Nonlinear orbital response across topological phase transition in centrosymmetricmaterials

Margarita Davydova,1, 2 Maksym Serbyn,2 and Hiroaki Ishizuka1, 3

1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA2IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria

3Department of Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo, 152-8551, Japan

Nonlinear optical responses are often used as a probe for studying electronic properties of mate-rials. For topological materials, studies so far focused on the photogalvanic electric current, whichrequires breaking inversion symmetry. In this work we present a theory of orbital current responsein inversion-symmetric topological insulators. We find a symmetry-allowed orbital current responsethat occurs in centrosymmetric materials under illumination by linearly polarized light. The sign ofthe dc nonlinear conductivity reflects the Z2 index and the conductivity changes sign at the tran-sition between trivial and topological insulator phases. We derive an expression for the nonlinearorbital photocurrent for a general class of models with two doubly degenerate bands, and discussits specific applications in the cases of the Bernevig-Hughes-Zhang model and the 1T’ phase oftransition metal dichalcogenides. Experimental setups for observation of the orbital current are alsodiscussed.

INTRODUCTION

The topology of band structure manifests itself in a richvariety of transport phenomena. A key quantity in thesephenomena is the Berry curvature of electronic bands [1].The Berry curvature modifies the equations of motion inthe low energy limit and is responsible for the emergenceof the anomalous velocity which is perpendicular to theexternal electric field [2]. The effect of modified elec-tron dynamics manifests as intrinsic [2, 3] and extrin-sic [4, 5] anomalous Hall effects. Recent studies deter-mined the effect of the Berry curvature on low-frequencynonlinear responses, such as the nonlinear Hall effect [6],photovoltaic effects [7, 8], and magnetoresistance [9–11].For degenerate bands, the Berry curvature becomes non-Abelian [12, 13] and also affects the dynamics of Blochwave packets [14]. It is responsible for the emergence ofa dissipationless intrinsic spin Hall current in p-dopedsemiconductors [15, 16].

Optical responses involving interband electron transi-tions also reflect the topological nature of band struc-ture. In photovoltaics, the shift current [17–22] has expe-rienced a revival of interest in recent years. Recent stud-ies in Weyl semimetals found large second harmonic [23]and photovoltaic [24, 25] responses; first-principle calcu-lations demonstrated that they are related to the Weylnodes [26]. On the other hand, the quantization of an-other kind of photovoltaic response, the injection current,has been observed both theoretically [27] and experimen-tally [28]. All these phenomena, which take place in non-centrosymmetric materials, were shown to be related tothe Berry curvature of electronic bands.

In contrast, the role of topology in light-matter interac-tions in centrosymmetric materials remains largely un-explored. In such systems, which possess both time-reversal and inversion symmetries, photovoltaic andsecond-harmonic responses of electric current are sym-

metry prohibited. In this work, we construct an orbitalcurrent whose response reflects the topological natureof inversion-symmetric topological insulators [29, 30]; inparticular, the sign of the orbital current depends on thetopological index of 2D topological insulators.

To explicate the nature of this photo-induced orbital cur-rent, we present a general theory for semiconductors withtwo valence and two conduction bands. Focusing on thecase of linearly polarized light, we find at the topologicalphase transition, the nonlinear conductivity correspond-ing to the orbital current changes sign, which reflects thechange in the Z2 topological index [Fig. 1a]. For theBernevig-Hughes-Zhang (BHZ) model [31], an effectivemodel for HgTe/CdTe quantum wells, the orbital cur-rent corresponds to a transport of intracell electric po-larization. Thus, accumulation of the orbital current willproduce ferroelectric polarization at the two ends of thematerial [Fig. 1b-c]. Similar physics also appears in tran-sition metal dichalcogenides MX2. These examples showthat the orbital response reflects topological propertiesof centrosymmetric materials and can be connected toexperimentally observable quantities.

RESULTS

Model. We consider an electron model with fourbands. The most general form of the two-band 4 × 4Bloch Hamiltonian is

H = d0(k)I4 +

5∑a=1

da(k)Γa, (1)

where I4 is 4 × 4 identity matrix and Γa (a = 1, . . . , 5)are the Dirac matrices that satisfy the Clifford alge-bra

Γa,Γb

= 2Iδa,b. We choose them as Γa =

τz, τy, τxσx, τxσy, τxσz, where Pauli matrices σα andτα (α = x, y, z) are for spin and orbital degrees of free-

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(d)

topological phase trivial phase

+ =

(b) (c)

FIG. 1. (a) The orbital current changes its sign as the massterm (gap) reverses sign at the topological phase transition.Schematics of the τx-orbital separation as a result of the flowof orbital current due to irradiation are illustrated below (b)and above (c) the topological phase transition. (d) Construc-tion of an orbital with a fixed eigenvalue of τx operator re-quires an addition of ws and wx±iy ≡ wpx±ipy Wannier orbitalstates. This results in displacement of the electron density inx-direction with respect to the atomic center.

dom, respectively; τα can be atomic orbitals, differentlayers in layered two-dimensional materials, or differentsublattices. The model (1) has two doubly-degeneratebands (1, 1′) and (2, 2′) with energies ε1(k) = ε1′(k),ε2(k) = ε2′(k), respectively:

ε1,2(k) = d0(k)∓

√√√√ 5∑a=1

da(k), (2)

where the degeneracy is a consequence of the Kramerstheorem in the presence of the inversion symmetry.

The time reversal and inversion symmetries are imple-mented by the operators

T = iσyK, P = τzP, (3)

where K denotes complex conjugation and P is theinversion operator acting on the real-space coordinateP f(r) = f(−r). The Hamiltonian (1) is invariant underboth T and P operators, provided that functions d0,1(k)[d2,...,5(k)] are even [odd] with respect to the inversion ofmomentum.

Orbital current. Systems with inversion symmetryallow for a nonzero second-order dc response for certainkinds of orbital currents; more specifically, these currentscorrespond to the transport of orbitals that transform

into each other under inversion (See Materials and Meth-ods for details). For concreteness, we consider the orbitalcurrent given by

Jorb = 〈u〉 = 〈τx,v〉 , (4)

where u = P 1xvP

1x − P 2

xvP2x = τx,v is the staggered

velocity operator, P 1,2x = 1

2 (I2 ± τx) are the orbital pro-jection operators, and 〈· · · 〉 is the thermal average. Incontrast to the current of the orbital angular momen-tum [32, 33], the operator u does not change sign underP . Thus, the corresponding second-order nonlinear opti-cal response is allowed by the symmetry.

An example of the orbital degree of freedom correspond-ing to the current in Eq. (4) is the electric polarization.Suppose τz = 1 is an s orbital |s〉 and τz = −1 is a p or-bital state |pσ〉 = |px + iσpy〉 (σ = ±). In this case, theeigenstates of τx are 1√

2(|s〉 ± |pσ〉). In particular, this

reflects the case for the BHZ model [31]. Such a super-position will cause a displacement of the electron densityin the x-direction, as is shown in Fig. 1d. This displace-ment corresponds to the intracell electric polarization.The flow of the polarization can lead to an accumulationof the polarization at the edges of the sample or in thecontact leads.

We note that, in addition to the one introduced above,one can consider other components of orbital and spincurrents. The τx orbital current operator is the sim-plest one that by construction involves only orbital de-grees of freedom in the definition and possesses the nec-essary symmetry properties. In the Methods section weuse symmetry considerations to classify all such opera-tors and discuss their properties. We also show that theorbital current introduced above can be split into con-served and nonconserved parts, similarly in spirit to theRef. [32].

Second-order orbital response. We calculate thenonlinear orbital current by combining Floquet bandformalism with the Keldysh Green’s functions tech-nique [34, 35], see the Methods section. In the caseof the linearly polarized light, the orbital current readsJorb = −iTr (uG<), where G< is the lesser Green’s andthe trace denotes the combination of integrating over thefrequencies ω, integrating over the Brillouin zone, and thematrix trace in the spin + orbital basis. For the modelin (1) and assuming a 2D system, the calculation gives:

Jorb =

∫d2k

(2π)2

(j

(1)orb + j

(2)orb + j

(3)orb

), (5)

where

j(1)orb= i

Γ|A|2

∆2Ω + Γ2

Tr[V +Uk − (Uk)+V

], (6)

j(2)orb=

(ε2(k)− ε1(k)− Ω) |A|2

∆2Ω + Γ2

Tr[V +Uk + (Uk)+V

](7)

j(3)orb=u11+2τ

|A|2Γ

∆2Ω + Γ2

detV [u22−u11]+(1, 2↔1′, 2′)(8)

3

Here ∆2Ω = (ε2 − ε1 − Ω)2 + 4|A|2

(|v0

12|2 + |v01′2|2

), Ω

is the light frequency, Γ−1 is the decay time, and A isthe vector potential that corresponds to the intensity ofthe light. The prefactor Γ/(∆2

Ω + Γ2) becomes a deltafunction when Γ → 0. The matrices V and Uk are theinterband components of the velocity operator,

V =

(v0

12 v012′

v01′2 v0

1′2′

),

and the derivatives of the staggered velocity operator

Uk =

( ∂u∂kα

)12

(∂u∂kα

)12′(

∂u∂kα

)1′2

(∂u∂kα

)1′2′

,

respectively. Here, v0 = ∂H/∂kα and kα ≡ k ·A/A.

The first term in Eq. (5) corresponds to the shift-current-type response, which will be the main focus of our work.Equation (6) has the same structure as the shift currentin nonlinear response theory [17, 19]. The current inEq. (7) vanishes in the presence of time reversal symme-try, whereas the last contribution, Eq. (8), is the injectioncurrent proportional to the relaxation time. The injec-tion current vanishes for the case of linearly polarizedlight. However, it gives a non-zero current in the caseof circularly-polarized light; this will be discussed else-

where. In the rest of this work, we study j(1)orb focusing

on the case of linearly polarized light.

The presence of U(2) × U(2) gauge symmetry may leadto a non-Abelian Berry phase in the system. It en-

ters the expression for orbital shift current j(1)orb explic-

itly through covariant derivative in k-space(∂u∂kα

)ij

=

∂uij∂kα

+ i[aα,u]ij , where the matrix elements for non-Abelian Berry connection for the quasimomentum in thedirection of A are aαij = −i 〈i| ∂

∂kα|j〉. For more details,

we direct the reader to the Supplementary section.

Orbital current in four-band model. For thefour-band model in Eq. (1), the current parallel to thelight polarization reads

J(1)orb =

∫∆Ω(k)=0

dk

2π

4|A|2

ε|∇k∆Ω|[d′′0(d2d

′1−d1d

′2) + εlsrglg

′sg′′r

],

(9)

where the primes denote derivatives over kA, ε =√∑5i=1 d

2i , and ε is the Levi-Civita tensor. We have

introduced a short-hand notation for the spin-orbit cou-pling coefficients g(k) = d3, d4, d5. The integral is overthe surface in the Brillouin zone that satisfies ∆Ω = 0[Fig. 2b-c]. For a direct-gap semiconductor when Ω isclose to the band gap, the integral is taken over theboundary of a small pocket around the band bottom (e.g.contour (ib) in Fig. 2b). This is the case for topologicalinsulators in the vicinity of the topological phase transi-tion.

FIG. 2. (a) The dependence of the total orbital current

J(1)orb ≡ Jx on the light frequency for BHZ model. εs/tss = 3.5

is below and εs/tss = 4.5 is above the phase transition fromtopological to the trivial insulator that occurs as εs/tss crossesthe value 4. The dashed gray line corresponds to the minimumfrequency. The bandstructure is shown below (b) and above(c) the topological phase transition where the lines (ib, c) and(iib, c) are the equal energy difference contours at which theoptical transitions (shown by arrows) occur. The correspond-ing frequencies are indicated in panel (a). (d), (e) - mapsof the orbital current density jx(k) at (b) εs/tss = 3.5 and(c) εs/tss = 4.5. The other parameters are εp = tpp = 0,tsp/tss = 1.

To uncover the relation between the topological index

and j(1)orb, we consider a situation where d1(k) smoothly

changes sign at an odd number of time-reversal-invariantmomenta (TRIM) Γi (i = 0, 1, 2, . . . ). Due to this prop-erty d2,3,4,5(k) vanish at Γi, but their first derivativesremain nonzero. Therefore, Eq. 9 for Ω ≈ ε2(k =

4

Γi)− ε1(k = Γi) reads

j(1)orb(k ≈ Γi) ∝ −d′′0d′2sign(d1)k=Γi , (10)

where we have used ε(Γi) = |d1(Γi)|.The signs of d1(k) at at Γi-points can be used to expressthe Z2 invariant of Hamiltonian (1) as

δZ2=∏i

√Det[B(Γi)]

Pf[B(Γi)]=∏i

sign[d1(k = Γi)], (11)

where Bmn(k) = 〈um(−k)| T |un(k)〉 is the sewingmatrix[36]. The Z2 indicator δZ2 corresponds to the Z2

topological index in 2D or to the strong topological num-ber in 3D if the sum is over all TRIMs in the half of theBrillouin zone (BZ). If the summation runs over TRIMsin the same plane in a half of BZ, the index becomes aweak topological number in 3D. Comparing the Z2 indexto Eq. (10), we obtain that∏

i

sign[j

(1)orb(k ≈ Γi)

]∝ δZ2

. (12)

This links the behavior of a physical observable, the or-bital current, to the indicator of topological phase tran-sition.

The quantity defined above changes the sign when the Z2

index changes as d′′0 and d′2 are finite at the phase tran-sition. In many relevant examples, the gap closes at oneTRIM, therefore measuring the sign of the orbital cur-rent when inducing transitions near that specific pointwould yield the topological index. Note that neither d′′0nor d′2 vanishes at this point unless it is required by someadditional symmetry. Therefore, a smooth change of thesign as d1(Γi) leads to a change in the sign of the or-bital current, which is a robust effect regardless of thematerial.

One can see that the sensitivity of the orbital current tothe change of the topological phase is physically mean-ingful from the following considerations. At the phasetransition the change of the parity of the wavefunctionsoccurs at the corresponding TRIMs[37]. Under crossingthe phase transition, the wavefunctions undergo the fol-lowing change:

ψ1(k)∣∣∆

= −Pψ2(−k)∣∣−∆

,

ψ1′(k)∣∣∆

= −Pψ2′(−k)∣∣−∆

,(13)

where ∆ = 2d1(k = 0) is value of the bandgap that ac-counts for the possibility of band inversion due to thechange in the sign of d1. Equation (13) implies that thewavefunctions above the transition are related to anotherband’s parity-inverted wavefunctions below the transi-tion. We use that Pψ2 = −T ψ2′ and Pψ2′ = T ψ2 torelate the action of parity and time-reversal symmetriesat the same point k in the BZ. In simple terms, flippingof the eigenfunction parity reverses the arrow of time for

transition processes, thus changing the sign of the cur-rent. Let us use Eq. (13) to argue that the expressionfor the orbital current in Eq. (6) changes sign. Both thev and vz velocity operators that enter the expression forthe orbital current density are odd under T . Using thisfact and the above transformation of the wavefunctionswe obtain v12(k)↔ v∗1′2′(−k), v12′(k)↔ −v∗12′(−k) and

v1′2(k) ↔ −v∗1′2(−k). It follows that Tr[V +Uk

]be-

comes complex conjugated at the point −k upon crossingthe phase transition. Since the expression for the shiftcurrent in Eq. (6) is the imaginary part of the trace andonly the k-even part of the current density survives af-ter integration, the shift current must necessarily changesign. Thus, the orbital current probes the change in theparity of wavefunctions at the topological phase transi-tion.

A similar argument explains why the current vanishesnear the gap-closing TRIM exactly at the phase transi-tion point. At this point the eigenfunctions of the Hamil-tonian also become parity eigenstates. In this case, thereis no difference between electron density distributions inthe initial and final states, and thus, there can be no shiftorbital current.

Bernevig-Hughes-Zhang model. As an example,we consider Bernevig-Hughes-Zhang (BHZ) model on asquare lattice, [31] an effective model for HgTe/CdTequantum wells. The model corresponds to the Hamilto-nian in Eq. (1) with the wave function basis |s, ↑〉, |s, ↓〉,|px + ipy, ↑〉, |px − ipy, ↓〉 and di(k) shown in Table I [36].By varying the thickness of the quantum wells, the con-stants εs and εp can be tuned to the transition pointbetween a trivial and topological insulator [31]. UsingEq. (9), the orbital current density in the direction oflight polarization reads:

j(1)orb =

2Γ|A|2

∆2Ω + Γ2

4a3tsp (tss − tpp)ε

f(kx, ky), (14)

with f(kx, ky) = − cos kxa[(εp − εs) cos kxa + 2(tss +tpp)(1 + cos kxa cos kya)], and a being the lattice con-stant. In the vicinity of the center of the BZ, the currentalong x-direction reads

j(1)orb

∣∣Γ→0

≈ −8πa3|A|2δ(∆Ω)tsp(tss − tpp) sign [d1(0)] .

(15)

Equation (14) explicitly shows the sign change at d1(k =0) = εs − εp − 4(tss + tpp)/2 = 0, reflecting the topo-logical phase transition associated with the gap closing

d0 (εs + εp)/2− (tss − tpp)(cosk · a1 + cosk · a2)

d1 (εs − εp)/2− (tss + tpp)(cosk · a1 + cosk · a2)

d2 2tsp sink · a1

d3, d4 0

d5 2tsp sink · a2

TABLE I. Coefficients of the BHZ Hamiltonian from Ref. [36].

5

at the Γ0 point. In Fig. 2a, we plot the total orbitalshift current as a function of the light frequency belowand above the phase transition. At small frequencies thelight probes the current density near the Γ0 point, wherethe gap closing occurs at the topological phase transition.At low frequencies, the sign of the current in Fig. 2a isthe opposite below and above the topological phase tran-sition (the relevant area is highlighted in yellow). Thevalue of the current remains nearly constant when thesurface of the integral is the boundary of a pocket. Thedifference in the sign of the current density, that is in-dicative of the topological phase transition, is also shownin Fig. 2d-e.

Exactly at the transition point, the current vanishes ink · p approximation, which predicts the behavior of thecurrent at small frequencies. For the BHZ model in thissection, the k · p Hamiltonian has the form

Hk·p = c0k2 + δτz + 2tspa (kx τy + ky τxσz) . (16)

Here we neglected a constant contribution, the mass termis δ = d1(k = 0) and the coefficient c0 comes from theseries expansion of d0(k). The Hamiltonian in (16) pro-duces an orbital current jorb ∝ δ, which equals to zeroat the phase transition where δ = 0. In our particularexample [Fig. 2a, εs/tss = 4] it acquires a correction onlyat quite large frequencies, greater than ∼ 2.5tss.

On the other hand, the magnitude and the sign of currentbehaves distinctly for high frequencies, when the vicinityof other TRIMs contributes to the current. At these fre-quencies the topology of the equal energy-difference con-tours (shown in Fig. 2b-c and d-e) changes as the lightfrequency increases. At Ω ≥ εs− εp the contour encirclesTRMs (0, π) and (π, 0), in addition to (0, 0). This corre-sponds to contours (iib) and (iic) in Figs. 2b-e, causinga jump of the sign of the current as the current densitynear these points is the same as that near Γ0 but is op-posite in sign, whereas the JDOS is finite regardless ofthe radius of the circle, as is expected in two dimensions.The contour encounters van Hove singularities upon fur-ther increase the frequency, which manifests as a growthof the response.

In experiment, the orbital current can give rise to an elec-tric polarization at the edges of the material. In the BHZmodel, the average of the operator τx is proportional tothe intracell polarization in x direction, Px = −(e/V )x(see Methods) where V is the volume of the unit cell.Therefore, the orbital polarization due to the photo-induced orbital current is related to the electrical intra-cell polarization [Fig. 1c-d]. The spatial distribution ofthe polarization is observable using, for example, opti-cal techniques [38] that can be extended to the photo-induced orbital current as well.

For an estimate of the magnitude of the current inHdTe/CdTe quantum wells, we use εs = 3.99 eV, εp =tpp = 0, tsp = tss=1 eV, which corresponds to a bandgap

d0∑

i=x,y

~2k2i

4

(1

mdi− 1

mpi

)d1 −δ −

∑i=x,y

~2k2i

4

(1

mdi+ 1

mpi

)d2 −v2~kxd3 v1~kyd4, d5 0

TABLE II. Coefficients of the BHZ Hamiltonian for MX2 fam-ily near the Γ0-point [44, 45]. md,p

x,y are the effective massesof d and p- bands along different directions, respectively, v1,2are the velocities and δ is the value of the band inversion.

of 10 meV. Eq. (9) can be further simplified to

J(1)orb = − 4e2

~3Ω2

I

cκε0

∫∆Ω(k)=0

dk

2π

1

ε|∇k∆Ω|d′′0d′2d1. (17)

Here, we restored the constants e, ~ and c. The lightintensity is I = ε0κc|E|2, and κ is the effective di-electric permittivity. Plugging the above parameters forthe effective model for HdTe/CdTe quantum wells intoEq. (17) and assuming ε = 10, I = 10 GW cm−2, andthat the frequency Ω equals to the bandgap, we obtainJorb ≈ 1024 m−1 s−1. An intensity equal to 10 GW/cm2

was chosen because such values recently became achiev-able in THz experiments [39–41].

1T’ topological phase of MX2. Another exam-ple of 2D TI is transition metal dichalcogenides MX2

(where M stands for a metal and X is a chalcogen atom)in 1T’ phase, [42, 43] such as MoS2, WSe2 and WTe2.The Hamiltonian of MX2 in 1T’ phase is well describedby the four-band Hamiltonian in Eq. (1). [44, 45] Ink · p approximation, the coefficients of the Hamiltonianare given in Table II. In MoS2, the effective masses forpx (py)orbitals are[42] mp

x = 0.48me (mpy = 0.29me),

and mdx(y) = 2.32(0.92)me for the two d-orbitals. δ =

−0.27 eV corresponds to d−p band inversion, and veloc-ities are v1 = 0.23 × 105 m/s and v2 = 3.38 × 105 m/s.Plugging the numbers into Eq. (17) and performing theintegral, for the light frequency equal to the band gap atΓ0-point (Ω = 2|δ|), the order of magnitude estimationof the orbital current density is Jorb ≈ 1022 m−1 s−1.Here, we assumed I = 10 GW cm−2 and the dielectricconstant of κ = 10.

DISCUSSION

In this work, we studied the optical dc response of annewly constructed orbital current in time-reversal andinversion symmetric materials. We find that the sign oforbital current reflects the Z2 index of centrosymmetrictopological insulators, which makes the orbital currentan experimental probe for the topological properties ofelectronic bands. At the topological phase transition, theorbital current exhibits a robust sign change. We illus-

6

trated these properties by considering the BHZ Hamilto-nian and an effective model for MoS2. In these materials,our estimate for the orbital current gives 1022−1024 m−1

s−1 for a light power of 10 GW cm−2.

The orbital current generated in a semiconductor accu-mulates intracell electric polarization as schematicallyshown in Fig. 1. We expect that the accumulated po-larization decays over timescales on the order of the or-bital momentum relaxation time τ ≈ 10−14 − 10−15 s,which is determined by the value of the bandgap (see dis-cussion in the Materials and Methods section). Takingthe relaxation into account, the number of carriers with〈τx〉 = 1 that can be transported per unit cross-sectionlength and subsequently accumulated is roughly equal toτJorb. The dipole moment of a fully orbitally polarizedunit cell (〈τx〉 = 1) equals 10−29 C m, and an orbital cur-rent of magnitude 1022 − 1024 m−1 s−1 produces τJorb

on the order of 1010− 107 m−1. The accumulated dipolemoment per meter of cross-section length therefore equals10−19 − 10−22 C.

A similar effect may also take place in other kinds ofsystems. For instance, by applying time-periodic drivingto a BHZ model, one can induce a phase transition toa Floquet topological insulator state [46]. It was shownthat even when the driving potential breaks time-reversalsymmetry, the system is still described by the BHZ modelwith different effective parameters when one works in anappropriate rotating frame of reference. We expect theorbital current to be a suitable probe for such a phasetransition. Application of optically-driven orbital cur-rents to the detection of other topological phases andstrongly correlated states is another interesting problemthat remains an open question.

ACKNOWLEDGEMENTS

We are grateful to Takahiro Morimoto and ZhanybekAlpichshev for fruitful discussions. MD was supported byAustrian Agency for International Cooperation in Educa-tion and Research (OeAD-GmbH) and by the John SeoFellowship at MIT. HI was supported by JSPS KAK-ENHI Grant Numbers JP19K14649 and JP18H03676,and by UTokyo Global Activity Support Program forYoung Researchers.

MATERIALS AND METHODS

Derivation of nonlinear shift current. The coef-ficients d0,1,2,3,4,5(k) of the Hamiltonian (1) have to beeither even or odd in the BZ:

d0,1(k) = d0,1(−k),

d2,3,4,5(k) = −d2,3,4,5(−k).(M1)

The energy eigenvalues are

ε1,2(k) = d0(k)±

√√√√|d1(k)|2 +

5∑i=2

|di(k)|2 = d0(k)±ε(k).

(M2)The term d1(k) has special place in this model because

Γ1 is the only of five matrices that is even under P andT . At TR-invariant momenta d2,3,4,5 all vanish and theparameter d1(k) represents the mass term that governsthe topological phase transition. When it changes signat odd number of points, this changes the (strong) topo-logical index in 2D (3D).

To derive the expression for the shift current, we use theFloquet bands formalism combined with Keldysh Green’sfunctions [35]. We choose the wave functions 1 and 1’ (2

and 2’) to be the T P -partners, which partially fixes theU(2)× U(2) gauge redundancy.

Let us write the vector potential A(t) = iAeiΩt −iA∗e−iΩt and assume that incident electromagnetic waveis linearly polarized. We consider the Floquet bands withindices −1 (valence bands) and 0 (conduction bands).The Floquet Hamiltonian reads:

HF =

ε1 + Ω 0 −iA∗v0

12 −iA∗v012′

0 ε1 + Ω −iA∗v01′2 −iA∗v0

1′2′

iAv021 iAv0

21′ ε2 0

iAv02′1 iAv0

2′1′ 0 ε2

(M3)

where v0 = ∂H0(k)∂k ·A/A is the velocity along the polar-

ization direction. The dc orbital current is given by

Jα = −i∫dω

2π

∫dk Tr

(vFαG

<), (M4)

where vFα is the orbital velocity operator defined in theprevious section taken in Floquet representation. Thelesser Green’s function of the Floquet Hamiltonian isG<(ω) = GR(ω)Σ<GA(ω), where Σ< is the self-energy.In the gauge that demands the spin components withinthe pairs 1, 2 and 1′, 2′ to be parallel, the self-energy ac-quires diagonal form Σ< = iΓ 1

2 (I2 + sz), where sz is aPauli matrix that acts on the band degree of freedom.

Evaluating the integral over ω and collecting the termswith the similar structure, we arrive at Eq. (5).

Connection between orbital degrees of freedomand intracell polarization. In BHZ model thebasis consists of states |s, ↑〉, |s, ↓〉, |px + ipy, ↑〉 and|px − ipy, ↓〉. Let us consider a simplified argument andassume that the Wannier functions for p-orbitals havethe form wpx±ipy (r) = c(x± iy)ws(r), where c is a con-stant that is assumed to be real. The matrix of the x-component of the electrical polarization operator is

7

Px = − e

V2

∫UC

dr

(I2 xI2 + iyσz

xI2 − iyσz(x2 + y2

)I2

)cx|ws(r)|2 = aτx ⊗ I2, (M5)

where integral is performed over unit cell, V2 is the areaof 2D unit cell, and a is a real constant obtained afterintegration a = − e

V2

∫drcx2|ws(r)|2 and 2× 2 matrix I2

acts on spins. In a more general case, expression for Pxwill contain a linear combination of matrices τx,y withcomplex coefficients. However, by applying an appropri-ate rotation we retrieve Eq. (M5) again, confirming thatthis is a general result.

Symmetry-based classification of allowed re-sponses for two-dimensional layer groups andthree-dimensional centrosymmetric groups.Let us consider a second-order response in the currentassociated with some operator O. Phenomenologically,the dc response is :

ji(O) = σijk(O)EjE∗k , (M6)

where σijk(O) can be a third rank tensor is O is scalar(pseudoscalar) or a highter-rank tensor if the operator Ohas tensor structure. We assume that ji(O) transformsunder a representation Γ(vi × O) ∼ Γ(vec) ⊗ Γ(O). Be-cause we are considering centrosymmetric materials, forthe response (M6) to be nonzero, O has to transformunder an inversion-odd representation.

Assuming that O transforms under an irreducible repre-sentation (irrep) of the corresponding point group, onecan deduce the amount of independent response coef-ficients entering the tensor σijk(O) that is symmetricin the last two indices as in the case of shift current.In Tables III and IV we summarize the results of thesymmetry-based analysis for 2D and 3D centrosymmet-ric groups, respectively. Naturally, only inversion-oddrepresentations of O give rise to nonzero response. Thesecond columns of Tables III and IV can serve as guidancefor choosing the appropriate observables for each symme-try group. Consider, for example, group C6h. The irrepAu may correspond to pz-orbital or a sublayer degreeof freedom perpendicular to z-direction, E1u is a two-dimensional representation of that corresponds to px±ipyorbitals, and Bu and E2u comprise parts of an electric oc-tupole [47].

Conservation of the orbital current. The orbitalpolarization operator τx can be separated into conserved

(τ(c)x ) and nonconserved (τ

(n)z ) parts, where conserved

part obeys continuity equation and can be used to definea conserved current. By analogy with Refs. [15 and 32],

we write τ(c)x = P vτxP

v + P cτxPc, where P v,c are the

projection operators on the subspaces of states belong-ing to valence and conduction bands, respectively. The

non-conserved part is the difference between the full op-

erator and its conserved component, τ(n)x = τx − τ

(c)x .

We find that the second term in Eq. (9) corresponds tothe conserved orbital current, while the first term is thePoint Representations of O Ntot

group

Ci Au(12) 12

C2h Au(4); Bu(8) 12

D2h Au(2); B1u(2); B2u(4); B3u(4) 12

C4h A1u(2); A2u(2); Eu(8) 12

D4h A1u(1); A2u(1); B1u(1); B2u(1); Eu(4) 8

S6 Au(4); Eu(8) 12

D3d A1u(2); A2u(2); Eu(4) 8

C6h Au(2); Bu(2); E1u(6); E2u(2) 12

D6h A1u(1); A2u(1); B1u(1); B2u(1); E1u(3); E2u(1) 8

TABLE III. Independent invariant tensor coefficients for 2Dcentrosymmetric point groups. The second column gives theirreducible representation for O where the response is allowedby symmetry and the number of independent tensor coef-ficients corresponding to a given representation. The thirdcolumn provides the total possible number of nonzero tensorcoefficients Ntot.

non-conserved part.

To examine the dynamics of the non-conserved part oforbital charge, we find the equation of motion for theorbital polarization density T = ψ†τxψ to obtain:

∂

∂tT −∇Jorb + B = Q. (M7)

Here Jα = 12ψ† τα,vψ is the orbital current associated

with the flow of α-th orbital and Q = T × F , F =0, 0,∆, is the torque acting on the orbital polarization.The mass term ∆ = 2d1(k = 0) corresponds to the valueof the gap at Γ0-point. The third term in the expressionis Bα = ψ†i∇ · b[τα, τy]ψ, where b = ∂d2/∂k

∣∣k=Γi

and

we are considering the continuous limit in the vicinity ofΓi. The last term in the LHS of Eq. (M7) couples ∇Tz todynamics of Tx and vice versa. In a nearly-homogeneoussetup this term is negligible.

Lastly, let us discuss the torque term Q. Near the topo-logical phase transition the gap closes (∆ → 0), thus bybeing close enough to the transition allows making thetorque Q arbitrarily small. Away from the transition,the torque can be viewed as determining the rate of thedissipation of the orbital polarization in the material.

[1] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects onelectronic properties, Rev. Mod. Phys. 82, 1959 (2010).

[2] R. Karplus and J. M. Luttinger, Hall effect in ferromag-

8

Point Representations of O Ntot

group

Ci Au(18) 18

C2h Au(8); Bu(10) 18

D2h Au(3); B1u(5); B2u(5); B3u(5) 18

C4h Au(4); Bu(4); Eu(10) 18

D4h A1u(1); A2u(3); B1u(2); B2u(2); Eu(5) 13

S6 Au(6); Eu(12) 18

D3d A1u(2); A2u(4); Eu(6) 12

C6h Au(4); Bu(2); E1u(8); E2u(4) 18

D6h A1u(1); A2u(3); B1u(1); B2u(1); E1u(4); E2u(1) 12

Th Au(1); Eu(2); Tu(5) 8

Oh A1u(1); Eu(1); T1u(3); T2u(2) 7

TABLE IV. Same as Table III but for centrosymmetricgroups in three dimensions.

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[26] Y. Zhang, H. Ishizuka, J. van den Brink, C. Felser,B. Yan, and N. Nagaosa, Photogalvanic effect in weylsemimetals from first principles, Phys. Rev. B 97, 241118(2018).

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10

SUPPLEMENTARY

Symmetries and the velocity operator

The time-reversal operator is an antiunitary operator, thus the adjoint of it is defined as:⟨φ∣∣∣Tψ⟩ = 〈T †φ|ψ〉. (S1)

In the presence of both TR- and P - symmetries, one can choose the wavefunctions to be T I-partners and we denotethe bands as 1, 1′, 2 and 2′, or explicitly:

|ψ1′〉 = T I |ψ1〉 , |ψ1〉 = −T I |ψ1′〉 ,|ψ2′〉 = T I |ψ2〉 , |ψ2〉 = −T I |ψ2′〉 .

(S2)

Now we want to derive a relation between the matrix elements 〈ψ| v |ψ〉 and⟨T Iψ

∣∣∣ v ∣∣∣T Iψ⟩; it was also used that

operator v is invariant under T I symmetry.⟨T Iφ

∣∣∣ v ∣∣∣T Iψ⟩ = 〈(T I)†v(T I)ψ|φ〉 = 〈ψ|

(T I)v(T I)†|φ〉 = 〈ψ| v |φ〉 . (S3)

The last statement leads to the following relations:

v12 = v∗1′2′ , v1′2 = −v∗12′ , v11 = v1′1′ , v22 = v2′2′ . (S4)

Another property of the velocity matrix, which is not related to the time-reversal, is that:

v11′ = v22′ = 0. (S5)

The proof goes as follows

H |ψ1〉 = ε1 |ψ1〉∂H

∂k|ψ1〉+H

∂ |ψ1〉∂k

=∂ε1∂k|ψ1〉+ ε1

∂ |ψ1〉∂k

〈ψ1′ | ∂H∂k|ψ1〉+ ψ1′H

∂ |ψ1〉∂k

= 〈ψ1′ | ∂ε1∂k|ψ1〉+ ε1ψ1′

∂ |ψ1〉∂k

v1′1 + ε1′

⟨ψ1′

∣∣∣∣∂ψ1

∂k

⟩=∂ε1∂k〈ψ1′ |ψ1〉+ ε1

⟨ψ1′

∣∣∣∣∂ψ1

∂k

⟩v1′1 =

∂ε1∂k〈ψ1′ |ψ1〉+ (ε1 − ε1′)

⟨ψ1′

∣∣∣∣∂ψ1

∂k

⟩= −i (ε1 − ε1′) a1′1 = 0.

(S6)

Relation between the second-order response and Berry connection

The expression for the shift current j(1)orb in Eq. (6) can be related to the non-Abelian Berry connections a`j =

−i 〈`| ∂∂k |j〉:

j(1)orb = 2

Γ|A|2

∆2Ω + Γ2

Re[v0

21u12

](∂ϕ21

∂k+ a22 − a11

)− Im

[v0

21vα12

] ∂ ln | 〈1|u| 2〉|∂k

+ Re[v0

12 (a22′u2′1 − u21′a1′1)]

+

+ 〈1↔ 1′〉+ 〈2↔ 2′〉+ 〈1, 2↔ 1′, 2′〉 ,(S7)

where (ϕ)i is the phase of the corresponding i-th vector component of the staggered velocity u. One can perform anSVD transformation of the velocity matrix in Eq. (6) in order to diagonalize the velocity matrix. In this case one ofthe unitary matrices act in the space of wavefunctions (1, 1′) and another in space (2, 2′). Then the expression (6)transforms into:

j(1)orb = −2

Γ|A|2

∆2Ω + Γ2

Im

[v0

21

(∂u

∂k

)12

+ v02′1′

(∂u

∂k

)1′2′

],

j(2)orb = 2

(ε2 − ε1 − Ω) |A|2

∆2Ω + Γ2

Re

[v0

21

(∂u

∂k

)12

+ v02′1′

(∂u

∂k

)1′2′

].

(S8)

11

This resembles the result from Ref. 35. In terms of the Berry connection Eq. (S8) can be rewritten as

j(1)orb =

Γ2 |A|

2

∆2Ω + Γ2

4

Re[v0

21u12

](∂ϕ21

∂k+ a22 − a11

)− Im

[v0

21u12

] ∂ ln | 〈1|u| 2〉|∂k

+ 〈1, 2↔ 1′, 2′〉 (S9)

Neglecting spin-orbit coupling (i.e. g1,2,3 = 0), we find that the shift current is given by the last expression in the curly

brackets: j(1)orb ∝ Im

[v0

21u12

] ∂ ln |〈1|u|2〉|∂k + 〈1, 2↔ 1′, 2′〉. To relate the matrix elements in the last expression with

more familiar operators τx and v, one can further expand it to obtain j(1)orb ∝

(v0

11 + v022

)v21τ

x12∂ ln |u12|

∂k +〈1, 2↔ 1′, 2′〉.Arbitrary direction of polarization

In the case when the light is polarized in the direction that we denote as α and the orbital current is measured indirection β the expression yields:

j(1)orb = −

Γ2 |A|

2

∆2Ω + Γ2

4

4

ε

(∂kβ∂kαd0

)[(∂kαd1) d2 − d1 (∂kαd2)]− εlmngl (∂kαgm)

(∂kβ∂kαgn

)(S10)

BHZ model energies and wavefunctions

The requirements that are imposed on the coefficients in the Hamiltonian due to TR- and I-symmetries are:

d0,1(k) = d0,1(−k)

d2,3,4,5(k) = −d2,3,4,5(−k)(S11)

The energies and wavefunctions of the Hamiltonian (1):

ε1,2(k) = d0(k)±

√√√√ 5∑i=1

|di(k)|2 = d0(k)± ε(k) (S12)

ψ1 =1√

2ε (ε+ d1)

d5 − id2

d3 + id4

−(ε+ d1)

0

, ψ1′ = T Iψ1 =1√

2ε (ε+ d1)

d3 − id4

−(d5 + id2)

0

−(ε+ d1)

,

ψ2 =1√

2ε (ε− d1)

d5 − id2

d3 + id4

ε− d1

0

, ψ2′ = T Iψ2 =1√

2ε (ε− d1)

d3 − id4

−(d5 + id2)

0

ε− d1

.

(S13)

At the topological phase transition, where upon smoothly varying some of the constants in the Hamiltonian theparameter d1 crosses zero and other stay approximately unchanged, the property (13) is easy to see from the explicitform of the wavefunctions.

Joint density of states

Here we show the joint density of states for parameters similar to those in Fig. 2a. The JDOS reproduces some ofthe features that are seen in Fig. 2a, namely a jump at εs − εp, and two van Hove singularities at Ω/tss ≈ 4.2 andΩ/tss ≈ 7.

12

FIG. 3. The dependence of the joint density of states on the light frequency for the BHZ model. εs/tss = 3. The dashed graylines correspond to the minimum and maximum frequencies. The other parameters are εp = tpp = 0, tsp/tss = 1.