Dynamical and current-induced Dzyaloshinskii-Moriya interaction: Role for damping, gyromagnetism, and current-induced torques in noncollinear magnets
aa r X i v : . [ c ond - m a t . o t h e r] J un Dynamical and current-induced Dzyaloshinskii-Moriya interaction: Role for damping,gyromagnetism, and current-induced torques in noncollinear magnets
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 and Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
Both applied electric currents and magnetization dynamics modify the Dzyaloshinskii-Moriya in-teraction (DMI), which we call current-induced DMI (CIDMI) and dynamical DMI (DDMI), respec-tively. We report a theory of CIDMI and DDMI. The inverse of CIDMI consists in charge pumpingby a time-dependent gradient of magnetization ∂ M ( r , t ) /∂ r ∂t , while the inverse of DDMI describesthe torque generated by ∂ M ( r , t ) /∂ r ∂t . In noncollinear magnets CIDMI and DDMI depend onthe local magnetization direction. The resulting spatial gradients correspond to torques that needto be included into the theories of Gilbert damping, gyromagnetism, and current-induced torques(CITs) in order to satisfy the Onsager reciprocity relations. CIDMI is related to the modification oforbital magnetism induced by magnetization dynamics, which we call dynamical orbital magnetism(DOM), and spatial gradients of DOM contribute to charge pumping. We present applications ofthis formalism to the CITs and to the torque-torque correlation in textured Rashba ferromagnets. I. INTRODUCTION
Since the Dzyaloshinskii-Moriya interaction (DMI)controls the magnetic texture of domain walls andskyrmions, methods to tune this chiral interaction byexternal means have exciting prospects. Application ofgate voltage [1–3] or laser pulses [4] are promising ways tomodify DMI. Additionally, theory predicts that in mag-netic trilayer structures the DMI in the top magneticlayer can be controlled by the magnetization directionin the bottom magnetic layer [5]. Moreover, methods togenerate spin currents may be used to induce DMI, whichis predicted by the relations between the two [6, 7]. Inmetallic magnets one expects that also electric currentsmodify DMI. However, a rigorous theoretical formalismfor the investigation of current-induced DMI (CIDMI) inmetallic magnets has been lacking so far, and the devel-opment of such a formalism is one goal of this paper.Recently, a Berry phase theory of DMI [6, 8, 9] hasbeen developed, which formally resembles the moderntheory of orbital magnetization [10–12]. Orbital mag-netism is modified by the application of an electricfield, which is known as the orbital magnetoelectric re-sponse [13]. In the case of insulators it is straightfor-ward to derive the expressions for the magnetoelectricresponse directly. However, in metals it is much easierto derive expressions instead for the inverse of the mag-netoelectric response, i.e., for the generation of electriccurrents by time-dependent magnetic fields [14]. The in-verse current-induced DMI (ICIDMI) consists in chargepumping by time-dependent gradients of magnetization.Due to the analogies between orbital magnetism and theBerry phase theory of DMI one may expect that in metalsit is convenient to obtain expressions for ICIDMI, whichcan then be used to describe the CIDMI by exploiting thereciprocity between CIDMI and ICIDMI. We will showin this paper that this is indeed the case. In noncentrosymmetric ferromagnets spin-orbit inter-action (SOI) generates torques on the magnetization –the so-called spin-orbit torques (SOTs) – when an elec-tric current is applied [15]. The Berry phase theory ofDMI [6, 8, 9] establishes a relation to SOTs. The for-mal analogies between orbital magnetism and DMI havebeen shown to be a very useful guiding principle in thedevelopment of the theory of SOTs driven by heat cur-rents [16]. In particular, it is fruitful to consider the DMIcoefficients as a spiralization, which is formally analo-gous to magnetization. In the theory of thermoelectriceffects in magnetic systems the curl of magnetization de-scribes a bound current, which cannot be measured intransport experiments and needs to be subtracted fromthe Kubo linear response in order to obtain the mea-surable current [17–19]. Similarly, in the theory of thethermal spin-orbit torque spatial gradients of the DMIspiralization, which result from the temperature gradi-ent together with the temperature dependence of DMI,need to be subtracted in order to obtain the measurabletorque and to satisfy a Mott-like relation [8, 16]. In non-collinear magnets the question arises whether gradientsof the spiralization that are due to the magnetic texturecorrespond to torques like those from thermal gradients.We will show that indeed the spatial gradients of CIDMIneed to be included into the theory of current-inducedtorques (CITs) in noncollinear magnets in order to sat-isfy the Onsager reciprocity relations [20].When the system is driven out of equilibrium by mag-netization dynamics rather than electric current one mayexpect DMI to be modified as well. The inverse effect ofthis dynamical DMI (DDMI) consists in the generationof torques by time-dependent magnetization gradients.In noncollinear magnets the DDMI spiralization variesin space. We will show that the resulting gradient cor-responds to a torque that needs to be considered in thetheory of Gilbert damping and gyromagnetism in non-collinear magnets.This paper is structured as follows. In section II Awe give an overview of CIT in noncollinear magnets andintroduce the notation. In section II B we describe theformalism used to calculate the response of electric cur-rent to time-dependent magnetization gradients. In sec-tion II C we show that current-induced DMI (CIDMI)and electric current driven by time-dependent magneti-zation gradients are reciprocal effects. This allows usto obtain an expression for CIDMI based on the formal-ism of section II B. In section II D we discuss that time-dependent magnetization gradients generate additionallytorques on the magnetization and show that the inverseeffect consists in the modification of DMI by magnetiza-tion dynamics, which we call dynamical DMI (DDMI). Insection II E we demonstrate that magnetization dynam-ics induces orbital magnetism, which we call dynamicalorbital magnetism (DOM) and show that DOM is relatedto CIDMI. In section II F we explain how the spatial gra-dients of CIDMI and DOM contribute to the direct andto the inverse CIT, respectively. In section II G we dis-cuss how the spatial gradients of DDMI contribute to thetorque-torque correlation. In section II H we completethe formalism used to calculate the CIT in noncollinearmagnets by adding the chiral contribution of the torque-velocity correlation. In section II I we finalize the theoryof the inverse CIT by adding the chiral contribution ofthe velocity-torque correlation. In section II J we fin-ish the computational formalism of gyromagnetism anddamping by adding the chiral contribution of the torque-torque correlation and the response of the torque to thetime-dependent magnetization gradients. In section IIIwe discuss the symmetry properties of the response totime-dependent magnetization gradients. In section IV Awe present the results for the chiral contributions to thedirect and the inverse CIT in the Rashba model and showthat both the perturbation by the time-dependent mag-netization gradient and the spatial gradients of CIDMIand DOM need to be included to ensure that they arereciprocal. In section IV B we present the results for thechiral contribution to the torque-torque correlation in theRashba model and show that both the perturbation bythe time-dependent magnetization gradient and the spa-tial gradients of DDMI need to be included to ensure thatit satisfies the Onsager symmetry relations. This paperends with a summary in section V.
II. FORMALISMA. Direct and inverse current-induced torques innoncollinear magnets
Even in collinear magnets the application of an electricfield E generates a torque T CIT1 on the magnetization when inversion symmetry is broken [15, 21]: T CIT1 i = X j t ij ( ˆ M ) E j , (1)where t ij ( ˆ M ) is the torkance tensor, which depends onthe magnetization direction ˆ M . This torque is calledspin-orbit torque (SOT), but we denote it here CIT1,because it is one contribution to the current-inducedtorques (CITs) in noncollinear magnets. Inversely, mag-netization dynamics pumps a charge current J ICIT1 ac-cording to [22] J ICIT1 i = X j t ji ( − ˆ M )ˆ e j · " ˆ M × ∂ ˆ M ∂t , (2)where ˆ e j is a unit vector that points into the j -th spa-tial direction. Generally, J ICIT1 can be explained bythe inverse spin-orbit torque [22] or the magnonic chargepumping [23]. We denote it here by ICIT1, because itis one contribution to the inverse CIT in noncollinearmagnets. In the special case of magnetic bilayers one im-portant mechanism responsible for J ICIT1 arises from thecombination of spin pumping and the inverse spin Halleffect [24, 25].In noncollinear magnets there is a second contributionto the CIT, which is proportional to the spatial deriva-tives of magnetization [26]: T CIT2 i = X jkl χ CIT2 ijkl E j ˆ e k · " ˆ M × ∂ ˆ M ∂r l . (3)The adiabatic and the non-adiabatic [27] spin trans-fer torques are two important contributions to χ CIT2 ijkl ,but the interplay between broken inversion symmetry,SOI, and noncollinearity can lead to a large number ofadditional mechanisms [20, 28]. Similarly, the currentpumped by magnetization dynamics contains a contri-bution that is proportional to the spatial derivatives ofmagnetization [20, 29, 30]: J ICIT2 i = X jkl χ ICIT2 ijkl ˆ e j · " ˆ M × ∂ ˆ M ∂t ˆ e k · " ˆ M × ∂ ˆ M ∂r l . (4) T CIT2 i and J ICIT2 i can be considered as chiral contribu-tions to the CIT and to the ICIT, respectively, becausethey distinguish between left- and right-handed spin spi-rals. Due to the reciprocity between direct and inverseCIT [20, 22] the coefficients χ ICIT2 ijkl and χ CIT2 jikl are relatedaccording to χ ICIT2 ijkl ( ˆ M ) = χ CIT2 jikl ( − ˆ M ) . (5) B. Response of electric current to time-dependentmagnetization gradients
In order to compute J ICIT2 based on the Kubo linearresponse formalism it is necessary to split it into two con-tributions, J ICIT2a and J ICIT2b . The first contributionis determined by the second derivative of magnetizationwith respect to time and space variables and can be writ-ten as J ICIT2a i = X jk χ ICIT2a ijk ∂ ˆ M j ∂r k ∂t . (6)A nonzero second derivative ∂ ˆ M j ∂r k ∂t is what we refer toas a time-dependent magnetization gradient . The secondcontribution is proportional to both the time-derivativeand the spatial derivative of magnetization and is givenby J ICIT2b J ICIT2b i = X jkl χ ICIT2b ijkl ˆ e j · " ˆ M × ∂ ˆ M ∂t ˆ e k · " ˆ M × ∂ ˆ M ∂r l . (7)These two contributions need to be treated differentlywithin the Kubo formalism: While J ICIT2a i is obtained aslinear response to the perturbation by a time-dependentmagnetization gradient in a collinear magnet, J ICIT2b i isobtained as linear response to the perturbation by mag-netization dynamics in a noncollinear magnet. We willshow below that ∂ ˆ M j ∂r k ∂t can be expressed in terms of theproducts ∂ ˆ M l ∂r k ∂ ˆ M l ∂t , which will allow us to rewrite J ICIT2a i in the form of Eq. (4) in the cases relevant for the chiralICIT. In the remainder of this section we discuss the cal-culation of the contribution J ICIT2a . The contribution J ICIT2b is discussed in section II H below. J ICIT2a occurs in two different situations, which needto be distinguished. In one case the magnetization gra-dient varies in time like sin( ωt ) everywhere in space. Anexample is ˆ M ( r , t ) = η sin( q · r ) sin( ωt )01 , (8)where η is the amplitude and the derivatives at t = 0 and r = 0 are ∂ ˆ M ( r , t ) ∂r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = t =0 = ∂ ˆ M ( r , t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = t =0 = 0 (9)and ∂ ˆ M ( r , t ) ∂r i ∂t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = t =0 = ηq i ω . (10)In the other case the magnetic texture varies like apropagating wave, i.e., proportional to sin( q · r − ωt ). An example is given byˆ M ( r , t ) = η sin( q · r − ωt )01 − η sin ( q · r − ωt ) , (11)where the derivatives at t = 0 and r = 0 are ∂ ˆ M ( r , t ) ∂r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = t =0 = ηq i , (12) ∂ ˆ M ( r , t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = t =0 = − ηω (13)and ∂ ˆ M ( r , t ) ∂r i ∂t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = t =0 = η q i ω . (14)In the latter example, Eq. (11), the second derivative,Eq. (14), is along the magnetization ˆ M ( r = 0 , t = 0),while in the former example, Eq. (8), the second deriva-tive, Eq. (10), is perpendicular to the magnetizationwhen r = 0 and t = 0.We assume that the Hamiltonian is given by H ( r , t ) = − ~ m e ∆ + V ( r ) + µ B ˆ M ( r , t ) · σ Ω xc ( r )++ 12 ec µ B σ · [ ∇ V ( r ) × v ] , (15)where the first term describes the kinetic energy, the sec-ond term is a scalar potential, Ω xc ( r ) in the third term isthe exchange field, and the last term describes the spin-orbit interaction. Around t = 0 and r = 0 we can de-compose the Hamiltonian as H ( r , t ) = H + δH ( r , t ),where H is obtained from H ( r , t ) by replacing ˆ M ( r , t )by ˆ M ( r = 0 , t = 0) and δH ( r , t ) = ∂H ∂ ˆ M x η sin( q · r ) sin( ωt )= µ B Ω xc ( r ) σ x η sin( q · r ) sin( ωt ) (16)in the case of the first example, Eq. (8). In the case ofthe second example, Eq. (11), δH ( r , t ) ≃ ∂H∂ ˆ M x η sin( q · r − ωt )+ ∂H∂ ˆ M z η sin( q · r ) sin( ωt ) , (17)where for small r and t only the second term on theright-hand side contributes to ∂ H ( r ,t ) ∂r k ∂t .The perturbation by the time-dependent gradient canbe written as δH = ∂H∂ ˆ M · ∂ ˆ M ∂r i ∂t sin( q i r i ) q i sin( ωt ) ω , (18)which turns into Eq. (16) when Eq. (10) is inserted.When Eq. (14) is inserted it turns into the second termin Eq. (17).In Appendix A we derive the linear response to pertur-bations of the type of Eq. (18) and show that the corre-sponding coefficient χ ICIT2a ijk in Eq. (6) can be expressedas χ ICIT2a ijk = ie π ~ Z d d k (2 π ) d Z d E f ( E )Tr h v i Rv k RR O j R + v i RRv k R O j R + − v i RR O j Rv k R − v i Rv k R O j AA + v i R O j Av k AA + v i R O j AAv k A − v i Rv k RR O j A − v i RRv k R O j A + v i RR O j Av k A + v i Av k A O j AA − v i A O j Av k AA − v i A O j AAv k A i , (19)where R = G R k ( E ) and A = G A k ( E ) are shorthands for theretarded and advanced Green’s functions, respectively,and O j = ∂H/∂ ˆ M j . e > ∂ ˆ M ∂r i ∂t is perpendicular to M . In thiscase it is convenient to rewrite Eq. (6) as J ICIT2a i = X jk χ ICIDMI ijk ˆ e j · " ˆ M × ∂ ˆ M ∂r k ∂t , (20)where the coefficients χ ICIDMI ijk are given by χ ICIDMI ijk = ie π ~ Z d d k (2 π ) d Z d E f ( E )Tr h v i Rv k RR T j R + v i RRv k R T j R + − v i RR T j Rv k R − v i Rv k R T j AA + v i R T j Av k AA + v i R T j AAv k A − v i Rv k RR T j A − v i RRv k R T j A + v i RR T j Av k A + v i Av k A T j AA − v i A T j Av k AA − v i A T j AAv k A i , (21)and T = ˆ M × ∂H∂ ˆ M (22)is the torque operator. In Sec. II C we will explain that χ ICIDMI ijk describes the inverse of current-induced DMI(ICIDMI). In the case of the perturbation of the type of Eq. (11)the second derivative ∂ ˆ M j ∂r k ∂t may be rewritten as productof the first derivatives ∂ ˆ M l ∂t and ∂ ˆ M l ∂r k . This may be seenas follows: ∂H∂ ˆ M · ∂ ˆ M ∂r i ∂t = ∂ H∂t∂r i == ∂∂t "(cid:18) ˆ M × ∂H∂ ˆ M (cid:19) · ˆ M × ∂ ˆ M ∂r i ! == " ∂ ˆ M ∂t × ∂H∂ ˆ M ! · ˆ M × ∂ ˆ M ∂r i ! == " ˆ M × ∂ ˆ M ∂t ! × ˆ M ! × ∂H∂ ˆ M · " ˆ M × ∂ ˆ M ∂r i == − " ˆ M × ∂ ˆ M ∂t · " ˆ M × ∂ ˆ M ∂r i ˆ M · ∂H∂ ˆ M (cid:21) == − ∂ ˆ M ∂t · ∂ ˆ M ∂r i (cid:20) ˆ M · ∂H∂ ˆ M (cid:21) . (23)This expression is indeed satisfied by Eq. (12), Eq. (13)and Eq. (14): ∂ ˆ M ∂r i · ∂ ˆ M ∂t = − ∂ ˆ M ∂r i ∂t · ˆ M (24)at r = 0, t = 0. Consequently, Eq. (6) can be rewrittenas J ICIT2a i = X jk χ ICIT2a ijk ∂ ˆ M j ∂r k ∂t == − X jkl χ ICIT2a ijk ∂ ˆ M l ∂r k ∂ ˆ M l ∂t [1 − δ jl ]= X jkl χ ICIT2a ijkl ˆ e j · " ˆ M × ∂ ˆ M ∂t ˆ e k · " ˆ M × ∂ ˆ M ∂r l , (25)where χ ICIT2a ijkl = − X m χ ICIT2a iml [1 − δ jm ] δ jk . (26)Thus, Eq. (25) and Eq. (26) can be used to express J ICIT2a i in the form of Eq. (4). C. Direct and inverse CIDMI
Eq. (21) describes the response of the electric currentto time-dependent magnetization gradients of the typeEq. (16). The reciprocal process consists in the current-induced modification of DMI. This can be shown by ex-pressing the DMI coefficients as [8] D ij = 1 V X n f ( E k n ) Z d r ( ψ k n ( r )) ∗ D ij ψ k n ( r )= 1 V X n f ( E k n ) Z d r ( ψ k n ( r )) ∗ T i ( r ) r j ψ k n ( r ) , (27)where we defined the DMI-operator D ij = T i r j . Usingthe Kubo formalism the current-induced modification ofDMI may be written as D CIDMI ij = X k χ CIDMI kij E k (28)with χ CIDMI kij = 1 V lim ω → h e ~ ω Im hhD ij ; v k ii R ( ~ ω ) i , (29)where hhD ij ; v k ii R ( ~ ω ) = − i ∞ Z dte iωt h [ D ij ( t ) , v k (0)] − i (30)is the Fourier transform of a retarded function and V isthe volume of the unit cell.Since the position operator r in the DMI operator D ij = T i r j is not compatible with Bloch periodic bound-ary conditions, we do not use Eq. (29) for numericalcalculations of CIDMI. However, it is convenient to useEq. (29) in order to demonstrate the reciprocity betweendirect and inverse CIDMI.Inverse CIDMI (ICIDMI) describes the electric currentthat responds to the perturbation by a time-dependentmagnetization gradient according to J ICIDMI k = X ij χ ICIDMI kij ˆ e i · " ˆ M × ∂ ˆ M ∂t∂r j . (31)The perturbation by a time-dependent magnetizationgradient may be written as δH = − X j m · ∂ ˆ M ∂t∂r j r j Ω xc ( r ) sin( ωt ) ω == X j T · " ˆ M × ∂ ˆ M ∂t∂r j r j sin( ωt ) ω = X ij D ij ˆ e i · " ˆ M × ∂ ˆ M ∂t∂r j sin( ωt ) ω . (32)Consequently, the coefficient χ ICIDMI kij is given by χ ICIDMI kij = 1 V lim ω → h e ~ ω Im hh v k ; D ij ii R ( ~ ω ) i . (33)Using hhD ij ; v k ii R ( ~ ω, ˆ M ) = −hh v k ; D ij ii R ( ~ ω, − ˆ M ) (34) we find that CIDMI and ICIDMI are related through theequations χ CIDMI kij ( ˆ M ) = − χ ICIDMI kij ( − ˆ M ) . (35)In order to calculate CIDMI we use Eq. (21) for ICIDMIand then use Eq. (35) to obtain CIDMI.The perturbation Eq. (17) describes a different kindof time-dependent magnetization gradient, for which thereciprocal effect consists in the modification of the ex-pectation value h σ · ˆ M r j i . However, while the modifi-cation of hT i r j i by an applied current can be measuredfrom the change of the DMI constant D ij , the quantity h σ · ˆ M r j i has not been considered so far in ferromagnets.In noncollinear magnets the quantity h σ r j i can be usedto define spin toroidization [31]. Therefore, while theperturbation of the type of Eq. (16) is related to CIDMIand ICIDMI, which are both accessible experimentally,in the case of the perturbation of the type of Eq. (17)we expect that only the effect of driving current by thetime-dependent magnetization gradient is easily accessi-ble experimentally, while its inverse effect is difficult tomeasure. D. Direct and inverse dynamical DMI
Not only applied electric currents modify DMI, butalso magnetization dynamics, which we call dynamicalDMI (DDMI). DDMI can be expressed as D DDMI ij = X k χ DDMI kij ˆ e k · " ˆ M × ∂ ˆ M ∂t . (36)In Sec. II G we will show that the spatial gradient ofDDMI contributes to damping and gyromagnetism innoncollinear magnets. The perturbation used to describemagnetization dynamics is given by [22] δH = sin( ωt ) ω ˆ M × ∂ ˆ M ∂t ! · T . (37)Consequently, the coefficients χ DDMI kij may be written as χ DDMI kij = − V lim ω → (cid:20) ~ ω Im hhD ij ; T k ii R ( ~ ω ) (cid:21) . (38)Since the position operator in D ij is not compatiblewith Bloch periodic boundary conditions, we do not useEq. (38) for numerical calculations of DDMI, but insteadwe obtain it from its inverse effect, which consists in thegeneration of torques on the magnetization due to time-dependent magnetization gradients. These torques canbe written as T IDDMI k = X ij χ IDDMI kij ˆ e i · " ˆ M × ∂ ˆ M ∂t∂r j , (39)where the coefficients χ IDDMI kij are χ IDDMI kij = 1 V lim ω → (cid:20) ~ ω Im hhT k ; D ij ii R ( ~ ω ) (cid:21) , (40)because the perturbation by the time-dependent gradientcan be expressed in terms of D ij according to Eq. (32)and because the torque on the magnetization is describedby − T [21]. Consequently, DDMI and IDDMI are relatedby χ DDMI kij ( ˆ M ) = − χ IDDMI kij ( − ˆ M ) . (41)For numerical calculations of IDDMI we use χ IDDMI ijk = i π ~ Z d d k (2 π ) d Z d E f ( E )Tr h T i Rv k RR T j R + T i RRv k R T j R + −T i RR T j Rv k R − T i Rv k R T j AA + T i R T j Av k AA + T i R T j AAv k A −T i Rv k RR T j A − T i RRv k R T j A + T i RR T j Av k A + T i Av k A T j AA −T i A T j Av k AA − T i A T j AAv k A i , (42)which is derived in Appendix A. In order to obtain DDMIwe calculate IDDMI from Eq. (42) and use the reciprocityrelation Eq. (41).Eq. (39) is valid for time-dependent magnetization gra-dients that lead to perturbations of the type of Eq. (16).Perturbations of the second type, Eq. (17), will inducetorques on the magnetization as well. However, the in-verse effect is difficult to measure in that case, because itcorresponds to the modification of the expectation value h σ · ˆ M r j i by magnetization dynamics. Therefore, whilein the case of Eq. (16) both direct and inverse responseare expected to be measurable and correspond to ID-DMI and DDMI, respectively, we expect that in the caseof Eq. (17) only the direct effect, i.e., the response of thetorque to the perturbation, is easy to observe. E. Dynamical orbital magnetism (DOM)
Magnetization dynamics does not only induce DMI,but also orbital magnetism, which we call dynamical or-bital magnetism (DOM). It can be written as M DOM ij = X k χ DOM kij ˆ e k · " ˆ M × ∂ ˆ M ∂t , (43)where we introduced the notation M DOM ij = eV h v i r j i DOM , (44)which defines a generalized orbital magnetization, suchthat M DOM i = 12 X jk ǫ ijk M DOM jk (45) corresponds to the usual definition of orbital magnetiza-tion. The coefficients χ DOM kij are given by χ DOM kij = − V lim ω → h e ~ ω Im hh v i r j ; T k ii R ( ~ ω ) i , (46)because the perturbation by magnetization dynamics isdescribed by Eq. (37). We will discuss in Sec. II F thatthe spatial gradient of DOM contributes to the inverseCIT. Additionally, we will show below that DOM andCIDMI are related to each other.In order to obtain an expression for DOM it is conve-nient to consider the inverse effect, i.e., the generation ofa torque by the application of a time-dependent magneticfield B ( t ) that acts only on the orbital degrees of freedomof the electrons and not on their spins. This torque canbe written as T IDOM k = 12 X ijl χ IDOM kij ǫ ijl ∂B l ∂t , (47)where χ IDOM kij = − V lim ω → h e ~ ω Im hhT k ; v i r j ii R ( ~ ω ) i , (48)because the perturbation by the time-dependent mag-netic field is given by δH = − e X ijk ǫ ijk v i r j ∂B k ∂t sin( ωt ) ω . (49)Therefore, the coefficients of DOM and IDOM are relatedby χ DOM kij ( ˆ M ) = − χ IDOM kij ( − ˆ M ) . (50)In Appendix A we show that the coefficient χ IDOM ijk canbe expressed as χ IDOM ijk = − ie π ~ Z d d k (2 π ) d Z d E f ( E )Tr h T i Rv k RRv j R + T i RRv k Rv j R + −T i RRv j Rv k R − T i Rv k Rv j AA + T i Rv j Av k AA + T i Rv j AAv k A −T i Rv k RRv j A − T i RRv k Rv j A + T i RRv j Av k A + T i Av k Av j AA −T i Av j Av k AA − T i Av j AAv k A i . (51)Eq. (51) and Eq. (21) differ only in the positions ofthe two velocity operators and the torque operator be-tween the Green functions. As a consequence, IDOMare ICIDMI are related. In Table I and Table II we listthe relations between IDOM and ICIDMI for the Rashbamodel Eq. (84). We will explain in Sec. III that IDOMand ICIDMI are zero in the Rashba model when the mag-netization is along the z direction. Therefore, we discussin Table I the case where the magnetization lies in the xz plane, and in Table II we discuss the case where the mag-netization lies in the yz plane. According to Table I andTable II the relation between IDOM and ICIDMI is ofthe form χ IDOM ijk = ± χ ICIDMI jik . This is expected, becausethe index i in χ IDOM ijk is connected to the torque operator,while the index j in χ ICIDMI ijk is connected to the torqueoperator.
TABLE I: Relations between the inverse of the magnetization-dynamics induced orbital magnetism (IDOM) and inversecurrent-induced DMI (ICIDMI) in the 2d Rashba model whenˆ M lies in the zx plane. The components of χ IDOM ijk (Eq. (51))and χ ICIDMI ijk (Eq. (21)) are denoted by the three indices ( ijk ). ICIDMI IDOM(211) (121)(121) (211)-(221) (221)(112) (112)-(212) (122)-(122) (212)(222) (222)(231) (321)(132) (312)-(232) (322)
TABLE II: Relations between IDOM and ICIDMI in the 2dRashba model when ˆ M lies in the yz plane. ICIDMI IDOM(111) (111)-(211) (121)-(121) (211)(221) (221)-(112) (112)(212) (122)(122) (212)-(131) (311)(231) (321)(132) (312)
F. Contributions from CIDMI and DOM to directand inverse CIT
In electronic transport theory the continuity equationdetermines the current only up to a curl field [32]. Thecurl of magnetization corresponds to a bound currentthat cannot be measured in electron transport experi-ments such that J = J Kubo − ∇ × M (52)has to be used to extract the transport current J from thecurrent J Kubo obtained from the Kubo linear response. The subtraction of ∇ × M has been shown to be impor-tant when calculating the thermoelectric response [32]and the anomalous Nernst effect [18]. Similarly, in thetheory of the thermal spin-orbit torque [8, 16] the gra-dients of the DMI spiralization have to be subtracted inorder to obtain the measurable torque: T i = T Kubo i − X j ∂∂r j D ij , (53)where the spatial derivative of the spiralization arisesfrom its temperature dependence and the temperaturegradient.Since CIDMI and DOM depend on the magnetizationdirection, they vary spatially in noncollinear magnets.Similar to Eq. (53) the spatial derivatives of the current-induced spiralization need to be included into the theoryof CIT. Additionally, the gradients of DOM correspondto currents that need to be considered in the theory ofthe inverse CIT, similar to Eq. (52).The response of electric current and torque need to besupplemented as follows: J ICIT i = J Kubo i − X j ∂ ˆ M ∂r j · ∂M DOM ij ∂ ˆ M (54)and T CIT i = T Kubo i − X j ∂ ˆ M ∂r j · ∂D CIDMI ij ∂ ˆ M , (55)where J ICIT i is the current driven by magnetization dy-namics, and T CIT i is the current-induced torque.Interestingly, we find that Eq. (54) is the proper wayto extract the transport current in noncollinear magnetswhile Eq. (52) cannot be used for this purpose. Addi-tionally, we find that also the diagonal elements M DOM ii are nonzero. This shows that the generalized defini-tion Eq. (44) is necessary, because the diagonal elements M DOM ii do not contribute in the usual definition of M i ac-cording to Eq. (45). These differences in the symmetryproperties between equilibrium and nonequilibrium or-bital magnetism can be traced back to symmetry break-ing by the perturbations. Also in the case of the spi-ralization tensor D ij the nonequilibrium correction δD ij has different symmetry properties than the equilibriumpart (see Sec. III).The contribution of DOM to χ ICIT2 ijkl can be written as χ ICIT2c ijkl = −
12 ˆ e k · " ˆ M × ∂χ DOM jil ∂ ˆ M (56)and the contribution of CIDMI to χ CIT2 ijkl is given by χ CIT2b ijkl = −
12 ˆ e k · " ˆ M × ∂χ CIDMI jil ∂ ˆ M . (57) G. Contributions from DDMI to gyromagnetismand damping
The response to magnetization dynamics that is de-scribed by the torque-torque correlation function con-sists of torques that are related to damping and gyro-magnetism [22]. The chiral contribution to these torquescan be written as T TT2 i = X jkl χ TT2 ijkl ˆ e j · " ˆ M × ∂ ˆ M ∂t ˆ e k · " ˆ M × ∂ ˆ M ∂r l , (58)where the coefficients χ TT2 ijkl satisfy the Onsager relations χ TT2 ijkl ( ˆ M ) = χ TT2 jikl ( − ˆ M ) . (59)Since DDMI depends on the magnetization direction, itvaries spatially in noncollinear magnets and the resultinggradients of DDMI contribute to the damping and to thegyromagnetic ratio: T TT i = T Kubo i − X j ∂ ˆ M ∂r j · ∂D DDMI ij ∂ ˆ M . (60)The resulting contribution of the spatial derivatives ofDDMI to the coefficient χ TT2 ijkl is χ TT2c ijkl = −
12 ˆ e k · " ˆ M × ∂χ DDMI jil ( ˆ M ) ∂ ˆ M . (61) H. Current-induced torque (CIT) in noncollinearmagnets
The chiral contribution to CIT consists of the spatialgradient of CIDMI, χ CIT2b ijkl in Eq. (57), and the Kubolinear response of the torque to the applied electric fieldin a noncollinear magnet, χ CIT2a ijkl : χ CIT2 ijkl = χ CIT2a ijkl + χ CIT2b ijkl . (62)In order to determine χ CIT2a ijkl , we assume that the magne-tization direction ˆ M ( r ) oscillates spatially as describedby ˆ M ( r ) = η sin( q · r )01 q η sin ( q · r ) , (63)where we will take the limit q → ∂ ˆ M ( r ) ∂r i = ηq i cos( q · r )00 + O ( η ) , (64) the chiral contribution to the CIT oscillates spatially pro-portional to cos( q · r ). In order to extract this spatiallyoscillating contribution we multiply with cos( q · r ) andintegrate over the unit cell. The resulting expression for χ CIT2a ijkl is χ CIT2a ijkl = − eV η lim q → lim ω → " q l Z cos( q l r l ) Im hhT i ( r ); v j ( r ′ ) ii R ( ~ ω ) ~ ω d rd r ′ , (65)where V is the volume of the unit cell, andthe retarded torque-velocity correlation function hhT i ( r ); v j ( r ′ ) ii R ( ~ ω ) needs to be evaluated in thepresence of the perturbation δH = T k η sin( q · r ) (66)due to the noncollinearity (the index k in Eq. (66) needsto match the index k in χ CIT2a ijkl ).In Appendix B we show that χ CIT2a ijkl can be written as χ CIT2a ijkl = − e ~ Im h W (surf) ijkl + W (sea) ijkl i , (67)where W (surf) ijkl = 14 π ~ Z d d k (2 π ) d Z d E f ′ ( E )Tr " T i G R k ( E ) v l G R k ( E ) v j G A k ( E ) T k G A k ( E )+ T i G R k ( E ) v j G A k ( E ) v l G A k ( E ) T k G A k ( E ) −T i G R k ( E ) v j G A k ( E ) T k G A k ( E ) v l G A k ( E )+ ~ m e δ jl T i G R k ( E ) G A k ( E ) T k G A k ( E ) (68)is a Fermi surface term ( f ′ ( E ) = df ( E ) /d E ) and W (sea) ijkl = 14 π ~ Z d d k (2 π ) d Z d E f ( E ) " − Tr [ T i Rv l RRv j R T k R ] − Tr [ T i Rv l R T k RRv j R ] − Tr [ T i RRv l Rv j R T k R ] − Tr [ T i RRv j Rv l R T k R ]+ Tr [ T i RRv j R T k Rv l R ] + Tr [ T i RR T k Rv j Rv l R ]+ Tr [ T i RR T k Rv l Rv j R ] − Tr [ T i RRv l R T k Rv j R ] − Tr [ T i Rv l RR T k Rv j R ] + Tr [ T i R T k RRv j Rv l R ]+ Tr [ T i R T k RRv l Rv j R ] + Tr [ T i R T k Rv l RRv j R ] − ~ m e δ jl Tr [ T i RRR T k R ] − ~ m e δ jl Tr [ T i AAA T k A ] − ~ m e δ jl Tr [ T i AA T k AA ] (69)is a Fermi sea term. I. Inverse CIT in noncollinear magnets
The chiral contribution J ICIT2 (see Eq. (4)) to thecharge pumping is described by the coefficients χ ICIT2 ijkl = χ ICIT2a ijkl + χ ICIT2b ijkl + χ ICIT2c ijkl , (70)where χ ICIT2a ijkl describes the response to the time-dependent magnetization gradient (see Eq. (19), Eq. (26),and Eq. (25)) and χ ICIT2c ijkl results from the spatial gra-dient of DOM (see Eq. (56)). χ ICIT2b ijkl describes the re-sponse to the perturbation by magnetization dynamics ina noncollinear magnet. In order to derive an expressionfor χ ICIT2b ijkl we assume that the magnetization oscillatesspatially as described by Eq. (63). Since the correspond-ing response oscillates spatially proportional to cos( q · r ),we multiply by cos( q · r ) and integrate over the unit cellin order to extract χ ICIT2b ijkl from the retarded velocity-torque correlation function hh v i ( r ); T j ( r ′ ) ii R ( ~ ω ), whichis evaluated in the presence of the perturbation Eq. (66).We obtain χ ICIT2b ijkl = 2 eV η lim q → lim ω → " q l Z cos( q l r l ) Im hh v i ( r ); T j ( r ′ ) ii R ( ~ ω ) ~ ω d rd r ′ , (71)which can be written as (see Appendix B) χ ICIT2b ijkl = 2 e ~ Im h V (surf) ijkl + V (sea) ijkl i , (72)where V (surf) ijkl = 14 π ~ Z d d k (2 π ) d Z d E f ′ ( E )Tr h v i G R k ( E ) v l G R k ( E ) T j G A k ( E ) T k G A k ( E )+ v i G R k ( E ) T j G A k ( E ) v l G A k ( E ) T k G A k ( E ) − v i G R k ( E ) T j G A k ( E ) T k G A k ( E ) v l G A k ( E ) i (73)is the Fermi surface term and V (sea) ijkl = 14 π ~ Z d d k (2 π ) d Z d E f ( E )Tr h − Tr [ v i Rv l RR T j R T k R ] − Tr [ v i Rv l R T k RR T j R ] − Tr [ v i RRv l R T j R T k R ] − Tr [ v i RR T j Rv l R T k R ]+ Tr [ v i RR T j R T k Rv l R ] + Tr [ v i RR T k R T j Rv l R ]+ Tr [ v i RR T k Rv l R T j R ] − Tr [ v i RRv l R T k R T j R ] − Tr [ v i Rv l RR T k R T j R ] + Tr [ v i R T k RR T j Rv l R ]+ Tr [ v i R T k RRv l R T j R ] + Tr [ v i R T k Rv l RR T j R ] i (74)is the Fermi sea term.In Eq. (71) we use the Kubo formula to describe theresponse to magnetization dynamics combined with per-turbation theory to include the effect of noncollinearity. Thereby, the time-dependent perturbation and the per-turbation by the magnetization gradient are separatedand perturbations of the form of Eq. (16) or Eq. (17)are not automatically included. For example the flat cy-cloidal spin spiralˆ M ( x, t ) = sin( qx − ωt )0cos( qx − ωt ) (75)moving in x direction with speed ω/q and the helical spinspiral ˆ M ( y, t ) = sin( qy − ωt )0cos( qy − ωt ) (76)moving in y direction with speed ω/q behave like Eq. (11)when t and r are small. Thus, these moving domain wallscorrespond to the perturbation of the type of Eq. (11)and the resulting contribution J ICIT2a from the time-dependent magnetization gradient is not described byEq. (71) and needs to be added, which we do by adding χ ICIT2a ijkl in Eq. (70).
J. Damping and gyromagnetism in noncollinearmagnets
The chiral contribution Eq. (58) to the torque-torquecorrelation function is expressed in terms of the coeffi-cient χ TT ijkl = χ TT2a ijkl + χ TT2b ijkl + χ TT2c ijkl , (77)where χ TT2c ijkl results from the spatial gradient of DDMI(see Eq. (61)), χ TT2a ijkl describes the response to a time-dependent magnetization gradient in a collinear magnet,and χ TT2b ijkl describes the response to magnetization dy-namics in a noncollinear magnet.In order to derive an expression for χ TT2b ijkl we as-sume that the magnetization oscillates spatially accord-ing to Eq. (63). We multiply the retarded torque-torquecorrelation function hhT i ( r ); T j ( r ′ ) ii R ( ~ ω ) with cos( q l r l )and integrate over the unit cell in order to extract thepart of the response that varies spatially proportional tocos( q l r l ). We obtain: χ TT2b ijkl = 2
V η lim q l → lim ω → " q l Z cos( q l r l ) Im hhT i ( r ); T j ( r ′ ) ii R ( ~ ω ) ~ ω d rd r ′ . (78)In Appendix B we discuss how to evaluate Eq. (78) infirst order perturbation theory with respect to the per-turbation Eq. (66) and show that χ TT2b ijkl can be expressed0as χ TT2b ijkl = 2 ~ Im h X (surf) ijkl + X (sea) ijkl i , (79)where X (surf) ijkl = 14 π ~ Z d d k (2 π ) d Z d E f ′ ( E )Tr " T i G R k ( E ) v l G R k ( E ) T j G A k ( E ) T k G A k ( E )+ T i G R k ( E ) T j G A k ( E ) v l G A k ( E ) T k G A k ( E ) −T i G R k ( E ) T j G A k ( E ) T k G A k ( E ) v l G A k ( E ) (80)is a Fermi surface term and X (sea) ijkl = 14 π ~ Z d d k (2 π ) d Z d E f ( E )Tr " − ( T i Rv l RR T j R T k R ) − ( T i Rv l R T k RR T j R ) − ( T i RRv l R T j R T k R ) − ( T i RR T j Rv l R T k R )+ ( T i RR T j R T k Rv l R ) + ( T i RR T k R T j Rv l R )+ ( T i RR T k Rv l R T j R ) − ( T i RRv l R T k R T j R ) − ( T i Rv l RR T k R T j R ) + ( T i R T k RR T j Rv l R )+ ( T i R T k RRv l R T j R ) + ( T i R T k Rv l RR T j R ) (81)is a Fermi sea term.The contribution χ TT2a ijkl from the time-dependent gra-dients is given by χ TT2a ijkl = − X m χ TT2a iml [1 − δ jm ] δ jk , (82)where χ TT2a iml = i π ~ Z d d k (2 π ) d Z d E f ( E )Tr h T i Rv l RR O m R + T i RRv l R O m R + −T i RR O m Rv l R − T i Rv l R O m AA + T i R O m Av l AA + T i R O m AAv l A −T i Rv l RR O m A − T i RRv l R O m A + T i RR O m Av l A + T i Av l A O m AA −T i A O m Av l AA − T i A O m AAv l A i , (83)with O m = ∂H/∂ ˆ M m (see Appendix A). III. SYMMETRY PROPERTIES
In this section we discuss the symmetry properties ofCIDMI, DDMI and DOM in the case of the magneticRashba model H k ( r ) = ~ m e k + α ( k × ˆ e z ) · σ + ∆ V σ · ˆ M ( r ) . (84) Additionally, we discuss the symmetry properties of thecurrents and torques induced by time-dependent magne-tization gradients of the form of Eq. (11).We consider mirror reflection M xz at the xz plane,mirror reflection M yz at the yz plane, and c2 rotationaround the z axis. When ∆ V = 0 these operations leaveEq. (84) invariant, but when ∆ V = 0 they modify themagnetization direction ˆ M in Eq. (84), as shown in Ta-ble III. At the same time, these operations affect thetorque T and the current J driven by the time-dependentmagnetization gradients (see Table III). In Table IV andTable V we show how ˆ M × ∂ ˆ M /∂r k is affected by thesymmetry operations.A flat cycloidal spin spiral with spins rotating in the xz plane is mapped by a c2 rotation around the z axis ontothe same spin spiral. Similarly, a flat helical spin spiralwith spins rotating in the yz plane is mapped by a c2 ro-tation around the z axis onto the same spin spiral. There-fore, when ˆ M points in z direction, a c2 rotation aroundthe z axis does not change ˆ M × ∂ ˆ M /∂r i , but it flips thein-plane current J and the in-plane components of thetorque, T x and T y . Consequently, ˆ M × ∂ ˆ M /∂r i ∂t doesnot induce currents or torques, i.e., ICIDMI, CIDMI, ID-DMI and DDMI are zero, when ˆ M points in z direction.However, they become nonzero when the magnetizationhas an in-plane component (see Fig. 1).Similarly, IDOM vanishes when the magnetizationpoints in z direction: In that case Eq. (84) is invariantunder the c2 rotation. A time-dependent magnetic fieldalong z direction is invariant under the c2 rotation aswell. However, T x and T y change sign under the c2 rota-tion. Consequently, symmetry forbids IDOM in this case.However, when the magnetization has an in-plane com-ponent, IDOM and DOM become nonzero (see Fig. 2).That time-dependent magnetization gradients of thetype of Eq. (8) do not induce in-plane currents andtorques when ˆ M points in z direction can also be seen di-rectly from Eq. (8): The c2 rotation transforms q → − q and M x → − M x . Since sin( q · r ) is odd in r , Eq. (8) is in-variant under c2 rotation, while the in-plane currents andtorques induced by time-dependent magnetization gradi-ents change sign under c2 rotation. In contrast, Eq. (11)is not invariant under c2 rotation, because sin( q · r − ωt )is not odd in r for t >
0. Consequently, time-dependentmagnetization gradients of the type of Eq. (11) inducecurrents and torques also when ˆ M points locally intothe z direction. These currents and torques, which aredescribed by Eq. (25) and Eq. (83), respectively, need tobe added to the chiral ICIT and the chiral torque-torquecorrelation. While CIDMI, DDMI, and DOM are zerowhen the magnetization points in z direction, their gra-dients are not (see Fig. 1 and Fig. 2). Therefore, the gra-dients of CIDMI, DOM, and DDMI contribute to CIT, toICIT and to the torque-torque correlation, respectively,even when ˆ M points locally into the z direction.1 TABLE III: Effect of mirror reflection M xz at the xz plane,mirror reflection M yz at the yz plane, and c2 rotation aroundthe z axis. The magnetization M and the torque T transformlike axial vectors, while the current J transforms like a polarvector. M x M y M z J x J y T x T y T z M xz − M x M y − M z J x − J y − T x T y − T z M yz M x − M y − M z − J x J y T x − T y − T z c2 - M x - M y M z - J x − J y − T x − T y T z TABLE IV: Effect of symmetry operations on the magneti-zation gradients. Magnetization gradients are described bythree indices ( ijk ). The first index denotes the magnetiza-tion direction at r = 0. The third index denotes the di-rection along which the magnetization changes. The secondindex denotes the direction of ∂ ˆ M /∂r k δr k . The direction ofˆ M × ∂ ˆ M /∂r k is specified by the number below the indices( ijk ). (1,2,1) (1,3,1) (2,1,1) (2,3,1) (3,1,1) (3,2,1)3 -2 -3 1 2 -1 M xz (-1,2,1) (-1,-3,1) (2,-1,1) (2,-3,1) (-3,-1,1) (-3,2,1)-3 -2 3 -1 2 1 M yz (1,2,1) (1,3,1) (-2,-1,1) (-2,3,1) (-3,-1,1) (-3,2,1)3 -2 -3 -1 2 1c2 (-1,2,1) (-1,-3,1) (-2,1,1) (-2,-3,1) (3,1,1) (3,2,1)-3 -2 3 1 2 -1. TABLE V: Continuation of Table IV (1,2,2) (1,3,2) (2,1,2) (2,3,2) (3,1,2) (3,2,2)3 -2 -3 1 2 -1 M xz (-1,-2,2) (-1,3,2) (2,1,2) (2,3,2) (-3,1,2) (-3,-2,2)3 2 -3 1 -2 -1 M yz (1,-2,2) (1,-3,2) (-2,1,2) (-2,-3,2) (-3,1,2) (-3,-2,2)-3 2 3 1 -2 -1c2 (-1,2,2) (-1,-3,2) (-2,1,2) (-2,-3,2) (3,1,2) (3,2,2)-3 -2 3 1 2 -1 A. Symmetry properties of ICIDMI and IDDMI
In the following we discuss how Table III, Table IV, andTable V can be used to analyze the symmetry of ICIDMIand IDDMI. According to Eq. (20) the coefficient χ ICIDMI ijk describes the response of the current J ICIT2a i to the time-dependent magnetization gradient ˆ e j · [ ˆ M × ∂ ˆ M ∂r k ∂t ]. Sinceˆ M × ∂ ˆ M ∂r k ∂t = ∂∂t [ ˆ M × ∂ ˆ M ∂r k ] for time-dependent magnetiza-tion gradients of the type Eq. (8) the symmetry proper-ties of χ ICIDMI ijk follow from the transformation behaviourof ˆ M × ∂ ˆ M ∂r k and J under symmetry operations.We consider the case with magnetization in x direc-tion. The component χ ICIDMI132 describes the current in x direction induced by the time-dependence of a cycloidalmagnetization gradient in y direction (with spins rotating FIG. 1: ICIDMI in a noncollinear magnet. (a) Arrows illus-trate the magnetization direction. (b) Arrows illustrate thecurrent J y induced by a time-dependent magnetization gra-dient, which is described by χ ICIDMI221 . When ˆ M points in z direction, χ ICIDMI221 and J y are zero. The sign of χ ICIDMI221 andof J y changes with the sign of M x .FIG. 2: DOM in a noncollinear magnet. (a) Arrows illustratethe magnetization direction. (b) Arrows illustrate the orbitalmagnetization induced by magnetization dynamics (DOM).When ˆ M points in z direction, DOM is zero. The sign ofDOM changes with the sign of M x . in the xy plane). M yz flips both ˆ M × ∂ ˆ M ∂y and J x , butit preserves ˆ M . M zx preserves ˆ M × ∂ ˆ M ∂y and J x , but itflips ˆ M . A c2 rotation around the z axis flips ˆ M × ∂ ˆ M ∂y ,ˆ M and J x . Consequently, χ ICIDMI132 ( ˆ M ) is allowed bysymmetry and it is even in ˆ M . The component χ ICIDMI122 describes the current in x direction induced by the time-dependence of a helical magnetization gradient in y di-rection (with spins rotating in the xz plane). M yz flipsˆ M × ∂ ˆ M ∂y and J x , but it preserves ˆ M . M zx flips ˆ M × ∂ ˆ M ∂y and ˆ M , but it preserves J x . A c2 rotation around the z axis flips J x and ˆ M , but it preserves ˆ M × ∂ ˆ M ∂y . Conse-quently, χ ICIDMI122 is allowed by symmetry and it is odd inˆ M . The component χ ICIDMI221 describes the current in y direction induced by the time-dependence of a cycloidalmagnetization gradient in x direction (with spins rotat-ing in the xz plane). M zx preserves ˆ M × ∂ ˆ M ∂x , but it flips2 J y and ˆ M . M yz preserves ˆ M , J y , and ˆ M × ∂ ˆ M ∂x . Thec2 rotation around the z axis preserves ˆ M × ∂ ˆ M ∂x , butit flips ˆ M and J y . Consequently, χ ICIDMI221 is allowed bysymmetry and it is odd in ˆ M . The component χ ICIDMI231 describes the current in y direction induced by the time-dependence of a cycloidal magnetization gradient in x di-rection (with spins rotating in the xy plane). M zx flipsˆ M × ∂ ˆ M ∂x , ˆ M , and J y . M yz preserves ˆ M × ∂ ˆ M ∂x , ˆ M and J y . The c2 rotation around the z axis flips ˆ M × ∂ ˆ M ∂x , J y ,and ˆ M . Consequently, χ ICIDMI231 is allowed by symmetryand it is even in ˆ M .These properties are summarized in Table VI. Due tothe relations between CIDMI and DOM (see Table I andTable II), they can be used for DOM as well. When themagnetization lies at a general angle in the xz plane or inthe yz plane several additional components of CIDMI andDOM are nonzero (see Table I and Table II, respectively). TABLE VI: Allowed components of χ ICIDMI ijk when ˆ M pointsin x direction. + components are even in ˆ M , while - compo-nents are odd in ˆ M .
132 122 221 231+ - - +Similarly, one can analyze the symmetry of DDMI. Ta-ble VII lists the components of DDMI, χ DDMI ijk , which areallowed by symmetry when ˆ M points in x direction. TABLE VII: Allowed components of χ DDMI ijk when ˆ M points in x direction. + components are even in ˆ M , while - componentsare odd in ˆ M .
222 232 322 332- + + -
B. Response to time-dependent magnetizationgradients of the second type (Eq. (11) ) According to Eq. (14) the time-dependent magneti-zation gradient is along the magnetization. Therefore,in contrast to the discussion in section III A we can-not use ˆ M × ∂ ˆ M ∂r k ∂t in the symmetry analysis. Eq. (25)and Eq. (26) show that χ ICIT2a ijjl describes the response of J ICIT2a i to ˆ e j · h ˆ M × ∂ ˆ M ∂t i ˆ e j · h ˆ M × ∂ ˆ M ∂r l i while χ ICIT2a ijkl =0 for j = k . According to Eq. (24) the symmetry prop-erties of h ˆ M × ∂ ˆ M ∂t i · h ˆ M × ∂ ˆ M ∂r l i agree to the symmetryproperties of ˆ M · ∂ ˆ M ∂r l ∂t . Therefore, in order to under-stand the symmetry properties of χ ICIT2a ijjl we considerthe transformation of J and ˆ M · ∂ ˆ M ∂r l ∂t under symmetryoperations.We consider the case where ˆ M points in z direction. χ ICIT2a1 jj describes the current driven in x direction, when the magnetization varies in x direction. M xz flips ˆ M ,but preserves J x and ˆ M · ∂ ˆ M / ( ∂x∂t ). M yz flips ˆ M , J x ,and ˆ M · ∂ ˆ M / ( ∂x∂t ). c2 rotation flips ˆ M · ∂ ˆ M / ( ∂x∂t )and J x , but preserves ˆ M . Consequently, χ ICIT2a1 jj is al-lowed by symmetry and it is even in ˆ M . χ ICIT2a2 jj describes the current flowing in y direction,when magnetization varies in x direction. M xz flips ˆ M and J y , but preserves ˆ M · ∂ ˆ M / ( ∂x∂t ). M yz flips ˆ M ,and ˆ M · ∂ ˆ M / ( ∂x∂t ), but preserves J y . c2 rotationflips ˆ M · ∂ ˆ M / ( ∂x∂t ) and J y , but preserves ˆ M . Conse-quently, χ ICIT2a2 jj is allowed by symmetry and it is odd inˆ M .Similarly, one can show that χ ICIT2a1 jj is odd in ˆ M andthat χ ICIT2a2 jj is even in ˆ M .Analogously, one can investigate the symmetry prop-erties of χ TT2a ijjl . We find that χ TT2a1 jj and χ TT2a2 jj are oddin ˆ M , while χ TT2a2 jj and χ TT2a1 jj are even in ˆ M . IV. RESULTS
In the following sections we discuss the results for thedirect and inverse chiral CIT and for the chiral torque-torque correlation in the two-dimensional (2d) Rashbamodel Eq. (84), and in the one-dimensional (1d) Rashbamodel [33] H k x ( x ) = ~ m e k x − αk x σ y + ∆ V σ · ˆ M ( x ) . (85)Additionally, we discuss the contributions of the time-dependent magnetization gradients, and of DDMI, DOMand CIDMI to these effects.While vertex corrections to the chiral CIT and tothe chiral torque-torque correlation are important in theRashba model [33], the purpose of this work is to showthe importance of the contributions from time-dependentmagnetization gradients, DDMI, DOM and CIDMI. Wetherefore consider only the intrinsic contributions here,i.e., we set G R k ( E ) = ~ [ E − H k + i Γ] − , (86)where Γ is a constant broadening, and we leave the studyof vertex corrections for future work.The results shown in the following sections are ob-tained for the model parameters ∆ V = 1eV, α =2eV˚A,and Γ = 0 . . z direction, i.e., ˆ M = ˆ e z . The unit of χ CIT2 ijkl is charge times length in the 1d case and charge in the2d case. Therefore, in the 1d case we discuss the chiraltorkance in units of ea , where a is Bohr’s radius. In the2d case we discuss the chiral torkance in units of e . Theunit of χ TT2 ijkl is angular momentum in the 1d case andangular momentum per length in the 2d case. Therefore,we discuss χ TT2 ijkl in units of ~ in the 1d case, and in unitsof ~ /a in the 2d case.3 -2 -1 0 1 2Fermi energy [eV]-0.02-0.0100.010.020.030.040.05 χ ij k l C I T [ ea ] FIG. 3: Chiral CIT in the 1d Rashba model for cycloidal gra-dients vs. Fermi energy. General perturbation theory (solidlines) agrees to the gauge-field approach (dashed lines).
A. Direct and inverse chiral CIT
In Fig. 3 we show the chiral CIT as a function of theFermi energy for cycloidal magnetization gradients in the1d Rashba model. The components χ CIT22121 and χ CIT21121 arelabelled by 2121 and 1121, respectively. The component2121 of CIT describes the non-adiabatic torque, while thecomponent 1121 describes the adiabatic STT (modifiedby SOI). In the one-dimensional Rashba model, the con-tributions χ CIT2b2121 and χ CIT2b1121 (Eq. (57)) from the CIDMIare zero when ˆ M = ˆ e z (not shown in the figure). For cy-cloidal spin spirals, it is possible to solve the 1d Rashbamodel by a gauge-field approach [33], which allows us totest the perturbation theory, Eq. (67). For comparisonwe show in Fig. 3 the results obtained from the gauge-field approach, which agree to the perturbation theory,Eq. (67). This demonstrates the validity of Eq. (67).In Fig. 4 we show the chiral ICIT in the 1d Rashbamodel. The components χ ICIT21221 and χ ICIT21121 are labelledby 1221 and 1121, respectively. The contribution χ ICIT2a1221 from the time-dependent gradient is of the same order ofmagnitude as the total χ ICIT21221 . Comparison of Fig. 3 andFig. 4 shows that CIT and ICIT satisfy the reciprocityrelations Eq. (5), that χ CIT21121 is odd in ˆ M , and that χ CIT22121 is even in ˆ M , i.e., χ CIT22121 = χ ICIT21221 and χ CIT21121 = − χ ICIT21121 .The contribution χ ICIT2a1221 from the time-dependent gradi-ents is crucial to satisfy the reciprocity relations between χ CIT22121 and χ ICIT21221 .In Fig. 5 and Fig. 6 we show the CIT and the ICIT, re-spectively, for helical gradients in the 1d Rashba model.The components χ CIT22111 and χ CIT21111 are labelled 2111 and1111, respectively, in Fig. 5, while χ ICIT21211 and χ ICIT21111 are labelled 1211 and 1111, respectively, in Fig. 6. Thecontributions χ CIT2b2111 and χ CIT2b1111 from CIDMI are of the -2 -1 0 1 2Fermi energy [eV]-0.0200.020.04 χ ij k l I C I T [ ea ] χ FIG. 4: Chiral ICIT in the 1d Rashba model for cycloidalgradients vs. Fermi energy. Dashed line: Contribution fromthe time-dependent gradient. same order of magnitude as the total χ CIT22111 and χ CIT21111 .Similarly, the contributions χ ICIT2c1211 and χ ICIT2c1111 fromDOM are of the same order of magnitude as the to-tal χ ICIT21211 and χ ICIT21111 . Additionally, the contribution χ ICIT2a1111 from the time-dependent gradient is substantial.Comparison of Fig. 5 and Fig. 6 shows that CIT and ICITsatisfy the reciprocity relation Eq. (5), that χ CIT22111 is oddin ˆ M , and that χ CIT21111 is even in ˆ M , i.e., χ CIT21111 = χ ICIT21111 and χ CIT22111 = − χ ICIT21211 . These reciprocity relations be-tween CIT and ICIT are only satisfied when CIDMI,DOM, and the response to time-dependent magnetiza-tion gradients are included. Additionally, the compar-ison between Fig. 5 and Fig. 6 shows that the contri-butions of CIDMI to CIT ( χ CIT2b1111 and χ CIT2b2111 ) are re-lated to the contributions of DOM to ICIT ( χ ICIT2c1111 and χ ICIT2c1211 ). These relations between DOM and ICIT areexpected from Table I.In Fig. 7 and Fig. 8 we show the CIT and the ICIT,respectively, for cycloidal gradients in the 2d Rashbamodel. In this case there are contributions from CIDMIand DOM in contrast to the 1d case with cycloidal gra-dients (Fig. 3). Comparison between Fig. 7 and Fig. 8shows that χ CIT21121 and χ CIT22221 are odd in ˆ M , that χ CIT21221 and χ CIT22121 are even in ˆ M , and that CIT and ICIT sat-isfy the reciprocity relation Eq. (5) when the gradientsof CIDMI and DOM are included, i.e., χ CIT21121 = − χ ICIT21121 , χ CIT22221 = − χ ICIT22221 , χ CIT21221 = χ ICIT22121 , and χ CIT22121 = χ ICIT21221 . χ CIT21121 describes the adiabatic STT with SOI, while χ CIT22121 describes the non-adiabatic STT. Experimentally, it hasbeen found that CITs occur also when the electric fieldis applied parallel to domain-walls (i.e., perpendicular tothe q -vector of spin spirals) [34]. In our calculations, thecomponents χ CIT22221 and χ CIT21221 describe such a case, wherethe applied electric field points in y direction, while the4 -2 -1 0 1 2Fermi energy [eV]-0.04-0.0200.020.040.06 χ ij k l C I T [ ea ] χ χ FIG. 5: Chiral CIT for helical gradients in the 1d Rashbamodel vs. Fermi energy. Dashed lines: Contributions fromCIDMI. -2 -1 0 1 2Fermi energy [eV]-0.0200.020.040.06 χ ij k l I C I T [ ea ] χ χ χ FIG. 6: Chiral ICIT for helical gradients in the 1d Rashbamodel vs. Fermi energy. Dashed lines: Contributions fromDOM. Dashed-dotted line: Contribution from the time-dependent magnetization gradient. magnetization direction varies with the x coordinate.In Fig. 9 and Fig. 10 we show the chiral CIT andICIT, respectively, for helical gradients in the 2d Rashbamodel. The component χ CIT22111 describes the adiabaticSTT with SOI and the component χ CIT21111 describes thenon-adiabatic STT. The components χ CIT22211 and χ CIT21211 describe the case when the applied electric field pointsin y direction, i.e., perpendicular to the direction alongwhich the magnetization direction varies. Comparisonbetween Fig. 9 and Fig. 10 shows that χ CIT21111 and χ CIT22211 are even in ˆ M , that χ CIT21211 and χ CIT22111 are odd in ˆ M andthat CIT and ICIT satisfy the reciprocity relation Eq. (5)when the gradients of CIDMI and DOM are included, i.e., -2 -1 0 1 2Fermi energy [eV]-0.00200.0020.0040.006 χ ij k l C I T [ e ] χ χ χ FIG. 7: Chiral CIT for cycloidal gradients in the 2d Rashbamodel vs. Fermi energy. Dashed lines: Contributions fromCIDMI. -2 -1 0 1 2Fermi energy [eV]-0.00200.0020.0040.006 χ ij k l I C I T [ e ] χ χ χ χ χ FIG. 8: Chiral ICIT for cycloidal gradients in the 2d Rashbamodel vs. Fermi energy. Dashed lines: Contributions fromDOM. Dashed-dotted lines: Contributions from the time-dependent gradients. χ CIT21111 = χ ICIT21111 , χ CIT22211 = χ ICIT22211 , χ CIT21211 = − χ ICIT22111 , and χ CIT22111 = − χ ICIT21211 . B. Chiral torque-torque correlation
In Fig. 11 we show the chiral contribution to thetorque-torque correlation in the 1d Rashba model forcycloidal gradients. We compare the perturbation the-ory Eq. (79) plus Eq. (83) to the gauge-field approachfrom Ref. [33]. This comparison shows that perturba-tion theory provides the correct answer only when thecontribution χ TT2a ijkl (Eq. (83)) from the time-dependent5 -2 -1 0 1 2Fermi energy [eV]-0.00200.0020.0040.006 χ ij k l C I T [ e ] χ χ χ χ FIG. 9: Chiral CIT for helical gradients in the 2d Rashbamodel vs. Fermi energy. Dashed lines: Contributions fromCIDMI. -2 -1 0 1 2Fermi energy [eV]-0.004-0.00200.0020.0040.006 χ ij k l I C I T [ e ] χ χ χ χ χ χ FIG. 10: Chiral ICIT for helical gradients in the 2d Rashbamodel vs. Fermi energy. Dashed lines: Contributions fromDOM. Dashed-dotted lines: Contributions from the time-dependent gradient. gradients is taken into account. The contributions χ TT2a1221 and χ TT2a2221 from the time-dependent gradients are compa-rable in magnitude to the total values. In the 1d Rashbamodel the DDMI-contribution in Eq. (61) is zero for cy-cloidal gradients (not shown in the figure). The compo-nents χ TT22121 and χ TT21221 describe the chiral gyromagnetismwhile the components χ TT21121 and χ TT22221 describe the chi-ral damping [33, 35, 36]. The components χ TT22121 and χ TT21221 are odd in ˆ M and they satisfy the Onsager relationEq. (59), i.e., χ TT22121 = − χ TT21221 .In Fig. 12 we show the chiral contributions to thetorque-torque correlation in the 1d Rashba model forhelical gradients. In contrast to the cycloidal gradients -2 -1 0 1 2Fermi energy [eV]-0.00500.0050.01 χ ij k l TT [ h _ ] χ χ FIG. 11: Chiral contribution to the torque-torque correla-tion for cycloidal gradients in the 1d Rashba model vs. Fermienergy. Perturbation theory (solid lines) agrees to the gauge-field (gf) approach (dotted lines). Dashed lines: Contributionfrom the time-dependent gradient. (Fig. 11) there are contributions from the spatial gra-dients of DDMI (Eq. (61)) in this case. The Onsagerrelation Eq. (59) for the components χ TT22111 and χ TT21211 issatisfied only when these contributions from DDMI aretaken into account, which are of the same order of mag-nitude as the total values. The components χ TT22111 and χ TT21211 are even in ˆ M and describe chiral damping, whilethe components χ TT21111 and χ TT22211 are odd in ˆ M and de-scribe chiral gyromagnetism. As a consequence of theOnsager relation Eq. (59) we obtain χ TT21111 = χ TT22211 = 0for the total components: Eq. (59) shows that diagonalcomponents of the torque-torque correlation function arezero unless they are even in ˆ M . However, χ TT2a1111 , χ TT2c1111 ,and χ TT2b1111 = − χ TT2a1111 − χ TT2c1111 are individually nonzero.Interestingly, the off-diagonal components of the torque-torque correlation describe chiral damping for helical gra-dients, while for cycloidal gradients the off-diagonal ele-ments describe chiral gyromagnetism and the diagonalelements describe chiral damping.In Fig. 13 we show the chiral contributions to thetorque-torque correlation in the 2d Rashba model for cy-cloidal gradients. In contrast to the 1d Rashba modelwith cycloidal gradients (Fig. 11) the contributions fromDDMI χ TT2c ijkl (Eq. (61)) are nonzero in this case. Withoutthese contributions from DDMI the Onsager relation (59) χ TT22121 = − χ TT21221 is violated. The DDMI contribution isof the same order of magnitude as the total values. Thecomponents χ TT22121 and χ TT21221 are odd in ˆ M and describechiral gyromagnetism, while the components χ TT21121 and χ TT22221 are even in ˆ M and describe chiral damping.In Fig. 14 we show the chiral contributions to thetorque-torque correlation in the 2d Rashba model for he-lical gradients. The components χ TT21211 and χ TT22111 are even6 -2 -1 0 1 2Fermi energy [eV]-0.00500.0050.01 χ ij k l TT [ h _ ] χ χ χ χ χ χ FIG. 12: Chiral contribution to the torque-torque correla-tion for helical gradients in the 1d Rashba model vs. Fermienergy. Dashed lines: Contributions from DDMI. Dashed-dotted lines: Contributions from the time-dependent gradi-ents. -2 -1 0 1 2Fermi energy [eV]-0.000500.00050.001 χ ij k l TT [ h _ / a ] χ χ χ χ FIG. 13: Chiral contribution to the torque-torque correla-tion for cycloidal gradients in the 2d Rashba model vs. Fermienergy. Dashed lines: Contributions from DDMI. Dashed-dotted lines: Contributions from the time-dependent gradi-ents. in ˆ M and describe chiral damping, while the compo-nents χ TT21111 and χ TT22211 are odd in ˆ M and describe chiralgyromagnetism. The Onsager relation Eq. (59) requires χ TT21111 = χ TT22211 = 0 and χ TT22111 = χ TT21211 . Without thecontributions from DDMI these Onsager relations are vi-olated. -2 -1 0 1 2Fermi energy [eV]-0.000500.00050.001 χ ij k l TT [ h _ / a ] χ χ χ χ χ FIG. 14: Chiral contribution to the torque-torque correla-tion for helical gradients in the 2d Rashba model vs. Fermienergy. Dashed lines: Contributions from DDMI. Dashed-dotted lines: Contributions from the time-dependent gradi-ents.
V. SUMMARY
Finding ways to tune the Dzyaloshinskii-Moriya inter-action (DMI) by external means, such as an applied elec-tric current, holds much promise for applications in whichDMI determines the magnetic texture of domain walls orskyrmions. In order to derive an expression for current-induced Dzyaloshinskii-Moriya interaction (CIDMI) wefirst identify its inverse effect: When magnetic texturesvary as a function of time, electric currents are driven byvarious mechanisms, which can be distinguished accord-ing to their different dependence on the time-derivative ofmagnetization, ∂ ˆ M ( r , t ) /∂t , and on the spatial deriva-tive ∂ ˆ M ( r , t ) /∂ r : One group of effects is proportionalto ∂ ˆ M ( r , t ) /∂t , a second group of effects is propor-tional to the product ∂ ˆ M ( r , t ) /∂t ∂ ˆ M ( r , t ) /∂ r , anda third group is proportional to the second derivative ∂ ˆ M ( r , t ) /∂ r ∂t . We show that the response of the elec-tric current to the time-dependent magnetization gradi-ent ∂ ˆ M ( r , t ) /∂ r ∂t contais the inverse of CIDMI. Weestablish the reciprocity relation between inverse and di-rect CIDMI and thereby obtain an expression for CIDMI.We find that CIDMI is related to the modification oforbital magnetism induced by magnetization dynamics,which we call dynamical orbital magnetism (DOM). Weshow that torques are generated by time-dependent gra-dients of magnetization as well. The inverse effect con-sists in the modification of DMI by magnetization dy-namics, which we call dynamical DMI (DDMI).Additionally, we develop a formalism to calculate thechiral contributions to the direct and inverse current-induced torques (CITs) and to the torque-torque correla-7tion in noncollinear magnets. We show that the responseto time-dependent magnetization gradients contributessubstantially to these effects and that the Onsager reci-procity relations are violated when it is not taken into ac-count. In noncollinear magnets CIDMI, DDMI and DOMdepend on the local magnetization direction. We showthat the resulting spatial gradients of CIDMI, DDMIand DOM have to be subtracted from the CIT, fromthe torque-torque correlation, and from the inverse CIT,respectively.We apply our formalism to study CITs and the torque-torque correlation in textured Rashba ferromagnets. Wefind that the contribution of CIDMI to the chiral CIT isof the order of magnitude of the total effect. Similarly, wefind that the contribution of DDMI to the chiral torque-torque correlation is of the order of magnitude of thetotal effect. Acknowledgments
We gratefully acknowledge computing time on thesupercomputers of J¨ulich Supercomputing Center andRWTH Aachen University as well as funding by DeutscheForschungsgemeinschaft (MO 1731/5-1).
Appendix A: Response to time-dependent gradients
In this appendix we derive Eq. (19), Eq. (21), Eq. (42),and Eq. (83), which describe the response to time-dependent magnetization gradients, and Eq. (51), whichdescribes the response to time-dependent magnetic fields.We consider perturbations of the form δH ( r , t ) = B b qω sin( q · r ) sin( ωt ) . (A1)When we set B = ∂H∂ ˆ M k and b = ∂ ˆ M k ∂r i ∂t , Eq. (A1) turns intoEq. (18), while when we set B = − ev i and b = ǫ ijk ∂B k ∂t we obtain Eq. (49). We need to derive an expression forthe response δA ( r , t ) of an observable A to this pertur-bation, which varies in time like cos( ωt ) and in space likecos( q · r ), because ∂ ˆ M ( r ,t ) ∂r i ∂t ∝ cos( q · r ) cos( ωt ). There-fore, we use the Kubo linear response formalism to obtainthe coefficient χ in δA ( r , t ) = χ cos( q · r ) cos( ωt ) , (A2)which is given by χ = i ~ qωV h hh A cos( q · r ) , B sin( q · r ) ii R ( ~ ω ) −hh A cos( q · r ) , B sin( q · r ) ii R ( − ~ ω ) i , (A3)where hh A cos( q · r ) , B sin( q · r ) ii R ( ~ ω ) is the retardedfunction at frequency ω and V is the volume of the unitcell. The operator B sin( q · r ) can be written as B sin( q · r ) = 12 i X k nm h B (1) k nm c † k + n c k − m − B (2) k nm c † k − n c k + m i , (A4)where k + = k + q / k − = k − q / c † k + n is the cre-ation operator of an electron in state | u k + n i , c k − m is theannihilation operator of an electron in state | u k − m i , B (1) k nm = 12 h u k + n | [ B k + + B k − ] | u k − m i (A5)and B (2) k nm = 12 h u k − n | [ B k + + B k − ] | u k + m i . (A6)Similarly, A cos( q · r ) = 12 X k nm h A (1) k nm c † k + n c k − m + A (2) k nm c † k − n c k + m i , (A7)where A (1) k nm = 12 h u k + n | (cid:2) A k + + A k − (cid:3) | u k − m i (A8)and A (2) k nm = 12 h u k − n | (cid:2) A k + + A k − (cid:3) | u k + m i . (A9)It is convenient to obtain the retarded response func-tion in Eq. (A3) from the corresponding Matsubara func-tion in imaginary time τ V hh A cos( q · r ) , B sin( q · r ) ii M ( τ ) == 14 i Z d d k (2 π ) d X nm X n ′ m ′ h A (1) k nm B (2) k n ′ m ′ Z (1) k nmn ′ m ′ ( τ ) − A (2) k nm B (1) k n ′ m ′ Z (2) k nmn ′ m ′ ( τ ) i , (A10)where d = 1 , Z (1) k nmn ′ m ′ ( τ ) = h T τ c † k + n ( τ ) c k − m ( τ ) c † k − n ′ (0) c k + m ′ (0) i = − G M m ′ n ( k + , − τ ) G M mn ′ ( k − , τ ) , (A11) Z (2) k nmn ′ m ′ ( τ ) = h T τ c † k − n ( τ ) c k + m ( τ ) c † k + n ′ (0) c k − m ′ (0) i = − G M m ′ n ( k − , − τ ) G M mn ′ ( k + , τ ) , (A12)and G M mn ′ ( k + , τ ) = −h T τ c k + m ( τ ) c † k + n ′ (0) i (A13)8is the single-particle Matsubara function. The Fouriertransform of Eq. (A10) is given by1 V hh A cos( q · r ) , B sin( q · r ) ii M ( i E N ) == i ~ β Z d d k (2 π ) d X nm X n ′ m ′ X p h A (1) k nm B (2) k n ′ m ′ G M m ′ n ( k + , i E p ) G M mn ′ ( k − , i E p + i E N ) − A (2) k nm B (1) k n ′ m ′ G M m ′ n ( k − , i E p ) G M mn ′ ( k + , i E p + i E N ) i , (A14)where E N = 2 πN/β and E p = (2 p + 1) π/β are bosonicand fermionic Matsubara energy points, respectively, and β = 1 / ( k B T ) is the inverse temperature.In order to carry out the Matsubara summation over E p we make use of1 β X p G M mn ′ ( i E p + i E N ) G M m ′ n ( i E p ) == i π Z d E ′ f ( E ′ ) G M mn ′ ( E ′ + i E N ) G M m ′ n ( E ′ + iδ )+ i π Z d E ′ f ( E ′ ) G M mn ′ ( E ′ + iδ ) G M m ′ n ( E ′ − i E N ) − i π Z d E ′ f ( E ′ ) G M mn ′ ( E ′ + i E N ) G M m ′ n ( E ′ − iδ ) − i π Z d E ′ f ( E ′ ) G M mn ′ ( E ′ − iδ ) G M m ′ n ( E ′ − i E N ) , (A15)where δ is a positive infinitesimal. The retarded function hh A cos( q · r ) , B sin( q · r ) ii R ( ω ) is obtained from the Mat-subara function hh A cos( q · r ) , B sin( q · r ) ii M ( i E N ) by theanalytic continuation i E N → ~ ω to real frequencies. Theright-hand side of Eq. (A15) has the following analyticcontinuation to real frequencies: i π Z d E ′ f ( E ′ ) G R mn ′ ( E ′ + ~ ω ) G R m ′ n ( E ′ )+ i π Z d E ′ f ( E ′ ) G R mn ′ ( E ′ ) G A m ′ n ( E ′ − ~ ω ) − i π Z d E ′ f ( E ′ ) G R mn ′ ( E ′ + ~ ω ) G A m ′ n ( E ′ ) − i π Z d E ′ f ( E ′ ) G A mn ′ ( E ′ ) G A m ′ n ( E ′ − ~ ω ) . (A16)Therefore, we obtain χ = − i π ~ qω Z d d k (2 π ) d [ Z k ( q, ω ) − Z k ( − q, ω ) − Z k ( q, − ω ) + Z k ( − q, − ω )] , (A17) where Z k ( q, ω ) == Z d E ′ f ( E ′ )Tr h A k G R k − ( E ′ + ~ ω ) B k G R k + ( E ′ ) i + Z d E ′ f ( E ′ )Tr h A k G R k − ( E ′ ) B k G A k + ( E ′ − ~ ω ) i − Z d E ′ f ( E ′ )Tr h A k G R k − ( E ′ + ~ ω ) B k G A k + ( E ′ ) i − Z d E ′ f ( E ′ )Tr h A k G A k − ( E ′ ) B k G A k + ( E ′ − ~ ω ) i . (A18)We consider the limit lim q → lim ω → χ . In this limitEq. (A17) may be rewritten as χ = − i π ~ Z d d k (2 π ) d ∂ Z k ( q, ω ) ∂q∂ω (cid:12)(cid:12)(cid:12)(cid:12) q = ω =0 . (A19)The frequency derivative of Z k ( q, ω ) is given by1 ~ ∂Z k ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 = Z d E ′ f ( E ′ )Tr " A k ∂G R k − ( E ′ ) ∂ E ′ B k G R k + ( E ′ ) − Z d E ′ f ( E ′ )Tr " A k G R k − ( E ′ ) B k ∂G A k + ( E ′ ) ∂ E ′ − Z d E ′ f ( E ′ )Tr " A k ∂G R k − ( E ′ ) ∂ E ′ B k G A k + ( E ′ ) + Z d E ′ f ( E ′ )Tr " A k G A k − ( E ′ ) B k ∂G A k + ( E ′ ) ∂ E ′ . (A20)Using ∂G R ( E ) /∂ E = − G R ( E ) G R ( E ) / ~ we obtain ∂Z k ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 = − Z d E ′ f ( E ′ )Tr h A k G R k − G R k − B k G R k + i + Z d E ′ f ( E ′ )Tr h A k G R k − B k G A k + G A k + i + Z d E ′ f ( E ′ )Tr h A k G R k − G R k − B k G A k + i − Z d E ′ f ( E ′ )Tr h A k G A k − B k G A k + G A k + i . (A21)Making use of lim q → ∂G R k + ∂q = 12 G R k v · q q G R k (A22)9we finally obtain χ = − i π ~ Z d d k (2 π ) d lim q → lim ω → ∂ Z ( q, ω ) ∂q∂ω == − i π ~ q q · Z d d k (2 π ) d Z d E f ( E )Tr h A k R v RR B k R + A k RR v R B k R − A k RR B k R v R − A k R v R B k AA + A k R B k A v AA + A k R B k AA v A − A k R v RR B k A − A k RR v R B k A + A k RR B k A v A + A k A v A B k AA − A k A B k A v AA − A k A B k AA v A i , (A23)where we use the abbreviations R = G R k ( E ) and A = G A k ( E ). When we substitute B = ∂H∂ ˆ M j , A = − ev i , and q = q k ˆ e k , we obtain Eq. (19). When we substitute B = T j , A = − ev i , and q = q k ˆ e k , we obtain Eq. (21). Whenwe substitute A = −T i , B = T j , and q = q k ˆ e k , we obtainEq. (42). When we substitute B = − ev j , A = −T i ,and q = q k ˆ e k , we obtain Eq. (51). When we substitute B = ∂H∂ ˆ M j , A = −T i , and q = q k ˆ e k , we obtain Eq. (83). Appendix B: Perturbation theory for the chiralcontributions to CIT and to the torque-torquecorrelation
In this appendix we derive expressions for the retardedfunction hh A cos( q · r ); C ii R ( ~ ω ) (B1)within first-order perturbation theory with respect to theperturbation δH = B η sin( q · r ) , (B2)which may arise e.g. from the spatial oscillation of themagnetization direction. As usual, it is convenient to ob-tain the retarded response function from the correspond-ing Matsubara function hh cos( q · r ) A ; C ii M ( τ ) = −h T τ cos( q · r ) A ( τ ) C (0) i . (B3)The starting point for the perturbative expansion isthe equation − h T τ cos( q · r ) A ( τ ) C (0) i == − Tr (cid:2) e − βH T τ cos( q · r ) A ( τ ) C (0) (cid:3) Tr [ e − βH ] == − Tr (cid:8) e − βH T τ [ U cos( q · r ) A ( τ ) C (0)] (cid:9) Tr [ e − βH U ] , (B4) where H is the unperturbed Hamiltonian and we con-sider the first order in the perturbation δH : U (1) = − ~ Z ~ β d τ T τ { e τ H / ~ δHe − τ H / ~ } . (B5)The essential difference between Eq. (A3) and Eq. (B4) isthat in Eq. (A3) the operator B enters together with thefactor sin( q · r ) sin( ωt ) (see Eq. (A1)), while in Eq. (B4)only the factor sin( q · r ) is connected to B in Eq. (B2),while the factor sin( ωt ) is coupled to the additional op-erator C .We use Eq. (A4) and Eq. (A7) in order to express A cos( q · r ) and B sin( q · r ) in terms of annihilation andcreation operators. In terms of the correlators Z (3) k nmn ′ m ′ n ′′ m ′′ ( τ, τ ) = h T τ c † k − n ( τ ) c k + m ( τ ) c † k + n ′ ( τ ) c k − m ′ ( τ ) c † k − n ′′ c k − m ′′ i (B6)and Z (4) k nmn ′ m ′ n ′′ m ′′ ( τ, τ ) = h T τ c † k − n ( τ ) c k + m ( τ ) c † k + n ′ ( τ ) c k − m ′ ( τ ) c † k + n ′′ c k + m ′′ i (B7)and Z (5) k nmn ′ m ′ n ′′ m ′′ ( τ, τ ) = h T τ c † k + n ( τ ) c k − m ( τ ) c † k − n ′ ( τ ) c k + m ′ ( τ ) c † k + n ′′ c k + m ′′ i (B8)and Z (6) k nmn ′ m ′ n ′′ m ′′ ( τ, τ ) = h T τ c † k + n ( τ ) c k − m ( τ ) c † k − n ′ ( τ ) c k + m ′ ( τ ) c † k − n ′′ c k − m ′′ i (B9)Eq. (B4) can be written as hh cos( q · r ) A ; C ii M ( τ ) == ηV i ~ Z d d k (2 π ) d Z ~ β dτ X nm X n ′ m ′ X n ′′ m ′′ " − B (2) k nm A (1) k n ′ m ′ C k − n ′′ m ′′ Z (3) k nmn ′ m ′ n ′′ m ′′ ( τ, τ ) − B (2) k nm A (1) k n ′ m ′ C k + n ′′ m ′′ Z (4) k nmn ′ m ′ n ′′ m ′′ ( τ, τ )+ B (1) k nm A (2) k n ′ m ′ C k + n ′′ m ′′ Z (5) k nmn ′ m ′ n ′′ m ′′ ( τ, τ )+ B (1) k nm A (2) k n ′ m ′ C k − n ′′ m ′′ Z (6) k nmn ′ m ′ n ′′ m ′′ ( τ, τ ) (B10)within first-order perturbation theory, where we de-fined C k − n ′′ m ′′ = h u k − n ′′ | C | u k − m ′′ i and C k + n ′′ m ′′ = h u k + n ′′ | C | u k + m ′′ i .Note that Z (5) can be obtained from Z (3) by replac-ing k − by k + and k + by k − . Similarly, Z (6) can beobtained from Z (4) by replacing k − by k + and k + by k − . Therefore, we write down only the equations for0 Z (3) and Z (4) in the following. Using Wick’s theoremwe find Z (3) k nmn ′ m ′ n ′′ m ′′ ( τ, τ ) == − G M m ′ n ( k − , τ − τ ) G M mn ′ ( k + , τ − τ ) G M m ′′ n ′′ ( k − , G M mn ′ ( k + , τ − τ ) G M m ′′ n ( k − , − τ ) G M m ′ n ′′ ( k − , τ )(B11)and Z (4) k nmn ′ m ′ n ′′ m ′′ ( τ, τ ) == − G M mn ′ ( k + , τ − τ ) G M m ′ n ( k − , τ − τ ) G M m ′′ n ′′ ( k + , G M mn ′′ ( k + , τ ) G M m ′ n ( k − , τ − τ ) G M m ′′ n ′ ( k + , − τ ) . (B12)The Fourier transform hh cos( q · r ) A ; C ii M ( i E N ) == Z ~ β d τ e i ~ E N τ hh cos( q · r ) A ; C ii M ( τ ) (B13)of Eq. (B10) can be written as hh cos( q · r ) A ; C ii M ( i E N ) == ηV i ~ Z d d k (2 π ) d X nm X n ′ m ′ X n ′′ m ′′ " − B (2) k nm A (1) k n ′ m ′ C k − n ′′ m ′′ Z (3 a ) k nmn ′ m ′ n ′′ m ′′ ( i E N ) − B (2) k nm A (1) k n ′ m ′ C k + n ′′ m ′′ Z (4 a ) k nmn ′ m ′ n ′′ m ′′ ( i E N )+ B (1) k nm A (2) k n ′ m ′ C k + n ′′ m ′′ Z (5 a ) k nmn ′ m ′ n ′′ m ′′ ( i E N )+ B (1) k nm A (2) k n ′ m ′ C k − n ′′ m ′′ Z (6 a ) k nmn ′ m ′ n ′′ m ′′ ( i E N ) (B14)in terms of the integrals Z (3 a ) k nmn ′ m ′ n ′′ m ′′ ( i E N ) = Z ~ β d τ Z ~ β d τ e i ~ E N τ ×× G M mn ′ ( k + , τ − τ ) G M m ′′ n ( k − , − τ ) G M m ′ n ′′ ( k − , τ ) == 1 ~ β X p G M k + mn ′ ( i E p ) G M k − m ′′ n ( i E p ) G M k − m ′ n ′′ ( i E p + i E N )(B15)and Z (4 a ) k nmn ′ m ′ n ′′ m ′′ ( i E N ) = Z ~ β d τ Z ~ β d τ e i ~ E N τ ×× G M mn ′′ ( k + , τ ) G M m ′ n ( k − , τ − τ ) G M m ′′ n ′ ( k + , − τ ) == 1 ~ β X p G M k + mn ′′ ( i E p ) G M k − m ′ n ( i E p ) G M k + m ′′ n ′ ( i E p − i E N ) , (B16)where E N = 2 πN/β is a bosonic Matsubara energy pointand we used G M ( τ ) = 1 ~ β ∞ X p = −∞ e − i E p τ/ ~ G M ( i E p ) , (B17) where E p = (2 p + 1) π/β is a fermionic Matsubara point.Again Z (5 a ) is obtained from Z (3 a ) by replacing k − by k + and k + by k − and Z (6 a ) is obtained from Z (4 a ) inthe same way.Summation over Matsubara points E p in Eq. (B15) andin Eq. (B16) and analytic continuation i E N → ~ ω yields2 πi ~ Z (3 a ) k nmn ′ m ′ n ′′ m ′′ ( ~ ω ) = − Z d E f ( E ) G R k + mn ′ ( E ) G R k − m ′′ n ( E ) G R k − m ′ n ′′ ( E + ~ ω )+ Z d E f ( E ) G A k + mn ′ ( E ) G A k − m ′′ n ( E ) G R k − m ′ n ′′ ( E + ~ ω ) − Z d E f ( E ) G A k + mn ′ ( E − ~ ω ) G A k − m ′′ n ( E − ~ ω ) G R k − m ′ n ′′ ( E )+ Z d E f ( E ) G A k + mn ′ ( E − ~ ω ) G A k − m ′′ n ( E − ~ ω ) G A k − m ′ n ′′ ( E )(B18)and2 πi ~ Z (4 a ) k nmn ′ m ′ n ′′ m ′′ ( ~ ω ) = − Z d E f ( E ) G R k + mn ′′ ( E ) G R k − m ′ n ( E ) G A k + m ′′ n ′ ( E − ~ ω )+ Z d E f ( E ) G A k + mn ′′ ( E ) G A k − m ′ n ( E ) G A k + m ′′ n ′ ( E − ~ ω ) − Z d E f ( E ) G R k + mn ′′ ( E + ~ ω ) G R k − m ′ n ( E + ~ ω ) G R k + m ′′ n ′ ( E )+ Z d E f ( E ) G R k + mn ′′ ( E + ~ ω ) G R k − m ′ n ( E + ~ ω ) G A k + m ′′ n ′ ( E ) . (B19)In the next step we take the limit ω → − V lim ω → Im hh A cos( q · r ); C ii R ( ~ ω ) ~ ω == η ~ Im h Y (3) + Y (4) − Y (5) − Y (6) i , (B20)1where we defined Y (3) = 1 i ~ Z d d k (2 π ) d X nm X n ′ m ′ X n ′′ m ′′ B (2) k nm A (1) k n ′ m ′ C k − n ′′ m ′′ ×× ∂ Z (3 a ) k nmn ′ m ′ n ′′ m ′′ ( ~ ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω =0 , Y (4) = 1 i ~ Z d d k (2 π ) d X nm X n ′ m ′ X n ′′ m ′′ B (2) k nm A (1) k n ′ m ′ C k + n ′′ m ′′ ×× ∂ Z (4 a ) k nmn ′ m ′ n ′′ m ′′ ( ~ ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω =0 , Y (5) = 1 i ~ Z d d k (2 π ) d X nm X n ′ m ′ X n ′′ m ′′ B (1) k nm A (2) k n ′ m ′ C k + n ′′ m ′′ ×× ∂ Z (5 a ) k nmn ′ m ′ n ′′ m ′′ ( ~ ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω =0 , Y (6) = 1 i ~ Z d d k (2 π ) d X nm X n ′ m ′ X n ′′ m ′′ B (1) k nm A (2) k n ′ m ′ C k − n ′′ m ′′ ×× ∂ Z (6 a ) k nmn ′ m ′ n ′′ m ′′ ( ~ ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω =0 , (B21)which can be expressed as Y (3) = Y (3 a ) + Y (3 b ) and Y (4) = Y (4 a ) + Y (4 b ) , where2 π ~ Y (3 a ) = 1 ~ Z d d k (2 π ) d Z d E f ( E ) ×× Tr " A k G R k − ( E ) C k − G A k − ( E ) B k G A k + ( E ) G A k + ( E )+ A k G R k − ( E ) G R k − ( E ) C k − G A k − ( E ) B k G A k + ( E )+ A k G R k − ( E ) C k − G A k − ( E ) G A k − ( E ) B k G A k + ( E ) = Z d d k (2 π ) d Z d E f ′ ( E ) ×× Tr h A k G R k − ( E ) C k − G A k − ( E ) B k G A k + ( E ) i (B22)and 2 π ~ Y (3 b ) = − ~ Z d d k (2 π ) d Z d E f ( E ) ×× Tr " A k G A k − ( E ) C k − G A k − ( E ) B k G A k + ( E ) G A k + ( E )+ A k G R k − ( E ) G R k − ( E ) C k − G R k − ( E ) B k G R k + ( E )+ A k G A k − ( E ) C k − G A k − ( E ) G A k − ( E ) B k G A k + ( E ) . (B23) Similarly,2 π ~ Y (4 a ) = 1 ~ Z d d k (2 π ) d Z d E f ( E ) ×× Tr " A k G R k − ( E ) B k G R k + ( E ) C k + G A k + ( E ) G A k + ( E ) − A k G R k − ( E ) G R k − ( E ) B k G R k + ( E ) C k + G A k + ( E ) − A k G R k − ( E ) B k G R k + ( E ) G R k + ( E ) C k + G A k + ( E ) = Z d d k (2 π ) d Z d E f ′ ( E ) ×× Tr h A k G R k − ( E ) B k G R k + ( E ) C k + G A k + ( E ) i (B24)and 2 π ~ Y (4 b ) = − ~ Z d d k (2 π ) d Z d E f ( E ) ×× Tr " A k G A k − ( E ) B k G A k + ( E ) C k + G A k + ( E ) G A k + ( E )+ A k G R k − ( E ) G R k − ( E ) B k G R k + ( E ) C k + G R k + ( E )+ A k G R k − ( E ) B k G R k + ( E ) G R k + ( E ) C k + G R k + ( E ) . (B25)We call Y (3 a ) and Y (4 a ) Fermi surface terms and Y (3 b ) and Y (4 b ) Fermi sea terms. Again Y (5) is obtained from Y (3) by replacing k − by k + and k + by k − and Y (6) isobtained from Y (4) in the same way.Finally, we take the limit q → − ~ V η
Im lim q → lim ω → ∂∂ω ∂∂q i hh A cos( q · r ); C ii R ( ~ ω )= 12 ~ lim q → ∂∂q i Im h Y (3) + Y (4) − Y (5) − Y (6) i = 12 ~ Im h X (3) + X (4) − X (5) − X (6) i , (B26)where we defined X ( j ) = ∂∂q i (cid:12)(cid:12)(cid:12)(cid:12) q =0 Y ( j ) (B27)for j = 3 , , ,
6. Since Y (4) and Y (6) are related bythe interchange of k − and k + it follows that X (6) = − X (4) . Similarly, since Y (3) and Y (5) are related by theinterchange of k − and k + it follows that X (5) = − X (3) .Consequently, we needΛ = 1 ~ Im h X (3 a ) + X (3 b ) + X (4 a ) + X (4 b ) i , (B28)where X (3 a ) and X (4 a ) are the Fermi surface terms and X (3 b ) and X (4 b ) are the Fermi sea terms. The Fermi2surface terms are given by X (3 a ) = − π ~ Z d d k (2 π ) d Z d E f ′ ( E )Tr " A k G R k ( E ) v k G R k ( E ) C k G A k ( E ) B k G A k ( E )+ A k G R k ( E ) C k G A k ( E ) v k G A k ( E ) B k G A k ( E ) − A k G R k ( E ) C k G A k ( E ) B k G A k ( E ) v k G A k ( E )+ A k G R k ( E ) ∂ C k ∂k G A k ( E ) B k G A k ( E ) (B29)and X (4 a ) = − h X (3 a ) i ∗ . (B30)The Fermi sea terms are given by X (3 b ) = − π ~ Z d d k (2 π ) d Z d E f ( E )Tr " − ( A RvRR C R B R ) + ( A A C AA B AvA ) − ( A RRvR C R B R ) − ( A RR C RvR B R )+ ( A RR C R B RvR ) − ( A AvA C A B AA ) − ( A A C AvA B AA ) + ( A A C A B AvAA )+ ( A A C A B AAvA ) − ( A AvA C AA B A ) − ( A A C AvAA B A ) − ( A A C AAvA B A ) − ( A RR ∂ C ∂k R B R ) − ( A A ∂ C ∂k AA B A ) − ( A A ∂ C ∂k A B AA ) (B31)and X (4 b ) = − h X (3 b ) i ∗ . (B32)In Eq. (B31) we use the abbreviations R = G R k ( E ), A = G A k ( E ), A = A k , B = B k , C = C k . It is importantto note that C k − and C k + depend on q through k − = k − q / k + = k + q / q derivative thereforegenerates the additional terms with ∂ C k /∂k in Eq. (B29)and Eq. (B31). In contrast, A k and B k do not dependlinearly on q .Eq. (B28) simplifies due to the relations Eq. (B30) andEq. (B32) as follows:Λ = 2 ~ Im h X (3 a ) + X (3 b ) i . 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