Adiabatic and Nonadiabatic Spin-transfer Torques in Antiferromagnets
AAdiabatic and Nonadiabatic Spin-transfer Torques in Antiferromagnets
Junji Fujimoto ∗ Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China (Dated: September 8, 2020)Electron transport in magnetic orders and the magnetic orders dynamics have a mutual dependence, whichprovides the key mechanisms in spin-dependent phenomena. Recently, antiferromagnetic orders are focusedon as the magnetic order, where current-induced spin-transfer torques, a typical effect of electron transporton the magnetic order, have been debatable mainly because of the lack of an analytic derivation based onquantum field theory. Here, we construct the microscopic theory of spin-transfer torques on the slowly-varyingstaggered magnetization in antiferromagnets with weak canting. In our theory, the electron is captured bybonding/antibonding states, each of which is the eigenstate of the system, doubly degenerates, and spatiallyspreads to sublattices because of electron hopping. The spin of the eigenstates depends on the momentum ingeneral, and a nontrivial spin-momentum locking arises for the case with no site inversion symmetry, withoutconsidering any spin-orbit couplings. The spin current of the eigenstates includes an anomalous componentproportional to a kind of gauge field defined by derivatives in momentum space and induces the adiabaticspin-transfer torques on the magnetization. Unexpectedly, we find that one of the nonadiabatic torques has thesame form as the adiabatic spin-transfer torque, while the obtained forms for the adiabatic and nonadiabatic spin-transfer torques agree with the phenomenological derivation based on the symmetry consideration. This findingsuggests that the conventional explanation for the spin-transfer torques in antiferromagnets should be changed.Our microscopic theory provides a fundamental understanding of spin-related physics in antiferromagnets.
I. INTRODUCTION
Manipulation of antiferromagnetic orders by electric meansis one of the most important topics [1–4] because of its ap-plicational potentials, such as producing no stray fields andshowing ultrafast dynamics, compared to ferromagnets. Al-though the application is an essential driving factor, the phe-nomena induced by the interplay of the magnetic order withelectron transport contains rich physics, which is not fully un-derstood yet. The antiferromagnetic order is purely quantummechanical, and the electronic eigenstate coupled with the an-tiferromagnetic order is no longer bare electron, in contrastto the ferromagnetic order. Here, we call the eigenstate inantiferromagnets the bonding/antibonding state.The spin-transfer torques in antiferromagnets without spin-orbit couplings have been studied more than for a decade forspin-valve-like structures [5–11] and for slowly-varying an-tiferromagnetic texutres [12–18], as a typical effect of theelectron transport on the antiferromagnetic order. Conven-tionally, the spin-transfer torque in antiferromagnets has beenconsidered as the spin-transfer torques acting on two coupledferromagnets, which means that the spin torques are obtainedfrom the summation of the torques on each sublattice magneti-zation [9, 12, 18]. However, this explanation is still debatable,mainly because there is no analytic derivation in the adia-batic regime based on an eigenstate picture which clarifies thephysics. Hence, such a fundamental microscopic theory haslong been desired.In this paper, we present the microscopic theory based on theeigenstate picture for the spin-transfer torque in slowly-varyingstaggered antiferromagnets with weak canting. To derive themodel based on the eigenstate picture, we use two unitary trans-formations as schematically shown in Fig. 1. One is the real ∗ Email:[email protected] localized spinelectron spin + spin gauge field + spin gauge field (i) (ii) (iii) (iv)laboratory frame rotated frame eigenstate frame
FIG. 1. Schematic description of the process for obtaining the ef-fective model by using two unitary transformations U ( r , t ) and V k .(i) The laboratory frame: the quantization axis of the electron spin isindependent from space and time. (ii) The rotated frame: by usingthe unitary transformation U ( r , t ) , the electron spin is described bythe spin coherent state for the Néel vector. In addition, the spatialvariation of the Néel vector is encoded to the spin gauge field in thisframe. (iii) and (iv) The eigenstate frame: after the unitary transfor-mation by V k , the electron is no longer localized at a single sublatticebut spreads into both sublattices as (iii) the bonding state and (iv) theantibonding state, which are the eigenstates of electrons coupled tothe Néel vector. space transformation, U ( r , t ) , in which the quantization axisof the electron spin becomes along the local Néel vector [19],where we call this frame the rotated frame (Fig. 1 (ii)). Inthe rotated frame, the spatial variation of the Néel vector isencoded to the real space gauge field A r , i = − iU † ∂ i U , whichcouples to the spin current, and the magnetization due to cant-ing couples to the electron spin. The other transformation V k is the momentum space transformation to diagonalize theHamiltonian, where we call the frame after this transforma-tion the eigenstate frame. In the eigenstate frame, we have theHamiltonian described by the eigenstates, the bonding and an-tibonding states, each of which doubly degenerates and spreadsto sublattices (Fig. 1 (iii) and (iv)).The perturbation Hamiltonians are also transformed by theunitary matrices U ( r , t ) and V k . In the eigenstate frame, the a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p electron spin is found to be generally depending on the mo-mentum of the eigenstate, and a nontrivial spin-momentumlocking arises for the case with no site inversion symmetry,without considering any spin-orbit couplings. Note that thesite inversion symmetry, or site-centered inversion symmetry,is usually used in the one dimensional (1D) quantum spin sys-tems and is defined as the invariance under the transformationof site index i changing to − i in the center of the i = A r , i . In the eigenstate frame, wefind that the spin current consists of two components; one is theordinary spin current given by the combination of the velocityand spin. The other is an anomalous spin current given by themomentum space gauge field defined by A k , i = − iV † k ∂ i V k . Asimilar momentum space gauge field was discussed by Chengand Niu [21], but it seems to be different from A k , i .We then evaluate the effect of the conduction electron on thedynamics of the antiferromagnets, which is the spin-transfertorque on the order parameters; the torques on the Néel vectordenoted by τ n , and the torque on the magnetization denoted by τ m . Considering the antiferromagnetically ordered localizedspin system, we derive the equation of motion of the orderparameters in the presence of the sd exchange coupling to theconduction electron spin. In the adiabatic regime, the spintorque τ m is given by the divergence of the anomalous spincurrent, which is a novel expression, and the other torque τ n isproportional to the perpendicular component of the conductionelectron spin, which is the same form as the spin torque in theferromagnets. By evaluating the linear responses of anomalousspin current and spin to the electric field, we obtain the current-induced spin-transfer torques.As mentioned above, the spin-transfer torque in antifer-romagnets has been considered conventionally as the spin-transfer torques acting on two coupled ferromagnets. Forthe adiabatic spin-transfer torques defined as the spin-transfertorque to which the adiabatic processes only contribute, weconfirm that the above explanation is valid. This agree-ment is because, in the adiabatic regime, each of the bond-ing/antibonding states conducts each sublattice.We also evaluate the nonadiabatic spin-transfer torques de-fined as the torques to which the nonadiabatic processes suchas the mixing of the bonding and antibonding states contribute.We find that one of the nonadiabatic spin-transfer torques τ na n (the superscript ‘na’ denotes the nonadiabatic) has the samedependence as the adiabatic torque τ n , which means that theabove conventional explanation does not work for the nonadia-batic torques. This deviation is understood by the fact that thenonadiabatic processes are characteristic of electrons coupledto the antiferromagnetic order, which are not equivalent to theelectrons coupled to two coupled ferromagnetic orders.We here give a comment on the relation of the adia-batic/nonadiabatic torques to the reactive/dissipative torques.The reactive and dissipative spin-transfer torques are defined as the even and odd terms under the time-reversal transfor-mation in the equation of motion of the magnetic orders. Inferromagnets, the reactive torque is equivalent to the adiabaticone, and the dissipative is the same as the nonadiabatic. How-ever, in antiferromagnets, the correspondences do not realize,since one of the nonadiabatic torques has the same form as theadiabatic torque.The paper is organized as follows. In Sec. II, we definethe electron system we consider, and Sec. III is devoted to thederivation of the effective Hamiltonian in the eigenstate frame.In Sec. IV, we present the main topic of the adiabatic andnonadiabatic spin-transfer torques. Then, Sec. V concludesthis paper. II. MODEL
We begin with the tight-binding model coupled to the stag-gered magnetization slowly-varying spatially through the sd -type exchange coupling, in which the Hamiltonian is given by H e = H t + H sd + V imp . The first term is hopping Hamilto-nian given by H t = ˝ i , j ( t ij c † i c j + H . c . ) with t ij the hoppingintegral, which is assumed to be finite only for the nearest-neighbors of the intra-sublattices and of inter-sublattices,where c i is the spinor form of the annihilation operator onthe i -th site. The second term is the exchange coupling givenby H sd = − J sd ˝ i S i · ( c † i σ c i ) with the strength J sd ( > ) andthe localized spin S i which consists of the staggered magne-tization with weak canting. Here, σ = ( σ x , σ y , σ z ) is thePauli matrices for spin space, and we use σ as the unit matrixfor spin space. V imp denotes an impurity potential, which isto be approximated as a nonmagnetic potential acting on theeigenstates for simplicity, and leads to k -independent lifetimeof the bonding/antibonding states. The impurity effects areout of focus in this paper, and we will discuss them anotherpaper.In antiferromagnets with the slowly-varying staggered mag-netization and with weak canting, the localized spin S i isexpressed by using two smooth functions; the Néel vector N ( r , t ) and the magnetization M ( r , t ) as S i = (− ) P i N ( r i , t ) + M ( r i , t ) , (see Fig. 2 (a)), where r i is the position of the i -th site,and P i describes the sign change depending on the sublattice; P i = i ∈ A and P i = i ∈ B. For the case of weakcanting, the Néel vector is much larger than the magnetization; M / N (cid:28) N ( r , t ) = N n ( r , t ) and M ( r , t ) = M m ( r , t ) ,where n and m are the unit vectors. We here assume N and M as well as S = | S i | are constant for time and space, whichleads n · m = III. EFFECTIVE HAMILTONIAN
We now derive the Hamiltonian for the bonding/antibondingstates (Fig. 1). Firstly, we take the unitary transformation U ( r , t ) such that the electron spin takes the spin coherent statealong the Néel vector; U † ( n · σ ) U = σ z with c i = U ˜ c i .Using the spin gauge field [22–24] A r ,µ = − iU † ( ∂ U / ∂ r µ ) = N i ( t ) M i ( t ) S i − N j ( t ) M j ( t ) S j AA BB ˆ e µ ˆ e ν ABunit cellunit cell ˆ e µ ˆ e ν AB t ' t unit cellunit cell (b)(a) (c) t ' t FIG. 2. (a) The decomposition of the localized spin into the Néel vector and the magnetization. From the decomposition, we can define theadiabatic motion of electron for the antiferromagnetic ordered state, since N i ( t ) = N ( r i , t ) and M i ( t ) = M ( r i , t ) are slowly-varying vector,while the vector S i is rapidly changing. (b) and (c) Schematic figures of (two dimensional) antiferromagnetic models; (b) the square lattice,(c) honeycomb lattice. Then intra-sublattice nearest neighbor vector is described by ˆ e ν , and inter-sublattice nearest neighbor vector is denotedby ˆ e µ . A α r ,µ ( r , t ) σ α / ( r , r , r , r ) = ( t , x , y, z ) and α = x , y, z ,we find H e = H + H A + H f + O(A ) (1)with H = ˝ k Ψ † k H k Ψ k + V imp and H A = ~ ∫ ω (cid:213) q j α s , i (− q ) A α r , i ( q , ω ) , (2) H f = − MN J sd ∫ ω (cid:213) q s α (− q ) (cid:0) R − m (cid:1) α q ,ω , (3)where we introduced Ψ k as two sets of spinors of annihilationoperators, H k describes the unperturbed Hamiltonian givenby H k = T k ρ σ − ( Re η k ) ρ σ − ( Im η k ) ρ σ − J sd ρ σ z with ρ µ being the Pauli matrix for sublattice space for µ = , , µ =
0. Here, T k and η k are hoppings of intra- and inter-sublattices, respectively,and we introduced J sd = N J sd . In this work, we assumedthat the electron is described by the inter- and intra-sublatticenearest neighbor hoppings as shown Fig. 2 (b) and (c), we find T k = − t ˝ ν Re [ exp (− i k · ˆ e ν )] and η k = t ˝ µ exp (− i k · ˆ e µ ) ,where t and t are inter- and intra-sublattice nearest neighborhopping parameters, respectively, and ˆ e µ and ˆ e ν are inter- andintra-sublattice nearest neighbor vectors. The Hamiltonian H A represents the couplings of the spin current j α s , i ( q ) to the spingauge fields, where α = x , y, z is the spin index, and i = x , y, z is spatial index. The Hamiltonian H f describes the coupling ofthe magnetization and the spin in the conventional way of theexchange coupling, where M / N (cid:28) H f perturbatively. Here, R is the rotational matrix defined by U † ( r , t ) σ U ( r , t ) = R σ . The frame described by Ψ k is calledthe rotated frame (Fig. 1).Secondly, H k is not yet a diagonal matrix, hence necessary tobe diagonalized to obtain the effective model. We diagonalize H k so as to hold the equation V † k H k V k = T k ρ σ − ∆ k ρ σ z = '›››« T k − ∆ k T k + ∆ k T k + ∆ k T k − ∆ k “fififi‹ with ∆ k = (| η k | + J sd ) / . The explicit form of V k is given inSupplemental Material (SM) [25]. From this, we see that theouter 2 × × V k .Thirdly, we use the projection operator P ± = ( ρ σ ± ρ σ z )/ s = + and to the antibondingstate for s = − , which leads P s ( V † k Ψ k ) = ψ k , s , where ψ k , s isannihilation operator of the bonding/antibonding state, whichcan be written by using that of the spin coherent states as ψ k , s = s cos ( ϑ k / ) c A k , s + se − i ϕ k sin ( ϑ k / ) c B k , s cos ( ϑ k / ) c B k , ¯ s + se i ϕ k sin ( ϑ k / ) c A k , ¯ s , ! (4)with ¯ s = − s , where c X k ,ξ is the annihilation operator of sub-lattice X = A , B with the spin coherent state ( ξ = + ) and thatantiparallel to the state ( ξ = − ), and we introducedsin ϑ k = | η k |/ ∆ k , cos ϑ k = J sd / ∆ k (5)and ϕ k = tan − ( Im η k / Re η k ) . (6)Here, ϕ k depends on Im η k , which is only finite in the casewith no site inversion symmetry. We see below that the spinoperator of the eigenstates has an essentially different formdepending on whether the system has the site inversion sym-metry.We finally obtain the effective model in the adiabatic regimewhere the mixing of the bonding and antibonding states isnegligible, which is given by Eq. (1) with H = (cid:213) s = ± (cid:213) k (cid:15) k s ψ † k , s ψ k , s + V imp , (cid:15) k s = T k − s ∆ k (7)and Eqs. (2) and (3). Note that the bonding/antibonding statesdoubly degenerates, and hence the spinor form can be capturedby the another Pauli matrices, which is hereafter denoted by τ µ ( µ = , x , y, z ). We call the frame described by ψ k , s the eigenstate frame. We here consider the impurity potential as V imp = u i ˝ k , k , s ρ ( k − k ) ψ † k , s ψ k , s for simplicity, where u i is the potential strength, and ρ ( q ) is the Fourier componentof the impurity density. The spin in Eq. (3) in the eigenstateframe with adiabatic approximation is s α ( q ) = (cid:213) s = ± (cid:213) k ψ † k − q , s (cid:16) s τ α k , s (cid:17) ψ k + q , s , (8)where τ k , s = ( τ x k , s , τ y k , s , τ z k , s ) is given as τ k , s = '›« sin ϑ k ˆ e ϕ k · τ ⊥ s sin ϑ k (cid:0) ˆ z × ˆ e ϕ k (cid:1) · τ ⊥ τ z “fi‹ (9)with τ ⊥ = ( τ x , τ y , ) and ˆ e ϕ k = ( cos ϕ k , sin ϕ k , ) . The spinbeyond the adiabatic regime is given in SM [25].Here, we consider a specific configuration, such as thesquare lattice (Fig. 2 (b)), where the inter-sublattice hopping η k becomes real, η k = η ∗ k , hence ϕ k = e ϕ k = ˆ x , we findthe spin of the eigenstate as τ k , s = '›« sin ϑ k τ x s sin ϑ k τ y τ z “fi‹ . From this, we find the following three points: (i) spin directionis independent from the momentum in this case, and the cor-respondences between the electron picture and the eigenstatepicture arises; σ x ↔ τ x , σ y ↔ τ y , σ z ↔ τ z , (ii) the transverse spin shrinks depending on k as sin ϑ k = | η k |/ ∆ k , and (iii) the chirality for the bonding state ( s = + ) isopposite for the antibonding state ( s = − ) since the sign of the y component depends on the bonding or antibonding states.For the case with no site inversion symmetry, such as thehoneycomb lattice (Fig. 2 (c)), where the inter-sublattice hop-ping does not become real, the vector ˆ e ϕ k changes dependingon k , which leads to the momentum-dependent spin directionchanging (see Eq.(9)). We emphasise that since we does notconsider any spin-orbit couplings, this spin-momentum lock-ing is a nontrivial result. This spin-momentum locking withoutany spin-orbit couplings is one of the important findings in this work. The spin-momentum locking is expected to connect tothe nontrivial spin polarization [26, 27] and the anomalousHall effect [28] in noncollinear antiferromagnets.The spin current in Eq. (2) is given as j s , i = ( j x s , i , j y s , i , j z s , i ) with j s , i ( q ) = j s0 , i ( q ) + j sA , i ( q ) , where j s0 , i ( q ) is the ordinaryspin current given by j s0 , i ( q ) = (cid:213) k , s ψ † k − q , s (cid:18) s ~ ∂(cid:15) k s ∂ k i τ k , s (cid:19) ψ k + q , s , (10)where the spin operator is given by Eq. (9), and j sA , i ( q ) is ananomalous spin current given by j sA , i ( q ) = J sd ~ (cid:213) k , s ψ † k − q , s '›« ( ˆ z × A ⊥ k , i ) · τ ⊥ − s A ⊥ k , i · τ ⊥ “fi‹ ψ k + q , s . (11)Here, A α k , i in Eq. (11) is a kind of gauge filed defined byderivatives in momentum space A k , i = − iV † k ( ∂ V k / ∂ k i ) = ( A z k , i ρ σ + A ⊥ k , i · ρ σ z )/ A ⊥ k , i ( k ) = ∂ϑ k ∂ k i ˆ z × ˆ e ϕ k − ∂ϕ k ∂ k i sin ϑ k ˆ e ϕ k , (12a) A z k , i ( k ) = ∂ϕ k ∂ k i ( − cos ϑ k ) . (12b)The expression of the anomalous spin current is also one ofthe important points of this work, since this spin current con-tributes to the spin torque as shown below in Eq. (14a). Notethat A z k , i ( k ) is the Berry phase in the momentum space and isonly finite in the case with no site inversion symmetry. IV. SPIN-TRANSFER TORQUES
Here, we evaluate the spin-transfer torques on the Néel vec-tor and magnetization. First, we derive the equation of motionfor the two order parameters in the presence of the sd exchangecoupling. The part related to the localized spin in the La-grangian is given as L s = L B −H s −H sd , where the first termin the right hand side is calculated in the continuum limit [29]as L B = − ~ ∫ ( d r / V )[ M · ( n × ( ∂ n / ∂ t ))] , where V is the vol-ume of the system, and the second term H s is the Hamiltonianof the localized spin. The sd exchange coupling in the contin-uum limit is rewritten as H sd = ∫ ( d r / V )[− N J n · ˜ Σ − J M · ˜ s ] ,where ˜ Σ and ˜ s are staggered electron spin and ferromagneticelectron spin in the laboratory frame, respectively.The Eular-Lagrange equation is calculated as M (cid:219) m = n × ~ δ H s δ n + τ m + n × ~ δ W δ (cid:219) n , (13a) (cid:219) n = n × ~ M δ H s δ m + τ n + n × ~ M δ W δ (cid:219) m , (13b)where W is the phenomenologically introduced dampingfunction. We find that the sd exchange coupling inducesthe following spin torques in the rotated frame, R − τ m = ( JN / ~ )( ˆ z × h Σ i neq ) and R − τ n = ( J / ~ )( ˆ z × h s i neq ) , which areobtained as (R − τ m ) q = iq j h j sA , j ( q , t )i neq , (14a) (R − τ n ) q = J sd ~ ˆ z × h s ( q , t )i neq (14b)in the adiabatic regime, where j sA , j ( q ) is the anomalous spincurrent of the eigenstate given by Eq. (11), and the spin s ( q ) is given by Eq. (8). Here, h · · · i neq means the statistical av-erage in nonequilibrium. (See SM [25] for the derivation ofEq. (14a).) Equation (14a) suggests that the spin torque onthe magnetization m is given by the divergence of the anoma-lous spin current. Equation (14b) is the same form as the spintorque in ferromagnets. These two expressions (14) are one ofthe important findings in this work. Note that the spin torque τ m is already the first order of q and has no zeroth order termswith respect to q .Then, we evaluate the spin torques (14) in the pres-ence of the electric field. Following the linear responsetheory, we have h j α sA , i ( q , ω )i neq = X α ij ( q , ω ) A em j ( ω ) and h s α ( q , ω )i neq = Y α j ( q , ω ) A em j ( ω ) , where A em j ( ω ) is the vec-tor potential, and the linear response coefficients X α ij ( q , ω ) and Y α i ( q , ω ) are obtained from the corresponding Matsubara func-tions X α ij ( q , i ω λ ) = − e hh j α sA , i ( q ) , j j ( )ii and Y α j ( q , i ω λ ) = − e hh s α ( q ) , j j ( )ii with the canonical correlation hh A , B ii = V − ∫ β d τ e i ω λ τ h T τ A ( τ ) , B ( )i by taking the analytic contin-uation i ω λ → ~ ω + i
0. Here, j j ( q ) is the electric current andthe explicit form in the eigenstate frame is given in SM [25],and β = / k B T is the inverse temperature. Rewriting the Mat-subara functions in terms of the thermal Green function, weexpand the Green function up to the first order of the Hamil-tonian H f for the coefficient X α ij ( q , i ω λ ) . For the coefficient Y α j ( q , i ω λ ) , we expand the first order of the spin gauge field A r , i as in the calculation of the spin-transfer torques in ferro-magnets [24, 30, 31]. Note that the terms proportional to thespin gauge field in X α ij ( q , i ω λ ) become the spin torques in thesecond order of q since the spin gauge field is the first orderof q and Eq. (14a) is already the first order of q , so that weneglect the terms.After some straightforward calculations [25], the resultantexpressions for the adiabatic spin-transfer torques in the labo-ratory frame are obtained as τ m ( r , t ) = MN (cid:18) P m e j c · ∇ (cid:19) m ( r , t ) , (15a) τ n ( r , t ) = N (cid:18) P n e j c · ∇ (cid:19) n ( r , t ) , (15b)where − e is the elementary charge, j c is the charge current,and P m and P n are nondimensional coefficients, which aregiven in SM [25]. Equations (15) are the first main results ofthis work. This result does not depend on the lattice symmetry.The obtained spin-transfer torques τ m and τ n are similarforms to that in the ferromagnets ∝ ( j s · ∇ ) m , where j s is thespin current and m is the magnetization in the ferromagnet.Especially, Eq. (15b) is the same form as Eq. (20) in Ref. [18], which is obtained by considering the antiferromagnet as thetwo coupled ferromagnets. This agreement suggests that theconventional explanation is valid for the adiabatic spin-transfertorque. It is mainly because each of the doubly-degeneratedbonding/antibonding state conducts each sublattice in the adi-abatic regime. Note that there is one difference from the caseof ferromagnets; in antiferromagnets, P m j c and P n j c are notequivalent to the spin current. In ferromagnets, the charge cur-rent accompanies with spin polarization, hence charge currentcan be rewritten as j s = P j c by means of the spin polariza-tion P , while the charge current in antiferromagnets does notaccompany with spin polarization. The eigenstate is doublydegenerated as Eq. (4), and both of two degenerated sates con-tribute additively to the spin-transfer torques, so that chargecurrent induces the spin torques.In order to treat a nonadiabatic contribution to the spintorques, we need to consider nonadiabatic processes, which aretransitions from the bonding state to the antibonding state andvice versa. Note that the nonadiabatic torques are not obtainedfrom Eqs. (14) even with the spin relaxation mechanism. Thederivation and calculation of the nonadiabatic spin-transfertorques are given in SM [25], and we find τ na m ( r , t ) = i ωτ sd n × (cid:18) P na m e j c · ∇ (cid:19) n , (16a) τ na n ( r , t ) = − N (cid:18) P na n e j c · ∇ (cid:19) n , (16b)where τ sd = ~ / J sd , and P na m and P na n are nondimensionalcoefficients given in SM [25]. Here, ω is the frequency ofthe external electric field E = E e − i ω t and the charge currentis given by j c = σ c E , where σ c is the conductivity. Equa-tions (16) are the second main results of this work. Note that P na m = P na n = T k is zero. In contrast, for the case of ∆ k ’ J sd , wefind P na m = P na n ’ alternating currentin ferromagnets. Equation (16a) indicates that the alternating current induces the nonadiabatic torque in antiferromagnetsand the direct current ( ω =
0) does not give rise to the nonadi-abatic torque on m for the case of the simple impurity poten-tial we consider. By analogy of the case in ferromagnets [31], i ωτ sd could be replaced by i ωτ sd + ζ s , where ζ s is the spinrelaxation rate, and the direct current induces the nonadiabaticspin-transfer torque in that case.Although the nonadiabatic spin-transfer torque (16a) is al-ready expected from the symmetry consideration [14], thetorque (16b) is unexpectedly obtained. Equation (16b) sug-gests that the nonadiabatic processes also contribute to theordinary spin-transfer torque (15b), which is essentially differ-ent from ferromagnets, where the nonadiabatic process onlycontributes to the nonadiabatic spin-transfer torques. The re-sult (16b) indicates that the conventional explanation for thespin-transfer torque does not work for the nonadiabatic torque.Furthermore, the correspondences between adiabatic (nonadi-abatic) torque and reactive (dissipative) torque, which is validin ferromagnets, are no longer realized due to Eq. (16b).For further discussion, we need to solve the equations (13)for a specific configuration, such as a domain wall [32–34],and the analysis is one of the future works. V. CONCLUSION
In conclusion, we have constructed the microscopic theoryof the spin-transfer torques on the slowly-varying staggeredmagnetization in antiferromagnets with weak canting. Theeffective model is obtained by using the two unitary trans-formations; one is the real space unitary transformation inwhich the electron spin is to be along the Néel vector, andthe other is the momentum space unitary transformation inwhich the unperturbed Hamiltonian is to be diagonalized. Bythese transformations, we have two kinds of gauge fields; oneis the spin gauge field which is well-known in ferromagneticspintronics, and the other is the momentum space gauge field,which is an extension of the momentum space Berry phase.The spin operator of the eigenstates depends on the momen-tum in general, and a nontrivial spin-momentum locking arisesin the case with no site inversion symmetry. The spin currentoperator of the eigenstates has two components; one is the ordi- nary spin current operator and the other is the anomalous spincurrent which is proportional to the momentum space gaugefield. The divergence of the anomalous spin current inducesthe spin torque on the magnetization in the adiabatic regime,and the spin torque on the Néel vector is given as the sameform as the spin torque in ferromagnets. The obtained formsfor the adiabatic and nonadiabatic spin-transfer torques agreewith the phenomenological derivation based on the symmetryconsideration. For the adiabatic torques are understood by theconventional explanation for the spin-transfer torques, but theexplanation fails for the nonadiabatic spin-transfer torque. Ourmicroscopic theory provides a fundamental understanding ofspin-related physics in antiferromagnets, which paves the wayfor developing the antiferromagnetic spintronics.
ACKNOWLEDGMENTS
The author would like to thank G. Tatara and Y. Yamane forvaluable advices in the early stage of this work. The authoralso thank M. Matsuo, A. Shitade, C. Akosa, S.C. Furuya, andY. Ominato for their stimulating comments and suggestions.This work is partially supported by the Priority Program ofChinese Academy of Sciences, Grant No. XDB28000000. [1] A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A , 3098(2011).[2] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat.Nanotechnol. , 231 (2016).[3] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, andY. Tserkovnyak, Rev. Mod. Phys. , 015005 (2018).[4] J. Železný, P. Wadley, K. Olejník, A. Hoffmann, and H. Ohno,Nat. Phys. , 220 (2018).[5] A. S. Núñez, R. A. Duine, P. Haney, and A. H. MacDonald,Phys. Rev. B , 214426 (2006).[6] Z. Wei, A. Sharma, A. S. Nunez, P. M. Haney, R. A. Duine,J. Bass, A. H. MacDonald, and M. Tsoi, Phys. Rev. Lett. ,116603 (2007).[7] S. Urazhdin and N. Anthony, Phys. Rev. Lett. , 046602 (2007).[8] P. M. Haney and A. H. MacDonald, Phys. Rev. Lett. , 196801(2008).[9] H. V. Gomonay and V. M. Loktev, Phys. Rev. B , 144427(2010).[10] H. B. M. Saidaoui, A. Manchon, and X. Waintal, Phys. Rev. B , 174430 (2014).[11] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett. ,057601 (2014).[12] Y. Xu, S. Wang, and K. Xia, Phys. Rev. Lett. , 226602(2008).[13] A. C. Swaving and R. A. Duine, Phys. Rev. B , 054428 (2011).[14] K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Phys. Rev. Lett. , 107206 (2011).[15] E. G. Tveten, A. Qaiumzadeh, O. A. Tretiakov, and A. Brataas,Phys. Rev. Lett. , 127208 (2013).[16] Y. Yamane, J. Ieda, and J. Sinova, Phys. Rev. B , 054409(2016).[17] J. Barker and O. A. Tretiakov, Phys. Rev. Lett. , 147203(2016). [18] H.-J. Park, Y. Jeong, S.-H. Oh, G. Go, J. H. Oh, K.-W. Kim,H.-W. Lee, and K.-J. Lee, Phys. Rev. B , 144431 (2020).[19] J. J. Nakane, K. Nakazawa, and H. Kohno, Phys. Rev. B ,174432 (2020).[20] Y. Fuji, F. Pollmann, and M. Oshikawa, Phys. Rev. Lett. ,177204 (2015).[21] R. Cheng and Q. Niu, Phys. Rev. B , 245118 (2012).[22] V. Korenman, J. L. Murray, and R. E. Prange, Phys. Rev. B ,4032 (1977).[23] Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B ,R3213 (1998).[24] H. Kohno and J. Shibata, J. Phys. Soc. Jpn. , 063710 (2007).[25] See Supplemental Material at [URL will be inserted by pub-lisher] for details of calculations.[26] S. Hayami, Y. Yanagi, and H. Kusunose, Phys. Rev. B ,220403 (2020).[27] L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger,Phys. Rev. B , 014422 (2020).[28] M. Onoda, G. Tatara, and N. Nagaosa, J. Phys. Soc. Jpn. ,2624 (2004).[29] S. Sachdev, Quantum Phase Transitions , 2nd ed. (CambridgeUniversity Press, Cambridge, 2011).[30] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. , 213 (2008).[31] J. Fujimoto and M. Matsuo, Phys. Rev. B , 220402 (2019).[32] N. Papanicolaou, Phys. Rev. B , 15062 (1995).[33] R. Jaramillo, T. F. Rosenbaum, E. D. Isaacs, O. G. Shpyrko,P. G. Evans, G. Aeppli, and Z. Cai, Phys. Rev. Lett. , 117206(2007).[34] T. Okuno, D.-H. Kim, S.-H. Oh, S. K. Kim, Y. Hirata,T. Nishimura, W. S. Ham, Y. Futakawa, H. Yoshikawa,A. Tsukamoto, Y. Tserkovnyak, Y. Shiota, T. Moriyama, K.-J. Kim, K.-J. Lee, and T. Ono, Nat. Electron. , 389 (2019). upplemental Material:Adiabatic and Nonadiabatic Spin-transfer Torques in Antiferromagnets Junji Fujimoto
Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China (Dated: September 8, 2020)
CONTENTS
I. Diagonalization matrix 1II. Projection operator and projected spaces 2A. Commutative class 3B. Noncommutative class 4III. Nonadiabatic components of spin and spin current 5IV. Spin torques in eigenstate frame 6V. Electric current operator in eigenstate frame 7VI. Adiabatic spin-transfer torques 7A. Linear response coefficient for adiabatic component of anomalous spin current 8B. Linear response coefficient for adiabatic component of spin 10VII. Nonadiabatic spin-transfer torques 12
I. DIAGONALIZATION MATRIX
In this section, we show the explicit form of the diagonalization matrix V k . First, we introduce the following vector form X k = '›« Re η k σ Im η k σ J sd σ z “fi‹ = ∆ k '›« σ sin ϑ k cos ϕ k σ sin ϑ k sin ϕ k σ z cos ϑ k “fi‹ , (S1.1)and hence the Hamiltonian is expressed as H k = T k ρ σ − X k · ρ . Here, we define the V k as V k = ˜ χ k · ρ , (S1.2)where ˜ χ k = '›« σ sin ( ϑ k / ) cos ϕ k σ sin ( ϑ k / ) sin ϕ k σ z cos ( ϑ k / ) “fi‹ = χ ⊥ k σ + χ z k ˆ z σ z . (S1.3)By using V k , one obtains V † k (cid:0) X k · ρ (cid:1) V k = ∆ k ( χ ⊥ k · ρ ⊥ σ + χ z k ρ σ z ) (cid:0) sin ϑ k ( ˆ e ϕ k · ρ ⊥ ) σ + cos ϑ k ρ σ z (cid:1) ( χ ⊥ k · ρ ⊥ σ + χ z k ρ σ z ) = ∆ k (cid:16) cos ( ϑ k / ) ρ σ + i sin ( ϑ k / )( ˆ z × ˆ e ϕ k ) · ρ ⊥ σ z (cid:17) ( χ ⊥ k · ρ ⊥ σ + χ z k ρ σ z ) = ∆ k ρ σ z , (S1.4)where ˆ e ϕ k = ( cos ϕ k , sin ϕ k , ) . Hence, we have V † k H k V k = T k ρ σ − ∆ k ρ σ z . a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p II. PROJECTION OPERATOR AND PROJECTED SPACES
Here, we show the projection operator into the eigenstates and the projected spaces. The projection operator into the bondingstate ( s = + ) and into the antibonding state ( s = − ) are given by P s = ρ σ + s ρ σ z , (S2.1)or, in the matrix form, P + = '›››« “fififi‹ , P − = '›››« “fififi‹ . (S2.2)Since the operator P s ( s = ± ) holds the following relation, P s = P s , (S2.3)the operator P s is a projection operator. We also see P + + P − = ρ σ , (S2.4)so that we have the orthogonality: P + P − = P − P + = . (S2.5)Next, we derive the formulas for the projections of ρ µ σ ν ( µ = , , , ν = , x , y, z ). We first divide P s ρ µ σ ν inttwo classes: one is commutative, P s ρ µ σ ν = ρ µ σ ν P s for ( µ, ν ) = ( , ) , ( , z ) , ( , ) , ( , z ) , ( , x ) , ( , y ) , ( , x ) , ( , y ) , and theother is noncommutative P s ρ µ σ ν = ρ µ σ ν P ¯ s for ( µ, ν ) = ( , x ) , ( , y ) , ( , ) , ( , z ) , ( , ) , ( , z ) , ( , x ) , ( , y ) , where ¯ s = − s . Thecommutative class is equivalent to the adiabatic contribution; (cid:213) k ( V k Ψ k ) † ρ µ σ ν V k Ψ k = (cid:213) k (cid:213) s , s ( P s V k Ψ k ) † (cid:0) P s ρ µ σ ν P s (cid:1) P s V k Ψ k = (cid:213) k (cid:213) s , s ψ † k , s (cid:0) P s ρ µ σ ν P s (cid:1) ψ k , s = (cid:213) k (cid:213) s , s ψ † k , s (cid:0) P s P s ρ µ σ ν (cid:1) ψ k , s = (cid:213) k (cid:213) s ψ † k , s (cid:0) P s ρ µ σ ν (cid:1) ψ k , s , (S2.6)where we inserted 1 = ρ σ = ˝ s = ± P s and use P s = ( P s ) in the first equal, and P s V k Ψ k = ψ k , s is the eigenstate operator. Thesimilar calculation shows that the noncommutative class is equivalent to the nonadiabatic contribution, which is a mixing of thebonding and antibonding states, (cid:213) k ( V k Ψ k ) † ρ µ σ ν V k Ψ k = (cid:213) k (cid:213) s , s ψ † k , s (cid:0) P s ρ µ σ ν P s (cid:1) ψ k , s = (cid:213) k (cid:213) s , s ψ † k , s (cid:0) P s P ¯ s ρ µ σ ν (cid:1) ψ k , s = (cid:213) k (cid:213) s ψ † k , s (cid:0) P s ρ µ σ ν (cid:1) ψ k , ¯ s . (S2.7)Then, we show the explicit forms for each class. A. Commutative class
For the commutative class ( µ, ν ) = ( , ) , ( , z ) , ( , ) , ( , z ) , ( , x ) , ( , y ) , ( , x ) , ( , y ) , we have P + ρ µ σ ν , P + ρ σ = P + ρ σ z = '›››« + + “fififi‹ ≡ τ ++ , (S2.8a) P + ρ σ x = P + ( i ρ )( i σ y ) = '›››« + + “fififi‹ ≡ τ x ++ , (S2.8b) P + ρ ( i σ y ) = P + ( i ρ ) σ x = i '›››« − i + i “fififi‹ ≡ i τ y ++ , (S2.8c) P + ρ σ = P + ρ σ z = '›››« + − “fififi‹ ≡ τ z ++ , (S2.8d)and P − ρ µ σ ν , P − ρ σ = '›››« + + “fififi‹ = + τ −− , P − ρ σ z = '›››« − − “fififi‹ = − τ −− , (S2.9a) P − ρ σ x = '›››« + + “fififi‹ ≡ + τ x −− , P − ( i ρ )( i σ y ) = '›››« − − “fififi‹ ≡ − τ x −− , (S2.9b) P − ( i ρ ) σ x = i '›››« − i + i “fififi‹ ≡ + i τ y −− , P − ρ ( i σ y ) = i '›››« + i − i “fififi‹ ≡ − i τ y −− , (S2.9c) P − ρ σ = '›››« + − “fififi‹ ≡ + τ z −− , P − ρ σ z = '›››« − + “fififi‹ ≡ − τ z −− . (S2.9d)Here, we defined τ µ ss ( s = ± and µ = , x , y, z ).To summarize these, we have P s ρ µ = , σ = τ µ = , zss , (S2.10a) P s ρ ⊥ σ x = τ ⊥ ss , (S2.10b) P s ρ ⊥ σ y = − s ˆ z × τ ⊥ ss = − is τ zss τ ⊥ ss , (S2.10c) P s ρ µ = , σ z = s τ µ = z , ss . (S2.10d) B. Noncommutative class
For the noncommutative class ( µ, ν ) = ( , x ) , ( , y ) , ( , ) , ( , z ) , ( , ) , ( , z ) , ( , x ) , ( , y ) , we have P + ρ µ σ ν , P + ρ σ = P + ( i ρ ) σ z = '›››« + + “fififi‹ ≡ τ x + − , (S2.11a) P + ( i ρ ) σ = P + ρ σ z = '›››« + − “fififi‹ ≡ i τ y + − , (S2.11b) P + ρ σ x = P + ρ ( i σ y ) = '›››« +
10 0 + “fififi‹ ≡ τ + − , (S2.11c) P + ρ σ x = P + ρ ( i σ y ) = '›››« +
10 0 − “fififi‹ ≡ τ z + − , (S2.11d)and P − ρ µ σ ν , P − ρ σ = '›››« + + “fififi‹ ≡ + τ x − + , P − ( i ρ ) σ z = '›››« − − “fififi‹ ≡ − τ x − + , (S2.12a) P − ( i ρ ) σ = '›››« + − “fififi‹ ≡ + i τ y − + , P − ρ σ z = '›››« − + “fififi‹ ≡ − i τ y − + , (S2.12b) P − ρ σ x = '›››« + + “fififi‹ ≡ + τ − + , P − ρ ( i σ y ) = '›››« − − “fififi‹ ≡ − τ − + , (S2.12c) P − ρ σ x = '›››« + − “fififi‹ ≡ + τ z − + , P − ρ ( i σ y ) = '›››« − + “fififi‹ ≡ − τ z − + . (S2.12d)Here, we defined τ µ s ¯ s ( s = ± and µ = , x , y, z ).To summarize these, we have P s ρ ⊥ σ = τ ⊥ s ¯ s , (S2.13a) P s ρ ⊥ σ z = s τ zs ¯ s τ ⊥ s ¯ s = − is ( ˆ z × τ ⊥ s ¯ s ) , (S2.13b) P s ρ µ = , σ x = τ µ = , zs ¯ s , (S2.13c) P s ρ µ = , σ y = − is τ µ = z , s ¯ s . (S2.13d)In the main text, we denote τ µ ss as τ µ for readability, and in this supplemental material we use the explicit expressions. III. NONADIABATIC COMPONENTS OF SPIN AND SPIN CURRENT
Here, we show the nonadiabatic components of spin and spin current operators, which are terms mixed between the bondingand antibonding states. In the rotated frame, the spin operator is given by s α ( q ) = (cid:213) k Ψ † k − q ρ σ α Ψ k . (S3.1)After the unitary transformation by V k and the projection operation, we have the adiabatic and nonadiabatic components of thespin: s x ( q ) = (cid:213) k , s ψ † k − q , s (cid:0) s sin ϑ k ˆ e ϕ k · τ ⊥ ss (cid:1) ψ k + q , s + (cid:213) k , s ψ † k − q , s (cid:16) − cos ϑ k τ s ¯ s (cid:17) ψ k + q , ¯ s , (S3.2a) s y ( q ) = (cid:213) k , s ψ † k − q , s (cid:0) sin ϑ k ( ˆ z × ˆ e ϕ k ) · τ ⊥ ss (cid:1) ψ k + q , s + (cid:213) k , s ψ † k − q , s (cid:0) is cos ϑ k τ zs ¯ s (cid:1) ψ k + q , ¯ s , (S3.2b) s z ( q ) = (cid:213) k , s ψ † k − q , s (cid:0) s τ zss (cid:1) ψ k + q , s . (S3.2c)Note that s z does not have the nonadiabatic component in the lowest order of q . By neglecting the nonadiabatic components,we obtain the Eq. (8) with (9) in the main text: s α ( q ) = (cid:213) s = ± (cid:213) k ψ † k − q , s (cid:16) s τ α k , s (cid:17) ψ k + q , s , (S3.3)where τ k , s = ( τ x k , s , τ y k , s , τ z k , s ) is given as τ k , s = '›« sin ϑ k ˆ e ϕ k · τ ⊥ ss s sin ϑ k (cid:0) ˆ z × ˆ e ϕ k (cid:1) · τ ⊥ ss τ zss “fi‹ (S3.4)(In the main text, we denote τ µ ss as τ µ for readability.) We here define the nonadiabatic components as s ⊥ na = ( s x na , s y na , ) ; s ⊥ na ( q ) = (cid:213) k , s ψ † k − q , s '›« − cos ϑ k τ s ¯ s is cos ϑ k τ zs ¯ s “fi‹ ψ k + q , ¯ s . (S3.5)This expression are used in Sec. VII.The spin current operator in the rotated frame is given by j α s , i ( q ) = (cid:213) k Ψ † k − q (cid:18) ~ ∂ H k ∂ k i σ α (cid:19) Ψ k + q . (S3.6)After the unitary transformation by V k and the projection operation, we have the spin current operator in the eigenstate frame; j x s , i ( q ) = (cid:213) k , s ψ † k − q s (cid:26) s ~ ∂(cid:15) k s ∂ k i τ x k , s + J sd ~ ( ˆ z × A ⊥ k , i ) · τ ⊥ ss (cid:27) ψ k + q s + (cid:213) k , s ψ † k − q s (cid:20) − ~ ∂(cid:15) k s ∂ k i cos ϑ k τ s ¯ s − is ∆ k sin ϑ k ~ n A ⊥ k , i · ˆ e ϕ k τ zs ¯ s + i ( A ⊥ k , i × ˆ e ϕ k ) z τ s ¯ s o(cid:21) ψ k + q ¯ s , (S3.7) j y s , i ( q ) = (cid:213) k , s ψ † k − q s (cid:18) s ~ ∂(cid:15) k s ∂ k i τ y k , s − s J sd ~ A ⊥ k , i · τ ⊥ ss (cid:19) ψ k + q s + (cid:213) k , s ψ † k − q s (cid:20) is ~ ∂(cid:15) k s ∂ k i cos ϑ k τ zs ¯ s − ∆ k sin ϑ k ~ n A ⊥ k , i · ˆ e ϕ k τ s ¯ s + i ( A ⊥ k , i × ˆ e ϕ k ) z τ zs ¯ s o(cid:21) ψ k + q ¯ s , (S3.8) j z s , i ( q ) = (cid:213) k , s ψ † k − q s (cid:18) s ~ ∂(cid:15) k s ∂ k i τ zss (cid:19) ψ k + q s + (cid:213) k , s ψ † k − q s (cid:18) is ∆ k ~ A ⊥ k , i · τ ⊥ s ¯ s (cid:19) ψ k + q ¯ s , (S3.9)where τ x k , s and τ y k , s are given in Eq. (S3.3). By neglecting the nonadiabatic components and dividing into the ordinary spincurrent j s0 , i and the anomalous spin current j sA , i , we have Eq. (10) and (11) in the main text: j s0 , i ( q ) = (cid:213) k , s ψ † k − q , s (cid:18) s ~ ∂(cid:15) k s ∂ k i τ k , s (cid:19) ψ k + q , s (S3.10)and j sA , i ( q ) = J sd ~ (cid:213) k , s ψ † k − q , s '›« ( ˆ z × A ⊥ k , i ) · τ ⊥ ss − s A ⊥ k , i · τ ⊥ ss “fi‹ ψ k + q , s . (S3.11)Here, we define the nonadiabatic components of the spin current as j nas , i ( q ) = (cid:213) k , s ψ † k − q s J i , k , s ψ k + q ¯ s + ˆ z (cid:213) k , s ψ † k − q s (cid:18) is ∆ k ~ A ⊥ k , i · τ ⊥ s ¯ s (cid:19) ψ k + q ¯ s (S3.12)with J i , k , s = (cid:18) ~ ∂(cid:15) k s ∂ k i cos ϑ k (cid:19) '›« − τ s ¯ s is τ zs ¯ s “fi‹ + ∆ k sin ϑ k ~ (cid:16) A ⊥ k , i · ˆ e ϕ k (cid:17) '›« − is τ zs ¯ s − τ s ¯ s “fi‹ + ( A ⊥ k , i × ˆ e ϕ k ) z '›« s τ s ¯ s − i τ zs ¯ s “fi‹ (S3.13)We use these expressions in Sec. VII. IV. SPIN TORQUES IN EIGENSTATE FRAME
In this section, we show the adiabatic and nonadiabatic components of the spin torques. Since the spin torques are given as (R − τ m ) q = J sd ~ ˆ z × h Σ ( q , t )i neq , (S4.1) (R − τ n ) q = J ~ ˆ z × h s ( q , t )i neq , (S4.2)we need to know the adiabatic and nonadiabatic components of Σ ( q ) = ˝ k Ψ k − q ρ σ α Ψ k + q and s ( q ) . The expression for s ( q ) is already obtained in Eqs. (S3.2). Hence, we calculate Σ ( q ) in the same manner as calculated s ( q ) , and then we have Σ x ( q ) = (cid:213) k (cid:213) s ψ † k − q s (cid:18) isq j A ⊥ k , j · τ ⊥ ss (cid:19) ψ k + q s + (cid:213) k (cid:213) s ψ † k − q s (cid:18) − τ zs ¯ s − iq j A z k , j τ s ¯ s (cid:19) ψ k + q ¯ s , (S4.3) Σ y ( q ) = (cid:213) k (cid:213) s ψ † k − q s (cid:18) iq j ( ˆ z × A ⊥ k , j ) · τ ⊥ ss (cid:19) ψ k + q s + (cid:213) k (cid:213) s ψ † k − q s (cid:16) is τ s ¯ s − sq j A z k , j τ zs ¯ s (cid:17) ψ k + q ¯ s , (S4.4) Σ z ( q ) = (cid:213) k (cid:213) s ψ † k − q s (cid:18) s cos ϑ k τ ss − isq j A z k , j τ zss (cid:19) ψ k + q s + (cid:213) k (cid:213) s ψ † k − q s (cid:0) sin ϑ k ˆ e ϕ k · τ ⊥ s ¯ s (cid:1) ψ k + q ¯ s , (S4.5)where we keep the first order with respect to q . By using the vector form, in the adiabatic regime, we find Σ ( q ) = ˆ z (cid:213) k (cid:213) s ψ † k − q s (cid:18) s cos ϑ k τ ss − iq j A z k , j τ zss (cid:19) ψ k + q s + iq j (cid:213) k (cid:213) s ψ † k − q s '›« s A ⊥ k , j · τ ⊥ ss ( ˆ z × A ⊥ k , j ) · τ ⊥ ss “fi‹ ψ k + q s = ˆ z (cid:213) k (cid:213) s ψ † k − q s (cid:18) s cos ϑ k τ ss − iq j A z k , j τ zss (cid:19) ψ k + q s − i ~ q j J sd ( ˆ z × j sA , j ( q )) . (S4.6)Hence, we have the Eq. (14a) in the main text.For the nonadiabatic component, we find Σ ⊥ na = ( Σ x na , Σ y na , ) with Σ ⊥ na ( q ) = (cid:213) k (cid:213) s ψ † k − q s '››« − τ zs ¯ s is τ s ¯ s “fifi‹ + O( q ) ψ k + q ¯ s . (S4.7)We use this expression in Sec. VII. V. ELECTRIC CURRENT OPERATOR IN EIGENSTATE FRAME
In order to calculate the current-induced spin-transfer torques based on the linear response theory, we derive the charge currentoperator in the eigenstate frame. The electromagnetic coupling in the rotated frame is given as H em = − e ∫ ω (cid:213) q j (− q ) · A em ( q , ω ) (S5.1)with the electric current operator j = j p + j A , where j p is the paramagnetic current given by j p = ( j p , x , j p ,y , j p , z ) with j p , i ( q ) = (cid:213) k Ψ † k − q (cid:18) ~ ∂ H k ∂ k i (cid:19) Ψ k + q , (S5.2)and j A is the anomalous current given by j A = ( j A , x , j A ,y , j A , z ) with j A , i ( q ) = ~ ∫ ω (cid:213) k , q A α r , j ( q + q , ω ) Ψ † k + q (cid:18) ~ ∂ H k ∂ k i ∂ k j (cid:19) σ α Ψ k − q . (S5.3)In the eigenstate frame, Eqs. (S5.2) and (S5.3) are obtained as j p , i ( q ) = (cid:213) k , s , s ψ † k − q s (cid:18) ~ ∂(cid:15) k s ∂ k i τ ss (cid:19) ψ k + q s + (cid:213) k , s , s ψ † k − q s (cid:26) − ∆ k ~ ( ˆ z × A ⊥ k , i ) · τ ⊥ s ¯ s (cid:27) ψ k + q ¯ s , (S5.4)and j A , i = j ( ) A , i + j ( ) A , i , j ( ) A , i ( q ) = ~ ∫ ω (cid:213) q A α r , j ( q + q , ω ) N α , ij (− q ) + j ( ) , naA , i ( q ) , (S5.5) j ( ) A , i ( q ) = ~ ∫ ω (cid:213) q A α r , j ( q + q , ω ) N α , ij (− q ) + j ( ) , naA , i ( q ) (S5.6)with N α , ij ( q ) = (cid:213) k , s ψ † k − q (cid:20)(cid:18) s ~ ∂ (cid:15) k s ∂ k i ∂ k j + ∆ k ~ A ⊥ k , i · A ⊥ k , j (cid:19) τ α k , s (cid:21) ψ k + q s , (S5.7) N α , ij ( q ) = ~ (cid:213) k , s ψ † k − q '›« cos ϑ k P ij · τ ⊥ ss s cos ϑ k ( ˆ z × P ij ) · τ ⊥ ss “fi‹ α ψ k + q s , (S5.8)where P ij = P ji is defined by P ij = ˆ z × ∂ ∆ k ∂ k j A ⊥ k , i + ∂ ∆ k ∂ k i A ⊥ k , j + ∆ k ∂ A ⊥ k , i ∂ k j + ∆ k ∂ A ⊥ k , j ∂ k i ! + ∆ k (cid:16) A z k , j A ⊥ k , i + A z k , i A ⊥ k , j (cid:17) . (S5.9)Here, j ( ) , naA , i ( q ) and j ( ) , naA , i ( q ) are the nonadiabatic components of the anomalous current, which are found to be negligible for thecalculation in the nonadiabatic spin-transfer torques. VI. ADIABATIC SPIN-TRANSFER TORQUES
Now, we show the calculation of the adiabatic spin-transfer torques. As shown in the main text, the adiabatic spin-transfertorques are obtained as the linear response of the anomalous spin current and the spin to the vector potential. h j α sA , i ( q , ω )i ne = X α ij ( q , ω ) A em j ( ω ) , (S6.1) h s α ( q , ω )i ne = Y α j ( q , ω ) A em j ( ω ) , (S6.2)where the linear response coefficients X α ij ( q , ω ) and Y α j ( q , ω ) are calculated from the corresponding correlation functions, X α ij ( q , i ω λ ) = − eV ∫ β d τ e i ω λ τ D T τ n j α sA , i ( q , τ ) j j ( , ) oE , (S6.3) Y α i ( q , i ω λ ) = − eV ∫ β d τ e i ω λ τ (cid:10) T τ (cid:8) s α ( q , τ ) j j ( , ) (cid:9)(cid:11) (S6.4)by taking the analytic continuation i ω λ → ~ ω + i X α ij ( q , ω ) = X α ij ( q , ~ ω + i ) , (S6.5) Y α j ( q , ω ) = Y α j ( q , ~ ω + i ) . (S6.6) A. Linear response coefficient for adiabatic component of anomalous spin current
First, we calculate the coefficient X α ij ( q , i ω λ ) . We divide X α ij ( q , i ω λ ) into the three terms, X α ij ( q , i ω λ ) = X ( ) α ij + X ( ) α ij + X ( ) α ij , (S6.7)where X ( ) α ij = − eV ∫ β d τ e i ω λ τ D T τ n j α sA , i ( q , τ ) j p , j ( , ) oE , (S6.8) X ( ) α ij = − eV ∫ β d τ e i ω λ τ D T τ n j α sA , i ( q , τ ) j ( ) A , j ( , ) oE , (S6.9) X ( ) α ij = − eV ∫ β d τ e i ω λ τ D T τ n j α sA , i ( q , τ ) j ( ) A , j ( , ) oE . (S6.10)Here, j p , j is given by Eq. (S5.4), and j ( ) A , j and j ( ) A , j are given by Eqs. (S5.5) and (S5.6). However, X ( ) α ij and X ( ) α ij are higherorder contributions with respect to q , since the anomalous current is proportional to the spin gauge field, which is first order of q . Hence, we neglect X ( ) α ij and X ( ) α ij .Rewriting X ( ) α ij by means of the Green function and expanding the first order of H f , we have X ( ) α ij = − MN e J sd ~ (R − m ) β q β V (cid:213) n (cid:213) k , s ~ ∂(cid:15) k s ∂ k j tr h C αγ k , i , s τ γ ss τ β k , s i n ( g + k , s ) g k , s + g + k , s ( g k , s ) o , (S6.11)where g + k , s = g k , s ( i (cid:15) n + i ω λ ) , g k , s = g k , s ( i (cid:15) n ) , and '›« ( ˆ z × A ⊥ k , i ) · τ ⊥ ss − s A ⊥ k , i · τ ⊥ ss “fi‹ = '›« − A y k , i A x k , i − sA x k , i − sA y k , i
00 0 0 “fi‹ αβ '›« τ xss τ y ss τ zss “fi‹ β = C k s , i τ β ss . (S6.12)Here, we keep the q dependence only of (R − m ) β q and set q = q contributions from the other partsare higher orders. Here, we calculate the trace astr h C αγ k , i , s τ γ ss τ β k , s i = ϑ k ( A ⊥ k , i × ˆ e ϕ k ) z δ αβ + s sin ϑ k ( A ⊥ k , i · ˆ e ϕ k ) (cid:15) αβ z = (cid:18) ˆ z · ∂ x k ∂ k i (cid:19) δ αβ − s (cid:26) ˆ z · (cid:18) x k × ∂ x k ∂ k i (cid:19)(cid:27) (cid:15) αβ z , (S6.13)where x k = '›« sin ϑ k cos ϕ k sin ϑ k sin ϕ k cos ϑ k “fi‹ , (S6.14)and we use the relation sin ϑ k ( A ⊥ k , i × ˆ e ϕ k ) z = (R k ˆ z ) · h(cid:16) R k A ⊥ k , i (cid:17) × R k ( sin ϑ k ˆ e ϕ k ) i = ˆ z · ∂ x k ∂ k i , (S6.15)sin ϑ k ( A ⊥ k , i · ˆ e ϕ k ) = (cid:16) R k A ⊥ k , i (cid:17) · R k ( sin ϑ k ˆ e ϕ k ) = − ˆ z · (cid:18) x k × ∂ x k ∂ k i (cid:19) (S6.16)with R αβ k = χ α k χ β k − δ αβ , where χ α k is defined by Eq. (S1.3). By using ∂(cid:15) k s ∂ k j n ( g + k , s ) g k , s + g + k , s ( g k , s ) o = ∂∂ k j (cid:16) g + k , s g k , s (cid:17) , (S6.17)and integrating by parts, we have X ( ) α ij = MN Φ ij ( i ω λ )(R − m ) α q − MN Φ ij ( i ω λ ) (cid:8) (R − m ) q × ˆ z (cid:9) α (S6.18)with Φ ij ( i ω λ ) = e J sd ~ β V (cid:213) n (cid:213) k , s ∂ cos ϑ k ∂ k i ∂ k j g + k , s g k , s , (S6.19) Φ ij ( i ω λ ) = − e J sd ~ β V (cid:213) n (cid:213) k , s s (cid:26) ˆ z · ∂∂ k j (cid:18) x k × ∂ x k ∂ k i (cid:19)(cid:27) g + k , s g k , s . (S6.20)Then, we take the analytic continuation, so that we obtain Φ ij ( ~ ω + i ) = − i ω P m e σ c δ i , j , (S6.21) Φ ij ( ~ ω + i ) = i ω e J sd ~ V (cid:213) k , s s (cid:26) ˆ z · ∂∂ k j (cid:18) x k × ∂ x k ∂ k i (cid:19)(cid:27) τδ ( µ − (cid:15) k s ) , (S6.22)where τ is the lifetime of the bonding/antibonding state, σ c is the conductivity given by σ c = e V (cid:213) k , s (cid:18) ~ ∂(cid:15) k s ∂ k i (cid:19) τδ ( µ − (cid:15) k s ) , (S6.23)and the nondimensional coefficient P m is given as P m = e σ c J sd ~ V (cid:213) k , s " J sd ∆ k (cid:18) ∂ ∆ k ∂ k i (cid:19) − J sd ∆ k ∂ ∆ k ∂ k i τδ ( µ − (cid:15) k s ) . (S6.24)For the case of T k =
0, we have the simple form P m ’ J sd /| µ | , ( T k = ) . (S6.25)For Φ ij , we calculate it in two cases; the case with site inversion symmetry and the honeycomb lattice for the simplest casewith no site inversion symmetry. For the case with site inversion symmetry, x k = '›« sin ϑ k ϑ k “fi‹ , ˆ z · ∂∂ k j (cid:18) x k × ∂ x k ∂ k i (cid:19) = , (S6.26)so that Φ ij = η = ± as the valley degree of freedom, x k = ∆ k '›« ηv k x v k y J sd “fi‹ , ˆ z · ∂∂ k j (cid:18) x k × ∂ x k ∂ k i (cid:19) = η ∂∂ k j (cid:18) v ∆ k ( k x δ i ,y − k y δ i , x ) (cid:19) , (S6.27)0and by summing η = ± , we have Φ ij = h j α sA , i ( q , ω )i ne = MN P m e j c , i (R − m ) α q , (S6.28)which leads to the result τ ( r , t ) = MN (cid:18) P m e j c · ∇ (cid:19) m ( r , t ) , (S6.29)where P m is given by Eq. (S6.24). B. Linear response coefficient for adiabatic component of spin
Then, we evaluate the linear response coefficient for the adiabatic component of spin Y α j ( q , ω ) . In the similar way of thecalculation X α ij ( q , i ω λ ) , we divide Y α j ( q , i ω λ ) into three parts, Y α j ( q , i ω λ ) = Y ( ) α j + Y ( ) α j + Y ( ) α j (S6.30)with Y ( ) α i = − eV ∫ β d τ e i ω λ τ (cid:10) T τ (cid:8) ˜ s α ( q , τ ) ˜ j p , i ( , ) (cid:9)(cid:11) , (S6.31) Y ( ) α i = − eV ∫ β d τ e i ω λ τ D T τ n ˜ s α ( q , τ ) ˜ j ( ) A , i ( , ) oE , (S6.32) Y ( ) α i = − eV ∫ β d τ e i ω λ τ D T τ n ˜ s α ( q , τ ) ˜ j ( ) A , i ( , ) oE . (S6.33)Rewriting them by means of the Green function and expanding the first order with respect to the spin gauge field, we have Y ( ) α i = e ~ β V A β r , j ( q ) (cid:213) n (cid:213) k , s s ~ ∂(cid:15) k s ∂ k i tr (cid:20) τ α k s (cid:18) s ~ ∂(cid:15) k s ∂ k j τ β k s + J sd ~ C βδ k s , j τ δ ss (cid:19)(cid:21) (cid:16) ( g + k s ) g k s + g + k s ( g k s ) (cid:17) , (S6.34) Y ( ) α i = e ~ β V A β r , j ( q ) (cid:213) n (cid:213) k , s s (cid:18) s ~ ∂ (cid:15) k s ∂ k i ∂ k j + ∆ k ~ A ⊥ k , i · A ⊥ k , j (cid:19) tr h τ α k s τ β k s i g + k s g k s , (S6.35) Y ( ) α i = e ~ β V A β r , j ( q ) (cid:213) n (cid:213) k , s ~ cos ϑ k tr h τ α k s D βδ ij , k s τ δ ss i g + k s g k s , (S6.36)where '›« P ij · τ ⊥ ss s ( ˆ z × P ij ) · τ ⊥ ss “fi‹ = '›« P xij P y ij − sP y ij sP xij
00 0 0 “fi‹ '›« τ xss τ y ss τ zss “fi‹ = D ij , k s τ ss . (S6.37)Calculating tr [· · · ] , we obtain tr h τ α k s τ β k s i = ϑ k δ αβ , (S6.38)tr h τ α k s C βδ k s , j τ δ ss i = ϑ k ( A ⊥ k , j × ˆ e ϕ k ) z δ αβ − s sin ϑ k ( A ⊥ k , j · ˆ e ϕ k ) (cid:15) αβ z , (S6.39)tr h τ α k s D βδ ij , k s τ δ ss i = ϑ k ( P ij · ˆ e ϕ k ) δ αβ + s sin ϑ k ( P ij × ˆ e ϕ k ) z (cid:15) αβ z . (S6.40)For Y ( ) α i , integrating by parts, we find Y α i = e ij A α r , j ( q ) + f ij (cid:15) αβ z A β r , j ( q ) , (S6.41)1where e ij = e ~ V (cid:213) k e k s , ij Q k s ( i ω λ ) , (S6.42) f ij = e ~ V (cid:213) k f k s , ij Q k s ( i ω λ ) (S6.43)with Q k s ( i ω λ ) = β (cid:213) n g + k , s g k , s → − i ωτδ ( µ − (cid:15) k s ) (S6.44)and e k , s , ij = s cos ϑ k ~ (cid:18) s ~ ∂(cid:15) k s ∂ k j (cid:19) ∂ cos ϑ k ∂ k i − s J sd ~ ∂∂ k i sin ϑ k ( A ⊥ k , j × ˆ e ϕ k ) z + s ∆ k ~ A ⊥ k , i · A ⊥ k , j sin ϑ k + ~ cos ϑ k sin ϑ k ( P ij · ˆ e ϕ k ) , (S6.45) f k s , ij = J sd ~ ∂∂ k i h sin ϑ k ( A ⊥ k , j · ˆ e ϕ k ) i + s ~ cos ϑ k sin ϑ k ( P ij × ˆ e ϕ k ) z . (S6.46)By using Eqs. (S6.15), (S6.16),sin ϑ k ( P ij · ˆ e ϕ k ) = J sd ∆ k ∂ ∆ k ∂ k i ∂ ∆ k ∂ k j − J sd ∆ k ∂ ∆ k ∂ k i ∂ k j + J sd (cid:18) ∂ x k ∂ k i · ∂ x k ∂ k j (cid:19) , (S6.47)and sin ϑ k ( P ij × ˆ e ϕ k ) z = ˆ z · (cid:18) ∆ k x k × ∂ x k ∂ k i ∂ k j + ∂ ∆ k ∂ k i x k × ∂ x k ∂ k j + ∂ ∆ k ∂ k j x k × ∂ x k ∂ k i (cid:19) , (S6.48)we have e k s , ij = − ϑ k ∆ k ~ ∂(cid:15) k s ∂ k j ∂ ∆ k ∂ k i − s ~ cos ϑ k (cid:18) ∆ k ∂ ∆ k ∂ k i ∂ ∆ k ∂ k j − ∂ ∆ k ∂ k i ∂ k j (cid:19) + ∆ k ~ (cid:16) s sin ϑ k + cos ϑ k (cid:17) ∂ x k ∂ k i · ∂ x k ∂ k j , (S6.49) f k s , ij = − J sd ~ ˆ z · ∂∂ k i (cid:18) x k × ∂ x k ∂ k j (cid:19) + s ~ cos ϑ k ˆ z · (cid:18) ∆ k x k × ∂ x k ∂ k i ∂ k j + ∂ ∆ k ∂ k i x k × ∂ x k ∂ k j + ∂ ∆ k ∂ k j x k × ∂ x k ∂ k i (cid:19) . (S6.50)Similar consideration as in Φ ( i ω λ ) [Eqs. (S6.26) and (S6.27)] leads to f k s , ij = Y α j ( q , ω ) = − i ω P n e J sd σ c A α r , j ( q ) , (S6.51)where P n = e J sd σ c V (cid:213) k , s e k , s , ii τδ ( µ − (cid:15) k s ) = e J sd σ c V (cid:213) k , s " J sd ∆ k s ~ ∂ ∆ k ∂ k i ∂ k i − J sd ∆ k ~ ∂ T k ∂ k i ∂ ∆ k ∂ k i + ~ s | η k | + J sd ∆ k ∂ x k ∂ k i · ∂ x k ∂ k i τδ ( µ − (cid:15) k s ) . (S6.52)For the simple case of T k = P n ’ J sd | µ | J sd µ − J sd − sign ( µ ) ! . (S6.53)We here note that P n does not have the particle-hole symmetry. We also note that P n → ∞ when | µ | → J sd , but the spin torqueis to be zero since σ c → h s α ( q , ω )i ne = P n e J sd j c , j A α r , j ( q ) , (S6.54)which leads to τ n = N (cid:18) P n e j c · ∇ (cid:19) n . (S6.55)2 VII. NONADIABATIC SPIN-TRANSFER TORQUES
Finally, we derive and calculate the nonadiabatic spin-transfer torques. In Eqs. (S3.5) and (S4.7), we have the nonadiabaticcomponents of s and Σ . Hence, the nonadiabatic spin torques in the rotated frame are given as (R − τ na m ) q = J sd N ~ ˆ z × h Σ ⊥ na ( q , t )i neq , (R − τ na n ) q = J sd ~ ˆ z × h s ⊥ na ( q , t )i neq , (S7.1)so that we calculate the linear response of Σ ⊥ na and s ⊥ na to the vector potential as h Σ α na ( q , ω )i neq = X na ,α j ( q , ω ) A em j ( ω ) , (S7.2) h s α na ( q )i neq = Y na ,α j ( q , ω ) A em j ( ω ) , (S7.3)where α = x , y , and the response coefficients are obtained from the corresponding Matsubara functions X na ,α j ( q , i ω λ ) = − eV ∫ β d τ e i ω λ τ (cid:10) T τ (cid:8) Σ α na ( q , τ ) j j ( , ) (cid:9)(cid:11) , (S7.4) Y na ,α j ( q , i ω λ ) = − eV ∫ β d τ e i ω λ τ (cid:10) T τ (cid:8) s α na ( q , τ ) j j ( , ) (cid:9)(cid:11) (S7.5)by taking the analytic continuation, i ω λ → ~ ω + i
0. We below evaluate X na ,α j ( q , i ω λ ) and Y na ,α j ( q , i ω λ ) . Σ α na ¯ s ss ¯ s ¯ s ¯ ss s ¯ s ss ¯ s ¯ ss i n + i ω λ i n + i ω λ i n + i ω λ i n + iω λ i n + iω λ i n + iω λ i n i n i n i n i n + i ω λ i n i n i n (a) (b)(c) (d) (e)(f) J βi, k , ¯ s j β s ,i j nap ,j k s ∂k j N na(1) ,βij, ¯ s ¯ ss i n + iω λ i n N na(2) ,βij, ¯ s FIG. S1. Feynman diagrams for (a)-(d) X na , ( ) ,α j , for (e) X na , ( ) ,α j and for (f) X na , ( ) ,α j . The diagrams (a) and (d) are main contributionswith respect to the lifetime of the bonding/antibonding state. First, we calculate X na ,α j ( q , i ω λ ) . We divide X na ,α j ( q , i ω λ ) into three parts; X na ,α j ( q , i ω λ ) = X na , ( ) ,α j + X na , ( ) ,α j + X na , ( ) ,α j , (S7.6)3where X na , ( ) ,α j = − eV ∫ β d τ e i ω λ τ h T τ { Σ α na ( q , τ ) j p , j ( , )}i , (S7.7) X na , ( ) ,α j = − eV ∫ β d τ e i ω λ τ h T τ { Σ α na ( q , τ ) j ( ) A , j ( , )}i , (S7.8) X na , ( ) ,α j = − eV ∫ β d τ e i ω λ τ h T τ { Σ α na ( q , τ ) j ( ) A , j ( , )}i . (S7.9)Rewriting them by means of the Green function, we have the Feynman diagrams shown in Fig. S1, which read X na , ( ) ,α j = ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr [E αγ s τ γ s ¯ s J β i , k ¯ s ] (cid:18) ~ ∂(cid:15) k ¯ s ∂ k j g + k , ¯ s g k , ¯ s g k , s Λ s + ~ ∂(cid:15) k s ∂ k j g + k , ¯ s g + k , s g k , s Λ s (cid:19) + ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr (cid:20) E αγ s τ γ s ¯ s (cid:18) − s ~ ∂(cid:15) k ¯ s ∂ k j τ β k ¯ s + J sd ~ C βδ k ¯ s , j τ δ ¯ s ¯ s (cid:19) − ∆ k ~ ( ˆ z × A ⊥ k , j ) · τ ⊥ ¯ ss (cid:21) ( g + k , ¯ s ) g k , s Λ s + ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr (cid:20) E αγ s τ γ s ¯ s − ∆ k ~ ( ˆ z × A ⊥ k , j ) · τ ⊥ ¯ ss (cid:18) s ~ ∂(cid:15) k s ∂ k j τ β k s + J sd ~ C βδ k s , j τ δ ss (cid:19)(cid:21) g + k , ¯ s ( g k , s ) Λ s , (S7.10) X na , ( ) ,α j = ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr h E αγ s τ γ s ¯ s N na ( ) ,β ij , ¯ s i g + k , ¯ s g k , s Λ s , (S7.11) X na , ( ) ,α j = ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr h E αγ s τ γ s ¯ s N na ( ) ,β ij , ¯ s i g + k , ¯ s g k , s Λ s , (S7.12)where (cid:18) − τ zs ¯ s is τ s ¯ s (cid:19) α = (cid:18) − is (cid:19) αβ (cid:18) τ s ¯ s τ zs ¯ s (cid:19) β ≡ E αβ s τ β s ¯ s , (S7.13) J i , k , s = ( J xi , k , s , J y i , k , s , ) is given by Eq. (S3.13), and Λ s is the vertex correction given by Λ s = + n i u (cid:213) k g + k , ¯ s g k , s Λ s . (S7.14)Here, N na ( ) ,β ij , ¯ s and N na ( ) ,β ij , ¯ s are the nonadiabatic components of the anomalous current. (We see below that X na , ( ) ,α j and X na , ( ) ,α j are negligible.)The vertex correction Λ s is calculated as Λ s = − n i u ˝ k g + k , ¯ s g k , s , (S7.15)and by using g + k , ¯ s g k , s = − g + k , ¯ s + g k , s i ω λ − s ∆ k − Σ + ¯ s + Σ s ’ − g + k , ¯ s + g k , s i ω λ − s J sd − Σ + ¯ s + Σ s , (S7.16)where Σ + s = Σ s ( i (cid:15) n + i ω λ ) = n i u ˝ k g + k , s and Σ = Σ ( i (cid:15) n ) = n i u ˝ k g k , s are the self energies, and we have approximate ∆ k ’ J sd , we have g + k , ¯ s g k , s Λ s = − g + k , ¯ s + g k , s i ω λ − s J sd . (S7.17)From this, we find the main contribution is the first term of X na , ( ) ,α j , which is evaluated as X na ,α j ( q , i ω λ ) = ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr [E αγ s τ γ s ¯ s J β i , k ¯ s ] (cid:18) ~ ∂(cid:15) k s ∂ k j g + k , s g k , s − ~ ∂(cid:15) k ¯ s ∂ k j g + k , ¯ s g k , ¯ s (cid:19) i ω λ − s J sd = ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr [E αγ s τ γ s ¯ s J β i , k ¯ s ] i ω λ − s J sd (cid:213) σ ~ ∂(cid:15) k σ ∂ k j g + k ,σ g k ,σ . (S7.18)4The trace tr [· · · ] is calculated astr [E αγ s τ γ s ¯ s J β i , k ¯ s ] = ϑ k ~ ∂(cid:15) k ¯ s ∂ k i is (cid:15) αβ z − ∆ k ~ sin ϑ k ( A ⊥ k , i · ˆ e ϕ k ) is δ αβ + ∆ k ~ sin ϑ k ( A ⊥ k , i × ˆ e ϕ k ) z i (cid:15) αβ z , (S7.19)which reads tr [E αγ s τ γ s ¯ s J β i , k ¯ s ] = ϑ k ~ ∂(cid:15) k ¯ s ∂ k i is (cid:15) αβ z − ϑ k ∂ ∆ k ∂ k i i (cid:15) αβ z = ϑ k ~ ∂ T k ∂ k i is (cid:15) αβ z . (S7.20)(the term proportional to δ αβ is zero from the similar consideration as in Eqs. (S6.26) and (S6.27).)Hence, we have X na ,α j ( q , i ω λ ) = e ~ V ( A ⊥ r , i × ˆ z ) α β (cid:213) n (cid:213) k , s i ω λ − s J sd (cid:213) σ ~ ∂ T k ∂ k i ∂(cid:15) k σ ∂ k j g + k ,σ g k ,σ , (S7.21)and taking the analytic continuation, we finally obtain X na ,α j ( q , ω ) = − i ω i ~ ω ~ ω − J sd P na m e σ c ( A ⊥ r , j × ˆ z ) α , (S7.22)where P na m = e σ c V (cid:213) k , s J sd ∆ k ~ ∂ T k ∂ k i ∂(cid:15) k σ ∂ k i τδ ( µ − (cid:15) k s ) . (S7.23)For the case of T k =
0, we find P na m = τ na m ( r , t ) = i ωτ sd − ω τ sd n × (cid:18) P na m e j c · ∇ (cid:19) n ’ i ωτ sd n × (cid:18) P na m e j c · ∇ (cid:19) n ( ωτ sd (cid:28) ) . (S7.24)The calculation of Y na ,α j ( q , i ω λ ) can be done in the similar way of X na ,α j ( q , i ω λ ) , and we have Y na ,α j ( q , i ω λ ) ’ ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr [F αγ s τ γ s ¯ s J β i , k ¯ s ] (cid:18) ~ ∂(cid:15) k s ∂ k j g + k , s g k , s − ~ ∂(cid:15) k ¯ s ∂ k j g + k , ¯ s g k , ¯ s (cid:19) i ω λ − s J sd = ~ eV A β r , i ( q ) β (cid:213) n (cid:213) k , s tr [F αγ s τ γ s ¯ s J β i , k ¯ s ] i ω λ − s J sd (cid:213) σ ~ ∂(cid:15) k σ ∂ k j g + k ,σ g k ,σ , (S7.25)where F αγ s is defined by (cid:18) − cos ϑ k τ s ¯ s is cos ϑ k τ zs ¯ s (cid:19) α = (cid:18) − cos ϑ k is cos ϑ k (cid:19) αβ (cid:18) τ s ¯ s τ zs ¯ s (cid:19) β ≡ F αβ s τ β s ¯ s . (S7.26)Here, the trace is evaluated astr [F αγ s τ γ s ¯ s J β i , k ¯ s ] = ϑ k ~ ∂(cid:15) k ¯ s ∂ k i δ αβ − ∆ k ~ sin ϑ k cos ϑ k ( A ⊥ k , i · ˆ e ϕ k ) (cid:15) αβ z + ∆ k ~ sin ϑ k cos ϑ k ( A ⊥ k , i × ˆ e ϕ k ) z δ αβ = ϑ k ~ ∂ T k ∂ k i δ αβ . (S7.27)(the term proportional to (cid:15) αβ z is zero from the similar consideration as in Eqs. (S6.26) and (S6.27).)Finally, we have Y na ,α j ( q , ω ) = − i ω ~ J sd ~ ω − J sd P na n e σ c A α r , j , (S7.28)5which leads to the nonadiabatic spin-transfer torque on the Neel vector, τ n ( r , t ) = N − ω τ sd (cid:18) P na n e j c · ∇ (cid:19) n ’ − N (cid:18) P na n e j c · ∇ (cid:19) n ( ωτ sd (cid:28) ) . (S7.29)Here, P na n = e σ c V (cid:213) k , s (cid:18) J sd ∆ k (cid:19) ~ ∂ T k ∂ k i ∂(cid:15) k σ ∂ k i τδ ( µ − (cid:15) k s ) , (S7.30)and For the case of T k =
0, we have P na n =
0, and ∆ k ’ J sd , we find P na n = P na m ’ , (S7.31)since ∂ T k / ∂ k i ’ ∂(cid:15) k s / ∂ k ii