↓). In Eq. (8), the first, second, and third summations represent the in- traorbital Coulomb interaction in the same orbital with opposite spins, the interorbital Coulomb interactions be- tween orbital uand orbital v, and the spin-flip and pair- hopping processes, respectively.C. Second-order perturbation For Ir5+ions, the t2gmanifold splits into a doublet and a quartet under the SOC, which are respectively charac- terized by the pseudospin jeff= 1/2 and 3 /2. In the ground state, the latter is fully occupied and the former is half filled, which is described by the Kramers doublet |jeff= 1/2,+⟩and|jeff= 1/2,−⟩[27, 74]: |jeff= 1/2,+⟩=1√ 3(|dyz↓⟩+i|dzx↓⟩+|dxy↑⟩),(9) |jeff= 1/2,−⟩=1√ 3(|dyz↑⟩ −i|dzx↑⟩ − | dxy↓⟩).(10) When the system is in the spin-orbit coupled Mott insulating state with the low-spin d5configuration, the low-energy physics can be described by the pseudospin jeff= 1/2 degree of freedom. In this case, the effective exchange interactions between the pseudospins can be estimated by using the second-order perturbation theory in the atomic limit, where the three terms in Eq. (3), Htri+Hsoc+HU, are regarded as unperturbed Hamil- tonian, and Hhopis treated as perturbation. The en- ergy correction for a neighboring pseudospin pair in the second-order perturbation is given by E(2) σ′ i,σ′ j;σi,σj=X n⟨σ′ iσ′ j|Hhop|n⟩⟨n|Hhop|σiσj⟩ E0−En,(11) where σiandσ′ idenote the pseudospin + or −at site i,|σiσj⟩and⟨σ′ iσ′ j|is the initial and final states dur- ing the perturbation process, respectively, and |n⟩is the intermediate state with 5 d4-5d6or 5d6-5d4electron con- figuration; E0is the ground state energy for the 5 d5- 5d5electron configuration, and Enis the energy eigen- value for the intermediate state |n⟩. Here, |n⟩andEnare obtained by diagonalizing the unperturbed Hamiltonian Htri+Hsoc+HU. The effective pseudospin Hamiltonian is written in the form of H=X γ=x,y,zX ⟨i,j⟩ST iJγ ijSj, (12) where i, jdenote the neighboring sites, and γdenotes the three types of Ir-Ir bonds on the MgIrO 6honeycomb layer that are related by C3rotation. The coupling constant Jγ ijis explicitly given, e.g., for the zbond as Jz ij= JΓ Γ′ ΓJΓ′ Γ′Γ′K , (13) where J,K, Γ, and Γ′represent the coupling con- stants for the isotropic Heisenberg interaction, the bond- dependent Ising-like Kitaev interaction, and two types of the symmetric off-diagonal interactions. Using the per- turbation energy E(2) σ′ i,σ′ j,σi,σjobtained by Eq. (11), the6 coupling constants are calculated as J= 2E(2) +,−;−,+, (14) K= 2 E(2) +,+;+,+−E(2) +,−;+,− , (15) Γ = 2Imn E(2) −,−;+,+o , (16) Γ′= 4Ren E(2) +,+;+,−o . (17) IV. ELECTRONIC BAND STRUCTURE A. LDA+SOC results for paramagnetic states Let us begin with the electronic band structures from LDA+SOC calculations. The results for MgIrO 3/ATiO 3 with A= Mn, Fe, Co, and Ni are shown in Figs. 2(a), 3(a), 4(a), and 5(a), respectively. Here we display the band structures in the paramagnetic state obtained by theab initio calculations (black solid lines) and the MLWF analysis (blue dashed lines), together with the atomic orbitals PDOS including 5 dof Ir atoms, 3 dofA atoms, and 2 pof O atoms related to IrO 6andAO6oc- tahedra. It is obvious that the systems are metallic for all the types of heterostructures, regardless of the choice ofAatoms. The strong SOC splits the t2gbands of Ir atoms into jeff= 1/2 and jeff= 3/2 bands, as depicted in the PDOS. Specifically, the jeff= 1/2 bands are pre- dominated to form the metallic bands in the proximity of the Fermi level, covering the energy region almost from −1.0 eV to 0 .2 (0.4) eV for A= Mn with type-II (type-I and III), from −0.8 eV to 0 .2 eV for A= Fe with all types, from −0.6 (−1.0) eV to 0 .2 eV for A= Co with type-II (type-I and III), and from −0.4 eV to 0 .4 eV for A= Ni with type-I and II, and −0.8 eV to 0 .2 eV with type-III. On the other hand, the jeff= 3/2 bands pri- marily occupy the energy region below the jeff= 1/2 bands. From the PDOS in right panels of Figs. 2(a), 3(a), 4(a), and 5(a), the 3 dbands of Aatoms and the O 2 p bands of AO6octahedra simultaneously across the Fermi level, with hybridization with the Ir 5 dbands. Notably, the energy range of the A3dbands closely overlaps with that of the Ir 5 d jeff= 1/2 bands for A= Mn and Fe, but it overlaps with both jeff= 1/2 and jeff= 3/2 manifold forA= Co and Ni. As to the O 2 pbands, the energy range of the PDOS overlaps with that for corresponding Ir 5dorA3dencapsulated in the octahedra, suggesting Ir-O and A-O hybridization. B. LDA+SOC+ Uresults for magnetic states 1. Magnetic ground states The bulk counterpart of each constituent of the het- erostructures exhibits some magnetic long-range ordersin the ground state. In the bulk MgIrO 3, Ir ions show a zigzag-type AFM order with the magnetic moments ly- ing almost within the honeycomb plane [58]. In the bulk ATiO 3, the A= Mn and Fe ions show N´ eel-type AFM orders with out-of-plane magnetic moments [75], while theA= Co and Ni ions support N´ eel orders with in- plane magnetic moments [75, 76]. It is intriguing to ex- amine how these magnetic orders in the bulk are affected by making heterostructures. We determine the stable magnetic ground states for each heterostructure through ab initio calculations by including the effect of electron correlations based on the LDA+SOC+ Umethod. To determine the potential magnetic ground state for each heterostructure, we compare the energy across a total of 16 magnetic configurations among all combinations of fol- lowing types of the magnetic orders: FM and N´ eel orders with in-plane and out-of-plane magnetic moments for A layer, and N´ eel and zigzag orders with in-plane magnetic moments, as well as FM with in-plane and out-of-plane magnetic moments for Ir layer, within a 2 ×2×1 supercell setup. TABLE II. Stable magnetic orders obtained by the LDA+SOC+ Ucalculations: FM, N´ eel, and zigzag denotes the ferromagnetic, N´ eel-type antiferromagnetic, and zigzag- type antiferromagnetic orders, respectively. While the direc- tions of the magnetic moments are all in-plane for the Ir lay- ers, those for Acan be in-plane (“in”) or out-of-plane (“out”) depending on Aand type of the heterostructure. A layer type-I II III MnIr in-zigzag in-zigzag in-zigzag Mn in-N´ eel out-FM in-N´ eel FeIr in-N´ eel in-N´ eel in-FM Fe out-N´ eel in-FM in-FM CoIr out-FM in-FM in-N´ eel Co in-N´ eel in-N´ eel in-N´ eel NiIr in-N´ eel in-zigzag in-zigzag Ni out-N´ eel in-FM in-N´ eel The results of the most stable magnetic state are listed in Table II. The details of the energy comparison are shown in Appendix A. In most cases, the Aions show N´ eel orders as in the bulk cases, but the direction of mag- netic moments is changed from the bulk in some cases. For instance, type-I and III with A= Mn and type-III with A= Fe switch the moment direction from out-of- plane to in-plane. While all the types with A= Co retain the in-plane N´ eel states, type-I with A= Ni is changed into the out-of-plane N´ eel state. The results indicate that the direction of magnetic moments are sensitively altered by making the heterostructures with MgIrO 3. Mean- while, the other cases, type-II with A= Mn, Fe, and Ni as well as type-III with A= Fe, are stabilized in the FM state. These results are in good accordance with the ef- fective magnetic couplings between the Aions estimated by a similar perturbation theory in Sec. III C [77, 78], attesting to the reliability of magnetic properties in het- erostructures (see Appendix A).7 !MK!-2-101Energy (eV) !MK!-2-101Energy (eV)!MK!-2-101Energy (eV) !MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)type-Itype-IItype-III !MK!-2-101Energy (eV)(b)jeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pIrMnOMn2pOIr2p2pIrMnOMnOIrIrMnOMnOIr FIG. 2. The band structures of MgIrO 3/MnTiO 3for type-I, II, and III with (a) the LDA+SOC calculations for the paramagnetic state and (b) the LDA+SOC+ Ucalculations for the stable magnetic orders (see Sec. IV B 2). The black lines represent the electronic structure obtained by the ab initio calculations, and the light-blue dashed curves represent the electronic dispersions obtained by tight-binding parameters using the MLWFs. The right panels in each figure denote the PDOS for different orbitals on specific atoms: The red and blue lines represent the jeffmanifolds of Ir atoms, the cyan and orange lines represent the 2 p orbitals of O atoms in IrO 6octahedra (O Ir) and MnO 6octahedra (O Mn), respectively, and the green line represents the 3 d orbitals of Mn atoms. The Fermi energy is set to zero. The magnetic states in the Ir layer are more complex due to the possibility of the zigzag state. For A= Mn, the magnetic ground states of the Ir layer in all three types prefer the in-plane zigzag state as in the bulk of MgIrO 3. In contrast, for A= Fe, type-I and II are stable in the in-plane N´ eel state, but type-III prefers the in-plane FM state. For A= Co, only type-III stabilizes the in-plane N´ eel state, while others exhibit the out-of-plane FM state for type-I and the in-plane FM state for type-II. Lastly, forA= Ni, both type-II and type-III prefer the in-plane zigzag state, while it changes into the in-plane N´ eel state in type-I. These results indicate that the magnetic state in the Ir honeycomb layer is susceptible to both Aions and the heterostructure type. We will discuss this point from the viewpoint of the effective magnetic couplings in Sec. V. 2. Band structures We present the band structures obtained by the LDA+SOC+ Ucalculations in Figs. 2(b), 3(b), 4(b), and 5(b) for A= Mn, Fe, Co, and Ni, respectively. In these calculations, we adopt the stable magnetic states in Ta- ble II, except for the cases with in-plane zigzag order inthe Ir layer. For the zigzag cases, for simplicity, we re- place them by the in-plane N´ eel solutions, keeping the Alayer the same as the stable one. This reduces signifi- cantly the computational cost of the MLWF analysis for the zigzag state with a larger supercell. We confirm that the band structures for the N´ eel state are similar to those for the zigzag state, and the energy differences between the two states are not large as shown in Appendix A. When we turn on Coulomb repulsions for both Ir and Aatoms, most of the type-I and III heterostructures be- come insulating, except for Mn type-III. The band gaps, obtained by Eg=Ec−Ev, are shown in Fig. 6, where Ec denotes the energy of conduction band minimum and Ev is that of valence band maximum. In all cases, except for type-I with A= Mn and Ni and type-III with A= Co, the gap is defined by the jeff= 1/2 bands of Ir ions, that is, both conduction and valence bands are jeff= 1/2, and thejeff= 1/2 bands is half filled. It is worth highlight- ing that there are four jeff= 1/2 bands, which originate from different sites of Ir atoms with opposite magnetic moments; in the bulk and monolayer cases they are de- generate in pair [57], but the degeneracy is lifted in the heterostructures and two out of four are occupied in the half-filled insulating state. Meanwhile, in the cases of type-I with A= Mn and Ni and type-III with A= Co,8 !MK!-2-101Energy (eV) !MK!-2-101Energy (eV)!MK!-2-101Energy (eV) !MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)type-Itype-IItype-III !MK!-2-101Energy (eV)(b)jeff =1/2 jeff =3/2jeff =1/2 jeff =3/2jeff =1/2 jeff =3/23d2pIrFeOFe2pOIrIrFeOFeOIrIrFeOFeOIr3d2p2p3d2p2p FIG. 3. The band structures of MgIrO 3/FeTiO 3for type-I, II, and III obtained by (a) the LDA+SOC calculations for the paramagnetic state and (b) the LDA+SOC+ Ucalculations for the stable magnetic orders (see Sec. IV B 2). The notations are common to Fig. 2. !MK!-2-101Energy (eV) !MK!-2-101Energy (eV)!MK!-2-101Energy (eV) !MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)type-Itype-IItype-III !MK!-2-101Energy (eV)(b)jeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pIrCoOCo2pOIr2p2pIrCoOCoOIrIrCoOCoOIr FIG. 4. The band structures of MgIrO 3/CoTiO 3for type-I, II, and III obtained by (a) the LDA+SOC calculations for the paramagnetic state and (b) the LDA+SOC+ Ucalculations for the stable magnetic orders (see Sec. IV B 2). The notations are common to Fig. 2.9 !MK!-2-101Energy (eV) !MK!-2-101Energy (eV)!MK!-2-101Energy (eV) !MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a)type-Itype-IItype-III !MK!-2-101Energy (eV)(b)jeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pjeff =1/2 jeff =3/23d2pNiONi2pIrOIr2p2pNiONiIrOIrNiONiIrOIr FIG. 5. The band structures of MgIrO 3/NiTiO 3for type-I, II, and III obtained by (a) the LDA+SOC calculations for the paramagnetic state and (b) the LDA+SOC+ Ucalculations for the stable magnetic orders (see Sec. IV B 2). The notations are common to Fig. 2. the 3 dbands of Aions hybridized with O 2 porbitals intervene near the Fermi level, and the gap is defined between the jeff= 1/2 and 3 dbands. In these cases, however, a larger gap is well preserved in the jeff= 1/2 bands, as shown in Figs. 2(b), 4(b), and 5(b). We note that the Co type-III is a further exception since the gap opens between the highest-energy jeff= 1/2 band and the Co 3 dband; the jeff= 1/2 bands are not half filled but 3 /4 filled (see Appendix B). We plot the band gap defined by the jeff= 1/2 bands by red asterisks in Fig. 6, including the 3 /4-filled case for the Co type-III. These results clearly indicate that the inclusion of both SOC and Ueffects results in the opening of a band gap in the jeff= 1/2 bands at half filling in type-I and III heterostructures excluding Mn type-III and Co type-III. This suggests the formation of spin-orbit coupled Mott insulators in the Ir honeycomb layers, which are corner- stone of the Kitaev candidate materials [27], motivating us to further investigate the effective exchange interac- tion in Sec. V. The Co type-III is in an interesting state with 3 /4 filling of jeff= 1/2 bands, but we exclude it from the following analysis of the effective exchange in- teractions in Sec. V. Distinct from the emergence of spin-orbit coupled Mott insulator, the LDA+SOC+ Uband structures show metallic states for type-II heterostructures. The jeff= 1/2 bands do not show a clear gap and cross the Fermi level, resulting in the spin-orbit coupled metals. Notably, in all cases, the upper jeff= 1/2 bands are partiallyTABLE III. Electronic states of each heterostructure obtained by the LDA+SOC+ Ucalculations. SOCI and SOCM denote spin-orbit coupled insulator and metal, respectively. e and h in the parentheses represent the carriers in the SOCM doped to the mother SOCI. The asterisk for the Co type-III indicates that the system is in the 3 /4-filled insulating state of the jeff= 1/2 bands. A type-I II III Mn SOCI SOCM(e) SOCM(h) Fe SOCI SOCM(e) SOCI Co SOCI SOCM(e) SOCI* Ni SOCI SOCM(e) SOCI doped, realizing electron-doped Mott insulators. The doping rate varies with Aatoms. We note that the type- III heterostructure of Mn also exhibits a metallic state, but in this case, holes are doped to the lower jeff= 1/2 band. See Appendix B for the orbital projected band structures. We summarize the electronic states in Table III. The type-I and III heterosuructures are all spin-orbit cou- pled Mott insulators (SOCI) except for the type-III Mn case, while the type-II are all spin-orbit coupled met- als (SOCM). For the SOCM, we also indicate the nature of carriers, electrons or holes; the type-II heterostruc- tures are all electron doped, while the type-III Mn is hole doped.10 Mn Fe Co Ni Atoms00.10.20.30.40.50.60.7Band gap (eV)type-I type-III type-I: jeff=1/2 type-III: jeff=1/2 FIG. 6. The band gap in the insulating states for type-I and III obtained by the LDA+SOC+ Ucalculations. The open circles represent the gaps opening in the jeff= 1/2 bands for Mn and Ni of type-I and Co of type-III. Note that the Co type-III is exceptional since the jeff= 1/2 bands are at 3 /4 filling, rather than half filling in the other cases; see the text for details. V. EXCHANGE INTERACTIONS The electronic band structure analysis reveals the emergence of the spin-orbit coupled Mott insulating state in the MgIrO 6layer of type-I and III heterostructures, except for the Mn and Co type-III. In these cases, the low-energy physics is expected to be described by effec- tive pseudospin models with dominant Kitaev-type inter- actions [27]. The effective exchange interactions can be derived by means of the second-order perturbation for the multiorbital Hubbard model (Secs. III B and III C). We setUIr= 3.0 eV, JH/UIr= 0.1, and λ= 0.4 eV in the perturbation calculations. The results are summarized in Fig. 7. For comparison, we also plot the estimates for monolayer and bulk MgIrO 3. For the bulk case, its po- tential for hosting Kitaev spin liquids was demonstrated in the previous study [57]. Regarding the monolayer, we obtain the results from the band structures shown in Appendix C, which illustrate the preservation of the jeff= 1/2 manifold and spin-orbit coupled insulating na- ture. In type-I heterostructures, the dominant interaction is the FM Kitaev interaction K < 0 for almost all A atoms, except for Mn. Particularly for A= Ni, the abso- lute value of Kis significantly larger than the others, even considerable when compared with the monolayer and bulk MgIrO 3. The subdominant interaction is the off-diagonal symmetric interaction Γ >0. The other off- diagonal symmetric interaction Γ′as well as the Heisen- berg interaction Jis weaker than them. In the Mn case, all the interactions are exceptionally weak, presumably because of the intervening Mn 3 dband and its hybridiza-tion with the Ir jeff= 1/2 bands. Meanwhile, for the type-III heterostructures, since the Mn and Co cases ex- hibit a metallic state and 3 /4 occupation of jeff= 1/2 bands, respectively, we only calculate the effective mag- netic constants for Fe and Ni. In these cases also, the dominant interaction is the FM K, accompanied by the subdominant Γ interaction, as shown in Fig. 7(a). Thus, in all cases except the Mn type-I heterostructure, the dominant magnetic interaction in the spin-orbit cou- pled Mott insulating state in the Ir honeycomb layer is effectively described by the FM Kitaev interaction. Since Γ′is smaller than the other exchange constants, the low- energy magnetic properties can be well described by the generic K-J-Γ model [79, 80], which has been widely and successfully applied to study the Kitaev QSLs. We sum- marize the obtained effective exchange interactions of K, J, and Γ by using the parametrization (K, J, Γ) =N(sinθsinϕ,sinθcosϕ,cosθ), (18) where N= (K2+J2+ Γ2)−1/2is the normalization fac- tor. Figure 7(b) presents the results except for Mn type-I. Our heterostructures distribute in the region near the FM Konly case ( θ=π/2 and ϕ= 3π/2). We find a general trend that larger Aatoms make the systems closer to the FMKonly case; the best is found for Ni type-I and III. In the previous studies for the K-J-Γ model [79, 80], a keen competition between different magnetic phases was found in this region, which does not allow one to conclude the stable ground state in the thermodynamic limit. Given that this region appears to be connected to the solvable point for the FM Kitaev QSL, our heterostructures pro- vide a promising platform for investigating the Kitaev QSL physics and related phase competition by finely tun- ing the magnetic interactions via the proximity effect in the heterostructures. VI. DISCUSSION Our systematic study of ilmenite heterostructures MgIrO 3/ATiO 3with A= Mn, Fe, Co, and Ni reveals their fascinating electronic and magnetic properties. The heterostructures in the paramagnetic state are metallic in terms of band structures obtained by LDA+SOC, re- gardless of types and Aatoms. When incorporating the effect of electron correlation by the LDA+SOC+ Ucal- culations, type-II heterostructures remain metallic across entire Aatoms, whereas type-I and III heterostrcutures turn into insulating states, except for Mn type-III. As a consequence, the electronic states of heterostructures are classified into the spin-orbit coupled insulators and metals, each holding unique properties. The insulating cases possess the jeff= 1/2 pseudospin degree of freedom, and furnish a fertile playground to investigate the Kitaev QSL. In these cases, however, due to the magnetic prox- imity effects from the Alayer, we may expect interesting modification of the QSL state, as discussed in Sec. VI A below. Meanwhile, the metallic cases open avenues for11 -150-100-50050Coupling const. (meV)(b) ◆◆◆◆★★▲▲▲▲♡♡♡♡AFM K only FM K onlyFM J onlyAFM J onlyFe-ICo-INi-IFe-IIINi-IIIbulkmonolayerΓ only𝜙 = 0𝜙 =𝜋/2𝜙 = 𝜋 𝜙 = 3𝜋/2(a) CoAtomFeNimonolayerbulkJKΓΓ′ Mn𝜃 = 𝜋/2𝜃 = 3𝜋/8𝜃 = 𝜋/4𝜃 = 𝜋/8 FIG. 7. The effective magnetic constants of heterostructures for different Aatoms ( A= Mn, Fe, Co, and Ni) of (a) type-I (solid line with pentagram) and type-III (dashed line with hol- low pentagram). For comparison, we also show the results for monolayer and bulk. In (b), we summarize the results of K, J, and Γ in (a) except for Mn type-I by using the parametriza- tion in Eq. (18). The parameters of the intraorbital Coulomb interaction, Hund’s coupling, and spin-orbit coupling are set toUIr= 3.0 eV, JH/UIr= 0.1, and λ= 0.4 eV, respectively, in the perturbation calculations. exploring spin-orbit coupled metals, relatively scarce in strongly correlated systems [81–85]. In Sec. VI B, we dis- cuss the possibility of exotic superconductivity in our self- doped heterostructures. In addition, we discuss the fea- sibility of fabrication of these heterostructures and iden- tification of the Kitaev QSL nature in experiments in Sec. VI C. A. Majorana Fermi surface by magnetic proximity effect In the pure Kitaev model, the spins are fractionalized into itinerant Majorana fermions and localized Z2gauge fluxes [25]. The former has gapless excitations at thenodal points of the Dirac-like dispersions at the K and K’ points on the Brillouin zone edges, while the latter is gapped with no dispersion. When an external magnetic field is applied, the Dirac-like nodes of Majorana fermions are gapped out, resulting in the emergence of quasipar- ticles obeying non-Abelian statistics [25]. Beyond the uniform magnetic field, the Majorana dispersions are fur- ther modulated by an electric field and a staggered mag- netic field [86, 87]. For instance, with the existence of the staggered magnetic field, the Dirac-like nodes at the K and K’ points are shifted in the opposite directions in energy to each other, leading to the formation of the Majorana Fermi surfaces. Moreover, the introduction of both uniform and staggered magnetic fields can lead to further distinct modulations of the Majorana Fermi sur- faces around the K and K’ points, which are manifested by nonreciprocal thermal transport carried by the Majo- rana fermions [87]. In our heterostructures of type-I and III, the Alayer supports a N´ eel order in most cases (Table II). It can generate an internal staggered magnetic field applied to the Ir layer through the magnetic proximity effect. This mimics the situations discussed above, and hence, it may result in the Majorana Fermi surfaces in the possible Ki- taev QSL in the Ir layer. The combination of the uniform and staggered magnetic fields could also be realized by applying an external magnetic field to these heterostruc- tures. Thus, the ilmenite heterostructures in proximity to the Kitaev QSL in the Ir layer hold promise for the formation of Majorana Fermi surfaces and resultant ex- otic thermal transport phenomena, providing a unique platform for identifying the fractional excitations in the Kitaev QSL. B. Exotic superconductivity by carrier doping QSLs have long been discussed as mother states of ex- otic superconductivity [19, 88, 89]. There, the introduc- tion of mobile carriers to insulating QSLs possibly in- duces superconductivity in which the Cooper pairs are mediated by strong spin entanglement in the QSLs. A representative example discussed for a long time is high- Tccuprates; here, the d-wave superconductivity is in- duced by carrier doping to the undoped antiferromag- netic state that is close to a QSL of so-called resonat- ing valence bond (RVB) type [89–91]. A similar ex- otic superconducting state was also discussed for an irid- ium oxide Sr 2IrO4with spin-orbital entangled jeff= 1/2 bands [92, 93]. Carrier doped Kitaev QSLs have also gar- nered extensive attention due to its potential accessibil- ity to unconventional superconductivity that may possess more intricate paring from the unique QSL properties. It was reported that doping into the Kitaev model with additional Heisenberg interactions ( K-Jmodel) led to a spin-triplet topological superconducting state [94], where the pairing nature is contingent upon the doping concen- tration. Furthermore, the competition between Kand12 Jalso significantly impacts the superconducting state; for example, Kprefers a p-wave superconducting state, whereas Jtends to favor a d-wave one [95, 96]. Even topological superconductivity is observed in an extended K-J-Γ model [97]. In the present work, we found metallic states in the spin-orbital coupled jeff= 1/2 bands in the Ir layer for all type-II heterostructures (electron doping) and the type- III Mn heterostructure (hole doping) (see Table III). Be- sides, in the type-III Co heterostructure, electron doping occurs in the Ir layer, resulting in the 3 /4-filled insulat- ing state in the jeff= 1/2 bands. These appealing results suggest that our ilmenite heterostructures offer a plat- form for studying exotic metallic and superconducting (even topological) properties with great flexibility by var- ious choices of materials combination, which have been scarcely realized in the bulk systems. C. Experimental feasibility The bulk compounds of ilmenite ATiO 3with A= Mn, Fe, Co, and Ni have been successfully synthesized and investigated for over half a century due to its fruitful magnetic and novel electronic properties [75, 76, 98–100]. Technically, the Fe case, however, is more challenging compared to the others, as its synthesis needs very high pressure and high temperature conditions [101, 102]. Be- sides, the iridium ilmenite MgIrO 3has also been syn- thesized as a power sample, where a magnetic phase transition was observed at 31 .8 K [56]. The experimen- tal lattice parameters are 5 .14˚A for ATiO 3with A= Mn [75, 103–105], 5 .09˚A for A= Fe [101, 106, 107], 5.06˚A for A= Co [76, 108, 109], and 5 .03˚A for A= Ni [75, 107, 110, 111], respectively, as well as that is 5.16˚A for MgIrO 3. The relatively small lattice mismatch between these materials also ensure the possibility of combining them to create heterostructure with different compounds. Indeed, we demonstrated this in Sec. II; see Table I. More excitingly, the IrO 6honeycomb lattice has been successfully incorporated into the ilmenite MnTiO 3 with the formation of several Mn-Ir-O layers [54]. This development lightens the fabrication of a supercell be- tween MgIrO 3andATiO 3. The verification of Kitaev QSL poses a significant chal- lenge even though the successful synthesis of aforemen- tioned heterostructures. First of all, it is crucial to iden- tify the spin-orbital entangled electronic states with the formation of the jeff= 1/2 bands in these heterostruc- tures, as they are essential for the Kitaev interactions between the pseudospins. Several detectable spectro- scopic techniques are useful for this purpose, applicable to both bulk and heterostructures [11, 34, 35, 39, 42, 43, 46, 112, 113]. Even the Kitaev exchange interaction can be directly uncovered in experiment [36, 37]. How- ever, the key challenge lies in probing the intrinsic prop- erties of Kitaev QSL, such as fractional spin excitations. Thus far, despite cooperative studies between theoriesand experiments on, for instance, dynamical spin struc- ture factors [44, 114–119] and the thermal Hall effect and its half quantization [45, 120–122], have been developed to identify the fractional excitations in Kitaev QSL, di- rectly applying them on the heterostructures is still a great challenge. A promising experimental tool would be the Raman spectroscopy, given its successful application to not only bulk [41, 123, 124] but also atomically thin layers [125, 126]. The signals might be enhanced by pil- ing up the heterostructures. Besides, many proposals for probing the Kitaev QSL in thin films and heterostruc- tures have been recently made, such as local probes like scanning tunneling microscopy (STM) and atomic force microscopy (AFM) [127–131] as well as the spin Seebeck effect [132]. Additionally, as mentioned in Sec. VI A, the observation of the Majorana Fermi surfaces by thermal transport measurements in some particular heterostruc- tures is also interesting. VII. SUMMARY To summarize, we have conducted a systematic inves- tigation of the electronic and magnetic properties of the bilayer structures composed by the ilmenites ATiO 3with A= Mn, Fe, Co, and Ni, in combination with the can- didate for Kitaev magnets MgIrO 3. We have designed and labeled three types of heterostructure, denoted as type-I, II, and III, distinguished by the atomic config- urations at the interface. Our analysis of the electronic band structures based on the ab initio calculations has re- vealed that the spin-orbital coupled bands characterized by the pseudospin jeff= 1/2, one of the fundamental component for the Kitaev interactions, is retained in the MgIrO 3layer for all the types of heterostructures. We found that the MgIrO 3/ATiO 3heterostructures of type- I and III are mostly spin-orbit coupled insulators, while those of type-II are spin-orbit coupled metals, irrespec- tive of the Aatoms. In the insulating heterostructures of type-I and III, based on the construction of the multior- bital Hubbard models and the second-order perturbation theory, we further found that the low-energy magnetic properties can be described by the jeff= 1/2 pseudospin models in which the estimated exchange interactions are dominated by the Kitaev-type interaction. We showed that the parasitic subdominant interactions depend on the type of the heterostructure as well as the Aatoms, offering the playground for systematic studies of the Ki- taev spin liquid behaviors. Moreover, the stable N´ eel or- der in the ATiO 3layer acts as a staggered magnetic field through the magnetic proximity effect, leading to the potential realization of Majorana Fermi surfaces in the MgIrO 3layer. Meanwhile, in the metallic heterostruc- tures of type-II as well as type-I Mn, we found that the nature of carriers and the doping rates vary depending on the heterostructures. This provides the possibility of systematically studying the spin-orbit coupled metals, in- cluding exploration of unconventional superconductivity13 due to the unique spin-orbital entanglement. In recent decades, significant progress has been made in the study of QSLs, primarily focusing on the discov- ery and expansion of new members in bulk materials. However, there has been limited exploration of creating and manipulating the QSLs in heterostructures despite the importance for device applications. Our study has demonstrated that the Kitaev-type QSL could be sur- veyed in ilmenite oxide heterostructures, displaying re- markable properties distinct from the bulk counterpart, such as flexible tuning of the Kitaev-type interactions and other parasitic interactions, and carrier doping to the Kitaev QSL. Besides the van der Waals heterostruc- tures such as the combination of α-RuCl 3and graphene, our finding would enlighten an additional route to ex- plore the Kitaev QSL physics including the utilization of Majorana and anyonic excitations for future topological computing devices. ACKNOWLEDGMENTS We thank Y. Kato, M. Negishi, S. Okumura, A. Tsukazaki, and L. Zh. Zhang, for fruitful discussions. This work was supported by JST CREST Grant (No. JP- MJCR18T2). Parts of the numerical calculations were performed in the supercomputing systems of the Insti- tute for Solid State Physics, the University of Tokyo. Appendix A: Detailed ab initio data for energy and magnetic coupling In this Appendix, we present the details of ab initio results for various types of heterostructures. Tables IV- VII list the energy differences between different magnetic states for MgIrO 3/ATiO 3heterostructures with A= Mn, Fe, Co, and Ni. The bold elements in these tables are the lowest-energy state in each type, utilized for the calcu- lations of band structures in Sec. IV B 1. We also show in Table VIII the effective magnetic coupling constants between Aatoms, in which negative and positive value indicates the FM and AFM coupling, respectively. Note that the Aatoms comprise a triangular lattice at the in- terface in type I, a honeycomb lattice at the ATiO 3layer, and a honeycomb lattice at the interface, as depicted in Fig. 1. Appendix B: Orbital projected band structures In this Appendix, we show the projection of the band structure to the Ir 5 dorbitals for type-II heterostructures in Fig. 8 and type-III of Mn and Co in Fig. 9. The green shaded bands include high-energy four jeff= 1/2 bands and low-energy eight jeff= 3/2 bands. The results in Fig. 8 indicate that electrons are doped to the half-filled jeff= 1/2 bands, realizing the spin-orbit coupled metallicTABLE IV. Energy differences between different magnetic or- dered states obtained by the LDA+SOC+ Ucalculations for MgIrO 3/MnTiO 3: FM, N´ eel, and zigzag denotes the ferro- magnetic, N´ eel-type antiferromagnetic, and zigzag-type anti- ferromagnetic orders, respectively. While the directions of the magnetic moments are all in-plane for the Ir layers, those for Acan be in-plane (“in”) or out-of-plane (“out”). The bold numbers denote the low-energy states used for calculating the band structures in Sec. IV B 1. magnetic state energy/Ir (meV) Ir Mn I II III in-FMinFM 323.5 4.868 975.4 N´ eel 1.492 46.52 1002 outFM 18.13 4.268 1065 N´ eel 3.695 44.554 978.2 out-FMinFM 4.266 7.343 990.8 N´ eel 0.773 43.68 993.2 outFM 35.71 1.584 1014 N´ eel 76.55 40.34 879.6 N´ eelinFM 27.18 8.127 78.52 N´ eel 26.98 42.83 48.51 outFM 20.45 4.474 121.0 N´ eel 2.953 39.39 22.45 zigzaginFM 0.307 2.923 78.80 N´ eel 0.000 21.55 0.000 outFM 20.05 0.000 120.49 N´ eel 4.893 39.62 22.95 TABLE V. Energy differences between different magnetic or- dered states for MgIrO 3/FeTiO 3. The notations are common to Table IV. magnetic state energy/Ir (meV) Ir Fe I II III in-FMinFM 9.090 1.437 0.000 N´ eel 17.00 15.75 13.14 outFM 15.71 21.80 2.631 N´ eel 1.170 27.02 14.25 out-FMinFM 10.68 0.973 0.092 N´ eel 12.12 0.948 13.01 outFM 55.58 21.13 1.739 N´ eel 8.115 26.29 8.833 N´ eelinFM 24.64 0.000 0.056 N´ eel 11.80 0.721 12.97 outFM 14.48 20.21 1.244 N´ eel 0.000 25.37 8.596 zigzaginFM 15.58 1.233 0.056 N´ eel 4.376 13.60 4.222 outFM 14.35 21.42 2.244 N´ eel 4.743 26.59 9.608 states for all Aatoms. The doping rates are large (small) forA= Mn and Co (Fe and Ni). Meanwhile, Fig. 9(a) shows that the jeff= 1/2 bands are slightly hole doped in the type-III with A= Mn. Figure 9(b) indicates that the type-III with A= Co achieves an insulating state with 3/4-filled jeff= 1/2 bands.14 TABLE VI. Energy differences between different magnetic or- dered states for MgIrO 3/CoTiO 3. The notations are common to Table IV. magnetic state energy/Ir (meV) Ir Co I II III in-FMinFM 188.2 20.02 0.795 N´ eel 191.3 0.000 0.253 outFM 40.23 25.63 72.39 N´ eel 1.201 235.7 87.28 out-FMinFM 0.174 20.06 0.800 N´ eel 0.000 0.019 0.071 outFM 295.3 25.85 260.4 N´ eel 47.55 0.044 49.64 N´ eelinFM 189.5 19.56 0.749 N´ eel 184.3 19.36 0.000 outFM 17.53 24.82 33.16 N´ eel 0.617 234.9 31.16 zigzaginFM 191.8 19.88 0.748 N´ eel 189.6 19.39 1.969 outFM 40.81 25.45 34.02 N´ eel 1.476 130.0 32.03 TABLE VII. Energy differences between different magnetic ordered states for MgIrO 3/NiTiO 3. The notations are com- mon to Table IV. magnetic state energy/Ir (meV) Ir Ni I II III in-FMinFM 2.990 0.097 37.80 N´ eel 4.998 70.93 35.99 outFM 3.536 1.255 26.76 N´ eel 0.156 69.45 35.74 out-FMinFM 6.901 2.039 37.48 N´ eel 7.311 70.74 35.98 outFM 20.94 2.385 24.79 N´ eel 6.220 69.61 61.76 N´ eelinFM 14.06 1.678 48.98 N´ eel 13.00 18.68 54.35 outFM 4.941 0.294 26.51 N´ eel 0.000 18.631 37.36 zigzaginFM 4.971 0.000 36.97 N´ eel 6.275 33.08 0.000 outFM 3.451 0.042 26.20 N´ eel 3.876 68.36 35.45 Appendix C: Band structure of monolayer MgIrO 3 In this Appendix, we show the electronic band struc- tures of monolayer MgIrO 3obtained through ab ini- tiocalculations with the LDA+SOC [Fig. 10(a)] and LDA+SOC+ Uscheme [Fig. 10(b)]. We set UIr= 3.0 eV and JH/UIr= 0.1 in the LDA+SOC+ Ucalcu-lations. In the LDA+SOC result, the system behaves as an insulating state with a tiny band gap of approx- imately ∼0.096 eV. However, the introduction of Uin the LDA+SOC+ Ucalculation results in a larger band gap, characteristic of the spin-orbit coupled insulator. We also calculate the PDOS of the jeff= 1/2 and 3 /2 TABLE VIII. Effective magnetic coupling constants between theAatoms for three types of heterostructures. The unit is in meV. A type-I II III Mn 0.667 -0.712 2.417 Fe 0.017 -0.607 -0.149 Co 0.065 0.262 6.458 Ni 0.088 -0.126 0.405 (a) Mn (b) Fe (c) Co(d) Ni1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓ1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓ1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓ1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓEnergy (eV)Energy (eV) FIG. 8. Projection to the Ir 5 dorbitals of the band structure for the type-II MgIrO 3/ATiO 3heterostructures with (a) A= Mn, (b) Fe, (c) Co, and (d) Ni. The gray lines depict the band structures shown in the middle panels of Figs. 2(b)-5(b), and the green shade represents the weight of Ir 5 dorbitals. The Fermi level is set to zero. manifolds for Ir atoms, as shown in the right panels of Figs. 10(a) and 10(b). The PDOS of Ir atoms certifies that the jeff= 1/2 and jeff= 3/2 manifold are retained to support the spin-orbit coupled Mott insulating state in the monolayer, as in the bulk case [57]. [1] N. F. Mott, Metal-Insulator Transition , Rev. Mod. Phys. 40, 677 (1968).[2] M. Imada, A. Fujimori, and Y. Tokura, Metal-Insulator Transitions , Rev. Mod. Phys. 70, 1039 (1998).15 (a)Mn jeff = 1/2jeff = 1/2jeff = 1/2jeff = 1/2Co 3d(b) Co1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓEnergy (eV)1.00.50.0-0.5-1.0-1.5-2.0ΓMKΓ FIG. 9. Projection to the Ir 5 dorbitals of the band structure for the type-III heterostructures of (a) MgIrO 3/MnTiO 3and (b) MgIrO 3/CoTiO 3. The notations are common to Fig. 8. [3] X.-L. Qi and S.-C. Zhang, Topological Insulators and Superconductors , Rev. Mod. Phys. 83, 1057 (2011). [4] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators , Rev. Mod. Phys. 82, 3045 (2010). [5] X. Wan, A. Vishwanath, and S. Y. Savrasov, Compu- tational Design of Axion Insulators Based on 5dSpinel Compounds , Phys. Rev. Lett. 108, 146601 (2012). [6] D. M. Nenno, C. A. Garcia, J. Gooth, C. Felser, and P. Narang, Axion Physics in Condensed-Matter Sys- tems, Nat. Rev. Phys. 2, 682 (2020). [7] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological Semimetal and Fermi-Arc Surface States in the Electronic Structure of Pyrochlore Iridates , Phys. Rev. B 83, 205101 (2011). [8] A. A. Burkov and L. Balents, Weyl Semimetal in a Topological Insulator Multilayer , Phys. Rev. Lett. 107, 127205 (2011). [9] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac Semimetals in Three-Dimensional Solids , Rev. Mod. Phys. 90, 015001 (2018). [10] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba- lents, Correlated Quantum Phenomena in the Strong Spin-Orbit Regime , Annu. Rev. Condens. Matter Phys. 5, 57 (2014). [11] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, Novel Jeff= 1/2Mott State Induced by Relativistic Spin- Orbit Coupling in Sr2IrO4, Phys. Rev. Lett. 101, 076402 (2008). [12] S. J. Moon, H. Jin, K. W. Kim, W. S. Choi, Y. S. Lee, J. Yu, G. Cao, A. Sumi, H. Funakubo, C. Bernhard, and T. W. Noh, Dimensionality-Controlled Insulator- Metal Transition and Correlated Metallic State in 5d Transition Metal Oxides Srn+1IrnO3n+1(n= 1,2, and ∞), Phys. Rev. Lett. 101, 226402 (2008). [13] M. K. Crawford, M. A. Subramanian, R. L. Harlow, J. A. Fernandez-Baca, Z. R. Wang, and D. C. Johnston, Structural and Magnetic Studies of Sr2IrO4, Phys. Rev. B49, 9198 (1994). [14] N. Read and B. Chakraborty, Statistics of the Excita- tions of the Resonating-Valence-Bond State , Phys. Rev. B40, 7133 (1989). [15] X.-G. Wen, Topological Orders and Chern-Simons The- ory in Strongly Correlated Quantum Liquid , Int. J. Mod. Phys. B 5, 1641 (1991). !MK!-2-101Energy (eV)!MK!-2-101Energy (eV)(a) (b)FIG. 10. Electronic band structure of monolayer MgIrO 3ob- tained by the (a) LDA+SOC and (b) LDA+SOC+ Ucalcula- tions for the in-plane N´ eel AFM state. In (b), the parameters of the Coulomb interaction and Hund’s coupling are respec- tively set to UIr= 3.0 eV and JH/UIr= 0.1. The notions are common to Fig. 2. [16] P. W. Anderson, Resonating Valence Bonds: A New Kind of Insulator? , Mater. Res. Bull. 8, 153 (1973). [17] L. Balents, Spin Liquids in Frustrated Magnets , Nature 464, 199 (2010). [18] Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum Spin Liq- uid States , Rev. Mod. Phys. 89, 025003 (2017). [19] C. Broholm, R. Cava, S. Kivelson, D. Nocera, M. Nor- man, and T. Senthil, Quantum Spin Liquids , Science 367, eaay0668 (2020). [20] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian Anyons and Topological Auantum Computation , Rev. Mod. Phys. 80, 1083 (2008). [21] P. W. Anderson, Ordering and Antiferromagnetism in Ferrites, Phys. Rev. 102, 1008 (1956). [22] S. Sachdev, Kagom´ e- and Triangular-Lattice Heisen- berg Antiferromagnets: Ordering from Quantum Fluc- tuations and Quantum-Disordered Ground States with Unconfined Bosonic Spinons , Phys. Rev. B 45, 12377 (1992).16 [23] A. Ramirez, Strongly Geometrically Frustrated Magnets , Annu. Rev. Mater. Sci. 24, 453 (1994). [24] Z. Nussinov and J. van den Brink, Compass Models: Theory and Physical Motivations , Rev. Mod. Phys. 87, 1 (2015). [25] A. Kitaev, Anyons in an Exactly Solved Model and Be- yond, Ann. Phys. 321, 2 (2006). [26] A. Y. Kitaev, Fault-Tolerant Auantum Computation by Anyons , Ann. Phys. 303, 2 (2003). [27] G. Jackeli and G. Khaliullin, Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models , Phys. Rev. Lett. 102, 017205 (2009). [28] H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and Realization of Kitaev Quan- tum Spin Liquids , Nat. Rev. Phys. 1, 264 (2019). [29] Y. Motome and J. Nasu, Hunting Majorana Fermions in Kitaev Magnets , J. Phys. Soc. Jpn. 89, 012002 (2020). [30] Y. Motome, R. Sano, S. Jang, Y. Sugita, and Y. Kato, Materials Design of Kitaev Spin Liquids Beyond the Jackeli–Khaliullin Mechanism , J. Phys. Condens. Mat- ter32, 404001 (2020). [31] S. Trebst and C. Hickey, Kitaev Materials , Phys. Rep. 950, 1 (2022). [32] Y. Singh and P. Gegenwart, Antiferromagnetic Mott In- sulating State in Single Crystals of the Honeycomb Lat- tice Material Na2IrO3, Phys. Rev. B 82, 064412 (2010). [33] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Relevance of the Heisenberg-Kitaev Model for the Honeycomb Lattice Iri- dates A2IrO3, Phys. Rev. Lett. 108, 127203 (2012). [34] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veen- stra, J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker, J. N. Hancock, D. van der Marel, I. S. Elfimov, and A. Damascelli, Na 2IrO3as a Novel Relativistic Mott Insulator with a 340 meV Gap, Phys. Rev. Lett. 109, 266406 (2012). [35] C. H. Sohn, H.-S. Kim, T. F. Qi, D. W. Jeong, H. J. Park, H. K. Yoo, H. H. Kim, J.-Y. Kim, T. D. Kang, D.- Y. Cho, G. Cao, J. Yu, S. J. Moon, and T. W. Noh, Mix- ing between Jeff=1 2and3 2Orbitals in Na2IrO3: A Spec- troscopic and Density Functional Calculation Study , Phys. Rev. B 88, 085125 (2013). [36] S. Hwan Chun, J.-W. Kim, J. Kim, H. Zheng, C. C. Stoumpos, C. Malliakas, J. Mitchell, K. Mehlawat, Y. Singh, Y. Choi, etal.,Direct Evidence for Dominant Bond-Directional Interactions in a Honeycomb Lattice Iridate Na2IrO3, Nat. Phys. 11, 462 (2015). [37] S. D. Das, S. Kundu, Z. Zhu, E. Mun, R. D. McDon- ald, G. Li, L. Balicas, A. McCollam, G. Cao, J. G. Rau, H.-Y. Kee, V. Tripathi, and S. E. Sebastian, Magnetic Anisotropy of the Alkali Iridate Na2IrO3at High Mag- netic Fields: Evidence for Strong Ferromagnetic Kitaev Correlations , Phys. Rev. B 99, 081101 (2019). [38] H. Zhao, B. Hu, F. Ye, M. Lee, P. Schlottmann, and G. Cao, Ground State in Proximity to a Possible Ki- taev Spin Liquid: The Undistorted Honeycomb Iridate NaxIrO3(0.60≤x≤0.80), Phys. Rev. B 104, L041108 (2021). [39] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, α-RuCl 3: A Spin-Orbit Assisted Mott Insulator on a Honeycomb Lattice , Phys. Rev. B 90, 041112 (2014).[40] Y. Kubota, H. Tanaka, T. Ono, Y. Narumi, and K. Kindo, Successive Magnetic Phase Transitions in α- RuCl 3: XY-like Frustrated Magnet on the Honeycomb Lattice , Phys. Rev. B 91, 094422 (2015). [41] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J. Kim, and K. S. Burch, Scattering Continuum and Possible Fractionalized Excitations in α-RuCl 3, Phys. Rev. Lett. 114, 147201 (2015). [42] A. Koitzsch, C. Habenicht, E. M¨ uller, M. Knupfer, B. B¨ uchner, H. C. Kandpal, J. van den Brink, D. Nowak, A. Isaeva, and T. Doert, JeffDescription of the Honey- comb Mott Insulator α-RuCl 3, Phys. Rev. Lett. 117, 126403 (2016). [43] S. Sinn, C. H. Kim, B. H. Kim, K. D. Lee, C. J. Won, J. S. Oh, M. Han, Y. J. Chang, N. Hur, H. Sato, etal.,Electronic Structure of the Kitaev Material α- RuCl 3Probed by Photoemission and Inverse Photoemis- sion Spectroscopies , Sci. Rep. 6, 39544 (2016). [44] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome, Y. S. Kwon, D. Adroja, D. Voneshen, K. Kim, T.-H. Jang, etal.,Majorana Fermions in the Kitaev Quantum Spin System α-RuCl 3, Nat. Phys. 13, 1079 (2017). [45] Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, S. Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Mo- tome, etal.,Majorana Quantization and Half-Integer Thermal Quantum Hall Effect in a Kitaev Spin Liquid , Nature 559, 227 (2018). [46] H. Suzuki, H. Liu, J. Bertinshaw, K. Ueda, H. Kim, S. Laha, D. Weber, Z. Yang, L. Wang, H. Taka- hashi, etal.,Proximate Ferromagnetic State in the Ki- taev Model Material α-RuCl 3, Nat. Commun. 12, 4512 (2021). [47] R. Sano, Y. Kato, and Y. Motome, Kitaev-Heisenberg Hamiltonian for High-Spin d7Mott Insulators , Phys. Rev. B 97, 014408 (2018). [48] H. Liu and G. Khaliullin, Pseudospin Exchange Inter- actions in d7Cobalt Compounds: Possible Realization of the Kitaev Model , Phys. Rev. B 97, 014407 (2018). [49] R. Jang, Seong-Hoon and, Y. Kato, and Y. Motome, Antiferromagnetic Kitaev Interaction in f-Electron Based Honeycomb Magnets , Phys. Rev. B 99, 241106 (2019). [50] S.-H. Jang, R. Sano, Y. Kato, and Y. Motome, Compu- tational Design of f-Electron Kitaev Magnets: Honey- comb and Hyperhoneycomb Compounds A2PrO 3(A= Alkali Metals) , Phys. Rev. Mater. 4, 104420 (2020). [51] Y. Sugita, Y. Kato, and Y. Motome, Antiferromagnetic Kitaev Interactions in Polar Spin-Orbit Mott Insulators , Phys. Rev. B 101, 100410 (2020). [52] S. Biswas, Y. Li, S. M. Winter, J. Knolle, and R. Va- lent´ ı, Electronic Properties of α-RuCl 3in Proximity to Graphene , Phys. Rev. Lett. 123, 237201 (2019). [53] S. Mandal, D. Mallick, A. Banerjee, R. Gane- san, and P. S. Anil Kumar, Evidence of Charge Transfer in Three-Dimensional Topolog- ical Insulator/Antiferromagnetic Mott Insulator Bi1Sb1Te1.5Se1.5/α-RuCl 3Heterostructures , Phys. Rev. B 107, 245418 (2023). [54] K. Miura, K. Fujiwara, K. Nakayama, R. Ishikawa, N. Shibata, and A. Tsukazaki, Stabilization of A Honeycomb Lattice of IrO6Octahedra by Formation of Ilmenite-Type Superlattices in MnTiO 3, Commun. Mater. 1, 55 (2020).17 [55] B. Kang, M. Park, S. Song, S. Noh, D. Choe, M. Kong, M. Kim, C. Seo, E. K. Ko, G. Yi, J.-W. Yoo, S. Park, J. M. Ok, and C. Sohn, Honeycomb Oxide Heterostruc- ture as a Candidate Host for A Kitaev Quantum Spin Liquid , Phys. Rev. B 107, 075103 (2023). [56] Y. Haraguchi, C. Michioka, A. Matsuo, K. Kindo, H. Ueda, and K. Yoshimura, Magnetic Ordering with an XY-like Anisotropy in the Honeycomb Lattice Iri- dates ZnIrO 3andMgIrO3Synthesized via a Metathesis Reaction , Phys. Rev. Mater. 2, 054411 (2018). [57] S.-H. Jang and Y. Motome, Electronic and Mag- netic Properties of Iridium Ilmenites AIrO3(A= Mg,Zn,and Mn), Phys. Rev. Mater. 5, 104409 (2021). [58] X. Hao, H. Jiang, R. Cui, X. Zhang, K. Sun, and Y. Xu, Electronic and Magnetic Properties of Spin– Orbit-Entangled Honeycomb Lattice Iridates MIrO 3 (M = Cd ,Zn,and Mg), Inorg. Chem. 61, 15007 (2022). [59] K. Momma and F. Izumi, VESTA 3 for Three- Dimensional Visualization of Crystal, Volumetric and Morphology Data , J. Appl. Crystallogr. 44, 1272 (2011). [60] P. Giannozzi, S. Baroni, N. Bonini, M. Calan- dra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, etal.,QUAN- TUM ESPRESSO: A Modular and Open-Source Soft- ware Project for Quantum Simulations of Materials , J. Phys.: Condens. Matter 21, 395502 (2009). [61] P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas, Phys. Rev. 136, B864 (1964). [62] J. P. Perdew and A. Zunger, Self-Interaction Correc- tion to Density-Functional Approximations for Many- Electron Systems , Phys. Rev. B 23, 5048 (1981). [63] P. E. Bl¨ ochl, Projector Augmented-Wave Method , Phys. Rev. B 50, 17953 (1994). [64] A. Dal Corso, Pseudopotentials Periodic Table: From H toPu, Comput. Mater. Sci. 95, 337 (2014). [65] H. J. Monkhorst and J. D. Pack, Special Points for Brillouin-Zone Integrations , Phys. Rev. B 13, 5188 (1976). [66] A. I. Liechtenstein, V. I. Anisimov, and J. Zaa- nen, Density-Functional Theory and Strong Interac- tions: Orbital Ordering in Mott-Hubbard Insulators , Phys. Rev. B 52, R5467 (1995). [67] B. L. Chittari, Y. Park, D. Lee, M. Han, A. H. MacDon- ald, E. Hwang, and J. Jung, Electronic and Magnetic Properties of Single-Layer MPX3Metal Phosphorous Trichalcogenides , Phys. Rev. B 94, 184428 (2016). [68] M. Arruabarrena, A. Leonardo, M. Rodriguez-Vega, G. A. Fiete, and A. Ayuela, Out-of-Plane Magnetic Anisotropy in Bulk Ilmenite CoTiO 3, Phys. Rev. B 105, 144425 (2022). [69] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, An Updated Version of Wannier90: A Tool for Obtaining Maximally-Localised Wannier Functions , Comput. Phys. Commun. 185, 2309 (2014). [70] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M. Imada, First-Principles Study of the Honeycomb- Lattice Iridates Na2IrO3in the Presence of Strong Spin- Orbit Interaction and Electron Correlations , Phys. Rev. Lett. 113, 107201 (2014). [71] B. H. Kim, G. Khaliullin, and B. I. Min, Electronic Ex- citations in the Edge-Shared Relativistic Mott Insulator: Na2IrO3, Phys. Rev. B 89, 081109 (2014).[72] A. M. Ole´ s, Antiferromagnetism and Correlation of Electrons in Transition Metals , Phys. Rev. B 28, 327 (1983). [73] L. M. Roth, Simple Narrow-Band Model of Ferromag- netism Due to Intra-Atomic Exchange , Phys. Rev. 149, 306 (1966). [74] G. Khaliullin, Orbital Order and Fluctuations in Mott Insulators , Prog. Theor. Phys. Suppl. 160, 155 (2005), https://academic.oup.com/ptps/article- pdf/doi/10.1143/PTPS.160.155/5162453/160-155.pdf. [75] G. Shirane, S. J. Pickart, and Y. Ishikawa, Neutron Diffraction Study of Antiferromagnetic MnTiO 3and NiTiO 3, J. Phys. Soc. Japan 14, 1352 (1959). [76] R. Newnham, J. Fang, and R. Santoro, Crystal Structure and Magnetic Properties of CoTiO 3, Acta Crystallogr. 17, 240 (1964). [77] A. I. Liechtenstein, M. Katsnelson, V. Antropov, and V. Gubanov, Local Spin Density Functional Approach to the Theory of Exchange Interactions in Ferromag- netic Metals and Alloys , J. Magn. Magn. Mater. 67, 65 (1987). [78] X. He, N. Helbig, M. J. Verstraete, and E. Bousquet, TB2J: A Python Package for Computing Magnetic In- teraction Parameters , Comput. Phys. Commun. 264, 107938 (2021). [79] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Generic Spin Model for the Honeycomb Iridates beyond the Kitaev Limit , Phys. Rev. Lett. 112, 077204 (2014). [80] J. Rusnaˇ cko, D. Gotfryd, and J. c. v. Chaloupka, Kitaev- like Honeycomb Magnets: Global Phase Behavior and Emergent Effective Models , Phys. Rev. B 99, 064425 (2019). [81] M. Hanawa, Y. Muraoka, T. Tayama, T. Sakakibara, J. Yamaura, and Z. Hiroi, Superconductivity at 1 K in Cd2Re2O7, Phys. Rev. Lett. 87, 187001 (2001). [82] K. Ohgushi, J.-i. Yamaura, M. Ichihara, Y. Ki- uchi, T. Tayama, T. Sakakibara, H. Gotou, T. Yagi, and Y. Ueda, Structural and Electronic Properties of Pyrochlore-Type A2Re2O7(A= Ca ,Cd,and Pb ), Phys. Rev. B 83, 125103 (2011). [83] J. Harter, Z. Zhao, J.-Q. Yan, D. Mandrus, and D. Hsieh, A Parity-Breaking Electronic Nematic Phase Transition in the Spin-Orbit Coupled Metal Cd2Re2O7, Science 356, 295 (2017). [84] Z. Hiroi, J.-i. Yamaura, T. C. Kobayashi, Y. Matsub- ayashi, and D. Hirai, Pyrochlore Oxide Superconduc- torCd2Re2O7Revisited , J. Phys. Soc. Jpn. 87, 024702 (2018). [85] S. Tajima, D. Hirai, T. Yajima, D. Nishio-Hamane, Y. Matsubayashi, and Z. Hiroi, Spin–Orbit-Coupled Metal Candidate PbRe 2O6, J. Solid. State. Chem. 288, 121359 (2020). [86] R. Chari, R. Moessner, and J. G. Rau, Magnetoelectric Generation of a Majorana-Fermi Surface in Kitaev’s Honeycomb Model , Phys. Rev. B 103, 134444 (2021). [87] K. Nakazawa, Y. Kato, and Y. Motome, Asymmetric Modulation of Majorana Excitation Spectra and Nonre- ciprocal Thermal Transport in the Kitaev Spin Liquid Under a Staggered Magnetic Field , Phys. Rev. B 105, 165152 (2022). [88] P. W. Anderson, The Resonating Valence Bond State inLa2CuO 4and Superconductivity , Science 235, 1196 (1987).18 [89] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott Insulator: Physics of High-Temperature Superconductiv- ity, Rev. Mod. Phys. 78, 17 (2006). [90] J. G. Bednorz and K. A. M¨ uller, Possible High TcSu- perconductivity in the Ba-La-Cu-OSystem , Z. Phys. B 64, 189 (1986). [91] P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Resonating-Valence-Bond Theory of Phase Transitions and Superconductivity in La2CuO 4-Based Compounds , Phys. Rev. Lett. 58, 2790 (1987). [92] H. Watanabe, T. Shirakawa, and S. Yunoki, Monte Carlo Study of an Unconventional Superconducting Phase in Iridium Oxide Jeff=1/2Mott Insulators In- duced by Carrier Doping , Phys. Rev. Lett. 110, 027002 (2013). [93] Y. J. Yan, M. Q. Ren, H. C. Xu, B. P. Xie, R. Tao, H. Y. Choi, N. Lee, Y. J. Choi, T. Zhang, and D. L. Feng, Electron-Doped Sr2IrO4: An Analogue of Hole- Doped Cuprate Superconductors Demonstrated by Scan- ning Tunneling Microscopy , Phys. Rev. X 5, 041018 (2015). [94] Y.-Z. You, I. Kimchi, and A. Vishwanath, Doping a Spin-orbit Mott Insulator: Topological Superconductiv- ity from the Kitaev-Heisenberg Model and Possible Ap- plication to (Na2/Li2)IrO 3, Phys. Rev. B 86, 085145 (2012). [95] T. Hyart, A. R. Wright, G. Khaliullin, and B. Rosenow, Competition between d-Wave and Topological p-Wave Superconducting Phases in the Doped Kitaev-Heisenberg Model , Phys. Rev. B 85, 140510 (2012). [96] S. Okamoto, Global Phase Diagram of a Doped Kitaev- Heisenberg Model , Phys. Rev. B 87, 064508 (2013). [97] J. Schmidt, D. D. Scherer, and A. M. Black-Schaffer, Topological Superconductivity in the Extended Kitaev- Heisenberg Model , Phys. Rev. B 97, 014504 (2018). [98] Y. Ishikawa and S.-i. Akimoto, Magnetic Properties of theFeTiO 3-Fe2O3Solid Solution Series , J. Phys. Soc. Japan 12, 1083 (1957). [99] H. Kato, M. Yamada, H. Yamauchi, H. Hiroyoshi, H. Takei, and H. Watanabe, Metamagnetic Phase Tran- sitions in FeTiO 3, J. Phys. Soc. Japan 51, 1769 (1982). [100] B. Yuan, I. Khait, G.-J. Shu, F. C. Chou, M. B. Stone, J. P. Clancy, A. Paramekanti, and Y.-J. Kim, Dirac Magnons in a Honeycomb Lattice Quantum XY Magnet CoTiO 3, Phys. Rev. X 10, 011062 (2020). [101] B. A. Wechsler and C. T. Prewitt, Crystal Structure of Ilmenite (FeTiO 3)at High Temperature and at High Pressure , Am. Mineral. 69, 176 (1984). [102] A. Raghavender, N. H. Hong, K. J. Lee, M.-H. Jung, Z. Skoko, M. Vasilevskiy, M. Cerqueira, and A. Saman- tilleke, Nano-Ilmenite FeTiO 3: Synthesis and Charac- terization , J. Magn. Magn. Mater. 331, 129 (2013). [103] H. Yamauchi, H. Hiroyoshi, M. Yamada, H. Watan- abe, and H. Takei, Spin Flopping in MnTiO 3, J. Magn. Magn. Mater. 31, 1071 (1983). [104] J. Ko and C. T. Prewitt, High-Pressure Phase Transi- tion in MnTiO 3from the Ilmenite to the LiNbO 3Struc- ture, Phys. Chem. Miner. 15, 355 (1988). [105] N. Mufti, G. R. Blake, M. Mostovoy, S. Riyadi, A. A. Nugroho, and T. T. M. Palstra, Magnetoelectric Cou- pling in MnTiO 3, Phys. Rev. B 83, 104416 (2011). [106] Y. Ishikawa and S. Sawada, The Study on Substances Having the Ilmenite Structure I. Physical Properties of Synthesized FeTiO 3andNiTiO 3Ceramics , J. Phys. Soc.Jpn.11, 496 (1956). [107] Y. Ishikawa and S.-i. Akimoto, Magnetic Property and Crystal Chemistry of Ilmenite (MeTiO 3)and Hematite (αFe2O3)System I. Crystal Chemistry , J. Phys. Soc. Jpn.13, 1110 (1958). [108] T. Acharya and R. Choudhary, Structural, Dielec- tric and Impedance Characteristics of CoTiO 3, Mater. Chem. Phys. 177, 131 (2016). [109] B. Yuan, I. Khait, G.-J. Shu, F. Chou, M. Stone, J. Clancy, A. Paramekanti, and Y.-J. Kim, Dirac Magnons in a Honeycomb Lattice Quantum XY Mag- netCoTiO 3, Physical Review X 10, 011062 (2020). [110] G. S. Heller, J. J. Stickler, S. Kern, and A. Wold, An- tiferromagnetism in NiTiO 3, J. Appl. Phys. 34, 1033 (1963). [111] J. K. Harada, L. Balhorn, J. Hazi, M. C. Kemei, and R. Seshadri, Magnetodielectric Coupling in the Ilmenites MTiO 3(M= Co ,Ni), Phys. Rev. B 93, 104404 (2016). [112] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y. Singh, S. Manni, P. Gegenwart, J. Kim, A. H. Said, D. Casa, T. Gog, M. H. Upton, H.-S. Kim, J. Yu, V. M. Katukuri, L. Hozoi, J. van den Brink, and Y.- J. Kim, Crystal-Field Splitting and Correlation Effect on the Electronic Structure of A2IrO3, Phys. Rev. Lett. 110, 076402 (2013). [113] X. Zhou, H. Li, J. A. Waugh, S. Parham, H.-S. Kim, J. A. Sears, A. Gomes, H.-Y. Kee, Y.-J. Kim, and D. S. Dessau, Angle-Resolved Photoemission Study of the Kitaev Candidate α-RuCl 3, Phys. Rev. B 94, 161106 (2016). [114] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess- ner, Dynamics of a Two-Dimensional Quantum Spin Liquid: Signatures of Emergent Majorana Fermions and Fluxes , Phys. Rev. Lett. 112, 207203 (2014). [115] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess- ner, Dynamics of Fractionalization in Quantum Spin Liquids , Phys. Rev. B 92, 115127 (2015). [116] A. Banerjee, C. Bridges, J.-Q. Yan, A. Aczel, L. Li, M. Stone, G. Granroth, M. Lumsden, Y. Yiu, J. Knolle, etal.,Proximate Kitaev Quantum Spin Liquid Be- haviour in a Honeycomb Magnet , Nat. Mater. 15, 733 (2016). [117] J. Yoshitake, J. Nasu, and Y. Motome, Fractional Spin Fluctuations as a Precursor of Quantum Spin Liquids: Majorana Dynamical Mean-Field Study for the Kitaev Model , Phys. Rev. Lett. 117, 157203 (2016). [118] J. Yoshitake, J. Nasu, Y. Kato, and Y. Motome, Ma- jorana Dynamical Mean-Field Study of Spin Dynamics at Finite Temperatures in the Honeycomb Kitaev Model , Phys. Rev. B 96, 024438 (2017). [119] J. Yoshitake, J. Nasu, and Y. Motome, Tempera- ture Evolution of Spin Dynamics in Two- and Three- dimensional Kitaev Models: Influence of Fluctuating Z2 flux, Phys. Rev. B 96, 064433 (2017). [120] J. Nasu, J. Yoshitake, and Y. Motome, Thermal Trans- port in the Kitaev Model , Phys. Rev. Lett. 119, 127204 (2017). [121] T. Yokoi, S. Ma, Y. Kasahara, S. Kasahara, T. Shibauchi, N. Kurita, H. Tanaka, J. Nasu, Y. Mo- tome, C. Hickey, etal.,Half-Integer Quantized Anoma- lous Thermal Hall effect in the Kitaev Material Candi- dateα-RuCl 3, Science 373, 568 (2021). [122] L. E. Chern, E. Z. Zhang, and Y. B. Kim, Sign Structure of Thermal Hall Conductivity and Topological Magnons19 for In-Plane Field Polarized Kitaev Magnets , Phys. Rev. Lett. 126, 147201 (2021). [123] J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and R. Moessner, Fermionic Response from Fractionaliza- tion in an Insulating Two-Dimensional Magnet , Nat. Phys. 12, 912 (2016). [124] A. Glamazda, P. Lemmens, S.-H. Do, Y. Choi, and K.- Y. Choi, Raman Spectroscopic Signature of Fractional- ized Excitations in the Harmonic-Honeycomb Iridates β- and γ-Li2IrO3, Nat. Commun. 7, 12286 (2016). [125] B. Zhou, Y. Wang, G. B. Osterhoudt, P. Lampen- Kelley, D. Mandrus, R. He, K. S. Burch, and E. A. Henriksen, Possible Structural Transformation and En- hanced Magnetic Fluctuations in Exfoliated α-RuCl 3, J. Phys. Chem. Solids 128, 291 (2019). [126] J.-H. Lee, Y. Choi, S.-H. Do, B. H. Kim, M.-J. Seong, and K.-Y. Choi, Multiple Spin-Orbit Excitons in α- RuCl 3from Bulk to Atomically Thin Layers , npj Quan- tum Mater. 6, 43 (2021). [127] J. Feldmeier, W. Natori, M. Knap, and J. Knolle, Local Probes for Charge-Neutral Edge States in Two-Dimensional Quantum Magnets , Phys. Rev. B 102, 134423 (2020). [128] R. G. Pereira and R. Egger, Electrical Access to Ising Anyons in Kitaev Spin Liquids , Phys. Rev. Lett. 125, 227202 (2020). [129] E. J. K¨ onig, M. T. Randeria, and B. J¨ ack, Tunneling Spectroscopy of Quantum Spin Liquids , Phys. Rev. Lett. 125, 267206 (2020). [130] M. Udagawa, S. Takayoshi, and T. Oka, Scanning Tun- neling Microscopy as a Single Majorana Detector of Ki- taev’s Chiral Spin Liquid , Phys. Rev. Lett. 126, 127201 (2021). [131] T. Bauer, L. R. D. Freitas, R. G. Pereira, and R. Eg- ger,Scanning Tunneling Spectroscopy of Majorana Zero Modes in a Kitaev Spin Liquid , Phys. Rev. B 107, 054432 (2023). [132] Y. Kato, J. Nasu, M. Sato, T. Okubo, T. Misawa, and Y. Motome, Spin Seebeck Effect as a Probe for Majo- rana Fermions in Kitaev Spin Liquids , arXiv preprint arXiv:2401.13175 (2024).

1304.7968v2.Can_electrostatic_field_lift_spin_degeneracy_.pdf

arXiv:1304.7968v2 [quant-ph] 7 Aug 2014Can electrostatic field lift spin degeneracy? T. G. Tenev1,∗and N. V. Vitanov1 1Department of Physics, Sofia University, 5 James Bourchier B lvd, Sofia 1164, Bulgaria (Dated: October 31, 2021) There are two well known mechanisms which lead to lifting of e nergy spin degeneracy of single electron systems - magnetic field and spin-orbit coupling. W e investigate the possibility for existence of a third mechanism in which electrostatic field can lead to l ifting of spin-degeneracy directly without the mediation of spin-orbit coupling. A novel argum ent is provided for the need of spin- orbit coupling different from the usual relativistic consid erations. It is shown that due to preserved translational invariance spin splitting purely by electro static field is not possible for Bloch states. A possible lifting of spin degeneracy by electrostatic field c haracterized by broken inversion and broken translational invariance is considered. I. INTRODUCTION The field of spintronics1, a term coined by Wolf2, has attracted considerable attention in the past 20 years. This interest was initially triggered by the discovery of the effect of giant magnetoresistance (GMR)3,4which now has established commercial applications. A paral- lel interest in semiconductor spintronics has arisen due to the proposal for a ballistic spin-transistor5which has recently been realized6. A generalization of the device operating in diffusive regime has been proposed7as well as other devices1. They depend on the spin splitting caused by the spin-orbit coupling through several differ- ent mechanisms usually classified in two big groups. His- torically the first group often referred to as Dresselhaus spin-orbit coupling8–10, or alternatively, bulk inversion asymmetry11–13(BIA), has as a necessary condition the broken space inversion invariance of a bulk crystal. It also appears in a modified form1,11,12,14,15in quasi two- dimensional (Q2D) structures grown in crystals lacking inversion symmetry. The second group referred to as either Rashba spin-orbit coupling16–18or structure in- version asymmetry1,11,12(SIA) leads to spin splitting in systems lacking macroscopic inversion symmetry. In all these cases the inversion asymmetry is merely a neces- sary condition and not a sufficient one. The other neces- sary condition is taking spin-orbit coupling into account in various ways but usually through the folding down procedure11–13,18from 8 band or 14 band models. Here we consider whether the breaking of inversion invariance of electrostatic field in several classes of systems can also be a sufficient condition for lifting of spin degeneracy. We examine a symmetry argument11–13,19due to Kittel19. We provide novel derivation of the symmetry argument11–13,19relying only on the commutation prop- erties of the time-reversal ˆKand space inversion ˆIop- erators with the translation ˆTRnand spin operators ˆSu. This is unlike the usual derivation19relying on explicit form of Bloch states. We show a novel argument for the need for introduction of a spin-orbit coupling term simi- lar to the argument motivating Maxwell to introduce the displacement current term in the Maxwell’s equations. We note that the symmetry argument11–13,19involvingtime-reversal and space inversion invariance does not re- quire the presence of a spin-orbit coupling term in the Hamiltonian but merely to take the spin degree of free- dom of the electron into account. The application of this symmetry argument, to a nonrelativistic model of elec- tron with spin moving in electrostatic field characterized by discrete translational invariance, naturally suggests the hypothesis that electrostatic field with broken space inversionsymmetry can lead to lifting ofspin degeneracy. We explorethe hypothesisby perturbationmethod treat- ment and show that if discrete translational invariance is preserved electrostatic field can not lift spin-degeneracy. If both translational and space inversion symmetry of a perturbing electrostatic field are broken a possible con- tribution to spin splitting by electrostatic field is indi- cated as far as the perturbation method is applicable. Physically this would be naturally explained since we know21that every non-accidental degeneracy stems from symmetries of the underlying system and in general re- duction of symmetries leads to lifting of degeneracies. In Sec. II we introduce the theoretical models. The basic symmetry argument is presented in Sec. III. The transformation properties of Bloch states in a model ne- glecting spin-orbit coupling but taking into account the spin degree of freedom are given in Sec. IIIA. They are used in Sec. IIIB to show the appearance of at least four- fold degeneracy in the spectrum as a consequence of the combination of time-reversal and space inversion invari- ance and in Sec. IIIC to show the lifting of spin degen- eracy of Bloch states as a consequence of broken space inversion invariance. How the presented symmetry ar- gument differs from the usual one11–13,19is discussed in Sec. IIID. In Sec. IV a new argument for the introduc- tion of spin-orbit coupling is presented different from the usual relativistic arguments12,21,23,24. We then explore the hypothesis formulated in Sec. IIID by perturbation method treatment in Sec. V. In Sec. VI we discuss cer- tain aspects of the utilized model and methods and we present our conclusions in Sec. VII.2 II. THEORETICAL MODELS We first consider two basic models of a nonrelativistic electron moving in a pure electrostatic field characterized by discrete translational invariance. We focus our atten- tionon3Dmodelsofthetricliniccrystalsystem. Thetwo classes of models differ from each other by the properties of the electrostatic potential with respect to space inver- sion symmetry. In both models we take spin degree of freedom into account but neglect the spin-orbit coupling term. The eigenstates of both models are Bloch states19, which can be characterizedby two quantum numbers: (i) crystal wavevector and (ii) spin index which is the eigen- value of the u-component ˆSuof the spin vector operator ˆS. Furthermore we employ the standard Born-von Kar- man periodic boundary conditions. The first class of models represents 3D crystals of the triclinic pinacoidal symmetry class. Its Hamiltonian takes the form ˆH0=ˆp2 2mˆσ0+V0(r)ˆσ0. (1) This represents a nonrelativistic electron moving in a pure electrostatic potential φ0(r) with potential energy V0=−eφ0(r) characterized by translational invariance, [ˆTRn,φ0(r)] = 0, time-reversal invariance [ ˆK,φ0(r)] = 0, and space inversion invariance [ ˆI,φ0(r)] = 0. The second class of models represents 3D crystals of the triclinic pedial symmetry class in which the overall potential does not possess space inversion invariance. A general potential not possessing space inversion invari- ance can be split into parts symmetric with respect to spaceinversion,that isonethatcommuteswith the space inversion operator ˆI, and a part antisymmetric with re- spect to space inversion, that is one that anticommutes withˆI. We consider the electrostatic field of the second class of models as made of the symmetric part φ0(r) and an antisymmetric one φ(r), satisfying ˆIφ(r)ˆI+=−φ(r). The Hamiltonian takes the form ˆH=ˆp2 2mˆσ0+V0ˆσ0−eφ(r)ˆσ0, (2) While the Hamiltonian in such systemspossessesdiscrete translational invariance [ ˆH,ˆTRn] = 0, and time-reversal invariance, [ ˆH,ˆK] = 0, it is no longer invariant with respect to space inversion, [ ˆH,ˆI]/ne}ationslash= 0. Inordertoemphasizethatwehavetakenspindegreeof freedom into account we have written the Hamiltonians (1) and (2) with the 2 ×2 identity matrix σ0. III. SYMMETRY ARGUMENT A. Transformation of Bloch States WeconsiderageneralBlochstate |k,su/an}b∇acket∇i}httheproperties of which are identical in the two models considered. Itsatisfies the Bloch theorem19, ˆTRn|k,su/an}b∇acket∇i}ht=e−ik·Rn|k,su/an}b∇acket∇i}ht, (3) where we use the active convention20,21for the space translation operator ˆTRn. The time-reversaltransformed state|k′,s′ u/an}b∇acket∇i}ht=ˆK|k,su/an}b∇acket∇i}htof a Bloch state |k,su/an}b∇acket∇i}htis still a Bloch state because the time-reversal operator commutes21with the spatial translation operators ˆTRn. Applying the time-reversal operator ˆKto the Bloch the- orem (3), taking into account that [ ˆTRn,ˆK] = 0 and that ˆKas antilinear operator does not commute with com- plex scalars cbut satisfies the identity21ˆKc=c∗ˆK, one obtains the identity ˆTRn|k′,s′ u/an}b∇acket∇i}ht=eik·Rn|k′,s′ u/an}b∇acket∇i}ht. Comparing it with the Bloch theorem, ˆTRn|k′,s′ u/an}b∇acket∇i}ht= e−ik′·Rn|k′,s′ u/an}b∇acket∇i}ht, shows that k′=−k. Applying ˆKto the relation ˆSu|k,su/an}b∇acket∇i}ht=su|k,su/an}b∇acket∇i}htand using the rela- tionˆKˆSu=−ˆSuˆK, which follows directly from the definition21of the time-reversal operator ˆK, one obtains ˆSuˆK|k,su/an}b∇acket∇i}ht=−suˆK|k,su/an}b∇acket∇i}ht. Thus the ˆK-transformed state|k′,s′ u/an}b∇acket∇i}ht=ˆK|k,su/an}b∇acket∇i}htis an eigenstate of ˆSuwith an eigenvalue −su, therefore s′ u=−su. Summarizing, the time-reversal operator ˆKtransforms a Bloch state |k,su/an}b∇acket∇i}htrepresenting an electron moving with a wavevec- torkand a spin pointing ”up” the axis uinto the Bloch state|−k,−su/an}b∇acket∇i}ht, ˆK|k,su/an}b∇acket∇i}ht=|−k,−su/an}b∇acket∇i}ht, (4) representing an electron moving in the opposite direction with acrystalwavevector −kanda spinpointing ”down” the axis u. The space translationand space inversionoperatorsdo not commute but satisfy20the identity ˆIˆTRn=ˆT−RnˆI. (5) Applying the spaceinversionoperator ˆIto the Blochthe- orem, (3), using Eq.(5) and the fact that ˆIdoes not act on the phase factor e−ik·Rnone obtains ˆT−RnˆI|k,su/an}b∇acket∇i}ht=e−ikRnˆI|k,su/an}b∇acket∇i}ht. (6) Since the ˆI-transformed Bloch state I|k,su/an}b∇acket∇i}htsatisfies the Blochtheorem it is still a Blochstate, but in generalwith different quantum numbers |k′,s′ u/an}b∇acket∇i}ht=ˆI|k,su/an}b∇acket∇i}ht. Com- paring Eq.(6) with the Bloch theorem ˆT−Rn|k′,s′ u/an}b∇acket∇i}ht= eik′Rn|k′,s′ u/an}b∇acket∇i}htone obtains that k′=−k. By definition21 the space inversion operator ˆIcommutes with any com- ponent of the spin vector operator ˆs. As a consequence, using the usual procedure applied above to ˆSu|k,su/an}b∇acket∇i}ht= su|k,su/an}b∇acket∇i}htgives that s′ u=su. Therefore, the space in- version operator ˆImaps a Bloch state |k,su/an}b∇acket∇i}ht, describ- ing electron motion with crystal wavevector kand spin pointing in the direction of axis u, into the state ˆI|k,su/an}b∇acket∇i}ht=|−k,su/an}b∇acket∇i}ht, (7)3 representing an electron motion with the same orienta- tion of spin but moving in the opposite direction with a wavevector −k. Using the definition of the conjugation operator19ˆC=ˆKˆIand Eqs. (4) and (7), one obtains the action of the ˆCon Bloch states, ˆC|k,su/an}b∇acket∇i}ht=|k,−su/an}b∇acket∇i}ht. (8) B. Spin Degeneracy Supposing the spectrum problem of the Hamiltonian ˆH0with space-inversion invariant electrostatic potential φ0(r) solved its eigenvalue-eigenvectorproblem takes the form of the identity ˆH0|k,su/an}b∇acket∇i}ht ≡E0 k,su|k,su/an}b∇acket∇i}ht. (9) The eigenvalues E0 k,suofˆH0are labeled with quantum numbers k,suand the eigenstates of H0and|k,su/an}b∇acket∇i}htsat- isfying the Bloch theorem possess all the properties of Bloch states, in particular Eq. (4) and Eq. (6). Applying the time-reversal operator ˆKto Eq.(9) and using the time-reversal invariance of the Hamiltonian [ˆH,ˆK] = 0, we obtain ˆH0ˆK|k,su/an}b∇acket∇i}ht ≡E0 k,suˆK|k,su/an}b∇acket∇i}ht. (10) Due to the time-reversalinvarianceof ˆH0the twolinearly independent states |k,su/an}b∇acket∇i}htand| −k,−su/an}b∇acket∇i}ht=ˆK|k,su/an}b∇acket∇i}ht correspond to the same eigenvalue of ˆH0, which we now denote as E0 θ≡E0 k,su=E0 −k,−su. The pairs of linearly independent states ( |k,su/an}b∇acket∇i}ht,| −k,−su/an}b∇acket∇i}ht) and (|−k,su/an}b∇acket∇i}ht,|k,−su/an}b∇acket∇i}ht) span respectively the 2D subspaces ε0 θandε0 −θof the Hilbert space of the single-particle system. The 2-fold degeneracy of the eigenenergies E0 θ andE0 −θ≡E0 −k,su=E0 k,−su, to which the subspaces ε0 θandε0 −θcorrespond, is a consequence of the time- reversalinvarianceof ˆH0andistherealizationofKramers degeneracy21,22in the system described by Eq. (9). By their construction the subspaces ε0 θandε0 −θare invari- ant with respect to the time reversal symbolically writ- ten asˆKε0 ±θ=ε0 ±θ. Comparing the spectrum equation ˆH0|−k,−su/an}b∇acket∇i}ht=E0 −k,−su|−k,−su/an}b∇acket∇i}htfor astate |−k,−su/an}b∇acket∇i}ht with Eq. (10) allows us to express the Kramers degener- acy in the studied system in the form E0 k,su=E0 −k,−su. (11) Acting on the left of Eq. (9) with the space-inversion operator ˆI,using the hypothesis [ ˆH,ˆI] = 0 and the result |−k,su/an}b∇acket∇i}ht ≡ˆI|k,su/an}b∇acket∇i}htfrom Eq.(7), oneobtainsthe identity ˆH0|−k,su/an}b∇acket∇i}ht ≡E0 k,su|−k,su/an}b∇acket∇i}ht. (12) It showsthat the space-inversioninvarianceofthe Hamil- tonianˆH0requires that the Bloch states |k,su/an}b∇acket∇i}htand |−k,su/an}b∇acket∇i}htbelong to the same eigenenergy, E0 k≡E0 k,su=E0 −k,su. (13)Expression (12) shows that the states |k,su/an}b∇acket∇i}htand |−k,su/an}b∇acket∇i}ht, respectively belonging to the subspaces εθand ε−θ, must belong to the same degenerate eigenvalue E0 k as a consequence of the space inversion invariance of ˆH0. Takingintoaccounttheconsequencesofthetime-reversal invariance of ˆH0given in Eqs. (10) and (11), all eigenen- ergiesE0 kofˆH0must be 4-fold degenerate. To every en- ergy value E0 kcorresponds a four-dimensional subspace ε0 k, which is a direct sum, ε0 k=ε0 θ+ε0 −θ, of the two sub- spacesε0 θandε0 −θ. The subspace ε0 kis invariant and re- ducible with respect to space inversion and time-reversal written symbolically as ˆKε0 k=ε0 kandˆIε0 k=ε0 k. Combination of the space inversion and time-reversal invarianceofthe Hamiltonian ˆH0ofthe considered trans- lational invariant system is equivalent to spin degener- acy. Formally this is illustrated using the conjugation operator ˆC=ˆKˆIwhich commutes with the Hamiltonian ˆH0, [ˆH0,ˆC] = 0 if it commutes separately with ˆKand ˆI. Using the usual procedure of applying the operator ˆCto Eq.(9) and taking into account Eq.(8) one obtains the identity ˆH0|k,−su/an}b∇acket∇i}ht=E0 k,su|k,−su/an}b∇acket∇i}ht. Therefore the Bloch states |k,su/an}b∇acket∇i}htand|k,−su/an}b∇acket∇i}htdescribing an electron with opposite spins belong to the same degenerate en- ergy value E0 k. The subspace ε0 kcorresponding to E0 kis invariant with respect to ˆC. The spin degeneracy can be viewed also as a consequence of the SU(2) invariance of the Hamiltonian ˆH0. C. Broken Spin Degeneracy The problem for the spectrum of the Hamiltonian ˆH shown in Eq (2) is given by the identity ˆH|κ,σu/an}b∇acket∇i}ht ≡Eκ,σu|κ,σu/an}b∇acket∇i}ht. (14) The Bloch states |κ,σu/an}b∇acket∇i}htare the common set of eigen- states of the commuting operators ˆHandˆTRnandEκ,σu are the corresponding eigenvalues of the Hamiltonian ˆH. They transform among each other according to relations (4), (7) and (8) because the Hamiltonian ˆHgiven in Eq (2) is translational invariant. The spectrum (14) of the Hamiltonian ˆHpossesses the time-reversal induced properties derived from Eq. (10) and Eq. (11). It is Kramers degenerate and each of its eigenenergies are two-fold degenerate, Eκ,σu=E−κ,−σu. However because the space inversion invariance of the electrostatic potential φ(r), and therefore of ˆHis bro- ken, the degeneracy due to space inversion invariance is lifted,Eκ,σu/ne}ationslash=E−κ,σu. This requires the lifting of spin degeneracy Eκ,σu/ne}ationslash=Eκ,−σu. (15) The detailed proof follows in the next paragraph. The broken spin degeneracy is proved by applying the conjugation operator ˆCto Eq. (14). However, because4 [ˆH,ˆI]/ne}ationslash= 0 and hence [ ˆH,ˆC]/ne}ationslash= 0 we can not interchange the positions of ˆCandˆH. Instead, using ˆ1 =C−1Cand |κ,−σu/an}b∇acket∇i}ht=ˆC|κ,σu/an}b∇acket∇i}htone obtains from Eq. (8) that ˆCˆHˆC−1|κ,−σu/an}b∇acket∇i}ht=Eκ,σu|κ,−σu/an}b∇acket∇i}ht.(16) The Bloch state |κ,−σu/an}b∇acket∇i}ht, as an eigenfunction of ˆTRn, and because [ ˆH,ˆTRn] = 0, is still an eigenstate of ˆH with an eigenvalue Eκ,−σu. However, because of broken inversion invariance, ˆCˆHˆC−1=ˆIˆKˆHˆK−1ˆI−1=ˆIˆHˆI−1/ne}ationslash=ˆH |κ,−σu/an}b∇acket∇i}htis not anymore an eigenstate of ˆHwith the eigenvalue Eκ,σu. Instead the Bloch state |κ,−σu/an}b∇acket∇i}htis eigenstate of some other operator ˆH′=ˆIˆHˆI−1with the eigenvalue Eκ,σu. Therefore the states |κ,σu/an}b∇acket∇i}htand |κ,−σu/an}b∇acket∇i}htdo not correspond to the same energy. Elec- trons in Bloch states characterized by the same wavevec- torκbut having opposite spin orientations do not posses the same energy, the result given in Eq. (15). D. Discussion of Symmetry Argument Inthewellknowntreatment8,11,12,19itissupposedthat the spin-orbit coupling is part of the model Hamiltonian and that the spin degeneracy is lifted by the spin-orbit coupling term HSO=¯h 4m2c2ˆσ·(∇V(r)׈p) (17) if the electrostatic potential V(r) =V0(r)−eφ(r) does not possess space inversion invariance. However, close examination of the symmetry analysis developed in the text above shows that there is no such requirement. The symmetryanalysisisvalidalsoforanonrelativisticmodel that contains just electrostatic fields φ0(r) andφ(r) and does not contain spin-orbit coupling or magnetic field. It suggests that electrostatic field alone without the media- tion of spin-orbit coupling can lead to lift of spin degen- eracy given that φ0(r) andφ(r) possess discrete transla- tional invariance and are characterized by preserved and broken space inversion symmetry respectively. IV. NOVEL ARGUMENT FOR INTRODUCTION OF SPIN-ORBIT COUPLING TERM Taking spin into account suggests examining for SU(2) symmetry. Since we consider models of a nonrelativistic electron in electrostatic field neglecting spin-orbit cou- pling, the Hamiltonians ˆH0andˆHcommute with ev- ery component of the spin vector operator ˆSand there- fore the Hamiltonians ˆH0andˆHare SU(2) invariant. They commute, [ ˆRs u(φ),ˆH] = 0, with every spin rotation operator21ˆRs u(φ) =e−iφˆSuforarbitraryaxis uandangleof rotation φ. As a consequence of the SU(2) invariance, the spin degeneracy for the model with Hamiltonians ˆH0 andˆHmust be preserved, in particular for ˆH Eκ,σu=Eκ,−σu (18) This result, however, contradicts the result (15) stem- ming from symmetry analysis based solely on time- reversalandbrokenspaceinversioninvariancewhenspin- orbit coupling is neglected. Apossiblewaytoresolvethisinconsistencyistheintro- duction of a term in the model Hamiltonian that breaks theSU(2) invariance and at the same time is consis- tent with the different cases of symmetry analysis in- volving just time-reversal and space-inversion operators. The spin-orbit coupling term (17), which does not com- mute with any spin-rotation operator21ˆRs u(φ), satisfies the above requirements and resolves the noted inconsis- tency. We interpret this as a novel argument for intro- duction of the spin-orbit coupling term different from the usual21,23,24purely relativistic considerations, which give an incorrect numerical factor by 1 /2. This difference is accounted for by Thomas precession or by taking the nonrelativistic limit of the Dirac equation21,23,24. Thus a realistic model of electron dynamics requires taking into account the spin-orbit coupling term as a minimum; oth- erwise we would encounter the above mentioned incon- sistencies. Therefore in all subsequent models treated within perturbation method the spin-orbit coupling term is part of the considered Hamiltonian. As a consequence SU(2) symmetryis alwaysbrokenand there isno require- ment for preservation of spin-degeneracy. V. PERTURBATION METHOD TREATMENT A model with spin-orbit coupling does not exclude the possibility that electrostatic field by itself leads to lift- ing of spin degeneracy suggested by the symmetry anal- ysis based on time-reversal and space-inversion invari- ance in Sec. IIID. It merely shows that if such splitting exists it will lead to additional numerical factor in the spin splitting alreadycaused by spin-orbit coupling when the space-inversion invariance of the electrostatic field is broken. Since the crystal wavevector kvaries in discrete stepsweinvestigatethisoptionusingstandardstationary perturbation method for degenerate levels. The unperturbed Hamiltonian is ˆH0=ˆp2 2m+ V0(r), where V0(r) possesses space inversion invariance, [V0,ˆI] = 0, while the perturbation δˆVconsists of an electrostatic potential φ(r) which is odd with respect to space inversion, {φ(r),ˆI}= 0, and the spin-orbit cou- pling terms ˆU1=¯h 4m2c2ˆσ·(∇V0(r)׈p), (19a) ˆU2=−e¯h 4m2c2ˆσ·(∇φ(r)׈p).(19b)5 Unlike in the standard k·ˆpmethod where the k= 0 stationary states are used as unperturbed basis, we use the stationary states |k,su/an}b∇acket∇i}htofˆH0for arbitrary k/ne}ationslash= 0. This choice is naturally suggested by the symmetry anal- ysis above since for k= 0 we have just two-fold Kramers degeneracy that is not lifted as far as time-reversal in- variance is preserved. We consider 3D model of triclinic pedial and triclinic pinacoidal systems in which cases the degeneracy of every energy level E0 kof the unperturbed Hamiltonian ˆH0is exactly four-fold. The corresponding subspace ε0 kis spanned by the four Bloch states |k,su/an}b∇acket∇i}ht, |k,−su/an}b∇acket∇i}ht,|−k,su/an}b∇acket∇i}htand|−k,−su/an}b∇acket∇i}ht. The first-ordercorrection E(1) ktothe energyeigenvalue E(0) kand the zeroth-order states |0/an}b∇acket∇i}htare determined from the eigenvalue equation ˆP0 kδˆVˆP0 k|0k/an}b∇acket∇i}ht=E(1)|0k/an}b∇acket∇i}ht, (20) whereˆP0 kis the projector to the subspace ε0 kandδˆV= −eφ(r) +ˆU1+ˆU2.The first-order correction E(1) kto the energy is determined by the solution of the secular equation det/bracketleftBig ˆP0 kδVˆP0 k−E(1) k/bracketrightBig = 0 corresponding to the eigenvalue problem (20). The condition for time-reversal invariance simplifies the secular equation by introducing relationships between its matrix elements, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea−E(1) kc d 0 c∗b−E(1) k0 d d∗0b−E(1) k−c 0 d∗−c∗a−E(1) k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0,(21) wherethe matrixelements a=a1+a2+α,b=b1+b2+α, c=c1+c2andd=d1+d2+βhavecontributionsfromthe three different perturbing terms. The matrix elements due to the spin-orbit coupling term ˆU1are denoted as a1,b1,c1,d1, the ones due to ˆU2asa2,b2,c2,d2, and the ones due to the electrostatic field φ(r) asαandβ, a1=/an}b∇acketle{tsu,k|ˆU1|k,su/an}b∇acket∇i}ht, a2=/an}b∇acketle{tsu,k|ˆU2|k,su/an}b∇acket∇i}ht,(22a) b1=/an}b∇acketle{t−su,k|ˆU1|k,−su/an}b∇acket∇i}ht,b2=/an}b∇acketle{t−su,k|ˆU2|k,−su/an}b∇acket∇i}ht,(22b) c1=/an}b∇acketle{tsu,k|ˆU1|k,−su/an}b∇acket∇i}ht, c2=/an}b∇acketle{tsu,k|ˆU2|k,−su/an}b∇acket∇i}ht,(22c) d1=/an}b∇acketle{tsu,k|ˆU1|−k,su/an}b∇acket∇i}ht, d2=/an}b∇acketle{tsu,k|ˆU2|−k,su/an}b∇acket∇i}ht,(22d) α=/an}b∇acketle{tk|δV(r)|k/an}b∇acket∇i}ht, β=/an}b∇acketle{tk|δV(r)|−k/an}b∇acket∇i}ht.(22e) Taking into account the properties of the electrostatic potential φ(r) and the terms ˆU1andˆU2leads to further simplification of the above result. The inversion invari- anceˆIˆU1ˆI+=ˆU1requires that a1=b1,c1= 0 and that d1is a real number. The condition that the term ˆU2 must be odd ˆIˆU2ˆI+=−ˆU2with respect to space inver- sion invariance, requires a2=−b2andd2to be purely imaginary, while it does not place any restrictions on c2. The fact that the electrostatic field φ(r) is odd with re- spect to inversion invariance, ˆIφ(r)ˆI+=−φ(r), requires α=−αand thus α= 0, while βneeds to be purelyimaginary. Taking into account these simplifications the first order corrections to the energy take the form E(1)± k=a1±/radicalBig a2 2+|c2|2+|d1+d2+β|2.(23) Eq. (23) shows that the spin-orbit coupling term ˆU1 possessing space inversion invariance leads to a shift of the first-order energy levels by the amount a1= /an}b∇acketle{tsu,k|ˆU1|k,su/an}b∇acket∇i}ht. A. Perturbation with Translational Invariance The expression (23) shows that the four-fold degen- eracy of E0 kcan be lifted up to two twofold degenerate levels in first-order perturbation method. The spin split- ting caused by the spin-orbit coupling term ˆU2, which lacks inversion invariance, is embodied in the matrix ele- mentsa2,c2andb2withinthisapproach. Thepresenceof the matrix element βin the energy correction (23) leaves open the optionthat the electrostaticfield leadstoan ad- ditionalcontributiontothespinsplitting. However,since the condition ˆIφ(r)ˆI+=−φ(r) requires that either βis purely imaginary or 0, the consideration of time-reversal invariance and broken space inversion invariance within perturbation method does not offer conclusive evidence thatβ/ne}ationslash= 0. We have supposed that the nonperturbed and per- turbed electrostatic potentials possess discrete transla- tional invariance, ˆTRnφ(r)ˆT+ Rn=φ(r). Consequently, the terms ˆU1andˆU2also possess discrete translational invariance, ˆTRnˆU1ˆT+ Rn=ˆU1, andˆTRnˆU2ˆT+ Rn=ˆU2 respectively. Using this and the Bloch theorem for d2=/an}b∇acketle{tsu,k|ˆU2|−k,su/an}b∇acket∇i}htwe find /an}b∇acketle{tsu,k|ˆU2|−k,su/an}b∇acket∇i}ht=/an}b∇acketle{tsu,k|ˆT+ RnˆU2ˆTRn|−k,su/an}b∇acket∇i}ht= =e2ik·R/an}b∇acketle{tsu,k|ˆU2|−k,su/an}b∇acket∇i}ht,(24) which for arbitrary kand translation vector Rcan be satisfied only if d2= 0. Similar reasoning shows that d1= 0 and β= 0. Thus the first-order energy correction (23) takes the form E(1)± k=a1±/radicalBig a2 2+|c2|2. (25) Therefore it is not possible to have lifting of spin degen- eracy of Bloch states purely by translationally invariant electrostatic field, which is odd with respect to space in- version invariance, but the reason for this is the discrete translational invariance of the perturbing electrostatic fieldφ(r). This is interesting since in the usual sym- metry argument translational invariance is not explicitly considered, nor any attention is devoted to it. Further- more,allenergycorrectionsofhigherorderdisappeardue to translational invariance since all /an}b∇acketle{tsu,k|ˆU|k′,su/an}b∇acket∇i}ht= 0. This makes Eq. (25) exact to all orders in the perturba- tion method treatment.6 B. Perturbation without Translational Invariance Natural continuation of the above line of thought is to consider perturbing electrostatic field φ′(r) with corre- sponding energy V′(r) =−eφ′(r) not possessing a center of inversion, ˆIV′(r)ˆI+/ne}ationslash=V′(r), andlacking translational invariance ˆTRnV′(r)ˆT+ Rn/ne}ationslash=V′(r), (26) to which we will refer as external. Its correspondingfirst- order matrix element which is not necessarily zero due to time-reversal invariance is β′=/an}b∇acketle{tsu,k|ˆV′| −k,su/an}b∇acket∇i}ht= /an}b∇acketle{t−su,k|ˆV′| −k,−su/an}b∇acket∇i}ht. It is defined between zeroth- order states with spin, but since the electrostatic field does not contain operator acting directly on the spin- degree of freedom it can be written shorthanded as β′=/an}b∇acketle{tk|ˆV′|−k/an}b∇acket∇i}ht. Because V′(r) is not invariant with re- spect to space inversion, the disappearance of the matrix elementβ′is not guaranteed. Examples of electrostatic fields possessing these properties are the built-in poten- tial in p-n junctions and externally applied electric fields. These are usually orders of magnitude smaller than the bulk electrostatic fields. By using the Bloch theorem and the inequality (26) we obtain the inequality /an}b∇acketle{tsu,k|V′(r)|−k,su/an}b∇acket∇i}ht /ne}ationslash=e2ik·R/an}b∇acketle{tsu,k|V′(r)|−k,su/an}b∇acket∇i}ht, (27) for the matrix element β′. Therefore for any given kand arbitrary translation vector Rwe must have β′/ne}ationslash=e2ik·Rβ′. This condition excludes the possibility the matrix element to be equal to zero and thus β′/ne}ationslash= 0. In calculating the first-order corrections to the en- ergy we neglect the matrix elements a′,b′and c′stemming from the spin-orbit coupling term ¯h 4m2c2ˆσ·(∇V′(r)׈p)sincetheyshouldbemuchsmaller thanβ′=/an}b∇acketle{tsu,k|ˆV′|−k,su/an}b∇acket∇i}htbecause of the prefactor ¯h 4m2c2. Since β′/ne}ationslash= 0 it will appear in the expressions for the energy splitting. For the triclinic pedial system in which the bulk potential does not possess space in- version invariance, the first-order corrections to the en- ergy,E(1)± k=a1±/radicalbig a2 2+|c2|2+|β′|2, contain contribu- tions from the spin-orbit coupling a2andc2and a contri- bution purely from electrostatic field in β′. The original fourfold degeneracy is lifted to two distinct two-fold de- generate levels. The energy splitting between them is given by ∆E(1) k= 2/radicalBig a2 2+|c2|2+|β′|2. (28) For the triclinic pinacoidal system where the bulk elec- trostatic potential possesses space inversion invariance, a2andc2are identically zero. The expression for the first-order energy corrections takes the form E(1)± k= a1±|β′|. The energy splitting, ∆E(1) k= 2|β′|, (29)between the two twofold degenerate levels has contribu- tions only directly from the electrostatic field through β′. C. The lifted degeneracy A heuristic argument supporting the interpretation of the energy splittings (28) and (29) as spin splitting is the usual interpretation of the energy splitting due to spin-orbit coupling in systems lacking space inversion invariance8,11,12,19as spin splitting. This is embodied in the matrix elements a2andc2in Eq. (28). The addi- tional effect due to electrostatic field with broken space inversion and translational invariance is embodied in β′ in Eq. (28). Of course because of broken translational in- variance of V′(r) the perturbed states |Ψ/an}b∇acket∇i}htare no longer Bloch states. The energy splittings (28) and (29) can be interpreted as pure spin-splitting for the zero-th order Bloch states. This is easily demonstrated using the invariances of the subspaces ε(0) k±as shown in the following paragraphs. Since the fourfold degeneracy of E(0) kis not completely removed,eachofthefirst-ordercorrectionstoenergy E(1) k± corresponds21to a two-dimensional subspace ε(0) k±of the four-dimensional subspace ε(0) kof the unperturbed prob- lem (9). The subspaces ε(0) k±are mutually exclusive and their direct sum ε(0) k++ε(0) k−=ε(0) kis the four-dimensional subspace ε(0) k. The zeroth-order state |0k/an}b∇acket∇i}ht, which is the projection of the perturbed state |Ψ/an}b∇acket∇i}htonto the subspace ε(0) k, cannot be determined uniquely, only its belonging tooneofthe subspaces ε(0) k±canbe inferredfromEq.(20). Symmetry arguments suggest that Kramers degener- acy is preserved and therefore the remaining twofold de- generacy in first order is the Kramers degeneracy. In or- der to show this we suppose the equation for first-order energy correction to be solved and consider it as identity ˆP(0) kV′(r)ˆP(0) k|0±/an}b∇acket∇i}ht ≡E(1) k±|0±/an}b∇acket∇i}ht, (30) where the states |0±/an}b∇acket∇i}htare arbitrarily chosen zeroth-order states belonging to the subspaces ε(0) k±, respectively, and therefore to ε(0) k. Applying the time reversal opera- tor to Eq.(30) and using that [ ˆK,ˆP(0) k] = 0, because ε0 kis invariant with respect to ˆK, [ˆK,V′(r)] = 0 by hypothesis and E(1) k±is real, we obtain the identity ˆP(0) kV′(r)ˆP(0) kˆK|0±/an}b∇acket∇i}ht ≡E(1) k±ˆK|0±/an}b∇acket∇i}ht. TheˆK-transformed stateˆK|0±/an}b∇acket∇i}htis orthogonal21to|0±/an}b∇acket∇i}htsince we consider a single-electronsystemtakingintoaccountthespindegree of freedom . As shown above, ˆK|0±/an}b∇acket∇i}htalso identically sat- isfies Eq. (30) with the eigenvalues E(1) k±and therefore be- long to the subspaces ε(0) k±, respectively. Since we choose |0±/an}b∇acket∇i}htarbitrarily, and by the above argument every ˆK|0±/an}b∇acket∇i}ht7 belongs to ε(0) k±, the subspaces ε(0) k±are invariant with respect to the time-reversal operator ˆK,ˆKε(0) k±=ε(0) k±. Thus the remaining two-fold degeneracy in first order is precisely the Kramers degeneracy. The two-dimensional subspaces ε(0) k±are, however, not invariantwith respect to the space-inversionoperatorbe- cause the perturbation V′(r) does not commute with it, [ˆI,V′(r)]/ne}ationslash= 0. This is provedby againapplying the time- reversal operator ˆIto the identity (30) satisfied by arbi- trary state |0±/an}b∇acket∇i}ht ∈ε(0) k±. However, since [ ˆI,V′(r)]/ne}ationslash= 0 and by using, ˆI−1ˆI= 1, Eq.(30) transforms into the identity, ˆP(0) kˆIV′(r)ˆI−1ˆP(0) k|0±/an}b∇acket∇i}ht=E(1) k±ˆI|0±/an}b∇acket∇i}htfor theˆI- transformed state ˆI|0±/an}b∇acket∇i}ht. This shows that ˆI|0±/an}b∇acket∇i}htdoes not satisfy Eq.(30) with eigenvalue E(1) k±but a different equation with the eigenvalue E(1) k±becauseV′/ne}ationslash=ˆIV′ˆI−1. Therefore the ˆI-transformed states ˆI|0±/an}b∇acket∇i}htdo not belong to the subspaces ε(0) k±. The choice of |0±/an}b∇acket∇i}htis arbitrary apart from the condition |0±/an}b∇acket∇i}ht ∈ε(0) k±and therefore for every state |0±/an}b∇acket∇i}htbelonging to ε(0) k±theˆI-transformed stateˆI|0±/an}b∇acket∇i}htdoes not belong toε(0) k±. However ,the four- dimensional subspace ε(0) k=ε(0) k++ε(0) k−is invariant with respect to ˆI,ˆIε(0) k=ε(0) kand therefore ˆI|0±/an}b∇acket∇i}htmust be- long toε(0) k. Since ˆI|0±/an}b∇acket∇i}htdoes not belong to ε(0) k±the only remaining option is that it belongs to the other two-dimensional subspace ε(0) k∓. Thus the space inver- sion operator ˆImaps every state |0+/an}b∇acket∇i}ht ∈ε(0) k+to a state ˆI|0+/an}b∇acket∇i}ht ∈ε(0) k−belonging to the other two-dimensional subspace ε(0) k−and vice versa, symbolically written as ˆIε(0) k±=ε(0) k∓. In other words the zeroth-order states |0±/an}b∇acket∇i}ht andˆI|0±/an}b∇acket∇i}htbelong to the two different first-order energy corrections E(1) k±separated from each other by the energy difference ∆ E(1) k. Now consider any arbitrary zeroth-order state with some spin polarization belonging to the subspace ε(0) k+ which we denote as |0+/an}b∇acket∇i}htand apply the spin-flip operator19ˆCto obtain ˆC|0+/an}b∇acket∇i}ht=ˆIˆK|0+/an}b∇acket∇i}ht=ˆI|0+′/an}b∇acket∇i}ht. From the previous results we know that |0+′/an}b∇acket∇i}ht=ˆK|0+/an}b∇acket∇i}htalso be- longs to the subspace ε(0) k+, while the ˆI-transformed state ˆC|0+/an}b∇acket∇i}ht=ˆI|0+′/an}b∇acket∇i}htbelongstotheothertwo-dimensionalsub- spaceε(0) k−. Therefore the spin-flip operator ˆC, Eq. (8), maps any zeroth-order state |0k/an}b∇acket∇i}ht ∈ε(0) k±to a state be- longing to the other two-dimensional set |0k/an}b∇acket∇i}ht ∈ε(0) k∓sim- ilarly to the space inversion operator ˆI. By definition a zeroth-order state corresponds to the subspace ε(0) k±if it satisfies Eq.(20) with the corresponding eigenvalue E(1) k±. So the zeroth-order states |0k/an}b∇acket∇i}ht ∈ε(0) k±andˆC|0k/an}b∇acket∇i}ht ∈ε(0) k∓ characterizedby identical quantum numbers but describ- ing opposite spin orientations belong to the two differentfirst-order energy corrections E(1) k±. This constitutes the proof. A question might arise as to why should a matrix el- ementβ′which is diagonal in spin indices be thought of as a spin splitting even just for the zero-th order states. A perspective to understand the issue is that indeed if the secular problem to be solved reduces to 2 ×2 matrix then matrix element diagonal in spin indices can not be interpreted as spin splitting. However we consider a 4 ×4 matrix and the matrix element β′while diagonal in spin indices is not diagonal in the 4 ×4 matrix. They are on the third diagonal and they lead to lifting of the origi- nal four-fold degeneracy into two two-fold degeneracies. Above we have presented detailed argument for the in- terpretation of these as pure spin splitting for the zeroth order Bloch states. However the perturbed states |Ψ/an}b∇acket∇i}htare no longer Bloch states because the perturbation V′(r) breaks the space translation invariance. As a consequence the crystal wavevector kis no longer a good quantum number for theperturbedstatesandinhigherordersintheperturba- tion method the lifted degeneracy can not be interpreted purely as spin degeneracy. Instead it can be interpreted as lifting of some sort of orbital-spin degeneracy similar tothefinestructuresplittingduetospin-orbitcouplingin atomic systems. This is within the perturbation method treatment. VI. DISCUSSION The primary goal of the study has been to explore the hypothesis whether an electrostatic field can lift the spin degeneracyof Bloch states. Integral new contributions of this have been the novel point for introduction of spin- orbit coupling presented in Sec. IV and the null result of Sec. VA, which have naturally shaped the presenta- tion. The hypothesis presented in mathematical detail in Sec. III and discussed in subsection IIID has been in- spired by a symmetry argument which to the best of our knowledge is due to Kittel19. The original argument has been used to predict the lifting of spin degeneracy in the presence of spin-orbit coupling. In subsection IIID we noted that it is valid also for a model of electron moving in an electrostatic field in which we take spin degree into account but neglect spin-orbit coupling. Such a model is easily justified on the ground that in nature there is no electron without spin. The trivial consequence of such a model would be dou- ble degeneracy of all levels due to the spin-degree of free- dom. In such a model, where spin-orbit coupling is ne- glected, SU(2) symmetryis preservedasnotedin Sec. IV. Thus the double spin degeneracy of such a model can be viewed also as a consequence of SU(2) symmetry. Of course when spin-orbit coupling is taken into account, SU(2) symmetry is broken21, and it is not possible to introduce spin quantum number as quantum number of type constant of motion. However we treat spin-orbit8 coupling terms as perturbations and in the zeroth-order model the spin can be introduced as a separate quantum number. The original symmetry argument due to Kittel19does not use the irreducible representations of the point groups. We considered formulating the symmetry argu- ment using the irreducible representations of the point groups, but for the particular purpose we concluded that working with the symmetry operators themselves is suffi- cient. This is so, because we consider a conceptual ques- tion, thereforewetestthehypothesisonthesimplestpos- sible systems which posses the required characteristics of the problem. These are3Dcrystalsofthe triclinic crystal system: the triclinic pedial and triclinic pinacoidal crys- tal classes. These are the least symmetric classes, which haveincommononlyarotationby2 πandatime-reversal symmetry. In addition the pedial class is characterized with a broken space inversion symmetry, while the only other symmetry element of the pinacoidal class is the space inversion. The compatibility relations of the irreducible repre- sentations can be used to calculate the effects, including lifting of degeneracy, by electric fields which do not break the translational invariance. However, this approach would not offer an answer to the hypothesis whether an electrostatic field alone would break the spin degeneracy. This is so because based solely on the symmetry argu- ment we cannot say whether the spin splitting is caused by the electrostatic field alone or by the spin-orbit cou- pling. This is the reasonto use the perturbation method. Indeed, when we use the perturbation method it turns outthat ifthe translationalinvarianceoftheelectrostatic field is preserved no spin-splitting occurs. On the other hand the perturbation method treatment indicates a lift- ing of the spin degeneracy when both the translation and inversion invariances are broken. However, if the trans- lational invariance is broken we can no longer talk about space symmetry groups and their subgroups - the pointgroups. This is so because the discussion of the space groups requires the translational invariance of the crys- tal lattice. VII. CONCLUSIONS We have scrutinized the symmetry argument based on space inversion and time-reversal invariance predicting the appearance of spin splitting in case of broken space inversion symmetry. A novel argument for the need for introduction of a term breaking SU(2) invariance like the spin-orbit coupling term has been presented. This ar- gument is different from the usual arguments based on special relativity used for the introduction of spin-orbit coupling term. We have shown that in systems possess- ing discrete translational invariance it is not possible to have spin splitting solely by electrostatic field with bro- kenspace inversionsymmetrydue to the preservedtrans- lational invariance while a spin splitting exists due to the combinationofspin-orbitcoupling and electrostaticfield. The possibility for lifting of spin degeneracy due to elec- trostatic field without the mediation of spin-orbit cou- pling has been investigated using perturbation method treatment suggesting its possible existence in systems characterized by both broken space inversion invariance and broken translational invariance, as far as the pertur- bation method is applicable. There is a possibility that the last result is a quirk of the application of the per- turbation method to the case of electrostatic field with broken translational invariance. The problem might be further clarified and the results tested theoretically by treating the same scenario in the Dirac equation. If they are confirmed the possibility needs to be tested experi- mentally, most suitable for which maybe systems of the triclinic pinacoidal symmetry system. ∗tenev@phys.uni-sofia.bg 1I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 2S. A. Wolf, A. Y. Chtchelkanova, and D. M. Treger, IBM Journal of Research and Development 50, 101 (2006). 3M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). 4G. Binasch, P. Gr¨ unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989). 5S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). 6H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and M. Johnson, Science 325, 1515 (2009), http://www.sciencemag.org/cgi/reprint/325/5947/1515 .pdf. 7J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003). 8G. Dresselhaus, Physical Review 100, 580 (1955). 9E. O. Kane, J. Phys. Chem. Solids. 1, 249 (1957).10M. H. Weiler, R. L. Aggarwal, and B. Lax, Phys. Rev. B 17, 3269 (1978). 11W. Zawadski and P. Prfeffer, Semicond. Sci. Technol. 19, R1 (2004). 12R. Winkler, Spin-Orbit Coupling Effects in Two Dimen- sional Electron and Hole Systems (Springer Verlag, 2002). 13J. Fabian, A. Matos-Abiague, C. Erther, P. Stane, I. Zutic, Acta Physica Slovaca 57(2007). 14R. Eppenga and M. F. H. Schuurmans, Phys. Rev. B 37, 10923 (1988). 15E. A. de Andrada e Silva, Phys. Rev. B 46, 1921 (1992). 16Y. A. Bychkov and E. I. Rashba, Journal of Physics C: Solid State Physics 17, 6039 (1984). 17F. J. Ohkawa and Y. Uemura, Journal of the Physical So- ciety of Japan 37, 1325 (1974). 18R. Lassnig, Phys. Rev. B 31, 8076 (1985). 19C. Kittel, Quantum Theory of Solids , 2nd ed. (John Wiley & Sons, 1987) p. 528.9 20W.-K. Tung, Group Theory in Physics (World Scientific, 2008). 21A. Messiah, Quantum Mechanics (Dover Publications, 1999). 22J. J. Sakurai, Modern Quantum Mechanics (Addison- Wesley Publishing Company, 1994).23J. J. Sakurai, Advanced Quantum Mechanics (Addison- Wesley Publishing Company, 1967). 24Bernd Thaller, The Dirac Equation (Springer-Verlag, 1992).

1310.7847v1.Self_Quenching_of_Nuclear_Spin_Dynamics_in_Central_Spin_Problem.pdf

arXiv:1310.7847v1 [cond-mat.mes-hall] 29 Oct 2013Self-Quenching of Nuclear Spin Dynamics in Central Spin Pro blem Arne Brataas1and Emmanuel I. Rashba2 1Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2Department of Physics, Harvard University, Cambridge, Mas sachusetts 02138, USA We consider, in the framework of the central spin s= 1/2 model, driven dynamics of two electrons in a double quantum dot subject to hyperfine interaction with nuclear spins and spin-orbit coupling. The nuclear subsystem dynamically evolves in response toLa ndau-Zener singlet-triplet transitions of theelectronic subsystemcontrolled byexternalgate volta ges. Withoutnoise andspin-orbit coupling, subsequent Landau-Zener transitions die out after about 104sweeps, the system self-quenches, and nuclear spins reach one of the numerous glassy dark states. W e present an analytical model that captures this phenomenon. We also account for the multi-nuc lear-specie content of the dots and numerically determine the evolution of around 107nuclear spins in up to 2 ×105Landau-Zener transitions. Without spin-orbit coupling, self-quenchin g is robust and sets in for arbitrary ratios of the nuclear spin precession times and the waiting time bet ween Landau-Zener sweeps as well as under moderate noise. In presence of spin-orbit coupling of a moderate magnitude, and when the waiting time is in resonance with the precession time of one o f the nuclear species, the dynamical evolution of nuclear polarization results in stroboscopic screening of spin-orbit coupling. However, small deviations from the resonance or strong spin-orbit co upling destroy this screening. We suggest that the success of the feedback loop technique for building nuclear gradients is based on the effect of spin-orbit coupling. PACS numbers: I. INTRODUCTION Electrical operation of electron spins in semiconduc- tor double quantum dots (DQD) is one of the cen- tral avenues of semiconductor spintronics1and quantum computing.2–5There are three basic types of electronic spin qubits, (i) the Loss-DiVincenzo2qubits operating single-electron spins, (ii) singlet-triplet qubits operating a two-electron system and (iii) three electron qubits.6 The second type is the center of our attention. Most widely explored singlet-triplet DQD qubits7are based on GaAs8,9andInAs.10–12BothinGaAsandInAs, thereare three species of nuclei possessing non-vanishing angular momenta, and the coupling between electron and nuclear spins (mostly through contact interaction) strongly in- fluences electron-spin dynamics. Primarily, this coupling has a destructive effect causing electron spin relaxation, and many theoretical studies have focused on the chal- lenging problem of determining the relaxation rate of an electron spin interacting with about N≈106nuclear spins.13–20The problem of an electron spin interacting with a bath of nuclear spins is known as the central spin problem. However, a controllable nuclear spin polarization, act- ing as an effective magnetic field, can also become a re- source for manipulating electron spins.21,22In particu- lar, the difference (gradient) in the nuclear polarization of the left and right dots can be used for σxrotations of aS-T0singlet-triplet qubit on the Bloch sphere.8Ef- ficient control of a vast ensemble of nuclear spins is very challenging, and many analytical and numerical works have been carried on this subject.23–31The principal ex- perimental tool for polarizing nuclear spins and building gradients is based on driving a two-electron DQD elec-trically through the avoided crossing ( S-T+anticrossing) of its singlet level Sand theT+component of its elec- tronic triplet T= (T+,T0,T−).T+is the lowest energy triplet component because the electron g-factor is nega- tive,g <0, both in GaAs and InAs. The width of the anticrossing is controlled by hyperfine and spin-orbit32 interactions. When the electron state changes from S toT+orvice versa by passaging through the S-T+an- ticrossing, and there is no spin-orbit coupling, up to one quantum of the angular momentum is transferred to the nuclear subsystem, and such transfer facilitates polariz- ing the nuclear bath by performing multiple passages. Unfortunately, experimental data show that the nu- clear polarization saturates at a rather low level, typi- cally of about 1%.33The origin of this low saturation level remains unclear and constitutes the critical obsta- cle for achieving higher levels of nuclear spin polariza- tion. We recently uncovered a mechanism of dynamic self-quenching which, in absence of spin-orbit (SO) cou- pling, results in fast suppression ofthe transversenuclear polarization under stationary pumping.30This is caused by screening the random field of the initial nuclear spin fluctuation by the nuclear polarization produced through pumping and closing the anticrossing. This conclusion is in a qualitative agreement with the data of Refs. 25,31 in the strong magnetic field limit, and the states with vanishing transverse magnetization are known as “dark states”. Meantime, by applying feedback loops, exper- imenters managed to achieve considerable and control- lable nuclear spin polarizations.34This poses a challeng- ing question in which way closing the S-T+intersection due to the self-quenching mechanism could be avoided. Our data of Ref. 30 indicate that SO coupling changes thepatternsofself-quenchingdramatically,whichimplies that it is the spin-orbit coupling that might resolve the2 problem. The main goal of the current paper is to shed more light on the mechanisms controlling the transfer of angularmomentumfromtheelectronqubittothenuclear bath. For this purpose, we solve numerically the equa- tions describing coupled electron and nuclear spin dy- namics for DQDs of a realistic size of more than N∼106 nuclear spins and a shape of two overlapping Gaussian distributions. These simulations follow up to 2 ×105 sweeps and unveil intimate patterns of transferring spin polarizationfromtheelectronictothenuclearsubsystem. To get quantitative insight onto the long time dynam- ics of spin pumping by multiple passagesacross the S-T+ anticrossing, we restrict ourselves to the strong magnetic field regime when the Zeeman split-off T0andT−compo- nents of the electron spin triplet are well separated from theSandT+states, hence, transitionsto these statesare disregarded. Therefore, with a semiclassical description of nuclear spins, electrons form a two-level system, and passages across the S-T+anticrossing are described by the Landau-Zenertype theory.35,36The detailed patterns depend on the shape on the pulses on the gates and the instantaneousnuclearconfiguration. Inturn, duringeach LZ sweep the nuclear configuration changes due to the direct transfer of the angular momentum and shake-up processes.29,30Between the LZ sweeps, this configuration changes because of the difference in the Larmor preces- sion rates of different nuclear species. To follow the long term evolution, we solve the problem of the coupled dy- namics self-consistently. From the mathematical point of view, we arrive to a central spin s= 1/2 problem with a driven dynamics of the electron spin. Hence, beyond the application to the spin pumping problem, our results are of general interest for coupled dynamics of many body systems. In this paper we prove, both analytically and numer- ically, that self-quenching into dark states is a generic property of the pseudospin s= 1/2 model in absence of SO coupling, and that self-quenching sets in after only about 104sweeps. We also demonstrate that this result stands under a moderate noise. However, the main fo- cus of the paper is on the effect of SO coupling. Because the SO field is static while the hyperfine Overhauserfield oscillates in time with the Larmor frequencies of nuclei, self-quenching cannot set in. Nevertheless, if the wait- ing time between LZ sweeps coincides with the Larmor period of one of the species, self-quenching sets in stro- boscopically (as was demonstrated in our previous paper for a single-specie model30). More specifically, the Over- hauser field of the resonant specie screens the SO field during the LZ sweeps (whose duration is small compared with Larmor periods). Therefore, during the sweeps the electron and nuclear subsystems become decoupled. As distinct from self-quenching in absence of SO coupling, the stroboscopic self-quenching is fragile. Even a small deviationfromtheresonance,about1%, destroysthedel- icatecompensationoftheSOandhyperfinecontributions during the LZ sweeps. Moreover, we were able to observe the stroboscopic self-quenching only for moderate valuesof the SO coupling that do not exceed considerably the random fluctuations of the hyperfine field. We conclude that it is the SO coupling that endows the nuclear subsystem with a long term dynamics under the stationary LZ pumping. Therefore, we suggest that SO coupling is critical for efficient operating the feedback loops that requireaccumulation of largepolarizationgra- dients at the scale of about 106sweeps.34Effect of SO coupling at a shorter time scale has been recently un- veiled by Neder et al.37by comparing with experimental data of Ref. 38. II. OUTLINE AND BASIC RESULTS In Sec. III, following two introductory sections, we present the basic equations of the driven coupled electron-nuclear spin dynamics of the central spin-1 /2 problem that is the basis for all following calculations. In Sec. IV, we find an analytical solution for a simple model demonstrating the phenomenon of self-quenching which reveals basic factors controlling its rate. Our numerical technique that allows following the coupled dynamics of the electron 1 /2-pseudospin and about 107nuclear spins during up to 2 ×105LZ sweeps is described in Sec. V. It also includes parameters of the double quantum dot and LZ pulses used in simulations. Sec. VI is the central one. It opens with the nuclear parameters of InAs and GaAs used in simulations, and includes the results of simulations and their discussion. In this section, we demonstrate that self-quenching is a generic and robust property of the coupled dynamics in absence of spin-orbit coupling, and analyze its specific features in systems consisting of two and three nuclear species. Next, we introduce SO coupling and demon- strate that it eliminates self-quenching and causes the nuclear subsystem to exhibit a persistent, but irregu- lar, oscillatory dynamics. We also demonstrate the phe- nomenonofstroboscopicself-quenchingthat sets in when the waiting time between LZ sweeps is in resonance with the Larmor period of one of the nuclear species and show that it is very sensitive to deviations from the exact res- onance. Finally, we suggest that the SO induced nuclear dynamicsiscriticalforthe feedbacklooptechniquedevel- oped by Bluhm et al.34for building controllable nuclear polarization gradients. We summarize our results in Sec. VII and estimate the strength of SO coupling in InAs and GaAs double quantum dots in Appendix A. III. BASIC EQUATIONS Hyperfine electron-nuclear interactions and SO cou- plinggovernthecouplingbetweenelectronstatesin A3B5 quantum dots utilized for quantum computing purposes. Nuclear spins are dynamic and can be controlled by ma- nipulating magnetic fields and electronic states.3 We consider electrons in double quantum dots inter- acting with nuclear spins via the hyperfine interaction. When there are two electrons in the dot, the orthogonal basis consists of singlet and triplet spin states. Hyperfine and SO interactionscouple these states. By changing the gate voltages that confine electrons and determine sin- glet and triplet energies, a transition from a singlet Sto a triplet electron state T+(or vice versa) is accompanied by a change in the nuclear spin states. Our focus is on what happens to the nuclear spins as we repeat Landau- Zener (LZ) transitions many times, up to 2 ×105, and how the changes in the nuclear spin states in turn affect electrons in the quantum dot. We define a LZ sweep in the following way. We assume that the quantum dot is first set in the singlet state, then a change in the gate voltages drives a (partial) transition to the triplet state, and finally one electron is taken out of the system and re-inserted so that the system again is in its singlet state. During the sweep, the dynamics of the electronic qubit is controlled by the electric field produced by the gates and the nuclear polarization as described by Eq. (5) below. In turn, semiclassical dy- namics of nuclear spins is driven by the Knight fields ∆jλ(t) arising from electron dynamics /planckover2pi1dIjλ dt=∆jλ×Ijλ, (1) where the sub index jλdenotes a nuclear specie λat a lattice site j. Assuming the time-scale TLZof the LZ sweeps is much shorter than the nuclear precession times tλin the external magnetic field, the total effect of the time-dependent fields ∆jλ(t) on each nuclear spin can be integrated over the LZ sweep. Then the change of an individual nuclear spin during a sweep is △Ijλ=Γjλ×Ijλ, (2) whereΓjλaccounts for the effective magnetic field in- duced by the hyperfine interaction during the LZ sweep and depends on the configuration of all the nuclear spins before the sweep. Landau-Zener sweeps are repeated many times. Be- tween consecutive LZ sweeps, electrons are in the singlet state and do not interact with nuclear spins. During this waiting time Twbetween consecutive sweeps, nu- clear spins precess in an external magnetic field Bap- plied along the z-direction. The changes of the nuclear spins between LZ sweeps are △Ix jλ= cosφjλIx jλ−sinφjIy jλ, (3a) △Iy jλ= cosφjλIy jλ+sinφjλIx jλ, (3b) △Iz jλ= 0, (3c) where the superscripts x,y, andzdenote Cartesian com- ponentsofthenuclearspins,thetransversephasechanges areφjλ=−2πTw/tλin terms of the spin precession timestλ= 2π/planckover2pi1/gλµIB, wheregλ=µλ/Iλis theg-factor for a nuclear specie λ,µλis its magnetic moment, and µI= 3.15×10−8eV/T is the nuclear magneton.We also model the influence of noise by adding phe- nomenologically a random magnetic field along the z- direction for each nuclear spin so that the accumu- lated phases in Eq. 3 change to φeff jλ=φjλ+φnoise jλ→ −2πTw(1/tλ+rjλ/τ), whererjλare random numbers in the interval from −1 to 1. This procedure simulates a randomization of the transverse components of nuclear spinsafteratime ofthe order τ, andτistermedthe noise correlation time in what follows. While simulations de- scribed below were performed by using random sets of rjλ, we mention that averaging over the noise results ef- fectivelyinchangingthephase-dependentfactorsinEq.3 as∝an}bracketle{tcosφeff jλ∝an}bracketri}htn= cosφjλsin(2πTw/τ)/(2πTw/τ), and sim- ilarly for ∝an}bracketle{tsinφjλ∝an}bracketri}htn. Therefore, this model of transverse noise leads to a semiclassical dephasing of the transverse components of nuclear spins on the time scale τ. Let us next review how electronic Landau-Zener sweeps influence nuclear spins via Γjλ.29The hyperfine electron-nuclear interaction is Hhf=Vs/summationdisplay λAλ/summationdisplay j∈λ/summationdisplay m=1,2δ(Rjλ−rm)(Ijλ·s(m)), (4) wheres(m) =σ(m)/2 are the electron-spin operators in terms of the vector of Pauli matrices σ(m) for each electronm= (1,2),Ijλare the nuclear spin operators, Aλis the electron-nuclear coupling constant for a specie λ, andVsis the volume per single nuclear spin. We consider GaAs or InAs quantum dots below; hyperfine coupling parameters for them can be found in Sec. VI. Assuming that gate voltages keep the system close to the singlet S- tripletT+transition, the effective Hamil- tonian describing the electron qubit is H(ST+)=/parenleftbigg ǫSv+ v−ǫT+−η/parenrightbigg , (5) whereǫSis the singlet energy and ǫT+is the triplet T+ energy in the external magnetic field B=Bˆzwhen nu- clear spins are unpolarized. By retaining only SandT+ states, the problem is reduced to a 1/2 pseudospin prob- lem, and we apply the term the central spin problem in this sense. The energies ǫSandǫT+are controlled by the gate voltages. The off-diagonal components v±=v± n+v± SO, coupling the singlet Sand triplet T+states, contain con- tributions from nuclear spins v± n=Vs/summationdisplay λAλ/summationdisplay j∈λρjλI± jλ, I± jλ= (Ix jλ±iIy jλ)/√ 2,(6) and SO coupling v± SO.26,39When nuclear spins are po- larized, the energy of the triplet state is affected by the Overhauser shift η=−Vs/summationdisplay λAλ/summationdisplay j∈λζjλIz jλ. (7) The singlet-triplet electron-nuclear couplings are ρjλ=/integraldisplay drψ∗ S(r,Rjλ)ψT(r,Rjλ), (8)4 and the electron-nuclear couplings in the T+state are ζjλ=/integraldisplay dr/vextendsingle/vextendsingleψT(r,Rjλ)2/vextendsingle/vextendsingle, (9) whereψS(ψT) is the orbital part of the singlet (triplet) wavefunction. Beyondthe2 ×2S-T+model, itispossible to define a hyperfine term that determines the singlet S- tripletT0coupling(asinEq.4in Ref. 29), but it isnot at the centerof ourattention and will not be discussed here. The effect of the electron spin T0andT−components is critical for the development of nuclear polarization gra- dients and has been investigated in Refs. 25,31. In terms of these parameters, the changes of the nu- clear spins △Ijλ=Γjλ×Ijλduring a Landau-Zener sweep are determined by coefficients Γ(x) jλ=−VsAλρjλ(Pvy+Qvx)/(2v2),(10a) Γ(y) jλ=VsAλρjλ(Pvx−Qvy)/(2v2), (10b) Γz jλ=VsAλζjλR/(2v), (10c) withv2=|v+|2=/parenleftbig v2 x+v2 y/parenrightbig /2. In these expressions, 0≤P≤1 is theS-T+transition probability, a real numberQis the shake-up parameter defined via29 P+iQ=−i2v−/integraldisplayTLZ −TLZdt /planckover2pi1cS(t)c∗ T+(t) (11) in terms of the singlet (triplet) amplitude cS(t) (cT(t)), and R= 2vTLZ/integraldisplay −TLZdt/vextendsingle/vextendsinglecT+(t)/vextendsingle/vextendsingle2//planckover2pi1 (12) accounts for the Overhauser shift due to the triplet T+component of the electron state during the interval (−TLZ,TLZ). In the absence of SO coupling, v± SO= 0, the change ∆ Izof the total angular momentum of nuclei Iz=/summationtext jIz jduring a single sweep equals ∆ Iz=−P, as follows from the angular momentum conservation.29 Using the amplitudes ( cS(t),cT+(t)) found from solv- ing the time-dependent Schr¨ odinger equation with the Hamiltonian H(ST+)of Eq. (5) in combination with the dynamical equations for nuclear spins of Eq. (2) makes our approach completely self-consistent. We assume that electrons are loaded into the singlet (0,2) state with energies far away from the S-T+an- ticrossing. After loading electrons, gate voltages are changed to bring the system closer to the level anticross- ing, and this change is performed fast enough to keep the system in the singlet state.40From there on, an LZ sweep brings the system through the anticrossing. After the slow sweep, the system is moved back to the recharg- ing point where it exchanges electrons with the reservoir. This back motion is fast at the scale of the narrow anti- crossingand therefore does not influence the nuclear spin subsystem, but slow at the scale of the electron Zeeman splitting to keep the system inside the S−T+subspace.IV. SIMPLE MODEL WITH ANALYTICAL SOLUTION Let us first present a simplified model that can be solved analytically and that manifests basic features of self-quenching30in the absence of SO coupling, v± SO= 0. Different versions of this “box” or “giant spin” model were applied to various problems, see Refs. 25,26,41,42. Subsequently, we will in Section V outline a more com- plex and extensive numerical procedure and discuss nu- merical results for realistic models in Section VI. Re- stricting ourselves to a single nuclear specie, we sim- plify the system by modelling it as a box inside which the electron wave functions are independent on posi- tion, all nuclei possess the same Larmor frequency, and all hyperfine coupling constants Aλand electron-nuclei couplignsρjλare equal, Aλ=¯Aandρjλ= ¯ρ; typi- cally,¯A≈10−4eV. Then Eqs. (6) and (8) simplify to vα=A0Iα,α= (+,−,z) withA0=Vs¯A¯ρ∼¯A/N. Here Nis the number of nuclei in the box, and Iα=/summationtext jIα j are components of the “giant” collective angular momen- tum of nuclei. In the framework of this model, nuclear spin precession in the Zeeman and Overhauserfields does not influence the coupled electron-nuclearspin dynamics, ∆z= 0. With these assumptions, Eq. (1) becomes dI+ dt=−i /planckover2pi1∆+Iz,dI− dt=i /planckover2pi1∆−Iz,dIz dt=i /planckover2pi1(∆+I−−∆−I+). (13) ForLZ pulses, the energylevel difference changeslinearly witht,ǫS(t)−ǫT+(t) =β2t//planckover2pi1, for−TLZ≤t≤TLZ, and thedynamicsofthequbit iscontrolledbyadimensionless parameterγ=v+v−/β2. The equation of motion for γ following from (13) is dγ dt=−A2 0 2β2d dt(Iz)2. (14) During each sweep, Izchanges by ∆ Iz=−Pandγ changes by ∆ γ= (A2 0/β2)PIz. Precession of the collec- tive nuclear spin Iin the external magnetic field during the waiting times between sweeps changes neither Iznor γand is disregarded. The discrete number of sweeps ncan be considered as a continous variable since the changes ∆γand ∆Izduring a single sweep are small as compared to γandIz. We then arrive at the differential equations that determine the evolution of the γ(n) and Iz(n): dγ dn=A2 0P β2Iz, (15a) dIz dn=−P. (15b) This central result for the simple model clarifies the dif- ferent modes of self-quenching. The evolution of the LZ parameterγthat controls the LZ probability Pdiffers in two scenarios that manifest themselves for opposite signs ofIz. (i) When Izis initially negative, it continues5 to decrease (becoming more negative) and magnitudes of bothγand the LZ probability Pdecrease, hence, the process slows down. Finally, self-quenching sets in ex- ponentially, see Eq. 19 below. (ii) When Izis initially positive,γfirst increases so that the LZ probability P becomes larger. However, since Izonly can be reduced, it eventually becomes negative and self-quenching of sce- nario (i) sets in. So, self-quenching ultimately sets in generically independent on the original sign of Iz. We can get a more detailed insight into the self- quenchingdynamicsbyusingthe firstintegralofEq.1443 γ=−(A2 0/2β2)(Iz)2+γ0, (16) whereγ0is an integration constant that depends on the initial values of γandIz,γiandIz i. Obviously, Eq. 16 dictates that γ≤γ0, andγ0≥0 becauseγ≥0 by definition. Therefore, Iz=±√ 2(β/A0)√γ0−γ, (17) and dγ/dn=±√ 2(A0/β)√γ0−γP(γ).(18) In scenario (i), when Iz i<0, the minus sign should be chosen in Eqs. 17 and 18, and γ(n) decreases monoton- ically. In scenario (ii), when Iz i>0, the dynamics first follows the plus branches of Eqs. 17 and 18, and γ(n) increases until it reaches its maximum value γ=γ0. At this point, Iz(n) vanishes, changes sign, and continues to decrease as follows from Eq. 16 and Eq. 15b [because P(γ0)>0]. At the same point, the signs in Eqs. 17 and 18 switch from plus to minus, and afterwards γ(n) decreases monotonically as follows from Eq. 15a. The detailed asymptotic behavior of γ(n) forn→ ∞ depends on P(γ). For long LZ sweeps with 2 TLZ≫/planckover2pi1/v, PLZ(γ) = 1−e−2πγ≈2πγ, and γ(n)∝exp[−2π/radicalbig 2γ0(A0/β)n]. (19) Equation(19) describesanexponentialdecaywith anon- universal exponent. The rate of decay increases with de- creasingβ, when sweeps become more adiabatic. There- fore, in absence of SO coupling the large- nbehavior of γ(n) is exponential, and self-quenching sets on for arbi- trary initial conditions. Let us make a rough estimate of the number of sweeps n∞before self-quenching sets in based on Eq. 19. A typical original fluctuation includes N1/2spins, hence, v∼A0√ N. For LZ pulses with an amplitude of about v and duration of about /planckover2pi1/v,βis aboutβ∼v. Therefore, n∞∼β/A0∼√ N, i.e., about the number of nuclear spins in a typical fluctuation. The dependence of n∞ onβdemonstrates the effect of the sweep duration TLZ, n∞is smaller for longer sweeps. A similar estimate for the length ∆ nof the exponential tail in Eq. (19), with 2π√2γ0≈10, results in ∆ n∼n∞/10, i.e., it is shorter thann∞by a numerical factor.More detailed estimates for both regimes require spe- cificassumptionsabouttheshapeanddurationofsweeps. For sufficiently long sweeps, PLZ(γ) can be used for P(γ) and Eq. (18) can be integrated. The number of sweeps n=n(γi,γf), inunitsof β/(√ 2A0), betweentheinitial γi and finalγfvalues ofγis plotted in Fig. 1 for two modes; the value of γ0has been chosen equal to γ0= 2. Fig. 1(a) is plotted for Iz i<0 and Fig. 1(b) for Iz i>0. Front sec- tions ofn(γi,γf) surfaces by γf= 0 planes demonstrate n∞(γi), the number of sweeps before self-quenching. For Iz i<0, the curve increases fast with γiand reaches its maximum value at γ=γ0. It is achieved at a ridge at then(γi,γf) surface that originates from the square-root singularity in the dn/dγdependence and is well seen in Fig. 1(a). For Iz i>0, then∞(γi) dependence is much slower and becomes fast only near γ=γ0. In both cases, n∞∼β/A0, in agreement with the previous estimate. Therefore,themodelnotonlyprovidesanalyticaljusti- fication of the self-quenching phenomenon found numer- ically in Ref. 30 for systems without SO coupling but relates, for single-specie systems, two modes of behavior (monotonicandnonmonotonic)tothedifferenceininitial conditions. It is the first analytical solution of the cen- tral spin problem (i) describing dynamical evolution of a pumped system into a “dark state”25,44and (ii) estab- lishing a connection between the initial and final states of the system. V. NUMERICAL PROCEDURE During a sweep, the difference in the singlet and triplet energiesǫS−ǫT+varies linearly in time within the sweep- ing interval −TLZ≤t≤TLZ. We impose no restrictions ontherelativemagnitudeofthesweepduration2 TLZand the inverse S-T+coupling /planckover2pi1/v, but, as stated above, TLZ is long as compared to the inverse electron Zeeman en- ergy. Furthermore, it is assumed that the variationof the energies of both the upper and lower spectrum branches is symmetric with respect to the S-T+anti-crossing for thefirsttransition when the initial position of the T+ level isη=ηi. We denote the amplitude of the change in the energy difference between the singlet and triplet energies as ǫmax. In other words, we use ǫS(t) =ǫmaxt/2TLZ (20a) ǫT+(t)−η=−ǫmaxt/2TLZ−(η−ηi) (20b) in Eq. (5).45Note that as a result of the dynamical nu- clear polarization, LZ sweeps become asymmetric with respect to the anticrossing point because of the changing Overhausershift η. There is no longer any traditional LZ passagewhenever |η−ηi|>ǫmax, i.e. after the anticross- ing point passes across one of the ends of the sweeping interval. This naturally implies a slowdown in accumu- lating dynamical nuclear polarization. Maintaining the LZ passages requires additional feedback mechanisms by changing the energy level difference, which we introduce below by shifting the edges of the integration interval.6 Figure 1: Number of sweeps nbetween the initial and final values of the Landau-Zener parameter γfor two modes; nin unitsofβ/(√ 2A0). (a)Initial nuclear polarization is negative, Iz i<0. (b) Initial nuclear polarization is positive, Iz i>0. In theplots, thelower boundsof γiandγfwere chosen tobe 0.01 to cut off logarithmic singularities in n(γ) developing because ofthePLZ(γ)factor inEq.(18). Curves n∞(γi)infrontpanels show the number of sweeps before the self-quenching sets in. See text for details. Assuming |η−ηi| ≤ǫmax, theSandT+states are degenerate at t∗=−TLZ(η−ηi)/ǫmax. To avoid trivial quenching due to the shift in ηcaused by the accumu- lating polarization far away from the degeneracy point, the electronic energies were renormalized after every 100 sweeps keeping η−ηi≈0 and ensuring the S-T+anti- crossing be passed during all LZ sweeps, −TLZ< t∗< TLZ. As a result, the center of the sweep was perma- nently kept close to the anticrossingpoint. Such aregime can be achieved experimentally by applying appropriate feedback loops. In order to relate the properties of the sweeps to the conventional notations of the LZ transition proba- bilities in the limit TLZ→ ∞, it is helpful to intro- duce the dimensionless initial τi=−TLZ[1+t∗/TLZ]β//planckover2pi1 and finalτf=TLZ[1−t∗/TLZ]β//planckover2pi1times, where β= (ǫmax/planckover2pi1/TLZ)1/2. The Landau-Zener parameter is γ= (v/β)2. When −τi≫√γandτf≫√γ, the transition probability converges towards the Landau-Zener result PLZ= 1−exp(−2πγ). We consider a simple model for the electron wavefunc- tions. The orbital part of the singlet wave function is ψS(1,2) = cosνψR(1)ψR(2) +sinν[ψL(1)ψR(2)+ψL(2)ψR(1)]/√ 2 (21)and the triplet part is ψT(1,2) = [ψL(1)ψR(2)−ψ(2)ψR(1)]/√ 2,(22) whereψL(ψR) denotes the wave function in the left (right) dot. The angle νdepends on the electron Zeeman energy. We assume the electrons are in the lowest orbital harmonic oscillator state so that the wave functions are ψ(x,y,z) =exp/bracketleftbig −(x2+y2)/l2−z2/w2/bracketrightbig /radicalbig wl2(π/2)3/2,(23) wherelis the lateral size of each dot and wis its height. For two dots that are separated by a distance dwe form an orthonormal basis set based on the functions ψ(x− d/2,y,z) andψ(x+d/2,y,z), that defines the above ψL andψR. While both dots are chosen of the same size, hyperfine couplings in them differ due to the dependence ofρjλof Eq. 8 on the mixing angle ν. We solve the nuclear dynamics numerically by using Mathematica 9. First, we include all nuclear spins that are in the vicinity of the double quantum dot and sat- isfy the condition that the electron-nuclear coupling con- stantsζjλ≥κMax{ζjλ}, whereκis a small parameter. We checked, by changing κ, that our results converged and have found that reducing κbelowκ= 0.01 does not produce any visible changes in the plots we present. Ini- tial configurations of the nuclear spin directions are cho- sen by a pseudo-random number generator. The initial nuclear spin configuration determines the 2 ×2 electron S-T+Hamiltonian. We solve the time-dependent 2 ×2 differentialequationnumericallyforlinearLZsweepsand compute the probability P, the shake-up parameter Q, and the time-integrated effect of the Overhauser shift of the triplet state T+described by the parameter R. We then let the nuclear spins precess in the external mag- netic field and a random noise field before the next LZ sweep takes place. We record all electron singlet-triplet coupling parameters as a function of the sweep number, as well asP,Q, and the change in the total magnetiza- tion. We choose realistic parameters for a double quantum dot of a height w= 3˚A, sizel= 50˚A, and distance d= 100˚A. We consider an external magnetic field of B=10 mT. Using a cut-off κ= 0.01 implies that we explicitly include in our calculations around ten millions spins. A single such calculation takes about one week on our state-of-the-art workstation. We have studied the evolution of the nuclear spin dynamics durin up to 2 × 105LZ sweeps for 107spins and used various pseudo- random initial configurations of nuclear spins. While the detailed pattern of the dynamics depends on the initial conditions, all basic regularities were exactly the same in all simulations. Hence, our results are representative for the generic behavior of a pumped electron-nuclear system.7 VI. DYNAMICAL NUCLEAR POLARIZATION We are now ready to discuss numerical results for dy- namical polarization of nuclear spins. In all our simu- lations we consider double dots of the size w= 3 nm, l= 50 nm, and d= 100 nm. GaAs (InAs) has 8 nuclear spins per cubic unit cell so that the effective volume per site is Vs=a3/8, where the lattice constant is a= 5.65˚A (a= 6.06˚A). When all nuclear spins are fully polarized in GaAs (InAs), the Overhauser field seen by the electrons is 5 .3 T (0.86 T). We accept the following values of electron g-factors, gGaAs=−0.44 (gInAs=−8). The other parameters re- flecting the abundance, nuclear g-factors, and hyperfine coupling constants are listed in Table I for GaAs and Ta- ble II for InAs. From these values, it can be understood that in our simulations GaAs behaves as a three-specie system, whereas InAs behaves as a two-specie system. Although there are three distinct species in InAs, two of them behave in the same way with respect to the preces- sionrateinanexternalmagneticfieldandthecouplingto electrons so that InAs is an effective two-specie system. 69Ga71Ga75As p30%20%50% g1.31.70.96 A(µeV)779994 I3/23/23/2 Table I: Nuclear abundances p, nuclear g-factors, hyperfine coupling constants A, and nuclear spin in GaAs.46,47 113In115In75As p2%48%50% g1.21.20.96 A(µeV)14014076 I9/29/23/2 Table II: Nuclear abundances p, nuclear g-factors, hyperfine coupling constants A, and nuclear spin in InAs.48,49 We will consider systems with different number of nu- clear species to deduce coupled electron-nuclear dynam- icsphenomenathat arerobustwith respecttoorstrongly influenced by the number of species. To this end, we choose InAs and GaAs as model systems. These systems havedifferentmagnitudesoftheSOsplitting; itismodest in GaAs but strong in InAs, see Appendix A for details. In Sec. VIA, where calculations for nuclear parameters of InAs of Table II are carried out, we use modest val- ues of SO coupling v± SOto illustrate how the dynamics becomes increasinglycomplex and irregularwith increas- ing strength of the spin-orbit interaction. Nevertheless, this allows making conclusions about the expected nu- clear dynamics in InAs for realistic values of v± SO, see the end of Sec. VIA. The SO coupling constant v± SOis a com- plex number. Without loss of generality, we will assumein the remainder of the paper that it is real and positive, as well as use a simplified notation, v± SO=vSO. A. Two-specie systems: InAs Let us first consider InAs which effectively consists of two species because the parameters of113In and115In practically coincide. Therefore, species113In and115In behave as a single specie and75As as a second specie. We first demonstratethat, in absenceofSO coupling, the dynamical evolution of nuclear spins in InAs is similar to the dynamics in GaAs reported earlier.30In all our InAs simulations, we startin the sameinitial (pseudo-random) configuration of the nuclear spins. We have checked that similarresultsareobtainedwhenwestartinseveralother configurations. In all our simulations in this section, the waiting time between LZ sweeps equals the precession time of specie75As in the external magnetic field, Tw= t75As. We start by presenting results for a system without spin-orbitcoupling, vSO= 0, toprovethatself-quenching occurs and investigate its stability with respect to nu- clear noise. Fig. 2(a) shows the evolution of the magni- tude of the singlet-triplet coupling |v± n|with increasing number of sweeps n. For a sweep duration of TLZ= 40 ns, the initial LZ probability for the first few sweeps is P∼0.5, see Fig. 3, and the singlet-triplet coupling is self-quenched already after about 20000 sweeps. The number of sweeps nrequired to reach self-quenching is about the same as for GaAs.30The appearance of sev- eral peaks of v± nin the rangeof 5000-20000sweeps before the self-quenching sets in is typical of multi-specie sys- tems. In contrast to single-specie systems, and especially the model of Sec. IV, in multi-specie systems final self- quenching is usually preceeded by partial self-quenchings followed by revivals. We attribute this behavior to com- petition between subsystems with the different Zeeman precession times. As seen in Fig. 3, in each peak of |v± n| theS-T+transitionprobability Pincreasesstrongly,near itIzshows a step-like behavior (not shown), and accom- panying peaks of Qindicate massive shakeups which flop many nuclear spins per LZ sweep. The model of Sec. IV that only deals with the total magnetization Izdoes not describe such events and provides a smoothened picture of the nuclear spin evolution. Transverse noise transforms the dynamical evolution into a dissipative one. Fig. 2 shows the effect of the in- crease of the level of noise from (a) through (b) to (c). In Fig. 2(a), the noise correlation time is of the order of the self-quenching time τ/t75As= 10000. In this case, transverse noise only modestly perturbs the nuclear spin evolutionascomparedtothenon-dissipativeregime(sim- ulated and analyzed, but not shown). Note the presence of a long slightly visible tail with irregular oscillations along it. In contrast, Fig. 2(b) and (c) demonstrate that when the transverse noise correlation time τis shorter than the typical self-quenching set-in time in un-noisy8 50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1a/RParen1 50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1b/RParen1 50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1c/RParen1 Figure 2: Hyperfine-induced singlet-triplet coupling |v± n|for a double quantum dot with the nuclear parameters of InAs in absence of SO coupling with increasing level of noise as a function of sweep number n. (a)τ/t75As= 10000, (b) τ/t75As= 1000, and (c) τ/t75As= 100. The other param- eters are Tw=t75As,TLZ= 40 ns. systems, self-quenchingissuppressedandeventuallydoes not happen at all; in particular, Fig. 2(b) demonstrates a possibility of revivals. We conclude that for high noise levels the chaotic evolution of the nuclear spins persists, but themagnitudesofthe peaksof |v± n|seemtogradually decrease in time. Fig. 2(a) suggests a glassy behavior of the nuclear system with an extensive manifold of dark states sepa- rated by low barriers. In the absence of noise, repeated LZ sweeps cause the system to end in one of the dark states (usually after passing through several peaks of |v± n|). Weak noise produces slow diffusion between adja- centdarkstatesacrosslowsaddle points. Duringthis dif- fusion, the magnetization Izchanges only slightly. With increasing noise, the system experiences revivals as seen inFig.2(b)asasharppeakin |v± n|. Duringsuchevents P50000 100000n0.51.P 50000 100000n48Q Figure 3: (a) Landau-Zener transition probability Pas func- tion of the sweep number nfor a double quantum dot with the nuclear parameters of InAs in absence of SO coupling. (b) Shake-up parameter Qas a function of sweep number n. The parameters are as in Fig. 2(a) increasesstrongly, Izshowsstep-likebehavior, and peaks inQ(not shown) indicate massive shakeups, similarly to the patterns discussed as applied to Fig. 2(a) above. We demonstrated earlier that SO coupling is screened stroboscopicallyin asingle-speciesystem.30Next, wewill demonstrate that SO coupling can be screened strobo- scopically also in multi-specie systems, and investigate this phenomenon in more detail. Fig. 4 shows simula- tions of the singlet-triplet coupling v± nforvSO= 62 neV and three LZ sweep durations TLZ. In comparison, the straight black line indicates the value of the spin-orbit couplingvSO= 62 neV (which is independent of the sweep number n). We see that in all these simulations, the spin-orbit coupling eventually becomes screened so that all the colored lines approach the black line which impliesthat |v± n|=|vSO|. ForlongerLZsweepdurations, oscillations of v± nare more rapid, but screening eventu- ally occurs faster because nuclear spins are more strongly affected during each sweep. Screening of the SO coupling even in multi-specie sys- tems sounds counter-intuitive at first glance. Indeed, the spin-orbitcoupling vSOisstaticwhilethetransversecom- ponents of the nuclearspins contributing to v± nprecessin time. In InAs, twonuclearspecies113Inand115In precess at the same frequency and behave effectively as a single spin specie whereas the third spin specie,75As, precesses at a different frequency. So, while screening indicates that the magnitude of the singlet-triplet coupling v± nre- mains finite, it must inevitably precess in time. There-9 50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1 Figure 4: Transverse nuclear polarization for a double quan - tum dot with the nuclear parameters of InAs as a function of sweep number nfor the spin-orbit coupling vSO= 62 neV (black line) and the Landau-Zener sweep durations (red curve)TLZ= 80 ns, (green curve) TLZ= 40 ns, and (blue curve)TLZ= 20 ns. Resonant pumping with Tw=t75As, the polarization is plotted at multiples of Tw. See text for details. fore, it cannot compensate the spin-orbit coupling vSO at all instants of time. The screening we observe is only possible because the waiting time Twis exactly equal to the precession time of the75As specie,Tw=t75As. The data used in our plots of |v± n|were taken at exact mul- tiples of the the waiting time, which was equal to the precession time of the75As specie. Therefore, the self- quenching that manifests itself in Fig. 4 is a stroboscopic self-quenching. Stroboscopic self-quenching can be understood in the following way. The dynamical evolution of nuclear spins causes self-quenching of the sum of the contributions fromthetransversecomponentsofspecies113Inand115In (that are out of resonance with the pumping period Tw, hence, their contribution to v± nvanishes). In contrast, the contribution from the specie75As tov± nexactly com- pensatesthespin-orbitcoupling v± SOateverytimeinstant when a LZ sweep happens. In other words, the matrix elementsv± n(t) changein time harmonicallywith the am- plitudev± SOand a period t75As: v± n(t) =v± SOcos(2πt/t75As). (24) This generalizes our previous findings of the screening of SO coupling in single-specie systems.30For a single- specie imitation of GaAs, we found that the SO coupling was screened in such a way that that the matrix element changed harmonically with the amplitude vSOand a pe- riodtGaAs, wheretGaAsis the average precession time of the three nuclear spin species in GaAs.30 Let us now demonstrate explicitly that when self- quenching sets in, the sum of the contributions from the transversecomponents of113In and115In to|v± n|vanishes while the contribution from75As equalsvSO. We show in Fig. 5(a) the contribution from113In and114In to|v± n| as a function of the number of sweeps n. Clearly, it van- ishes for large n. On the other hand,75As whose nuclear precession time equals the waiting time Tw, makes a con- tribution to |v± n|that exactly compensates |v± SO|at allintegers ofTw, see Fig. 5(b). Hence, Eq. (24) is satisfied. 50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1/LParen1a/RParen1Sumof specie1 and2 50000 100000n100200300/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1/LParen1b/RParen1Specie3 Figure 5: (a) Sum of contributions from113In and115In to hyperfine-induced singlet-triplet coupling v± nfor spin-orbit coupling vSO= 62 neV (black line) as a function of sweep number n. (b) Contribution from75As to hyperfine-induced singlet-triplet coupling v± nfor spin-orbit coupling vSO= 62 neV (black line) as a function of sweep number n. The LZ sweep duration is TLZ= 80 ns. We note that the contributions from113In and115In tov± nvanish not only stroboscopically but identically, at each instant of time (not shown). We have also checked that in systems without spin-orbit coupling, self- quenching sets in for all species and for an arbitraryratio betweenTwand the precession times of the species (not shown). For three-specie systems the last statement is proven below, see Sec. VIB. Now we will illustrate that stroboscopic screening of SO coupling can be practically achieved only for small and moderate magnitudes of vSO. Since stroboscopic screening implies that the contribution from75As tov± n compensates vSOwhile the combined contribution from species113In and115In vanishes, we show in Fig. 6 the evolution of the contribution of75As tov± nas a function of sweep number for three values of spin-orbit coupling vSO= 31,62 and 91 neV.50While all the results in Fig. 6 were found for the same value of TLZand the same initial conditions, screening sets in at n≈75000 forvSO= 31 neV, is delayed to n≈125000 for vSO= 62neV, and is far from complete even at n= 200000 for vSO= 93 neV. These data suggest that stroboscopic self-quenching sets in whenvSO/lessorsimilarv0 n, wherev0 n≈A/√ Nis a typical fluctu- ation of the Overhauser field, and cannot be practically achieved for vSO/greaterorsimilarv0 n; see estimates of the magnitude10 of the spin-orbit coupling in Appendix A. This criterion resembles the criterion of the phase transition of Ref. 26. One should keep in mind that with the interval be- tween LZ pulses of about 1 µs, a set of n∼106pulses takes about 1 s which is a typical scale of nuclear spin diffusion51, which is not taken into account in the above considerations. Weexpect, buthavenotchecked, thatin- homogeneity of magnetic field should have a detrimental effect on stroboscopic self-quenching. Therefore, we con- clude that stroboscopic quenching of SO coupling is less generic and more fragile than self-quenching in systems without spin-orbit coupling. 50000 100000 150000 200000n20406080100/VertBar1vn/PlusMinus/VertBar1/LParen1eV/RParen1 Figure 6: Contribution of the specie75As to the singlet- triplet coupling v± nas a function of sweep number nfor differ- ent values of vSO: 31 neV (red curve), 62 neV (black curve), and 91 neV (blue curve). Duration of LZ pulses TLZ= 40 ns. Waiting time between consecutive LZ pulses equals the precession time of75As,Tw=t75As. Nuclear parameters of InAs. We estimatein Appendix Athat SOcouplingisweaker or comparable to (stronger than) the typical nuclear polarization induced singlet-triplet coupling in GaAs (InAs). As a consequence, we expect the SO coupling might be stroboscopically screened in GaAs systems, but that stroboscopic screening is improbable in InAs sys- tems. B. Three-specie systems: GaAs In this section, we present new results for GaAs that complete the picture of the generic nature of self- quenching in multi-specie systems. Furthermore, we show that screening of the SO coupling requires that the waiting time Twis in resonance with the precession time of one of the nuclear species. When the resonance con- dition is not satisfied, screening of the SO coupling is partial and irregular. In Sec. VIA and in Ref. 30, self-quenching in multi- specie systems in absence of SO coupling was demon- started only under the conditions when the waiting time Twwas in resonance with the precession time t75Asof the 75As specie,Tw=t75As. We demonstrate here that whileself-quenching is generic and independent of the waiting time, the evolution towards the self-quenched states de- pends on the waiting time. To this end, we plot in Fig. 7 the evolution of the singlet-triplet coupling v± nfor two different values of the waiting time Tw. Fig. 7(a) displays results of simulations for the resonant case when Tw=t75As, in which pro- nounced oscillations are distinctly seen. For n/greaterorsimilar3000, the plot consists of five branches that reflect coupled dy- namics of three species. In contrast, in the absence of the resonance, Fig. 7(b), the evolution is chaotic. Nev- ertheless, self-quenching sets-in in both cases and, what is most remarkable, at the same time scale of n≈104. Remarkably, the processes of Figs. 7(a) and 7(b) ended in states with the same Iz(not shown). While the set of dark states is vast (as follows from our discussion in Section IV), this observation indicates that the number of strong attraction centers in which self-quenching ends is more scant. We conclude that self-quenching in systems without spin-orbit coupling is generic and robust, at least in the framework of S-T+scheme. We checked that not only does the total matrix ele- mentv± nvanish, but also the matrix elements for all of three species contributing to it. Because between the LZ sweeps the electron subsystem is in its singlet state, the Knight shift vanishes, and according to Eq. (3) all nuclei belonging to some specie precess with the same speed. Therefore, the self-organization of the nuclear subsystem that annihilates its coupling to the electron spin persists during the free precession periods. Finally, we demonstate that while self-quenching is a genericfeatureintheabsenceofSOcouplingregardlessof theratiobetweenthewaitingtimebetweentheLZsweeps Twand the nuclear precession times tλ, in presence of SO coupling the stroboscopic self-quenching is not generic andhighlysensitivetothisratio. Onlymodestdeviations fromtheresonancedestroysthescreeningofSOcoupling. In Fig. 8, we plot |v± n|under the conditions when the waiting time is in exact resonance with the precession time of75As (red curve), and when there is a 1% devi- ation from the resonance (black curve). While the SO coupling is clearly screened in resonance, only a tiny de- viation from resonance destroys screening. More insights into the sensitivity of the screening of SO coupling to the deviation from the resonance can be gained from Fig. 9 that displays the contributions to v± n from eachofthe species. Using the same intial conditions as in Fig. 8, we plot the evolution of the matrix elements |v± n|for both the resonant and slightly off-resonance regimes. Initially they follow each other closely. How- ever, after a couple of thousand sweeps, the deviations become significant. Ultimately, the contributions from 69Ga and71Ga do not vanish in the non-resonant case, and the contributions from75As does not screen the SO coupling. The critical sensitivity of stroboscopic self-quenching to small deviations from resonance looks indicative of11 2000 4000 6000 8000 10000n10203040/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1 2000 4000 6000 8000 10000n10203040/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1 Figure 7: Transverse nuclear polarization as a function of sweep number nfor a GaAs double quantum dot in absence of spin-orbit coupling and transverse noise. Duration of LZ pulsesTLZ= 80 ns. (a) The waiting time is in resonance with the75As precession time, Tw=t75As= 13.7µs. (b) The waiting time is incommensurate with the75As precession time,Tw= 1.39t75As= 19.1µs. 10000 20000 30000n255075/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen169Ga,71Ga, and75As Figure 8: Transverse nuclear polarization as a function of sweep number nfor a GaAs double quantum dot with vSO=8 neV. The duration of LZ pulses TLZ= 80 ns. The red curve shows results for the waiting time in resonance with the pre- cession time of75As,Tw=t75Ar, and the black curve for a 1% deviation from the resonance. a chaotic behavior of the system.52This is not surpris- ingbecausethe system ofintegro-differentialequationsof Eq. (1) is highly nonlinear because the coefficients ∆jλ depend through Eqs. (10) on the electronic amplitudes cS,cT+that, in turn, depend on all nuclear angular mo- mentaIjλ. In this context, we speculate that a strong revival of all black curves in Fig. 9 near n≈15000 where all red curves saturate, and the return of black curves close to their initial values near n≈28000, is reminis-cent of the strange attractor pattern.52These signatures ofchaoticnucleardynamicinSOcoupledsystemsrequire a more detailed study. WeconcludethatstroboscopicscreeningoftheSOcou- pling is not a robust phenomenon. While the above simulations are focused on the large nregion, we mention that commensurability oscillations in the polarization accumulation per sweep were ob- served experimentally38and described theoretically37in the smallnregion,n/lessorsimilar104. 10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1a/RParen169Ga 10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1b/RParen171Ga 10000 20000 30000n1020/VertBar1vn/PlusMinus/VertBar1/LParen1neV/RParen1/LParen1c/RParen175As Figure 9: Transverse nuclear polarization as a function of sweep number nfor a GaAs double quantum dot with vSO=8 neV. The duration of LZ pulses TLZ= 80 ns. Red curves show results for the waiting time Twin resonance with the precession time of75As,Tw=t75As, while black curves the data for 1% off-resonance regime. In addition to the regular investigation of the trans- verse magnetization, we also followed the time de- pendence of the longitudinal magnetization vz n= Vs/summationtext λAλ/summationtext j∈λρjλIz jλ. In presence of SO coupling, it shows an oscillating sign-alternating behavior, and we were unable to detect any signatures of its accumulation.12 Summarizing the results of Secs. VIA and VIB, we conclude that SO coupling eliminates self-quenching and causes the nuclei of a pumped system to exhibit a per- sistent irregular dynamics. We speculate that this phe- nomenon is closelyrelated to the feedback looptechnique for building controllable nuclear gradients which is inher- ently based on employingsuch a dynamics.34Indeed, any long-term control of the nuclear ensemble by alternating S→T+andT+→Ssweeps is impossible after the self- quenching set-in time that is of a millisecond scale in absence of SO coupling. Our data, especially Fig. 8, sug- gest that near the resonance between the waiting time of LZ pulses and the Larmor frequency of one of the nuclear species the quasi-periods of nuclear fluctuations become longer and are controlled by the deviations from the exact resonance. We also expect that under these conditions the nuclear gradients should be dominated by the resonant specie. VII. CONCLUSIONS An analytical solution of a simplified model, and ex- tensive numerical simulations for a realistic geometry, provethat self-quenchingis a genericpropertyofthe cen- tral spin-1/2 problem in absence of spin-orbit coupling. As applied to a double quantum dot of a GaAs type, where electron and nuclear spins are coupled viahyper- fine interaction, pumping nuclear magnetization across a S-T+avoided crossing through successive Landau-Zener sweeps ceases after about 104sweeps. This is a result of the screening of the initial fluctuation of the nuclear magnetization by the injected magnetization and van- ishing of the S-T+anticrossing width, and this sort of self-quenchingisrobust. Under the influence ofmoderate noise, thesystemwandersthroughasetofdarkstatesbe- longingtoawastlandscapeofthesystemincludingabout 106nuclear spins coupled through inhomogeneous elec- tron spin density. With time intervals depending on the level of the noise, the system experiences revivals when additional magnetization is injected, and afterwards it wanders through a new set of dark states. Due to the violation of the angular momentum conser- vation, spin-orbit coupling changes the situation drasti- cally. Self-quenching sets in only stroboscopically under the condition that the waiting time between consecutive Landau-Zener sweeps is in resonance with the Larmor precession time of one of the nuclear species. Then the precessing Overhauser field of the resonant specie com- pensates the spin-orbit field vSOduring the sweep, while contributions of other species vanish. This sort of self- quenching is fragile and sensitive even to minor deviation from the resonance. Generically, injection of nuclear po- larization oscillates in time and changes sign. Therefore, spin-orbit coupling causes the nuclear magnetization of a pumpedS-T+doublequantumdottoexhibit apersistent dynamics. We suggest that the feedback loop technique for build-ing controllable nuclear field gradients34is based on the oscillatory behavior of the nuclear spin magnetization caused by spin-orbit coupling. Spin-orbit coupling is a natural mechanism of overcoming self-quenching. The technique employs persistent oscillations and selects the sign of the pumping response to the changing magneti- zation gradient. Acknowledgments A.B.wouldliketothankB.I.Halperinforhishospital- ity at Harvard University where this work was initiated. We are grateful to B. I. Halperin, C. M. Marcus, L. S. Levitov, H. Bluhm, K. C. Nowack, M. Rudner, and L. M. K. Vandersypen for useful discussions. E. I. R. was supported by the Office of the Director of National In- telligence, Intelligence Advanced Research Projects Ac- tivity (IARPA), through the Army Research Office grant W911NF-12-1-0354and by the NSF through the Materi- als Work Network program DMR-0908070. Appendix A: Spin-orbit coupling The Rashba spin-orbit Hamiltonian is Hso=α/summationdisplay n=1,2[σx(n)ky(n)−σy(n)kx(n)],(A1) whereαis the strength of SO interaction, and kx(n) and ky(n) are the in-plane momenta for the electron n. The singlet and triplet states are Ψ S(1,2) =ψS(1,2)χS(1,2) and Ψ T+(1,2) =ψT(1,2)χT+(1,2), where the orbital components of the singlet and triplet wave functions have been defined in Sec. V and the spin parts of the wave functions are χS(1,2) = (| ↑1∝an}bracketri}ht| ↓2∝an}bracketri}ht−| ↓ 1∝an}bracketri}ht ↑2∝an}bracketri}ht)/√ 2 andχT+(1,2) =| ↑1∝an}bracketri}ht| ↑2∝an}bracketri}ht. In terms of the orbital wave functions, the SO induced S-T+coupling is then v+ SO=iαcosν∝an}bracketle{tψR|kx+iky|ψL∝an}bracketri}ht. This matrix element can be estimated similarly to Refs. 26 and 39. It depends exponentially on the overlap between the wave functions of the dots, ψLandψR. Therefore, the SO coupling v+ SO can be tuned and strongly decreases with the interdot distanced. In InAs quantum wires the SO coupling pa- rameter is around α∼10−11eVm53corresponding to a spin precession length of lSO=/planckover2pi12/(2m∗α) of around 100 nanometers. With typical parameters of d= 100 nm andl= 50 nm and cos ν= 1/√ 2, we find that vSOis around 4 ×10−5eV. This is about two orders of mag- nitude larger than the typical S-T+coupling 10−7eV induced by the hyperfine interaction, but decreases with dexponentially. Theoretical estimates of vSOhold only with exponential accuracy. Pre-exponential factors are model dependent, and a somewhat different estimate was proposed in Ref. 54. In GaAs quantum dots,55the SO coupling constant αis two orders of magnitude smaller than in InAs with lSO≈30µm, so that the SO coupling13 may be comparable to the hyperfine induced coupling, and is usually considered as weaker than it. These esti- mates should be treated with caution since the SO cou- pling is not only a function of the material but is sample specific. For GaAs, an estimate of a typical fluctuation as v0 n≈ A/√ NwithAfrom Table II and N≈106results inv0 n≈ 100 neV. Our estimates of vSOof Ref. 30 gave vSO≈50 neV, while the estimate of Ref. 37 is vSO≈15 neV.56Recently Shafiei et al.57managed to resolve the SO and hyperfine components of electric dipole spin resonance (EDSR)58in the same system, a GaAs double quantum dot. Their data suggest that both components were of a comparable magnitude, with the hyperfine component maybe somewhat stronger. Keeping in mind the uncertainty in the values of vSO, we carry out simulations both with SO coupling and without it. 1I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76 323 (2004) 2D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 3L. P. Kouwenhoven and C. Marcus, Physics World 11(6), 35 (1998). 4J. Levy, Phys. Rev. Lett. 89, 147902 (2002). 5Awschalom D. D., D. Loss, and N. Samarth (2002) Semiconductor Spintronics and Quantum Computation (Springer, Berlin) 6D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard and K. B. Whaley, Nature (London) 408, 339 (2000). 7R. Hanson, L. P. Kouwenhoven, J. R. Petta, S.Tarucha, and L. M. K. Vandersypen, Rev. Mod. Phys. 70, 1217 (2007). 8J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 309, 2180 (2005). 9R. Brunner, Y.-S. Shin, T. Obata, M. Pioro-Ladrire, T. Kubo, K. Yoshida, T. Taniyama, Y. Tokura, and S. Tarucha, Phys. Rev. Lett. 107, 146801 (2011) 10A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys. Rev. B76, 161308 (2007). 11S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Nature 468, 1084 (2010). 12K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung, J. M. Taylor, A. A. Houck, and J. R. Petta, Nature 490, 380 (2012). 13I. A. Merkulov, Al. L. Efros, and M. Rosen, Phys.Rev. B 65, 205309 (2002). 14A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev.B 67, 195329 (2003). 15S. I. Erlingsson and Yu. V. Nazarov, Phys. Rev. B 70, 205327 (2004). 16W. A. Coish and D. Loss, Phys. Rev. B 72, 125337 (2005) 17W. Yao, R.-B. Liu, and L. J. Sham, Phys. Rev. B 74, 195301 (2006). 18G. Chen, D. L. Bergman, and L. Balents, Phys. Rev. B 76, 045312 (2007) 19L. Cywi´ nski, W. M. Witzel, and S. Das Sarma, Phys. Rev. Lett.102, 057601 (2009). 20D. Stanek, C. Raas, and G. S. Uhrig, arXiv:1308.0242. 21D. Klauser, W. A. Coish, and D. Loss, Phys. Rev. B 78, 205301 (2008). 22E. A. Chekhovich, M. N. Makhonin, A. I. Tartakovskii, A. Yacoby, H. Bluhm, K. C. Nowack, and L. M. K. Vander- sypen, Nature Materials 12, 494 (2013). 23G. Ramon and X. Hu, Phys. Rev. B 75, 161301(R) (2007). 24M. Stopa, J. J. Krich, and A. Yacoby, Phys. Rev. B 81,041304(R) (2010). 25M. Gullans, J. J. Krich, J. M. Taylor, H. Bluhm, B. I. Halperin, C. M. Marcus, M. Stopa, A. Yacoby, and M. D. Lukin, Phys. Rev. Lett. 104, 226807 (2010). 26M. S. Rudner and L. S. Levitov, Phys. Rev. B 82, 155418 (2010). 27M. S. Rudner, I. Neder, L. S. Levitov, and B. I. Halperin, Phys. Rev. B 82, 041311 (2010). 28H. Ribeiro and G. Burkard, Phys. Rev. Lett. 102, 216802 (2009). 29A. Brataas and E. I. Rashba, Phys. Rev. B 84, 045301 (2011). 30A. Brataas and E. I. Rashba, Phys. Rev. Lett. 109, 236803 (2012). 31M. Gullans, J. J. Krich, J. M. Taylor, B. I. Halperin, M. D. Lukin, Phys. Rev. B 88, 035309 (2013). 32R. Winkler, Spin-orbit Coupling Effects in Two- Dimensional Electron and Hole Systems (Springer Berlin Heidelberg, 2010). 33D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 104, 236802 (2010). 34H. Bluhm, S. Foletti, D. Mahalu, V. Umansky, and A. Yacoby, Phys. Rev. Lett. 105, 216803 (2010). 35L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York) 1977. 36For brevity, we use this traditional notation for the Landau-Majorana-St¨ uckelberg-Zener problem.59,60 37I. Neder, M. S. Rudner, and B. I. Halperin, arXiv:1309.3027. 38S. Foletti, H. Bluhn, D. Mahalu, V. Umansky, ans A. Ya- coby, Nature Physics 5, 903 (2009). 39D. Stepanenko, M. Rudner, B. I. Halperin, and D. Loss, Phys. Rev. B 85, 075416 (2012). 40For thedetails ofthechoice of initial conditions see Ref. 2 9. 41S. M. Ryabchenko and Y. G. Semenov, Zh. Eksp. Teor. Fiz.84, 1419 (1983) [Sov. Phys. JETP 57, 825 (1983). 42A. C. Durst, R. N. Bhatt, and P. A. Wolff, Phys. Rev. B 65, 235205 (2002). 43This equationcanbealso derivedbychangingthevariables in Eq. 15 so that γdepends on Izinstead of nby using dγ/dIz= (dγ/dn)/(dIz/dn) which results in dγ/dIz= −A2 0Iz/β2. Because this equation is independent of the sweep number nand the probability P, it can be readily integrated. 44J. M. Taylor, A. Imamoglu, and M. D. Lukin, Phys. Rev. Lett91, 246802 (2003). 45Because our procedures are numerical, they can be applied to an arbitrary tdependencies of ǫS−ǫT+. We choose a14 linear dependence only to stay closer to the terms of the standard Landau-Zener theory. 46J. Schliemann, A. Khaetskii, and D. Loss, J. Phys. Con- dens. Matter 15, R1809-R1833 (2003). 47J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus, and M. D. Lukin, Phys. Rev. B 76, 035315 (2007). 48M. Gueron, Phys. Rev. 135, A200 (1964). 49M. Syperek, D. R. Yakovlev, I. A. Yugova, J. Misiewicz, I. V. Sedova, S. V. Sorokin, A. A. Toropov, S. V. Ivanov, and M. Bayer, Phys. Rev. B 84, 085304) (2011). 50It is seen from Figs. 4 - 6 that SO interaction makes the coupled electron-nuclei dynamics highly volatile. Oscill a- tions of|v± n|are accompanied with strong shakeups. E.g., the shakeup factor Qis typically Q∼15 in the peaks of |v± n|for thevSO= 62 neV curve of Fig. 6. 51E. A. Chechnovich, M. N. Makhonin, A. I. Tartakovskii, A. Yacoby, H. Bluhm, K. C. Nowack, and L. M. K. Van- dersypen, Nature Physics 12, 494 (2013). 52E. N. Lorenz, The Essence of Chaos , (U. WashingtonPress, Seattle) 1993. 53S. E. Hern´ andez, M. Akabori, K. Sladek, Ch. Volk, S. Alagha, H. Hardtdegen, M. G. Pala, N. Demarina, D. Gr¨ utzmacher, Th. Sch¨ apers, Phys. Rev. B 82, 235303 (2010). 54C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and D. Loss, Phys. Rev. Lett. 98, 266801 (2007). 55K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarov, and L. M. K. Vandersypen, Science 318, 1430 (2007). 56Caption to their Fig. 6. 57M. Shafiei, K. C. Nowack, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, Phys. Rev. Lett. 110, 107601 (2013). 58E. I. RashbaandV. I.Sheka, in: Landau Level Spectroscpy , edited by E. I. Landwehr and G. Rashba (North-Holland, Amsterdam, 1991), Vol. 1, p. 131. 59E. Majorana, Nuovo Cimento 9, 43 (1932). 60E. C. G. St¨ uckelberg, Helv. Phys. Acta 5, 369 (1932).

1007.5037v1.Current_induced_torques_in_the_presence_of_spin_orbit_coupling.pdf

arXiv:1007.5037v1 [cond-mat.mes-hall] 28 Jul 2010Current-induced torques in the presence of spin-orbit coup ling Paul M. Haney and M. D. Stiles Center for Nanoscale Science and Technology, National Inst itute of Standards and Technology, Gaithersburg, Maryland 20899-6 202, USA In systems with strong spin-orbit coupling, the relationsh ip between spin-transfer torque and the divergence of the spin current is generalized to a relati on between spin transfer torques, total angular momentum current, and mechanical torques. In ferro magnetic semiconductors, where the spin-orbit coupling is large, these considerations modify the behavior of the spin transfer torques. One example is a persistent spin transfer torque in a spin val ve: the spin transfer torque does not decay away from the interface, but approaches a constant val ue. A second example is a mechanical torque at single ferromagnetic-nonmagnetic interface. Introduction — Since the prediction [1–3] of spin trans- fer torques in non-collinear ferromagnetic metal circuits, they have been the subject of extensive research [4, 5]. The possibility of using spin transfer torque to im- prove the commercial viability of magnetic random ac- cess memory (MRAM) [6], and the rich non-equilibrium physics involved establish the topic as one of practical and fundamental interest. These torques arise from the exchange interaction between non-equilibrium, current- carrying electrons and the spin-polarized electrons that make up the magnetization. In systems where the spin- orbit coupling is weak, the torque on the magnetization can be computed from the change in the spins flowing through the region containing the magnetization. This relation is a consequence of conservation of total spin. Here, we consider systems in which the spin-orbit cou- pling cannot be neglected (and hence total spin is no longer conserved). In systems where spin angular momentum is not con- served, the relationship between the spin transfer torque and the flow of spins needs to be generalized. Conserva- tion oftotalangular momentum implies that mechanical torques on the lattice of the material accompany changes in the magnetization [7, 8]. This effect has been used for decades to measure the g-factor of metals. More recent theoretical [9, 10] and experimental [11] work considers the current-inducedmechanicaltorquespresentatthe in- terfaceofaferromagnetandnon-magnet, similarin spirit to the spin transfer torques on the magnetization present in spin valves. In this article we develop a theory for current-induced torques (both spin transfer torques and mechanical torques) in systems with strong spin-orbit coupling, and apply it to a model of dilute magnetic semiconductors. We find that by accounting for the orbital angular mo- mentum of the electrons, we can relate the change in total angular momentum flow to spin transfer torques and mechanical torques. We study two system geome- tries where these torques play important roles. The first is a spin-valve geometry, which is used to study the fea- tures ofspin transfertorques in the presence ofspin-orbit coupling. The second is a single interface between a fer- romagnet and non-magnet, which elucidates the physics underlying current-induced mechanical torques.Formalism — We consider a Hamiltonian consisting of a spin-independent kinetic and potential energy H0= −¯h2∇2 2m+V(r), an exchange splitting ∆, and an atomic- like spin-orbit interaction parameterized by α: H=H0+∆ ¯h(M·ˆs) Ms+α ¯h2/parenleftBig ˆL·ˆs/parenrightBig ,(1) whereˆLandˆsare the electron angular momentum and spin operators, respectively [12]. The exchange splitting arises from a magnetization M, with magnitude Ms. We treat the magnetization within mean field theory. We consider the torque on the magnetization due to electric current flow. The spin transfer torque τSTT at position rfrom electronic states with spin density s(r) is proportional to the component of spin trans- verse to the magnetization [14]: τSTT(r) =dM(r) dt= −∆ ¯h2(M(r)×s(r)). In the absence of spin-orbit coupling, this torque can be related to the divergence of a spin current, which offers conceptual and computational sim- plicity [15]. In the following we analyze how spin-orbit coupling changes this simple result. One consequence is an expression for the mechanical torque τlat. We develop an expression for τSTTby evaluating the time-dependence of the electron spin and angular momentum densities. To do so, we adopt a Heisen- berg picture of time evolution, and evaluatedˆO(r) dt= i ¯h/bracketleftBig H,ˆψ†(r)ˆOˆψ(r)/bracketrightBig ,whereˆψ(r) is the position operator, for the operators ˆO=ˆ s,ˆL. This procedure leads to [4]: dˆ s dt=∇·ˆQs(r)−ˆτSTT+α ¯h2/parenleftBig ˆL׈s/parenrightBig (2) whereˆQs(r) =ˆψ†(r)ˆv⊗ˆ sˆψ(r), and the velocity operator is given by ˆ v=i¯h 2m/parenleftBig← −∇−− →∇/parenrightBig ; here the arrowsuperscript specifies the direction in which the gradient acts. In ad- dition: dˆL dt=∇·ˆQL(r)−ˆτlat+α ¯h2/parenleftBig ˆs׈L/parenrightBig (3) whereˆQL(r) =ˆψ†(r)1 2/parenleftBig ˆvˆL+ˆLˆ v/parenrightBig ˆψ(r) (the product2 FIG. 1: Left and right panels shows GaMnAs band structure without and with spin-orbit, respectively (for γ2=γ3= 2.4). (arrows indicate spin direction of eigenstates). The inset shows the direction of bulk magnetization, and spin, veloc- ity, and k vectors for a single state (in black, red, blue, and green). The torque from the misalignment between magne- tization and spin equals the torque from the misalignment between velocity and k vectors. of non-commuting operators ˆLandˆ vis symmetrized). We’ve defined ˆτlat(r) =i ¯hˆψ†(r)/bracketleftBig H0,ˆL/bracketrightBig ˆψ(r), which is nonzeroforapotentials V(r)whichbreakrotationalsym- metry [16]. We define a total angularmomentum ˆJ=ˆL+ˆs, a total angular momentum current ˆQJ=ˆQL+ˆQs, and combine Eqs. (2) and (3) to obtain: dˆJ dt−∇·ˆQJ=−ˆτSTT−ˆτlat. (4) Finally, we take the expectation value of Eqs. (2-4), re- placing operatorsby densities. Eq. (4) is ourmain formal result. When spin-orbit coupling is important, the total angular momentum in the conduction electrons couples both to the magnetization and the lattice. The coupling ofelectronspintothelatticerequiresbothspin-orbitcou- pling and crystal field potential. The term τlatchanges the physical picture of spin transfer torque substantially, as is illustrated by considering Eq. (4) for a single bulk eigenstate:dJ dtand∇·QJvanish, however τSTTand τlatmay both be non-zero, implying a coupling from the angular momentum of the lattice to the magnetization. This coupling flows from the lattice to the orbital sub- system through the crystal field, which then couples to the spin through spin-orbit coupling, and finally to the magnetization through the exchange interaction. Application to DMS — We apply this general formal- ism to a model of a dilute magnetic semiconductor (DMS). DMSs are semiconductor host materials which become ferromagnetic when doped with magnetic atoms.Ga1−xMnxAs is the archetype for these materials, and can be described as a system of local moments of Mn d-electrons, whose interaction is mediated by holes in the semiconductor valence band [17]. The valence states are described by the Kohn-Luttinger Hamiltonian HKL 0, which represents a small- kexpansion for a periodic H0, acting in the ℓ= 1 subspace (describing valence states). It is given by: HKL 0=¯h2 2m/parenleftbigg (γ1+4γ2)k2−6γ2 ¯h2(L·k)2 −6 ¯h2(γ3−γ2)/summationdisplay i/negationslash=jkikjLiLj ,(5) whereLare the spin-1 matrices for the p-state orbitals, γ1, γ2, γ3are Luttinger parameters, and kis the Bloch wave-vector. Figure 1 shows how the presence of spin- orbit coupling affects the band structure. For periodic systems the velocity operator can be writ- ten as:ˆ v=1 ¯h∂H ∂k, and spin and angular momentum cur- rent densities are again defined as symmetrized products ofˆvandˆL, andˆvandˆs. The dynamics of the magne- tization occur on a much longer time scale than that of the electronic states, so we compute the dynamics from a sum over scattering states, for whichds dt=dL dt= 0. For the Luttinger Hamiltonian, the z-component of ˆτlatis: ˆτz lat= (ˆvׯhk)z+6(γ2−γ3) ¯hm{(kxLy+kyLx), (kxLx−kyLy)} (6) where the brackets on the second term indicate an an- ticommutator. Other components are given by cyclic permutation of indices. The first term of Eq. (6) can be written as ˆvׯhk=d dt(ˆrׯhk). This term can be interpreted as a torque on the crystal angular momen- tumˆrׯhk, and results from the misalignment between wave vector and velocity. It is generically nonzero for any material with a non-spherical Fermi surface. In the spherical approximation ( γ2=γ3), Eq. (4) implies that the net flux of total angular momentum into a volume is equal to the change of magnetization plus the crystal angular momentum inside the volume. STT in spin-valves — We firstconsiderasystem tostudy theτSTTterm of Eq. (4). Figure 2(a) shows the geome- try; current flows in the ˆ z-direction, perpendicular to the magnetization of both layers. We focus on the compo- nent of torque which is in the plane spanned by the two magnetization directions. This in-plane torque is deter- mined by the out-of-plane (or ˆ z-component) spin density [14]. For the results presented here, we use the param- eter values: ( γ1,γ2,γ3) = (6.85,2.1,2.9), ∆ = 0.27 eV, α= 0.11 eV,EF= 0.16 eV (EFis measured from the top of the valence band). The tunnel barrier is described by Eqs. (1) and (5), with ∆ = 0, and with an energy offset so that the top of the valence band is 0.1 eV below EF. We calculate the eigenstates numerically and apply3 FIG. 2: (a)Spin valve geometry: FM layers’ magnetization points in the ˆ xand ˆy(out-of-page) directions. (b) The spin transfer torque versus position away from the left normal metal-FM interface, which decays to zero in the absence of spin-orbit coupling, and does not in its presence. (c) Plot o f the total spin transfer torque, the net flux of spin current, and net flux of total angular momentum current versus FM thickness. The linear dependence for large thickness is due to a persistent spin transfer torque. boundary conditions as described in Ref. [18]. Figure 2(b) shows the spin transfer torque density as a function of distance away from the interface. We find that forα= 0 (no spin-orbitcoupling), the torque decays to zero away from the interface, as expected [15]. For α/ne}ationslash= 0, the torque oscillates around a nonzero value, and extends into the bulk. Figure 2(c) shows that the total spin transfer torque as a function of ferromagnetic (FM) layer thickness LFMis proportional to thickness for large LFM. This is in contrast to the metallic spin valve, where the torque is an interface effect and becomes constant for largeLFM. This persistent spin transfer torque arises because the spins of individual eigenstates are not aligned with the magnetization (see Fig. 1) in the presence of spin-orbit coupling. The misalignment gives rise to a torque be- tween the lattice and the magnetization. In equilib- rium, these torques cancel when summed over all occu- pied states. However, the presence of a current changes the occupation of the bulk states and can give rise to a torque [19, 20] in systems without inversion symme- try. Inversion symmetry is only very weakly broken in bulk GaMnAs, and is not included in the Kohn Luttinger Hamiltonian, Eq. (5). Here, interfaces between materials breaks inversion symmetry. The combination of an interface and a current flow changes the occupation of the bulk states near the Fermi energy (depending on the transmission probabilities of individual states across the interface) and induces coher- ence between these states. The change in the occupation probabilities gives rise to a persistent transverse spin ac- cumulation, which only decays through other scattering mechanisms not included here ( e.g.defect scattering).0 0.5 1−0.500.511.5 α/α0STT/I (h/e) STT total QJout−QJin −τlat persistent STT FIG. 3: The total spin transfer torque, the net flux of total angular momentum, −τlat, and the persistent component of the spin transfer torque on a FM layer with LFM= 30 nm as αis increased from 0 to α0= 0.11 eV. This spin accumulation gives rise to the persistent spin transfer torque. The coherence between the states modi- fies the spin accumulation and the torque near the inter- face but these corrections decay away from the interface due to dephasing. Figure 3 shows, as a function of the spin-orbit coupling constantα, the values of total spin transfer torque, the angular momentum current flux, - τlat, and the persistent contribution to spin transfer torque (for LFM= 30 nm). We determine the persistent contribution from the slope of the integrated total versus FM width LFMat large LFM(see Fig. 2c). This procedure neglects the contri- butions from coherence near the interface. In this exam- ple, the spin transfer torque increases with the addition of spin-orbit coupling, largely because of the addition of the persistent term. This qualitative behavior depends on system parameters: for EF= 0.34 eV, for example, the spin-orbit coupling decreases the total torque. Nanomechanical torques in wires — We next consider a system which exemplifies that physics of the τlatterm of Eq. (4): a single interface between GaMnAs and GaAs, with the direction of the magnetization parallel to the current flow (see Fig. 4a). This is similar to the geom- etry considered in previous theoretical and experimental work [9–11]. The vanishing magnetization in GaAs im- pliesτSTT= 0, so that τlat=∇·QJ, and its total value can be deduced from Qin J−Qout J. We use the same pa- rameters as before, except EF= 0.06 eV, and the top of the valence band of both layers coincide. Figure 4c shows τlatin the GaAs layer as a function of distance away from the interface (assuming electron particle flow from left to right). The total torque (dark curve) shows oscillatory decay, while the torque from a particular channel (light curve) shows simple oscilla- tion. The behavior of the single channel is illustrated in Fig. 4b. We assume specular scattering, so that the in- cident state chosen (black circle) transmits into the four states of GaAs with equal kx, ky(also shown with black4 FIG. 4: (a) shows the system geometry. We take electron particle flow from left to right, and consider the mechanical torqueτlatin the z direction. (b) shows slices of the Fermi surface for the different layers, with /angb∇acketleftJ(k)/angb∇acket∇ightsuperimposed, and also shows the J-character of the states specified by the black circle. Also shown is the transmission probability for each of the states in the GaAs. (c) shows the total mechanical torque density in the GaAs as a function of distance from the interface (dark curve), and the contribution from the singl e incoming state specified in (b) (dashed curve). circles). The character of these states, along with the transmissionprobability, is shown in Fig. 4b. The incom- ing state couples most strongly to the state with similar Jcharacter, but also partially transmits into other states with different Jcharacter and wave vector kz. These dif- ferent scattering channels interfere with each other, lead- ing to an oscillatory J(z), with an oscillation period in- versely proportional to the splitting of kzwave-vectorsof the different sheets of the Fermi surface. This splitting is from the lattice crystal field and spin-orbit coupling, the agents responsible for τlat. Different channels have dif- ferent oscillation periods, so that their total decays away from the interface, ashappens for spin transfertorquesin ferromagnets [15]. For the parameters used here, we findQin Jz= 1.20¯hI e, due to the polarization of the states from the magnetization, while Qout Jz= 0.46¯hI e. Mechanisms not considered here, such as spin-flip scattering, ensure thatQout Jzdecays to zero away from the interface. Themechanicaltorqueis Qin Jz−Qout Jz= 0.74¯hI e. Forap- propriate experimental conditions, this torque is greater than the thermal fluctuations and is a measurable ef- fect. We refer the reader to Ref. [9–11] for details of treatment of the torsion dynamics and experimental de- tails. The formalism developed here generalizes previous work to allow for microscopic evaluation of the electronic structure contribution to the current-induced mechanical torque. For systems with nonzero magnetization, the mi- croscopic form of τlatis necessary to determine the parti- tioning of total angular momentum flux between torques onthemagnetizationandtorquesonthelattice. Ourthe- ory neglects other mechanisms of spin relaxation, such as disorder-induced spin-flip scattering, so that full calcula- tions will require microscopic calculations like these to be embedded in diffusive transport calculations. Conclusion — We have shown how atomic-like spin-orbit coupling affects current-induced torques: both the spin transfer torque on the magnetization and the mechanical torque on the lattice. In GaMnAs spin valves, we find a contribution to the spin transfer torque that persists throughout the bulk. This result may explain experi- ments which find critical currents which are up to an order of magnitude smaller than the value expected from a simple accounting of the net spin current flux [21, 22]. For a single interface between GaMnAs and GaAs, we microscopically compute the mechanical torque due to scattering from the interface. These results highlight im- portant, qualitativelydifferentphysicsatplaywhenspin- orbit coupling is strong. TheauthorsacknowledgehelpfulconversationswithA. H. MacDonald. [1] L. Berger, J. Appl. Phys. 3, 2156 (1978); ibid. 3, 2137 (1979). [2] J. Slonczewski, J. Magn. Magn. Mat. 62, 123, (1996). [3] L. Berger, Phys. Rev. B 54, 9353 (1996). [4] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2007). [5] M. D. Stiles and J. Miltat, Top. Appl. Phys. 101, 225 (2006). [6] J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater. 320, 1217 (2007). [7] O. W. Richardson, Phys. Rev. 26, 248 (1908). [8] A. Einstein and A. de Hass, Verhandlungen der Deutschen Physikalischen Gesellschaft, 17, 152 (1915). [9] P. Mohanty et al., Phys. Rev. B 70, 195301 (2004). [10] A. A. Kovalev et al., Phys. Rev. B 75, 014430 (2007). [11] G. Zolfagharkhani et al., NatureNanotech. 3, 720 (2008). [12] In addition to atomic-like angular momentum, there is a contribution to the total orbital angular momentum from itinerant motion through the lattice. The distinction be-tween “local” and “itinerant” orbital angular momentum is discussed in Ref. [13] . In this work, we consider only the atomic-like contribution. [13] T. Thonhauser et al., Phys Rev. Lett. 95, 137205 (2005). [14] A. S. N´ u˜ nez and A. H. MacDonald, Solid State. Comm. 139, 31 (2006). [15] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 (2002). [16] For H0=−¯h2∇2/2m+V(r), our definition of ˆ τlatis equivalent to/bracketleftbig V(r),ˆL/bracketrightbig . We use ˆ τlat=/bracketleftbig H0,ˆL/bracketrightbig in antici- pation of other forms of H0, in particular the k·pform of the Luttinger Hamiltonian. [17] T. Jungwirth et al., Rev. Mod. Phys. 78, 809 (2006). [18] A. M. Malik et al., Phys. Rev. B 59, 2861 (1999). [19] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). [20] Ion Garate and A. H. MacDonald, Phys. Rev. B 80, 134403 (2009). [21] D. Chiba et al., Phys. Rev. Lett. 93, 216602 (2004).5 [22] M. Elsen et al., Phys. Rev B 73, 035303 (2006).

1106.4349v3.Effective_one_body_Hamiltonian_of_two_spinning_black_holes_with_next_to_next_to_leading_order_spin_orbit_coupling.pdf

arXiv:1106.4349v3 [gr-qc] 5 Sep 2013Effective one body Hamiltonian of two spinning black-holes w ith next-to-next-to-leading order spin-orbit coupling Alessandro Nagar Institut des Hautes Etudes Scientifiques, 91440 Bures-sur- Yvette, France (Dated: May 24, 2018) Building on the recently computed next-to-next-to-leadin g order (NNLO) post-Newtonian (PN) spin-orbitHamiltonian for spinningbinaries [1] we improv e theeffective-one-body(EOB) description of the dynamics of two spinning black-holes by including NNL O effects in the spin-orbit interaction. The calculation that is presented extends to NNLO the next-t o-leading order (NLO) spin-orbit Hamiltonian computed in Ref. [2]. The presentEOB Hamiltoni an reproduces the spin-orbit coupling through NNLO in the test-particle limit case. In addition, i n the case of spins parallel or antiparallel to the orbital angular momentum, when circular orbits exist , we find that the inclusion of NNLO spin-orbit terms moderates the effect of the NLO spin-orbit c oupling. PACS numbers: 04.25.-g,04.25.Nx I. INTRODUCTION Coalescing black-hole binaries are among the most promising gravitational wave (GW) sources for the cur- rently operating network of ground-based interferomet- ric GW detectors. Since the spin-orbit interaction can increase the binding energy of the last stable orbit, and thereby leading to large GW emission, it is reasonable to think that the first detections will concern binary sys- tems made of spinning binaries. For this reason, there is a urgent need of template waveforms accurately describ- ing the GW emission from coalescing spinning black-hole binaries. These template waveforms will be functions of at least eight intrinsic real parameters: the two masses m1andm2and the two spin-vectors S1andS2. Be- cause of the multidimensionality of the parameter space, it seems unlikely for state-of-the-art numerical simula- tions to densely sample this parameter space. This gives a boost to develop analytical methods for computing the needed, densely spaced, bank of accurate template wave- forms. Among the existing analytical methods for com- puting the motion and the dynamics of black hole (and neutron star) binaries, the most complete and the most promising is the effective-one-body approach (EOB) [3– 8]. Several recent works have shown the possibility of getting an excellent agreement between the EOB ana- lytical waveforms and the outcome of numerical simula- tions of coalescing black-hole (and inspiralling neutron- star [9, 10]) binaries. A considerable part of the current literature deals with nonspinning black-holesystems [11– 18], with different (though not extreme) mass ratios (see in particular [19, 20]) or in the (circularized) extreme- mass-ratio limit [21–25] (notably including spin [26]). The work at the interface between numerical relativity and the analytical EOB description of spinning binaries has been developing fast in recent years. The first EOB Hamiltonian which included spin effects was conceived in Ref. [6]. It was shown there that one could map the 3PN dynamics, together with the leading-order (LO) spin- orbit andspin-spin dynamical effects ofa binarysystems, onto an effective test-particle movingin a Kerr-typemet-ric, together with an additional spin-orbit interaction. In Ref. [27] the use of the nonspinning EOB Hamilto- nian augmented with PN-type spin-orbit and spin-spin terms allowed to carryout the first (and up to now, only) analytical exploratory study of the dynamics and wave- forms from coalescing spinning binaries with precessing spins. Recently, Ref. [2], building upon the PN-expanded Hamiltonian of [28], extended the EOB approach of [6] so to include the next-to-leading-order (NLO) spin-orbit couplings (see also Refs. [29, 30] for a derivation of these couplings in the harmonic-coordinates equations of mo- tion and Ref. [31] for a derivation using an effective field theory approach). Using this model (with the addition of EOB-resummed radiation reaction force [7, 22, 32]), Ref. [33] performed the first comparison with numerical- relativity simulations of nonprecessing, spinning, equal- mass, black-holes binaries. Then, building on Ref. [2, 6] and Ref. [34], Ref. [35] worked out an improved Hamil- tonian for spinning black-hole binaries. Recently, Hartung and Steinhoff [1] havecomputed the PN-expanded spin-orbit Hamiltonian at next-to-next-to- leading order (NNLO), pushing one PN order further the previous computation of Damour, Jaranowski and Sch¨ afer [28]. The result of Ref. [1] completes the knowl- edge of the PN Hamiltonian for binary spinning black- holes up to and including 3.5PN. This paper belongs to the lineage of Refs. [2, 6] and it aims at exploiting the PN-expanded Hamiltonian of Ref. [1] so as to obtain the NNLO-accurate spin-orbit interaction as it enters the EOB formalism. Note that, by contrast to Refs. [35] and [2], we shall not discuss here spin-spin interactions, nor shall we try to propose a specific way to incorporate our NNLO spin-orbit results into some complete, resummed EOB Hamiltonian. Al- though the Hamiltonian that we shall discuss here does not resum all the spin-orbit terms entering the formal “spinning test-particle limit”, we shall check that it con- sistently reproduces the “spinning test-particle” results of Ref. [35]. The paper is organized as follows: in Sec. II we re- call the structure of the PN-expanded spin-orbit Hamil-2 tonian(inArnowitt-Deser-Misner(ADM) coordinates)of Ref.[1]andthenweexpressitinthecenterofmassframe. Section III explicitly performs the canonical transforma- tion from ADM coordinates to EOB coordinates and fi- nally computes the effective Hamiltonian, and, in partic- ular, the effective gyro-gravitomagneticratios. In Sec. IV wediscuss the caseofcircularequatorialorbits, wederive the test-mass limit and we exploit the gauge freedom to simplify the expression of the final Hamiltonian. We adopt the notation of [2] and we use the letters a,b= 1,2 as particle labels. Then, ma,xa= (xi a),pa= (pai), andSa= (Sai) denote, respectively, the mass, the position vector, the linear momentum vector, and the spin vector of the ath body; for a∝ne}ationslash=bwe also define rab≡xa−xb,rab≡ |rab|,nab≡rab/rab,|·|stands here for the Euclidean length of a 3-vector. II. PN-EXPANDED HAMILTONIAN IN ADM COORDINATES We closely follow the procedure of Ref. [2]. The start- ing point of the calculation is the PN-expandend two- body Hamiltonian Hwhich can be decomposed as the sum of an orbital part, Ho, a spin-orbit part, Hso(linear in the spins) and a spin-spin term Hss(quadratic in the spins), that we quote here for completeness but that weare not going to discuss in the paper. It reads H(xa,pa,Sa) =Ho(xa,pa)+Hso(xa,pa,Sa) +Hss(xa,pa,Sa). (1) The orbital Hamiltonian Hoincludes the rest-mass con- tribution and is explicitly known (in ADM-like coordi- nates) up to the 3PN order [36, 37]. It has the structure Ho(xa,pa) =/summationdisplay amac2+HoN(xa,pa) +1 c2Ho1PN(xa,pa)+1 c4Ho2PN(xa,pa) +1 c6Ho3PN(xa,pa)+O/parenleftbigg1 c8/parenrightbigg .(2) The spin-orbit Hamiltonian Hsocan be written as Hso(xa,pa,Sa) =/summationdisplay aΩa(xb,pb)·Sa.(3) Here, the quantity Ωais the sum of three contributions: the LO ( ∝1/c2), the NLO ( ∝1/c4), and the NNLO one (∝1/c6), Ωa(xb,pb) =ΩLO a(xb,pb)+ΩNLO a(xb,pb)+ΩNNLO a(xb,pb). (4) The 3-vectors ΩLO aandΩNLO awere explicitly computed in Ref. [28], while ΩNNLO acan be read off Eq.(5) of Ref. [1]. We write them here explicitly for completeness. For the particle label a= 1, we have ΩLO 1=G c2r2 12/parenleftbigg3m2 2m1n12×p1−2n12×p2/parenrightbigg , (5a) ΩNLO 1=G2 c4r3 12/parenleftBigg/parenleftbigg −11 2m2−5m2 2 m1/parenrightbigg n12×p1+/parenleftbigg 6m1+15 2m2/parenrightbigg n12×p2/parenrightBigg +G c4r2 12/parenleftBigg/parenleftbigg −5m2p2 1 8m3 1−3(p1·p2) 4m2 1+3p2 2 4m1m2−3(n12·p1)(n12·p2) 4m2 1−3(n12·p2)2 2m1m2/parenrightbigg n12×p1 +/parenleftbigg(p1·p2) m1m2+3(n12·p1)(n12·p2) m1m2/parenrightbigg n12×p2+/parenleftbigg3(n12·p1) 4m2 1−2(n12·p2) m1m2/parenrightbigg p1×p2/parenrightBigg , (5b) ΩNNLO 1=G r2 12/bracketleftbigg/parenleftbigg7m2(p2 1)2 16m5 1+9(n12·p1)(n12·p2)p2 1 16m4 1+3p2 1(n12·p2)2 4m3 1m2 +45(n12·p1)(n12·p2)3 16m2 1m2 2+9p2 1(p1·p2) 16m4 1−3(n12·p2)2(p1·p2) 16m2 1m2 2 −3(p2 1)(p2 2) 16m3 1m2−15(n12·p1)(n12·p2)p2 2 16m2 1m2 2+3(n12·p2)2p2 2 4m1m3 2 −3(p1·p2)p2 2 16m2 1m2 2−3(p2 2)2 16m1m3 2/parenrightbigg n12×p1+/parenleftbigg −3(n12·p1)(n12·p2)p2 1 2m3 1m23 −15(n12·p1)2(n12·p2)2 4m2 1m2 2+3p2 1(n12·p2)2 4m2 1m2 2−p2 1(p1·p2) 2m3 1m2+(p1·p2)2 2m2 1m2 2 +3(n12·p1)2p2 2 4m2 1m2 2−(p2 1)(p2 2) 4m2 1m2 2−3(n12·p1)(n12·p2)p2 2 2m1m3 2−(p1·p2)p2 2 2m1m3 2/parenrightbigg n12×p2 +/parenleftbigg −9(n12·p1)p2 1 16m4 1+p2 1(n12·p2) m3 1m2 +27(n12·p1)(n12·p2)2 16m2 1m2 2−(n12·p2)(p1·p2) 8m2 1m2 2−15(n12·p1)p2 2 16m2 1m2 2 +(n12·p2)p2 2 m1m3 2/parenrightbigg p1×p2/bracketrightbigg +G2 r3 12/bracketleftbigg/parenleftbigg −3m2(n12·p1)2 2m2 1+/parenleftbigg −3m2 2m2 1+27m2 2 8m3 1/parenrightbigg p2 1+/parenleftbigg177 16m1+11 m2/parenrightbigg (n12·p2)2 +/parenleftbigg11 2m1+9m2 2m2 1/parenrightbigg (n12·p1)(n12·p2)+/parenleftbigg23 4m1+9m2 2m2 1/parenrightbigg (p1·p2) −/parenleftbigg159 16m1+37 8m2/parenrightbigg p2 2/parenrightbigg n12×p1+/parenleftbigg4(n12·p1)2 m1+13p2 1 2m1 +5(n12·p2)2 m2+53p2 2 8m2−/parenleftbigg211 8m1+22 m2/parenrightbigg (n12·p1)(n12·p2) −/parenleftbigg47 8m1+5 m2/parenrightbigg (p1·p2)/parenrightbigg n12×p2 +/parenleftbigg −/parenleftbigg8 m1+9m2 2m2 1/parenrightbigg (n12·p1)+/parenleftbigg59 4m1+27 2m2/parenrightbigg (n12·p2)/parenrightbigg p1×p2/bracketrightbigg +G3 r4 12/bracketleftbigg/parenleftbigg181m1m2 16+95m2 2 4+75m3 2 8m1/parenrightbigg n12×p1−/parenleftbigg21m2 1 2+473m1m2 16+63m2 2 4/parenrightbigg n12×p2/bracketrightbigg (5c) The expressions for ΩLO 2,ΩNLO 2andΩNNLO 2can be ob- tained from the above formulas by exchanging the parti- cle labels 1 and 2. Let us consider now the dynamics of the relative mo- tion of the two body system in the center of mass frame, which is defined by setting p1+p2= 0. Following [2], we rescale the phase-space variables R≡x1−x2and P≡p1=−p2of the relative motion as r≡R GM,p≡P µ≡p1 µ=−p2 µ,(6) whereM=m1+m2andµ≡m1m2/M. In addition, we rescale the original time variable Tand any part of the Hamiltonian as t≡T GM,ˆHNR≡HNR µ,(7) whereHNR≡H−Mc2denotes the “nonrelativistic” Hamiltonian, i.e. the Hamiltonian withouth the rest- mass contribution. As in [2] we work with the following two, basic combinations of the spin vectors: S≡S1+S2=m1ca1+m2ca2, (8) S∗≡m2 m1S1+m1 m2S2=m2ca1+m1ca2,(9)where we have also introduced the Kerr parameters of the individual black-holes, a1≡S1/(m1c) anda2≡ S2/(m2c). We recall that in the formal1“spinning test mass limit” where, for example, m2→0 andS2→0, while keeping a2=S2/(m2c) fixed, one has a “back- ground mass” M≃m1, a “background spin” Sbckgd≡ Mcabckgd≃S1=m1ca1, a “test mass” µ≃m2, and a “test spin” Stest=S2=m2ca2≃µcatest[with atest≡Stest/(µc)]. Then, in this limit the combina- tionS≃S1=m1ca1≃Mcabckgd=Sbckgdmea- sures the background spin, while the other combination, S∗≃m1ca2≃Mcatest=MStest/µmeasures the (spe- cific) test spin atest=Stest/(µc). Finally, since the use of the rescaled variables corresponds to a rescaling of the action by a factor 1 /(GMµ), it is also natural to work with the corresponding rescaled variables ¯SX≡SX GMµ, (10) for any label X (X= 1 ,2, ,∗). 1As noted in Ref. [2] this formal limit is not relevant for the p hysi- cally mostimportant case of binary black holes, forwhich a2→0 andm2→0.4 Using the definitions (6)-(10), the center-of-mass spin- orbit Hamiltonian (divided by µ) in terms of the rescaled variables has the structure ˆHso(r,p,¯S,¯S∗)≡Hso(r,p,¯S,¯S∗) µ(11) =1 c2ˆHso LO(r,p,¯S,¯S∗) +1 c4ˆHso NLO(r,p,¯S,¯S∗) +1 c6ˆHso NNLO(r,p,¯S,¯S∗)+O/parenleftbigg1 c8/parenrightbigg , (12) and it can be written as ˆHso(r,p,¯S,¯S∗) =ν c2r2/parenleftbig gADM s(¯S,n,p)+gADM S∗(¯S∗,n,p)/parenrightbig , (13)with the following definitions: ν≡µ/Mis the sym- metric mass ratios and ranges from 0 (test-body limit) to 1/4 (equal-mass case); the notation ( V1,V2,V3)≡ V1·(V2×V3) =ǫijkVi 1Vj 2Vk 3stands for the Euclidean mixed products of 3-vectors; n≡r/|r|;gADM SandgADM S∗ are the two (dimensionless) gyro-gravitomagnetic ratios as introduced (up to NLO accuracy) in [2]. These two coefficients parametrize the coupling between the spin vectors and the apparent gravito-magnetic field seen in the rest-frame of a moving particle. Their explicit ex- pressions including the NNLO contribution read gADM S= 2+1 c2/parenleftbigg /parenleftbigg/parenleftbigg19 8νp2+3 2ν(n·p)2−/parenleftBig 6+2ν/parenrightBig1 r/parenrightbigg /parenrightbigg/parenrightbigg +1 c4/braceleftBigg −9 8ν/parenleftBig 1−22 9ν/parenrightBig p4−3 4ν/parenleftBig 1−9 4ν/parenrightBig p2(n·p)2+15 16ν2(n·p)4 +1 r/bracketleftbigg −157 8ν/parenleftBig 1+39 314ν/parenrightBig p2−16ν/parenleftBig 1+45 256ν/parenrightBig (n·p)2+1 r21 2/parenleftBig 1+ν/parenrightBig/bracketrightbigg/bracerightBigg , (14a) gADM S∗=3 2+1 c2/parenleftbigg /parenleftbigg/parenleftbigg/parenleftBig −5 8+2ν/parenrightBig p2+3 4ν(n·p)2−/parenleftBig 5+2ν/parenrightBig1 r/parenrightbigg /parenrightbigg/parenrightbigg +1 c4/braceleftBigg 1 16/parenleftBig 7−37ν+39ν2/parenrightBig p4+9 16ν(2ν−1)p2(n·p)2 +1 r/bracketleftbigg1 8/parenleftBig 27−129ν−39 2ν2/parenrightBig p2−6ν/parenleftBig 1+15 32ν/parenrightBig (n·p)2+1 r/parenleftbigg75 8+41 4ν/parenrightbigg/bracketrightbigg/bracerightBigg . (14b) The label “ADM” on the gyro-gravitomagnetic ra- tios (14) is a reminder that, although the LO values are coordinate independent, both the NLO and NNLO con- tributions to these ratios actually depend on the defi- nition of the phase-space variables ( r,p). In the next Section we shall introduce the two, related, effective gyro-gravitomagneticratios that enter the effective EOB Hamiltonian, written in effective (or EOB) coordinates, according to the prescriptions of [2]. III. EFFECTIVE HAMILTONIAN AND EFFECTIVE GYRO-GRAVITOMAGNETIC RATIOS Following Ref. [2], two operations have to be per- formed on the Hamiltonian written in the center of massframe so to cast it in a form that can be resummed in a way compatible to previous EOB work. First of all, one needs to transform the (ADM) phase-space coordi- nates (xa,pa,Sa) by a canonical transformation compat- ible with the one used in previous EOB work. Second, one needs to compute the effective Hamiltonian corre- sponding to the canonically transformed realHamilto- nian. Following the same procedure adopted in [2], we start by performing the purely orbital canonical trans- formation which was found to be needed to go from the ADM coordinates used in the PN-expanded Hamiltonian to the coordinates used in the EOB dynamics. Since in Ref. [2] one was concerned only with the NLO spin-orbit interaction, it was enough to consider the 1PN-accurate transformation. In the present study, because one is workingat NNLO in the spin-orbitinteraction, oneneeds to take into account the complete 2PN-accurate canon-5 ical transformation introduced in [3]. The transforma- tion changes the ADM phase-space variables ( r,p,¯S,¯S∗) to (r′,p′,¯S,¯S∗) and it is explicitly given by Eqs. (6.22)- (6.23) of [3]. To our purpose, we actually need to use theinverserelations r=r(r′,p′) andp=p(r′,p′), soto replace ( r,p) with (r′,p′) in Eq. (13). The needed transformation is easily found by solving, by iteration, Eqs. (6.22)-(6.23) of [3], and we explicitly quote it here for future convenience. It reads ri−r′ i=1 c2/bracketleftBigg −/parenleftBig 1+ν 2/parenrightBigr′i r′+ν 2p′2r′ i+ν(r′·p′)p′ i/bracketrightBigg +1 c4/braceleftBigg/bracketleftBigg 1 4r′2/parenleftbig −ν2+7ν−1/parenrightbig +3ν 4/parenleftBigν 2−1/parenrightBigp′2 r′−ν 8(1+ν)p′4−ν/parenleftbigg 2+5 8ν/parenrightbigg(r′·p′)2 r′3/bracketrightBigg r′ i +/bracketleftBigg ν 2/parenleftBig −5+ν 2/parenrightBigr′·p′ r′+ν 2(ν−1)p′2(r′·p′)/bracketrightBigg p′ i/bracerightBigg , (15) pi−p′ i=1 c2/bracketleftBigg −/parenleftBig 1+ν 2/parenrightBigr′·p′ r′3r′ i+/parenleftBig 1+ν 2/parenrightBigp′ i r′−ν 2p′2p′ i/bracketrightBigg +1 c4/braceleftBigg/bracketleftBigg 1 r′2/parenleftbigg5 4−3 4ν+ν2 2/parenrightbigg +ν 8(1+3ν)p4−ν 4/parenleftbigg 1+7 2ν/parenrightbiggp2 r′+ν/parenleftBig 1+ν 8/parenrightBig(r′·p′)2 r′3/bracketrightBigg p′ i +/bracketleftBigg/parenleftbigg −3 2+5 2ν−3 4ν2/parenrightbiggr′·p′ r′4+3 4ν/parenleftBigν 2−1/parenrightBig p2r′·p′ r′3+3 8ν2(r′·p′)3 r′5/bracketrightBigg r′i/bracerightBigg . (16) As pointed out in [3], in the test-mass limit ( ν→0) one hasr′i=/bracketleftbig 1+1/(2c2r)/bracketrightbig ri, which is the relation be- tween Schwarzschild ( r′) and isotropic ( r) coordinates in a Schwarzschild spacetime2. When this transforma- tion is applied to to the spin-orbit Hamiltonian in ADMcoordinates, Eq. (13), one gets a transformed Hamilto- nian of the form ˆH′(r′,p′,¯S,¯S∗) =ˆH′ o(r′,p′,¯S,¯S∗) + ˆH′so(r′,p′,¯S,¯S∗), with the NNLO spin-orbit contribu- tion that explicitly reads ˆH′so NNLO(r′,p′,¯S,¯S∗) =ν r′2/braceleftBigg (¯S∗,n′,p′)/bracketleftBigg ν r′2/parenleftBig −8+ν 2/parenrightBig +1 r′/bracketleftbigg/parenleftbigg −13 4ν−3 4ν2/parenrightbigg p′2+/parenleftbigg43 4ν−75 16ν2/parenrightbigg (n′·p′)2/bracketrightbigg +/parenleftbigg −3 8ν+9 16ν2/parenrightbigg p′4+/parenleftbigg9 4ν−3 16ν2/parenrightbigg p′2(n′·p′)2+135 16ν2(n′·p′)2/bracketrightBigg , +(¯S∗,n′,p′)/bracketleftBigg −1 r′2/parenleftbigg1 2+53 8ν+5 8ν2/parenrightbigg +1 r′/bracketleftbigg/parenleftbigg1 4−53 16ν+3 8ν2/parenrightbigg p′2+/parenleftbigg5 4+121 8ν−3ν2/parenrightbigg (n′·p′)2/bracketrightbigg +/parenleftbigg7 16−3 16ν+ν2 4/parenrightbigg p′4+/parenleftbigg57 16ν−3 4ν2/parenrightbigg p′2(n′·p′)2+15 2ν2(n′·p′)2/bracketrightBigg/bracerightBigg ,(17) 2As a check of the transformation (15)-(16) one can explicitl y verify that it preserves the orbital angular momemntum at 2P Norder, i.e. r′×p′=r×p+O/parenleftbigg1 c6/parenrightbigg .6 where we introduced the radial unit vector n′=r′/|r′|. With this result in hands, we can further perform on it a secondary purely spin-dependent , canonical transfor- mation that affects both the NLO and NNLO spin orbit terms. This transformation can be thought as a gauge transformation related to the arbitrariness in choosing a spin-supplementary condition and in defining a local frame to measure the spin vectors. Such gauge condition can then be conveniently chosen so to simplify the spin- orbit Hamiltonian. This procedure was pushed forward, at NLO accuracy in Ref. [2]. In that case, the canonical transformation was defined by means of a 2PN-accurate generating function, that was chosen proportional to the spins and with two arbitrary ( ν-dependent) dimension- less coefficients a(ν) andb(ν). Using rescaled variables, the NLO generating function of [2] reads ¯Gs2PN=1 c4ν(n′·p′) r′/parenleftbig a(ν)(¯S,n′,p′)+b(ν)(¯S∗,n′,p′)/parenrightbig . (18) In Ref. [2] the parameters a(ν) andb(ν) were selected so to remove the terms proportional to p2in the final (effective) Hamiltonian. Letusrecallthat, atlinearorder in the¯Gs2PN, that was enough for the NLO case, the new Hamiltonian was computed as ˆH′′so(y′′) =ˆH′so(y′′)− {ˆH′,¯Gs2PN}(y′′), were we address collectively with y′′= (r′′,p′′,¯S′′,S′′∗) the new phase space-variables. We wish now to introduce a more general gauge trans- formation such to act also on the NNLO terms of the Hamiltonian. To do so, in addition to the NLO part ¯Gs2PNof the spin-dependent generating function men- tioned above, one also needs to introduce a NNLO con- tribution of the form ¯Gs3PN=1 c6ν/braceleftBigg (n′·p′) r′/bracketleftbiggα(ν) r′+β(ν)(n′·p′)2+γ(ν)p′2/bracketrightbigg ×(¯S,n′,p′) +(n′·p′) r′/bracketleftbiggδ(ν) r′+ζ(ν)(n′·p′)2+η(ν)p′2/bracketrightbigg ×(¯S∗,n′,p′)/bracerightBigg , (19) with six, arbitrary, ν-dependent dimensionless coeffi- cients. We shall then consider the effect of a spin- dependent generating function of the form ¯Gs=¯Gs2PN+ ¯Gs3PN. Since ¯Gsstarts at 2PN order, it turns out that possible quadratic terms in the generating function are of order c−8, i.e. at 4PN and thus are of higher order than the NNLO accuracy that we are currently consid- ering in the spin-orbit Hamiltonian. The consequence is that the purely spin-dependent gauge transformation at NNLO will involve onlythe contribution linear in ¯Gs. In other terms, we only need to consider the following transformation on the Hamiltonian ˆH′′(y′′) =ˆH′(y′′)−{ˆH′,¯Gs}(y′′).(20)Extracting from this equation the spin-dependent terms, we find that the relevant terms in the new spin-orbit Hamiltonian up to NNLO are then given by ˆH′′so LO(r′′,p′′,¯S′′,¯S′′∗) =H′so LO(y′′), ˆH′′so NLO(r′′,p′′,¯S′′,¯S′′∗) =H′so NLO(y′′)−{H′ oN,¯Gs2PN}(y′′), ˆH′′so NNLO(r′′,p′′,¯S′′,¯S′′∗) =ˆH′so NNLO(y′′) −/bracketleftBig {ˆH′ oN,¯Gs3PN} +{ˆH′ o1PN,¯Gs2PN} +{H′so LO,¯Gs2PN}/bracketrightBig (y′′).(21) Note that the single prime in these equations explic- itly addresses the various contribution to the spin-orbit Hamiltonian as computed after the purely orbital canon- ical transformation mentioned above (note however that only the functional form of ˆH′ o1PNis modified by the ac- tion of the orbital canonical transformation). Further simplifications occur in the third Poisson bracket of Eq. (21). First of all, since we are inter- ested in computing only the contribution to the spin- orbit interaction, the terms quadratic in spins are ne- glected. In addition, from the basic relation {Si,Sj}= ǫijkSkonecanshowbyastraightforwardcalculationthat {Hso′ LO,¯Gs2PN}= 0 (always at linear order in the spin). Consequently, the effect of the purely spin-dependent canonical transformation is fully taken into account by the two Poisson brackets involving the generating func- tions¯Gs2PNand¯Gs3PN, and the purely orbital contribu- tions to the Hamiltonian, ˆH′ oNandˆH′ o1PN. For simplicity of notation, we shall omit hereafter the double primes from the transformed Hamiltonian. We now need to connect the real Hamiltonian Hto the effec- tive oneHeff, which is more closely linked to the descrip- tion of the EOB quasigeodesic dynamics. The relation between the two Hamiltonians is given by [3] Heff µc2≡H2−m2 1c4−m2 2c4 2m1m2c4(22) wheretherealHamiltonian Hcontainstherest-masscon- tributions Mc2. In terms of the nonrelativistic Hamilto- nianˆHNR, this equation is equivalent to ˆHeff c2= 1+ˆHNR c2+ν 2(ˆHNR)2 c4, (23) where it is explicitly ˆHNR=/parenleftBigg ˆHoN+ˆHo1PN c2+ˆHo2PN c4+ˆHo3PN c6/parenrightBigg +/parenleftBiggˆHso LO c2+ˆHso NLO c4+ˆHso NNLO c6/parenrightBigg .(24) By expanding in powers of 1 /c2up to 3PN fractional ac- curacy (and in powers of the spin) the exact effective7 Hamiltonian, one easily finds that the spin-orbit part of the effective Hamiltonian ˆHeff(i.e., the part which is linear-in-spin) reads ˆHso eff=1 c2ˆHso LO+1 c4/parenleftBig ˆHso NLO+νˆHoNˆHso LO/parenrightBig +1 c6/bracketleftBig ˆHso NNLO+ν/parenleftBig ˆHoNˆHso NLO+ˆHo1PNHso LO/parenrightBig/bracketrightBig . (25) Combining this result with the effect of the generating function discussed above, we get the transformed spin- orbit part of the effective Hamiltonian in the form as ˆHso eff=ν c2r2/parenleftbig geff S(¯S,n,p)+geff S∗(¯S∗,n,p)/parenrightbig .(26)Theeffectivegyro-gravitomagneticratios geff Sandgeff S∗dif- fer from the ADM ones introduced above because of the effect of the (orbital+spin) canonical transformation and becauseofthe transformationfrom HtoHeff. They have the structure geff S= 2+1 c2geffNLO S(a)+1 c4geffNNLO S(a;α,β,γ) (27) geff S∗=3 2+1 c2geffNLO S(b)+1 c4geffNNLO S∗(b;δ,ζ,η),(28) where we made it apparent the dependence on the ( ν- dependent) NLO and NNLO gauge parameters. Includ- ing the new NNLO terms, they read geff S= 2+1 c2/bracketleftBigg/parenleftbigg3 8ν+a/parenrightbigg p2−/parenleftbigg9 2ν+3a/parenrightbigg (n·p)2/parenrightBigg −1 r(ν+a)/bracketrightBigg +1 c4/bracketleftBigg −1 r2/parenleftbigg 9ν+3 2ν2+a+α/parenrightbigg +1 r/bracketleftbigg (n·p)2/parenleftbigg35 4ν−3 16ν2+6a−4α−3β−2γ/parenrightbigg +p2/parenleftbigg −17 4ν+11 8ν2−3a 2+α−γ/parenrightbigg/bracketrightbigg +/parenleftbigg9 4ν−39 16ν2+3a 2+3β−3γ/parenrightbigg p2(n·p)2+/parenleftbigg135 16ν2−5β/parenrightbigg (n·p)4 +/parenleftbigg −5 8ν−a 2+γ/parenrightbigg p4/bracketrightBigg , (29) geff S∗=3 2+1 c2/bracketleftBigg/parenleftbigg −5 8+1 2ν+b/parenrightbigg p2−/parenleftbigg15 4ν+3b/parenrightbigg (n·p)2−1 r/parenleftbigg1 2+5 4ν+b/parenrightbigg/bracketrightBigg +1 c4/bracketleftBigg −1 r2/parenleftbigg1 2+55 8ν+13 8ν2+b+δ/parenrightbigg +1 r/bracketleftBigg (n·p)2/parenleftbigg5 4+109 8ν+3 4ν2+6b−4δ−3ζ−2η/parenrightbigg +p2/parenleftbigg1 4−59 16ν+3 2ν2−3b 2+δ−η/parenrightbigg/bracketrightBigg +/parenleftbigg57 16ν−21 8ν2+3b 2+3ζ−3η/parenrightbigg p2(n·p)2+/parenleftbigg15 2ν2−5ζ/parenrightbigg (n·p)4 +/parenleftbigg7 16−11 16ν−ν2 16−b 2+η/parenrightbigg p4/bracketrightBigg . (30) This is the central result of the paper. The NNLO con- tribution to the gyro-gravitomagnetic ratios computed here is the crucial, new, information that it is needed to improve to the next PN order the spin-dependent EOB Hamiltonian (either in the version of Ref. [2] or [35]). Let us recall in this respect that in the EOB approach of [2] the relative dynamics can be equivalently represented by the dynamics of a spinning effective particle with effec- tive spin σmoving onto a ν-deformed Kerr-type metric.The gyro-gravitomagnetic ratios enter the definition of the test-spin vector σas σ=1 2/parenleftbig geff S−2/parenrightbig S+1 2/parenleftbig geff S∗−2/parenrightbig S∗,(31) that can then be inserted in Eqs. (4.16) of Ref. [2] to get the spin-orbit interaction additional to the leading Kerr- metric part. Together with Eqs. (4.17), (4.18) and (4.19) of Ref. [2] this defines the real EOB-improved,resummed8 Hamiltonian for spinning binaries at NNLO in the spin- orbit interaction. IV. LIMITS, CHECKS AND GAUGE FIXING A. The extreme-mass-ratio limit The effective spin-orbit Hamiltonian (26) is naturally connected to the test-mass ( ν→0) Hamiltonian explic- itly obtained3in [34]. To show this in a concrete case, let us consider the spin-orbit Hamiltonian of a spinning test-particle on Schwarzschild spacetime written explic- itly using isotropic coordinates, as given by Eq. (5.12) of Ref. [34]. By considering the Schwarzschild metric writ- ten as ds2=−f(r)dt2+h(r)(dx2+dy2+dz2),(32) whererlabelsheretheisotropicradius4,r2=x2+y2+z2, (that is meant to be expressed in rescaled units, where nowM≃m1is the background mass and µ≃m2is the test-particle mass), with h=/parenleftbigg 1+1 2c2r/parenrightbigg4 , (33) and using rescaled variables (and making explicit the speed of light) Eq. (5.12) of Ref. [34] can be written as ˆHso ISO=ν c2r2gISO 0/parenleftbig n,p,¯S∗ 0/parenrightbig . (34) In this equation, ¯S∗ 0is the (rescaled) spin of the test-mass and we have introduced the test-mass gyro- gravitomagnetic ratio in isotropic coordinates gISO 0, that is known in closed form [34] and reads gISO 0=h−3/2 √Q/parenleftbig 1+√Q/parenrightbig/bracketleftbigg 1−1 2c2r+/parenleftbigg 2−1 2c2r/parenrightbigg/radicalbig Q/bracketrightbigg , (35) where Q= 1+1 c2p2 h. (36) By transforming the Hamiltonian (34) from isotropic to Schwarzschild coordinates using the ν→0 limit of the (purely orbital) canonical transformation given by 3Note in passing that the simple procedure described in Ref. [ 28] to obtain the spin-orbit Hamiltonian is totally general and can be applied, in particular, to the test-mass case. 4Note that we use the same notation for the isotropic radius on Schwarzschild spacetime and the ADM radial coordinates. Th ere is no ambiguity here since for the Schwarzschild spacetime A DM coordinates do actually coincide with isotropic coordinat esEqs. (15)-(16), expanding in powers of 1 /c2, (and drop- ping again the primes for simplicity) one obtains ˆHso Schw=ν c2r2gSchw 0/parenleftbig n,p,¯S∗/parenrightbig . (37) with gSchw 0=3 2−1 c2/parenleftbigg1 2r+5 8p2/parenrightbigg +1 c4/bracketleftbigg −1 2r2+1 r/parenleftbigg5 4(n·p)2+1 4p2/parenrightbigg +7 16p4/bracketrightbigg . (38) In theν→0 (Schwarzschild) limit, one has lim ν→0(H− const.)/µ= limν→0ˆHeff(when dropping inessential con- stants), ¯S∗=¯S∗ 0and¯S= 0. One then finds that the result (38) agrees in the ν→0 limit with Eq. (30) when the gauge parameters ( b,δ,ζ,η) are simply zero. In addition, in the ν→0 limit where the background is a Kerr black hole, i.e. ¯S∝ne}ationslash= 0, Eq. (29) consistently exhibits that both the NLO and NNLO contributions become pure gauge, that can just be set to zero by de- manding ( a,α,β,γ) to vanish. B. Circular equatorial orbits Let us consider now the situation where both individ- ualspinsareparallel(orantiparallel)tothe (rescaled)or- bital angular momentum vector ℓ=rn×p. [Note that in this Section the quantity rdenotes the EOB radial coordinate (further modified by spin-dependent gauge terms, see below)]. In this case, circular orbits exists (but in the general case, when the spin vectors are not aligned with ℓ, there are no circular orbits). One can then consistently set everywhere the radial momentum to zero, pr≡n·p= 0 and express the total (orbital plus spin-orbit part) real, PN-expanded and canonically transformed Hamiltonian, H(y′′)≡H′′ o(y′′) +H′′so(y′′) (dropping hereafter the primes for simplicity) as a func- tion ofr,ℓ(using the link p2=ℓ2/r2, whereℓ≡ |ℓ|) and of the two scalars ˆ aand ˆa∗measuring the projection of the basic spin combinations SandS∗along the direction of the orbital angular momentum ℓ. Following the same notation of [2], we introduce here the dimensionless spin variables corresponding to SandS∗ ˆa≡cS GM2,ˆa∗≡cS∗ GM2, (39) and we define the projections as ˆa·ℓ= ˆaℓ,ˆa∗·ℓ= ˆa∗ℓ, (40) with the scalars ˆ aand ˆa∗positive or negative depending on whether say ˆais parallel or antiparallel to ℓ. The se- quence of circular (equatorial) orbits5is then determined 5To avoid confusion, let us stress that we are here considerin g the circular orbits of the PN-expanded real Hamiltonian and notthe9 by the constraint ∂H(r,ℓ,ˆa,ˆa∗)/∂r= 0, (41) (or equivalently by ∂Heff/∂r= 0). To start with, let us consider first the link between the nonrelativistic en- ergy (per unit mass µ) and the orbital angular momen- tum along circular orbits. The relevance of this quantity in the nonspinning case, say Ecirc(ℓ)≡HNR o(ℓ)/µ, was pointed out in Ref. [38], since it provides a completely gauge-invariant characterization of the dynamics of cir- cular orbits. When the black holes are spinning, the same property of gauge-invariance is maintained when the spins are parallel (or antiparallel) to the orbital an-gular momentum, so that it is meaningful to explicitly compute Ecirc(ℓ,ˆa,ˆa∗)≡HNR o(ℓ)/µ+HNR so(ℓ,ˆa,ˆa∗)/µin thiscase. Since itis agauge-invariantquantity, theresult is independent of the canonical transformations that we have performed on the two-body Hamiltonian in ADM coordinates, so that it gives a reliable check of the pro- cedure we followed. As a first operation, we need to solve, iteratively, the constraint (41) so to obtain the EOB coordinate radius rin function of ( ℓ,ˆa,ˆa∗). This function (that is not invariant and depends explicitly on the gauge parameters) reads (putting back the explicit double primes on ras a reminder that this is the EOB radial coordinate) r′′(ℓ,ˆa,ˆa∗) =ℓ2/braceleftBigg 1+1 c2/bracketleftbigg −3 ℓ2+1 c1 ℓ3/parenleftbigg 6ˆa+9 2ˆa∗/parenrightbigg/bracketrightbigg +1 c4/bracketleftBigg (−9+3ν)1 ℓ4+1 c1 ℓ5/parenleftBigg ˆa/parenleftbigg 33−17 8ν+a(ν)/parenrightbigg +ˆa∗/parenleftbigg157 8−5 2ν+b(ν)/parenrightbigg/parenrightBigg/bracketrightBigg +1 c6/bracketleftBigg/parenleftbigg −54+257 3ν−41 16π2ν/parenrightbigg1 ℓ6+1 c1 ℓ7/parenleftBigg ˆa/parenleftbigg1197 4−1973 16ν+3 4ν2+6a(ν)+α(ν)+γ(ν)/parenrightbigg +ˆa∗/parenleftbigg2777 16−1633 16ν+7 16ν2+6b(ν)+δ(ν)+η(ν)/parenrightbigg/parenrightBigg/bracketrightBigg/bracerightBigg . (42) The function Ecirc(ℓ,ˆa,ˆa∗) is obtained by inserting this relation in the expression of H(r,ℓ,ˆa,ˆa∗), and it reads Ecirc(ℓ,ˆa,ˆa∗) =−1 2ℓ2/braceleftBigg 1+1 c2/parenleftbigg1 4(9+ν)1 ℓ2−1 c1 ℓ3(4ˆa+3ˆa∗)/parenrightbigg +1 c4/bracketleftBigg 1 8/parenleftbig 81−7ν+ν2/parenrightbig1 ℓ4−1 c1 ℓ5/parenleftbigg/parenleftbigg 36+3 4ν/parenrightbigg ˆa+99 4ˆa∗/parenrightbigg/bracketrightBigg +1 c6/bracketleftBigg 2 ℓ6o1(ν)+1 c1 ℓ7/parenleftbigg ˆa/parenleftbigg −324+54ν−5 8ν2/parenrightbigg +ˆa∗/parenleftbigg −1701 8+195 4ν/parenrightbigg/parenrightbigg/bracketrightBigg/bracerightBigg (43) where we defined 2o1(ν) =3861 64−8833 192ν+41 32π2ν−5 32ν2+5 64ν3,(44) for the 3PN-accurate orbital part, with a slight abuse of the notationofRef. [38]. Note that, asit should, Eq. (43) circular orbits of the EOB-resummed real Hamiltonian, as do ne in Sec. V of Ref. [2]. This analysis is postponed to future wor k.is totally independent of the eight gauge parameters. We have further verified that that the same result (43) is ob- tained starting from the PN-expanded Hamiltonian writ- ten in ADM coordinates and in the center of mass frame, Eqs. (13)-(14). As a last remark, let us note that, as it was the case at NLO [2], the effective gyro-gravitomagnetic ratios for circular orbits are gauge independent also at NNLO. To see this explicitly, one just imposes in Eqs. (29)-(30) the10 condition ( n·p) = 0 and the (approximate) link p2=1 r+1 c23 r2+O(ˆa,ˆa∗), (45) that is obtained by inverting Eq. (42) at 1PN accuracy and neglecting the linear-in-spin terms (that would give quadratic-in-spin contributions). At NNLO, one obtains geff Scirc= 2−1 c25 8ν1 r−1 c4/parenleftbigg51 4ν+1 8ν2/parenrightbigg1 r2,(46) geff S∗ circ=3 2−1 c2/parenleftbigg9 8+3 4ν/parenrightbigg1 r −1 c4/parenleftbigg27 16+39 4ν+3 16ν2/parenrightbigg1 r2.(47) These equations indicate that the inclusion of NNLO spin-orbit coupling has the effect of reducing the magni- tude of the gyro-gravitomagneticratios. The NNLO and NLO spin-orbit contributions act then in the same direc- tion, so to reduce the repulsive effect ofthe LO spin-orbit coupling which is, by itself, responsible for allowing the binary system to orbit on very close, and very bound, or- bits (see also Ref. [6] and the discussion in Sec. VI of [2]). We postpone to future work a detailed quantitative anal- ysis of the properties of the binding energy entailed by Eqs. (46)-(47). C. Gauge fixing We can finally exploit the flexibility introduced by the spin-dependent gauge transformation so to consid-erably simplify the expression of the effective gyro- gravitomagnetic ratios, Eqs. (29)-(30). This is helpful in the study of the dynamics of a binary system with gener- ically oriented spins. Reference [2] found it convenient to fix the NLO gauge parameters ( a(ν),b(ν)) to a(ν) =−3 8ν, b(ν) =5 8−ν 2(48) so to suppress the dependence on p2at NLO. One can follow the same route at NNLO, i.e., by choosing the six gauge parameters so to suppress the terms proportional top2,p4andp2(n·p)2. Inthiswaythespin-orbitHamil- tonian is expressed in a way that the circular (gauge- invariant) part is immediately recognizable. With ( a,b) fixed as per Eq. (48), one easily sees that the aforemen- tioned NNLO terms are removed by the following choices of the NNLO gauge parameters α(ν) =11 8ν(3−ν), (49) β(ν) =1 16ν(13ν−2), (50) γ(ν) =7 16ν, (51) δ(ν) =1 16(9+54ν−23ν2), (52) η(ν) =1 16/parenleftbig −2+7ν+ν2/parenrightbig , (53) ζ(ν) =1 16/parenleftbig −7−8ν+15ν2/parenrightbig . (54) The effective gyro-gravitomagnetic ratios are then sim- plified to geff S= 2+1 c2/braceleftbigg −1 r5 8ν−27 8ν(n·p)2/bracerightbigg +1 c4/braceleftbigg −1 r2/parenleftbigg51 4ν+ν2 8/parenrightbigg +1 r/parenleftbigg −21 2ν+23 8ν2/parenrightbigg (n·p)2+5 8ν(1+7ν)(n·p)4/bracerightbigg , (55) geff S∗=3 2+1 c2/braceleftbigg −1 r/parenleftbigg9 8+3 4ν/parenrightbigg −/parenleftbigg9 4ν+15 8/parenrightbigg (n·p)2/bracerightbigg +1 c4/braceleftbigg −1 r2/parenleftbigg27 16+39 4ν+3 16ν2/parenrightbigg +1 r/parenleftbigg69 16−9 4ν+57 16ν2/parenrightbigg (n·p)2+/parenleftbigg35 16+5 2ν+45 16ν2/parenrightbigg (n·p)4/bracerightbigg .(56) This result extends the information of Eqs. (3.15a) and (3.15b) of Ref. [2] at NNLO accuracy. The circular- orbit result mentioned above is immediately recovered “at sight” by imposing ( n·p) = 0. With this result in hand, one can proceed similarly to Sec. IV of Ref. [2] (as outlined above) to introduce the spin-dependent EOB-resummed real Hamiltonian including NNLO spin-orbit couplings.11 V. CONCLUSIONS Building on the recently-computed next-to-next-to- leading order PN-expanded spin-orbit Hamiltonian for two spinning compact objects [1], we computed the ef- fective gyro-gravitomagnetic ratios entering the EOB Hamiltonian at next-to-next to-leading order in the spin- orbit interaction. This result is obtained by a straightfor- ward extension of the procedure followed in [2] to derive the NLO spin-orbit EOB Hamiltonian. We discussed in detail the test-particle limit and the case of equatorial circular orbits, when the spins are parallel or antiparallel to the orbital angular momentum. In this case, one finds that the NNLO spin-orbit terms moderate the effect ofthe spin-orbit coupling (as the NLO terms was already doing [2]). Finally, while this paper was under review process, Ref. [39] appeared in the archives as a preprint: that study uses the Lie method to obtain effective gyro- gravitomagnetic coefficients that are physically equiva- lent to the ones presented here. In addition, it also works out two classes of EOB Hamiltonians that are different from the one considered here. ACKNOWLEDGMENTS I am grateful to Thibault Damour for suggesting this project and for maieutic discussions during its develope- ment. [1] J. Hartung and J. Steinhoff, arXiv:1104.3079 [gr-qc]. [2] T. Damour, P. Jaranowski, G. Schaefer, Phys. Rev. D78, 024009 (2008). [arXiv:0803.0915 [gr-qc]]. [3] A. Buonanno, T. Damour, Phys. Rev. D59, 084006 (1999). [gr-qc/9811091]. [4] A. Buonanno, T. Damour, Phys. Rev. D62, 064015 (2000). [gr-qc/0001013]. [5] T. Damour, P. Jaranowski, G. Schaefer, Phys. Rev. D62, 084011 (2000). [gr-qc/0005034]. [6] T. Damour, Phys. Rev. D 64, 124013 (2001) [arXiv:gr- qc/0103018]. [7] T. Damour, B. R. Iyer, A. Nagar, Phys. Rev. D79, 064004 (2009). [arXiv:0811.2069 [gr-qc]]. [8] T. Damour, A. Nagar, Phys. Rev. D81, 084016 (2010). [arXiv:0911.5041 [gr-qc]]. [9] L. Baiotti, T. Damour, B. Giacomazzo, A. Nagar, L. Rezzolla, Phys. Rev. Lett. 105, 261101 (2010). [arXiv:1009.0521 [gr-qc]]. [10] L. Baiotti, T. Damour, B. Giacomazzo, A. Nagar and L. Rezzolla, arXiv:1103.3874 [gr-qc]. [11] A. Buonanno, G. B. Cook, F. Pretorius, Phys. Rev. D75, 124018 (2007). [gr-qc/0610122]. [12] A. Buonanno, Y. Pan, J. G. Baker, J. Centrella, B. J. Kelly, S. T. McWilliams and J. R. van Meter, Phys. Rev. D76, 104049 (2007) [arXiv:0706.3732 [gr-qc]]. [13] T. Damour, A. Nagar, Phys. Rev. D77, 024043 (2008). [arXiv:0711.2628 [gr-qc]]. [14] T.Damour, A.Nagar, E. N.Dorband, D.Pollney, L.Rez- zolla, Phys. Rev. D77, 084017 (2008). [arXiv:0712.3003 [gr-qc]]. [15] M. Boyle, A. Buonanno, L. E. Kidder, A. H. Mroue, Y. Pan, H. P. Pfeiffer, M. A. Scheel, Phys. Rev. D78, 104020 (2008). [arXiv:0804.4184 [gr-qc]]. [16] T. Damour, A. Nagar, M. Hannam, S. Husa and B. Bruegmann, Phys. Rev. D 78, 044039 (2008) [arXiv:0803.3162 [gr-qc]]. [17] T. Damour and A. Nagar, Phys. Rev. D 79, 081503 (2009) [arXiv:0902.0136 [gr-qc]]. [18] A. Buonanno, Y. Pan, H. P. Pfeiffer, M. A. Scheel, L. T. Buchman, L. E. Kidder, Phys. Rev. D79, 124028 (2009). [arXiv:0902.0790 [gr-qc]].[19] Y. Pan, A. Buonanno, M. Boyle, L. T. Buchman, L. E. Kidder, H. P. Pfeiffer and M. A. Scheel, arXiv:1106.1021 [gr-qc]. [20] A. L. Tiec, A. H. Mroue, L. Barack, A. Buo- nanno, H. P. Pfeiffer, N. Sago and A. Taracchini, arXiv:1106.3278 [gr-qc]. [21] A. Nagar, T. Damour and A. Tartaglia, Class. Quant. Grav.24, S109 (2007) [arXiv:gr-qc/0612096]. [22] T. Damour and A. Nagar, Phys. Rev. D 76, 064028 (2007) [arXiv:0705.2519 [gr-qc]]. [23] S. Bernuzzi and A. Nagar, Phys. Rev. D 81, 084056 (2010) [arXiv:1003.0597 [gr-qc]]. [24] S. Bernuzzi, A. Nagar, A. Zenginoglu, Phys. Rev. D83, 064010 (2011). [arXiv:1012.2456 [gr-qc]]. [25] N. Yunes, A. Buonanno, S. A. Hughes, M. Coleman Miller, Y. Pan, Phys. Rev. Lett. 104, 091102 (2010). [arXiv:0909.4263 [gr-qc]]. [26] N. Yunes, A. Buonanno, S. A. Hughes, Y. Pan, E. Ba- rausse, M. C. Miller and W. Throwe, Phys. Rev. D 83, 044044 (2011) [arXiv:1009.6013 [gr-qc]]. [27] A. Buonanno, Y. Chen and T. Damour, Phys. Rev. D 74, 104005 (2006) [arXiv:gr-qc/0508067]. [28] T. Damour, P. Jaranowski and G. Schaefer, Phys. Rev. D77, 064032 (2008) [arXiv:0711.1048 [gr-qc]]. [29] G. Faye, L. Blanchet, A. Buonanno, Phys. Rev. D74, 104033 (2006). [gr-qc/0605139]. [30] L. Blanchet, A. Buonanno, G. Faye, Phys. Rev. D74, 104034 (2006). [gr-qc/0605140]. [31] M. Levi, Phys. Rev. D82, 104004 (2010). [arXiv:1006.4139 [gr-qc]]. [32] Y. Pan, A. Buonanno, R. Fujita, E. Racine, H. Tagoshi, Phys. Rev. D83, 064003 (2011). [arXiv:1006.0431 [gr- qc]]. [33] Y.Pan, A.Buonanno, L.T. Buchman, T. Chu, L.E. Kid- der, H. P. Pfeiffer and M. A. Scheel, Phys. Rev. D 81, 084041 (2010) [arXiv:0912.3466 [gr-qc]]. [34] E. Barausse, E. Racine and A. Buonanno, Phys. Rev. D 80, 104025 (2009) [arXiv:0907.4745 [gr-qc]]. [35] E. Barausse and A. Buonanno, Phys. Rev. D 81, 084024 (2010) [arXiv:0912.3517 [gr-qc]]. [36] T. Damour, P. Jaranowski and G. Schaefer, Phys. Rev. D62, 021501 (2000) [Erratum-ibid. D 63, 029903 (2001)]12 [arXiv:gr-qc/0003051]. [37] T. Damour, P. Jaranowski and G. Schaefer, Phys. Lett. B513, 147 (2001) [arXiv:gr-qc/0105038]. [38] T. Damour, P. Jaranowski and G. Schaefer, Phys. Rev. D62, 044024 (2000) [arXiv:gr-qc/9912092].[39] E. Barausse, A. Buonanno, [arXiv:1107.2904 [gr-qc]].