Conversion Between Electron Spin and Microscopic Atomic Rotation

Home , Orbital magnetization, Spin polarization

PHYSICAL REVIEW RESEARCH 2, 023275 (2020)

Conversion between electron spin and microscopic atomic rotation

Masato Hamada1 and Shuichi Murakami 1,2 1Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan 2TIES, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8551, Japan

(Received 19 March 2020; accepted 13 May 2020; published 3 June 2020)

We theoretically investigate the microscopic mechanism of conversion between the electron spin and the microscopic local rotation of atoms in crystals. In phonon modes with angular momenta, the atoms microscopically rotate around their equilibrium positions in crystals. In a simple toy model with phonons, we calculate the spin expectation value by using the adiabatic series expansion. We show that the time-averaged spin magnetization is generated by the microscopic local rotation of atoms via the spin-orbit interaction. On the other hand, in the system with a simple vibration of atoms, time-averaged spin magnetization becomes zero due to the time-reversal symmetry. Moreover, the magnitude of the time-averaged spin magnetization depends on the inverse of the difference of instantaneous eigenenergy, and we show that it becomes smaller in band insulators with a larger gap.

DOI: 10.1103/PhysRevResearch.2.023275

I. INTRODUCTION been proposed, such as a magnetic field [9,23], a Coriolis forceinarotatingframe[24,25], circularly polarized light The conversion between magnetization and mechanical [21,22], infrared excitation [18,26], a temperature gradient rotation, such as the Einstein–de Haas effect [1] and the in the system without inversion symmetry [27], and an elec- Barnett effect [2], has been known for more than a hundred tric field in the magnetic insulator [28]. Here, one question years, and it has been focused on in spintronics recently. These arises, whether electron spins couple with a microscopic effects have been understood by means of the spin-rotation local rotation of atoms in crystals. Because the microscopic coupling [3–5], which means that mechanical rotation acts on local rotation is quite different from the macroscopic rotation electron spins as an effective magnetic field. In spintronics, studied in the Einstein–de Haas effect and the Barnett effect, mechanical generation of spin current has been theoretically whether one can apply the spin-rotation coupling to coupling proposed or experimentally reported by several means, such between the electron spins and a microscopic local rotation is as the surface acoustic wave [3,6], the twisting vibrational a crucial issue. mode in carbon nanotubes [7], and the flow of liquid metals In this paper, we theoretically propose a microscopic [8] via the spin-rotation coupling. However, the microscopic mechanism of conversion between an electron spin and a mechanism of conversion between electron spins and the microscopic local rotation of atoms in crystals. We first rotational motion of atoms in solids is not well known. introduce a simple toy model, which is a two-dimensional Recently, an angular momentum of phonons, which is a system forming the honeycomb lattice with a microscopic microscopic local rotation of atoms in crystals, was formu- local rotation of atoms. We set the rotation of atoms to be lated [9], and various phenomena related to it have been periodic in time, and we calculate the spin expectation value. energetically investigated, such as the phonon Hall effect Here we assume that the rotational motion is much slower [10–14], spin relaxation [15–17], the orbital magnetic mo- than the characteristic energy scale of electrons, and we use ment of phonons [18], and conversion between phonons and the adiabatic series expansion in two previous theories, the magnons [19,20]. In a system without inversion symmetry, Berry phase in the adiabatic process [29] and the iteration the phonons have chirality [21]. In particular, at the valleys of the unitary transformation [30,31], in order to calculate a in momentum space, the angular momentum of phonons is time-averaged spin expectation value. By using the adiabatic quantized, and such phonons are called chiral phonons. Chiral series expansion, we show that the microscopic local rotation phonons can be excited by a circularly polarized light via of atoms leads to a nonzero spin magnetization even after intervalley scattering of electrons, and this has been exper- averaging over one period in our toy model. Meanwhile, a imentally observed in monolayer tungsten diselenide [22]. simple vibration of atoms induces spin magnetization as a Methods of generation of the phonon angular momentum have function of time, but after time averaging it becomes zero. Moreover, we discuss the dependence on the parameters of time-averaged spin magnetization.

Published by the American Physical Society under the terms of the II. TOY MODEL WITH PHONON ANGULAR MOMENTUM Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) We introduce a two-dimensional tight-binding model on and the published article’s title, journal citation, and DOI. the honeycomb lattice with a microscopic local rotation of

2643-1564/2020/2(2)/023275(10) 023275-1 Published by the American Physical Society MASATO HAMADA AND SHUICHI MURAKAMI PHYSICAL REVIEW RESEARCH 2, 023275 (2020)

dij is the nearest neighbor bond vector, and sα is the Pauli † = † , † = matrices of electron spin. The operatorsc î (ci↑ cî↓ ),c î T (ˆci↑, cî↓) represent the creation and annihilation operators for an electron at the ith site, respectively. By using the Bloch wave function |n(k), this Hamiltonian Hˆ0 can be diagonalized, and the Bloch Hamiltonian Hˆ0(k)ink space is represented as

1 12 2 Hˆ0(k) = t0[FR (k) − FIm (k) ] + λv 4 3 24 23 − λR[F1(k) + F2(k) + F3(k) − F4(k) ], (4) FIG. 1. Schematic ﬁgures of (a) the honeycomb lattice with a microscopic local rotation and (b) the ﬁrst Brillouin zone. where

FR (k) = 2 cos x cos y + cos 2y, atoms, as shown in Fig. 1(a).InFig.1(b), we show the first F (k) = 2 cos x sin y − sin 2y, Brillouin zone. The two-dimensional honeycomb lattice with- Im √ out inversion symmetry has chiral phonons at valleys K and K F1(k) = 3sinx sin y, in momentum space [21]. The K and K phonons have a chi- F (k) = cos x cos y − cos 2y, rality of ±1, and in these modes, the atoms circularly rotate. 2 √ On the other hand, at the zone center , the longitudinal and F3(k) = 3sinx cos y, transverse optical phonon modes are degenerate, and these F (k) = cos x sin y + sin 2y, (5) modes do not have phonon angular momentum. However, we 4 can get the circular polarized phonon modes by superposition √ = / = / of these degenerate modes at the point. Here, we focus with x 3kxa0 2, y kya0 2. To express H0(k) in a com- 1,2,3,4,5 = on a microscopic local rotation of atoms corresponding to pact form, we introduced five Dirac matrices: σ ⊗ ,σ ⊗ ,σ ⊗ ,σ ⊗ ,σ ⊗ the phonon by superposition of two optical degenerate ( x s0 z s0 y sx y sy y sz ), where the Pauli σ modes. In Fig. 1(a), the red and blue balls represent the A matrices α and sα represent the sublattice index and the σ , and B atoms at the two sublattices, respectively, and atoms electron spin, respectively, and 0 s0 are unit matrices. We ab = a,b / A and B circularly rotate in the xy plane, with their phases also introduced their ten commutators, [ ] (2i). Next, we consider a perturbation term due to the micro- of rotation different by π. The√ primitive vectors are written scopic local rotation of atoms. Since the nearest-neighbor as a1 = a(1, 0), a2 = a(1/2, 3/2) with a lattice constant bond lengths change with time t by the microscopic local rota- a, and the vectors√ between the nearest-neighbor√ atoms are tions, we take these rotations as the modulation of the nearest- written as d1 = a0( 3/2, 1/2), d2 = a0(− 3/2, 1/2), d√3 = neighbor hopping and the Rashba parameters. We assume a0(0, −1), with the nearest-neighbor bond length a0 = a/ 3. We set the angular velocity of atoms A and B to be , that the lattice deformation u modifies the nearest-neighbor → + δ = , , the displacement vectors of atoms A and B at time t are hopping parameter as t0 t0 ta (a 1 2 3) [33]. We represented as define the Hamiltonian representing the modulation of the nearest-neighbor hopping term as = A , , =− B , , uA u0 (cos t sin t ) uB u0 (cos t sin t ) (1) Hˆ = δt (t )ˆc†cˆ . (6) and the displacement vector from atom A to atom B is h ij i j ij u = uB − uA =−u+(cos t, sin t ), (2) We assume that the modulation of the nearest-neighbor hop- A B with u+ = u + u . 0 0 ping parameter δta (a = 1, 2, 3) is proportional to the change Next, we consider the following nearest-neighbor tight- in the nearest-neighbor bond length, which is given by u(t ) · binding Hamiltonian for electrons, which is similar to the (da/a0 ). Thus, the modulation of the nearest-neighbor hop- Kane-Mele model, representing a two-dimensional topolog- ping parameter δta(t ) is given by ical insulator [32]: δ λ δ =− t0 · . = † + λ ξ † + R † × . ta(t ) u(t ) da (7) Hˆ0 t0 cˆ cˆ j v icˆ cî i cˆ (s dij)zcˆ j a2 i i a i 0 ij i 0 ij Therefore, the Hamiltonian for the modulation of the hopping (3) term for the wave vector k is represented as The first term is a nearest-neighbor hopping term, and t is a 0 ˆ , =−δ 1 + 12 hopping parameter. The second term is a staggered potential, Hh(k t ) t(F1(k) F3(k) ) cos t ξ ξ = 1 12 which we include to break inversion symmetry. i is A(B) − δt(F2(k) − F4(k) )sint, (8) +1(−1) for the A and B atoms, and λv is an on-site potential. The third term is a nearest-neighbor Rashba term, which leads with δt = δt0u+/a0. Similarly, we consider the modulation of to a spin-split band structure, λR is the Rashba parameter, the nearest-neighbor Rashba term. We assume that the Rashba

023275-2 CONVERSION BETWEEN ELECTRON SPIN … PHYSICAL REVIEW RESEARCH 2, 023275 (2020) term is modulated as of an operator Xˆ (t ). To this end, we introduce the Hamiltonian Hˆ (t,λ) such that d ˆ = λ † × ij , HR(t ) i R cî s cˆ j (9) |d | Hˆ (t,λ)|λ=0 = Hˆ (t ), (14) ij ij z ∂λHˆ (t,λ)|λ=0 = Xˆ (t ). (15) where dij is the nearest-neighbor bond vector with the modulation u(t ). This can be expanded in Taylor series in powers For example, in static systems, when the parameter λ is an of the displacement vector u, external magnetic field Bz, the quantity Xˆ is magnetization Mz. λ In particular, when the magnetic field enters the Hamiltonian = i R † × HˆR (t ) cˆ (s dij)zcˆ j Xˆ a i as a vector potential, the quantity is the orbital magnetiza- 0 ij tion. On the other hand, when the magnetic field enters the −iλ Hamiltonian as a Zeeman field, Xˆ is the spin magnetization. + R [d · u(t )]ˆc†(s × d ) cˆ a3 ij i ij z j Next, we calculate the time-averaged expectation values of 0 ij the quantity Xˆ (t ), under the time-periodic Hamiltonian Hˆ (t ) | iλR with time period T .Let n(t ) be an instantaneous eigenstate + cˆ†[s × u(t )] cˆ + O(u2 ). (10) a i z j of the band n at time t with an instantaneous eigenvalue n(t ), 0 ij where n is the band index. We assume that the time dependence of the Hamiltonian is slow enough that the state hardly The first term is the Rashba term without the microscopic lo- transits to other states, which means that the gap between cal rotation, and the second and third terms are the modulated nm bands n and m satisfies T h¯ with time period T .We Rashba terms by the displacement vector u. The modulated nm introduce the von Neumann equation ih¯∂ ρˆ (t ) = [Hˆ (t ), ρˆ (t )] Rashba term H (t ) at the wave vector k is represented as t R with the density matrixρ ˆ (t ). By using the density matrixρ ˆ (t ), ˆ , =−δλ { 23 + 3 we define the time average of the expectation value Xˆ (t )as HR (k t ) R F3(k) F1(k) T − + 24 1 [2FIm (k) 3 cos x sin y] X ≡ dt tr[ˆρ(t )Xˆ (t )]. (16) 4 T 0 + [2FR (k) + 3 cos x cos y] } cos t 23 We suppose that in the absence of the time evolution the − δλR{[2FIm (k) + F5(k)] density matrix is identical to |n(t )n(t )|, corresponding to 3 − [2FR (k) + F6(k)] the band n. The density matrixρ ˆ (t ) can be expanded to first −1 24 4 order with respect to ( nmT ) as − F3(k) − F1(k) } sin t, (11) −2 ρˆ (t ) =|n(t )n(t )|+δρˆ (t ) + O[( nmT ) ]. (17) where The first term is the projector onto the instantaneous eigen- F5(k) = cos x sin y − 2sin2y, state in the absence of the time evolution. The second term F (k) = cos x cos y + 2 cos 2y, (12) describes transitions to the excited states through the time 6 evolution. The matrix element between bands n and m is given λRu+ with δλR = . Therefore, the Hamiltonian with a periodic by 2a0 microscopic local rotation of atoms for the wave vector k is m(t )|∂t n(t ) m(t )| ρˆ (t ) |n(t )=ih¯ (18) represented as m(t ) − n(t ) from the von Neumann equation and Eq. (17). From Eq. (15), Hˆ (k, t ) = Hˆ (k) + Hˆ (k, t ) + Hˆ (k, t ). (13) 0 h R the matrix elements of Xˆ (t ) between bands n and m are written as III. ADIABATIC SERIES EXPANSION m(t )| Xˆ (t ) |n(t )=m(t )| ∂λHˆ (t,λ)|λ=0 |n(t ) Now we calculate spin expectation values for the time- = − |∂ ,λ | . periodic Hamiltonian. In general, the motions of atoms due [ n(t ) m(t )] m(t ) λ n(t ) λ=0 to phonons are much slower than those of electrons because (19) the masses of atoms are much larger than those of electrons. ∂ ˆ ,λ Therefore, in this section, in order to calculate the spin expec- Here, we used the Sternheimer identity [ λH(t )] | ,λ = ,λ − ˆ ,λ |∂ ,λ +∂ ,λ tation values we employ the adiabatic series expansion in two n(t ) [ n(t ) H(t )] λ n(t ) λ n(t ) | ,λ ways: the Berry-phase method in the adiabatic process and n(t ) . By combining these equations, Eq. (16) is rewritten as iteration of unitary transformation. The use of the adiabatic expansion is justified when the band gap is much larger than T = 1 ∂ ,λ + ,λ | , h¯. Xn dt[ λ n(t ) Bn(t )] λ=0 (20) T 0

where Bn(t,λ) = ih¯∂t n(t,λ)|∂λn(t,λ)−ih¯∂λ n(t,λ)| A. Berry-phase method in the adiabatic process ∂t n(t,λ) is the Berry curvature of a band n in the In this section, we explain the Berry-phase method to (t,λ) space. Moreover, in time-periodic systems, the calculate the dynamical response in the adiabatic process instantaneous eigenstates can be chosen to satisfy proposed in Ref. [29]. We will calculate the expectation values |n(t,λ)=|n(t + T,λ). Therefore, the time average

023275-3 MASATO HAMADA AND SHUICHI MURAKAMI PHYSICAL REVIEW RESEARCH 2, 023275 (2020) of the expectation value is rewritten as where = inst + geom, ˜ (0) τ = ˆ† ˆ τ ˆ = ε τ |ε ε | , Xn Xn Xn (21) H ( ) R0H( )R0 n( ) n(0) n(0) (29) n T 1 (0) † inst ≡ | ˆ | V˜ (τ ) = Rˆ (−ih¯∂τ )Rˆ . (30) Xn dt n(t ) X (t ) n(t ) 0 0 T 0 ˜ (0) τ T Note that time dependence of H ( ) is contained only in the 1 ε τ = dt∂λn(t,λ)|λ=0, (22) eigenvalues n( ). T 0 In the first-order approximation of the adiabatic series (0) (0) geom 1 expansion, V˜ is much smaller than H˜ (τ ), and we drop X ≡− ∂λϕ (λ)|λ= , (23) n T n 0 the V˜ (0) term. Here, to first order, we put the unitary oper- T ator Rˆ2 = Rˆ3 =···=Iˆ and calculate Rˆ1.FromEq.(28), the ϕ λ ≡ ,λ | ∂ ,λ , (0) n( ) dt n(t ) ih¯ t n(t ) (24) solution of H˜ (τ )Rˆ1(τ ) − ih¯∂τ Rˆ1(τ ) = 0 is given by 0 τ inst 1 (0) where Xn is the time average of the expectation value Rˆ1(τ ) ≡ R¯1(τ ) = exp dsH˜ (s) . (31) geom ih¯ of band n for the instantaneous eigenvalues, Xn is the 0 geometric contribution of band n due to the adiabatic time In the first-order approximation, the state at τ is represented ϕ λ evolution, and n( ) is the Berry phase associated with the as adiabatic time evolution. ψ τ = ˆ τ ψ = ˆ τ ¯ τ ψ . When the parameter λ is the external magnetic field Bz,the ( ) U ( ) (0) R0( )R1( ) (0) (32) time-averaged expectation values X correspond to the orbital geom Next, we consider the second-order approximation of the or spin magnetization. Xn in Eq. (23) is the geometrical ˜ τ = ˜ (0) τ + inst adiabatic series expansion. Here, we put H1( ) H ( ) magnetization, and X in Eq. (22) is the magnetization from (0) n V˜ , and we set the instantaneous eigenvector |ε˜ , (τ ) the snapshot of the eigenstates averaged over one period. 1 n with the instantaneous eigenvaluesε ˜1,n(τ ), H˜1(τ ) |ε˜1,n(τ )= ε τ |ε τ B. Iteration of unitary transformation ˜1,n( ) ˜1,n( ) . We perform the perturbation expansion for H˜1(τ ), andε ˜1,n(τ ) and |ε˜1,n(τ ) are represented, to the first In this section, we explain the method of the adiabatic order of ,as series expansion proposed by Berry [30,31], and then we (0) show the expectation value of each state in the second-order ε˜1,n(τ ) = εn(τ ) + εn(0)|V˜ (τ ) |εn(0) , (33) adiabatic approximation. (0) First, we consider the time-periodic Hamiltonian Hˆ (t ) εm(0)|V˜ (τ ) |εn(0) d |ε˜ , (τ )=|ε (0)+ |ε (0) . and its time-dependent Schrödinger equation, ih¯ ψ(t ) = 1 n n m ε (τ ) − ε (τ ) dt m( =n) n m Hˆ (t )ψ(t ), where ψ(t ) is a wave function. This Hamiltonian is periodic in time with a period T = 2π/, Hˆ (t + T ) = Hˆ (t ). (34) τ = Here, we introduce the rescaled time t and rewrite Note that we assume that the instantaneous eigenvalues ε (τ ) ψ = ψ τ , ˆ = ˆ τ , ˆ τ + n the functions of time as (t ) ( ) H(t ) H( ) H( for the instantaneous eigenstates |ε (τ ) have no degeneracy. 2π ) = Hˆ (τ ). Hence, the time-dependent Schrödinger equa- n We define the unitary operator Rˆ1(τ )as tion is rewritten as d ˆ τ = |ε τ ε | . ih¯ ψ(τ ) = Hˆ (τ )ψ(τ ). (25) R1( ) ˜1,n( ) ˜1,n(0) (35) dτ n Next, we introduce the time-evolution operator Uˆ , which From Eq. (28), one has is unitary. In terms of the time-evolution operator Uˆ (τ ), the † wave function is written as ψ(τ ) = Uˆ (τ )ψ(0). By substitut- Uˆ [H(τ ) − ih¯∂τ ]Uˆ ing it into the Schrödinger equation, we get † † † = ···Rˆ [Rˆ H˜ (τ )Rˆ − ih¯Rˆ ∂τ Rˆ − ih¯∂τ ]Rˆ ··· † † 2 1 1 1 1 1 2 [Uˆ (τ )Hˆ (τ )Uˆ (τ ) − ih¯Uˆ (τ )∂τUˆ (τ )]ψ(0) = 0. (26) = ··· ˆ† ˜ (1) τ + ˜ (1) τ − ∂ ˆ ··· , R2[H ( ) V ( ) ih¯ τ ]R2 (36) Here, we introduce the instantaneous eigenstate |εn(τ ) with the instantaneous eigenvalue εn(τ ), Hˆ (τ ) |εn(τ )= where εn(τ ) |εn(τ ), where n is the band index. In the following, we ˜ (1) τ = ˆ† ˜ τ ˆ = ε τ |ε ε | , will calculate Uˆ (τ )intheformUˆ (τ ) = Rˆ0(τ )Rˆ1(τ )Rˆ2(τ ) ···, H ( ) R1H1( )R1 ˜1,n( ) ˜1,n(0) ˜1,n(0) (37) where Rî(τ )(i = 1, 2, 3,...) is a unitary operator. As a n ˆ τ ˜ (1) τ = ˆ† − ∂ ˆ . zeroth-order approximation of U ( ), we define the unitary V ( ) R1( ih¯ τ )R1 (38) operator Rˆ0(τ )as Note that the time dependence of H˜ (1)(τ ) is contained only in ˆ τ = |ε τ ε | . R0( ) n( ) n(0) (27) the eigenvaluesε ˜1,n(τ ). n In the second-order approximation of the adiabatic series Then we rewrite Eq. (26), expansion, we assume that V˜ (1)(τ ) is of the order of O(2 ), † Uˆ [Hˆ (τ ) − ih¯∂τ ]Uˆ and this term is sufficiently smaller than H˜ (1)(τ ). Hence, we (1) τ = ··· ˆ† ˆ† ˆ† ˆ τ ˆ − ˆ†∂ ˆ − ∂ ˆ ˆ ··· drop the V˜ ( ) term. Therefore, to second order, we put the R2R1[R0H( )R0 ih¯ R0 τ R0 ih¯ τ ]R1R2 unitary operators Rˆ3 = Rˆ4 =···=Iˆ and calculate Rˆ2.From = ··· ˆ† ˆ† ˜ (0) τ + ˜ (0) τ − ∂ ˆ ˆ ··· , ˜ (1) τ ˆ τ − ∂ ˆ τ = R2R1[H ( ) V ( ) ih¯ τ ]R1R2 (28) Eq. (36), the solution of H ( )R2( ) ih¯ τ R2( ) 0is

023275-4 CONVERSION BETWEEN ELECTRON SPIN … PHYSICAL REVIEW RESEARCH 2, 023275 (2020) given by τ 1 (1) Rˆ2(τ ) ≡ R¯2(τ ) = exp dsH˜ (s) . (39) ih¯ 0 In the second-order approximation, the state at time τ is represented as n=1 n=2 ψ(τ ) = Uˆ (τ )ψ(0) = Rˆ0(τ )Rˆ1(τ )R¯2(τ )ψ(0). (40) n=3 n=4 Similarly, we can iterate this expansion n times, and then the state at τ is given as

ψ(τ ) = Rˆ0(τ )Rˆ1(τ ) ···R¯n(τ )ψ(0). (41) Then, the expectation value of an operator Xˆ in the second order is given by FIG. 2. Electronic energy band structure along the high- X (τ ) =ψn(τ )| Xˆ |ψn(τ ) n symmetry line in the system without phonons. We set the parameter =ψ | ¯† ˆ† ˆ† ˆ ˆ ˆ ¯ |ψ , λ = . λ = . n(0) R2R1R0XR0R1R2 n(0) (42) values to be v 0 2t0 and R 0 4t0. |ψ where n(0) is the initial state. First, the unitary operator Since we have assumedε ˜ , /h¯ 1, the phase term in ¯ τ 1 n R2( ) is rewritten as Eq. (44) rapidly oscillates, and this phase term has been ne- glected after time averaging. This time-averaged expectation −i τ R¯ (τ ) = exp ds ε˜ , (s) |ε˜ , (0)ε˜ , (0)| value is equal to Eqs. (22) and (23) in the Berry-phase method 2 h¯ 1 n 1 n 1 n 0 n in the adiabatic process. −i τ dsε˜1,n (s) = e h¯ 0 |ε˜1,n(0)ε˜1,n(0)| (43) IV. TIME-AVERAGED SPIN MAGNETIZATION n In this section, we calculate the time-averaged spin mag- ε τ |ε τ =δ from the orthogonality relation ˜1,n( ) ˜1,m( ) n,m.By netization for our toy model. The spin operator is expressed substituting Eqs. (27), (35), and (43) into Eq. (42), the ex- as pectation value is rewritten as h¯ h¯ Sˆα = σ0 ⊗ sα ≡ sˆα (α = x, y, z). (46) i τ 2 2 ε , −ε , τ = τ + − h¯ 0 ds[˜1 n (s) ˜1 m (s)] X ( ) n Xnn( ) 1 e From Eq. (15), the spin magnetization is represented as m( =n) gμ Hˆ =−mˆ · B = B Sˆ · B = μ sˆ · B, (47) (0) B spin B εm(0)|V˜ (τ ) |εn(0) h¯ × Xnm(τ ) + c.c. with g factor g = 2 and μB being the Bohr magneton. First, εn(τ ) − εm(τ ) we calculate the spin expectation values without phonons. We = Xnn(τ ) + h¯ show the electronic energy band structure for the Hamiltonian m( =n) without phonons, Eq. (4), in Fig. 2.InFig2, we assume that ε τ the Fermi energy F is zero, and the temperature is zero, i ds{ε (s)−ε (s)−h¯[γ (s)−γ (s)]} × 1 − e h¯ 0 n m n m and then this toy model is a band insulator. The staggered potential term breaks inversion symmetry, and the electron εm(τ )|(−i∂τ )εn(τ ) spin is split by the Rashba spin-orbit interaction term. In the × Xnm(τ ) + c.c. , (44) εn(τ ) − εm(τ ) following we consider the spin expectation values below the Fermi energy εF (=0). τ =ε τ | |ε τ γ τ = where Xnm( ) n( ) Xˆ m( ) and n( ) We introduce the operator = i(σ0 ⊗ sy )K, where K is εn(τ )|(−i∂τ )εn(τ ) is the Berry connection. The ﬁrst term the complex-conjugate operator, and this corresponds to the shows the instantaneous expectation value at time τ, and the time-reversal operator in static systems without phonons. The second term is proportional to the inverse time period 1/T . relations between the operator and matrices are given by a −1 a ab −1 ab The time-averaged expectation value X¯n is given as = , =− , with a, b = 1, 2, 3, 4, 5. |ε 2π From Eq. (4), the Hamiltonian H0(k) and its eigenstate n(k) 1 ε −1 = X¯ = dτ X τ with eigenenergy n(k) are represented as H0(k) n π ( ) n 2 0 H0(−k) and |εn(k)=|εn(−k). Since the spin operator Sˆα ⎡ −1 satisﬁes Sˆα =−Sˆα, the spin expectation values Sˆ (k) 2π n 1 ⎣ are an odd function of the wave vector k in our toy model dτ Xnn(τ ) + h¯ Xnm(τ ) 2π without phonons, and the total spin expectation values is zero 0 m( =n) ⎤ due to cancellation between k and −k. ε τ | − ∂ |ε τ Next, we show the spin expectation values with the mi- × m( ) ( i τ ) n( ) + . .⎦. ε τ − ε τ c c (45) croscopic local rotation of atoms. By applying Eqs. (21)– n( ) m( ) (23) to our toy model, the time-averaged spin magnetizations

023275-5 MASATO HAMADA AND SHUICHI MURAKAMI PHYSICAL REVIEW RESEARCH 2, 023275 (2020) are given by ⎡ T occ 1 ⎣ M = dt ε (k, t )| ∂ Hˆ | = |ε (k, t ) spin VT n B B B 0 n 0 k n ⎤ ε (k, t )| ∂ Hˆ | = |ε (k, t )ε (k, t )|(−ih¯∂ )ε (k, t ) + n B B B 0 m m t n + c.c.⎦ ε (k, t ) − ε (k, t ) m( =n) n m 1 2π occ = dτ minst (τ ) + mgeom (τ ) , (48) 2π n n 0 n with and μ B ,τ =ε ,τ | − ∂ ε ,τ minst (τ ) = sˆ (k,τ), (49) Anm(k ) n(k ) ( i τ ) m(k ) n V nn k =εn(−k,τ)|(i∂τ )εm(−k,τ) μ h¯sˆ (k,τ)A (k,τ) geom τ = B nm mn + . . , =− − ,τ . mn ( ) c c Anm( k ) (55) V εn(k,τ) − εm(k,τ) k m( =n) By substituting these relations into Eq. (48), the instantaneous (50) inst τ spin magnetization mn ( ) becomes zero since its sum over the first Brillouin zone cancels between k and −k at time where |ε (k, t ) is the instantaneous eigenstate with n τ. On the other hand, the geometrical spin magnetization instantaneous eigenenergy ε (k, t ) for the snapshot n mgeom (τ ) can be nonzero since it is an even function of the Hamiltonian in momentum space Hˆ (k, t ), sˆ (k,τ) = n nm wave vector k at time τ. Therefore, only the geometrical spin ε (k,τ)| sˆ |ε (k,τ) is the instantaneous matrix element n m magnetization contributes to the time-averaged spin magneti- for spin between bands n and m at time τ = t, and zation. Amn(k,τ) =−i εm(k,τ)|∂τ εn(k,τ) is the Berry connection geom We calculate the geometrical spin magnetization mα, for defined by the instantaneous eigenstates between bands n n each band below the Fermi energy ε (=0), and it is nonzero, and m. Here, the summation over n represents that over F inst τ as shown in Figs. 3(a)–3(c). In this case, the generators of the occupied bands. mn ( ) is the instantaneous spin geom the space-time symmetries are {C |T/3} and {mτ m |T/2}, magnetization for the energy band n at time τ; m (τ )is 3z x n where {C |T/3} is the threefold rotation around the z axis, the geometrical spin magnetization in the adiabatic process at 3z followed by T/3 time translation, and {mτ m |T/2} is a time τ, and this term is proportional to the phonon frequency x product of the operation, the mirror reflection with respect . We note that whiles ˆnm and Aˆnm (m = n) depend on a geom to the yz plane, and the mτ operation, which represents gauge choice, m (τ ) is gauge invariant. n inversion of the sign of time τ, followed by T/2 time Here, we apply the operator to our toy model with phonons. From Eq. (13), the time-dependent Hamiltonian is rewritten as (a) H(k,τ) = H0(k) + Hc(k) cos τ + Hs(k)sinτ. (51) (d) Therefore, the time-dependent Hamiltonian satisfies − (b) H(k,τ) 1 = H(−k,τ) (52)

−1 because Hc(s)(k) = Hc(s)(−k). Thus, within our dynamical model, the operator does not ﬂip the sign of τ, and (c) therefore, to avoid confusion, we call it a operation instead of time-reversal operation. Hence, the instantaneous eigenstates |εn(k,τ) and the instantaneous eigenenergy εn(k,τ) satisfy

|εn(k,τ)=|εn(−k,τ) ,εn(k,τ) = εn(−k,τ). (53) FIG. 3. Geometrical spin magnetization for one cycle in the From these relations, the instantaneous matrix elements for system with microscopic rotation versus the rescaled time τ for (a) x, spin sˆnm(k,τ) and the Berry connection Anm(k,τ)aregiven (b) y,and(c)z components. (d) Evolution of spin magnetization by for one cycle in the xy plane. The small circles show the values at τ = 0, and the arrows show the time-evolution direction. We set the sˆnm(k,τ) =εn(k,τ)| sˆ |εm(k,τ) parameters as λv = 0.2t0,λR = 0.4t0,δt = 0.1t0,andδλR = 0.1λR. In (a)–(d), the blue line shows the lowest band (n = 1), the red line =ε (−k,τ)| (−sˆ) |ε (−k,τ) n m shows the second-lowest band (n = 2), and the black line shows the =−sˆnm(−k,τ) (54) sum of the bands below the Fermi energy.

023275-6 CONVERSION BETWEEN ELECTRON SPIN … PHYSICAL REVIEW RESEARCH 2, 023275 (2020) translation. We note that the dynamical symmetry is defined using the discrete time-translation-invariance Floquet systems (a) (d) [34,35]. These symmetries, {C3z|T/3} and {mτ mx|T/2}, = − π σ ⊗ are represented by unitary operators U1 exp[ i 3 ( 0 sz )] and U2 = (σ0 ⊗ sx ), respectively. Here, the time-dependent Hamiltonian satisfies (b) 2π ˆ ,τ −1 = ˆ ,τ + , U1H(k )U1 H Rk (56) 3 (c) ˆ , ,τ −1 = ˆ , − , −τ + π , U2H(kx ky )U2 H(kx ky ) (57) where R is the threefold√ rotational√ matrix around the z axis, Rk = (−kx/2 − 3ky/2, 3ky/2 − kx/2). From these relations, the instantaneous matrix elements for spin sˆnm(k,τ) and the Berry connection A (k,τ)aregivenby nm FIG. 4. Geometrical spin magnetization for one cycle in the 2π system with the simple vibration versus rescaled time τ for (a) x, sˆ (k,τ) = R−1sˆ Rk,τ + , (58) nm nm 3 (b) y,and(c)z components. (d) Evolution of spin magnetization for one cycle in the xy plane. The small circles show the values at ,τ = ,τ + 2π , τ = 0, and the arrows show the time-evolution direction. We set the Anm(k ) Anm Rk 3 (59) parameters as λv = 0.2t,λR = 0.4t0,δt = 0.1t0, and δλR = 0.1λR. = and In (a)–(d), the blue lines show the lowest band (n 1), the red lines ⎛ ⎞ ⎛ ⎞ show the second-lowest band (n = 2), and the black lines show the sˆx,nm(kx, ky,τ) −sˆx,nm(kx, −ky, −τ + π ) sum of the bands below the Fermi energy. ⎝ ⎠ ⎝ ⎠ sˆy,nm(kx, ky,τ) = sˆy,nm(kx, −ky, −τ + π ) , (60) sˆ , (k , k ,τ) sˆ , (k , −k , −τ + π ) z nm x y z nm x y Next, we compare two cases: a simple vibration of atoms Anm(kx, ky,τ) = Anm(kx, −ky, −τ + π ). (61) and a microscopic local rotation of atoms. For this compari- son, we consider a case where atoms A and B mutually vibrate Hence, the geometrical spin magnetization satisfies with the displacement vector u given by π geom −1 geom 2 m (τ ) = R m τ + , (62) u = u − u =−u+(cos τ,0). (65) n n 3 B A geom τ =− geom −τ + π , Then the time-dependent Hamiltonian (13) is rewritten as mx,n ( ) mx,n ( ) (63) H(k,τ) = H0(k) + Hc(k) cos τ. We calculate the geometric mgeom (τ ) = mgeom (−τ + π ). (64) spin magnetization in this system with the simple vibration of y(z),n y(z),n atoms, and it is shown in Fig. 4.InFigs.4(a)–4(c),weshow The geometrical spin magnetization of the z components the x, y, and z components of geometrical spin magnetization oscillates with a T/3 period. On the other hand, the geo- versus time τ, respectively. In Fig. 4(d), we show the evolution metrical spin magnetization in the xy plane rotates, with its of the geometrical spin magnetization for one period in the xy trajectory forming a trianglelike form for one cycle, as shown plane. Similar to the results with the microscopic local rota- in Fig. 3(d).FromEqs.(63) and (64), at τ = π/2, 3π/2, tion of atoms, the geometrical spin magnetization can be finite the geometrical spin magnetization for each band lies in at time τ. Meanwhile, the time-averaged spin magnetization the yz plane. Moreover, from the {C3z|T/3} symmetry, the vanishes. geometrical spin magnetization in the xy plane rotates once We discuss the symmetry reason for why the time-averaged within the period T , and its time average over one period van- spin magnetization vanishes. In our toy model with a simple ishes. In addition, when the direction of the microscopic local vibration of atoms, the Hamiltonian and instantaneous eigen- rotation is reversed, the geometrical spin magnetizations are states satisfy not only Eqs. (52) and (53) but also the relations reversed. Therefore, the microscopic local rotation works like −1 an effective magnetic field, and the spin magnetization along H(k,τ) = H(−k, −τ ), (66) the rotational axis is proportional to the phonon frequency. |εn(k,τ)=|εn(−k, −τ ) (67) V. DISCUSSION because H(k,τ) = H(k, −τ ). The instantaneous matrix ele- ,τ ,τ In this section, we discuss how the microscopic local ment for spin sˆnm(k ) and the Berry connection Anm(k ) rotation couples with electron spins. First, when the Rashba satisfy λ parameter R is zero in our toy model, the time-averaged sˆ (k,τ) =−sˆ (−k, −τ ), (68) spin magnetization is zero since the energy bands are spin nm nm degenerate and the instantaneous matrix element for spin Anm(k,τ) = Anm(−k, −τ ). (69) sˆnm(k,τ) becomes zero for arbitrary wave vector, time, and bands. Therefore, within our model, the spin-orbit coupling is From Eqs. (54), (55), (68), and (69), the geometrical spin required for spin polarization induced by microscopic atomic magnetization is an even function of the wave vector k and, at rotations. the same time, an odd function of time τ, and therefore, the

023275-7 MASATO HAMADA AND SHUICHI MURAKAMI PHYSICAL REVIEW RESEARCH 2, 023275 (2020)

(a) (a) (d) (b) (e) (d) (b) (c)

(c)

(b) (d) n=4 3 n=4 0.2 3 Ē/t₀ Ē/t₀ : FIG. 5. Geometrical spin magnetization for one cycle in the : 0.1 δλ = system with microscopic rotation as R 0 versus the rescaled n=3 n=3 time τ for (a) x,(b)y,and(c)z components. (d) Evolution of spin 0 0 0 0 magnetization for one cycle in the xy plane. The small circles show n=2 n=2 τ = 0, and the arrows show the time-evolution direction. We set -0.1 the parameters as λ = 0.2t ,λ = 0.4t ,δt = 0.1t ,andδλ = 0. -3 -0.2 -3 v 0 R 0 0 R n=1 n=1 Time-averaged energy Time-averaged In (a)–(d), the blue lines show the lowest band (n = 1), the red lines energy Time-averaged show the second-lowest band (n = 2), and the black lines show the sum of the bands below the Fermi energy. (c) (e) 0.4 3 n=4 3 n=4 Ē/t₀ 0.4 Ē/t₀ : : geometrical spin magnetization has opposite signs between τ and −τ. In this case, the generators of the space-time n=3 n=3 0 0 0 0 symmetry of the system are {} and {m |T/2}, which is x n=2 n=2 the mirror reﬂection with respect to the yz plane mx and / the time translation by half the period T 2. This symmetry -3 -0.4 -3 { | / } = σ ⊗ n=1 n=1 -0.4 Time-averaged energy Time-averaged mx T 2 is represented by the unitary operator U3 0 sx. energy Time-averaged ˆ , ,τ −1 = ˆ − , ,τ + π From U3H(kx ky )U3 H( kx ky ), the instantaneous matrix element for spin and the Berry connection satisfy λ the following relations: FIG. 6. Dependence on the staggered potential v of geometrical spin magnetization. (a) Geometrical spin magnetization with or without the modulated Rashba parameter δλR versus staggered potential sˆx,nm(kx, ky,τ) = sˆx,nm(−kx, ky,τ + π ), (70) λv. The blue line shows the geometrical spin magnetization without the modulated Rashba parameter δλR = 0, and the red line shows it sˆy(z),nm(kx, ky,τ) =−sˆy(z),nm(−kx, ky,τ + π ), (71) with δλR = 0.04t. (b)–(e) Time-averaged energy expectation values and the geometrical spin magnetization. For (b)–(e), the color map , ,τ = − , ,τ + π . Anm(kx ky ) Anm( kx ky ) (72) shows the time-averaged geometrical spin magnetization on the high-symmetry line. We set the parameters as λR = 0.4t0,δt = 0.1t0, By combining these relations for the operator [Eqs. (68) and (b) λv = 0.2t0,δλR = 0, (c) λv = 0.2t0,δλR = 0.04t0,(d)λv = { | / } and (69)] and symmetry operator mx T 2 [Eqs. (70)–(72)], t0,δλR = 0, and (e) λv = t0,δλR = 0.04t0. geom the geometrical spin magnetization mα,n (τ ) satisﬁes

geom τ =− geom −τ =− geom −τ + π , mx,n ( ) mx,n ( ) mx,n ( ) (73) along the microscopic rotational axis becomes nonzero, as geom τ =− geom −τ = geom −τ + π . my(z),n( ) my(z),n( ) my(z),n( ) (74) shown in Fig. 5(c). Figures 5(a), 5(b), and 5(d) show that the geometrical spin magnetization in the xy plane isotropically At τ = 0,π, the geometrical spin magnetization becomes rotates for one cycle. When the modulation of the Rashba zero, and at τ = π/2, 3π/2, the x component of the geo- term is absent, the electron spin couples with the microscopic metrical spin magnetization becomes zero, while the y and local rotation indirectly via the Rashba spin-orbit term, and z components are symmetric with respect to τ = π/2, 3π/2. the magnitude of the geometrical spin magnetization becomes This symmetry analysis coincides with the numerical results smaller than that with the modulation of the Rashba term shown in Fig. 4. Therefore, the time-averaged spin magnetiza- (Fig. 3). Therefore, the microscopic local rotation of atoms tion vanishes for a simple vibration of atoms by time-reversal couples with the electron spins via the Rashba spin-orbit symmetry. interaction, and then the time-averaged spin magnetization Next, we discuss an effect of the modulation of the Rashba along the rotational axis is generated. term δλR. In our model, even without the modulation of the Next, we discuss how the spin magnetization depends on Rashba term δλR = 0, the time-averaged spin magnetization the gap. The geometrical spin magnetization can be rewritten

023275-8 CONVERSION BETWEEN ELECTRON SPIN … PHYSICAL REVIEW RESEARCH 2, 023275 (2020)

(a) ×10-2 (b) ×10-2 dynamically excite electrons across the gap. The excitation depends on spin via the Rashba term, and as a result, the n=1 1.6 n=1 2 n=2 n=2 system acquires a spin magnetization. Figures 6(b)–6(e) show sum 1.2 sum the distribution of the contribution to the spin magnetization in Eq. (76) as a color map. When the staggered potential 1 0.8 λv is small [Figs. 6(b) and 6(c)], the gap is small, and the 0.4 time-averaged geometrical spin magnetization around the K 0 and K points for n = 2 mainly contributes to the spin mag- 0 netization. On the other hand, when the staggered potential 0 0.1 00.010.02 λv becomes larger [Figs. 6(d) and 6(e)], the time-averaged geometrical spin magnetization around the K and K points cancels between the n = 1 and n = 2 bands, and then the FIG. 7. Dependence on the modulated hopping parameter δt and spin magnetization becomes small. Moreover, since the ge- the modulated Rashba parameter δλR of the geometrical spin magnetization. (a) Geometrical spin magnetization without the modulated ometrical spin magnetization is of the second order of the displacement by phonons, the spin polarization is generated Rashba parameter δλR = 0 versus the modulated hopping parameter δt 2. (b) Geometrical spin magnetization without the modulated hop- by interactions between the dynamically excited states for ping parameter δt = 0 versus the modulated Rashba parameter δλ2 . electrons. This clearly shows that the dynamically excited R We set the parameters as λv = 0.2t0,λR = 0.4t0. The blue lines show electrons across the gaps around the K and K points generate the lowest band (n = 1), the red lines show the second-lowest band spin polarization. Therefore, as the band gap becomes larger, (n = 2), and the black lines show the sum of the bands below the the geometrical spin magnetization becomes smaller. Fermi energy. VI. CONCLUSION as In summary, we have theoretically shown that a micro- geom τ mn ( ) scopic local rotation of atoms induces electron spins in the n system with Rashba spin-orbit interaction. Using our toy occ sˆnm(k,τ)Amn(k,τ) model, which is a two-dimensional tight-binding model on = h¯ + c.c. the honeycomb lattice with Rashba spin-orbit interaction and εn(k,τ) − εm(k,τ) k n m( =n) phonons, we calculate the time-averaged spin magnetization occ unocc using the adiabatic approximation, which is valid because sˆ (k,τ)A (k,τ) = h¯ nm mn + c.c. . (75) the phonon frequency is much smaller than the frequency εn(k,τ) − εm(k,τ) k n m scale from the electron bandwidth. The time-averaged spin magnetization has two terms: the instantaneous spin mag- In the second line, we separate the sum over all the bands m netization and the geometrical spin magnetization. The in- into that over the occupied bands and that over the unoccupied stantaneous spin magnetization is given by the instantaneous bands, and the former sum vanishes. From Eq. (75), since eigenstates for the snapshot Hamiltonian, and this term van- the geometrical spin magnetization is proportional to the − ishes in the snapshot system with time-reversal symmetry. inverse of the difference in energy [ (k,τ) − (k,τ)] 1, n m The geometrical spin magnetization is represented in terms the magnitude of the geometrical spin magnetization be- of the product of the Berry connection and the instantaneous comes larger when the gap is small, namely, when the matrix element for spin, and it is proportional to the phonon staggered potential λ is small, as shown in Fig. 6(a).In v frequency . When the system has both the Rashba spin- Figs. 6(b)–6(e), we show the time-averaged energy expecta- orbit interaction and the microscopic local rotation of atoms, tion value E¯ and the time-averaged geometrical spin mag- geom the geometrical spin magnetization becomes nonzero. The netization mn,z on the high-symmetry line, where E¯n(k) = π geometrical spin magnetization for each band perpendicular 1 2 dτ ε k,τ | Hˆ k,τ |ε k,τ 2π 0 n( ) ( ) n( ) is the time-averaged to the rotational axis has a finite value as a function of time, energy expectation value and the time-averaged geometrical geom and its time average over one period is zero. Meanwhile, spin magnetization mz,n (k) is derived from Eq. (48)as the time-averaged spin magnetization along the rotational geom mn (k) axis has a finite value, and it is proportional to the phonon μ h¯ 2π sˆ (k,τ)A (k,τ) frequency . On the other hand, when the motions of atoms = B dτ nm mn + c.c. . are vibrations, the geometrical spin magnetization for each 2π ε (k,τ) − ε (k,τ) 0 m( =n) n m band has a finite value as a function of time, but the time- (76) averaged spin magnetization vanishes due to the time-reversal symmetry. Therefore, we have shown that the electron spins Moreover, we show the dependence on the modulated hop- couple to the microscopic local rotation of atoms via the ping and the modulated Rashba parameters of the geometrical spin-orbit interaction, and the time-averaged spin magnetiza- spin magnetization in Fig. 7.FromFigs.7(a) and 7(b),the tion along the rotational axis is proportional to the phonon geometrical spin magnetization is proportional to the square frequency. of the modulated parameters δt and δλR in the lowest order. In the present paper, we studied phonons at the point. We Thus, the physical picture of the present effect is as follows. expect that phonons at general k points lead to a qualitatively Our model is insulating in the static case, and the phonons similar result. This is because the effect which we discuss is

023275-9 MASATO HAMADA AND SHUICHI MURAKAMI PHYSICAL REVIEW RESEARCH 2, 023275 (2020) of the second order in the phonon displacement. Therefore, also note that a related phenomenon was studied for an orbital phonons with wave vector k can give a uniform magnetization magnetization in Ref. [29]. It was shown that a geometric in the second order because k + (−k) = 0. Detailed behaviors orbital magnetization appears in a periodic adiabatic process. of the spin magnetization sensitively depend on the systems It does not require spin-orbit coupling. Meanwhile, our result and phonons considered and are left as future problems. shows that generation of spin magnetization via rotations of The generation of spin polarization due to rotational phonons requires spin-orbit coupling. motion looks similar to the effect due to spin-rotation coupling discussed in Refs. [3–5]. Nonetheless there ACKNOWLEDGMENTS is one important difference between our work and the effects due to spin-rotation coupling. The effect in the We thank K. Saito for useful discussions. This work was present requires spin-orbit coupling, and the resulting spin partly supported by a Grant-in-Aid for Scientiﬁc Research on polarization depends on the spin-orbit coupling, whereas the Innovative Area, “Nano Spin Conversion Science” (Grant No. spin-rotation coupling does not depend much on spin-orbit JP26103006), by MEXT Elements Strategy Initiative to Form coupling. Thus, the result in the present paper cannot be Core Research Center (TIES), and also by JSPS KAKENHI attributed to the spin-rotation coupling in Refs. [3–5]. We Grants No. JP17J10342 and No. JP18H03678.

[1] A. Einstein and W. J. de Haas, Verh. Dtsch. Phys. Ges. 17, 152 [18] D. M. Juraschek and N. A. Spaldin, Phys. Rev. Mater. 3, 064405 (1915). (2019). [2] S. J. Barnett, Phys. Rev. 6, 239 (1915). [19] J. Holanda, D. S. Maior, A. Azevedo, and S. M. Rezende, [3] M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa, Phys. Nat. Phys. 14, 500 (2018). Rev. B 87, 180402(R) (2013). [20] S. C. Guerreiro and S. M. Rezende, Phys.Rev.B92, 214437 [4] M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Phys. Rev. Lett. (2015). 106, 076601 (2011). [21] L. Zhang and Q. Niu, Phys. Rev. Lett. 115, 115502 (2015). [5] M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Phys.Rev.B [22] H. Zhu, J. Yi, M.-Y. Li, J. Xiao, L. Zhang, C.-W. Yang, R. A. 84, 104410 (2011). Kaindl, L.-J. Li, Y. Wang, and X. Zhang, Science 359, 579 [6] D. Kobayashi, T. Yoshikawa, M. Matsuo, R. Iguchi, S. (2018). Maekawa, E. Saitoh, and Y. Nozaki, Phys. Rev. Lett. 119, [23] D. M. Juraschek, M. Fechner, A. V.Balatsky, and N. A. Spaldin, 077202 (2017). Phys. Rev. Mater. 1, 014401 (2017). [7] M. Hamada, T. Yokoyama, and S. Murakami, Phys.Rev.B92, [24] Y.-T. Wang, P.-G. Luan, and S. Zhang, New J. Phys. 17, 073031 060409(R) (2015). (2015). [8] R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo, S. [25] Y. Liu, Y. Xu, S.-C. Zhang, and W. Duan, Phys.Rev.B96, Okayasu, J. Ieda, S. Takahashi, S. Maekawa, and E. Saitoh, 064106 (2017). Nat. Phys. 12, 52 (2016). [26] T. F. Nova, A. Cartella, A. Cantaluppi, M. Först, D. Bossini, [9] L. Zhang and Q. Niu, Phys. Rev. Lett. 112, 085503 (2014). R. V. Mikhaylovskiy, A. V. Kimel, R. Merlin, and A. Cavalleri, [10] C. Strohm, G. L. J. A. Rikken, and P. Wyder, Phys. Rev. Lett. Nat. Phys. 13, 132 (2017). 95, 155901 (2005). [27] M. Hamada, E. Minamitani, M. Hirayama, and S. Murakami, [11] A. V. Inyushkin and A. N. Taldenkov, JETP Lett. 86, 379 Phys. Rev. Lett. 121, 175301 (2018). (2007). [28] M. Hamada and S. Murakami, Phys. Rev. B 101, 144306 [12] K. Sugii, M. Shimozawa, D. Watanabe, Y. Suzuki, M. Halim, (2020). M. Kimata, Y. Matsumoto, S. Nakatsuji, and M. Yamashita, [29] L. Trifunovic, S. Ono, and H. Watanabe, Phys. Rev. B 100, Phys. Rev. Lett. 118, 145902 (2017). 054408 (2019). [13] Y. Kagan and L. A. Maksimov, Phys. Rev. Lett. 100, 145902 [30] M. V. Berry, Proc. R. Soc. London, A 414, 31 (1987). (2008). [31] G. Rigolin, G. Ortiz, and V. H. Ponce, Phys. Rev. A 78, 052508 [14] L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys.Rev.Lett.105, (2008). 225901 (2010). [32] C. L. Kane and E. J. Mele, Phys.Rev.Lett.95, 146802 (2005). [15] D. A. Garanin and E. M. Chudnovsky, Phys.Rev.B92, 024421 [33] K.-i. Sasaki and R. Saito, Prog. Theor. Phys. Suppl. 176, 253 (2015). (2008). [16] J. J. Nakane and H. Kohno, Phys.Rev.B97, 174403 (2018). [34] T. Morimoto, H. C. Po, and A. Vishwanath, Phys.Rev.B95, [17]S.Streib,H.Keshtgar,andG.E.W.Bauer,Phys. Rev. Lett. 121, 195155 (2017). 027202 (2018). [35] S. Xu and C. Wu, Phys.Rev.Lett.120, 096401 (2018).

023275-10