Optical responses induced by spin gauge field at the second order
OOptical responses induced by spin gauge field at the second order
E. Karashtin ∗ University of Nizhny Novgorod, 23 Prospekt Gagarina,603950, Nizhny Novgorod, Russia andInstitute for Physics of Microstructures RAS,GSP-105, 603950, Nizhny Novgorod, Russia
Gen Tatara
RIKEN Center for Emergent Matter Science (CEMS),2-1 Hirosawa, Wako, Saitama, 351-0198 Japan (Dated: May 13, 2020) a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y bstract Optical responses of ferromagnetic materials with spin gauge field that drives intrinsic spin currenis theoretically studied. The conductivity tensor is calculated based on a linear response theory tothe applied electric field taking account of the non-linear effects of the spin gauge field to the secondorder. We consider the case where the spin gauge field is uniform, realized for spiral magnetizationstructure or uniform spin-orbit interaction. The spin gauge field, or an intrinsic spin current, turnsout to give rise to anisotropic optical responses, which is expected to be useful to experimentaldetection of magnetization structures.
I. INTRODUCTION
Spin current plays central roles in spintronics. Moreover, intrinsic spin current in solidshas been shown to induce several peculiar interactions. As spin current breaks inversionsymmetry keeping the time reversal symmetry, the interactions induced by spin currenthave the same symmetry , like the antisymmetric Dzyaloshinskii-Moriya (DM) interactionbetween spins. It was theoretically shown that the DM interaction constant is D ai = (cid:126) a j a s ,i ,where i and a represent directions in space and spin, respectively, and a is the lattice con-stant. Namely, it is proportional to the expectation value of the spin current j a s ,i intrinsicallyinduced by the spin-orbit interaction when inversion symmetry is broken . In this picture,the origin of the DM interaction is the Doppler shift due to the intrinsic flow of spin. Thepicture naturally explains the Doppler shift of spin waves in the presence of the DM inter-action observed in Refs. . The spin current picture was explored further in Refs. . Therelation between the DM constant and spin current has been experimentally confirmed byinjecting external spin current . In the case of insulator magnets, Ref. pointed out that anelectric polarization is induced by the vector chirality, which is equivalent to an intrinsic spincurrent. Similar flexo-magnetoelectric effects due to intrinsic spin current were discussed inRefs. . The second harmonic generation was discussed to arise from intrinsic spin currentunder the action of optical electromagnetic wave .The Doppler shift picture can be applied for discussing anomalous optical properties ofsolids. When a spin-orbit interaction breaking inversion symmetry, like the Rashba spin-orbitinteraction, coexists with interactions breaking time-reversal invariance, like the couplingto a magnetization, intrinsic charge flow is allowed, resulting in a directional dichroism2f light . The dichroism here is induced by the effective gauge field proportional to u ≡ α R × M , where α R and M denote the Rashba field and the magnetization, respectively.The effective gauge field induces charge current, and the effect is viewed as Doppler shiftof light, mentioned in Ref. . It was in fact demonstrated theoretically that the Rashbainteraction and magnetization leads to a coupling proportional to u · ( E × B ) between theelectric and magnetic fields, E and B . This coupling is equivalent to switching to a movingframe with velocity u . As seen above, various intrinsic flow or effective gauge field causingDoppler shift in solids have been detected as asymmetric transport effects .Spin current is manipulated by spin gauge fields , and spin-orbit interaction argued aboveis an example of spin gauge field driving the spin current. Slowly-varying spin textures alsoact as spin gauge field for conduction electron . In this paper, we study nonlinear effectsof intrinsic spin currents induced by spin gauge field on the optical properties of metallicsystems. We include the spin gauge field to the second order, and derive linear responseexpression of the optical conductivity. We consider the case of uniform spin current, i.e., spingauge field uniform in space and time. We have in mind the case where one spin gauge fieldis due to an intrinsic spin-orbit interaction like Rashba interaction and another generated bymagnetization structures. As magnetization structure, we consider a spiral spin structurewhere the spins change with a constant pitch. II. FORMALISM
The spin gauge field approach for spin structures is valid in the adiabatic regime wherethe electron spin follows the magnetization structure due to sd exchange interaction . Thepresent manuscript therefore studies a strong sd exchange case, different from the perturba-tive treatment of the sd exchange interaction carried out in Ref. . The model we consideris the sd exchange model described by a Hamiltonian H = (cid:90) d rc † (cid:18) − ∇ m − (cid:15) F − M n ( r ) · σ (cid:19) c (1)where c and c † are electron field operators, (cid:15) F is the Fermi energy, m is the electron mass, M is the spin splitting energy due to the sd exchange interaction, and n ( r ) is a unit vectorfield representing the magnetization direction, σ being a vector or Pauli matrices. We applya unitary transformation in the spin space to diagonalize the exchange interaction . The3ew electron field in the transformed rotated frame is ˜ c ≡ U − ( r ) c , where U ( r ) is a 2 × U − ( n ( r ) · σ ) U = σ z . The Hamiltonian in therotated frame reads H = (cid:90) d r ˜ c † (cid:18) − m ( ∇ i + iA s ,i ) − (cid:15) F − M σ z (cid:19) ˜ c (2)where A s ,i ≡ − iU − ∇ i U is the spin gauge field of a 2 × A α s ,i , where i and α are directions inspace and spin space, respectively, and derive the expression for the conductivity tensor.Later in the next section various spin gauge fields are introduced, and the results in thissection should be read replacing A α s ,i by the sum of all the spin gauge fields of interest. Theconductivity tensor is σ ij (Ω) = 1 − i Ω (cid:90) dω π (cid:88) k tr[ v i G k ω v j G k ,ω +Ω + δ ij G k ω ] < (3)Here the velocity operator is v i = k i m + A α s ,i σ α (4)and the Green’s function includes spin gauge field A s and < denotes the lesser component.The last term of Eq. (3) arise from the ’diamagnetic’ contribution to the electric current,namely the second term of j i = v i + A i , where A represents the gauge field of electromag-nitism. Retarded Green’s function is G r k ω = 1 ω − k m − (cid:80) iα ( γ k ) α σ α + iη (5)where ( γ k ) α ≡ M α + (cid:88) i k i A α s ,i (6) η represents small positive imaginary part due to the electron damping, and M ≡ M σ z isthe diagonalized spin splitting. (Being in the rotated frame, M is diagonalized along the z σ ij (Ω) = 1 − i Ω (cid:90) dω π (cid:88) k tr (cid:20) ( f ( ω + Ω) − f ( ω )) v i G r k ω v j G a k ,ω +Ω + f ( ω ) v i G a k ω v j G a k ,ω +Ω − f ( ω + Ω) v i G r k ω v j G r k ,ω +Ω + δ ij f ( ω ) ( G a k ω − G r k ω ) (cid:21) (7)where f ( ω ) ≡ [ e βω + 1] − is the Fermi distribution function ( β = ( k B T ) − is the inversetemperature). Trace of spin ( a, b = r , a), (cid:88) k tr[ v i G a k ω v j G b k ,ω (cid:48) ] ≡ (cid:88) k K abij ( k , ω, ω (cid:48) ) (8)is written defining G a k ω = π akω + γ k · σ Π akω (9)where ( (cid:15) k = k m − (cid:15) F ) π r kω ≡ ω − (cid:15) k + iη, Π akω ≡ ( π akω ) − γ k , (10)as K abij ( k , ω, ω (cid:48) ) = 1Π akω Π bk,ω (cid:48) × tr (cid:20)(cid:18) k i m + A s ,i · σ (cid:19) ( π akω + ( M + k k A s ,k ) · σ ) (cid:18) k j m + A s ,j · σ (cid:19) (cid:0) π bk,ω (cid:48) + ( M + k l A s ,l ) · σ (cid:1)(cid:21) (11)Focusing on the second order contribution in the spin gauge field, we obtain (cid:88) k K abij ( k , ω, ω (cid:48) ) = 2 (cid:20) m ( κ (2) abij + M κ (0) abij ) + 1 m (cid:0) κ abi ( A s ,j · M ) + κ abj ( A s ,i · M ) (cid:1) + 1 m ( κ (1) abik ( A s ,j · A s ,k ) + κ (1) abjk ( A s ,i · A s ,k )) + 2 m κ abijk ( A s ,k · M ) + 1 m κ abijkl ( A s ,k · A s ,l )+ (cid:2) λ (2) ab − M λ (0) ab (cid:3) ( A s ,i · A s ,j ) + 2 λ (0) ab ( A s ,i · M )( A s ,j · M ) − i Ω λ (0) ab ( A s ,i × A s ,j ) · M (cid:21) (12)5here the coefficients are (cid:88) k akω Π bk,ω (cid:48) ≡ λ (0) ab ( ω, ω (cid:48) ) , (cid:88) k π akω π bkω (cid:48) Π akω Π bk,ω (cid:48) ≡ λ (2) ab ( ω, ω (cid:48) ) (cid:88) k k i k j Π akω Π bk,ω (cid:48) ≡ κ (0) abij ( ω, ω (cid:48) ) , (cid:88) k k i k j π akω + π bkω (cid:48) Π akω Π bk,ω (cid:48) ≡ κ (1) abij ( ω, ω (cid:48) ) , (cid:88) k k i k j π akω π bkω (cid:48) Π akω Π bk,ω (cid:48) ≡ κ (2) abij ( ω, ω (cid:48) ) (cid:88) k k i π akω + π bkω (cid:48) Π akω Π bk,ω (cid:48) ≡ κ abi ( ω, ω (cid:48) ) , (cid:88) k k i k j k k akω Π bk,ω (cid:48) ≡ κ abijk ( ω, ω (cid:48) ) , (cid:88) k k i k j k k k l akω Π bk,ω (cid:48) ≡ κ abijkl ( ω, ω (cid:48) )(13)Note that contributions odd in k such as κ abi and κ abijk are finite because the energy (cid:15) k ± γ k contains contribution odd in k when M (cid:54) = 0. The summations over k are evalulatedexpanding γ k in Π akω with respect to the spin gauge field. The odd terms turn out to be κ abi ( ω, ω (cid:48) ) = κ ab ( ω, ω (cid:48) )( A s ,i · M ) κ abijk ( ω, ω (cid:48) ) = κ ab ( δ ij A s ,k + δ ik A s ,j + δ jk A s ,i ) · M (14)where κ ab and κ ab do not depend on the spin gauge field to the lowest order. Other coefficientsdepend on the gauge field to the lowest order as ( µ = 0 , , λ ( µ ) ab = λ ( µ ) ab + λ ( µ ) ab (cid:88) k ( A s ,k · M ) κ ( µ ) abij = δ ij [ κ ( µ ) ab + κ ( µ ) ab d2 (cid:88) k ( A s ,k · M ) ] + κ ( µ ) ab ( A s ,i · M )( A s ,j · M ) κ abijkl = κ ab ( δ ij δ kl + δ ik δ jl + δ il δ jk ) (15)where d denotes diagonal and λ ( µ ) abν , κ ( µ ) abν ( ν = 0 , , d2) and κ ab do not depend on the gaugefield to the lowest order. III. RESULT
From the above consideration, the conductivity tensor to the second order in the spingauge field is written as σ ij = δ ij [ χ + χ d2 (cid:88) k ( A s ,k · A s ,k ) + χ (ad)d2 (cid:88) k ( A s ,k · M )( A s ,k · M )]+ χ ( A s ,i · A s ,j ) + χ (ad)2 ( A s ,i · M )( A s ,j · M ) + χ ( A s ,i × A s ,j ) · M (16)6here χ i ’s are functions of the external angular frequency Ω. The scalar product ( A s ,i · M ) represents the adiabatic component (denoted by (ad) ) of the spin gauge field, i.e., thecomponent along the magnetization. The nonadiabatic (perpendicular) components of thegauge field affects the terms with χ d2 , χ and χ .Although the form in Eq. (16) is natural from the symmetry consideration, the ex-pressions for the coefficients are in principle known by the present microscopic study. Forexample, we have χ (Ω) = 1 − i Ω (cid:90) dω π (cid:20) ( f ( ω + Ω) − f ( ω )) ˜ χ ( ω, ω + Ω) + f ( ω ) ˜ χ ( ω, ω + Ω) − f ( ω + Ω) ˜ χ ( ω, ω + Ω) (cid:21) χ (Ω) = (cid:90) dω π (cid:20) ( f ( ω + Ω) − f ( ω )) λ (0)ra ( ω, ω + Ω) + f ( ω ) λ (0)aa ( ω, ω + Ω) − f ( ω + Ω) λ (0)rr ( ω, ω + Ω) (cid:21) (17)where ˜ χ ab ( ω, ω (cid:48) ) ≡ (cid:20) λ (2) ( ω, ω (cid:48) ) − M λ (0) ( ω, ω (cid:48) ) + 2 m κ (1)0 ( ω, ω (cid:48) ) + 2 m κ ( ω, ω (cid:48) ) (cid:21) ab (18)The coefficients χ i ’s are finite at Ω = 0, in spite of the factor of Ω − in the definition,Eqs. (7)(17). This is checked easily based on Eq. (7). In fact, the square bracket in Eq.(7) vanishes linearly at Ω →
0, as (cid:80) k tr[ v i G a k ω v j G a k ω ] = (cid:80) k tr[( G a k ω ) − ( ∂ k i G a k ω ) v j G a k ω ] = − δ ij (cid:80) k tr[ G a k ω ], where we used ( ∂ k i G a k ω ) = G a k ω v i G a k ω and integral by parts with respect to k . In the low frequency limit (Ω → σ ij (Ω →
0) = (cid:90) dω π f (cid:48) ( ω ) (cid:88) k (cid:20) K ra ij ( k , ω, ω ) − (cid:0) K aa ij ( k , ω, ω ) + K rr ij ( k , ω, ω ) (cid:1)(cid:21) (cid:39) − π (cid:88) k (cid:20) K ra ij ( k , , − (cid:0) K aa ij ( k , ,
0) + K rr ij ( k , , (cid:1)(cid:21) (19)where we used f (cid:48) ( ω ) (cid:39) − δ ( ω ) assuming low temperatures in the last line.The parameter χ in Eq,. (16) characterizes the magnitude of anisotropy of opticalresponse. Let us derive explicit expression for χ . Using1Π akω = 1 γ k (cid:88) σ = ± σg a k ωσ (20)where g a k ωσ ≡ π akω − σγ k (21)7s the spin-polarized Green’s function, we obtain λ (2) ab ( ω, ω (cid:48) ) = 14 (cid:88) k σ ( g a k ωσ g b k ω (cid:48) σ + g a k ωσ g b k ω (cid:48) , − σ ) λ (0) ab ( ω, ω (cid:48) ) = 14 (cid:88) k σ γ k ) g a k ωσ ( g b k ω (cid:48) σ − g b k ω (cid:48) , − σ ) κ (1) ab ( ω, ω (cid:48) ) = 112 (cid:88) k σ k γ k σg a k ωσ g b k ω (cid:48) σ κ ab ( ω, ω (cid:48) ) = 160 (cid:88) k σ k ( γ k ) g a k ωσ ( g b k ω (cid:48) σ − g b k ω (cid:48) , − σ ) (22)Those coefficients with a = r and b = a turn out to be dominant for η/M (cid:28)
1. Moreover,contributions containing the Green’s functions with different spins are neglected for η/M (cid:28)
1. Considering low frequency (Ω →
0) limit, the coefficient χ is χ (Ω) = − i (cid:88) σ ν σ τ σ (cid:18) k σ ) mM + ( k σ ) m M (cid:19) (23)where ν σ , k σ and τ σ ( ≡ / (2 η σ )) are spin-resolved electron density of states, Fermi wavevector and elastic lifetime, respectively. Noting that the Boltzmann conductivity is σ ∼ (cid:80) σ ν σ ( k σ ) τ σ , the anisotropic terms induced by spin gauge field is of the relative orderof ( A s /k F ) ( k F being the Fermi wave vector) compared to σ . For spiral magentizationstructure with a pitch Q , this ratio is A s /k F ∼ ( Q/k F ) and for Rashba spin gauge field, A R /k F ∼ α R ( k F ) /(cid:15) F , as will be discussed in the next section. For a short spiral wavelength (several nanometers like in Ho ) and for large Rashba coupling like in BiTeI , theanisotropy would be easily detected experimentally. IV. APPLICATION TO SPIRAL STRUCTURES
We consider examples of spiral magnetization structures (Fig. 1).
A. N´eel type spiral
The first one is a N´eel type spiral along the x -direction; n ( r ) = ˆ x sin Qx + ˆ z cos Qx (24)8 IG. 1. Magnetization structure of two spirals, N´eel (left) and Bloch (right). The direction of thespiral denoted by Q is the x -axis. where ˆ denotes the unit vector along the coordinate axis and Q is the pitch of the spiral.Equilibrium spin current induced by magnetization textures is given by j s ,i = n × ∇ i n ,which in the present case is perpendicular to the magnetization plane ( xz -plane); j s ,i = δ i,x Q ˆ y . The unitary transformation to diagonalize the sd exchange interaction for thismagnetization structure is U = m · σ , where m = (cid:18) sin Qx , , cos Qx (cid:19) (25)The spin gauge field arising from the structure is A α s ,i = − i tr[ σ α U − ∇ i U ], which for the N´eelstructure is A N ,α s ,i = Q δ i,x δ α,y (26)The direction of the spin polarization of the gauge field is y , which is consistent with theequilibrium spin current flow. B. Bloch type spiral
The second one is a bloch type spiral along the x -direction, where magnetization rotatesin the plane perpendicular to the x -direction, n ( r ) = ˆ y sin Qx + ˆ z cos Qx (27)The vector m is m = (cid:18) , sin Qx , cos Qx (cid:19) (28)and the equilibrium spin current is j s ,i = − δ i,x Q ˆ x , and the spin gauge field arising from thestructure is A B ,α s ,i = − Q δ i,x δ α,x (29)9oth N´eel and Bloch type of spiral, the spin gauge field is finite for spatial direction ofthe spiral, i.e., x -axis, and so the second-order contribution to the conductivity tensor (Eq.(16)) has only the diagonal components. If the direction of the spiral deviates from thecoordinate axis, symmetric off-diagonal components σ ij = σ ji ∝ A s ,i · A s ,j appear, where ij denotes the directions in the plane containing the spiral direction. The optical response canthus detect the direction of the intrinsic spin current induced by spin gauge field.However Bloch and N´eel spirals cannot be distinguished by the present optical response.This is because the optical response does not see the spin polarization direction (denoted by α of A α s ,i ) but only the scalar product or the trace in the spin index (Eq. (16)). Spin directionaffects optical response if an additional spin polarization is introduced by an external fieldor a spin-orbit interaction, which we consider in next two subsections. C. Spirals in an external magnetic field
We consider the case of a magnetic field applied along the x -axis for Bloch and N´eelspirals. We simply assume that the magnetization structure has a constant component ofmagnitude β along the field without deriving solutions including magnetic field, and so theargument may not be applicable for large β . The magnetization profiles with β are n N = ˆ x β + sin Qx (cid:112) β + 2 β sin Qx + ˆ z cos Qx (cid:112) β + 2 β sin Qx , (30) n B = ˆ x β (cid:112) β + ˆ y sin Qx (cid:112) β + ˆ z cos Qx (cid:112) β . (31)The tilted Bloch case is the one representing the excitation around a magnetic skyrmionlattice . Note that the limit β → β → ∞ stands for auniformly magnetized medium. The vector m which generates the unitary transformationto diagonalize the exchange interaction now has the form m N = sin Qx (cid:118)(cid:117)(cid:117)(cid:116) − cos Qx √ β +2 β sin Qx − cos Qx , , cos Qx (cid:118)(cid:117)(cid:117)(cid:116) cos Qx √ β +2 β sin Qx Qx , (32)10 B = β sin Qx (cid:118)(cid:117)(cid:117)(cid:116) − cos Qx √ β (1 − cos Qx ) (cid:0) β + sin Qx (cid:1) , sin Qx (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:18) − cos Qx √ β (cid:19) sin Qx (cid:0) β + sin Qx (cid:1) , cos Qx (cid:118)(cid:117)(cid:117)(cid:116) cos Qx √ β Qx (33)for N´eel and Bloch magnetization.In the N´eel case, the vector-potential takes the form A N ,α s ,i = δ i,x δ α,y Q Qxβ + sin Qx (cid:115) ( β + sin Qx ) sin Qx β sin Qx β + 2 β sin Qx . (34)Obviously, it has the same component as it was with no mangetic field applied (see (26)).After averaging over the coordinates we arrive at (cid:68) A N ,α s ,i (cid:69) = δ i,x δ α,y Q (cid:0) − π arctan | β | (cid:1) , | β | < δ i,x δ α,y Q π arcsin | β | β , | β | > . (35)where (cid:104) (cid:105) stands for spatial averaging. This function continuously decreases from Q to zeroas β grows.Gauge field component having a perpendicular spin emerges if we apply an out-of planemagnetic field for the Bloch case (33): A B ,α s ,i = δ i,x Q − β cos Qx + (cid:112) β sin Qx (cid:112) β (cid:0) β + sin Qx (cid:1) , β (cid:112) β (cid:112) sin Qx cos Qx √ β , β tan Qx cos Qx (cid:18) − cos Qx √ β (cid:19)(cid:113) tan Qx (cid:0) β + sin Qx (cid:1) . (36)After averaging over coordinates, (cid:10) A B ,z s ,x (cid:11) = 0, while two other components are finite: (cid:10) A B ,x s ,x (cid:11) = − Q (cid:32) − | β | (cid:112) β (cid:33) , (37) (cid:10) A B ,y s ,x (cid:11) = Q βπ (cid:112) β log (cid:112) β + 1 (cid:112) β − . (38)One can see that (37) is the same as (29) that decays with the magnitude of magneticoscillations. The y -component (cid:10) A B ,y s ,x (cid:11) is odd with respect to β . It is zero at β = 0 but hasan infinite derivative at this point, thus growing very fast at small applied field. It reachesits maximum at β ∗ ≈ .
66 with the value (cid:10) A B ,y s ,x (cid:11) ( β ∗ ) ≈ . (cid:10) A B ,x s ,x (cid:11) ( β = 0).11s we saw, response of the gauge field to a magnetic field depends much on the magneticstructure. Observation of optical response with an applied field is therefore expected to beuseful to distinguish the structure. D. Spin-orbit interaction
Let us consider spin-orbit interactions that break the inversion symmetry. The first oneis the Rashba interaction, whose Hamiltonian is H R = − i (cid:90) d rc † α R · ( ↔ ∇ × σ ) c (39)We consider first the case the Rashba field vector α R is along the z -axis. In the rotatedframe, the interaction reads H R = − i (cid:90) d r ˜ c † α R · ( ↔ ∇ × ˜ σ )˜ c + spin density part (40)where the first terms is the interaction between the spin current and the Rashba spin gaugefield, while the last term describing the spin density is neglected. The electron field in therotated frame is ˜ c ≡ U − c and ˜ σ = U − σ U is the spin operator in the rotated frame. TheRashba spin gauge field read from the interaction is A R ,i = − im(cid:15) ijk α R ,j ˜ σ k (41)Explicit form for each magnetization profile is calculated using ˜ σ k = 2 m k ( m · σ ) − σ k .For the N´eel type spiral, Eq. (24),˜ σ k = ( − cos( Qx ) σ x + sin( Qx ) σ z , − σ y , sin( Qx ) σ x + cos( Qx ) σ z ) (42)and we have (in the vector representation with respect to spatial direction i ) A NR = − imα R ( σ y , − cos( Qx ) σ x + sin( Qx ) σ z ,
0) (43)whose Fourier transform is A NR ( q ) = − imα R δ q ⊥ , (cid:34) δ q x , σ y ˆ x − (cid:88) ± δ q x , ± Q ( σ x ± iσ z ) ˆ y (cid:35) (44)where q ⊥ ≡ (0 , q y , q z ). Uniform component is A NR ( q = 0) = − imα R σ y ˆ x .12 s yx ,xy A s yx ,yy FIG. 2. Schematic figure showing spin polarized flow induced by A y s ,x and A y s ,y . For the Bloch type spiral, Eq. (27),˜ σ k = ( − σ x , − cos( Qx ) σ y + sin( Qx ) σ z , sin( Qx ) σ y + cos( Qx ) σ z ) (45)and we have (in the vector representation with respect to spatial direction i ) A BR ,i = − imα R (cos( Qx ) σ y − sin( Qx ) σ z , − σ x ,
0) (46)and A BR ( q ) = − imα R δ q ⊥ , (cid:34) − δ q x , σ x ˆ y + 12 (cid:88) ± δ q x , ± Q ( σ y ± iσ z ) ˆ x (cid:35) (47)Uniform component is A BR ( q = 0) = imα R σ x ˆ y . Similar calculations for α R along ˆ x or ˆ y with the use of (42) and (45) give uniform components shown in Table I.For the Weyl type spin-orbit interaction, H W = − λ W i (cid:90) d rc † ( ↔ ∇ · σ ) c (48)the gauge field is A R ,i = λ W ˜ σ i and uniform component is as in Table I.The spin gauge field and Rashba and Weyl spin gauge fields are summarized in Table I.The Rashba spin gauge field for different directions of α R in the magnetization plane hasdifferent coordinate components that are determined by the choice of coordinate systemreference point and do not differ physically; the third α R direction gives zero value. Hencethe most representative case is α R || ˆ z that is considered in detail above. Besides, it is seenthat A s ,i and uniform component of the Rahsba gauge field have the same spin polarizationdirection, i.e., perpendicular to the vector m and diagonalization axis ˆ z . The adiabaticcomponents of the conductivity, χ (ad) i therefore do not arise from the spin structure anduniform contribution of Rashba gauge field. The antisymmetric term χ does not ariseeither.Having two spin gauge fields from different origins offers interesting possibilities of ma-nipulation of spin and charge. A spin gauge field A α s ,i induces a flow in the direction i ashba ( q = 0), A R ,i Spin structure, A s ,i α R || ˆ x α R || ˆ y α R || ˆ z Weyl ( q = 0), A W ,i N´eel spiral Q δ i,x σ y imα R σ y δ i,z − imα R σ y δ i,x − λ W σ y δ i,y Bloch spiral − Q δ i,x σ x − imα R σ x δ i,z imα R σ x δ i,y − λ W σ x δ i,x TABLE I. Table of uniform components of spin gauge field and Rashba spin gauge fields for N´eeland Bloch type spirals. polarized along spin direction α , namely spin current j α s ,i (Fig. 2). Such spin polarized flowdoes not directly trigger optical responses of material as those responses are governed bycharge sector. The same spin gauge field maps the spin current to the charge one but theeffect on the conductivity tensor is diagonal in simple settings, as the resultant charge flowis along the original direction of spin flow. Rich possibilities appear if there is another spingauge field with different symmetry. For instance, if we have A β so ,j arising from spin-orbitinteraction (like A R or A W ), the cross product of the two gauge fields (cid:80) α A α s ,i A α so ,j can induceoff-diagonal charge correlation, i (cid:54) = j , as a result of conversion of spin polarization along α -direction to charge flow in spatial direction j . From Table I, we see that such off-diagonaloptical response arises for the Weyl spin-orbit interaction with N´eel spiral and Rashba in-teraction with Bloch spiral structure. Optical response can thus be used to identify spinstructures. Particularly, sudden change of magnetization structures and formation of do-mains would be detected as emergence of anisotropic and/or off-diagonal optical responseswhen external field or temperature is varied. V. DIRECTIONAL EFFECTS
As the spin current and the corresponding spin gauge field breaks inversion symmetry,the gauge field appears in the uniform component of the conductivity tensor from the secondorder. The information on the direction of the spin current flow induced by the spin gaugefield is therefore smeared in the uniform optical response considered so far. Direct effectsdue to the spin current flow are contained in the directional effects which depend on thewave vector q of the external electric field. The effects linear in q turn out to be linear (orhigher-odd order) in the spin gauge field. Let us briefly study these directional effects. For14he non-uniform component of the conductivity tensor, we need to calculate ( a, b = r , a) K abij ( q , ω, Ω) ≡ (cid:88) k tr[ v i ( k ) G a k − q ,ω v j ( k ) G b k + q ,ω +Ω ] (49)The Green’s function is expanded with respect to q as G a k ∓ q ,ω = G a k ,ω ∓ q k ∂ k k G a k ,ω ) + O ( q ) , (50)to obtain K abij ( q , ω, Ω) = (cid:88) k q k [ K abijk − K bajik ] + K abij ( q = 0 , ω, Ω) (51)where K abijk ( ω, Ω) ≡ (cid:88) k tr[ v i G a k ,ω v j G b k ,ω +Ω v k G b k ,ω +Ω ] (52)The trace is calculated in the same way as Eq. (12). To the first order in the gauge field,the result is ( ω (cid:48) = ω + Ω) K abijk ( ω, Ω) = (cid:88) k akω (Π bk,ω (cid:48) ) (cid:20) δ ij ( A s ,k · M ) (cid:20) k ( π bkω (cid:48) ) + 25 k π akω (cid:48) (cid:21) + δ ik ( A s ,j · M ) (cid:20) k [( π bkω (cid:48) ) + M ] + 25 k π akω (cid:48) (cid:21) + δ jk ( A s ,i · M ) (cid:20) k [( π bkω (cid:48) ) + M ] + 25 k π akω (cid:48) (cid:21)(cid:21) (53)Here A s denotes the total spin gauge field including the one due to magnetization structureand spin-orbit interaction. We therefore obtain the conductivity tensor linear (denoted by (1) ) in both q and the spin gauge field as σ (1) ij ( q , Ω) = δ ij q k ( A s ,k · M ) γ + [ q i ( A s ,j · M ) + q j ( A s ,i · M )] γ (54)where γ i ’s are functions of Ω. Contribution linear in q changes sign for opposite lightinjection, resulting in directional effects like directional dichroism. The directional featurearises in the symmetric components in the conductivity to the linear order in the spin gaugefield. As seen from Eq. (54), directional effects arise from the adiabatic component ofthe gauge field, A s ,i · M , which vanishes for the spin configurations considered in TableI. The directional effects predicted by Eq. (54) emerges when A s is due to the spin-orbitinteraction and when the magnetization M is uniform. (Note that Eq. (54) applies to15rbitrary direction of M if M is uniform.) In fact, directional effect was pointed out inRef. for the case of the Rashba spin-orbit interaction treating uniform M perturbatively.For the Rashba gauge field, Eq. (41), its uniform component is A α R ,i ( q = 0) = im(cid:15) ijα α R ,j and thus the diagonal term of Eq. (54) is proportional to q k ( A R ,k · M ) ∝ q · ( α R × M ). Thevector α R × M , sometimes called a troidal moment, describes intrinsic velocity of charge asnoted in Refs. . For Weyl type, the gauge field is A α W ,i = λ W δ iα , connecting the spaceand spin diagonally, and so the directional dichroism is with respect to the magnetizationdirection, q k ( A W ,k · M ) ∝ q · M . VI. SUMMARY
We have theoretically explored optical properties induced by the second-order effects ofspin gauge fields. The conductivity matrix was calculated in the slowly-varying limit withvanishing wave vector and angular frequency of the spin gauge field. Possibility of opticaldetection of spin structure was pointed out. Additional information is provided by studyingthe behaviour of optical response under the action of an external magnetic field. Wave-vector ( q )-resolved optical response, partially studied here as directional effects, is expectedto provide detailed information on the magnetization structures, and this is to be studiedin a future work. ACKNOWLEDGMENTS
This investigation was supported by a Grant-in-Aid for Exploratory Research (No.16K13853)and a Grant-in-Aid for Scientific Research (B) (No. 17H02929) from the Japan Society forthe Promotion of Science and a Grant-in-Aid for Scientific Research on Innovative Areas(No.26103006) from The Ministry of Education, Culture, Sports, Science and Technology(MEXT), Japan. E.K. would like to thank the Russian Science Foundation (Grant No.16-12-10340). ∗ [email protected] G. Tatara, Physica E: Low-dimensional Systems and Nanostructures , 208 (2019). T. Kikuchi, T. Koretsune, R. Arita, and G. Tatara, Phys. Rev. Lett. , 247201 (2016). Y. Iguchi, S. Uemura, K. Ueno, and Y. Onose, Phys. Rev. B , 184419 (2015). S. Seki, Y. Okamura, K. Kondou, K. Shibata, M. Kubota, R. Takagi, F. Kagawa, M. Kawasaki,G. Tatara, Y. Otani, and Y. Tokura, Phys. Rev. B , 235131 (2016). F. Freimuth, S. Bl¨ugel, and Y. Mokrousov, Phys. Rev. B , 054403 (2017). T. Koretsune, T. Kikuchi, and R. Arita, Journal of the Physical Society of Japan , 041011(2018), https://doi.org/10.7566/JPSJ.87.041011. G. V. Karnad, F. Freimuth, E. Martinez, R. Lo Conte, G. Gubbiotti, T. Schulz, S. Senz,B. Ocker, Y. Mokrousov, and M. Kl¨aui, Phys. Rev. Lett. , 147203 (2018). N. Kato, M. Kawaguchi, Y.-C. Lau, T. Kikuchi, Y. Nakatani, and M. Hayashi, Phys. Rev.Lett. , 257205 (2019). H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. , 057205 (2005). P. Bruno and V. K. Dugaev, Phys. Rev. B , 241302 (2005). A. P. Pyatakov and A. K. Zvezdin, Physics-Uspekhi , 557 (2012). J. Wang, B.-F. Zhu, and R.-B. Liu, Phys. Rev. Lett. , 256601 (2010). L. K. Werake and H. Zhao, Nature Phys. , 875 (2010). E. A. Karashtin and A. A. Fraerman, J. Phys. Condens. Matter , 165801 (2018). E. A. Karashtin, Phys. Solid State , 2189 (2017). J. Shibata, A. Takeuchi, H. Kohno, and G. Tatara, Journal of the Physical Society of Japan , 033701(5pages) (2016), http://dx.doi.org/10.7566/JPSJ.85.033701. J. Shibata, A. Takeuchi, H. Kohno, and G. Tatara, Journal of Applied Physics , 063902(2018), https://doi.org/10.1063/1.5011130. K. Sawada and N. Nagaosa, Phys. Rev. Lett. , 237402 (2005). H. Kawaguchi and G. Tatara, Phys. Rev. B , 235148 (2016). G. E. Volovik, Journal of Physics C: Solid State Physics , L83 (1987). G. Tatara, H. Kohno, and J. Shibata, Physics Reports , 213 (2008). W. C. Koehler, Journal of Applied Physics , 1078 (1965), https://doi.org/10.1063/1.1714108. K. Ishizaka, M. S. Bahramy, H. Murakawa, M. Sakano, T. Shimojima, T. Sonobe, K. Koizumi,S. Shin, H. Miyahara, A. Kimura, K. Miyamoto, T. Okuda, H. Namatame, M. Taniguchi,R. Arita, N. Nagaosa, K. Kobayashi, Y. Murakami, R. Kumai, Y. Kaneko, Y. Onose, andY. Tokura, Nat Mater , 521 (2011). O. Petrova and O. Tchernyshyov, Phys. Rev. B , 214433 (2011). G. Tatara and H. Fukuyama, Journal of the Physical Society of Japan , 104711 (2014),http://dx.doi.org/10.7566/JPSJ.83.104711., 104711 (2014),http://dx.doi.org/10.7566/JPSJ.83.104711.