Nonlinear electric transport in odd-parity magnetic multipole systems: Application to Mn-based compounds
NNonlinear electric transport in odd-parity magnetic multipole systems:Application to Mn-based compounds
Hikaru Watanabe ∗ and Youichi Yanase
1, 2 Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Institute for Molecular Science, Okazaki,444-8585, Japan (Dated: October 19, 2020)Violation of parity symmetry gives rise to various physical phenomena such as nonlinear transportand cross-correlated responses. In particular, the nonlinear conductivity has been attracting a lotof attentions in spin-orbit coupled semiconductors, superconductors, topological materials, and soon. In this paper we present theoretical study of the nonlinear conductivity in odd-parity magneticmultipole ordered systems whose PT -symmetry is essentially distinct from the previously studiedacentric systems. Combining microscopic formulation and symmetry analysis, we classify the non-linear responses in the PT -symmetric systems as well as T -symmetric (non-magnetic) systems, anduncover nonlinear conductivity unique to the odd-parity magnetic multipole systems. A giant non-linear Hall effect, nematicity-assisted dichroism and magnetically-induced Berry curvature dipoleeffect are proposed and demonstrated in a model for Mn-based magnets. I. INTRODUCTION
Nonlinear responses have been giving rise to a lot ofresearch interest in condensed matter physics. For in-stance, the nonlinear optical response provides a power-ful tool for spectrometry. It has been used to obtain areal-space imaging of the parity-violating magnetic orderin insulators [1, 2] and to explore exotic order in spin-orbit coupled metals and superconductors [3–6]. In theoptical nonlinear responses, energy of irradiating light isusually larger than that of electron bands, and the ob-served signals are attributed to interband transitions [7].On the other hand, intraband transitions are also im-portant in conductivity measurements in which the fre-quency of the probe is usually lower than that of opticalprobes and comparable to the electronic energy scale. Itis therefore expected that nonlinear responses are infor-mative for investigating metallic compounds where theintraband transitions are relevant.Regarding the nonlinear response in metals, thesecond-order nonlinear conductivity (NLC) measurementhas attracted much attention. Previous studies aremainly divided into two streams; field-induced NLC andfield-free NLC. The former can be traced back to Rikken’sseminal works [8, 9]. They realized the longitudinalNLC under an external magnetic field, and significant en-hancement has recently been discovered in strongly spin-orbit coupled semiconductors and superconductors [10].The microscopic origin of the longitudinal NLC is at-tributed to a semiclassical contribution which we callDrude term [11–13]. On the other hand, a lot of theoret-ical and experimental efforts have recently been devotedto the transverse NLC, that is, nonlinear Hall effect [14–16]. The nonlinear Hall effect realized without the mag-netic field is rooted in a geometric quantity named Berrycurvature dipole (BCD) [14]. ∗ [email protected] According to the symmetry argument, the second-order NLC requires violation of parity symmetry P . Thecondition is satisfied by the acentric property of crystalswhich were previously studied [8–11, 14–16]. In contrast,the P -symmetry breaking can also be accompanied bythe magnetic order, that is called odd-parity magneticmultipole order [17–19]. It is expected that counterpartsof the NLC exist in magnetic metals.A key to the odd-parity magnetic multipole order islocally-noncentrosymmetric property of crystals. Withsuch structure of crystals, the local site-symmetry ofatoms does not have P -symmetry although the global P -symmetry is preserved owing to the sublattice degreeof freedom [20, 21]. Then, the anti-symmetric spin-orbitcoupling (ASOC) emerges in a sublattice dependent wayand gives rise to exotic responses such as the antiferro-magnetic Edelstein effect [21–24]. Supposing the anti-ferromagnetic order preserving the translational symme-try, both of the P and T -symmetries may be violatedwhile the combined symmetry, namely PT -symmetry,is preserved. Such parity-violating but PT -symmetricmagnetic order is called odd-parity magnetic multipoleorder and has been discussed in the context of mul-tipole physics [18, 19] and antiferromagnetic spintron-ics [25–27]. More than 100 candidate materials such asBaMn As and EuMnBi have been identified [18].In this work, the NLC in odd-parity magnetic multi-pole metals are investigated. We present a general sym-metry classification of NLC based on a quantum mechan-ical calculation. Supported by the microscopic analysis,we clarify NLC characteristic of magnetic metals withand without external magnetic field. We find that theNLC at H = 0 is a measure of the ASOC. Furthermore,we reveal two types of field-induced NLC; the nematicity-assisted dichroism, and Berry curvature dipole effect in-duced by what we call magnetic ASOC. These phenom-ena originate from locally-noncentrosymmetric crystalstructures and magnetic order, and hence have strikingdifference from the NLC in noncentrosymmetric (non- a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t magnetic) crystals. We show the correspondence between T -symmetric and PT -symmetric systems in Table I. TABLE I. Second-order NLC in T / PT -symmetric systemswith/without magnetic field H . Dominant contributions suchas the BCD term are shown. The boldfaced terms are studiedin this work. T PT H = 0 BCD Drude H (cid:54) = 0 magnetic Drude magnetic BCDII. QUANTUM THEORY AND SYMMETRYANALYSIS A theoretical treatment of the second-order NLC hasbeen established in Sipe and his coworkers’ works [28–30] where the nonlinear response functions are derivedfrom straightforward extension of the linear response the-ory [31]. A detailed calculation of the NLC is shown inAppendix A. Here we only describe the outline of deriva-tion.A spatially-uniform electric field E ( t ) is introduced soas to be compatible with calculations based on the Blochstates in the length gauge framework [28–30, 32–34]. Us-ing the density matrix formalism, we derive the second-order electric current in the frequency domain as J µ (2) ( ω ) = (cid:88) νλ (cid:90) dω dω π σ µ ; νλ ( ω ; ω , ω ) × E ν ( ω ) E λ ( ω ) δ ( ω − ω − ω ) . (1)Assuming the clean limit where the relaxation time τ →∞ within the transport regime ωτ (cid:28)
1, the NLC is clas-sified by the dependence on the phenomenological relax-ation time τ = γ − as σ µ ; νλ = σ µ ; νλ D + σ µ ; νλ BCD + σ µ ; νλ int , (2)in which the indices ν, λ of applied electric fields aresymmetric. The first two components are obtained as σ µ ; νλ D = − e γ (cid:90) d k (2 π ) d (cid:88) a ∂ µ ∂ ν ∂ λ (cid:15) k a f ( (cid:15) k a ) , (3) σ µ ; νλ BCD = e γ (cid:90) d k (2 π ) d (cid:88) a (cid:15) µνκ f ( (cid:15) k a ) ∂ λ Ω κa + [ ν ↔ λ ] , (4)= e γ (cid:15) µνκ D λκ + [ ν ↔ λ ] , (5)which are the Drude [11] and BCD [14, 35, 36] terms,respectively. The index a represents the band index. Weintroduced the electron charge e > D µν = (cid:90) d k (2 π ) d (cid:88) a f ( (cid:15) k a ) ∂ µ Ω νa . (6) These two terms are finite only in the metal state anddivergent in the clean limit since both are Fermi surfaceterms. The remaining term σ µ ; νλ int comes from interbandtransitions, and it is not divergent in the clean limit [37].This term, therefore, gives a negligible contribution tothe NLC in good metals.In the group-theoretical classification of quantumphases, the parity-violating phases are classified into odd-parity electric/magnetic multipole phases where T / PT -symmetry is preserved [18, 19]. It is known that thesepreserved symmetries impose strong constraints on theresponse functions [24, 38] in addition to equilibriumproperties of the systems [39]. Thus, the symmetry anal-ysis enables us to classify the NLC allowed in either T -symmetric or PT -symmetric systems based on the relax-ation time dependence. The result is shown in Table II.In T -symmetric systems, all the terms in NLC arescaled by odd-order O ( τ n +1 ). Recalling the linear re-sponse theory, the scattering rate γ can be replaced bythe adiabaticity parameter whose sign represents irre-versibility due to external fields [31]. Thus, the NLCshould be accompanied by a dissipative response. This isconsistent with previous theories [40, 41]. In contrast tothe familiar linear conductivity, the Drude term is pro-hibited because it is even-order with respect to τ . Theleading order term is the BCD term for the transverseNLC.On the other hand, the T -symmetry is broken bythe magnetic order in the parity-violating PT -symmetricsystems which we focus on. Therefore, the relaxationtime dependence is even-order O ( τ n ), and intrinsic con-tributions O ( τ ) are allowed. The leading order term isthe Drude term O ( τ ). We will show that the Drude termis a measure of the hidden ASOC characteristic of locally-noncentrosymmetric systems. The BCD term is prohib-ited to be consistent with the fact that the Berry curva-ture itself disappears due to the PT -symmetry. Althoughthe effect of the T -symmetry breaking in acentric systemshas been discussed in previous works [14, 37, 42, 43], ourclassification has clarified the contrasting role of T and PT -symmetries in NLC. Below we will see that the PT -symmetry gives a clear insight into the NLC.In our classification, extrinsic contributions such asthe side jump and skew scattering are not taken intoaccount [42–45]. We, however, note that the extrinsiccontributions may be similarly classified by the symme-tries. Indeed, for nonmagnetic impurities with δ -functionpotential, we show that while extrinsic terms are allowedin the T -symmetric systems [43], they are strongly sup-pressed by the PT -symmetry (See Appendix C). Thissuppression is highly contrasting to the fact that theimpurities play an important role in the NLC in T -symmetric materials such as WTe [46]. When we focuson the PT -symmetric magnetic systems, the classifica-tion in Table II is meaningful beyond the relaxation timeapproximation for impurity scattering.All the terms in NLC are allowed in the absence of both T and PT -symmetry. For instance, the Drude term be- TABLE II. Relaxation time dependence of the second-orderNLC in T / PT -symmetric systems. ‘N/A’ denotes that thecomponent is forbidden by symmetry. σ D σ BCD σ int T N/A O ( τ ) O ( τ − ) PT O ( τ ) N/A O ( τ ) comes finite when we apply magnetic fields to originally T -symmetric systems [8–11], that is described as ‘mag-netic Drude’ in Table I. Similarly, we expect a magnetic-field-induced NLC in originally PT -symmetric systems;the BCD term indeed arises from the PT -symmetrybreaking (called ‘magnetic BCD’ in Table I). This termis clarified in this work below. In the following, we con-sider the PT -preserving antiferromagnetic metal with orwithout the magnetic field, and discuss the Drude andBCD terms which are dominant in clean metals. III. NLC IN ODD-PARITY MAGNETICMULTIPOLE SYSTEMS
We introduce a minimal model of BaMn As which un-dergoes odd-parity magnetic multipole order [24]. Manymagnetic compounds in the list of Ref. [18] belong to thesame class. The Hamiltonian reads H ( k ) = (cid:15) ( k ) τ + g ( k ) · σ τ z + h · σ τ + V AB ( k ) τ x , (7)where σ and τ are Pauli matrices representing thespin and sublattice degrees of freedom, respectively.In addition to the intra-sublattice and inter-sublatticehoppings, (cid:15) ( k ) and V AB ( k ), we introduce the stag-gered g -vector g ( k ) = g ( k ) + h AF consisting of thesublattice-dependent ASOC g ( k ) [21–23] and the molec-ular field h AF = h AF ˆ z due to antiferromagnetic order inBaMn As [47–49]. The detailed material property ofBaMn As and expressions of (cid:15) ( k ), V AB ( k ), and g ( k )are available in Appendix B 1. We also consider an ex-ternal magnetic field h to discuss field-induced NLC. A. Field-free nonlinear Hall effect
First, we show the NLC at zero magnetic field ( h = ). Then, the NLC is mainly given by the Drudeterm (see Table II), and it is determined by the anti-symmetric and anharmonic property of the energy dis-persion [see Eq. (3)]. Such dispersion is known to bea pronounced property of the odd-parity magnetic mul-tipole systems [18, 21, 23, 24, 50]. In the case ofBaMn As , the anti-symmetric component was identi-fied to be a cubic term k x k y k z [24]. Indeed, the energyspectrum of the model Eq. (7) is obtained as E ± k = (cid:15) ( k ) ± (cid:112) V AB ( k ) + g ( k ) . (8) The anti-symmetric distortion in the band structurearises from the coupling term g ( k ) · h AF which is approx-imated by ∼ k x k y k z near time-reversal-invariant momen-tum. Thus, σ z ; xy and its cyclic components of NLC ten-sor are allowed. This indicates the nonlinear Hall effect,namely, the second-order electric current J z generatedfrom the electric field E (cid:107) [110]. For the strong antifer-romagnet, | h AF | (cid:29) | (cid:15) ( k ) | , | V AB ( k ) | , | g ( k ) | , the Drudecomponent is analytically obtained as σ z ; xy D = σ x ; yz D = σ y ; zx D = e α (cid:107) n γ sgn ( h AF ) , (9)in the lightly-hole-doped region. Here n denotes the car-rier density of holes and α (cid:107) represents the strength ofASOC parallel to the staggered magnetization h AF . Itis noteworthy that Eq. (9) does not depend on the an-tiferromagnetic molecular field and therefore it is use-ful to evaluate the sublattice-dependent ASOC. Thus,the NLC provides a way to experimentally deduce thesublattice-dependent ASOC, although it was called ”hid-den spin polarization” [51, 52] because it is hard to bemeasured. Equivalence of σ z ; xy D = σ x ; yz D = σ y ; zx D holdsindependent of parameters and it can be tested by ex-periments. Numerical calculations of Eq. (3) are con-sistent with the above-mentioned symmetry argumentand analytic formula as shown in Appendix B 2. Atypical value of the nonlinear Hall response is obtainedas σ z ; xy D / [( σ xx ) σ zz ] ∼ − [A − · V · m ] and it ismuch larger than the experimental value of bilayer WTe , σ y ; xx / ( σ xx ) ∼ − [A − · V · m ] [16]. Because theDrude term is more divergent with respect to τ than theBCD term, we may see a giant nonlinear Hall responsein the PT -symmetric antiferromagnet.The NLC is a useful quantity not only to evaluate thesublattice-dependent ASOC but also to detect domainstates in antiferromagnetic metals [18]. Indeed, the signof the NLC depends on the antiferromagnetic domainand hence it may promote developments in the antifer-romagnetic spintronics [25, 26]. In fact, the read-out ofantiferromagnetic domains has been successfully demon-strated by making use of the NLC [53]. For BaMn As and related materials listed in Ref. [18], the nonlinearHall effect can be used to identify antiferromagnetic do-main states. So far we considered intrinsic contributions.We have shown that the extrinsic contributions from im-purity scattering are suppressed due to the preserved PT -symmetry, and therefore, they are not relevant tothe above discussions. B. Nematicity-assisted dichroism
In the absence of the external field, BaMn As -typemagnetic materials do not show the longitudinal NLCalong the high symmetry axes, namely, σ µ ; µµ = 0. Be-low, we show that the longitudinal NLC can be in-duced by magnetic fields. Since the BCD term con-tributes to only the transverse response, we have onlyto consider the Drude term. Generally speaking, to ob-tain a finite longitudinal electronic dichroism, the sys-tem is required to possess an anti-symmetric disper-sion such as k µ or higher-order one. According to thegroup-theoretical classification, the ‘polarization’ in themomentum-space denoted by k µ may share the samesymmetry as k µ [18, 24]. Thus, the momentum-spacepolarization is a key to realize the longitudinal dichro-ism.In BaMn As and related materials, the momentum-space polarization can be induced by the nematicity. Wecan understand this by the discussion of the magne-topiezoelectric effect [24, 54–57]. A magnetopiezoelec-tric effect means that the planer (electronic) nematicityis induced by the out-of-plane electric current. That iswritten as ε xy = e xy ; z J z , (10)where ε µν represents the strain tensor. It was experimen-tally discovered in EuMnBi [55, 57] and CaMn Bi [56]in accordance with theoretical prediction. The responseis derived from the anti-symmetrically distorted Fermisurface and hence realizable in the odd-parity magneticmultipole systems. Similar to the conventional piezoelec-tric effect, we may expect an inverse effect. Given the in-plane nematic order or strain, the system should obtainthe momentum-space polarization P k z whose symmetryis the same as the electric current J z , P k z = ˜ e z ; xy ε xy . (11)Accordingly, the longitudinal dichroism σ z ; zz is allowed.Thus, nematicity-assisted dichroism which is unique tothe odd-parity magnetic multipole systems is implied.The nematicity can be induced by the magnetic fieldthrough the spin-orbit coupling. In the model forBaMn As the sublattice-dependent ASOC plays an es-sential role. By h (cid:54) = 0, the energy spectrum of the lowerbands E − k in Eq. (8) is modified as E − k = (cid:15) ( k ) − (cid:112) V AB ( k ) + g ( k ) + h ± | λ | , (12)where λ = V AB ( k ) h + [ g ( k ) · h ] . The magnetic fieldnot only lifts the Kramers degeneracy but also causesthe nematicity through the coupling [ g ( k ) · h ] in λ ,although linear terms in h are canceled out betweensublattices in sharp contrast to acentric systems stud-ied before [11]. For BaMn As with Dresselhaus-typestaggered ASOC [26], the nematicity denoted by ε xy ismaximally induced by the magnetic field h parallel to[110] or [1¯10].We expect nematicity-assisted dichroism in BaMn As under the magnetic field h (cid:107) [110] from the above discus-sions. In numerically calculated NLC σ z ; zz D with rotatingthe magnetic field in the azimuthal plane, the dichro-ism with two-fold field-angle dependence is clearly seen(Fig. 1). In this case the magnetic field is a bipolar field rather than a vector field, in sharp contrast to the mag-netic Drude term for which the observed field-angle de-pendence is one-fold [8, 9, 11]. Although the field-inducedNLC is tiny as evaluated in Appendix B 3, it was actuallydetected in a recent experiment for BaMn As [58]. D ] L P X W K D Q J O H > G H J @ z ; zz > $ 9 @ × FIG. 1. Drude term of a longitudinal NLC σ z ; zz D as a func-tion of the azimuthal angle of external magnetic fields h = h (cos φ, sin φ, h = 0 . T = 0 .
01, chemical potential µ = − .
5, relax-ation time γ − = 10 , and Brillouin zone mesh N = 135 areadopted. The other parameters and adopted energy scale aredescribed in Appendix B. C. Magnetic ASOC and Berry curvature dipole
Now we consider the counterpart of the magneticDrude term [11], that is the magnetic BCD term. The PT -symmetry ensures Kramers doublet at each momen-tum k , and Berry curvature is completely canceled inthe odd-parity magnetic multipole systems. The dou-blet, however, should be split when the PT -symmetryis broken by the external magnetic field. Let us con-sider BaMn As -type magnet under the magnetic field h = h z ˆ z for an example. Then, while the total Berrycurvature (cid:82) d k Ω z is trivially induced, the BCD alsoemerges. Using the allowed symmetry operations, theinduced BCD is identified as D xy = D yx . (13)Because the BCD has the same symmetry as theASOC [26], emergence of one indicates the presence ofthe other. Therefore, the field-induced BCD is under-stood by discussing magnetically-induced ASOC in thefollowing way. Although the sublattice-dependent ASOCis compensated with h = 0, combination of the staggeredexchange spitting h AF · σ τ z and uniform Zeeman field h · σ τ leads to imbalance between the sublattices with-out Brillouin zone folding (Fig. 2). One of sublatticesobtains an increased carrier density, and consequentlythe sublattice-dependent ASOC is not compensated. Theemergent ASOC has distinct properties compared to theconventional crystal ASOC since the former originatessolely from the magnetic effects. We therefore name thisfield-induced ASOC ‘magnetic ASOC’. Interestingly, themagnetic ASOC is tunable by external magnetic fields.Thus, the concept of magnetic ASOC may be useful todesign spin-momentum locking in more controllable waythan the crystal ASOC which is determined by the crystalstructure [59]. In the model for BaMn As the magneticASOC and BCD with the same symmetry as Eq. (13) areactually obtained. FIG. 2. Mechanism of the magnetic ASOC and field-inducedBCD. The blue-colored arrows denote the spin-polarizationor Berry curvature at each k . (Left panel) A magnetic fieldalong the z -axis splits the Fermi surface depending on the an-tiferromagnetic molecular field h AF . (Right panel) The splitFermi surface is viewed in the xy -plane which indicates theDresselhaus-type ASOC and BCD. The field-induced BCD allows nonlinear Hall conduc-tivity in accordance with Eq. (5), which satisfies the re-lation σ z ; xx BCD = − σ z ; yy BCD = − σ x ; xz BCD = 2 σ y ; yz BCD . (14)For example, we show the numerical result for σ z ; xx BCD inFig. 3, which reveals the dependence on the elevation an-gle of h . The induced BCD is inverted when the externalfield is flipped. Therefore, the field-angle dependence isone-fold in contrast to the nematicity-assisted dichroism(Fig. 1). H O H Y D W L R Q D Q J O H > G H J @ z ; xx > $ 9 @ × FIG. 3. BCD term of a nonlinear Hall conductivity σ z ; xx BCD asa function of the elevation angle of external magnetic fields h = h (sin θ, , cos θ ). Parameters and unit are the same asFig. 1. Finally, we comment on a linear Hall response. Be-cause the systems under the external magnetic field pos-sess neither the T - nor PT -symmetry, a linear Hall response is also allowed. This is in contrast to thepreviously studied acentric systems [14–16, 36] wherethe linear Hall response is forbidden because of the T -symmetry. However, the nonlinear Hall response can bedistinguished from the linear one by symmetry. For ex-ample, the NLC, σ z ; xx BCD and σ z ; yy BCD , in Eq. (14) representsthe Hall response for which the linear response is forbid-den.
IV. CONCLUSION AND DISCUSSIONS
This work presents symmetry classification of thesecond-order NLC, and explores the NLC of odd-paritymagnetic multipole systems. The Drude term gives riseto a giant nonlinear Hall conductivity at zero magneticfield, and provides an experimental tool for a probe ofthe sublattice-dependent ASOC. Thus, the hidden spinpolarization in centrosymmetric crystals can be clarified.It also enables us to elucidate domain states in antifer-romagnetic metals, and hence the NLC will be useful inthe field of antiferromagnetic spintronics. Interestingly,the NLC induced by magnetic fields is significantly dif-ferent from those studied in previous works. We clarifiedthe nematicity-assisted dichroism and the BCD-inducedNLC due to the magnetic ASOC.In accordance with our theoretical result, a recent ex-perimental study actually detected nematicity-assistedelectric dichroism under the magnetic field [58]. We be-lieve that further studies of the nonlinear response inparity-violated magnetic systems will be motivated byour work.
Acknowledgments —The authors are grateful to A. Shitade, A. Daido, Y.Michishita, M. Kimata, and R. Toshio for valuable com-ments and discussions. Especially, the authors thankM. Kimata for providing experimental data and motivat-ing this work. This work is supported by a Grant-in-Aidfor Scientific Research on Innovative Areas “J-Physics”(Grant No. JP15H05884) and “Topological Materials Sci-ence” (Grant No. JP16H00991, No, JP18H04225) fromthe Japan Society for the Promotion of Science (JSPS),and by JSPS KAKENHI (Grant No. JP15K05164,No. JP15H05745, and No. JP18H01178). H.W. is aJSPS research fellow and supported by JSPS KAKENHI(Grant No. 18J23115).
Appendix A: Derivation of nonlinear conductivity
We reproduce the expression for second-order electricconductivity [28–30]. We take the unit (cid:126) = 1 below. Ingeneral, we have several choices of gauging to introducean electric field. In an spatially uniform electric field,the Hamiltonian is modified by introducing the vectorpotential A ( t ) into the canonical momentum p as p → p − q A ( t ) , (A1)where q is the charge of carriers and the electric field isobtained as E ( t ) = − ∂ t A ( t ). This choice is called thevelocity gauge. On the other hand, the electric field canbe taken into account in the Hamiltonian by including H E = − q r · E ( t ) , (A2)where r is the position operator. H E is called the dipoleHamiltonian and the gauge choice is called the lengthgauge. These two choices should give equivalent resultsbecause of the gauge invariance [32, 33].The dipole Hamiltonian breaks the translational sym-metry in crystals. Therefore, it seems that the Blochstates | ψ k a (cid:105) = exp ( i k · ˆ r ) | u a ( k ) (cid:105) labeled by the crystalmomentum k and the band index a are not good basisfor the total Hamiltonian. However, making use of theBlount’s prescription [60], the position operator in theBloch representation r k is given by (cid:104) ψ k a | r | ψ k (cid:48) b (cid:105) = δ ( k − k (cid:48) ) [ i∂ k δ ab + ξ ab ] , (A3)= δ ( k − k (cid:48) ) [ r k ] ab . (A4)We define the Berry connection ξ ab = i (cid:104) u a ( k ) | ∂ k u b ( k ) (cid:105) .The length gauge is adopted in the following calculations.
1. Density matrix formalism
Following the Sipe’s seminal work [28–30] and subse-quent theoretical studies [32, 61], we derive the nonlin-ear conductivity (NLC) based on the density matrix for-malism. Time evolution of the density matrix operator P ( t ) = e − H ( t ) / ( k B T ) / Tr[ e − H ( t ) / ( k B T ) ] is described by thevon-Neumann equation, i∂ t P ( t ) = [ H ( t ) , P ( t )] . (A5)For convenience in the perturbative calculations, we in-troduce the reduced density matrix, ρ k ,ab ( t ) = Tr[ c † k b c k a P ( t )] , (A6)where c k a is the annihilation operator of the Bloch state | ψ k a (cid:105) . In the following, momentum dependence of thereduced density matrix ρ k ,ab ( t ) is implicit unless other-wise mentioned. The Hamiltonian consists of the non-perturbative part H and the dipole Hamiltonian H E ( t )in the Schr¨odinger picture. The Bloch state satisfies theequation H ( k ) | u a ( k ) (cid:105) = (cid:15) k a | u a ( k ) (cid:105) in which H ( k ) isthe Bloch representation of H . Thus, Eq. (A5) is recastas i∂ t ρ ab ( t ) − ( (cid:15) k a − (cid:15) k b ) ρ ab ( t ) = − qE µ [ r µ k , ρ ( t )] ab . (A7) Using the Fourier transformation defined as ρ ab ( t ) = (cid:90) dω π e − iωt ρ ab ( ω ) , (A8)we obtain( ω − (cid:15) ab ) ρ ab ( ω ) = − q (cid:90) d Ω2 π E µ (Ω) [ r µ k , ρ ( ω − Ω)] ab , (A9)where (cid:15) ab = (cid:15) k a − (cid:15) k b . We expand the reduced densitymatrix ρ = (cid:80) n ρ ( n ) by powers of the electric field ρ ( n ) = O ( | E | n ), and obtain the recursion formula for the densitymatrix,( ω − (cid:15) ab ) ρ ( n +1) ab ( ω ) = − q (cid:90) d Ω2 π E µ (Ω) [ r µ k , ρ ( n ) ( ω − Ω)] ab . (A10)In particular, the zero-th order term is obtained as ρ (0) ab ( ω ) = 2 πδ ( ω ) f ( (cid:15) k a ) δ ab , (A11)where f ( (cid:15) ) = [1 + exp (( (cid:15) − µ ) / ( k B T ))] − is the Fermidistribution function. The expression is simplified as ρ ( n +1) ab ( ω ) = − q (cid:90) d Ω2 π d ωab E µ (Ω) [ r µ k , ρ ( n ) ( ω − Ω)] ab , (A12)by using the matrix ˆ d ω [32, 33] defined as d ωab = 1 ω − (cid:15) ab . (A13)The above derivations are natural extension of the lin-ear response theory [31]. The position operator in theBloch representation r µ k is divided into the diagonal andoff-diagonal parts, that is, the first and second terms ofEq. (A4). Denoting these two components as r i and r e ,respectively [29], the perturbation due to the electric fieldis classified into the intraband effect − q r i · E and inter-band effect − q r e · E . We phenomenologically introducethe scattering rate by replacing the matrix ˆ d ω with d ωab → ω + iγ − (cid:15) ab , (A14)where γ denotes the scattering rate which is the inverseof the relaxation time τ = γ − [33]. This assumptionmay be satisfied in the presence of the nonmagnetic im-purities.We sequentially obtain corrections to the reduced den-sity matrix ρ ( n ) ( n > ρ (2) is explicitly written by ρ (2) ab ( ω ) = ρ ( ii ) ab ( ω ) + ρ ( ie ) ab ( ω ) + ρ ( ei ) ab ( ω ) + ρ ( ee ) ab ( ω ) , (A15)where components in the right hand side are labeled bytwo kinds of the perturbations denoted by intraband ( i )and interband ( e ) effects. Each component is obtainedas ρ ( ii ) ab ( ω ) = ( − iq ) (cid:90) d Ω d Ω (cid:48) (2 π ) E µ (Ω) E ν (Ω (cid:48) ) d ωab d ω − Ω ab ∂ µ ∂ ν f ( (cid:15) k a ) × πδ ab δ ( ω − Ω − Ω (cid:48) ) , (A16) ρ ( ie ) ab ( ω ) = − iq (cid:90) d Ω d Ω (cid:48) (2 π ) E µ (Ω) E ν (Ω (cid:48) ) d ωab (cid:2) ∂ µ (cid:0) d ω − Ω ab f ab ξ νab (cid:1) − i ( ξ µaa − ξ µbb ) d ω − Ω ab f ab ξ νab (cid:3) × πδ ( ω − Ω − Ω (cid:48) ) , (A17) ρ ( ei ) ab ( ω ) = − iq (cid:90) d Ω d Ω (cid:48) (2 π ) E µ (Ω) E ν (Ω (cid:48) ) d ωab d ω − Ω aa ξ µab ∂ ν f ab × πδ ( ω − Ω − Ω (cid:48) ) , (A18) ρ ( ee ) ab ( ω ) = q (cid:88) c (cid:90) d Ω d Ω (cid:48) (2 π ) E µ (Ω) E ν (Ω (cid:48) ) d ωab (cid:2) d ω − Ω cb ξ µac ξ νcb f bc − d ω − Ω ac ξ µcb ξ νac f ca (cid:3) × πδ ( ω − Ω − Ω (cid:48) ) , (A19)where f ab = f ( (cid:15) k a ) − f ( (cid:15) k b ). The summation of the re-peated Greek indices such as µ = x, y, z is implicit and ∂ µ = ∂/∂k µ . For perturbative calculations of nonlinearresponses, we should respect the intrinsic permutation symmetry between the applied external fields E µ and E ν [34]. We hence symmetrize the indices and frequen-cies of electric fields. For instance, Eq. (A16) is modifiedas ρ ( ii ) ab ( ω ) = ( − iq ) (cid:90) d Ω d Ω (cid:48) (2 π ) E µ (Ω) E ν (Ω (cid:48) ) d ωab d ω − Ω ab ∂ µ ∂ ν f ( (cid:15) k a ) × πδ ab δ ( ω − Ω − Ω (cid:48) ) + [( µ, Ω) ↔ ( ν, Ω (cid:48) )] . (A20)The expectation value of the current density is givenby J ( t ) = Tr[ q v (E) P ( t )] , (A21)where v (E) is the velocity operator in the length gauge.In this way, we should express the velocity operator in agiven gauge. Starting from the first quantization in theHeisenberg picture, the velocity operator is given by (cid:104) v (E) ( t ) (cid:105) µ = (cid:104) ∂ t r (E) ( t ) (cid:105) µ = 1 i [ r µ ( t ) , H ( t )] . (A22)Since the electric field is introduced by taking the dipoleHamiltonian [Eq. (A2)] into account, the velocity op-erator is not modified in the length gauge and shouldbe identical with the unperturbed velocity operator [32].The unperturbed velocity operator in the Bloch repre-sentation is given by v ab (cid:104) = v (E) ab (cid:105) = ∇ k (cid:15) a δ ab + i(cid:15) ab ξ ab . (A23)Note that the velocity operator does not coincide withthe unperturbed velocity operator in the velocity gauge[Eq. (A1)] because of the non-commutative property be-tween the position operator and the perturbative part ofthe Hamiltonian arising from Eq. (A1) [33, 34].
2. Second-order nonlinear conductivity
Here, we derive the second-order NLC by making useof the results in the previous section. The expectation value of the current density proportional to | E | is J µ (2) ( ω ) = (cid:90) d k (2 π ) d (cid:88) a,b qv µab ρ (2) ba ( ω ) , (A24) ≡ (cid:90) dω dω (2 π ) ˜ σ µ ; νλ ( ω ; ω , ω ) E ν ( ω ) E λ ( ω ) . (A25)Since all the components of the reduced density matrix ρ (2) [Eq. (A15)] have the coefficient 2 πδ ( ω − ω − ω ),we take the convention of the second-order NLC tensor σ µ ; νλ as˜ σ µ ; νλ ( ω ; ω , ω ) = 2 πδ ( ω − ω − ω ) σ µ ; νλ ( ω ; ω , ω ) . (A26)Substituting the right hand side of Eq. (A15) for ρ (2) ab ( ω ),we express σ µ ; νλ as σ µ ; νλ = σ µ ; νλ D + σ µ ; νλ G + σ µ ; νλ e . (A27)The first term (Drude term) is derived from the compo-nent ρ ( ii ) in Eq. (A15) and arises from only the intrabandeffects. The expression is given by σ µ ; νλ D ( ω ; ω , ω )= − q (cid:90) d k (2 π ) d (cid:88) a v µaa d ωaa d ω aa ∂ ν ∂ λ f ( (cid:15) k a )+ [( ν, ω ) ↔ ( λ, ω )] , (A28)= − q (cid:90) d k (2 π ) d (cid:88) a ω + iγ )( ω + iγ ) ∂ µ ∂ ν ∂ λ (cid:15) k a f ( (cid:15) k a )+ [( ν, ω ) ↔ ( λ, ω )] , (A29)where we use Eq. (A23). The Drude term can be cap-tured by the conventional Boltzmann’s transport the-ory in which only the intraband effect is semiclassicallytreated without geometric effects [11].The second term σ µ ; νλ G (Geometric term) is derived from the component ρ ( ei ) and obtained as σ µ ; νλ G ( ω ; ω , ω )= q (cid:90) d k (2 π ) d (cid:88) a (cid:54) = b d ωba d ω aa (cid:15) ab ξ µab ξ νba ∂ λ f ba + [( ν, ω ) ↔ ( λ, ω )] . (A30)The third term σ µ ; νλ e due to ρ ( ie ) and ρ ( ee ) is written as σ µ ; νλ e ( ω ; ω , ω ) = q (cid:90) d k (2 π ) d (cid:88) ab v µab d ωba (cid:2) − i∂ ν (cid:0) d ω ba f ba ξ λba (cid:1) − ( ξ νbb − ξ νaa ) d ω ba f ba ξ λba (cid:3) + v µab d ωba (cid:34)(cid:88) c (cid:0) d ω ca ξ νbc ξ λca f ac − d ω bc ξ νca ξ λbc f cb (cid:1)(cid:35) + [( ν, ω ) ↔ ( λ, ω )] . (A31)Now we obtained the full expression of the second-order NLC. Although the expression seems complicated,it is simplified by making use of the symmetry. In thenext section, we clarify the constraints from the basic T -and PT -symmetries.
3. Symmetry constraints on nonlinear conductivitytensor
It is well known that the crystal symmetries imposestrong constraints on physical quantities such as equilib-rium properties and transport coefficients [39]. Further-more, by combining with expressions derived from micro-scopic calculations, the symmetry also simplifies physicalquantities expressed in the Bloch representation [24, 38].We can hence distinguish which intraband effect or inter-band effect is relevant in a given response function.Let us consider the spin polarization induced by theelectric field for example. Under the T -symmetry the re-sponse coefficients are determined by intraband contribu-tions, whereas the response arises from interband contri-butions in PT -symmetric systems [18]. These responsesare hence distinguished and classified as Edelstein ef-fect and magnetoelectric effect in the T -symmetric and PT -symmetric systems, respectively. In the frameworkof the multipole-based classification, the parity-violating T -/ PT -symmetric systems are called odd-parity elec-tric/magnetic multipole systems [18, 19]. Thus, the rep-resentation theory of multipole is also useful to asso-ciate response functions with symmetry. In a similarmanner, we conduct a symmetry analysis of NLC inEqs. (A29), (A30), and (A31). a. Drude term First, we discuss the Drude term. Equation (A29)shows that the second-order Drude conductivity σ D is de-termined by the Fermi surface effect as denoted by ∂f /∂(cid:15) and that this term is finite only when the system has ananti-symmetric component in the energy spectrum (cid:15) k a .According to the representation theory of multipole de-grees of freedom in solids [18, 19, 24], the asymmetricdispersion is a striking property of the PT -symmetricodd-parity magnetic multipole systems [21, 23, 50, 62].The bases of multipoles in the momentum space for suchsystems are spin-independent and anti-symmetric as k x , k x k y k z , (A32)and these bases imply the anti-symmetric modulation inthe energy spectrum of elementary excitations such aselectrons and magnons [21, 62]. Thus, we may see asecond-order Drude conductivity in odd-parity magneticmultipole systems.On the other hand, in the T -symmetric odd-parityelectric multipole systems, the momentum-space basesare spin-dependent such as k x ˆ y − k y ˆ x, (A33)in which ˆ x and ˆ y are T -odd pseudo-vectors represent-ing spin polarization, Berry curvature, and so on. Thesebases represent spin-momentum locking arising fromthe parity violation [26]. Indeed, odd-parity electricmultipole systems include familiar noncentrosymmetriccrystals. Meanwhile, the spin-independent and anti-symmetric basis does not exist in T -symmetric systems.In fact, the Kramers doublet {| u a ( k ) (cid:105) , | u ¯ a ( − k ) (cid:105)} pro-tected by the T -symmetry gives rise to the degeneracybetween the ± k points as (cid:15) k a = (cid:15) − k ¯ a . Therefore, the en-ergy spectrum is symmetric, and the second-order Drudeconductivity vanishes in the odd-parity electric multipolesystems. However, external magnetic fields may inducethe Drude term [11].Summarizing the above-mentioned symmetry analy-sis, the second-order Drude conductivity is finite if andonly if both of the P and T -symmetries are violated,and the allowed components due to asymmetric disper-sions are indicated by the momentum-space basis of mul-tipoles shown in the classification results [18, 19, 24].This symmetry requirement is satisfied in the odd-paritymagnetic multipole systems with and without externalfields [18, 19] as well as in the noncentrosymmetric sys-tems under the external magnetic field [8–10]. In thosesystems, the Drude term may give rise to a sizable NLCin the clean limit. Taking the static limit ( ω, ω , ω → σ µ ; νλ D sta. −−→ q γ (cid:90) d k (2 π ) d (cid:88) a ∂ µ ∂ ν ∂ λ (cid:15) k a f ( (cid:15) k a ) , (A34)whose relaxation time dependence is O ( τ ). Thus, theDrude term is the dominant contribution to the NLC inclean metals. b. Geometric term Next, we consider the geometric contribution σ G . Inthe case of the PT -symmetric systems, the energy spec-trum at each k has two-fold degeneracy, that is, Kramersdoublet. Explicitly denoting the Kramers doublet ofBloch states, | u a ( k ) (cid:105) = | u A,ρ ( k ) (cid:105) , (A35)we introduce the Pauli matrices ρ spanned by theKramers degrees of freedom. Then, a Bloch state is char-acterized by the index of energy band A and the Kramersdegrees of freedom ρ = ± . The transformation propertyof Kramers doublet can be taken as [24] PT | u A,ρ ( k ) (cid:105) = (cid:88) ρ (cid:48) | u A,ρ (cid:48) ( k ) (cid:105) ( − iρ y ) ρ (cid:48) ρ , (A36)= | u A, ¯ ρ ( k ) (cid:105) ( − iρ y ) ¯ ρρ , (A37)where ¯ ρ = − ρ . Accordingly, matrix elements of the in-terband Berry connection are transformed as ξ µab ( k ) = ξ µAρ,Bρ (cid:48) ( k ) = − (cid:88) τ,τ (cid:48) ξ µBτ,Aτ (cid:48) ( k )( − iρ y ) † ρ (cid:48) τ ( − iρ y ) τ (cid:48) ρ . (A38)Of course, the Kramers doublet denoted by | u A,ρ ( k ) (cid:105) with ρ = ± have the same energy, (cid:15) k Aρ = (cid:15) k A ¯ ρ . Hence, we obtain the following relation for Eq. (A30) (cid:88) a (cid:54) = b d ωba (cid:15) ab ξ µab ξ νba ∂ λ f ba = (cid:88) a (cid:54) = b (cid:88) c,d,e,f d ωba (cid:15) ba ξ µcd ξ νef ∂ λ f ab × ( − iρ y ) † bc ( − iρ y ) da ( − iρ y ) † ae ( − iρ y ) fb , (A39)= (cid:88) ¯ a (cid:54) =¯ b d ω ¯ b ¯ a (cid:15) ¯ a ¯ b ξ µ ¯ b ¯ a ξ ν ¯ a ¯ b ∂ λ f ¯ b ¯ a , (A40)= (cid:88) a (cid:54) = b d ωba (cid:15) ab ξ µba ξ νab ∂ λ f ba , (A41)in which we use the abbreviated label ¯ a = ( A ¯ ρ ). Becauseof the PT -symmetry, the product of the Berry connec-tions, ξ µab ξ νba and ξ µba ξ νab , are related to each other in thesymmetric way under ν ↔ λ . By using the obtainedrelations, Eq. (A30) is simplified as q (cid:90) d k (2 π ) d (cid:88) a (cid:54) = b d ωba d ω aa (cid:15) ab ξ µab ξ µba ∂ ν f ba = q (cid:90) d k (2 π ) d ω + iγ ) × (cid:88) a (cid:54) = b d ωba (cid:15) ab ξ µab ξ νba ∂ λ f ba + (cid:88) ¯ a (cid:54) =¯ b d ω ¯ b ¯ a (cid:15) ¯ a ¯ b ξ µ ¯ b ¯ a ξ ν ¯ a ¯ b ∂ λ f ¯ b ¯ a , (A42)= q (cid:90) d k (2 π ) d ω + iγ ) (cid:88) a (cid:54) = b ( d ωba + d ωab ) (cid:15) ab ξ µab ξ νba ∂ λ f ba . (A43)Taking the static limit, we have d ωba + d ωab ω + iγ ) = 1 ω + iγ ω + iγ ( ω + iγ ) − (cid:15) ab , (A44) sta. −−→ − γ + (cid:15) ab . (A45)We safely take the clean limit ( γ →
0) since the expres-sion converges. Finally, we obtain the geometric contri-bution σ µ ; νλ G in the PT -symmetric systems as σ µ ; νλ G → σ µ ; νλ e’ = q (cid:90) d k (2 π ) d (cid:88) a (cid:54) = b ∂ λ f ab ξ µab ξ νba (cid:15) ab + [ ν ↔ λ ] , (A46)= q (cid:90) d k (2 π ) d (cid:88) a (cid:54) = b ∂ λ f ( (cid:15) k a ) (cid:18) ξ µab ξ νba (cid:15) ab + ξ µba ξ νab (cid:15) ab (cid:19) + [ ν ↔ λ ] , (A47)which is an intrinsic contribution in the sense that thisterm is O ( τ ) and insensitive to the relaxation time.Therefore, we denote the geometric term in the PT -symmetric systems as σ µ ; νλ e’ . In particular, for the two-band Hamiltonian, the products of Berry connections arerewritten by the quantum metric [37, 63].0Here we consider the T -symmetric systems, where the T -symmetry ensures a relation similar to Eq. (A38), ξ µAρ,Bρ (cid:48) ( k ) = (cid:88) τ,τ (cid:48) ξ µBτ,Aτ (cid:48) ( − k )( − iρ y ) † ρ (cid:48) τ ( − iρ y ) τ (cid:48) ρ . (A48)Following the parallel discussion, we obtain the expres-sion of the geometric term in the static limit as σ µ ; νλ G = q (cid:90) d k (2 π ) d ω + iγ ) (cid:88) a (cid:54) = b ( d ωba − d ωab ) (cid:15) ab ξ µab ξ νba ∂ λ f ba + [ ν ↔ λ ] , (A49) sta. −−→ σ µ ; νλ BCD = q iγ (cid:90) d k (2 π ) d (cid:88) a (cid:54) = b ∂ λ f ba ξ µab ξ νba + [ ν ↔ λ ] , (A50)= iq γ (cid:90) d k (2 π ) d (cid:88) a (cid:54) = b ∂ λ f ( (cid:15) k a ) ( ξ µab ξ νba − ξ µba ξ νab ) + [ ν ↔ λ ] , (A51)= q γ (cid:90) d k (2 π ) d (cid:88) a (cid:15) µνκ ∂ λ f ( (cid:15) k a )Ω κa + [ ν ↔ λ ] , (A52)= − q γ (cid:90) d k (2 π ) d (cid:88) a (cid:15) µνκ f ( (cid:15) k a ) ∂ λ Ω κa + [ ν ↔ λ ] , (A53)where we define the Berry curvature Ω µa = (cid:15) µνλ ∂ ν ξ λaa and make use of Eq. (A48) in Eq. (A49). The integral inEq. (A53), D λκ = (cid:90) d k (2 π ) d (cid:88) a f ( (cid:15) k a ) ∂ λ Ω κa , (A54)is called Berry curvature dipole (BCD) [14]. Therefore,we call the geometric contribution BCD term and it iswritten as σ µ ; νλ BCD = − q γ (cid:15) µνκ D λκ + [ ν ↔ λ ] , (A55)which has been captured by a semiclassical theory modi-fied to take into account geometrical effects in solids [14,35, 36]. The BCD term linearly depends on the relaxationtime as O ( τ ) and it is the leading component of NLCin the clean and T -symmetric systems. An importantproperty of the BCD-induced NLC is that the responseis always transverse because of Eddington epsilon (cid:15) µνκ inEq. (A55) [14, 36].As shown in Eqs. (A42) and (A49), T / PT -symmetryforbids either of symmetric and anti-symmetric part ofthe Berry connections given by ξ µab ξ νba . The above classi-fication of the NLC based on the T and PT -symmetriesis therefore complementary. Supposing a system without T - and PT -symmetries, all the terms in the above analy-sis are present and we can use Eqs. (A47), (A53) withoutmodification. c. Summary Similarly, we can identify the relaxation time depen-dence of σ µ ; νλ e , the last term of Eq. (A27). Supposingthe clean and static limit, σ µ ; νλ e is O ( τ − ) and O ( τ ) in T -symmetric and PT -symmetric systems, respectively.Here, we rewrite the NLC of Eq. (A27) in accordancewith the relaxation time dependence, σ µ ; νλ = σ µ ; νλ D + σ µ ; νλ G + σ µ ; νλ e , (A56)= σ µ ; νλ D + σ µ ; νλ BCD + σ µ ; νλ e’ + σ µ ; νλ e , (A57)= σ µ ; νλ D + σ µ ; νλ BCD + σ µ ; νλ int . (A58) σ µ ; νλ int = σ µ ; νλ e’ + σ µ ; νλ e yields the intrinsic NLC in theclean limit [37]. The decomposition is shown in Eq. (2).The symmetry analysis in this section is summarized inTable II. Substituting the electron’s charge q = − e inEqs. (A34) and (A53), we obtain Eqs. (3) and (5). Appendix B: Calculations of NLC in Mn-basedodd-parity magnetic multipole system
We show the detail of the calculations of the NLC inthe odd-parity magnetic multipole systems. In this sec-tion, q = − e is taken.
1. Model Hamiltonian (a) (b)FIG. 4. (a) Crystal and magnetic structures of BaMn As .(b) Two Mn sublattice in BaMn As depicted with surround-ing As atoms. Two Mn atoms are not related by the P -symmetry in the antiferromagnetic state while they are re-lated in the paramagnetic state. It is known that we have a broad range of candi-date materials for the odd-parity magnetic multipole sys-tems, where both of the P and T -symmetries are brokenwhile the combined PT -symmetry is preserved [17, 18].In particular, a series of Mn-pnictide compounds arepromising candidates and several experimental evidenceshave been recently reported [55–57]. Thus, we performmicroscopic calculations based on the model Hamilto-nian for one of the candidate materials, BaMn As [24].1The crystal structure of BaMn As is ThCr Si -type(space group: I /mmm , No. 139) and Mn atoms arelocated at the locally-noncentrosymmetric Wyckoff posi-tion [47, 48]. This system undergoes the G-type antifer-romagnetic order and magnetic moments at Mn sites arealigned along the z -axis as shown in Fig. 4 (a). Althoughthe magnetic structure is apparently a simple antiferro-magnetic, it breaks both of the P and T -symmetries in-stead of the translational symmetry because of the sub-lattice degree of freedom depicted in Fig. 4 (b). Indeed,it has been shown that the magnetic order is regarded asan odd-parity magnetic multipole order [24]. The mag-netic structure is denoted by the magnetic point group I (cid:48) /m (cid:48) m (cid:48) m which has neither P nor T -symmetry butrespects the PT -symmetry. This compound is semicon-ducting with narrow energy gap [47, 48, 64]. Doping holecarriers, however, have successfully realized the metal-lic state of BaMn As without significant modificationof antiferromagnetic order and band structure [49, 65].Thus, the hole-doped BaMn As is a good example tostudy the itinerant phenomena such as nonlinear elec-tric transport in odd-parity magnetic multipole orderedsystems.The model Hamiltonian captures the electronic struc-ture observed in the lightly hole-doped BaMn As . Thesingle-orbital model is represented by the Bloch Hamil-tonian [24], H ( k ) = (cid:15) ( k ) + g ( k ) · σ τ z + h · σ + V AB ( k ) τ x , (B1)= (cid:15) ( k ) + [ g ( k ) + h AF ] · σ τ z + h · σ + V AB ( k ) τ x , (B2)where σ and τ are Pauli matrices representing the spinand sublattice degrees of freedom, respectively. We intro-duce the sublattice-dependent anti-symmetric spin-orbitcoupling (sASOC), g · σ τ z , the molecular field of anti-ferromagnetic order, h AF · σ τ z , and the external mag-netic field, h · σ . In particular, the sASOC charac-terizes the locally-noncentrosymmetric crystal structureof BaMn As and respects the local symmetry of Mnatoms [24].The components of the Hamiltonian are given by (cid:15) ( k ) = − t (cos k x + cos k y ) − t cos k x k y k z , (B3) V AB ( k ) = − t cos k x k y − t cos k z , (B4) g ( k ) = α sin k y + α cos k x sin k y cos k z α sin k x + α sin k x cos k y cos k z α sin k x sin k y sin k z , (B5) h AF = (0 , , h AF ) . (B6)Hopping parameters t i , ˜ t i and the sASOC strength α i are introduced. In accordance with the reported mag-netic structure of BaMn As , the molecular field is takenas h AF (cid:107) z [48, 49]. The model reproduces the experi-mentally observed Fermi surface in the lightly hole-doped compounds [65, 66], when the parameters are chosen as t = − . , t = − . , ˜ t = 0 . , ˜ t = 0 . , (B7)and α = − . , α = 0 . , α = 0 . , h AF = 1 . (B8)In this work, we adopted these parameters. The remain-ing parameter h is set in the following sections.
2. Nonlinear Hall conductivity at zero magneticfield
In this section, we calculate the Drude term of NLC inthe absence of the external magnetic field, and hence wetake h = in Eq. (B1).The odd-parity magnetic multipole systems preserveneither the P -symmetry nor T -symmetry owing to theparity-violating antiferromagnetic order. The electronicstructure, therefore, shows a peculiar property. Diago-nalizing the Bloch Hamiltonian of Eq. (B1), the energyspectrum is obtained as E ± k = (cid:15) ( k ) ± (cid:112) V AB ( k ) + g ( k ) , (B9)where the spectrum has the two-fold degeneracy pro-tected by the PT -symmetry. The coupling between thesASOC and the molecular field written as g ( k ) · h AF gives rise to an anti-symmetric component in the energydispersion, and thus E ± k (cid:54) = E ±− k . This anti-symmetriccomponent is reduced to k x k y k z around a time-reversal-invariant momentum, which is consistent with the group-theoretical classification theory [18, 24]. Accordingly,the Drude terms of the second-order conductivity, σ z ; xy D , σ y ; xz D , and σ x ; yz D are allowed in BaMn As , since theDrude term is determined by the anti-symmetric andanharmonic property of the energy spectrum denotedby ∂ µ ∂ ν ∂ λ E k in Eq. (A29). These components are in-deed nonlinear Hall conductivity. It is noteworthy thatthe second-order Drude conductivity is totally-symmetricwith respect to the permutation of indices ( µ, ν, λ ). Thus,the following relation is satisfied σ x ; yz D = σ y ; xz D = σ z ; xy D , (B10)in spite of the intrinsic anisotropy of the tetragonal sym-metry. We conduct analytical and numerical calculationsof the nonlinear Hall conductivity σ z ; xy D below.To obtain an analytical expression of the second-orderDrude conductivity, we assume low carrier density andapproximate the energy spectrum up to O ( | k | ). The as-sumption is reasonable for the observed electronic struc-ture in the hole-doped BaMn As [65, 66]. The micro-scopic parameters in Eqs. (B7) and (B8) imply | h AF | (cid:29) | t i | , | ˜ t i | , | α i | . (B11)2We hence evaluate the Drude contribution in the staticlimit as σ z ; xy D (cid:39) − q (cid:90) d k (2 π ) (cid:88) a γ [ ∂ x ∂ y ∂ z g ( k )] · h AF | h AF | f ( (cid:15) k a )(B12)= − q (cid:90) d k (2 π ) (cid:88) a γ α h AF | h AF | f ( (cid:15) k a ) (B13)= − × q α n γ sgn ( h AF ) , (B14)where n denotes the carrier density. The coefficient 2in the last line arises from the two-fold degeneracy dueto the PT -symmetry. Substituting the electron’s charge q = − e , we obtain Eq. (9).Next, we show the numerical result. Numerical inte-gration for k is carried out by adopting N = L dis-cretized cells. Taking the chemical potential µ as a pa-rameter, we plot the Drude term σ z ; xy D in Fig. 5. Weconfirm that the Drude term is finite only in the metallicstate and it is proportional to the square of the relaxationtime τ . This is the dominant contribution in clean met-als because the other allowed contribution, that is, σ int ,is O ( τ ) in the PT -symmetric systems (See Table. II).For a realistic parameter of the hole-doped BaMn As ,we should consider a chemical potential near the top ofthe lower band, that is µ ∼ − . z ; xy > $ 9 @ FIG. 5. Drude term of nonlinear Hall conductivity σ z ; xy D as afunction of the chemical potential µ . Green-colored shaded re-gion indicates the metallic regime where the density-of-statesis finite. The parameters are T = 0 .
01 and γ − = 1 . × .We take N = 135 . Here, we quantitatively estimate the nonlinear Hallconductivity. By taking the energy scale | t | = 1 eV ,the Drude term is evaluated to be σ z ; xy D ∼ − [A · V − ].We also numerically calculate the linear conductivity σ µν with the same parameters (not shown), and the value isestimated to be σ xx ∼ [A · V − · m − ] with a latticeconstant a = 10 [˚A]. Note that the Drude term in thelinear conductivity is O ( τ ) while the second-order Drudeconductivity is O ( τ ). Thus, nonlinear Hall current maynot be negligible in spin-orbit coupled clean metals. Finally, we compare the nonlinear Hall effect calculatedfor BaMn As with the experimental result of WTe [16].Recently, the nonlinear Hall effect has been observed inWTe at zero magnetic field [15, 16] and many relatedstudies have been reported. For comparison, we considera similar measurement geometry of the Hall response. Inthe case of BaMn As , a nonlinear Hall current J NL (cid:107) [001] gives rise to a Hall electric field E H in the paralleldirection E H = J NL σ zz = σ z ; xy D σ zz E , (B15)where E ext is the applied electric field along the [110]-direction. Here σ zz represents the out-of-plane lin-ear conductivity, which is numerically estimated to besmaller than σ xx by one order of magnitude. Convert-ing the electric field into the electric current by E ext = J ext /σ xx , we obtain the relation between E H and J ext , E H = σ z ; xy D ( σ xx ) σ zz J . (B16)Assuming the values estimated above, σ z ; xy D ∼ − [A · V − ], σ xx ∼ [A · V − · m − ], and σ zz ∼ [A · V − · m − ], we obtain σ z ; xy D ( σ xx ) σ zz ∼ − [A − · V · m ] , (B17)for a model of BaMn As .In the case of the bilayer WTe , the observed voltagedrop due to the nonlinear Hall effect is V H ∼
10 [ µ V]with the applied current I ext ∼ µ A] [16]. Consider-ing the experimental setup in Ref. [16] and the thick-ness of the bilayer WTe , d ∼ E H = 10 [V · m − ] and J ext = 10 [A · m − ] , respec-tively. Thus, we estimate σ y ; xx ( σ xx ) ∼ − [A − · V · m ] , (B18)for the bilayer WTe . The coordinates of (non)linearconductivity tensors are chosen to be the same as thosein Ref. [16].Comparing Eq. (B17) with Eq. (B18), we expect thatthe nonlinear Hall response in BaMn As may be muchlarger than that observed in WTe . It should be noticedthat the nonlinear Hall response of BaMn As is due tothe Drude term proportional to the square of the relax-ation time O ( τ ), while that of WTe is considered tobe determined by the BCD term or extrinsic contribu-tions, which are O ( τ n +1 ) (See Table II and Ref. [43]).Hence, cleanness of the sample enhances the nonlinearHall conductivity in BaMn As much more than it doesin WTe .
3. Nematicity-assisted dichroism
Next, we investigate the Drude contribution to theNLC induced by the magnetic field. Diagonalizing the3effective Hamiltonian in Eq. (B1) with h (cid:54) = 0, we obtainthe energy spectrum E k = (cid:15) ( k ) ± (cid:112) V AB ( k ) + g ( k ) + h ± | λ | , (B19)where λ = V AB ( k ) h + [ g ( k ) · h ] . We see that theenergy spectrum is modified from Eq. (B9), and theKramers degeneracy is split due to violation of the PT -symmetry. As we discussed in Sec. III B, the in-planemagnetic field gives rise to the nematicity in the xy -plane.The corresponding term is represented by the couplingbetween the g -vector and the magnetic field given by[ g ( k ) · h ] . The energy spectrum is therefore symmetricagainst the transformation h → − h .For comparison, we consider the two-band Hamilto-nian for noncentrosymmetric systems H ( k ) = ˜ (cid:15) ( k ) + ˜ g ( k ) · σ , (B20)where the g -vector consists of the ASOC and the mag-netic field, ˜ g ( k ) = ˜ g ( k ) + h . The energy spectrum isobtained as E = ˜ (cid:15) ( k ) ± | ˜ g ( k ) | . Because of the couplingterm ˜ g ( k ) · h , the dispersion is not invariant when theexternal field is inverted [11]. Thus, the effect of magneticfield is significantly different between the noncentrosym-metric systems and locally-noncentrosymmetric systemswith parity-violating magnetic order.As we discussed in Sec. III, a longitudinal NLC σ z ; zz D isinduced by the nematicity caused by the in-plane mag-netic field. We present a numerical result of σ z ; zz D inFig. 6. The parameters are the same as Fig. 5 except foran additional magnetic field, h (cid:107) [110] with | h | = 0 . As a realistic parame-ter of the chemical potential should be near the top ofthe lower band ( µ ∼ − .
5) [65]. In this region, the lon-gitudinal NLC is in the order of σ z ; zz D ∼ − which ismuch smaller than σ z ; zz D ∼ − in the heavily-doped re-gion. This is partly because lightly-doped holes lead toa Fermi pocket near k = . In the vicinity of k = the inter-sublattice hopping term V AB ( k ) surpasses thesASOC g ( k ), and therefore, the effect of the sASOC isnot significant [20]. In fact, when the sign of the hop-ping integrals t i in Eq. (B7) is inverted, the hole pocketappear around k = ( π, π, π ), where the magnitude ofsASOC is comparable to the inter-sublattice hopping en-ergy. Then, the NLC is significantly enhanced. The field-induced dichroism is strongly enhanced in materials hav-ing strongly spin-orbit coupled Fermi surfaces.The magnitude of the nonlinear response is estimatedto be σ z ; zz D ∼ − [A · V − ]. Because a controllableparameter in the experiment is the electric current ratherthan the electric field, we rewrite the nonlinear responseby the electric field generated by the electric current E z = ρ zz J z + ρ z ; zz ( J z ) . (B21)The linear term is usually much larger than the second-order term, and hence resistivity tensors are approxi-mated by [11] ρ zz = 1 σ zz , ρ z ; zz = − σ zz σ z ; zz ( σ zz ) . (B22) Then, we obtain ρ zz ∼ − [Ω · m] and ρ z ; zz ∼− − [Ω · m · A − ] when we assume σ zz ∼ [A · V − · m − ] and σ z ; zz D ∼ − [A · V − ] . Taking J z ∼ [mA · mm − ] , the electric field responsible for thelinear and nonlinear conductivity are estimated to be E (1) ∼ [V · m − ] and E (2) ∼ − − [V · m − ] , re-spectively. Although the voltage drop due to the non-linear conductivity is tiny, it may be detected via theAC measurement [11]. Furthermore, the nonlinear re-sponse is enhanced by applying a large electric currentin a microscopic sample. A recent experimental study ac-tually detected a longitudinal NLC in BaMn As underthe magnetic field [58]. z ; zz > $ 9 @ × (a) z ; zz > $ 9 @ × (b)FIG. 6. (a) Drude term of a longitudinal NLC σ z ; zz D under theexternal magnetic field as a function of the chemical potential.(b) Enlarged plot near the top of the band. Green-coloredshaded area indicates the metallic regime. The parametersare T = 0 . γ − = 1 . × , and h = 0 .
01. We take N = 135 .
4. Nonlinear Hall conductivity due to Berrycurvature dipole
The BCD term of nonlinear Hall conductivity can beunderstood by considering the spin-momentum locking[Fig. 7 (a)]. In the presence of the spin-momentum lock-ing in electronic states, an electric field causes a shift of4Fermi surfaces and accordingly changes the momentum-resolved spin polarization near Fermi surfaces. As a re-sult, a net spin polarization is induced in a steady state.This is called Edelstein effect [67] and has been inten-sively discussed in recent spintronics research [68]. Wecan intuitively understand the nonlinear Hall responsearising from the BCD in an analogous way, as discussedbelow and illustrated in Fig. 7.For instance, let us consider a system with BCD, D xy = D yx , which is investigated in Sec. III C. Berrycurvature on a Fermi surface can be illustrated as inFig. 7 (a), where the total Berry curvature of occupiedstates, Ω = (cid:82) d k (cid:80) a f ( (cid:15) k a ) Ω a ( k ), is completely canceledin the momentum space. At the first step, an appliedelectric field E (cid:107) ˆ x induces a shift of the Fermi surfaceand a total Berry curvature along the y axis, Ω y , conse-quently emerges in a steady state [Fig. 7 (b)]. At the sec-ond step, because of the electric field E (cid:107) ˆ x , the flowingelectric current is bent towards the z -direction as it is bythe anomalous Hall effect due to the dynamically-inducedBerry curvature Ω y [Fig. 7 (c)]. As a consequence, thenonlinear Hall response denoted by the component σ z ; xx BCD occurs. All the other nonlinear Hall response coefficientsinduced by the BCD, D xy = D yx , can be explained bya similar argument. The allowed components satisfy therelation, σ z ; xx BCD = − σ z ; yy BCD = − σ x ; xz BCD = 2 σ y ; yz BCD . (B23)Note that a similar argument can be found in Ref. [69].From the above-mentioned argument, we find that thechiral BCD given by Tr[ ˆ D ] does not contribute to thenonlinear Hall response. In a system with a chiral BCD, D xx = D yy = D zz , the Berry curvature induced by anelectric current is always parallel to the current. Then,the Hall response illustrated in Fig. 7 does not occur.The chiral BCD is, therefore, irrelevant to the nonlinearHall response while the BCD itself can be finite in chiralcrystals.
5. Analytical and numerical results of BCD term
In this section we calculate the BCD term of nonlinearHall conductivity σ BCD . First, we derive an analyticalexpression in simplified models. In the two-band Hamil-tonian in Eq. (B20), the Berry curvature is analyticallyobtained asΩ λ ± = ∓ (cid:15) µνλ [ ∂ µ ˆ g ( k ) × ∂ ν ˆ g ( k )] · ˆ g ( k ) , (B24)where the subscript ± represents the upper/lower bandand ˆ g ( k ) = ˜ g ( k ) / | ˜ g ( k ) | is an unit vector [37]. Com-pared to this case, the Berry curvature of the four bandHamiltonian [Eq. (B1)] for odd-parity magnetic multi-pole systems does not have a simple expression because ofthe inter-sublattice hopping term V AB ( k ). Therefore, wehere adopt V AB ( k ) = 0 for simplicity to present analyti-cal results. Although this simplification is not reasonable for BaMn As , it may be appropriate in other magneticsystems possessing valley or layer degree of freedom [70].Anyway an intuitive understanding for the BCD of mag-netic multipole systems is obtained below.Ignoring V AB ( k ), we obtain two pairs of bands labeledby the sublattice degree of freedom. The Berry curvatureis given byΩ λ ( i ) ± = ∓ (cid:15) µνλ (cid:2) ∂ µ ˆ g ( i ) ( k ) × ∂ ν ˆ g ( i ) ( k ) (cid:3) · ˆ g ( i ) ( k ) , (B25)where i = ( A, B ) denotes the sublattice degree of free-dom. The g -vector consists of the sASOC, the molecularfield of antiferromagnetic order, and the Zeeman field, g (A) ( k ) = g ( k ) + h AF + h , (B26) g (B) ( k ) = − g ( k ) − h AF + h . (B27)A symmetry analysis shows that the BCD, D xy = D yx ,appears under the external magnetic field along the z -axis, h = h ˆ z . We therefore consider x and y componentsof the Berry curvature. For the model Hamiltonian inEq. (B1) with the assumption that V AB ( k ) = 0 and α =0, the x -component is obtained asΩ x ( A ) − = + 12 (cid:2) ∂ y ˆ g (A) ( k ) × ∂ z ˆ g (A) ( k ) (cid:3) · ˆ g (A) ( k ) , (B28)= − | g (A) ( k ) | α α k x sin k x k y sin k y k z . (B29)The model parameters in Eqs. (B7) and (B8) imply | h AF | (cid:29) | t i | , | α i | , | h | , (B30)and hence we perturbatively deal with the external mag-netic field as1 | g (A) ( k ) | (cid:39) | h AF | (cid:18) − g ( k ) + h ] · h AF h (cid:19) . (B31)We neglect the term g · h AF /h − at the right hand sidesince the term gives a small correction to the leadingterm. In the low density region, we haveΩ x (A) − (cid:39) − α α | h AF | (cid:18) − hh AF (cid:19) k x k y . (B32)Calculating Ω x (B) − in the same way, we express the BCDsummed over lower bands by D yx ( A ) − + D yx ( B ) − = (cid:90) d k (2 π ) d (cid:104) ∂ y Ω x (A) − f ( (cid:15) (A) − ) + ∂ y Ω x (B) − f ( (cid:15) (B) − ) (cid:105) (B33)= − α α | h AF | (cid:90) d k (2 π ) d k x (cid:34)(cid:8) f ( (cid:15) (A) − ) − f ( (cid:15) (B) − ) (cid:9) − hh AF (cid:8) f ( (cid:15) (A) − ) + f ( (cid:15) (B) − ) (cid:9)(cid:35) , (B34)5 (a) (b) (c)FIG. 7. A schematic picture of the nonlinear Hall effect due to the BCD. Orange-colored circles and blue-colored arrowsrepresent a Fermi surface and Berry curvature, respectively. (a) In equilibrium, the Berry curvature on the Fermi surface iscompletely compensated. (b) When an electric field E (cid:107) ˆ x is applied, the Fermi surface is shifted to the field direction, and thenet Berry curvature Ω (cid:107) ˆ y is induced as in the case of the Edelstein effect. (c) Responding to the second electric field E (cid:107) ˆ x ,the electric current flowing along the x -axis is bent towards the z -axis owing to the anomalous Hall effect by the induced Berrycurvature Ω y . This is an intuitive explanation of the BCD term, σ z ; xx BCD . where the momentum dependence of energy (cid:15) ( i ) − ( i =A, B) is implicit. The impact of the magnetic field is two-fold. One is the Fermi surface term [ f ( (cid:15) (A) − ) − f ( (cid:15) (B) − )]determined by the Zeeman energy, and the other is theFermi sea term [ f ( (cid:15) (A) − ) + f ( (cid:15) (B) − )] derived from a cor-rection to the Berry curvature proportional to the exter-nal field h . The former contribution is evaluated by (cid:90) d k (2 π ) d k x (cid:8) f ( (cid:15) (A) − ) − f ( (cid:15) (B) − ) (cid:9) (cid:39) (cid:90) d k (2 π ) d k x ∂f ( (cid:15) (A) − ) ∂(cid:15) (cid:12)(cid:12)(cid:12) h =0 ( − h ) , (B35)= mh π (2 m | (cid:15) F | ) / , (B36)where the energy spectrum at h = 0 is approximated bythe parabolic band dispersion, (cid:15) (A) − = − k / m in thelast line, and (cid:15) F is the Fermi energy of hole carriers. TheFermi sea term is evaluated as − hh AF (cid:90) d k (2 π ) d k x (cid:0) f ( (cid:15) (A) − ) + f ( (cid:15) (B) − ) (cid:1) (cid:39) − hh AF (cid:90) d k (2 π ) d k x [ f ( (cid:15) (A) − )] (cid:12)(cid:12) h =0 , (B37)= 2 mh π (2 m | (cid:15) F | ) / | (cid:15) F | h AF . (B38)When the molecular field h AF is much larger than theFermi energy (cid:15) F , the Fermi sea term is negligible com-pared to the Fermi surface term. Taking only the Fermisurface term, we obtain the magnetic-field-induced BCD, D yx ( A ) − + D yx ( B ) − (cid:39) − α α | h AF | mh π (2 m | (cid:15) F | ) / . (B39)From this expression the nonlinear Hall conductivity σ z ; xx BCD is given by σ z ; xx BCD = − q γ (cid:16) D yx ( A ) − + D yx ( B ) − (cid:17) . (B40) The magnetic-field-induced nonlinear Hall response isproportional to the external field and vanishes at h = 0where the PT -symmetry is preserved. In other words,this nonlinear response is tunable by using the magneticfield.The presence of the magnetic-field-induced BCD andnonlinear Hall response is supported by numerical cal-culations. In the calculations, we take into account theinter-sublattice hopping term V AB ( k ) and assume the pa-rameters in Eqs. (B7) and (B8), and h = 0 .
01. We con-firmed that the numerical result is consistent with thesymmetry analysis in Eq. (B23). Note that the nonlin-ear Hall conductivity is comparable between the cases V AB ( k ) = 0 and V AB ( k ) (cid:54) = 0. Thus, above discussionsfor a simplified model are qualitatively appropriate.In Fig. 8 we show the numerical result of σ z ; xx BCD as afunction of the chemical potential ranging around thelower energy bands [Eq. (12)]. Because of the same rea-son as that for the nematicity-assisted dichroism shown inFig. 6, the nonlinear conductivity is small in the lightlyhole-doped region. The magnitude is furthermore sup-pressed by the factor α α / | h AF | revealed in Eq. (B39).A typical value is estimated as σ z ; xx D ∼ − [A · V − ] inour unit. The response may be enhanced when the ex-change splitting is smaller than sASOC or comparable toit. Such situation realizes in antiferromagnetic materialsincluding BaMn As near N´eel temperatures. Appendix C: Extrinsic contribution to nonlinearconductivity
The formulation presented in Sec. A is based on theclean limit ( τ → ∞ ) with the phenomenological scat-tering rate, and thus intrinsic contributions are studiedthroughout our work. On the other hand, extrinsic con-tributions such as impurity scattering and electron corre-lations may be non-negligible in a realistic situation. Forinstance, it is well-known that the anomalous Hall effectis affected by impurities and that such extrinsic contribu-tions may overwhelm the intrinsic contributions [71, 72].6 z ; xx > $ 9 @ × FIG. 8. BCD term of nonlinear Hall conductivity σ z ; xx BCD underthe magnetic field along the z -axis. Green-colored shadedregion indicates the metallic region. The parameters are T =0 . γ − = 1 . × , and h = 0 .
01. We take N = 135 . Moreover, a recent experiment pointed out that the im-purity scattering contributes to the nonlinear Hall re-sponse in WTe as well as the Berry curvature dipole [46].Therefore, it is necessary to examine whether our clas-sification shown in Table II is modified when extrinsiccontributions included.According to a recent theoretical work [43], the formulafor NLC is modified by the skew scattering and side jumpeffects arising from the impurity scattering. Ref. [43]clearly shows that the NLC in T -symmetric systems in-cludes such extrinsic terms in addition to the Berry cur-vature dipole effect in Eq. (A55) and that the Drude termappears as a result of the T -symmetry breaking. On theother hand, it has not been studied whether the nonlinearconductivity formula in PT -symmetric systems is mod-ified in the presence of the impurity scattering. Thus,we below consider the effect of impurity scattering in the PT -preserved systems. Interestingly, we will see that the PT -symmetry leads to strong suppression of the extrinsiccontributions.For simplicity, we consider spinless systems and takeinto account on-site impurities having the delta-function-shaped potential. The potential energy of the randomly-distributed impurities is written by V imp (ˆ r ) = (cid:88) i v δ (ˆ r − R i ) , (C1)where v denotes the strength of impurity potential, i la-bels the impurities, and R i denotes the position of the i -th impurity. Owing to the random distribution of im-purities, the random average (cid:104)(cid:105) imp satisfies the relation (cid:104) V imp (cid:105) imp = 0 . (C2)We below omit the subscript ‘imp’ of V imp unless oth-erwise mentioned. Based on the results in Ref. [43],the leading extrinsic contributions to the NLC σ µ ; νλ ext arethree-fold: σ µ ; νλ ext = σ µ ; νλ sj,1 + σ µ ; νλ sj,2 + σ µ ; νλ sk , (C3) in which the abbreviated labels ‘sj‘ and ’sk’ represent‘side jump’ and ‘skew scattering’ effects [43], respectively.These contributions are added to the Berry curvaturedipole term in the T -symmetric systems and may explainthe observed nonlinear Hall response in WTe [46].First, we investigate the side jump contributions. Theside jump terms are determined by the side jump velocitygiven by v µ (sj) a = (cid:88) b W sy ab δr µ ba , (C4)where a = ( a, k a ) [ b = ( b, k b )] denotes Bloch states la-beled by the band index a ( b ) and crystal momentum k a ( k b ). W sy ab = W ab / W ba / W ab . W ab is definedwith the T-matrix as W ab = 2 π (cid:126) | T ab | δ ( (cid:15) ab ) , (C5)in which we introduced (cid:15) ab = (cid:15) k a a − (cid:15) k b b . Importantly,the side jump velocity depends on the positional shift δr µ ab given by δr µ ab = (cid:28) u a ( k a ) (cid:12)(cid:12)(cid:12)(cid:12) i ∂u a ∂k µa ( k a ) (cid:29) − (cid:28) u b ( k b ) (cid:12)(cid:12)(cid:12)(cid:12) i ∂u b ∂k µb ( k b ) (cid:29) − (cid:18) ∂∂k µa + ∂∂k µb (cid:19) arg ( V ab ) , (C6)which represents the coordinate shift during the scatter-ing process a ← b [73]. With the impurity potential inEq. (C1), the matrix element of V is given by V ab = cv (cid:104) u a ( k a ) | u b ( k b ) (cid:105) (cid:88) i e − i ( k a − k b ) · R i , (C7)= cv I ab ρ k a − k b , (C8)where c denotes a scalar constant, I ab = (cid:104) u a ( k a ) | u b ( k b ) (cid:105) , and ρ Q = (cid:80) i exp ( − i Q · R i ). Since thematrix element V ab depends on k a − k b , the positionalshift is simplified as δr µ ab → (cid:28) u a ( k a ) (cid:12)(cid:12)(cid:12)(cid:12) i ∂u a ∂k µa ( k a ) (cid:29) − (cid:28) u b ( k b ) (cid:12)(cid:12)(cid:12)(cid:12) i ∂u b ∂k µb ( k b ) (cid:29) − (cid:18) ∂∂k µa + ∂∂k µb (cid:19) arg (cid:104) u a ( k a ) | u b ( k b ) (cid:105) , (C9)which is similar to the so-called shift vector [7]. Now, weconsider the constraint due to the PT -symmetry. TheBloch state is transformed by PT -symmetry into | u a ( k a ) (cid:105) → | u a ( k a ) (cid:105) = | u a ( k a ) (cid:105) e − iφ a ( k a ) , (C10)where φ a denotes the phase factor determined by theadopted gauge. Accordingly, we obtain the relation of7the connection term (cid:104) u a ( k a ) | i∂ µ u a ( k a ) (cid:105) given by (cid:28) u a ( k a ) (cid:12)(cid:12)(cid:12)(cid:12) i ∂u a ∂k µa ( k a ) (cid:29) = i (cid:28) ∂u a ∂k µa ( k a ) (cid:12)(cid:12)(cid:12)(cid:12) u a ( k a ) (cid:29) , (C11)= − (cid:28) u a ( k a ) (cid:12)(cid:12)(cid:12)(cid:12) i ∂u a ∂k µa ( k a ) (cid:29) − ∂φ a ( k a ) ∂k µa . (C12)Similarly, using the PT -symmetry, we can seearg (cid:104) u a ( k a ) | u b ( k b ) (cid:105) = − arg (cid:104) u a ( k a ) | u b ( k b ) (cid:105) + φ b ( k b ) − φ a ( k a ) . (C13)Thus, the positional shift satisfies δr µ ab = − δr µ ab = 0 , (C14)which indicates that the side jump velocity v µ (sj) vanishesby the PT -symmetry. As a result, the side jump contri-butions σ sj,1 and σ sj,2 in Eq. (C3) are forbidden in the PT -symmetric systems. We can also understand thisconclusion intuitively from the fact that the positionalshift δr µ ab occurring in the forward scattering roughlycorresponds to ∼ (cid:15) µνλ Ω ν ( k λa − k λb ) where we introducethe Berry curvature Ω [73]. We can see immediatelythat δr µ ab = 0 since Ω = 0 due to the PT -symmetry.Although the derivation has assumed the delta-function-shaped impurity potential, the positional shift is gener-ally suppressed by the PT -symmetry in the case of thespherically-symmetric impurity potential whose Fouriercomponent V ab is a function of k a − k b .Next, we consider the skew scattering contribution de-noted by σ sk in Eq. (C3). Although we do not show theexpression of the skew scattering contribution (Ref. [43]for details), an important ingredient taking the wholeexpression is the anti-symmetric component of the scat-tering amplitude W as ab = W ab / − W ba /
2. Thus, theskew scattering contribution is closely related to the anti-symmetric scattering between the states a and b .To evaluate W as , perturbation expansion of the T-matrix is performed here. The T-matrix is defined as T ab = (cid:68) ψ k a a (cid:12)(cid:12)(cid:12) ˆ V (cid:12)(cid:12)(cid:12) Ψ b (cid:69) , (C15)where the ket vector | Ψ b (cid:105) is obtained by solving Lipman-Schwinger equation with the impurity potential ˆ V . Theequation is written as | Ψ b (cid:105) = | ψ k b b (cid:105) + ˆ V(cid:15) k b b − ˆ H + iη | Ψ b (cid:105) , (C16)where ˆ H denotes the unperturbed Hamiltonian and η ( >
0) is an infinitesimal number. We therefore obtain per-turbation expansion of the T-matrix in the power of thematrix element V ab . The lowest-order contribution is ob-tained by replacing T ab with V ab , that is, | Ψ b (cid:105) → | ψ k b b (cid:105) .The resulting contribution to W ab is given by W (2) ab = 2 π (cid:126) | V ab | δ ( (cid:15) ab ) , (C17) which is O ( V ). The expression is symmetric under thepermutation a ↔ b and therefore this term does not yielda skew scattering. The third-order contribution to W ab is given by (cid:126) π W (3) ab = (cid:88) c (cid:32) (cid:104) V ∗ ab V ac V cb (cid:105) imp (cid:15) bc + iη + (cid:104) V ∗ ac V ∗ cb V ab (cid:105) imp (cid:15) ac − iη + c.c (cid:33) δ ( (cid:15) ab ) , (C18)= (cid:88) c (cid:32) (cid:104) V ∗ ab V ac V cb (cid:105) imp (cid:15) bc + iη + (cid:104) V ∗ ac V ∗ cb V ab (cid:105) imp (cid:15) bc − iη + c.c (cid:33) δ ( (cid:15) ab ) . (C19)Considering the impurity potential defined in Eq. (C1),we have (cid:104) V ∗ ab V ac V cb (cid:105) imp ∝ I ba I ac I cb (cid:10) ρ ∗ k a − k b ρ k a − k c ρ k c − k b (cid:11) imp . (C20)By taking the average over random distribution of impu-rities [74], we obtain (cid:104) ρ k ρ k ρ k (cid:105) imp = N imp δ k + k + k , + N imp ( N imp − × ( δ k , δ k + k , + δ k , δ k + k , + δ k , δ k + k , )+ N imp ( N imp −
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