Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

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Citation Sodemann, Inti, and Liang Fu. “Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials.” Physical Review Letters 115, no. 21 (November 2015). © 2015 American Physical Society

As Published http://dx.doi.org/10.1103/PhysRevLett.115.216806

Publisher American Physical Society

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Citable link http://hdl.handle.net/1721.1/100020

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 115, 216806 (2015) PHYSICAL REVIEW LETTERS 20 NOVEMBER 2015

Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

Inti Sodemann and Liang Fu Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 11 August 2015; published 20 November 2015) It is well known that a nonvanishing Hall conductivity requires broken time-reversal symmetry. However, in this work, we demonstrate that Hall-like currents can occur in second-order response to external electric fields in a wide class of time-reversal invariant and inversion breaking materials, at both zero and twice the driving frequency. This nonlinear Hall effect has a quantum origin arising from the dipole moment of the Berry curvature in momentum space, which generates a net anomalous velocity when the system is in a current-carrying state. The nonlinear Hall coefficient is a rank-two pseudotensor, whose form is determined by point group symmetry. We discus optimal conditions to observe this effect and propose candidate two- and three-dimensional materials, including topological crystalline insulators, transition metal dichalcogenides, and Weyl semimetals.

DOI: 10.1103/PhysRevLett.115.216806 PACS numbers: 73.43.-f, 03.65.Vf, 72.15.-v, 72.20.My

Introduction.—The Hall conductivity of an electron nonequilibrium current-carrying distribution is not sym- system whose Hamiltonian is invariant under time-reversal metric under k → −k, the integral of the Berry curvature symmetry is forced to vanish. Crystals with sufficiently low weighed by it can be finite, leading to a net anomalous symmetry can have resistivity tensors which are aniso- velocity and hence a transverse current. tropic, but Onsager’s reciprocity relations [1] force the Our study builds upon a seminal work by Moore and conductivity to be a symmetric tensor in the presence of Orenstein [3], which predicted a dc photocurrent in time-reversal symmetry. Hence, when the electric field is quantum wells without inversion symmetry due to the along its principal axes the current and the electric field are anomalous velocity associated with the Berry phase. The collinear, at least to the first order in electric fields. quantum nonlinear Hall effect presented here can be However, this constraint is only about the linear response regarded as a generalization of this effect. We predict that and does not necessarily enforce the full current to flow an oscillating electric field can generate a transverse current collinearly with the local electric field. at both zero and twice the frequency in two- and three- In this Letter we study a special type of such nonlinear dimensional materials with a large class of crystal point Hall-like currents. We will demonstrate that metals without group symmetries. In particular, the second harmonic inversion symmetry can have a nonlinear Hall-like current generation is a distinctive signature that may facilitate arising from the Berry curvature in momentum space. The the experimental detection of the quantum nonlinear Hall conventional Hall conductivity can be viewed as the zero- effect. Additionally, the effect does remain finite in the dc order moment of the Berry curvature over occupied states, limit of the applied electric field. namely, as an integral of the Berry curvature within the General theory.—The electric current density is given by metal’s Fermi surface. The effect we discuss here is the integral of the physical velocity of the electrons va determined by a pseudotensorial quantity that measures weighed by their occupation function fðkÞ: a first-order moment of the Berry curvature over the Z occupied states, and hence we call it the Berry curvature ja ¼ −e fðkÞva: ð1Þ dipole. This nonlinear Hall effect has a quantum origin k arising from the anomalous velocity of Bloch electrons generated by the Berry curvature [2], but it is not expected For simplicity, we imagine a single bandR systemR but allow it d d to be quantized. to be two or three dimensional: k ≡ d k=ð2πÞ . The In a time-reversal invariant system, the Berry curvature is velocity contains two contributions, namely, the group odd in momentum space, ΩaðkÞ¼−Ωað−kÞ, and hence its velocity of the electron wave and the anomalous velocity integral weighed by the equilibrium Fermi distribution is arising from the Berry curvature [2] (ℏ ¼ 1): forced to vanish, because Kramers pair states at k and −k v ∂ ϵ k ε Ω k_ ; are equally occupied. However, the second-order response a ¼ a ð Þþ abc b c ð2Þ is determined by the integral of the Berry curvature ∂ ∂=∂ ϵ Ω evaluated in the nonequilibrium distribution of electrons where a ¼ ka , and b are the energy dispersion and computed to first order in the electric field. Since the the Berry curvature of the electrons in question:

Ωa ≡ εabc∂bAc;Ac ≡ −ihukj∂cjuki: ð3Þ The presence of the factor ∂bf0 will guarantee that only states close to the Fermi surface will contribute to the Within the Boltzmann picture of transport, the canonical integral in the low temperature limit, so that this response is momentum of electrons changes in time in response to the a Fermi liquid property [6]. Equation (8) can be rewritten as external electromagnetic fields. In the absence of external follows: Z magnetic fields, the change of momentum is e3τ χ −ε f ∂ Ω : abc ¼ adc 0ð b dÞ ð9Þ _ 2ð1 þ iωτÞ k kc ¼ −eEcðtÞ; ð4Þ This expression [Eq. (9)] for the nonlinear conductivity E t E eiωt E ∈ C where cð Þ¼Ref c g, with c the driving tensor, χabc, is the first main result of this work. It shows electric field which oscillates harmonically in time but that χabc is proportional to the dipole moment of the Berry is uniform in space. In the relaxation time approximation, curvature over the occupied states, defined as the Boltzmann equation for the distribution of electrons Z is [4] Dab ¼ f0ð∂aΩbÞ: ð10Þ k −eτEa∂af þ τ∂tf ¼ f0 − f; ð5Þ It is interesting to note that this tensor is dimensionless in where f0 is the equilibrium distribution in the absence of three dimensions. At frequencies above the width of the ωτ ≫ 1 external fields. We are interested in computing the response Drude peak and below the interband transition χ to second order in the electric field; hence, we expand the threshold, the prefactor in abc becomes independent of the scattering time, so that χabc directly measures the quantum distribution up to second order: f ¼ Reff0 þ f1 þ f2g, n geometry of the Bloch states. In the dc limit or for linearly where the term fn is understood to vanish as E . One finds a recursive structure: polarized electric fields, the Berry curvature dipole term always produces a current that is orthogonal to the electric field jaEa ¼ 0 [7]. ω iωt ω eτEa∂af0 f1 ¼ f1 e ;f1 ¼ ; To close this section, we wish to remark that there exist 1 þ iωτ additional second-order corrections to the current arising eτ 2EE ∂ f 0 2ω 2iωt 0 ð Þ a b ab 0 from modifications to Eq. (2) that are intrinsic to the band f2 ¼ f2 þ f2 e ;f2 ¼ ; 2ð1 þ iωτÞ structure, containing no powers of the scattering time τ [8]; 2 however, these contributions vanish for time-reversal invari- 2ω ðeτÞ EaEb∂abf0 f2 ¼ : ð6Þ ant systems. Other type of rectifications might arise in 2ð1 þ iωτÞð1 þ 2iωτÞ systems with an inversion asymmetric scattering rate, 0 0 2ω 2iωt namely, when the scattering from k to k has a different Writing the current as ja Re ja ja e , one obtains ¼ f þ g rate than that from −k to −k0, which produces a kind of Z Z ratchet effect [9]. These semiclassical Berry-phase indepen- e2 j0 ε Ω Efω − e f0∂ ϵ k ; dent contributions are distinguished from the quantum a ¼ 2 abc b c 1 2 a ð Þ Zk Zk nonlinear Hall effect discussed in this work because they e2 are expected to scale as τ2. j2ω ε Ω E fω − e f2ω∂ ϵ k : a ¼ abc b c 1 2 a ð Þ ð7Þ — 2 k k Berry curvature dipole in three dimensions. Let us explore the constraints imposed by crystal point sym- 0 D The term ja describes a rectified current while the term metries on the Berry curvature dipole tensor ab. A point 2ω symmetry is described by an orthogonal matrix S. Because ja describes the second harmonic. The second terms that appear in Eq. (7) are completely semiclassical and do not the Berry curvature is a pseudovector, the Berry curvature require the presence of Berry curvature. However, within dipole transforms as a pseudotensor. Hence, crystal sym- the approximation of a constant τ, one finds that these metries impose constraints of the form nonlinear terms are proportional to the integral of a three- D ¼ detðSÞSDST: ð11Þ index tensor, ∂aϵðkÞ∂bcf0ðkÞ, which is odd under time reversal and, hence, they are forced to vanish. Therefore, To determine which components of this tensor are nonzero, the only surviving terms are those associated with the Berry 0 2ω it is convenient to decompose it into symmetric and curvature. By writing ja ¼ χ EbEc, ja ¼ χ EbEc, one T abc abc antisymmetric parts, D ¼ðD D Þ=2, which transform has [5] independently under symmetry operations. The antisym- Z e3τ metric part of a pseudotensor transforms as a vector, as can χ ε ∂ f Ω : abc ¼ adc ð b 0Þ d ð8Þ be verified from Eq. (11). The components of this vector 2ð1 þ iωτÞ k d ≡ ϵ D− =2 can be taken to be a abc bc . Therefore, for it to be

216806-2 week ending PRL 115, 216806 (2015) PHYSICAL REVIEW LETTERS 20 NOVEMBER 2015 nonzero the crystal must have a polar axis. From the 32 positive eigenvalue and the orientation of the corresponding crystallographic point groups, 10 allow for a polar axis, eigenvector within the rotoreflection plane. namely, fCn;Cnvg, with n ¼ 1, 2, 3, 4, 6. The vector da Berry curvature dipole in two dimensions.—In two- will be oriented along such axis. The contribution to the dimensional crystals the Berry curvature behaves as a current from this antisymmetric part can be written in pseudoscalar (only the out-of-plane component is nonzero); vector notation as hence, the Berry curvature dipole behaves as a pseudovector contained in the two-dimensional plane: e3τ ~j0 E~ d~ E~ ; Z ¼ 2 1 iωτ × ð × Þ ð þ Þ Da ¼ f0ð∂aΩzÞ: ð13Þ 3 k 2ω e τ ~ ~ ~j ¼ E × ðd~ × EÞ: ð12Þ 2ð1 þ iωτÞ This vector has units of length. Therefore, symmetry constraints are more severe in two dimensions. In fact, Let us now determine which crystals allow for a nonzero the largest symmetry of a 2D crystal that allows for a symmetric part Dþ. We require the crystal to be inversion nonvanishing Berry curvature dipole is a single mirror line asymmetric, for otherwise the Berry curvature would be (i.e., a mirror plane that is orthogonal to the 2D crystal), identically zero due to time-reversal symmetry. Any real which would force Da to be orthogonal to it. In vector symmetric matrix can be diagonalized and has a real notation, the current can be written as spectrum. Let us denote its eigenvalues and eigenvectors P 3 þ 3 T e τ δi ei D δieie ~0 ~ ~ ~ by , , respectively: ¼ i¼1 i . All inversion j ¼ zˆ × E ðD · EÞ; asymmetric crystals without left-handed symmetries allow 2ð1 þ iωτÞ for Dþ to be nonzero, but might impose constraints on its 3 ~2ω e τ ~ ~ ~ eigenvectors to lie along the principal symmetry axis and j ¼ zˆ × EðD · EÞ: ð14Þ 2ð1 þ iωτÞ some of its eigenvalues to be degenerate, much in the same way they constrain an ordinary tensor. Such noncentro- The presence of a single mirror symmetry would force the symmetric crystal point groups without left-handed sym- linear conductivity tensor to have its principal axes aligned metries are fO; T; C1;Cn;Dng, with n ¼ 2,3,4,6. with the mirror line. Consequently, according to Eq. (14), However, under left-handed symmetries (det S ¼ −1) when the driving electric field is aligned with the direction Dþ the transformations of differ from those of an ordinary of the Berry curvature dipole vector D~ , all the current that tensor. Equation (11) implies that under a left-handed flows orthogonal to it would arise solely from the Berry symmetry operation the spectrum goes to minus itself: curvature dipole term. fδ1; δ2; δ3g → f−δ1; −δ2; −δ3g. Therefore, for it to remain Candidate materials.—Berry curvature often concen- invariant as a set, it must have the form trates in small regions in momentum space where two or fδ1; δ2; δ3g¼fδ; 0; −δg. In such a case the eigenvectors more bands cross or nearly cross. Therefore, Dirac and would transform as Se1 ¼e3, Se3 ¼e1, Se2 ¼e2. Weyl materials are excellent candidates to observe the Therefore, any crystal with a left-handed symmetry and an quantum nonlinear Hall effect predicted in this work. n-fold rotation axis with n ≥ 3 will force the tensor Dþ to Moreover, since this effect requires a Berry curvature identically vanish, since such n-fold rotation would addi- dipole, it is advantageous to choose low-symmetry crystals tionally force the eigenvectors contained within the invari- with tilted Dirac or Weyl point (see below). We propose ant plane to be degenerate. For a mirror symmetry, the null three classes of candidate materials: topological crystalline eigenvector has to be parallel to the mirror plane, and the insulators (TCIs), two-dimensional transition metal dichal- eigenvectors with opposite eigenvalues must be at π=4 cogenides, and three-dimensional Weyl semimetals. angles from such plane, so that they are swapped under the The surface of topological crystalline insulators hosts mirror operation. Therefore, the only noncentrosymmetric massless Dirac fermions protected by mirror symmetries crystals with mirror symmetries that allow for a nonzero [11,12]. In particular, the [001] surface of TCIs SnTe, þ þ D are C1v and C2v [10].ForC1v symmetry, D has two Pb1−xSnxTe, and Pb1−xSnxSe hosts four massless Dirac independent parameters which can be taken to be the fermions [13] protected by two mirror symmetries. Pairs of positive eigenvalue and the orientation of the null eigen- Dirac cones with spin-momentum locking are located near ¯ vector within the mirror plane. For C2v there is only one the X points of the surface Brillouin zone, forming a independent parameter, which can be taken to be the Kramers pair. At low temperatures the surface undergoes a positive eigenvalue, since the null eigenvector is forced structural transition into a ferroelectric state and one of the to lie along the rotation axis. Finally, the point group S4, mirror symmetries is spontaneously broken [14,15], while which contains a single left-handed fourfold rotoreflection the other remains intact. As a result, two of the surface symmetry, allows for a nonzero Dþ, whose null eigenvector Dirac cones become massive, while the other two remain is forced to lie along the rotoreflection axis. Dþ has two massless [16] (see inset in Fig. 1). Since the remaining independent parameters for S4, which can be taken to be the massless Dirac points have vanishing Berry curvature, it is

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02 2 0 an ellipse in the primed coordinates, kx =γx þðkyþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 02 0 sk0Þ =γy ¼ 1, where γx ¼ γ=v, γy ¼γ= v −α , k0 ¼μα = ffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 −α02 v pv v γ μ2 v2 − α02 k2 − β2 ð pÞ,ffiffiffiffiffiffiffiffiffiffiffiffi¼ x y, ¼ þð Þ 0 , 0 2 02 α ¼ α vx=vy. The condition v > α is equivalent to 2 2 vy > α and is needed for the stability of the tilted Dirac 2 2 02 2 2 cones. The condition μ þðv − α Þk0 > β states that the chemical potential is outside the gap, so that there is a finite density of massive Dirac fermions. The surviving mirror symmetry, which takes ky → −ky, dictates that only the y component of the Berry curvature dipole is nonzero, and is found to be

3v2nβα μ 1 u2 D j jð þ Þ ; FIG. 1 (color online). Berry curvature dipole dependence on y ¼ 2 2 2 2 2 5=2 ð18Þ chemical potential μ. Upper inset: Surface Brillouin zone of TCI ½μ ð1 þ u Þð1 þ 2u Þ − u β SnTe or (Pb,Sn)Se. The blue arrow indicates the direction of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ferroelectric distortion. Lower inset: Brillouin zone of monolayer u α0= v2 − α02 n d2k= 2π 2 where ¼ and ¼ jεsðkÞjorbit coupling and where s ¼1. β is the size of the gap opened by the lack of an inversion center produces substantial local Berry ferroelectric distortion. This low energy theory coincides curvatures [21,22]. The C3v symmetry of these crystals with that previously considered in the literature [13,14,16], would force the Berry curvature dipole to vanish. However, except for the term proportional to α, which produces a tilt uniaxial strain can reduce this symmetry so that a single in the Dirac cones. This tilt is allowed by symmetry and has mirror operation survives, in which case the effect is been observed in ARPES studies [17], and is required for a allowed. In fact, two copies of the model of Eq. (15), each nonzero Berry curvature dipole (see below). The dispersion with a different gap, can describe the states near charge relation for the titled Dirac cone is neutrality within a k · p model [21], and when the shear 2 2 2 2 2 1=2 strain is applied along high-symmetry lines (see inset of εs k sαky sgn μ β vxkx vyky ; 16 ð Þ¼ þ ð Þð þ þ Þ ð Þ Fig. 1). s ¼ would label valleys K and K0 in this α μ > 0 μ < 0 case. The anisotropic velocity term parametrized by where ( ) for conduction (valence) band. The would be proportional to the strain, much in the same Berry curvature can be found, from Eq. (3),tobe way as in strained graphene [23]. For TMDCs one sv v β obtains a scale ℏα=β ∼ 0.2 Å, using a Fermi velocity of sgnðμÞ x y 5 Ωs : 17 v ≈ 4.5 × 10 m=s, a gap β ≈ 1.5 eV, and α 0.1v. ¼ 2 β2 v2k2 v2k2 3=2 ð Þ ¼ ð þ x x þ y yÞ Last but not least, the Berry curvature dipole induced nonlinear Hall effect should be present in a large class of At zero temperature the Berry curvature dipole, computed three-dimensional noncentrosymetric crystals. Interesting from Eq. (13), reduces to an integral over the region candidates are the recently discovered Weyl semimetals in ε k < μ sð Þ . This integral can be computed byp performingffiffiffiffiffiffiffiffiffiffiffiffi the TaAs material class [24–27]. These materials are k0 v =v k an areapffiffiffiffiffiffiffiffiffiffiffiffi preserving transformation, y ¼ y x y, noncentrosymmtric and have a polar axis, which allows 0 kx ¼ vx=vykx, and by noting that the Fermi surface is the quantum nonlinear Hall effect described by Eq. (12).

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When tilted, a Weyl point generates a singular configura- [11] T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, tion of Berry curvature, with a finite dipole moment whose Nat. Commun. 3, 982 (2012). magnitude can be easily estimated from band structure [12] Y. Ando and L. Fu, Annu. Rev. Condens. Matter Phys. 6, 361 (2015). calculations. In addition, other polar materials such as 88 BiTeI with a strong Rashba-type spin-orbit coupling [28] [13] J. Liu, W. Duan, and L. Fu, Phys. Rev. B , 241303(R) (2013). may also have large Berry curvature dipole moments. [14] Y. Okada, M. Serbyn, H. Lin, D. Walkup, W. Zhou, These three-dimensional Weyl and Rashba materials pro- C. Dhital, M. Neupane, S. Xu, Y. J. Wang, R. Sankar, vide promising platforms for the observation of the F. Chou, A. Bansil, M. Zahid Hasan, S. D. Wilson, L. Fu, quantum nonlinear Hall effect. and V. Madhavan, Science 341, 1496 (2013). [15] B. M. Wojek, M. H. Berntsen, V. Jonsson, A. Szczerbakow, We wish to thank T. Gulden for valuable discussions and P. Dziawa, B. J. Kowalski, T. Story, and O. Tjernberg, D. Pesin for bringing Ref. [5] to our attention. I. S. is arXiv:1505.03414. supported by the Pappalardo Fellowship. L. F. is supported [16] M. Serbyn and L. Fu, Phys. Rev. B 90, 035402 by the DOE Office of Basic Energy Sciences, Division (2014). of Materials Sciences and Engineering under Award [17] Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, No. DE-SC0010526. T. Takahashi, K. Segawa, and Y. Ando, Nat. Phys. 8, 800 (2012). [18] n is not the full carrier density since there will also be a contribution from the massless Dirac cones. [1] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, in [19] Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, Statistical Physics, Course of Theoretical Physics Vol. 5 X. Chen, X. Ma, and Q.-K. Xue, Phys. Rev. Lett. 108, 3rd. ed. (Pergamon Press, Oxford, 1999) 016401 (2012). [2] G. Sundaram and Q. Niu, Phys. Rev. B 59, 14915 (1999); 2 [20] For an area of 1 mm and radiation of 1 W, one gets typical For a review, see D. Xiao, M.-C. Chang, and Q. Niu, Rev. currents of 20 nA for parameters of SnTe. Mod. Phys. 82, 1959 (2010), and references therein. [21] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. [3] J. E. Moore and J. Orenstein, Phys. Rev. Lett. 105, 026805 Rev. Lett. 108, 196802 (2012); G.-B. Liu, W.-Y. Shan, (2010). Y. Yao, W. Yao, and D. Xiao, Phys. Rev. B 88, 085433 [4] G. D. Mahan, Many-Particle Physics, 3rd. ed. (Kluwer (2013). Academic/Plenum Publishers, New York, 2000). [22] For a recent review, see X. Xu, W. Yao, D. Xiao, and T. F. [5] After this work was completed and posted, we learned that a Heinz, Nat. Phys. 10, 343 (2014). result similar to Eq. (8) appeared in, E. Deyo, L. E. Golub, [23] T. L. Linnik, J. Phys. Condens. Matter 24, 205302 E. L. Ivchenko, and B. Spivak, arXiv:0904.1917v1 (unpub- (2012). lished). [24] H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, [6] F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004). Phys. Rev. X 5, 011029 (2015). [7] Note that in the pure dc case writing EcðtÞ¼Ec, with 0 [25] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, Ec ∈ R, the current is ja ¼ 2limω→0ja. 112 B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, [8] Y. Gao, S. A. Yang, and Q. Niu, Phys. Rev. Lett. , S. Jia, A. Bansil, H. Lin, and M. Z. Hasan, Nat. Commun. 6, 166601 (2014). 7373 (2015). [9] P. Olbrich, L. E. Golub, T. Herrmann, S. N. Danilov, H. [26] S.-Y. Xu et al., arXiv:1502.03807. Plank, V. V. Belkov, G. Mussler, Ch. Weyrich, C. M. [27] B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Schneider, J. Kampmeier, D. Grutzmacher, L. Plucinski, 113 Huang, L. X. Zhao, G. F. Chen, C. Matt, F. Bisti, V. Strokov, M. Eschbach, and S. D. Ganichev, Phys. Rev. Lett. , J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, 096601 (2014). C arXiv:1503.09188. [10] Although 2h has only twofold rotations, the fact that the [28] M. S. Bahramy, B.-J. Yang, R. Arita, and N. Nagaosa, Nat. rotation axis is perpendicular to the mirror plane forces all Commun. 3, 679 (2012). the elements to vanish.

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